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Full text of "Bicycles & tricycles; an elementary treatise on their design and construction, with examples and tables"

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sa iL^i'-^' /7 




HARVARD 
COLLEGE 
LIBRARY 



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7 



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BICYCLES AND TRICYCLES 



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Bicycles & Tricycles, 

AN ELEMENTARY TREATISE ON THEIR 
DESIGN AND CONSTRUCTION 



WITH EXAMPLES AND TABLES 



BY 

ARCHIBALD ^lARP, B.Sc. 

WHITWORTH SCHOLAR 

ASSOCIATE MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS 

MITGLIED DES VERBINS DEUTSCHER INGENIEURB 

rjrSTRUCTOR IM ENGINEERING DESIGN AT THE CKNTRAI. TECHNICAL COLLF^K 

SOUTH KRNSINGTON 



WITH NUMEROUS ILLUSTRATIONS 



LONGMANS, GREEN, AND CO. 

LONDON, NEW YORK, AND BOMBAY 
1896 

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All rights reserved 



^ 



^ C^ \'oV. tr-'^"^ 



^ HARVARD COLLEGE UBRARY 

6IFT0F 
tSyVARD WORTHINGTON SARGENT 






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PREFACE 



A BICYCLE or a tricycle is a more or less complex machine, 
and for a thorough appreciation of the stresses and strains 
to which it is subjected in ordinary use, and for its efficient 
design, an extensive knowledge of the mechanical sciences 
is necessary. Though an extensive literature on nearly all 
other types of machines exists, there is, strange to say, 
very little on the subject of cycle design ; periodical 
cycling literature being almost entirely confined to racing 
and personal matters. In the present work an attempt 
is made to g^ve a rational account of the stresses and 
strains to which the various parts of a cycle are sub- 
jected ; only a knowledge of the most elementary portions 
of algebra, geometry, and trigonometry being assumed, 
while graphical methods of demonstration are used as far 
as possible. It is hoped that the work will be of use to 
cycle riders who take an intelligent interest in their 
machines, and also to those engaged in their manufacture. 
The present type of rear-driving bicycle is the outcome of 
about ten years* practical experience. The old * Ordinary,' 
with its large front driving-wheel, straight fork, and curved 
backbone, was a model of simplicity of construction, 



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vi Bicycles and Tricycles 

but with the introduction of a smaller driving-wheel, 
driven by gearing from the pedals, and the consequent 
greater complexity of the frame, there was more scope for 
variation of form of the machine. Accordingly, till a 
few years ago, a great variety of bicycles were on the 
market, many of them utterly wanting in scientific design. 
Out of these, the present-day rear-driving bicycle, with 
diamond-frame, extended wheel-base, and long socket 
steering-head — the fittest — has survived. A better techni- 
cal education on the part of bicycle manufacturers and 
their customers might have saved them a great amount 
of trouble and expense. Two or three years ago, when 
there seemed a chance of the dwarf front-driving bicycle 
coming into popular favour, the same variety in design of 
frame was to be seen ; and even now with tandem bicycles 
there are many frames on the market which evince on the 
part of their designers Utter ignorance of mechanical 
science. If the present work is the means of influencing 
makers, or purchasers, to such an extent as to make the 
manufacture and sale of such mechanical monstrosities in 
the future more difficult than it has been in the past, the 
author will regard his labours as having been entirely 
successful. 

The work is divided into three parts. Part I. is on 
Mechanics and the Strength of Materials, the illustrations 
and examples being taken with special reference to bicycles 
and tricycles ; Part II. treats of the cycle as a complete 
machine ; and Part 1 1 L treats in detail of the design of its 
various portions. 

The descriptive portions are not so complete as might 
be wished ; however, the * Cyclist Year Books,' published 

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Preface vii 

early in each year, enable anyone interested in this part of 
the subject to be well informed as to the latest novelties 
and improvements. 

The author would like to express his indebtedness to 
•the following works : 

The ' Cyclist Year Books ' ; 

* Bicycles and Tricycles of the Year/ by H. H. Griffin, 

a valuable series historically, which extends from 
1878 to 1889; 
'Cycling Art, Energy, and Locomotion,' by R. P. 
Scott ; 

* Traits des Bicycles et Bicyclettes,* par C. Bourlet ; 
The * Cyclist ' weekly newspaper ; 

and to the various cycle manufacturers mentioned in 
the text, who have, without exception, always afforded 
information and assistance when asked. He has also to 
thank Messrs. Ackermann and Farmer for assistance in 
preparing drawings, and Messrs. Ackermann and Hummel 
for reading the proofs. 

In a work like the present, containing many numerical 
examples, it \^ improbable that the first issue will be 
entirely free from error ; corrections, arithmetical and 
otherwise, will therefore be gladly received by the author. 



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CONTENTS 



PART I 
PRINCIPLES OF MECHANICS 

CHAPTER I 
FUNDAMENTAL CONCEPTIONS OF MECHANICS 

PAGE 

I. Divbion of the Subject. — 2. Space. — 3. Time. — 4. Matter . 1-3 



CHAPTER II 

SPEED, RATE OF CHANGE OF SPEED, VELOCITY, 
ACCELERATION, FORCE, MOMENTUM 

5. Speed 6. Uniform Speed. — 7. Angular Speed. — 8. Relation 

between Linear and Angular Speeds. — 9. Variable Speed. — 10. 
Velocity. — 1 1. Rate of Change of Speed. — 12. Rate of Change 
of Angular Speed. — 13. Acceleration. — 14. Force. — 15. Momen- 
tum. — 16. Impulse. — 17. Moments of Force, of Momentum, &c. 4-14 



CHAPTER III 

KINEMATICS : ADDITION OF VELOCITIES 

18. Graphic Representation of Velocity, Acceleration, &c. — 19. Addi- 
tion of Velocities. — 20. Relative Velocity. — 21. Parallelogram of 
Velocities. — 22. Velocity of Point on a Rolling U heel. — 23. Re- 
solution of Velocities. — 24. Addition and Resolution of Accelera- 
tions. — 25. Hodograph. — 26. Uniform Circular Motion ^ . ^5-22 

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X Bicycles and Tricycles 



CHAPTER IV 
KINEMATICS : PLANE MOTION 

TAGS 

27. DeBnition of Plane Motion. — 28. General Plane Motion of a Rigid 
Body. — 29. Instantaneous Centre. — 30. Point-paths, Cycloidal 
Curves. — 31. Point-paths in Link Mechanisms. — 32. Speeds in 
Link Mechanisms. — 33. Speed of Knee-joint when pedalling a 
Crank. — 34. Simple Harmonic Motion. — 35. Resultant Plane 
Motion. — 36. Simple Cases of Relative Motion of two Bodies in 
Contact. — 37. Combined Rolling and Rubbing. . . .23-38 

CHAPTER V 

KINEMATICS : MOTION IN THREE DIMENSIONS 

38. Resultant of Translations. — 39. Resultant of two Rotations about 
Intersecting Axes. — 40. Resultant of two Rotations about Non- 
intersecting Axes. — 41. Most General Motion of a Rigid Body. — 
42. Most Genera Motion of two Bodies in Contact . . . 39-42 

CHAPTER VI 

STATICS 

43. Graphic Representation of Force. — 44. Parallelogram of Forces. — 
45. Triangle of Forces. — 46. Polygon of Forces. — 47. Resultant of 
any number of Co-planar Forces. — 48. Resolution of Forces. — 
49. Parallel Forces. — 50. Mass-centre. — 51. Couples. — 52. 
Stable, Unstable, and Neutral Equilibrium. — 53. Resultant of 
any System of Forces 43-55 

CHAPTER VII 

DYNAMICS : GENERAL PRINCIPLES 

54. Laws of Motion. — 55. Centrifugal Force. — 56. Work. — 57. 
Power. — 58. Kinetic Energy. — 59. Potential Energy. — 60. 
Conservation of Energy. — 61. Frictional Resistance. — 62. Heat. 56-64 

CHAPTER VIII 

DYNAMICS [continued) 

63. Dynamics of a Particle. — 64. Circular Motion of a Particle. — 
65. Rotation of a lamina about a fixed axis perpendicular to itB 



Contents xi 

PAGE 

Plane. — 66. Pressure on the Fixed Axis. — 67. Dynamics of a Rigid 
Body. — 68. Slarting in a Cycle Race. — 69. Impact and Collision. 
— 70. Gyroscope. — 71. Dynamics of any system of Bodies . 65-77 

CHAPTER IX 

FRICTION 

72. Smooth and Rough Bodies. — 73. Friction of Rest. — 74. Coefficient 
of Friction. — 75. Journal Friction. — ^(i, Collar Friction. — 77. 
I'iviH Friction. — 78. Rolling Friction 78-84 

CHAPTER X 

STRAINING ACTIONS : TENSION AND COMPRESSION 

79" Action and Reaction. — 8a Stress and Strain. — 81. Elasticity. — 
82. Work done in stretching a Bar. — 83. Framed Structures. — 
84. Thin Tubes subjected to Internal Pressure . . . . 85 92 

CHAPTER XI 

STRAINING ACTIONS : BENDING 

8$. Intioductory. — 86. Shearing-force. — 87. Bending-moment. — 
88. Simple Example of Beams. — 89. Beam supporting a number 
of Loads. — 90. Nature of Bending Stresses. — 91. Position of 
Neutral Axis. — 92. Momentof Inertia of an Area. — 93. Moment 
of Bentling Resistance. — 94. Modulus of Bending Resistance of 
a Section. — 95. Beams of Unilorm Strength. — 96. Modulus of 
Bending Resistance of Circular Tubes. — 97. Oval Tubes. — 
98. O Tubes. — 99. Square and Rectangular Tubes . . 93-119 

CHAPTER XII 

SHEARING, TORSION, AND COMPOUND STRAINING 
ACTION 

100. Compression. — lot. Compression or Tension combined with 
Bending. — 102. Columns. — 103. Limiting Load on Long 
Columns. — 104. Gordon's Formula for Columns. — 105. Shear- 
ing. — 106. Torsion. — 107. Torsion of a Solid Bar. — 108. Tor- 
sion of Thick Tubes. — 109. Lines of Direct Tension and Com- 
pression on a Bar subject to Torsion. — no. Compound 
Stress. — III. Bending and Twisting of a Shaft . ' r^ ' 120-131 

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xii Bicycles and Tricycles 



CHAPTER XIII 
STRENGTH OF MATERIALS 

PAGE 

112. Stress r Breaking and Working. — 113. Elastic Limit. — 114. 
Stress-strain Diagram. — 115. Mild Steel and Wrought Iron. — 
116. Tool Steel. — 117. Cast Iron. — 118. Copper. — 119. 
Alloys of Copper. — 120. Aluminium, — 121. Wood. — 122. 
Raising of the Elastic Limit — 123. Complete Stress-strain 
Diagram. — 124. Work done in breaking a Bar. — 125. Me- 
chanical Treatment of Metals. — 126. Repeated Stress . . 132-143 



PART 11 
CYCLES lAT GENERAL 

CHAPTER XIV 

DEVELOPMENT OF CYCLES : THE BICYCLE 

127. Introduction. — 128. The Dandyhorse. — 129. Early Bic>des. — 
130. The * Ordinary.'— 131. The * Xtraordinary. ' — 132. The 
♦Facile/ — 133. The 'Kangaroo.*— 134. The Rear-driving 
Safety.— 135. The 'Geared Facile.'— 136. The Diamond- 
frame Rear-driving Safety. — 137. The * Rational Ordinary.* — 
138. The 'Geared Ordinary * and Front -driving Safety. — 139, 
The * Giraffe* and 'Rover Cob.* — 140. Pneumatic Tyres. — 
141, Gear-cases% — 142.. Tandem Bicycles « « . , 145-164 

CHAPTER XV 

DEVELOPMENT OF CYCLES : THE TRICYCLE 

143. Early Tricycles. — 144. Tricycles with Differential Gear. — 
145. Modem Single driving Tricycles. — 146. Tandem Tri- 
cycles. — 147. Sociables. — 148^ Convertible Tricycles. — 
149. Quadricycles . r . . • r . . . 165-182 

CHAPTER XVJ 

CLASSIFICATION OF CYCLES 

150. Stable and Unstable Equilibrium. — -151. Method of Steering. — 
152. Bicycles : Front-drivers. — 153. Bicycles; Rear-drivers. — 

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FACE 

154. Tricycles. — 1 55. Front-steering Front -driving Tricycles. — 
156. Front -steering Rear-driving Tricycles. — 157. Rear-steering 
Fioot-driving Tricycles.— 15& Quadricycles.— 1591 Multicycles 183-196 

CHAPTER XVII 

STABILfTY OF CYCLES 

i6a Subflity of Tricydesw — i5i. Stability of Quadricycles. — 
162. Balancing on a Bicycle. — 163. Balancing on the * Otto ' 
Dicyde. — 164. Wheel Load in Cycles when driving ahead. — 
165. Stability of a Bieyde moving in a Circle. — 166. Friction 
between Wheels and the Ground. — 167. Banking of Racing 
Tracks. — 168. Gyroscopic Action. — 169, Stability of a Tricycle 
moving in 2 circle. — 170. Side-slipping. — 171. Influence of 
Speed on Si^e-slipping. — 172. Pedal Effort and Side-slipping. 

— 173, Headevs . 197-220 

CHAPTER XVin 

STEERING OF CYCLES 

174* Steerii^ in GeneraL — J75, Bicycle Steering. — 176. Steering 
of Tricydes. — 177. Weight on Steering- wheef. — 178. Motion of 
Cycle Wlieel. — 179. Steering without Hands. — 180. Tendency 
of an Obstacle on the Road to cause Swerving. — 181. * Cripper ' 
Tricyde-— 182. • Royal Crescent' Tricycle. — 183. 'Humber' 
Tricycle. — 184. * Olympia * Tricycle and * Rudge ' Quadricycle. 

— 185. * Coventry ' Rotary Tricycle. — 186. * Otto* Dic>'cle. — 
187. Single and Dbuble-driving Tricycles. — 188. Qutch-gear 

for Tricycle Axles. — 189. Difierential Gear for Tricycle Axle . 221 -242 

CHAPTER XIX 
^rOTION OVER UNEVEN SURFACES 

190. Motion over a Stone. — 191. Influence of Size of Wheel. — 
192. Influence of Saddle Position. — 193. Motion over Uneven 
Road. — 194. Loss of Eneigy 243-249 

CHAPTER XX 

RESISTANCE 0¥ CYCLES 

J95- Expenditure of Eneigy. — 196. Resistance of Mechanism. — 
197. Rolling Resistance. — 198. Loss of Energy by Vibration. 

— 199. Air Resistance. — 200, Total Resistance , ' C^ ' 250-256 

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xiv Bicycles and Tricycles 



CHAPTER XXI 
GEARING IN GENERAL 

PACK 

20 1. Machine. — 202. Higher and Lower Pairs. — 203. Classification 
of Gearing. — 204. Efficiency of a Machine. — 205. Power. — 
206. Variable-speed Gear. — 207. Perpetual Motion. — 208. 
Downward Pressure. — 209. Cranks and Levers. — 210. Variable 
Leverage Cranks. — 211. Speed of Knee-joint during Pedalling. — 
212. Pedal-clutch Mechanism. — 213. Diagrams of Crank Eflfort. 
— 214. Actual Pressure on Pedals.— 215. Pedalling. — 216. 
Manumotive Cycles. — 217. Auxiliary Hand- Power Mechanisms 257-273 



PART III 
DETAILS 

CHAPTER XXII 

THE FRAME : DESCRIPTIVE 

2 1 8. Frames in General. — 219. Frames of Front-drivers. — 22a 
Frames of Rear-drivers. — 221. Frames of Ladies' Safeties. — 
222. Tandem Frames. — 223. Tricycle Frames. — 224. Spring- 
frames. — 225. The Front-frame ..•,.. 275-302 



CHAPTER XXIII 

THE FRAME : STRESSES 

226. Frames of Front-drivers. — 227. Rear-driving Safety Frame. — 
228. Ideal Braced Safety Frame. — 229. Humber Diamond 
Frame. — 230. Diamond-frame with no Bending on Frame 
Tubes. — 231. Open Diamond -frame. — 232. Cross-frame. — 
233. Frame of Ladies' Safety. — 234. Curved Tubes. — 235. 
Influence of Saddle Adjustment. — 236. Influence of Chain Ad- 
justment. — 237. Influence of Pedal Pressure. — 238. Influence of 
Pull of Chain on Chain-struts. — 239. Tandem Bicycle Frames. 
— 240. Stresses on Tricycle Frames. — 241. The Front-frame. 
— 242. General Considerations Relating to Design of Frame . 303-336 



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Contents xv 

CHAPTER XXIV 
WHEELS 

rAGB 

243. Introductory. — 244. Compression-spoke Wheels. — 245. Ten- 
sion-spoke Wheels. — 246. Initial Compression on Rim. — 247. 
Direct-spoke Driving-Wheel. — 248. Tangent-spoke Wheel. — 
249. Direct-spokes. — 250. Tangent -spokes. — 251. Sharp's 
Tangent Wheel. — 252. Spread of Spokes. — 253. Disc Wheels. 

— 254. Nipples. — 255. Rims. — 256. Hubs. — 257. Spindles. 

— 258. Spring Wheels 337-36$ 

CHAPTER XXV 

BEARINGS 

259. Definition of Bearings. — 260. Journals, Pivot and Collar 
Bearings. — 261. Conical Bearings. — 262. Roller-bearings. — 
263. Ball-bearings. — 264. Thrust Bearings with Rollers. — 
265. Adjustable Ball-bearing for Cycles. — 266. Motion of Ball 
in Bearing. — 267. Magnitudes of the Rolling and Spinning of 
the Balls on their Paths. — 268. Ideal Ball-bearing. — 269. 
Mutual Rubbing of Balls in the Bearing. — 27a * Meneely ' 
Tubular Bearing. — 271. Ball-bearingfor Tricycle Axle. — 272. 
Ordinary Ball Thrust Bearing. — 273. Dust-proof Bearings. — 
274. Oil-retaining Bearings. — 275. Crushing Pressure on Balls. 

— 276. Wear of Ball-bearings. — 277. Spherical Ball-races. — 

278. Universal Ball-bearing 366-395 

CHAPTER XXVI 

CHAINS AND CHAIN GEARING 

279. Transmission of Power by Flexible Bands. — 280. Early Tri- 
cycle Chain. — 281. Humber Chain. — 282. Roller Chain. — 
283. Pivot Chain. — 284. Roller Chain- wheel. — 285. Huml^er 
Chain- wheel. — 286. Side-clearance, and Stretching of Chain. — 
287. Rubbing and Wear of Chain and Teeth. — 288. Common 
Faults in Design of Chain- wheels. — 289. Summary of Conditions 
determining the Proper Form of Chain-wheels. — 290. Form of 
Section of Wheel Blanks — 291. Design of Side-plates of Chain. 

— 292. Rivets. — 293. Width of Chain, and Bearing Pressure on 
Rivets. — 294. Speed-ratio of Two Shafts connected by Chain 
Gearing. — 295. Size of Chain-wheels. — 296. Spring Chain- 
wheel. — 297. Elliptical Chain-wheel. — 298. Friction of Chain 
Gearing. — 299. Gear-case. — 300. Comparison of Different 

Forms of Chain. — 301. Chain-tightening Gear . . . 396-433 

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xvi Bicycles and Tricycles 



CHAPTER XXVII 
TOOTHED-WHEEL GEARING 

PACE 

302. Transmission by Smooth Rollers. — 303. Friction Gearing. — 
304. Toothed- wheels. — 305. Train of Wheels. — 306. Epicyclic 
Train. — 307. Teeth of Wheels. — 308. Relative Motion of 
Toothed - wheels. — 309. Involute Teeth. — 310. Cycloidal 
Teeth. — 311. Arcsof Approach and Recess. — 312. Friction of 
Toothed-wheels.— 313. Circular Wheel-teeth.— 314. Strength 
of Wheel-teeth. — 315. Choice of Tooth Form. — 316. Front- 
driving Gears. — 317. Toothed-wheel Rear-driving Gears. — 
318. Compound Driving Gears. — 319. Variable Speed Gears . 434-471 

CHAPTER XXVIII 
LEVER-AND-CRANK GEAR 

320. Introductory. — 321. Speed of Knee-joint with * Facile ' Gear. — 

322. Pedal and Knee-joint .Speeds with * Xtraordinary * Gear. — 

323. Pedal and Knee-joint Speeds with • Geared Facile' 
Mechanism. — 324. Pedal and Knee-joint Speeds with * Geared 
Claviger* Mechanism. — 325. * Facile' Bicycle. — 326. 'Xtra- 
ordinary.' — 327. Claviger Bicycles. — 328. Early Tricycles . 472-^84 

CHAPTER XXIX 

TYRES 

329. Definition. — 330. Rolling Resistance on Smooth Surface. — 
331. Metal Tyre on Soft Road. — 332. Loss of Energy by Vibra- 
tion. — 333. Rubber Tyres. — 334. Pneumatic Tyres in (General. 

— 335. Air-tube. — 336. Outer-cover 337. Classification of 

Pneumatic Tyres. — 338. Tubular Tyres. — 339. Interlocking 

Tyres. — 340. Wire-held Tyres 341. Devices for Preventing, 

and Minimising the Effect of Punctures. — 342. Non-slipping 

Covers. -^ 343. Pumps and Valves 485-506 

CHAPTER XXX 

PEDALS, CRANKS, AND BOTTOM BRACKETS 

344. Pedals. ^ 345. Pedal-pins. — 346. Cranks. — 347. Crank-axles. 

-^ 348. Crank-brackets.— 349. Pressure on Crank-axle Bearings 507-516 

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Contents xvii 



CHAPTER XXXI 

SPRINGS AND SADDLES 

PAGB 

50. Spring under the Action of suddenly applied Load. — 351. Spring 
Supporting Wheel. — 352. Saddle Springs. — 353. Cylindrical 
Spiral Springs. ~ 354. Flat Springs. — 355. Saddles. — 356. 
Pneumatic Saddles 517-52$ 



CHAPTER XXXn 

BRAKES 

57. Brake Resistance on the Level. - 358. Brake Resistance Down- 
hill — 359. Tyre and Rim Brakes. — 360. Band Brakes . 526-530 

INDEX 531 



LIST OF TABLES 

TABLE 

L Work done in Foot-lbs. per Stroke of Pedal in raising 100 lbs. 

Weight against Gravity 60 

n. Work done in Foot-lbs. per Minute in pushing 100 lbs. Weight 

Up-hill 61 

HL Sectional Areas and Moduli of Binding Resistance of Round 

Bars 109 

IV. Sectional Areas, Weights per Foot run, and Moduli of Bend- 
ing Resistance of Steel Tubes 1 12-3 

V. Ultimate and Elastic Strengths of Materials, and Coefficients 

of Elasticity 134 

VI. Specific Gravity and Strength of Woods '139 

VII. Tensile Strength of Helical and Solid-drawn Tubes . 142 

VIIL Banking of Racing Tracks 205 

IX. Banking of Racing Tracks 206 

X, Air Resistance to Safety Bicycle and Rider , , , '253 

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xviii Bicycles and Tricycles 

TABLE rAGE 

XL Resistance of Cycles on Common Roads .... 256 

XII. Sectional Areas and Weights per 100 feet Length of Steel 

.Spokes 346 

XIII. Weights, Approximate Crushing Loads, and Safe Working 

Loads of Diamond Cast Steel Balls 394 

XIV. Chain Gearing 397 

XV. Chain-wheels 405 

XVI. Variation of Speed of Crank-axle 4^5 

XVII. Greatest Possible Variation of Speed -ratio of Two Shafts 

Geared Level 426 

XVIII. Circular Wheel-teeth, External Gear 452 

XIX. Circular Wheel-teeth, Internal Gear 452 

XX. Safe Working Pressure on Toothed-wheels .... 454 



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^ 



PART I 
PRINCIPLES OF MECHANICS 

CHAPTER I 

FUNDAMENTAL CONCEPTIONS OF MECHANICS 



1. Divinon of the Sabject. — Geometry is the science which 
treats of relations in space. Kinematics treats of space and 
time, and may be called the geometry of motion. Dynamics is 
the science which deals with force, and is usually divided into 
two parts— statics, dealing with the forces acting on bodies which 
are at rest ; kinetics, dealing with forces acting on bodies in motion. 
Mechanics includes kinematics, statics, kinetics, and the applica- 
tion of these sciences to actual structures and machines. 

2. Space. — The fundamental ideas of time and space form 
part of the foundation of the science of mechanics, and their 
accurate measurement is of great importance. The British unit 
of length is the imperial yard^ defined by Act of Parliament to be 
the length between two marks on a certain metal bar kept in the 
office of the Exchequer, when the whole bar is at a temperature of 
60° Fahrenheit. Several authorised copies of this standard of 
length are deposited in various places. The original standard is 
only disturbed at very distant intervals, the authorised copies 
serving for actual comparison for purposes of trade and commerce. 
The yard is divided into three y^^/, and the foot again into twelve 
inches. Feet and inches are the working units in most general use 
by engineers. The inch is further subdivided by engineers, by a 
process of repeated division by two, so that y\ J", ^", ^^\ &c., 
are the fractions generally used by them. A more convenient 

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2 Principles of Mechanics chap. i. 

subdivision is the decimal system into yV» i^» roW* ^c. ; this is 
the subdivision generally used for scientific purposes. 

The unit of length generally used in dynamics is th^foot. 
Metric System, — The metric system of measurement in 
general use on the Continent is founded on the metrCy originally 
defined as the ,o.«^.o^ part of a quadrant of the earth from the 
pole to the equator. This length was estimated, and a standard 
constructed and kept in France. The metre is subdivided into 
ten parts called decimetres, a decimetre into ten centimetres, 
and a centimetre into ten millimetres. For great lengths a 
kilometre, equal to a thousand metres, is the unit employed. 

I metre = 39*371 inches = 3*2809 feet. 

I kilometre = 0*62138 miles. 

I inch = 25-3995 millimetres. 

I mile = I '6093 1 kilometres. 

3. Time. — The measurement of time is more difficult theo- 
retically than that of space. Two different rods may be placed 
alongside each other, and a comparison made as to their lengths, 
but two different portions of time cannot be compared in this 
way. * Time passed cannot be recalled.' 

The measurement of time is effected by taking a series of 
events which occur at certain intervals. If the time between any 
two consecutive events leaves the same impression as to duration 
on the mind as that between any other two consecutive events, 
we may consider, tentatively at least, that the two times are equal. 
The standard of time is the sidereal day^ which is the time the 
earth takes to make one complete revolution about its own axis, 
and which is determined by observing the time from the apparent 
motion of a fixed star across the meridian of any place to the 
same apparent motion on the following day. The intervals of 
time so measured are as nearly equal as our means of measure- 
ment can determine. 

The solar day is the interval of time between two consecutive 
apparent movements of the sun across the meridian of any phce. 
This interval of time varies slightly from day to day, so that for 
purposes of everyday life an average is taken, called the mean 
solar day. The mean solar day is about four minutes longer than 

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CHAP. I. Fundamental Conceptions of Mechanics 3 

the sidereal day, owing to the nature of the earth's motion round 
the sun. 

The mean solar day is subdivided into twenty-four hours^ 
one hour into sixty minutes^ and one minute into sixty seconds. 
The second is the unit of time generally used in dynamics. 

4. Katter. — Another of our fundamental ideas is that 
relating to the existence of matter. The question of the measure- 
ment of quantity of matter is inextricably mixed up with the 
measurement of force. The mass^ or quantity of matter, in one 
body is said to be greater or less than that in another body, 
according as the force required to produce the same effect is 
greater or less. The mass of a body is practically estimated by 
its weight, which is, strictly speaking, the force with which the 
earth attracts it. This force varies slightly from place to place 
on the earth's surface at sea level, and again as the body 
is moved above the sea level. Thus, the mass and the weight 
of a body are two totally different things ; and many of the 
difficulties encountered by the student of mechanics are due to 
want of proper appreciation of this. The difficulty arises from the 
fact that ih^ pound \^ the unit of matter, and that the weight of 
this quantity of matter, i,e, the force by which the earth attracts 
it, \^ used often as a unit of force. A certain quantity of lead 
will have a certain weight, as shown by a spring-balance, in 
London at high level water-mark, and quite a different weight if 
taken twenty thousand feet above sea level, although the mass is 
the same in both places. 

The British unit of mass is the imperial pounds defined by 
Act of Parliament to be the quantity of matter equal to that of a 
certain piece of platinum kept in the office of the Exchequer. 

t The unit of mass in the metrical system of measurement is the 
gramme^ originally defined to be equal to the mass of a cubic 
centimetre of distilled water of maximum density. This is, how- 
ever, defined practically, like the British unit, as that of a certain 
piece of platinum kept in Paris. 



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Principles of Mechanics 



CHAPTER II 

SPEED, RATE OF CHANGE OF SPEED, VELOCITY, ACCELERATION, 
FORCE, MOMENTUM 

5. Speed. — A body in relation to its surroundings may either 
be at rest or in motion. Linear speed is the rate at which a body 
moves along its path. 

Speed may be either uniform or variable. With uniform 
speed the body passes over equal spaces in equal times ; with 
variable speed the spaces passed over in equal times are unequal. 
The motion may be either in a straight or curved path, but in 
both cases we may still speak of the speed of a point as the rate 
at which it moves along its path. 

6. Uiiifonii Speed is measured by the space' passed over in 
the unit of time. The unit of speed is one foot per second. Let 
s be the space moved over by the body moving with uniform 
speed in the time /, then if v be the speed, we have by the above 
definition. 

^ = 7 0) 

Example. — If a bicycle move through a space of one mile in 
four minutes we have, reducing to feet and seconds, 

z; = \ =22 feet per second. 

4 X 60 ^ 

It will be seen that the unit of speed is a compound one, in- 
volving two of the fundamental units, space and time. 

In the above example, the same speed is obtained whatever 
be the time over which we make the observations of the space 
described. For example, in one minute the bicycle will move 

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J 



CHAP. II. Speedy Rate of Change of Speed, &c. 5 

through a distance of a quarter of a mile, that is 440 yards, or 
3 X 440 feet. Using formula (i) we get 

r =x , =22 feet per second, 

the same result as before. 

Now, consider the space described by the bicycle in a small 
fraction of a second, say -j*(jth, if the speed is uniform, this will be 
2-2 ft. Using formula (i) again, we have 

2 * 2 
V =^ , - = 22 feet per second. 

Proceeding to a still smaller fraction of a second, say -n/^'h, 
if our means of observation were sufficiently refined, the distance 
passed over in the time would evidently be found to be the 
xHrxth part of a foot, i.e. = 022 feet. Again using formula (i) 
we have 

*022 

V = - J =22 feet per second. 

Uniform Motion in a Circle. — Another familiar example of 
uniform motion is that of a point moving in a circular path ; a 
point on the rim of a bicycle wheel has, relative to the frame of 
the bicycle, such a motion, uniform when the speed of the 
bicycle is uniform. The linear speed, relative to the frame, of a 
point on the extreme outside of the tyre will be the same as the 
linear speed of the bicycle along and relative to the road, while 
that of any point nearer the centre of the wheel will be less. 

7. Angular Speed.— When a wheel is rotating about its axis, 
the linear speed of any point on it depends on its distance from 
the centre, is greatest when the point is on the circumference of 
the wheel, and is zero for a point on the axis. The number of 
complete turns the wheel, as a whole, makes in a second gives a 
convenient means of estimating the rotation. Let O (fig. i) be 
the centre of a wheel, and A a point on its circumference ; O A 
may thus represent the position of a spoke of the wheel at a 
certain instant At the end of one second, suppose the spoke 
which was initially in the position O A^ to occupy the position 
OA^;if the motion of rotation of the wheel is uniform, the linear 



/^""T?^ 


^^cK M 


Kjy 



6 Principles of Mechanics chap. h. 

s peed of the point A on the rim is measured by the arc A^ A^, 
while the angular speed of the wheel is measured by the angle 
A I OAi. Generally,theangularspeedof a body rotating uniformly 
is the angle turned through in unit of time. 

The angular speed may be expressed in various ways. For 
example, the number of degrees in the angle A^ OA2 swept out per 
second may be expressed ; this method, however, is little used prac- 
tically. The method of expressing angular velocity most in use 
by engineers, is to give the number of revolutions per minute, n. 
One revolution = 360° ; revolutions per 
minute can be converted into degrees 
per second by multiplying by 360 and di- 
\ viding by 60, that is, by multiplying by 6. 

For scientific purposes another 
method is used. Mathematicians find 
that the most convenient unit angle to 
adopt is not obtained by dividing a 
right angle into an arbitrary number of 
parts ; they define the unit angle as that which subtends a circular 
arc of length equal to the radius. Thus, in figure i, if the arc -4, A^ 
be measured off equal to the radius (9 ^,, the angle A^ O A^^ will 
be the unif angle. This is called a radian. 

The ratio of the length of the circumference of a circle to its 
diameter is usually denoted in works on mathematics and 
mechanics by the Greek letter jt (pronounced like the English 
word *pie'), and is 3 . 14159 .... This number is * incom- 
mensurable,* which means that it cannot be expressed exactly in 
our ordinary system of numeration. It may, however, be ex- 
pressed with as great a degree of accuracy as is desired ; a very 
rough value often used for caculations is 3}. It is easily 
seen that there are nr radians in an angle of half a revolution, 
and therefore the angle of one revolution, that is, four right 

angles, is 2n- radians. Therefore, i radian = ^ — = 57*28*. 

The angular speed w of a rotating body is expressed in radians 
turned through per second, and 

27r« 

"=-60- •>••;■ ^^> 

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CHAP. II. Speed, Rate of Change of Speed, &c, y 

8. Selation between Linear and Angnlar Speeds.— The 
connection between the angular speed of a rotating body 
and the linear speed of any point in it may now be easily ex- 
pressed. Let O (fig. i) be the centre of the rotating body, and A 
a point on it, distant r from the centre, which moves in unit of 
time from Ax to A^^ the number of units in the linear speed oi A 
is equal to the number of units in the length of the arc A^ -4?, 
similarly the angular speed of the rotating body is numerically 
equal to the angle A^ O A^ in radians. But this by definition 
must be equal to the arc A^ A.^ divided by the radius OA^^ hence 
if CD (omega) be the angular speed of a rotating body, v the linear 
speed of any point on it distant r from the centre, we have 

^ = J! (3) 

The sf)eed of a bicycle is conveniently expressed in miles per 
hour, and the angular speed of the driving-wheel in revolutions 
per minute. Let V be the speed in miles per hour, D the 
diameter of the driving-wheel in inches, and n the number of 
revolutions of the driving-wheel per minute ; then feet and 
seconds being the units in (3), 

2x n V X 5280 D 

60 3000 2 X 12 

Substituting in (3) we get 

2T « _ F X 5280 X 2 X 12 
60 "" 3600 X D ' 

from which F = - - — (4) 

33613 

that is, the speed of the bicycle in miles per hour is equal to the 
number of revolutions per minute of the driving-wheel, multiplied 
by the diameter of the driving-wheel in inches, and divided by 

336- 13- 

A more convenient rule than the above for finding the speed 
of a bic>'cle can be deduced. Let N be the number of revolu- 
tions of the driving-wheel made in / seconds ; then 

,;. n X t , 6oiV 

N =s — - — , and n = 



60 ' ''Digitized by Google 



8 Principles of Mechanics chap. u. 

Substituting in (4), we get 

33613 i 

Now, suppose that iV be chosen equal to V\ that is, / is 
chosen such that the number of revolutions in / seconds is equal 
to the number of miles travelled in one hour. Substituting above 
we get 

/=-^-, (5) 

5-502 

which is equivalent to the following convenient rule. Divide the 
diameter of the driving-wheel in inches by 5*502, the number of 
revolutions of the driving-wheel made in the number of seconds 
equal to this quotient is equal to the speed of the cycle in miles 
per hour. 

If, in a geared-up cycle, D be taken as the diameter Xx> which 
the driving-wheel is geared, N will be the number of revolutions 
per minute of the crank-axle, and formula (5) will still apply. 

9. Variable Speed. — The numerical example in section 6 
may help towards a clear understanding of the measurement of 
variable speed. When the speed of a moving body is changing 
from instant to instant, if we want to know the speed at a certain 
point, it would be quite incorrect to observe the space described 
by the body in, say, one hour or one minute after passing the 
point in question ; but the smaller the interval of time chosen, 
the more closely will the average speed during that interval 
approximate to the speed at the instant of passing the point of 
observation. 

Now, suppose the body after passing the point to move with 

exactly the same speed it had at the point, and that in / seconds 

it moves over s feet, its speed at the point of observation would 

s 
be T feet per second. In a very small fraction of a second, say 

WoTT^j the amount of change in the speed of the body is very 
small, and by taking a sufficiently small period of time the average 
speed during the period may be considered equal to the speed at 
the beginning of the period, without any appreciable error. The 



CHAP. II. Speed, Rate of Change of Speed, &c. 9 

speed at any instant will thus be expressed by equation (i), 
provided t be chosen small enough. 

Suppose a bicyclist just starting to race, and that we wish to 
observe his speed at a point 5 feet from the starting-point We 
obsenre the instant he passes the point, and the distance he 
travels in a period of time reckoned from that instant. If in a 
minute he travel 2,400 feet, his average speed during that time 

= , =40 feet per second. But in a quarter-minute, reckoned 

from the same instant, he may only travel 420 feet, giving an 

average speed of =28 feet per second ; while in five 

seconds he may only have travelled no feet, in one second 
15 feet, in one-tenth of a second 105 foot, in one-hundredth part 
of a second one-tenth part of a foot, with average speeds during 
these periods of 22, 15, 10-5, and 10 feet per second. The last 
of these values may be taken as a very close approximation to his 
speed when passing the point in question. 

10, Velocity. — If the speed of a point and the direction of 
its motion be known, its velocity is defined : thus, in the concep- 
tion * velocity,' those of * speed ' and * direction ' are involved. 
Velocity has been defined as * speed directed,' or * rate of change 
of position.' Again, speed may be defined as the magnitude of 
velocity. 

Velocity, involving as it does the idea of direction, can there- 
fore be represented by a straight line, the direction of which 
indicates the direction of the motion, and, by choosing a suitable 
scale, the length of the line may represent the speed, or the 
magnitude of the velocity. A quantity which has not only 
magnitude and algebraical sign, but also direction, is called a 
vector quantity. Thus, velocity is a vector quantity. A quantity 
which has magnitude and sign, but is independent of direction, is 
called a scalar quantity. Speed is a scalar quantity. 

Velocity may be linear or angular ; it may also be uniform or 
variable. A point on a body rotating with uniform angular speed 
about a fixed axis has its linear speed uniform, but since the 
direction of its motion is continually changing, its linear velocity 
is variable, its angular velocity is uniform. Angular rveloci^c3Ln 



lo Principles of Mechanics chap. ii. 

also be represented by a vector, the direction of the vector being 
parallel to the axis of rotation, and the length of the vector being 
equal to the angular speed. 

II. Bate of Change of Speed. — If a moving body at a 
certain instant has a speed of 3 feet per second, and a second 
later a speed of 4 feet per second, two seconds later a speed 
of 5 feet per second, three seconds later a speed of 6 feet 
per second, and so on ; in one second the speed increases by 
I foot per second. In other words, its rate of change of speed 
is I foot per second per second. 

Rate of change of speed may be either uniform or variable. 
An important example of uniform rate of change of speed is that 
of a body falling freely under the action of gravity. If a stone be 
dropped from a height, its speed at the instant of dropping is 
zero ; at the end of one second, as determined by experiment, 
32*2 feet per second approximately ; at the end of two seconds, 
64*4 feet per second ; at the end of three seconds, 96*6 feet per 
second— at least, these would be the speeds if the air offered no 
resistance to the motion. Thus the rate of change of speed of a 
body falling freely under the action of gravity is 32*2 feet per 
second per second. 

If a be the rate of change of speed of a body starting from 
rest, at the end of / seconds its speed will be 

v^iit (6) 

Its average speed during the time will be ^ a /, and therefore the 
space it passes over in time / is 

s^\at y^ t^\vLO . . . . . (7) 

A cyclist starting in a race affords a good example of variable 
rate of change of speed. At the instant of starting the speed of 
the machine and rider is zero ; at the end of two seconds it may 
be five miles an hour ; at the end of three seconds, nine miles an 
hour ; at the end of four seconds, thirteen ; at the end of five, seven- 
teen ; at the end of six, twenty ; at the end of seven, twenty-two ; 
at the end of eight, twenty-three ; at the end of nine, twenty- 
three and threequarters — the increase in the speed with each 
second becoming smaller and smaller until, say^fteenj or twenty 



CHAP. II. Speed, Rate of Change of Speedy &c. 1 1 

seconds from the start, the maximum speed is reached, the speed 
remains constant, and the rate of change becomes zero. In this 
case not only the speed, but also its rate of change, is variable. 
The rate of change probably increases at first, and reaches its 
maximum soon after the start, then diminishes, and ultimately 
reaches the value zero. If the speed of a body diminish, its 
rate of change of speed is negative. A cyclist while pulling up 
previous to stopping is moving with negative rate of change of 
speed. 

The unit of rate of change of speed, like that of speed, is a 
compound one, into which the fundamental units of time and space 
enter. In expressing rate of jchange of speed we have used the 
phrase * feet per second per second ' ; this deserves careful study 
on the part of the beginner, as a proper understanding of the 
ideas involved in these units is absolutely necessary for satisfactory 
progress in mechanics. This rate of change is often loosely 
spoken of in some of the earlier text books as so many * feet per 
second ' ; this method of expression is quite wrong. For instance, 
considering the rate of change of speed due to gravity, we have 
stated above that it is 32 feet per second per second. This means 
that at the end of one second the speed of a freely falling body 
is increased by an additional speed of 32 feet per second, or 
1,920 feet per minute. In one minute the speed would be in- 
creased by sixty times the above additional speed — that is, by 
1,920 feet per second, or 115,200 feet per minute. This rate of 
change of speed may therefore be expressed either as * 32 feet 
per second per second,' * 1,920 feet per minute per second,' or 
* 115,200 feet per minute per minute.' 

The relation between the units of rate of change of speed, 

space, and time is expressed by the formula (7), which may be 

written 

2 s 

« = ,^. 

which shows that the magnitude of the unit rate of change of 
speed is proportional to that of unit space, and inversely propor- 
tional to the square of that of unit time. 

12. Bate of Change of Angular Speed.— The angular speed 
of a rotating body may be either constant or variable ; in the 



1 2 Principles of Mechanics chap. ii. 

latter case the rate of change of angular speed is the increment 
in one unit of time of the angular speed. Let ^ be the rate of 
change of angular speed, a the rate of change of linear speed of 
any point on the body distant r from the centre, then 

6 = ^ (8) 

r 

13. Acceleration is rate of change of velocity ; it may be 
zero, uniform, or variable. When it is zero the velocity remains 
constant, and the motion takes place in a straight line. 

When a point is moving with uniform speed in a circle, though 
its speed does not change, the direction of its motion changes, 
and therefore its velocity also changes. It must therefore be sub- 
jected to acceleration. An acceleration which does not change 
the speed of the body on which it acts must be in a direction at 
right angles to that of the motion, and is called radial accelera- 
tion. An acceleration which does not change the direction of a 
moving body must act in the direction of motion, and is called 
tangential acceleration. The magnitude of the tangential accele- 
ration is the rate of change of speed. 

14. Force. — The definition and measurement of force has 
afforded scope for endless metaphysical disquisitions. Force has 
been defined as * that which produces or tends to produce motion 
in a body.' The unit of force is defined as * that force which, 
acting for one unit of time on a body initially at rest, produces at 
the end of the unit of time a motion of one unit speed.' If the 
units of space, mass, and time be one foot, one pound, and 
one second respectively, the unit of force is called a poundal. 
In the centimetre-gramme-second system of units, the unit of 
force is called a dyne. The measurement of the unit of mass 
involves the idea of force, so that perhaps no satisfactory logical 
definition can be given. 

The unit of force above defined is called the absolute unit. 
The magnitude of a force in absolute units is measured by the 
acceleration it would produce in unit of time on a body of unit 
mass. The force with which the earth attracts one pound of 
matter is equal to 32*2 poundals, since in one second it produces 
an acceleration of 32*2 feet per second per second. , Generally, 

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CHAP. 11. Speedy Rate of Change of Speedy &c. 1 3 

if a force f acting on a mass m produces an acceleration a, we 

have r / V 

/= ^ « (9) 

The unit of force used for practical purposes is the weight of 
on^ pound of matter ; this is called the gravitation unit of force. If 
/be the number of absolute, and F the number of gravitation units 
in a force, / = gF^ or / 

F^^ (10) 

g 

The acceleration due to gravity is usually denoted by the 
letter g. The value of g, or, in other words, the weight of unit 
mass in absolute units of force, as has already been stated above, 
Taries from place to place on the earth's surface. For Britain its 
value is approximately 32*2, the foot-pound-second system of units 
being used. 

Great care must be exercised in distinguishing between one 
pound quantity of matter and i lb. weight, the former being a 
unit of mass, the latter an arbitrary unit of force. 

15. Momentum. — The product of the mass of a body into 
its velocity i^'ncalled its quantity of motion or momentum. The 
momentum of a body of mass one pound moving with a velocity 
of ten feet per second, is thus the same as that of a body of mass 
ten pounds moving with a velocity of one foot per second. 

16. ImpnlBe. — Multiply both sides of equation (9) by /, we 

^^^" set // - ^ / 

J t ^ m €1 1, 

But if the body start from rest, a t = v, its velocity at the end of 
/seconds, therefore //=^,2, (xx) 

Equation (ii) asserts therefore that the momentum, mv, of a 
body initially at rest is equal to the product of the force acting 
on it and the time during which the force acts. The product // 
is called the impulse of the force. 

Equation (ii) is true, however small /, the time during which 
the force acts, may be. Now a momentum of 10 foot-pounds 
per second may be generated by the application of a force of 
I lb. acting for ten seconds, or a force of ten poundals for one 
second, or a force of 1000 poundals acting for -ji^th part of a 




14 Principles of Mechanics chap. n. 

second ; and so on. When two moving bodies collide, or when 
a blow is struck by a hammer, the surfaces are in contact for a 
very small fraction of a second, and the mutual force between 
the bodies is very great. Neither the force nor the time during 
which it acts can be directly measured, but the momentum of the 
bodies before and after collision can be easily measured. Such 
forces of great magnitude acting for a very short space of time 
are called impulsive forces ; they differ only in degree, but not in 
kind, from forces acting for appreciably long periods. 

17. Moments of Force, of Momentum, &c.— Let figure 2 
represent a body fastened by a pin at 6>, so that it is free to turn 

about 6> as a centre, but is otherwise 
constrained. Let it be acted on by the 
forces jP, and F^* Now, it is a matter 
of every day experience that the turning 
effect of such a force as P^ depends not 
only on its magnitude, but also, in popu- 
lar language, on its leverage, that is, on the length of the perpen- 
dicular from the centre of rotation to the line of action of the force. 
For example, in screwing up a nut, if a long spanner be used the 
force required to be exerted at its end is much smaller than if a 
short spanner be used. The product P^ l\ of the force into this 
distance is called the moment of the force about the given centre. 
The force /*, tends to turn the body in the direction of the hands 
of a watch, while P^ tends to turn the body in the opposite direc- 
tion. Therefore, if the moment P<i l^ be considered positive, the 
moment P^ l\ must be considered negative. The positive direction 
is usually taken contra-clockwise. 

If the body be at rest under the action of the forces /*, and 
P2 their moments must be equal in magnitude but of opposite 
sign ; that is, their algebraic sum must be zero. 

The moment of momentum about a given point O oi 2i body of 
mass m moving with velocity v is the product of its momentum 
m V, and the length of the perpendicular / from the given point to 
the direction of motion — />., ;// v /. In the same way the moment 
of an impulse// is the product of the impulse and the length of 
the perpendicular from the given point to the line of action of the 
impulse— />.,///. ^ , 

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15 



CHAPTER III 

KINEMATICS ; ADDITION OF VELOCITIES 

1 8. Graphic Bepresentation of Telocity. — For the complete 
specification of a velocity two elements— its magnitude and 
direction— 2JQ necessary. If a body be moving at any instant 
with a certain velocity, the direction of the motion may be repre- 
sented by the direction of a straight line drawn on the paper, and 
the speed of the body by the length of the straight line. For this 
purpose the unit of speed is supposed to be represented by any 
convenient length on the paper ; the number of times this length 
is contained in the straight hne drawn will be numerically equal 
to the speed of the body. For example, the 
line a b (fig. 3) represents a velocity of three t d 

feet per second in the direction of the arrow, ' ' 

while the line c d represents a velocity of two 
feet per second in a direction at right angles fL 'i, ^ ■ f 



to that of the former. The scale of velocity ^^^ 

in this diagram has been taken i foot per 

second = \ inch. In the same way, any quantity which involves 

direction as well as magnitude can be represented by a straight 

line having the same direction and its length proportional to 

some scale to the magnitude of the quantity. Such a straight 

line is called a vector. 

Example, — If a wheel be turning about its axis with uniform 
speed, the velocities of all points on its rim are numerically equal, 
but have all different directions. Thus, the velocities of the 
points A^ By and C on the rim (fig. 4) are represented by the three 
equal lines, A a^ B b, and Cc respectively at right angles to the 
TzdiiOA, O By and O C. 

19. Addition of Velocities.— A body may be subjected at 

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i6 



Principles of Mechanics 



the same instant to two or more velocities, and its aggregate 
velocity may be required. For example, take a man climbing the 
mast of a ship. Let the ship move horizontally in the direction 
ab (fig. 5), and let the length ab indicate the space passed over 
by it in one second. Let a r be the mast, and as it passes the 
point a let the man commence climbing. At the end of one 
second suppose he has climbed the distance a d. The line a d 
will represent the velocity of the man climbing up the mast, the 
line a b the velocity of the ship. But if a r be the position of the 
mast at the beginning of the second, at the end of the second it 
will be in the position b ^', and the man will be at </^ the length 
b d ^ being, of course, the same as a d. The actual velocity, in 





Fig. 



Fig. 5. 



magnitude and direction, of the man is represented by the line 
a d^. At the end of half a second the foot of the mast would be 
zXe^ a e being equal to ^ a ^, and the man would have ascended 
the mast a distance af \ the actual position of the man would be 
/*, midway between a and d^. Thus his actual motion in space 
will be along the line a // L 

The two velocities a b and a d above are called the component 
velocities, and the velocity ad^ the resultant velocity of the 
man. 

20. Belative Velocity. — We have spoken above of the 
motion of a body, meaning thereby the motion of the body in 
relation to the objects in its immediate neighbourhood, but these 
objects themselves may be in motion in relation to some other 
objects. For example, a man walking from window to window of 

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CHAP. m. Kinematics; Addition of Velocities 17 

a railway carriage in rapid motion has a motion of a certain 
velocity relative to the carriage. But the carriage itself is in 
motion relative to the earth, and the motion of the man relative 
to the earth is quite different from that relative to the carriage. 
Again, the earth itself is not at rest, but rotates on its own axis, 
so that the man's motion relative to a plane of fixed direction 
passing through the earth's axis is still more complex. But besides 
a motion round its own axis, the earth has a motion round the 
sun. The sun itself, and with it the whole solar system, has a 
motion of tiaAslation relative to the visible universe ; in fact, 
there is no such thing as absolute rest in nature. Therefore, 
having no body at rest to which we can refer the motion of any 
body, we know nothing of absolute motion. The motions we 
deal with, therefore, are all relative, and the velocities are also 
relative. It will thus often be necessary, in specifying a velocity 
to express the body in relation to which it is measured. 

21. Parallelopcam of Telocities. — Given two component 
uniform velocities to which a body is subjected, the resultant 
velocity of the body may therefore be found as follows : — 
Draw a parallelogram with two adjacent sides, a and b (fig. 6), 




Fig. 6. Fig. 7. 

representing in magnitude and direction the component velocities^ 
The resultant velocity is represented in magnitude and direction 
by the diagonal o c oi the parallelogram. This proposition is 
known as the parallelogram of velocities. Since velocity involves 
the two ideas of speed and direction, but not position, the 
resultant of two velocities may also be found by the following 
method : — Let o b (fig. 7) be one of the given velocities ; from b 
draw b c equal and parallel to the other ; o c will represent the 
resultant velocity. 

Vector Addition, — The geometrical process used above is 

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1 8 Principles of Mechanics chap. m. 

called * vector addition/ and is used in compounding any physical 
quantities that can be represented by, and are subject to the same 
laws as, vectors. Accelerations, forces acting at a point, rotations 
about intersecting axes, are treated in this way. In general, the 
sum of any number of vectors is obtained by placing at the final 
point of one the initial point of another, and so forming an 
unclosed irregular polygon ; the vector formed by joining the 
initial point of the first to the final point of the last is the required 
sum, the result being independent of the order in which the com- 
ponent vectors are taken. Thus, the sum of the vectors ob^ bc^ 
c dy d e, and ^/(fig. 7) is the vector of, 

22. Velocity of any Point on a Boiling Wheel— Let a 
wheel roll along the ground, its centre having the velocity v. 
The wheel as a whole partakes of this velocity, which may be 



represented by the line O a (fig. 8). The relative motion of the 
wheel and ground will be the same if we consider the centre of 
the wheel fixed and the ground to move backwards with velocity 
V, In this way it is seen that the linear speed of any point on 
the rim of the wheel relative to the frame is equal to v. We can 
now find the velocity, relative to the earth, of any point A on the 
rim of the wheel. The point A is subjected to the horizontal 
forward velocity A a^ with speed v, and to the velocity with speed 
V, in a direction Aa^ z-i right angles to O A, due to the rotation of 
A round O. The resultant velocity is obtained by completing 
the parallelogram A a^ A^ a2. The diagonal A A^ represents the 
velocity of A in magnitude and direction. If the point on the 
rim be taken at-^oi the point of contact with the ground, it will be 
seen that the parallelogram in the above construction reduces to 

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(HAP. III. Kinematics; Addition of Velocities 19 

two coincident straight lines: In this case, however, it is easily 
seen that the velocity of A^ due to the rotation of the wheel, is 
backwards, and, therefore, when added to the forward velocity due 
to the translation of the wheel, the resultant velocity is zero. On 
the other hand, if the point be taken at A^y the top of the wheel, 
the velocity due to rotation is in the forward direction. Thus, the 
velocity of the uppermost point of the wheel is 2 z/— that is, twice 
the velocity of the centre. 

In the same way the velocity of any point B on one 
of the spokes may be found. Join O a^ and draw B b^ 
parallel to A ai, meeting O ai at by The velocity of By due to 
rotation, is represented by B b^. Draw B b^ equal and parallel to 
A «2, and complete the parallelogram B b^ B^ b^. The velocity of 
B is represented by B BK It will be found that the velocity of B 
is greatest when passing its topmost position B^, and least when 
passing its lowest position Bq, 

The above problem can be dealt with by another method. 
The motion of the wheel has been compounded of two motions, 
the linear motion of the bicycle and the motion of rotation of the 
wheel about its axis. But the resultant motion of the wheel — 
that is, its motion relative to the ground — can be more simply 
expressed. If the wheel rolls on the ground without slipping, its 
point of contact Aq is, at the moment in question, at rest. The 
linear velocity of the wheel's centre O is evidently the same as that 
of the bicycle Vy and is in a horizontal direction. The centre of 
the wheel, therefore, may be considered to rotate about the point 
Aq. But as the wheel is a rigid structure, every point on 
it must be rotating about the same centre. The point Aq 
is called the instantaneous centre of rotation of the wheel. The 
linear velocity of any point on the wheel is, by (3) (chap, ii.), equal 
to w r, where r is the distance of the point from the centre of 

rotation Aq, But w is equal to ^, where tq is the radius of the 

wheel ; therefore, the linear velocity of any point B on the wheel 

is equal to —.AqB, and is in the direction BB^ at right angles 

to Aq B, 

The centre of rotation Aoo( the wheel is called an instan- 

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20 Principles of Mechanics chap. m. 

taneous centre of rotation, as distinguished from a fiooed centre of 
rotation, since when the wheel is rolled through any distance 
however small, its point of contact with the ground, and therefore 
its centre of rotation, is changed. 

23. Besolntion of Velocities is the converse of the addi- 
tion of velocities, and has for its object the finding of com- 
ponents in two given directions, whose resultant motion shall be 
equal to the given motion. Hoc (fig. 6) be the given velocity, 
o b and a the directions of the required components, the latter 
are found by drawing from c straight lines, c b and c a, parallel 
respectively Xo oa and o b^ cutting them at b and a: o b and o a 
represent the required components. 

Example, — Suppose a cyclist to ride up an incline of one in 
ten at the rate of ten miles an hour. To find at what rate he 

rises vertically, draw a horizon- 
tal line A B (fig. 9) ten inches 

A — ^ 2J — jp* long, and a vertical line B C 

p,^^ ^ one inch long ; join A C. 

Along this line to any conve- 
nient scale mark o^ A Z>, the velocity ten miles an hour (14I feet 
per second). Draw D E 2X right angles to A B, meeting A B^ 
produced, if necessary, at ^. D E is the required vertical velocity 
of the cyclist. By measurement this is found to be i '46 feet per 
second (less than i mile per hour). 

Example, — A cyclist is riding along the road with a velocity 

indicated in direction and magnitude by O A{^%, 10). The wind is 

blowing with velocity (9^, and is therefore partially in the direction 

in which the cyclist is riding. To find the apparent 

^_-4 ^J direction of the wind, that is, its direction relative 

\ /i i *^ *^^ moving bicycle, join A B and draw O C 

\ / \ I equal and parallel to A B ; O C will be the 

\ / \ ' velocity of the wind apparent to the cyclist, which is 

W \j thus apparently blowing partially against him. The 

Jr ^ velocity O C can be resolved into two, O D dead 

Fig. 10. against the cyclist, and D C sideways, CD being 

drawn at right angles to A O. For, from the 

parallelogram of velocities it is seen that the actual velocity, O By 

of the wind relative to the earth is compounded of its velocity 

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CHIP. in. Kinematics; Addition of Velocities 



21 



relative to the bicycle O (7, and the velocity of the bicycle, O Ay 
relative to the earth. 

The above figure may explain why cyclists seldom seem to 
feel a back wind, while head winds seem always to be present. 

24. Addition and Besolntion of Acoelerations.— An accele- 
ration involves the idea of magnitude and direction, but not 
position ; it may, therefore, be represented by a vector. Figs. 6 
and 7 are, therefore, directly applicable to the compounding and 
resolving of accelerations. 

25. Hodograph — If a body move in any path, its velocity at 
any instant, both as to direction and magnitude, can be con- 
veniently represented by a vector drawn from a fixed point ; the 
curve formed by the ends of such vector is called the hodograph 
of the motion. 

26. Vniform Ciroolar Motion. — The hodograph for uniform 
circular motion can easily be found as follows : — When the 
body is at A (fig. 11), its velocity is in the direction A A^, 
From a fixed point 
(fig. 12) set off 
oa equal and paral- 
lel to ^^». When 
the body is at ^ its 
velocity is repre- 
sented by B B\ 
equal in length to 
A A^ ; the corre- 
sponding line b 
on the hodograph 
(fig. 12) is equal and 
parallel to B B^. Repeating this construction for a number of 
positions of the moving body, it is seen that the hodograph abc 
is a circle. 

Since the direction of motion changes from instant to instant, 
the moving body must be subjected to an acceleration, which 
can be determined as follows : — When the body is at A^ its 
velocity is represented by a, and when 2itBhy ob ; therefore, in 
the interval of passing from A to B 2Ln additional velocity, repre- 
sented by a b, has been impressed on it. If the point B be taken 




Fig. II. 



Fig. 



22 Principles of Mechanics chap. ni. 

very close to A^ i.e, if a very short interval of time be taken, 
b will be very close to a, and therefore a b^ the direction of the 
impressed velocity, will be parallel to A (9, i,e, directed towards 
the centre of rotation. If the interval of time is taken suffi- 
ciently small, the additional velocity ab \% also very small, and 
the resultant velocity o b does not sensibly differ in magnitude 
from a \ thus the only effect of the additional velocity is to 
change the direction of motion from o a io o b (fig. 1 2). 

When at B suppose the body to undergo the same operation, 
at the end of it the direction of the motion will be c. After a 
number of such operations the body will be at Z? (fig. 11), and its 
velocity will be represented by od {fig, 12). The total additional 
velocity imparted to it between the positions A and D has only 
had the effect of changing the (Jirection, but not the magnitude 
of its velocity. This total additional velocity is represented by 
the arc a d. 

Now, suppose the body to take one second to pass from A to 
Z>, then a d represents the increase of velocity in unit of time, and 
is, therefore, numerically equal to the acceleration a. Let v be 
the linear speed of the body, and r the radius of the circle in 
which it moves ; then the arc A D is numerically equal to v, o a 
is by definition equal to z/, and since o a and o d are resf>ec- 
tively parallel to the tangents at A and Z>, the angle a od is equal 
to the angle A O D ', therefore, 

a __ 2JQ ad __ zxc AD ^v 
V "" radius ao"^ radius A O ~ r 

^'^' « = ^- (0 

That is, in uniform circular motion, the radial acceleration is 
proportional to the square of the speed, and inversely propor- 
tional to the radius. 



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23 



CHAPTER IV 

KINEMATICS— PLANE MOTION 

27. BefinitioDS of Plane Motion. — If a body move in such 
a manner that each point of it remains ahvays in the same 
plane, it is said to have plane motion. Plane motion can be 
perfectly represented on a flat sheet of paper ; and, fortunately 
for the engineer, most moving parts of machines have only plane 
motion. In cycling mechanics there are more examples of 
motion in three dimensions. The motion of the wheels as the 
machine is moving in a curve and the motion of a ball in its 
bearing are examples of non-plane motion. 

Each particle of a body having plane motion will describe a 
plane curve, which is called 2i point-path, 

28. General Plane Motion of a Bigid Body.— The plane 
motion of a rigid body may be — 

(i) Simple translation^ without rotation. In this case any 
straight line drawn on the rigid body always remains in the same 
direction. The motion of the body will be completely determined 
if that one point of it is known. 

(2) Rotation about a fixed axis, — In this case the path of any 
point is a circle of radius equal to the distance of the point from 
the axis of rotation. 

(3) Translation combined with a motion of rotation, — We 
shall see later that in this case it is possible to represent the 
motion at any instant by a rotation of the body about an axis 
perpendicular to the plane of motion, the position of the axis, 
however, changing from instant to instant. 

If the paths of two points of a rigid body be known, the path 
of any other point on the rigid body is determined. Let A^ i?, 
and 6* (fig. 13) be three points rigidly connected, ^i^nioving on 

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24 



Principles of Mechanics 




the curve aa^ B on the curve b b. The path of the point C can 
evidently be found as follows : — Let A ^ be any position of the 

point on the curve a a ; 
the corresponding posi- 
tion -^i is found by draw- 
ing an arc with centre 
Ax and radius A By 
cutting the curve bb in 
Bi, With centres A^ 
and B^y and radii A C 
and B C respectively, 
draw two arcs intersect- 
ing at C,. Cwill be a 
point on the path de- 
scribed by C 

29. Instantaneous Centre.— Let A and B (fig. 13) be two 
points of a rigid body, a a and b b their respective point-paths. 
In the position shown the direction of the motion of -^ is a 
tangent at the point A^ to the curve a a. The point A may 
therefore be considered to rotate about any point, w, lying on the 
normal at -^1 to the curve a a. For, if A be considered to rotate 
either about nix or ^2> ^^^ direction of the motion at the instant 
is in either case the same tangent, A^ ax* In the same way, 
since the tangent Bx bx is also the tangent to any circle through 
B having its centre on the normal -5| «,, the point B may be 
considered to rotate about any point in the normal at Bx to the 
curve bb. If the normals Ax nix and B^ «, intersect at /, A and 
B may be both considered to be rotating at the instant about the 
centre /. No other point in the plane satisfies this condition, 
/ is therefore called the instantaneous centre of rotation of the 
rigid body. Every point on the rigid body is at the instant 
rotating about the centre /, therefore the tangent at Cx to the 
point-path ^ r is at right angles to Cj /. 

30. Point-paths, Cycloidal Cnrves.— A few point-paths de- 
scribed in simple mechanisms are ofgreat importance in mechanics. 
We will briefly notice the most important. 

Cycloid,— li a circle roll, without slipping, along a straight 
line, the curve described by a point on its circumference is called 

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CHAP. IV. Kinematics — Plane Motion 25 

a cycloid. Let a circle roll along the straight . line XX {^%, 14). 
The curve described by the point C on its circumference can be 
readily drawn as follows : — Divide the circumference of the circle 
into a number of equal parts (twenty-four will be convenient^ as 
this division can be effected by the use of the 45° and 60° set 
squares), and number the divisions as shown. Through the 
centre draw a straight line parallel \.o XX\ this will be the path 
of the centre of the circle. Along this line mark off a number of 
divisions, each equal in length to those on the circumference of 
the circle, and number them correspondingly. When any point, 




Fig. 14. 



say 9, on the circumference of the circle is rolled into contact 
with the line X X^ the centre of the circle will be on the corre- 
sponding point, 9, of the straight line. Draw the circle in this 
position. The corresponding position C^ of C is evidently ob- 
tained by projecting over from the point 9 of the circumference. 
By repeating this process for each of the points of division, twenty 
four points on the cycloid will be obtained j through these a fair 
curve may be drawn freehand. The curve C© C C, shows one 
portion of the cycloid. The point-path is a repetition, time after 
time, of this curve. 

Prolate and Curtate Cycloid, — The path described by a point, 
Dy inside the rolling circle is called a prolate cycloid. D^ D Z>, 
shows one complete portion of the curve. The method of draw- 
ing it is exhibited in figure 14, and hardly requires any further 
explanation. 

The curve described by a point lying outside the rolling circle 
is called a curtate cycloid. E^EE^ (fig. 14) shows one complete 
portion. ^ i 

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26 



Principles of Mechanics 



CHAP. IV. 



A point on the circumference of a bicycle wheel describes a 
cycloid as the machine moves in a straight line. Any point on 
the spokes, or any point on the crank, describes a prolate cycloid. 
Epicycloid and Hypocycloid, — If one circle roll on the cir- 
cumference of another, the curve described by a point on the 
circumference of the rolling circle is called an epicycloid or a 
hypocycloid^ according as the rolling circle lies outside or inside 
the fixed circle. These curves are of great importance in the 
theory of toothed- wheels. 

In figure 15, -SjE is an epicycloid and H H z. hypocycloid, in 
each of which the diameter of the rolling circle is one-third that of 
the fixed circle. The method of draw- 
ing these curves is similar to that of 
drawing the cycloid, the only difference 
being that the divisions along the path 
of the centre of the rolling circle will 
not be equal to those along the cir- 
cumference of the rolling circle, but the 
divisions along the fixed and rolling 
circles will correspond. 
^*"- '5- ^ particular case occurs when the 

diameter of the rolling circle is equal to the radius of the fixed 





Fig. 16. Fig. 17. 

circle ; the hypocycloid in this case reduces to a straight line, a 
diameter of the fixed circle. ..^.^.^^^ ^^ Google 



CHAP. IV. 



Kinematics — Plane Motion 



27 



Epitrochaids and Hypoirochoids, — If the tracing point does not 
lie on the circumference of the rolling circle, the curve traced is 
called an epitrochoid or a hypotrochoid. Figures 16, 17, and 18 
show some examples of epitrochoids and hypotrochoids. 

Involute, — Let a string be wrapped round a circle and have a 
pencil attached at some point ; as it is unwound from the circle 





Fig. 18. 



Fig. 19. 



the pencil will describe a curve on the paper, called an involute 
(fig. 19). This curve is also of great importance in the theory ot 
toothed-wheels. 

The involute is a particular case of an epicycloid. If the 
rolling circle be of infinitely great radius its circumference will 
become a straight line. The curve traced out by a point /^(fig. 19) 
of a straight line, which rolls without slipping on a circle, is an 
involute. 

31. Point-paths in Link Mechanisms. — We have already 
shown how to find the path described 
by any point of a rigid body of 

which two point-paths are known. If / x _,^— -t-\j^j5^ p 

the paths a a and b b (fig. 1 3) be cir- 
cular arcs, the bar A B may be con- 
sidered as the coupling link between 
two cranks. The variety of curves 
described by points rigidly connected ^»g. 20. 

to such a coupling link is very great ; some of them have been 
of great practical use. Figure 20 shows a point-path described 




28 



Principles of Mechanics 



CnAP. IT. 



by a tracing point, /*, which does not lie on the axis of the link 
AB, 

In Singer's * Xtraordinary ' bicycle the motion given to the 
pedal was such a curve. The mechanism and the path described 
by the pedal are discussed in chapter xxix, 

32. Speeds in Link Meohanisms.— If the speed of any point 
in a mechanism be known, it will in general be possible to de- 
termine that of any other point. In a four-link mechanism, 
A B CD (fig. 21), in which CZ>is the fixed link, the nature of the 
motion will depend on the relative length of the links. H D A 

be the shortest, A B -\- D Ahe less 
than C Z> + ^ C, and ^ i^ - DA 
be greater than C D -- B Cy D A 
will rotate continuously, and C ^ os- 
cillate. The speeds of points on the 
lever C B 2sq proportional to their 
distances from the fixed centre of 
rotation C\ similarly for points on 
the lever D A. Now in any position 
of the mechanism the link A B may 
be considered to have a rotation about the instantaneous centre /, 
the point of intersection of AD and CB, produced if necessary ; and 
thus the linear speed of any point of the link is proportional to its dis- 
tance from /. If the point A rotates with uniform speed, the point 
B will oscillate in a circular arc with a variable speed. Let v^ be the 
uniform speed of A^ and v^ the corresponding speed of B. Then, 
since the body A B is rotating at the instant about the centre /, 

V, IB 

Draw Z>^ parallel to C B^ meeting A B^ produced if necessary, 
at e. Then the triangles A Dcy A I B SLve similar, and therefore 

/A ^DA 
IB De' 




Fic. 21. 



and 



Vf, _ D e 
Va DA' 



or 



v^ 



DA 



De 



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(•) 



CTA?.iv. Kinematics — Plane Motion 29 

Now DA is constant whatever be the position of the mechanism, 
and therefore if Voi the speed of A^ be constant, the speed Of the 
point B is proportional to the intercept D e, 

Mark off 2?/along D A equal to D e. The length Df is thus 
proportional to the speed of the point B when the crank DA is in 
the corresponding position. If this construction be repeated for 
all positions of the crank D A, the locus of the point/ will be the 
fo/ar curve of the speed of the point B, 

33. Speed of Knee-joint when Pedalling a Crank.— In 
pedalling a crank-driven cycle, the motion of the leg from the hip 





Aoi Ai 

Fig. 23. 

to the knee is one of oscillation about the hip-joint. If the ankle 
be kept quite stiff during the motion, as, unfortunately, is too 
often the case with beginners, the leg from the knee-joint down- 
wards practically constitutes the coupling-link of a four-link 
mechanism. The pedal-pin (fig. 22) rotating with uniform speed, 
figure 23 shows the curve of speed of the knee-joint It may be 
noticed that the maximum speed of the knee during the up-stroke 
is less than during the down-stroke. Also, the point B is at the 
upper end of its path when the pedal-pin is in the position ^,, 
some considerable distance after the vertical position D A^ of the 
crank ; while B is in its lowest position when the pedal pin is at 

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30 Principles of Mechanics chap. it. 

A<^, The angle A^ D A^ passed through by the crank dunng the 
down-stroke of the knee is less than the angle passed through 
during the up-stroke ; consequently, since the speed of the pedal- 
pin is uniform, the average speed of the knee during the down- 
stroke is less than during the upstroke. If the rider can just 
barely reach the pedal when at its lowest point, the speed of the 
knee-joint is very great immediately before and after coming to 
rest at the lowest point of its path. 

34. Simple Harmonic Motion. — If /' be a point moving 

with uniform speed in a circle of radius r of which a d is 3. 

diameter, and Pp be a perpendicular let fall on a d (fig. 24), 

while /'moves in the circle, the point/ will move backwards and 

forwards along the straight line a b. The point / is then said to 

have simple harmonic motion. The motion of a point on a 

jh ^. vibrating string, and of a particle of air 

l^_^^ ^ in an organ-pipe when the simplest pos- 

j/^"'/S\\ ^^^^^> '^ °^ '^^^ character. The speed of 

/ ^/y\^\^ P will vary with its varying position. At 

/ oj^ \ w any instant the velocity of the point P is 

r / P j in the direction P /», the tangent at P, 

j\y^ / Setting off v=P m to scale along this line 

l/\^^^^^^^>/ it may be resolved into two components 
^ Pn and n m respectively parallel, and at 

^'°- **• right angles, to a d. The parallel com- 

ponent Pn is, of coiu^e, equal to the speed of the point /. If the 
scale of z^ be taken such that P m is equal to r, the triangles 
Pm n and P op are equal, and therefore P p is equal to P n. 
That is, in any position of the point p moving along a b with simple 
harmonic motion, its speed may be represented by the ordinate 
p Pio the circle on a ^ as diameter. 

If P moves uniformly in the circle, its acceleration is constant 

in magnitude and equal to - , and is in the direction of the radius 

Po. The scale of acceleration may be chosen such that the 
vector P o represents the acceleration of P, which may be decom- 
posed into Pp and / o respectively at right angles, and parallel to, 
a b. The parallel component p o is, of course, equal to the 
acceleration of the point/ along a b — that is, in simple harmonic 

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CHAP. IT. Kinematics — Plane Motion 3 1 

motion the acceleration is proportional to the distance of the 
moving point from the centre of its motion. If an ordinate/ Q 
be set off equal to o /, the locus of Q will be the acceleration 
diagiara of the motion ; this locus is a straight line A B passing 
through 0^ the centre of the motion. 

The motion of the knee-joint when pedalling approximates to 
simple harmonic motion, the approximation being closer the 
shorter the crank D A(^%. 22) is in comparison with the lever C B 
and connecting-link A B, If the motion were exactly simple har- 
monic motion, the polar curve of speed of knee-joint (fig. 23) 
would consist of two circles passing through D, 

35. Besnltant Plane ILo^ou.— Resultant of Two Transla- 
tions.— 1{ a rigid body be subjected to two motions of translation 
simultaneously, the resultant motion will evidently be a motion of 
simple translation, which can be found by an application of the 
parallelogram of velocities. 

Resultant of Two Rotations about Parallel Axis, — Let a body 
be subjected to two rotations, w, and wg* about the axes A and B 

J C, 

1^/ 




A C B 

Fig. 26. 

(fig. 25). If the motion be plane, the axes must be parallel, and at 
right angles to the plane of the motion. Let P be any point in the 
body. Join P to A and B, and draw P a and P d at right angles 
to /i4 and /'^respectively. The resuUant linear velocity of Pwill 
be the resultant of the velocity w^xA Pin the direction P a, and 
of «2 X B~P in the direction Pd. l( P a and Pdhe marked off 
respectively equal to these velocities to any convenient scale, 
the resultant P c can be found by the parallelogram of velocities. 
From P draw a perpendicular P Q on the line, produced if 
necessary, joining the centres A and B, Draw a a^ and b b^ per- 
pendicular to PQ. Then, the velocity of /'due to the rotation w, 
about A may be resolved into the velocity a* a parallel to, and the 
velocity P a^ at right angles to, A B. Similarly, the velocity of P 

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32 Principles of Mechanics chap. it. 

due to the rotation wj about B may be resolved into the two 
components P b^ and b^ b. The triangles A Q P and P a^ a are 
similar ; so, also, are the triangles B Q /*and Pb^ b. It is, there- 
fore, easy to show that the components of Ps velocity due to «,, 
at right angles, and parallel, to A B, are respectively (w, x A~Q) and 
(w, xQ P)' Similarly, the components due to wj are (wg xB Q) 
and (.12 X (2 /"). Therefore, the components of Ps resultant 
velocity at right angles, and parallel, to A Bsire respectively : — 

Vi={i^^xA Q)-{-Oo^xB Q) .... (2) 

and 

V2 = {i02^^\)^Q (3) 

Let C be a point on the straight line A B, dividing it in the 
inverse ratio of the angular speeds wi and m^, then 

A C 0).2 

CB ~ ;7, 

and 

A C = —^-AB, C Ji = '—A B 

Substituting A Q^A C-\-C Q, and B Q=C Q-C B in (2), it 
is easily deduced that 

f'l = (w, 4- W2) ^^ (4) 

From (3) and (4) it is evident that the resultant velocity of P 
is (wi + W2) ^^- That is, any point P, and therefore the whole 
body, is rotating with angular speed equal to the sum of the 
component angular speeds, about a parallel axis in the same 
plane, and distant from the axis of the component rotations 
inversely as the component angular speeds. 

The above result can be more simply attained by an applica- 
tion of the principle of * addition of vectors.' Let p be the vector 
A P^ from the axis A to any point P of the rotating body, and 
let a be the vector A B, Then Pa\%^ vector of magnitude q>, />, 
at right angles to p; B P '\s the vector (p — a) ; and Pb is a 
vector of magnitude 012(f) — a), at right angles to (p — a). 

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CHA?. nr. Kinematics — Plane Motion 33 

Vector Pc = vector Pa + vector Pb 

= a),p + a>2(p — a) 

= (wi + wj) (p — — 5!!l_ a ) 

= (o), + 0)2) (p — -4 C) 

= («, + o),) C-P, and at right angles to C P, 

That is, any point P rotates about the axis C (where 
AC\ CB = Ola : w,) with angular speed equal to the sum of the 
component angular speeds. 

Let figure 26 be a view of the body taken in a direction at 
right angles to that of figure 2$^ A B now representing the plane 
of the motion. The rotation (i>| may be represented by a line 
AAyZi right angles to AB, its length representing, to some 
scale, the magnitude of the rotation w,. In the same way B B^ 
may represent the rotation w^. The resultant rotation, C C, is 
equal to the sum of the rotations w, and wg, and takes place about 
an axis whose distances from A and B are inversely proportional 
to the rotations 01 , and o>.2. 

Thus, rotations about parallel axes can be represented in the 
same way as parallel forces, and their resultant found by the 
methods used to find the resultant of parallel forces (see 
chapter vi.). 

Example, — Find the instantaneous centre of rotation of the 
crank of a front-driver geared two to one. Let n be the 

number of revolutions the cranks make in a ^ ^ 

minute, the wheel makes 2 n revolutions, 

and the crank must make n revolutions 

backward relative to the wheel— />. makes 

- n revolutions per minute. The crank's 

motion may be considered as the resultant 

of a rotation 2 n about B, the point of 

contact of the wheel with the ground, and ^^j-- 

a rotation — n about the wheel centre A ^'°" ^'^' 

(fig. 27). Applying the preceding results, the instantaneous centre 

is on the line A By and 

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34 Principles of Mechanics chap. r?. 

A C _ 2jn _ _ 
CB ^n 

That is, AC^-zCB 

or AC^2BC=2AB. 

The motion of the cranks relative to the ground is, therefore, 
the same as if they were fixed to a wheel twice the size of the 
driving-wheel, and running on a flat surface below the ground. 

Translation and Rotation. — Let a body be subjected to a 
rotation a», about an axis A (fig. 25), and to a translation with 
velocity z; in a direction /i/ in the same plane as that of the 
motion. From A draw Af at right angles to//",. Let /'be 
any point on the body. From P draw P Q ^t right angles to 
Af, Then proceeding as before, the components of Ps resultant 
velocity at right angles and parallel to Af^Q respectively 

z;, = (co, X ^ - f/ (5) 

2^2 = 0)1 X QP (6) 

Let C be a point on Af such that (u), x -4 C) = » ; then (5) 
becomes 

v^—ia^x{AQ — AC) — iii^xCQ ... (7) 

By comparing (6) and (7), it is evident that the resultant 
velocity of P is one of rotation about the centre C with angular 
velocity w,. Thus, the resultant of a rotation and a translation is 
a rotation of the same magnitude about a parallel axis, the plane 
of the two axes being at right angles to the direction of transla- 
tion. 

Example. — A cycle wheel, relative to the frame, has a motion 
of rotation about the axle ; the frame, and therefore the axle, has 
a motion of translation. The instantaneous motion of the wheel 
is the resultant of these two motions. The resultant axis of rota- 
tion of the wheel is the point of contact with the ground. 

36. Simple Cases of Belative Motion of Two Bodies in 
Contact — In the theory of bearings it is important to know the 
relative motion of the portions of two bodies in the immediate 
neighbourhood of the point of contact, the motion of the bodies 

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CHAP. nr. Kinematics — Plane Motion 35 

being such that they remain always in contact. Before discussing 
the general case we will notice a few simpler examples. It will 
be convenient to consider one of the bodies as fixed, we will 
then have to speak only of the motion of one of the bodies ; this 
may be done without in any way altering the relative motion. 

Sliding, — If the motion of the body can be expressed as a 
simple translation, * sliding ' is said to take place at the point of 
contact With this definition, pure sliding can only exist con- 
tinuously when the surface of either the fixed or moving body is 
cylindrical ; the elements of the surfaces at the point of contact 
will constitute a ' sliding pair.' An example is afforded by the 
motion of a pump-plunger in its barrel. 

Rolling. — If the instantaneous axis of rotation passes through, 
and lies in the tangent plane at, the point of contact of the fixed 
and moving bodies, the motion is said to be * rolling ' ; the 
rolh'ng is therefore the same as the relative rotation. At the 
point of contact of a wheel rolling along the ground, the motion 
is pure rolling. The position of the instantaneous axis con- 
tinually changes ; but in plane motion it always preserves the 
same direction. 

Spinning. — If the instantaneous axis of rotation passes 
through, and it is at right angles to the tangent plane at, the 
point of contact, the motion is similar to the spinning of a top, 
and may be called spinning. An example of pure spinning is 
found at the centre of a pivot-bearing. 

Rubbit^. — In a turning pair, the motion can be expressed as a 
simple rotation about the axis of the pair. For example, the 
motion of a shaft of radius r in a plain cylin- 
drical bearing is a rotation, w, about the centre 
o of the bearing (fig. 28). The motion can also 
be expressed as an equal rotation, w, about a 
parallel axis through P^ a point on the surface 
of the bearing, and a translation with speed ^'^' '^* 

t? = <i» r in the direction PT2X right angles to O P. The motion at 
Z^is kinematically more complex than * sliding,' as above defined, 
and yet there is nothing of what is commonly understood as 
rolling; we may give it the name rubbing. Thus, rubbing at 
any point on the surface of contact of a cylindrical shaft of radius 




y--^ •^ T^ 



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36 



Principles of Mechanics 



r is equivalent to a translation v and a rotation — about an axis, 

parallel to that of the shaft, passing through the point in question. 
More generally, let A and B be two bodies in contact at the 
point P (fig. 29), let r^, and r^ be their respective radii of curva- 
ture at P^ and let / be the instantaneous axis of rotation of 
angular speed u). / must lie on the common normal at P^ since 
the bodies remain in contact during the motion. Suppose A 
fixed, and that the same point of the body B rubs along A with 
speed Viox at least two consecutive instants. The motion^ of B 
on A may then be said to be pure rubbing. In this case / must 







Fig. 99. 



A i;)4>fc 



Fig. 30. 



^ 



^ 



evidently coincide with the centre of curvature of the body A at 

the point P \ then C^^, the rubbing of B on A^ takes place with 

speed, ^ = 0) r^, and is therefore equivalent to a translation 

y 
Va and a rotation — , or 



£/,= />; and-" 

'a 



(8) 



Similarly, if the position of / be such that the same point of A 
rubs on B for at least two consecutive instants, 



K,= K and 



(9) 



37. Combined Rolling and Rnbbing.— In figure 29 let ^ be 
fixed, and let the motion of the body B be kinematically a 
translation F„ = F, and a rotation w^ = w about the point of 
contact P. The motion at P is compounded of tubbing and 

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CHAP. IT. Kinematics — Plane Motion 37 

rolling. The rubbing has already been defined ; Rf^y the rolling 
of B on Ay will be the total motion less the rubbing, />. — 

i?„ = (K,andcu,) -^F^and^") 

V V 

= Wa —'*=<»> — (10) 

'a 'a 

In using the formula (10) the positive directions of the axis 
of ci>^ of r^, and of ^ should be taken so that, in the order 
named, they form a right-handed system of rectangular axes. 
That is, looking along the positive direction of the axis of o>, a 
positive rotation, o>, will appear clock-wise, and the positive direc- 
tion of r if rotated a right angle in the positive direction of w, will 
come into the positive direction of F. r^ and r^ may be taken 
positive for convex surfaces, negative for concave surfaces. The 
positive directions of oi^, r^ and K« are shown in figure 29. 

In figure 30 let the relative motion of the bodies be exactly the 
same as in figure 29, but let B be fixed. Then V^ and co,, will be 
oppositely directed in space to V^ and (o„ respectively. But with 
the above conventions as to positive directions, taking r^ positive, 
V^ will be positive and equal to V^ (05 will be negative and equal 
to — o). Therefore 

i?, = «o, - ^^=-(0-^" (11) 

From formulae (10) and (11) it is seen that when rolling and 
rubbing combined take place, the * rollings ' 
of the two bodies are not reciprocal. The 
actions at the points of contact in the two 
bodies are not reciprocal, as may be illus- 
trated by a few examples. 

Example I, — Let the bodies A and -5 be a //////?, 
plane and cylinder of radius r respectively, in 
contact at P {^g. 31). Let the instantaneous 
axis of rotation coincide with the axis of the cylinder, and let w be 
the angular speed of B relative to A, Then at P\— ra= <x> ; rf, = r; 
the speed of rubbing F= K„, — - w r. 

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38 



Princifdes of Mechanics 



ra 

I? ^ - ^r ^ 

^ft = — <i> — = — O) = O. 

n r 

That is, the cylinder's motion on the plane is compounded of a 
rubbing of speed co r, and a rollmg of 
angular speed cu. The plane's motion 
on the cylinder is one of pure rubbing 
with speed co r. 

Example IL — Let the bodies A and 
j9 be a circular bearing and shaft respec- 
f iG. 32. tively, of the same radius r (fig. 32), w 

being the angular speed of the shaft. Then at /*, r« = — r, r^ = r, 

F=K«= — w r, and 

Ka. = 01— -™=w— =: O 




^1,=:— W— — ^— toi — 



Thus the definitions given in (10) and (11) of the magni- 
tudes of the rollings of one body on the other are consistent 
with our usual conceptions in these simple cases. 



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39 



CHAPTER V 



KINEMATICS : MOTION IN THREE DIMENSIONS 

38. Besnltant of Trandatioiui.— If a body be subjected to a 
number of translations in different directions in space, the re- 
sultant velocity can be found by finding the resultant of any two 
of the given translations, which resultant must evidently lie in the 
same plane as the two given translations. The resultant of a 
third given translation with the resultant of the first two can again 
be found by the same method ; and so on for any number of 
given translations. Thus the resultant of any number of transla- 
tions in space is a motion of translation. 

39. Resultant of Two Eotations aboat Intersecting Axes.— 
Let the axes O A and O B oi the rotations intersect at the point 
(fig. 33). The rotations wi and w^ may be represented by the 
length of the lines O A and O B respectively, and since rotations 
are resolved -and com- 
pounded like forces, the 
resultant rotation will be 
represented by the dia- 
gonal OC qA the paral- 
lelogram of which O A 
and OB are adjacent 
sides. This proposition 
is called the parallelogram ofrotatiom. In using this proposition, 
attention must be paid to the sense of the rotation. The lengths 
of the lines representing the magnitudes of the rotations must be 
set off along the axes of the rotations in such directions that when 
looking in the positive directions the motions both appear either 

in watch-hand direction, or both in contra watch-hand direction. 

In figure 33, the rotations are both in watch- hand direction ; the 




i—g^B 



Fig. 33. 



40 



Principles of Mechanics 



resultant rotation about the axis O C will therefore be in the 
direction indicated by the arrow. 

The above proposition is so important that a separate proof 
depending on first principles will be instructive. Let O A and 
O Bh^ the axes of rotation, and let /* be a point on the body 
lying in the plane A OB, Draw Fa and Fb perpendicular to 
OA and OB respectively. If F lie in the angle between the 
positive directions OA and O B^ the linear velocity of F^ which 
is in a direction at right angles to the plane of the axes, will be 



wj aF — to^b F 



(0 



If /*lie on the axis of resultant rotation its velocity is zero, and 
(3) becomes ia^aF — w^b F ^^ O^ 

^^bF 
a'o a F' 



or 



Draw Fc and Fd parallel respectively to O B and O Ay meeting 
O A and O B 2Xc and d respectively. Then, the triangles Fac 
and Fbd diXQ similar, and therefore — 



bF ^Fd ^ 
aF Fc' 



Oc 
Od 



(2) 



That is, O Fis the diagonal of a parallelogram whose adjacent 
sides coincide with the direction of the axis of rotation, and are 
of lengths respectively proportional to the component angular 
velocities about these axes. 

40. Besultant of Two Botations about Non-interseotin^ 

Axes.— Let A A and 
BB {fig. 34) be the 
two axes, and let ^ A 
be the common per- 
pendicular to A A 
and BB. Through 
h draw a line CC 
parallel to A A, 
Then by section 35, 
the rotation (U) about 
the axis ^ ^ is equivalent to an equal rotation about the axis C C, 




Fig. 34. 



caAP. V, Kinematics: Motion in Three Dimensions 41 

together with a translation in the direction ^^ at right angles 
to the plane containing A A and C C, The resultant of the rota- 
tions about the axes B B and C C is, by section 39, a rotation 
about an axis D D passing through h. Thus, the given motion is 
equivalent to a rotation about an axis D Z>, and a translation in 
the direction h k. The translation in the direction // k may be 
resolved into two components, h I along D D and / ^ at right 
angles to D D. By section 35, the rotation about D D and the 
translation in the direction/^ are equivalent to an equal rotation 
about a parallel axis E E* Thus, finally, the resultant motion is 
a rotation about an axis E E and a translation in the direction of 
that axis. Such a motion is called a screw motion. 

41. Koft Oeneral Motion of a Eigid Body. — In the same 
way it can be shown that the resultant of any number of transla- 
tions and of any number of rotations about intersecting or non - 
intersecting axes may be reduced to a rotation about an axis and 
a translation in the direction of that axis. If a common screw 
I)olt be fixed and its nut be moved, the motion imparted is of 
this character. The motion of the nut can be specified by giving 
the pitch of the screw and its angular speed of rotation about 
its axis. In the same way, the motion of a rigid body at any 
instant can be expressed by specifying the axis and pitch of its 
screw, and its angular speed. 

42. Most Oeneral Motion of Two Bodies in Contact. We 
have seen that the most general motion of a rigid body can be 
resolved into a rotation w and a translation v in the direction of 
the axis of rotation. Also that a rotation about any axis is 
equivalent to an equal rotation about a parallel axis through any 
point, together with a translation at right angles to the plane of 
the parallel axes. Hence, if two bodies move in contact, the 
relative motion at any point of contact can be resolved into a 
translation, and a rotation about an axis passing through the point 
of contact. The direction of the translation must be in the 
tangent plane at the point ; since, if the two bodies move in con- 
tact, there can be no component of the translation in the 
direction of the normal. 

Let figure 35 be a section of the two bodies A and -5 by a 
plane, passing through the point of contact /*, at riglu-angkp to 



42 



Principles of Mechanics 



CHAP. V. 



the instantaneous axis of rotation //. The body A may be con- 
sidered fixed, the body B to have a rotation w round, and a 
translation v along, //. If PI be perpendicular to //, the 
motion of B is equivalent to a rotation w about the axis Pa^ 
parallel to //, together with a translation w . IP along Pb 2X 
right angles to the plane di PI and Pa^ plus a translation v 
along Pa. The resultant of these two translations is a translation 

F along Z^^. Pcm\isX lie in 
the common tangent plane to 
the surfaces at P. 

Let PNhe, the normal at 
P, and Pd the intersection 
of the tangent plane with the 
plane containing PNdJid Pcl 
Then, the rotation <■» about 
Pa can be resolved into rota- 
tions w, and 0)^ about P N 
and Pd respectively. Thus, 
the motion at P consists of 
translation with velocity V in 
the direction -Pr, a spinning, 
ai„ about the normal PN^ 
and a rolling, ai„ about the 
Therefore the most general 
contact is compounded of 




Fig. 35. 



axis P flying in the tangent plane, 
relative motion of two bodies in 
* rubbing,' * rolling,' and * spinning.' 

We have in the chapter on Plane Motion given examples of 
the pure motions just mentioned. We shall see, in the chapter 
on Bearings, that the motion of a ball on its path in the ordinary 
form of adjustable bearing is compounded of rolling and 
spinning ; while, in some special ball-bearings, the motion at 
the point of contact of a ball with its path is compounded of 
rubbing, rolling, and spinning. 



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43 



CHAPTER VI 

STATICS 

43. OrapMc Bepresentation of Force. — For the complete 
specification of a force acting on a body, its direction, line of 
application, and magnitude are required. A force can therefore 
be represented completely by a straight line drawn on a diagram, 
the length of the line representing to scale its magnitude, the 
direction and position of the line giving the direction and positions 
of application of the force. Thus a force can be represented by 
a localised \tciox, 

44. Parallelogram of Forces.— When two or more forces 
are applied at the same point, a single force can be found which 
is equivalent to the original forces. This is called the resultant 
force, and the original forces are called the components. If the 
forces act in the same direction, the resultant is, of course, equal 
to the sum of the component forces. If two forces act in opposite 
directions, the resultant is the difference of the two. Generally, if 
a number of forces act along a straight line, some in one direc- 
tion, others in the opposite direction, the resultant of the whole 
system is equal to the difference between the sum of the forces 
acting in one direction and that of the forces acting in the 
opposite direction- 
Suppose two forces acting at a point in different directions 

are represented \yj oa and ob respectively (fig. 36), then it is 
evident that some force such as ^ ^ in a direction between o a and 
ob will be the resultant. The resultant ^^ is found by completing 
the parallelogram oacb and drawing the diagonal ar, exactly as 
in the case of the parallelogram of velocities. 

Want of space prevents a strict elementary mathematical 
proof of this proposition, but it can be easily verified experi- 



44 



Principles of Mechanics 



mentally as follows : Fasten two pulleys, A and B (fig. 37), on a 
wall, the pulleys turning with as little friction as possible on their 
spindles. Take three cords jointed together at O with weights 
JF,, Jf^2> ^^3 ^t their ends. Let the heaviest weight hang 
vertically downwards from (9, and let the other two cords be 
passed over the pulleys A and B respectively. Then, if the 
heaviest weight, I^F^, underneath O be less than the sum of the 
other two, the whole system will come to rest in some particular 



r---/> 





Fig. 36. 



Fig. 37. 



position. While in this position make a drawing on the wall of 
the three cords meeting at 0, Produce the vertical cord upwards 
to any point r, and from c draw parallels c a and cb \.o the other 
two cords. It will be found on measurement that the lengths 
O a^ Ob, and Oc are exactly proportional to the weights Jf^„ /F'j, 
and /F3. Thus the resultant of the forces along O a and Ob'vs 
given by the diagonal O c oi the parallelogram whose sides 
represent the component forces. 

Example, — The crank spindle of a bicycle is pressed vertically 
downwards by the rider with a force of 25 lbs., while the horizontal 

pull of the chain is 50 lbs. What is 
the magnitude and direction of the 
resultant pressure on the bearing? 
Set off O A (fig. 38) vertically down- 
wards equal to 25 lbs. and O B hori- 
zontally equal to 50 lbs. Complete 
the parallelogram OABC, The re- 
sultant is equal in magnitude and direction to the diagonal O C, 
which by measurement is found to be 55*9 lbs. 

45. Triangle of Forces.— Suppose that in addition to the 
two forces oa and ob (fig. 36) a third force, co, aqts at the point ; 

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Fig. 38. 



CHAP. Ti. Statics 45 

this third force being exactly equal, but opposite to, the resultant 
of the two forces. If these three forces act simultaneously no 
effect will be produced, and the body will remain at rest, be \s 
equal and parallel to o a, and may therefore represent in magni- 
tude and direction the force o a acting at A, The three sides 
oby bcy and co oi the triangle obCy therefore, taken in order, 
represent the three forces acting at the point and producing 
equilibrium. The proposition of the parallelogram of forces 
may therefore be put in the following form, which is often con- 
venient : 

If three forces act at a point and produce equilibrium they 
can be represented in magnitude and direction by the three sides 
of a triangle taken in order round the triangle. The converse 
of this proposition is also true. 

A very important proposition which can be deduced im- 
mediately from the triangle of forces is, that if three forces act 
on a body and produce equilibrium they must all act through 
the same point. 

46. Polygon of Forces. — Since forces acting at a point can 
be represented by vectors, the resultant ^ of a number of forces, 



a 





Fig. 39. Fig. 40. 

/I, ^, r, //, and ^, acting at the same point (fig. 39) can be found 
by drawing a vector polygon (fig. 40) whose sides represent the 
given forces ; the resultant vector R represents the resultant 
force. If a force equal, but oppositely directed, to R acted at 
the same point as the forces a, ^, c^ d^ and tf, they would be in 
equilibrium. Therefore, if a number of forces acting at a point 
are in equilibrium, they can be represented in magnitude and 
direction by the sides of a polygon, taken in order round the 
polygon. Conversely, if a number of forces acting at a point 

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46 Principles of Mechanics chap. ti. 

are represented in magnitude and direction by the sides of a 
polygon taken in order, they are in equilibrium. 

In the preceding paragraph it must be clearly understood 
that the sides of the polygon represent the forces in magnitude 
and direction, but not in position. Thus the sides of the polygon 
a, ^, Cy d^ e (fig. 40) represent in magnitude and direction the five 
forces acting at the same point. If a body were acted on by 
forces represented by the sides of a polygon, in position as well 
as in magnitude and direction, a turning motion would evidently 
be imparted to it. 

47. Besultant of any Number of Co-planar Forces. — The 
resultant of any number of forces all lying in the same plane 
acting on a rigid body, and which do not necessarily all act at 
the same point, may be found by repeated applications of the 
principle of the parallelogram of forces. The resultant -^2 of 
any two of the given forces P^ and /g passes through the point 
of intersection of the latter ; the resultant -^3 of -^2 and a third 
force, /a, passes through the point of intersection of R^^ and -^3 ; 
and so on. This process is very tedious when a great number 
of forces have to be dealt with. The following method is more 
convenient : 

Let figure 41 represent the position of the given forces, and 
figure 42 the corresponding force-polygon P^ P^. . . . The 
resultant P of all the given forces is evidently represented in 
magnitude and direction by the line a/ forming the closing side 
of the polygon ; for if a force of magnitude and direction /a 
were added to the given forces, the resultant would be of zero 
magnitude. It only remains therefore to determine the position 
of the resultant P on figure 41. 

No difference will be made if two equal and opposite forces 
be added to the system. We will add a force Qy represented by 
Oa in the force-polygon, which acts along any line a (fig. 41). 
The resultant of Q and P^ is Od (fig. 42), and it passes through 
^,, the point of intersection of Q and P^ (fig. 41). Draw from 
the point />, the line ^ parallel to O d (fig. 42), cutting the line of 
action of P^ at /a- The resultant of Q^ Pi, and P^is Oc (fig. 42), 
and it passes through /j- Draw from the point /, the line c 
parallel io O c (fig. 42). Continuing this process, the resultant 

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CHIP. TI. 



Statics 



47 



of Q, /*!, F^ jP^, F^, and F^ is Of (fig. 42), and acts through 
the point /j. From /j draw the line / parallel to Of (fig. 42), 





Fig. 41. 



Fig. 42. 



cutting the line a, the line of action of the added force Q, at/^- 
The resultant of Of and — Q is «/= ^ (fig. 42), and it acts 
through the point /^ 

The above construction may be expressed thus : Take any 
pole O and from it draw radius vectors to the comers of the force- 
polygon. Draw another polygon, which may be called the iink- 
pofygofiy having its comers /i, /« . . . on the lines of action of 
the given forces /*,, F^, . . . and having its sides a, b^ . , , 
parallel to the radius vectors O a^ Ob . .' . of the force-polygon ; 
the sequence of sides and corners a, /„ ^, /a • • • in the link- 
polygon being the same as that of the corners and sides 
a, /^j, b, Fi, ... of the force -polygon. The point of inter- 
section of the first and last sides of the link -polygon determines 
the position of the resultant 7?. 

It is readily seen from the above, that if a system of forces 
acting on a rigid body are in equilibrium, both the force- and link- 
polygons must be closed. 

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48 



Principles of Mechanics 




Fig. 43. 



T^ 



48. Besolntion of Forces.- A single force may be resolved 
into two components in given lines which intersect on the line of 
action of the given force. The principle of the parallelogram of 

forces is, of course, used again here. 
Let o c (fig. 43) be the given force 
acting at <?, and let its components in 
the directions a and ob\}^ required. 
From c draw c a and c b respectively 
parallel io bo and a <?, meeting o a 
and bm a and b respectively : o a 
and o b are the required components 
of the given force in the two given directions. 

Example. — Given the vertical pressure on the hub of the 
driving-wheel of a Safety bicycle, to find th^ forces acting along 

the top and bottom forks, O A and O B 

(fig. 44). 

Draw O c vertical and equal to 
the given pressure on the hub. This 
is the direction and magnitude of the 
force with which the wheel presses on 
the hub spindle. From c draw c a 
and c b parallel to O B and A O 
respectively, meeting O A and B O 
produced in a and b respectively, oa 
and o b are the forces acting along the 
top and bottom forks respectively. It 
will be seen that the top fork O A is 
compressed and the bottom fork O^ is in tension. 

Resolution of a Force into Three Components in given Directions 
and Positions. — Let ^ be a force whose components acting 
along the given lines /*,, P^, and P^ (fig. 45) are required. Let 
P and /*, intersect at A, P^ and /*3 intersect at B. Then R may 
be resolved into two forces acting along /*, and A B respectively, 
the latter into two forces acting along /j ^"^ Pz respectively. 
The constructions necessary are indicated in fig. 46. 

Any force, R^ acting on a rigid body can be resolved into two, 
one acting along a given line /*,, the other passing through a 
given point B, The latter force must pass through A^ the point 

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Fig. 44. 



CHAP. VI. 



Statics 



49 



of intersection of R and /*,. The construction is clearly shown 
in figures 45 and 46. 

If the point of intersection A 
be inaccessible, as in figure 47, the 
link-polygon method may be used 
with advantage. In the force dia- 
gram (fig. 48) set off af equal to R 
to any convenient scale, draw fb 
parallel to Pj. Commence the 
link-polygon at B, by drawing the 
side a parallel to the vector O a, 
then draw the side / parallel to 
the vector O/, cutting the line of 
action of J^i at /i. The closing 
side b of the link-polygon is the 
straight line /i B, Draw the 
vector O b parallel to the side b 

of the link-polygon, cutting the side P^ of the force triangle at b. 
The force P^ is represented in magnitude and direction by the 




Fig. 45. 



Fig. 46. 




Fig. 47- 



Fig. 48. 



third side ab o\ the force triangle. Comparing with figures 41 
and 42, the truth of the above construction is obvious. 

49. Parallel Forces. — I^t two parallel forces P^ and P.^ act 
on a body (fig. 49). Required to find their resultant. It is 
evident that the resultant force R is equal to the sum /*, + 
P^ ; the only element to be found is the point at which it acts. 
I^ A B hea line in the body at right angles to the directions of 
Pi and P2, and let C be the point at which the resultant R acts. 

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Principles of Mechanics 




Fig. 49. 



Let another force, Q^ equal and opposite to R^ be applied to the 
body ; then since it is equal and opposite to the resultant of Px 
and /*2> the body is in equilibrium under the action of the three 

forces Px'i A> and Q, Consider the 
moments of the forces about the point 
C \ that of Q is zero, and, therefore, 
the algebraic sum of the moments of 
/*i and /a must also be zero, since the 
body is in equilibrium. Therefore, 
P^ X C~B ^ PyXs AC . (i) 
that is, the point C divides A B into two parts inversely propor- 
tionate to the forces P^ and /*2- 

If the forces P^ and P^ acted in opposite directions (fig. 50), 
paying attention to the sign of the moments, it is seen that the 

point C will lie beyond A^ the point 
of application of the larger force. 
Here again 

P^ X C^= Px X ATc . (i) 

The above is often referred to 
as the principle of the lever. The 
experimental verification is easy. 

The resultant of any number of parallel forces /*„ Z^, can 

be found by the method of figures 41 and 42 ; the force-polygon 
(fig. 42) becoming in this case a straight line. 

50. Hass-oentre. — An important case of finding the resultant 
of a number of parallel forces is finding the centre of gravity of 
a body. The earth exerts an attraction on every part of the 
body, and therefore the resultant force of gravity on the body is 
the resultant of a great number of parallel forces. 

Considering a body as made up of an indefinite number of 
small particles of equal mass, the mass-centre of the body is a 
point such that its distance from any plane is the mean distance 
of all the particles from that plane. If the body is subjected to 
gravitational attraction, every particle is acted on by a force, the 
total force acting on the body is the resultant of all such forces. 
The centre of gravity is a point at which the total mass of the 
body may be considered to be concentrated, in considering its 

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Fig. 50. 



CHAP. Tl. 



Statics 



SI 



attraction by other bodies. When the attractions on the particles 
of a body are proportional to their mass, as is practically the case 
on the surface of the earth, the mass-centre and the centre of 
gravity of a body are coincident. 

If the density of the body is uniform, the mass centre will also 
be the geometrical centre of figure ; in fact, it is the geometrical 
centre of figure that is of importance in problems on mechanics. 

The mass-centres for a few important cases may be given here. 

Circular^ Square;^ or Rectangular Disc, — If these discs be cut 
out of metal plate of uniform thickness, it is evident that the 
mass-centre will also be at the geometrical centre of the figure. 

Triangle, — Let A B C {^%. 51) be a triangle, which we may 
consider cut out of thin metal plate. Consider any narrow strip, 
//, parallel to the side -5 C ; the 
mass-centre /i of this strip is at 
the middle of its length. Divid- 
ing up the triangle into a number 
of such slips, their mass-centres 
will all lie on the line A a, joining 
A to the middle point of B C. 
In the same way, by dividing the 
triangle up into a number of 
strips parallel to A B^ it may be 
seen that the mass-centres of all the strips will lie on the line Cc 
joining C, the middle point of A B, The mass-centre of the 
whole triangle must lie somewhere on the line A a ; it must also 
lie somewhere on the line Cc; (9, the 
point of intersection of these lines, is 
therefore the mass-centre. It can easily 
be proved that aO is one-third of a Ay 
and Co one-third of c C. 

Circular Arc— L^i AB (fig. 52) be 
a portion of a circular arc with centre O, 
Consider the moment about any dia- 
meter OX. L-et Z'/* be a portion 
of the arc so short that it does not 
sensibly differ from the straight line F F^j and its length is 
n^ligible in comparison with the radius. The mass may be 





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by Google 



52 



Principles of Mechanics 



considered proportional to the length of the line, and we may 
therefore say that the moment oi PP^ about 0X\% PP^ x Pp^ ; 
Pp^ being drawn perpendicular to O X \ and P^ Q being neg- 
ligible compared with Pp^. 

Draw YYdL tangent to the circle and parallel to the axis O A'; 
from A^ P^ P^ and B project a, /, /* and b on this tangent, the 
projectors being at right angles to it. Draw PQ parallel to, and 
P^ Q at right angles to O X, the two lines meeting at Q. Join 
OP. Then, since the triangles PP^ Q and O Pp\ are similar, 

PP^ PO 



PQ pp: 



the 
the 



Therefore, P P^ x Pp^ = PQ x PO=pp^ x pp\—i>e^ 
moment of the arc PP^ about the axis O X is equal to 
moment of the straight line pp^ about the same axis. 

This holds for all the elements of which the arc A B may be 
considered made up. Therefore, by summing the moments of 
these elements we get the important result, that the moment of 
the arc A B about the axis ^ AT is equal to the moment of the 
straight line ab^ its projection on the tangent parallel to the axis. 

If the arc under consideration be a semicircle of radius r, 
and G be its mass-centre, its length is 7rr, the length of its pro- 
jection on the tangent is 2 r, and we get 

•!rrxOG = 2rxr. 



Therefore 



(9G' = 



{2) 



€ B 



Sector of a Circle. — The mass-centre of 
a sector of a circle OAB (fig. 53) is found by 
dividing it up into a number of smaller sectors, 
O C B, the arc B C being so short as not to 
differ sensibly from a straight line. The sector 
O CB may then be considered a triangle, its 
mass-centre will be at a distance from O equal to 
two-thirds O B. Thus, the mass-centres of the 
small sectors into which OAB can be divided 
all lie on the arc a b^ whose radius is two-thirds that of the arc 

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Fig. 53 



Statics 



S3 




A B \ and therefore the mass-centre of the sector O A B is the 
same as that of the arc a b. 

In particular, the centre of area included between a semi- 

arcle and its diameter is at a distance — from the centre of the 

drcle. 

51, Couples. — If two parallel but opposite forces, -P, and P^ 
(fig- 54)> ai*^ ^so equal, their resultant is zero, they tend to turn 
the body without giving it a motion of translation. 
Two equal, parallel, but oppositely directed forces 
constitute a coupky whose magnitude is measured 
by the product PI of one of the equal forces 
into the perpendicular distance between their 
lines of action. A couple may be regarded as 
equivalent to a zero force acting at an infinite 
distance ; with this point of view they form no 
exception to the general case of finding the resultant of given 
forces. 

In the construction of figures 41 and 42, if the points a and/ 
of the force-polygon coincide, the resultant of the given forces is 
zero. If, in addition, the line/5/, is parallel to O a^ the link- 
polygon is also closed, and the given forces are in equilibrium. 
If, however, /j/, is not parallel to Oa^ the 
resultant of the given forces is a couple. 

Let two parallel forces /*, and P^ (fig. 55), 
each equal to /*, at a distance / apart, con- 
stitute a couple. The sum of the moments 
of the two forces about any point O in the 
plane of P^ and P^^ distant x from P,, is 



Fig. 54. 




Fig. 55. 



that is, the turning effect of a couple depends only on its moment 
PI, and not on the position of its constituent forces relative to 
the axis of turning. The axis of the couple is at right angles to 
its plane. 

Let a single force P act on a body at A (fig. 54). Introduce 
at B two opposite forces Py^ and 7^2* each equal to, and distant / 
from, P. No change in the condition of the body is. effected by 

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54 



Principles of Mechanics 



this procedure, since Px and P^ neutralise each other. But the 
system of forces may now be expressed as a single force P^ 
acting at B^ together with a couple PI formed by the forces P 
and /a. Thus, a force acting on a body at A is equivalent to an 
equal force acting at B^ together with a couple of transference PL 

A couple may be graphically represented by a vector parallel 
to its axis— ;.^. at right angles to its plane ; the length of the 
vector being equal, to some scale, to the moment PI of the 
couple. 

52. Stable, TTiistable, and Neutral Equilibrium. — If a heavy 
body be situated so that a vertical line through its mass-centre passes 
within its base it is in equilibrium. If the vertical line through 
the mass-centre fall outside the base, the body is not 
in equilibrium, and will fall unless otherwise supported. 
If a body, supported in such a way that it is free to 
turn about an axis O (fig. 56), be left to itself it will 
come to rest in such a position that its mass-centre 
G will be vertically underneath the axis of suspension 
O, If the body be displaced slightly, so that its mass- 
centre is moved to G^^ when left to itself it will 
return to its original position. In fact, the forces 
now acting on the body are, its weight acting downwards through 
6^*, and the reaction at the support O acting vertically ; these 
two forces form a couple evidently tending to restore the body 
to its original position. In this case the body is said 
to be in stable equilibrium. 

If now the body be placed with its mass-centre 
above O (fig. 57), though in equilibrium, the smallest 
displacement will move G sideways, and the body 
will fall. The equilibrium in this case is said to be 
unstable. 

If the mass-centre of the body coincide with 
the axis of suspension, the body will remain at rest in any position, 
and the equilibrium in this case is said to be neutral 

A body may have equilibrium of one kind in one direction, 
and of another kind in another direction : thus a bicycle resting 
on the ground in its usual position is in stable equilibrium in a 
longitudinal direction, and is in unstable equilil 




Fig. 56. 




Fig. 57. 



"'^"?J5?)^c 



a trans- 



CHAP. n. Statics 5 5 

verse direction. A bicycle wheel resting on the ground is in 
neutral equilibrium in a longitudinal direction, and in unstable 
equilibrium in a transverse direction. 

53. Besnltant of any System of Poroes. — Concurrent forces,— 
If the given forces all pass through the same point, but do not 
all lie in the same plane, the method of section 46 can be ex- 
tended to them ; their resultant will be represented as before, 
by the closing side of the vector-polygon, the only difference from 
the case of coplanar forces being that the vector-polygon is no 
longer plane. Thus, the resultant of a system of concurrent forces 
is either zero or a single concurrent force. 

Non-concurrent^ non-planar forces, — Let P,, P^^ ... be the 
given system of forces. Take any point O as origin and introduce 
two opposite forces, /i and — /i, each equal and parallel to Fy 
No change is made by this procedure, since /i and — /, neutralise 
each other. The force /*, is therefore equivalent to a single force 
px acting at (9, and a couple of transference P^ l\ ; A being the 
length of the perpendicular from O to P,, and the axis of the 
couple being perpendicular to the plane of P, and /|. Similarly, 
Pj is equivalent to an equal and parallel force /j acting at O, 
together with a couple of transference P^ 1% \ and so on for all the 
given forces. The resultant of the concurrent forces /,, /a • • • 
is either zero or a single concurrent force, /. Since the couples 
^\ Ai A 4j • • • are vector quantities, their resultant is also a 
similar vector quantity — />. a couple C Hence the resultant of 
any system of forces can be expressed as the sum of a single force 
P and a couple C, 

The magnitude of / does not depend on the position of the 
origin O^ while that of C does. The couple C can be resolved 
into two couples C and C, having their axes respectively parallel 
to, and at right angles to, the direction of /. The resultant of / 

and C is a force /', equal to, parallel to, and at a distance - 

/ 
in a direction at right angles to the plane of / and the axis C 
from, /. Thus, finally, the resultant of any system of forces can 
be expressed as a single force /' and a couple C" having its axis 
parallel to/. 

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56 Principles of Mechanics chap, nu 



CHAPTER VII 

DYNAMICS — GENERAL PRINCIPLES 

54. Laws of Hotion. — In section 13 we have seen that the 
measurement of force is closely associated with that of motion. 
The general phenomena of force and motion have been summed 
up by Newton in his well-known laws of motion : 

I. Every body continues in its state of rest or of uniform 

motion in a straight line, except in so far as it may be 

compelled by applied forces to change that state. 

II. Change of motion is proportional to the force applied, 

and takes place in the direction in which the force acts. 

III. The mutual actions of any two bodies are always equal 

and oppositely directed in the same straight line ; or, 

action and reaction are equal and opposite. 

These laws apply to forces acting in the direction of the 

motion, and also to forces acting in any other direction. A force 

like the latter will alter the direction of the body's motion, and 

may, or may not, increase or diminish its speed. It follows from 

Newton's first law that any body moving in a curved path must 

be continually acted on by some force so long as its motion in 

the curved path continues. 

55. Centrifugal Force. — An important case of motion, es- 
pecially to engineers and mechanicians, is uniform motion in a 
circle. If a stone at the end of a string be whirled round by 
hand, the string is drawn tight and a pull is exerted on the hand. 
This pull is called centrifugal force. At the other end the string 
exerts a pull on the stone tending to pull it inwards towards the 
hand. This pull is called the centripetal force, and it is the con- 
tinual exercise of this force that gives the stone its circular pwith. 
If this force ceased to act at any instant the stonje would continue 

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CHAP. Til. Dynamics — General Principles 57 

its motion, neglecting the influence of gravity, in a straight h'ne 
in the direction it had at the instant the centripetal force ceased 
to act 

The distinction between the two forces must be carefully kept 
in mind 

Every point on the rim of a rapidly rotating bicycle wheel is 
acted on by a centripetal force which is supplied partially by the 
tension of the spokes. If the speed of rotation gets abnormally 
high, the centripetal force required to give the particles in the rim 
their curvilinear motion may be so great that the strength of the 
material is insufficient to transmit it, and the wheel bursts. The 
flywheels of steam engines are often run so near the speed limited 
by these considerations, that it is not uncommon for them to 
burst under the action of the centrifugal stress. 

Let m be the mass in lbs. of the body moving with speed v 
feet per second in a circle of radius r. It has been shown (sec. 26) 

that the radial acceleration n is — . But if / be the radial force 
acting, by section 14, 

/ = /^a = poundals, or/ = lbs. . . (i) 

56. Work. — When a force acts on a body and produces 
motion it is said to do work. If a force acts on a body at rest, 
and no motion is produced, no work is done. The idea of 
motion is essential to work. If a man support a load without 
moving it, although he may become greatly fatigued, he cannot 
be said to have done mechanical work. The load, as regards 
its mechanical surroundings, might as well have been supported 
by a table. If the applied force be constant throughout the 
motion, the work done is measured by the product of the force 
into the distance through which it acts. The practical unit of 
work is \hQ foot-pound^ which is the work done in raising a weight 
of one pound through a vertical distance of one foot. 

It should be noted particularly that the idea of time does not 
enter into work ; the work done in raising one ton ten feet high 
being the same whether a minute or a year be taken to perform 
it In the same way, the work done by a cyclist in Jtiding^up a 

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58 



Principles of Mechanics 



CHAP. Til. 



hill of a given height is the same whether he does it slowly or 
quickly. 

The work done in raising a body through a definite height is 
quite independent of the manner or path of raising, neglecting 
frictional resistance and considering only the work done against 
gravity. The work a cyclist does against gravity in ascending a 
hill of a certain height is quite independent of the gradient of the 
road over which he travels. 

Example, — Let the machine and rider weigh 200 lbs., then 
the work done by the rider in rising 100 feet vertically is 
20,000 foot-lbs. If the gradient of the road be known, this can 
be calculated in another way, which, for the present purpose, is 
roundabout but instructive. Consider an extreme gradient of 
one vertical to two on the slope (fig. 58), the length of the hill 
will be 200 feet. The work done in 
ascending the hill may be estimated by 
the product of the force required to 
push the machine and rider up the hill, 
into the length of the hill. The machine 
and rider weigh 200 lbs. ; this force acts 
vertically downwards, and can be re- 
solved into two, one parallel to the 
road's surface, and one at right angles 
to it. \i Oa be set off equal to 
200 lbs., and the construction of section 48 be performed, it will 
be found that the component b O required to push the machine 
and rider up the hill is 100 lbs. The work done will be the 
product of this force into the distance through which it acts, 
200 feet ; the result, 20,000 foot-lbs., being the same as before. 

This is only the work done against gravity. In riding along 
a level road there is no work done against gravity, any resistance 
being made up of the rolling friction of the wheels on the road, 
air resistance, and the friction of the bearings. These resistances 
will remain, to all intents and purposes, the same on an incline 
as on a level. The work done in riding along 200 feet of level 
road would have to be added to the 20,000 foot-lbs. of work 
done against gravity, in order to get the total work done by the 
cyclist in ascending the hill. ^ i 

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Dynamics — General Principles 59 



1 



Generally, the work done by, or against, a force is the product 
of the force into the projection on the direction of action of the 
force of the path of the moving body. Thus, if a 
body move from A to B^ and be acted on by the force 
/ which always retains the same direction, the work 
done is / A~C \ B C being perpendicular X.o A C V'5 

(fig. 59). ^^^-^^ 

The centripetal force acting on a body moving in a circle is 
always at right angles to the direction of motion ; consequently 
in this case the projection of the path is zero, and no work is 
done. 

In the Simpson lever-chain the pressure of the chain rollers 
on the teeth of the hub sprocket wheel is at right angles to the 
surface of the teeth, and consequently makes a considerable angle 
with the direction of motion of the rollers. In this case, there- 
fore, the projection A C (fig. 59), on the line of action of the 
pressure, of the distance A B moved through, is very much less 
than A B, The claims of its promoters vutually amount to saying 
that the work done on the hub by the pull of the chain is/. A B, 
whereas the correct value is /. AC. 

In driving a cycle up-hill, the work done against gravity by 
the rider at each stroke of the pedal is the product of the total 
weight and the vertical distance moved through during half a turn 
of the crank axle. Let the gradient be x parts vertical in 100 on 
the slope, D the diameter in inches to which the driving-wheel is 
geared, and W the total weight of machine and rider in lbs. 
The vertical distance passed through per stroke of pedal is 

^-.'^^ inches. 

100 2 

The work done per stroke of pedal is therefore 

'^^ ^inch-lbs. 
200 

= '001309 ^Z> ^foot-lbs (2) 

Table I., on the following page, is calculated from equa- 
tion (2). ^ T 

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6o 



Principles of Mechanics 



fMAV, -VII. 



Table I. — Work done in Foot-lbs. per Stroke of 
Pedal, in Raising ioo lbs. Weight against Gravity. 



Diameter, 


1 




Gradient. Darts n 100 






to which 


1 
















driving-wheel 
















8 


is geared 


■. 


2 


3 


4 


5 


6 


7 


Inches 


















40 


5-24 


IO-47 


15-70 


20-94 


26-18 


31-41 


46-65 


41-89 


t 45 


5-89 


1 1 78 


1767 


23-56 


29-45 


35-34 


41 23 


47-12 


1 50 


6-55 


13-09 


19-63 


26-18 


3272 


39-27 


47-97 


52-36 


55 


7 -20 


14-40 


21-60 


2879 


36-00 


43-20 


5039 


57-59 


60 


7-86 


15-71 


2356 


31-42 


39-27 


47-12 


54-97 


6283 


65 


8-51 


1 7 02 


25-52 


34-04 


42-54 


51-05 


59-56 


68-08 


70 


916 


18-32 


27-49 


36-65 


45-81 


54-98 


64-14 


73-30 


1 75 


9-82 


19-64 


29-45 


39-27 


49-09 


58-90 


6872 


l^'H 


1 80 


10-47 


20-94 


31-41 


41-89 


52-36 


62-83 


73*30 


8378 



57. Power. — The rate of doing work is called the pmver of 
an agent, and into its consideration time enters. The standard of 
power used by engineers is the horse-power. Any agent which 
performs 33,000 foot-lbs. of work in one minute is said to be 
of I H.P. This, Watt's estimate, is in excess of the average 
power of a horse, but it has been retained as the unit of power for 
engineering purposes. The average power of a man is about 
one-tenth that of a horse- that is, equal to 3,300 foot-lbs. per 
minute. 

If Fbe the speed, in miles per hour, of a cyclist riding up a 
gradient oi x parts in 100, the vertical distance moved through in 
one minute is 



100 



60 



(3) 



and the power expended is 

•88 xVW foot-lbs. per minute .... 

Table II. is calculated from equation (3). 

58. Kinetic Energy.— So far we have dealt with the work 
done by a force which gives motion to a body against a steady 
resistance, the speed of the body having no influence on the 
question, further than it must be the same at the end as at the 
beginning. If a body free to move be acted on by a force, the 
work done will be expended in increasing its speed. The work ia 

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CHAP. Til. 



Dynamics — General Principles 



6i 



Table II. — Work Done, in Foot- lbs. per Minute, in 
Pushing ioo lbs. Weight Up-hill. 











Slope, 


parts in 100 






Speed. 
Miles per 


































hour 


I 

352 


2 
704 


* 3 
1056 


4 
1408 


5 
1760 


6 
2112 


7 
2464 


8 


4 


2S16 


5 


440 


880 


1320 


1760 


2200 


2640 


3080 


3520 


6 


528 


1056 


1584 


2112 


2640 


3168 


3696 


4224 


7 


616 


1232 


1848 


2464 


3080 


3696 


4312 


4928 


8 


704 


1408 


2112 


2816 


3520 


4224 


4928 


5632 


9 


792 


1584 


2376 


3168 


3960 


4752 


5544 


6336 


10 


880 


1760 


2640 


3520 


4400 


5280 


6160 


7040 


II 


968 


1936 


2904 


3872 


4840 


5808 


6776 


7744 


12 


1056 


2112 


3168 


4224 


5280 


6336 


7392 


8448 


13 


1 144 


2288 


3432 


4576 


5720 


6864 


8008 


9152 


M 


1232 


2464 


3696 


4928 


6160 


7392 


8624 


9856 


15 


1320 


2640 


3960 


5280 


6600 


7920 


9240 


10560 


i6 


1408 


2816 


4224 


5632 


7040 


8448 


9856 


1 1 264 


17 


1496 


2992 


4488 


5984 1 7480 


8976 


10472 


1 1968 


i8 


1584 


3168 


4752 


6336 


7920 


9504 t 


11088 


12672 


19 


1672 


3344 


5016 


6688 


8360 1 


10032 


1 1704 


13376 


20 


1760 


3520 


5280 


7040 


8800 


10560 


12320 


14080 



Stored in the moving body, and can be restored in bringing the 
body again to rest This stored work is called kinetic energy, 

59. Potential Energy. — Newton's first law of motion expresses 
the idea of permanence of motion of a body unless altered by 
applied forces. If the speed of a body on which no force acts 
remains constant, its kinetic energy must also remain constant. 
If a body free to move is acted on by a force, the work done by 
the force is stored up as kinetic energy. If work is done by 
moving the body against the resistance of a force which is 
constant in magnitude and direction, whatever be the direction 
of motion, the work is expended in changing the position of the 
body. For example, in raising a body from the ground, the 
resistance overcome is its weight, which always acts vertically 
downwards, whether the body be at rest or moving upwards or 
downwards. If the body be lowered by suitable means to the 
ground, the work done in raising it is again restored. The body 

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62 Principles of Mechanics chap. vn. 

at rest a certain height above the ground possesses therefore an 
amount of energy due to its position ; this is called potential energy. 
If the body be allowed to fall freely under the action of gravity, 
at the instant of reaching the ground it possesses no potential 
energy, but kinetic energy due to its speed. Its initial store of 
potential energy has been all converted into kinetic energy. 

60. Conservation of Energy. — The great principle of con- 
servation of energy is an assertion that energy cannot be created 
or destroyed. This is one of the most comprehensive generalisa- 
tions that has been deduced from our observations of natural 
phenomena. Applied to the case of a body moving under the 
action of force without any frictional resistance, it asserts that 
the sum of the kinetic and potential energies is constant. A 
cyclist riding down a short hill with his feet off the pedals- and 
not using the brake, will have a greater speed at the bottom than 
at the top, part of the potential energy due to the high position at 
the top of the hill being converted into kinetic energy at the 
bottom. If another short hill of equal height has to be ascended 
immediately, the kinetic energy at the bottom gets partially con- 
verted into potential energy at the t9p ; the rider arriving at the 
top of the second hill with the same speed as he left the first. 
The friction of the air, tyres, and bearings has been neglected in 
the above discussion. If the rider just work hard enough to over- 
come these resistances as on a level road, the above statement 
will be strictly true. 

Applied to mechanism used to transmit and modify power, the 
principle of the conservation of energy is sometimes quoted, * No 
more work can be got out at one end of a machine than is put in 
at the other.' The work got out will be exactly equal to that put 
into the machine, provided the friction of the machine is zero, 
an ideal state of things sometimes closely approached, but never 
actually attained in practice. The chronic inventor of cycle 
driving-gears might save himself a great deal of trouble by master- 
ing this principle. 

61. Frictional Besistance.— It is a matter of every-day ex- 
perience that a moving body left to itself will ultimately come to 
rest, thus apparently contradicting Newton's first law. A flat 
stone moved along the ground comes to rest very soon. If the 

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CHAP. ?n. Dynamics — General Principles 63 

stone be round, it may roll along the ground a little longer, while 
a bicycle wheel with pneumatic tyre set off with the same speed 
will continue its motion for a still longer period. A wheel set 
rapidly rotating on its axis will gradually come to rest. If the 
wheel be supported on ball-bearings, the motion may continue for 
a considerable fraction of an hour, but ultimately the wheel will 
come to rest. In all these cases there is a force in action 
opposing the motion, the force of friction^ which is always called 
into play when two bodies move in contact with each other. The 
amount of friction depends on the nature of the surfaces in con- 
tact. The friction is very great with the flat stone sliding along 
the ground, is less with the rolling stone, and still less with the 
pneumatic-tyred wheel. The friction of a ball-bearing may be 
reduced to a very small amount, but cannot be entirely abolished ; 
the less the friction, the longer the motion persists. The air 
also offers a considerable resistance to the motion, which varies 
with the speed. If a wheel with ball-bearings could be set in 
rapid rotation under a large bell-jar from which the air had been 
exhausted by an air-pump, the motion of the wheel might persist 
for several hours, and thus give a close approximation to an 
experimental verification of Newton's first law of motion. The 
movement of the planets through space affords the best illustration 
of the permanence of motion. 

62. Heat. — The force of friction is thus seen to diminish the 
kinetic energy of a moving body, while if the body move in a 
horizontal plane, its potential energy remains the same throughout, 
and energy is said to be dissipated. The energy dissipated is not 
destroyed, but is converted into hcai^ the temperature of the 
bodies in contact being raised by friction. Heat is a form of 
energy, and the conversion of mechanical work by friction into 
heat is a matter of every-day experience ; conversely, heat can be 
converted into mechanical work. Steam engines, gas-engines, 
and oil-engines are machines in which this conversion is effected. 
Heat due to friction is energy in a form which cannot be utilised 
in the machine in which it arises ; hence popularly engineers 
speak of the work lost in friction, such energy being in a useless 
form. 

In riding down-hill the potential energy of the machine and 

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64 Principles of Mechanics chap. vii. 

rider gets less ; if the speed remains the same, the kinetic energy 
remains the same, and the potential energy is dissipated in the 
form of heat. If a brake be used, the heat appears at the brake- 
block and the wheel on which it rubs. If back-pedalling be em- 
ployed, the same amount of heat is expended in heating the 
muscles of the legs, though the other physiological actions going 
on may be such as to render the detection or measurement of this 
heat difficult. 

Mechanical Equivalent of Heat, — The conversion of heat into 
work, and work into heat, takes place at a certain definite rate. 
780 foot-pounds of work are equivalent to one unit of heat ; the 
unit of heat being the quantity of heat required to raise the 
temperature of one pound of water one degree Fahrenheit. Thus, 
in descending a hill 100 feet high, a rider and machine weighing 
200 lbs. would convert 20,000 foot-lbs. of work into ^Uo^ ~ ^5*^ 
units of heat. If this could all be collected at the brake-block, it 
would be sufficient to raise the temperature of one pound of water 
25-6 degrees. 



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65 



CHAPTER VIII 
DYNAMICS {continued), 

63. Dynamics of a Particle. — A particle, an ideal conception 
in the Science of Mechanics, is a heavy body of such small 
dimensions that it may be considered a point. If a particle of 
mass m initially at rest, but free to move, be acted on for time / 
by a constant force/ we have seen (sec. 16) that the speed v 
imparted is such that 

ft^mv 

or 

/=7 (0 

f=i ma (2) 

o being the acceleration, or rate of change of speed, m v is the 

momentum acquired in time /, hence ''' ^ is the momentum ac- 

qmred in unit of time, and (i) is equivalent to defining force as 
' rate of change of momentum.' 

Let s be the distance traversed in the time / ; then since the 
average speed is half the speed at the end of the period, 

J = i z; / = i a /2 (3) 

The work done during the period is /y, and 

fs^\vft^\mv' (4) 

If the particle has initially a speed z/q, equations (i), (3) and (4) 
become 

ft^mi^v — v^) (5) 

s z=z\(p ■\' v^t (6) 

fs = {m {v^ - v\) (7) 

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(^ Principles of Mechanics chap. vm. 

Kinetic Energy, — The work done by the force has been ex- 
pended in giving the body its speed v^ and the body in coming to 
rest can restore exactly the same amount of work. The product 
\mv^\^ called the kinetic energy of the moving body ; it may be 
denoted by the symbol E, 

The units employed above are all absolute units. The unit of 
kinetic energy in (4) is the foot-poundal ; in foot-pounds the kinetic 
energy is 

^ = ^'^^' (8) 

Falling Bodies, — A body falling freely under the action of 
gravity is a special case of the above. Let the mass in be one 
pound, the force acting on the body is i lb. weight, i.e, g 
poundals. Writing g instead of/, and ;//=i, in equations (i)-(4) 
the formulae for falling bodies are obtained. 

64. Circular Motion of a Particle.— Let the particle be con- 
strained to move in a circle of radius r, and be acted on by a force 
of constant magnitude / which is always in the direction of the 
tangent to the path of the particle ; then since the radial force 
does no work, equations (i) to (7) still hold. Multiply both sides 
of (i) by r, then 

r mv r / V 

A=-^- (9) 

/r is the moment of the applied force about the axis of rotation, 
/// V is the momentum, m v r the moment of momentum or angular 
momentum ; hence the moment of a force is equal to the rate of 
change of angular momentum it produces. 

If 01 be the angular speed and d the angular acceleration of the 

particle about the axis at the end of the time t, v = ta r, = *^, 

and (9) may be written 

/m (J) /"^ o /J , \ 

r = z= m r^ 6 (10) 

The product ;// r^ is the moment of inertia of the particle about 
the axis of rotation, and may be denoted by /; (10) may then be 
written 

fr^.iQ (10) 

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CHIP. ym. Dynamics 6y 

That is, the moment of the force is equal to the product of the 
moment of inertia of the body on which it acts and the angular 
acceleration it produces. 

Equation (4) becomes, for this case, 

e=/s = ^mv^ = ^mr^Q}^ = ^iii}^ . . . (11) 

That is, the kinetic energy of a particle moving in a circle is half 
the product of its moment of inertia about the centre and the 
square of its angular speed. 
(9) may be written 

/fr=^mvr=mr^is} = i(o .... (12) 

/ / is the impulse of the force ; therefore the moment of the 
impulse is equal to the product of the moment of inertia of the 
I^rticle and the angular speed produced by the impulse. 

65. Rotation of a Lamina about a Fixed Axis Perpendicular 
to its Plane. — A rigid body of homogeneous material may be 
considered to be made up of a great number of particles, all of 
equal mass uniformly distributed. A rigid 
lamina is a rigid body of uniform, but inde- 
finitely small, thickness lying between two 
parallel planes ; a flat sheet of thin paper is a 
physical approximation to a lamina. Let O 
(fig. 60) be the fixed axis of rotation, perpen- 
dicular to the plane of the paper ; let ^ be 
any particle of the lamina distant r from O. 
Then using the same notation, equations (9) 
to (12) hold for the particle A, the acting 
force / being always at right angles to the 
radius O A. Now the rigid lamina may be 
considered made up of a number of heavy 
particles like A, embedded in a rigid weightless frame. Instead 
of the force / acting directly at A, suppose a force / act at a 
point B of the frame in a direction at right angles to O B, Let 
B ^l, then if 

Pl^fr (13) 

the effects of the forces / and p in turning the weightless frame 
and heavy particle A about the centre O are exactly the same ; 
the motion of A is unaltered by the substitution. 

F2 

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68 Principles of Mechanics chap. tih. 

Also, if ^ be the space passed over during the period by the 
point B^ 

^=ioTS = ^f and 

I r I y 

therefore 



. pi dr ^ . 
r I 



Substituting in (lo), (ii) and (12) they may be written 

pl^iO (14) 

tf=///=^la>2 (15) 

plt^ii^ (16) 

Let /■„ I'a • . • • t>e the moments of inertia of the heavy 
particles Ai, A^ . . . - of which the lamina is composed ; /,, 
/oj • • • • the corresponding forces at the point B required to 
give them their actual motions ; then for all the particles, (14), (15) 
and (16), may be written 

(/i + /i + . . . ) / = (A + ^2 + . . . )0 



(/i + A + • . ) ^ = H'l + ^2 • . • ) 



.« 



(A -^A + . • . )/^=(A + /2 + . . . )« 
/, /, and <i> being the same for all the particles. Let / = 
(i\ + /2 + • • • )> then / is the moment of inertia of the 
lamina about the axis (^ ; let (/i f />, + . . . ) = /*, then 
B is the actual force applied at the point B of the lamina ; let 
(<?i + <?2 + • • • ) = -^» then B is the kinetic energy of the 
lamina ; and the above equations may be written 

Bl=^/e (17) 

^=^/oi2 (18) 

Plt^Ii^ (,9) 

/* / is the magnitude of the applied turning couple. 

66. Pressure on the Fixed Axis.— In the above investigation 
the pressure on the axis at O has been neglected, since whatever 
be its value, its moment about O is zero, and it does not, therefore, 
influence the speed of rotation. It is, however, desirable to know 
the pressure on the bearings of the rotating body ; we therefore 
proceed to investigate it. Consider only the particle A^ connected 
by the rigid weightless frame to B and O ; if the force p dX B 

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CHAP. Till. Dynamics 69 

gives A its tangential acceleration, the weightless frame must press 
on the particle A with a force/ in the direction at right-angles to 
r, and the particle A must react on the frame with an equal and 
opposite force — / But the particle A also presses on the frame 



2 



with the centrifugal force ^= = w w^r, in the direction of 

the radius r. The frame being weightless must be in equilibrium 
under the forces acting on it ; since, by (2), a finite force, however 
small, acting on a body of zero mass would produce infinite 
acceleration. These forces are : —/at Ay p at B^ the reaction q 
of the axis at O^ and the centrifugal force r, which also acts 
through O, But the forces —/at ^, and/ at By are equivalent 
to equal and parallel forces at O^ and the couples — fr and / /. 
The couples equilibrate each other, therefore the four forces 
-/, /*, q and ^ at 6^ are in equilibrium. Therefore, 

vector q = vector/— vector p — vector c . . . (20) 

Let Q be the resultant reaction of the fixed axis on the lamina, 
due to the particles -^„ ^2> . • • of which it is composed, i.e. — 

vector Q = sum of vectors q\y q^ . . . 
Similarly, let 

vector jP= sum of vectors /,,/2 • • • 
vector 7^= sum of vectors/,, /a . . . 
vector C = sum of vectors r,, Ciy ^3 . . 

Then, adding equations (20) for all the particles A^y A2 . . ., 
vector Q = vector J^— vector P-- vector C . . (21) 

But by (10)— 

vector J*'^ mO X vector sum (^i -I- ^2 + . . . .) 

And the vector sum (^i + ^2 + . . . r) is the vector n . 01^ ; 
G being the mass-centre of the lamina (fig. 61), and n the number 
of particles, each of mass w, it contains. 
Therefore, 

vector 7^= MO . (TG (22) 

iV being the total mass of the lamina. The component forces/,, 
/i . . . acting at right-angles to the corresponding vectors r,. 

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70 



Principles of Mechanics 



CH4P. Tin. 



^2 . . ., the resultant force F will act at right angles to the re- 
sultant vector O G, Similarly, 

vector C = Mm^ . VG (23) 

the force C acting along CTG. 

Now, from (13) and (10)/ =-^^ = ^^^-^. 

The vectors/ are all in the same direction, at right angles to 
O B^ and are therefore added like scalars. Therefore, 



vector /*= X sum {m^ r^^-\'m<^r^ . . .) = 



le 



(24) 




Substituting these values in (21), the reaction Q (fig. 61) of 
the fixed axis is the resultant of: — A force at O equal and parallel 
to that required to accelerate the mass M 
supposed concentrated at 6^ ; a force at 
equal, opposite and parallel to the applied 
force P \ the centripetal force J/. <i>*^ . (TSi 
acting along G O, 

From (21) many important results can 
be deduced. Let a couple act on a rigid 
lamina quite free to move in its plane ; then 
P= o, ^ = o ; and (21) becomes 

Fig. 61. vector F — vector C = o. 

But the vectors F and — C are at right angles ; their sum can 
only be zero when each is zero. This is the case when 0~G = o 
— see (22) and (23) — that is, when the mass-centre and the axis 
of rotation coincide. Hence a couple applied to a lamina free to 
move causes rotation about its mass-centre. 

67. Dynamics of a Bigid Body.— Equations (17), (18) and 
(19) are applicable to the rotation of any rigid body about a fixed 
axis. Equations (21) to (24) are applicable if the rigid body is 
symmetrical about a plane perpendicular to the axis of rotation ; 
this includes most cases occurring in practical engineering. But 
in a non- symmetrical body, e^, a pair of bicycle cranks and their 
axle, the resultant pressure on the bearings cannot be expressed 



CHAP. vni. Dynamics 7 1 

as a single force, but is a couple. Thus, such a rigid body, if per- 
fectly free, will turn about an axis, in general, not parallel to that 
of the acting couple. 

From (23), the centrifugal pressure on the fixed axis of any 
rigid body is the same as if the whole mass were concentrated at 
the mass-centre G. If the mass-centre lie^ on the axis of rotation, 
the centrifugal pressure is zero. Hence the necessity of accu- 
rately balancing rapidly revolving wheels. In this case also (21) 
becomes Q= — P^ i.e, the pressure on the bearing is equal and 
parallel to the applied force, provided Q can be expressed as a 
single force. If only a couple be applied, P = o, and the pressure 
on the bearings is zero. In a rapidly rotating wheel with hori- 
zontal axis, P is the weight of the wheel ; with vertical axis /* = o, 
the weight acting parallel to the axis. 

The motion of a rigid body can be expressed (sec. 41) as 
a translation of its mass-centre, and a rotation about an axis 
passing through its mass-centre. Any applied force is equivalent 
to an equal parallel force at the mass-centre and a couple of 
transference. The rotation about the mass-centre is the effect of 
this couple. Hence, the turning effect of any system of forces 
acting on a free rigid body is the same as if its mass-centre were 
fixed. Since the resultant couple does not influence the motion 
of the mass-centre, the motion of the mass-centre of a rigid body 
under the action of any system of forces is the same as if equal 
parallel forces were applied at the mass- centre. 

The kinetic energy of any moving body is the sum of the 
energy due to the speed of its mass-centre, and the energy due 
to its rotation about the mass-centre. 

Moments of Inertia, — If M be the total mass of a rigid body, 
its moment of inertia may be expressed /= Mk*^ \ and k is 
called the radius of gyration. The / about an axis through the 
mass-centre is least : let it be denoted by /q ; that about any 
parallel axis distant h is 

I^I^^MH' (25) 

The values of / for a few forms may be given here. For a thin 
ring of radius r and mass M rotating about its geometric axis, 
/q = Mr^. This is approximately the case of the rim and tvre of 

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72 Principles of Mechanics chap. vin. 

a bicycle wheel. For the same ring rotating about an axis at its 
circumference, as in rolling along the ground, 7=2 Mr^, 

For a bar of length / rotating about an axis through its end 

perpendicular to its own axis, 7= This is approximately 

3 
the case of the spokes of a bicycle wheel. 

For a circular disc of uniform thickness and radius r rotating 

M r^ 
about its geometric axis, I^ = - — For the same disc rolling 

2 

along the ground, 7=5 Mr^, 

2 

68. Starting in a Cycle Race. — The work done by a rider at 
the beginning of a race is nearly all expended in giving himself 
and machine kinetic energy, the frictional resistances being small 
until a high speed is attained. If the winning-post be passed at 
top speed, the kinetic energy is practically not utilised. In a 
short distance race, this kinetic energy may be large in comparison 
to the energy employed in overcoming frictional resistances. The 

kinetic energy of translation of the machine and rider is — 

gr 

foot-lbs., W being the total weight. Hence, a light machine, 
other things being equal, is better than a heavy one for short races. 
P^urther, there is the kinetic energy of rotation of the wheels and 
cranks. For the rims and tyres this is nearly equal to their trans- 
lational kinetic energy ; therefore, at starting a race, one pound in 
the rim and tyres is equivalent to two pounds in the frame. In 
comparing racing machines for sprinting, the weight of the frame, 
added to twice that of the rims and tyres, would give a better 
standard than the weight of the complete machine. The pneu- 
matic tyre, with its necessarily heavier rim, is, in this respect, 
inferior to the old narrow solid tyre. Of course, once the top 
speed 's attained, the weight of the parts has no direct influence, 
but only so far as it affects frictional resistances. 

6(). Impact and Collision. — If two bodies moving in opposite 
directions collide, their directions of motions are apparently 
changed instantaneously ; but, as a matter of fact, the time during 
which the bodies are in contact, though extremely short, is still 
appreciable. The magnitude of the force required to generate 



CHAP. Tin. Dynamics 73 

velocity in a body, or to destroy velocity already existing, is in- 
versely proportional to the time of action ; if the time of action 
be very short, the acting force will he very large. Such forces 
are called impulsive forces. 

Now in the case of colliding bodies, such as a pair of billiard 
balls, it is impossible either to measure /or / ; but the mass m of 
one of the balls, and its velocities v^ and v before and after colli- 
sion, may easily be measured. The expression on the right-hand 
side of (5) denotes the increase of momentum of the body due to 
the collision ; the product// on the left-hand side is called the 
impulse ; therefore, from (5), the impulse is equal to the change 
of momentum it produces. 

We shall now have to examine more minutely the nature of the 
forces between two bodies in collision : At the instant that the 
bodies first come into contact they are approaching each other 
with a certain velocity. Suppose A (fig. 62) 
to be moving to the right, and B to the left ; 
immediately they touch, the equal impulsive 
forces /, and f^ will be called into action, 
and will oppose the motions of A and B 
respectively. The parts of the bodies in the 
neighbourhood of the place of contact will be flattened, and this 
flattening will increase until the relative velocity of the bodies is 
zero. The time over which this action extends is called the period 
of compression. If the bodies are elastic, they will tend to recover 
their original shapes, and will therefore still press against each 
other; the forces now tending to give the bodies a relative 
velocity in the direction opposite to their original relative velocity. 
These impulsive forces will be in action until the original shape 
has been recovered and the bodies leave each other. The time 
over which this action extends is called the period of restitution ; 
and the total impulse may be conveniently divided into two parts, 
the impulse of compression and the impulse of restitution. 

Index of Elasticity, ^l^oyf it is an experimental fact that in 
bodies of given material the impulse of restitution bears a constant 
ratio to the impulse of compression ; this ratio is called the index 
of elasticity, A perfectly elastic material has its index of elasticity 
unity ; in an inelastic body the index of elasticity is zero ; if the 

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74 Principles of Mechanics chap. vm. 

index of elasticity lies between zero and unity, the body is imper- 
fectly elastic. The index of elasticity e is, for balls of glass |f, 
for balls of ivory ^, and for balls of steel 4- These are the values 
given by Newton, to whom the theory of collision of bodies is due. 
Conservation of Momentum, — In figure 62, the force /i at any 
instant acting on A is exactly equal to the force /2 acting on B ; 
the total impulse on A is therefore equal to the total impulse on 
B \ and as they are in opposite directions their sum is zero. Thus, 
the momentum of the system is the same after collision as before 
it. This is true whether the bodies are inelastic, imperfectly 
elastic, or perfectly elastic. If two bodies of mass »i, and Wj, 
moving with velocities v^ and v^' respectively, collide, their 
velocities after collision can be easily determined, if the index of 
elasticity e is given. For cyclists, the most important case is 
when one of the bodies is rigidly fixed ; in other words, when m^ 
is infinite and v^' zero. Let, as before, the mass of the finite 
body be ///, its velocities before and after collision with the 
infinite body be v^ and v ; then before collision its momentum is 
;// z/q' Let C be the impulse of compression ; then since at the 
end of the compression period the velocity is zero, we get by 
substitution in (i) 

C =. m Va^ (26) 

The impulse of restitution, by definition, is ^ C ; therefore, if 
V be the velocity of the body after collision, we have 
^ C = — mvo 
Substituting the value of C from (26), we get 

v=-'evQ (27) 

That is, the speed of rebound is equal to the speed of impact 
multiplied by the index of elasticity. The speed of rebound is 
therefore always less than the speed of impact. 

This result at first sight seems to be contradictory to the prin- 
ciple of the conservation of momentum, but remembering that the 
mass of the fixed body may be considered infinite, and its velocity 
zero, its momentum is 

00x0, 

an expression which may represent any finite magnitude. We 
may say the fixed body gains the momentum lost by the moving 

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CHAP. VIII. 



Dynamics 



75 



body by the collision. For example, when a ball falls vertically 
:uid rebounds from the ground, the earth as a whole is displaced 
by the collision. 

Loss of Energy, — The kinetic energy of the moving body 
before impact is 

^^«' foot-lbs. ; 

the kinetic energy after impact is 

^^.^;,^^^^, 

2^ 

The loss of energy due to collision is thus 

(i-^^)^o 



(29) 

70. Gyroscope.— I^t a wheel W (fig. 63), of moment of 
inertia /, be set in rapid rotation on a spindle S, which can be 
balanced by means of a counterweight w^ on a pivot support T 
(fig. 63). If a couple C, formed by two equal and opposite 
vertical forces F^^ and F^ acting at a distance /, be applied to the 
spindle, tending to make it turn about a horizontal axis, it is found 
that the axis of the spindle turns slowly in a horizontal plane. 
This motion is called * precession.' This phenomenon, which, 





Fig. 63. 

when observed for the first time, appears startling and paradoxical, 
can be strikingly exhibited by removing the countenveight iv^ so 
that statically the spindle is not balanced over its support. The 
explanation depends on the composition of rotations. Figure 64 
is a plan showing the initial direction O A^ of the axis of rotation 
of the wheel W, The initial angular momentum of the wheel 
can be represented to any convenient scale by the length O Aq. 
The couple C tends to give the wheel a rotation about the axis 



76 Principles of Mechanics chap. vni. 

O B zt right angles to O Aq. If this couple C acts for a very 
short period of time, /,, the angular momentum it produces about 
the axis OB is C/,. This may be represented to scale by O ^q. 
The resultant angular momentum of the wheel at the end of the 
time, /,, may therefore be represented in magnitude and direction 
by OAK If the time /, be taken very small, OA^ is practically 
equal to O Aq, and the only effect of the couple C is to alter the 
direction of the axis of rotation. At the end of a second short 
interval of time, t^y it may be shown in the same manner that 
the axis of rotation is O A"y A' A" being at right angles to OA'. 
At the end of one second the increment of the angular momentum 
is numerically equal to C, and may be represented by the arc 
AqA^', thus at the end of one second the axis of rotation isOAy, 
I^t « be the angular speed of precession, then is numerically 
equal to the angle A^O A^^ />., 

Q_ arc ^0^1 _ C , , 

radius (9^0 /« 

or 

^MvkO (31) 

where M is the mass and k the radius of gyration of the wheel, 
and V the linear speed of a point on the wheel at radius k. 

In drawing the diagram (fig. 64) care should be taken that 
the quantities O Aq and O b^ are marked off in the proper direction. 
If the rotation of the wheel when viewed in the direction OA^ 
appear clock-wise, it may be considered positive ; similarly, the 
rotation which the couple C tends to produce, appears clock -wise 
when measured in the direction O ^0, and is therefore also con- 
sidered positive. If the couple C were of the opposite sign, the 
increment of angular momentum O b^ would be set off in the 
opposite direction, and the precession would also be in the oppo- 
site direction. 

The geometrical explanation of this phenomenon is almost the 
same as that given for centrifugal force in the case of uniform 
motion in a circle. 

A cyclist can easily make an experiment on precession without 
any special apparatus as follows : Detach the front wheel from a 

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CHAP. THi. Dynamics 77 

biq^cle, and, supporting the ends of the hub spindle between the 
thumb and first fingers of each hand, set it in rotation by striking 
the spokes with the second and third fingers of one hand. On 
withdrawing one hand the wheel will not fall to the ground, as it 
would do if at rest, but will slowly turn round, its axis moving in a 
horizontal plane. As the speed of rotation gradually gets less 
owing to filction of the air and bearings, the speed of precession 
gets greater, until the wheel begins to wobble and ultimately falls. 

71. Dynamics of any System of Bodies. — The forces acting 
on any given system of bodies may be conveniently divided into 
' external * and * internal ' ; the former due to the action of bodies 
external to the given system, the latter made up of the mutual 
actions between the various pairs of bodies in the given system. 
The latter forces are in equilibrium among themselves ; that is, 
the force which any body A exerts on any other body B of the 
system is equal and opposite to the force exerted by B on A, 
The motion of the mass-centre of the given system is therefore 
unaffected by the internal forces, and some of the results of sec- 
tion 67 can be extended to any system of bodies, thus : 

The motion of the mass-centre of a system of bodies under 
the action of any system of forces is the same as if equal parallel 
forces were applied at the mass-centre. 

The turning effect of a system of forces acting on any system 
of bodies is the same as if the mass-centre of the system were 
fixed. 

The kinetic energy of any system of bodies is the sum of the 
kinetic energies due to : {a) the total mass collected at, and 
moving with the same speed as, the mass-centre of the system ; 
(^) the masses of the various bodies concentrated at their respec- 
tive mass-centres, and moving round the mass-centre of the sys- 
tem ; {c) the rotations of the various bodies about their respective 
naass-centres. 

Example. — If a retarding force be applied to the side wheel of 
a tricycle, the diminution of speed is the same as if the force were 
applied at the mass-centre of the machine and rider, while the 
taming eflfect on the system is the same as if the machine were at 
rest (See chap, xviii.) 

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78 Principles of Mecftanics chap. ix. 



CHAPTER IX 

FRICTION 

72. Smooth and Eongh Bodies. — If two perfectly smooth bodies 
are in contact, the mutual pressure is always in a direction at right 
angles to the surface of contact. Thus a smooth stone resting on 
the smooth frozen surface of a pond presses the ice vertically 
downwards, and the reaction from the ice is vertically upwards. 
If a horizontal force be applied to the stone it will move hori- 
zontally, the mutual pressure between it and the ice offering little 
resistance to this motion. A smooth surface may be defined as 
one which offers no resistance to the motion of a body upon it. 
No perfectly smooth surface exists in nature, but all are more or 
less rough, and offer resistance to the motion of a body upon 
them. This resistance is cdXit^ friction. 

Friction always acts in the direction opposed to the motion of 
a body, and thus tends to bring it to rest. In all machinery, 
therefore, great efforts are made to reduce the friction of the 
moving parts to the least possible value. In bearings of machinery 
friction is a most undesirable thing, but in other cases it may be 
a most useful agent. Without friction, no nut would remain tight 
after being screwed up on its bolt ; railways would be impossible ; 
and in cycling, not only would it be impossible to ride a bicycle 
upright on account of side-slip, but not even a tricycle could be 
driven by its rider along the ground, as the driving-wheels would 
simply skid. 

73. Friction of Eest. — The greatest possible friction between 
two bodies is measured by the force parallel to the surface of con- 
tact which is just necessary to produce sliding. If a force acting 
parallel to the surface be less than this amount, the bodies will 
remain at rest. 

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CHAP. IX. Friction 79 

It is found by experiment that friction varies with the nature 
of the surfaces of contact ; is proportional to the mutual nor- 
mal pressure, and is independent of the area of the surface of 
contact so long as the pressure remains the same. When 
sliding motion actually takes place, the friction is often less 
than when the bodies are at rest in a state just bordering on 
motion. 

74. Coefficient of Friction. — Let F be the force perpendicular 
to the surface of contact- with which two bodies are pressed to- 
gether, and /^the force parallel to the surface which is just neces- 
sary to make one slide on the other. Then, as stated above, it is 
found experimentally that F is proportional to F, The ratio of 
F\.o F'\^ called the coefficient of friction for the particular surfaces 
in contact ; this is usually denoted by the Greek letter /a. The 
coefficient of friction for iron on stone varies from '3 to 7 ; for 
wood on wood from -3 to *5 ; for metal on metal from -15 to '25 ; 
while for india-rubber on paper the author has observed values 
greater than i 'o. 

Angle of Friction, — If two bodies be pressed together with a 
force /*, making an angle B with the normal to the surface, its com- 
ponents /'i, perpendicular to, and ^2* parallel to, the surface can 

P 
be readily obtained by drawing. If ^ be less than /i, no slid- 

F\ 
p 
mg will take place, but if J* be greater than /x, sliding will occur. 
F\ 

The angle B at which sliding just occurs is called the angle oj 
friction. 

If one of the bodies be an inclined plane and the other a body 
of weight W resting op it, the force F pressing them together is 
vertical, and therefore inclined at an angle ^ to the normal to the 
surface ; the angle Q of the inclined plane at which the body will 
first slide down is evidently the same as the angle of friction, and 
is sometimes called the angle of repose. 

75. Jonmal Friction. — It has been established by experiment 
that the friction of two bodies sliding on each other at moderate 
speeds, under moderate pressures, and with the surfaces either 
dry or very slightly lubricated, is independent of the speed of 
sliding and of the area of the surfaces of contact, and is simply 

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8o Principles of Mechanics chap. ix. 

proportional to the mutual pressure. The experiments on which 
the laws of friction rest were made by Morin in 1831. With well- 
lubricated surfaces, such as in the bearings of machinery, the laws 
of friction approximate to those relating to the friction of fluids. 
Mr. Tower made experiments, for the Institution of Mechanical 
Engineers, on the friction of cylindrical journals, which showed 
that when the lubrication of the bearing was perfect, the total 
friction remained constant for all loads within certain limits. 
The coefficient of friction is therefore inversely proportional to the 
load. The total friction also varies directly as the square root of 
the speed. The coefficient of friction may therefore be repre- 
sented by a formula 

i^=c^; (I) 

These experiments clearly show that with perfect lubrication the 
journal does not actually touch the bearing, but floats on a thin 
film of oil held between the two surfaces. The most perfect form 
of lubrication is that in which the journal dips into a bath of oil. 
The ascending surface drags with it a supply of oil, and so the 
film between the journal and its bearing is constantly renewed. 
If the lubrication is imperfect the coefficient of friction rises con- 
siderably, the conditions approaching then those which hold with 
regard to solids. 

The journal experimented on was 4 in. diameter by 6 in. long. 
With oil-bath lubrication, running at 200 revolutions per minute, 
and with a total load on the journal of 12,500 lbs., the total 
friction at the surface of the journal was 12*5 lbs., giving a coeffi- 
cient of friction of 'ooio. With a total load of 2,400 lbs. the total 
friction at the surface of the journal was 13*2 lbs., giving a coeffi- 
cient of friction of "0055. 

76. Collar Friction. — The research committee of the Institu- 
tion of Mechanical Engineers also carried out some experiments 
on the friction of a collar bearing. The collar was a ring of mild 
steel, 12 in. inside and 14 in. outside diameter, and bore against 
gun-metal surfaces. The pressure per square inch which such a 
bearing could safely carry was far less than in a cylindrical journal ; 
the lowest coefficient of friction was '031, corresponding to a 

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Friction 



8i 



pressure of 90 lbs. per square inch, and a speed of 50 revolutions 
per minute, ft was practically constant, its average value being 
about 036. 

The much higher coefficient of friction in a collar than in a 
cylindrical bearing is no doubt due to the fact that a thin film of 
oil cannot be held between the surfaces, and be continually 
renewed. 

77. Pivot Friction. — The relative motion of the surfaces of 
contact in a pivot bearing is one of rotation about an axis at right 
angles to the common surface of contact. Let 
figure 65 represent plan and elevation of a pivot 
bearing, being the axis of rotation and co the 
angular speed. The linear speed of rubbing of 
any point at a radius r from the centre will be 
w r. Let W be the total load on the pivot, 
D its diameter, and R its radius. If we assume 
the pressure to be uniformly distributed over 
the surface of contact, the pressure per square 
inch will be, 



W 

^ 

( ^ > 







Fig. 65. 



The area of a ring of mean radius rand width f is 2 vr f. The 
frictional resistance due to the pressure on this ring is 2 ft tt r //, 
and the moment about the centre O is 2 fiir r^tp. Summing 
the moments for all the rings into which the bearing surface of 
the pivot may be divided, the moment of the frictional resistance 
of the pivot is 

2iLit R^ p a WD , V 

~-^=^^ (2) 

3 3 ^ ' 

That is, the frictional resistance due to the load W may be sup- 
posed to act at a distance from the centre of one-third the dia- 
meter of the pivot. 

If the diameter be very small, the average linear speed of 
rubbing, and therefore also the total work lost in friction, will 
be small. The work lost in friction is converted into heat, and 
the heat must be carried away as fast as it is generated, or the 
temperature of the bearing will rise and the surface will seize. 

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82 



Principles of Mechanics 




The pressure per square inch a bearing may safely carry will thus 
depend on the quantity of heat generated per unit of surface, and 
therefore on the speed of rubbing. This speed being small in 
pivot bearings, they may safely work under greater pressure than 
collar bearings. 

It will be shown (chapter xxv.) that the motion of a ball in 
a ball bearing is compounded of rolling and spinning. Rolling 
friction is discussed in section 78. 

Spinning friction of a ball on its path is analogous to pivot 
friction, with the exception that the surfaces have contact only 
at a point when no load is applied. When the 
ball is pressed on its path by a force W (fig. 
66) the surfaces in the immediate neighbour- 
hood of the geometrical point of contact are 
deformed, and contact takes place over an area 
a o b. The intensity of pressure is probably 
greatest at <?, and diminishes to zero at a and b. 
The frictional resistance thus ultimately depends 
on the diameter of the ball, its hardness, the 
radius of curvajure of its path, the load W as 
well as the coefficient of friction. No experi- 
ments on the spinning friction of balls have 
been made, to the author's knowledge, though they would be of 
great use in arriving at a true theory of ball-bearings. 

78. Soiling Friction.— When a cylindrical roller rolls on a 
perfectly horizontal surface there is a resistance to its motion, 
called rolling friction. Professor Osborne Reynolds has investi- 
gated the nature of rolling resistance, and he finds that it is due 

to actual sliding of 
the surfaces in con- 
tact. No material in 
nature is absolutely 
rigid, so that the 
roller will have an 
area of contact with 
the surface on 'which it rolls, the extent of which will vary with 
the material and with the curvature of the surfaces in contact. 
Figure 67 shows what takes place when an iron roller rests on 

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Fig. 66. 




Fig. 67. 



Friction 



83 



a flat thick sheet of india-rubber. The roller sinks into the 
rubber and has contact with it from C to D, Lines drawn on the 
india-rubber originally parallel and equidistant are distorted as 
shown. The motion of the roller being from the left to the right, 
contact begins at D and ceases at C The surface of the 
rubber is depressed at /*, the lowest point of the wheel, and is 
bulged upwards in front of, and behind, the roller. The vertical 
compression of the layers of the rubber below P causes them to 
bulge hterally, whilst the extension vertically of the layers in 
front of D causes them to get thinner laterally. This creates a 
tendency to a creeping motion of the rubber along the roller. If 
the resistance to sliding friction between the surfaces be great, no 
relative slipping may take place, but if the frictional resistance be 
small, slipping will take place, and energy will be expended, e r 
and/r limit the surfaces over which there is no slipping ; between 
^rand Z>, and again between/rand C, there is no relative slipping. 
This action is such as to cause the distance actually travelled 
by a roller in one revolution to be different from the geometric 
distance. Thus, an iron roller rolled about two per cent, less 
per revolution when rolling on rubber than when rolling on 
wood or iron. The following table shows the actual slipping of 
a rubber tyre three-quarters of an inch thick, glued to a roller. 



Nature of surface 



Steel bar . . . . 

India-rubber 0*156 in. thick 1 
(clean) ... J 

Ditto (black -leaded) 
Ditto o*o8 in. thick (clean) 
Ditto (black-leaded) 
Ditto 0-36 in. thick (clean) 
Ditto (black -leaded) 
Ditto 075 in. thick (clean) 
Ditto (black-leaded) 



Distance 

travelled in 

one revolution 

22*55 J"- 

22*55 „ 

22-55 M , 
22*5 „ I 
22*52 „ ; 

22*39 „ 
22*42 „ ' 

22*4 „ , 
22*4 „ I 



Circumference 
of the ring 



22*5 »n. 

22*5 „ 

22*5 „ 

22-5 „ 

22*5 .. 

22*5 „ 

22*5 ». 

22*5 „ 

22*5 »» 



Amount of 
slipping 



-0*05 in. 

-0*05 

-0*05 

O'O 

—0*02 
0*11 
o*o8 
o-i 
o*i 



With regard to the work lost in rolling friction, a little con- 
sideration will show that a soft substance like rubber will waste 
more work, and therefore have a greater rolling resistance than 
a harder substance such as iron or steel. Professor Osborne 

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G 2 



84 



Principles of Mechanics 



Reynolds has shown that the rolling resistance of rubber is about 
ten times that of iron. Experiments were made on a cast-iron 
roller and plane surfaces of different materials, the plane being 
inclined sufficiently to cause the roller to start from rest. The 
following table shows the mean of results for various conditions 
of surface and manner of starting, the figures tabulated giving the 
vertical rise in five thousand parts horizontal. 



Nature of surface 



Cast-iron 
Glass . 
Brass . 
Boxwood 
India-rubber 



Starts from rest 



Starts from rest in the 
opposite direction 



Clean 

5*66 
6 32 

775 
10*05 

35 37 



Oiled or 
black-leaded 

5-61 
5-96 

6-53 
925 

3875 



{ Qean 

I 2-57 
I 1-93 
I 2*07 

' 571 
31-87 



I Oiled or 
I black-leaded j 

I 236 I 
2-56 
2-587 ' 

I 2-34 

I 28 'OO 



Mean 



4-05 
4-19 
4*73 
7-09 

3324 



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85 



CHAPTER X 

STRAINING ACTION : TENSION AND COMPRESSION 

79. Action and Beaotion. — Newton's third law of motion is 
thus enunciated : 

" To every action there is always an equal and contrary re- 
action ; or, the mutual action of any two bodies are always equal, 
and oppositely directed in the same straight line ; or, action and 
reaction are equal and opposite." 

We have in the preceding chapters spoken of single forces, 
but remembering that force can only be exerted by the mutual 
action of two bodies, the truth of Newton's third law is apparent. 
If a rider press his saddle downwards with a force of 150 lbs., the 
saddle presses him upwards with an equal force ; if he pull at his 
handles, the handles exert an equal force on his hands in the 
opposite direction. The passive forces thus called into existence 
are quite as real as what are apparently more active forces. For 
example, suppose a man to pull at the end of a rope with a force 
of 100 lbs., the other end of which is fastened to a hook in a 
wall, the hook exerts on the rope a contrary pull of 100 lbs. 
Suppose now that two men at opposite ends of the rope each 
exert a pull of 100 lbs., the * active ' pull of the second man in the 
second case is exactly equivalent ^ 

to the * passive ' pull of the hook a ^ B ^ 

in the first case. ^ rp 5^ ^^^^cu^\^lllUu\lll gmgg^ ^ 

The different forces must be "v 7'7 "^ 2 ^ 

carefully distinguished in such ' * ^ 

cases. Thus, m figure 68 the ^''" ^^' 

force exerted by the rope on the hook in the wall is in the direc- 
tion a, the force exerted by the hook on the rope is in the 
direction b^ the pull exerted by the man on the end of the rope 

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S6 Principles of Mechanics chap. x. 

is in the direction r, and the pull of the rope on the man is in 
the direction d, 

80. Stress and Strain. — Consider the rope divided at C into 
two parts, A and B, The part A will exert a pull in the direc- 
tion /, on B^ and similarly the part B will exert a pull in the 
direction t^ on A, The two forces /, and t^ constitute a straining 
action at C 

In the case of a rope the forces b and c acting on its ends are 
directed outwards, and the straining action is called a tension, 

^ If a bar (fig. 69) be subjected 

M» ^ » l ^\^ "" k ^ < w to equal forces, a and ^, at its ends 

• acting inwards, the straining action 

^^^' ^' is called a compression. 

In figures 68 and 69 the parts A and B tend to separate from 

or approach each other in a direction at right angles to the 

plane C If the parts A and B tend to slide relative to each 

!c 



c::^s:d 



Fig. 70. Fig. 71. 

Other in the direction of the plane (fig. 70), the straining action 
is called shearing. 

If the parts A and B tend to rotate about an axis perpendi- 
cular to the axis of the bar (fig. 71), the straining action is called 
bending. 

If the parts A and B tend to rotate in opposite directions 
about the axis of the bar (fig. 72), the 
straining action is called torsion. 

Compound straining actions con- 
sisting of all or any of the simple 
straining actions may take place. 

Fig. 72. rr^x. . ' ' .' J 

These strammg actions are resisted 
by the mutual action between the particles of the material, this 
mutual action constituting the stress at the point. 

Tensile Stress, — If a bar be subjected to forces as in figure 68, 
every transverse section throughout its length is subject to a tensile 

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JM' J 



CHAP. X. Straining Action : Tension and Cofnpression 87 

stress. If P be the magnitude of the forces b and c (fig. 68), and 
A the area of the transverse section at C, the force acting on 
each unit of transverse section — that is, the tensile stress per unit 
of area— is 

^=s <'> 

Compressive Stress, — In the same way, if the bar be subjected 
to forces directed inwards (fig. 69), every transverse section of it 
is subjected to a compressive stress. The compressive stress per 
unit of area will also be in this case 

^ = ^ (•> 

81. Elasticity. — If a bar of unit area (fig. 73) be fixed at one 
end, and subjected at the other end to a load, /, it is found that 
its length is increased by a small quantity. If the 
load does not exceed a certain limit, when it is re- 
moved the bar recovers its original length. It is I 
found experimentally that with nearly all bodies, i 
metals especially, this increase in length, x, is propor- • 
tional to the load, and to the original length of the ^ 
bar, so that we may write • 

or, 

/ = ^ (2) 

E t ^ ^ 

where E is sl constant, the value of which depends ^^^' ^^' 
on the nature of the material. The ratio of this elongation to the 
original length — that is, the extension per unit of length— is 
called the extension, and denoting it by e we have 

e^} (3) 



///y/ 



X/^yy 




substituting in (2) we have 

P = v. ^ 

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^ = E .^. ..(4) 



e 



88 Principles of Mechanics chap. x. 

E is called the modulus of elasticity of the material. A general 
idea of its nature may be had as follows : Conceive the material 
to be infinitely strong, and to stretch under heavy loads at the 
same rate as under small loads. Let the load be increased until 
the change of length, x^ is equal to /, the original length of the 
bar. Substituting ^ =s / in (2) we have p =^E. That is, the 
modulus of elasticity is the stress which would be required to 
extend the bar to twice its original length, provided it remained 
perfectly elastic up to this limit. 

The value of E for cast iron varies from 14,000,000 to 
23,000,000 lbs. per sq. in. ; for wrought-iron bars, from 
27,000,000 to 31,000,000 lbs. per sq. in. ; for steel plate 31,000,000 
lbs, per sq. in. ; for cast steel, tempered, 36,000,000 lbs. per sq. in. 

Example. — The spokes of a wheel are No. 16 W.G., 12 inches 
long ; the nipples are screwed up till the spokes are stretched 
1 J^ in. What is the pull on each spoke ? 

Taking E = 36,000,000 lbs. per sq. in., and substituting in (2), 
we get 



from which, 



36,000,000 12 



p = 30,000 lbs. per sq. in. 



A^ the sectional area of each spoke (Table XII., p. 346), is 
•00322 sq. in. ; P, the total pull on the spoke, is p A, There- 
fore, 

P = 30,000 X '00322 = 96*6 lbs. 

82. Work done in Stretching a Bar.— In section 81 we have 
found the stress, /, corresponding to an extension, a*, of a bar ; we 
can now find the work done in stretching the bar. It will be con- 
venient to draw a diagram to represent graphically the relation 
between / and x. Let A B^ (fig. 74) be the bar, fixed at A^ and 
let B^ be the position of the lower end when subjected to no 
load. Under the action of the load P let the lower end be 
stretched into position B^ then BqB ^=^x. Let B N he^ drawn 
at right angles to the axis of the bar, representing to any con- 
venient scale the load P, If these processes h^ repeated for a 

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CHAP. X. Straining Action : Tension and Compression 89 



number of different values of P^ the locus of the point iV'will be 
a straight line passing through B^^ and the area of the triangle 
B^B N will represent the work done in 
stretching the bar the distance B^ B, There- 
fore, ^ 
Work done = i P:tr ... (5) 

Substitute the value of x from (2) in (5), 
and remembering that F = Ap^ we get 

Work done = C — = -^ x volume of 
E 2 2E 




the bar 



(6) 




Fig. 74. 



Therefore the quantities of work done in 
producing a given stress, /, on different bars 
of the same material are proportional to the 
volumes of the bars. On bars of equal 
volume but of different materials the quan- 
tities of work done in producing a given 
stress, /, are inversely proportional to the moduli of elasticity. 
The work done in stretching a given bar is proportional to the 
square of the stress produced. 

If the bar be tested up to its elastic limit, /, the work done is 

f% 

£__. X volume of bar. 
2E 

This gives a measure of the work that can be done on the bar 

without permanendy stretching it. The quantity ^^ depends only 

on the material, is called its modulus of resilience^ and gives a 
convenient measure of the value of the material for resisting im- 
pact or shock. 

Example, — The work done in stretching the spoke in the 
example, section 81, is 

\. X 96*6 X 10^ = '483 inch-lb. or '04 foot-lb. 

83. Framed Struotures.— A framed structure is formed by 
jointing together the ends of a number of bars by pins in such a 
manner that there can be no relative motion of theirs without 

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90 



Principles of Mecltanics 



distorting one or more. If each bar be held at only two points, 
and the external forces be applied at the pins, the stress on any 
bar must be parallel to its axis, and there will be no bending. In 

figure 75 let the external 
forces 7^,, y^j . . . be ap- 
plied at the pins A^^A^, . . 
Let the frame be in equi- 
librium under the forces, and 
let F,, R,,,, (fig. 76) be 
the sides of the force-poly- 
gon. If all the forces 7^„ 
/^i ... be known, it will be 
possible, in general, to find 
the stress on each bar of the 
frame by a few applications 
of the principle of the force- 
triangle. In a trussed beam 
{e,g. a bridge, roof, or bicycle 
frame) the external forces 
are the loads carried by the 
structure, whose magnitude 
and lines of action are gene- 
rally known, and the re- 
actions at the supports. If 
there are two supports the 
reactions can be determined by the methods of section 1 7, so that 
they shall be in equilibrium with the loads. 

To find the stresses on the individual members of the frame 
we begin by choosing a pin at which two bars meet and one 
external load acts ; the magnitude and direction of the latter, and 
the direction of the forces exerted by the bars on the pin, being 
known, the force-triangle for the pin can be drawn. Thus, 
beginning at the pin -4,, on which three forces (the external force 
7^1, and the thrusts of the bars Ai A^ and A^ A,,,) act, the force- 
triangle can be at once drawn. Before proceeding with this 
drawing it will be convenient to use the following notation : I^t 
the spaces into which the bars divide the frame be denoted by a, 
d, . . . , and the spaces between the external forces -/^,» R^ . , , 

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cBAP. X. Straining Action : Tension and Compression 91 

by ^, /, . . . , then the bar ^, A^ which divides the spaces a and 

k will be denoted by a k, the stress on this bar will also be denoted 

by tf k. The force-triangle for the pin A^y at which point the 

spaces tf, and ^ meet, is a o ^ (fig. 76). Proceeding now to the 

pin A^y at which four forces act, the external force F2 and that 

exerted by the bar a k are known, and the direction of the forces 

exerted by the bars a b and b I are known. Two sides, a k and k /, 

of the force-polygon for the pin A^ are already drawn, the polygon 

is completed by drawing a b and / ^ (fig. 76) respectively, parallel 

to the bars a b and lb (fig. 75). Proceeding now to the pin A-^^ 

only two forces are as yet unknown, and of the force-polygon two 

sides, b I and / //i, are already drawn. The remaining sides, b c 

and m r, are drawn parallel to the corresponding bars (fig. 75). 

At the pin A^y four of the forces acting are already known, 

and the corresponding sides, n Oy o Uy a by and b Cy of the force - 

polygon are already drawn. The side n c oi the force-diagram 

must therefore be parallel to the corresponding bar of the 

frame-diagram, and a check on the accuracy of the drawing is 

obtained. 

With the above notation, the letters A^ A^ . . • and F^ F^ 
. . . may be suppressed. 

Figure 75 is called the frame-diagram and figure 76 the 
stress-diagram, ox force-diagram. In the force-diagram, the polygon 
of external forces is drawn in thick lines, and the direction of each 
force is indicated by an arrow. From these arrows it will be easy 
to ddtermine whether the stress on any member of the frame is 
tensile or compressive. 

The total force on any member of a framed structure being 
obtained, its sectional area can be obtained at once by formula (i). 
84. Thin Tubes subjected to Internal Pressure. An im- 
portant case of simple tension is that of a hollow cylinder subjected 
to fluid pressure ; e,g, the internal shell of a steam boiler, or the 
pneumatic tyre of a cycle wheel. In long cylindrical boilers the 
flat ends have to be made rigid in order to preserve their form 
under internal pressure, while the cylindrical shell is in stable 
equilibrium under the action of the internal pressure. A pneu- 
matic tyre of circular section is also of stable form under internal 
pressure ; a deformation by external pressure at any point will 

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92 



Principles of Mechanics 



be removed as soon as the external pressure at the point be 
removed. 

I^t / be the internal pressure in lbs. per sq. in., d the diameter 
and / the thickness of the tube (fig. 77). Consider a section by a 

plane, A A, passing through the 
axis of the tube. The upper 
half, A B Ay is under the action 
of the internal pressure /, dis- 
tributed over its inner surface, 
and the forces T due to the pull 
of the lower part of the tube ; 
therefore 2 T = the resultant 
of pressure / on the half tube. 
This resultant can be easily 
found by the following artifice : 
Consider a stiff flat plate joined 
at A A to the half tube, so as to form a D tube. If this tube 
be subjected to internal pressure, /, and to no external forces, it 
must remain at rest ; if otherwise, we would obtain perpetual 
motion. Therefore, the resultant pressure -^, on the curved part 
must be equal and opposite to the resultant pressure -^2 on the 
flat portion of the tube. If we consider a portion of the tube i in. 
long in the direction of the axis. 




Fig. 77. 



and therefore 



R^=. p dy 
2 T — pd 



(7) 



But if / be the intensity of the tension on the sides of the tube, 
T^ft 

(8) 



2 / 



Example. — K pneumatic tyre \\ in. inside diameter, outer cover 
^Q in. thick, is subjected to an air pressure of 30 lbs. per square 
inch. The average tensile stress on the outer cover is 



/ = 



2 



175 _ 



420 lbs. per sq. in. 



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93 



CHAPTER XI 

STRAINING ACTIONS : BENDING 

85. Introductory. — We have in chapter x. considered the 
stresses on a bar acted on by forces parallel to its axis. We now 
proceed to consider the stresses on a bar due to forces the lines 
of action of which pass through the axis, but do not coincide with 
it Each force may be resolved into two components, respectively 
parallel to, and at right angles to, the axis. The components 
parallel to the axis may be treated as in the previous chapter. Of 
the transverse forces, the simplest case is that in which they all lie 
in the same plane, a beam supporting vertical loads being a 
typical example. Such a beam must be acted on by at least 
three forces, the load and the two reactions at the supports. 

^, Shearing-force on a Beam. — If a bar in equilibrium be 
acted on by three parallel forces at right angles to its axis (fig. 78), 
every section by a plane parallel to the direction of the forces will 
be subjected to a bending stress. 

Consider the body divided into two portions by a plane at X. 
Under the action of the force -A*, the part A will tend to move 
upwards relative to the part B, The part A therefore acts on the 
part B with a force R^ equal and parallel to ^,, and the part B 
reacts on the part A with an equal opposite force R^'* The two 
forces R^' and R\' at A' constitute a shearing at the section. It 
will easily be seen that the shearing- force will be the same for all 
sections of the beam between the points of application of the 
forces R^ and Wy and that the shearing- force on the section X^ 
wiU be the algebraic sum of the forces to the left-hand side, or to 
the right-hand side, of the section. This is true for a beam acted 
on by any number of parallel forces. 

In particular, if a beam be supported at its ends (fig. 78) and 



94 



Principles of Mechanics 



loaded with a weight, W^ the reactions R^ and R^ at the supports 
will, by section 49, be equal to 

bW aW 



(I) 



where a and b are the segments in which the length of the beam 
is divided at the point of application of the load. The shearing- 
force on the part A will be equal to ^„ and the shearing-force on 
the part B will be equal to ^1 - /F= — R^^ 

Shearing-force Diagram, — The value of the shearing-force at 
any section of a beam is very conveniently represented by draw- 
ing an ordinate of length 
equal to the shearing- 
force at the correspond- 
ing section, any conve- 
nient scale being chosen. 
The shaded figure (fig. 
79) is the shearing-force 
diagram for a beam sup- 
ported at the ends and 
loaded with a single 
weight. 

The shearing- force at 
the section X (fig. 78) is 
of such a nature that the 
part on the left-hand side 
tends to slide up7vards 
relative to the part on the 
right-hand side of the 
section. The shearing- 
force at X^ is of such a 
'^' ^^' nature that the part on 

the left tends to slide dowmvards relative to the part at the right 
of the section. Thus shearing-forces may be opposite in sign ; if 
that at X be called positive, that at X^ will be negative. The 
diagram (fig. 79) is drawn in accordance with this convention. 

87. Bending-moment.— If a bar of length, /, be fixed horizon- 
tally into a wall (fig. %2>\ and be loaded at the other end with a 

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CHIP. XI. Straining Actions: Bending 95 

weight W, the said weight will tend to make bar turn at its sup- 
port, the tendency being measured by the moment Wl of the 
force. This tendency is resisted by the reaction of the wall on 
the beam. The section of the beam at the support is said to be 
subjected to a bending-moment of magnitude Wl, 

From this definition a weight of 50 lbs. at a distance of 
20 inches will produce the same bending-moment as a weight of 
100 lbs. at a distance of 10 inches; the bending-moment being 
50x20, or 100 X 10 = 1000 inch-lbs. 

Returning to the discussion of figure 78, it will be seen that 
the part A is acted on by two equal, parallel, but opposite forces, 
^1 and R^'y constituting a couple 
of moment -^j x tending to turn 
the part A, But the part A is 
actually at rest ; it must, there- 
fore be acted on by an equal and 
opposite couple. The only other 
forces acting on A are those 
exerted by the part B at the sec- 
tion X. The upper part of por- 
tion^ (fig. 81, which is part of 
figure 78 enlarged) acts on the 

portion A with a number of forces, c^ r,, diminishing in intensity 
from the top towards the middle of the beam ; the resultant of 
these may be represented by C^, The lower part of B acts on 
A with the forces /, /,, whose resultant may be represented by 7^. 
Since the part A is in equilibrium, the resultant of all the hori- 
zontal forces acting on it must be zero ; therefore T\ and Cj are 
equal in magnitude, and constitute a couple which must be equal 
^■oR^x. If d be the distance between 7", and Ci, we must 
therefore have 

T^d — R^x, 

The part A acts on the part B with forces ^2 at the top, and forces 
1 2 at the bottom of the beam ; the resultants being indicated by 
Cj and 7^2 respectively. The two sets of forces c^ and c^ consti- 
tute a set of compressive stresses on the upper portion of the beam 
at Xy and the two sets of forces /, and t^ constitute a set of tensile 

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96 



Principles of Mec/tanics 



CBAP. XI. 




Stresses on the lower portion of the beam. The moment of the 
couple J?i X is called the bending-moment at the section X ; while 
the moment of the couple T', d is called the moment of resistance 
of the section. 

The existence of the shearing-force and bending-moment at 
any section of a beam can be experimentally demonstrated by 

actually cutting the beam, and re- 
placing by suitably disposed fasten- 
ings the molecular forces removed 
by the cutting. Figure 82 shows 
diagram mat ically a cantilever treated 
in this manner. The shearing-force 
at the section is replaced by the 
upward pull IV of a, spiral spring, 
and the couple acting on the part 
B formed by the load IV, and the 
pull of the spring is balanced by the equal and opposite couple 
formed by the pull T^ of the fastening bands at the top and the 
thrust 6*2 of the short strut at the bottom of the section. 

Bending-moment Diagram, — The bending-moment at any 
section of a beam can be conveniently represented by a diagram, 
the ordinate being set up equal in length to the bending-moment 
at the corresponding section. 

Since the bending-moment at the section X (fig. 78) is the pro- 
duct of the force -^1 into the distance x of the section from its 
point of application, the further the section X be taken from the 
end of the beam the greater will be the bending-moment. In the 
case of a beam supported at the ends and loaded at an interme- 
diate point with a weight W, the bending-moment M on the section 
over which W acts will be given by — 



Fig. 82. 



M^R.a^- 



ab 



a^b 



W 



(2) 



and the bending-moment on any section between R^ and /Fwill 
be represented by the ordinate of the shaded area in figure 80. 

The bending-moment at the section X^ (fig. 78) is the sum of 
the moments of the forces ^, and Jf^ about X^ ; or is equal to the 
moment of the force Ri about X^, 

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CHAP. XI. 



Straining Actions: Bending 



97 




Fig. 83. 



88. Sinple Examples of Beams. — A few of the most commonly 
occurring examples of beams may be discussed here. Figure 83 
shows a cantilever of length, /, 
supporting a weight, W, at its 
end. The bending-moment at 
a section very close to the sup- 
port is Wiy that at a section 
distant x from the outer end of 
the cantilever is Wx. The 
bending-moment diagram is, 
therefore, a straight line, the 
maximum ordinate, Wi^ being 
at the support, that at the end 
zero. The shearing-force Is 
equal to W for all sections ; the shearing-force diagram is, there 
fore, a straight line parallel to the axis. 

Figure 84 shows a cantilever loaded uniformly, the total weight 
being W, The resultant weight acts at the middle of the canti- 
lever distant - from the support, I 

the bending-moment at the 

IV i 
support is, therefore, — At 
2 

any section distant x from the 
end of the cantilever, we find 
the bending-moment as fol- 
lows : Consider the portion of 
the cantilever lying to the right 
of the section, the resultant of 
the load resting on it 'iswx,w 
being the weight per unit of 

length, and acts at a distance - 



E 

^ 



W 1 



wx 



^ 







Fig. 84 

from the section. The bending- 



moment on the section is therefore 

,, wx^ W x'^ I X 

^=--=--^- ....... (3) 

Plotting these values for diflferent values of x^ the bending-moment 
curve is a parabola. ■ 

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98 



Principles of Mechanics 



The shearing -force on the section distant x from the end is 

W X 
wx ss, — — -. Plotting these values for different values of x, we 

get the shearing-force curve a straight line, having the ordinate IV 

at the support and zero ordinate at the end. 

Figure 85 shows a beam of span, /, supporting a load, ^ at the 

middle. The reactions at the support are evidently each equal to 

fV 
, the bending-moment at any 

section distant x from the end 

IS therefore - ' , x being less 
2 







w 


1 

1 


1 : 1 


' «- — 


-/- 


^*^<n>w 





Ml 



than 



Hi 



At the middle of the 



beam the bending-moment is a 
maximum, and equal to 






Fig. 85. 



JV 

2 



IVl 
4 



(4) 



\mx 



UilllilUJlllilllll 
-'^^TTTTTTTnimMTIT ^ " 



then 
2 



The bending-moment curve is a triangle, the maximum ordinate 

being in the middle. The shearing-force is constant and equal to 

W 
' from one end up to the 
2 

middle of the beam, 

changes sign and becomes 

over the other half. 

Figure 86 shows a beam 
supporting a load, IV, uniformly 
distributed. The reaction at 

Tl/' 

each support is evidently ; 

2 

the bending-moment at a sec- 
tion distant x from the end is 
the sum of the moments due to 



Fig. S6. 



IF 



the reaction — , and of the resultant load w x acting on the 

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CEAP. M. Straining Actions : Bending 99 

right-hand side of the action at a distance - from the 

2 

section, 

... .if= »:._«,.. -=^/.--') . . (5) 

2 2 2 \ / / 

If ^ be made equal to -, the above formula gives the bending- 
2 

W 1 1 l\ W I 
moment at the middle of the beam, M^ = ( I = - o • 

2 V2 4/ 8 
The bending-moment curve is a parabola with its maximum 

ordinate — at the middle of the beam. 

o 

89. Beam supporting a Number of Loads at Different 
Points. — ^The loads and their positions along the beam being 
given, the reaction R^ at one support can be found by taking 
moments about the other support ; the bending-moment at any 
section can then be calculated by adding algebraically the moments 
of all the forces on either one side or other of that section. 
The reactions^ I and R^ at the supports can also be found by the 
method of sections 47 and 48. * Since in this case the forces are all 
parallel, the construction is simplified ; the force-polygon becomes 
a straight line, and the corners of the link-polygon lie on the verti- 
cal lines of action of the loads and reactions. 

Figure 87 shows a beam supporting a number of weights, ^,, 
Jf^j, W^3, ^4, and figure 88 the force-polygon tf, ^, c, d^ e. The 
construction of figure 41 becomes as follows : From any point fi^ 
on the line of action of W^ draw a straight line b parallel to the line 
Ob (fig. 88). From p^, where this line cuts the line of action of 
IV^ draw a straight line, <r, parallel to the line O c \ continuing this 
process until the point ^4 on the line of action W^ is reached. 

Through/, and/4 draw /j/^ and /4/r parallel to Oa and Oe 
respectively, intersecting each other at p^ and the lines of action 
oi Rx and R^ at r, and r^ respectively. The resultant of the 
loads IV^y J^2, W^ and Wj^ passes through/,. Through O draw 
Or parallel to r, r<^ ; then the reactions ^i and ^2 are equal to 
ra and e r respectively. 

Link-polygon as Bending-moment Diagram, — If the pole O be 
chosen at random, the closing line r^ r^ of the link-polygon will 

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lOO 



Principles cf Mechanics 



CHXP. Xf. 



not, in general, be parallel to the axis of the beam. Let a new 
pole, (9^ be taken by drawing O O^ parallel to, and r (?* at right 
angles to, the lines of action of the loads ^,, W^2 • • •> and let a 



Fig. 87. 



"i 




Fia» g7» 



Fig. 88 



new link-polygon (fig. 89) be drawn. If a thin wire be made to 
the same outline as this same polygon and be attached to the 
beam, and the loads W^, IV2 . . . attached at the angles, it is 
evident that the compound structure formed by the bar and 

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CHA?. XI. Straining Actions: Bending loi 

wire is subjected to the same bending stresses as the beam (fig. 87). 
In both cases the dispositions of the loads and reactions are 
identical ; but in the compound structure the bar is subjected to 
a thrust, T', represented in the force-diagram (fig. 88) by (?* r. 
Considering the corner of the wire at which IV^ acts, the tensions 
on the two portions of the wire^ aixi the force W^ are in equi- 
iibriuna, and are represented by the fodrce-triangle O^ab (fig. 88) ; 
similarly for the other portions of the wire. It will be noticed 
that at each part of the wire the horizontal component of the pull 
is equal to (?* r ; that is, equal to 7! Taking any vertical section 
of the compound structure (fig. 89) the mutual actions consist of 
a thrust, Ty on the bar, an equal horizontal pull, 7J on the wire, and 
the vertical component of the pull on the wire. The two former 
constitute the bending-couple at the section, the latter the shearing- 
force. The bending-moment on any section of the beam is there- 
fore equal to Th^ h being the ordinate of the link-polygon ; the 
link-polygon can therefore be used as a bending-moment diagram. 

The shearing-force on any section of the beam (fig. 87) is 
equal to the vertical component of the pull on the wire (fig. 89), 
which is equal to the vertical component of the corresponding 
line from the pole O^ (fig. 88). A shearing-force diagram (fig. 90) 
can therefore be constructed by projecting over a base line from 
r, and straight lines from a, b , . , (fig. 88) to the corresponding 
divisions of the beam. 

Example, — Calculate, and draw, a bending-moment diagram 
for the frame of a tandem bicycle carrying two riders, each 150 
lbs. weight (30 lbs. of which is assumed to be applied at the 
crank-axle) ; the wheel-base being 64 inches long, the rear crank- 
axle being 19 inches in front of the rear wheel centre, the crank- 
axles 22 inches apart, and the saddles 10 inches behind their 
respective crank-axles. 

The figures of illustrations are given in chapter xxiii., page 327. 

To calculate the reactions on the wheel spindles, take moments 
about the centre of the rear wheel — 

(120 X 9) + (30 X 19) + (120 X 31) + (30 X41) — (R X 64) = o 

from which, ^i = 103*1 lbs., 

M ^3 = 1969 lbs. r noolp 

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I02 



Principles of Mechanics 



The greatest bending-moment, which occurs on the vertical 
section passing through the front seat, is 

M=:{io^'i X 33) — (30 X 10) = 3,102 inch-lbs. 

The frame, or beam (fig. 321) is drawn ^^"^ ^"^^ size ; the 
scale of the force-diagram (fig. 323) is i inch to 400 lbs., and the 
pole distance O^ corresponds to 125 lbs. ; i inch ordinate of the 
bending-moment diagram (fig. 324) therefore represents 32 in, x 
125 lbs., i.e. 4,000 inch-lbs. 

The results got by the graphical and arithmetical methods must 
agree ; thus a check on the accuracy of the work is obtained. 

90. Nature of Bending Stresses. — We must now consider 
more minutely the nature of the stresses / and c (fig. 81) on any 
section subject to bending. 

Let a beam be acted on by two equal and opposite couples at 
its ends ; it will be bent into a form, shown greatly exaggerated 

in figure 91. It can be easily seen 
that the bending-moment on the 
middle portion of the beam will be 
of the same value throughout, and 
if the section is uniform, the amount 
of bending will be the same at all 
sections ; that is, the beam, origi- 
nally straight, will be bent into a 
circular arc. 

Consider the portion of the beam 
included between two parallel sec- 
tions A and B, After bending, 
these sections are inclined, and if 
produced, will meet at the centre 
of curvature of the beam. The top 
fibres of the beam will be shortened 
and the lower fibres lengthened, 
while those at some intermediate layer, NJVf will be unaltered in 
length. The surface in which the centres of the fibres JVJV lie 
is called the neutral surface of the beam, while its line of inter- 
section with a transverse plane is called the neutral axis of the 
section. Now, suppose that the fibres could bedaid out flat and 

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CHAP. XI. 



Straining Actions: Bending 



103 



of exactly the same length as they are after bending. If the left- 
hand ends all lay in the plane A A (fig. 92) at right angles to 
NNy the other ends must evidently lie in a plane B^ B^ ; B B 
representing the plane in which the ends of the unstretched fibres 
would lie. The distance, parallel to NN^ included between the 




Fig. 93. 



Fig. 93. 



lines -^ -^and B^ B^ gives the amount of the contraction or elonga- 
tion of the corresponding fibres. The elongation or contraction 
of any fibre is thus seen to be proportional to its distance from 
NN. Now the stress on a bar or fibre is proportional to the 
extension produced ; therefore the stress on the fibres of a beam 
varies as the distance from the neutral axis. 

Let O be the centre, and R the radius of curvature oi NN 
(fig. 91), y the distance of any fibre /above the neutral axis, 
the angle NON subtended at the centre O by the portion of 
the fibre considered. The radius of curvature of the fibre / is 
(-^— jk), the length of the arc /, f^ (fig. 92) is therefore {R—y) ; 
and the length of the arc Ni N^ is R G, A fibre at the neutral 
axis is unaltered in length by bending, so the length N^ N^ is the 
same as in the straight position. The length of the fibre/, /a was 
originally equal to that of N^ N^ ; the decrease in its length is 
therefore 

RQ-{R''y)d=:ye] 

its compression per unit of length is therefore 

RO R 
By section 81, the stress producing this compression is 

^_-^. . . ■ . • • • 

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(6) 



I04 Principles of Mechanics chap. xi. 

That is, the intensity of stress on any fibre of a beam subject to 
bending is proportional to its distance from the neutral axis, and 
inversely proportional to the radius of curvature of the neutral 
axis. If a fibre below the neutral axis be taken, y will be nega- 
tive, the fibre will be stretched, and the stress on it will be 
tensile. 

Since the material near the neutral axis is subjected to a low 
stress, it adds very little to the strength of the beam, while it adds 
to the weight. It is therefore economical to place the material as 
far as possible from the centre of the section. The framework of 
the earliest bicycles was made of solid bars ; but a great saving 
of weight, without sacrificing strength, was effected by using hollow 
tubes. The same principle is carried out to a fuller extent in a 
well -designed Safety frame ; the top- and bottom-tubes together 
forming a beam, in which practically all the material is at a 
great distance from the neutral axis. If the frame be badly 
designed, however, the top- and bottom-tubes may form merely 
two more or less independent beams, instead of one very deep 
beam. 

91. Position of Keutral Axis.— Consider the equilibrium of 
the portion of the beam to the left hand of section A (fig. 91). 
There are no external horizontal forces acting on this portion, and 
therefore the resultant of the horizontal forces due to the internal 
reaction of the particles at the section A must be zero. 

Let figure 93 be the transverse section at A (fig. 91), N N 
being the neutral axis. The part of the section above N N 
is subjected to compression, that below N N to tension ; the 
resultant compressive force must therefore be equal to the re- 
sultant force of tension. Consider a strip of the section of 
breadth ^, and thickness /, at a distance y from the neutral axis ; 

the area of this strip is b /, the stress per square inch is J ; the 
total force on it is therefore 

The total force on the whole section will be the sum of the forces 
on all such strips ; compression being considered positive an4 

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CHAP. XI. Straining Actions : Bending 105 

tension negative. is the same for all the strips, therefore the 
R 

resultant force on the section may be written 

l»b ty indicating the sum of all the products b t y. Since the 
resultant force on the section is zero, we must have 

2^/7 = (7) 

Referring to section 50, it will be seen that this condition is 
equivalent to saying that tlie neutral axis must pass through the 
mass-centre of the section. 

92. Moment of Inertia of an Area.— In figure 93, ^ / is the 
area of a narrow strip parallel to, and distant y from, the axis 
N N\ b t y'^'x^ therefore the product of a small element of area 
into the square of its distance from the axis. The sum of such 
products for all the elementary strips into which the given area 
can be divided is called the moment of inertia of the area^ and, as 
shall be shown in the next section, is of fundamental import- 
ance in the theory of bending. 

The calculation of moments of inertia for areas of given shape 
is beyond the scope of an elementary work like the present ; a 
few of the most important results will be given for convenience 
of reference. 

Let / denote the moment of inertia about an axis passing 
through the mass centre. Then, for a square of side ^, 

/=i>^ (8) 

For a circle of diameter d^ 

^=6^^ (9) 

For a rectangular section of breadth b and depth h (perpendicular 
to the neqtral ^is), 

I^^ bh^ (10) 



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io6 Principles of Mechanics chap. xi. 

For an elliptical section of breadth b and depth h^ 

^=6V'^' <") 

Por a hollow circular section of outside and inside diameters, d^ 
and d^ respectively, 

/=6'^W-'//) (12) 

Let A be the area, the moment of inertia of which is being 
considered. Then for a rectangular section A ^ b h, and (lo) 
may be written 

^=l2^^" <^3) 

For a circle A = d^, and (9) may be written 

^=^6^'^' (M) 

Similarly, for an ellipse of breadth b and depth A, A = b d ; 
(11) may therefore be written 

I^l^Ah* (,5) 

That is, for each of the three sections considered, the moment of 
inertia is equal to the product of the area, and the square of the 
depth at right angles to the axis of inertia, multiplied by a constant 
factor, which depends on the shape of the section. It can be 
shown that this is true for sections of all shapes, the value of the 
constant factor being different for different shapes of section, but 
the same for large or small sections of the same shape. 

Moment of Inertia of an Area about Parallel Axes, — The 
moment of inertia of an area is least about an axis passing through 
the centre of area. 

Let /o be the moment of inertia of an area A about any axis 
through the centre of area. Then it can be easily shown that the 
moment of inertia about a parallel axis distant jo fro"^ the centre 
of area is /© + ^ jo *. 

Moment of Inertia of an Area about different Axes passing 
through the centre of figure, — The moment of inertia jof an area 

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CHAP. M. Straining Actions : Bending 107 

about different axes passing through the centre of figure are in 
general different, but however complex be the outline of the area, 
an ellipse can be drawn with its centre coinciding with the centre 
of the area, such that the moment of inertia relative to any axis 
drawn through the centre varies inversely as the square of the 
corresponding radius-vector of the ellipse. This ellipse is called 
the ellipse of inertia^ or the momenial ellipse^ of the area. The 
axes corresponding to the major and minor axes of the ellipse are 
called the principal axes of the figure. 

The momental ellipse for a rectangle, if drawn to a suitable 
scale, touches its sides. Similarly, for a triangle it can be shown 
that the ellipse touching the three sides at their middle points can 
be taken as the momental ellipse. 

If the major and minor axes of the momental ellipse are equal, 
the ellipse becomes a circle, and the moments of inertia about all 
axes through the centre are equal. For example, since from 
symmetry the momental ellipse for a square is a circle, the moment 
of inertia of a square is the same for all axes passing through its 
centre. 

93. Moment of Bending Resistance. — The moment about the 
neutral axis of all the forces / on the fibres of the cross section is 
called the moment of resistance to bending of the section, and is of 
course equal to the bending-moment on the section due to the 
external forces. 

The moment of the force on the strip b t (fig. 93) is 

^bty y.y 



and the moment of all the forces on all the strips is 
which may be written 



M=^j^-S.bty^ (16) 



M^^rI (17) 

Substituting the value of— from (6) in (17) it may be written 

f =j <.«) 

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io8 Principles of Mechanics , chap. xi. 

(17) and (18) may be conveniently written together thus : 

f-^=i <■" 

94. Kodnliu of Bending Besistance of a Section.— The 
greatest stress on a section occurs, as has already been shown, on 
the fibre furthest away from the neutral axis. Let /be this stress, 
then, denoting the corresponding of^ by^„ (18) may be written 

M^Lf (20) 

The quantity which is a geometrical quantity depending on 

yv 

the area and shape of the section, and not in any way on the 
material, is called the modulus of bending resistance of the section, 
and will be denoted by the letter Z. (20) may then be written 

M=zZf (21) 

From (21) it is evident that the modulus of a section bears the 
same relation to the bending-moment on it, as the area of a section 
bears to the total direct tension or compression on it. The 
total pull on a bar is equal to the product of its area into the 
tensile strength per square inch. The bending-moment on any 
section of a beam is equal to the modulus of the section multiplied 
by the greatest stress on the section. 
For a rectangular section 



Z=^^-^lAh (22) 



For a circular section,, 
or approximately, 



Z^'^d^^lAd (23) 

-^=-5 (24) 



For a hollow circular section, 

Z= ""-i^^'.-J^') (2C) 

32 d, ^ ^^ 

Table III. gives the sectional areas and moduli foj; round bars. 

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CBAP. XI. 



Straining Actions : Bending 



109 



From (20) and (23) it is evident that the bending-moment a 
round bar can resist, t,e, its transverse strength, is proportional to 
the cube oi its diameter. 

Table III. — Sectional Areas and Moduli of Bending 
Resistance of Round Bars. 



i Diameter 


Sectional area 
Sq. in. 


_ z 

In.* 


Diameter 
Inches 


Sectional area 


Z 


Inches 


Sq. in. 


In.* 




•0031 
•0123 
•0276 
•0491 


•000024 
•000192 
•000647 
•001534 


I 

Te 

I 


•5185 
•6013 
•6903 
•7854 


•0526 
•0658 
•0809 
•0982 


6 

f 


•0767 
•IIO4 
•1503 
•1964 


•00300 
•00517 
•00822 
•01227 


4 ' 


•9940 
I -2272 
14849 
1 767 1 


•1398 
•1917 
•2552 

•3313 


If 


•2485 
•3068 
•3712 
•4418 


•0175 
•0240 
•0319 
•0414 


2 1 


20739 

24053 
2-7611 
3*1416 


VJ2II 
•5261 
•6471 
•7854 



95. Beams of XTniform Strength. — The bending-moment on a 
beam generally varies from section to section along the axis ; 
consequently, if of uniform section throughout it will be weakest 
where the bending-moment is greatest. A beam of uniform 
strength is one in which the section varies with the bending- 
moment in such a manner that the tendency to break is the same 
at all sections. This means that f the maximum stress on the 
section, has the same value throughout, and therefore that M is 
proportional to Z, 

For a thin hollow tube of constant external diameter through- 
out its length, Z is approximately proportional to the thickness ; 
therefore for a tubular beam in which the bending-moment varies 
continuously the thickness should also vary continuously, if the 
beam is required to be of uniform strength. For example, the 
bending-moment on the handle-bar of a bicycle, due to the pull 
of the rider, increases from zero at the end to its maximum value 
at the handle-pillar. If the external diameter of the handle-bar be 
the same throughout, the lightest possible bar would vary in 

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no 



Principles of Mechanics 



CHAP. ZI. 



thickness from the middle to the ends. This ideal handle-bar 
cannot be conveniently made, but an approximation thereto is 
sometimes made by inserting a liner at the middle, where the 
bending-moment is greatest ; there will in this case be three weak 
sections, the middle section and those just beyond the ends of the 
liners. 

96. Kodnlns of Circular Tubes.— On account of the extensive 
use of tubes in bicycle making, it will be desirable to give some 
additional formula relating to the moment 
of inertia and the modulus of a tubular 
section. 

Let ^/i, d^ and d^ (fig. 94) be the out- 
side, mean, and inside diameters respec 
tively, / the thickness, and A the area of 
the transverse section of the tube. From 
(12) for this section 




Fig. 94. 



/= l^ (d,< - <//) = l-^ {d, - d^){d, + d.^(d,^ + </,^) . (26) 

Now, dx— d^^2ty ^, + i/g = 2 ^/, ^l = </ + /, //g = </— /. 

Therefore, 

(d,^ + d^^)^{d +/)^ + {d^ fi) =2(^2 +^2). 

Substituting in (26) 

7=7 .2/.2^.2(d^2 + /2) 

64 



But Tzdt^A^ therefore. 



Now, 



/=j^W + ^2^)= g-(^^ + 0. 



7^1 ^A (d,^ + d^^) ^ A (2dl - 4^, / 4- 4 O 
^, 8 </, 8 //, 



(27) 



(28) 



IF the tube be //««, t^ will be small in comparison with ^', and 



CHAP. M. Straining Actions : Bending ill 

2 /^ 

- will be small in ( 

may then be written 



^ - will be small in comparison with ^2- Equations (27) and (28) 



▼ ., . ^^^ 



/= - ^' /= - — approximately . . . (29) 

o o 

Z = I ^/2 / = "^/^^ approximately .... (30) 
4 4 

The error introduced by using the approximate formula (30) 
for Z is on the safe side, and is very small for the ordinary tube 
sections used in cycle construction. Thus for a tube i inch 
diameter, 16 W.G., the exact value of Z is 04140, that given by 
(30) is -04102, the error being less than i per cent, in this case. 
If, however, d or dy^ be used instead of d^ in formula (30) the 
error will be on the wrong side. 

Table IV. gives the sectional areas, weights per foot run, and 
moduli of bending resistance for the ordinary sections of steel 
tubes used in cycle construction, the moduli having been calcu- 
lated from the exact formula (28). 

From (30) the transverse strength of a tube is proportional to 
its sectional area and to its internal diameter. If the internal 
diameter be kept constant, the transverse strength is proportional 
to the thickness. If the sectional area be kept constant, the 
transverse strength is proportional to the internal diameter. If the 
thickness be kept constant the strength is approxi- . 
mately proportional to the square of the diameter. 

97. Oval Tubes. — We have already seen that the 
moment of inertia of an ellipse with major and minor 

axes h and b respectively is J d h^, 

64 

Let a second ellipse {fig. 95) be drawn outside the first 

and concentric with it, having its semi-axes the length 

/ greater. The axes of the second ellipse will be ^ + 2 / and 

A + 2 / respectively, and its moment of inertia will be 

^ [bh^ -V {2h^ '\- 6 dh^)t'\- {12 h^-V \2bh)^ 
64 L 




Fig. 95. 



iI2 



Principles of Mechanics 



CAAP. zi. 



TABI 
Sectional Areas, Weights per Foot Run, a. 



Outside diameter 
of tube 






No. lo 
II 

12 

13 
14 

15 

lb 

J7 
18 

19 



21 
22 

23 
24 

2; 
26 
28 
30 
32 



•128 
•116 
•104 
•092 
•080 

•072 
•064 
•056 
•048 
'040 

•036 
•032 
•028 
•034 
•022 

*020 
•018 
•0148 
•0124 

•oip8 



Outside diameter 
of tube 



No. 10 
II 
12 
13 
14 

15 

16 

17 
18 

19 



23 
24 



•128 
•116 
•104 
•092 
•080 

•072 
•064 
•056 
•048 
•040 

•036 
•032 
•028 
•024 

*022 




1// 

9 



w I I 

lbs. per ; A 1 Z 

foot \ sq. in. t in " 

length I 



•34 
•33 
•31 
•28 



•0993 
•0944 
•0885 
•0818 



•26 \ '0742 

•0685 
•0625 
0561 

•0493 
0421 

•0383 

0345 

•0305 

I 0265 

•08 I '0244 



•24 

*22 
•19 
•17 

•13 

•la 
•w 
•09 



•08 
•07 
•06 

•05 
•04 



•0223 
•0202 
•0167 
•0141 
'01 24 



ir 



156 
1*43 
I 29 
i-i6 

I '02 

•92 
•82 
•73 
•63 
•53 

•47 
•42 

•37 
•32 
•29 



•CO5I 
•0051 
•0049 
'OO48 
0046 

•0044 
•0042 
•0039 
•0036 
•0032 

0030 
•0027 
•0025 
•0022 

0020 

•0019 
•0017 
•0015 
0012 

•ooii 



w 




lbs. per 


A 


foot 


sq. in. 


length 




•52 


•1496 


•48 


•1399 


•45 


•1294 


41 


•I 180 


•36 


•1056 


'Zl 


•0968 


•30 


•0877 


•27 


•0781 


•24 


•0682 


•20 


•057« 


•18 


•0525 


•16 


•0470 


•14 


*c^IS 


•12 


•0359 


•II 


•0330 


•10 


•0302 


•09 


•0273 


•08 


•0226 


•07 


•0190 


•06 


•0166 



8 



. lbs. per ! A 
I fool sq. in. 



•01 16 
•0113 I 
•0108 
0103 
•0096 

•0091 
•0085 
•0078 
•0070 
0062 

•0057 
•0052 
0046 
•0041 
•0038 

•0035 
0032 
•0027 
•0023 

*0O2O 



,8// 
*8 



•451 1 


•II5I 


173 


•4132 


•IP74 


i'59 


•3744 


•0992 


''•^1 


•3347 
•2940 


•0903 
•0809 


1-28 

I'I2 


•2664 
•2384 

•2IOI 


•0742 
•0673 
•0600 


I "02 
•91 
•80 


•1813 
•I52I 


•0525 
•0446 


•69 


•1373 
•1224 


•0405 
•0364 


•52 
•47 


•1075 
■0924 
•0849 


•0321 
0278 
•0256 


•41 
•35 
•32 



•5013 
•4587 
4152 
•3708 
•3255 
•2947 
•2635 

•2320 
•2001 

•1679 

•I5I4 
•1350 

•II85 
•IOI9 

•0935 



I 



M33 
1334 
•1228 
•1115 
•0996 

0912 
•0825 

•0735 
•0642 

•0544 

•CM94 

•0443 
•0391 

•0338 
•03 1 1 



•69 
•64 
•59 
•53 
•47 

•43 
•39 
•35 
•30 
•25 

*23 

'21 
■18 
•16 
•14 

'J3 

'12 
•10 
•08 
•07 



I 91 
174 
1-58 
1-41 
1-23 

I"I2 

I '00 

•88 
•76 
•63 

•57 

•45 
•38 
•35 



•1998 i "02 
•1855 '°* 

1702 I or 
•1541 , 01 
•1370 I 01 

•1251 *oi 

•I 1 28 'Ol 
•lOOI 'o/ 
•0870 , 'OI 

•0736 I 01 
•0666 00 

•0596 od 

•0525 : 'oo 
•0453 oo 

0417 'oo 
0380 i 00 

•0343 ^ 

'0284 I 'OG 

0239 'OC 
*0208 *OC 



1// 

^2 



•5516 

•5043 
•4560 
•4069 
•3569 

•3230 
•2887 

•2540 

•2190 

•1836 

•1656 
•1476 
•1295 
•II13 

'I022 



•»7 
•lb 

•14 
•13 

•la 



•09 
•08 

•07 
•06 

•OS 
•03 



Digitized by CjOOQIC 



CHAP. XI. 



Straining Action : Bending 



113 



rv 

Moduli of Bending Resistance of Steel Tubes 





s// 




111' 




t'' 






,1// 






4 




8 




I 


1 


U 


w 






W 


1 


W 


1 


1 w 




lb(.per 


A 


Z 


lbs. per 


A Z 


lbs. per 


A 1 Z 


lbs. per 


A Z 


fool 


sq. in. 


in.* 


(cot 


sq. in. ^ in.' 


foot 


sq. in. ' in.^' 


foot 


sq. in. in.' 


run 






run 






run 


1 


run 


1 


•86 


•2501 


•0336 


I-04 


•3003 ; -0493 


1*21 


•3506 1 -0681 


138 


■4009 0900 


•80 


•2310 


•0320 


96 


•2766 ; 0466 


I^ii 


•3222 ! '0640 


127 


">M7 -0843 


73 


■ '21 1 1 


0301 


•87 


•2519 


•0436 


I'Ol 


•2927 i '0595 


1*15 


, 3335 1 -0781 


60 


[ -1902 ! '0280 


•78 


•2263 


•0402 


•91 


•2624 ^ "0546 


1-03 


•2981 1 0714 


•58 


•1684 0256 


•69 


•1998 


•0365 


•80 


•2312 1 ^0493 


•91 


•2626 -0641 


•53 


•1534 -0238 


•63 


•I8I6 


•0337 


•73 


•2099 ^0455 


•82 


•2382 0589 


•48 


•1379 0218 


•56 


•I63I 


•0308 


•65 


•1882 \ '0414 


•74 


•2133 -0535 


■42 


•1221 ^ '0197 


•50 


•I44I 


•0277 


•57 


•1661 1 0371 


•65 


•1881 0479 


•37 


, 1059 0175 


'^} 


•1247 


•0244 


•50 


•1436 . 0326 


•56 


•1624 '0419 


•31 


X)893 0150 


•36 


•1050 


•0210 


•42 


•1207 -0279 


•47 


•'364 0357 


•28 


•0807 I 0137 


•33 


•0949 


•0191 


•38 


•1090 0253 


•43 


•1232 ; 0325 


■25 


i -0722 


'0124 


•29 


•0847 ^0172 


•34 


•0973 "0228 


•38 


•1099 0292 


22 


•063s 


Olio 


•26 


•0745 0152 


•30 


•0855 '0202 


•33 


I -0965 -0258 


19 


•0547 


'0096 


•22 


0642 0133 


•25 


•0736 0175 


•29 


1 0830 0223 


•17 


•0503 


•0089 


•20 


•0590 '0122 


•23 


•0676 "0161 


•26 


•0762 0206 


•16 


0459 ' -0082 


•19 


0537 *0'I2 


•21 


•0616 •0148 


•24 


' -0694 1 0188 


M 


•0414 1 0074 


•i7 


•0485 •0102 


•19 


•0555 -0134 


'22 


•0626 1 -0170 


12 


•0342 0062 


•14 


•0400 '0085 


•16 


0458 1 01 1 1 


•18 


1 0516 ^0140 


•10 


•0287 1 -0052 


•12 


0336 0071 


•13 


•0385 1 0094 


•15 


1 '0433 •0J19 


.09 


•0251 0046 


•10 


•0293 0063 


•12 


•0336 0082 


•13 


1 0378 0104 




-6'/ 




-8// 




-7// 




_ V 




Is 




I4 




Is 




260 


2 


208 


•6019 1 2090 


2*25 


•6521 


•2467 


243 


•7024 


•2874 


•7526 -3312 


iqo 


•5499 1938 


a '06 


•5954 


•2283 


2^21 


•6409 


•2657 


2*37 


•6865 -3058 


172 


•4969 1777 


1-86 


•5377 


•2090 


2*00 


•5785 


•2428 


214 


•6194 , 2792 


I '53 


•44^1 


•1608 


1-66 


•4792 


•1888 


1-78 


•5'53 


'2191 


1-91 


•5515 


•2516 


'34 


•3883 


•1430 


»'45 


•4197 


•1676 


1-56 


•45" 


•1942 


1-67 


•4826 


•2228 


I -21 


•3513 


•1306 


1-31 


•3795 -1530 


1-41 


•4078 


•1771 


I -51 


•4361 


•2029 


1 08 


•3138 


•I 179 


I -17 


•3390 '1379 


1-26 


•3641 


•1594 


1*34 


•3892 


•1825 


95 


•2760 


•1047 


103 


•2980 


•1223 


ITl 


•3200 


•1413 


i^i8 


•3420 


•1617 


•82 


•2378 


•0911 


•'9 


•2566 


•1063 


•95 


•2755 


•1227 


I -02 


•2944 -14021 


69 


•1992 


•0770 


■74 


•2150 


•0898 


•80 


2307 


•1036 


•85 


•2465 


•1184 


•62 


•1797 


•0698 


•67 


•1938 


•0814 


•72 


•2080 


•0938 


•77 


•2221 


•1071 


•55 


•1601 


•0625 


•60 


•1727 


•0728 


•64 


•I8S3 


•0839 


•68 


•1978 


•0958 


49 


•>405 


•0551 


•S3 


•1515 


•0642 


•56 


•1625 


•0739 


•60 


•1735 


•0843 


•42 


•1207 


•0476 


•45 


•1301 


•0554 


•48 


■1396 


•0638 


•51 


•1490 


•0727 


•38 


•1108 


•0438 


•41 


•119* 05101 


•44 


•I 28 1 -0586 


'47 ' 


•1367 0669 ' 



Digitized 



byGoogk 



114 Principles of Mechanics chap. xi. 

Therefore the moment of inertia of the area included between the 
ellipses is 

/=^-|(2>43+ (ibh^)t -{■ {12 h} ^ 12 bh)^ 

4-(24^ + 8^)/ + i6/^| (31) 

If/ is small in comparison with b and h^ the second, third, and 
fourth terms in the expression for / are smaller and smaller com- 
pared with the first, and may be neglected. Therefore, the moment 
of inertia of the figure is approximately 

7=J^>4V(>4 + 3^) (32) 

The modulus of bending resistance is approximately 

Z='^^ht(h^Zb) (33) 

\Vhen a tube of circular section is flattened to form an oval 
tube, its thickness will be nearly uniform throughout, but in the 
oval tube section, above discussed, the thickness is not constant 
throughout, but is a little less than / except at the ends of the 
major and minor axes. The strength of an oval tube of uniform 
thickness will therefore be under-estimated if the formula (33) be 
used, so that the error is on the safe side. 

The area of the ellipse is -«^; and in the same way as above 

4 
it can be shown that the area included between the two ellipses is 

A^'^-(b-^h)t (34) 

2 

Therefore, (32) and (33) may be respectively written, 

^-tlfit^" w 

An oval tube of uniform thickness will be stronger than indi- 
cated by formula (36). This is clearly shown by figure 96, which 
represents a quarter section ; the modulus giveii in (36) is that of 

Digitized by VjOOQ 



CH^P. XI. 



Straining Action : Bending 



"5 



the tube whose inner surface is represented by the dotted line. The 

inner continuous line represents a tube of the same sectional area 

A, and of uniform thickness. It has the area 

b in excess of the dotted tube, and the areas 

a and c deficient, a + r = ^. It is evident 

from the figure that the moment of inertia of 

b about the minor axis is greater than that of a 

and Cy and therefore the tube of uniform 

thickness is slightly stronger than the section 

above discussed. 

98. D Tubes.— Tubes of D section have 
been recently introduced for the lower back 
fork of a bicycle ; it will be instructive to in- 
vestigate their bending resistances here. We 
will assume that the outline of the D tube is 
made up of a semicircle and its diameter 
(fig- 97)- Let r be the radius and h the dia- 
meter of the semicircle, / the thickness, which 
we will consider very small in comparison with 
r, and A the sectional area of the D tube. First, consider the 
moment of inertia about the axis a a 2X right angles to the flat 
side of the tube. The moment of inertia of 
the rectangle of depth h and width / is 




12 



h^ /, the moment of inertia of the semicircle 



is ^^/, therefore the moment of inertia of 
16 

the D tube about the axis a a\s 



'={L'-:y^- ■ ■ <37) 



1 
1 


^ 


1 


n 


at 


ic X, 


-Nt- 


y 



The modulus of the section, about the same axis, is 



Fig. 97. 



=(^s)^" 



W 



Consider now the moment of inertia about the axis b b^ coin- 
ciding with the flat side of the D tube. The moment of inertia of 

Digitized by CjOOQ I 2 



Ii6 Principles of Mechanics chap. xi. 

the flat side \% ~ h /', that of Jhe semicircle is ^ ^^ / ; therefore 

12 lO 

the moment of inertia of the section of the D tube is 

12 10 

If / be small in comparison with ^, the first term in this expres- 
sion may be neglected in comparison with the second, and there- 
fore, 

/ = — i4^ /, = ^ t^ t^ approximately . . . (39) 
16 2 

'But the /just found is not about an axis through the centre of 
figure ; this we now proceed to find. Let G be the centre of 
figure ; the distance O G can be found as follows : The moment 
of the semicircle about the axis ^ i^ is 2 r* / (see sec. 50), that of 
the straight side about the same axis is zero, the total moment of 
the D tube about the axis ^ ^ is therefore 2 r* /. But the total 
moment is also equal to the total area multiplied by the distance 
O G ', therefore 2 r» /= (2 r + Trr) / x OG, 

^"'i^^ = (.l^=air^'- = -3«9- . . . (40) 

Let /o be the moment of inertia about an axis gg passing 
through G parallel to ^^ ; then by section 92 

Therefore./. = ;H/-^^=(;-4^y,. . (4.) 
But A = {2 + Tt)rf; therefore we may write 

Z, the modulus of bending resistance about the axis gg is equal 
to " A" being the extremity of the radius through O G, Now, 

GX=:OX'-'OG=r— ^- r= "" r. 



2 -f TT 2 + TT 

4 

(2 + 



^={;'.(2V.)} ^^=-^5^4^>; . (43) 



Digitized by CjOOQIC 



CHAP. XI. Straining Action : Bending 117 

Let d be the diameter of a round tube equal in perimeter to the 
P tube. Then ir ^ = (2 4- ^y, 

/. r =-^ d=: '6iiod. 

2 + TT 

Substituting this value of r in (43), we get, 

The Z of the original round tube is approximately A d, so 

4 

the strengths of a round tube and the D tube into which it can 

be pressed are in the ratio of 2500 to 1542, i.e. 1000 to 617. 

But since a D tube is used when the space O X is limited, it 
would seem fairer to compare it with a round tube of equal 
weight and of diameter O X, The Z of a round tube of diameter 
O X '\s '2^Ar. Comparing this value with 
that in (43), it is seen that the strength of the D 1 ^ 

tube is slightly greater than that of a round r ^ 
tube of equal weight, and of diameter equal to 
the smallest diameter of the D tube, the ratio 
being "252 to '2500, a difference of less than 
one percent, in favour of the Dtube. 

99. Square and Eeotangolar Tabes.— Con- '''*' ^' 

sider the / of a square tube of section A B C D (fig. 98), about 
an axis a a parallel to the side A B. The / of each of the sides 

BC2Jv6.DA\% --, that of each of the sides ^^ and CZ> is 
12 

ht.\ therefore, for the whole section 
4 

I^2.'^A^2.htM^'-h^t (45) 

12 4 3 

The total sectional area is 4 ^ /, therefore 

J^\Ah^ (46) 

also, Z^i^^ Ah (47) 

^ 3 



Digitized by CjOOQIC 



1 1 8 Principles of Mechanics chap. xi. 

Let d be the diameter of a round tube of the same perimeter 
as the square tube ; then 4 ^ = tt </ 

.'. h^'^ d=z 7854 </, and 
4 

Z =- Ad=z'26i2>Ad (48) 

12 

hence, comparing with (30), the moduH of bending resistance of 
the square tube and of the original round tube are in the ratio of 

— to -, or of TT to 3, i,e, 1047 to 1000, in favour of the square 
12 4 

tube. Compared with a round tube of equal sectional area, but 

of the same diameter as the side of the square tube, the ratio is 

Ad Ad I I 

- to — , i,e. - to , or i33'3 to 100 : i,e. the square tube is 33*3 

3 4 3 4 

per cent, stronger than the round tube of equal area and diameter. 

Rectangular Tubes. — If a round tube be drawn into a rectan- 
gular tube of the same thickness, perimeter, and sectional area, it 
can be shown that the strength of the latter will be greatest when 
its depth h is three times its width b. * 

For any rectangular section, approximately 

Z = m(^+^) (50) 

For the strongest rectangular tube, (49) becomes 

/=9/^=^=I^>i' (51) 

^"^» Z^^Ah (52) 

4 

Comparing (33) and (50), it is seen that a thin rectangular 
tube is stronger than an elliptical tube of the same depth, width, 
and thickness in the ratio 16 : 3 tt. Now the ratio of the peri- 
meters, and therefore the weights, is never greater than 4 : ir ; 
this being the value when theeUipse and rectangle become a circle 

Digitized by CjOOQIC 



CBAP. XI. 



Straining Action : Bending 



119 



and square respectively. Weight for weight, then, the rectangular 
has at least ^ times the strength of the elliptical tube. 

That the rectangular is stronger than the elliptical tube of 
equal depth, width, and sectional area, can be easily shown from 
first principles, as follows : Figure 99 shows 
quadrants of rectangular and elliptical tubes of 
equal sectional area. Since the perimeter of the 
ellipse is less than that of the rectangle, its 
thickness is greater. Let a portion a of the 
ellipse be marked off equal in width to the 
corresponding part of the rectangle, so that the 
moments of inertia about the axis OXzxt equal. 
The part b is common to both ellipse and rect- 
angle, and there remain only the parts c. That 
belonging to the rectangle is at a much greater 
distance from the axis O X than that belonging 
to the ellipse ; its moment of inertia is therefore greater, and the 
rectangular is stronger than the elliptical tube to resist bending. 




Fig. 99. 



.Digitized by CjOOQIC 



1 20 Principles of Mechanics chap. xrt. 



CHAPTER XII 

SHEARING, TORSION, AND COMPOUND STRAINING ACTION 

loo. Compression. — The laws relating to simple compressive 
stress are exactly the same as those of simple tension, the formula 
(i), (2), (3), and (4), of chapter X. will apply, / being in this 
case the compressive stress, e the compression per unit of length, 
and E the modulus of elasticity for compression. For a homo- 
geneous material with perfect elasticity, as above defined, E would 
be the same for tension and compression. 

On a bar which is short in comparison to its diameter, if the 
compressive stress be increased above the elastic limit of compres- 
sion, the bar gives way ultimately by lateral yielding. If the 
material be hard, the bar may actually split up into several pieces. 
If of a soft, ductile material it will bulge gradually in the middle 
while being shortened in length. 

loi. Compression or Tension combined with Bending. — If 
a bar be simultaneously subjected to bending, and a pull or thrust 
parallel to its axis, the maximum stress on the section is the sum 
of the separate stresses due to the separate straining actions. If 
the bar be subjected to a pull Fy and a bending-moment M, A 
being the area and Z the modulus of the section, the maximum 
tensile stress is 

^-A^-2 • . (0 

and the minimum tensile stress is 

/'=3-f <•) 

For circular tubes of small thickness, substituting the value of Z 
from (30), section 96, 

f^A^-A-^ ^- • .• • (3) 



A Ad 



Digitized by CjOOQIC 



CHAP. ZII. 



Shearing, Torsion^ drc 



121 



The bending-moment may be produced by applying the pull 
/'at a distance x from the neutral axis of the section (fig. loo). 
In this case M =^ Fx and (3) may be written 



^'^'^ Ad 



(4) 




>P-. 



Fig. 101. 



If the bar be subjected to a compression P and a bending- 
moment My equations (i), (3), and (4) give the maximum compres- 
sive stress on the section, equation (2) 
the minimum compressive stress. 

102. Columns. — If a long bar be 
subjected to tension, any slight devia- 
tion from straightness (fig. 100) will, 
under the action of the forces, tend to 
get less. If, on the other hand, the 
bar be subjected to compression, the 
deviation from straightness will tend to 
get greater, and the bar will give way 
by bending (fig. 10 1). 

The stresses on a straight short 
column supporting a load, placed 
eccentrically, are given by formulae (i) 
and (2). 

Example, — A bicycle tube i in. diameter, 16 W.G., is subjected 

to a compressive force, the axis of which is in. from the axis of 

4 
the tube. Find the breaking load, the breaking stress of the material 
being 30 tons per sq. in. From Table IV., A = . 1882 sq. in., 

Z = '0414 in.^, also M -^^ P inch-lbs. /= 30 x 2240 lbs. per 

4 
sq. in. ; substituting in (i) 

P ^ P 

30x2240= QQ -h , 

^ -1882 4 X -0414 

from which, /> = 592 1 lbs. 

If the load were placed exactly co-axial with the tube, it would 

reach the value given by, 

— --. = 30 X 2240 
•1882 ^ ^ 

/>., P = 1 2650 lbs. Digitized by GoOglC 



122 



Principles of Mechanics 



CHA1». Xlt. 



103. Limiting Load on Long Colomns.— If, under the action 
of the load, the deviation x becomes greater, the bending-moment 
also becomes greater without any addition being made to the 
load ; thus the deviation once started, may rapidly increase until 
fracture of the column takes place. 

Let the section of the column be such that, under the action 
of the load, its neutral axis bends into a circular arc AC B (fig. 
Id) of radius R, Let A D B he the chord, CZ> the greatest 
deviation, and C E2l diameter of the circle. Then, by the well- 
known proposition in elementary geometry, 

CD xDE^AD X DB. 
i,e. neglecting the difference between CE and DE^ 



2 Rx^s- - approximately. 



EI 



But R = -^Tj^, from (17), chap, xi., and M^Fx. Substituting, 
M 



P^ 



ZEI 



(5) 



If the load be less than that given by (5), no deviation will take 
place. 

If the column be of constant section throughout its length 

its neutral axis bends into a curve of sines, and it can 

be shown that the limiting load is 



P^ 



'EI 



(6) 



Fig. 102. 



If the middle section of the column be prevented from 
deviating laterally, it will bend into the form shown in 
figure 102. In this case the length of the segment of 
the curve corresponding to figure loi is half the total 
length, and the corresponding load will be 

r^ 



jP = 



(7) 



Again, if the ends of the column be held in such a 
manner that the directions of the axis at the end are always the same, 
it will give way by bending as shown in figure lot^ The segment 

Digitized by VjOOQ 



CBAP. ZII. 



Shearings Torsion^ &c. 



123 



bd in this case is of the same shape as the curve in figure loi, while 
the portions a b and e d are of the same form as ^ ^ and dc. In this 

case, therefore, the length of the segment ^ ^ is , and the corre- 

2 

sponding limiting load is given by the formula 



P^ 



^ic^EI 



(8) 



If the column be fixed at one end (fig. 104), held laterally but 
free to turn at the other, 

/2 



F = 



(9) 



^- 


tn: 


it- 

1 1 
1 ' 


••V 




1/ 

iLl_ 





Fig. 103. 



Fig. 104. 



Fig. X05. 



If the column be fixed at one end and quite free at the other 
end (fig. 105), 

^= -4/2- ('°) 

These are known as Euler's formulae, and are only applicable to 
bars or columns in which the length / is great as compared with 
ihe least transverse dimension. / is the length before bending ; 
though in the figures, in which the bending is greatly exaggerated, 
it is marked as after bending. 

104. Gordon's Formula for Coliiiim8.--The pieces of tube 
used in bicycle building are too long to have the simple com- 
pression formula applied to them, and too short for the application 

Digitized by V^j 



1 24 Principles of Mechanics chap. xn. 

of Euler's formula. A great many experiments on columns, 
principally cast iron, have been made by Hodgkinson, and Gordon 
has suggested an empirical formula which agrees very closely with 
his experiments. For thin tubes, Gordon's formula becomes 

W^ / 

A , .1/^ (") 

'^ cd' 

/and c being constants depending on the material. 

Actual experiments on the compressive strengths of weldless 
steel tubes are wanting, but taking / = 30 tons per sq. in., and 
c = 32,000, Gordon's formula becomes 

W 67200 

A "■ 7^ r^ .... (12) 

32000 (P 

Example, — A tube is i in. diameter. No. 16 W.G., 20 in. 
long ; required the crushing load by Gordon's formula. 
From Table IV., page 113, ^ = -1882 ; 

W 67200 67200 

•1882 400 1*0125 

32000 X I 
from which, 

JF= 12490 lbs., 

slightly less than for a short length of the same tube (sec. 102). ' 

105. Shearing.— Let A B C D (fig. 106) be a small square 
prism of unit width perpendicular to the paper, subjected to 

shearing stress on the planes 

^ ^ and C D. If the planes 

A B and C Z> be very close 

to each other, the shearing 

I A stress will be the same on 

/\q both. If q be the shearing 

^ < > ' ' stress per unit of area, the 

downward force acting 2X A B 

Fic. ,06. F,c. X07. and the upward force at C Z> 

will each he q x A B. But since the portion A B C Diszi rest, 

the couple formed by the forces at -^4 ^ and C Z} must be 

Digitized by VjOOQIC 



U^ 






CHIP. xir. Shearing, Torsion, &c. 125 

balanced by an equal and opposite couple, formed by forces 
acting at -^ Z> and B C, since no force acts normally at the 
surfaces A B and C D, Thus the shearing^ stress on the sides 
A D and B C'v^ equal to that on A B and Z> C ; or the shearing 
stress on a plane is always accompanied by an equal shearing 
stress on a plane at right angles to the former, and to the direction 
of the shearing stress on the former plane. 

Transverse Elasticity, — Under the action of the shearing forces 
the square A B C D (fig. 106) will be distorted into a rhombus, 
A^ B^ CD, the angle of distortion A D A^ being proportional to 
the shearing stress. Let ^ be this angle and q the shearing stress 
producing it ; then 

• q^ Ci^ (13) 

C being the modulus of transverse elasticity, or the coefficient of 
rigidity of the material. 

Shearing Stress equivalent to Simultaneous Tension and Com- 
pressive Stresses, —Dmw a diagonal B D (fig. 106) ; the triangular 
prism A D B 'y& m equilibrium under the action of the three 
forces, / gy and h, acting on its sides, which can therefore be repre- 
sented by the sides of a triangle {^g, 107). /and g being equal, 
the force h is evidently at right angles to the side B D, The 
triangles A B D and fgh^Lie similar ; that is, the forces/ g and A 
are proportional to the lengths of the sides on which they act ; the 
stress per unit area must therefore be the same for the three sides 
A B, B D, and D A, Thus, the stress on the plane B D'\s 2i 
compressive stress of the same intensity as the shearing stress on 
the planes A B and A D, 

In the same way it may be shown that a tensile stress of equal 
magnitude exists on the plane A C, Thus, in any body a pair of 
shearing stresses on two planes at right angles are equivalent to a 
pair of compressive and tensile stresses respectively on two planes 
mutually at right angles, and inclined 45° to the former planes. 

106. Torrion. — If a long bar be subjected to two equal and 
opposite couples acting at its ends, the axes of the couples being 
parallel to the axis of the bar (fig. 108), it is said to be subjected 
to torsion. The moment of the couple applied is called the 
tivisting-moment on the bar. If one end be rigidly fixed, the 

Digitized by CjOOQIC 



126 



Principles of Mechanics 



other end will, under the action of the twisting-moment, be dis- 
placed through a small angle, and any straight line on the surface 
of the bar originally parallel to the axis will be twisted into a 
spiral curve a a. If the twisting-moment be increased indefinitely, 
the bar will ultimately break, the total angle of twist before break- 
ing depending on the nature of the material. 

Let figure 109 be the longitudinal elevation of a thin tube of 
mean radius r and thickness /, subjected to a twisting -moment 



3: 




Fig. 108. 



Fig. 109. 



T' foot-lbs. A square, abcd^ drawn on the surface of the tube 
becomes distorted while in a strained condition into the rhombus 
abd cH, Thus, every transverse section of the tube is subjected 
to a shearing stress. If the tube be of uniform diameter and 
thickness, this shearing stress, ^, will be the same throughout, 
provided the thickness is very small in comparison with the 
diameter. 

The sectional area of the tube is nrtr \ and since q is the 
shear on unit area, the total shear on the section is nrgtr. 
The shearing-force on each element of the section acts at a 
distance r from the centre of the tube ; the moment of the total 
shearing-force is therefore 2irqtr^, This must be equal to the 
twisting-moment T', applied to the end ; therefore 

T.^iirqtr' (14) 

Thus the twisting-moment which can be transmitted by a thin 
tube of circular section is proportional to the square of its radius 
or diameter and to its thickness. 

107. Torsion of a Solid Bar. — In a solid cylinder of radius r,, 
imagine the square abcd(^%, 109) drawn on a concentric cylin- 
drical surface of radius r \ it is easily seen that the angle of 
distortion of the fibres, <t>, or da d\ is proportip^ial to/*. If ^i be 

Digitized b 



)rtip|ial to r. 
d by Google 



CHAP. XH. Shearing, Torsion^ &c, 127 

the angle of distortion for a square drawn on the surface of the 
cylindrical rod, q and qx the shearing stresses at radii r and r, 
respectively, then evidently 



and therefore 



*=*•- 






If now the solid rod be considered to be divided into a 
number of thin concentric tubes, all of the same thickness, /, the 
area of the tube of radius r is 2 ir /r, the total shear on this tube is 

and the twisting-moment resisted is 

The sum of the moments of all the concentric tubes into 
which the rod is divided is easily found, by one of ^}ie simplest 
examples in the integral calculus, to be 

or, r=^^^^V. =^^^approx. . (15) 

108. Torsion of Thick Tubes.— If r^ and r^ be the external 
and internal radii of a hollow tube, the sum of the twisting-moments 
(45) o^ ^^^ v^T ^^*" concentric tubes into which it may be divided 
— and, therefore, the twisting- moment such a tube can resist— is 

or 

T^-^ WjnMfi .... (16) 

16 d^ 

The quantity — "T — ^ depends simply on the dimensions 

Digitized by CjOOQIC 



128 Principles of Mechanics chap. xu. 

of the section of the tube, and may be called the modulus of 
resistance to torsion ; it may be denoted by the symbdl Z^, Then 

Comparing Zp with Z, chapter xi., it will be seen that the 
modulus of resistance of a circular tube or solid bar to torsion is 
twice its modulus of resistance to bending. The strength of any 
tube to resist bending can therefore be obtained by multiplying 
the modulus from Table IV., page 112, by twice the maximum 
shear ^|. 

109. Lines of Direct Tension and Compression on a Bar 
subject to Torsion. — From what has been said in section 105, 
there will be a compressive stress on the plane a r, and a tensile 
stress on the plane b d (fig. 109). This holds for every point on 
the surface of the tube. Now if the tube be split up into a 
number of narrow strips by the spiral lines / /, inclined 45** to the 
axis (fig. 10), the tensile stresses can be transmitted just as before. 
The spiral lines / / are said to be tension lines, and the spiral lines 
c cdX right angles compression lines. If the twisting-moment be in 
the opposite direction, however, the pressure and tension spiral 
lines will be interchanged, and the split tube will not be able to 
transmit the twisting-moment. 

no. Compound Stress. — If the straining actions on any part 
of a structure be all parallel to one plane, the stress on any plane 
section, at right angles to the plane of the straining actions, can 
be resolved into a normal stress, tension or compression — and a 
tangential stress, shearing. It can be shown that any system of 
stress in two dimensions is equivalent to a pair of normal stresses 

on two planes mutually at right 
angles, and that the stress on one 
of these planes is greater than, that 
on the other plane less than, on 
any other plane section of the ma- 
terial. On any other plane the 
stress will have a tangential com- 
^'^- "°- ponent 

An important case of compound stress is that of a shaft sub- 
jected to bending and torsion ; a section at right angles to the 




CHAP. XII. Shearings Torsion, &c, 1 29 

axis of the shaft is subjected to a nonnal stress, / and simul- 
taneously to a torsional shearing stress, q. Consider a small 
portion of a material (fig. no) subjected to stresses parallel to the 
plane of the paper. Let A B Che a, small prism, of unit depth 
at right angles to the paper, the face B C being subjected to a 
normal stress, / and a tangential stress, q. From section 106 we 
know that an equal shearing stress, ^, must exist on the face A B. 
Let us find the magnitude of the stress p on the face A C, on 
which the stress shall be wholly normal. 

Considering the equilibrium of the prism ABC, and resolv- 
ing the forces on the three faces parallel to the side A B, we 
have 

p.AC.sinB-q.AB-'f.BC^o 
or 

{p^f)tane=q (17) 

Similarly resolving the forces parallel to B C, we get, 

p,AC,cose-q,BC=o 

or 

p — qtanB (t8) 

Multiplying (17) and (18) together, we get 
from which 

^ = i{/± >/7^TT^«} (19) 

the two values of / in (19) are the maximum and minimum 
normal stresses on the material. That is, the tension / and the 
shear q, on the face B C, produce on some plane A C the 
maximum tensile stress ^ {/ + s/P + 4 q^] , and on another plane 
the minimum tensile stress \{f — 'JP + 4 ^^} ; the latter plane 
being at right angles to the former. 

If the stresses on two planes at right angles be wholly normal 
and of equal intensity, it can easily be shown that the stress on 
any other plane is wholly normal and of the same intensity. If 
the normal stress be compression, the whole system of stress is of 
the nature of fluid pressure. If there be a tensile stress on one 
plane and an equal compressive stress on the plane at right angles, 

Digitized by V^jOOQK 



130 Principles of Mechanics chap. xn. 

it has already been shown that this is equivalent to shearing 
stresses of the same intensity on two planes at angles of 45° with 
the planes of the normal stresses. This pair of shearing stresses 
tends to distort the body but not to alter its volume, whereas fluid 
pressure tends to alter the volume but not the shape of the body. 
Any set of stresses in two dimensions can be expressed as the 
sum of a fluid stress and a shearing stress. Let two planes, A and 
B^ at right angles be subjected to normal tensile stresses of in- 
tensity, p and ^, respectively. Then this state of stress is equivalent 
to the sum of two states of stress, the first being a tensile stress 

-^ ^ on both planes A and B^ the second a tensile stress - ^""^ 
2 2 

on A and an equal compressive stress on the plane B. For 

/ =/_+.^. ^P-^lJ, and q =^^-±-? - t^.J, This principle will 
2 2 2 2 

be made use of when discussing the outer cover of a pneumatic 
tyre. 

III. Bending and Twisting of a Shaft— -In a circular shaft 
of diameter, //, subjected to a bending-moment, J/, and a twisting- 
moment, Ty the normal stress due to the bending-moment is 

32 
and the shearing-stress due to the twisting-moment is 

T 

q= -. 

'^ (P 
16 

Substituting these values in (19), 

16 

if the shaft be subjected to a twisting- moments 7*^, which would 
produce the same stress, p^ 

T 

16 



Digitized by CjOOQIC 



CHAP. XII. Shearings Torsion^ &c. 1 3 1 

and 7i is said to be the twisting-moment equivalent to the given 
bending-moment and twisting-moment acting simultaneously. 
Comparing the two expressions for /, we get 

T,^M ^ A/J/mr» (20) 

Similarly, the equivalent bending-moment is 

M,^\T,^\{M^s/-AP^rT^^, . . (21) 



Digitized by CjOOglS 



132 Principles of Mechanics chap. xht. 



CHAPTER XIII 

STRENGTH OF MATERIALS. 

112. Stress, Breaking and Working.— Each part of a machine 
must be capable of resisting the greatest straining actions that 
may come on it. This condition fixes, as a rule, the smallest 
possible section of the part below which it is not permissible to 
go. In ordinary machines, where mere mass is sometimes re- 
quisite, the section actually used may often with advantage be 
considerably greater than the minimum ; but in bicycles, since 
* lightness' is always sought after, though it should always be 
secondary to * strength,' the actual section used must not be very 
much greater than the minimum consistent with safety. The 
magnitude of the stress on any piece depends on the general con- 
figuration of the machine and of the arrangement of the external 
forces acting on it. The strength of the various parts depends on 
the physical qualities of the materials of which they are made, as 
well as on their section ; this we will now proceed to discuss. 

Breaking Stress, — If a load be applied at the end of a bar and 
be gradually increased, the bar will ultimately break under it. If 
the bar be of unit section—one square inch— the load on it at the 
instant of breaking is called the breaking tensile strength of the 
material. A great number of experiments have been made from 
time to time on the strength of materials, and the values of the 
breaking tensile strength for all materials used in construction are 
fairly accurately known. 

Factor of Safety, — One method of designing parts of a machine 
or structure is to fix arbitrarily on a working stress which shall not 
be exceeded. This working stress is got by dividing the breaking 
stress of the material, as determined by experiment, by an arbitrary 

Digitized by VjOOQ 



CHAP. xin. Strength of Materials 133 

number called 2: factor of safety. This factor of safety varies with 
the nature of the material used, and with the conditions to which 
the structure is subjected. Professor Unwin, in * Elements of 
Machine Design/ gives a table of factors of safety, the factor vary- 
ing from 3 for wrought iron and steel supporting a dead load, to 
30 for brickwork and masonry subjected to a varying load. The 
factor of safety should be large for a material that can be easily 
broken by impact, and may be low for a material that undergoes 
considerable deformation before fracture actually takes place. 

113. Elastic Limit — We have already seen (sec. 81) that the 
application of a load to a bar of what might be popularly called a 
rigid material produces an elongation, and that this elongation is 
proportional to the load applied up to a certain limit. If not 
loaded beyond this limit, on removing the load the bar returns to 
its original length, and no permanent alteration has been made. 
If, however, the load applied be greater than the above limit, the 
elongation produced by it becomes greater proportionally, and 
on the load being removed the bar is found to be permanently 
increased in length. The stress beyond which the elongation is 
no longer proportional to the load, is called the elastic limit. 

Since the elongation is in most metals proportional to the load 
applied up to this point, it has also been called the proportional 
limit (German, * Proporiionalitatsgrenze '). In a few metals — 
cast iron, brass—there is no well-defined proportional limit. 

The total elongation of a bar loaded up to a stress just inside 
the elastic limit is a very small fraction of its original length. On 
increasing the load beyond the elastic limit and up to the break- 
ing point, the elongation before fracture occurs, in the case 
of most materials, is a very much greater fraction of the original 
length. 

Table V. gives the breaking and elastic strengths and coefficients 
of elasticity of most of the materials used in cycle making ; the 
figures are taken from Professor Unwinds * Elements of Machine 
Design.* 

114. Stress-strain Diagram. — The relation between the 
elongation and the load producing it can be conveniently exhi- 
bited in the form of a diagram. Let the stress be represented by 
an ordinate O y drawn vertically (not shown on the diagram), and 

Digitized by V^j 



134 



Principles of Mechanics 



CHAP. XIII. 



1 I 



\8 fOO "^ o 






OOO*'* "^ ooo 



i 



ro W CO ^ »^ <» CS 



il I I ! I I I 



u 

D 
O 



3 

M 



ri 



i§§§ § §1 






f^ ' 



o »^ 



o »^o o 



I I 



I I 



I III 



*n u^OD 0^5vrJ000000O I 000*^000000 



. C5 



•3 

•c 



•I ti 

•a iu = 

o c y 
> Ac/) 



a 






1 



o 



Googk 



3 



-8 



Digitized by 



CHAP. XIII. Strength of Materials 1 3 5 

the corresponding extension be a lineJ^'/ drawn horizontally from 
y. The locus of the point / will be the stress-strain curve of the 
material. Stress-strain curves for a number of different materials 
subjected to tension are shown in figure iii. 

It has been proposed to represent the comparative values of 
materials for constructive purposes by figures derived from their 
stress-strain curves. The work done in breaking a test piece, 
reckoned per cubic inch of volume, may be used. This is pro- 
portional to the area included between the base and the stress- 
strain curve. Tetmajer's * value- figure' for a material is the 
product of the maximum stress and the elongation per unit length. 
It is the area of the rectangle formed by drawing from the final 
point of the stress-strain curve lines parallel to the axes. Of the 
materials represented in figure 1 1 1, * Delta ' metal and aluminium 
bronze have the highest * value-figures.' 

115. Hild Steel.— Figure m shows the stress-strain curve for 
mild steel, such as the material from which weldless steel tubes 
are made. The straight portion O a represents the action within 
the elastic limit. If the load be increased beyond that represented 
by a, the extension takes place at a more rapid rate, as shown by 
the slightly curved portion a b. At a point, b^ somewhat above 
the elastic limit, a, a sudden lengthening of the bar takes place 
without any increase of load, this being represented by the portion 
b c oi the curve. The stress at which this occurs is called the 
yield-point of the material. On further increasing the load, exten- 
sion again takes place, at first comparatively slowly, but afterwards 
more rapidly, until the maximum stress at the point d is reached. 
Under this stress the bar elongates until it breaks. If, however, 
the stress be partially removed after the maximum stress, d^ is 
reached, as can be done in a testing machine, the curve falls 
gradually, as at d e, then more rapidly until fracture occurs at / 
The elongation represented by the curve up to E takes place 
uniformly over the whole length of the bar, that represented by 
efonXy on a small portion in the neighbourhood of the fracture. 

In wrought iron, the yield-point is not so distinctly marked as 
in mild steel ; the stresses at the elastic limit and at breaking are 
less, the elongation before fracture is also less. The specific 
gravity of wrought iron and mild steel is, on an average, 77. 

.oogle 



136 



Principles of Mechanics 



CHAP. xin. 



1 1 6. Tool SteeL — For a tool steel of good quality, containing 
about one per cent, carbon, the maximum stress may be much 
higher ; the stress-strain curve takes the form shown in figure in, 




PERCENTAGE ELONGATION 
Fig. zxx. 



the extension being smaller, though the tenacity is very much 
greater, than that of mild steel. 

117. Cart Iron has no well-defined elastic limit ; in fact, the 
stress-strain curve is not straight for any part of its length, so that 
for cast iron the term * elastic limit,' though often used, has no 
definite meaning. 

118. Pore Copper varies greatly in tensile strength, according 
to the mechanical treatment to which it has been subjected. 
Rolling and wire-drawing both increase its tenacity. The stress- 
strain curve for rolled copper (fig. iii) is from Professor Unwinds 
* The Testing of Materials of Construction.' 

119. The Alloys of Copper with other metals form a most 

Digitized by CjOOQIC 



CHAP. XIII. Strength of Materials 137 

important series. Their mechanical properties are most fully 
discussed in Professor Thurston's 'Brasses, Bronzes, and other 
Alloys.' 

Brass contains 66-70 per cent, copper, and 34-30 per cent, 
zinc ; sometimes a little lead. The stress-strain diagram (fig. iii) 
shows that the stress at the elastic limit is very low in comparison 
with the ultimate breaking stress. 

Gun-metal is an alloy of copper and tin. The stress-strain 
diagram (fig. iii) is from a metal containing 98 per cent, copper, 
2 per cent. tin. 

Ternary alloys of copper, zinc, and tin have been exhaustively 
investigated by Professor Thurston. He finds the best proportion, 
when toughness as well as tenacity is important, is copper 55, tin 
0-5, zinc 44-5. 

Aluminium Bronze, — Copper and aluminium form a most 
useful series of alloys. The stress-strain curve (fig. iii) is from 
an alloy containing about 10 percent, aluminium ; it shows clearly 
the great strength and ductility of the material. 

Alloys containing a much larger proportion of aluminium are 
valuable where lightness is the first consideration, but since they 
possess little strength and ductility, they can only be sparingly 
used in structural work. 

Delta metal is a copper-zinc-iron alloy, which can be cast and 
worked hot or cold. It possesses great strength and ductility, as 
is shown by the stress-diagram (fig. iii) from a bar 79 sq. in. 
sectional area, tested by Mr. A. S. E. Ackermann at the Central 
Technical College. 

1 20. AluminiiiTn. — A specimen of squirted aluminium, con- 
taining 98 per cent, of the pure metal, was tested at the Central 
Technical College by Mr. Ackermann ; the tenacity was 6*32 tons 
per sq. in. ; the elongation in 8" was i*i2", of which '53" was in 
the immediate neighbourhood of the fracture ; the general elonga- 
tion may, therefore, be taken as 10 per cent. For comparison 
this result is plotted in figure 1 1 1 . 

Pure aluminium has not sufficient strength and toughness to 
be of much value as a structural material, though its lightness as 
compared with other metals is a desirable quality. Some alloys, 
containing a small percentage of aluminium, possess great strength, 



138 Principles of Mechanics chap, xm 

but they are, of course, heavy. It remains to be seen whether a 
strong alloy, containing a large percentage of aluminium, and 
therefore light, can be discovered. Such an alloy may possibly be 
of value in cycle making. 

The specific gravity of sheet aluminium is 2*67, of mild steel 

77- 

121. Wood is not so homogeneous as most metals ; it is, as a 
rule, much stronger along than across the grain. The fact that 
wood joints are generally of low efficiency is against its use in 
tension members of a frame. For compression members, where 
there is no loss of strength at the joints, it may be used with 
advantage in some cases, its compressive strength (see Table VI.) 
being not much inferior, weight for weight, to that of the metals. 
In beams of short span subjected to bending, it is, in some im- 
portant cases, immensely superior, weight for weight, to metal. 
The strength of a rectangular beam is proportional to its width, 
the square of its depth, and the strength of the material from 
which it is made (sec. 94), i,e, proportional to b z^f. If beams of 
equal weight be made from wood and steel, the width b being the 
same in both, the depth d of the wood beam will be greater than 
that of the steel beam ; and the product z^fmW be much greater 
for the wood than the steel beam. 

The rim of a bicycle wheel is subjected to compression and 
bending (sec. 255). Since its width must be made to suit the tyre, 
a wood rim will be much stronger than a solid steel rim of the 
same weight ; or, for equal strengths, the wood rim will be the 
lighter. A holloiv steel rim will possibly be stronger than a wood 
rim of equal weight. 

Table VI., taken from Laslett's * Timber and Timber Trees, 
gives the weights and strengths of a few woods. 

122. Eaising of the Elastic Limit. — Let a bar be subjected 
to a stress — represented by the point k (fig. in) — consider- 
ably above its elastic limit. If the load be removed and the 
bar be again tested, it wfU be found that it is elastic up to a stress 
as high as that indicated by k. Thus the elastic limit in tension 
of a material like mild steel can be raised by simply applying an 
initial stress a little above the limit required. 

An important application of this principle occurs in the case 

Digitized by V^jOOQ 



CHAP. XIII. 



Strength of Materials 



139 



Table VI. 
Specific Gravity and Strength of Woods. 



Name of wood 

1 


Specific 

^ water being 
taken I'ooo 


Transverse 
load on 
pieces 

2"X2"X72" 


Tensile 

stress on 

pieces 

2"xa"x3o" 


Vertical 
sire^ on 
pieces 

2" X 2" X 2" 


Ash, English . 

] ,, American 

Elm, English . 

,, Canadian 


1 

-480 
1 .558 
1 748 


lbs. 

862 
638 

393 
920 


lbs. per sq. in. 
3,780 

5,495 
5.460 
9,182 


lbs. per sq. in. 
5U94 

5.78| 

7,418 


Fir, Dantzic . 
„ Spruce, Canada 
Kauri, New Zealand 
Larch, Russian 


1 -484 

•646 


877 

670 

816 
626 


3,231 
3,934 
4,543 
4,203 


7,104 

5,985 


Oak, English . 

„ French . 

,y White, American 
Pine, Yellow . 

„ Pitch, American 


'735 
•976 

' -983 

1 659 


776 

878 
804 

505 
1,049 


7,571 
8,102 
7,021 
2,027 
4,666 


7,640 
7,942 
6,964 
4,172 
6,462 



of Southard's twisted cranks. Here the cranks are given a con- 
siderable initial twist in the direction in which they are strained 
while driving ahead ; their strength is considerably increased 
thereby. A twist (sec. 109) is equivalent to a direct pull along 
certain fibres, and a direct compression along other fibres at right 
angles. The initial twist in Southard's crank is, therefore, equi- 
valent to raising the elastic limit of tension of the fibres under 
tensile stress, and the elastic limit of compression of the fibres 
under compressive stress. 

123. Complete Stress-strain Diagram. — ^The complete stress- 
strain diagram of a material should extend below the axis O X ; 
in other words, it should give the contractions of the bar under 
compressive stresses, as well as elongations under tensile stresses. 
Figure 112 represents such a curve, the point a denoting the 
elastic limit in tension, and b the elastic Hmit in compression. If 
the bar has had its elastic limit in tension raised artificially to the 
point k (fig. Ill), it is found experimentally that the elastic limit 
in compression has been lowered, and thus the new stress-strain 
curve would be somewhat as represented in figure 1 13^^ 

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140 Principles of Mechanics chap. xiu. 

These considerations, when applied to the case of Southard's 
cranks, detract from the value of the initial twist. The line / / 
(fig. 109), which is the tension line when the 
rider is pedalling ahead, has had its elastic 
limit in tension artificially raised, and its elastic 
limit in compression artificially lowered by the 
initial twist. When back-pedalling, / / becomes 
the compression line. A twisted crank is 
therefore weaker for back-pedalling than an 
untwisted crank of the same material. 

124. Work done in Breaking a Bar. — A 
o\ material that gives very little extension before 
breaking is said to be wanting in toughness^ 
and is not so suitable for structural purposes 
I as a material with a larger extension. The total 

Fig. XX2. Fig. 113. elongation of a material is usually expressed 
as a percentage of its original length. If the actual instead of 
the percentage elongations be set off horizontally (fig. iii), the 
area included between the stress-strain curve, its end ordinate, 
and the axis O X, represents the work done in breaking the bar. 
A bicycle is a structure subjected not to steadily applied forces 
but to impact The relative value of a hard and a tough mate- 
rial for resisting such straining actions may be illustrated by an 
example. 

Example, — ^Take a material like hardened steel, elastic up to 
its breaking-point, so that its stress-strain diagram is as shown at 
figure 114. Let its breaking-stress be 60 tons per square inch, 
and jE = 1 2,000 tons per square inch. Then the extension at 
breaking-point is 

Z7 60 
12000 

If the original length of the bar be 10 inches, the total elonga- 
tion O X (fig. 1 14) will be -05 inches, and the work done will be 
the area of the triangle Oax^ 

= i X 60 X '05 = 1*5 inch-tons. 

Take now a material like mild steel, and consider that its 
stress-strain curve is quite straight up to the yield-point d (fig. 115). 

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CHAP. XIII. 



Strength of Materials 



141 




X 
Fig. 114. 



XX 



Fig. 115. 



Let the yield-point occur at 15 tons per square inch ; then, taking 
E^ as before, 12,000 tons per square inch, and the original length 
of the bar ib inches, O jp will be '0125 inches. 
The work done in stretching the bar up to the 
yield-point will be 

^ X 60 X -0125 = o*375 inch-tons. 

Consider both bars to be acted on by a 
force of impact equivalent to a weight of 10 tons 
falling through a height of \ inch. The work 
stored up in this falling weight will be 

10 X i = 2 inch-tons. 

This must be taken up by the bar. But the 
work done in breaking the hard steel bar of high 
tenacity is only i 5 inch-tons ; it would therefore be broken by 
such a live load. The mild-steel bar would be stretched an 
additional length, xx^^ until the total area, Ob b^ .r,, was equal to 
2 inch-tons. The area, b b^ x^ x, is therefore 

2 — o'375 = 1*625 inch-tons. 
The distance x Xi will be 

^•^^.5 = .108 inch. 

Thus the only effect of the impulsive load on the mild steel bar 
is to stretch it a small distance, though the same load is sufficient 
to break the bar of much higher tensile strength but with little or 
no elongation before fracture. 

The above examples show that the elongation before fracture 
of a material is almost as important as its breaking strength in 
determining its value as a material for bicycle building. 

125. Mechanical Treatment of Metals.— The tenacity of a 
metal is almost invariably increased by rolling, or by drawing 
through dies. A metal to be drawn into wire or tube must be 
strong and ductile. The finest wire is made from a metal in 
which the ratio of the elastic to the ultimate strength is low. A 
metal with very high tenacity has not generally the ductility neces- 
sary for drawing into tubes or wire. The Premier Cycle Company, 

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142 



Principles of Mechanics 



CHAP. xni. 



instead of using drawn tubes, which must be made from a steel 
having a comparatively low tenacity, build up their tubes from flat 
sheets bent into spirals, each turn of the spiral overlapping the 
adjacent one, so that there are two thicknesses of plate at every 
part ol the tube (fig. ii6). A steel of much higher tenacity can 




Fig. ii6. 

be used for this process than could be successfully drawn into 
tubes. These 'helical' tubes, therefore, have greater tenacity 
but less ductility than weldless steel tubes, as is shown by the 
comparative tests of helical and solid-drawn tubes i inch external 
diameter, recorded in Table VII. For comparison with other 
materials, the results of these tests are plotted on figure 1 1 1 ; the 
final points of the stress-strain diagrams being the only ones 
obtainable from the data, the curves are drawn dotted. 

Table VII. 
Tensile Strength of Helical and Solid-drawn Tubes. 



Description 



Helical 14A 

„ 20A 

,, 20c 

Sol id -drawn c, 

»» H, 



Sectional 
area 

Sq. in. 
0-105 

0'107 
0-134 

o*io6 
o-io6 



Extension 

in 
' 10 inches 



Ultimate 
stress 

lbs. per sq. in. 

117,000 j 31 
122,000 y 15 

130*000 , 3*4 

80,000 I 187 

94,000 8-0 



Appearance of 
fracture 



r 12 per cent, silky 
[ 88 per cent, granular 

Granular 

Granular 

Silky 

Silky 



126. Bepeated Stresses. — If a bar be subjected to a steady 
load just below its breaking load, it will support it for an indefinite 
period provided the load remains constant, neither being increased 
or diminished. If the load is variable, however, the condition is 
quite different. Wohler has shown that if the load vary from a 
maximum T^ to a minimum T^, fracture will occur when T^^ is 
less than the statical breaking load T^ after a certain number of 
alterations from T^ to T^. The number of repetitions of the load 

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CHAP. xin. 



Strength of Materials 



H3 



before fracture takes place depends not only on jT, but on the 
difference T'l — T^^ between the maximum and minimum loads. 
With a great range of stress the number of repetitions before 
fracture is less than with a smaller range. 

A steel axle tested by Bauschinger, which had a statical tensile 
strength of 40 tons per square inch, stood at least two or three 
million changes of load before breaking, with the following ranges 
of stress : 



Maximum stress 
tons per sq. in. 




Range of stress 
tons per sq. in. 



21 'O 

197 

I2-I 

O 



A fuller discussion of this subject is given . in Professor 
Unwinds * Machine Design ' and ' The Testing of Materials of 
Construction.' 

The running parts of a bicycle — the wheels, chain, pedal-pins, 
cranks, and crank-axle— are subjected, during riding, to varying 
stresses. The range of stress on the spokes is probably small, so 
that a high maximum stress may be used without running any 
risk of fracture after the machine has been in use a considerable 
time. The stress on a link or rivet of the chain varies from zero, 
when on the slack side, to the maximum on the tight side. The 
double change of stress on the pedal-pins, cranks, and crank-axle 
takes place once during each revolution of the latter. A distance 
of 5,000 miles ridden on a bicycle geared to 60" corresponds to 
1,500,000 double changes of stress on the cranks and axle. If 
these be made light (see chapter xxx.), no surprise need be ex- 
pressed if fracture occurs at any time, after having run satisfac- 
torily for one or two years. 



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PART II 
CYCLES IN GENERAL 

CHAPTER XIV 

DEVELOPMENT OF CYCLES : THE BICYCLE. 

127. Introductory. — Wheeled vehicles drawn by horses have 
probably been used by all civilised nations. The chariot of the 
ancients was two-wheeled, the wheels revolving upon the axle. 
Coming down to later times, the coachy a covered vehicle for 
passengers, appears to have been first made in the thirteenth 
century, the earliest record relating to the entry of Charles of 
Anjou and his queen into Naples in a small carretta. The first 
coaches in England are said to have been made by Walter Rippon 
for the Earl of Rutland in 1555, and for Queen Elizabeth in 1564. 
The weight of these early coaches was probably so great that for 
centuries it seemed utterly impracticable to make a vehicle that 
could be propelled by the rider. With the growth of the 
mechanical arts, at the beginning of this century, more attention 
was given to the subject. Starting from the four-wheeled vehicles 
drawn by a horse, the most obvious step towards getting a pedo- 
motive vehicle was to make one of the axles cranked, and let the 
rider drive it either direct or by a system of levers, the wheels 
being rigidly fastened to the ends of the axle. Such a cycle is 
illustrated in figures 117, |i8. If this cycle had to travel in 
straight lines or curves of large radius, as on a railway, it might 
have been, apart from its weight, fairly satisfactory. A grave 
mechanical defect was that in moving round a sharp curve one or 
both driving-wheels slipped, as well as rolled, on the giound, with 
a corresponding waste of energy in friction. 

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146 



Cycles in General 



CHAP. XIV. 



The first attempts at overcoming this difficulty consisted in 
fastening only one wheel rigidly to the driving-axle, the other 
running freely. This gave, however, a machine which did not 
always respond to the steering gear as the rider wished ; in fact, 
while a driving effort was being exerted, the machine tended to 
turn to the side opposite to the driving-wheel (see chap. xviiL). 
The introduction of the differential driving-axle, which allows both 




Fig. 118. 



wheels to be driven at different speeds, overcame this difficulty 
completely without introducing any new ones. 

The weight of the four-wheeler, and even of the three-wheeler, 
was, however, so great that it was not in this direction that cycles 
were at first developed. A wooden frame for supporting two 
wheels was, of course, much lighter than one for three wheels ; for 
this reason principally, bicycles were brought to a fair degree of 
perfection before tricycles. The use of steel tubes for the various 
parts of the frame made it possible to combine the strength and 
lightness necessary for a practicable cycle, and laid on a sure basis 
the foundations of the cycle-making industry. 

Without attempting to give an exhaustive history of the de- 

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CHAP. XIV. Development of Cycles : the Bicycle 



H7 



velopment of bicycles and tricycles, a short account of the various 
types that have from time to time obtained public favour may be 
given here. 

128. The Bandy-horse. — Figure 119 may be taken as the first 
velocipede man-motor carriage. This was patented in France in 
1818 by Baron von Drais. In * Ackermann's Magazine/ 1819, an 
account of this pedestrian hobby-horse is given. " The principle 




Fig. 119. 

of the invention consists in the simple idea of a seat upon two 
wheels propelled by the two feet acting on the ground. The 
riding seat or saddle is fixed upon a perch on two short wheels 
running after each other. To preserve the balance a small board 
covered and stuffed is placed before, on which the arms are laid, 
and in front of which is a little guiding pole, which is held in the 
hand to direct the route. The swiftness with which a person well 
practised can travel is almost beyond belief, 8, 9, and even 10 
miles may, it is asserted, be passed over within the hour on good 
level ground." 

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148 Cycles in General chap. xiv. 

129. Early Bicycles. — Messrs. Macredy and Stoney, in *The 
Art and Pastime of Cycling,' write : " To Scotland, it appears, 
belongs the honour of having first affixed cranks to the bicycle ; 




Fig. X20. 



and, still stranger to relate, it was not to the * hobby-horse,' but 
to a low-wheeled rear-driver machine, the exact prototype of 
the present-day Safety. The honour of being the inventor has 
now been fixed on Kirkpatrick M*Millan, of Courthill, Dumfries- 




Fig. 121. 



shire, though prior to 1892 Gavin Dalzell of I^smahagow was the 
reputed inventor. It seems, however, that Dalzell only copied 
and probably improved on a machine which he saw with McMillan. 

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CHAP. XIV. Development of Cycles : the Bicycle 149 

McMillan first adapted crank-driving to the * hobby-horse ' about 
the year 1840, and it was not earlier than 1846 that Dalzell built a 
replica of McMillan's machine, a woodcut of which we reproduce 
(fig. 120). McMillan is said to have frequently ridden from Court- 
hill to Dumfries, some fourteen miles, to market on his machine, 
keeping pace with farmers in gigs." Figure 121 illustrates the 
* French ' bicycle or ' Bone-shaker,' which was in popular favour 
during the sixties. The improvement on the Dandy-horse con- 
sisted principally in the addition of cranks to the front wheel, so 
that the rider was supported entirely by the machine. 

In * Velocipedes, Bicycles, and Tricycles,' published by George 
Routledge & Sons in 1869, descriptions and illustrations of the 
bicycles, tricycles, and four-wheelers then in use are given. The 
concluding paragraph of this little book may be quoted : " Ere I 
say farewell, let me caution velocipedists against expecting too 
much from any description of velocipede. They do not give 
power, they only utilise it ; there must be an expenditure of power 
to produce speed. One is inclined to agree with the temperate 
remarks of Mr. Lander, C.E., of Liverpool, rather than with the 
extravagant enthusiasm of American or French riders. As a means 
of healthful exercise it is worthy of attention. Certainly not more 
than forty miles in a day of eight hours can be done with ease ; 
Mr. Lander thinks only thirty. If this is correct, it does not beat 
walking, though velocipedists affirm that double the distance can 
be done with ease. Much will and must depend on the skill of 
the rider, the state of the roads, and the country to be travelled." 

130. The Ordinary.— What has since been called the * Ordi- 
nary 'bicycle came into use early in the seventies. Figure 122 
illustrates one made by Messrs. Humber & Co., Limited. The 
great advance on the bicycle illustrated in figure 121 consisted 
mainly in the use of indiarubber tyres, thus diminishing vibration 
and jar, and consequently diminishing the power necessary to 
propel the machine. As a direct consequence of this, a larger 
driving-wheel could be driven with the same ease as the com- 
paratively small driving-wheel of the French bicycle. The design 
of the * Ordinary ' is simplicity itself, and it still remains the embodi- 
ment of grace and elegance in cycle construction, though super- 
seded by its more speedy rival, the rear-driving^--Safetj». The 



ISO 



Cycles in General 



motive power of the rider is applied direct to the driving-wheel ; 
wheel, cranks and pedal-pins forming one rigid body. In this 
respect it has the advantage over bicycles of later design, with 
gearing of some kind or other between the pedals and driving- 
wheel. 

In the 'Ordinary ' the mass-centre of the rider was nearly directly 
over the centre of the wheel, so that any sudden obstruction to 
the motion of the machine frequently had the effect of sending 




Fig. 122. 



the rider over the handle-bar. This element of insecurity soon 
led to the introduction of other patterns of bicycles. 

131. The * Xtraordinaxy ' (fig. 123), made by Messrs. Singer 
& Co., was one of the first Safety bicycles. The crank-pin was 
jointed to a lever, one end of which vibrated in a circular arc 
(being suspended by a short link from near the top of the fork), 
the other end was extended downwards and backwards, and 
supported the pedal. A smaller wheel could thus be used, 
and the saddle placed further back than was possible in the 
* Ordinary.' 

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CHAP. XIV. 



Development of Cycles : the Bicycle 



151 




Fig. 123. 

132. The Facile. — In the * Facile' bicycle a smaller driving- 
wheel was used, and the mass-centre of the rider brought further 
behind the centre of the driving-wheel. This was accomplished 




Fic. 124. 

by driving the crank by means of a short coupling-rod from 
a point about the middle of a vibrating lever^ the end of 

Digitized by VjOOQ 



152 



Cycles in General 



this vibrating lever forming the pedal. The fork of the front 
wheel was continued downwards and forwards to provide a fulcrum 
for the lever. The motion of the pedal relative to the machine 
was thus one of up-and-down oscillation in a circular arc, and was 
quite different from that of the uniform circular motion in the 
* Ordinary.* From the position of the mass-centre of the rider rela- 
tive to the centre of the driving-wheel, it is evident that this bicycle 
possessed a much greater margin of safety than the ' Ordinary.' 
Also, from the fact that the machine and rider offered a less surface 
to wind resistance, the machine was easier to propel under certain 
circumstances than the * Ordinary.' In 1883, Mr. J. H. Adams 
rode 242^ miles on the road within twenty-four hours ; this was at 
that time the best authentic performance on record. 

133. Kangaroo. — Figure 125 illustrates the * Kangaroo ' type of 
front wheel crank-driven Safety introduced by Messrs. Hillman, 




Fig. 125. 

Herbert, and Cooper, 1884. A smaller driving-wheel is used than 
in the * Ordinary ' ; the crank-axle is placed beneath and a little 
behind the centre of the driving-wheel. The crank-axle is divided 
into two parts, since its axis passes through the driving-wheel ; the 
front- wheel fork is continued downwards to support the crank- 

.oogle 



CHAP. XI7. Development of Cycles : the Bicycle 153 

axle bearings ; the motion of each portion of the crank- axle is 
transmitted by chain-gearing to the driving-wheel. In a 100-mile 
road race on September 27, 1884, organised by the makers of the 
machine, the distance was covered by Mr. G. Smith in 7 hours 
7 minutes and 11 seconds, the fastest time on record for any 
cycle then on the road. 

A geared dwarf bicycle is superior to an * Ordinary ' in two 
important respects, which more than compensate for the friction of 
the extra mechanism. Firstly, the rider being placed lower, the 
total surface exposed by the machine and rider is much less, the 
air resistance is therefore less, this advantage being greatest at 
high speeds. Secondly, since the speeds of the driving-wheel 
and crank-axle may be arranged in any desired ratio, the speed 
of pedalling and length of crank can be chosen to suit the 
convenience of the rider, irrespective of size of driving-wheel ; 
while in an * Ordinary ' the length of crank is less, and the speed of 
pedalling greater, than the best possible values. 

As regards safety, the * Kangaroo * is a little better than the 
* Ordinary,* but not so good as the * Rover ' or * Humber ' Safety. 
Two serious defects, which ultimately made it yield in popular 
favour to the rear-driving Safety, existed. A narrow tread must 
be kept between the pedals, and the consequent narrow width of 
bearing of the crank-axle gives a bad design mechanically. 
Again, the chains, however carefully adjusted initially, will, after 
a time, get a trifle slack. In pressing the pedal downwards the 
front side of the chain is tight, but when the pedal is ascending, 
since it cannot be lifted direct by the rider, it is pulled up by the 
chain, the rear side of which gets tightened. This reversal, taking 
place twice every revolution, throws a serious jar on the gear. 
This defect cannot, as in the * Humber ' with only one driving- 
chain, be overcome by skilful pedalling. 

134. The Bear-diinng Safety was invented by Mr. H. J. 
Lawson in 1879, but it was a few years later before it was in great 
demand. The * Rover ' Safety (fig. 1 26), made by Messrs. Starley 
and Sutton in 1885, was the first rear-driving bicycle that attained 
popular favour. The cranks and pedals are placed on a separate 
axle, the motion of which is transmitted by a single driving-chain to 
the driving-wheel. This type is absolutely safe as regards headers 



154 



Cycles in General 



CHAP. XIT. 



over the handle-bar. Compared with the * Kangaroo ' gearing, the 
single driving-chain is a great improvement, as its driving side 




Fig. 136 



may be kept tight continuously. The steering-head ot the front 
wheel was vertical, and an intermediate handle-pillar was used, 
with coupling-rods to the front fork. In a later design (fig. 127) 




Fig. 177. 



the front fork was sloped, and the steering made direct ; this 
machine thus formed the protot>'pe of the modern rear-driving 
bicycle. 

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CHAP. XIV. Developntent of Cycles : tlie Bicycle 155 

Figure 128 is an illustration of the *Humber* Safety dwarf- 
roadster, made in 1885. In this all the arrangements of the 
' Ordinary * may be said to be reversed ; the proverbial Irishman's 
description of it being " The big wheel is the smallest, and the 
hind wheel is in front." The driving-wheel is changed from front 
to back, the small wheel is placed in front, and the mass-centre 
of the rider is brought nearer the centre of the rear wheel. 

The * Humber ' Safety of 1885 is essentially the same machine 
as that in greatest demand at the present day. The improvements 




Fig. 128. 

effected since 1885, though undoubtedly of very great practical 
importance, are merely improvements in details. Change in the 
relative size of the front and back wheels, different design of frame, 
and last, but not least, the introduction of pneumatic tyres, 
account for the different appearances of the earliest and latest 
Safeties. 

Rear-driving Safeties were made by all the makers, the differ- 
ence in bicycles by different makers being merely in detail. About 
this time (1886) the number of Safety bicycles made per annum 
began to increase very rapidly, while a few years later the number 
of * Ordinaries ' began to diminish. 

135. Oeared Facile.— The * Facile ' bicycle, with its small driv- 
ing-wheel and direct link-driving from the pedal lever, necessitated 



156 



Cycles in General 



CHAP. XIV. 



very fast pedal action on the part of the rider. The * Geared 
Facile* (fig. 124) enabled the pedalling to be reduced to any 
desired speed. The connecting link in the * Geared Facile ' did 
not work directly on the driving-wheel, but the crank shaft ran 
loose co-axially with the driving-wheel, a sun-and-planet gear 
being inserted between the crank and the wheel. Figure 129 




Fig. X29. 

shows a * Geared Facile * rear-driving bicycle, the usual sun-and- 
planet gear being modified to suit the altered conditions. 

136. Diamond-frame Rear-driving Safety.— From the date 
of its introduction, the rear-driving Safety advanced steadily in 
popular favour until, in 1887, it was the bicycle in most general 
demand. In the preface to ' Bicycles and Tricycles of the Year 
1888,' Mr. H. H. Griffin says : "We made careful inquiries of all 
those in a position to know as to the proportion of Dwarf 
Safeties and Ordinary bicycles, and were not a little surprised 
to hear that, taking the average through the trade, at least six 
Dwarf Safeties are made to one Ordinary." Up to the year 
1890 the greatest possible variety existed in the frames of the 
rear-driving Safety, but they all agreed in having the distance 
between the rear and front wheels reduced to a minimum. The 
crank-bracket was placed just sufficiently in front of the driving- 
Digitized by CjOOqIc 



CHAP. XIV. 



Development of Cycles: the Bicycle 



157 



wheel to have the necessary clearance, the steering-wheel suf- 
ficiently far in front to allow it in steering to swing clear of the 
pedals and the rider's foot. The down-tube, from the saddle to 




Fig. 130. 



the crank-bracket, was usually curved, both in the diamond-frame 
and the cross-frame, or omitted altogether, as in the open-frame. 
Up till 1890 the nearest approach to the now universally adopted 




Fig. 131. 



frame was that made by Humber & Co. (fig. 130). During these 
years the diamond-frame was being more and more generally 
adopted, and after Messrs. Humber introduced their rear-driving 

Digitized by CjOOQIC 



158 



Cycles in General 



CHAP. XIV. 



Safety, with long wheel-base and diamond-frame (fig. 131), it 
became almost universal. By having several inches clearance 
between the crank-bracket and the driving-wheel, it was possible 
to use a straight tube from the saddle to the crank- bracket, while 
the long wheel-base rendered the steering more reliable. In the 
chapter on * Frames ' the reasons for the survival of the diamond- 
frame and the practical extinction of all others will be given. 

137. Sational Ordinary.— The admirers of the 'Ordinary' 
bicycle were loth to let their favourite machine fall into disuse, 
and attempts were made to make it safer and more comfortable, 
by placing the saddle further behind the driving-wheel centre, 
by sloping the front fork, and by making the rear wheel larger 
than was usual in the * Ordinary.' Such a machine was called a 
* Rational Ordinary.' 

138. Geared Ordinary and Front-driving Safety.— In 1891, 
the Crypto Cycle Company — with whom Messrs. Ellis & Co., the 

makers of the * Facile ' and 

* Geared Facile ' had amal- 
gamated — brought out a 
Geared Ordinary, This 
bicycle was in external 
appearance just like a 

* Rational ' ; but the cranks, 
instead of being rigidly 
connected to the driving- 
wheel, drove the latter by 
means of an epicyclic gear 
(see sec. 306) concealed 

in the hub. The number of revolutions of the driving-wheel 
could thus be made greater than those of the crank ; in fact, the 
machine could be geared up, just like a rear-driving Safety. The 
size of the driving-wheel being reduced, a front-driving Safety was 
obtained. Figure 132 shows the 'Bantam,' the latest develop- 
ment of the front-driver in this direction, with the front wheel 
24 inches in diameter, and geared to 66 inches. The resem- 
blance, in general arrangement at least, to the French bicycle 
(fig. 121) will be apparent, though as regards efficiency of action the 
two machines are as wide apart as the poles. Figure 243 shows 

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Fig. 132. 



CHAP. XIV. Development of Cycles : the Bicycle 



1 59 



the * Bantamette,' in which the frame is so arranged that the bicycle 
may be ridden by a lady. 

139. The Giraffe and Bover Cob.— The * Ordinary' had un- 
doubtedly many good points which are missing in the modern 
Safety, among which may be mentioned greater lateral stability 
and steadiness in steering due to the high mass-centre. The 
'Giraffe' (fig. 133), by the New Howe Machine Company, is a 
high-framed Safety, the saddle being raised as high as in the 




Fig. 133. 

* Ordinary.' In the introduction to Leechman's * Safety Cycling,' 
Mr. Henry Sturmey gives an enthusiastic account of the * Giraffe, 
and a comparison with the low-framed Safety. 

The * Rover Cob' (fig. 134), made by Messrs. J. K. Starley & Co., 
is at the opposite extreme, the frame being made so low that the 
pedals will just clear the ground when rounding a comer at slow 
speed. It is intended for those who may have fear of falling ; 
the mounting can be done by simply pushing off from the ground. 

140. Pnenmatic Tyres. — Whether judged by speed perform- 
ances on the road or racing track, or from additional comfort 
and ease of propulsion to the tourist, the greatest advance in 
cycle construction due to a single invention must be credited to 

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i6o Cycles in General chap. xiv. 

Mr. James Dunlop, the inventor of the pneumatic tyre. A patent 
for a pneumatic tyre had been taken out by Thompson in 1848, 
but there is no record that he made a commercial success of 
his invention. In 1890, Mr. James Dunlop, of Dublin, made 
a pneumatic tyre for his son, and the results obtained by its 
use being so astounding, arrangements were very soon made 
for their manufacture. While in 1889 a pneumatic tyre was 
unheard of, at the Stanley Bicycle Club Show, November- 
December, 1 89 1, from an analysis* of the machines exhibited, it 
appears that 40 per cent, of the tyres exhibited were pneumatic^ 




Fig. 134. 

32 J per cent, cushion^ i6i per cent, solid^ 10 per cent, inflated^ 
and the remainder, about i per cent., were classed as nondescript. 
In the above classification, under pneumatic tyres are included 
only those with a separate inner tube, the inflated being really 
single-tube pneumatic tyres. Cushion tyres were made and used 
as a kind of compromise between solids and pneumatics. The 
proportion of pneumatic tyres to the total has grown greater year 
by year, until now there is hardly a cycle made, for use in Britain 
at least, with any other than pneumatic tyres. 

141. Oear-oases. — The most troublesome portion of a modem 
rear-driving bicycle is undoubtedly the chain and the accompany- 
ing gear. The chain, however well made originally, is found to 
stretch slightly under the heavy stresses to which it is subjected 

* The Cyclist's Annual and Year-book for 189a. 

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CHAP. xiT. Development of Cycles : the Bicycle 



i6i 



in ordinary working. If the distance between the centres of the 
two chain-wheels — on the crank-axle and driving-wheel hub respec- 
tively—over which the chain passes is unalterable, the chain will 
ultimately get so slack that there will be a great risk of it over- 
riding the teeth of the wheels, to the danger of the rider. All 
chain-driven cycles are consequently provided with some means 
of tightening the chain, />. of increasing the distance between 
the centres of the two chain-wheels. Again, in an exposed chain, 
it is practically impossible to lubricate perfectly the rubbing 
parts, very little of the oil applied to the outside surface finding 



il. r^'^l. 




Fig. 135. 

its way in between the rivet-pins and the blocks of the chain. 
Dust and grit from the road soon adhere to the chain and chain- 
wheel, so that the frictional resistance of the chain as it is wound 
on and off the chain -wheel is rapidly increased. 

These considerations led Mr. Harrison Carter to introduce 
the gear-case, jthe function of which is to exclude dust and 
mud, and provide an oil-bath in which the lowest portion of the 
chain may dip. The reduction of frictional resistance is perhaps 
one of the least of the advantages pertaining to the use of the 
gear-case ; one great advantage is that less trouble is given to the 
rider, and chain adjustments need not be made so frequently. In 

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1 62 



Cycles in General 



CHAP. XIV. 



fact, some makers claim that with an oil -tight gear-case the chain 
does not stretch perceptibly, and no chain adjustments are neces- 
sary. The author is not aware, however, that any maker has 
ventured to place on the market a bicycle with gear-case but no 
chain adjustment. 

142. Tandem Bicycles.— When the success of the bicycle for 
a single rider was assured, attempts were soon made to make a 
bicycle for two riders. Figure 135 shows the * Rucker ' Tandem 
bicycle, made in 1884, one of the first successful tandem bicycles. 
This consists practically of two ' Ordinary ' driving-wheels and forks 
connected together by a straight tubular backbone. At the front 




Fig. 136. 

end of this backbone there is an * Ordinary ' steering centre ; at the 
other end it is connected to the head of the rear-wheel fork by a 
frame which permits it to twist sideways. Figure 136 shows a later 
andem bicycle, also made by Mr. Rucker — probably the first 
practicable machine of this type. It is practically a tandem 
* Kangaroo.' In a paper on * Construction of Cycles,' read before 
the Institution of Mechanical Engineers in 1885, Mr. R. E 
Phillips says, " This tandem bicycle . . . eclipses the earlier, and 
bids fair to prove the fastest cycle yet produced. The weight is 
only 55 lbs., and it is, therefore, the lightest machine yet made to 
carry two riders " 

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CHAP. XIV. Development of Cycles : the Bicycle 163 

Figure 137 shows a front-driving chain-driven Safety Tandem, 
made by Hillman, Herbert, and Cooper, 1887. Both riders 
drive the front wheel, and both wheels are moved in steering. 




Fig. 137. 



The * Invincible' Tandem Safety (fig. 138), and the *Iver 
Tandem Safety (fig. 139), which was made convertible so that it 




Fig. 138. 



could be used as a single Safety, were among the first approxima- 
tions to the present popular type of Tandem Safetv^ both riders 

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164 



Cycles in General 



CHAP. XIT. 



being placed between the wheels, and both driving the rear 
wheel. It will be noticed that the front crank-axle is connected 
by chain gearing to the rear crank-axle, the two axles rotating at 
the same speed ; the second chain passes over the larger wheel 
on the rear crank-axle and the chain-wheel of the driving-axle. 




Fig. 139 

Both riders have control of the steering, a light rod connecting 
the front fork to the rear steering-pillar. The long wheel-base 
of these bicycles adds to the steadiness of the steering at high 
speeds, since (see fig. 202), for the same deviation of the handle- 
bars, a machine with long wheel-base will move in a curve of 
larger radius than one with a shorter wheel-base. The distance 
between the wheel centres being much greater than in the 

single machine, the 
frame is subjected to 
very much greater strain- 
ing actions, and imper- 
fect design will be much 
more serious than in the 
single machine. 

Figure 140 is an ex- 
ample of the present 
popular type of Tandem bicycle made by Messrs. Thomson and 
James. The machine is kinematically the same as that of figure 
138, the particular difference being in the rear frame, which is of 
the diamond type, completely triangulated. 

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Fig. 140. 



i65 



CHAPTER XV 

DEVELOPMENT OF CYCLES : TRICYCLES, QUADRICYCLES, &C. 

143. Early Tricycles. — No sooner was a practicable bicycle 
made than attention was turned to the three-wheeler as being the 
safer of the two machines, and offering some advantages, such as 
the possibility of sitting while the machine is at rest. It was very 
early found that the greater safety of the three-wheeler was more 
apparent than real. * Velox,' writing in 1869, says, ** Strange as it 
may appear to the un- 
initiated, the tricycle is 
far more likely to upset 
the tyro than the bicycle." 

Figure 141 (from 
* Velox's ' book) represents 
a simple form of tricycle 
made in the sixties by Mr. 
Lisle, of Wolverhampton, 
known as the 'German' 
tricycle. It was, in fact, a 
converted * Bone-shaker ' 
bicycle, with the rear wheel 
removed and replaced by 
a pair of wheels running 
free on an axle two feet 
long. The motive power ^'^" '*' 

was applied by pedals and cranks attached to the axle of the front 
wheel. A number of tricycles were made on the same general 
principle ; but the weight of the rider being applied vertically over 
a point near the front corner of the wheel-base triangle, the margin 
of lateral stability was small. Mr. Lisle also made a ladies' double- 
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i66 



Cycles in General 



driving tricycle (fig. 142), in which the power was applied by 
treadles and levers acting on cranks on the axle of the rear wheels. 
Nothing is said about the axle of the rear wheels being divided, 




Fig. 142. 

SO it is probable that in turning round a corner the rear wheels 
skidded, just as is the case with railway rolling stock. 

In the * Dublin ' tricycle (fig. 143) the driving-whed was behind, 
and two steering-wheels placed in front ; the margin of stability 

in case of a stoppage 
was much greater 
than in the * German ' 
tricycle (fig. 141). 
Another point of 
difference consisted 
in the application 
of the lever gear- 
ing ; the pedals were 
fixed on oscillating 
levers, the motions 
of which were com- 
municated by crank 
and connecting-rods 
to the driving-wheel. 
The * Coventry ' bicycle was at first made with lever gearing, but 
chain gearing was very soon afterwards applied to it. The 
* Coventry Rotary ' (S\g, 144) was the most succ^ful of the early 

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CHAP. XV. Development of Cycles : Tricycles ^ Src. 167 

single-driving tricycles. It may be interesting to note that this 
type has been revived recently, the Princess of Wales having 
selected a tricycle of this type. 




Fig. 144. 

If the mass-centre be vertically over the centre of the wheel- 
base triangle, the pressure on each wheel will be one- third of the 




Fig. 145. 

total weight. Under certain circumstances this pressure is in- 
sufficient for adhesion for driving, hence arose the^necessity for 

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1 68 



Cycles in General 



double-driving tricycles. In the * Devon' tricycle, made in 1878 
(fig. 145), which is fitted with chain gearing, the cog-wheels co- 
axial with the driving-wheels are fitted loose on their axles, and 
each cog-wheel drives its axle by means of a ratchet and pawl. 
In rounding a corner, the inside wheel is driven by the chain, 
while the outside wheel overruns its cog-wheel, the pawls of the 
ratchet-wheel being arranged so as to permit of this. 

In the *Club' tricycle (fig. 146), made by the Coventry 
Machinists Company in 1879, one of the wheels was thrown 




Fig 146. 

automatically out of gear when turning to one side or the other. 
Later, the same company used a clutch gear, somewhat similar in 
principle to the ratchet gear, but which had the advantage that 
the clutch could come into action at any point of the revolution, 
instead of only at as many points as there were teeth in the 
ratchet-wheel. The tricycle illustrated in figure 146 has only two 
tracks, which, in the early days of tricycles, was supposed to be ot 
some advantage, in so far that it was easier to pick out two good 
portions along a bad piece of road than three. 

A number of single and side-driving, rear-steering tricycles 
(fig. 147) were made about the years 1879 ^"^ 1880, but on 
account of their imperfect steering they were sometimes found 
extremely dangerous, and their manufacture was soon abandoned 

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CHAP. XV. Development of Cycles : TricycUs^ &c, 169 

in favour of double-driving rear-steerers, of which the * Cheyles- 
more ' (fig. 148), made by the Coventry Machinists Company, was 




Fig. 147. 



one of the most successful. Tradesmen's carrier tricycles are still 
made of this type. 

144. Tricycles with Differential Gear. — The front-steering, 
double-driving tricycle with loop frame, as in figure 145, next 




Fig. Z48. 



became more and more popular. The invention by Mr. Starley 
of the * Differential ' tricycle axle or balance-gear marks a great 
step in the development of the three-wheeler. Thi§^ gc^>, or its 

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I70 



Cycles in General 



equivalent, has been ever since used for double-drivers, clutch and 
ratchet gears having been abandoned. 

As improvements in detail were slowly introduced, the lever 
gear fell into disuse (which is easily accounted for by the fact that 
with it gearing either up or down is impossible), and chain gearing 
became universal. With chain gear, and the possibility of gearing 
up, the driving-wheels were made gradually smaller and smaller, 
with a consequent reduction in the weight of the machine. 

The * Humber ' tricycle met with great success on the racing 
path, but, on account of its tendency to swerve on passing over a 




Fio. 149. 

stone, its success as a roadster was not so marked. When used as 
a tandem (fig. 149), with one rider seated on the front-frame sup- 
porting the driving-axle, the tendency to swerve was reduced and 
the safety increased (see sec. 183). In a later type this difficulty 
was overcome by converting the machine into a rear-steerer, the 
steering-pillar being connected by light levers and rods to the 
steering-wheel. 

The loop-frame tricycle was gradually superseded by one with 
a central frame, in which the steering-wheel was actuated direct 
by the handle-bar, the result being the * Cripper ' tricycle (fig. 
150). In this, as made by Messrs. Humber & Co., the chain lies 

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CHAP. XV. Development of Cycles : Tricycles, &c. 1 7 1 

in the same plane as the backbone ; the crank-bracket being 
suspended from the backbone and the gear being exactly central. 




Fio. 150. 



The axle is supported by four bearings, though the axle-bridge, 
with four bearings, had already been used in the * Humber ' 
tricycles. 




Fig. 151. 



Among the successful tricycles of this period may be men- 
tioned the * Quadrant,' in which the steering- wh^el was not 

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[72 



Cycles in General 



mounted in a fork, but the ends of the spindle ran on guides in 
the frame (see fig. 254), and the *Rudge Royal Grescent' (fig. 
151), in which the fork of the steering-wheel was horizontal, and 
the steering-axis intersected the ground some considerable distance 
between the point of contact of the steering-wheel. 

Up to the year 1886 the * Ordinary * bicycle had a very great 
influence on tricycle design, the driving-wheels of tricycles being 
usually made very large (in fact, sometimes they were geared 
down instead of up) and the steering-wheel small. The weight 
of two large wheels was a serious drawback, while the excessive 
vibration from the small steering-wheel was a source of great 




Fig. 152. 

discomfort to the rider. The distance between the wheel centres 
was usually made as small as possible, the idea being that the 
tricycle should occupy little space. Common measurements for 
, Cripper * tricycles at this time were : Driving-wheels, 40 in. 
diam. ; steering-wheel, 18 in. diam. ; distance between driving- 
and steering-wheel centres, 32 in. ; driving-wheel tracks, 32 in. 
apart. Weight : Racers, 40 lbs. ; roadsters, 70-80 lbs. 

The size of the driving-wheel has been gradually diminished, 
that of the steering-wheel increased, until now (1896) 28 in. may 
be taken as the average value for the diameter of each of the 
three wheels. The wheel centres have been put further apart. 

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CHAP. XT. Development of Cycles : Tricycles, &c, 173 

42-45 in. being now the usual distance, the comfort of the rider 
and the steadiness of steering being both increased thereby. 




Fig. 153. 



The design of frame has also been greatly improved, so that the 
weight of a roadster has been reduced to 40-45 lbs. without in 
any way sacrificing strength. 




Fig. T54. 

Figure 152 shows a tricycle by the Premier Cycle Company, 
Ltd., embodying these improvements. The frame and chain gearing 
is almost identical with that of the bicycle ; the balance-gear and 
axle-bridge, with its four bearings, being added. 

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174 



Cycles in General 



CHAP. XT. 



Figure 153 shows a tricycle by Messrs. Starley Bros., in which 
the bridge is a tube surrounding, and concentric with, the axle, 




Fig. 155. 



and the gear is exactly central ; so that the frame is considerably 
simplified, and the appearance of the machine vastly improved. 




Fig. 156. 



This may be taken as the highest point reached in the develop- 
ment of the * Cripper ' type of tricycle^ 

145. Modem Single-driving Tricycles. — Several successful 

d by Google 



Digitized t 



CHAP. XV. Development of Cycles : Tricycles^ Src, 175 

single-driving rear-driver tricycles have been made, among them 
being the * Facile Rear- Driver' (fig. 154) and the * Phantom' 




Fig. 157. 



(fig. 155). In these the two idle (or non-driving) wheels run 
freely on an axle supported by the front frame. These tricycles 




Fig. 158. 



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176 



Cycles in General 



are subject to the same faults of swerving as the * Humber ' 
tricycle. 




Fig. 159. 

An important improvement is effected by mounting each 
wheel on a short axle, which can turn about a vertical steering- 




head placed as close as possible to the wheel, as in the * Olympia ' 
(fig. 160), one of the most successful of modern tricycles. 

146. Tandem Tricycles. — Tricycles for two riders were soon 
brought to a relatively high state of perfection, and were almost, 

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CHAP. XV. Development of Cycles: Tricycles^ &c, 177 

if not quite, as popular as tricycles for single riders. Among the 
earliest may be mentioned the * Rudge Coventry Rotary ' (fig. 156), 




Fig. x6i. 



the *Humber' (fig. 149), the * Invincible ' rear-steerer {^g. 15 7), and 
the 'Centaur' (fig. 158). Later, the *Cripper' (fig. 1 59) andthe* Royal 




Fig. i6a. 



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byGoOgl^ 



178 



Cycles in General 




Fig. 164. 



Crescent' (fig. 151) 
were made as tandems. 
In all these tandems, 
with the exception of 
the * Coventry Rotary,' 
one of the riders over- 
hangs the wheel-base, 
so that the load on the 
steering-wheel is actu- 
ally less than when a 
single rider used the 
machine. The * Coven- 
try Rotary' is a single- 
driver, the others are 
double-drivers. 

The most successful 
modern tandem tricycle 
is the *01ympia' (fig. 
160), a single-driver. 

147. Sociables, or 
tricycles for two riders 
sitting side by side, 
were at one time 
comparatively popular. 
Figure 161 shows one 
with a loop frame made 
by Messrs. Rudge & 
Co., which, by the re- 
moval of certain parts, 
could be converted into 
a single tricycle ; figure 
162, a * Sociable' formed 
by adding another driv- 
ing-wheel, crank-axle, 
and seat to the 'Co- 
ventry Rotary' (fig. 144). 

In the * One-track 
Sociable ' (fig. 163), 

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CHAP. XV. Development of Cycles : Tricycles^ &c, 179 

made in t886 by Mr. J. S. Warman, the weight of the rider 
rested mainly on the two central wheels, the small side wheels 
merely preventing the machine overturning when starting and 
stopping. It was, in fact, a sociable bicycle with two side safety- 
wheels added. 

In the * Nottingham Sociable * tricycle (fig. 164), made by the 
Nottingham Cycle Co. in 1889, each rider sat directly over the 




Fig. 165. 

rear portion of a * Safety ' bicycle, and the heads of the two 
frames were united by a trussed bridge to a central steering- 
head. 

148. Convertible Tricycles. — A great many machines for 
two riders were at one time made by adding a piece to a tricycle 
so as to form a four-wheeler. Of these convertible tricycles^ as 
they were called, the * Royal Mair two-track machine (fig. 165) 
and the * Coventry Rotary Sociable ' (fig. 162) may be noticed. 

Figure 166 shows the * Regent' tandem tricycle, formed by 

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i8o 



Cycles* in General 



CHAP. XV. 




Fiu. 167 



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CHAP. XV. Development of Cycles: Tricycles, &c, i8i 

coupling the front wheel and backbone of a * Kangaroo ' bicycle 
to the rear portion of a * Cripper ' tricycle ; affording an example 
of a treble-driving cycle. 

Figure 167 shows a four-wheeler formed by coupling together 
the driving portions of a *Humber' and a * Cruiser' tricycle, 
affording an example of a quadruple-driving cycle, all four wheels 
being used as drivers. 

149. Qnadricycles. — With the exception of the convertible 
tricycles above referred to, comparatively few four-wheeled cycles 
have been made. In 1869 * Velox' wrote : "No description of 
velocipedes would be perfect without some allusion to the favourite 




Fig. 168. 



our-wheeler of the past generation of mechanics." Figures 1 1 7 
and 118 show one of the best as manufactured by Mr. Andrews, 
of Dublin. The frame was made of the best inch square 
iron 7 feet long between perpendiculars, and was nominally rigid, 
so that in passing over uneven ground either the frame was 
severely strained or c^nly three wheels touched the ground. The 
two driving' wheels were fixed at the ends of a double cranked-axle 
driven by lever gear, the path of each pedal being an oval curve 
with its longer axis horizontal. While moving in a circle, the 
driving-wheels skidded as well as rolled, since the outer had to 
xsi/Cfte over a greater distance than the inner. 

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1 82 Cycles in General chap. xv. 

Bicycles and tricycles have almost monopolised the attention 
of cycle makers, and no practicable quadricycle was made until 
Messrs. Rudge & Co. produced their * Triplet ' quadricycle (^^%, 
1 68) in 1888. The front-frame supporting the two side steer- 
ing-wheels can swing transversely to the rear-frame, so that the 
four wheels always touch the ground, however uneven, without 
straining the frame. The same design was applied to a quadri- 
cycle for a single rider. 



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183 



CHAPTER XVI 

CLASSIFICATION OF CYCLES 

150. Stable and Unstable Equilibrium.— Cycles may be 
divided into two great classes, according as the static equilibrium 
during the riding is stable or unstable. The former class may be 




Fig. 169. 

further separated into three divisions : (a) Tricycles, in which the 
frame, supported as it is at three points, is a statically determinate 
structure ; {b) Multicycles, having four or more wheels, the frame 

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1 84 



Cycles in General 



generally having a hinge or universal joint, so that the wheels may 
adjust themselves to any inequalities of the ground. If the frame 
be absolutely rigid it will be a statically indeterminate structure. 
(c) Dicycles of the * Otto ' type, with two wheels, in which the 
mass -centre of the machine and rider is lower than the axle. No 
machine of this class has ever been made, to the author's knowledge. 

Cycles with unstable equi- 
librium may be divided into 
three classes, according to 
the direction in which the 
unstable equilibrium exists : 
MonocycIeSy having only one 
wheel ; Bicycles^ having two 
wheels forming one track ; and 





Fig. 170. 



Fig. 171. 



Dicycles^ having two wheels mounted on a common axis. In all 
monocycles the transverse equilibrium is unstable ; they may be 
subdivided into two sub-classes, according as the longitudinal equi- 
librium is stable or unstable. An example of the former sub- 
class is shown in figure 169, in which the frame, carrying seat, pedal- 
axle, and handle, runs on an inner annular wheel, d^ on the driving- 
wheel A \ the central opening, By being large enough for the body 

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CHAP. XVI. 



Classifitation of Cycles 



185 




of the rider, while his legs hang on each side of the main wheel. 
An example of the latter is shown in figure 170, and a sociable 
monocycle of the former class for two riders in figure 171. 

In bicycles, the transverse equilibrium is unstable and the 
longitudinal equilibrium stable. In dicycles, the transverse 
equilibrium is stable. They may be subdivided into two sub- 
classes, according as 
their longitudinal 
equilibrium is stable 
or unstable. 

The *Otto' di- 
cycle (fig. 172) is 
the only example 
of the former sub- 
class, while none of 
the latter class, as 
already remarked, 
have attained any 
commercial import- 
ance. A dicycle of 
the latter type would 
be made with very 
large driving-wheels, and the mass-centre of machine and rider 
lower than the axis of the driving-wheel. 

151. Hefhod of Steering.— Proceeding to the further division 
and classification of bicycles, the first subdivision that suggests 
itself takes account of the method of steering ; a bicycle being 
said to be 2i front- or r^ar-steerer, according as the steering-wheel 
is in front or behind, while among tricycles there are also side- 
steerers. A few bicycles have been made with double-steering. 

The complete frame of the machine is usually divided into two 
parts, called respectively the front-frame and the rear-frame^ 
united at the steering centre ; though sometimes that part to 
which the saddle is fixed is called the * frame,' to the exclusion 
of the other portion carrying the steering-wheel. It should be 
pointed out that the steering portion will sometimes be the larger 
and heavier of the two, the * H umber ' tricycle (fig. 149) affording 
an example of this. In the 'Chapman Automatic-Steering' 

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Fig. 172. 



1 86 



Cycles in General 



CHAP. XVI. 



Safety (fig. 173) the saddle is not fixed direct to the rear-fi:ame, 
but moves with the steering fork. The complete frame is in this 
case divided into three parts, which can move relative to each 




Fig. 173. 



Other, on which are fixed the driving-gear, the steering-wheel, and 
the saddle respectively. 

Exami)les of double-steering are afforded by the * Adjustable ' 
Safety (fig. 174), made by Mr. J. Hawkins in 1884, and by the 




Fig. 174. 



* Premier' Tandem Safety (fig. 137), in each of which the forks 
of both wheels move relative to the backbone. 



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CHAP. XVI. 



Classification of Cycles 



187 



There have been very few rear-steering bicycles made, though 
their only evident disadvantage is, that in turning aside to avoid 
an obstacle, the rear-wheel may foul, though the front-wheel has 
already cleared. Nearly all successful types of bicycles have been 
front-steerers. 

152. Bicycles, Front-drivers. — Bicycles may be divided into 
front-drivers and rear-drivers, according to which wheel is used 
for driving. The * Rucker * Tandem (fig. 135) is an example of a 
bicycle in which both wheels are used as drivers ; but generally 
only one wheel is used for driving. Each of these divisions may 
again be subdivided into ungeared and geared. 

Among ungeared front -drivers we have the * Bone-shaker,' the 

* Ordinary,' the * Rational,' the ' Facile,' the * Xtraordinary,' and 
the *Claviger' (fig. 504). In this classification we regard as 
ungeared those machines in which one revolution of the driving- 
wheel is made for each complete cycle of the pedal's motion. 
Thus, any bicycle with only lever gearing will be classed as un- 
geared, since with such mechanism it is, in general, impossible to 
gear up or gear down. 

Geared bicycles may be subdivided into toothed-wheel geared, 
chain geared, and clutch geared. Among wheel geared front- 
drivers we have the * Geared Ordinary,' * Front-Driver,' the 

* Bantam,' the 'Geared 
Facile,' the * Sun-and- 
Planet ' bicycle (fig. 479), 
and the * Premier ' Tan- 
dem Dwarf Safety (fig. 
137). Among chain 
geared safeties we have 
the * Kangaroo,' with 
two driving chains, one 
on each side of the 
driving-wheel, the * Ad- 
justable ' Safety Road- 
ster (fig. 1 74), and the * Shellard Dwarf ' Safety Roadster (fig. 175)- 

A combination of toothed-wheel and chain gear was used in 
the * Marriott and Cooper ' Front- Driver. 

Clutch geared bicycles have never been very successful, the 

Digitized by CjOOQIC 




Fig. 175. 



1 88 



Cycles in Generai 



CHAP. XVI. 



Brixton Merlin Safety (fig. 176) being about the only example of 
this type. In the Merlin gear, a drum rotates on the axle at each 
side of the wheel, round which is coiled a leather strap, the other 




Fig. 176. 



end being fastened to the pedal lever. When the pedal is pushed 
outwards by the rider the drum is locked by a clutch to the axle, 
and the effort is transmitted to the wheel. On the upstroke a 

spring raises the 
pedal lever. With 
this gear any length 
of stroke may be 
taken, but the imper- 
fect action of the 
clutch is such that 
the great advantages 
due to the possibility 
of varying the length 
Fig. 177. of stroke are more 

than neutralised. 
Figure 177 shows a possible front-driving rear-steering geared 
bicycle, the front hub having a * Crypto ' or equivalent gear. 

153. Bicycles, Sear-drivers. — Among ungeared rear-drivers 

Digitized by CjOOQIC 




CHAP. XVI. 



Classification of Cycles 



189 



may be mentioned the Rear-driving * Facile ' and the American 
*Star'(fig. 178). 

Among toothed-wheel geared rear-drivers we have the * Burton/ 
the Geared * Facile* Rear-driver (fig. 129), the *Claviger' Geared 
(fig- 507)? the * Femhead * Chainless Safety, driven by bevel- 
gearing. Of chain geared rear-drivers, the present popular Safety 




Fig. 178. 

of the * Humber ' or * Rover * type is the most important repre- 
sentative. 

In the * Boudard-geared ' Safety a combination of toothed- 
wheels and chain gear is used, while the same may be said of the 
two-speed gears that are applied to the ordinary type of chain- 
driven safety. 

This classification is represented diagrammatically on page 194. 
From this diagram it will be seen that no successful type of rear- 
steering bicycle has been evolved. Experimenters might with 
advantage direct their energies to this comparatively untrodden 
domain. 

Digitized by CjOOQIC 



I90 



Cycles in General 



CHAP. XVI. 




Fig. 179. 



154. Tricycles, Side-steering.— The classification of tricycles 
may go on on similar lines to that of bicycles. There would be three 

types — front-steer- 
ing, side-steering, 
and rear-steering. 
Of side-steering tri- 
cycles there are two 
subdivisions : the 
* Rudge Coventry 
Rotary' (fig. 156) 
being a side-driver, 
while the * Dublin ' 
(fig. 143) and the *01ympia' (fig. 160) are back-drivers. No 
side-steering, front-driving tricycle has been made, to our know- 
ledge; though we 
can see nothing at 
present to prevent 
tandem tricycles 
of this type (figs. 
1 79-181) from 
being successful 
roadsters. That 
shown in figure 
179 could be ridden by a lad/ in ordinary costume on the 
front seat ; it would, perhaps, be slightly deficient in lateral 

stability, as the mass- 
centre would be near 
the forward corner 
of the wheel-base tri- 
angle. That shown in 
figure 180 would be 
y superior in this re- 
^y^ spect, while the weight 
on the driving-wheel 
would still probably 
be sufficient for all ordinary requirements. A type inter- 
mediate (fig. 181) might be made with a ' Crypto ' gear on the 
front wheel hub, the two crank-axles being connected by a 




Fig. 180. 




Fig. 181. 



Digitized by CjOOQIC 



CHAP. XVI. Classification of Cycles 191 

chain ; the frame would be simpler than in figures 179 and 
180. 

Tricycles are either single-driving or double-driving, according 
as there are one or two driving-wheels. The only treble-driving 
tricycle which has been yet put on the market is the tandem 
made by Messrs. Trigwell and Co., by coupling the front wheel 
and backbone of a * Kangaroo ' to the rear portion of a * Cripper ' 
(fig. 166). The driving-wheels of a double-driving tricycle are 
invariably mounted on the same axle, and since in going 
round a corner the wheels, if of equal size, must rotate at 
different speeds, the driving-axle must be in two parts. In the 

* Cheylesmore ' tricycle two separate driving chains were used 
between the crank- and wheel-axles, the cog-wheel on the wheel- 
axle being held by a clutch when driving in a straight line, 
while in rounding a comer the wheel which tended to go the 
faster overran the clutch, and all the driving effort was transmitted 
through the more slowly moving wheel. Starley's differential gear 
(see sec. 189), allowing, as it does, both wheels to be drivers under 
all circumstances, is now universally used for double-driving. 

155. Front-steering Front-driving Tricycles —The eariy 

* Bone-shaker' tricycle (fig. 141) is an ungeared example of this 
class, while the *Humber' tricycle (fig. 149) is a geared tricycle of 
this same class. The * Humber * is a double-driver. 

Single-driving tricycles of this division may be made by taking 
a * Crypto' or * Kangaroo ' bicycle, and having two back wheels at the 
end of a long axle. They would, however, be deficient in lateral 
stability, unless used as tandems, on account of the load being ap- 
plied over a point near the forward apex of the triangular wheel-base. 

156. Front-steering Eear-driving Tricycles.— Of ungeared 
cycles, Lisle's early Ladies' tricycle (fig. 142) and the *Club' 
(fig. 146) are examples. 

The geared tricycles may be subdivided into single-drivers and 
double- drivers. Of the former class the * Olympia ' (fig. 160), the 
'Phantom' (fig. 155), the * Facile' (fig. 154), the 'Claviger,' and 
the 'Trent ' convertible (fig. 182) are examples. 

The double-drivers may be conveniently subdivided into 
direct-steerers and indirect-steerers. The 'Cripper' (figs. 150, 
152, 153), of which probably more examples have been made 

Digitized by V^jOOQ 



192 



Cycles in General 



CHAP. XVI. 



than all the other types put together, is a direct-steerer ; 
so also is the Merlin (fig. 183). Among indirect-steerers we 
may mention the 'Devon* tricycle (fig. 145), the *Club' (fig 




Fig. 182. 



146). The * Nottingham Sociable' (fig. 164) formed by conver- 
sion of two bicycles, and Singer's Omnicycle with clutch gear 
(fig. 184), made in 1879, ^^so belong to this division. 




Fig. 183. 

This classification of tricycles is shown diagram matically on 
page 195. 

157. Bear-steering Front-driviiig Tricycles.— The *Veloci- 

Digitized by CjOOQIC 



CHAP. XVI. 



Classification of Cycles 



193 




Fig. 184. 



man,' a hand-tricycle made by Messrs. Singer & Co., of which 
figure 241 represents an improved design for 1896, is an example 
of this class. The * Cheylesmore ' (fig. 148), made by the Coventry 
Machinists Co., was one of the 
most successful of the early 
tricycles. Several tandem tri- 
cycles were made on this type, 
one of the most popular being 
tne * Invincible ' (fig. 157), made 
by the Surrey Machinists Co., 
Limited. 

The tandem tricycles in 
figures 1 79-18 1, if made with 
both rear wheels running freely 
on the same axle, fixed to a rear 
frame, would afford examples of 
single-drivers of this class. 

A rear-steering side-driving 
tricycle was the * Challenge ' (fig. 147), made by Messrs. Singer in 
1879. 

158. Quadricycles. — A great many quadricycles were made at 
one time by adding a piece to a tricycle, so as to form a machine 
for two riders (see sec. 148). The attachment of the extra 
portion was usually made by means of a universal joint. The 
one track Sociable (fig. 163) may really be classified as a four- 
wheel cycle, though from the lack of the universal joint in the 
frame it differs essentially from those mentioned above. 

Rudge's quadricycle (fig. 168), giving only two tracks and a 
rectangular wheel base, is a very well designed machine of this 
type. The steering gear is similar in principle to that used in the 
* Olympia ' tricycle. The front portion of the frame supporting the 
two side-steering wheels is connected to the rear portion by a 
horizontal joint at right angles to the driving-axle, so that the four 
wheels may each touch the ground, however uneven, without 
straining the frame. It is made as a single, tandem, and triplet. 
Its stability is discussed in section 161. 

The' quadricycle with two tracks has some advantages as com- 
pared with the tricycle, and may well repay further consideration 

Digitized by Vj O 



194 



Cycles in General 



CHAP. XVI. 




Si 



Co 



•3 

I 
1 

8 



^ 1 



«0 



^Is.^ 



cjs: 


^ 


i^- 


-4-1 


5s 


<< 


^\ 




•^3 




0Qf3 




^3 




§37 









so* 






III 



«0 






-I? 

6.S 

1 




Digitized by CjOOQIC 



CHAP. XVI. 



Classification of Cycles 



195 



1 1 



C/3 



i 



^5 

ft; 



i 
S 



§ 



lit 



-2 



J 



11 

OO 



1 



' -HI' 



I 



5-^ 
5 §s 



§ 

ft; 



IN 



J'2 

fc« hJ w 



-If si -I? 



1|i 



■II 



11 



*^g 







•« 




M 




M 




tC 




) 




. B 




B *> B 





5 -c « 






■Ml 




ss 






52 1 






Digitized by CjOOQIC 



O 2 



196 Cycles in General chap. xvi. 

by cycle makers and designers. If a satisfactory mode of support- 
ing the frame on the wheel axles by springs could be devised, the 
horizontal joint might be omitted, the design of frame simplified, 
the stability of the machine increased, and additional comfort 
obtained by the rider. If the two steering-wheels revolved inde- 
pendently on a common axle, as in the 'Phantom* tricycle 
(fig. 155), the design of the machine would be further simplified ; 
the relation of the wheels to the frame being exactly the same as 
is a four-wheeled vehicle drawn by a horse. This type of quadri- 
cycle would, however, possess the same objection- 
able features as to swerving as the tricycles shown 
in figures 149, 154, and 155. In a horse vehicle 
the front axle is fixed to the shafts to which the 
horse is harnessed, so that the axle cannot swerve 
when one wheel meets an obstacle without dragging 
the horse sideways. In this respect the horse 
performs the same function as the front wheel of 
Fig 18 ^ * Cripper ' tricycle. A hansom cab is equivalent 

to a * Cripper' tricycle, and a four-wheeler to a 
pentacycle (fig. 185), in which the rear portion trails after the 
front. 

159. Multicycles. — By stringing together a number of 
* Humber ' or * Cripper ' frames with their crank -axles and pairs of 
driving-wheels, a cycle of 4, 6, 8, or any even number of wheels 
may be obtained. The steering of such a multicycle should be 
effected by the front rider, the intersection of the first two axles 
determining the radius of curvature of the path. The following 
wheels should be merely trailing wheels, so that they may follow 
in the required path. 




Digitized by CjOOQIC 



197 



Fm. 



CHAPTER XVII 

STABILITY OF CYCLES 

1 60. Stability of Tricycles.— If abc {^g, 186) be the points 
of contact of the three wheels of a tricycle with the ground, it will 
be in equilibrium under the action of the rider's weight, provided 
the perpendicular from the mass-centre 
of the rider and machine falls within 
the triangle a be. If this perpendicu- 
lar fall at the point d, the pressures of 
the wheels on the ground can easily be 
found by the principle of moments. 
Let W be the total weight of the rider 
and machine, z£/„, Wi^ and w^ the pres- 
sures of the wheels at a, b^ and c on 
the ground. Then taking moments 
about the line b r, draw perpendiculars 
a a, and d d^ \o b c. We then have 



W X d dx =Wa X a a^ 
dd, 



or 



Wn = 



\V 



(I) 




Similar expressions for Wi, and w^ can 

be found. Fig. 187. 

If the point d fall outside the triangle abc^ the tricycle will 
topple over. 

161. Stability of Quadricydes. — If the quadricycle be made 
with the steering-axle capable of turning only round a vertical 
axis, as in the case of an ordinary four-wheeled carriage drawn by 
horses, the mass-centre of the machine and rider may lie vertically 

Digitized by CjOOQIC 



198 Cycles in General chap. xvn. 

above the rectangle abed (fig. 188), ^, ^, c and d being the 
points of contact of the wheels with the ground. But if one of 
the axles be hinged to the frame, so as to allow the four wheels 
to be always in contact with the ground, how- 
ever uneven — as in the case of the * Rudge ' 
quadricycle (fig. 168)— the mass-centre of 
machine and rider, exclusive of front portion 
• f \ j a by must lie vertically above the triangle 
I / \ I e c dy e being the intersection of the plans of 

S\ fl the steering-axle and hinge joint. If the 
-^ U perpendicular from the mass-centre of ma- 
^|j chine and rider fall between e c and b r, the 
Fig. 188. wheel at d will lift from the ground, and the 

portion e c d di the machine will continue 
to overturn until stopped by coming in contact with the portion 
ab. 

In a tandem quadricycle formed by attaching a trailing wheel, 
d (fig. 189), to a * Cripper ' tricycle, a b c,by means of a universal 
joint at <r, the mass-centre of the machine and 
riders must lie vertically above and inside the 
quadrilateral abed. If the joint e be behind 
the axle, b r, another condition must be satisfied, 
viz. the vertical downward pressure at ^, due to 
the weight on the trailing frame, must not be suflS- 
cient to tilt the triangle a b c about the axle b c. 
This condition will in general be satisfied if the 
joint e be not far behind the axle. 

162. Balanoing on a Bicycle.— A bicycle has 
only two points of contact with the ground, and 
*°* *^' a perpendicular from the mass -centre of machine 
and rider must fall on the straight line joining them. If the 
bicycle and rider be at rest, the position is thus one of unstable 
equilibrium, and no amount of gymnastic dexterity will enable 
the position to be maintained for more than a few seconds. 
If the mass-centre get a small displacement sideways, the dis- 
placement w^ill get greater, and the machine and rider will fall 
sideways. In riding along the road with a fair speed the mass- 
centre is continually receiving such a displacement. If the rider 

Digitized by CjOOQIC 




CHAP. XVII. Stability of Cycles 199 

steer his bicycle in an exact straight line this displacement will get 
greater, and he and his bicycle will be overturned, as when at 
rest. But, as every learner knows, when the machine is felt to be 
falling to the left-hand side, the rider steers to the left— that is, 
he guides the bicycle in a circular arc, the centre of which is 
situated at the left-hand side. In popular language, the centri- 
fugal force due to the circular motion of the machine and rider 
now balances the tendency of the machine to overturn ; in fact, 
the expert rider automatically steers the bicycle in a circle of 
such a diameter that the centrifugal force slightly overbalances 
the tendency to overturn, and the machine again regains its 
perpendicular position. The rider now steers for a short interval 
of time exactly in a straight line. But probably the perpendicular 
position has been slightly overshot, and the machine falls slightly 
to the right-hand side. The rider now unconsciously steers to 
the right hand, that is, in a circle having its centre to the right- 
hand side. 

If the track of a bicycle be examined it will be found to be, 
not a straight line, but a long sinuous curve. With beginners the 
waviness of the curve will be more marked than with expert 
riders; but even with the latter riding their straightest the 
sinuosity is quite apparent. A patent had actually been taken 
out for a lock to secure the steering-wheel of an * Ordinary ' bicycle, 
the purpose being to make it move automatically in a straight 
line. The above considerations will show, as clearly as the actual 
trial of his device probably did to the inventor, the absurdity of 
such a proceeding. 

It would be possible to ride a bicycle in a perfectly straight 
line with the steering-wheel locked, by having a fly-wheel capable 
of revolving in a vertical plane at right angles to that of the 
bicycle wheels, and provided with a handle which could be turned 
by the rider. If the bicycle were falling to the right, the fly-wheel 
should be driven in the same direction ; the reaction on the rider 
and frame of bicycle would be a couple tending to neutralise that 
due to gravity causing the machine to fall. 

Lateral Oscillation of a Bicycle, — From the above explana- 
tion of the balancing on a bicycle, it will be seen that the 
machine and rider are continually performing small oscillations 

Digitized by CjOOQIC 



2CX) 



Cycles in General 



CHAP. XTII. 



sideways — the axis of oscillation being the line of contact with 
the ground— simultaneously with the forward motion. The bi- 
cyclist and his machine may thus be roughly compared to an 
inverted pendulum. The time of vibration of a simple pendulum 
is proportional to the square root of its length, a long pendulum 
vibrating more slowly than a short one. In the same way, the 
oscillations of a high bicycle are slower than those of a low one ; 
ue, the time taken for the mass-centre to deviate a certain angle 
from the vertical is greater the higher the mass-centre ; a rider 
equally expert on high and low bicycles will thus be able to keep 
a high bicycle nearer the exact vertical position than he will a low 
bicycle. In other words, the angle of swing from the vertical is 
greater in the * Safety ' than in the * Ordinary.' 

The track of an * Ordinary ' will therefore be straighter — that 
is, made up of flatter curves — than that of a * Safety,* both bicycles 
being supposed ridden by equally expert riders. 

163. Balancing on the Otto Dicyde.— In an *Otto' dicycle 
at rest the mass-centre of the frame and rider is, in its normal 

position, vertically above the axle 
of the wheels ; the machine is thus 
in stable equilibrium laterally and 
in unstable position longitudinally. 
In driving along at a uniform speed 
against a constant wind resistance, 
F (neglecting at present other re- 
sistances), the mass-centre, Gy is in 
its normal position, a short dis- 
tance, /, in front of the axle (fig. 
190). ^Vhile the rider exerts the 
driving effort the wheel exerts the 
force Fx on the ground, directed 
backwards, and the reaction of the ground on the wheel is an equal 
force, /^J, in the direction of motion. The force F^^ is equivalent 
to an equal force F^ at the axle and a couple Fr^ r being the 
radius of the driving-wheel. The couple Fr is applied by the 
pull of the chain to the rigid body formed by the driving-wheel 
and axle ; therefore, if T be the magnitude of this pull and r, the 
radius of the cog-wheel on the axle, Tr^^^ Fr 

Digitized by CjOOQIC 




Fig. 19a 



CHAP. xvir. Stability of Cycles 201 

Consider now the forces acting on the rigid body formed by 
the frame and rider : these are, the reaction at the bearing C, the 
weight J^ acting downwards, the wind-resistance, F^ and the pull 
of the chain T. Since the frame is in equilibrium, the moment 
of all these forces about any point must be zero. Taking the 
moments about C we get 

Wl^ Fl, = Tr.^Fr (2) 

Suppose now the mass-centre, G, to fall a little forward of the 
position of equilibrium, so that the moment of IV about C be- 
comes IV l^ y in order that equilibrium may be established the 
pull of the chain must have a greater value, 7^\ thus Wl^ — Fl^ 
= T^ r,. This increased pull on the chain is produced by the 
rider pressing harder on the pedals ; in other words, by driving 
harder ahead. 

In the same way, should the mass-centre, G, fall a little behind 
the position of equilibrium, the tendency to fall backward is 
checked by the rider easing the pressure on the pedals, i,e, by 
slightly back-pedalling. 

The frame and rider in an * Otto ' dicycle thus perform oscilla- 
tions about the axle of the machine ; the length of the inverted 
pendulum is much less than in the * Ordinary' or even the * Safety' 
bicycle, and the backward or forward oscillation is greater than 
the lateral oscillation in a bicycle. 

164. Wheel load in Cycles when driving ahead. — A great 
deal of misconception exists as to the modification of the wheel 
loads, due to driving ahead. If the cycle move uniformly, and the 
several resistances be neglected, the wheel loads will, of course, be 
the same as if the cycle were at rest, and therefore will depend 
only on the position of the mass-centre of machine and rider 
relative to the wheels. If the only resistance considered is the 
wind pressure F^ (fig. 191), the load on the front wheel will be 
decreased, and that on the rear wheel increased, by the amount F, 
determined by the equation 

F.h.^Rl, (3) 

/ being the wheel-base, and h^ the distance of the centre of wind 
pressure above the ground. Frictional resistances, including the 

Digitized by V^jOOQ 



202 Cycles in General chap. xvn. 

friction of the bearings and gearing and the rolling friction of the 
wheels on the ground, make no modification of the distribution of 
wheel load ; the former, because they are internal forces, and do 
not in any way affect the external forces, the latter because they 
act tangentially to the ground, and must be balanced by an equal 
and opposite reaction of the ground on the driving-wheel. 

If the speed of the cycle be increased, the forces due to 
acceleration can be easily shown as follows : Consider the mass 
of the machine and rider to be concentrated at the mass-centre 
G^ and that the wheels and frame are weightless ; then, to produce 

the acceleration, the frame 
must act on the mass, and 
the mass react on the 
frame with an equal but 
opposite force, / Intro- 
duce at the point of con- 
tact of the driving-wheel 
with the ground two equal 
and opposite forces, /, 
and /a (fig. 191), each 
equal and parallel to /; 
then / is equivalent to 
^"'- '9'- the force/,, and the couple 

formed by the equal and opposite forces / and Z- The force /, 
must be equilibrated by the reaction P of the ground on the 
driving-wheel, the couple tends to diminish the weight on the 
front wheel, and increases that on the rear wheel, by an amount, 
Ry given by the equation 

Rl^fh^ (4) 

h<i being the height of the mass-centre, G^ above the ground. 

In the most general case, the external forces acting on the 
system of bodies formed by the machine and rider are shown in 
figure 191. These are the resistance / due to the increase of 
speed, the wind pressure F^^ the resistance of the wheels to 
rolling, /^2, the reaction of the ground on the driving-wheel, Py the 
weight, Wy of the machine and rider, and the vertical reactions, 
R^ and R^y on the wheels. /*, ^1 and ^2 are determined so as to 

Digitized by CjOOQIC 




-T-r^T 



CHAP. xni. 



Stability of Cycles 



203 



produce equilibrium with the other forces. Pressure exerted on 
the pedal does not in any way modify the reactions -^i and ^21 
except so far as it affects, or is affected by, the resistances -F,, F^, 
and/; i,e. work spent in overcoming resistances of the mechanism 
does not in any way affect the wheel loads, 

165. Stability of Bicycle moving in a Circle.— Let r be the 
radius of the circle in which the cycle is moving, W the weight of 
the rider and machine, and G the 
position of the mass- centre (fig. 
192). We have already seen that a 
body of mass, ^Ibs., moving in a 
circle of radius, r, with speed v^ has 



and must 
on by a radial force 
Now, considering the 



a radial acceleration, - 
r 

be acted 
-^Ibs. 

weight of the rider and bicycle 
concentrated at G, and that it is 
transmitted from G to the ground 
by a weightless frame, the only 
forces acting on the frame are the 
weight Wy acting vertically down- 
wards at Gy and the reaction from 
the ground, J^. The resultant, C, of the two forces, W and Ry 
must therefore be equal to the horizontal radial force 




Fig. 192. 



gr 



is) 



required to give the mass the circular motion, and the line of action 

of R must therefore pass through G, Draw a b equal to W 

W v^ 
(fig- 193) vertically downwards, and b c equal to horizontal. 

Then the reaction, R^ is represented in magnitude and direction 
by c a. When the rider is moving steadily in a circle the machine 
must be inclined at the angle c ab\Q the vertical, so that the re- 
action, Ry from the ground may pass through G (see sec. 45). 
166. Friction between Wheel and Oronnd.— When there 

Digitized by CjOOQIC 



204 Cycles in General chap. xvn. 

is no friction between two surfaces in contact the mutual pressure 
is at right angles to the surfaces. Any component of force 
parallel to the common surface of contact can only be due to 
friction. In the case of a bicycle moving in a circle, the centri- 
petal force is supplied by the friction between the wheel and the 
ground. If the surface of the road be greasy, the friction is in- 
sufficient to provide the proper amount of force, and the force of 
reaction of the ground, F^ together with the weight of the machine 
and rider, W^ form a couple (fig. 192) tending to overturn the 
machine. 

Now when a couple acts on a rigid body free to move, the 
body turns about its mass-centre (see sec. 66). In the case of the 
bicycle (fig. 192), the mass-centre, (9, will have a simultaneous 
motion downwards, so that the final result will be that the wheel 
will slip to the right. 

Figure 192 also illustrates the forces acting on a bicycle which 
is being steered in a straight line, and which has already attained 
a slight inclination to the vertical ; the weight, W^ of the rider and 
the reaction of the ground, F^ form a couple tending to increase 
still further the deviation from the vertical. 

167. Banking of Baoing Tracks.— In racing tracks, the surface 
of the ground at the corners is sloped, as at -4 -4 (fig. 192), so as 
to be perpendicular to the average slope of the bicycles going 
round the corner. From (5) it is evident that this slope depends 
on the speed of the cyclists and the radius of the track. Table 
VIII. gives the necessary slopes for different speeds and radii of 
track. 

Example, — Taking a speed of twenty-four miles per hour and 

the radius of the track 160 feet, v = -^ ^ ^ — ^ = 35-2 ft. per 

3600 ^ 

second, - ^- becomes 35 2 \v ^ -24 W \ that is, b c -=. 
gr 32*2 X 160 

•24 a b (fig. 193), and therefore the surface of the track must be laid 

at a slope of 24 vertical to 100 horizontal. If the track be laid at 

this slope, the wheel of a bicycle moving at a less speed than 

twenty-four miles an hour will tend to slip downwards towards the 

inside of the track, that of a bicycle moving at a higher speed will 

tend to move upwards towards the outside. 

Digitized by CjOOQIC 



CHAP. XVII. 



Stability of Cycles 



205 



Table VIIL— Banking of Racing Tracks. 

Parts Vertical Rise in 100 Parts Horizontal, 



^ Mean 
radius of 






Speed 


miles per 


hour. 




track 


"^ ^ 


95 


— 1- 


30 


:^5 


' 40 


50 ft. 


53*4 


83-4 




1 20 -2 


1637 


2137 


, 100 ft. 


267 


417 




6oi 


817 


I06-8 


i 150 ft. 


17-8 


27-8 




40-I 


54-5 


71-2 


200 ft. 


13-3 


20 '9 




30-0 


409 


53-4 


250 ft. 


107 


167 




24-0 


, 327 


427 


300 ft. 


8-9 


13-9 




20 -o 


1 ^7-2 


35-6 



If the width of the track be considerable, the slope should be 
greater at the inner than at the outer edge, for a given speed. In 




ioo Feet 



Fig. 1Q4. 



this case it can be shown by an easy application of the integral 
calculus, that if R be the [radius at any point of the track and 

Digitized by CjOOQIC 



206 



Cycles in General 



CHAP. XVII. 



V the corresponding height above a certain horizontal datum 
level 

>' = ^log,i^ (6) 

feet and seconds being the units. 

If Fbe the speed in miles per hour, 

J =15383 V^\o%R^ (7) 

y and R being in feet, and log R being the ordinary tabular 
logarithm. 

Table IX. contains the values of y for different values of R 
from 40 to 300 feet, and at various speeds from 20 to 40 miles per 
hour, and figure 194 shows cross sections of tracks for these various 
speeds. 

Table IX. — Banking of Racing Tracks. 

Elevation above a datum Uvely in feet. 



Radius 




Speed, miles per hour 






fe«t 


20 


25 


30 


35 


40 


40 


98-4 


153*8 


221*5 


301-4 


393*7 


50 


104 5 


163*3 


235-2 


3201 


418-1 


60 


109-4 


170-9 


246-2 


335*0 


437*6 


70 


113-5 


177*4 


255*4 


347*6 


454*1 


80 


II7-I 


183-0 


263-5 


358-6 


468-4 


90 


120 -2 


187*9 


270-5 


368-2 


481-0 


\QO 


I23-I 


192-3 


276-9 


3769 


492-2 


IIO 


125 -6 


196-3 


2826 


384*6 


502-4 


120 


127-9 


199*9 


287-8 


391*8 


511*7 


130 


130-5 


203-2 


292-6 


3983 


520-2 


140 


1320 


2063 


297-1 


404*3 


528-1 


150 


133*9 


209-2 


3013 


410-0 


535*6 


175 


138-0 


215-6 


310-5 


422-6 


552-0 


200 


141-6 


221-2 


318-6 


433*6 


566-3 


225 


144-7 


226 I 


325-6 


443*2 


578-9 


250 


147*5 


2305 


332-0 


451*8 


590-1 


275 


1501 


234*5 


337*7 


459*4 


600-3 


300 


152-4 


238-1 


342-9 


4667 


6096 



Since the circumference of the inner edge of the track is less 
than that of the outer edge, when record-breaking is attempted, 
the rider keeps as close as he safely can to the inner edge • conse- 



odSie 



CHAP. XVII. Stability of Cycles 207 

quently the average speed of riding is greatest at the inner edge. 
On this account, the convexity of the cross-section is, with advan- 
tage, made greater than shown in figure 194. 

168. Oyrosoopio Action.— In the above investigation, it has 
been assumed that the weight of the wheels is included in that of 
the rider and machine, and no account has been taken of their 
gyroscopic action. We have already seen (sec. 70) that if a wheel, 
of moment of inertia /, have a rotation, w, about a horizontal 
axis, and a couple, C, be applied to the axle tending to make it 
turn in a vertical plane, the axle will actually turn in a horizontal 
plane with an angular velocity of precession 

0=F (8) 

7(1) 

Thus, in estimating the stability of a wheel rolling along a circular 
arc, both centrifugal and gyroscopic actions must be considered. 
Let R be the radius of the track described by the bicycle, 
r the outside radius and r^ the radius of gyration of the wheels, 
F the speed of the cyclist, and w the weight of the wheels ; then 

^ = -^ /=ci)ri^ ci)= . 
Substituting in formula (8) we get 

^^-Rr (9) 

i,e, the gyroscopic couple required, in addition to the centrifugal 
couple, is proportional to the square of the speed, inversely pro- 
portional to the radius of the track, and approximately propor- 
tional to the radius of the cycle wheels. 

Example, — If the total weight of the machine and rider be 
180 lbs., the weight of the wheels 8 lbs., speed 30 miles per hour 
=44 feet per sec, the radius of the track 100 feet, r the radius 
of the wheel 14 in. = \% feet, and r, = 13 in. = V3^*set, we 
get K= 44 ft. per sec, and 

C = S X44\x i3» ^ g foot-poundals 
100x14x12 

= 4-84 foot-lbs. 

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2o8 Cycles in General chap. xvn. 

i,e, the mass-centre of the machine and rider will have to be 

^ "* = -027 feet, or -32 inches further from the vertical than if the 
180 

wheels were weightless, and gyroscopic action could be neglected. 

From the above example it will be seen that gyroscopic action 
in bicycles of the usual types is negligible, except at the highest 
speeds attainable on the racing-path, and on tracks of small radius. 
If a fly-wheel were mounted on a bicycle and geared higher than 
the driving-wheel, the gyroscopic action might be, of course, in- 
creased. If the fly-wheel were parallel to, and revolved in the 
same direction as the driving-wheel, the rider, while moving in a 
circle, would have to lean further over than would be necessary 
without the fly-wheel. If, on the other hand, the fly-wheel revolved 
in the opposite direction, the rider would have to lean over a less 
distance ; in fact, by having the /w of the fly-wheel large enough 
it might be possible for a bicyclist to keep his balance while lean 
ing towards the outside of the curve being described. 

The same gyroscopic action takes place when a tricycle moves 
in a circle. 

169. Stability of a Tricycle moving in a Circle.— A tricycle 
moving round a curve is subjected to the same laws of centrifugal 
force as a bicycle, the only difference being that the frame of the 
machine cannot tilt so as to adjust itself into equilibrium with the 
forces acting. 

Let figure 186 be the plan and figure 187 the elevation of a 
tricycle moving in a circle, the centre of which lies to the left. 
Let G be the mass-centre of the machine and rider, a, b and c 
the points of contact of the wheels with the ground. Considering 
the mass of the machine and rider concentrated at G, a horizontal 
force, -^2* applied at G is necessary to give the body its circular 
motion. This force is supplied by the horizontal component of 
the reaction of the saddle on the rider. There will be an equal 
horizontal force, -^3, exerted on the frame at (9, by the rider. This 
force tends to make the wheels slip sideways on the ground, an 
equal but oppositely directed force, -F,, will be exerted by the 
ground on the wheels. The force F^ gives the body its necessary 
radial acceleration, while the forces F^ and F^ acting on the 
machine form a couple tending to overturn it. If the resultant R 

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CHAP. XVII. Stability of Cycles 209 

of the forces F-^ and W cut the ground at a point, /, outside the 
wheel base, abc^ the machine will overturn. Hence the necessity 
for tricychsts leaning over towards the inside of a curve when 
moving round it. 

Again, if the force F^ be greater than /i W^ the tricycle will 
slip bodily sideways, /i being the co-efl5cient of sliding friction 
between the tyres and the ground. This slipping is often experi- 
enced on greasy asphalte or wood paving. 

1 70. Side-slipping. — ^The side-slipping of a bicycle depends 
on the coeflficient of friction between the wheels and the ground, 
and the angle of inclination of the bicycle to the vertical. The 
coefficient of friction varies with the condition of the road, being 
very low when the roads are greasy ; when the roads are in this 
condition the bicyclist, therefore, must ride carefully. The con- 
dition of the roads is a matter beyond his control, but the other 
factor entering into side-slipping is quite within his control. In 
order to avoid the chance of side-slipping, no sharp turns should 
be made on greasy roads at high or even moderate speeds. To 
make such turns, we have seen (sec. 165) that the bicycle must be 
inclined to the vertical, this slope or inclination increasing with 
the square of the speed and with the curvature of the path. At 
even moderate speeds this inclination is so great that on greasy 
roads there would be every prospect of side-slipping taking place. 
If a turn of small radius must actually be effected, the speed of 
the machine must be reduced to a walking pace or even less. 

A well-made road is higher at the middle than at the sides. 
When riding straight near the gutter the angle made by the plane 
of the bicycle with the normal to the surface of the ground is 
considerable. If the rider should want to steer his bicycle up 
into the middle of the road, in heeling over this angle is increased. 
This may be safely done when the road is dry, but on a wood 
pavement saturated with water it is quite a dangerous operation 
With the road in such a condition the cyclist should ride, if 
traffic permit, along its crest. 

The explanation given above (sec. 162) that in usual riding the 
lateral swing of a * Safety ' is greater than that of an * Ordinary,' 
explains why side-slipping is more often met with in the lower 
machines. The statement of some makers that their particular 

Digitized by V^j P 



2IO Cycles in General chap. xrn. 

arrangement of frame, gear, or tread of pedals, &c., prevents side- 
slipping is utterly absurd ; the only part of the machine which 
can have any influence on the matter being the part in contact 
with the ground — that is, the tyres. Again, the statement of 
riders that their machines have side-slipped when going straight 
and steadily cannot be substantiated. A rider may be going 
along quite carefully, yet if his attention be distracted for a moment, 
and he give an unconscious pull at the handles, his machine may 
slip. 

Side-slipping with Pneumati< T^res. — A pneumatic tyre has a 
much larger surface of contact with the ground than the old solid 
tyre of much smaller thickness. This fact, which is in its favour 
as regards ease of riding over soft roads, is a disadvantage as 
regards side-slipping on greasy surfaces. The narrow tyre on a 
soft road sinks into it, the bicycle literally ploughing its way 
along the ground ; and on hard roads the narrow tyre is at least 
able to force the semi-liquid mud from beneath it sideways, until 
it gets actual contact with the ground. The pressure per square 
inch on the larger surface of a pneumatic tyre, in contact with the 
ground being very much smaller, the tyre is unable to force the 
mud from beneath it ; it has no actual contact with the ground, 
but floats on a very thin layer of mud, just as a well lubricated 
cylindrical shaft journal does not actually touch the bearing on 
which it nominally rests, but floats on a thin film of oil between 
it and the bearing. The coefficient of friction in such a case is 
very small, and a slight deviation of the bicycle from the vertical 
position — ie, steering in any but a very flat curve — may cause 
side-slip. 

The non-slipping covers, now almost entirely used on roadster 
pneumatic tyres, are made by providing projections of such small 
area that the weight of the machine and rider presses them 
through the thin layer of mud into actual contact with the ground. 
The coefficient of friction under these circumstances is higher, 
and the risk of side-slip correspondingly reduced. 

Apparent Reduction of Coefficient of Friction, — While the 
driving-wheel rests on a greasy road a comparatively small driving 
force may cause the wheel to slip circumferentially on the road, 
instead of rolling on it. This skidding of the wheel, though 

Digitized by CjOOQIC 



CHAP. XVII. Stability of Cycles 2 1 1 

primarily making no difference in the conditions of stability, in a 
secondary manner influences side-slipping considerably. 

Let a body J/ (fig. 195) of weight W^ resting on a horizontal 
plane, be acted on by two horizontal forces, a and b^ at right 
angles. Let ^ be the coefficient of friction, and , , 

let at first only one of the forces, b^ be in action. ' U 

To produce motion in the direction M X^ b must j / 

be greater than /x W, Now, suppose the body M \ 

is being driven, under the action of a force fl, in __i — -f—, j^ 
the direction M Y^ in this case a much smaller "b"^ — * — ' 
force, by will suffice to give the body a component p 

motion in the direction MX. The actual motion 
will be in the direction M R^ and since friction '°* '^^ 

always acts in a direction exactly opposite to that of the motion, 
the resultant force on the body M must be in the direction 
M R. Let F be this resultant force ; its components in the 
directions Ai X and M Y must be b and a respectively. Now, if 
the force a be just greater than /i W^ it will be sufficient to cause 
the body to move in the direction M K, and any force, ^, however 
small, will give J/ a component motion in the direction M X, 

A familiar example illustrating the above principle, which has 
probably been often put into practice by every cyclist, is the 
adjusting of the handle-pillar in the steering-head. If the handle- 
pillar fits fairly tightly, as it ought to do, a direct pressure or pull 
parallel to its axis may be insufficient to produce the required 
motion, but if it be twisted to and fro— as can easily be done on 
account of the great leverage given by the handles — while a slight 
upward or downward pressure is exerted, the required motion is 
very easily obtained. 

In the * Kangaroo ' bicycle the weight on the driving-wheel was 
less than in either the ' Rover Safety ' or in the * Ordinary.' On 
greasy roads it was easy to make the driving-wheel skid circum- 
ferentially by the exercise of a considerable driving pressure. This 
circumferential slipping once being established, the very smallest 
inclination to the vertical would be sufficient to give the wheel a 
sideway slip, which would, of course, rapidly increase with the 
vertical inclination of the machine. 

171. Influence of Speed on Side-slipping.— The above dis- 

Digitized by V^j P 3 



212 



Cycles in General 



CHAP. XVII. 



cussion on side-slipping presumes that the speed of the machine 
and rider is not very great, so that the momentum of moving 
parts does not seriously influence the question. If the speed be 
very great, however, the momentum of the reciprocating parts, 
due principally to the weight of the rider's legs, pedals, and part 
of the weight of the crank, may have a decided influence on side 
slipping. 

Let G be the mass-centre of the machine and rider (fig. 196), 
let the total mass be Jf^lbs., let the linear speed of the pedals 
relative to the frame of the machine be v^ 
and let w be the mass in lbs. of one of the 
two bodies to which the vertical components 
of the pedals' velocity is communicated : iv 
will approximately be made up of the pedal, 
half the crank, the rider's shoe, foot, and 
leg from the knee downwards, and about 
one-third of the leg from the knee to the 

1^ T||J3 ^l I hip-joint. If the rider's ankle-action be 

'-il|g ifil - perfect, the mass w may be considerably 
less, depending on the actual vertical speeds 
communicated to the various portions 01 
the leg. Let the centre of the mass w be 
distant /, from the central plane of the 
bicycle. When the pedal is at the top of its 
path this mass possesses no velocity in a 
vertical direction, and therefore no vertical momentum. When 
the crank is horizontal and going downward, the vertical velocity 
is at its maximum, and the momentum is iv v. Let / be the time 
in seconds taken to perform one revolution of the crank, the time 

taken to impress this momentum is - ; and if/^ be the average 

4 
force in poundals acting during this time to produce the change, 
we must have (sec. 63) : 




Fig. 196. 



Therefore 



4 



Digitized by CjOOQIC 



CHAP. XVII. Stability of Cycles 2 1 3 

If/ be the average force in lbs., f^ =igf^ and the above 
equation may be written, 

/=4^^^ (10) 

If / be the length of the crank, the length of the path de- 
scribed in one revolution by the pedal-pin is 2 jt /, and the time 

taken to perform one revolution is — - . Substituting in (10) 

we get, 

/=^ (") 

Now leaving out of consideration for an instant the action 
of any force at the point of contact of the machine with the 
ground, and considering the machine and rider as forming one 
system, the above force / is an internal force, and can thus have 
no action on the mass-centre, G, of the whole system. But two 
parts of the sjistem have each been impressed with a moment of 
momentum, wz;/,, about the mass-centre G, the remaining part 
{W — 2w) will be impressed with a momentum numerically 
equal but of opposite sense. Let G} be the mass-centre of this 
remaining part. Then the up-and-down motion of the two 
pedals being as indicated by the arrows /, and /a* the point 6^, 
must move to the left with a velocity, z/,, such that 

2WVlx = ( JF— 2W)Vx X GGx* 
Thus, if there be absolutely no friction between the wheel 
and the ground, the point of contact of the wheel must slip side- 
ways to the right. 

Let F be the average frictional resistance, in lbs., required to 
prevent this slipping, then 

Fh=2fl,, 

or 

F =4J^^V, (,,) 

gn Ih 

If n be the number of turns per second made by the crank, 
t; = 2 TT ;i /, and (12) may be written 

gh ' ^ ^^ 

Digitized by CjOOQIC 



214 Cycles in General chap. xm. 

From (12) and (13) the lateral force F^ or what may be called 
the * tendency ' to side-slip, is proportional to the masses which 
partake of the vertical motion of the pedals, to the width of the 
tread, and inversely proportional to the height of the mass-centre 
from the ground; from (12) it is proportional to the square of 
the speed of the pedals, and inversely proportional to the length 
of the crank ; from (13) it is proportional to the square of the 
number of revolutions of the crank-axle and to the length of the 
crank. 

The force /^changes in direction twice during one revolution 
of the crank-axle. It is equivalent to an equal force acting at 
Gy and a couple Fh, The force acting at G, changing in 
direction, will therefore cause the mass-centre of the bicycle and 
rider to move in a sinuous path, even though the track of the 
wheel be a perfectly straight line. The less this sinuosity, other 
things being equal, the better ; ue, in this respect a high bicycle 
is better than a low one for very high speeds. 

It must be carefully noted that in the above investigation the 
pressure exerted on the pedal by the rider does not come into 
consideration. When moving at a given speed the tendency to 
side-slip is therefore quite independent of whether pressure is 
being exerted on the pedal or not. 

172. Pedal Effort and Side-slip.— The idea that the pressure 
on the pedal causes a tendency to side-slip is so general that it 
may be worth while to study in detail the forces acting on the 
rider, the wheel and pedals, and the frame of the machine. For 
simplicity we will consider an * Ordinary,' in which the rider is 
vertically over the crank-axle. The investigation will be of the 
same nature, but a little longer, for a rear-driving * Safety.' The 
weight of the machine will be neglected. 

Let W be the weight of the rider, 7^, the vertical thrust on 
the pedal, F^^ the upward pull on the handle-bar, F^ the vertical 
pressure on the saddle ; let /, and Li be the distances of the 
lines of action of F^ and F<i respectively, and 4 the distance of 
the crank axle-bearing from the central plane of the machine 
(fig. 196). 

Consider first the forces acting on the rider ; these are, his 

weight, Wy acting downwards at G \ the pull, ^1 of the handle- 
Digitized by VjOOQ 



CHAP. XTII. 



Stability of Cycles 



215 



bar downwards ; the reaction, /^„ from the pedal upwards ; and 
the reaction, /^g, of the saddle. These forces are all parallel, and 
since the rider is in equilibrium we must have 



^ - i^i + /?i - i?i = o 



(14) 



Also, the moments of these forces about any point is zero ; there- 
fore, taking moments about the mass-centre, tr, if the rider has 
not shifted sideways when exerting the pressure F^ on the pedals. 



F,l, - F^l^^o 



(15) 



If the rider does not pull at the handles he must either grip tightly 
on to the saddle, or shift sideways, so that the moment of the force 
Fi is balanced. 

Consider next the forces acting on the frame, which, for clear- 
ness of illustration, is shown isolated (fig. 197) ; these are, the 
pull, F^y on the handle-bar upwards ; the pressure, -F3, of the 
rider on his saddle downwards ; and the upward reaction of the 
bearings /, and/2. These forces 
are all parallel, and since they 
are in equilibrium, 

^2 - ^3 + fi +/2 = o ; 

that is, 

/, +/2 = ^3- ^2. . (16) 




Since the force {F^ — F^) has 

no horizontal component, neither 

will the force (/, -f /j). By taking 

moments of all the forces about 

the point of application of /a, 

the value of / may be found, fig. 197. Fig. X98. 

and then/2 can be determined. 

Now, consider the forces acting on the wheel (fig. 198), in- 
cluding cranks and pedal-pin, which together form one rigid body. 
Besides the forces -^1,/, and /a, there is only the reaction of the 
ground, R^ and since the wheel is in equilibrium vertically, 



^ - ^1 -/i -A = o. 



Digitized by CjOOQIC 



2l6 



Cycles in General 



CHAP. XVIT. 



Substituting the value of/, + /^ from (i6) we get 
R^ F^-- F^Ar Ft,^W . 



(17) 



R being vertical, there is no tendency to side-slip. 

The above result can be more simply obtained, thus : con- 
sidering the bicycle and rider as forming one system of bodies, 
the external forces acting are in equilibrium ; and since these 
consist only of the weight, W^ and the reaction, R^ R must be 
(sec. 71) equal, parallel but opposite to W, W being vertical, 
R must also be vertical. The force Fy^ exerted by the rider on 
the pedal is an internal force, and has not the slightest influence 
on the external forces acting on the system. 

173. Headers. — Taking a * header' over the handle-bar was 
quite an every-day occurrence with riders of the * Ordinary ' bicycle. 
In the * Ordinary,' the mass-centre of the rider and machine was 
situated a very short distance behind a vertical through the centre 
of the front wheel, so that the margin of stability in a forward 
direction was very small ; any sudden check to the progress of 
the machine by an obstruction on the road, by the rider applying 
the brake, or back-pedalling, was in many cases sufficient to send 
him over the handle-bar. Two classes of headers have to be dis- 
tinguished : (I) That in which the front wheel may be considered 
rigidly fixed to the frame ; the header being caused either by the 

application of the brake to 
the front wheel, or by back- 
pedalling in a Front-driver. 
(II) That in which the 
front wheel is quite free to 
revolve in its bearings ; the 
header being caused by an 
obstruction on the road, 
application of the brake to 
the back wheel, or back- 
pedalling in a Rear-driver. 
(I) Let /, (fig. 199) be 
the distance of the mass- 
centre, G^ from a vertical through the wheel centre, c ; then, in 
order that the wheel, frame, and rider may turn as one body about 

Digitized by V^jOOQ 







Fig. 199. 



CHAP. xm. Stability of Cycles 217 

the point a as centre, a moment, W l^^ must be applied. If d be 
the diameter of the driving-wheel, /i the coefficient of friction of 
the brake, and P the pressure of the brake just necessary to lock 
the frame on the wheel and so cause a header, 

'■^ = ^A (18) 

If the pressure actually applied to the brake be equal to or 
greater than P^ determined by the above equation, the wheel will 
be locked to the frame. 

Let the circle through G with centre a cut the vertical through 
c at h. From G draw a horizontal to cut c h in f. In taking a 
header, the weight of the machine and rider has to be lifted a 
distance /A. If v be the speed of the machine, the kinetic energy 

stored up m it is -i-t", and the work done in lifting it through 

o 

the height /A \s IV x fi ; therefore, if the speed v be greater 
than that determined by the formula 

v^ — 

-2^=/>^ (19) 

a header will occur if the brake-pressure be applied strongly. 

If the check to the speed of a Front-driver be made by back- 
pedalling, r be the radius of the crank, and P^ the back-pedalling 
force applied, we have, 

Pxr= Wl, (20) 

The action of back-pedalling in a Front-driver is the same as 
that of applying the brake to the front wheel, as regards the lock- 
ing of the front wheel to the frame. The speed at which a header 
will occur if vigorous back-pedalling be applied is in this case also 
given by equation (19). 

Example I. — In a 54-inch * Ordinary,* the point G (fig. 199) may 
be 60 inches above the ground and 10 inches behind the wheel- 
centre c. The height, /A, will then be about 1*2 inch = ^^ foot. 
Substituting in (19) 

v^ I 

= , from which 7; = 2*5 feet per second, 

2 X 32*2 10* J r > 

= 1*9 mile per hour. 

Digitized by CjOOQIC 



2l8 



Cycles in General 



CHAP. xm. 



Example II, — In a 'Safety' (fig. 200) the height,//^, may be 
2 feet. Substituting in (19), 

= 2, from which f/ = ii'i feet per second, 



2 X 322 



= 7*6 miles per hour. 




Fig. 200. 



The subject may be looked at from another point of view. 
Let Fy be the horizontal force of retardation which must be 

supplied by the action of 
the ground on the wheel 
This is transmitted 
through the wheel, 
so that an equal force, 
F^y acts on the mass at 
G^ and the mass reacts 
on the frame with an 
equal and oppositeforce, 
F^, Then, in order that 
stability may be main- 
tained, the resultant R 
of W and -^3 must not cut the ground in advance of the point of 
contact a. If R cuts the ground in front of a, the machine 
will evidently roll over about a as centre. 

(II) Brake on Back Wheel — If the brake be applied to the 
rear, instead of the front wheel, the bicycle is much safer as re- 
gards headers. If the brake, in this case, be applied too suddenly, 
the retarding force causes an incipient header, the frame turning 
about the front wheel centre c as axis, and the rear wheel im- 
mediately rises slightly from the ground. The retarding force 
being thus removed, the development of the header is arrested, 
the rear wheel again falls to the ground, and the process is re- 
peated, a kind of equilibrium being established. 

Headers through Obstructions on the Road, — If the check to 
the progress of the machine be caused by an obstruction on the 
road, the only difference from the case treated above is that the 
front wheel is free to revolve in its bearings ; the header is taken 
about the point ^ as a centre, and the resultant R of the weight 
W and the force F^ must not pass in front of the wheel centre c. 

Digitized by VjOOQ 



CHAP. XVII. 



Stability of Cycles 



219 



The direction of the forces between two bodies in contact is 
(neglecting friction) at right angles to the surface of contact. In 
a bicycle wheel with no friction at the hub, the direction of the 
pressure exerted by a stone at the rim must therefore pass through 
the wheel centre. This condition enables us to determine the 
size of the largest stone which can be ridden over at high speed 
without causing a header. Join the mass-centre, G, to the front 
wheel centre, c (fig. 201), and produce the line to cut the circum- 




Fig. 30I. 



ference of the wheel at b. A stone touching the rim at a point 
higher than b may cause a header at high speed ; a stone touch- 
ing at a lower point may be ridden over at any speed. Figure 
200 is the same diagram for a * Safety' bicycle, a glance at which 
shows that with this machine a much larger stone can be safely 
surmounted than with an * Ordinary.' 

The above discussion presupposes that at the instant the 
front wheel strikes the stone no driving force is being exerted. 
If the rider is driving the front wheel forward at the instant, a 
larger obstacle may be safely surmounted. Let e (fig. 201) be the 
point of contact of a large stone ; the reaction ^1 is in the direc- 
tion e c. The resultant force R on the mass at G must be equal 
and parallel but opposite to ^j. The forces R and -^i form a 

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220 Cycles in General chap. xvii. 

couple R /, tending to turn the frame and rider about the centre 
<:, / being the length of the perpendicular from 6^ on ^ ^ pro- 
duced. If the rider apply to the front wheel a turning moment 
in the forward direction equal to or greater than R /, there will be 
a couple of equal magnitude acting on the frame tending to turn 
it in the opposite direction, which will neutralise the couple R L 
The final result is that the wheel safely surmounts the obstacle, 
turning about e as centre. 



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221 



CHAPTER XVIII 

STEERING OF CYCLES 

174. Steering in General. — When a bicycle moves in a 
straight line, the axes of its wheels are parallel to each other. The 
steering is effected by changing the direction of one of the wheel 
spindles relatively to the other. In order to effect this change of 
direction, the frame carrying the wheels is made in two parts ; 
jointed to each other at the steering-head, the parts being called 
respectively the rear- and front-frames. One of these parts, that 
carrying the saddle, is usually much larger than the other (and is 
often called the frame, to the exclusion of the other part called 
the fork) ; the wheel— or wheels —mounted on the other (smaller) 
part of the frame is called the steering-wheel — or wheels. 
According to this definition, the driving wheel of an 'Ordinary' is 
also the steering-wheel. In side-steering tricycles (see chap, xvi.) 
the frame is in three parts, and there are two steering-heads. 

Cycles are front- or r^^r-steerers, according as the steering- 
wheel is mounted on the front- or rear-frame. All bicycles that 
have attained to any degree of public favour are front-steerers : 
The * Ordinary,' the * Kangaroo,' the * Rover Safety,' the * American 
Star,' and the * Geared Ordinary.' A few successful tricycles have, 
however, been rear-steerers. 

175. Bicycle Steering. — Let a (fig. 202) be the wheel fixed to 
the rear-frame, b the steering-wheel, and d the intersection of the 
steering-axis with the ground ; this, in most cases, is at or near 
the point of contact of the wheel with the ground, though in the 
* Rover Safety,' with straight front forks, it occurs some little 
distance \n front. Let the plan of the axes of the wheels a and b 
be produced to meet at 0, then if the wheels roll, without slipping 
sideways, on the ground, the bicycle must move in a circle having 

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222 Cycles in General chap. xvm. 

as its centre. The steering-wheel, ^, will describe an arc of 
larger radius than that described by the wheel a \ consequently 
if in making a sharp turn to avoid an obstacle the front wheel 
clears, the rear wheel will also clear. In a rear-steering bicycle, on 

the other hand, it may 

happen that the rear 

wheel may foul an object 

^ which has been cleared 

^'^^^ by the front wheel. 

^'^^ The actual sequence 

^ ^ .^ ^ of operations in steering 

"^^^ a bicycle is not com- 

'"'^K ^ monly understood. If a 
beginner turn the steer- 
ing-wheel to one side 
P,^. 203 before his body and the 

bicycle have attained the 
necessary inclination, the balance will be lost. On the other hand, 
the beginner is often told to lean sideways in the direction he wants 
to steer. This operation cannot, however, be directly performed ; 
since, if he lean his body to the right, the bicycle will lean to the 
left, and the sideway motion of the mass-centre cannot be con- 
trolled in this way. It has been shown (sec. 162) that the path 
described by a bicycle, even when being ridden as straight as 
possible, is made up of a series of curves, the bicycle being 
inclined alternately to the right and to the left. If at the instant 
of resolving to steer suddenly to one side the bicyclist be inclined 
to that side, he simply delays turning the steering-wheel until his 
inclination has become comparatively large. The radius of curva- 
ture of the path corresponding to the large inclination being small, 
the steering-wheel can then be turned, and the bicycle will 
describe a curve of short radius. If, on the other hand, he be 
inclined to the opposite side, the steering-wheel is at first turned 
in the direction opposite to that in which he wishes to steer, so as 
to bring the bicycle vertical, and then change its inclination ; 
the further sequence of operations is the same as in the former 
case. Thus, to avoid an object it is often necessary to steer for a 
small fraction of a second towards it, then steer aw^y from it ; this 

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CS^T. ZYIII. 



Steering of Cycles 



223 



IS probably the most difficult operation the beginner has to master. 
In steering, the rider's body should remain quite rigid in relation 
to the frame of the bicycle. 

176. Steering of Tricycles.— The arrangement of the steering 
gear of a tricycle should be such that in rounding a comer the 
axes of the three wheels all intersect at the same point. In the 
* Humber,' the * Cripper,' and any tricycle with a pair of wheels 
mounted on one axle this condition is satisfied. 

Let O be the intersection of the axes, a, b^ r, of the three 
wheels. The tricycle as a whole rotating round (7 as a centre, the 



T1 rv^ 



I 

/ I 






/ -'' — o 



<--/.-v 



>s I 




Fig. 203. 

linear speed of the rim of wheel c will be greater than that of 
wheel b nearer the centre of rotation. If b and c are not driving- 
wheels, and are mounted independently on the axle, they will run 
automatically at the proper speeds. If b and c are driving-wheels, 
as in the * Humber,' *Cripper,* and * Invincible' tricycles, some 
provision must be made to allow the wheel on the outside of the 
curve to travel faster than the inner. This is described in sections 
188, 189. 

177. Weight on Steering-wheel.— We have already seen that 
a considerable portion of the total weight of the machine must be 

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224 Cycles in General chap. xvin. 

placed on the driving-wheel, so as to prevent skidding under the 
action of the driving effort. A certain amount of weight must 
also rest on the steering-wheel in order that it may perform its 
functions properly. 

If the machine be moving at a high speed in a curve 
of short radius, the motion of the frame and rider can be ex- 
pressed either as one of rotation about the point (7, or as a 
translation equal to that of the mass-centre of the machine and 
rider, combined with a rotation about a vertical axis through 
the mass-centre G, If the rider should want to change from a 
straight to a curved course, the linear motion of the machine 
remains the same, but a rotation about an axis through the mass- 
centre must be impressed on it. To produce this a couple must 
act on the machine. The external forces, jP, and P^^ constituting 
this couple can evidently only act at the points of contact of the 
wheel and the ground, and, presuming that the rolling friction may 
be neglected, can only be al right angles to the direction of 
rolling. The magnitudes of the forces P^ and P^ depend on the 
speed at which the cycle is running, and also on the general 
distribution of weight of the machine and rider — in mathematical 
language, on the moment of inertia of the system. The weight, 
w^ on the steering-wheel must be equal to, or greater than, 

P 

— ^ u. being the coefficient of friction. The moment of inertia, 

about its mass-centre, of a system consisting of a machine and 
two riders is very much greater than twice that of a system con- 
sisting of a machine and one rider ; consequently the pressure 
required on the steering-wheels of tandems is much greater than 
twice that required on the steering-wheel of a single machine. 

A simple analogy may help towards a better understanding of 
this. Suppose two persons of equal weight be seated at opposite 
ends of a see-saw, and that the up-and-down motion is imparted 
by a person standing on the ground, and applying force at one 
end of the see-saw. If now only one person be left on the see- 
saw, and he be placed at the middle exactly over the support, the 
person standing on the ground will have to supply a much smaller 
force than in the former case to produce swings of equal speed and 
amplitude. The swinging up and down of the see-saw corresponds 

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CHAP. XVIII. Steering of Cycles 225 

to the change of steering of the cycle from left to right, the forces 
applied by the person standing on the ground to the forces, Py 
and /^2, of reaction of the ground on the wheels. The single 
person on the middle of the see-saw corresponds to a single rider 
on a cycle, the two persons at the ends to the riders on a tandem. 
Sensitiveness of Steering, —V^e have continually spoken of the 
point of contact of a wheel with the ground, thereby meaning the 
geometrical point of contact of a circle of diameter equal to that 
of the wheel. The actual contact of a wheel with the ground 
takes place over a considerable 

surface, the lower portion of the ^:^,. ^ 

tyre getting flattened out as j^'Mw^Tjny , t r r" 

shown, somewhat exaggerated in 



iir »<■/ I 



mmmnKii^mii^m^, 



figure 204. The total pressure of 
the wheel on the ground is dis- 
tributed over this area of contact. Considering tyres of the same 
thickness, it is evident that a wheel of large diameter will have 
the length of its surface of contact in the direction of the plane 
of the wheel greater than that of a wheel of smaller diameter. 

Consider now the resistance to turning such a wheel, pivot- 
like, on the ground, as must be done in steering. Let A be the 
area of the surface of contact, and suppose the pressure of 
intensity,/, distributed uniformly over it, as will be very approxi- 
mately the case with pneumatic tyres ; then 

^ A 

Consider a small portion of the area of width, /, included between 
two concentric circular arcs of mean radius, r. Let a be the area 
of this piece, the total pressure on this will be / a^ and the 
frictional resistance to spinning motion of this portion of the tyre 
on the ground will he fipa. The moment of this force about the 
geometrical centre, O, is 

H'/'ar . . (i) 

and the total moment of resistance of the wheel to spinning on the 
ground is the sum of all such elements. If we consider the 
surface of contact to be a narrow rectangle, whose width is very 

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226 Cycles in General caxv, xVni. 

small in comparison with its length, /, the average value of r in (i) 
will be -, and the total moment of resistance to spinning will be 

tJ^J (.) 

4 

Thus a greater pull will be required at the handle-bar to steer a 
large wheel than a small one ; in other words, a small steering- 
wheel is more sensitive than a large one. The assumption made 
above, that the width of the surface of contact is very small com- 
pared with its length, is not even approximately true for pneumatic 
tyres. The moment of resistance in this case will, however, 
increase with /, and, therefore, the conclusion as to the relative 
sensitiveness of small and large wheels holds. 

The above expression gives the moment of resistance to turning 
the steering-wheel on the ground when the bicycle is at rest. This 
moment is quite considerable, and is much greater than the actual 
moment required to steer when the bicycle is in motion, as can be 
easily verified by experiment The explanation of this phenome- 
non is practically of the same nature as the explanation, given in 
section 170, of the small force necessary to overcome friction in 
one direction, provided motion in a direction at right angles exists. 
In the present case the wheel is rotating about a horizontal axis 
during its forward motion ; the steering is effected by giving it a 
motion about a vertical axis. On account of the motion about a 
horizontal axis already existing, a comparatively small moment is 
sufficient to overcome the frictional resistance to motion about a 
vertical axis. 

178. Motion of Cycle Wheel— It is a popular notion that the 
motion of a vehicle wheel is one of pure rolling on the ground, 
but a little consideration will show that this is not always the case. 
So long as a tricycle moves in a straight line, the wheels merely 
roll on the ground, the instantaneous axis of rotation being a line 
through the point of contact of the wheel and ground, parallel to 
the axis. When the vehicle is moving in a curve, in addition to 
this rotation about a horizontal axis, the wheel possesses a motion 
round a vertical axis, and some parts of the tyre in contact with the 
ground slide over the ground, as described in section 177. The 

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CHAP. XVIIf. 



Steering of Cycles 



227 



instantaneous axis of rotation is now a line inclined to the 
ground. 

Suppose that the plane of the wheel can be inclined to the 
vertical when the cycle is moving in a curve, as in the case of a 
bicycle or steering-wheel of a * Cripper ' tricycle. Let the axis of 
the wheel be produced to cut the ground at K, then if the cycle 
be at the instant turning about the point V as centre, the motion 
of the wheel on the ground will be one of pure rolling, no sliding 
being experienced by any point of the tyre in contact with the 
ground. The part of the wheel in contact with the ground may 
be considered part of a right circular cone, having its vertex at V, 
Such a cone would roll without slipping on a plane surface, the 
vertex, V, of the cone remaining always in the same position. 

The intersection of the axis of the wheel with the ground is 
determined by the inclination of the wheel to the vertical. This 
inclination depends on the radius of the curve in which the 
bicycle is moving, and also its speed. For a curve of a given 
radius there is, therefore, one particular speed at which V will 
coincide with (9, the centre of turning of the bicycle. At this 
speed there will be no spinning 
of the tyre on the ground, 
while at greater or less speeds 
spinning occurs to a greater 
or less degree. 

179. Steering Without 
Hands. — In a front-driving 
bicycle, the saddle and crank- 
axle being carried by the rear- 
and front-frames respectively, 
there is theoretically no diffi- 
culty in steering without using 
the handle-bar. If it be de- 
sired to turn towards the right, 
a horizontal thrust at the left 
pedal as it passes its top 
p)osition, or a pull at the right pedal as it passes its lowest position, 
will effect the desired motion. 

In a rear-driving bicycle, the saddle and crank-axle being 

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d*i.^i 



Fic. 305. 



228 



Cycles in General 



CHAP. rvui. 



carried by the rear-frame, there is no direct connection between 
the rider and the steering-wheel axle except by the handle-bar. 

Let tto be the angle the steering-axis makes with the horizontal 
when the bicycle is vertical (fig. 205) ; h the distance of the 
wheel centre from the steering-axis ; k^ the distance between b^ the 
point of contact of the wheel with the ground, and d the point of 
intersection of the steering-axis with the ground, when the bicycle 
is vertical and the steering-wheel in its middle position ; / the 




Fig. 2C7. 



distance of the mass-centre of the steering-wheel and front-frame 
(including handle-bar, &c.) from the steering-axis ; the inclina- 
tion of the middle plane of the rear-frame to the vertical ; the 
angle the handle-bar is moved from its middle position, i.e, the 
angle between the middle planes of the front and rear wheels ; 
and a the angle the steering-axis makes with the horizontal, cor- 
responding to the values of and <?>. Figs. 206 and 207 are 
elevation and plan of a bicycle heeling over. The forces acting 
on the front wheel and frame which may tend to turn it about the 

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CHAP. xnii. 



Steering of Cycles 



229 



Fig. 208. 



'^^ 



Steering-axis are— the reaction of the ground, and the weight, a/, of 
the front wheel and frame. The reactions at the ball-head inter- 
sect the steering-axis, and therefore cause no tendency to turn. 
The reaction of the ground can be resolved into three com- 
ponents — W, acting vertically upwards ; F^ the resistance in the 
direction of motion of the wheel ; and C, the centripetal force at 
right angles to R The line of action of F passes very near the 
steering-axis for all values of and ^, and since F is itself small 
in comparison with W and (7, 
its moment may be neglected. 
Figs. 208 and 209 are elevation 
and plan enlarged from figs. 206 
and 207, showing the relation of W 
to the steering-axis, b d^ is the 
plan and b^^ d^ the elevation of 
the shortest line between JF and 
the steering-axis. W can be re- 
solved into a force, 6", parallel to 
the steering-axis, and a force, T^ at 
right angles to the plane containing 
.Sand the steering-axis. If b^ ^i* 
represent W to scale, q^ ^/ and 
p^ b^^ are the elevations of the forces 
T and 5, while Q <J/ and b.^^ Q 
show to scale the true magnitudes 
of T'and S respectively ; i,e, b^^ b^ 
q^ is the elevation of the force- 
triangle, and b^^ b^ Q is its true 
shape. Also it may be noticed 
that the line b d^ in plan measures 
the true length of the perpendicular between W and the steering- 
axis ; and IV tends to turn the steering-wheel still further, its 
moment about the steering-axis being Q by^ x b d\. The centri- 
petal force C tends to turn the steering-wheel back into its middle 
position. The effect of the weight w in tending to turn the steering- 
wheel can be shown in exactly the same way as that of the vertical 
reaction W, The tendency is in general to increase the devia- 
tion of the steering-wheel, but when a straight fork is used the 

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Fig. 209. 



230 Cycles in General chap. xvm. 

tendency is to reduce it, on account of the mass-centre of the 
handles being behind the steering-axis. 

We shall now determine the analytical expressions for the 
moments of JF, C, and w^ assuming that the angles and ^ are 
small, and that, therefore, we may use the approximations 

sin e = e = fan d 
sin (ii=i<p=s. tan <p. 

We have seen above that the moment of W is 

Qj^ X bd\' 
Now "C^i*= Wcos a, 

also sin a = sin ao cos 0, 

Therefore <2 ^ i^ = ^^ >/ i ^^/« ^ oTcos «T 

= IV cos ao approximately. 

Now ^ Ji = Fdsin b d </,. The angle b d d^ is made up of 
the two angles a d b and a d d^. The former is zero if ^ is zero, 
and the latter is zero if is zero. For small values of Q and 0, 
the angle a d b -^^ ^ sin ao, and a d d^-ss-^ tan ao. 

Therefore bd^^=> Td sin (9 tan ao + ^ sin a©). 

Therefore, if we assume that b d remains constant, we have 
'b d^^k {d tan ao + ^ sin ao) approximately, and moment of Jf-^is 

IVk sin aj (d + <l> cos ao) (3) 

The moment of .C for small values of and ^ will be approxi- 
mately C X b d X sin ao. 

Now, if the angle 6 remains constant 

C= ^^A ^ = -r-^-^ = ~ ^ — approximately, 
g /c sinadb i^ sin n^ 

V being the speed of the bicycle, R the radius of the circle 
described by the front wheel, and / the length of the wheel-base. 
Therefore the moment of C is 

Wv^ k ^ sin'^ ap / x 

g i ' ■ ^"^^ 

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CHAP. xvni. 



Steering of Cycles 



231 



The moment of tv can be found as follows : Resolving w 
into two components parallel to and at right angles to the steer- 
ing-axis, the latter is w cos a. Figure 210 shows side and end 
elevations of the steering-axis and mass-centre, G, The perpen- 
dicular distance Bi B.^ between w and the steering-axis for a 
small value of is 

/e 



ZTSTx 6 = 



cos tto 



W COS 



" \cos ao "^ V 



while for a small value of ^ it is / f. Therefore moment of w is 

\COS tto 

=^w/{B + <l>cosao) (5) 

Hence, finally adding (3), (4), and (5), the moment tending to 
turn the steering-wheel still further from its middle position is 



IVJIi sin ao (a + cos ao) - l!L±$J^ 



Wk <fk v^ sin'^ Op 



gl 



+ a;/(a + ^r^^ao) 



^{Wksina,^wf){Q^^cosa,)^}^'^-Yl^v^ . (6) 

To maintain equilibrium the expression (6) should have the 
value zero, to steer further to one side or other it should have 
a small positive value, and to steer 
straighter a small negative value. 

For given values of v and <p there 
remains an element 0, the inclination 
of the rear-frame, at the command of 
the rider ; but even with a skilled rider 
the above moment varies probably so 
quickly that he could not adjust the 
inclination Q quickly enough to pre- 
serve equilibrium. 

In the above expressions we have 
taken no account of the gyroscopic action of the wheel, though 
probably this is the most important factor in the problem 
Taking account of the gyroscopic action, the above moment 
about the steering-axis would produce a motion of precession 
about an axis at right angles to those of the Jball-head and 

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Fig. a 10. 



232 Cycles in General chap, xtul 

steering-wheel ; while to turn the steering-wheel about the steer- 
ing-axis, a couple, with its axis at right angles to the steering-axis, 
would be required. This is produced by the side pressures on 
the steering tube ; so that in steering without hands, if the rider 
wishes to turn to the right, he inerely leans over slightly to the 
right, and the steering-wheel receives the required motion, pro 
vided the value of the expression (6) is small. 

Example, — With the same data as in section i68, to turn the 
steering-wheel at the speed indicated, a couple of 2*42 foot-lbs. 
is required, />. if the ball-head be 8 inches long, side pressures of 
3*63 lbs. would suffice to turn the front wheel at the speed 
indicated. To turn the steering-wheel more quickly, a greater side 
pressure mus^ be exerted on the steering-head. 

From section 168 the gyroscopic couple required is proportional 
to the square of the speed,. and approximately proportional to the 
weight and to the diameter of the front wheel ; therefore, steering 
without hands should be easier the higher the speed, the larger 
the steering-wheel, and the heavier the rim of the steering-wheel. 
This agrees with the fact that a fair speed is necessary to perform 
the feat, that the feat is easier with pneumatic than with solid 
tyres, the former with rim being heavier than the latter ; it also 
accounts for the easy steering with large front wheels, and for the 
fact that the * Bantam ' is more difficult to steer without hands 
than the * Ordinary.* 

It may be noticed that if this explanation be correct, it should 
be possible to ride without hands a bicycle in which the steering- 
axis cuts the ground at the point of contact of the front wheel. 
M. Bourlet, who discusses the subject at considerable length, says 
this is impossible ; he also says that the mass-centre of the front 
wheel and frame must lie in front of the steering-axis ; but this 
would mean that a bicycle with straight forks could not be ridden 
without hands ; whereas some of the earliest * Safety ' bicycles, 
made with straight forks, were easily ridden without hands. 

180. Tendency of an Obstacle on the Boad to Cause Swerv- 
ing. — If a bicycle run over a stone, the force exerted by the stone 
on the steering-wheel acts in a direction intersecting the steering- 
axis, and has thus no tendency to cause the steering-wheel to turn 
in either direction. In the same way, the steering-wheel of a 

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CHAP, xviii. Steering of Cycles 233 

* Cripper ' or * Invincible * tricycle in running over a stone experiences 
no tendency to turn, and therefore no resistance need be applied 
by the rider at the handle-bar. The line of action of the force 
exerted on the machine cuts a vertical line through the mass- 
centre ; the force therefore only tends to reduce the speed of the 
machine, but not to deviate it from its path. If the obstacle meet 
one of the side wheels of a tricycle, the force exerted by the stone 
and the force of inertia of the rider form a couple tending to turn 
the machine and rider as a whole about their common mass- 
centre. In some tricycles the force exerted by the stone tends 
also to change the position of the steering gear, and so cause 
sudden swerving. A few of the chief types of tricycles are dis- 
cussed in detail, with reference to these points, in the following 
sections. 

181. Cripper Tricycle.— Let one of the driving-wheels meet 
with an obstacle. Introducing at 6^, the mass-centre, two opposite 
forces, F^ and F^y each equal to /^„ no change is made in the 
static condition of the system. The force, F^ (fig. 203), exerted by 
the stone on the machine is equivalent to an equal force, 7^,, 
acting at the mass-centre of the machine and rider, and retarding 
the motion, and a couple formed by the forces F^ and /^, tending 
to turn the machine about its mass-centre, G, This turning is 
prevented by the side friction of the wheels on the ground. To 
actually turn about G, the driving-wheels must roll a little and the 
front steering-wheel slip sideways. 

Let /be the resistance to slipping sideways of the front wheel, 
/j and 4 the lengths of the perpendiculars from G on the lines of 
action of the forces F^ and / w the load on the steering-wheel, 
and ft the coefficient of friction between the steering-wheel and 
the ground. Then fl^ must be equal to or greater than F^ l^, 
Also/= ft Wy therefore ft zf/ /j ^ F/, or 

If, in the * Cripper ' tricycle, the steering-axis produced passes 
exactly through the point of contact of the steering-wheel with the 
ground (fig. 211), the reacrion from the ground on the steering- 
wheel has no tendency to cause it to turn ; no resistance is necessary 

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234 



Cycles tn General 



CHAP. XVIII. 



at the handle-bar when one of the driving-wheels strikes an 
obstacle. If, as in all modern tricycles, the steering-axis produced 
passes in front of the point of contact of the steering-wheel with 
the ground (fig. 212), the force,/ will tend to turn the steering- 
wheel sideways, and must be resisted by a force, -^4, at the handle- 
bar, such that F^ l^ =//j, h toeing the length of the perpendicular 




from the point of contact with the ground to the steering-axis, and 
l^ the half-length of the handle-bar. 

In a tricycle with a straight fork, the distance /j, and therefore 
also the necessary force F^^ at the handle-bar to prevent swerving, 
is greater than with a curved fork (fig. 212). 

182. Eoyal Crescent Tricycle.— In the * Royal Crescent * tri- 
cycle (fig. 151), made by Messrs. Rudge & Co., the steering-axis 
intersected the ground at a point ^ (fig. 213), some distance behind 
the point of contact of the wheel. The force, / would therefore 
tend to turn the steering-wheel about the steering-axis, in the oppo- 
site direction to that in the * Cripper.' The distance, 4, being much 
greater than in the * Cripper,^ the force, F^^ necessary at the handle- 
bar to prevent swerving was also greater. A spring control was 
used for the steering, so that a considerable force was necessary to 
move the steering-wheel from its middle position. 

183. Hmmlier Tricycle^ — In a * Humber' tricycle, an obstacle in 
front of one of the driving-wheels tends to turn the driving-axle 
round the steering-axis, a (fig. 214). This must be resisted by a 
force, /^i, applied by the rider at the handle-bar given by the 
equation F^ l\ = /< ^4ror the obstacle will change the direction of 
motion suddenly and a spill may occur. If the rider supply the 

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CHAP. XVIII. 



Steering of Cycles 



235 



necessary force, F^, the conditions as to the machine as a whole 
turning about the mass-centre G, and as to the weight necessary 
on the steering-wheel to prevent this turn- 
ing, are the same as discussed in section 
181. 

It will be seen from the above that 
the arrangement of the steering in the 

* Humber ' tricycle is less satisfactory than 
in some of the other types. 

Any cycle in which there are a pair of 
independent wheels mounted on a com- 
mon axle, pivoted to the frame at its 
middle point, will be subject to the same 
defect of steering. Examples are afiforded 
in figures 154, 155, and 182. 

184. Olympia Tricycle and Sudge 
ftuadricycle. — The wheel plan of an 

* Olympia' tricycle is shown at figure 215. A single rear driving- 
wheel is used ; the two front wheels are side-steerers. In some 
of the earlier patterns of this tricycle made by Marriott & 




Fic. 214. 






Oi O Oi 



Fig. 215. 




Cooper, the steering-wheels ran free on the same axle, which 
was pivoted at a to the rear-frame of the machine ; the action in 
steering was therefore the same as in. the *Humber' tricycle. In 

Digitized by CjOOQIC 



236 Cycles in General chap. xtih. 

the modern patterns of the ' Olympia ' tricycle the steering is effected 
by providing the steering-wheel spindles with separate steering- 
heads at «, and a^. Short bell-cranks are formed on the spindles, 
and the ends of these cranks are connected by links to the end of 
a crank at the bottom of the steering-post a. The distance, /j, 
between the steering-axis and the point of contact of the steering- 
wheel with the ground being much less than in the *Humber' 
tricycle, the influence of an obstacle in causing swerving is corre- 
spondingly less, though in this respect the * Olympia* is inferior to 
the * Cripper/ The arrangement of this gear should be such that 
the axes of the steering-wheels in any position intersect at a point, 
(9, situated somewhere on the axis of the driving-wheel. This 
cannot possibly be effected by any arrangement of linkwork, but 
the approximation to exactness may be practically all that can be 
desired for road riding. The gear should be arranged so that the 
bell-crank of the outer steering-wheel swings through a less angle 
from its middle position than that of the inner wheel. 

If the axes of the wheels a^ and tzg intersect the axis of the 
driving-wheel at O^ and O^ (fig. 215), the machine as a whole may 
be supposed to turn about a point, (9, somewhere between O^ and 
O^, Let c be the point of contact of wheel a, with the ground 
when the tricycle is moving round centre O^ and let the linear velo- 
city of a point on the frame vertically above c be represented by c dy 
drawn perpendicular to Oc, From c draw ce perpendicular, and 
from ^ draw ^ ^ parallel, to the axis O^Cy these two lines inter- 
secting at <?, the actual velocity ^^ is compounded of a velocity of 
rolling ccoi the wheel on the ground, and a velocity of side-slip, 
e d. The existence of this side-slip in running round curves neces- 
sitates careful arrangement of the steering mechanism, so that the 
centres O^ and O^ may never be widely separate. This side-slip 
must also add appreciably to the effort required to propel the 
* Olympia' tricycle in a curved path, such as a racing track; and for 
such a purpose might possibly appreciably handicap it as com- 
pared with a * Cripper.' 

The steering gear of the * Rudge ' quadricycle is the same as 
that of the ' Olympia ' tricycle. 

185. Eudge Coventry Eotary.— In the * Rudge Coventry 
Rotary' two-track tricycle, with single driving-wheel and two 

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CHAP. XTCll. 



Steering of Cycles 



237 




Fig. 216. 



Steering-wheels (fig. 216), the reaction from the ground in driving 
being at F, there was continually a couple, Fi^^ in action tending 
to turn the machine, and which was resisted by the reactions, /, 
and /a, of the ground on the sides of 
the two side wheels. For equilibrium. 

The steering-wheels were pivoted 
about axes passing through their points 
of contact with the ground and con- 
nected by short levers, connecting-rods, 
and a toothed-rack, to a toothed-wheel 
controlled by the rider. The arrange- 
ment, in this case, should again be such 
that in any position of the steering-gear 
the three axes intersect at a point O ; 
the machine would then turn about O 
as a centre. 

If either of the steering-wheels pass 
over an obstacle, it is evident that 
since the direction of the force acting on the wheel intersects 
the steering-axis there will be no tendency to turn the wheel, 
and therefore no resistance need be offered at the handle by the 
rider. The tendency of an obstacle to turn the machine as a 
whole about the mass-centre, G, is discussed in exactly the same 
way as for the * Cripper ' tricycle. 

186. Otto Bicycle. — In the ' Otto ' dicycle, the steering was 
effected by connecting each of the driving-wheels, by means of a 
smooth pulley and steel band, to the crank-axle. To run round 
a corner, the tension on one of the bands was reduced by the 
motion of the steering-handle, the band slipped on its pulley, and 
the other wheel being driven at a faster rate, the machine de- 
scribed the curve required. In a newer pattern with central gear 
(fig. 172) the motion was transmitted by a chain from the crank- 
axle to the common axle of the two wheels. The wheel-axle was 
divided into two portions, a differential gear being used, as ex- 
plained in section 189. In steering, one of the driving-wheels 
was partially braked by a leather-lined metal strap, thereby making 
it more difficult to run than the other wheel : one wheel was 



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238 Cycles in General chap, xnn. 

thus driven faster than the other, and the machine described a 
curve. 

If an obstacle met one of the wheels, its tendency was to 
retard the machine and to make it turn about its mass-centre. In 
performing this motion of rotation, neither of the wheels slipped 
sideways, and therefore no resistance was offered to the swerving ; 
consequently some other provision had to be made to prevent this 
motion. This was accomplished by locking the gear when 
running straight, so that the two driving-wheels were, for the time 
being, rigidly fixed to the axle, and ran at the same speed. If the 
horizontal force, F^ actually caused the machine to swerve, one or 
other of the wheels actually slid on the ground. The frictional 

resistance to this sliding was '^ - , JF being the weight of the 

2 

machine and rider. If /^was less than this, and the mechanism 

acted properly, the machine moved straight ahead over the 

obstacle. 

187. Single and Double-driving Tricyclee.— A tricycle, in 
which only one of the three wheels is driven, is said to be single- 
driving. The * Rudge ' two-track and the * Olympia ' are familiar 
examples. In single-driving tricycles the two idle wheels are 
supported independently, so that the three wheels have perfect 
freedom to rotate at different speeds. 

If the two driving-wheels of a double-driving tricycle are (as 
is almost invariably the case) of the same diameter, while driving 
in a straight line they rotate at the same speed. They could, 
therefore, be rigidly fixed on the same axle, if only required to 
run straight ; but in running round a curve the outer wheel must 
rotate faster than the inner, unless one or other of the wheels 
skid, as well as roll, on the ground. Some arrangement of me- 
chanism must be used to render possible the driving of the two 
wheels at different speeds. 

188. Clutch Gear for Tricycle Axles.— Besides the *Otto* 
double-driving gear above described, two others, the clutch gear 
and the differential (or balance) gear, have been used to a consi- 
derable extent, though at present the differential gear is the only 
one used. In the * Cheylesmore ' clutch gear (fig. 2 1 7), made by 
the Coventry Machinists Co., Limited, a sprocket wheel, w, in the 

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CHAP. XVIII. 



Steering of Cycles 



^39 




Fig. 217. 



form of a shallow box, was mounted loosely near each end of the 
pedal crank-axle, and was connected by a chain to the corre- 
sponding driving-wheel. A cam, r, was fixed near each end of the 
crank-axle, and between the cam and the inner surface of the 
wheel, w^ four balls, ^, were placed ; the four spaces between the 
cam and the rim of the toothed-wheel being narrower at one end, 
and wider at the other, than the ball. In driving the axle in the 
direction of the arrow, 
the balls, ^, were 
jammed between the 
wheel and the cam, the 
wheel consequently 
turned with the axle. 
If the axle were turned 
in the opposite direc- 
tion, or if the wheel 
tended to move faster 
than the axle in the 
direction of the arrow, 
the balls, ^, were liberated, and the cog-wheel revolved quite in- 
dependently of the axle. While moving in a straight line both 
driving-wheels were driven ; but when running in a curve the inner 
wheel was driven by the clutch, while the outer wheel running 
faster than the inner overran the axle and liberated the balls, the 
outer wheel being thus left quite free to revolve at the required 
speed. 

189. Differential Gear for Tricycle Axle.— Let two co-axial 
shafts, m and n (fig. 218), be geared to a shaft, ^, the axis of which 
intersects that of the shafts, m and ;/, at right angles. The gearing 
may consist of three bevel wheels, at, ^, and r, fixed respectively 
to shafts, w, ^, and n. The three shafts are carried by bearings, 
tn^y ki, and «| respectively. Let the shaft, /^, be rotated in its 
bearings, it will communicate equal but opposite rotations to the 
shafts m and n. If w, be the angular speed of the shaft ;;/, that 
of n will be — wj, and the relative angular speed of the shafts m 
and n will be 2 cui. 

Now, let the shaft, ky carrying with it its bearings, k^,he rotated 
about the axis, m n, with an angular speed, co ; the te^th o£ the 

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240 



Cycles in General 



CHAP. XVIll. 



wheel, by engaging with those of a and r, will cause the shafts, m 
and «, to rotate with the same speed, w, about their common 
axis ; the shaft, ^, being at rest relative to its bearings, k^. If 
driving-wheels be mounted at the ends of the shafts, m and n^ 
they will both be driven with the same angular speed w about the 
axis m n. 

Let now the shaft, k^ be rotated in its bearings, giving a rotation 
(Oi to the shaft w, and a rqtation — wj to the shaft w, while k and 
its bearings are being simultaneously rotated about the axis m n 




Fig. 2i8. 

with the angular speed, co. The resultant speed of the shaft m 
will be (a)+«i), that of the shaft n will be (w — wj). Thus, 
finally, the average angular speed of the shafts m and n is the 
same as that of the bearings, ^,, while the difference of their 
angular speeds is quite independent of the angular speed of ^,. In 
Starley's differential tricycle gear, or balance gear, a chain -wheel is 
formed on the same piece of metal as the bearings, /^,, and is 
driven by a chain from the crank-axle. The driving effort of the 
rider is thus transmitted to the driving-wheels at the end of the 
shafts m and //. The shafts have still perfect freedom to rotate 
relatively to each other, and thus if in steering one wheel tends 
to go faster or slower than the other, there is nothing in the 
mechanism to prevent it. 

In figure 218, the bevel-wheels, a and r, in gear with the wheel b 
are shown of equal size. In Starley's gear (fig. 2 19) a second wheel 

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CHJLF. XVIII. 



Steering of Cycles 



241 



near the other end of the spindle, k^ gears with those on the ends of 
the two half axles, so that the driving effort is transmitted at two 
points to each of these wheels. This forms, perhaps, the neatest 
possible gear, but a great variety could be made if necessary. 
Such a differential gear consists essentially of the chain-wheel, k^y 
carrying a shaft, ^, which gears in any manner with the shafts m 
and n. The particular form of gearing is optional ; provided that 
it allows m and n to rotate relatively to each other. Thus in 
Singer's double-driving gear, the wheel, ^, was a spur pinion, with 




Fig. 219. 

its axis parallel to m «, and engaging with a spur-wheel and 
an annular-wheel fixed respectively to the shafts, m and n. This 
gear had the slight disadvantage that equal efforts could not be 
communicated to the driving-wheels, that connected to the annular- 
wheel of the gear doing most of the work. 

The balance gear being only used differentially for steering, the 
relative motion of the bevel-wheels, a^ b^ c (fig. 218), is very slow, 
and there is not the same absolute necessity for excessive accuracy 
as in toothed-wheel driving gear. 

Example. — A tricycle with 28-in. driving-wheels, tracks 32 in. 
apart, being driven in a circle of 100 feet radius at a speed of 20 
miles an hour, required the speed of the balance-gear. 

While the centre of the machine moves in a circle 1200 inches 
radius, the inner and outerwheels move in circles (1200— 16) and 
( 1 200 -h 1 6) inches radii respectively. The circumferences of these 

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242 Cycles in General chat*, xtoi. 

circles are respectively 27r x 1 200, 27r x 1 1 84, and 27r x 1 2 1 6 inches. 
While the centre of the machine moves over 27r x 1 200 inches, the 
outer wheel moves over 27r x 32 inches more than the inner. The 

relative linear speed is therefore -??^^?- x 20 

27rX 1200 

= '5333 miles per hour 

^ 3333 X 5280 X 12 ^ ^^^.^ .^^j^gg minute. 
60 

The circumference of a 28-in. wheel is 87*96 in. The number 
of revolutions made by the outer part of the axle in excess of those 
made by the inner is therefore 

5^-^ ^ = 6*40 per minute. 
8796 

The number of revolutions of the axle divisions relative to 
the balance box, ^, is therefore 3*20 per minute. 



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243 



CHAPTER XrX 

MOTION OVER UNEVEN SURFACES 

190. Motion over a Stone. — If a cycle be moving along a 
perfectly smooth, flat road, neglecting the slight horizontal side- 
way motion due to steering, the motion of every part of the 
frame of the machine is in a straight line. Suppose a bicycle 
to move over a stone which is so narrow that its top may be 
considered a point. The motion being in the direction of the 
arrow, the path of the centre of the driving-wheel will be a 
straight line OA (fig. 220) parallel to the ground until the tyre 



comes in contact with the obstacle at »S, when the further motion 
of the wheel centre will be in a circular arc, A B, having S as 
centre. The further path of the wheel centre is the straight line, 
B C, parallel to the ground. The path of the centre of the rear 
wheel is of the same nature : a straight line, a, until the tyre 
meets the obstacle 5, the circular arc, a b^ with S as centre, and 
then the straight line b c. 

The motion of any point rigidly connected to the frame of 

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244 



Cycles in General 



CHAP. XIX. 



the bicycle can now be easily found. Let P and Q be the 
centres of the front and rear wheels respectively, and let it be 
required to find the form of the path of the point R lying on the 
saddle and rigidly connected to P and Q, Having drawn on the 
paper the paths of P and Q (fig. 220), take a small piece of 
tracing paper, and on it trace the triangle PQR, Move this 




Fig. 231. 

sheet of tracing paper over the drawing paper so that the points 
P and Q lie respectively on the curves O A B C and oabc. In 
this position prick through the point Py and a point on its path 
will be obtained. By repeating this process a number of points 
on the required path can be obtained sufficiently close together 
to draw a curve through them. Figures 220, 221, and 222 




Fig. 222. 

respectively show the curves described by a point a short distance 
above the saddle of an * Ordinary,' of a * Rear-driving Safety ' with 
wheels 28 in. and 30 in. diameter, and of a * Bantam' with both 
wheels 24 in. diameter, the point being midway between the wheel 
centres. A number of such curves are given and exhaustively 
discussed in R. P. Scott's * Cycling Art, Energy, and Loco- 

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CHAP. XIX. 



Motion over Uneven Surfaces 



245 



motion/ though it should be noticed that the curved portions of 
the saddle paths, due to the front and rear wheels passing over 
the obstruction, are shown placed in wrong positions. 

191. Influence of Size of Wheel — In figure 220 it will be 
noticed that the total heights of the curved portions of the paths 
of the wheel centres above the straight portions are the same, 
whatever be the diameter of the wheel ; but the greater the 
diameter of the rolling wheel, the greater is the horizontal distance 
moved over by the wheel centre in passing over the stone. 
Thus with a large wheel the stone is mounted and passed over 
more gradually, and therefore with less shock, than with a small 
wheel. Therefore, other things being the same, large wheels are 
better than small for riding over loose stones lying on a good 
flat road. 

192. Influence of Saddle Position. — ^The motion of the saddle 
may be conveniently resolved into vertical and horizontal com- 
ponents. In riding along a level 
road the vertical motion is zero 
and the horizontal motion uniform. 
When the front wheel meets an 
obstacle the motion of the frame 
may be expressed as a motion of 
translation equal to that of the 
rear wheel centre, Q^ together with 
a motion of rotation of the frame 
about Q as centre. Let w be the 
angular speed of this rotation at 
any instant The linear motions 
of P and R relative to Q will be 
in directions at right angles ioQP and Q R respectively, and their 
speeds will be w x QPTm^ia x <2^ respectively ; the lines QP 
and QR (fig. 223) may therefore represent the magnitudes of the 
velocities, the directions being at right angles to these lines. 
Through Q draw a horizontal line, and to it draw perpendiculars 
Pp and Rr. Then Qp and ^r will represent the vertical com- 
ponents of the motions of P and Q respectively, Pp and Rr 
the horizontal components. 

In the same way, if the front wheel be moving along the level, 




Fig. 223. 



246 Cycles in General chap. ux. 

and the back wheel be passing over an obstacle, by drawing 
perpendiculars Rr^ and ^^* to a horizontal line through /*, it can 
be shown that Pq^ and Pr^ represent the vertical components of 
the motions of Q and R respectively relative to /*, Qq^ and Rr^ 
the horizontal components. 

Therefore, in a bicycle with equal wheels, the vertical 'jolting' 
communicated to the saddle by one of the wheels passing over 
an obstacle is proportional to the horizontal distance of the 
saddle from the centre of the other wheel, the horizontal 

* pitching ' to the vertical distance from the centre of the other 
wheel. With wheels of different sizes the average angular speeds 
w are inversely proportional to the chords A B and ab (^^, 220) ; 
this ratio must be compounded with that mentioned above. 

If the saddle of a tricycle be vertically over the centre of the 
wheel-base triangle, its vertical motion will be one-third that of 
one of the wheels passing over a stone. In the * Rudge ' quadri- 
cycle the vertical motion would be one-fourth, with similar con- 
ditions as to position of saddle. 

From the above discussion it is readily seen that the most 
comfortable position for the saddle, as regards riding over rough 
roads, is midway between the wheel centres, the vertical motion 
of the saddle being then half that of a wheel going over a stone. 
In a tandem, with one seat outside the wheel centres, the vertical 
jolting of this seat is greater than that of the nearer wheel. 
Again, as regards horizontal pitching, the high bicycle compares 
unfavourably with the low ; the rider on the top seat of the 

* Eiffel ' bicycle would have to hold on hard to avoid being pitched 
clean out of his seat while riding fast over a rough road. A long 
wheel-base is a decided advantage as regards horizontal pitching 
in riding over stones. The angular speed di of the frame in 
mounting over a stone is, other conditions remaining the same, 
inversely proportional to the length of the wheel-base. There- 
fore, the pitching is also inversely proportional to the length of 
the wheel-base. 

A curious point may be noticed in the case of the ' Ordinary.* 
From the saddle path shown (fig. 220) it will be seen that when 
the rear wheel, after surmounting the obstacle, is descending 
again to the level, the saddle actually moves backwards. This 

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CHIP. ZIX. 



Motion over Uneven Surfaces 



247 



can only happen at slow speeds ; at higher speeds the rear wheel 
actually leaves the stone before touching the ground, and the 
backward kink in the saddle path may be eliminated. 

J 93. Motion over Uneven Eoad. — If the surface of the road 
be undulating, but free from loose stones, the paths of the wheel 
centres, P and Q^ will be curves parallel to that of the road surface, 
and the path of any point rigidly 
fixed to the frame can be found by 
the same method. In a very bad 
case, the undulations being very 
close together (fig. 224), it may 
happen that the radius of curvature 
of one of the holes is less than 
the radius of a large bicycle wheel. 
In this case the path, //, of the large wheel will have abrupt angles, 
while that of the smaller wheel, q q^ may be continuous, the large 
wheel being actually worse than the small one. 

194. L088 of Energy. — If the motion of a wheel over an 
obstacle took place very slowly, there would theoretically be no 
loss of energy in passing over it, since the work done in raising 




Fig. 224. 





Fig. 225. 



Fig. 226. 



the weight would be restored as the weight descended ; but at 
appreciable speeds the loss of energy by impact and shock may 
be considerable. Let a wheel moving in the direction of the 
arrow (fig. 225) pass over an obstacle of such a form that the 
wheel rises without sudden jerk or shock to a height ^ the speed 
being so great that at its highest point the wheel is clear both of 



248 Cycles in General chap. xix. 

the obstacle and the ground. If W be the weight (including 
that of the wheel) resting on the axle, the energy lost will be Wh^ 
since the kinetic energy in position b is this amount less than 
that in position a. The energy due to the fall from ^ to ^ is 
wasted in shock, there being no means of obtaining a forward 
effort from the work done during the descent. 

If the wheel strike the obstacle suddenly (fig. 226) and then 
rises to the height ^, clear of the ground and obstacle, the energy 
lost may be greater than Wh^ the amount depending on the 
nature of the surface of the wheel tyre and the obstacle struck. 

If the horizontal speed of the wheel be such that it does not 
leave contact with the obstacle in passing over it, the nature of 
the losses of energy can be shown as follows : 

The centre of the wheel at the instant of coming into contact 
with the stone, S (fig. 227), is moving with velocity «/ in a hori- 

zontal direction. This can be 

/'^^^ ^^N. resolved into a velocity v^ in the 

/ NvK . \ direction c^ S, joining the wheel 

^ ^ ^ I /^ N^^ / .1- centre to the stone, and a velocity 

i VI \^\yT / ^2 at right angles to this direction. 

yjt}T^ V X / The velocity, «/,, is the velocity 

j^t^, ^ ^^ of impact of the wheel on the 

^777777777777r7m779mTrf^77m777? stone S, and the energy due to 

*°' ^^^' this velocity may be entirely lost 

If e be the index of elasticity, the velocity of rebound is e r,, 

and with suitable elastic tyres the energy due to this velocity may 

be saved. The loss of energy due to the impact on the stone 

will be at least (sec. 69) 

(^-^^)?f (0 

and may be as great as 

-^^ (2) 

where m is the weight of the portion of the machine rigidly con- 
nected to the wheel tyre. 

The motion of the wheel continuing, the wheel centre mounts 
over the stone, describing a circle, Cy^ ^2> with centre, ^S", and the 

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CHAP. XIX. Motion over Uneven Surfaces 249 

tyre will again touch the ground on a point in front of the 
stone. If the speed of the machine be uniform, the velocity of 
the wheel centre as the wheel again just touches the ground may 
be equal in magnitude to z/j* the same as immediately after 
impact on the stone. This velocity, v^ (fig. 227), can be resolved 
into horizontal and vertical components, v^ and v^. v^ is the 
velocity of impact on the ground, and the energy due to it is 
either partially or entirely lost, and the final velocity of the wheel 
centre is v^. 

The assumption made above, that the speeds of the wheel 
centre, C, when in positions c^ and c^ are equal, is equivalent to 
assuming that the reactions of the stone on the wheel in any 
position before passing the vertical line through the stone is 
exactly equal to the reaction when at an equal distance past the 
stone ; or, briefly, the reactions as the wheel rolls on and off the 
stone are equal. With a hard unyielding tyre this is not even 
approximately true, except at very low speeds, consequently the 
positive forward effort exerted on the wheel as it rolls oflf the 
stone is less than the backward eflbrt exerted as it rolls on, and 
the speed is seriously diminished. With a tyre that can adapt 
itself instantaneously to the inequalities of the road, the reactions 
during rolling on and off a stone are equal, and there is no loss 
of energy. The pneumatic tyre is the closest approximation to 
such an ideal tyre, while rubber is much better than iron. 

If the road surface be undulating, the undulations being so 
long that the path of the wheel centre is a curve with no sudden 
discontinuities, there may be no loss of energy due to the undu- 
lations. If the undulations, however, be so short, and the speed 
of the machine so great, that the wheel after ascending an undula- 
tion actually leaves contact with the ground, there will be a loss 
of energy due to the impact on reaching the ground. 



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250 Cycles in General 



CHAPTER XX 

RESISTANCE OF CYCLES 

195. Expenditure of Power. — The energy a cyclist generates 
while riding along a level road is expended in overcoming the 
various resistances to motion. These may be classed as follows : 
(i) Friction of bearings and gearing of the machine. (2) Rolling 
resistance of the wheels on the ground. (3) Resistance due to 
loss of energy by vibration. (4) Resistance of the air. The 
power expended in overcoming these resistances is the power 
actually communicated to the machine, and may be called the 
brake power of the rider. The power actually generated in the 
living heat-motor (the rider's body) may be called the indicated 
power ; the difference between the indicated ^nA the brake powers 
will be the power spent in overcoming the frictional resistance of 
the motor — i.e. the friction of the rider's joints, muscles, and 
ligaments. At very high pedal speeds the brake power is small 
compared with the indicated ; in fact, by supporting the bicycle 
conveniently, taking off the chain, and pedalling as fast as he 
can, a rider may possibly develop more indicated power than 
when racing on a track, though the brake power is practically zero. 
The gearing of the bicycle, therefore, must not be made too low, 
or the greater part of the rider's energy will be spent in heating him- 
self The estimation of the work so wasted lies in the domain of the 
physiologist rather than in that of the engineer ; we proceed, there- 
fore, to the consideration of the brake power and its expenditure. 

196. Besistance of Mechanism. — The frictional resistance of 
the bearings is very small compared with the other resistances to 
be overcome ; the resistance due to friction of the bearings of a 
bicycle moving on a smooth track is practically J;he same at all 

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CHAP. XX. Resistance of Cycles 251 

speeds. Professor Rankin estimated this at -joVit P^^^ o^ the 
weight of the rider, but exact experiments are wanting. [ 

The frictional resistance of the chain possibly varies with the 
pull on it, and as, other things being equal, the pull of the chain 
increases with the speed, the resistance will also vary with the 
speed. However, in comparison with the resistance due to roll- 
ing and with the air resistance, that of the chain is small, and 
may be included in the internal resistance of the machine, which 
we may say is approximately constant at all speeds. 

197. Eolling Eesistance. — The resistance to rolling is, accord- 
ing to the experiments of Morin, composed of two terms, one 
constant, the other proportional to the speed. With a pneumatic 
tyre on a smooth road the second term is negligible in comparison 
with the first, according to M. Bourlet. The rolling resistance is 
inversely proportional to the diameter of the wheel. 

In * Traits des Bicycles et Bicyclettes,* C. Bourlet says that 
the rolling resistance with pneumatic tyres is small, independent 
of the speed, and on a dry road it varies from 

•005 ^to '01 W (i) 

while on a racing track the probable value for the resistance is 
•004 W^ W being the total weight of machine and rider. 

The resistance of a solid rubber tyre varies with the speed, 
and may possibly be expressible by a formula of the form 

jR = A -^ Bv, (2) 

A and B being constants. 

The power B required to overcome the rolling resistance 
•005 JVvit the speed v is 

B = '00$ ^t; units (3) 

If IV he expressed in lbs. and v in miles per hour, 

B = '44 Wv foot-lbs. per min. . . (4) 

1^8. L088 of Energy by Vibration.— One of the great ad- 
vantages of a pneumatic tyre is that little or no vibration is com- 
municated to the machine and rider. On a smooth road or track 
with pneumatic tyres the loss due to vibration is probably 
negligible ; but on i^ rough road it may be very large, and is 

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252 Cycles in General ohap. xx. 

possibly proportional to the speed. With solid tyres, a consider- 
able amount of energy is lost in vibration. Bourlet's experiments 
on the road show that the work wasted in vibration is about one- 
sixth of the total. 

The use of a pneumatic tyre enables the tremulous vibration 
to be almost eliminated, no vibration being communicated to any 
part of the machine. For riding over very rough roads the intro- 
duction of springs into the wheel or frame may still further 
diminish vibration. The an ti- vibrators should be placed so that 
they protect as great a portion of the machine from vibration as 
possible. In this respect a spring wheel should be better than a 
spring frame, and a spring frame, in turn, better than a spring 
saddle. The machine, as a whole, should be made sufficiently 
strong and rigid that none of its parts yield under the stresses to 
which they are subjected. Of course, when a spring yields and 
again extends, a certain amount of energy is lost ; it thus becomes 
a question as to when springs are advantageous or otherwise. 
Probably the rougher the road, the more can springs be used with 
advantage in the wheels, frame, and saddle ; whereas, on a smooth 
racing track, their continual motion would simply provide means 
of wasting a rider's energy. 

199. Eesistance of the Air. — M. Bourlet discusses the air 
resistance of a rider and machine, and concludes that it may be 
represented by a formula 

R^kSv' (5) 

R being the air resistance, S the area of the surface exposed, v the 
speed, and k a constant. If the resistance be measured in kilo- 
grammes, the area in square metres, and the speed in metres per 
secopd, k = '06. The area of surface exposed will depend on 
the size of the rider and his attitude on the bicycle. A mean 
value for ^S" is "5 square metre j then 

^ = -03 z;2 (6) 

If the resistance be measured in lbs., and the speed V in miles 
per hour, 

R = .013 F^ ...... (7) 

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CHAP. XX. Resistance of Cycles 

The power required to overcome this resistance is 
1*144 y^ foot-lbs. per minute . 



253 



... (8) 

Table X. gives the air resistances and the corresponding powers 
at different speeds calculated from these formula. 

Table X. — Air Resistance to 'Safety' Bicycle and 
Rider. 



Speed 


Resistance 


Power 


Miles per 


lbs. 


Foot-lbs. per 


hour 




min. 


5 


•32 


143 


6 


•47 


247 


7 


•64 


392 


8 


•83 


586 


9 


105 


834 


10 


1-30 


1,144 


II 


1-57 


1,522 


12 


1-87 


1,977 


13 


2 -20 


2,513 


14 


2-55 


3.139 


15 


2-92 


3.861 


16 


3-33 


4.685 


17 


376 


5,620 




6,672 

7.846 
9,152 

io,6c» 
12,180 
13.920 
15,820 
17,870 
20,100 
22,520 
25,110 
27,900 
30,890 



If the wind be blowing exactly with or against the cyclist, his 
speed relative to the air must be used in the above formula. 
Thus, if the wind be blowing at the rate of 10 miles per hour, and 
the rider be moving at the rate of 20 miles per hour, while going 
against the wind, the air resistance is that due to a speed of 30 
miles per hour, while going with the wind there is still a resist- 
ance due to a speed of 20 — 10 = 10 miles per hour. 

If V be the speed of the cyclist, V that of the wind, while 
riding against the wind the relative speed is {v -f- V), If the 
cyclist rides at a high speed, a very slight breeze against him may 
increase the air resistance considerably. Whilst riding with the 
wind the relative speed is {v — V), In this case, if the speed of 
the wind be greater than that of the cyclist, there will be no 
resistance, but, on the contrary, assistance will be afforded by the 
wind. If the speed of the wind be less than that of the cyclist, 
there will be air resistance due to the speed {v — F). 

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254 Cycles in General chap. xx. 

The power required to overcome air resistance in driving at 
V miles per hour against a wind blowing V miles an hour is 

/'= i'i44 z' (z' + F)* foot-lbs. per minute r . (9) 

that required in going with the wind, 

jP= 1*144 V {^ — Vy foot-lbs. per minute . . (10) 

This equation gives also the power expended in overcoming air 
resistance by a rider behind pace-makers ; the principal beneficial 
effect of pace-makers being to create a current of wind of speed V 
assisting the rider. 

With a side wind blowing, the air resistance is greater than 
that due to the relative speed. In moving through still air, or 
against a head wind, the cyclist drags with him a certain quantity 
of air. A side wind has the effect of changing very rapidly the 
actual particles dragged by the cyclist, so that in a given period of 
time the mass of air which has to be impressed with the rider's 
speed is greater than with a head wind of the same speed. 
Hence an increased resistance is experienced by the rider. 

A consideration of the figures in Table X. will show that" 
bicycle record-breaking depends more on pace-making arrange- 
ments than on any other single factor. For example, to ride unpaced 
at twenty-seven miles an hour requires the expenditure of more 
than two-thirds of a horse-power to overcome only the air resist- 
ance. Though an average speed of 27^ miles per hour was kept 
up by Mr. R. Palmer and by Mr. F. D. Frost in the Bath Road Club 
loo-miles race, 1896, it is most improbable that they worked at any- 
thing like this rate during the whole period, the difference being due 
to the decrease in the air resistance caused by the pace-makers in front. 

200. Total Eesistance. — Summing up, the total resistance of 
the bicycle can be expressed by the formula 

R:=^A^Bv^Cv^ (11) 

and the power required to drive it by 

F^Av-^-Bv^ ->t Cif" (12) 

A^ B^ and C being co-efficients depending on the nature of the 
mechanism and the condition of the road, but which are constant 
for the same machine on the same road at different speeds. 

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CHAP. XX. 



Resistance of Cycles 



255 



Figure 228 shows graphically the variation of the power required 
to propel a cycle as the speed increases. The speeds are set off as 
abscissae. For any speed, O Sy the power required to over- 
come the frictional resistance of the mechanism is set off as an 
ordinate S M -, the power required to overcome rolling resist- 
ance is M T (W^ being taken at 180 lbs.) ; the power required 




S .a 20 
miles per hour 

Fig. 228. 

to overcome air resistance is TR ; and the total power required 
is the ordinate SR, The curve M can be lowered by improve- 
ments in the mechanism, the curve T by improvements in the 
tyres and track-surface, and the curve R by improvements in 
pace-making. 

Experiments on the total resistance of a cycle can be carried 

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256 



Cycles in General 



CHAP. XX. 



out in two ways. Firstly, by towing the machine and rider along 
a level road by means of another machine, the pull on the tow- 
line being read off from a spring-balance. Secondly, by letting 
the machine and rider run down a hill, the gradient of which is 
known, until a uniform speed is attained ; the ratio of the 
resistance at the speed attained to the total weight of machine 
and rider is the sine of the angle of inclination of the road. The 
second method is not convenient for a series of experiments at 
different speeds, since a number of hills of different gradients are 
required ; but since no extra assistance is required, a rider uiay 
use it when unable to use the first method. 

Table XI., taken from * Engineering,' January 10, 1896, giving 
results of experiments by Mr. H. M. Ravenshaw, serves to show 
the variation of the resistance according to the state of the road. 

Table XI. — Resistance of Cvcles on Common Roads. 



Machine 



Tandem Tri- 
cycles, Pneu- { 
matic Tyres \ 



Road 



neu-j 



Tandem 

cycles, Pneu--. 
matic Tyres 

Single Tri- f 
cycles, Solid] 
Tyres . . [ 



I 



Flint 



Asphalte pavement 
>» »» 

Heavy mud . 
Wet mud • • 

Flint . 

»» • • 
Hea\7 mud . 

»» »» • 

Flag pavement 

Flint . 

»» • • 
Flag pavement 
Heavy mud . 



Toul 


Pounds 


weight, 


per 


Lbs. 


ton 


120 


37 


290 


31 


290 


31 


290 


31 


440 


35 


440 


35 


290 


31 


440 


30 


440 


30 


290 


IZ 


290 


65 


200 


33 


370 


30 


2CX) 


9'> 


370 


78 


200 


33 


220 


60 


220 


60 


220 


60 


200 


146 



Miles 
hour 



4 

4 

io*4 

7 

4 

8-3 

4 

4 

6 

4 
12 

5 
5 
5 
5 
5 

4 
8 

5 
4 



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257 



CHAPTER XXI 

GEARING IN GENERAL 

20 1. A Machine is a collection of bodies designed to transmit 
and modify motion *and force. The moving parts of a machine 
are so connected, that a change in the position of one piece in- 
volves, in general, a certain definite change in the position of the 
others. A bicycle or tricycle is a machine in which work done by 
the rider's muscles is utilised in changing the position of the 
machine and rider. Coming to narrower limits, we may say a 
cycle is a machine by which the oscillatory movement of the rider's 
legs is converted into motion of rotation of a wheel or wheels 
rolling along the ground, on which is mounted a frame carrying 
the rider. Still more narrowly, we may consider a cycle as a 
mechanism for converting the motion of the pedals, which may 
be either oscillatory or circular, into motion of rotation of the 
driving wheel. 

202. Higher and Lower Pairs.— Each part of a machine 
must be in contact with at least one other part ; two parts of a 
mechanism in contact and which may have relative motion 
forming a pair. If the two parts have contact over a surface, as 
is necessary when heavy pressures are transmitted, the pair is said 
to be lower. From this definition there can only be three kinds 
of lower pairs — turning pairs, sliding pairs, and screw pairs ; as in 
a shaft and its journal, a cylinder and piston, a bolt and its nut, 
respectively. If the elements of a pair do not have contact over 
a surface, or if one of the elements is not rigid, the pair is said to 
be higher^ the relative motion of the pair being, as a rule, much 
more complex than that of lower pairs. A pair of toothed-wheels 
in contact, a flexible band and drum, a ball and its bearing-case, 
are examples of higher jxiirs. 

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258 Cycles in General chap. xxi. 

Link or Connector. — Two elements of consecutive pairs may 
be connected together by a link. An assemblage of pairs con- 
nected by links constitute a kinematic chain^ or a mechanism^ or a 
gear. The simplest kinematic chain contains four pairs con- 
nected by four links ; it is therefore called a four-link mechanism. 
If one link be fixed, a motion given to a second link will produce 
a determinable motion of the two remaining links. Three pairs 
united by three links constitute a rigid triangle, while a five-link 
chain requires further constraint for movement of a definite 
character to be produced. The four-link kinematic chain is the 
basis of probably 99 per cent, of all linkwork mechanisms. 

203. Classification of Gearing. — Professor Rankine defines 
an elementary combination in mechanism as a pair of primary 
moving pieces so connected that one transmits motion to the 
other ; that whose motion is the cause is called the driver, the 
other the follower. The connection between the driver and 
follower may be : 

(i) By rolling contact of their surfaces, as in toothless wheels- 

(2) By sliding contact of their surfaces, as in toothed-wheels 
and cams, &c. 

(3) By flexible bands, such as belts, cords, and gearing chains. 

(4) By linkwork, such as connecting-rods, &c. 

(5) By reduplication of cords, as in the case of ropes and 
pulleys. 

(6) By an intervening fluid. 

The driving gear of cycles has been made from classes (2), (3), 
and (4), each of which will form the subject of a separate chapter. 
An example of (i) is found in the *Rotherham ' cyclometer, the 
wheel of which is driven by rolling contact from the tyre of the 
front wheel. The pump of a pneumatic tyre is an example 
of (6). We cannot recollect an example in cycle construction 
corresponding to (5), though it would be easy to design one to 
work in connection with a pedal clutch gear, such as the * Merlin.' 

204. Efficiency of a Machine. — If the pairs of a mechanism 
could perform their relative motion without friction, the work 
done by the prime mover at the driving end of the machine 
would be transmitted intact to the driven end ; in other words, 
the work got out of the machine would be equal, to that put into 

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CHAP. XXI. Gearing in General 259 

it. But however skilfully the parts be designed to reduce friction 
to the lowest possible amount, there is always some frictional 
resistance which consumes energy, so that the work got out of 
the machine is less than that put into it, by the amount of work 
spent in overcoming the frictional resistance of the pairs. 

The ratio of the work transmitted by the machine to that 
supplied to it is called the efficiency of the machine. The efficiency 
of a machine will be higher according as the number of its pairs 
is small ; an increase in the number of pairs increases the oppor- 
tunities for work to be wasted away. Thus, in general, the 
simpler the mechanism used, the better will be the results 
obtained. 

It seems perhaps unnecessary to say that no advantage can be 
derived from mere complexity of mechanism, but the number of 
driving gears for cycles that are being patented shows either that 
the perpetual motion inventor has plenty of vitality, or that the 
technical common sense of a large number of cycle purchasers is 
not of a very high standard. 

205. Power. — We have already seen that the work done by 
an agent is the product of the applied force, into the distance 
through which the point of application of the force is moved in 
the direction of the applied force. The power of an agent is equal 
to the rate of doing work — that is, power may be defined as the 
work done per unit of time. If E be the work done in / seconds, 
and P the power of the agent, then 

But E is equal to F s^ where F is the force acting and s the 
distance moved ; therefore 

t 

But - is equal to the speed ; therefore 

P^Fv (i) 

That is, the power of the agent is equal to the product of the 
acting force and the speed of its point of application. The same 

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26o Cycles in General chap, xxi, 

principle is expressed in the maxim, *What is gained in power 
is lost in speed ' ; the word * power * in this maxim having 
the meaning we have associated with * force ' throughout this 
book. 

In a frictionless machine the power is transmitted without loss. 
The above equation shows that any given horse-power may be 
transmitted by any force F^ however small, provided the speed v 
can be made sufficiently great. On the other hand, if the speed of 
transmission be very small, a very large force, Fy may correspond 
to a very small transmission of power. An example of the former 
case occurs in transmitting power to great distances by means of 
wire rope. Here the speed of the rope is made as large as it 
is fcAind practicable to run the pulleys, so that a rope of com- 
parative small diameter may transmit a considerable amount of 
power. An example of the latter case occurs in a hydraulic 
forging press, where the pressure exerted on the ram is, in many 
cases, 10,000 tons ; but the speed of the ram being small— only a 
few inches per minute— the horsepower required to work such a 
press may be comparatively small. 

These principles are of direct application to the gearing of 
cycles. 

Example I.- Suppose two rear-driving bicycles each to have 
28-inch driving-wheels geared to 56 inches ; let the bicycles be 
equal in every lespect, except that in one the numbers of teeth in 
the wheels on the crank -axle and hub are 16 and 8 respectively, 
while in the other the numbers are 18 and 9 respectively. When 
going along the same gradient at \k\^ same speed, the speeds of 
the chain relative to the machine are in the ratio of 8 to 9 ; 
consequently, the pulls on the chain will be in the ratio 9 to 8, 
that on the chain of the bicycle having the smaller wheels being 
the greater. 

Example II. — Let two bicycles be the same in every respect, 
except that in one the cranks are 6 inches long, in the other 
7 inches. When running along the same road at the same speed, 
the work done in overcoming the resistance will be the same in 
the two cases, and, therefore, the work done by the pressure of 
the feet on the pedals is the same in both cases. But the pedals' 
speeds are in the ratio of 6 to 7, therefore the average pressures 

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CHAP. XXI. Gearing in General 261 

to be applied to the pedals are in the ratio 7 to 6, the shorter 
crank requiring the greater pressure. 

Example TIL — Suppose two Safety bicycles to be equal in 
every respect, except that one is geared to 56 inches, the other to 
63 inches. With equal riders, running along the same road at the 
same speed, the work done in both cases will be equal. But the 
distances moved over during one revolution of the crank are in 
the ratio of 56 to 63, that is, 8 to 9. The numbers of revolutions 
required to move over a given distance will therefore be in the 
ratio of the reciprocals of the distance— that is, 9 to 8. Conse- 
quently, the average pressures to be applied to the pedals in the 
two cases will be in the ratio of 8 to 9, the bicycle with the low 
gear requiring the smaller pressure on the pedals 

The whole question of gear for a bicycle thus resolves itself 
into a question of what will suit best the convenience of the rider. 
Assuming that the maximum power of two riders is exactly the 
same, one may be able to develop his maximum power by a com. 
paratively light pressure on the pedals and a high speed of revolu- 
tion of the cranks, the other may develop his maximum power 
with a heavier pressure and a smaller speed of revolution of the 
crank-axle. The former would therefore do his best work on a 
lower geared machine than the latter. The question of length of 
crank depends also on the same general principles, different riders 
being able to develop their maximum powers on different lengths 
of crank. 

The maximum power a rider can develop by pedalling a 
crank-axle is probably at low speeds proportional to the speed of 
driving ; at higher speeds the power does not increase so rapidly 
as the speed, and soon reaches an absolute maximum ; at still 
higher speeds the rapidity of pedalling is too great, and the power 
actually communicated to the crank-axle rapidly falls to zero. 
These variations of the power with the speed are graphically 
represented by the curves Pand P^ (fig. 228), /^, being for longer 
sustained effort than P\ a certain speed of the crank-axle corre- 
sponding to a definite speed of the cycle on the path, so long as 
the gearing remains unaltered. The height of the ordinates will 
depend on the duration of the ride, and the maximum power a b 
for an effort of short duration may be developed at> a less axle 

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262 Cycles in General chap. xxi. 

speed than the maximum a^b^ for a longer effort. By increasing 
the amount of gearing-up, the abscissae of the curve would be all 
proportionately increased, while the ordinates remain as before. 
The best gearing-up possible for the rider will be such that the 
power curve of the machine intersects the rider's power curve at 
the highest point of the latter. From ^, the highest point of the 
rider's power curve with a certain gearing-up, draw b b^ to intersect 
at b^ the power curve R of the machine, then the rider will 
develop the greatest speed c b^ on the machine if the gearing-up 
be increased in the ratio of ^ ^ to cb^. If, as seems to the author 

most probable, the ratio for the shorter effort is greater than 

c b^ 
the ratio -L— ' for the longer effort, the gearing-up should be 
C\bx 

greater for the former than for the latter. That is, to attain in 

all races his highest possible speed, the shorter the distance the 

higher should be the gear used by the rider. 

Very little is known as to the maximum power that can be 
developed by a cyclist, no accurate experiments, to the author's 
knowledge, having been made. Rankine gives 4,350 foot-lbs. 
per minute as the average power of a man working eight hours 
raising his own weight up a staircase or ladder, and 1 7,200 foot-lbs. 
per minute in turning a winch for two minutes. Possibly racing 
cyclists of the front rank develop for short periods two-thirds of 
a horse-power — i,e, 22,000 foot-lbs. per minute. If this estimate 
and that of the air resistance (sec. 199) be correct, from figure 228 
it is evident that a speed of 28 miles per hour could not be 
attained on a single bicycle, in still air, without pace-makers, even 
though the mechanism and the tyres were theoretically perfect 
It should be noted that the conventional horse power, 33,000 foot- 
lbs, per minute, introduced by Watt, and employed by engineers 
as the unit of power, is considerably in excess of the average 
power of a draught horse. 

206. Variable-speed Oear. — The maximum power of any rider 
is exerted at a particular speed of pedal and with a particular 
length of crank. The best results on all kinds and conditions of 
roads would probably be attained if the pedal could always be 
kept moving at this particular speed whatever the- resistance : the 

Digitized by Vj 



CHAP. XXI. Gearing in General 263 

gearing would then have to vary the distance travelled over per 
stroke of pedal, until equilibrium between the effort and resistance 
was established. An ideal variable gear would be one which 
could be altered continuously and automatically, so that when 
going uphill a low gear was in operation, and when going down- 
hill a high gear. A number of two-speed gears have been used 
with success, and are described in chapter xxvii., but no con- 
tinuously varying gear has been used for a cycle driving gear, 
though such a combination is well known in other branches of 
applied mechanics. 

207. Perpetual Motion. — Many inventors and schemers do 
not appreciate the importance of the principle of * what is gained 
ia force, or effort, is lost in speed.' Since for a given power 
the effort or force can be increased indefinitely by suitable 
gearing, and likewise the speed, they appear to reason that by 
a suitably devised mechanism it may be possible to increase 
both together, and thus get more power from the machine 
than is put into it. A crank of variable length, the leverage 
being greater on the down than on the up-stroke, is a favourite 
device. The Simpson lever-chain is another device having the 
same object in view. The angular speeds of the crank-axle and 
back hub are inversely proportional to their numbers of teeth ; 
with an ordinary chain the distances of the lines of action from 
the centres are directly proportional to these numbers. By driving 
the back hub chain- wheel from pins on the chain links at a greater 
distance from the wheel centre, it was claimed that an increased 
leverage was obtained, and that the lever-chain was therefore 
greatly superior to the ordinary. It is possible, by using an 
algebraic fallacy which may easily escape the notice of anyone 
not sufficiently skilled in mathematics, to prove that 2 x 2 = 5 ; 
but though the human understanding may be deceived by the 
mechanical and algebraic paradoxes, in neither case are the laws 
of Nature altered or suspended. ^Vhen once the doctrine of the 
'conservation of energy' is thoroughly appreciated, plausible 
mechanical devices for creating energy will receive no more atten- 
tion than they deserve. 

208. Downward Pressure. — In all pedomotive cycles the 
general direction of the pressure exerted by the rider on the pedals 

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264 Cycles in General r.-HAr. xxi. 

is vertically downwards. If P be the average vertical pressure 
and d the vertical distance between the highest and lowest points 
of the pedal's path, the work done by the rider per stroke of pedal 
is Pd. This is quite independent of the form of the pedal path. 
209. Cranks and Levers.— If the pedals are fixed to the 
ends of cranks revolving uniformly, the vertical component of 
the pedal's motion will be a simple harmonic motion, and, 
neglecting ankle action, the motion of the rider's knee will be 
approximately simple harmonic motion along a circular arc. 

When the crank is vertical, its direction coincides with that of 

the vertical pressure, and consequently no pressure, however great, 

will tend to drive the crank in either direction. The crank is 

then said to be on a 'dead-centre.' In steam-engines, and 

mechanisms in which the crank is employed to convert oscillating 

into circular motion, a fly-wheel is used to carry the crank over 

, the dead-centre. In cycles, when speed has been got up, the 

' whole mass of the machine and rider tends to continue the 

I motion, and thus acts as a fly-wheel carrying the crank over the 

' dead-centre, so that in riding at moderate or high speeds the 

I existence of the dead-centre is hardly suspected. In riding at 

'' a very slow speed, however, the existence of the dead-centre is 

more manifest. If two cranks are placed at right angles to each 

other on the same shaft, while one is on the dead-centre the other 

1 is in the best position for exerting the downward effort, and 

; there is no tendency of the shaft to stop. 

' In the above discussion we have assumed that the connecting- 

rod which drives the crank can only transmit a simple thrust or 
pull ; if, in addition to this, the connecting-rod can transmit a 
transverse effort there may be no dead-centre. In turning the 
handle of a winch by hand, the arm acts as a connecting-rod 
which can transmit, thrust, pull, and transverse effort, so that no 
dead-centre exists. In Fleming & Ferguson's marine-engine 
two cylinders are connected by piston-rods and intermediate links 
to two corners of a triangular connecting-rod, the third comer of 
which is at the crank ; with this arrangement there is no dead- 
centre, the single crank and triangular connecting-rod being in 
this respect equivalent to two cranks at right angles. 

The existence of the dead-centre is supposed^ by some to be 

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CHAP. XXI. Gearing in General 265 

a disadvantage inherent to the crank, but the efficiency of the 
mechanism is not in any way directly affected by it. 

210. Variable Leverage Cranks.— One favourite notion of 
those inventors who have no clear and exact ideas of mechanical 
principles, is to have a crank of variable length arranged so 
that the leverage may be great during the down-stroke of the 
pedal and small during the up-stroke ; their idea evidently being 
to obtain all the mechanical advantages of a long crank, and yet 
only make the foot travel through a distance corresponding to a 
short crank. We have shown above that, presuming the pressure 
is vertical, the work done per stroke of pedal depends only on the 
pressure applied, and the vertical distance between the highest and 
lowest points of the pedal path ; the distance of the pedal from the 
centre of the crank-spindle having no direct influence whatever. 
The pedal path in most of the variable crank gears that have ap- 
peared from time to time is simply an epicycloidal curve which doeS 
not differ very much in shape from a circle, but which is placed 
nearer the front of the machine than an equal circle concentric 
with the crank-axle. Thus, the gear only accomplishes in a 
clumsy manner what could be done by a simple crank, having its 
axle placed a little further forward than that of the variable crank. 

Let O (fig. 229) be the centre of a variable crank, and r//the 
pedal path during the upstroke. Let the length of the crank 
become greater, the path of the pedal during 
this extension being da^ and let the arc a b 
be the pedal path during the down-stroke. 
The crank will then shorten, be being the 
pedal path. If the pressure be vertically 
downward, work will be done only while the 
pedal moves from a to ^, and the angle of 
driving will be the small angle aob. Thus 
while with a variable crank a greater turning effort may be exerted 
than with a fixed crank, the arc of action is correspondingly less. 
211. Speed of Knee-joint during Pedalling.— Regarding that 
part of the leg between the knee and the foot as a connecting-rod, 
that between the knee and the hip-joint as a lever vibrating about 
a fixed centre, the speed of the knee corresponding to a uniform 
speed of the pedal can easily be determined by the^methad of 




266 Cycles in General chap. xxi. 

^ section ^tZ' Figure 23 is a polar curve showing the varying 
speed of the knee for different positions of the crank. From this 
curve it will be seen that on the down-stroke the maximum speed 
is attained when the crank is nearly horizontal, but on the up-stroke 
the maximum speed is not attained till the crank is nearly 45° 
above the horizontal. The speed then rapidly diminishes, and is 
nearly zero when the crank is vertical. The shorter the crank, in 
comparison with the rider's leg, the more closely does the motion 
of the knee approximate to simple harmonic motion ; with simple 
harmonic motion the polar curve is two circles. 

In any gear in which a crank connected to the driving-wheel 
is used, the speed of the knee-joint will vary approximately as 
above described — i.e, it will gradually come to rest as it ap- 
proaches its highest and lowest positions, then gradually increase 
in speed until a maximum is attained. 

212. Pedal-olutoh Meohanism.— Instead of cranks, clutch 
gears have been used for the driving mechanism. In these a cylin- 
drical drum is placed at each side of the axle and runs freely on it 
A long strap, with one end firmly fixed to the drum, is coiled 
once or twice round it, the other end is fastened to the pedal 
lever. When the pedal is depressed, the drum is automatically 
clutched rigidly to the shaft ; when the pressure is removed from 
the pedal, the pedal lever is raised by a spring and the drum 
released from the axle. One of the most successful clutch gears 
was that used on the * Merlin' bicycles (fig. 176) and tricycles 
made by the Brixton Cycle Company. 

The general advantage which a clutch gear was supposed to 
have as compared with a crank was that any length of stroke 
could be taken from a pat of an inch up to the full throw of the 
gear. However, even supposing that the clutches which lock 
the drums to the axle and the springs which lift the pedal levers 
! are perfect in action, the gear has the serious defect that the down- 
I stroke of the pedal begins quite suddenly and is performed at a 
constant speed ; thus the legs must have a considerable speed 
imparted suddenly to them. At moderate and high speeds this 
is a decided disadvantage as against the gradual motion required 
for the crank -geared cycle. There is the further serious practical 
disadvantage that no clutch that has been hitherto designed is 



CHAP. XXI. Gearing in General 267 

perfectly instantaneous in its action of engaging and disengaging. 
When a clutch is used for continual driving, as in the clutch 
driving gears of some of the early tricycles, and where no great 
importance need be attached to the delay of a second or two in 
the action of the clutch gear, the case is quite different. Mr. 
Scott, in * Cycling Art, Energy, and Locomotion,' has put the 
comparison between the crank gear and clutch gear for pedals in 
a nutshell thus : "In the crank- clutch cycle, as in other uses, 
the immediate solid grip is a matter of very little concern ; if a 
half turn of the parts takes place before clutching, it does very 
little harm, since it is so small a fraction of the entire number of 
revolutions to be made before the grip is released. But if a grip 
is to be taken at every down-stroke of the foot, as in a lever-clutch 
cycle, the least slip or lost motion is fatal." 

These two objections are so weighty, that in spite of the 
immense advantage of providing a simple variable gear, pedal- 
clutch gears have never been much used. 

213. Diagrams of Crank Effort.— Though the pressure on 
the pedal may be constant during the down-stroke, the effort 
tending to turn the crank will vary with the 
varying crank position. The actual pressure on 
the pedal may be resolved into two components, 
parallel and at right angles to the crank ; the 
former, the radial component, merely causes pres- 
sure on the bearing, and, since no motion takes 
place in its direction, no work is done by it ; the 
latter, the tangential component, constitutes the 
active effort tending to turn the crank. HOC 
(fig. 230) be the crank in any position, and P the 
total pressure on the pedal, the radial and tan- p^^ 
gential components, R and 7] are equal to the 
projections of P respectively parallel to, and at right angles to 
the crank O C, If the tangential component T be set off along 
the corresponding crank direction, a polar curve of crank effort 
will be obtained. 

If the pressure, P, be constant during the down-stroke, and 
be directed vertically downwards, the polar curve of crank effort 
will be a circle. Let / be the effort exerted by the^ider at any 




268 Cycles in General chap. xxi. 

instant at his knee-joint in the direction of the motion of the 
latter, let / be the corresponding tangential effort on the pedal, 
let J be a very small space moved through by the pedal, and s^ 
the corresponding space moved through by the knee-joint 
Then the work done at the knee-joint is/j^, the corresponding 
work done at the pedal / s ; these two must be equal, presuming 
there is no appreciable loss in the transmission. Therefore 

'=7^ (^) 

But - is the ratio of the speeds of the knee-joint and pedal 

respectively, and is represented by the intercept Z>/ (fig. 21). 
If, therefore, the effort at the knee-joint be constant during the 
down-stroke of the pedal, figure 23 is the curve of crank effort as 
well as the speed curve of the knee. 

If, starting from any position, the distance moved through 
by the pedal relative to the machine be set off along a horizontal 

line, and the corre- 
sponding tangential 
effort on the crank 
be erected as an or- 
dinate, a rectangular 
Fig. 231. curve of crank effort 

will be obtained. 
Corresponding to the circle as the polar curve of crank effort, 
the rectangular curve will be a curve of sines. Figure 231 shows 
the rectangular curve corresponding to the down- stroke polar 
curve in figure 23. 

The area included between the base line and the rectangular 
curve of crank effort represents the amount of work done. The 
mean height of the rectangular curve therefore represents the 
mean tangential effort to be applied at the end of the crank in 
order to overcome the resistance of the cycle. 

214. Actuar Pressure on Pedals.— The actual pressure on 
the pedal during the motion of the cycle is not even approxi- 
mately constant. Mr. R. P. Scott investigated the actual 
pressure on the pedal by means of an instrument which he calls 
the * Cyclograph,' the description of which we take from * Cycling 




CHAP. XXI. 



Gearing in • General 



269 




Art, Energy, and Locomotion.' " A frame, A A (fig. 232), is pro- 
vided with means to attach it to the pedal of any machine. A 
table, j9, supported by springs, E E^ has a vertical movement 
through the frame A A, and car- _^«, — <^ j. A — --^ 

ries a marker, C. The frame carries 
a drum, D, containing within me- 
chanism which causes it to revolve 
regularly upon its axis. The cylin- 
drical surface of this drum D is 
wrapped with a slip of registering 
paper removable at will. When 
we wish to take the total foot 
pressure, the cyclograph is placed 
upon the pedal and the foot upon 
the table. The drum having been wound and supplied with the 
registering slip, and the marker C with a pencil bearing against 
the slip, we are ready to throw the trigger and start the drum, by 
means of a string attached to the trigger, which is held by the 
rider so that he can start the apparatus at just such time as he 
desires a record of the pressure." 

Figure 233 shows a cyclograph from a 52-inch * Ordinary ' on a 
race track, speed 18 miles per hour ; figure 234 that from the same 

AAAA/wfrVlAAJ^A/ 



Fig. 232. 



Fig. 233. 

machine ascending a gradient i in 10, speed 4 miles an hour ; 
and figure 235 is from the same machine back-pedalling down a 




Fig. 234. 



gradient i in 12. Figure 236 is from a rear-driver geared to 
54 inches up a gradient i in 20 at a speed of 9 miles an hour ; and 

y Google 



digitized by V 



270 



Cycles in General 



CHAP. XXI. 



figure 237 is from the same machine going up a gradient of i in 
7 at a speed of 10 miles per hour. The figures on the diagrams 



Fig. 235. 

are lbs. pressure on the pedal. These curves and many others 
are discussed in the work above referred to. 

These curves give no notion as to the varying tangential effort 



20c 

ssro 

960 , 




Fig. 236. 

on the crank, which is, of course, of more importance than the 
total pressure. Mallard & Bardon's dynamometric pedal, referred 




Fig. 237. 

to by C. Bourlet, is an instrument in which the tangential com- 
ponent of the pedal pressure is measured and recorded. 

215. Pedalling. — A vertical push during the down-stroke of 
the pedal is the most intense effort that the cyclist can com- 
municate, and unfortunately it is the only one that many cyclists 
are capable of exerting. From Scott's cyclograph diagrams it 
will be seen that in only one case is the pedal pressure zero 
during the up-stroke. The first improvement, therefore, that 
should be made in pedalling is to lift the foot during the up- 
stroke, though not actually allowing it to get out of contact with 
the pedal. Toe-clips will be of advantage in acquiring this. 

Next, just before the crank reaches its upper dead-centre a 
horizontal push should be exerted on the pedal, and before it 
reaches the lower dead-centre the pedal should be clawed back- 
wards. These motions, if performed satisfactorily, will consider- 
ably extend the arc of driving. 

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CHAP. XXI. Gearing in General 271 

Ankle Action. — To perform these motions satisfactorily the 
ankle moist be bent inwards when the pedal is near the top, and 
fully extended when near the bottom. Figures 238, 239, and 
240, from a booklet describing the * Sunbeam ' cycles issued by 
Mr. John Marston, show the positions of the ankle when the 
crank is at the top, the middle of the down-stroke, and the 
bottom respectively. The method of acquiring a good ankle 
action is well described in the * Sunbeam' booklet and in 
Macredy's * The Art and Pastime of Cycling.' Besides increasing 




Fig. 238. Fig. 239. Fig. 240. 

the arc of driving, ankle action has the further advantage of 
diminishing the extent of the motion of the leg. With a good 
ankle action the speed curves shown in figures 23, 501, and 
511 may be considerably modified ; in fact, the addition of a fifth 
link (between the foot and ankle) to the kinematic chain in 
figure 22 makes the motion of the leg indeterminate. 

If the shoe of the rider be fastened to the pedal an upward 
pull may be exerted, and the action of pedalling becomes more 
like that of turning a crank by hand, the arc of action being 
extended to the complete revolution. With pulling pedals more 
work is thrown on the flexor muscles of the legs, to the corre- 
sponding relief of the extensors. 

216. Manomotive Cycles. — A few cycles, principally tricycles, 
have been designed to be driven by the action of the hand and 
arms. 

Singers' ' Velociman ' has been for a number of years the 
best example of this type of machine. Figure 241 shows an up- 
to-date example. The effort is applied by the hands to two long 



272 



Cycles in General 



CHAP. XXI. 



levers, which, by sliding joints in place of connecting-rods, drive 
cranks at opposite ends of an axle ; this axle is connected by chain 




gearing to the balance gear on the driving-axle. The steering is 
done by the back pressing against a cushion supported at the end 
of a long steering bar. 

2x7. Anxiliary Hand-power Mechanisms.— A number of 
cycles have been made from time to time with gearing operated 
by hand, having the intention of supplementing the effort com- 
municated by the pedals. The idea of the inventors is that the 
greater the number of muscles concerned in the propulsion, the 
greater will be the speed, or a given speed will be obtained with 
less fatigue \ but though this may be true for extraordinary efforts 
of short duration, it is probably quite erroneous for long-con- 
tinued efforts. Whatever set of muscles be employed to do work, 
a man has only one heart and one pair of lungs to perform the 
functions required of them. It is a matter of ever>'day experience 
that the cyclist can tax his heart and lungs to their utmost, 
using only pedals and cranks ; so that, unless inventors can pro- 
vide a method of stimulating these organs to do more than they 
are at present capable of, it seems worse than useless to compli- 
cate the machine with auxiliary hand-power mechanism. Re- 

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CHAP. XXI. Gearing in General 273 

garded as a motor, the human body may be compared to a 
number of engines deriving steam from one boiler, supplied 
with feed-water by one feed-pump. If one engine is capable 
of using all the steam generated in the boiler, no additional, but 
rather less, useful work will be obtained by setting additional 
engines running. It is a fact well known to engineers that 
a steam-engine works most economically when running under 
its heaviest load. One engine, therefore, will utilise the steam 
generated in the boiler more efficiently than several. The lungs 
may be compared to the furnace of the boiler, the blood to the 
feed-water, the heart to the feed-pump which circulates the feed- 
water, the muscles of the legs to an engine capable of utilising all 
the energy supplied by the combustion of the fuel in the furnace, 
the arms to a small engine. If the analogy can be pushed so far, 
less work will be got from the body by using both legs and arms 
simultaneously than by using the legs only ; and this quite inde- 
pendently of the frictional resistance of the additional mechanism. 
The * Road-sculler * and * Oarsman ' tricycles were designed so 
that the rider might exercise the muscles of his legs, back, chest, 
and arms, as in rowing. The speed attained was less than in the 
crank-driven tricycle, the mechanism being more complex and 
therefore less efficient, while from the foregoing discussion it 
seems probable that the rider, though using more muscles, 
actually developed less indicated power. 



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PART III 
DETAILS 

CHAPTER XXII 

THE FRAME (DESCRIPTIVE) 

2 1 8. Frames in (General. — The frame of a bicycle forms 
practically a beam which carries a load — the weight of the rider — 
and is supported at two points, the wheel centres. In order to 
allow of steering, this beam is divided into two parts connected 
by a hinge joint — the steering-head. The two parts are some- 
times referred to as the * front-frame ' and the * rear-frame ' ; the 
front-frame of a * Safety * including the front fork, head-tube, and 
the handle-bar. The rear-frame has assumed many forms, 
which will be discussed in some detail. In all bicycles that have 
attained to any degree of success the rear-frame has been the larger 
of the two ; hence sometimes when * the frame ' is mentioned 
without any further qualification, the rear-frame is meant. It is 
usually evident from the context whether * the frame * means the 
rear-fhime or the complete frame. 

219. Frames of Front-drivers.— The 'Ordinary' has the 
simplest, structurally, of all cycle frames, consisting of a single 
tube, called the backbone, forked at its lower extremity for the 
reception of the hind wheel, and hinged to the top of the 
fork carrying the front wheel. The frame of the 'Geared 
Ordinary ' is the same as that of the * Ordinary,* the dis- 
tance between the seat and the top of the driving-wheel 
being too small to admit of bracing the structure. With the 
further reduction of the size of the driving-wheel, and the greater 
distance obtained between the saddle and top of the driving- 
wheel, it becomes possible to use a braced frame. Figure 1242 

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276 



Details 



CHAP. XXII. 



shows a front-driving frame made by the Abingdon Works 

Company (Limited). Here the weight of the rider is taken up 

by the two straight tubes, 
each of which will be sub- 
ject to bending-moment due 
to half the total weight. 

Figure 132 shows one 
form of frame used by the 
Crypto Works Company 
(Limited), in their * Bantam.' 
The bracing in this is more 
apparent than real, since the 
weight of the rider is trans- 
ferred to the middle of a 
straight tube of very little less 
^"'- '*'• length than the total distance 

between the wheel centres. This tube must, therefore, be made 

strong enough to resist the bending-moment. 





Fig. 243. 



Figure 243 shows the frame of the ' Bantamette,' made by the I 
same company, and which can be ridden by a ladj with skirts, j 



CHAP. XXXI. 



The Frame 



277 



Here, of course, the backbone is subjected to bending- stresses, 
and a very strong tube must be used for it. Figure 291 shows a 
properly braced front-driving frame designed by the author, which 
is practically equivalent to a triangular truss. The short tube join- 
ing the steering-head to the seat-lug is made stout enough to resist 
the bending due to the saddle-pin attachment, while the seat-struts 
are subjected only to compression, and the lower stays to tension. 

220. Frames of fiear-drivers. — ^The rear-driving chain-driven 
* Safety ' introduced in 1885 is kinematically the same as the popu- 
lar machine of the present day. The greatest difference between 
them lies in the design of the frame. So many designs of frame 
have been used that we can only notice a few general types here. 

The original 'Humber' frame (fig. 128) has a general 
resemblance to the present-day diamond-frame, though from a 
structural point of view, the want of a tube joining the saddle-pillar 
to the crank-axle makes it greatly different as regards strength. 

Figure 244 shows the * Pioneer* dwarf Safety, made by H. J. 
Pausey, 1885. This is of the cross-frame type, and consists 
practically of two 
members, one join- 
ing the driving- 
wheel spindle to 
the steering-head, 
the other running 
from the saddle to 
the crank-axle. It 
will be noticed that 
the frame is not 
braced or stayed in 
any manner, so that 
the whole weight 
of the rider is trans- 
ferred to the back- 
bone. When driving, the pull of the chain tends to bring the crank- 
axle and driving-wheel centres nearer together, and there being no 
direct struts to resist this action, the frame is structurally weak. In 
this respect it is much worse than the *Humber' frame (fig. 128). 

Figure 126 shows the * Rover' Safety made by Messrs. Starley 
& Sutton, 1885. The frame is of the open diamond type, ihe 




Fig. 244. 



278 



Details 



CHAP. xxn. 



front fork is vertical, and the steering is not direct, but the handle- 
bar is mounted on a secondary spindle connected by short links 
to the front fork. 

Figure 245 shows a Safety made by the Birmingham Small 
Arms and Metal Co., Limited. The principal difference between 




Fig. 245. 



this frame and that of figure 244 consists in the substitution of 
indirect for direct steering. 

Figure 127 shows the * Rover ' Safety, made by Messrs. Starley 




\^ 



Fig. 246. 



& Sutton in 1886. The frame is of the open diamond type, with 
curved tubes, and direct steering is used. The approximation 

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CHAP. XXII. 



The Frame 



279 



to the present type of frame is closer than in any of the previous 
examples. 

Figure 246 shows the ' Swift ' Safety, made by the Coventry 




Fig. 247. 



Machinists Co., 1887. This frame is of the open diamond type ; 
the top and bottom tubes from the steering head are curved. 

The first improvement on the elementary cross-frame (fig. 244) 
was to insert struts, or a lower fork, between the crank-bracket 




Fig. 248. 

and driving-wheel spindle (fig. 247), so that the pull of the chain 
could be properly resisted. Another improvement was to connect 
the steering-head and the top of the saddle-post by a light stay 
(fig. 248). In the * Ivel ' Safety of 1887 (fig. 249) a stay ran from 
the steering-head to the crank-bracket, but the chain-struts were 

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28o Details chap. xxu. 

omitted. The * Humber ' Safety of this period (fig. 250) had the 
crank-bracket stay and chain-struts. The * Invincible* Safety, 
made by the Surrey Machinists Co. in 1888 (fig. 251), had, in 




Fig. 349. 



addition, a stay between the steering-head and top of saddle-post ; 
while a later machine (fig. 252), by the same firm, had stay-rods 
from the driving-wheel spindle to the top of saddle-post. This 




Kk:. 250. 

bicycle was made forkless, the wheel-spindles being supported 
only at one end ; but in this respect the design is not to be 
recommended. 

The frame of the * Sparkbrook' Safety, 1887 (fig. 253), may be 

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CHAP. ZXII. 



The Frame 



281 



noticed. It is a kind of compromise between the cross-frame and 
the open diamond ; the crank-bracket and driving-wheel spindle 




Fig. 251. 



are directly connected, but the crank-axle is connected to a point 
about the middle of the upper tube of the frame. The bending- 




FiG. 252. 



moment, which attains nearly its maximum value at this point, 
is resisted by this single tube, which consequently must be rather 
heavy. 

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282 



Details 



CHAP. xzn. 



The frame of the * Quadrant ' bicycle (fig. 254) differs essen- 
tially from either the diamond- or cross-frame. In this bicycle 
the main frame is continued forward on each side of the steering- 




FiG. 253. 



wheel ; the spindle of the steering-wheel is not held in a fork, but 
its ends are mounted on cases which roll on curved guides or 
• quadrants.* From each case a light coupling-rod gives connec- 




FlG. 254. 



tion to a double lever at the bottom of the steering-pillar. The 
frame in front of this steering-pillar consists practically of two 
tubes with no bracing, while the bracing of the rear portion is 

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CHAP. xxn. 



The Frame 



283 



very imperfect. This arrangement for controlling the motion of 
the steering-wheel . is the same as used in the * Quadrant ' 
tricycle. 

The frame of the * Rover* Safety of 1888 (fig. 255) shows a 




Fig. 255. 

great advance on any of the earlier frames above described. It 
may be described as a combination of the cross- and diamond- 




FlG. 256. 

frames. The main tube trom the steering-head is joined on about 
the middle of the down-tube from the saddle to the crank-bracket, 
which thus may be considered to be supported at its ends and 
loaded in the middle, and must therefore be fairly he^vy to resist 

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284 



Details 



CHAP. XXII. 



the bending-moment on it. Another weak point in the design is 
the making of the top tube curved instead of straight. 

The * Referee ' frame (fig. 256) was one of the earliest with 




Fig. 257. 



practically perfect bracing. The crank-bracket being kept as near 
as possible to the rim of the driving-wheel, the diamond was 
stiffened by a curved down-tube. A short vertical saddle-tube was 




Fic;. 258. 



continued above the top tube, thus allowing the saddle and pin to 
be turned forward or backward— a good point which has been 
abandoned in later frames. Ball-socket steering was used. 

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CHAP. XXII. 



The Frame 



285 



Figure 257 show? the Safety made by Singer & Co., 1888, 
the frame of which differs very little essentially from that of figure 
255- 




Fig. 259. 

Figure 258 shows the 'Singer' Safety of 1889, the frame of 
which differs considerably from all types hitherto described. The 




Fig. 260. 



remarks applied to the design of the frame in figure 255 may also 
be applied to this frame. 

The * Ormonde ' (fig. 259) and the * Mohawk ' (fig. 260) frames 
may be noticed, the latter having the down-tube fropi saddle to 
crank-bracket in duplicate. Digitized by Goog 



286 



Details 



CHAP. xxn. 



Figure 130 shows the *Humber' Safety of 1889. This frame 
gives the first close approximation to the present almost univer- 
sally used * Humber * frame. 

In 1890 the * Humber' Safety, with extended wheel-base, was 
introduced. In this machine the distance between the crank - 
axle and the driving-wheel was increased, thereby increasing the 
distance between the points of contact of the two wheels with the 
ground. With this increased distance it was possible to join the 
seat-lug to the crank-bracket by a straight down-tube, thereby 
giving the well-known * Humber* frame (fig. 131). The stem of 
the saddle-pin goes inside this tube, and a neater appearance is 
obtained thereby. This frame is not a perfectly braced structure, 

the introduction of a 
tube to form one of the 
diagonals of the dia- 
mond being necessary 
to convert it into a per- 
fectly framed structure. 
This has been done 
in the * Girder' Safety 
frame (fig. 296). With a 
well designed * Humber ' 
frame, however, the pos- 
sible bending-moment 
on the tubes, due to the omission of the diagonal, is so 
small that it is practically not worth while to introduce the extra 
tube. 

Quite recently a * pyramid '-frame (fig. 261) has been introduced 
in America. It remains to be seen whether the excessive rake 
of the steering head, necessary with this design, will allow of the 
easy steering we are accustomed to wixh the diamond-frame. 

Bamboo Frames. — From the discussion of the stresses on the 
frame (chap, xxiii.) it will be seen that when the frame is properly 
braced, and its members so arranged that the stresses on them 
are along their axes, the maximum tensile or compressive stress 
on the material is small. If a steel tube were made as light as 
possible, with merely sufficient sectional area to resist these 
principal stresses, it would be so thin that it would be 

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Fig. 261 



CHAP. XXII. 



The Frame 



287 



unable to resist rough handling, and would speedily become 
indented locally. A lighter material with greater thickness, 
though of less strength, would resist these local forces better. 
The bamboo frame (fig. 262) is an effort in this direction, the 




Fig. 262. 

bamboo tubes being stronger locally than steel tubes of equal 
weight and external diameter. 

Aluminium Frames, — From the extreme lightness of aluminium 
compared with iron or steel, many attempts have been made to 
employ it in cycle construction. The tenacity of the pure metal 
is, however, very low, and its ductility still lower, compared with 
steel ; while no alloy containing a large percentage of aluminium, 
and therefore very light, has been found to combine the strength 
and ductility necessary for it to compete favourably with steel. 
Of course, for parts which are not subjected to severe stresses it 
may probably be used with advantage. 

221. Frames of Ladies* Safeties.— The design of the frame of 
a Ladies' Safety is more difficult than the design of the frame 
for a man's Safety. In the early Ladies' Safeties the frame was 
usually of the single tube type, and may be represented by the 
* Rover' Ladies' Safety (fig. 263). The single tube from the crank- 
bracket to the steering-head is subjected to the entire bending- 
moment, and must therefore be of fairly large section. If the 
lady rider wears skirts, the top tube, as used in a man's bicycle, 
must be omitted ; and if a second bracing tube be introduced, it 
must be very low down. Figure 264 shows one of the usual 

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288 



Details 



CHAP. XXII. 



forms of Ladies' Safety, a tube being taken from the top of the 
steering-head to a point on the down-tube a few inches above 
the crank-bracket. By this arrangement, of course, the down- 




FlG. 263. 

tube is subjected to a bending stress, while the frame, as a whole, 
is weakest in the neighbourhood of the crank-axle. Sinc^ the 
bending-moment on the frame diminishes from the crank-axle 




Fig. 264. 



towards the front wheel centre, it is better to have the two tubes 
from the steering-head diverging (fig. 265) instead of convei^ng 
as they approach the crank-axle ; the depth of the frame would 

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CH4P. XXII. 



The Frame 



289 



then vary proportionally to the depth of the bending-moment 
diagram, and the bending stresses on the members of the frame 





Fig. 266. 



Fig. 265. 

would be least. Such an attempt at bracing the frame of a Ladies' 
Safety, as is illustrated in. figure 266, is useless, since at the point P 
the depth of the frame is zero, and the 
only improvement is that the bending 
at the point P is resisted by two tubes 
instead of one. 

222. Tandem Frames. — A great 
variety of frames are in use at present, the 
processes of natural selection not having 
gone on for such a long time as is the case with frames for single 
machines. A frame (fig. 267), resembling that of the ordinary 
diamond-frame, with 
the addition of a cen- 
tral parallelogram, has 
been used. It will be 
noticed at once that 
the middle portion is 
not arranged to the 
best advantage for re- 
sisting shearing-force, 
so that as regards strength, the middle portion of the frame is simply 
equivalent to two tubes lying side by side and subjected to bending. 

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Fig. 267. 



290 



Details 



CHAP. XXII. 



Figure 268 shows a tandem frame, by the New Howe Machine 
Company, in which three tubes resist the bending on any vertical 
section ; and figure 269 shows a frame, by the Coventry Machin- 




FlG. 968. 



ists' Company, with the front seat arranged for a lady. Both 
these frames should be stronger, weight for weight, than that in 
figure 267, but they are not perfectly braced structures, and the 
bending-moment on the tubes will be considerable. 




Fig. 269. 



The addition of a diagonal to the central parallelogram, indi- 
cated by the dotted line (fig, 267), converts the frame into a 
braced structure, and the strength is proportionately increased. 

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CHAP. XXII. 



The Frame 



291 



The -front quadrilateral of the frame (fig. 267) requires a 
diagonal to make the frame a perfectly braced structure, and, 
though riding along a level road, it is possible, by properly 
disposing the top and bottom tubes, to insure that there shall be 
no bending on them, it would seem advisable to provide against 
contingencies by inserting this diagonal in tandem frames. Such 
is done in the * Thompson & James's' frame (fig. 140). 

Figure 270 shows a tandem frame, made by Messrs. J. H. 
Brooks & Co., intended to take a lady on either the front or back 
seat. On the side of the machine on which the chain is placed, 
instead of a single fork-tube two tubes are used, one above and 
one below the chain, and both lying in the plane of the chain. 




Fh.. 270. 



Thus the lower part of the frame constitutes a beam to resist the 
bending-moment, and the upper portions are used merely to 
support the saddles. 

Figure 271 shows a tandem frame also intended to take a lady 
on either the front or back seat, designed by the author. The 
frame is dropped below the axle -the lower part is, in fact, a 
braced structure of exactly the same nature as that in figure 267. 
The crank-bracket is held by a pair of levers, the lower ends of 
which are hinged on the pin at the lower point of junction of the 
frame tubes. The upper ends can be clamped in position on the 
tubes which form the chain-struts. The driving-wheel spindle is 

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292 



Details 



CHAP. XXII. 



thus permanently fastened to the frame, and therefore remains 
always in track. A single screw is used to adjust the crank-bracket, 
on releasing the top clamping screws of the supporting levers. 
Although this is a reversion to the hanging crank-bracket, it may 




be pointed out that it is connected rigidly to the frame at four 
points, and may therefore be depended upon not to work loose. 
223. Tricycle Frames.— In the early tricycles Y-sbaped 




FtG. 272. 

frames for front-driving rear-steerers and loop-frames for front- 
steerers were usually employed, while in side-drivers, such as the 
Coventry Rotary, the frame was fshaped, the top of the 7" being 
in a longitudinal direction. The frame of the * Cripper ' tricycle 
(fig. 150) was also J -shaped, the top of the f forming a bridge 

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CHAP. xxii. The Frame 293 

supporting the axle, and the vertical branch of the T running 
forward from the middle of the axle to the steering-head and 
supporting the crank-axle and seat These frames were almost 
entirely unbraced, and their strength depended only on the 
diameter and thickness of the tubes. 

The diamond-frame for tricycles, on the same general lines as 
the diamond-frame used in bicycles, marks a great improvement 




Fig. 273. 

in this respect, figure 272 illustrating the frame of the * Ivel ' 

tricycle. Figure 152 illustrates a tricycle with diamond-frame 

made by the Premier Cycle Company (Limited). It will be 

noticed that the frame is the same as that for a bicycle, with the 

addition of a bridge and four brackets supporting the axle. The 

next improvement, as regards the proper bracing of the frame, is 

the spreading of the seat-struts, so that they run towards the ends 

of the bridge, the bending stresses on the axle-bridge being slightly 

reduced by this arrangement. Figure 273 shows a tricycle with 

this arrangement, by Messrs. Singer & Co., but with the front part 

dropped, so that it may be ridden by a lady. 

In nearly all modern tricycles the driving-axle has been 

supported by four bearings, two near the chain-wheel, so that the 

pull of the chain can be resisted as directly as possible, and two 

.oogle 



294 



Details 



CHAP. xxn. 



at the outer ends, as close to the driving-wheels as possible, each 
bearing being held in a bracket from the bridge. The whole 
arrangement of driving-axle, bridge, and brackets looks rather 
complex, while the chain-struts are subjected to the same severe 
bending stresses as those of a bicycle (sec. 238). A great 
improvement is Starley's combined bridge and axle, the bridge 
being a tube concentric with, and outside, the axle. Figure 274 is 



TvJ=U 



4=^=^^ 




^11=^ 



Fig. 274. 



a plan showing the arrangement of the combined bridge and axle, 
crank-bracket and chain-struts, as made by the Abingdon Com- 
pany, the lug for the seat-strut being shown at the left-hand side 
of the figure. The driving cog-wheel on the axle is inclosed in an 
enlarged portion of the outer tube, in which two spaces are made 
to allow the chain to pass out and in. The chain adjustment is 
got by lengthening or shortening the chain-struts by means of a 
right- and left-handed screw, the hexagonal tubular nuts being 
clearly shown in the figure, an arrangement patented by the 
author in 1889. Messrs. Stariey Brothers have still further im- 
proved the tricycle frame by making the chain gear exactly central, 
so that the design of the frame is simplified by using only one 

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CHAP. XXII. 



The Frame 



295 



tube as a chain-strut, while the bending stresses caused by the 
pull of the chain are eliminated. The crank-bracket (fig. 153) is 
enlarged at the middle to form a box encircling a chain-wheel, 
two openings being provided for the chain to pass in and out, as 
in the axle-box, while three lugs are made on the outside of the 
box to take the three frame tubes. The narrowest possible tread 
is thus obtained. This, in the author's opinion, marks the 
highest level attained in the design of frame for a double-rear- 
driving tricycle. 

224. Spring-frames. — In the days of solid tyres many attempts 
were made to support the frame of a bicycle or tricycle on springs, 




Fig. 275. 

so that joltings due to the inequalities of the road might not be 
transmitted to the frame. The universal adoption of pneumatic 
tyres has led to the almost total abandonment of spring-frames. 
The springs should be so disposed that the distances between 
saddle, handle-bar, and crank-axle remain unaltered. In Har- 
rington's vibration check, which was typical of a number of 
appliances that could be fitted to the non-driving wheel of a 
bicycle, the wheel spindle was not fixed direct to the fork ends, 
but to a pair of short arm^ fastened to the fork ends and con- 
trolled by springs. This allowed the front wheel to move over an 
obstruction without communicating all the vertical motion to the 
frame. 

Figure 275 shows the * British Star' spring- frameJSafety, made 

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296 



Details 



CHAP. XXII. 



by Messrs. Guest & Barrow, the rear wheel being isolated by 
a powerful spring from the part of the frame carrying the saddle. 




Fig. 276. 



Figure 276 shows the *Cremome* spring-frame Safety, the springs 
being introduced near the spindle of the driving-wheel. In 




Fig. 277. 



both these spring-frames the lower fork is jointed to the frame at 
or near the crank-bracket. In the * EUand ' spring-frame, made 

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CHAP.'ZXII. 



The Franu 297 



by Cooper, Kitchen & Co., the spring was introduced just below 
the seat lug, and the lower fork was hinged to the crank- 
bracket. 

Figure 277 shows the * Whippet ' spring-frame bicycle, the 
most popular of the type, in which the driving-wheel and 
steering-wheel forks are carried in a rigid frame. The portion 
of the frame carrying saddle, crank-axle, and handle-bar is sus- 
pended from the main frame by a powerful spiral spring and a 
system of jointed bars, the arrangements of which are shown 
clearly in the drawing. 

Figure 278 shows a spring-frame bicycle now made by Messrs. 
Humber & Co. (Limited), the rear fork being jointed to the frame 




Fig. 278. 

at the crank-bracket, and the front wheel being suspended by a 
pair of anti-vibrators. The rear fork is subjected to a consider- 
able bending moment, and must therefore be made heavy ; in 
this respect the design is inferior to many of the earlier spring- 
frames. 

225. The Front-frame. — The front-frames of bicycles and 
tricycles show great uniformity in general design, any differences 
between those of different makers being in the details. The front- 
frame consists of the fork sides, which are now usually tubes of 
oval section tapered towards the wheel centre ; the fork crown ; 
the steering-tube ; and the handle-bar. The doublerplatej fork 

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298 



Details 



CHAP« TTIl. 



crown (fig. 279) is now almost universally used. The fork sides 
are brazed to the crown-plates. In the best work it is usual to 
insert a liner at the foot of the steering-tube, shown projecting in 
figure 279, so as to strengthen the part. The fork tubes are again 





Fig. 279. 



Fig. 280. 



strengthened by a liner, the top of which also forms a convenient 
finish for the fork crown. 

The top adjustment cone (fig. 280) of the ball-head is slipped 
on near the top of the steering-tube, the latter having been pre- 
viously placed in position through the ball-head. The end of the 
tube is screwed, to provide the necessary adjustment of the cone. 
The end of the tube and the tubular portion of the adjustment 
cone are slit, and the handle-bar having been fixed in the neces- 
sary position, the three are clamped together by a split ring and 
tightening screw. The lamp-bracket is often made a projection 
from this tightening ring, as shown in figure 280. Figure 280 
illustrates the ball-head made by the Cycle Components Manu- 
facturing Company (Limited), and shows the adjustment cone, 
lamp-bracket, and the adjusting nut apart on the steering-tube, 
while figure 281 shows the ball-head complete, with the parts 
assembled in position. 

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CHAP. XXII. 



The Frame 



299 



The steering-head of the Falcon' bicycle, made by the 
Yost Manufacturing Company, Toledo, U.S.A., differs from that 
by the Cycle Components Company, in having the adjusting cone 
screwed on the steering-tube. The top bearing cup is butted 





Fig. 282. 



Fig. 281. 



against the frame tube of the steering-head, the top lug embracing, 
and being brazed to, both. 

It is becoming more usual not only to make the handle-pillar 
adjustable in the steering-tube, but also to make the handle-bar 
adjustable in the socket at the head of the handle-pillar. One of 
the best designs for accomplishing this (fig. 282) is that used in 
the * Dayton * bicycles, made by the Davis Sewing Machine 
Company, Dayton, U.S.A A conical surface is formed on the 
handle-bar, and fits a corresponding surface on the socket at the 
top of the handle-pillar. A short portion of the handle-bar is 
screwed on the exterior ; the handle-bar is fixed in the required 
position by screwing up a thin nut, and thus wedging the two 
conical surfaces together. ^ i 

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300 



Details 



CHAP. xxri. 



The handle-bar is most severely stressed at its junction with 
the handle-pillar. A handle-bar liner (fig. 283), as made by the 




Fig. 283. 



Cycle Components Manufacturing Company, is used to strengthen 
it. 

The front-frame of the usual type of the present day is essen- 
tially a beam subjected to bending, showing in this respect no 
improvement on that of the earliest tricycles. In tandems and 
triplets many accidents have resulted from the 
collapse of the front frame ; additional strength 
is therefore desirable for this, generally the 
weakest part of these machines. This can be 
attained by making the steering-tube and fork 
sides of sufficient section, and also by entirely 
new designs for the front-frame. 

The * Referee ' front-frame (fig. 284) is made 
by continuing the fork sides up through the 
crown to the top of, and outside, the steering- 
head. The maximum bending-moment is thus 
resisted by the two fork sides and the steering- 
tube, instead of only by the latter, as in the 
ordinary pattern. There should be no possi- 
bility, therefore, of the steering-tube giving way. 
Duplex fork sides (fig. 285) continued to the 
top of the steering-head are a still further im- 
provement in the same direction ; the forward 
Fig. 284. ^^1^^ acting as a strut, the rear tube as a tie, 

though both are subjected, in some degree, to direct bending. 
A braced front frame has been made in the * Furore ' tandem. 
In a bicycle designed by the author in 1888, with the object 
of eliminating, as far as possible, all bending stresses on the frame 
tubes, the steering-head was behind the steering-wheel, and 

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CHAP. XXII. The Frame 30 1 

consequently the latter could be supported by a trussed frame. 
The complete frame (fig. 286) had a general resemblance to a 




Fig. 285. 



queen-post roof-truss. This design answered all requirements as 
regards lightness and strength , but as an expert rider experienced 



almost as much difficulty in learning to ride this machine as a 

novice in learning to ride one of the usual type, it was abandoned. 

In tandems steered by the rear rider, the front-frame could 

be immensely strengthened by taking stay-tubes fronithe ends of 

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302 



Details 



CHAP. xxn. 



the front wheel spindle to a double-armed lever near the bottom 
of the steering-pillar (fig. 287). These stay-tubes would have to 
be bent, as shown in plan (fig. 288), to clear the steering-wheel 

Fjg. 287. 




Fig. aSa, 

when turning a comer. The front fork would then be made 
straight, as it would act as a strut, while the stress would be 
almost entirely removed from the steering-tube. 



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303 



CHAPTER XXIII 

THE FRAME (STRESSES) 

226. Frames of Pront^rivers. — a b c (fig. 290) shows the 
hending-moment curve on the frame of an * Ordinary ' (fig. 289) 
due to the weight, W^ of the rider. The weight of the rider does 
not come on the backbone at one point, but, by the arrange 
ment of the saddle spring, at two 
points, /, and p^. If perpendi- 
culars be drawn from/, and p^ to 
meet this curve at d^ and ^2» 
^, b d^ will be the bending-inoment 
curve of the spring, and the re- 
mainder a dx d^c of the original 
ben ding-moment curve will give 
the hending-moment on the back- 
bone and rear fork. The bending- 
moment on the backbone is 
greatest near the head, and dimi- 
nishes towards the lower end. Ac- 
cordingly, the backbones of * Ordi- 
naries ' were invariably tapered. 

In the 'Ordinary' the front fork 
was vertical, and consequently the 
hending-moment on the frame just 
at the steering-head was zero. In the * Rational,* however, the front 
fork was sloped, and a bending-moment, R^ /, existed at the steering- 
head, / being the horizontal distance of the steering-head behind 
the front wheel centre. There would consequently be two equal 

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304 Details 



CHAP, llili . 



forces, -^j and F^^ acting at right angles to the head, at the top 
and bottom centres, such that 

F^h^RJ (i) 

where >4 is the distance between the top and bottom centres. The 
greater the distance h^ the smaller would be the force -^,, and 
thus a long head might be expected to work more smoothly and 
easily than a short one. 

It is easily seen that the side pressure on the steering-head of 
a * Safety ' bicycle or * Cripper ' tricycle arises in exactly the same 
way. The arrangement of the frame of a * Safety ' is such as 
permits of a much longer steering-head than can be used in the 
* Ordinary,' and as the pressure on the front wheel is much less 
than in the ' Ordinary,' the side pressure on the steering-head is 
also very much smaller. 

Example /.—In a * Geared Ordinary ' the rake of the front fork 
is 4 inches, the distance between the top and bottom rows of balls 
in the head is 3 inches, the weight of the rider is 150 lbs., and the 
saddle is so placed that two-thirds of the weight rest on the front 
wheel ; find the side pressure on the ball-head. The reaction, R^ 
(fig. 289), in this case is 

— X 150= 100 lbs., 
3 

the bending-moment at the head is 

100 X 4 = 400 inch-lbs., 

the force F is therefore 

^^^ = 133-3 lbs. 
3 

Example II, — In a * Safety ' bicycle the ball steering-head is 
9^ inches long, the horizontal distance of the middle of the head 
behind the front wheel centre is 9 inches, the rider weighs 1 50 lbs., 
and one-fourth of his weight rests on the front wheel ; find the 
side pressure on the steering-head. In this case, the reaction 
R^ is 

^ X 150 = 37*5 lbs., 
4 

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OTAP. xxin. The Frame 305 

the bending-moment on the head is 

37*5 X 9 = 33 7 '5 inch-lbs., 
the side pressure on the steering-head is therefore 

^^V^ = 35*5 lbs. 
9i 

Example III, — In Example I., if the point /| (fig. 289) of 
maximum bending-moment on the backbone be 6 inches behind 
the front wheel centre, find the necessary section of the backbone. 
The bending-moment at/i will be 

100 X 6 = 600 inch-lbs. 

If the maximum tensile stress be taken 15,000 lbs. per sq. in., 
substituting in the formula J/= Z/, we get 

Z= ^^ = '04 inch-units. 
15,000 

From Table IV., p. 112, a tube i^ in. diameter, number 20 W.G. 
would be sufficient. 

The section necessary at any other point of the backbone may 
be found in the same manner, but where the total weight of the 
part is small, it is usual to make the section at which the straining 
action is greatest sufficiently strong, and if the section be kept 
uniform throughout, all the other parts will have an excess of 
strength. In the backbone of an * Ordinary,' the section should 
not diminish by the tapering so quickly as the bending-moment. 

Example IV, — In a front-driver in which the load and the 
relative position of the wheel centres and seat are as shown in 
figure 291, the stresses can be easily calculated as follows : 

Taking the moment about the centre of the rear wheel, we get 
^, X 36 = (120 X 23) + (30 X 36) ; therefore 

^i = 1067 lbs., R<i = 43*3 lbs. 

The maximum bending-moment (fig. 293) on the frame occurs on 
the section passing through the seat, and is — 

J/ = 43*3 X 23 = 996 inch-lbs. 

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3o6 



Details 



CHAP. xxin. 



If the frame simply consists of a backbone formed by a tube 
i^^ in. diameter, i8 W.G., we find from Table IV., p. 112 — 



Z= '0525, and/= ^ = ^^ = 19,000 lbs. per sq. 
Z -0525 



m. 



Braced Frame for Front-driver. — A simple form of braced 
frame is shown diagrammatically in figure 291. The short tube 

Fig. agi. 




Stnss Dhqram 

Scmk.aO^'tmi 



Bending Mommt 
Seak,iO00mek'&S'lmdi 

Fig. 293. Fig. 292. 

from the steering-head to the seat-lug is made stout enough to 
resist the bending-moment due to the saddle adjustment, while the 
seat-struts are subjected to pure compression, and the lower stays 
to pure tension. Figure 292 shows the stress- diagram for this 
braced structure ; from which the thrust on the seat-struts, a ^, is 
1 16 lbs. If they are made of two tubes | in. diameter, 28 W.G., 
from Table IV., p. 112, ^ = 2 x 0284 = -057, and the com- 



pressive stress IS 



116 

•057 



= 2,000 lbs. per sq. in. 



The pull on the lower stay, a r, is 95 lbs., land if the stays are 
made of tubes of the same diameter and thickness as the seat- 
struts, the tensile stress will be correspondingly low. 

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CHAP. xzin. 



The Frame 307 



The greatest stress on the top-tube will be due to the saddle 
adjustment. With the horizontal branch of the L pii^ 3 i^* long* 
a total horizontal adjustment of 6 in. can be provided; the 
maximum bending-moment on the tube will be 

M=: ISO X 3 = 450 inch-lbs. 

If the tube be i in. diameter, 20 W.G., Z = '0253, and the 
maximum stress on the tube will be 

/= = _15o_. _ 17^800 lbs. per sq. in. 
Z "0253 

227. Reax-driving Safety Frame. — The bending-moment 
curve for the frame (taken as a whole) of any bicycle is inde- 
pendent of the shape of the frame, and depends on the weight to 
be carried, and the position of the mass-centre relative to the 
centres of the wheels. The actual stresses on the individual 
members of the frame, however, depend on the shape of the 
frame. The frame of a rear-driving chain-driven Safety must 
provide supports for the wheel spindle at IVj the crank-axle at C, 
the saddle at S, and the steering-head at Ify and If^ (fig* 296). 
Two principal types of frame are to be distinguished. In the 
cross-frame the point jfiT, and ^2 were very close together, and 
the opposite corners of the quadrilateral W C H Sv^^x^ united by 
tubes. In the diamond-frame^ adjacent comers of the pentagon, 
Hx H^ C JV S, are united by tubes. In both the diamond- and 
the cross-frames additional ties and struts are inserted, the object 
being to make the frame as rigid as possible, and, of course, to 
reduce its weight to the lowest possible consistent with strength 
The weight of a bar necessary to resist a given straining action 
depends on the magnitude of the straining action and its direction 
in relation to the bar. We have already seen that a force applied 
transversely to a bar and causing bending, to be effectually re- 
sisted, will require a bar of much greater sectional area than if 
the force be either direct compression or tension. It may thus be 
laid down as a guiding principle in designing cycle frames, that the 
various members should be so disposed^ that as far as possible they 
are all subjected to direct compression or tension^ but not bending^ 
It follows from this that each member should be attached to 

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3o8 



Details 



CHAP. ZXIU. 



Other members at only two points. A bar on which forces can 
only be applied at two points — its ends — cannot possibly be sub- 
jected to bending. If a third * support ' be added, the possibility 
and probability of subjecting the bar to bending arises. The 
early Safety frames and some Tandem frames of the present day 
show many examples of bad design, a long tube often being 
* supported ' at one or more intermediate points, the result being 
to throw a transverse strain on it, and therefore weaken, instead 
of strengthen the structure. 

228. The Ideal Braced Safety Frame.— In a Safety rear- 
frame the external forces act on five points ; the weight of the 





Fig. 394. 

rider bemg applied partially at the saddle 5, and at the crank- 
axle Cy the reaction of the back wheel at W^ and the pressure on 
the steering-head at ZT, and H^ (fig. 296). If the five points 
Zr„ H^y Sy IVy and C be joined by bars or tubes 
dividing the space into triangles, the frame will 
be perfectly braced, and there will be only direct 
tensile or compressive stresses on the bars. 
Figure 296 shows the arrangement used in the 
« Girder ' Safety frame, while figure 294 shows a 
number of possible arrangements of perfectly braced rear-frames. 
Figure 295 shows another perfectly braced rear-frame, in which 
the lower back fork between the crank-axle and driving wheel 
spindle is omitted. Comparing this with figure 320, it will be seen 
that a very narrow tread may be obtained with this frame, a saving 
of at least the diameter of one lower fork tube being effected. 

Example. — The rider weighs 150 lbs., 30 lbs. of which is 
applied at the crank-axle, the remainder, 120 lbs., at the saddle 5. 

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Fig. 295. 



CHAP. xxni. 



The Frame 



309 



From the given dimensions of the machine (fig. 296), the re- 
action, ^1 and ^2, on the front and back wheels can be calculated. 
Considering the complete frame, and with the dimensions 
marked, taking moments about the centre of the front wheel, we 
have 

(30 X 23) + (120 X 33) = ^2 X 42 
from which 

R^ = 1 107 lbs. 
and 

^, = 150 — 1 107 = 39-3 lbs. 

Consider now the front-frame consisting of the fork and steering- 
tube ; it is acted on by three forces, the reaction, ^1, upwards, and 

Fig. 296. 




7777T/ 

lllllllllllllllll}mrmL • 



• SkeariiHi' Force, 




Fig. 2q8. 



Setk./00/ks'lmJ, 
Fig. 297. 



Fig. 29Q. 

the reactions between it and the rear-frame, at H^ and H<^, 
These three forces must therefore (see sec. 45) pass through the 
same point. With the ordinary arrangement of ball-head the 
vertical pressure of the front portion of the frame acts4)n the rear 

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3IO Details 



CH/iP. xxm. 



portion at ZT,, and the resultant force at H^ may be assumed 
at right-angles to the steering-head, ZT, ^g- Therefore, from H^ 
draw H^ h, intersecting the vertical through the front wheel centre 
at h ; join ff^ ^t giving the direction of the force at H^. 

The stress-diagram (fig. 297) can now be constructed by the 
method of section 83. From this diagram it is easily seen that , 
the lower back fork is in tension, and also the tube from the 
lower end of the steering-head ) the other members of the frame 
are in compression. By measurement from figure 297 the thrust 
k c along the seat-struts is 1 1 7 lbs. ; that along the down-tube, b c, 
about 19 lbs. ; along the top-tube, h l^ about 41 lbs. ; along the 
steering-head, am, 12 lbs. ; along the diagonal, ad, from the top 
of the steering-head to the crank-bracket, 7 lbs. ; and the tension 
on the lower back fork, co, is 58 lbs. ; on the bottom-tube, a n, 
64 lbs. These values will, of course, vary slightly according to 
the dimensions of the frame. 

Taking a working stress of 10,000 lbs. per square inch, the 
sectional area of the two tubes constituting the seat-struts would 
only require to be y^^^^ = *oii7 sq. in., provided the diameter 
was great enough to resist buckling. The section of the other 
tubes would be correspondingly small. We shall see later, how- 
ever, that many of the frame tubes are subjected to bending, and 
that the maximum stresses due to such bending are much greater 
than those considered above. 

229. Humber Diamond-frame. — The force on the tube 
between the spaces a and b (fig. 296) is very small, and by careful 
design may be made zero (see sec. 230). In the * Humber ' diamond- 
frame this tube is suppressed, and thus if the frame tubes were 
connected by pin-joints at C, S, H^ and H.^, the frame would be 
no longer able to retain its form when subjected to the applied 
forces. That the frame actually retains its shape is due to the 
fact that the frame joints are rigid, and that the individual 
members are capable of resistance to bending. If all the frame 
joints are rigid, the stress in any member cannot be determined 
by statical methods, but the elasticity and deformation of the 
parts under stress must be considered. However, by making 
certain assumptions, results which may be approximately true can 
be obtained by statical methods. 

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CHAP. xxni. 



The Frame 311 



Example /. — Suppose the tubes C H^ and Hx H^ to be 
fastened together at H^^ so as to form one rigid structure, which 
we may consider connected by pin-joints to the frame at C and 
ZTa, the other joints of the frame being pin-joints. The distance 
of Hx from the axis of the suppressed member, H<i C, is 6 in. 
The bending-moment at H, on the part C H^ H^ is therefore 
7 lbs. X 6 in. = 42 inch-lbs. ; the bending-moment diminishes 
towards zero at C and H, If the tube Hx C be i in. diameter, 
and a working stress of 10,000 lbs. per sq. in. be allowed, sub- 
stituting in formula (3), section loi, we get 

10,000 = ^4 + 4 X 42 
A A X I 

or, A = -0232 sq. in. 

Consulting Table IV., p. 112, we see that the thinnest 
there given, No. 32 W.G., has an excess of strength. If the 
tube Hi C had been retained, the sectional area of the tube 
Hx C need only have been 

A=i^ = -0064 sq. in. 
10,000 

Example II, — Suppose the tubes Hx H^ and H^ S rigidly 
fastened at ZT^, and connected at Hx and 5 by a pin-joint to the 
rest of the frame. The part Hx H^ S may then be considered as 
a beam carrying a load of 7 lbs. at ZTj. The perpendicular dis- 
tances of Hx and S from the line of action of this force are 6 in. 
and 18 in. respectively. The bending-moment at -^2 is therefore 
(see sec. 87) 

7 X 6 X 18 • I. lu 

• ' = 31 5 mch-lbs. 

24 ^ ^ 

The compressions along Hx H^ and H^ S will be increased by 
the components of the original force, 7 lbs., along the suppressed 
bar at H^ C, Similarly, the forces along C Hx and C S will be 
altered. The thickness of tube required can be worked out as in 
Example J. above. 

230. Diamond-frame, with no Bending on the Frame Tubes. 
— Consider the complete frame divided by a plane, FP (fig. 296) 
immediately behind the steering-head, HxH^- If the frame 

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312 Details chap, xxnt 

tubes Hi S and H^ C are not subjected to bending, the forces 
exerted by the front part of the frame on the rear part must be 
in the direction of the tubes. The forces acting at /*, and J^^ on 
the front portion of the frame are equal in magnitude but re- 
versed in direction. The only other external force acting on the 
front portion of the frame is the reaction, ^„ of the wheel on the 
spindle ; these three forces are in equilibrium, and therefore 
must all pass through the same point. The condition then that 
the tubes in a diamond-frame should be subjected to no bending 
is that the axes of the top- and bottom-tubes should^ if prodtued^ 
intersect at a point vertically aver the front wheel centre. This is 
very nearly the case in figure 296 ; if it was exactly, the force b a 
along 11^ C would be zero. 

231. Open Diamond-frame.— The open diamond-frame (figs. 
127, 246), though in external appearance very like the * Humber' 
frame, is subjected to totally different straining actions. In the 
first place, if the joints at C, -^„ JI^^ S and W be pin-joints, 
under the action of the forces the frame would at once collapse. 
Practically, the top-tube, Hf^ S, and the seat-struts, S W, form one 
rigid beam, which must be strong enough to resist the bending- 
moment due to the load at S, Taking the same dimensions as in 
figure 296, the distances of Hi and IV from the line of action of 
the load at S are 2 1 in. and 9 in. respectively, and the weight of 
the rider 150 lbs., the bending-moment at S will be 

150x21x9^9^5^^^^.11,3, 
30 

Taking /= 20,000 lbs. per sq. in., and substituting in the (ot- 
mula. M= Zf 

Z = -^^^ = '0472 in.^ 
20,000 

From Table IV., p. 112, a tube i in. diam., 14 W.G., would be 
required ; or a tube i^ diam., 17 W.G. 

When the rider is going easily his whole weight rests on the 
saddle, and must be supported by the beam H^S W, On the 
other hand, when working hard, as in riding up a steep hill, his 
whole weight may be applied to the pedals, and, therefore, will 
come on the frame at C The bottom-tube, IL^ C, and the lower 

Digitized by VjOOQ 



OHAP. XZIIT. 



The Frame 



313 



back fork, C fV, must be rigidly jointed together at (7, and form 
a beam sufficiently strong to resist this bending-moment. Taking 
the same dimensions as in figure 296, the bending-moment at C is 



19 _ 



= 1,257 inch-lbs. 



y M 1,257 r o • 

^ = -^ = ' ^' = -0628 m. 
/ 20,000 

and a i in. tube, 11 W.G., or a i^ in. tube, 14 W.G., would be 
required. A comparison of these results with those of sections 
228-9 will reveal the weakness of the open diamond-frame. 

232. Cross-frame. —In the cross-frame {^%. 300), the forked 
backbone a runs straight from the steering-head to the back 




Fig. 300. 



wheel spindle. The crank-bracket and seat-lug are connected by 
the down-tube b. The earlier cross-frames consisted only of 
these two members ; but in the later ones, bottom stays ^, from 
the crank-bracket to the back wheel spindle, and stays c and dy 
from the steering-head to the crank-bracket and seat-lug respec- 
tively, were added. With this arrangement, the down-tube b is 
subjected to thrust, the stays r, d^ and e to tension, and the 
backbone a to thrust, combined with bending, due to the forces 
acting on the steering-head. 

The stress-diagram can be drawn as follows : The loads W^, 
and Wi^ at the seat-lug and crank-axle respectively being given, 
the reactions R^ and -^2 ^^ the wheel centres can be calculated, as 
in section 89. At the back wheel spindle three forces, act. The 

Digitized by CjOOQIC 



314 Details chat. xxm. 

magnitude and direction of ^2 ^^e known, the pull of the lower 
fork e is in the direction of its axis, and is, therefore, known, but 
the thrust on the backbone a is not along its axis ; its direction is 
not known, and we cannot, therefore, begin the stress-diagram 
with the forces acting at this point. If the initial tension on the 
stays c and d be assumed such that there is no straining action at 
the junction of the down-tube d and the backbone a, the former 
will be subjected to a thrust along its axis, and, therefore, the direc- 
tions of the three forces acting at the seat-lug are known. Draw 
^F| (fig. 301), equal and parallel to the weight acting at the seat-lug, 
and complete the force-triangle PV^ b r, the sides being parallel to the 
correspondingly lettered members of the frame (fig. 300). Proceed- 
ing to the crank-bracket, the forces acting are the weight W^ the 
thrust of the down-tube ^, and the pulls of the stays d and ^, the 
directions of which are known ; the force-polygon b W^ e d can 
therefore be drawn. Proceeding now to the back wheel spindle, 
the pull of the stay e and the upward reaction R^ are known. 
Setting off R^ (fig. 301) from the extremity of the side e^ and 
joining the other extremities of e and R^^ the direction and 
magnitude a} of the thrust on the backbone are obtained. This 
thrust does not act along the axis of the backbone, which is, 
therefore, in addition to a thrust along its axis, subjected to a 
bending-moment varying from zero at the back wheel spindle to a 
maximum PI2X the steering centre, /* being the thrust measured 
from figure 301, and / the distance of a perpendicular on the 
line of action of the thrust from the centre of the backbone at the 
steering-head. 

The forces acting on the backbone a are : the pull of the lower 
stay e and the reaction ^j, having the resultant a} \ the pulls 
c and d^ with resultant / acting, of course, through the point of 
intersection of c and d ; the pressures F^ and F^ on the steering 
centres. Since ^„ -^1 and F^ reversed are the only forces acting 
on the front-frame, the resultant of jp, and F^ must be R^ Thus 
the forces acting on the backbone a are equivalent to the three 
forces ^,, d^ and / which must, therefore, all pass through the 
point O. A check on the accuracy of the stress-diagram is thus 
obtained. 

The point O being determined, by joming it to the top and 

Digitized by CjOOQIC 



CHAP. XXIII. 



The Frame 



315 



bottom steering centres the directions of F^ and F^ are ob- 
tained, and their magnitudes by drawing the force-triangle 
^,i?;7?i(fig. 301). 

233. Frame of Ladies' Safety.— Figure 302 is the frame- 
diagram of a Ladies' Safety. Having given the loads at the seat-lug 
and crank-axle, the reaction of the wheels can be calculated, as 
in section 89. The stresses on the seat-struts a and the back fork 
e may be found in exactly the same manner as for the diamond 
frame ; viz. by drawing the force-triangle R^a e (fig. 303). The 
best arrangement of the two tubes c and d from the steering-head 



\IZOlk 




Stress -Diagram ^.>/NX> 

Soik. HfOAsl/md ^'"->^^ 



Fig. 303. 



will be when their axes intersect at a point vertically over the 
front wheel centre. Assuming that the forces on these two 
tubes are parallel to their axes, they are determined by drawing 
the force-triangle R^dc for the three forces {fig, 303) acting on 
the front-frame of the machine. The down-tube b is acted on by 
three forces — (i) The thrust of the tube c ; (2) the resultant / of 
the thrust a and load W^ acting at the seat-lug; (3) the re- 
sultant g of the pulls d and e and the load Wg, acting at the 
crank-axle. These three forces form the force-triangle c f g 
(fig. 303). A check on the accuracy of the work is obtained by 

Digitized by V^jOOQ 



3i6 Details 



CHAP. xxni. 



the fact that the forces /and g (fig. 302) must intersect at a point 
/ on the axis of the tube c. 

From figure 303, the thrust of ^ on the down-tube ^ is 145 lbs., 
while its component c^ at right angles to b is 142 lbs. The down- 
tube b is 22 in. long, divided by c into two segments, 7 and 
15 in. The greatest bending-moment on it is therefore 

22 

The lower part of the down-tube is subjected to a thrust ^' (the 
component of the force g parallel to the down-tube) of 62 lbs. 

234. Curved Tubes.— About the years 1890-2 a great number 
of Safety frames were made with the individual tubes curved in 
various ways. The curving of the tubes was made on aesthetic 
grounds, and possibly the tremendous increase in the maximum 
stress due to this curving was not appreciated. The maximum 
stress on a curved tube subjected to compression or tension at 
its ends is discussed in section 10 1. 

Example. — Let a tube be bent so that its middle point is a 
distance equal to four diameters from the straight line joining its 
ends ; the maximum stress is, by (4) section loi, 

P^ 4 X i^ X 4 ^ — ^7 ^ 
A^ Ad ~ "" y^ "• 

The tube would, therefore, have to be seventeen times the sec- 
tional area of a straight tube subjected to the same thrust. 

235. Influence of Saddle Adjustment — So far we have con- 
sidered the mass-centre of the rider to be vertically over the 
point S (fig. 296) \ this is approximately the case when the saddle 
is fixed direct, a^ in some racing machines, to the top of the 
back fork without the intervention of an adjustable saddle-pin. 
But when an adjustable saddle-pin is used, the weight on the 
saddle acts at a distance /, usually from 3 to 6 inches behind 
the point S, The weight ^acting at this distance is equivalent 
to an equal weight acting at 5, together with a couple JF/, 
producing a bending-moment Wy^ I at S, From the manner in 
which the adjustable pillar is usually fixed at 5, this bending- 
moment is generally transmitted to the down -tube S C, which 
must therefore be stout enough to resist it. Since, however, the 

Digitized by V^jOOQ 



The Frame 



317 



joint at S is rigid, a small part of this bending-moment may. 
be transmitted to the tube S H^, 

Example. — Taking /=5 in., and ^=120 lbs., as in 
figure 296, M=^i2ox 5 = 600 inch-lbs. Taking the direct 
thrust along 5 C 19 lbs., as in section 228, and a working stress 
20,000 lbs. per sq. in., the diameter of the tube i in., and sub- 
stituting in (3) section loi, 

19 . 4 X 600 

20.000 = -^ + --^—. 

we find A = -1210 sq. in. From Table IV., p. 113, the tube 
would require to be 18 W.G. 

If the saddle were placed vertically over 5, and no bending 



19 



came on the tube S C, its sectional area would be 

20,000 

= '00095 sq. in. : one-hundredth part of the section necessary 

with the saddle placed sideways from 5. 

This example is typical of the enormous additions which must 
be made to the weights of the tubes of a frame when the forces 
do not act exactly along the axes of the tubes. 

By the use of the T-shaped seat-pillar (fig. 304) the range of 
horizontal adjustment can be increased without increasing unduly 





Fig. 304. 



Fig. 305. 



the stresses due to bending ; or, for a given range of horizontal 
adjustment, the bending stresses are lower with the T-shaped 
than with the L-shaped seat-pillar. The adjustment got by 
L pin with horizontal and vertical limbs is much better 



an 



(figs. 256, 260), since by turning the L pin round, the saddle 
may be adjusted either before or behind the seat-lug 6'. Thus, 

Digitized by VjOOQ 



3i8 Details chap. xxm. 

for a horizontal adjustment of 6 inches, the maximum eccentricity 
/ need not be greater than 3 inches. By combining such an 
L pin with the * Humber ' frame it would be possible to further 
reduce the stresses on the frame. Figure 305 shows a seat-lug 
for this purpose, designed by the author. 

For racing machines of the very lightest type possible the 
best result is obtained by fastening the saddle direct at S; this, 
of course, does not allow of any adjustment, and a machine that 
might suit one rider admirably might not be suitable for others. 

236. InfluenceofChain AcyuBtment— In chain-driven Safeties 

it is found that chains stretch, no matter how carefully made, 

, V after being some time in use, and 

V\\ ' , therefore some provision must be 

\ \\ j^ / ^ made for taking up the slack. This 

V\\; 1 is usually done by making the dis- 

__j|l^v^V }->. tance between the centres of the 

j- - - - ^ ( : crank-axle and the driving-wheel ad- 

^ ' justable. Figure 306 shows a common 

J^ faulty design for the stamping at the 

driving-wheel spindle. The force 

iG. 306 j^^ ^^g ^^gj .g equivalent to an equal 

force acting at W plus a bending-moment R^ /, which is trans- 
mitted to the upper and lower forks. 

Example,— li the distance / (fig. 306) be \ inch, and R^ 
be, as in the example of section 228, 11 1 lbs., the bending- 
moment transmitted to the forks is 55-5 inch-lbs. The direct 
compression along the seat-struts 5 ^ is 41 lbs. (fig. 297), that 
along the lower fork W C \s 58 lbs. Taking 10,000 lbs. per 
sq. in. as the working stress of the material, a section of -0041 
sq. in. for the top fork, and '0058 sq. in. for the bottom fork 
would be sufficient, if they were not subjected to bending. 
Suppose the bending to be taken up entirely by the lower fork, 
made of two tubes J in. diameter, and of total area A ; then, 
when subjected to bending as well as to direct compression or 
tension, the maximum stress to which they are subjected is 
given by the formula (3) of section loi. Substituting the above 
numerical values of / /*, J/, and d^ we have 

10.000 = 58 ^ 4X ss-5, 

A A X -ysjbyCoogle 



cpAP. x^ii. The Frame 319 

or -^ = '035 sq. in. Thus the maximum stress, when the force 
R^ is applied \ in. from the point of intersection of the forks, is 
nearly seven times as great as when it is applied in the best 
possible position. 

Swinging Back Fork, — The centre of the driving-wheel may 
be always at the intersection of the top and bottom forks if the 
top fork be attached to the frame at 5 by a bolt — the bolt used 
for tightening the saddle-pin may serve for this purpose — and 
its lower ends be provided with eye-holes for the reception of 
the spindle of the driving-wheel. This arrangement, now almost 
universal, was first designed by the author in 1889. The lower 
fork may then be provided with a plain straight slot (fig. 307), 
along which the wheel spindle can be pulled by an adjusting 
screw. During a small adjustment of this nature the angle 






Fig. 307 Fig. 308. Fig. 309. 

S C W^(fig. 296) will vary slightly, so that theoretically the lower 

forks should be attached by a pin-joint at C ; but practically the 

elasticity of the tubes is sufficiently great to allow of the use of 

a rigid joint. In the form of this adjustment used by Messrs. 

Humber & Co. the slot is not in the direction of the axis of the 

lower fork, but curved (fig. 308) to a circular arc struck from 5 

as centre. In this way there is no tendency to alter the angle 

sew (fig. 296) ; but the fact that the centre of the wheel 

spindle does not always lie on the axis of the lower fork C W 

throws a combined tension and bending on it, the bending- 

moment being equal to P/, where P is the direct force on the 

lower fork parallel to its axis, and / is the distance of the centre 

of the wheel from the axis of the lower fork. 

d 
Example, — Let / = , that is, the centre of the wheel is just 

on a line with the top of the tube of the lower fork. 

Digitized by CjOOQIC 



320 



Details 



csla:p. xxiii. 




Substituting in (3), section loi, 

^^ A^ 2Ad'' A' 

If the centre of the wheel lay on the axis of the tube the stress 

p 
would be uniformly distributed and equal to -j. Thus the stress 

on the lower fork is increased by the eccentricity of the force 
acting on it to three times its value with no eccentricity. 

A better arrangement for the slot would be that shown in 
figure 309, where the spindle is adjusted equally above and 
below the centre line of the lower fork tubes. 

237. Influence of Pedal Pressure. — In the foregoing dis- 
cussion we have considered the forces to be applied in the 
middle plane of the bicycle frame ; but 
the rider applies pressure on the pedals 
at a considerable distance from the 
middle plane, and thus additional 
transverse straining actions are intro- 
duced. We now proceed to investigate 
the corresponding stresses. 

Figure 310 is a transverse sectional 
elevation, showing the pedals, cranks, 
crank-bracket, saddle, and down-tube, 
to the foot of which the crank-bracket 
is fixed. A force, P^ applied to the 
pedal will cause a bending of the crank- 
bracket, which will be transmitted to 
the down-tube. From the arrange- 
ment of the lower fork in relation to 
the crank-bracket it is seen that prac- 
tically none of this bending-moment 

^ can be transmitted to the lower fork. 

^ I A small portion of the bending-moment 
^''^- 3"- may be transmitted to the bottom-tube 

H^ C (fig. 296), but the greater part will be transmitted to the 
down-tube. 

The magnitude of the bending-moment is Pd^ ^ being the 

Digitized by CjOOQIC 



tt^ 



V- 



CHIP. XXIII. 



Tlu Frame 321 



length of the perpendicular from the centre of the crank-bracket 
on to the line of action of the force P, The narrower the tread 
the smaller will be d^ and therefore the smaller the transverse 
stresses on the frame. Hence the importance of obtaining a 
narrow tread. 

Example /. — I^t -P be 150 lbs., the tread, i.e, the distance from 
centre to centre of the pedals measured parallel to the crank-axle>' 
1 1 inches. The distance d may be taken equal to half the tread, 
/>. 5 1 inches. The bending-moment on the foot of the down- 
tube will be 150 x 5^ = 825 inch-lbs. Let the down-tube 
be Iff in. diameter, 20 W.G. From Table IV., p. 113, the Z 
for the section is '0325 in.*; substituting these values in the 
formula J!/= Z/, we get 

825 = -0325/ 

/>.y=: 25,400 lbs. per sq. in. 

Compared with the result on page 310, got by considering the 
forces applied in the middle plane of the frame, it is seen that on 
the down-tube the stress due to transverse bending is the most 
important. 

In the double diamond-frame the single down-tube of 
figure 310 is replaced by the two tubes which support the crank- 
bracket near its ends (fig. 311). This gives a 
much better construction to resist the trans- 
verse stresses, but unfortunately it is not so 
neat in appearance as the single tube, and 
its use has been practically abandoned of 
recent years. The maximum stress produced 
in this case can be easily calculated and may 
be illustrated by an example. 

Example IL — I^t the tubes be | in. diame- 
ter, 20 W.G., with their ends 3 in. apart. Under 
the action of the force F the nearer tube will 
be subjected to tension, the further one to compression. Taking 
moments about a (fig. 311), the point of attachment of the 
further tube to the crank-bracket, we get 

T F^^^F, i.e. F^ 1 F = 350 lbs. 

Digitized by CjOOg^C 




322 Details chap. xxm. 

The sectional area of the tube, from Table IV., p. 113, is 
•0807 sq. in., therefore the stress on the tube is 

/ = -3??- = 4,336 lbs. per sq. in. 
•0007 

238. Influence of Pull of Chain on Chain-struts. — In riding 
rasily along a level road, when very little effort is being exerted, 
the tension on the chain is small, and the stresses on the lower back 
fork, or chain-struts, will be as discussed in section 228. But when 
considerable effort is being exerted on the pedal, the tension on 
the chain is considerable, and since the chain does not lie in the 
middle plane of the frame, additional straining actions are intro- 
duced. 

The tension -^on the chain (fig. 312) can be easily found by 
considering the single rigid body formed by the pedal-pins, cranks, 

crank-axle, and chain-wheel. 
This rigid body is free to turn 
about the geometric axis of 
the crank -bracket, and it is 
acted on by three forces : 
P the pressure on the pedal- 
pin, the pull F of the chain, 
and the reaction of the balls 
on the crank-axle. Taking 
^'°' ^''' moments about the geometric- 

axis of the axle, that of the latter force vanishes, and we get 
Fl^Fr] /being the length of crank, and r the radius of the 
sprocket-wheel. 

Example L — Let jP= 150 lbs., /= 6^ in., and let the chain- 
wheel have eighteen teeth to fit the * Humber ' pattern chain. 
From Table XV., p. 405, we get r = 2*87 in. ; therefore 

2-877^= 6ijP, and 7^— - ^- x 150 = 340 lbs 
207 

Figure 313 is a plan showing the crank-bracket and the lower 
back fork. Consider the horizontal components of the forces 
acting on the crank-bracket. If the pressure on the pedals be 
vertical there will be no horizontal component due to it, and we 

Digitized by CjOOQIC 




— -*/- 



CHAP. ZXIII. 



The Frame 



323 



are left with the force F^^ the horizontal component of the pull on 
the chain. This is equilibrated by the horizontal components 




b 


-K 


*■ i 


t 


rt 




»0 

1 




C 


+ 





Fig. 314, 



r, 



r. 



Fig. 313. 

of the reactions at the bearings, therefore the crank-bracket i 
acted on by the forces at the bearings and the forces F<^ and F^ 
exerted by the ends of the lower back 
fork. 

Example II, — Let the chain-line be 2 J in. 
(/>. the distance from the centre of the 
chain to the centre of the fork is 2|^ in.), 
let the fork ends at the crank-bracket be 
3 in. apart ; then the forces to be con- 
sidered are shown in figure 314. 

To find the pull F^^ take moments about b, 

f F, = 3 /s, therefore /^3 = ^ 34© = 70*8 lbs. 

3 

To find the compression /^2 on the near tube, take moments 
about r, and we get 3J F^ 3/^, 

/. i^,=^*^^^34o = 4iolbs. 
3 

Comparing with the results of section 228, the compression on 
the near tube of the fork is much greater than the tension due to 
the weight of the rider applied centrally. The near tube, therefore, 
must be designed to resist compression. 

Bending of Chain-struts.—ThQ sides of the lower back 

Digitized by V^j00^2 



324 



Details 



CEAP. xxni. 




fork, the crank-bracket, and the back wheel spindle together 
form an open quadrilateral without bracing (fig. 315), a b and 
d c being the fork sides, b c the crank-bracket, and a d the 

wheel spindle. If 
this structure be 
acted on by forces 
there will be in 
general a tendency 
to distortion. The 
tension of the chain, 
efy is such that the 
points e and/ on the spindle and crank-bracket respectively, in 
the plane of the chain, tend to approach each other, and the 
structure is distorted into the position a} b c d^. The action 
can be easily imagined by supposing the structure jointed at 
the corners ^, b^ r, and d. In the actual structure this distortion 
is only resisted by the stiffness of the joints, and the bending- 
moment can be investigated thus : Consider the equilibrium of 
the wheel spindle ^^/(fig. 316). It is acted on by the pressure 
J a on the two bearings 

(the resultant of 
which is the pull of 
the chain €f\ and 
the forces exerted by 
the fork sides at the 
points a and d re- 
spectively. The spindle is acted on by three forces, which, being 
in equilibrium, must all pass through the same point /, lying 
somewhere on the line ^/produced indefinitely in both directions. 
Thus, the force acting on the fork side abi^'va the direction a L 
If F^ be the magnitude of this force, and Vj the perpendicular 
from b onal^ there will be a bending-moment J/j = F^ /j. With 
a similar notation for the fork side c d^ there will be a bending- 
moment J/3 = 7^3/3 at the point c of the fork side. If / coin- 
cided with the point of intersection oi ab and ef^ M^ would be 
zero, but M^ would be very great. 

Example III, — We might assume such a position for / that M^ 




Fig. 316. 



Digitized by CjOOQIC 



CHAP. XXIII. The Frame 325 

and M^ would be approximately equal. In this position, taking 
the data of the above examples, l^ would be about % in., and 

M.i = /^2 /j = 410 X I = 154 inch-lbs. 

If the lower fork be of round tube \ in. diameter, 20 W.G., 
we find, from Table IV., p 113, Z= '0137 in.^ Substituting in 
the formula J/= Z/we get 

/= '54l =s 11,200 lbs. per sq. in. 
•0137 

The sectional area of the tube, from Table IV., p. 113, is 
•0807 sq. in. ; therefore the stress due to the compression of 
410 lbs. is 

/= ^l^~ ■= s,o8o lbs. per sq. in. 
•0807 

Thus, the maximum compressive stress on the fork at b is 
/= 11,200 •+• '5,080 = 16,280 lbs. per sq. in. 

Section of Chain-struts, — The tubes from which the chain- 
struts are made are usually of round section. Occasionally 
tubes of oval section are used, the larger diameter of the 
tube being placed vertically. Since the plane of bending of the 
fork tubes is horizontal, if the fullest advantage be desired 
the oval tubes should be placed with the larger diameter hori- 
zontal. But the horizontal diameter is limited by the necessity of 
getting a narrow tread. For a given sectional area (or 



weight) of tube, and horizontal diameter, the bending ^ 1 
resistance will be greater, the greater the vertical dia- ^ 
meter and the less the thickness of the tube ; since a ^ 
larger proportion of the material will be at the greatest ^ 
distance from the neutral axis. ^fiaa^ 

D tubes have also been used with the flat side '°' ^'^* 
vertical. The discussion in section 98 has shown a difference 
of about one per cent, in favour of the D tube consisting of a 
semicircle and its diameter. Square or rectangular tubes have 
not been used to any great extent for the chain-struts, but the 
discussion in section 99 shows that for equal sectional area 
and diameter they are much stronger than the round^be. If the 

Digitized by V^jOOQ 



326 



Details 



CHAP. XXI ir. 



horizontal diameter b be constant, and the vertical unlimited, a 

rectangular tube with great vertical diameter will be stronger, 

weight for weight, than a square tube ; Z approaching the value 

A b 

— , corresponding to the whole sectional area being concen- 

2 

trated at the two sides parallel to the neutral axis, the other 

two sides being indefinitely thin. 

A still more economical section for the lower fork tubes would 

be a hollow rectangle, the vertical sides being longer and thicker 

than the horizontal. This might be attained by drawing a thin 
rectangular tube of uniform thickness, and 
brazing two flat strips on its wider faces 

(fig- 317). 

Figure 318 shows the sections of round, 
D, and square tubes of equal perimeter. 

Loiver Fork with Bridge Bracket. — If the 
cog-wheel on the crank- axle be placed be- 
tween the two bearings, as in the ' Ormonde ' 
bicycle (fig. 259), the chain will run between 
the two lower fork sides (fig. 319), and there 
will be no bending stresses on the fork tubes due to the pull of the 
chain. The objection to this arrangement is that the tread must 

be increased con- 
siderably in order 
to have a bearing 
outside the cog- 
wheel on the crank- 
axle. 

^Referee* Lmver 




Fig. 3t8. 



/- 



t 



d 



Fig. 319. 



^^^>t i^^/->^.— In the * Referee' bicycle the bending on the fork 




Fig. aao. 



sides is eliminated by an ingenious arrangemenUshown in figure 

Digitized by VjOOQIC 



CHAP. XXIII. 



The Frame 



327 



320. The fork tubes are parallel to the plane of the chain, and 
instead of running forward to the crank-bracket, they end at an 
intermediate bridge piece connected to the crank-bracket by two 
parallel tubes lying closer together than the fork sides. If the 
end lugs to which the ends of the driving-wheel spindle are 
fastened were central with the tubes, the bending stresses might 
be entirely confined to the bridge piece. 

239. Tandem Bicycle Frames. — The design of tandem frames 
is much more difficult than that of single bicycle frames, since 

Fic. 321. 




yV^y'|ff??^/r/////^/yV////VJt^////^/ ^/^^//>*^> 



Fig. 33<. 



Strm Diagram 





Sktarhg Forci 
SMk. 400&i»/meh 

Fig. 323. 

the weight to be carried is double, and the span of the pre- 
sent popular type of tandem from centre to centre of wheels is 
also greater than that of the single machine. The maximum 
bending-moment on a tandem frame is therefore much greater 
than that on the frame of a single bicycle. If, however, one 
of the riders overhangs the wheel centre, the maximum bending- 
moment on the frame may actually be less than on that a)f the 
single machine. ° 9' '^"^ '^y Google 



328 Details chap, xxhl 

In the *Rucker' tandem bicycle (fig. 135) each rider was 
nearly vertically over the centre of his driving-wheel, and the 
inaximum bending-moment on the backbone was not very great. 

Example,-:-y^'\\}ci 120 lbs. applied at .the rear saddle, with an 
overhang of 10 in., the maximum bending-moment was 

M = 120 X 10 = 1,200 inch-lbs. 

If the maximum stress on the backbone had not to exceed 
20,000 lbs., 

'\ Z = 1,200 = '060 in.' 

20,000 

A tube i^ in. diameter, 17 W.G., would have been sufficient 

It may be noticed that with one rider overhanging the wheel- 
base the bending-moment changes sign about the middle of the 
frame^/>. if the backbone were originally straight, while carrying 
the riders the rear portion would be slightly bent with its centre 
of curvature downwards, the front portion with its centre of 
curvature upwards. 

Figure 321 shows the frame of a rear-driving tandem Safety with 
both riders between the wheel centres, similar to that of figure 296. 
The top- and bottom-tubes of the forward portion of the frame 
should be arranged so that they intersect on the vertical through 
the front wheel centre, but in order to make the stress-diagram 
more general they are not so shown in figure 321. Figure 322 
shows the stress-diagram, regarding the frame as a plane structure, 
while figures 323 and 324 are the shearijg-force and bending- 
moment diagrams respectively. 

The scale of the stress-diagram, 200 lbs. to an inch, has been 
chosen half that of the stress-diagram of the single machine 
(fig. 297), and a few comparisons may be made. The thrusts on 
the top-tubes, a d and b g, of the tandem are respectively about 2^ 
and 3^ times that on the top-tube of the single machine. The 
pull on the front bottom-tube, e k^ of the tandem is about 2^ times 
that for the single. The thrusts on the diagonal,/^, and the front 
down-tube, ef, are respectively 3^ and 6^ times that on the down- 
tube of the single machine ; while the pull on the rear down-tube, 
ghy of the tandem is about four times the thrust on the down-tube 
of the single machine. The pulls on the lower back fork, m h^ and 

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C.TAP. xxiif. The Frame 329 

on the middle chain- struts,//, are respectively about 2 and 3^ times 
that on the lower back fork of the single machine. 

In making the above comparisons it should be remembered 
that the single frame (fig. 296) is relatively higher than the tandem 
frame (figl 321) illustrated. If the latter were higher, the stresses 
on its members would be less. 

The scale of the bending-moment diagram (fig. 324) is 
4,000 inch-lbs. to an inch, twice that for the single machine 
(fig. 299). The maximum bending-moment is more than three 
times that for the single machine. 

A glance at the shearing-force diagrams (figs. 298 and 323) 
shows that on a vertical section passing through the rear down- 
tube of the tandem the shear is negative, while at the down-tube 
of the single machine the shear is positive. Hence the stress on 
the rear down-tube is tensile. This can also be shown by a 
glance at the force-polygon, I m h ^/(fig. 322), for the five forces 
acting at the rear crank-bracket (fig. 321) ; the force h g^ being 
directed away from the bracket, indicates a pull on the down-tube. 

The thrust on the tube de is small, and vanishes when the front 
top- and bottom-tubes intersect vertically above the front wheel 
centre. The thrust on the diagonal tube, /^, of the middle 
parallelogram is 60 lbs., smaller than the thrust or pull on any other 
member of the frame. This explains why the frame with open 
parallelogram (fig. 267) and those with no proper diagonal bracing 
are able to stind for any time under the loads to which they are 
subjected. 

The maximum stresses on the members of the frame due to 
the vertical loads will be krgely increased by the stresses due to 
the pull of the chain, the thrust of the pedals, and the seat 
adjustment, as already discussed. The magnitudes of these 
stresses will be proportionately greater in the tandem than in the 
single frame. 

Tandem frames may be also subjected to considerable twisting 
strains. If the front and rear riders sit on opposite sides of the 
central plane of the machine, the middle part will be subjected to 
torsion. This torsion can be best resisted by one tube of large 
diameter ; no arrangement of bracing in a plane can strengthen a 
tandem frame against twisting. 

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330 Details chap. xxm. 

240. Stresses on Tricycle Frames.— Nearly all the frames of 
early tricycles were unbraced, and their strength depended entirely 
on the thickness and diameter of the tubes used, one exception 
being that of the * Coventry Rotary ' (fig. 144), the side portion of 
which formed practically a triangular truss ; another, that of the 
* Invincible,' a central portion of which was fairly well braced. 

In the early *Cripper' tricycles the frame was usually of T 
shape, and consisted of a bridge supporting the axle, and a b€uk' 
bone supporting the saddle and crank-axle. With the usual 
arrangements of wheels and saddle, about three-eighths of the 
weight of the rider rested on each driving-wheel. The strength 
of the bridge can easily be calculated thus : 

Example L — If the weight transferred to the middle of the 
bridge be 120 lbs., the track of the wheels be 30 inches apart, the 
middle of the bridge is subjected to a bending-moment 

-J/ = = ^ = 000 mch-lbs. 

4 4 

If the maximum stress be 20,000 lbs. per sq. in., 

^ M 000 ^ . , 

J 20,000 

A tube i^ in. diameter, 17 W.G. (see Table IV.), will be 
sufficient. 

In calculating the strength of the backbone the worst case will 
be when the total weight of the rider is applied at the crank-axle. 
Taking the relative distances as in the Safety bicycle {^%. 296), 

3/=:'5o„>i-^3Xi9^ 6oinch.lbs. 

42 ^ 

Z^=. ^ ' V ^ = -078 m.^ 
20,000 

A tube 1 2 in. diameter, 16 W.G., will be sufficient. 

With frames made on the same general design as that of the 
Safety bicycle the stresses will be calculated as already discussed 
for the bicycle, the only important additional part being the bridge 
supporting the axle. Its strength may be calculated as in the 
above example. The stresses on the axle-bridge are diminished 
by taking the seat-struts to the outer end of the bridge, as in 

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CHAP. xriTi. 



The Frame 



331 



*Starley*s' frame (fig. 153), and in the * Singer ' frame (fig. 273). 
Figure 325 is plan and elevation of the rear portion of * Starley's ' 
frame. At the outer end of the bridge, which in this case is a 
tube concentric with the axle, there are three forces acting, which, 
however, do not all lie in the same plane. These are the re 
action of the wheel R, the thrust T along the seat-strut, and 
the pull A along the bridge. These forces in the plan are 
denoted by the corresponding small letters, and in the elevation 





i k, { 




Fig. 326; 



Fig. 325. 



with the corresponding small letter with a dash (*) attached. If 
a force, H^ parallel to the chain-struts be applied at the end of 
the axle, the four forces H^ A^ R, and T will be in equilibrium, 
and may be represented by four successive edges of a tetrahedron 
respectively parallel to the direction of the forces. The plan and 
elevation of this tetrahedron, k I m n^ is drawn in figure 326, the 
length of the side corresponding to the force R being drawn to 
any convenient scale. The magnitudes of the forces ZT, A^ and 

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332 



Details 



CHAP. 3nciii. 



T'can be measured off from the true lengths of the corresponding 
edges of the tetrahedron. These are shown in the plan. 

Example IL — Suppose ^ = 60 lbs., and the direction of the 
tubes is such that H^^ ^o lbs., the resultant of the three forces 
H A^ and T is equal and opposite to 11) thus the bridge is 
s.ubjected to a bending in the plane of the chain-strut. If 
the distance from the end to the centre of the bridge be 14 in., 

M=i 30 X 14 = 420 inch-lbs. 
If the bridge be i in. diameter and 20 W.G., 

Z= '0253 in.^ and/= "^^^ = 16,600 lbs. per sq. in. 
*o253 • 

The axle will also be subjected to a bending-moment in a vertical 
plane, due to the fact that the centre of the wheel is overhung 
some distance from the end of the bridge. If the overhang be 
3 in., the bending-moment = 60 x 3 = 180 inch-lbs., a smaller 
value than that found above. 

241. The Front-frame. — The front-frame {fig, 327) is acted on 
by three forces— the reaction R^ of the front wheel on its spindle, 

and the reactions Hi and H^ of the 
ball-head on the steering-tube. 
Since the front-frame is in equili- 
brium under the action of these 
forces, they must all pass through 
a point hy situated somewhere on 
the vertical line passing through 
the wheel centre. If we assume 
that the direction of the force H^ 
at the upper bearing of the ball- 
head is at right angles to the head, 
the point h will be determined ; 
the magnitudes of ZT, and ZT, can 
then easily be determined by an 
application of the triangle of forces. 
Let ri, ^„ h^ be the components of the forces -^1, H^y H^ at 
right angles to the ball-head, then the front-frame is subjected 
to a bending-moment due to these three forces, and the bending- 

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Fig. 327. 



CHAP. XXIII. The Frame 333 

moment diagram may be represented by the shaded triangle 

(fig- 327). 

Example L — \{ R^ = 40 lbs., the slope of the ball head be such 
that r = 20 lbs., and the distance between the lines of action of 
ri and ^, be 17 in., the greatest bending-moment will be 

i)/= 20 X 17 = 340 inch-lbs. 

If the steering-tube be i in. diameter, 20 W.G., we get from 
Table IV., p. 113, Zs='o253, and the maximum stress on the 
tube will be 



/= 1, = ^^^ =s 13,440 lbs. per sq. in. 



253 

It is now becoming usual to strengthen the steering-tube by a 
liner at its lower end. P'or the nearest approximation to uniform 
strength throughout its length it is evident, from the shape of the 
bending-moment diagram, that its section should vary uniformly 
from top to bottom. If the liner extend half the length of the 
ball-head the tube will be of equal strength at the bottom and 
the middle, and will have an excess of strength at other points. 

In a tandem bicycle the nature of the forces on the front- 
frame are exactly the same as above discussed, but are greater in 
magnitude. If in a tandem R^ = 100 lbs., with the same dimen- 
sions as given above, yl/will be 850 inch-lbs. 

Example II. — If the steering-tube be i in. diameter, 18W.G., 
and be reinforced by a liner, 18 W.G., the combined thickness of 
tube and liner is '096 inches, a little greater than that of a tube 
13 W.G. The Z of a i-in. tube 13 W.G. is '055, therefore the 
maximum stress on the tube is 

/= ^^ = 15,460 lbs. per sq. in. 
'055 

The Fork Sides, at their junction to the crown, have to resist 
nearly the maximum bending-moment (fig. 327). They are usually 
made of tubes of oval section, tapering towards the wheel centre. 
The discussion of tubes of oval and rectangular sections (sees. 97 
and 99) has shown the latter form to be the superior ; and, as 
there is no limitation of space to be considered in designing the 

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334 Details 



CHAP. xxin. 



front fork, the sides may with advantage be made of rectangular 
tube. If the rectangular tube be of uniform thickness, it has 
been stated (sec. 99) that for the greatest strength its depth should 
be three times its width. A still greater economy can be got by 
thickening the sides of the tube parallel to the neutral 
axis, either by brazing strips to a tube of uniform thickness 
(fig. 328), or during the process of drawing. 

Pressure on Crown-plates, — The forces acting on the 
fork (fig. 329) are -^1, F^y and F^^ the reactions of the 
crown-plates. 

Example III, — Let the crown-plates be | in. apart 
Taking the components of these forces at right angles to 
the steering-head, and taking moments about the centre 

Fig. 328. Qf ^j^g yppgj. pj^^g^ ^.g g^j 

/. = \ — 5 X 20 = 440 lbs., i,e, 220 lbs. on each side. 

75 

In the same way we get 

/a = 420 lbs., i.e, 210 lbs. on each side. 
The great advantage of the plate crown over the old solid 
crown is that the forces /j and /j are made to act as far apart as 
possible with a given depth of crown, whereas 
with the older solid crown the pressure was dis- 
tributed over quite an appreciable distance, so 
that the distance between the resultant pressures 
/, and /j was small ; the forces /, and f^ 
were therefore correspondingly larger, since the 
moment to be resisted was the same. 

In some recent designs of crowns the two 
plates are united by short tubes outside of the 
* ^^ fork sides. As regards the attachment of the 

fork sides, this arrangement is therefore practically equivalent to 
the old solid crown. If any strengthening is desired, it should be 
done by an inside liner. Triple crown-plates have been used for 
tandems ; but, as far as we can see, the middle plate contributes 
nothing to the strength of the joint, and may with advantage be 
omitted 

Handle bar, — The handle-bar, when pulled upwards by the 

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CHAP, xxiii. The Frame 335 

rider with a force P at each handle, is subjected to a bending- 
moment Pl^l being the distance from the handle to the centre of 
the handle-pillar. 

Example IV, — If / = 12 in., then J/" = 12 -Pinch-lbs. Let 
the handle-bar be ^ in. diameter, 18 W.G., Z= '0244 in.^, and let 
the maximum stress on the handle-bar,/ be 20,000 lbs. per sq. in.; 
substituting in the formula M = Z/, we get 

12 -P =5 '0244 X 20,000. 
.-. P = 41 lbs. 

That is, a total upward pull of 82 lbs. will produce a maximum 
stress of 20,000 lbs. per sq. in. 

If the handles be bent backwards, the handle-bar is also sub- 
jected to a twisting-moment, which, however, usually produces 
smaller stresses than the bending-moment. For example, if the 
handle be bent 4 in. backwards, the twisting-moment T = 4 F, 
The modulus of resistance to torsion of a | in. tube, 18 W.G., is, 
from Table IV., p. 113, -0488 in.^ ; and therefore, with the same 
value for -Pas in the above example, we get 4 x 41 = '0488^5 

or /= 3,360 lbs. per sq. in. 

242. General Considerations Belating to Design of Frame. — 
The importance of having the forces acting on a tie or strut 
exactly central cannot be over-estimated ; the few examples 
already given above show how the maximum stress is enormously 
increased by a very slight deviation of the applied force from the 
axis. In iron bridge or roof building, this point is thoroughly 
appreciated by engineers ; but in bicycle building the forces acting 
on each tube of a frame are, as a rule, so small that tubes of the 
smallest section theoretically possible cannot be conveniently 
made. The tubes on the market are so much greater in sectional 
area than those of minimum theoretical section that they are 
strong enough to resist the increased stresses due to eccentricity 
of application of the forces ; and thus little or no attention has 
been paid to this important point of design. 

The consideration of the shearing-force and bending-moment 
diagrams simultaneously with the outline of the frame is. instruc- 
tive, and reveals at a glance some weak points in various types of 

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336 Details chap. 33111. 

frames. The vertical section at any point of a properly braced 
frame will cut three members ; the moment of the horizontal com- 
ponents of the forces acting on these members will be equal to 
the bending-moment at the section, while the sum of the vertical 
components will be equal to the shearing-force. Therefore, in 
general, any part of a frame in which the vertical depth is small 
will be a place of weakness. The Ladies' Safety frames (figs. 264 
and 265) have already been discussed. That shown in figure 266 
is weakest at the point of crossing of the two tubes to the steering- 
head, the depth of the frame being zero at this point, so that only 
the bending resistance of these tubes can be relied on. The cross- 
frame (fig. 249) is very weak in the backbone, just behind the point 
where the down-tube crosses it. The Sparkbrook frame (fig. 253) 
is weakest at a point on the top-tube, just in front of the point of 
attachment of the tube from the crank -bracket. The frames 
(figs. 244 and 245) are practically equivalent to a single tube un- 
braced. The frames shown in figures 247-251 are weakest just 
behind the steering-head. 

The consideration of the shearing-force curve shows the 
necessity for the provision of the diagonal o{ the central paral- 
lelogram in a tandem frame. The top- and bottom-tubes are 
nearly horizontal, so that if they were acted on by forces parallel 
to their axes they could not resist the shearing-force. The 
shearing-force must therefore be resisted by an inclined member 
of the frame, or, failing this, the forces on the top- and bottom- 
tubes cannot be parallel to their axes, and they must be subjected 
to bending. The same remarks apply to a frame formed by the 
duplication of either the top- or bottom-tubes without the provision 
of a diagonal, as in figure 268. 



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337 



CHAPTER XXIV 

WHEELS 

243. Introductory. — Wheels may be divided into two classes — 
rolling wheels and non-rolling wheels. In rolling wheels the in- 
stantaneous axis of rotation is at the circumference ; examples 
are, bicycle wheels, vehicle wheels, railway carriage wheels, &c. 
Such rolling wheels have, in general, a fixed axis of rotation 
relative to the frame, which has a motion of translation when the 
wheel rolls. Non-rolling wheels are those not included in the 
above class ; they may be mounted on fixed axes, their circum- 
ferences being free, or in contact with other wheels. Such are 
fly-wheels, gear-wheels, rope- or belt-pulleys, &c. 

Wheels may again be subdivided, from a structural point of 
view, into solid wheels, wheels with arms, nave, and rim, cast or 
stamped in one piece, and built-up wheels. In a solid rolling 
wheel, the load applied at the centre of the wheel is transmitted 
by compression of the material of the wheel to its point of contact 
with the ground. 

244. Compression-spoke Wheels.— A built-up wheel usually 
consists of three portions— the hub (nave, or boss), at the centre 
of the wheel ; the rim or periphery of the wheel ; and the spokes 
or arms, connecting the rim to the hub. Built-up wheels may be 
divided again into two classes, according to the method of action 
of the spokes. A wheel may be conceived to be made without a 
rim, consisting only of nave and spokes {^g, 330). In this case 
the load applied at the centre of the wheel is evidently transmitted 
by compression of the spoke, which is at the instant in contact 
with the ground. If the spokes are numerous, the rolling motion 
over a hard surface may be made fairly regular. In the ordinary 

Digitized by Vj 2 



338 



Details 



CHAP. XXIV. 



wooden cart or carriage wheel (fig. 331), the ends of the 
spokes are connected by wooden felloes, / the felloes being 
mortised to receive the spoke ends, and an iron tjrre, /, encircles 
the whole. This iron tyre is usually shrunk on when hot, and in 
cooling it compresses the felloes and spokes. This construction 
is very simple, since only one piece —the iron tyre — is required 




Fig. 330, 




Fig. 331. 



to bind the whole structure together. The compression wheel 
compares favourably in this respect with the tension wheel. On 

■the other hand, the sectional area of the spokes must be great, in 
order to resist buckling under the compression ; very light wheels 
cannot, therefore, be made with compression spokes. The method 

. of transmitting the load from the centre of the wheel to the ground 
is practically the same as in figure 330. 

245. Tension-spoke Wheels. — The initial stresses in a bicycle 
wheel of the usual construction are exactly the reverse of those 




/7777777777777Z 

Fig. 332. 




Fig. 333. 



on the compression-spoke wheel (fig. 331). The method of action 
of the tension-spoke wheel may be shown as follows. Suppose 
the hub connected by a single wire, a, to a point on the top of the 

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■J 



CHAP. XXIT. 



Wheels 



339 



rim, a load applied at the centre of the wheel would be transferred 
to the top of the rim and would tend to flatten it, the sides would 
tend to bulge outwards, and the rim to assume the shape shown 
by the dotted lines (fig. 332). This horizontal bulging might be 
prevented by connecting the hub to the rim by two additional 
spokes, b and c. If, now, a load were applied at the centre of the 
wheel, the three spokes, a, ^, and ^, would be subjected to tension, 
and if the rim were not very stiff" it would tend to flatten at its 
lower part, as indicated in figure 333. Additional spokes, d and 
^, would restrain this bulging. Thus, by using a sufficient number 
of spokes capable of resisting tension, the load applied at the 
centre of the wheel can be transmitted to the ground without 
appreciable distortion of the rim. 

246. Initial Compression in Eim.— In building a bicycle- 
wheel the spokes are always screwed up until they are fairly tight. 





Fig. 334. 



Fig. 335 



The tension on all the spokes should be, of course, the same. 
This tightening up of the spokes will throw an initial compression 
on the rim, which may be determined as follows. Suppose the 
rim cut by a plane, A O B, passing through the centre of the 
wheel (fig. 334). Consider the equilibrium of the upper portion 
of the rim of the wheel : it is acted on by the pulls of the spokes 
a, d, c^ d , . . and the reactions Fy and -^2 o^ ^^^ lower part of 
the rim at A and B. If the tension / be the same in all the 
spokes, the force-polygon a^ b^ c^ d ^ . . {^%, 335) will be half of 
a regular polygon. The sum of the forces F2XA and B will be 
equal to the closing side, LM^oi the force-polygon. 

If the number of spokes in the wheel be great, the force- 
polygon (fig. 335) may be considered a circle. Then, if « be the 

z ^ 



340 Details ' chap. xht. 

number of spokes in the wheel, the circumference L M (fig. 335) 
is equal to — , the diameter ZM to — . 

2 IT 

But 2F=ZM=^^; 

v 

therefore i^= (i) 

27r 

Example, — The driving-wheel of a Safety has 40 spokes, 
No. 14 W.G., which are screwed up to a tension of 10,000 lbs. 
per sq. in. Find the compression on the rim. 

From Table XII., page 346, the sectional area of each spoke is 
•00503 sq. in. ; the pull / is therefore '00503 x 10,000 = 50-3 lbs. 
Substituting in (i), 

« 40 X w% ,1 

J7= ^ 2_p =r 320 lbs. 

2 X 31416 

247. Direct-spoke Driving-wheel.— The mode of transmission 
of the load from the centre of a bicycle wheel to the ground 
having been explained, it remains to show how the driving eflfort 
is transmitted from the hub to the rim. In 
a large gear-wheel the arms are rigidly fixed 
to the nave, and while a driving effort is 
being exerted, the arms press on the rim of 
the wheel in a tangential direction. Thus 
each arm may be considered as a beam 
rigidly fixed to the nave and loaded by a 
force at its end near the rim. . The spokes 
of a bicycle wheel are not stiff enough 
to transmit in this manner forces transverse to their axis, being 
to all -intents and purposes perfectly flexible. When a driving 
force is exerted the hub turns through a small angle without 
moving the rim, so that the spokes whose axes initially all passed 
through the centre of the wheel now touch a circle, s (fig. 336). 
Let r be the radius of this circle, and P the pull of the driving 
chain which is exerted at a radius V?. Considering the equilibrium 
of the hub, the moment of the force P about the centre is PR : 

. . Digitized by CjOOQIC , 




CHAP. XXIV. 



Wheels 341 



the moment of the forces due to the pull of the spokes on the 

hub is 

n t r. 
Thus, FJi=nfr, 

and r = ^ (2) 

nt 

Example, — Let the driving-wheel have 40 spokes, each with 
on initial tension of 50 lbs. ; let the pull of the chain be 
300 lbs., and be exerted at a radius of i^ in. Find the size of the 
circle j, and the angle of displacement of the hub. 

Substituting in (2) 

40 X 50 

Figure 337 is a drawing showing the displacement of the hub. 
I^t cdh^ the radius of the circle touched by the spokes, ba the 
initial position of a spoke, b^ a} the displaced position, 
and let the distance of the point of attachment of the ^[ ^1 
spokes from the centre of the hub be | in. ; the angle 
of displacement of the hub, aca^ will be approximately 



or, 



!g^= -257 radians, 
257_xjlo^,^.ydeg. 



-*^ 



I"- 



Fig. 337- 



If the driving effort be reversed, as in back-pedalling, the hub 
will first return to its original position relative to the rim, and 
then be displaced in the opposite direction before the reversed 
driving effort can be transmitted. 

Thus, a direct-spoke bicycle wheel is not a rigid structure, but 
has quite a perceptible amount of tangential flexibility between 
the hub and the rim. 

Lever Tension Driving-wheels, — In the early days of the 
* Ordinary,' wheels were often made with a pair of long levers 
projecting from the hub, from the ends of which wires went 
off to the rim. These tangential wires were adjustable, and' 
by tightening them the rim was moved round relative to .the 



342 



Details 



CHAP. XXIT. 



hub, and thus the tension on the spokes could be adjusted. The 
tangential driving effort was also supposed to be transferred from 
the hub to the rim by the lever and tangent wires, while the 




Fig. 338. 

radial spokes only transmitted the weight from the hub to the 
rim. Figure 338 shows the * Ariel ' bicycle with a pair of lever 
tension wheels. 

248. Tangent-spoke Wheels. — In a tangent wheel the spokes 
are not arranged radially, but touch a circle concentric with the 

hub (fig. 339). The pull on the 
tangent-spokes indicated by the full 
lines would tend to make the hub 
turn in the direction of the arrow. 
Another set of spokes, represented 
by the dotted lines, must be laid 
inclined in the opposite direction, 
^^^' ^^^ so that the hub may be in equili- 

brium. The initial tension should be the same on all the 
spokes. 

Let a driving effort in the direction of the arrow be applied at 
the hub. This will have the effect of increasing the tension on 
one half of the spokes and diminishing the tension on the other 
half. If r be the radius of the circle to which the spokes are 




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CHAP. xxnr. W/teels 343 

tangential, /j and t^ the tensions on the tight and slack sppkes 
respectively, 'the total tangential pull of the spokes at the hub is 



-At,-t,). 



2 



Therefore 
from which 



nr 






Example, — Let r be J in., the spokes 15 W.G., the modulus 
of elasticity of the spokes 10,000 tons per sq. in. ; then, 
taking the rest of the data as in section 247, find the angle of 
displacement of the hub relative to the rim under the driving 
effort. 

Substituting in (3), 

40 xj ^' 

The sectional area of each spoke (Table XII.) is '00407 sq. in. ; 
the increase or diminution of the tension due to the pull of the 
chain is therefore 

^-^ = 3,156 lbs. per sq. in. = 1*41 tons per sq. in. 

2 X '00407 

The extension of one set of spokes and the contraction of the 

other set will thus be - ^'-th part of their original length, which 
10,000 

length in a 28-in. driving-wheel is about 12 in. The displacement 

of a point on the circle of radius | in. is thus 



I'4IXI2 , . 

^-^ = '00169 m. 



10,000 
The angle the hub is displaced relative to the rim will be 



:H^69_x 180 ^ .„ ^ 

i X T 



Digitized by 



Google 



344 



Details chap. iL.Tin. 



Comparing this example with that of section 247, the 
superiority of the tangent wheel in tangential stiffness is apparent. 
In this example it should be noted that the initial pull on the 
spokes does not enter into the calculation. Consequently, the 
initial pull on tangent- spokes may with advantage be less than 
that on direct-spokes. 

249. Direct-spokes.— The spokes of a direct-spoke wheel are 
usually of the form shown in figure 340, the conical head at the 

, , . end engaging in the rim, and the 
\rm\ ' ' _ -L-j * ' ' ' — d other end being screwed into the 
Pj^j hub. For the sake of preserving the 

spoke of equal strength through- 
out, its end is often butted before being screwed (fig. 341), 
the section at the bottom of the thread in this case being 
^^^ at least as great as at the middle 

HH 1 i ^ '3 of the spoke. 

Fig. 3n. -^^ ^^"^^ 339 the spoke is 

shown making an acute angle with 
the hub. As a matter of fact, under the action of a driving effort 
the spokes near the hub will be bent, as shown exaggerated in 
figure 342. The continual flexure under the driving 
effort weakens and ultimately causes breakage of 
direct spokes, unless made of greater sectional area 
than would be necessary if they could be connected 
to the hub by some form of pin-joint. The conical 
head lies loosely in the rim, and being quite free to 
Fig. 342. adjust itself to any alteration of direction, the spoke 
near the rim is not subjected to such severe strain- 
ing actions as at the hub. 

250. Tangent-spokes. — Tangent-spokes cannot be con- 
veniently screwed into the hub, but are threaded through holes 

in a flange of the hub, the end of the 

4 — 1-^— -^ ^ spoke being made as indicated in figure 

" . •i'' ^jEEaS-Hj, 343. This sharp bend of the spoke 

p,^, ^^^ . seriously affects its strength. Let P be 

the pull on the spoke,, and d its diameter. 

On the section of the sp6ke at a there will be a bending- 

moment P x^ x being the distance between the middle of the 

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CHAP. XXIV. 



W/tee/s 



345 



section a and the hub flange ; this distance may be taken 
approximately equal to d. The bending-moment is then 

» 3 

J^dyZ^^-i and the maximum stress, /, due to bending will 
be found by substitution in the formula M=s Zf, Therefore 

and 



/= ^.a 



The tensile stress on the middle of the spoke is 

Thus the stress due to bending on the section at the corner is 
eight times that on the body of the spoke due to a straight 
pull. 

Figure 344 shows a tangent-spoke strengthened at the end by 
butting. 

The ends of tangent-spokes must be fastened to the rim by 
means of nuts or nipples. The nipple has its inner surface 
screwed to fit the screw on the end of the 

spoke, has a conical head which lies in a cor- / ^ 7- 

responding counter-sunk hole in the rim, and & 

a square or hexagonal body threaded through ^'°- ^^^ 

the hole in the rim for screwing up by means of a small 

spanner. 

A piece of wire threaded through the hub flange (fig. 345), and 
its ends fastened to the rim by nipples in the usual way, is often 
used to form a pair of tangent spokes. 

The objection to the spoke shown f^///^^^ > ^r^77777% 

in figure 343 still holds with regard 
to this form ; but the fact that no Fig. 345. 

head has to be formed at the hub 

probably makes it slightly stronger than a single spoke of the 
same diameter headed at the end. 

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346 



Details 



CHAP. XSIT. 



Fig. 346. 



Figure 346 shows the form of tangent-spoke used by the 
St. George's Engineering Co. in the * Rapid ' cycle wheels. The 
spoke is quite straight from end to end, and is 
" fastened to the rim in the usual way by a 
nipple. It is fastened to the hub by means of 
a short stud projecting from the hub flange, 
a small hole being drilled in the projecting head of the stud, and 
the spoke threaded through it. The headed end of the spoke is 
pulled up against the stud. Spokes of this form are not sub- 
jected to bending, and are therefore much stronger than tangent- 
spokes of the usual form of the same gauge. 

Table XII. — Sectional Areas and Weights per 100 ft. 
Length of Steel Spokes. 



Imperial 








standard 


Diameter 


Sectional area 


Weight of 100 ft. 


wire gauge 










In. 


Sq. in. 


Lbs. 


6 


•192 


•0289s 


10005 


7 


•176 


•02433 


8-409 


8 


•160 


•0201 1 


6950 


9 


•144 


•01629 


5-629 


10 


•128 


•01287 


4*447 J 


II 


•116 


•01057 


3-652 , 


12 


•104 


•00849 


2-936 i 


13 


> -092 


■00665 


2-298, 


14 


•080 


•00503 


1738 


15 


•072 


■00407 


1-407 


16 


•064 


•00322 


I-H2 


17 


•056 


•00246 


-850 


18 


•048 


•OOI81 


•625 1 


19 


•040 


•00126 


•434 


20 


•036 


•00102 


•352 



251. Sharp's Tangent WheeL — The distinctive features of 
this wheel, invented by the author, are illustrated in figure 347. 
The hub is suspended from the rim by a series of wire loops, one 
loop forming a pair of spokes. In figure 347, for the sake of 
clearness of illustration, one loop or pair of spokes is shown 
thickened. The ends are fastened to the rim by nuts or nipples 
in the usual way. There is no fastening of the spokes to the hub, 

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CHAP. ZXIV. 



WAee/s 



347 



beyond that due to friction. Figure 348 represents the appear- 
ance of the spokes in contact with the hub. The arc of contact 
of the spoke and hub is a spiral, so that all the ends of the 
spokes on one side of the middle plane of the wheel begin contact 
with the hub at the same distance from the middle, the other ends 
all leaving the hub nearer the middle plane. A wheel could be 
made with loops of wire having circular contact with the hub, but 
it would not be symmetrical, and the spokes would not all be of 





Fig. 347. 



Fig. 348. 



the same length. By making the spokes have a spiral arc of con- 
tact with the hub,. the positions of all the spokes relative to the 
hub are exactly similar, the wheel is symmetrical, and the spokes 
are all of the same length. It will be noticed that there are no 
sudden bends in the ookes, so that they are much stronger than 
in the ordinary tangent wheel, no additional bending stresses being 
introduced. For non-driving cycle wheels there can be no ques- 
tion as to the sufficiency of the hub fastening, but it may at 
first sight seem startling that the mere friction of the spokes 
on the hub should be sufficient to transmit the driving effort to 
the rim, though it is well known that by coiling a rope round a 
smooth drum almost any amount of friction can be obtained. 
This system of construction is applicable to all types of built-up 
metal wheels, and has been applied with success to fly-wheels and 
belt-pulleys, and to the * Biggest Wheel on Earth ' — the gigantic 
pleasure-wheel at Earl's Court. 

Let / be the initial tension on the spokes ; then, while the 
driving eflbrt is being exerted, the tension on one half of each loop 

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348 Details chap. x>civ. 

rises to /i, and on the other half falls to /j- ^^ A be very much 
greater than /g there will not be sufficient friction between the hub 
and the wire, and slipping will occur. Let 
fl be the angle of contact (fig. 349) and 
H the coefficient of friction between the 
spoke and hub. Then, when slipping 
takes place, 

>1 =€^^ (4) 

Fig. 349. h 

If -1 is less than determined by (4), slipping will not occur. 
Equation (4) may be written in the form, 

♦2 

the symbol log -1 denoting the logarithm, to the * Naperian ' or 
* h 

natural base, of the number — . Using a table of * common ' loga- 
rithras, a more convenient form is— 

log^^'MM\i^ (S) 

Example I,r-P^ driving-wheel 28 in., diameter, on this system, 
has 40 spokes wrapped round a cylindrical portion of the hub 
i^ in. diameter, the initial tension on each spoke is 60 lbs., the 
pull on the chain is 300 lbs., and is exerted at a radius of i J^ in. 
Find whether slipping will take place or not. 

Let the arc of contact be half a turn, as shown approximately 
in figure 347, then = tt, the coefficient of friction fi for metal 
on metal dry surface will be about from '2 to -35, but assuming 
that oil from the bearing may get between the surfaces, we may 
take a low value, say 0*15 ; substituting in (5) 

log ± = -4343 X -15 X 3-1416 = -2046, 

from which, consulting a table of logarithms, 

-1 s= I -602 

^ Digitized by Google 



CHAP. XXIV. Wheels 349 

when slipping takes place. But from (3) 

/._/,= 2_J^ = ? .^ 3oo_ x_Ll ^ lbs. 
n r 40 X $ 

Therefore /j = 6o + i5 = 7S 

/2 = 6o— 15 = 45 

and } = 1-5. 

Thus, with the above conditions, slipping will not occur. 

As a matter of experiment, the author finds that with such a 
smooth hub and an arc of contact of half a turn slipping takes 
place in riding up steep hills only when the spokes are initially 
slacker than is usual in ordinary tangent wheels. 

Arc of Contact between Spokes and Hub, — The pair of spokes 
(fig. 347) is shown having an arc of contact with the hub of nearly 
two right angles. The arc of contact may 
be varied. For example, keeping the end 
tf , fixed, the other end of the spoke may 
be moved from a\ to a^2> or even further, 
so that the arc of contact may be as 
shown in figure 350. In this case there 
are five spoke ends left between the ends Fig. 350. 

of one pair. In general, 4 « + i spokes must be left between 
the ends of the same pair, n being an integer. 

In this wheel, should one of the spokes break, a whole loop 
of wire must be removed. Of course the tendency to break is, as 
already shown, far less than in direct or tangent spokes of the 
usual type. If the arc of contact, however, is as shown in 
figure 347, and a pair of spokes are removed from the wheel, a 
great additional tension will be thrown on the spoke between the 
two vacant spaces. If the angle of contact shown in figure 350 
be adopted, there will still remain five spokes between the two 
vacant spaces, so that the additional tension thrown on any single 
spoke will not be abnormally great. 

Grooved Hubs. — The hub surface in contact with the spokes 
may be left quite smooth, with merely a small flange to preserve 
th6 spread of the spokes. The parts of the spokes wrapped round 

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350 Details 



CHAP. XZIT. 



<t3' 



the hub will lie in contact side by side (f\.%. 348) Should one 
break and be removed from the wheel, the remaining spokes in 
contact with the hub will close up the space vacated by the 
broken one. In putting in a new spoke they will have to be 
again separated. Spiral grooves may be cut round the hub, so 
that each spoke may lie in its own special groove, and if one breaks, 
the space will be left quite clear for the new spoke to replace it 

The grooves may be made so as to considerably increase the 
frictional grip on the nave. Figure 351 shows the section of a 

spoke in a groove, the 
\p spoke touching the sides, 
^ — J but not the bottom of the 
"^ ^^^^ ^^ groove. It is pressed to 

_. p the hub by a radial force, 

Fig. 351. Fig. 352. r> , ./ . „ 

jP, and the reactions Ry^ 
and R>i are at right angles to the side of the grooves. Figure 352 
shows the corresponding force-triangle. The sum of the forces 
-^1 and i?2, between the spoke and the hub, may be increased 
to any desired multiple of P by making the angle between 
the sides of the groove sufficiently small, and the frictional 
grip will be correspondingly increased. If the angle of the sides 
of the grooves is such that i?, •\- R^ ^ n P^ n 11 must be used 
instead of /* in equations (4) and (5). 

Example 11, — If the spokes in the wheel in tlie above example 
lie in grooves, the sides of which are inclined 60® ; find the 
driving effort that can be transmitted without slipping. 

In this example the force-triangle (fig. 352) becomes an 
equilateral triangle, and R^ + R^ =^ 2 P. Taking )w = '15 and 
(^ = TT as l>efore, « /ix = -3, and 

%V = *4343 X -3 X 3-141 = •4093* 
from which, consulting a table of logarithms, 
^' = 2'566. 

But /, + /2 = 120 lbs. Solving these two simultaneous simple 
equations, we get 

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CHAP. XXIV. 



WAee/s 



351 




/, = 86-3 
^2 = 337» 
the driving effort is /| — /a = 52*6 lbs. 

Thus, the effect of the grooves inclined 60° is to nearly double 
the driving effort that can be transmitted. 

252. Spread of Spokes. — If the spokes of a tension wheel all 
lay in the same plane, then, considering the rim fixed, any couple 
tending to move the spindle would distort 
the wheel, as shown in figure 353. The 
distortion would go on until the moment of 
the pull of the spokes on the hub was equal 
to the moment applied to the shaft. If 
the spindle remains fixed in position, any 
lateral force applied to the rim causes a de- 
viation of its plane, the relative motion of the 
rim and spindle being the same as before ; the 
wheel, in fact, wobbles. If the spokes are 
spread out at the hub (fig. 354), the rim being 
fixed and the same bending-moment being 
applied at the spindle, the tension on the 
spokes A at the bottom right-hand side, and on the spokes B at 
the top left-hand side, is decreased, and that on the spokes 
C at the left-hand bottom side, and on the spokes Z> at the right- 
hand top side is increased. This increase and 
diminution of tension takes place with a prac- 
tically inappreciable alteration of length of the 
spokes, and therefore the wheel is practically rigid. 

The lateral spreading of the spokes of a cycle 
wheel should be looked upon as a means of connecting 
ihe hub rigidly to the rim^ rather than of giving the 
rim lateral stability relative to the hub. The rim 
must be of a form possessing initially sufficient lateral 
stability, otherwise it cannot be built up into a good 
wheel. The lateral components of the pulls of the 
spokes on the rim, instead of preserving the lateral 
stability of the rim, rather tend to destroy it. 
They form a system of equal and parallel forces, but alternately 
in opposite directions (fig. 355), and thus cause bending^tof the 

Digitized by VjO*"^^ ^ 



Fig. 353. 



a 



I 

¥ 

2^ 



Fig. 354. 



%fl 



352 



Details 



COAT, XZIf. 



rim at right angles to its plane. If the rim be very narrow in the 
direction of the axis of the wheel, it may be distorted by the pull 
of the spokes into the shape shown exaggerated in figure 355. 

The *Westwood' rim (fig. 373), on 

account of its tubular edges, is very 

strong laterally. 

253. Disc Wheels. — Instead of wire 

spokes to connect the rim and hub, 
two conical discs of very thin steel plate have been used, the discs 
being subjected to an initial tension. It was claimed — and there 
seems nothing improbable in the claim — that the air resistance of 




Fig. 355. 




f.r|i|.ji^iiiL"i 

Fig. 355. 

these wheels was less than that of wheels with wire spokes. Later, 
the Disc Wheel Company (Limited) made the front wheel of a 
Safety with four arms, as shown in figure 356. 

Nipples, — The nipples used for fastening the ends of the 
spokes to the rim are usually of steel or gun-metal. Perhaps, on 
the whole, gun-metal nipples are to be preferred to steel, since 
they do not corrode, and being of softer metal than the spokes, 
they cannot cut into and destroy the screw threads on the spoke 
ends. Figure 357 is a section of an ordinary form of nipple 
which can be used for both solid and hollow rims, and figure 358 
is an external view of the same nipple, showing its hexagonal 
external surface for screwing up. The hole in><the nipple is not 



Digitized b 



,oog^ 



CHAP. XXIV. 



Wheels 353 




I 



tapped throughout its whole length, but the ends towards the 

centre of the wheel are drilled the full diameter of the spoke, so 

that the few extra screw threads left on 

the spoke to provide for the necessary 

adjustment are protected by the nipple. 

Figure 359 shows a square-bodied nipple, 

otherwise the same as that in figure 358. 

When solid rims are used, the nipple ^'^•'''- ^''''''^' ^''''''^' 
heads must be flush with the rim surface, so as not to damage 
the tyre ; but when hollow rims are used, the nipple usually bears 
on the inner surface of the rim, and is therefore 
quite clear of the tyre. Figures 360 and 361 show 
forms of nipples for use with hollow rims, the 
screw thread of the spoke being protected by the 
latter nipple. 

In rims of light section, such as the hollow ^ 

, ° , . , , , F'G. 360. Fig. 361 

rims in general use, the greatest stress is the /oral 
stress due to the screwing up of the spokes. With a very tliin 
rim, which otherwise might be strong enough to resist the forces 
on it, the bearing surfaces of the nipples shown above are so small 
that the nipple would be actually pulled through the 
rim by the pull due to tightening the spoke. To dis- 
tribute the pressure over a larger surface of the rim, 
small washers (fig. 362) may be used with advantage, fig. 362. 
With wood rims, washers should be used below the nipples, 
otherwise the wood may be crushed as the tension comes on the 
spokes. 

Figure 363 shows the form of steel nipple to be used with 
Westwood's rim when the spokes are attached, not at the middle, 
but at the sides of the rim. Figures 357-363 are taken 
from the catalogue of the Abingdon Works Company 
(Limited), Birmingham. 

254. Bims. — We have already seen that the rim is 
subjected to a force of compression due to the initial pull^'^- ^^^• 
on the spokes. Let us consider more minutely the stresses on 
the rim when the wheel is not supporting any external load. Let 
figure 364 be the elevation of a wheel with centre C, A B 
being the chord between the ends of two adjacent spokes. Then 

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I 



354 



Details 



CHAF. XXIT. 



the stress-diagram (see figs. 334 and 335) of the structure will be 
a similar regular polygon, the pull on each spoke being repre- 




<T> 



Fig. 365. 

sented by the side and the compression on the rim by the radius 
of the polygon. 

If the rim were polygonal, the axes of the rim and the compres- 
sive force on it would coincide, and the compressive stress would 
be equally distributed over the section. But since the rim is 
circular, its axis will differ from the axis of the compression, and 
there will be a bending-moment introduced. Since at any point X 
this bending-moment is equal to the product of the compres- 
sion P into the distance x between the axis of the rim and the 
line of action of /*, the bending-moment on the rim will be propor- 
tional to the intercept between the rim and the chord A B, formed 
by joining the ends of two adjacent spokes, provided that the 
bending-moment on the rim at the points where the spokes are 
fastened is zero. The shaded area (fig. 364) would thus form a 
bending-moment diagram. But if the rim initially had no bend- 
. ing stress on it, it is likely that at the points A and B the pull ol 
the spokes will tend to straighten the rim, and therefore a bend- 
ing-moment, m, of some magnitude will exist at these points. 
The bending-moment at any point X will be diminished by the 
amount m, and the diagram will be as shown in figure 365, the 
bending-moments being of opposite signs at the ends of, and 
midway between, the spokes. From an inspection of figure 365, 
it is clear that in a wheel with 32 to 40 spokes, the bending- 
moment on the rim due to the compression will be negligibly 
small in comparison with the latter. 

When the wheel supports a load the distribution of stress on 
the rim is much more complex, and a satisfactory treatment of 

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CHAP. XXIV. 



JVAee/s 



355 



the subject is beyond the scope of the present work. The 
simplest treatment — which, however, the author does not think 
will give even rough approximations to the truth— will be to 
assume that the segments of the rim are jointed together at the 
points of attachment of the spokes. With this assumption, if the 
wheel supports a weight W^ when the lowest spoke is vertical, the 
force-triangle at A^ the point of contact with the ground, will 
be made up of the two compressions along the adjacent segments 
of the rim, and the pull on the vertical spoke plus the upward 
reaction of the ground, W. The rest of the stress-diagram will be 
as in the former case ; consequently, if the pull on the vertical 
spoke is zero, that on the other spokes will be W \ if the pull on 
the vertical spoke is /, that on the other spokes will be ( W 4- /). 

When the two bottom spokes are equally inclined to the 
vertical, the lower rim segment is in the condition of a beam 
supported at the ends and carrying in the middle a load, W \ 

therefore the bending- moment is , / being the length of the 

4 
rim segment. 

The assumption made above does not agree, even approxi- 
mately, with the actual condition of things in the continuous rim 







Fig. 366. 




Fk;. 367. 



of a bicycle wheel. A general idea of the nature of the forces 
acting may be obtained from figure 366, which represents a small 
portion, X X^ of the rim near the ground. This is acted on by the 
known force W, the upward reaction of the ground ; by the un- 

Digitized by VjJpQglC 



356 Details 



CHAP. xxrr. 



known forces /,, /j, . . . the pulls on the spokes directed 
towards the centre, C, of the wheel ; by forces of compression, P^ 
on the rim, unknown both in direction and magnitude ; and by 
unknown bending-moments, w, at the section X, The portion of 
the rim considered is, therefore, somewhat in the condition of an 
inverted arch. If the forces /*, /j, /j, ... and the bending- 
moments, w, were known, the straining action at any point on the 
rim could be determined as follows : Figure 367 shows the 
force- polygon, on the assumption that the forces considered are 
symmetrically situated with regard to the vertical centre line. 
The horizontal thrust on the rim at its point of contact with the 
ground is Hy the resultant of the forces /*, /,, /j, . . . on one side 
of the vertical. This, however, acts at a point 5, at a vertical 
distance y below the rim, determined as follows : Produce the 
lines of action of P and t^ to meet at A ; their resultant, which is 
parallel to O a (fig. 367), passes through the point A. Draw, 
therefore, A B parallel to Oa, cutting the line of action of/, zXB. 
Through B draw a line parallel to Oh^ giving the resultant of 
Py /v, and /,, and cutting the vertical through the point of contact 
at S. The rim at its point of contact with the ground is thus 
subjected to a compression H^ and a bending-moment tn 4- H y\ 
To make the solution complete, the unknown forces P^ /, and /, 
should be determined ; this can be done by aid of the theory of 
elasticity. 

Steel Pfms.— Figure 368 shows a section of a rim for a solid 
tyre, figure 369 for a cushion tyre. The edges of the latter are 





Fit-.. 368. Fig. 369. 

slightly bent over, so that the tyre when it bulges out on touching 
the ground will not be cut by the rim edge. Figure 370 shows a 
section of Warwick's hollow nm, which is rolled from one strip of 
steel bent to the required section, its edges scarfed, and brazed 
together. The part of the rim of smallest radius is thickened, so 

Digitized by CjOOQIC 



CHAP. XXIY. 



WAee/s 



357 



that the local stresses due to the screwing-up of the spokes may be 
better resisted. Figure 371 shows the * Invincible ' rim which was 





Fig. 370. 



Ku;. 371. 



made by the Surrey Machinists Company, rolled from two distinct 
strips, the inner being usually much thicker than the outer. The strips 








Fig. 372. 

were brazed together right round the circumference. Figure 372 

shows the Nottingham Machinists' hollow rim. In this the local 

strength for the attachment of the 

nipple is provided by folding over the 

plate from which the rim is made, so 

that four thicknesses are obtained. 

Figure 373 shows the * Westwood ' rim, *^ » • 373. 

which is formed from one plate bent round at each edge to form 
a complete circle. The spokes can be attached at the edges of the 
rim as indicated, or at the middle of the rim in the u.sual way. 

All the above rims are rolled to different sections to fit 
the different forms of pneumatic tyres. They are all made 
from straight strips of steel, and have, therefore, one joint in the 
circumference, the ends being brazed together. This joint, how- 
ever carefully made, is always weaker than the rest of the rim, 
and adds to the difficulty of building the wheel true. The 
Jointless Rim Company roll each rim from a weldless steel 
ring, in somewhat the same way as railway tyres ^re rolled. 

Digitized by VjOOQ 



358 



Details 



CHAP^ XXIT. 



This rim, though perhaps more costly, is therefore much stronger 
weight for weight than a rim with a brazed joint. 

Wood /^ims.— The fact that the principal stress on the rim of 
a bicycle wheel is compression, and that, therefore, the material 
must be so distributed as to resist buckling or collapse, and not 
concentrated as in a steel wire, suggests the use of wood as a 
suitable material. Hickory, elm, ash, and maple are used. Two 
types are in use : in one the rim is made from a single piece of 
wood, the two ends being united by a convenient joint. Figure 
374 shows the 'Plymouth' joint. The other type is a built-up 




Fh;. 374 




Fig. 376. 



rim composed of several layers of wood. Figures 375 and 376 
show the * Fairbank ' laminated rim, for a solutioned tyre and for 
the Dunlop tyre respectively, the grain of each layer of wood 

running in an opposite di- 



rection to that next it. Each 
layer or ring is made with 
a scarfed joint, and the 
various rings are fastened 
together with marine glue 
under hydraulic pressure. The built-up rim is then covered with 
a waterproof linen fabric, and varnished. 

255. Hubs. — Figure 404 shows a section of the ordinary 
form of hub for a direct spoke-wheel, and figure 377 an external 
view of a driving hub. The hub proper in this is made as short 
as possible, and the spindle, with its adjusting cones, projects 
considerably beyond the hub, so as to allow the wheel to clear the 
frame of the machine. 

Figure 378 shows a driving hub, in which the hub proper is 
extended considerably beyond the spoke flanges, and the ball-races 

Digitized by CjOOQIC 



CHAP. XXIV. 



Wheels 



359 



kept as far apart as possible. This hub is intended for tangent- 
spokes, the flanges being thinner than in figure 377. 




Fig. 377. 



Hubs for direct-spokes are made either of gun-metal or steel ; 
tangent-spoke hubs should be invariably of steel, as the local 




Fig- 378. 



Stress due to the pull of the spoke cannot be resisted by the 
softer metal. 

Figure 379 shows a pair of semi-tangent hubs, as made by 
Messrs. W. A. Lloyd & Co., the flanges for the attachment of the 




Fig. 379 



hub in this case forming cylindrical drums instead of flat discs, as 
in figure 378. The spokes may leave the circumference of the 

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36o 



Details 



irtLKP. xxnr. 



drum at any angle between the radius and the tangent, hence the 
name semi-tangent. 

In all the hubs above described the adjusting cones are 
screwed on the spindle, and the hard steel cups are rigidly fixed 
to the hub. In the * Elswick ' hub (fig. 380), the adjusting cone 




33o. 



is screwed to the hub and the ball-races on the spindle are rigidly 
fixed. One important advantage of this form of hub is that the 
clear space which must always be preserved between the fixed 
spindle and the rotating hub is of much smaller radius than in 
the others. The area by which dust and grit may enter the bearing 




is smaller, the bearing should therefore be more dust-proof than 
the others. Another important feature is the fact that the hub is 
oil-retaining, and the balls may have oil-bath lubrication at the 
lowest point of their path. Figure 381 shows the * Centaur' hub, 
also possessing dust-proof oil-retaining properties. 

Digitized by CjOOQIC 



CHAP. XXIV. 



Wheels 361 



In recent years *barrer hubs of large diameter have been 
used, whereas the earlier hubs were made just large enough to 
clear the spindle inside. The * Centaur ' is an example of a 
barrel hub. 

The best hubs are turned out of solid steel bar, the diameter 
of which must be as great as that of the flanges for the attachment 
of the spokes. To avoid this excessive amount of turning, the 
' Yost ' hub is made of two end pieces and a middle tube. 

The hubs of Sharp's tangent wheel may, with advantage, be 
made of aluminium, since the pull of the spokes has not to be 
transmitted by flanges. 

The ' Gem ' hub, made by the Warwick and Stockton Com- 
pany, has the hard steel cup screwed to the end of the hub. The 
balls lie between the cup and an inner projecting lip of the hub, 
so that they remain in place when the spindle is removed. 

256. Fixing Chain-wheel to Hub.— The chain-wheel should 
not be fixed by a key or pin, as this will usually throw it slightly 
eccentric to the hub. In testing the resistance of the chain gearing 
of a Safety it is often noticed that the chain runs quite slack 
in some places and tight in others. This can only mean that 
the centres of the pitch-polygons of the chain-wheels do not 
coincide with the axes of rotation. The chain-wheel and the 
corresponding surface on the hub, being turned to an accurate 
fit, are often fastened by simply soldering. The temperature at 
which the solder melts is sufficiently low to prevent injury to the 
temper of the ball-races of the hub. Another method is to screw 
.the chain-wheel, N, on the hub ; the screw should then be arranged 
that the driving efibrt in pedalling ahead tends to screw the 
chain-wheel up against the projecting hub flange. This is done 
in the * Elswick ' hub (fig. 380). If the chain is at the right-hand 
side of the machine looking forwards, the screw on the chain- 
wheel should be right-banded. During back-pedalling the 
driving effort will tend to unscrew the chain-wheel. This is 
counteracted by having a nut, K^ with left-handed screw, screwed 
up hard against the chain-wheel. If the chain-wheel, N, tends to 
unscrew during back-pedalling, it will take with it the nut K, 
which will then be screwed more tightly against the wheel, and 
its further unscrewing prevented. 

Digitized by CjOOQIC 



362 



Details 



CHAP. 2lXIV. 



A method adopted by the Abingdon Company a few years 
ago was to have the chain-wheel and hub machined out to a 
polygonal surface of ten sides, and the wheel then soldered on. 

257. Spiadles. — The spindle, strictly speaking, is a part ot 
the frame, and serves to transfer the weight of the machine and 
rider to the wheel. Let the spindle be connected to the frame at 
A and B (fig. 382), C and D be the points at which it rests on the 





Fig. 382. 



balls of the bearing, and W be the total load on the wheel. 

Then the spindle may be considered as a beam loaded at A and 

W 
B with equal weights - , and supported at the points C and D ; 
2 

the direction of the forces of reaction F^ at C and Z>, coinciding 

with the radii of the balls to their points of contact with their 

paths. Let e and / be the points at which the forces F cxxX, the 

axis of the spindle ; then F can be resolved into vertical and 

\V 
horizontal forces, - and /T respectively, acting through e. The 
2 

horizontal forces, H^ produce a tension on the part e f o{ the 

spindle, the remaining forces produce bending stresses. The 

spindle may thus be considered as a beam supported at e and/ 

W 
and loaded at A and B with equal weights, - . The bending- 

2 

W i. 
moment on any section between c and/ is , / being the dis- 

2 

tance A e. It is evident that this bending-moment will be zero 

if the points A and e coincide, and will be greater the greater the 

distance A e ; hence the spindle in figure 378 is subjected to 

a far smaller bending stress than that in figure 377. 

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CHAP. XXIV. 



Wheels 363 



Example /.—In a bearing the distance A e (fig. 382) is \ in., 
and the total weight on the wheel is 120 lbs., what is the neces- 
sary size of spindle, the maximum stress allowed being 10 tons 
per sq. in. ? 

The bending-moment on the spindle will be 

L^-? X 4, = 52*5 inch-lbs. 
2 8 

^3 
Substituting in the formula J/= /(sec. 94), we get 

10 

52-5 = X 10 X 2240, 
to 

that is 

d^ = '0234, and d = '286 in. 

i'his gives the least permissible diameter of the spindle, that is, 
the diameter at the bottom of the screw threads. 

Step. — ^The most convenient step for mounting a Safety bicycle 
is formed either by prolonging the spindle itself, or by forming a 
long tube on the outer nut that serves to fasten the spindle to the 
frame and lock the adjusting cone in position. If the length of 
this step be i^ in., ^the weight of the rider, and if the rider in 
mounting the machine press on its outer edge, the bending- 
moment produced on the spindle will be \\ JF inch-lbs. 

Example IL — If JF= 150 lbs., Af ^= 225 inch-lbs. ; substitut- 
ing in the formula M ^=^ Zfy 
we get 

d^ 
225 = X 10 X 2240, 



10 



from which 



d^ = '100 and d = '464 in. 



A common diameter for the spindle is g in. ; if the § in. 
spindle resist the whole of the above bending-moment, the maxi- 
mum stress on it will be much greater than 10 tons per sq. in. ; 
it will be 



•4.64.^ 

^ i^ X 10 = i8*o tons per sq. m 

•375' 



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364 Details chap. xxiv. 

The tube from saddle-pin to driving-wheel spindle may take up 
some of the bending due to the weight on the step, in which case 
the maximum stress on the spindle may be lower than given above. 
258. Spring Wheels. — Different attempts have been made to 
make the wheels elastic, so that vibration and bumping due to 
the unevenness of the road may not be communicated to the 
frame. One of the earliest successful attempts in this direction 
was the corrugated spokes used in the * Otto ' dicycle. These 
spokes, instead of being straight, were made wavy or corrugated, 
and of a harder quality of steel than used in the ordinary straight 
spokes. Their elastic extension was great enough to render the 
machine provided with them much more comfortable than one 
with the ordinary straight spokes. 

A spring wheel has the advantage over a spring frame, that it 
intercepts vibration sooner, so that practically only the wheel rim 
partakes of the jolting due to the roughness of the road. On the 
other hand, the springs of a wheel extend and contract once every 
revolution, and as this cannot be done without the expenditure of 
energy, a spring wheel must require more power than a rigid 
wheel to propel it over a good road. The springs of a frame remain 
quiescent under a steady load while running over a smooth road, 
only extending or shortening when the wheel passes over a hollow 
or lump in the road. 

In the * Everett ' spring wheel the spokes, instead of being 
connected directly to the hub, are connected to short spiral 
springs, thus giving an elastic connection between the hub and 
the rim, so that the rim may run over an obstacle on the road 
without communicating much shock to the frame. One objection 
to a wheel with spring spokes is the want of lateral stiffness of 
the rim, it being quite easy to deflect the rim sideways by a lateral 
pressure. The author is inclined to think that this objection may 
be over-rated, since in a bicycle the pressure on the rim of a 
wheel must be in, or nearly in, the plane of the wheel. The * Everett ' 
wheel is satisfactory in this respect. In the * Persil ' spring wheel 
two rims are used, the springs being introduced between them. 
The introduction of such a mass of material near the periphery of 
the wheel will make the bicycle provided with * Persil ' wheels 
slower in starting than one with ordinary wheels (see sec. 68). 

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CHAP. XXIV. 



W/ieels 



365 



In the * Deburgo ' spring wheel the springs are introduced at 
the hub, which is much larger than that of an ordinary wheel. 
Figure 383 shows a section of the ' Deburgo ' hub, and figure 384 
an end elevation with the outer dust cover removed, so as to 
show the springs. The outer hub or frame i, to which the 
spokes are attached, is suspended from the inner hub or axle- 
box, J, by spiral springs, 11 and 12, Frames 2 and ^ forming 
rectangular guides at right angles to each other, are fixed 
respectively to the outer and inner hubs ; an intermediate slide, 
S, is formed with corresponding guides, the combination com- 





FlG. 383. 



Fig. 384. 



polling the outer to turn with the inner hub, while retaining 
their axes always parallel to each other, and allowing their 
respective centres perfect freedom of linear motion. To diminish 
friction a number of balls are introduced between the slides. 
Dust-caps, 14^ fixed to the inner hub enclose the springs and 
guides. 

This spring wheel is quite rigid laterally, the only possible 
relative motion of the outer and inner hubs being at right angles 
to the direction of their axes. 

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366 Details chap. xit. 



CHAPTER XXV 

BEARINGS 

259. Definition. — A bearing is the surface of contact of two 
pieces of mechanism having relative motion. In a machine 
the frame is the structure which supports the moving pieces, 
which are divided into primary and secondary^ the former being 
those carried direct by the frame, the latter those carried by- 
other moving pieces. In a more popular sense the bearing is 
generally spoken of as the portions of the frame and of the 
moving piece in the immediate neighbourhood of the surface 
of contact In this sense the word 'bearing' will be used in 
this chapter. The bearings of a piece which has a motion of 
translation in a straight line must have cylindrical or prismatic 
surfaces, the straight lines of the cylinder or prism being parallel 
to the direction of motion. The bearings of pieces having rotary 
motion about a fixed axis must be surfaces of revolution. A 
part of a mechanism may have a helical motion — that is, a 
motion of rotation together with a motion of translation in the 
direction of the axis of rotation ; in this case the bearings must 
be formed to an exact screw. 

The three forms of bearing above mentioned correspond to 
the three lower pairs in kinematics of machinery, viz. the sliding 
pair, the turning pair, and the screw pair. In each of these 
three cases the two parts having relative motion may have con- 
tact with each other over a surface. 

260. Journal, Pivot, and Collar Bearings.— Figure 385 
shows the simplest form oi journal bearing for a rotating shaft, 
the section of the shaft and journal being circular. In this 
bearing no provision is made to prevent motion of the shaft in 
the direction of its axis. A bearing in which provision is made 

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CHAP. XXT. 



Bearings 



367 



€ 






3 



Fig. 385. 



to prevent the longitudinal motion of the shaft is called a pivot 
or collar bearing. Figure 386 shows the simplest form of pivot 
bearing, figure 387 a combined journal 
and pivot bearing, the end of the shaft 
being pressed against its bearing by a force 
in the direction of the axis. Figure 388 
shows a simple form of collar bearing in 
which the same object is attained. A rotating shaft provided 
with journal bearings may be constrained longitudinally, either 
by fixing a pivot bearing at each 
end, or by having a double collar 
bearing at some point along the 
shaft. This double collar bearing 
is usually combined with one of 
the journals, as at ^^ (fig. 389), a 
collar being formed at each end of 
the cylindrical bearing. In a long 
shaft supported by a number of 
journals it is only necessary to have 
one double collar bearing; theother 
bearings should be quite free lon- 
gitudinally. Thus, in a tricycle axle with four bearings, the best 
result will be got by having the longitudinal motion of the axle 





Fig. 386. 



Fig. 387. 




controlled at only one of the bearings ; if more collars, or their 
equivalents, are placed on the axle, the only effect is to increase 
the pressure of the collars on their bearings, and so increase the 
frictional resistance. 

From the point of view of the constraint of the motion it 
would be quite sufficient for a journal bearing to have contact 
with the shaft at three points (fig. 390), but as there is usually a 
considerable pressure on the bearings they would soon be worn. 
The area of the surfaces of contact should be such that the 

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368 



Details 



CHAP. XXT. 




Fig. 390. 



pressure per square inch does not exceed a certain limit, de- 
pending on the material used and the speed of rubbing. 

The bearings of the wheel of an * Ordinary bicycle were 
originally made as at A (fig. 389), the bearing at each side of 
the wheel being provided with collars, since 
the lateral flexibility of the forks was so great 
that otherwise the bearings would have 
sprung apart. It was impossible to keep 
the lubrication of the bearings constantly per- 
fect, and with no film of oil between the sur- 
faces the coefficient of friction rose rapidly 
and the resistance became serious. 
Journal Friction, — In a well-designed journal the diameter of 
the surface of the fixed bearing should be a little greater than 
that of the rotating shaft (fig. 391). The direction of the motion 
being then as indicated by the arrow, \{ the 
pressure is not too great, the lubricant at a 
is carried by the rotating shaft, and held by 
capillary attraction between the metal sur- 
faces, so that the shaft is not in actual contact 
with its bearing, but is separated from it by 
a thin film of oil. From the experiments 
carried out by the Institution of Mechanical Engineers it appears 
that the friction of a perfectly lubricated shaft is very small, the 
coefficient being in some cases as low as '001. This compares 
favourably with the friction of a ball-bearing. 

Pivot Friction,— V^'\\}ci a. pivot or collar bearing the case is 
quite different. The rubbing surface of the shaft is continually 
in contact with the bearing, and cannot periodically get a fresh 
supply of oil (as in fig. 391) to keep between the two surfaces. 
The consequence is that, with the best form of collar bearing, 
the coefficient of friction is much higher. From the experiments 
of the Institution of Mechanical Engineers it appears that -03 to 
•06 may be taken as an average value of fi for a well lubricated 
collar bearing. 

261. Conical Bearings. — In machinery subjected to much 
fiiction and wear, after running some time a shaft will run loose in 
its bearing. When the slackness exceeds a certain amount the 

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Fig. 391. 



CHAP. XXV. 



Bearings 



369 



bearing must be readjusted. One of the simplest means for 
providing for this adjustment is shown in the conical bearing 
often used for the back wheel of an * Ordinary* (fig. 392). The 
hub, Hy ran loose on the spindle, 5, which was fastened to the 
fork ends, F^^ and F^, The surfaces of contact of the hub and 
spindle were conical, a loose cone, C, being screwed on near one 




end of the spindle. If the bearing had worn loose, the cone C 
was screwed one or two turns further on the spindle until the 
shake was taken up. The cone was then locked in position by 
the nut «i, which also fastened the end of the spindle to the fork. 
During this adjustment the other end of the spindle was held 
rigidly to the fork end F^^ by the nut «2- 

262. Boiler-bearings. — The first improvement on the plain 
cylindrical bearing was the roller-bearing. Figure 393 is a 




Fig. 393. 



Fig. 394. 



longitudinal, and figure 394 an end section of a roller- bearing. 
In this a number of cylindrical rollers, A^ are interposed between 

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370 



Details 



the cylindrical shaft and the bearing- case, the axes of the rollers. 
A, being parallel to that of the shaft. These rollers were some- 
times quite loose in the bearing-case, in which case as many 
rollers as could be placed in position round the shaft were 
used. More often, however, the ends of the rollers were turned 
down, forming small cylindrical journals, supported in cages r, one 
at each end of the roller. This cage served the purpose of 
keeping the distance between the rollers always the same, so that 
each roller revolved free of the others ; whereas, without the 
cage, two adjacent rollers would often touch, and a rubbing action 
would occur at the point of contact. 

The chief advantage of a roller-bearing over a plain cylindrical 
bearing is that the lubrication need not be so perfect. Wliile a 
plain bearing, if allowed to run dry, will very soon get hot; a 
roller- bearing will run dry with little more friction than when 
lubricated. 

A plain collar bearing must be used in conjunction with a 
roller-bearing, to prevent the motion of the shaft endways. 

263. Ball-bearings. — Instead of cylindrical rollers, a number 
of balls, B (fig. 395), might be used. The principal difference in 




Fig, 395. Fig. 396. 

this case would be that each ball would have contact with the 
shaft and the bearing-case at a poiiit^ while each cylindrical roller 
had contact along a line. As a matter of fact, the surface of con> 
tact in the case of the ball-bearing would be a circle of very small 
diameter (point contact), while in the case of the roller- bearing 
it would be a very small, narrow rectangle of length equal to that 



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CHAP. XXV. 



Bearings 



371 



F-iG. 398. 



of the roller (line contact). Other things being equal, the roller- 
bearing should carry safely a much greater load than the ball- 
bearing before crushing took place. 

The motion of the balls in the bearing shown in figure 395 
loaded at right angles to the axis, is one of pure rolling, the axis 
of rotation of the ball being always parallel to that of the axes of 
the rolling surfaces of the shaft and bearing-case. 

264. Thrust Bearings with Boilers.— If a ball- or roller- 
bearing be required to resist pressure along the shaft, as m 
figures 386 and 387, the 
arrangement must be 
quite different. Two 
conical surfaces, a v a 
and bv b, formed on the 
frame and the rotating 
spindle respectively (fig. 
397), having a common 
vertex at v, and a com- 
mon axis coincident with 
the axis of the spindle, 
with conical rollers, avb^ 
having the Same vertex, v, 
will satisfy the condition 
of pure rolling. If the 
axis, vc, of the conical 
roller be supposed fixed, 
and the spindle be 
driven, the cone b v b 
will drive the roller by friction contact, and it in turn will drive 
the cone ava. If the cone av a he fixed, and the spindle be 
driven, the relative motion of the three conical surfaces will 
remain the same ; but in this case the axis of the roller, v r, will 
also rotate about the axis XX, With perfectly smooth surfaces, 
the direction of the pressure is at right angles to the surface of 
contact, and very nearly so with well lubricated surfaces. On the 
conical roller, avb^ there will therefore be two forces, A and B, 
acting at right angles to its sides, va and vb, respectively. These 
have a resultant along the axis v c, and unless a third force, C, be 

Digitized by Vj H H 2 




^S^ 



Fig. 397. 



372 Details 



CHAP. ht. 



applied to the conical roller, it will be forced outwards during the 
motion. 

The magnitudes of the forces A^ B, and C can easily be found 
if the force, F, along the axis is given. In figure 398 draw Ik equal 
to 7^ and parallel to the axis X X^ draw /;// at right angles to r/^ 
and km 3X right angles to A'Jf ; im will give the magnitude of 
the force B ; draw ;;/ «^and In respectively at right angles to va 
and a />, meeting at « ; m n and n I will be the magnitudes of the 
forces A and C respectively. 

In figure 397 the conical roller is shown with a prolongation 
on its axis rubbing against the bearing case, so that its further 
outward motion is prevented. With this 
arrangement there will be considerable 
rubbing friction between the end of 
the roller and the bearing-case. In 
Purdon & Walters* thrust bearing for 
marine engines the resultant outward 
pressure on the roller is balanced by 
letting its edge bear against a part of 
the bearing-case (fig. 399). The gene- 
rating line, V rt, of the roller is produced 
to a point d ] dv^ is drawn perpendicular 
to V dy and forms the generating line of 
a second conical surface coaxial with the 
'^* ^^^' first. A small portion on each side of 

d is the only part of this surface that presses against the bearing- 
case. The instantaneous axis of rotation being vd^ there is no 
rubbing of the roller on the case, but only a relative spinning 
motion at d. In this case, the force -triangle (fig. 398) will 
have to be modified by drawing I n^ 2X right angles to w « ; mn^ 
will then be equal to the force A^ and /«, to the pressure D 
Sitd. 

Relative Speeds of Roller and Spifidle.—lj&i P (figs. 397 and 
399) be any point in the line of contact of the conical roller with 
the spindle; draw P a^^ and Pb^ at right angles \.o va and vX 
respectively, and let V be the linear speed of the point P at any 
instant. Since rt, is a point on the instantaneous axis of rotation 
of the roller, and b^ a point on the fixed axis of rotation of the 

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hi 

i 




CHAP. XXV. Bearings 373 

spindle, the angular speeds 012 ^^^ '•'i o^ ^^e roller and spindle 
are respectively 

a,, = ^^and.,=^,^^ (') 

Therefore ^2 ^ /"^ (2) 

Comparing figures 397, 399, and 398, the triangles Pvd^ and 
/ m k are similar ; the triangles Fva^ and /w «, are similar ; so 
also are the four-sided figures i kmn^ and P h^v a^. Therefore, 

U.2 Pb, Ik F ^^^ 

or, Fhi^ -^ DiD.^ (4) 

That is, if only one roller be used, the angular speeds of the roller 
and spindle are inversely proportional to the pressures along their 
instantaneous axes of rotation. 

1( V m (fig. 397) be set off along the axis of the spindle equal to 
Fai, and v n along va equal to Pdy, the vectors v m and v n will 
represent the rotations of the spindle and roller respectively, both 
in magnitude and direction, v «, the rotation of the roller, can be 
resolved into the rotations e;«, and vn>i about the axes of the 
shaft and roller respectively. It can easily be shown, from the 
geometry of the figure, that vn^-=\vm ) therefore the axis 
of the roller turns about the axis of the shaft at half the speed of 
the shaft. 

The rotation ?;«, =^(o,, is equivalent to an equal rotation 
about a parallel axis through ^(fig. 397), together with a translation 
i CO y. vc. This translation and rotation constitute a rubbing of the 
roller on the bearing at c. Thus, finally, the relative motion at c 
consists of a rubbing with speed z/ r x i w and a spinning with 
speed (1)3 = z; n^. 

From figures 397 and 398, "^^ ^—^'^ = (fig. 398). 

Oil 2 7' .7, C 

Digitized by CjOOQIC 



374 



Details 



CHAP. HT. 



Therefore, /^co, =. Cwg, (5^ 

and if n rollers be used, with the total thrust W along the shaft, 
JF(i)| = «C(03 (6) 

If a number of conical rollers are interposed between the 
two conical surfaces on the shaft and bearing respectively, as in 
figure 397, the radial thrust, C, on the rollers may be provided for 
by a steel live-ring against which the ends of the rollers bear. 
This live-ring will rotate at half the speed of the shaft, and there 
will be no rubbing of the roller ends relative to it. But it should 
be noted that the speed of rotation W3 of each roller relative to the 
live-ring will be as a rule greater than the speed of rotation of the 
shaft, and therefore with a heavy end thrust on the shaft, the risk 
of abrasion of the outer ends of the rollers will be great. In a 




Fig. 40.. 



Fig. 401. 



thrust bearing for marine engines, designed by the author, a 
number of lens-shaped steel discs were introduced between the 
outer end of each roller and the live-ring, so that the average 
relative spinning motion of two surfaces in contact is made 
equal to the relative speed between the roller and the live-ring, 
divided by the number of pairs of surfaces in contact. Figures 
400 and 401 show a modification of this design, in which the 
conical rollers are replaced by balls, a'\ rolling between hard steel 
rings, r, fixed on the shaft and the pedestal respectively. The 
small portions of these rings and of the balls in contact may he 
considered as conical surfaces with a common vertex, 7'. Anti- 

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CHAP. XXY. 



Bearings 



375 



friction discs, g^ are carried in a nut, fy which is screwed into and 
can be locked in position on the live-ring, b. This design 
(figs. 400 and 401) is arranged so that if one ball breaks it can 
be removed and replaced without disturbing any other part of the 
bearing. In this thrust-block a plain cylindrical bearing is used 
to support the shaft. 

This bearing may be simplified by the omission of the anti- 
friction discs, and allowing the balls to run freely in the space 




Fig. 402. 



enclosed by the two steel rings, f, and the live-ring, b. Figure 402 
is a part longitudinal section of such a simplified thrust bearing, 
and figure 403 a part cross section. 

In a journal bearing the work lost in friction is proportional 
to the product of the pressure and the speed of rubbing, pro- 
vided the coefficient of friction remains constant for all loads. In 
the same way, in a pivot bearing, the work lost in friction — other 
things being equal — is proportional to the product of the pressure 
and the angular speed. Equations (4) and (6), therefore, assert 
that it is impossible by any arrangement of balls or rollers to 
diminish the friction of a pivot bearing below a certain amount. 
If a shaft subjected to a longitudinal force can be supported by a 
plain pivot bearing (fig. 386), the work lost in friction will be a 
minimum. If, however, the circumstances of the case necessitate 
a collar bearing (fig. 388), an arrangement of balls or conical 
rollers may serve to get rid of the friction due to the rubbing of 
the collar on its bearings. In other words, the effective arm at 
which the frictional resistance acts may be reduced by a properly 
designed ball- or roller-bearing to a minimum, so that it may be 
equivalent to that illustrated in figures 400-1. The pressure on 
the pivot may sometimes be so great as to make it undesirable to 
support it by a bearing of the type shown in figure 3^6-^^^ use 



376 



Details 



of a bearing of either of the types shown in figures 400-1 and 
402-3 with a number of balls or conical rollers, is equivalent to 
the subdivision of the total pressure into as many parts as there 
are rollers in the bearing. 

265. Adjustable Ball-bearing for Cycles.— Figure 404 shows 
diagrammatically one of the forms of ball-bearing used almost 
universally for cycles. The external load on the bearings of a 
bicycle or a tricycle is always, with the exception of the ball 
steering-head, at right-angles to its axis ; any force parallel to the 
axis being simply due to the reaction of the bearing necessary to 
keep the spindle in its place. Figure 404 represents the section 
of the hub of a bicycle wheel ; the spindle, S 5, is fixed to the 



^ ^///////////////// 




H 




Fig. 404. 

fork ; hardened steel * cones,' C C, are screwed on its ends, and 
hardened steel cups, Z>, are fixed into the ends of the hub, H, 
which is of softer metal. The balls, By run freely between the 
cone C and the cup D, One of the cones C is screwed up tight 
against a shoulder of the spindle S, the other is screwed up until 
the wheel runs freely on the spindle without undue shake, it is 
then locked in position by a lock-nut N, which usually also serves 
to fasten the spindle to the fork end, F, 

266. Motion of Ball in Bearing. — Consider now the equi- 
librium of the ball B, It is acted on by two forces, /| and /s 
(fig. 405), the pressure of the wheel and the^reaction of the 

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CHAP. XXV. 



Bearings 



Z77 



spindle respectively. Since the ball is in equilibrium, these two 
forces must be equal and opposite ; therefore the points of con- 
tact, a and b, of the ball with the cup and cone must be at the 
extremities of a diameter. During the actual motion in the 
bicycle the cone C is at rest, the ball B rolls round it, and the 
cup D rolls on the balls. The relative motion will be the same, 



O^ 



'=.<i 



j<^\ 



A 






■-X 




s 


1 X 
I 1 \ 

1 1 

1 i. 






. 5 



a' 



Fic. 405. 



however, if a motion of rotation about the axis of the spindle, 
SS^ oe impressed on the whole system, equal in amount but 
opposite in direction to that of the centre of the balls round the 
axis, SS, The centre c of the ball B may thus be considered to 
be at rest, the ball to turn about an axis through its centre, 
the cup D and cone C to rotate in opposite directions about 
their common axis, S S, 

Draw dVi at right angles to a^, cutting the spindle .55 in z;, 
(fig. 405, which is part of fig. 404 to a larger scale) ; from z;, draw 
a tangent Vi b^ to the circle B^ of which ab\% the diameter. If 
the relative motion of the ball and cup at ^ be one of pure 
rolling, the portion of the ball in contact at b may be considered 
as a small piece of a cone b v^ ^2, and the portion of the ball-race 
at b part of a cone coaxial with S 5, both cones having the 
common vertex v^. The axis of rotation of the ball will pass 
through Vx and the centre c of the ball B, 

Now draw a v.y at right angles to b «, cutting SS at>f!o. ^ If the 

Digitized by VjCXjQ 



378 Details chap. txs. 

relative motion of the ball and cone at a be one of pure rolling, 
the portions of the ball and cone surfaces in contact may be con- 
sidered portions of cones having a common vertex z/g ; the axis 
of rotation of the ball will thus be cv^^. But the ball cannot be 
rotating at the same instant about two separate axes v^ c and v^c^ 
so that motions of pure rolling cannot exist at a and b simul- 
taneously. If the surfaces of the cone and the cup be not equally 
smooth, it is possible that pure roUing may exist at the point of 
contact of the ball with the rougher surface. Suppose the rougher 
surface is that of the cup, the axis of rotation of the ball would 
then be v^ r, and the motion at a would be rolling combined 
with a spinning about the axis ac2X right angles to the surface of 
contact. Draw v<^ d parallel to a r, cutting v^ c at d. Then, if 
cd represent the actual angular velocity of the ball about its axis 
of rotation v^ c, V2 d will represent the angular velocity of the ball 
about the axis ac\ since the rotation cd \^ the resultant of a 
rotation c v^ about the axis c v^^ and a rotation v<i d about the axis 
ca\ dcv2 is, in fact, the triangle of rotations about the three 
axes intersecting at c. 

If the surfaces of the cone and cup be equally smooth the 
axis of rotation of the ball will ht cv^ v being somewhere between 
7'i and V2' If the angular speeds of the spinning motions at 
a and b be equal, cv will bisect dv^^ If e be this point of inter- 
section, ecv^ and ecd will be the triangles of rotation at the 
points a and d respectively. 

The above investigation clearly shows that a grinding action 
is continually going on in all ball-bearings at present used in cycle 
construction. The grooves formed in the cone and cup after 

running some time are thus ac- 
counted for, while the popular 
notion that all but rolling fric- 
tion is eliminated in a well-de- 
signed ball-bearing is shown to 
P^^ be erroneous. The effect of this 

grinding action will depend on the 
closeness with which the balls fit the cone and cup. If the radii 
of curvature of the ball, cone, and cup be nearly the same 
(fig. 406), friction due to the spinning will be great ; while, if they 

Digitized by V^jOOQ 




CHAP. xjtv. Bearings 379 

are perceptibly different (fig. 407), the friction of the bearing will 
be much less. On the other hand, a ball in the bearing (fig. 406) 
will be able to withstand greater pressure than a ball in the 
bearing (fig. 407), the surface of contact with a given load being 
so very much less in figure 407 than in figure 406. 

267. Magnitudes of the RoUing and Spinning of the Balls 
on their Paths. — From a^ r, and b (fig. 405) draw perpendiculars 
to the axis SS^ and let w he the actual angular speed of the 
wheel on its spindle, T the sum of the angular speeds of the 
spinning motions of the ball on its two bearing surfaces, r the 
radius ca oi the ball, and R the radius cc^ of the circle in which 
the ball centres run. From a draw aa^ perpendicular to rr'. 
Considering the motion relative to a plane passing through the 
spindle SS and the line v c — that is, considering the point c to be 
at rest, as described in section 266 — let w, be the angular speed 
of rotation of the ball about the axis v r, which may be assumed 
at right angles to a b. The linear speeds of the points a and b of 
the ball will be w, r. The angular speeds of the spindle and the 
wheel will be respectively 

-"•''and'^'''. 
a a^ bb^ 



But the spindle is actually at rest ; so, if the angular speed 

'*'' ^ aboyt the axis .S.S be now added to the whole system, the 

a a' 

actual angular speed of the wheel will be 

w=(^ h — 1^1^ (7) 

\aa' bb^J ^ 

Denoting the length r^Zj by ^, 

a a} ^=- R — q^ and bb^ =: R ■\- q -, 

equation (7) may therefore be written 



But by section 266 



<;f" w 



^= "«l (9) 

€ c 

Digitized by CjOOQIC 



380 Details 






Combining (8) and (9) 

e c 
An inspection of the diagram (fig. 405) will show that the 

fractions -L and — ^ - are smaller the nearer the diameter 
ec ^^^ 

ab of contact of the ball with its bearings is to a perpendicular 

to the spindle SS. Also, the distance ec depends on the position 

of the actual axis of rotation, c v^ of the ball ; but it does not 

vary greatly, its maximum value being when it coincides with 

CV2J its minimum when it is perpendicular to ab. 

The above considerations show that a ball-bearing arranged 

as in the full lines (fig. 408) will be much better than the one 

arranged as shown by the dotted lines. 

The end thrust in bicycle bearings is 

always small, so that the line of contact 

ab need not be inclined 45** to the axis, 

but be placed nearer a perpendicular to 

the axis. 

The rolling of the balls on the 

bearings will be much less prejudicial 

than the spinning ; it may be calculated as follows : 

The linear speed of the point b of the ball (fig. 405) is 

iD X b~l^ :={R -\- q) u>. The angular speed of rolling of the ball 

about the axis a Vo is therefore — u>. Consider now the 

* 2 r 

outer path D to be fixed, and the inner path C to revolve with 

the angular speed — w ; the relative motion will be, of course, 

the same as before. The linear speed of the point a of the ball 

is u> X 'aa^ =: (R — q) w, and the angular speed of rolling of the 




ball about the 


axis bvy'i^ therefore 


.r- 


The 


sum 


of the 


rolling speeds 


of the ball at a and b 
R 


is therefore 






(II) 



a result independent of the angle that the diameter of contact a b 

Digitized by V^jOOQ 



CHAP. XXV. 



Bearings 



381 



makes with the axis of the bearing. The pressure on the ball, 
however, and therefore also the rolling friction depends on this 
angle. 

Example. — In the bearing of the driving-wheels of a Safety 
bicycle the balls are \ in. diameter, the ball circle —that is, the 
circle in which the centres of 
the balls lie— is -8 in. diameter, 
and the line of contact of the 
ball is inclined 45® ; find the 
angular speed of the spinning 
of the balls on their bearing. 
Figure 409 is the diagram for 
this case drawn to scale, from 
which 2/2 ^='21 in., ^^=-44 
in., and q = -09. Substituting 
these values in (10) 

^ -21 (-16 — -0081) 
•44 X 2 X '4 X -125 




That is, for every revolution of the hub, the total spinning 
of each ball relative to the bearings is nearly three-fourths of a 
revolution. 

The pressure on each ball in this case is n/2 times the 
vertical load on it. Hence the resistance due to spinning fric- 
tion of the balls will be 72>/2, = i-oi8 times that of a simple 
pivot- bearing formed by placing a single ball between the end of 
the pivot and its seat, the total load being the same in each 
case. 

The sum of the speeds of rolling of the ball is, by (11), 



•8 
•"25 



it) = 3*2 01. 



268. Ideal Ball-bearing.— The external load on the ball- 
bearing of a cycle is usually at right angles to the axis, but from 
the arrangement of the bearing (fig. 404) the pressure on the 
balls has a component parallel to the axis. This component has 
to be resisted by the bearing acting practically as a collar bearing, 

Digitized by V^jOOQ 



382 



Details 



CHAF. xrr. 




as described in section 260. Thus not only is the actual pressure 
on the balls increased, but instead of having a motion of pure 
rolling, a considerable amount of spinning motion under con- 
siderable pressure is introduced. The actual force in the direction 
of the axis necessary to keep the wheel hub in place is very small 
compared with the total external load ; a ball-bearing in which 
the load is carried by one set of balls, arranged as in figure 395, 
and the end thrust taken up by another set, might therefore be 
expected to offer less frictional resistance than those in -use at 
present. Such a bearing is shown in figure 410. The main 
balls, B (fig. 410), transmitting the load from the wheel to the 
/ x spindle run between 

?^^^^^^ coaxial cylindrical sur- 
faces on the spindle and 
hub respectively ; the 
motion of the balls, B^ 
relative to both surfaces, 
is thus one of pure roll- 
ing. The space in which 
the balls run is a little 
longer, parallel to the 
axis of the spindle, 
than their diameter, so that they do not bear sideways. The 
wheel is kept in position along the spindle by a set of balls, 
b^ running between two conical surfaces on the spindle and 
hub respectively, having a common vertex, and kept radially in 
place by a live-ring, r. One of these cones is fixed to the spindle, 
the other forms part of the main ball cup. This bearing is 
therefore a combination of the ball-bearing (fig. 395) and the 
thrust bearing (fig. 402-3). The motion of the main balls, B, 
being pure rolling, the necessity of providing means of adjustment 
will not be so great as with the usual form ; in fact, the bearing 
being properly made by the manufacturer may be sent out without 
adjustment. A play of a hundredth part of an inch might be 
allowed in the two main rows of balls, B^ and a longitudinal 
play of one-twentieth of an inch for the secondary rows, b. If the 
main row of balls ultimately run loose, a new hard steel ring, Ry 
can be easily slipped on the spindle. 

Digitized by CjOOQIC 



mm. 



Fig. 410. 



CHAFi XXV. 



Bearings 



383 




Fig. 411. 



If adjustments for wear are required in this type of bearing, 
they can be provided by making the hard steel ball ring, Ry slightly 
tapered (fig. 411), and screwing it on the spindle. It would be 
locked in position by 
the nut fixing the spindle 
to the frame. There 
would be an adjustment 
at each end. 

These bearings may 
be somewhat simplified 
in construction, though 
the frictional resistance 
under an end thrust will 
be theoretically increased, by omitting the live-ring confining the 
secondary balls, and merging it in either the cup or the conical 
disc (fig. 412). If this be done 
a single ball will probably be 
sufficient for each row of se- 
condary balls, b. If a double 
collar be formed near one end 
of the spindle, one row of 
secondary balls, b, would be 
sufficient for the longitudinal 
constraint. They could be put 
in place through a hole in the 
ball cup (fig. 411), or by screwing an inner ring on the cup 




Fig. 412. 




Fig. 413. 



(fig- 413)- The Other end of the bearing will have only the 
row of balls. 



main 



Digitized by CjOOQIC 



384 Details 



CHAP. XZT. 




269. Mntnal Subbing of Balls in the Bearing.— Figure 414 
may be taken to represent a section of a ball-bearing by a plane 
at right angles to the axis, the central spindle being fixed and 
the outer case revolving in the direction of the arrow a. The 
balls will therefore roll on the fixed spindle in the direction 
indicated. If two adjacent balls, B^ and B^^ touch each other 
there will be rubbing at the point of contact, and of course the 
friction resistance of the bearing will be increased. Now, in 
a ball-bearing properly adjusted the ad- 
justing cone is not screwed up quite tight, 
but is left in such a position that the 
balls are not all held at the same 
moment between the cones and cups ; 
in other words, there is a little play left 
in the bearing. Figure 414 shows such 
a bearing sustaining a vertical load, as in 
the case of the steering-wheel of a bicycle, 
Fig. 4m. ^jj^ ^^ pj^y greatly exaggerated for 

the sake of clearness of illustration. The cone on the wheel 
spindle will rest on the balls near the lowest part of the bearing, 
and the balls at the top part of the bearing will rest on the cone, 
but be clear of the cup of the wheel. Thus, a ball in its course 
round the bearing will only be pressed between the two surfaces 
while in contact at any point of an arc, r, c^y and will run loose 
the rest of the revolution. The balls should never be jammed 
tightly round the bearing, or the mutual rubbing friction will be 
abnormally great. The ascending balls will all be in contact, the 
mutual pressure being due merely to their own weight. A ball, 
B^y having reached the top of the bearing will roll slightly fon^-ard 
and downward, until stopped by the ball in front of it, B^, The 
descending balls will all be in contact, the mutual pressure being 
again due to their own weight. On coming into action at the arc 
^i ^2, the pressure on the balls tends to flatten them slightly in the 
direction of the pressure, and to extend them slightly in all direc- 
tions at right angles. The mutual pressure between the balls may 
thus be slightly increased, but it is probable that it cannot be 
much greater than that due to the weight of the descending balls. 
As this only amounts to a very small fraction of an ounce, in com- 

Digitized by CjOOQIC 



CHAP. rxv. 



Bearings. 385 




parison with the spinning friction above described under a total 
load of perhaps 100 lbs., the friction due to the balls rubbing on 
each other is probably negligibly small. 

Figure 414 represents the actions in the bearings of non-driving 
wheels of bicycles and tricycles, and in the driving-wheels of chain- 
driven Safety bicycles ; also, supposing the outer case fixed and 
the inner spindle to revolve, it represents the action in the crank- 
bracket of a rear- driving Safety. 

In the bearings of the front wheel of an * Ordinary,' or the front- 
driving Safety, the action is different, and is represented in 
figure 415. In these cases the balls near 
the upper part of the bearing transmit 
the pressure, the lower balls being idle. 
The motion being in the direction shown 
by the arrow «, the ball B^ is just about 
to roll out of the arc of action, and will 
drop on the top of the ball B^, The ball 
^3, ascending upwards, will move into the 
arc of action c^ c^y and will be carried 
round, while the ball behind it, Ba. will ^ 

' ' *' Fig. 415. 

lag slightly behind. In this way, it is 

possible that there may be no actual contact between the balls 

transmitting the pressure. 

It would be interesting to experiment on the coefficient of 
friction of the same ball-bearing under the two different conditions 
illustrated in figures 414 and 415. In some of the earlier ball- 
bearings the balls were placed in cages, so as to prevent their mutual 
rubbing. Figures 416 and 417 show the * Premier' bearing with 
ball-cage. It does not appear that the rubbing of the balls on the 
sides of the cage is less prejudicial than their mutual rubbing ; 
and as, with a cage, a less number of balls could be put into a 
bearing, cages were soon abandoned. 

Effect of Variation in Size of Balis. — If one ball be slightly 
larger than the others used in the bearing, it will, of course, be 
subjected to a greater pressure than the others ; in fact, the whole 
load of the bearings may at times be transmitted by it, and there 
will be a probabiHty of it breaking and consequent damage to the 
surface of the cone and cup. Let V be the linear speed of the 

Digitized by V^j q q 



386 



Details 



CHAP. XXV. 



point of the cup in contact with the ball (fig. 414), R the radius 
of the ball centre, and r the radius of the ball ; the linear speed 

of the ball centre is - , and its angular speed round the axis of 



Fig. 416. 





Fig. 4x7. 

the spindle is - . The radius R is the sura of the radii of a 
^ 2 R 

ball and of the circle of contact with the cone ; consequently the 
angular speed round the centre of the spindle of a ball slightly- 
larger than the others will be less than that of the others, the 
large ball will tend to lag behind and press against the following 

ball. 

\i P be the bearing pressure on the large ball, the mutual 
pressure, F^ between it and the following ball may amount to /i P, 
and the frictional resistance of the bearing will be largely increased. 

The mutual rubbing of the balls may be entirely eliminated 
by having the balls which transmit the pressure alternating with 
others slighdy smaller in diameter. The latter will be subjected 
only to the mutual pressure between them and the main balls, and 
will rotate in the opposite direction. They may rub on the 

Digitized by CjOOQIC 



Bearings 



387 



CHAP. XXV. 

bearing-case or spindle, but, since the pressure at these points 
approaches zero, there will be very little resistance. This device 
may be used satisfactorily in a ball-thrust bearing, but in a bicycle 
ball-bearing the number of balls in action at any moment may 
be too small to permit of this. 

270. The Meneely Tubular Bearing.— In the Meneely 
tubular bearing, made by Messrs. Siemens Brothers (fig. 418), the 
mutual rubbing of the rollers is entirely ehminated by an in- 
genious arrangement. " The bearing is composed of steel tubes, 
uniform in section, which are grouped closely, although not in 
contact with each other, around and in alignment with the 




Fig. 418. 

journal ; these rollers are enclosed within a steel-lined cylindrical 
housing. They are arranged in three series, the centre series 
being double the length of the outer series. Each short tube is 
in axial alignment with the corresponding tube of the opposite 
end series, while exactly intermediate to these end lines are 
arranged the axes of the centre series, thus making the lines of 
bearing equal. Each end tube overlaps two centre tubes, as 
shown in figure 418. To keep the long and short tubes in proper 
relative position, there are threaded through their insides round 
steel rods. These rods both lock the rollers together and hold 
them apart in their proper relative position, collars on the rods 
also serving to aid in maintaining the endwise positions. These 
connecting-rods share in the general motion, rolling without fric- 
tion in contact with the tubes. They intermesh the long and 
short tubes, and keep them rigidly in line with the axis." For a 

Digitized by CjOOQIC 



388 



Details 



CHAP. nv. 




Fig. 419. 



journal 3 in. diameter, the external and internal diameters of the 
rollers are 2 in. and i^ in. respectively. 

271. Ball-bearing for Tricycle Axle.— Figure 419 represents 
a form of ball-bearing often used for supporting a rotating axle, 

as the front axle of an * Or- 
dinary/ tricycle axles, &c 
This bearing supports the 
load at right angles to 
the axle and at the same 
time resists end-way mo- 
tion. A ball has contact 
with the ball-races at four 
points, fl, ^, r, d^ which for 
the best arrangement 
should be in pairs parallel 
to the axis ; the motion of 
the ball will then be one of rotation, the instantaneous axis 
being c d, its line of contact with the bearing case. The motion 
of the ball relative to any point of the surface it touches will, 
however, be one of rolling combined with spinning about an axis 
perpendicular to the surface of contact. 

Figure 420 shows this form as made adjustable by Mr. W. 
Bown. The outer ball-cup is screwed into the bearing case, and 
when properly adjusted is fixed in position by a plate 
and set screw. If this bearing be attached to the 
frame or fork by a bolt having its axis at right angles 
to the rotating spindle, it will automatically adjust 
itself to any deflection of the frame or spindle ; the 
axes of the spindle and bearing case always remaining 
coincident. 

Let a> be the angular speed of the axle, r the radius 
of the ball, -^i R^ and -^2 the distances of the points a, 
Bj and d from the axis S S, The linear speed of the 
point a common to the ball and the axle will be a> -^,. 
The angular speed of the ball about its instantaneous 
axis of rotation d c will be 

_co^, _ _ a,i?, 

ad R^- Rx ^ ' 

Digitized by CjOOQIC 




CHAP. «▼. Bearings 389 

The relative angular speed of the ball and axle about their 
instantaneous axis a b will be 

«- +-0 -r='r--W ('3) 

Draw c c^ at right angles to the tangent to the ball at d ; then at 
the point d the actual rotation of the ball about the axis c d can 
be resolved into a rolling about the axis d c^ and a spinning 
about an axis d B 2X right angles \ d c c^ will be the triangle of 
rotations at the point d. If the angular speed of spinning of the 
ball at d is 7^, we have 

Draw bb^ perpendicular to the tangent at a ; then, in the same 
way, it may be shown that the angular speed of the relative 
spinning at the point a is 

rp bb' R., . . 

^^^ba • (i?.--^,r ^'^' 

P'rom (14) and (15) the speeds of spinning at a and d are 

inversely proportional to the radii ; the circumferences of the 

bearings at a and b are also proportional to the radii. If the 

wear of the bearing be proportional to the relative spinning speed 

of the ball, and inversely proportional to the circumference — both 

of which assumptions seem reasonable — the wear of the inner and 

outer cases at a and d will be inversely proportional to the 

squares of their radii. If the bearing surfaces at a, b, r, and d 

c c^ bb^ 
are all equally inclined to the axis, = - ; then adding (14) 

and (15), the sum of the angular speeds of spinning at a, b, r, 
and d will be 

7-= 2''' ^2 + ^' 



cd R2 - Ri 

cc^ R ... 

Digitized by CjOOQIC 



390 



Details 



CHAP. ZXT. 



If ^ be the angle that the tangent d c^ makes with the axis of 



c c 



the bearing, ' ^ ^^ sin 6^ a ^ = 2 r cos 0, and (16) may be 
c a 



written, 



T = 



2 Rtan% 



(17) 



Equation (17), therefore, shows that the spinning motion in 
this form of bearing is proportional to the radius of the ball 
circle, inversely proportional to the radius of the ball, and directly 
proportional to the tangent of the angle the bearing surfaces 
make with the axis. 

Example. — Let the four bearing surfaces be each inclined 45° 
to the axis ; then tan ^ = i, and (17) becomes 



^ 2 R 



(18) 



If the diameter of the ball is -, r = ^, and if -^ is — ; 

substituting in (18), 
r= Sol. 

This gives the startling 
result that for ever)- 
turn of the axle each 
ball has a total spin- 
ning motion of eight 
turns relative to the 
surfaces it touches. 
This form of bearing, 
therefore, is much in- 
ferior to the double ball- 
bearing, which was 
much used for the 
front wheels of * Ordi- 
naries.' Figure 421 is 
a sectional view of a 
Messrs. Singer & Co. The motion 
is the same as that analysed in 




Fig. 4ai. 



double ball-bearing as used by 
of the balls in this bearing 
section 266. 



Digitized by CjOOQIC 



CHAP. XXV, 



Bearings 



391 



272. Ordinary Ball Thrust Bearing.— Figure 422 is a section 
of a form of ball thrust bearing which is sometimes used in light 
drilling and milling machines. The lower row in the ball-head 
of a cycle also forms such a bearing. 

The arrangement of the ball and its grooves, shown in 
figure 422, is almost as bad as it could possibly be. Let a, b, c, 




f -in 




7J>\^ 



---1^ 



Fig. 42a. 



Fig. 423. 



and d (fig. 423) be the points of contact of the ball with the sides 
of the groove, and the centre of the ball. If no rubbing takes 
place at the points a and ^, the instantaneous axis of rotation of 
the ball relative to the groove B must be the line a b ; that is, 
the motion of the ball is the same as that of a cone, with vertex 
z/j and semi-angle ov^a^ rolling on the disc of which the line a Vx 
is a section. Suppose now that there is no rubbing at the point 
d^ and let w be the angular speed of the spindle A, Drop a 
perpendicular d v^i on to the axis. Then F^, the linear speed of 
the point Z>, will be 

and Vci the linear speed of the point c of the spindle, will be 



W X 2^2 ^. 

The angular speed of the ball is 

da da 



Digitized by CjOOQIC 



392 Details 



CHAP. ZZT. 



The linear speed of the point c on the ball must be equal to 
the speed of the point d on the ball, since these two points are at 
the same distance from the instantaneous axis of rotation a Vx- 
Therefore the speed of rubbing at the point c is 

ft) X i^^d — v^ c) 

= ft) X c d. 

If the grooves are equally smooth it seems probable that the 
actual vertex, v, of the rolling cone will be about midway between 
Vx and v^^ and the rolling cone, equivalent to the ball, will be 
e vf\ the points a and d will lie inside this cone, the points b 
and c outside, and the rubbing will be equally distributed between 
the points a, ^, r, and d. 

In comparison with the rubbing, the rolling and spinning fric- 
tions will be small. A much better arrangement would be to 
have only one groove, the other ball-race being a flat disc. 

273. Dust-proof Bearings.— If the ball-bearing (fig. 404) be 
examined it will be noticed that there is a small space left be- 
tween the fixed cone C, and the cup Z>, fastened to the rotating 
hub. This is an essential condition to be attended to in the de- 
sign of ball-bearings. If actual contact took place between the 
cone and the cup, the rubbing friction introduced would require 
a greater expenditure of power on the part of the rider. Now, for 
a ball-bearing to work satisfactorily, the adjusting cone should not 
be screwed up quite tight, but a perceptible play should be left 
between the hub and the spindle ; the clearance between the cup 
C and cone D should therefore be a little greater than this. 

In running along dusty roads it is possible that some may 
enter through this space, and get ground up amongst tlie balls. 
In so-called dust-proof bearings, efforts are made to keep this 
opening down to a minimum, but no ball-bearing can be abso- 
lutely dust-proof unless there is actual rubbing contact between 
the rotating ^hub and a washer, or its equivalent, fastened to 
the spindle. Approximately dust-proof bearings can be made by 
arranging that there shall be no corners in which dust may easily 
find a lodgment. Again, it will be noticed that the diameter of 
the annular opening for tlie ingress of dust is snudler in the bear- 

Digitized by CjOOQIC 



CHAP. ZXV. 



Bearings 393 



ing figure 413 than in the bearing figure 404 ; the former bearing 
should, therefore, be more nearly dust-proof than the latter. 

The small back wheels of * Ordinaries ' often gave trouble from 
dust getting into the bearings, such dust coming, not only from 
the road direct, but also being thrown off from the driving-wheel. 
When a bearing has to run in a very dusty position a thin washer 
of leather may be fixed to the spindle and press lightly on the 
rotating hub, or vice versd. The frictional resistance thus intro- 
duced is very small, and does not increase with an increase of 
load on the bearing. 

274. Oil-retaining Bearings. — Any oil supplied through a 
hole at the middle of the hub in the bearing shown in figure 404 
will sooner or later get to the balls, and then ooze out between 
the cup and cone. In the bearing shown in figure 413, on the 
other hand, the diameter of the opening between the spindle and 
hub being much less than the diameter of the outer ball-race, oil 
will be retained, and each ball at the lowest part of its course be 
immersed in the lubricant. 

Figure 380 is a driving hub and spindle, with oil-retaining 
bearings, made by Messrs. W. Newton & Co., Newcastle-on-Tyne. 
Figure 381 is a hub, also with oil-bath lubrication, by the Centaur 
Cycle Company. 

275. Crasbing Pressure on Balls. — In a row of eight or nine 
balls, all exactly of the same diameter and perfectly spherical, 
running between properly formed races, it seems probable that 
the load will be distributed over two or three balls. If one ball 
is a trifle larger than the others in the bearing, it will have, at 
intervals, to sustain all the weight. In a ball thrust bearing with 
balls of uniform size, the total load is distributed amongst all the 
balls. The following table of crushing loads on steel balls is 
given by the Auto-Machinery Company (Limited), Coventry, 
from which it would appear that if -P be the crushing load in lbs., 
and d the diameter of the ball in inches, 

P = 82400 d^ (19) 



Digitized by CjOOQIC 



394 Details chap. m. 

Table XIII.t-Weights, Approximate Crushing Ix>ads, and 
Safe Working Loads of Diamond Cast Steel Balls. 



Diameter of ball 


Weight per gross 


Crushing load 
lbs. 


Working load 


in. 


, lbs. 


lbs. 


k 


•0415 


1,288 


160 


h 


•I40I> 


2,900 


360 


•3322 


5,150 


640 


J _ 


•6488 


8.050 


1,000 


I II213 


11,600 


1,450 


* 


; 26576 


20,600 


2,570 



276. Wear of Ball-bearings. — It is found that the races in 
ball-bearings are grooved after being some time in use. This 
grooving may be due partly to an actual removal of material owing 
to the grinding motion of the balls, and partly to the balls gradu- 
ally pressing into the surfaces, the balls possibly being slightly 
harder than the cups and cones. 

Professor Boys has found that the wear of balls in a bearing is 
practically negligible (* Proc. Inst. Mech. Eng.,' 1885, p. 510). 

277. Spherical Ball-races. — If by any accident the central 
spindle in a ball-bearing gets bent, the axes of the two ball-races 
will not coincide, and the bearing may work badly. Messrs. 
Fichtel & Sachs, Schweinfurt, Germany, get over this difficulty 




Fig. 434. 



by making the inner ball-race spherical (fig. 424), so that however 
the spindle be bent the ball-race surface will remain unaltered. 

Digitized by LjOOQIC 



I 



OttAP. XXV. 



Bearings 



395 



278. Vnivenal BaU-bearinff. — Figure 425 shows a ball- 
bearing designed by the author, in which either the spindle or the 
hub may be considerably bent 
without affecting its smooth run- 
ning. The cup and cone between 
which the balls run, instead of 
being rigidly fixed to the hub 
and spindle respectively, rest on 
concentric spherical surfaces. 
One of the spindle spherical sur- 
faces is made on the adjusting 
nut. This bearing, automatically adjusting itself, requires no 
care to be taken in putting it together. The working parts, being 
loose, can be renewed, if the necessity arises, by an unskilled 
person. 

In the case of a bicycle falling, the pedal-pin runs a great 
chance of being bent, a bearing like either of the two above 
described seems therefore desirable for, and specially applicable 
to, pedals. 




Fig. 425. 



Digitized by CjOOQIC 



396 Details chap. Tin. 



CHAPTER XXVI 

CHAINS AND CHAIN GEARING 

279. Tranamiiiioii of Power by Flexible BaiLcli.--A flexible 
steel band passing over two pulleys was used in the * Otto ' dicycle 
to transmit power from the crank-axle to the driving-wheels. The 
effort transmitted is the difference of the tensions of the tight and 
slack sides of the band ; the maximum effort that can be trans- 
mitted is therefore dependent on the initial tightness. Like belts 
or smooth bands, chains are flexible transmitters. If the speed 
of the flexible transmitter be low, the tension necessary to transmit 
a certain amount of power is relatively high. In such cases the 
available friction of a belt on a smooth pulley is too low, and 
gearing chains must be used. Projecting teeth are formed on the 
drums or wheels, and fit into corresponding recesses in the links 
of the chain. 

A chain has the advantage over a band, that there is, or should 
be, no tension on its slack side, so that the total pressure on the 
bearing due to the power transmitted is just equal to the tension 
on the driving side. 

For chain gearing to work satisfactorily, the pitch of the chain 
should be equal to that of the teeth of the chain-wheels over which 
it runs. Unfortunately, gearing chains subjected to hard work 
gradually stretch, and when the stretching has exceeded a certain 
amount they work very badly. 

Gear.—ThQ total effect of the gearing of a cycle is usually 
expressed by giving the diameter of the driving-wheel of an 
* Ordinary ' which would be propelled the same distance per turn of 
the pedals. Thus, if a chain-driven Safety has a 28-in. driving- 
wheel which makes two revolutions to one of the crank-axle, the 

Digitized by CjOOQIC 



CHAP. XZVI. 



Chains and Chain Gearing 



397 



machine is said to be geared to 56 in. Let -A^, and iVj be the 
numbers of teeth on the chain-wheels on driving-wheel hub and 
crank-axle respectively, and d the diameter of the driving-wheel 
in inches, then the machine is geared to 



N, 



//inches (i) 



The distance travelled by the machine and rider per turn of the 
crank-axle is of course 

— ? JT // inches (2) 

The following table of gearing may be found useful for 
reference : 

Table XIV, — Chain Gearing. 

Gear to which Cycle is Speeded, 



1 Number 


























1 of teeth 
on 










Diameter of driving 


wheel 










1= 


iS 


aa 


a4 


a6 


a8 


30 


32 


34 


36 


38 


40 


4a 


44 


16 
16 


I 


SO? 
44 


5J» 


59? 
5a 


64 
56 


^ 


'4 


S» 


8a| 
7a 


?? 


?:♦ 


t 


16 
16 


9 
xo 


%\ 


in 


:n 


St 


f 


h 


n 


% 


SI 


u* 


Vi\ 


;3 


X7 
17 


I 


% 


58I 
5z 


63t 
55 


68 

59i 

5at 


r,l 


w 


82» 

7a 


%\ 


d 


r 


zoa 

894 


•:3 


»7 


9 


4« 
37 


45 
40I 


49 


56 


t\ 


641 


68 


v^ 


754 


794 


834 


17 


10 


44i 


47! 


5' 


57l 


614 


68 


7if 


744 


18 
18 


1 


l^ 


61^ 
54 


66t 
58 t 


11 


?A 


8af 
72 


% 


If 


t 


loaf 


108 


"34 
99 


18 


9 


44 


48 


5a 


56 


60 


64 


68 


7a 


76 


88 


..8 


zo 


39l 


43i 


46} 


50I 


54 


57l 


614 


64I 


681 


7a 


75! 


79l 


'9 


8 


5»J 


57 


61 ? 


664 


714 


76 


80J 


854 


^ 


^» 


Hi 


I04* 


1 19 1 9 


46! 


53 


54t 


594 


631 


67, 


'4 


7S. 


80} 


VA\ 


19 1 zo 


41I 


49 


534 


57 


60I 


681 


7a4 


76 


794 


I ao I 8 


55 


63 


65 


70 


75 


80 


85 


t 


iJ» 


xoo 


105 


no 


ao 9 


48I 


534 


57i 


6a| 


66i 


714 


75t 


881 


9li 


S' 


r ao 10 


44 


1 48 


5a 


56 


60 


64 


68 


7a 


76 


80 


84 



280. Early Tricycle Chain.— Figure 426 illustrates the 
* Morgan ' chain, used in some of the early tricycles, which was 

Digitized by CjOOQIC 



398 



Details 



CBAP. XXTL 



composed of links made from round steel wire alternating with 
tubular steel rollers. There being only line contact between 




Fig. 426. 



adjacent links and rollers, the wear was great, and this form of 
chain was soon abandoned. 

281. Hnmber Chain. — Figure 427 shows the ' Humber ' chain, 
formed by a number of hard steel blocks (fig. 428) alternating 



iiii 



31^ -Mg 

ji 



Fig. 427. 



with side-plates (^g, 429). The side-plates are riveted together 
by a pin (fig. 430), which passes through the hole in the block. 
The rivet-pin is provided with shoulders at each end, so that the 
distance between the side-plates is preserved a trifle greater than 




Fig. 4:8. 




Fig. 430. 



the width of the block. In the * Abingdon- Humber' chain the 
holes in one of the side-plates are hexagonal, so that the pair of 
rivet-pins, together with the pair of side-plates they unite, form 
one rigid structure, and the pins are prevented from turning in the 
side-plates. 

Digitized by CjOOQIC 



CHAP. XXVI. 



Chains and Chain Gearing 



399 



Figure 43 1 shows a * Humber ' pattern chain, made by Messrs. 
Perry & Co. The improvement in this consists principally in the 
addition of a pen steel bush surrounding the rivet. The ends of 
the bush are serrated, and its total length between the points is 
a trifle greater than the distance between the shoulders of the 




Fig. 431. 

rivet-pin. The act of riveting thus rigidly fixes the bush to the 
side-plate, and prevents the rivet-pins turning in the side-plates. 
The hard pen-steel bush bears on the hard steel block, and there 
is, therefore, less wear than with a softer metal rubbing on the 
block. 

Messrs. Brampton & Co. make a ' self-lubricating * chain of 
the * Humber' type (fig. 432). The block is hollow, and made in 




Fig. 432. 

two pieces ; the interior is filled with lubricant — a specially 
prepared form of graphite — sufficient for several years. 

282. Roller Chain. - Figure 433 shows a roller or lons^-link chain^ 
as made by the Abingdon Works Co., the middle block of the 
* Humber ' chain being dispensed with, and the number of rivet- 
pins required being only one-half. Each chain-link is formed by 
two side-plates, symmetrically situated on each side of the centre 
line, and each rivet thus passes through four plates. The two 
outer plates are riveted together, forming one chain-link ; while 
the two inner plates, forming the adjacent chain-link, can rotate 

Digitized by CjOOQIC 



400 



Details 



CHAP. DTI. 



on the rivet-pin as a bearing. If the inner plates were left as 
narrow as the outer plates, the bearing surface on the rivet would 




Fig. 433. 

be very small, and wear would take place rapidly. Figure 434 
shows the inner plate provided with bosses, so that the bearing 

surface is enlarged ; and figure 435 

shows the plates riveted together. 

The rivet, shown separately (fig. 

436), thus bears along the whole 
width of the inner chain-link. Loose rollers surround the bosses ; 
these are not shown in figure 435, but are shown in figure 433. 




Fig. 




Fig. 435. 

Single-link Chain, — The chain illustrated in figure 433 is a 
two-link chain ; that is, its length must be increased or diminished 
by two links at a time. Thus, if the chain stretches and 
becomes too long for the cycle, it can only be shortened 
by two inches at a time. Figure 437 shows a single-link 
chain ; that is, one which can be shortened by removing 
one link at a time. The side-plates in this case are not 
straight, but one pair of ends are brought closer together 

436. than the other ; the details of boss, rivets, and rollers are 

same as in the double-link chain. 

The width of the space between the side-plates of figure 433 

Digitized by CjOOQIC 



csAF. xxrr. 



Chains and Cfiain Gearing 



401 



is di/Terent for two consecutive links. If the narrow link fit the 
side of the chain-wheel, the side of the wide link will be quite 




Fig. 437. 



clear ; in other words, the chain will be guided sideways on to 
the chain-wheel only at every alternate link. The single-link chain 




Fig. 438. 



is in this respect superior to the double-link chain. In the * R. F. 
Hall ' corrugated-link chain (fig. 438) the alternate side-plates were 
depressed, so that the inside width was the same for all links. 

283. Pivot-chain.— In the pivot-chain (fig. 439), made by 
the Cycle Components Manufacturing Company, (Limited) the 
pins and bushes of the * Humber ' or long-link chain are replaced 
by hard steel knife edges. The relative motion of the parts is 




Fig. 439. 

smaller, and therefore the work lost in friction may also be expected 
to be smaller than in the * Humber ' chain, though it remains 
to be seen whether the bearing surfaces will be able to stand for 
a few years the great intensity of pressure to which they are sub- 
jected in ordinary running. 

284. EoUer-ohaiii Chain-wheel. — The pitch-line of a lotig- 
link chain-wheel must be a regular polygon of as many sides as 

Digitized by V^ L) D 



402 Details chap, zzvl 

there are teeth in the wheel. Let a^b^c , . . (fig. 440) be con- 
secutive angles of the polygon. When the chain is wrapped 
round the wheel the centres of the chain rivets will occupy the 
positions a^ b^ c , . . The relative motion of the chain and wheel 
will be the same, if the wheel be considered fixed and the chain 
to be wound on and off. If the wheel be turning in the direction 
of the arrow, as the rivet a leaves contact with the wheel, it 
will move relative to the wheel in the circular arc a a,, having b 
as centre, a, lying in the line c b produced. Assuming that the 
chain is tight, the links a b and b c will now be in the same straight 
line, and the rivet a will move, relative to the chain-wheel, in the 
circular arc a, a^, with centre c \ a^ lying in the straight line dc 




0^'.'- i^ 

Fig. 440. 

produced. Thus, the relative path of the centre of the rivet A as 
it leaves the wheel is a series of circular arcs, having centres 
b^ Cy d . . . respectively. It may be noticed that this path is 
approximately an involute of a circle, the approximation beii^ 
closer the larger the number of teeth in the wheel In 
the same way, the relative path of the centre of rivet ^ as it 
moves into contact with the wheel is an exactly similar curve b by 
b.i , . ., which intersects the curve aa^a^ . . .at the point x. If 
the rivets and rollers of the chain could l)e made indefinitel}- 
small, the largest possible tooth would have the outline aa^ xbx b. 
Taking account of the rollers actually used, the outline of the 
largest possible tooth will be a pair of parallel curves ^ .Y and 

Digitized by CjOOQIC j. 




CHAP. XXVI. Chains and Chain Gearing 403 

^ X intersecting at X^ and lying inside a a^ , , , and b b^ . . ., 
a distance equal to the radius of the rollers. 

Kinematically there is no necessity for the teeth of a chain- 
wheel projecting beyond the pitch-line, as is absolutely essential 
in spur-wheel gearing. If the 
pitches of the chain and wheel 
could be made exactly equal, 
and the distance between the 
two chain-wheels so accurately 
adjusted that the slack of the 
chain could be reduced to zero, 
and the motion take place with- 
out side-swaying of the chain, '*^' ^**' 
the chain-wheel might be made as in figure 441. With this ideal 
wheel there would be no rubbing of the chain-links on it as they 
moved into and out of gear. 

But, owing to gradual stretching, the pitch of the chain is 
seldom exactly identical with that of the wheel ; this, combined 
with slackness and swaying of the chain, makes it desirable, and 
in fact necessary, to make the cogs project from the pitch-line. 
If the cogs be made to the outline A XB (fig. 440), each link of 
the chain will rub on the corresponding cog along its whole length 
as it moves into and out of gear ; or rather, the roller may roll on 
the cog, and rub with its inner surface on the bosses of the inner 
plates of the link. To eliminate this rubbing the outline of the 
cog should therefore be drawn as follows : Let a and b (fig. 
442) be two adjacent corners of --w-^. 

the pitch-polygon, and let the 
rollers, with a and b as centres, cut 
a b 2X /and g respectively. The 
centres /// and n of the arcs of out- 
line through / and g respectively 
should lie on a b, but closer to- p^^ 

gather than a and b ; in fact, /may 

conveniently be taken for the centre of the arc through g^ and 
ince versd. The addendum -circle may be conveniently drawn 
touching the straight line which touches, and lies entirely outside 
of, two adjacent rollers. 

Digitized by CjQPg^ 




404 Details 



CBAP. XXTL 



TAe 'Simpson' Lever-chain has triangular links, the inner 
corners, Ay B^ C . . . (fig. 443), are pin-jointed and gear in the 
ordinary way with the chain-wheel on the crank-axle. Rollers 
project from the outer corners, a, ^, . . . and engage with the 
chain-wheel on the driving-hub. As the chain winds off the 
chain-wheel the relative path, / /j /^ . . . of one of the inner 
corners is, as in figure 440, a smooth curve made up of circular 
arcs, while that of an outer comer has cusps, «i, aj, . . . corre- 



FlG. 443- 

sponding to the sudden changes of the relative centre of rotation 
from A to By from B io Cy . , . , As the chain is wound on 
to the wheel, the relative path of an adjacent comer, ^, is a 
curve, b by b^ • * * oi the same general character, but not of 
exactly the same shape, since the triangular links are not equal- 
sided. These two curves intersect at x, and the largest possible 
tooth outline is a curve parallel to a a^ x b. If the actual tooth 
outline lie a little inside this curve, as described in figure 442, 
the rubbing of the rollers on their pins will be reduced to a 
minimum, and the frictional resistance will not be greater than 
that of an ordinary roller chain. Thus there is no necessity for 
the cusp on the chain-wheel ; the latter may therefore be made 
with a smaller addendum-circle. 

Let ay py and q (Hg. 410) be three consecutive comers of the 

Digitized by CjOOQIC 



CHAP. XXVI. 



Chains and Chain Gearing 



405 



pitch-polygon of a long-link chain-wheel, one-inch pitch. The cir- 
cumscribing circle of the pitch-polygon may, for convenience of 
reference, be called the pitch-circle. Let R be the radius of the 
pitch-circle, and -A^ the number of teeth on the wheel. From 6>, the 
centre, draw O k perpendicular to / q. The angle pOq'\% evidently 

5-5^ degrees, and the angle pOk therefore i^ degrees. And 



R^Op = jj^^o^ = -:^;5^ -o inches 



(3) 



Tabli 


£ XV. — Chain-wheels, i-in. I 

A* _ 
Radius of circumscribing circle of 


*ITCH. 


N 


Radius of circle 


Number of teeth 


pitch-polygon 


whose circumference 


in chain-wheel 


^ 


is iV inches 




Long-link chain 
Inches 


Humber chain 
Inches 






Inches 


6 


I -000 


-967 


-955 


7 


II53 


I-I25 


I-II4 


8 


1*307 


1-283 


1-274 


9 


1-462 


I -441 


1-433 


10 


I 618 


1-599 


1-592 


II 


1775 


1-758 


■ 
I -75 1 


12 


1932 


1-916 


I-9IO 


13 ^ 


2-089 


2-074 


2069 


14 


2247 


2-233 


2-228 


'5 


2405 


2-392 


2-387 


16 


2-563 


2-551 


2-546 


17 


2-721 


2-710 


2-705 


18 


2-880 


2-870 


2-865 


19 


3-039 


3-029 


3-024 


20 


3-197 


3-188 


3-183 


21 


3-356 


3347 


3-342 


22 


3-514 


3-505 


3-501 


23 


3-672 


3-664 


3-660 


24 


3-831 


3-824 


3-820 


25 


3990 


3-983 


3-979 


26 


4-148 


4-142 


, 4-138 


27 


4-307 


4-301 


' 4-297 


28 


4-466 


4-460 


1 4-456 


1 29 


4-626 


4-620 


1 4-616 


30 

1 


4-785 


4-779 ,igi 


|zedbyGt)??^le 



4o6 Details cnxf. xxn. 

The values of R for wheels of various numbers of teeth are 
given in Table XV. 

285. Humber Chain-wheel. — The method of designing the 
form of the teeth of a * Humber ' chain-wheel is, in general, the 
same as for a long-link chain, the radius of the end of the hardened 
block being substituted for the radius of the roller; but the 
distance between the pair of holes in the block is different from 
that between the pair of holes in the side-plates, these distances 




Fig. 444. 

being approximately -4 in. and -6 in. respectively. The pitch- 
line of a * Humber ' chain-wheel will therefore be a polygon with 
its corners all lying on the circumscribing circle, but with its sides 
•4 in. and '6 in. long alternately. Figure 444 shows the method 
of drawing the tooth, the reference letters corresponding to those 
in figures 440 and 442, so that the instructions need not be 
repeated. 

Let fl, ^, r, d (fig. 444) be four consecutive corners of the 
pitch-polygon of a ' Humber ' chain-wheel. Produce the sides a b 
and ^^ to meet at e. Then, since a, ^, r, and d lie on a circle, 
it is evident, from symmetry, that the angles ebc and ecb are 
equal. If -A^ be the number of teeth in the wheel, there are 2 N 

Digitized by CjOOQIC 



CHAP. XXVI. Chains and Chain Gearing 407 

sides of the pitch-polygon, and the external angle cbc will be 

-160 180 J 

•^ — = -- decrees. 

Let a r = Dy then R = -:- . 

.180° 
^s^n — 



But n^ = ab* + bc^ + 2ab.bc cos 

= -36 + -16 + -48 cos '^° 



180° 

N 



a/ 52 + -48 cos 



180" 

^= — . 180° ^ (4) 

The radii of the pitch-circles of wheels having different 
numbers of teeth are given in Table XV. 

286. Side-olearance and Stretching of Chain.— With chain- 
wheels designed as in sections 284-5, ^^^^ *^^ pitch of the teeth 
exactly the same as the pitch of the chain, there is no rubbing of 
the chain links on the wheel-teeth, the driving arc of action is the 
same as the arc of contact of the chain with the wheel, and all 
the links in contact with the wheel have a share in transmitting 
the effort. But when the pitch of the chain is slightly different 
from that of the wheel-teeth the action is quite different, and 
the chain-wheels should be designed so as to allow for a 
slight variation in the pitch of the chain by stretching, with- 
out injurious rubbing action taking place. The thickness of the 
teeth of the long-link chain-wheel (fig. 440) is so great that it 
can be considerably reduced without impairing the strength. 
Figure 445 shows a wheel in which the thickness of the teeth has 
been reduced. If the pitch of the chain be the same as that of 
the wheel, each tooth in the arc of contact will be in contact 
with a roller of the chain, and there will be a clearance space x 
between each roller and tooth. Let N be the number of teeth 
in the wheel ; then the number of teeth in action will be in 

Digitized by CjOOQIC 



4o8 Details 



CHAP. XITL 



N 

general not more than — + i. The original pitch of the chain 

2 

may be made -^ — less than the pitch of the wheel-teeth, the 

+ I 

2 

wheel and chain will gear perfectly together. Figure 445 illus- 
trates the wheel and chain in this case. After a certain amount 
of wear and stretching, the pitch oflhe chStn will become exactly 
the same as that of the teeth, and each tooth will have a roller in 
contact with it. The stretching may still continue until the pitch 

of the chain is — greater than that of the wheel-teeth, with- 

- + I 

2 

out any injurious action taking place. 

The mutual action of the chain and wheel having dififerent 
pitches must now be considered. First, let the pitch of the chain 
be a little less than that of the teeth (fig. 445), and suppose the 



e-e-^ 



Fig. 445. 

wheel driven in the direction of the arrow. One roller, Ay just 
passing the lowest point of the wheel will be driving the tooth in 
front of it, and the following roller, B^ will sooner or later come 
in contact with a tooth. Figure 445 shows the roller B just 
coming into contact with its tooth, though it has not yet reached 
the pitch-line of the wheel. The motion of the chain and wheel 
continuing, the roller B rolls or rubs on the tooth, and the 

Digitized by Cj^OOQ IC 



CHAP. XXVI. 



Chains and Chain Gearing 



409 



0-^ 



roller A gradually recedes from the tooth it had been driving. 
Thus the total effort is transmitted to the wheel by one tooth, or 
at most two, during the short period one roller is receding from, 
and another coming into, contact. 

If the pitch of the chain be a little too great, and the wheel be 
driven in the same direction, the position of the acting teeth is at 
the top of the wheel (fig. 446). The roller, C, is shown driving 
the tooth in front of it, 
but as it moves outwards 
along the tooth surface 
the following roller, Z>, 
will gradually move up 
to, and drive, the tooth 
in front of it 

The action between 
the chain and the driving- 
wheel is also explained 
on the same general prin- 
ciples ; if the direction 
of the arrow be reversed, 
figures 445 and 446 will 
illustrate the action. 

In a chain-wheel made with side-clearance, assuming the 
pitches of chain and teeth equal, there will be two pitch-polygons 
for the two directions of driving. Let a and b (fig. 447) be two 
consecutive comers of one of the pitch-polygons, and let the roller 




-e--Hs 



Fig. 446. 




Fig. 447. 



with centre a cut a ^ at / The centre m of the arc of tooth out- 
line through / lies on a b. Let aa' ^bb^ be the side-clearance 
measured along the circumscribing circle ; a' and b' will therefore 
be consecutive comers of the other pitch-polygon. Let the roller 

Digitized by CjOOQIC 



410 Details chap. xzn. 

with centre b' cut a' b"\xig\ the centre n of the arc of tooth outline 
through /lies on a* b'. The bottom of the tooth space should be a 
circular arc, which may be called the root-circle, concentric with 
the pitch-polygons, and touching the circles of the rollers a and ol, 
287. Rubbing and Wear of Chain and Teeth.- If the outline 
of the teeth be made exactly to the curve /-Y (^%, 440), the roller 
A will knock on the top of the tooth, and will then roll or nib 
along its whole length. If the tooth be made to a curve ]>in§ 
inside fXy the roller will come in contact with the tooth at a 
point / (fig. 448), such that the distance of / from the curve /A' 
is equal to the difference of the pitches of the teeth and chain ; 
// will be the arc of the tooth over which contact takes place. 
The length of this arc will evidently be smaller (and therefore 
also the less will be the work lost in friction), the smaller the 
radius of the tooth outline. 

In the * Humber ' chain the block comes in contact with the 

teeth, and there is relative rubbing over the arc // (fig. 448). 

w The same point of the block always 

^ N comes in contact with the teeth, so 

^\ that after a time the wear of the 

/ j5s\ blocks of the chain and the teeth of 

/^^""^f i/^'^N ^^ wheel becomes serious, especially 

J ^-.— jL—.. J««^ J if the wheel-teeth be made rather 

V_y ^ ^\^ full. 

B /^ The chief advantage of a roller- 

P»«- 448. chain lies in the fact that the roller 

being free to turn on the rivet, different points of the roller 
come successively in contact with the wheel-teeth. If the chain 
be perfectly lubricated the roller will actually roll over its arc of 
contact, //, with the tooth, and will rub on its rivet-pin. The 
rubbing is thus transferred from a higher pair to a lower pair, and 
the friction and wear of the parts, other things being equal, will 
be much less than in the * Humber ' chain. Even with imperfect 
lubrication, so that the roller may be rather stiff on its rivet-pin, 
and with rubbing taking place over the arc //, the roller will at 
least be slightly disturbed in its position relative to its rivet-pin, 
and a fresh portion of it will next come in contact with the wheel- 
teeth. Thus, even under the most unfavourable conditions, the 

Digitized by CjOOQIC 



CHAP. XXVI. Chains and Chain Gearing 411 

wear of the chain is distributed over the cylindrical surface of the 
roller, consequently the alteration of form will be much less than 
in a * Humber ' chain under the same conditions. 

It must be clearly understood that the function performed by 
rollers in a chain is quite different from that in a roller-bearing. In 
the latter case rubbing friction is eliminated, but not in the former. 

288. Common Faults in Design of Chain-wheels.— The por- 
tions of the teeth lying outside the pitch-polygon are often made far 
too full, so that a part of the tooth lies beyond the circular arc 
f X (fig. 440) ; the roller strikes the corner of the tooth as 
it comes into gear, and the 
rubbing on the tooth is J. 

excessive. This feulty / \ 

tooth is illustrated in figure / \ 

449. ^'"^^i Y^\ 

In long-link chain-wheels ^\^^ — f \ — -n 5 

the only convex portion is •'^^^^ >^ ^^•.^^'^ 

very often merely a small 

circular arc rounding off the *^* ^'' 

side of the tooth into the addendum-circle of the wheel. This 

rounding off of the corner is very frequently associated with the 

faulty design above mentioned. If the tooth outline be made to 

//, a curve lying well within the circular arc/ X (fig. 448), this 

rounding off of the corners of the teeth is quite unnecessary. 

Another common fault in long-link chain-wheels is thit the 
bottom of the tooth space is made one circular arc of a little 
larger radius than the roller. There is in this case no clearly 
defined circle in which the 
centres of the rollers are com- 
pelled to lie, unless the ends 
of the link lie on the cylin- 
drical rim from which the ^ | 
teeth project. In back-hub \ 
chain-wheels this cylindrical Fig. 450. 
rim is often omitted. Care 

should then be taken that the tooth space has a small portion 
made to a circle concentric with the pitch-circle of the wheel. 
Again, in this case, the direction of the mutual force between 

Digitized by V^jOOQ 





412 Details chap, nn, 

the roller and wheel is not along the circumference of the pitch- 
polygon ; there is therefore a radial component tending to force 
the rollers out of the tooth spaces, that is, there is a tendency 
of the chain to mount the wheel (fig. 450). 

In *Humber' pattern chain-wheels the teeth are often quite 
straight (fig. 451). This tooth-form is radically wrong. If the 

teeth are so narrow at the top as 
to lie inside the curve f X^ the 
force acting on the block of the 
chain will have an outward com- 
ponent, and the chain will tend 
to mount the wheel. This faulty 
'°* *^'* design is sometimes carried to an 

extreme by having the teeth concave right to the addendum-circle. 
Either of the two faults above discussed gives the chain a 
tendency to mount the wheel, and this tendency will be greater 
the more perfect the lubrication of the chain and wheel. 

289. Summary of conditions determining^ the proper fjom 
of Chain-wheels. — i. Provision should be made for the gradual 
stretching of the chain. This necessitates the gap between two 
adjacent teeth being larger than the roller or block of the chain. 

2. The centres of the rollers in a long-link chain, or the blocks 
in a * Humber ' chain, must lie on a perfectly defined circle con- 
centric with the chain-wheel. When the wheel has no distinct 
cylindrical rim, the bottom of the tooth space must therefore be a 
circular arc concentric with the pitch-polygon. 

3. In order that there should be no tendency of the chain to 
be forced away from the wheel, the point of contact of a tooth 
and the roller or block of the chain should lie on the side of the 
pitch-polygon, and the surface of the tooth at this point should be 
at right angles to the side of the pitch-polygon. The centre of 
the circular arc of the tooth outline must therefore lie on the side 
of the pitch-polygon. 

4. The blocks or rollers when coming into gear must not strike 
the corners of the teeth. The rubbing of the roller or block on 
the tooth should be reduced to a minimum. Both these condi- 
tions determine that the radius of the tooth outline should be less 
than * length of side-plate of chain, minus radius of roller or blodt' 

Digitized by CjOOQIC 



CHAP. XXVI. Chains and Chain Gearing 413 

The following method of drawing the teeth is a rksume of 
the results of sections 285-8, and gives a tooth form which satisfies 
the above conditions : Having given the type of chain, pitch, and 
number of teeth in wheel, find R^ the radius of the pitch-circle c r, 
by calculation or from Table XV. On the pitch-circle c c (fig. 452), 
mark off adjacent corners a and b of the pitch-polygon. With 
centres a and ^, and radius equal to the radius of the roller (or the 
radius of the end of the block in a * Humber* chain), draw circles, 
that firom a as centre cutting a b 2Xf, Through /draw a circular 
arc,/^, with centre m on a b^ w/ being less than bf. Mark off, 



Fig. 452. 

along the circle c c^ a a^ ^ b b^ = side clearance required, and 
with centres a} and ^*, and the same radius as the rollers, draw 
circles, that from centre b^ cutting the straight line a^ b^ at g. 
With centre «' lying on a^ b^^ and radius equal to mf, draw a 
circular arc g kK Draw the root-circle r r touching, and lying 
inside, the roller circles. The sides /^ and^^* of the tooth 
should be joined to the root-circle r rhy fillets of slightly smaller 
radius than the rollers. Draw a common tangent / / to the roller 
circles a and ^, and lying outside them ; the addendum-circle may 
be drawn touching / /. 

It should be noticed that this tooth form is the same whatever 
be the number of teeth in the wheel, provided the side-clearance 
be the same for all. The form of the spaces will, however, vary 
with the number of teeth in the wheel. A single milling-cutter 
to cut the two sides of the same tooth might herefore serve for all 
sizes of wheels ; whereas when the milling-cutter cuts out the 

Digitized by CjOOQIC 



414 



Details 



CBAT. xxn. 



space between two adjacent teeth, a separate cutter is required few 
each size of wheel 




Fig. 453. 



joogle 



CHAP. zzyi. 



Chains and Chain Gearing 



415 



Figure 453 shows the outlines of wheels for inch-pitch long- 
links made consistent with these conditions, the diameter of the 
roller being taken g in. The radius of the side of the tooth is in 




Fig. 454. 



each case f in. (it may with advantage be taken less), and the 
radius of the fillet at root of tooth \ in. The width of the roller 
space measured on the pitch-polygon is (-375 -f '005 N)\n.\N 
being the number of teeth in the wheel. 

Digitized by CjOOQIC 



4i6 



Details 




Fig. 454A. 



Fig. 454b. 



Figure 454 shows the outlines of wheels for use with the 
* Humber' chain, the pitch of the rivet-pins in the side-plates being 
6 in. and in the blocks *4 in., and the ends of the blocks being 
circular, -35 in. diameter. 

290. Section of Wheel Blanks. — If the chain sways sideways, 
the side-plates may strike the tops of the teeth as they come 
into gear, and cause the chain to mount the wheel, unless each 

link is properly guided sideways 
on to the wheel. The cross sec- 
tion of the teeth is sometimes 
made as in figure 454A, the sides 
being parallel and the top comers 
rounded off. A much better 
form of section, which will allow 
of a considerable amount of 
swaying without danger, is that 
shown in figure 454B. The thickness of the tooth at the root is 
a trifle less than the width of the space between the side-plates of 
the link. The thickness at the point is very small — say, 3^ in. 
to ^V ^^* — ^'^d ^^ tooth section is a wedge with curved sides. 

If the side-plates of the chain be bevelled, as in Brampton's 
bevelled chain (fig. 432), an additional security against the chain 
coming oflT the wheel through side swaying will be obtained. 

291. Design of Side-plates of Chain.— The side-plates of a 

well-designed chain should be subjected to simple tension. If F 

be the total pull on the chain, and A the least sectional area of 

p 
the two side-plates, the tensile stress is - Such is the case 

with side-plates of the form shown in figure 429. 

Example /.—The section of the side-plates (fig. 429) is 'z in 
deep and '09 in. thick. The total sectional area is thus 

2 X 2 X 09 = 036 sq. in. 

The proof load is 9 cwt. = 1,008 lbs. The tensile stress is, 
therefore, 

/=s ''?? ^ 28,000 lbs. per sq. in. * 

•036 

= 12*5 tons per sq. in. 

Digitized by CjOOQIC 




CHAP. XXVI. Chains and Chain Gearing 417 

A considerable number of chains are being made with the 
side-plates recessed on one side, and not on the other (fig. 455). 

These side-plates are subjected to ^""^^ — f 

combined tensile and bending f C^ ^/yj^'^F^Q 
stresses. Let ^ be the width of \^^ ^ ^^^ y^ 
the plate, / its thickness, b^ be "'Ficrns! 

the depth of the recess, and let 

^2 = ^ — ^1 ; that is, ^2 would be the width of the plate if re- 
cessed the same amount on both sides. The distance of the 

centre of the section from the centre line joining the rivets is -1. 

The bending-moment M on the link is — *. The modulus of 

the section, Z, is . The maximum tensile stress on the section 
6 

is (sec. loi) 

•^ A^ Z 2b t 2.2tb* 

-^,\^*m <5> 

The stress on the side-plate if recessed on both sides would be 

^-r(fi-wt ^'^ 

The stress / calculated from equation (6) is always, within prac- 
tical limits, less than the stress calculated from equation (5) ; and, 
therefore, the recessed side-plates can be actually strengthened by 
cutting away material. This can easily be proved by an elemen- 
tary application of the differential calculus to equation (5). 

Example IL — Taking a side-plate in which / = '09 in., b = 
•3 in., ^, = •! in., and therefore b^ = '2 in., we get 

^ = 2 X '09 X '3 = '054 sq. in., 
and 

y — 2 X 09 X '3^ 

^ 6 

= '0027 in.' 
The distance of the centre of the section from the centre-line of 

Digitized by Cj ^ ^ 



41 8 Details ohap. 

the side-plate is '05 in., and the bending-moment under a proof 
load of 9 cwt. is 

J/'=9Xii2X'o5= 50-4 inch-lbs. 

The m&ximum stress on the section is 

/= i?^ + 5?^= 37,320 lbs. per sq. in. 

•054 -0027 

= 167 tons per sq. in. 

Thus this link, though having 50 per cent, more sectional area, is 
much weaker than a link of the form shown in figure 429. 

If the plate be recessed on both sides, A = '036 sq. in., and 

/= i^^ = 28,000 lbs. per sq. in. 
•036 

= 12*5 tons per sq. in. 

Thus the side-plate is strengthened, even though 33 per cent of 
its section has been removed. 

From the above high stresses that come on the side-plates of 
a chain during its test, and from the fact that these stresses may 
occasionally be reached or even exceeded in actual work when 
grit gets between the chain and wheel, it might seem advisable 
to make the side-plates of steel bar, which has had its elastic 
limit artificially raised considerably above the stresses that will 
come on the links under the proof load. 

The Inner Side-plates of a Roller Chain^ made as in figure 434, 
are also subjected to combined tension and bending in ordinary 
working. Assuming that the pressure between the rivet and inner 

link is uniformly distributed, the side-plate of the latter will be 

p 
subjected to a bending-moment il/" = — x /, / being the dis- 

2 

tance measured parallel to the axis of the rivet, between the 
centres of the side-plate and its boss, respectively. 

Example III, — Taking / = -08 in., and the rest of the data 
as in the previous example, the maximum additional stress on the 
side-plate due to bending is 

M 504 X -08 X 6 „ 

-V = '^-^ 5 = 149,000 lbs. per sq. m. 

= 667 tons per sq. in., 

Digitized by CjOOQIC 



CHAP. JLX¥l« 



Chains and Chain Gearing 



419 




which, added to the 12*5 tons per sq. in. due to the direct pull, 
gives a total stress of 79*2 tons per sq. in. Needless to say, 
the material cannot endure such a stress ; what actually happens 
during the test is, the side-plates slightly bend when the elastic 
limit is reached, the pressure on the inner edge of the boss is 
reduced, so that the resultant pressure between the rivet and 
side-plate acts nearly in line with the latter. Thus the extra 
bearing surface for the rivet, supposed to be provided by the 
bosses, is practically got rid of the first time a heavy pull comes 
on the chain. 

A much better method of providing sufficient bearing surfisice 
for the rivet-pins is to use __ r— 1, 
a tubular rivet to unite W^M^ ^ 
the inner side-plates 
(fig. 456), inside which 
the rivet-pin uniting the 
outer side-plates bears, 
and on the outside of 
which the roller turns. 
This is the method adopted by Mr. 
gearing chains. 

Side-plates of Single-link Chain, — In the same way, it will be 
readily seen that the maximum stress on the side-plates of the 
chain shown in figure 437 is much greater than on a straight 
plate with the same load. If the direction of the pull on the 
plate be parallel to the centre line of the chain, each plate will be 
subjected to a bending action. The ^transverse distance between 
the centres of the sections of the two ends is /, the bending- 
moment on the section will therefore ht PL A more favourable 
assumption will be that the pull on each plate will be in a line 
joining the middle points of its ends. The greatest distance 
between this line and the middle of section will be then nearly 

, and M= — . The bending in this case is in a plane at right 
2 2 

angles to the direction of the bending in the recessed side-plate 
(fig. 4ss). The modulus of the section Z is -^. 

Example IV, — Taking the same data as in the former examples, 

Digitized by Vj R B 2 



Fig. 4sd. 

Hans Renold for large 



^20 Detatis cbap. zzn. 

the load on the chain is 9 cwt, ^= "3 in., and /= '09 in., the 

pull on each plate is 504 lbs., the bending-moment is ^^ ?? 

2 

= 227 inch-lbs., A = -027 sq. in., Z= ~^~~(~~ = '000405 in.* 
The maximum stress on the section is therefore 

^ A^ Z 

a = 5?4 , __227_ 
•027 '000405 

= 18,670 + 56,050 Si 74,720 lbs. per sq. in. 
= 33*3 tons per sq. in. 

292. Bivets. — ^The pins fSsistening together the side-plates 
must be of ductile material, so that their ends may be riveted 
over without injury. A soft ductile steel has comparatively low 
tensile and shearing resistances. The ends of these pins are sub- 
jected to shearing stress due to half the load on the chain. If ^ be 

the diameter of the rivet, its area is — , and the shearing stress 

4 

on it will be 

^d- <7) 

Example L — If the diameter of the holes in the side-plate 
(fig. 429) be 'IS in., under a proof load of 9 cwt. the shearing stress 
will be 

1008 o 1U 

__ ^ = 28,500 lbs. per sq. m. 

2 X -01767 '^ V ^ 

= 1278 tons per sq. in. 

The end of the rivet is also subjected to a bending-momait 

— X -. The modulus of the circular section is approximately 
2 2 

— , the stress due to bending will therefore be 



Digitized by CjOOQIC 



CHAP. XXVI. Chains and Chain Gearing 421 

Example II, — ^Taking the dimensions in Example I. of sec- 
tion 291, and substituting in (8), the stress due to bending is 

/= 10 X 1008 x:o9 ^ ^ 1^3 j^ 

4 X 15* 

= 30 tons per sq. in. 

The rivet is thus subjected to very severe stresses, which 
cause its ends to bend over (fig. 457). 

The stretching of a chain is probably always due more to the 
yielding of the rivets than to actual stretching of the side-plates, 

if the latter are properly designed. A material 

that is soft enough to be riveted cold has not a ^ 3 D 

very high tensile or shearing resistance. It would fig. 457. 

seem advisable, therefore, to make the pins of 
hard steel with a very high elastic limit, their ends being turned 
down with slight recesses (fig. 458), into which the side plates, 

made of a softer steel, could be forced by pressure. 

The Cleveland Cycle Company, and the Warwick ^S$$y=^ 
and Stockton Company, manufacture chains on this 
system. 



293. Width of Chain and Bearing Pressure on ;^$$$^i:g^ 
Siyets. — In the above investigations it will be noticed pic. 458. 
that the width of the chain does not enter into con- 
sideration at all. The only effect the width of the chain has is on 
the amount of bearing surface of the pins on the block. If / be 

the width of the block, and d the diameter of the pin, the pro- 

p 
jected bearing area is Id^ and the intensity of pressure is - -, If 

the diameter of the pin (fig. 430) be '17 in., and the width of 
the block be -^ in. = '3125 in., the bearing pressure under the 
proof load will be 

= 18,980 lbs. per sq. in. 

•17 X -3125 

This pressure is very much greater than occurs in any other 
example of engineering design. Professor Unwin, in a table of 
* Pressures on Bearings and Slides,' gives 3,000 lbs. per sq. in. 
as the maximum value for '^bearings on which the load is inter- 

Digitized by VjOOQ 



422 



Details 



CHAP. xxn. 



mittent and the speed slow. Of course, in a cycle chain the 
period of relative motion of the pin on its bearing is small com- 
pared to that during which it is 
at rest, so thit the lubricant, if 
an oil-tight gear-case be used, 
gets time to find its way in be- 
tween the surfaces. 

294. Speed-ratio of Two 
Shafts Connected by Chain Gear- 
ing. — The average speeds of two 
shafts connected by chain gear- 
ing are inversely proportional to 
the numbers of teeth in the chain 
wheels; but the speed-ratio is 
not consianty as in the case of two 
shafts connected together by a 
belt or by toothed-wheels. Let Ox 
and O^ (fig. 459) be the centres 
of the two shafts, let the wheel 
O^ be the driver, the motion being 
as indicated by the arrow, and 
let A C he the straight portion 
of the chain between the wheels 
at any instant. The instantaneous 
angular speed-ratio of the wheels 
is the same as that of two cranks 
Ox A and O,^ C connected by 
the coupling-rod A C, Let B 
and D be the rivets consecutive 
to A and C respectively ; then, 
as the motion of the wheels 
continues, the rivet D will ulti- 
mately touch the chain-wheel at 
the point d^ — ^i, d^ and Cx 
being in the same straight line— 
and the angular speed-ratio of the 
wheels will be the same as ^t 
A and O^D connected by the straight coup- 
Digitized by CjOOqIc 




Fig. 459- 



of the two cranks Ox 



CHAP. XXVI. Chains and Chain Gearing 423 

ling-rod A Z>, shorter by one link than the coupling-rod A C. 
The motion continuing, the rivet A leaves contact with the chain- 
wheel at a^ and the virtual coupling-rod becomes B D ) the 
points ^s» ^29 Ai^d ^s lying in one straight line. The angular 
speed-ratio of the wheels is now the same as that of the two 
cranks O^B and O^D connected by the coupling-rod B D^ of 
the same length as -4 C 

Thus, with a long-link chain, the wheels are connected by a 
virtual coupling-rod whose length changes twice while the chain 
moves through a distance equal to the length of one of its links. 
The small chain-wheel, being rigidly connected to the driving- 
wheel of the bicycle, will rotate with practically uniform speed ; 
since the whole mass of the machine and rider acts as an accumu- 
lator of energy (or fly-wheel), keeping the motion steady. The 
chain-wheel on the crank-axle will therefore rotate with variable 
speed. The speed-ratio in any position, say O^A C O2, can be 
found, after the method of section 32, by drawing Oi e parallel to 
O^ C, meeting CA (produced if necessary) at e ; the intercept 
Ox e is proportional to the angular speed of the crank-axle. If 
this length be set off along OxA^ and the construction be 
repeated, a polar curve of angular speed of the crank-axle will 
be obtained. It will be noticed that in figure 459 the angular 
speed of the crank-axle decreases gradually shortly after passing 
the position O^ d^ until the position O^ d^ is reached, and the 
rivet A attains the position aj. The length of the coupling-rod 
being now increased by one link, the angular speed of the crank- 
axle increases gradually until the rivet C attains the position ^i. 
Here the length of the coupling-rod is decreased by one link, and 
the virtual crank of the wheel changing suddenly from O^ c^ to 
C?2^i, the length of the intercept Oe also changes suddenly, 
corresponding to a sudden change in the angular speed of the 
crank-axle. 

With a * Humber ' chain the speed will have four maximum 
and minimum values while the chain moves over a distance equal 
to one link. 

The magnitude of the variation of the angular speed of the 
crank-axle depends principally, as an inspection of figure 459 will 
show, on the number of teeth in the smaller wheel. If the crank- 
Digitized by V^jOOQ 



424 Details 



CHAP. zxn. 



axle be a considerable distance from the centre of the driving- 
wheel, and if the number of teeth of the wheel on the crank-axk 
be great, the longest intercept, O e, will be approximately equal to 
the radius of the pitch-circle, and the smallest intercept to the 
radius of the inscribed circle of the pitch-polygon. The variation 
of the angular speed of the crank can then be calculated as 
follows : 

Let ^1 and N^ be the numbers of teeth in the chain-wheels 
on the driving-hub and crank-axle respectively, -^i and R^ the 
radii of the pitch-circles, r, and r^ the radii of the inscribed circles 
of the pitch-polygon. Then for a long-link chain the average 

speed-ratio = _J. Assuming the maximum intercept t?i^(fig.459) 
to be equal to R^^ then from (3), the 

stn ^ 



R N 
maximum speed-ratio = ^ = ? approx. . (9) 

Assuming the minimum intercept O e (fig. 459) to be equal 
to rj, then from (3) 

stn 



r N 
minimum speed-ratio = -J- = ? approx. . (10) 



^« tanl.^ 



Then, for the crank-axle, 



No stn V 



maximum speed ^ No , x 

^,— = ^-approx (11) 

mean speed ^,j/„- 

minitnumsp^^ '"" I^, 
mean speed ^ ^^^ ^ 

maximum speed Op /r„ ^ \ r ^^^ / \ 

— r-^ ^ , = >»^-(% 440) = approx. (13) 

mmimum speed Ok^ ° ' .^. «• 
* cos 

^, 

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CBAJP, XXYI. 



Chains and Chain Gearing 



425 



In the same way, we get for a * Humber ' chain, 



maximum speed __ 



(fig. 444) 



« R^ 



(14) 



minimum speed cos a O b ^ R^ — 0-3* 

Table XVI. is calculated from formulae (13) and (14). 

Table XVI. — Variation of Speed of Crank-axle. 

Assuming that centres are far apart, and that the number of teeth on chain- 
wheel of crank -axle is great. 

Number of Teeth on Hub 

I Ratio of ' 



maximum 

to 

minimum 

speed 



Humber . 1*052 
Long-link. , 1*155 



7 i 8 


9 1 10 1 " 


1 


1-038 


1028 


I 022 


I 018 


I 015 


1 
I 012 


I-IIO 


1-082 


I 064 


1-051 


1-042 


1035 



Discarding the assumptions made above, when N^ is much 
greater than i\^„ the maximum intercept O e (fig. 459) may be 
appreciably greater than R^^ while the minimum intercept may 
not be appreciably greater than r,. The variation of the speed- 
ratio may therefore be appreciably greater than the values given 
in Table XVI. 

An important case of chain-gearing is that in which the two 
chain-wheels are equal, as occurs in tandems, triplets, quadruplets, 
&c Drawing figure 459 for this case, it will be noticed that if 
the distance between the wheel centres be an exact multiple of 
the pitch of the chain, the lines O^ A and O^ C are always 
parallel, the intercept O^ e always coincides with O^ A, and there- 
fore the speed-ratio is constant. If, however, the distance between 




Fig. 460. 

the wheel centres be {k + ^) times the pitch, k being any whole 
number, the variation may be considerable. 

In this case, the minimum intercept (9. ^ , (fig^. 460), the 

edbyXjOOQle 



Digitized b 



4^6 Details chap, htl ^ 

radius O^ 4\ and the chain line -^, C, form a trian^e <7i -4, ^„ ' 
which is very nearly right-angled at ^,. Therefore for a long-link 
chain, 

minimum speed d ft ir / , I 

The corresponding triangle 0| A2 e^ formed by the maximum 
intercept is very nearly right-angled at A^, Therefore, 

maximum speed _ 0| ^2 _ ' 
mean speed 



maximum speed __ i 
minimum speed 2 '^ 



approx. 



cos 



N 



approx. 



(16) 
(17) 



For a * Humber ' chain, 

maximum speed ^yP« 3^ 

mmimum speed J<^ — 03^ 

Table XVII. is calculated from formulae (17) and (18). 

Table XVII.— Greatest Possible Variation of Speed-ratio 
OF Two Shafts Geared Level. 



Number of Teeth . 

Ratio of maxi- 
mum to mini- 
mum speed- 
ratio 



Humber ' 

chain . 1*058 

Long-link , 

chain . i i'i90 



.0 


13 


16 


30 


30 


1036 


1-025 


1*014 


1*009 


1*004 


I 105 


1-072 


1*039 


1*025 


1*011 



The figures in Table XVI. show that the variation of speed, 
when a small chain-wheel is used on the driving-hub, is not small 
enough to be entirely lost sight of. The * Humber' chain is 
better in this respect than the long-link chain. 

Again, in tandems the speed-ratio of the front crank-axle and 
the driving-wheel hub is the product of two ratios. The ratio 
of the maximum to the minimum speed of the front axle may be 
as great as given by the product of the two suitable numbers 
from Tables XVI. and XVII. 

Example, — With nine teeth on the driving-hub, and the two 

Digitized by CjOOQIC 



C9AP. XXVI. Chains and Chain Gearing 427 

axles geared by chain-wheels having twelve teeth each, the 
maximum speed of the front axle may be 

1-064 X 1*072 = 1*14 times its minimum speed, 

with long-Jink <:haixis ; and 

I '022 X 1*025 =;i 1*047 times 

with * Humber ' chains. 

With triplets and quadruplets the variation may be still 
greater ; and it is open to discussion whether the crank-axles 
should not be fixed, without chain-tightening gear, at a distance 
apart equal to some exact multiple of the pitch. 

If a hypothetical point be supposed tb move with a uniform 
speed exactly equal to the average speed of a corresponding point 
actually on the pitch-line of the crank-axle chain-wheel, the distance 
at any instant between the two is never very great. Suppose the 
maximum speed of the actual point be maintained for a travel of 
half the pitch, and that it then travels the same distance with its 
minimum speed. For a speed variation of one per cent, the 
hypothetical point will be alternately ^J^yth of an inch before and 
behind the actual point during each inch of travel. This small 
displacement, occurring so frequently, is of the nature of a 
vibratory motion, superimposed on the uniform circular motion. 

295. Size of Chain-wheels. — The preceding section shows 
that the motion of the crank-axle is more nearly uniform the 
greater the number of teeth in the chain-wheels. Also, if the 
ratio of the numbers of teeth in the two whesels be constant, 
the larger the chain-wheel the smaller will be the pull on the 
chain. Instead of having seven or eight teeth on the back-hub 
chain-wheel it would be much better, from all points of view, to 
have at least nine or ten, especially in tandem machines. 

296. Spring Chain-wheel. — Any sudden alteration of speed, 
that is, jerkiness of motion, is directly a waste of energy, since bodies 
of sensible masses cannot have their speeds increased by a finite 
amount in a very short interval of time without the application of 
a comparatively large force. The chain-wheel on the crank-axle 
revolving with variable speed, if the crank be rigidly connected the 
pedals will also rotate with variable speed. In the cycle spring 
chain-wheel (fig. 461) a spring is interposed between th^ wheel and 

Digitized by V^jOOQ 



428 



Details 



CHAT. ZZTL 



the cranks. If, as its inventors and several well-known biqrde 
manufacturers claim, the wheel gives better results than the 
ordinary construction, it may be possibly due to the fact that the 




Fig. 461. 



spring absorbs as soon as possible the variations of speed due to 
the chain-driving mechanism, and does not allow it to be trans- 
mitted to the pedals and the rider's feet 

If direct spokes are used for the driving-wheel they act as a 
flexible connection between the hub and rim, allowing the former 
to run with variable, the latter with uniform speed. 

297. Elliptical Chain-wheel. — An elliptical chain-wheel has 
been used on the crank-axle, the object aimed at being an in- 
creased speed to the pedals when passing their top and bottom 
positions, and a diminution of the speed when the cranks are 
passing, their horizontal positions. The pitch-polygon of the 
chain -wheel in this case is inscribed in an ellipse, the minor axis 
of which is in line with the cranks (fig. 462). 




Fig. 462. 



The angular speed of the driving-wheel of the cycle being 
constant and equal to w, that of the crank-axle is approximately 



r, <o 



Digitized by CjOOQIC 



CHAP. xxYi. Chains and Chain Gearing 429 

where rj and rj are the radii from the wheel centres to the ends 
of the straight portion of the chain, r^ and w being constant, the 
angular speed of the crank is therefore inversely proportional to 
the radius from the centre of the crank-axle to the point at which 
the driving side of the chain touches the chain-wheeL The speed 
of the pedals will therefore be least when the cranks are horizontal, 
and greatest when the cranks are vertical, as indicated by the 
dotted lines (fig. 462). 

If both sides of the chain connecting the two wheels be 
straight, the total length of the chain as indicated by the full lines 
(fig. 462) is greater than that indicated by the dotted lines, the 
difference being due to the difference of the obliquities of the 
straight portions when the cranks are vertical and horizontal 
respectively. This difference is very small, and may be practically 
left out of account. If the wheel centres are very far apart, so 
that the top and bottom sides of the chain may be considered 
parallel, the length in contact with the elliptical chain-wheel in 
any position is evidently equal to half the circumference of the 
ellipse ; similarly, the length in contact with the chain-wheel on 
the hub is half its circumference, and the length of the straight 
portions is approximately equal to twice the distance between the 
wheel centres. Thus the total length is approximately the same, 
whatever be the position of the chain-wheel. 

A pair of elliptical toothed-wheels are sometimes used to con- 
nect two parallel shafts. The teeth of these wheels are all of 
different shapes ; there can be at most four teeth in each wheel 
of exactly the same outline. It has therefore been rather hastily 
assumed that the teeth of an elliptical chain-wheel must all be of 
different shapes ; but a consideration of the method of designing 
the chain-wheel (sec. 289) will show that this is not necessarily 
the case. The investigation there given is applicable to elliptical 
chain-wheels, and therefore all the teeth may be made from a 
single milling-cutter, though the form of the spaces will vary from 
tooth to tooth. 

298. Friction of Chain Oearing.—There is loss by friction 
due to the rubbing of the links on the teeth, as they move into, 
and out of, contact with the chain-wheel. We have seen (sec. 
286) that the extent of this rubbing depends on the difference of 

Digitized by V^j 



430 Details chap. xrn. 

the pitches of the chain and wheel ; if these pitches be exactly 
equal, and the tooth form be properly designed, theoretically there 
is no rubbing. If, however, the tooth outline fall at or near the 
curve/ A' (fig. 440), the rubbing on the teeth may be the largest 
item in the frictional resistance of the gearing. 

As a link moves into, and out of, contact with the chain-whed, 
it turns through a small angle relative to the adjacent link, there 
is therefore rubbing of the rivet-pin on its bush. While the pin 
A (fig. 459) moves from the point b^ to the point a^ the link 

A B turns through an angle of ?^, and the link AA^ moves prac- 

tically parallel to itself. The relative angular motion of the 
adjacent links, BA and AA^^ and therefore also the angle 
of rubbing of the pin A on its bush, is the same as that 
turned through by the wheel Ox. In the same way, while the 
pin D moves from d^ to Cx the relative angular motion of the 
adjacent links D C and C C* is the same as the angle turned 

through by the wheel d?^, viz. ^^-. The pressure on the pins A 

and C during the motion is equal to P^ the pull of the chain. On 
the slack side of the chain the motion is exactly similar, but takes 
place with no pressure between the pins and their bushes. There- 
fore, the frictional resistance due to the rubbing of the pins on 
their bushes is the sum of that of two shafts each of the same 
diameter as the rivet-pins, turning at the same speeds as the 
crank-axle and driving-hub respectively, when subjected to no 
load and to a load equal to the pull of the chain. 

299. Oear-Case. — From the above discussion it will be seen 
that the chain of a cycle is subjected to very severe stresses, and 
in order that it may work satisfactorily and wear fairly well it must 
be kept in good condition. The tremendous bearing presstire 
on the rivets necessitates, for the efficient working of the chain, 
constant lubrication. Again, the bending and shearing stresses 
on the rivets, sufficiently great during the normal working of the 
chain, will be greatly increased should any grit get between 
the chain and the teeth of the chain-wheel, and stretching of the 
chain will be produced. Any method of keeping the chain con- 
stantly lubricated and preserving it from dust and grit should add 

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CHAP. xxYi. Chains and Chain Gearing 43 1 

to the general efficiency of the machine. The gear-case intro- 
duced by Mr. Harrison Carter fulfils these requirements. The 
Carter gear-case is oil-tight, and the chain at its lowest point dips 
into a small pool of oil, so that the lubrication of the chain is 
always perfect. The stretching of the chain is not so great with, 
as without a gear-case ; in fact, some makers go the length of 
saying that with a Carter gear-case, and the chain properly adjusted 
initially, there is no necessity for a chain-tightening gear. A great 
variety of gear-cases have recently been put on the market ; they 
may be subdivided into two classes : (i) Oil-tight gear-cases, in 
which the chain works in a bath of oil ; and (2) Gear-cases which 
are not oil-tight, and which therefore serve merely as a protection 
against grit and mud. A gear-case of the second type is probably 
much better than none at all, as the chain, being kept compara- 
tively free from grit, will probably not be stretched so much as 
would be the case if no gear-case were used. 

300. Comparison of Different Forms of Chain.— The ' Roller ' 
has the advantage over the * Humber,' or block chain, that its 
rubbing surface is very much larger, and that the shape of the 
rubbing surface — the roller — is maintained even after excessive 
wear. The * Roller,' or long- pitched chain, on the other hand, gives 
a larger variation of speed-ratio than the * Humber,' or short- 
pitched chain, the number of teeth in the chain-wheels being the 
same in both cases ; but a more serious defect of the * Roller ' chain 
is the imperfect design of the side-plates (sec. 291). If, however, 
the side-pktes of a * Roller * chain be properly designed, there should 
be no difficulty in making them sufficiently strong to maintain 
their shape under the ordinary working stresses. Undoubtedly 
the weakest part of a cycle chain, as hitherto made, is the rivet, 
the bending of the rivets probably accounting for most of the 
stretch of an otherwise well-designed chain. A slight increase in 
the diameter of the rivets would enormously increase their strength, 
and slightly increase their bearing surface. 

In the * Humber ' there are twice as many rivets as in a * Roller ' 
chain of the same length. It would probably be improved by 
increasing the length of the block, until the distance between the 
centres of the holes was the same as between the holes in the 
side-plates. This would increase the pitch to 1*2 inches, without 

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432 



Details 



CHAP. XXfL 



in any way increasing the variation of the speed ratio. In order 
to still further reduce the number of rivets, the pitch might be 




Fig. 463* 

increased to one inch, giving a total pitch of two inches for the 
chain. If the side-plates be made to the same outline as the 
middle block (fig. 463), they may also be used to come in contact 
with the teeth of the chain-wheel. The chain-wheel would then 

have the form shown in 
figure 464, in which the 
alternate teeth are in 
duplicate at the edges of 
the rim. For tandems, 
triplets, &c., a still greater 
pitch, say i^ inches, may 
be used with advantage ; 
the back-hub chain- 
wheel, with six teeth of 
this pitch, would have 
the same average radius 
of pitch-polygon as a 
chain-wheel with nine 
teeth of I inch pitch ; 
the chain would have 
^^^' *^** only one-third the num- 

ber of rivets in an ordinary * Humber ' chain of the same 
length, and if the rivets were made slightly larger than usual, 
stretching of the chain might be reduced to zero. 

30 !• Chain-tightening Gear. — The usual method of providing 
for the chain adjustment is to have the back-hub spindle fastened 

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CHAP. ZXYl. 



Chains and Chain Gearing 



433 



to a slot in the frame, the length of slot being at least equal to 
half the pitch of the chain. In the swinging seat-strut adjustment, 
the slot is made in the lower back fork, and the lower ends of the 
seat-struts are provided with circular holes through which the 
spindle passes. These have been described in the chapter on 
Frames. 

The * eccentric ' adjustment is almost invariably used for the 
front chain of a tandem bicycle. The front crank-axle is carried 
on a block, the outer surface of which is cylindrical and eccentric 
to the centre of the axle. The adjustment is effected by turning 




the block in the bottom-bracket, and clamping it in the desired 
position. 

A loose pulley carried at the end of a rod controlled by a 
spring (fig. 465) is used in conjunction with Linley & Biggs' 
expanding chain-wheel. 




Figure 466 illustrates a method used at one time by Messrs. 
Hobart, Bird & Co. When the chain required to be tightened, 
the loose chain-wheel was placed nearer the hub chain-wheel. 



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434 



Details 



CHAP. XETB. 



CHAPTER XXVII 



TOOTHED-WHEEL GEARING 



302. Transmission by Smooth Boilers. — Before beginning the 
study of the motion of toothed-wheels, it will be convenient to 
take that of wheels rolling together with frictional contact ; since 
a properly designed toothed-wheel is kinematically equivalent to 
a smooth roller. 

Parallel Shafts.— \jt\. two cylindrical rollers be keyed to the 
shafts A and B (fig. 467) ; if one 
shaft revolves it will drive the 
other, provided the frictional resist- 
ance at the point of contact of the 
rollers is great enough to prevent 
slipping. When there is no slipping, 
the linear speeds of two points, one 
on the circumference of each roller, 
must be the same. Let o>, and 
<ii2 be the angular speeds of the shafts, r, and r^ the radii of the 
rollers \ then the above condition gives 




or 






<i)i 

Wj 



(0 



the negative sign indicating that the shafts turn in opposite direc- 
tions. Thus the angular speeds are inversely proportional to the 
radii (or diameters) of the rollers. 

If the smaller roller lie inside the larger, they are said to have 
internal contact, and the shafts revolve in the same direction. 

Intersecting Shafts.— T^fO shafts, the axes of which intersect, 

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CHAP. xxvu. Toothed-wheel Gearing 435 

may be geared together by conical rollers, the vertices of the two 

cones coinciding with the point of intersection of the shafts. 

Figure 468 shows diagrammatically two 

shafts at right angles, geared together 

by rollers forming short frusta of 

cones. If there be no slipping at the 

point of contact, the linear speeds of 

two points, situated one on each wheel, 

which touch each other during contact, 

jiiust be equal. Equation (i) will hold *°* ^ 

in this case, r^ and r^ being the radii of the bases of the cones. 

Two shafts whose axes are not parallel and do not intersect 
may be connected by rollers, the surfaces of which are hyper- 
boloids of revolution. The relative motion will, however, not be 
pure rolling, but there will be a sliding motion along the line of 
contact of the rollers, which will be a generating straight line of 
each of the hyperboloids. This form of gear, or its equivalent 
hyperboloid skew-bevel gear, has not been used to any great 
extent in cycle construction, and will therefore not be discussed \n 
the present work. 

303. Friction Gearing. — If two smooth rollers of the form 
above described be pressed together there will be a certain fric- 
tional resistance to the slipping of one on the other, and hence if 
one shaft is a driver the other may be driven, provided its resist- 
ance to motion is less than the frictional resistance at the surface 
of the roller. Friction rollers are used in cases where small 
driving efforts have to be transmitted, but when the driving effort 
is large, the necessary pressure between the rollers would be so 
great as to be very inconvenient. In * wedge gearing,' the surfaces 
are made so that a projection of wedge section on one roller fits 
into a corresponding groove on the other ; the frictional resistance, 
for a given pressure, being thereby greatly increased. 

304. Toofhed-wheeh.— When the effort to be transmitted is 
too large for friction gearing to be used, projections are made on 
one wheel and spaces on the other ; a pair of toothed-wheels are 
thus obtained. 

Toothed-wheels should have their teeth formed in such a 
manner that the relative motion is the same as that of a pair 

Digitized by V^ K F 2 



436 Details chap, xint 

of toothless rollers. The surfaces of the equivalent toothless 
rollers are called the pitch surfaces of the wheels. By the radios 
or diameter of a toothed-wheel is usually meant that of its pitch 
surface ; equation (i) will therefore be true for toothed-wheels. 
The distance between the middle points of two consecutive teeth 
measured round the pitch surface is called the pitch or the circular 
pitch of the teeth. The pitch must evidently be the same for two 
wheels in gear. Let / be the pitch, N^ and N^ the numbers of 
teeth in the two wheels, and n^ and n^ the numbers of revolutioDs 
made per minute ; then the spaces described by two points, one 
on each pitch surface, in one minute are equal ; therefore 

2 IT «i ^1 = 2 IT ;f2 r2. 
Since 

2 TT ri = -A^i /, and 2 n r^^N^ /, we get 

Ni n^p'=' N^ n^p 
or, 

^1 = ''-^ (2) 

That is, the angular speeds of the toothed-wheels in contact are 
inversely proportional to the numbers of teeth. 

If the pitch diameter be a whole number, the circular pitch 
will be an incommensurable number. The diametral pitch is 
defined by Professor Unwin as "A length which is the same 







Fig. 469. 



fraction of the diameter as the circular pitch is of the circam- 
ference." The American gear-wheel makers define the diametral 
pitch as ''The number of teeth in the gear divided by the pitch 
diameter of the gear.'' The latter may be called tht pitch-number. 

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CHAP. ZXYII* 



Toothed'Wheel Gearing 



437 



It is much more convenient to use the pitch-number than the 
circular pitch to express the size of wheel-teeth. Figure 469 shows 
the actual sizes of a few teeth, with pitch-numbers suitable for use 
in cycle-making. If/ be the circular pitch, s the diametral pitch, 
and F the pitch-number. 



s = l 



jP= - = — 



(3) 



305. Train of Wheels.— If the speed-ratio of two shafts to be 
geared together by wheels be large, to connect them by a single 
pair of wheels will be in most cases inconvenient ; one wheel of 
the pair will be very large and the other very small. In such a 
case one or more intermediate shafts are introduced, so that the 
speed- ratio of any pair of Wheels !n contact is not very great The 
whole system is then called a train of wheels. For example, in 
a watch the minute hand makes one complete revolution in one 
hour, the seconds hand in one minute ; the speed-ratio of the 
two spindles is 60 to i ; here intermediate spindles are necessary. 

If the two shafts to be connected are coaxial, it is kinematically 
necessary^ not merely convenient, to employ a train of wheels. 
This is the case of a wheel or pulley rotating loosely on a shaft, 
the two being geared to have different speeds. Figure 470 shows 
the simplest form of gearing 
of this description, univer- 
sally used to form the slow 
gearing of lathes, and which 
has been extensively used to 
form gears for front-driving 
Safeties. A is the shaft to 
which is rigidly fixed the 
wheel D^ gearing with the 
wheel E on the intermediate 
shaft B, The bearings of the shafts A and B are carried by the 
frame C. On the shaft B is fixed another wheel F^ gearing with 
the wheel G^ rotating loosely on the shaft A, 

Denoting the number of teeth in a wheel by the corresponding 

Digitized by Vj 




fc&a ^^^ 



T\G, 470. 



438 Details chaf. xxm. 

small letter, the speed-ratio of the shafts B and A will be ; 

the negative sign indicating that the shafts turn in opposite direc- 

tions. The speed-ratio of the wheels G and -^will be — <-, and 

g 
the speed-ratio of the wheels G and D will be the product 






(4) 



The wheels D and G (fig. 470) revolve in the same direction, 
the four wheels in the gear all having external contact If one of 
the pairs of wheels has internal contact, the wheels A and G will 
revolve in opposite directions. The speed-ratio will then be 

--^^ <'» 

306. Epieyolio Train.— The mechanism (fig. 470) may be 
inverted by fixing one of the wheels D or 6^ and letting the firame- 
link C revolve ; such an arrangement is called an epicyclic train. 
The speed-ratio of the wheels D and G relative to C will still be 
expressed by (4). Suppose D the wheel fixed, also let its angular 
speed relative to the frame-link C be denoted by unity, and that of 
G^ by «. When the frame-link C is at rest its angular speed about 
the centre A is zero. The angular speeds of D^ C, and G are 
then proportional to i, o, and n. Let an angular speed — i be 
added to the whole system ; the angular speeds of />, C, and G 
will then be respectively 

o, —I, and «-i (5) 

If one pair of wheels has internal contact, the angular speeds | 
of 2?, C, and G will be represented by — i, o, and n ; adding a 
speed + 1 to the system, the speeds will become respectively 

o, I, and «+i (6) 

An epicyclic train can be formed with four bevel-wheels (fig. 
471) ; also, instead of two wheels, E and F (fig. 470), only one 
may be used which will touch A externally and G internally (fig. 

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CHAP. XXYII. 



Toothed-wheel Gearing 



439 



472) ; this is the kinematic arrangement of the well-known * Crypto ' 
gear for Front-drivers. Again, in a bevel-wheel epicyclic train the 
two wheels on the interme- 
diate shaft B may be merged 
into one ; this is the kinema- 
tic arrangement of Starley's 
balance gear for tricycle axles 
(fig. 219). 

In a Crypto gear, let N^ 
and N^ be the numbers of 
teeth on the hub wheel and 




Fig. 471. 



the fixed wheel respectively, then n 






and from (6), the 



(7) 



speed-ratio of the hub and crank is 

N^^ N^~ 

From (7) it is evident that if a speed-ratio greater than 2 be 
desired, N^ must be greater than iV^„ and the annular wheel must 
therefore be fixed to the frame and the inner wheel be fixed to 
the hub. 

Example, — The fixed wheel JD o( a, Crypto gear has 14 teeth, 
the wheel JE mounted on the arm C has 12 teeth ; the number of 





Fig. ija, 

teeth in the wheel G fixed to the hub of the driving-wheel must 
then be 12 + 12 + 14 = 38. The driving-wheel of the bicycle 
is 46 in. diameter ; what is it geared to ? Substituting in (7), the 

speed-ratio of the hub and the crank is 5.^, and the bicycle is 
geared to ^ - ^ = 62*95 inches. 

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44© . Details chap« xxm. 

307. Teefh of Wheels.— The projection of the pitch-surface 
of a toothed-wheel on a plane at right angles to its axis is called 
the pitch-circle ; a concentric circle passing through the points of 
the teeth is called the addendum-circle \ and a circle passing 
through the bases of the teeth is called the root-circle (fig. 473). 
The part of the tooth surface b c outside the pitch-line is called 
the face^ and the part a b inside the pitch-circle the flank of 




Fig. 473- 

the tooth. The portion of the tooth outside the pitch-circle is 
called the point \ and the portion inside, the root. The line 
joining the wheel centres is called the line of centres. The top 
and bottom clearance is the distance r d measured on the line 
of centres, between the addendum-circle of one wheel and the 
root-circle of the other. The side-clearance is the difference ef 
between the pitch and the sum of the thicknesses of the teeth (rf 
the two wheels, measured on the pitch -circle. 

For the successful working of toothed-wheels forming part of 
the driving mechanism of cycles it is absolutely necessary not only 
that the tooth forms should be properly designed, but also that they 
be accurately formed to the required shape. This can only be done 
by cutting the teeth in a special wheel-cutting machine. In these 
machines, the milling-cutter being made initially of the proper 
form, all the teeth of a wheel are cut to exactly the same shape, 
and the distances measured along the pitch-line between con- 
secutive teeth are exactly equal. In slowly running gear teeth, as 
in the bevel-wheels in the balance gear of a tricycle axle, the 
necessity for accurate workmanship is not so great, and the teeth 
of the wheels may be cast. 

Digitized by CjOOQIC 



CHAP. zzyn. 



Toothed-wheel Gearing 



441 



308. Bolatiye Motion of Toothed-wheels.— Let a Fa and 

b Fb (fig. 474) be the outlines of the teeth of wheels, ^ being the 
point of contact of the two teeth. Let D be the centre of curva- 
ture of the portion of the curve a Fa which lies very close to the 
point F\ that is, D is the centre of a cir- 
cular arc approximating very closely to a 
short portion of the curve aFava the neigh- 
bourhood of the point F. Similarly, let C 
be the centre of curvature of the portion of 
the curve b Fb near the point F, Whatever 
be the tooth-forms a F a and b Fb/\i will 
in general be possible to find the points D and 
(7, but the positions of C and D on the re- 
spective wheels change as the wheels rotate 
and the point of contact F of the teeth 
changes. While the wheels A and B rotate 
through a small angle near the position 
shown, their motion is exactly the same as 
if the points C and D on the wheels were 
connected by a link C Z>. The instantaneous motion of the 
two wheels is thus reduced to that of the levers A C and B D 
connected by the coupler C D, 

In figure 21 (sec. 32) let B A and CZ> be produced to meet 

at/; then ne ^DJ ^, j,,^DJ 

CB—CJ' ""' ^' CJ' 

And it has been already shown that the speed-ratio of the two 

Therefore the speed-ratio may be written equal 




Fig. 474. 



CB, 



cranks is -- ^ . 
DA 



to 



CB DJ 
DA " Cf 

Therefore, since C B and D Azx^ constant whatever be the 
position of the mechanism (fig. 21), the angular speeds of the two 
cranks in a four-link mechanism are inversely proportional to the 
segments into which the line of centres is divided by the centre- 
line of the coupling-link. Therefore if the straight line CD cut 
A B dXe (fig. 474) the speed-ratio of the wheels A and B is 

l^ , . . . . . (8) 

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Ac 



442 



Details 



OHAP. xxm. 



For toothed-wheels to work smoothly together the angular 
speed-ratio should remain constant ; (8) is therefore equivalent to 
the following condition : Tfie common normal to a pair of teeth at 
their point of contact must always pass through a fixed paint on 
the line of centres. This fixed point is called the pitch-pointy and is 
evidently the point at which the pitch-circles cut the line of centres. 
If the form of the teeth of one wheel be given, that of the 
teeth of the other wheel can in general be found, so that the 
above condition is satisfied. This problem occurs in actual 
designing when one wheel of a pair has been much worn and has 
to be replaced. But in designing new wheels it is of course most 
convenient to have the tooth forms of both wheels of the same 
general character. The only curves satisfying this condition are 
those of the trochoid family, of which the cycloid and involute are 
most commonly used. 

309. Involute Teeth. — Suppose two smooth wheels to rotate 
about the centres A and B {^%, 475), the sum of the radii being 

less than the distance between 
the centres. Let a very thin 
cord be partially wrapped 
round one wheel, led on to 
the second wheel, and partially 
wrapped round it. Let randi 
respectively be the points at 
which the cord leaves A and 
touches the wheel B, Let a 
pencil, P^ be fixed to the cord, 
and imagine a sheet of paper 
fixed to each wheel. Then 
the cord not being allowed 
to slip round either wheel, 
while the point P of the 
string moves from r to ^ the 
wheels A and ^ will be driven, 
and the pencil will trace out 
on the paper fixed to A an arc of an involute a a^^ and on the 
paper fixed to B an involute arc b^ b. If teeth-outlines be made 
to these curves they must touch each other at some point on the 

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Fig. 475. 



CHAP. xxvu. Toothed-wheel Gearing 443 

line € dy which is their common normal at their point of contact ; 
and since ^^ intersects the line of centres at a fixed point,/, the 
tooth-outlines satisfy the condition for constant speed-ratio. 

The circles round which the cord was supposed to be wrapped 
are called the base-circles of the involute teeth ; the line cd \% 
called the path of tlte point of contact, or simply Xhtpath of contact 
The longest possible involute teeth are got by taking the adden- 
dum-circles of wheels A and B through d and c respectively ; for 
though the involutes may be carried on indefinitely outwards from 
the base-circles, no portion can lie inside the base-circle. Except 
in wheels having small numbers of teeth, the arcs of the involutes 
used to form the tooth outlines are much smaller than shown in 
figure 475 ; the path of contact being only a portion of the 
common tangent cd to the base-circles. 

The angle of obliquity of action is the angle between the 
normal to the teeth at their point of contact, and the common 
tangent at / to the pitch-circles. The angle of obliquity of involute 
teeth is constant, and usually should not be more than 15°. 

No portion of a tooth lying inside the base-circle has working 
contact with the teeth of the other wheel, but in order that the 
points of the teeth of the wheels may get past the line of centres, 
the space between two adjacent teeth must be continued inside 
the base-circle. If the teeth be made with no clearance the con- 
tinuation of the tooth outline b^ Fb, between the base- and root- 
circles, is an arc of an epitrochoid bf described on the wheel B 
by the point a^ of the wheel A. The continuation of the tooth 
outline a^ Fa between the base- and root-circles is an arc of an 
epitrochoid a e, described on the wheel A by the point b^ of the 
wheel B. This part of the tooth outline lying between the root-circle 
and the working portion of the tooth outline is sometimes called 
the fillet. The flanks are sometimes continued radially to the 
root-circle ; but where the strength of the teeth is of importance, 
the fillet should be properly designed as above. The fillet-circle 
is a circle at which the fillets end and the working portions of the 
teeth begin. When involute teeth are made as long as possible, 
the base- and fillet-circles coincide. In any case, the fillet-circle 
of one wheel and the addendum-circle of the other pass through 
the same point, at the end of the path of contact. 

Digitized by CjOOQIC 



444 



Details 



CBAP. XXfll. 



Let the centres A and B of the wheels (fig. 475) be moved 
farther apart ; the teeth will not engage so deeply, and the line 
dc will make a larger angle with the tangent to the pitch-circles 
at /. The form of the involutes traced out by the pencil will, 
however, be exactly the same though a longer portion will be 
drawn. Therefore the teeth of the wheels will still satisfy the 
condition of constant speed-ratio. Wheels with involute teeth 
have therefore the valuable property that the distance between 






:<i^. 



^r-^ 



^^ --1 



' 1 


\'-h^' ^^ 


g 




1 
1 


\^ 


1 




Fig. 476. 





their centres may be slightly varied without prejudicially affecting 
the motion. 

310. Cycloidal Teeth. — Let A and B (fig. 476) be the centres 
of two wheels, and let p be the pitch-point Let a third circle 
with centre C lying inside the pitch-circle of A roll in contact 
with the two pitch-circles at the pitch-point. Suppose a pencil P 
fixed to the circumference of the rolling-circle. If the three 
circles roll so that/ is always their common point of contact, the 

Digitized by CjOOQIC 



OHAP. XXVII. Toothed'wheel Gearing 445 

pencil will trace out an epicycloid on wheel B, and an hypocycloid 
on A, Let P be any position of the pencil, then the relative 
motion of the two circles A and C is evidently the same as if ^ 
were fixed and C rolled round inside it ; / is therefore the instan- 
taneous centre of rotation of the circle C, and the direction of 
motion of -P relative to the wheel A must be at right angles to the 
• line / P ; that is, p Pis the normal at P to the hypocycloid Pa, 

In the same way it can be shown that the line p P \% the 
normal to the epicycloid Pb, If tooth outlines be made to these 
curves they will evidently satisfy the condition for constant speed- 
ratio. 

The tooth Pa is all flank, and the tooth /^^ all face. Another 
rolling-circle O may be taken inside the pitch-circle of wheel J?, 
a tracing-point -P* on it will describe an epicycloid on wheel A 
and a hypocycloid on wheel B, The tooth outline P^ a\ is all 
face, and the tooth outline P b^ is all flank. They may be 
combined with the former curves, so that the tooth outlines Paa^ 
and Pbb^ may be used. 

The path of contact P^pP'm this case is evidently made up 
of arcs of the two rolling-circles. If the diameter of the rolling- 
circle be equal to the radius of the pitch-circle, the hypocycloid 
described reduces to a straight line a diameter of the pitch-circle. 
The flanks of cycloidal teeth may therefore be made radial. 

If contact begins and ends at the points Pznd. P respectively, 
the addendum-circles of B and A pass through these points. If 
the teeth are made without clearance, the fillet will be, as in 
involute teeth, an arc of an epitrochoid, Pe^ described on the 
wheel A by the point P of the wheel B, Similarly, the fillet of 
wheel B between P^ and the root-circle is an arc of an epitrochoid 
7^/ described on wheel B by the point P^ of the wheel A, 

If a set of wheels with cycloidal teeth are required, one wheel 
of the set to gear with any other, the same rolling-circle must be 
taken for the faces and flanks of all. 

An important case of cycloidal teeth is that in which the 
rolling-circle is equal to the pitch-circle of one of the wheels of 
the pair. The teeth of one wheel become points ; those of the 
other, epicycloids described by one pitch-circle rolling on the other. 

If two tooth outlines gear properly together with constant 

Digitized by V^jOOQ 



446 Details chap, xxhl 

speed-ratio, tcx)th outlines formed by parallel curves will in genenl 
also gear together properly. In the above case the paint teeth 
of one wheel may be replaced by round //w, the epicycloid teeth 
of the other wheel by a parallel curve at a distance equal to the 
radius of the pin. Loose rollers are sometimes put round the 
pins so that the wear is distributed over a larger surfisice. 

An example of pin-gearing is found in the early patterns of the 
* Collier' two-speed gear (sec. 319). 

311. Arof of Approach and Seoesa. — The arc of approach is 
the arc through which a point on the pitch-circle moves from 
the time that a pair of teeth come first into contact until they are 
in contact at the pitch-point. The arc of recess is the arc through 
which a point on the pitch-circle moves from the time a pair of 
teeth are in contact at the pitch-point until they go out of con- 
tact The arc of contact is, of course, the sum of the arcs of 
approach and recess. 

With cycloidal teeth (fig. 476), if P and P^ be the points of 
contact when the teeth are just beginning and just leaving contact 
respectively, ap or bp will be the arc of approach, and p a^ <x 
p ^2 the arc of recess, provided the wheel A is the driver, in 
watch-hand direction. From the mode of generation of the 
epicycloid and the hypocycloid it is evident that the arc of the 
rolling circle, Pp^ is equal to the arc of approach, and p P^ to 
the arc of recess. 

With involute teeth (fig. 475) the path of contact is the 

straight line cd. The arc of contact, measured along either of 

the base-circles, is equal to cd. The arc of contact, measured 

— ■ Ap 

along the pitch-circle, is equal X.o cd multiplied by -j~, the ratio of 

the radii of the pitch- and base-circles. Draw the tangent //at 
/ to the pitch-circles, and produce A c and Bdio meet pt zx c^ 
and d^ respectively, c^p and pd^ are the lengths of the arcs of 
approach and recess respectively, measured along the pitch-circle. 
For from similar triangles, 

pc Ac' ^ Ac ^ 

Similarly pd^=z4J^ ,pd. 

Digitized by CjOOQIC 



CHAP. XXVII. ' Toothed-wheel Gearing 447 

312. Friction of Toothed-wheels. — ^There is a widespread 
impression, even among engineers, that, if the form of wheel-teeth 
be correctly designed, the relative motion of the teeth is one of 
pure rolling. Probably the use of the term rolling-circles in con- 
nection with cycloidal teeth has given rise to this impression ; 
but a very slight inspection of figures 475 and 476 will show that 
the teeth rub as well as roll on each other. In figure 476 a pair 
of teeth are shown in contact at P, While the teeth are passing 
the pitch-point, the points a and b touch each other at /. Now 
the length Pa of one tooth is much less than the length P b oi 
the other. The teeth must therefore rub on each other a distance 
equal to the difference between these two arcs. The same thing 
is apparent from figure 475. 

The speed of rubbing at any point can be easily expressed as 

follows : Let a pair of wheels rotate about the centres A and B 

(fig. 477) ; let their pitch-lines touch at/ ; let a' pa" be the path 

of contact ; let r, and rj be the radii of the pitch-circles, and 

V be their common linear speed, the 

V V 

angular speeds will be - and — 

respectively. Suppose a pair of teeth 
to be in contact at a \ the relative 
motion of the two wheels will be the 

same if the whole system be given a 

y 
rotation — about B in the direction 

r% 

opposite to the rotation of the wheel 

B, Wheel B will now be at rest, and 

the pitch-line of wheel A will roll on the pitch-line of B, The 

V V 
angular speed of wheel A is now H — . The instantaneous 

centre of rotation of wheel A is the point ^, and therefore the 
linear speed of the point a on the wheel A is 

^/A + i.") X chord/fl (9) 

This is, of course, the same as the relative speed of rubbing of 
the teeth in contact at a. In particular, the speed of rubbing is 

Digitized by CjOOQIC 




448 Details 



CHAP. xxm. 



greatest when the teeth are just coming into or just leaving con- 
tact, and is zero when the teeth are in contact at the pitch- 
point. 

If the two wheels have internal contact, by the same reasoning 

their relative angular speed may be shown to be V (^ — ^ ), and 
the speed of rubbing 

FL' - i) X chord /tf (lo) 

Thus, comparing two pairs of wheels with external and internal 
contact respectively, if the pitch-circles and arc of contact be 
the same in both, the wheels with internal contact have much 
less rubbing than those with external contact. If r, = 3 r^ 
the rubbing with external contact is twice as great as with 
internal. 

Friction and Wear of Wheel-teeth, — The frictional resistance, 
and therefore the wear, of wheel-teeth will be proportional to the 
maximum speed of rubbing, and will therefore be greater the 
longer the path of contact. The arc of contact, therefore, should 
be chosen as short as possible ; the working length of the teeth 
will then be short, and it will be much easier to make the teeth accu- 
rately. The arc of contact must, of course, be at least equal to 
the pitch, so that one pair of teeth comes into contact before the 
preceding pair has left contact. It may be chosen a little greater, 
in order to allow a margin for the centres of the wheels being 
moved a little further apart than was intended. The rubbing of 
the teeth against each other during approach is said to be more 
injurious than during recess. In a pair of wheels in which the 
driver and driven are never interchanged (as in gear-wheels of 
cycles, which are always driven ahead and never backwards), the 
arc of recess may therefore be chosen a little larger than the arc 
of approach. 

If r be the length of the arc of contact, the average speed of 
rubbing will be approximately (with external contact) 



4 V^'i rj 



Digitized by CjOOQIC 



CHAP. xxvn. Toothed-wheel Gearing 449 

If /* be the average normal pressure on the teeth, and /u the co- 
efficient of friction, the work lost in friction will be 



4 V'^i rj '^ 



The useful work done in the same time will be approximately 
P Vy and the efficiency of the gear will be 

4 VI '%/ 4 VI f^v 

Example L — In a pair of wheels with 12 and 24 teeth re- 
spectively, assuming /* = '08, and r= 1*2/, ^= — ='524, and 

rj 12 

^ = -262, and the efficiency is 

I -f •3~x~o8(-^4~+^62)~ ^ 

Example IL—ln an internal gear with 12 and 36 teeth, with 
the same assumptions as above, the efficiency is 

.t:,: — .r--x= '992. 



I + •o24(-524- -175) 

313. Circular Wheel-teeth. — Since only a small arc is used 
to form the tooth outline it is often convenient to approximate 
to the exact curve by a circular arc. Involute or cycloidal teeth 
are first designed by the above methods, then the circular arcs, 
which fit as closely as possible, are used for the actual tooth 
outlines. When this is done there will be a slight variation of 
the speed-ratio during the time of contact of a pair of teeth. 
The variation may be reduced to a minimum by (instead of pro- 
ceeding as just described) finding the values of AC^ B D, and 
CZ> (fig. 474), such that the point <? will deviate the smallest possible 
amount from /, the pitch-point. The author has investigated 
this subject in a paper on * Circular Wheel-teeth,' published in the 
'Proceedings of the Institution of Civil Engineers,* vol. cxxi. 
The analysis is too long for insertion here, but the principal 
results may be given : 

Digitized by CjOOQiC 



4SO 



Details 



CHAP. XXTi:. 



For a given value of the speed-ratio R = -^^ three positions of 

the coupling-link CD can be found in which it passes through 
the pitch-point/: let Ci Z>„ C^D^ and C^D^ (fig. 478) be 
these positions. The distance of C2, the middle position of C, 




Fig. 478. 



from the pitch-point/ may be chosen arbitrarily ; but the greater 
this distance the less will be the speed variation and the greater 
the obliquity. 

/ ^a ^, arc of contact 
r 



T^t 



1 r\ 

then assuming that the other two positions of the equivalent link 
in which its intersection e with the line of centres coincides with 
the pitch-point / are at the beginning and end of contact of a 
pair of teeth, we may take approximately 

/ Ci = (w + «) r„ / C2 = >w r„ / Cj = (w - n) r^. 

Let A C = /i, B D = /2, and the length C D of the equiva- 
lent coupling-link = A. The values of /i, /„ and /„ for given 
values of I^, w, and «, are given by the following equations : 

Mm*" <■•' 



VJ 3 






Digitized byVjOOQlC 



(14) 



CHAP. xxvTi. Toothed-wheel Gearing 45 1 

Also F, the percentage speed variation above and below the 
average, is given by the equation 






(15) 



from which, for a constant value of m, the variation is inversely 
proportional to the cube of the number of teeth in the smaller 

wheel. The values of , , *, and V, for ;;/ = -3 and various 
r, r, ra 

values of ^ and «, are given in Table XVIII. 

Having calculated, or found from the tables, the values of /,, 4, 
and hf the drawing of the teeth may be proceeded with as 
follows : 

Draw the pitch-circles, with centres A and B^ touching at the 
pitch-point/ ; draw the link-circle Cy C^ Cg with centre A and 
radius /, ; likewise draw the link-circle Z>, D^ D^, With centre/ 
and radii equal to (m -f n)ry,mry, and (/w — n)ry respectively, 
draw arcs cutting the Hnk-circle C at the points Cj, Cj, and C3, 
respectively. With centres C^, Cj, and C3, and radius equal to ^, 
draw arcs cutting the link-circle D at Z>,, Z>2, and Z>3 respec- 
tively. A check on the accuracy of the drawing and calculation 
is got from the fact that the straight lines C| D^^ C^ Di, and 
C3 Z>3 must all pass through the pitch-point /. 

Assuming that the arcs of approach and recess are equal, 
Ca and D^ will be the centres of the circular tooth outlines in con- 
tact at p. Mark off along C^ Z>, and C3 D^ respectively, C^ F^ and 
C3-F3 each equal to C^p- Then F^ and 7^3 will be the extreme 
points on the path of contact ; the addendum-circle of wheel B 
will pass through /^„ and the addendum-circle of wheel A through 
F^. No working portion of the teeth will lie nearer the respective 
wheel centres than F^ and F^, Fillet-circles with centres A and B 
may therefore be drawn through Fy and /^g. 

The circular portion of the tooth will extend between the 
fillet- and addendum-circles ; the fillet, between the fillet- and 
root-circles, is designed as with involute or cycloidal teeth. 

Internal Gear. — With internal gearing the radius of the larger 
wheel may be considered negative, and the value of ^ will also 

Digitized b-^ ,r\<^n 



?by<5^4V 



452 



Details 



CHAP. XX^IL 



o 
o 

p 



p 

CO 

p 

CM 
JO 


b 

b 

8 





Ximbiiqo 
uinuifx^j^ 


t.?? 8^.?.8.8' 

b b b b o 

_b b b_b. 

■|-i b b b o 

S;^^^2?8 8 

b b o b o 
b b b o 

§ SI, ^'i^.^ 

M b b b b b b 

M^S».'n»?.8.8 

*H 'o b 


6 
b 
II 

<5 

c 
c 

td 

1 

D 
O 

t-H 

X 

1 

i5 


t *Xiinl 
P 

P 
CO 

P 

P 
io 


sdnit 
ijiqo 

1 


umumrepi 


° V » •• o« »>• 


1 


000.0 = ^ 


1 



b 


K 


o o o or;: 




Iffflff 

III m 


. „ 
•^ k 


boo o p_p_ 
_i:^o P o_<LP_ 

O VwVO NO *«« V V Vo 

"rr^'fftl 8 

M b b b b b 



M b b b b b 0. 

rv lo N "(S .0 8 

« M b b 

0_0 _o_o_o_o_ 

_M ob_b 0. 

N (*>v3 « 

« *M 'o b b b b 
V) b b b 

« O 


r*.** •- p c : .- 
b be c c . 


Xiinbiiqo 
uin'uiixi!}^ 


b p p p c c .: 


.8i§ ^m 

**••»• o : : 


:^ 


.^«.2 opTi 

o o : 3 


--Iv" 


M M S A f« « N 

p p p p p .: c 


^-'k 


.11 i 1111 




Xiinbiiqo 
uinuiprej^ 


*- •« « b 5 C 3 


O 

b 





•>?!k- 


o o o coca 


p 


1 

b 


^"Ik 




y\ p p 5- fT p 8 

V^ « M b b b b 


m 1111 

M M » o o : 


« b b b o 




' Digiti 


:edbY 


*i 






air's 8jj,£»8 


2?^ 8S^i 


s 


b b b b b b 


L.OOQ 


b b b c -•• 



CHAP. XX71I. Toothed'ivheel Gearing 453 

be negative. In (12), (13), (14), and (15) substitute R^ ^R \ 
they become respectively, 

* - ^j'j,- (■«) 



3 

' + 3 /?»(/? -if zK" " • • ^^^> 

6-4is(ff_-j)(ff-__=^)«' , . 



(^:)'= 



Table XIX., with m = -2, is calculated from these equations. 
The values of ^\ ""', in Tables XVIII. and XIX. change so 

slowly, that their values corresponding to any value of R and ;/ 
not found in the tables can easily be found by interpolation. 

314. Strength of Wheel-Teeth. — The mutual pressure i^ be- 
tween a pair of wheels is sometimes distributed over two or more 
teeth of each wheel \ but when one of the pair has a small number 
of teeth it is impossible to have an arc of contact equal to twice 
the pitch, and the whole pressure will be borne at times by a 
single tooth ; each tooth must therefore be designed as a cantilever 
fixed to the rim of the wheel and supporting a transverse load F 
at its point Let/ be the pitch of the teeth, b the width, h the 
thickness of a tooth at the root, and / the perpendicular distance 
from the middle of the root of the tooth to the line of action 
of F. Then the section at the root is subjected to a bending- 
moment F /, while the moment of resistance of the section 

is — - •^. Therefore, 

^^= "6 (20) 

The width of the teeth is usually made some multiple of the pitch ; 
let b ^=i kp. The height of the tooth may also be expressed as a 
multiple of/ ; it is often as much as 7 /, but since long teeth are 
necessarily weak, the teeth should be made as short as possible 
consistent with the arc of contact being at least equal to the pitch. 

** Digiffzed by Vj * 



454 



Details 



CHAP. ZXflL 



If the height be equal to -6/, the length / may be assumed equal 
to s /. If there be no side-clearance the thickness at the pitch- 
line will be '5 p^ and with a strong tooth form the thickness at the 
root will be greater. Even with side-clearance, we may assume 
h •=. '^p. Substituting in (20) we have 

^=-o8333/«^/ (21) 

or, writing p :=z p, P being the diametral pitch-number, 

7?*= -82246^ (22) 

The value that can be taken for / the safe working stress of the 
material, depends in a great measure on the conditions to which 
the wheels are subjected. If the teeth be accurately cut and run 
smoothly, they will be subjected to comparatively little shock. 
For steel wheels with machine-cut teeth, 20,000 lbs. per sq. in. 
seems a fairly low value for/ the safe working stress. 

Table XX. is calculated on the assumptions made above. 

Table XX. — Safe Working Pressure on Toothed 
Wheels. 

Calculated from equaiion (22). 



Pitch- 




Lbs. Pressure when k 


b 




number 












548 


14 


2 


24 


3 


5 


822 


1097 


1371 


1645 


6 


381 


571 


761 


952 


1 142 


7 


282 


421 


561 


1 703 


845 


8 


215 


323 


430 


1 538 


64s 


9 


170 


266 


341 


1 427 


5" 


10 


137 


206 


^74 


; 343 


1 

411 


II 


"4 


171 


228 


i ^75 


31^ 


12 


96 


144 


192 


239 


287 


13 


81 


122 


162 


203 


, 244 


14 


70 


106 


141 


1 176 


211 


15 


61 


91 


122 


1 152 


! 183 


16 


54 


80 


107 


134 


161 


18 


43 


64 


85 


107 


128 


20 


35 


51 


69 


86 


1 103 


22 


28 


42 


55 


! 71 


1 ^5 


24 


24 


36 


48 


; ^ 


72 



Digitized by CjOOQIC 



CHAP, xxvii. Toothed-wheel Gearing 455 

The arc of contact is sometimes made equal to two or three 
times the pitch, with the idea of distributing the total pressure 
over two or three teeth. But in this case, although the pressure 
on each tooth may be less than the total, they must be made 
longer in order to obtain the necessary arc of contact. It is 
therefore possible that when the pressure is distributed, the teeth 
may be actually weaker than if made shorter and the pressure 
concentrated on one. 

In cycloidal teeth, for a given thickness at the pitch-line, the 
thickness at the root is greater the smaller the rolling-circle ; 
where strength is of primary importance, therefore, a small rolling- 
circle should be adopted. In involute teeth, the angle of obliquity 
influences the thickness at the root in the same manner; the 
greater the angle of obliquity the greater the root thickness. In 
circular teeth, the greater m be taken, the thicker will be the teeth 
at the root. 

315. Choice of Tooth Form. — It has already been remarked 
that involute toothed-wheels possess the valuable property that their 
centres may be slightly displaced without injury to the motion. 
Involute tooth outlines are simpler than cycloidal outlines, the 
latter having a point of inflection at the pitch-circle ; involute 
teeth cutters are therefore much easier to make to the required 
shape than cycloidal. With involute teeth the direction of the 
line of action is always the same, but with cycloidal teeth it con- 
tinually changes, and therefore the pressure of the wheel on its 
bearing is continually changing. Taking everything into con- 
sideration, involute teeth seem to be preferable to cycloidal. The 
old millwrights and engineers invariably used cycloidal, but the 
opinion of engineers is slowly but surely coming round to the side 
of involute teeth. 

316. Front-driving Gears.— Toothed-wheel gearing has been 
more extensively used for front-driving bicycles than for rear- 
drivers. A few special forms may be briefly noticed. 

^ Sun-and'Plamt^ Gear, — In the * Sun-and- Planet ' Safety 
(fig. 479), the pedal-pins are not fixed direct to the ends of the 
main cranks, but to the ends of secondary links, hung from the 
crank-pin. A small pinion is fastened to each pedal-link and 
gears with a toothed-wheel fixed to the hub. This ^is^^ simple 

^ Digitized by VjUOQ ^ 



4S6 



Details 



COAT. XZTIL 




form of epicyclic train, and can be treated as in section 306. If 
JVi and Jv^ be the numbers of teeth on the hub and p)edal-liiik 
respectively, it can be shown (sec. 306) that the speed-ratio of the 

driving-wheel and main 

crank is — l_ i — 1, 

If the driving-wheel 
be 40 in. diameter, and 
the pinion and hub 
have 10 and 30 teeth 
respectively, the bicyde 

is geared to ^ — — - 

X 40 in. = 53*3 inches. 
It should be noticed 
that the pedal-link will 
^^^" ^'^' not hang vertically, 

owing to the pressure on the pinion. During the down-stroke 
the pedal will be behind the crank-pin ; while on the up- 
stroke, if pressure be applied to the pedal, it will be in front 
of the crank-pin. The pedal path is therefore an oval curve 
with its longer axis vertical. If the pressure on the pedal be 
always applied vertically, the pedal path will be an ellipse, with 
its minor axis equal to the diameter of the toothed-wheel on the 
driving-hub. 

This simple gear might repay a little consideration on the part 
of those who prefer an up-and-down to a circular motion for the 
pedals. 

T^e * Geared Facile ' is a combination of the * Facile ' and 
* Sun-and-Planet ' gears, the lower end of the pinion-link of the 
latter being jointed to the pedal-lever of the former. In figure 
124, the planet-pinion is 2 in. diameter, the hub-wheel 4 in. 
diameter, and the driving-wheel 40 in. diameter ; the bicycle is 
2) 



(4 + 
therefore geared to — 
4 



X 40 = 60 in. 



Perry's Front-driving Gear is similar in arrangement to the 
back gear of a lathe. The crank-axle (fig. 480) passes through 
the hub and is carried by it on ball-bearings. A toothed-wheel 

Digitized by V^jOOQ 



CHAP. XXVII. 



Toothed-wheel Gearinsr 



457 



fixed to the crank-axle gears with a wheel on a short intermediate 
spindle, to which is also fastened a wheel gearing in turn with one 
fastened to the hub of the driving-wheel ; the whole arrangement 
being the same as diagrammatically shown in figure 470. 




Fig. 460. 



The mutual pressure between the wheels D and E (fig. 470) 
is equal to the tangential effort on the pedal multiplied by the 
ratio of the crank length to the radius of wheel D, 

Example,— li the pedal pressure be 150 lbs., the crank length 
6^ in., and the radius of wheel D \\ in., the pressure on the teeth 

will be "^ X 150 = 780 lbs. 

The * Centric ' Front-driving Gear affords an ingenious example 
of the application of internal contact. A large annular wheel is 
fixed to the crank-axle and drives a pinion fixed to the hub of the 
driving-wheel, the arrangement being diagrammatically shown in 
figure 481 ; a and b being the centres of the crank-axle and the 
driving-wheel hub respectively. As the crank-axle has to pass 
right through the hub, the latter must be large enough to encircle 
the former, as shown in section (fig. 482). The hub ball-races are 
of correspondingly large diameter, the inner race being a disc set 

Digitized by CjOOQ IC 



458 



Details 



CHAP. invn. 



eccentrically to the crank-axle centre. The central part of the 
hub must be large enough to enclose the toothed-wheel on the 





Fig. 481. 



Fig. 482. 



crank-axle. Instead of being made continuous and enclosing the 
toothed-wheel completely, the hub is divided in the middle, and 

the end| portions are 
united by a triangu- 
lar frame. 

From figure 482 
it is evident that the 
* Centric' gear can 
only be used for 
speed-ratios of hub 
and crank-axle less 
than 2. 

The 'Crypto' 
Front-driving Gear 
is an epicyclic train, 
similar in principle 
to that shown in 
figure 472. Figure 
484 is a longitudinal section of the gear ; figure 483 an end view, 
showing the toothed-wheels ; and figure 485 an outside view of the 




Fig. 4S3. 



Digitized by CjOOQIC 



CBAP. XXYII. 



Toothed-wheel Gearing 



459 



hub, bearings and cranks complete. The arm C (fig. 472) in this 
case takes the form of a disc fastened to the crank-axle A^ and 
carrying four wheels E^ which engage with the annular wheel G, 
forming part of the hub of the driving-wheel, and with the small 
wheel 2?, rigidly fastened to the fork. The crank-axle is carried on 
ball-bearings attached to the fork, the hub runs on ball-bearings on 




Fig. 484. 

the crank-axle, while the small wheels E run on cylindrical pins B 
riveted to the disc C. 

The pressures on the teeth of the wheels are found as follows : 
Considering the equilibrium of the rigid body formed by the pedal- 
pin, crank, crank-axle A^ disc C, and pins B, the moment — 
about the centre of the crank-axle — of the tangential pressure P, 
on the pedal-pin is equal to that of the pressures of the wheels E 
on the pins B. Let / be the length of the crank, and r the 
distance of the centre of the pin B from the crank-axle ; the 
pressure of each wheel E on its pin will be 



r 4r 



(23) 



Digitized by CjOOQIC 



460 



Details 



CHAP. XIVII 



This pressure of the pin on the wheel E is resisted by the 
pressures of the wheels D and G ; each of these pressures must, 
therefore, be equal to 

IP , . 

S-r <^^^ 

If iVi, N^i ^^^ ^3 t>e the numbers of teeth on the hub- 
wheel G, fork-wheel Z>, and intermediate wheels JE, respectively, 

the speeds of the wheels and arm C, 
relative to the latter, are respectively 
proportional to 




I 



and • 



(25) 



while, relative to the fork, the speeds 
(o ,;, CD 2), 0) £, and o> c 'ire propor- 



tional to 



iVa'^'^i 



)■ 



and 



Also 



JV,= 



JV,-JV, 



.(26) 



(27) 



l'"rom (26) the speed-ratio of the hub 
and crank, relative to the fork, is 



I 



^i 



Fig. 485- 



From (28) it is evident that when 
the hub speed is to be more than 
twice that of the crank, JVy must be less than -A^, ; that is, the 
annular wheel must be fixed to the fork. 

From (25) and (27) the speed-ratio of wheels £ and Z>, rela- 
tive to the disc C, is 



_ ^2 _ 2M, 



AT. N^ - iV, 



= 2 



(^-2) 



(»9) 



But the wheel Z> makes — i turn relative to the crank while 
the latter makes i turn relative to the fork. Therefore, for every 

Digitized byVjOOgle 



CHAF. xxTn. Toothed-wheel Gearing 461 

turn of the crank, the wheels E make 2 ; "" ( turns in their 

bearings. Since these are plain cylindrical bearings, and the 
pressure on them is large, their frictional resistance will be the 
largest item in the total resistance of the gear. 

Example, — If /= 6i, r = i^in., and /*== 150 lbs. ; from (23) 

the pressure on the teeth is -^ 5_ = 97-5 lbs., and of the 

wheels E on their pins 195 lbs. Also if -/?=2*5, as in gearing 
a 28-inch driving-wheel to 70-inch, the wheels E each make 

^~2J> = 6 turns on their pins to one turn of the crank. 
•5 

317. Toothed-wheel Bear-driving Oears. — A number of gears 
have been designed from time to time with the object of replac- 
ing the chain, but none of them have attained any considerable 
degree of success. 

The ^ Burton^ Gear was a spur-wheel train, consisting of 

a spur-wheel on the crank-axle, a small pinion on the hub, 

and an intermediate wheel, gearing with both the former and 

running on an intermediate spindle on the lower fork. The 

intermediate wheel did not in any way modify the speed-ratio, so 

that the gearing up of the cycle depended only on the numbers 

of teeth of the wheels on the crank-axle and hub respectively. 

If r was the radius of the spur-wheel on the crank -axle and / 

the length of the crank, the upward pressure on the teeth of the 

IP 
intermediate wheel was — , and therefore the upward pressure of 

IP 

the intermediate wheel on its spindle was 2 . This upward 

pressure was so great, that an extra bracing member was required 
to resist it. 

Example.— If /*= 150 lbs., /= 6| in., ^ = 4, the pressure on 

the intermediate spindle = -^ ^^^ '5o ^ ^g^.^ ibg. 

4 

The Fearnhead Gear was a bevel-wheel gear, bevel- wheels being 

fixed on the crank-axle and hub respectively and geared together 

by a shaft enclosed in the lower frame tube. If bevel-wheels could 

be accurately and cheaply cut by machinery, it is possible that 

Digitized by CjOOQIC 



462 Details 



CBA.V. zxm. 




gears of this description might supplant, to a considerable extent, 
the chain-driving gear ; but the fact that the teeth of bevel-wheels 
cannot be accurately milled is a serious obstacle to their practical 
success. 

318. Componnd Driving Oears. — For front-driving, Messrs, 
Marriott and Cooper used an epicyclic train (fig. 486), formed 
from a pair of spur-wheels and a pair 
of chain-wheels. Two spur-wheels, I) 
and JS, rotate on spindles fixed to the 
crank C Rigidly fixed to -^ is a chain- 
wheel G, connected by a chain to a 
chain-wheel jF, fixed to the fork. If 
the arm C be fixed and the pinion 
D be rotated, the chain-wheel jF will be driven in the opposite 
direction. Let — « be the speed-ratio of the wheels 7^ and £> rela- 
tive to the arm C (in figure 486, — « = — i), then theangular speeds 
of Fy C, and D are respectively proportional to «, o, and - r. If 
a rotation + 1 be given to the whole system, their speeds will be 
proportional to (« + i), i and o respectively. The wheel JD is 
fixed to the fork, the wheel jF to the hub of the driving-wheel, and 
C is the crank. The driving-wheel, therefore, makes (n + i) turns 
to one turn of the crank. With this gear, any speed-ratio of 
driving-wheel and crank can be conveniently obtained. 

A number of compound rear- driving gears have been made, 
some of which have been designed with the object of avoiding 
the use of a chain. In * Hart's ' gear, a toothed-wheel was fixed 
on the crank-axle and drove through an intermediate wheel a 
small pinion ; a crank fixed on this pinion was connected by 
a coupling-rod to a similar crank on the back hub. In this gear, 
there was a dead-centre when the hub crank was horizontal, and 
when going up-hill at a slow pace the machine might stop. In 
* Devoirs ' gear the secondary axle was carried through to the 
other side of the driving-wheel, two coupling-rods and pairs of 
cranks were used, and the dead-centre avoided. 

T/ie ^Boudard' Gear (fig. 487) was the first of a number of 
compound driving gears in which the chain is retained. An 
annular wheel is fixed near one end of the crank-axle and gears 
with a pinion on a secondary axle ; at the other end of the 

Digitized by CjOOQIC 



CHAP. XXVII. 



Toothed-wheel Gearing 



463 



secondary axle a chain-wheel is fixed and is connected by a chain 
in the usual way to a chain-wheel on the back hub. A great deal 
of discussion has taken place on the merits and demerits of this 
gear ; probably its promoters at first made extravagant claims, 
and its opponents have overlooked some points that may be 
advanced in its favour. Of course, the mere introduction of an 
additional axle and a 
pair of spur-wheels is 
rather a disadvantage 
on account of the extra 
friction. In the chapter 
on Chain Gearing it has 
been shown that it is 
advantageous to make 
the chain run at a high 
speed ; this can be done 
with the ordinary chain 
gearing by making both 
chain-wheels with large 
numbers of teeth, but if 
the back hub chain- 
wheel be large, say with 
twelve teeth, that on the 
crank-axle must be so 
large as to interfere 
with the arrangement of the lower fork. The * Boudard ' gear is a 
convenient means of using a high gear with a large chain-wheel 
on the back hub. 

The * Healy ' Gear (fig. 488) is an epicyclic bevel gear having a 
speed-ratio of 2 to i, which has the advantage of being more 
compact than the * Boudard ' gear, but has the disadvantage which 
applies to all bevel-wheels, viz. the fact that they cannot be 
cheaply and accurately cut. 

Geared Hubs. — Compound chain gears have been used in which 
the toothed-wheel gearing is placed at the hub of the driving- 
wheel. In the * Platnauer ' gear (fig. 489) the small pinion is fixed 
to the hub and gears with a large annular wheel which runs on a 
disc set eccentrically to the hub spindle, a row of^balls being 

Digitized by V^jOOQ 




Fig. 487. 



464 



Details 



CHAP. xxm. 



introduced. The outer part of this wheel has projecting teeth to 
gear with the chain. 




Fig. 488. 

These hub gears, as far as we can see, have none of the 
advantages of the crank-axle gears to recommend them, since the 
speed of the chain cannot be increased unless a very large crank- 




FiG. 489. 

axle chain-wheel be used, and they possess the disadvantages of 
additional frictional resistance of the extra gear 

Digitized by CjOOQIC 



CHAP. XXVII. 



Toothed-wheel Gearing 



465 



319. Variable Speed Oears. — It has been shown, in Chapter 
XXI., that it is theoretically desirable to lower the gear of the 
cycle while riding up-hill. 

In the * Collier ' Two-Speed Gear, of which figure 490 is a 
section, and figure 491 a general sectional view, a stud-wheel D 
(that is, a wheel with pin teeth) fixed on the crank-axle gears with 
a toothed-pinion P attached to the chain-wheel C The crank- 
axle A is carried on a hollow axle B^ the axes of the two axles 




Kk;. 490. 

being placed eccentrically. The chain-wheel 6', and with it the 
toothed-pinion J\ revolves on a ball-bearing at the end of the 
hollow axle B, There are twelve and fifteen teeth respectively 
on the pinion and stud-wheel, so that the ratio of the high and 
low gears is 5 : 4. When the low gear is in use, the two axles 
are locked together by means of a slide bolt S in the hollow axle 
which engages with a hole in the stud-wheel />, the whole 
revolving together on ball-bearings in the bottom- bracket F. 
When the high gear is used, the bolt in the hollow axle is with- 

Digitized by LjO^QtC 



466 



Details 



CHAP. ZXTII. 



drawn from the hole in the stud-wheel and fits in a notch in the 
operating lever. The toothed-pinion, and with it the chain-wheel C, 
is then driven at a higher speed than the crank-axle. 




Fig. 491. 



The arrangement of the two axles is shown diagrammatically in 

figure 492. When the high gear is in use the centre b of the 

crank-axle is locked in position vertically about the centre a of 

^--^ the hollow axle. If the 



cranks are exactly in line 

at high gear, the virtual 

cranks a Cy and a c^ will 

be slightly out of line at 

*^" • 49^ low gear. The pedals, 

however, describe practically equal circles with either gear in 

use. 

The * Eite and Todd ' Two-Speed Gear (fig. 493) consists of a 
double-barrelled bracket carrying the crank-axle — on which is 
keyed a toothed-wheel — and a secondary axle, to which is fixed 
two small pinions at one end, and the chain-wheel at the other. 
The pinions on the secondary axle are in gear with intermediate 

Digitized by CjOOQ IC 



CHAP. XXTII. 



Toothed-wheel Gearing 



467 



pinions running on balls on adjustable studs attached to an arm 
which can swing round the secondary axle. One or other of the 
intermediate pinions can be thrown into gear with the spur- 
wheel on the crank-axle, by the shifting mechanism under the 
control of the rider, by means of a lever placed close to the 
handle-bar. 

The Cycle Gear Company's Two-Speed Gear has an epicyclic 
train somewhat similar in principle to that of the * Crypto ' front- 




FiG. 493. 



driving gear. When high speed is required, the whole of the 
gear rotates as one rigid body ; but when low speed is required 
the small central wheel is fixed and the chain-wheel driven by an 
epicyclic train. 

The same Company also make a two-speed gear, the change of 
gearing being effected at the hub of the driving-wheel. 

The *y. and R,^ Tivo-Speed Gear (fig. 494) consists of an 
epicyclic gear in the back hub ; the central pinion of the gear 
is fixed to the driving-wheel spindle when the low gear is 
used, the wheel hub then rotating at a slower speed than the 

H i^ 



468 



Details 



CHAP. XXTll. 



chain-wheel. When the high gear is used, the epicyclic gear— and 
with it, of course, the chain-wheel— is locked to the driving-wheel 
hub. C, is the main portion of the driving-wheel hub ; to this 
is fastened the end portion Ca, on which are formed a ball-race 
for the chain-wheel G, and an annular wheel D^ in which the 




Fic. 494. 



central pinion D can be locked. The intermediate pinions E^ 
four in number, revolve on pins fastened to the hub C, and C,. 
The annular wheel G^^ which gears with the intermediate pinions, 
is made in one piece with the chain-wheel G, AVhen the low 
gear is in use the central pinion D is held by the axle-clutch />, 




Fig. 495. 



fastened to the spindle A, To change the gear, the central 
pinion D is shifted longitudinally out of gear with the axle clutch 
and into gear with the annular wheel D^. This shifting is done 
by means of a rack r and pinion / ; the latter is supported in a 
shifter-case S fixed to the driving-wheel spindle, and is operated 
by the rider at pleasure. 

Digitized by CjOOQIC 



CHAP. XXVII. 



Toothed-wheel Gearing 



469 



Figure 495 shows an outside view of the hub with the spindle 
and shifter-case partially removed. 

The * Sharp * Two-Speed Gear (fig. 496) is an adaptation of the 
* Boudard' driving gear. On the crank-axle A the disc D^ carries 
a drum D.^ on which are formed two annular wheels w^ and w^ 
which can gear with pinions p^ and p^ fastened to the secondary 
axle. The secondary axle is in two parts \ the chain-wheel ^is 




Fig. 496. 



fixed to one part ^,, the pinions/, and /a to the other part aj. 
The ball-bearing near the end of a^ is carried by a secondary 
bracket b^ which can be moved longitudinally in the main bracket 
B^ so that the pinion /j may be moved into gear with the wheel 
?«;,, or pinion p^ into gear with wheel ^e'^ 'i while in the inter- 
mediate position, shown in the figure, the crank-axle may remain 
stationary while the machine runs down hill. A hexagonal 
surface on the portion a.i fits easily in a hollow hexagonal surface 
on the portion a^ of the secondary axle, so that the one cannot 
rotate without the other, although there is freedom of longitudinal 
movement. The longitudinal movement is provided by a stud j. 



470 Details chap. xxm. 

which passes through a small spiral slot in the main bracket B^ 
and is screwed to the inner movable bracket b. The end of the 
stud can be raised or lowered, and the sliding bracket simulta- 
neously moved longitudinally, by the rider, by means of suitable 
mechanism, and the gear changed from high to low, or vice versa. 

The drum D^ is wider than that on the ordinary * Boudard ' 
gear, the corresponding crank, C^, may therefore be fastened to 
the outside of the drum instead of to the end of the crank-axle. 
The other crank, Cj, is fastened in the usual way to the crank- 
axle. 

IJnley and Biggs^ Expanding Chain-wheel (fig. 465) provides 
for three or four different gearings, and though there is no 
toothed-wheel gear about it, it may be mentioned here, since it 
has the same function as the two- speed gears above described. 
The rim of the chain-wheel on the crank-axle can be expanded 
and contracted by an ingenious series of latches and bolts, so as 
to contain different numbers of cogs. When pedalling ahead the 
driving effort is transmitted direct from the crank -axle to the 
chain-wheel; but if the chain- wheel be allowed to overrun the 
crank-axle the series of changes is effected in the former. The 
right pedal being above, below, before, or behind the crank-axle, 
corresponds to one particular size of the chain-wheel ; if pedalling 
ahead be begun from one of these positions, the chain-wheel will 
remain unaltered. The length of chain is altered by the changes, 
therefore a loose pulley at the end of a light lever, controlled by a 
spring (fig. 465), is used to keep it always tight. Back-pedalling 
is impossible with this expanding chain -wheel, so a very powerful 
brake is used in conjunction with it. 

A two-speed gear, with the gearing-down done at the hub, will 
be better than one with the gearing-down done at the crank- 
bracket, in so far that when driving with the low gear the speed 
of the chain will be greater, and therefore the pull on it will be 
less, presuming that the number of teeth on the back-hub chain- 
wheel is the same in both cases. 

The frictional resistance of an epicyclic two-speed gear is 
probably much greater than that of an annular toothed-wheel 
gear, such as the * Collier ' or * Sharp,' on account of the inter- 
mediate pinions revolving on plain cylindricaL^bearings under 

Digitized by V^jOOQ 



CHAP. XXVII. Tootlted'wheel Gearing 4^1 

considerable pressure. The crank-axle of the former gear runs 
on plain qrlindrical bearings when the gear is in action. The 

* Sharp ' and the * Eite and Todd ' two-speed gears have the dis- 
advantage, compared with the others, that the additional gear and 
its consequent increased frictional resistance is always in action ; 
in this respect the former is exactly on a level with the ordinary 

* Boudard ' gear. 



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472 Detatls chap. xx^m. 



CHAPTER XXVIII 

LEVER-AND-CRANK GEAR 

320. Introductory. — A number of lever-and-crank gears have 
been used to transmit power from the pedal to the driving-axle of 
a bicycle ; the majority of them are based on the four-link kine- 
matic chain. In general, a lever-and-crank gear does not lend 
itself to gear up or down ; that is, the number of revolutions 
made by the driving-axle is always equal to the number of com- 
plete up-and-down strokes made by the pedal. When gearing up 
is required, the lever-and-crank gear is combined with a suitable 
toothed-wheel mechanism, generally of the * Sun-and-Planet ' 
type. The four-link kinematic chain generally used for this gear 
consists of: (i) the fixed link, formed by the frame of the 
machine ; (2) the crank, fastened to the axle of the driving-wheel, 
or driving the axle by means of a * Sun-and-Planet ' gear ; (3) the 
lever, which oscillates to and fro about a fixed centre \ (4) the 
coupling-rod, connecting the end of the crank to a point on the 
oscillating lever. 

Lever-and-crank gears may be subdivided into two groups, 
according as the pedal is fixed to the lever, or to the coupling-rod of 
the gear. In the former group, the best known example of which 
is the * Facile ' gear, the pedal oscillates to and fro in a circular 
arc, having a dead-point at the top and bottom of the stroke. In 
the latter group, of which the * Xtraordinary ' and the • Claviger ' 
were well-known examples, the pedal path is an elongated oval 
curve, the pedal never being at rest relative to the frame of the 
machine. 

With lever-and-crank gears it is easy to arrange that the down- 
stroke of the knee shall be either quicker or slower than the up- 
stroke. In the examples analysed in this chapter, where a 

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CHAP. XXVIII. 



Lever-and' Crank Gear 



473 



difference exists, the down-stroke is the quicker. Probably this 
is merely incidental, and has not been a result specially aimed at 
by the designers. Regarded merely as a mechanical question, it is 
immaterial whether the positive stroke be performed more quickly 
or slowly than the return stroke, though, possibly, physiological 
considerations may slightly modify the question. 

321. Speed of Knee- Joint with < Facile' Gear.— If the 
pedal be fixed to the oscillating lever, its varying speed can 
be found as in section 33, 
the speed of the crank-pin 
being considered constant. 
The speed of the knee-joint 
can be found as follows : Let 
A B C D (fig. 497) be the 
four - link kinematic chain, 
D C being the frame-link, 
D A the crank, C B the 
oscillating lever, and A B the 
coupling-rod. Let the pedal 
be fixed to a prolongation of- 
the oscillating lever at P. 
Let H and K be the rider's 
hip- and knee-joints respec- 
tively, corresponding to the 
points C and B of figure 21. 
In any position of the me- 
chanism produce D A and 
B C X.O meet at /; / is the 
instantaneous centre of rota- 
tion of A B, Let HK and 
P C, produced if necessary, meet at J, Since P is at the 
instant moving in a direction at right angles to C P^ it may be 
considered to rotate about any point in I P\ for a similar 
reason, K may be considered to rotate about any point in 
H K\ therefore J is the instantaneous centre of rotation of 
the rider's leg, P K, from the knee downwards. Let Va^ z/^, . . . 
be the speeds at any instants of the points A, B, . , , Draw 
D e, parallel to B C, meeting B A, produced if necessary, at e. 

Digitized by CjOOQIC 




Fig. 497. 



474 Details chap, xxtiil 

Then, since the points B and P are both rotating about the 
centre C, 

(I) 



But from section 32, 









De 
DA 



(2) 



(3) 



Draw D e^ parallel io P C and equal to D e. Draw D g and 
tf* ^, meeting at g^ respectively parallel to H KtltA P K. Since 
the triangles J K P and D ge^ are similar, 

JP De^' 
and 

v.^JK^D g 

V, JP De' 

Multiplying (i), (2), and (3) together, we get 

v^ v^ v^^C£ De Dj 
v,^ Va' v^ CB' DA ' De'' 

that is, remembering that D e and D e^ are equal, 

V. ^ ^^.^,Dg (4) 

v^ CB.DA ^ ^^' 

Therefore since the lengths C /*, C B, and D A are constant for 
all positions of the mechanisms, the speed of the knee-joint is 
proportional to the intercept D g. li D kht set off along D A 
equal to D gy the locus of k will be the polar speed-curve of the 
knee-joint. 

322. Pedal and Knee-Joint Speeds with ' Xtraordinary ' 
Gear. — If the pedal P be rigidly fixed to a prolongation of the 
coupling-rod B A^ the construction is as follows : Produce D A 
and C By to intersect at / (fig. 498), the instantaneous centre of 
rotation of the coupling-rod A B. Draw D e, parallel to I P, 
cutting A Py produced if necessary, at e. [In some positions of 
the mechanism the instantaneous centre / will be inaccessible, 
and the direction of /-P not directly determinable ; the following 

Digitized by V^jOOQ 



C1XAF. ZZYin. 



Lever-and'Crank Gear 



475 



modification in the construction may be used : Draw D e^ 
parallel to B C, meeting A B zX. e^ \ then draw <?* e parallel to 
JB Py meeting A P ^X e,] 
Then 

^p = ^J^= ^ (c) '^' 
v^ I A DA ' ^^^ 



or, 






Trb-^' • (6) 




D A is, of course, of 
constant length for all posi- 
tions of the mechanism, 
and if the speed of the 
bicycle be uniform, v^ is 
constant, and therefore the 
speed of the pedal P 
along its path is propor- 
tional to the intercept De, 
If Dp be set off along the 
crank D A, equal to this 
intercept, the locus of / 
will be the polar curve of 
the pedal's speed. ^'''' ^^' 

Produce H Kio meet IP at y, then / is the instantaneous 
centre of rotation of the rider's leg K P from the knee to the 
pedal. From D and e draw </^ and eg, meeting at g, respectively 
parallel to ^ZTand P K, Since the points A' and P are at the 
instant rotating about the centre y, 



JP De 



Multiplying (5) and (7) together we get 



(7) 



Va 



DA 



or, 



^^ = WA-^' 



(8) 



Digitized by CjOOQIC 



476 



Details 



cbjlP, zmn^ 



Therefore, since Va and D A are constant, the speed of the knee- 
joint is proportional to the intercept D g. If i? ^ be set off 
along the crank D A^ equal to D g^ the locus of k will be the 
polar curve of the speed of the knee-joint. 

323. Pedal and Enee-Joint Speeds with ' Oeared-FacQe * 
Mechanism. — If toothed gearing be used in conjunction with a 
lever-and-crank gear, the motion of the mechanism is altered 
considerably. The toothed gearing usually employed in such 
cases is the well-known * Sun-and-Planet ' wheels, one toothed- 
wheel being fixed to the hub of the driving-wheel, the other 
centred on the crank-pin, and rigidly fixed to the coupling-rod of 
the gear. The driving-wheel will, as before, rotate with practically 
constant speed, since the whole mass of the machine and rider, 
moving horizontally, acts as a flywheel steadying the motion. 
Thus the sun-wheel of the gear moves with constant speed 

relative to the frame, but the speed 
of the crank is not constant, on 
account of the oscillation of the 
coupling-rod and planet-wheel. 

\^t D A B C (fig. 499) be, as 
before, the lever-and-crank gear, and 
let the * Sun-and-Planet ' wheels be 
in contact at the point S^ which 
must, of course, lie on the crank 
JD A. Let / be the instantaneous 
centre of rotation of the coupling- 
rod A By and planet-wheel ; and 
let V, be the speed, relative to the 
frame, of the pitch-line of the sun-wheel ; this will be, of course, 
the speed of the points of the wheel in contact at 5. Draw 
D e parallel to C B, meeting B S, produced if necessary, at e. 
Then, 

v^^IB ^ De 

v] IS J?^ 
or, 




^'=^5-^' 



(9) 



That is, the speed of the pedal is proportional to the intercept 

Digitized byVjOOgle 



ciiA>r. xxviii. 



Lever-and-Cra7ik Gear 



477 



D €, since v, and D S are constant. Performing the remainder 
of the construction as in figure 497, we get 

V, CB.DS 



Dg 



(10) 



The variation in the speed of the crank can easily be shown 
thus : The points A and S of the planet-wheel are at the instant 
rotating about the point /. Therefore, 



V. 



lA 
IS 






AS 
IS 



(") 



J S being considered negative when iS lies between A and /. 

324. Pedal and Knee-joint Speeds with 'OearedClaviger ' 
Mechanism. — In this case the modification of the construction in 
figure 498 is the following : Let S (fig. 500) 
be the point of contact of the * Sun-and-Planet ' . I / 

wheels. Join P S^ and draw D e parallel to 
I P^ meeting P S\ne. Then, as in section 323, Bt 



or, 



Vn = 



IP 

IS 

= 2l^ 
D S 



De 
DS' 

De . 



(12) 




Fig. 50a 



That is, the pedal speed is proportional to 
the intercept D e. 

If the instantaneous centre / of the 
coupling-rod be inaccessible, the method of 
determining D e may be as follows : — Join 
^ B, and draw D e^ parallel to C By meeting SB at e^. Draw 
e^ e parallel to B P^ meeting S P dX e, 

325. * Facile* Bicycle. — Figure 497 represents the * Facile' 
mechanism. From the centre of the driving-wheel D with radius 
{D A -^ A B) draw an arc cutting the circular arc forming the 
path of B in the point By ; from D with radius {A B -^ D A) 
draw an arc cutting the path of B in B^ ; then B^ and B^ will be 
the extreme positions of the pedal. The motion being in the 
direction of the arrow, and the speed of the machine being 

Digitized by CjOOQIC 



478 



Details 



CHAP. zxniL 



uniform, the times taken by the pedal to perform its upward 
and downward movements are proportional to the lengths of the 
arcs -^1 9 ^2 and A^ 3 A^, With the arrangement of the 
mechanism shown in the figure, the down -stroke takes a little less 
time than the up-stroke. 

// (fig. 501) is the polar curve of pedal speed, found by the 
method of section 32, and k ky the polar curve of speed of knee-joint, 
found by the method of section 321, for the dimensions of the gear 
marked in figure 497. The speed of the knee-joint is greatest 
when the crank is about 30° from its lowest position, then very 
rapidly diminishes to zero, and rapidly attains its maximum speed 
in the opposite direction. It should be remembered that the 
speed curve, k k, is obtained on the assumption that the ankle is 




Fig. 50 t. 



Fig. 50a. 



kept stiff during the motion. Using ankle action freely, the curve 
k k may not even approximately represent the actual speed of the 
knee ; but the more rapid the variation of the radius- vectors to the 
curve k k, the greater will be the necessity for perfect ankle 
action. It should be noticed that with any mechanism a slight 
change in the position of the point H (fig. 497) may make a con- 
siderable change in the form of the curve k k (fig. 501). 

In some of the early lever-and-crank geared tricycles the pedal 
was placed at the end of a lever which, together with the osdl- 
lating lever of the four-link kinematic chain, formed a bell 

Digitized by CjOOQIC 



CHAP. XXVUI. 



Lti^er-and'Crank Gear 



479 



crank (see fig. 146). The treatment of the pedal motion in this 
case is the same as for the * Facile ' mechanism. 




Fig. 503. 



Geared Facile, — Figure 502 shows the polar curves of pedal 
speed, p /, and of speed of knee-joint, k k^ for a Geared Facile : 




Fig. 504. 



Digitized by CjOOQIC 



48o 



'Details 



CHAP. xmii. 



the dimensions of the mechanism being exactly the same as in 
figure 497, and the ratio of the diameters of the * Sun-and- Planet ' 
wheels being 2:1. 

326. The < Xtraordinary ' was, perhaps, the first successful 
Safety bicycle, the driving mechanism being arranged so that the 

rider could use a large front 
wheel while sitting consider- 
ably further back and lower 
than was possible with an 
* Ordinary.' 

PP (fig. 498) is the pedal 
path in the * Xtraordinary,' 
P P (fig- 503) ^^ polar curve 
of pedal speed, and k k the 
polar curve of speed of the 
knee-joint The down-stroke 
of the knee is performed 
much more quickly than the 
up-stroke, as is evident either 
from the polar speed curve, 
k k^ or from the correspond- 
ingly numbered positions (fig. 
498) of the knee and crank- 
pin. During the down-stroke 
of the knee, the crank-pin 
moves in the direction of the 
arrow, from 12 to 5 ; during 
the up-stroke, from 5 to 12. 

327. Claviger Bicycle. — In the Claviger gear, as applied to the 
* Ordinary ' type of bicycle (fig. 504), the crank-pin was jointed to 
a lever, the front end of which moved, by means of a ball-bearing 
roller, along a straight slot projecting in front of the fork. At the 
rear end of the lever a segmental slot was formed to provide a 
vertical adjustment for the pedal, to suit riders of different heights. 
The mechanism is equivalent to the crank and connecting-rod of a 
steam-engine, the motion of the ball-bearing roller being the same 
as that of the piston or cross-head of the steam-engine. The 
mechanism may be derived from the four-link kinematic chain by 

Digitized by CjOOQIC 




CHAP. ZXTUI. 



Lever-and' Crank Gear 



481 



considering the radius of. the arc in which the end, B (fig. 21), of 
the coupling-rod moves to be indefinitely increased. The con^ 



5»- 



--^9 



Fig. 506. 



structions of figure 2 1 will be applicable, the only difference being 
that the straight line B I will always remain in the same direction, 




Fig. 507. 



Digitized by VjOOQIc 



482 



Details 



CHAF. XXTin. 



that is, at right angles to the straight slot. By bending the pedal 
lever downwards as shown (fig. 504), the position of the saddle is 
further backward and downward than in the * Ordinary.' 

P P (fig. 505) is the pedal path, / / (fig. 506) the polar curve 
of pedal speed, and k k the polar curve of speed of knee-joint, for 
the mechanism to the dimensions marked in figure 505. 




Fig. 508. 

Geared Claviger, — PP (fig. 508) is the pedal path, / / (fig. 509) 
polar curve of pedal speed, and k k polar curve of speed of knee- 
joint, for a 'Geared Claviger' rear-driving Safety (fig. 507); the 
dimensions of the mechanism being as indicated in figure 508, 
and the ratio of the diameters of the * Sun-and-Planet ' wheels 2 : i. 
The construction is as shown in figure 500. 

A few peculiarities of the gear, as made to the dimensions 
marked in figure 508, may be noticed. The motion of the pedal 
in its oval path, is in the opposite direction to that of a pedal fixed 
to a crank. The speed of the pedal increases and diminishes three 

Digitized by CjOOQIC 



CHAP. ZZTUI. 



Lever-and-Crdnk Gear 



483 



times in each up-and-down stroke ; the speed-curve, // (fig. 509), 
shows this clearly. The pedal path (fig. 508) also indicates the 
same speed variation; the portions 2-3, 6-7, and lo-ii, being 




Fig. 5x0. 



Digitized by CjOO^ IC 



484 



Details 



CHAP. zzvnL 



each longer than the adjacent portions, are passed over at greater 
speeds. 

328. Early Tricycles.— In the Dublin quadricycle (fig. 1 1 7), and 
in some of the early lever-driven tricycles (^g, 142), the pedal was 
placed about the middle of the coupling-rod, one end of which 
was jointed to the crank-pin, the other to the end of the oscillatii^ 
lever. The pedal path was an elongated oval, the vertical axis of 
which was shorter than the horizontal ; the early designers aiming 




at giving the pedals a motion as nearly as possible like that of the 
foot during walking. PP (fig. 5 1 o) is the pedal path, / / (fig. 511) 
the polar curve of pedal speed, and k k the polar curve of speed 
of knee-joint, the dimensions of the mechanism being shown in 
figure 510. The construction is as shown in figure 498. 

It may be noticed either from figure 510 or the curve k k (fig. 
511) that the down-stroke of the knee is performed in one-third 
the time of one revolution of the crank, the up-stroke in two- 
thirds. Also, the knee is at the top of its stroke, when the crank 
is nearing the horizontal position, descending. C (^c^QXe 



485 



CHAPTER XXIX 

TYRES 

329. The Tyre is that outer portion of the wheel which 
actually touches the ground. The tyres of most road and railway 
vehicles are of iron or steel, and in the early days of the bicycle, 
when wooden wheels were used, their tyres were also of iron. 
The tyre of a wooden wheel serves the double purpose of keeping 
the component parts of the wheel in place, and providing a suitable 
wearing surface for rolling on the ground. 

330. Boilings Sesistance on Smooth Surfaces.— The rolling 
friction of a wheel on a smooth surface is small, and if the surfaces 
of the tyre and of the groiind be hard and elastic the rolling 
friction, or tyre friction, may be neglected in comparison with the 
friction of the wheel bearings. This is the case with railway 
wagons and carriages. A short investigation of the nature of 
rolling friction has been given in section 78. 

In Professor Osborne Reynolds' experiments the rolling took 
place at a slow speed. When the speed is great another factor 
must be considered. The tyre of a 
circular wheel rolling on a flat surface 
gets flattened out, and the mutual 
pressure is distributed over a surface. 
Let c (fig. 512) be the geometrical 
point of contact, a^ and a^ two points 
at equal distances in front of, and 

behind, c ; /, and /a the intensities of the pressures at <af, and a^ 
respectively. The pressure/, opposes, the pressure /g assists, the 
rolling of the wheel. If the rolling takes place slowly, it is possible 
that/9 may be equal to/|,ancJ the resultant reaction on the whe^l 

Digitized by VjOOQ 





486 Details chap. tttt. 

may pass through the centre. But in all reversible dynamical 
actions which take place quickly, it is found that there is a loss of 
energy, which varies with the quickness of 
the action. The term 'hysteresis,' first 
used by Professor Ewing in explaining the 
phenomenon as exhibited in the magnet- 
isation of iron, may be used for the general 
phenomena. In unloading a spring 
quickly, the load corresponding to a given 
deformation is less than when loading it ; 
F»G' 5«3. more work is required to load the spring 

than it gives out during the removal of 
the load. \i O a P (fig. 513) be the stress-strain curve during 
loading, that during unloading will be -P ^ (9, and the area 
O a P b O will be the energy lost by hysteresis. Thus, p-^ is less 

than/,, the ratio ^^ lying between i and ^, the index of elasticity. 
/„ varies with the distance of a, from r, and is a maximum when 

fli coincides with c. Assuming the ratio ^ to remain constant 

P\ 
for all positions of a^ and a^ relative to r, we may say that the 
energy lost is proportional to 

c d being the radial displacement of a point on the tread of the tyre. 
Comparing three tyres of rubber, air, and steel respectively 

rolling on a perfectly hard surface, ^i—""^* will possibly be 

P\ 
smallest for air, and largest for rubber ; while the displacement 

c d will be smallest for steel. The 

* ^ rolling resistance of the steel tyre will 

y f. be least, that of the rubber tyre greatest 

5^55^,,,^^^^^ 331. Metal Tyre on Soft Eoai- 

FiG. 514. The road surfaces over which cycles 

have to be propelled are not always 

hard and elastic, but are often quite the opposite. If a hard 

Digitized by CjOOQIC 



CHAP. XZIZ. 



Tyres 487 



metal tyre be driven over a soft road a a (fig. 514) it sinks 
into it and leaves a groove c of quite measurable depth. The 
resistance experienced in driving a cycle with narrow tyres 
over a soft road is mainly due to the work spent in forming this 
groove. 

332. Loss of Energy by Vibration.— The energy lost on 
account of the impact of the tyre on the ground is proportional 
to the total mass which partakes of the motion of impact (see 
chap. xix.). In a rigid wheel with rigid tyres, this will consist 
of the whole of the wheel, and of that part of the frame which may 
be rigidly connected to, and rest on, the spindle of the wheel. 
If no saddle springs be used, part of the mass of the rider will also 
be included. The energy lost by impact, and which is dissipated 
in jar on the wheel of the machine, must be supplied by the 
motive power of the rider ; consequently any diminution of the 
energy dissipated in shock, will mean increased ease of propulsion 
of the machine. 

The state of the road surface is a matter generally beyond the 
control of the cycHst or cycle manufacturer,. and therefore so also 
are the velocities of the successive impacts that take place. 
However, the other factor entering into the energy dissipated, the 
mass m rigidly connected with the tyre is under the control of 
the cycle makers. In the first bicycles made with wooden wheels 
and iron tyres, and sometimes without even a spring to the seat, 
the mass m included the whole of the wheel and a considerable 
proportion of the mass of the frame and rider ; so that the energy 
lost in shock formed by far the greatest item in the work to be 
supplied by the rider. The first improvement in a road vehicle is 
to insert springs between the wheel and the frame. This prac- 
tically means that the up and down motion of the wheel is per- 
formed to a certain extent independently of that of the vehicle 
and its occupants ; the mass m in equation (2), chapter xix., is 
thus practically reduced to that of the wheel. The effort required 
to propel a spring vehicle along a common road is much less than 
that for a springless vehicle. 

333. Bubber Tyres. — If the tyre of the wheel be made elastic 
so that it can change shape sufficiently during passage over an 
obstacle, the motion of the wheel centre may not be perceptibly 

Digitized by VjOOQ 



488 Details . chap. tux. 

affected, and the mass subjected to impact may be reduced to 
that of a small portion of the tyre in the neighbourhood of the 
poiht of contact. Thus, the use of rubber tyres on an ordinary 
road greatly reduces the amount of energy wasted ; in jar of the 
machine. Again, the rubber tyre being elastic, instead of sinking 
into a moderately soft road, is flattened out. The area of contact 
with the ground being much larger, the pressure pCT unit area is 
less, and the depth of the groove made is sroaUer ; the ^netgy 
lost by the' wheel sinking into the road is therefore greatly reduced 
by the use of a rubber tyre. " . 

Rolling Resistance of Rubber Tyres. — The resistance to rdHiiig 
of a rubber tyre is of the same nature as that discussed in section 
78, but the amount of compression of the tyre in contact with the 
ground being much greater than in the case of a metal wheel on a 
metal rail, the rolling resistance is also greater. This may appear 
startling to cyclists, but this slight disadvantage of rubber as com- 
pared with steel tyres is more than compensated by the yielding 
quality of the rubber, which practically neutralises the minor 
inequalities of the road surface. 

334. Pnetimatic Tyres in OeneraL— The good qualities of a 
rubber tyre, as Compared with a metal tyre for bicycles, are present 
to a still greater d^^ee -in. pneuriiatic tyres. In a |-inch rubber 
tyre, half of which is usually buried in the rim of the wheel, the 
maximum height of a stone that can be passed over without 
influencing the motion of the wheel as a whole, cannot be much 
greater than a quarter of an inch. With a 2-inch pneumatic tyre^ 
most of which lies outside the rim, a stone i inch high may be 
passed over without influencing the motion of the wheel to 
any great extent, provided the speed is great. The provision 
against loss of energy by impact in moving over a rough road 
is more perfect in this case. Again, the tyre being of larger 
diameter, its surface of contact with the ground is greater, and 
the energy lost by sinking into a road of moderate hardness is 
practically nil. 

Rolling Resistance of Pneumatic T^res, — Considering the tyre as 
a whole to be made of the material * air,' and applying the result 
of section 194, if the material be perfectly elastic, there would 

Digitized by CjOOQIC 



CEAP. znz. 



Tyres. 489 



. be absolutely no rolling resistance. Now for all practical purposes 
air may be considered perfectly elastic, and there will be no dissi- 
pation of energy by the air of the tyre. The indiarubber tube in 
-which the air is confined, and the outer-cover of the tyre, are, how- 
ever, made of materials which are by no means perfectly elastic. 
The work done in bending the forward part of the cover will 
be a little greater than that restored by the cover as it regains its 
original shape. Probably the only appreciable resistance of a 
pneumatic tyre is due to the difference of these two forces. The 
wcmJl expended in bending the tyre will be greater, the greater the 
angle through which it is bent. This angle is least when the tyre 
is pumped up hardest ; and therefore on a smooth racing track 
pneumatic tyres should be pumped up as hard as possible. 

Again, the work required to bend the cover through a given 
angle will depend on its stiffness ; in other words, on its moment 
of resistance to bending. For a tyre of given thickness d this re- 
sistance will be greatest when the tyre is of the single-tube type, and 
other things being equal, will be proportional to the square of the 
thickness d. If the cover could be made of n layers free to slide 

on each other, each of thickness - , the resistance of each layer to 

ft 

d^ 
bending would be proportional to ~ -, and that of the n layers con- 

d^ 
stituting the complete covering to - . Thus for a tyre of given 

n 

thickness its resistance is inversely proportional to the number of 
separate layers composing the cover. This explains why a single- 
tube tyre is slower than one with a separate inner air-tube ; it also 
explains why racing tyres are made with the outer-cover as thin 
as possible. 

Relation beHveen Air Pressure and Weight Supported, — Let a 
pneumatic tyre subjected to air pressure / support a weight W, 
The part of the tyre near the ground will be flattened, as shown 
in figure 515. Let A be the area of contact with the ground, and 
let q be the average pressure per square inch on the ground. 
Then, if we assume that the tyre fabric is perfectly flexible, 
since the part in contact with the ground is quite flat, the 



Digitized by CjOOQIC 



49^ Details chap. xzn. 

pressures on the opposite sides must be equal. Therefore ^ =/. 
But the only external forces acting on the wheel are W and 




Fig. 515. 

the reaction of the ground. These must be equal and opposite, 
therefore 

Ap — W (I) 

Let /o and V^ respectively be the pressure per sq. in., and 
the volume of air inside the tyre, before the weight comes on the 
wheel ; and let / and V be these quantities when the tyre is 
deformed under the weight. The air is slightly compressed ; ue, 
V\s slightly less than V^ and / is a little greater than/o- Now 
the pressure of a given quantity of gas is inversely proportional to 
the volume it occupies ; ix, 

po y ^*^ 

p and/o being absolute pressures. 

Example. — If a weight of 1 20 lbs. be carried by the driving- 
wheel of a bicycle, and the pneumatic tyre while supporting 
the load be pfkmped to an air pressure of 30 lbs. per sq. in. 

120 
above atmosphere, the area of contact with the ground = - - 

= 4 sq. in. 

If the diameter of the wheel be 28 inches and that of the inner 
tube be i|-inch, it would be easy by a method of trial and error 
to find a plane section of the annulus having the area required, 
4 sq. in. If we assume that the part of the tyre not in contact 
with the ground retains its original form, which is strictly true 
except for the sides above the part in contact with the ground, the 
diminution of the volume of air inside the tyre would be the 
volume cut off by this plane section. In the above example this 

Digitized by CjOOQIC 



CHAP. xanx. Tyres 49 1 

decrease is less than i cubic inch. The original volume of air 
is equal to the sectional area of the inner tube multiplied by its 
mean circumference. The area of a i|-inch circle is 2 405 sq. in., 
the. circumference of a circle 26 inches diameter is 81 -68 ins. 

/. To = 2 405 X 8 1 -68 = 196-5 cubic inches. 

Fmay be taken 195*5 cubic inches. Taking the atmospheric 
pressure at 147 lbs. per sq. in., / = 30 -f 147 = 447. Hence, 
substituting in (2) 

p =44_7 — JL95„5 ^ 44*47 lbs. per sq. in. absolute 
1965 

= 2977 lbs. per sq. in. above atmosphere ; 

and therefore the pressure of the air inside the tyre has been 
increased by 0*23 lb. per sq. in. 

335. Air-tube. — The principal function of the air-tube is to 
form an air-tight vessel in which the air under pressure may be 
retained. It should be as thin and as flexible as possible, 
consistent with the necessity of resisting wear caused by slight 
chafing action against the outer-cover. It should also be slightly 
extensible, so as to adapt itself under the air-pressure to the exact 
form of the rim and outer-cover. Indiarubber is the only material 
that has been used for the air-tube. 

Two varieties of air-tubes are in use : the continuous tube 
and the butt-ended tube. The latter can be removed from a com- 
plete outer tube by a hole a few inches in length, while the 
former can only be removed if the outer-cover is in the form of 
a band with two distinct edges. 

336. Outer-cover. — The outer-cover has a variety of functions 
to perform. Firstly, it must be sufficiently strong transversely 
and longitudinally to resist the air-pressure. Secondly, in a 
driving-wheel it must be strong enough to transmit the tangential 
effort from the rim of the wheel to the ground. Thirdly, the 
tread of the tyre should be thick enough to stand the wear and 
tear of riding on the road, and to protect the air-tube from 
puncture. Fourthly, though offering great resistance to elongation 
by 4irect tension, it should be as flexible as possible, offering very 

Digitized by CjOOQIC 



492 Details 

little resistance to bending as it comes into, and leaves, contact 
with the ground, and as it passed over a stone. 

Stress on Fabric, — We have already investigated (sec. 84) 
the tensile stress on a longitudinal section of a pneumatic tyre. 
We will now investigate that on a transverse section. Consider 
a transverse section by a plane passing through the axis of the 
wheel, and therefore cutting the rim at two places. The upper 
part of the tyre is under the action of the internal pressure, 
and the pull of the lower portion at the two sections. If we 
imagine the cut ends of the half-tyre to be stopped by flat 
plates, it is evident that the resultant pressure on the curved 
portion of the half-tyre will be equal and opposite to the re 
sultant pressure on the flat ends. If d and / be respectively 
the diameter and thickness of the outer-cover, and / be the air- 
pressure, the area of each of the flat ends is — , and therefore 

4 

the resultant pressure on the curved surface is 2 -^*/. 

The area of the two transverse sections of the outer-cover is 
2irdt; therefore the stress on the transverse section is 

2-d^p . , 
/=_4_=/^ (3) 

Comparing with section 84, the stress on a transverse section 

of the fabric is half that on a 
longitudinal section. 

Spiral JFibres,— The first 
pneumatic tyres were made with 
canvas having the fibres run- 
ning transversely and circum- 
ferentially (fig. 516). The 
fibres of a woven fabric, intermeshing with each other, are not 
quite straight, and offer resistance to bending as it comes into 
and leaves contact with the ground. Further, when the fibres 
are disposed transversely and circumferentially the cover cannot 
transmit any driving effort from the rim of the wheel to th^ 

Digitized by CjOOQIC 




Tyres 



493 



ground, until it has been distorted through a considerable angle, 
as shown by the dotted lines. 

In the * Palmer ' tyre the fabric is made up of parallel fibres 
embedded in a thin layer of indiarubber, the fibres being wound 




Fig. 517, 



spirally (fig. 517) round an inner tube. Two layers of this fabric 
are used, the two sets of spirals being oppositely directed. When 
a driving eflfort is being exerted, the portion of the tyre between the 
ground and the rim is subjected to a shear parallel to the ground, 
which is, of course, accompanied by a shear on a vertical plane. 
This shearing stress is equivalent 
to a tensile stress in the direc- 
tion cc (fig. s 1 8), and a compres- 
sive stress in the direction dd 
(see sec. 105) ; consequently the 
fabric with spiral fibres is much 
better able to transmit the driv- 
ing effort from the rim to the 
ground. This construction is 
undoubtedly the best for driving-wheel tyres; but in a non-driving 
wheel practically no tangential or shearing stress is exerted on the 
fabric of the tyre. Therefore, for a non-driving wheel the best 
arrangement is, possibly, to have the fibres running transversely 
and longitudinally ; the brake should then be applied only to the 
driving-wheel. 

The tyre with soirally^arranged fibres has another curious 

Digitized by CjOOQIC 




^/////////, 



494 DBtaib 



cHAP» znx. 



property. It has been shown that the tensile stress on the trans- 
verse section bb oi the tyre is half thait on the longitudinal 
section a a. Let the stress on the section ^ ^ be denoted by /, 
that on a a by 2/. This state of stress is equivalent to two 
simultaneously acting states of stress : the first, equal tensile 

stresses ^^ on both sections ; the second, a tension ^ onaa, 
2 2 

and a compression ^ on bb. The first system of stress tends to 
stretch the fibre equally in all directions ; the second state of 

stress is equivalent to shearing stresses ^on the planes cc and 

2 

dd parallel to the spiral fibres. If the tyre be inflated free from 

the rim of the wheel, the fabric cannot resist the distortion due 

to this shearing stress, so that the tension ±~ on the section a a 

2 

tends to increase the size of the transverse section of the tyre, 

and the compression ^ on bb tends to shorten the circumference 

of the tyre. Thus, finally, the act of inflation tends to tighten the 
tyre on the rim. 

337. Classification of Pneimiatie Tyres. — Pneumatic tyres 
have been subdivided into two great classes : Single-tube tyres, 
in which an endless tube is made air-tight, and sufficiently strong 
to resist the air-pressure ; Compound tyres, consisting of two 
parts — an inner air-tube and an outer-cover. Quite recently, a 
new type, the * Fleuss ' tubeless tyre, has appeared. Mr. Henry 
Sturmey, in an article on * Pneumatic Tjrres' in the 'Cyclist's 
Year Book* for 1894, divides compound tyres into five classes, 
according to the mode of adjustment of the outer-cover to the 
rim, viz. : Solutioned tyres. Wired tyres. Interlocking and Infla 
tion-held tyres, Laced tyres, and Band-held tyres. 

A better classification, which does not differ essentially from 
the above, seems to be into three classes, taking account of the 
method of forming the chamber containing the compressed air, 
as fellows : 

Class I., with complete tubular outer-covers. This would 
include all single-tube tyres, most solutioned tyres, and some 

Digitized by CjOOQIC 



CHAP. XXIX. 



Tyres 495 



laced tyres. Tyres of this class can be inflated when detached 
from the rim of the wheel ; in fact, the rim is not an integral 
part of the tyre, as in the two following classes. This class may 
be referred to as Tubular tyres. 

Class II., in which the transverse tension on the outer-cover 
is transmitted to the edges of the rim, so that the outer-cover and 
rim form one continuous tubular ring subjected to internal air- 
pressure. The 'Clincher' tyre is the typical representative of 
this class. With most tyres of this class the compression on the 
rim due to the pull of the spokes is reduced on inflation. This 
class will be referred to as Interlocking tyres. 

Class III., in which the transverse tension on the outer-cover 
is transmitted to the edges of the latter, and there resisted by the 
longitudinal tension of wires embedded in the cover. This class 
includes most wired tyres. With tyres of this class the initial 
compression on the rim is increased on inflation. This class will 
be referred to as Wired tyres, and may be subdivided into two 
sections, according as the wire is endless, or provided with means 
for bringing the two ends together, and so adjusting the wire on 
the rim. 

338. Tubular Tyres. — Single-tube tyresy which form an im- 
portant group in this class, are made up of an outer layer of rubber 
forming the tread which comes in contact with the ground, a 
middle layer of canvas, or other suitable material, to provide the 
necessary strength and inextensibility, and an inner air-tight layer 
of rubber. The * Boothroyd ' and the * Silvertown ' were among 
the most successful of these single-tube tyres. The * Palmer ' 
tyre (fig. 517) was originally made as a single-tube. 

Since a solid plate of given thickness offers more resistance to 
bending than two separate plates having the same total thickness, 
the resilience of a tyre is decreased by cementing the air- tube 
and outer-cover together. 

Solutioned tyres, — The original *Dunlop' tyre (fig. 519), which 
was the originator of the principle of air tyres for cycles, belongs 
to this class. The outer-cover consists of a thick tread of rubber 
A solutioned to a canvas strip B, A complete woven tube of 
canvas ZT, encircles the air-tube C, and is solutioned to the rim 
E^ which is previously wrapped round by a canvas strip D ; while 

Digitized by CjOOQIC 



496 



Details 



CSUl2, ttit 



the flaps of the outer-cover are solutioned to the inner surface of 
the rim, one flap being lapped over the other, the side' being slit 
to pass the spokes. A strip of canvas F, solutioned over the flaps, 
makes a neat finish. 

In the Morgan and Wright tyre, the air-tube is butt-ended, or 
rather scarf-ended, the two ends overlapping each other about 




Fig. 520U 



eight or ten inches. The outer-cover forms practically a tube slit 
for a few inches along its under side ; this opening serves for the 
insertion of the air-tube, and is laced up when the air-tube is in 
place. When partially inflated the tyre is cemented on to 
the rim. 

Laced tyres, — In Smith's * Balloon ' tyre (fig. 520) the 
outer-cover was furnished with stud hooks at its edges, and 
enveloped the rim completely; its two edges were then laced 
together. 

339. Interlocking Tyres.— In this class of tyres the circum- 
ferential tension near the edge of the outer-cover is transmitted 
direct to the rim of the wheel, by suitably formed ridges, which 
on inflation are forced into and held in corresponding recesses of 
the rim. 

Inflation-held Tyres. — In tyres which depend primarily on 
inflation for the fastening to the rim, the edgd of the outer-cover 
is continued inwards forming a toe beyond the ridge or heel, the 

Digitized by CjOOQIC 



.CXIAF. ZXIX. 



Tyres 



497 



air-pressure on the toe keeping the heel of the outer-cover in close 

contact with the recess of the rim. 

The 'Clincher' tyre was the first 
of this type. The * Palmer ' detachable 
tyre (fig. 521), so 'far as regards the 
fastening of the outer-cover to the rim, 
is identical with the * Clincher.' 





Fig. SSI. 



Fig. 522. 



The * Decourdemanche ' tyre (fig. 522) is of the 'Clincher' 
type, but it has a wedge thickening JV on the inner part of the 
air-tube, which on inflation is pressed between the ridges T of the 
outer-cover, and forces them into the recesses of the rim. 

77^ * Swiftsure ' tyre differs essentially from those previously 
described. The outer-cover is furnished at the edges with circular 
ridges which lie in a central deep narrow-mouthed groove of the 
rim. The mouth of the groove: is just large enough to admit the 
ridge of the cover, while the body of the groove is wide enough 
to let them lie side by side. On inflation, the tendency is to draw 
both ridges from the groove together, so that they lock each other 
at the mouth, and thus the tyre is held on the rim. 

Hook-tyres, — In this subdivision the positive fastening of the 
outer-cover to the rim does not depend merely on inflation ; but 
the pull of the cover can be transmitted to the rim in the proper 
direction, even though there be no pressure in the air-tube. 

Digitized by Gofe^le 



498 



Details 



xnuLT.ixa. 



In the original * Preston-Davies ' tyre eye-holes were formed 
near the edges of the outer-cover ; these were threaded on hooks 
turned slightly inwards, so that on inflation the cover was held 
securely to the rim. 

The ' Grappler ' tyre is a successful modern example of this 
same class. Near each edge of the outer-cover a series of turned- 
back hooks or grapplers are fastened. These engage with the in- 
turned edge of the rim, so that on inflation the tyre is securely 
fastened. 

Band-held tyres. — In the * Humber ' pneumatic tyre (fig. 523) 
the outer-cover A is held down on the rim D by means of a lock- 
ing plate C on which the air-hibe B rests. 




Fig. 593. 

In the * Woodley ' tyre (fig. 524) it is possible that the flap acts 
in somewhat the same way as the plate in the * Humber ' tyre. 

The * Fleuss ' tubeless tyre (fig. 525) is fixed to 
the rim on the * Clincher ' principle. The inner 
surface of the tyre is made air-tight, and thus a 
separate air-tube is dispensed with. A flap, per- 
manently fastened to one edge of the tyre, is 
pressed on the other edge, when inflation is com- 
pleted. The difficulty of keeping an air-tight joint 
between this loose flap and the edge of the tyre, 
right round the circumference (a length of over six feet), has been 
successfully overcome. 

Digitized by CjOOQIC 




Fig. 524. 



CHAP. xnx. 



Tyres 



499 



340. Wire-held Tyres. — The mode of fastening to the rim, used 
in this class of pneumatic tyre, differs essentially from that used 

in the other classes. 
Wires IV {fig. 526) are 
embedded near the 
edges of the outer- 
cover C. On infla- 




FiG. 525. 



Fig. sa6. 



tion, a transverse tension T is exerted on the outer-cover, and 
transmitted to the wire IV, tending to p'll it out of the rim ^. 
The wire is also pressed against the rim. the reaction from which 
JV is at right angles to the surface. The resultant J^ of the forces 
T and JV must lie in the plane of the wire AF, and constitutes a 
radial outward force acting at all points of the ring formed by 
the wire. Thus, the chamber containing the air under pressure 
is formed of two portions : the outer-cover, subjected to tension T; 
and the rim ^ subjected to bending by the pressures JV exerted 
by the wires IV. 

Let d be the diameter of the air-tube (not shown in figure 526), 
Z> the diameter of the ring formed by the wire IV, and / the air- 
pressure. Then, by (7) chap, x., the force T per inch length of 

the wire is ^ . The force ^ will be greater than T, depending 
2 

on the angle between them. In the * Dunlop ' detachable tyre, 
this angle is about 30°, and therefore J^ = i'i55 T, The longitu- 
dinal pull P on the wire IV is, by another application of the same 
formula, 



P=n55j:^=.,S9pdB 



(4) 



Digitized by VjO(J>^@ 



Soo Details chap. xm. 

Example, — A pneumatic tyre with air-tube i| in. diameter is 
fixed by wires forming rings 24 ins. diameter \ and has an air 
pressure of 30 lbs. per sq. in. ; the pull on each wire is therefore 
•289 X 30 X 175 X 24 = 364 lbs. 

If the wire be No. 14 W. G. its sectional area (Table XII.), 
p. 346, is '00503 sq. in., and the tensile stress is 

3-i = 72300 lbs. per sq. in. 
or 32-3 tons per sq. in. 

Wire-held tyres may be sub-divided into two classes ; in one 
the wire is in the form of an endless ring, and is therefore non- 
adjustable, in the other the ends of the wire are fastened by 
suitable mechanism, so that it can be tightened or released at 
pleasure. 

The 'Dunlop' Detachable Tyre (fig. 527) is the principal 
representative of the endless wired division. In it two endless 

wires are embedded near the 
edges of the outer-cover. 
These wires form rings of less 
diameter than the extreme 
diameter of the rim, and are 
lodged in suitable recesses of 
the rim. The rim is deeper 
at the middle than at the re- 
cesses for the w^ire. To detach 
the tyre, after deflation, one part of one edge of the outer-cover is 
depressed into the bottom of the rim, the opposite part of the 
same edge will be just able to surmount the rim, and one part of 
the wire being got outside the rest will soon follow. 

The * Woodley ' tyre (fig. 524) is formed from the ' Dunlop ' by 
adding a flap to the outer-cover, this flap extending from one of 
the main fixing wires to the other, and so protecting the air-tube 
from contact with the rim. 

In the original * Beeston ' tyre this flap was extended so far as 
to completely envelop the air-tube. In the newer patterns this 
wrapping has been discarded, and the * Beeston ' is practically the 
same as the * Dunlop ' detachable. 

Digitized by CjOOQIC 




CHAP. XZIZ. 



Tyres 



SOI 



The ' 1895 Speed ' tyre, made by the Preston-Davies Valve and 
Tyre Company, is fixed to the rim by means of a continuous wire 
of three coils on each side. At each side of the tyre a complete 
coil is enclosed in a pocket near the edge of the outer-cover ; one 
half of each of the other coils is outside, and the remaining halves 
inside, the pocket. By this device a wire, composed of two half 
coils, is exposed all round between the cover and the rim. When 
the tyre is deflated this exposed wire can easily be pulled up with 
the fingers, the detached coil is then brought over the edge of the 
rim, more of the slack pushed back into the pocket, enlarging the 
other coils, whereupon the outer-cover can be removed from the 
rim. 

Tyrts with Adjustable Wires, — The * 1894 Preston-Davies ' 
tyre was attached to the rim by means of a wire running through 
the edge of the outer-cover, one end of the wire having a knob 
which fitted into a corresponding slot in the rim, the other end 
having a screwed pin attached to the wire by an inch or two of a 
very small specially made chain. This chain was introduced to take 
the sharp bend where the adjusting nut drew up the slack of the 
wire in tightening it upon the rim. 

In the * Scottish ' tyre (fig. 528) the ends of the adjustable wire 
are brought together by a right- and left-handed screw. A short 
wire, terminating in a 
loop, forms a handle for 
turning the screw. When 
in position this handle 
fits between the rim and 




outer-cover. 

The * Seddon ' pneu- 
matic tyre was the first 
successful wired tyre. 
Figure 529 is a view 
showing a portion of the ^^^* ^^^' 

tyre with the fastening released. The ends of the wire were 
secured by means of a small screw which was passed through the 
rim and locked in place by a nut. The ends of the wire were 
pulled together by means of a special screw wrench. 

In the ' Michelin ' tyre the wires are of square tubular section. 

Digitized by V^jOOQ 



502 



Details 



CHAP. xzn. 



The outer-cover, which is very deep, is provided at its edges with 
thick beads turned outwards, and each rested in the grooves of the 
specially-formed rim. A tubular wire is placed round these beads 




Fig. 529. 

and its ends are secured in notches cut in the rim, a T bolt and 
screw securing the ends in position. 

In the * Drayton ' tyre (fig. 530) the wires are tightened on the 
rim by a screw-and-toggle-joint arrangement. 




Fig. 530. 



341. Devices for Preventing, and Minimising the Effisct of^ 
Punctures. — In the * Silvertown Self-closure ' tyre, which was of 
tlie single-tube variety, a semi-liquid solution of rubber was left on 
the inner surface of the tyre. When a small puncture was made, 
the internal pressure forced some of the solution into the hole, and 
the solvent evaporating, the puncture was automatically repaired. 

Digitized by VjOOQ 



CHA». x:zix. 



Tyres 



S03 




In the * Macintosh ' tyre a section of the air-tube when deflated 
took the form shown in figure 531. On inflation the part of the 
air-tube at S was strongly compressed, so that 
if a puncture took place the elasticity of the 
indiarubber and the internal pressure com- 
bined to close up the hole. 

In the * Self-healing Air-Chamber ' the same 
principle is made use of; the tread of an 
ordinary air-tube is lined inside with a layer 
of vulcanised indiarubber contracted in every 
direction. When the chamber is punctured on 
the tread, the lining of contracted indiarubber expands and fills 
up the hole, so preventing the escape of air. 

In the * Preston-Davies ' tyre a double air-chamber with a 
separate valve to each was used. If puncture of one chamber 
took place it was deflated and the second chamber brought into 
use. 

In the 'Morgan and Wright Quick-repair Tyre ' (fig. 532) the 
air-tub^ is provided with a continuous patching ply, which normally 
rests in contact with that portion near to the 
rim. To repair a puncture a cement nozzle 
is introduced through the outer casing and 





Fig. 53a. 



Fig. sy\. 



tread of the air-tube (fig. 533), and a small quantity of cement 
is left between the tread and the patching ply. On pressing down 
the tread the patching ply is cemented over the hole, and the tyre 
is ready for use as soon as the cement has hardened. 

A punctured air-tube is usually repaired on the outside, so that 
the air pressure tends to blow away the patch. In the * Fleuss ' 

Digitized by V^jOOQ 



504 



Details 



CHAP.'ZXIZ. 



tubeless tyre, on the other hand, a puncture is repaired from the 
inside, the tyre can be pumped up hard immediately, and the air 
pressure presses the patch closely against the sides of the hole. 

342. Non-slipping Coven have projections from the smooth 
tread that penetrate thin mud and get actual contact with the 
solid ground (see sec. 170). These projections have been made 
diamond-shaped and oat-shaped, in the form of transverse bands, 
longitudinal bands (fig. 521), and interrupted longitudinal bands 
(fig. 527). They should offer resistance to circumferential as well 
as to side-slipping, though the latter should be the greater. 
Probably, therefore, the oat-shaped projections and the interrupted 
bands (fig. 527) are better than continuous longitudinal bands, 
and the latter in turn better than transverse bands. 

343. Pomps and Valves. — Figure 534 shows diagrammatically 
the pump used for forcing the air into the tyre. The pump 
barrel ^ is a long tube closed at one 

end, and having a gland G screwed ^^^^^^^N ^ ^ ..^^^ 

on to the other, through which a tubular ' ^^v. .^^ 

plunger P works loosely. To the | C 

inner end of the plunger a cup-leather 

Z is fastened. When the air-pressure 

in the inner part B^ of the barrel is 

greater than in the outer part B^ the 

edge of the cup-leather is pressed firmly 

against the sides of the barrel ; but 

when the pressure in the space B^ is 

less than in the space B the cup leather 

leaves the sides of the barrel and 

allows the air to flow past it from B 

into B^, A valve V at the inner end 

of the plunger allows the air to flow 

from B^ through the hollow plunger 

and connecting tube to the tyre, but 

closes the opening immediately the 

air tends to flow in the opposite direction. , The action is as 

follows : The plunger being at the bottom, and just banning 

the outward stroke, the volume B^ is enlarged, the air-pressure in 

B^ falls, and the valve Fis closed by the air-pressure in the hollow 




Fig. 534. 



CHAP.-JJJJL 



Tyres 



505. 



plunger (the same as that in the tyre). The outward stroke of the 
plunger continuing, a partial vacuum is formed in ^*, the cpp- 
leather leaves the sides of the barrel, and air passes from the space 
^ to space B^^ until the outward stroke is completed. On 
beginning the inward stroke, the air in B^ is compressed, forcing 
the edge of the cup-leather against the sides of the barrel, and so 
preventing any air escaping. The inward stroke continuing, the 





Fig. 535. 



Fig. 536. 



air in B^ is compressed until its pressure reaches that of the air in 
the tyre, the valve V is lifted, and the air passes from B^^ along 
the hollow plunger, into the tyre. At the same time a partial 
vacuum is formed in the space B^ and air passes into this space 
through the opening left between the plunger and the gland G, 

A valve is always attached to the stem of the air-tube, so as to 
give connection, when required, between the pump and the 
interior of the tyre. A non-return valve is the most convenient, 
i,e, one which allows air to pass into the tyre when the pressure in 

Digitized by V^jOOQ 



506 Details chap. zzn. 

the pump is greater than that in the tyre, and does not allow the 
air to pass out of the tyre. In the * Dunlop ' valve (fig. 535) the 
valve proper is a piece of indiarubber tube /, resting tightly on a 
cylindrical * air-plug/ K. The air from the pump passes from the 
outside down the centre of the air-plug, out sideways at /, then 
between the air-plug and indiarubber tube / to the inside of the 
air-tube A of the tyre. Immediately the pressure of the pump 
is relaxed, the indiarubber tube / fits again tightly on the air-plug 
and closes the air-hole /, By unscrewing the large cap M, the 
tyre may be deflated. 

Wood rims are seriously weakened by the comparatively large 
hole necessary for the valve-body B, Figure 536 shows a valve 
fitting, designed by the author, in which the smallest possible hole 
is required to be drilled through the rim. 



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507 



CHAPTER XXX 



PEDALS, CRANKS, AND BOTTOM BRACKETS 

344. Pedals. — Figure 537 shows the ball rubber pedal, as made 
by Mr. William Bown, in ordinary use up to a year or two ago. 
The thick end of the pin is passed through the eye of the crank 
and secured by a nut 
on the inner side of the 
crank. The pedal-pin 
is exposed along nearly 
its whole length, there 
are therefore four places 
at which dust may enter, 
or oil escape from, the 
ball-bearings. 

If the two pedal- 
plates be connected by a tube, a considerable improvement is 
effected, the pedal-pin being enclosed ; while if in addition a dust 
cap be placed over the adjusting cone at the end of the spindle. 




Fig. 537. 




Fig. 538. 



Digitized by CjOOQIC 



So8 



Details 



CBAT. 



there is only one place at which dust may enter or oil escape from 
the bearings. 

Figure 538 illustrates the pedal made by the Cycle Components 
Manufacturing Company, Limited, in which there are only three 

pieces, viz. : the pedal 
frame, pin, and adjust- 
ment cone. The ad- 
justment cone is screwed 
on the crank end of 
the pedal-pin, a portion 
of the cone is screwed 
on the outside and 
split The cone is then 
screwed into the eye of 
the crank, the pedal- 
pin adjusted by means 
of a screw-driver ap- 
plied at its outer end ; 
^then, by tightening up 
the clamping screw in 
the end of the crank, 
the crank, pedal-pin, and 
adjustment cone are se- 
curely locked together. 
The * Centaur' pedal (fig. 539) differs essentially from the 

others ; the arrangement is 
such that an oil-bath is pos- 
sible for the balls, whereas in 
the usual form of pedal the 
oil drains out of the ball- 
bearings. 

Recently, a number of 
new designs for pedals have 
been placed on the market, 
of which the *^olus Butter- 
Ay ' (fig- 54o)> by William 
Bown, Limited, and that (fig. 541) by the Warwick and Stockton 
Company, Newark, U.S.A., may be noticed. 

Digitized by CjOOQIC 




Fig. 54a 




Fig. 541. 



CHAP. XXX. Pedals^ Cranks^ and Bottom Brackets 509 

345. Pedal-pins. — ^The pedal-pin is rigidly fixed to the end 
of the crank ; it may therefore be treated as a cantilever 
(fig. 542) supporting a load Py the pressure of the rider's 
foot. This load comes on at 
two places, the two rows of balls. [^ [^ . _ y _ It 

One of these rows is close to ■*»« *- L' 

the shoulder of the pin abutting 
against the crank, the other is near 
the extreme end of the pin. At 
any section between the balls and 

distant x from the outer row the bending moment '\%\P x. \{ d 
be the diameter of the pin at this section, and / the maximum 
stress on the material, we have, substituting in the formula M 

That is, for equal strength throughout, the outline of the pedal-pin 
should be a cubical parabola. On any section between the 
shoulder and the inner row of balls, and distant y from the centre 
of the pedal, the bending moment will be P y. 

It will in general be sufficient to determine the section of the 
pin at the shoulder and taper it outwards. 

Example, — If /*= 150 lbs., /= 20,000 lbs. per sq. in., and 
the distance of P from the shoulder be 2 in., then 

J/ss 150 X 2 = 300 inch-lbs. Z= -^ — = '015 in.^ 

20,000 

From Table III., p. 109, d = ^\ in. 

346. Cranks. — Figure 543 is a diagrammatic view showing the 
crank-axle a, crank <r, and pedal-pin ;>, the latter being acted on by 
the force P at right angles to the plane of the pedal-pin and 
crank. Introduce the equal and opposite forces P^ and P^^ at the 
outer end of the crank, and the equal and opposite forces P^ and 
P^ at its inner end ; /*i, P^^, P^, and 7*4 being each numerically 
equal to P, The forces P and /*, constitute a twisting couple T 
of magnitude Plx^ acting on the crank, l^ being the distance of P 
from the crank. The forces P^ and P^ constitute a bending couple 
My of magnitude P I 2X the boss of the crank. The force P^ 

Digitized by CjOOQIC 



5IO 



Details 



CHAP. TIT 



■P\ 



>S 



I' 



causes pressure of the crank-axle on its bearings. Thus tbe 

original force P is equivalent to the equal force P^^ a twisting 

couple P /|, and a bending couple P L No 

motion takes place along the line of action of 

7^4, nor about the axis of the twisting couple 

P /i, the only work done is therefore due to 

the bending couple P L 

At any section of the crank distant x 
from its outer end, the bending-moment is 
P X, The equivalent twisting-moment 7^ 
which would produce the same maximum '■**" 
stress as the actual bending- and twisting- 
moments J/ and T acting simultaneously, is 
given by the formula T^ ^ M + VHf^ -|- T^. 
equivalent bending-moment 



Fig. 543- 



Similarly, the 



Example. — If /, = 2} in., 7=6^ in., P = 150 lbs., and / = 
20,000 lbs. per sq. in. 

M =- 150 X 6| = 975 inch-lbs., 7^= 150 x 2^ = 337 inch-lbs. 

Then 

Tg = 2007 inch-lbs., oi M^^^ 1003 inch-lbs. 

Then 

z = ", == ^- s= '0501 m.' 



/ 



20,000 



From Table III., p. 109, the diameter of a round crank at its 
larger end should be -j^ in. 

If the cranks are rectangular, and assuming that an equivalent 
bending-moment is 1000 inch-lbs., we get 

y _ bjl^ _ 1,000 

6 20,000 

.-. dh'^ =^ -30. 

If ^ = i ^, Le, the depth of the crank be twice its thickness, 
we get i i4* = -30, h^ = -60, and 

h = -843 in., ^ = '421 in. 

Digitized by CjOOQIC 



CHAP. ZXZ. 



Pedals, Cranks, and Bottom Brackets 5 1 1 



The cranlcs were at first fastened to the axle by means of a 
rectangular key, half sunk into the axle and half projecting into the 
boss of the crank. A properly fitted and driven key gave a very 
secure fastening, which, however, was very difficult to take apart, 
and detachable cranks are now almost invariably used. Perhaps 
the most common form of detachable crank is that illustrated in 
figure 544. The crank boss is drilled to fit the axle, and a conical 




Fig. 544. 

pin or cotter, flattened on one side, is passed through the crank 
boss and bears against a corresponding flat cut on the axle. The 
cotter is driven tight by a hammer, and secured in position by a 
nut screwed on its smaller end. 

In the * Premier ' detachable crank made by Messrs. W. A. 
Lloyd & Co. a flat is formed on the end of the axle, and the hole 
in the crank boss made to suit. The crank boss is split, and on 
being slipped on the axle end is tightened by a bolt passing 
through it. 




Fig. 545. 



Figure 545 illustrates the detachable chain-wheel and crank 
made by the Cycle Components Manufacturing Company. A 

Digitized by CjOOQIC 



SI2 



Details 



au:p. XXX. 



long boss is made on the chain-wheel, over which the crank boss 
fits. Both bosses are split and are clamped to the axle by means 
of a screw passing through the crank boss. In addition to the 
frictional grip thus obtained, a positive connection is got by means 
of a small steel plate, applied at the end of the crank-axle and 
wheel boss, and retained in position by the clamping-screw. The 
pedal end of this crank is illustrated in figure 538. 

The * Southard' crank, which is round-bodied (fig. 544), 
receives during manufacture an initial twist in the direction of the 
twisting-moment due to the pressure on the pedal in driving ahead. 
The elastic limit of the material is thus artificially raised, the 
crank is strengthened for driving ahead, but weakened for back 
pedalling ; as already discussed in section 123. 

In the * Centaur * detachable crank and chain-wheel, the crank 
boss is placed over the chain-wheel boss. Both wheel and crank 
are fixed to the axle by a tapered cotter, driven tight through the 
bosses and retained in position by a nut. 

It has been shown that near the boss of the crank the bending- 
moment is greater than the twisting-moment. Round-bodied 
cranks have the best form to resist twisting, rectangular-bodied to 
resist bending. A crank rectangular towards the boss and round 
towards the eye would probably be the best. Hollow cranks of 
equal strength would of course be theoretically lighter than solid 
cranks, but the difficulty of attaching them firnily to the axle has 
prevented them being used to any great extent. In some of the 

early loop-framed tricycles, the axle, 
cranks, and pedal-pins were made of 
a single piece of tubing. 

347. Crank-axle.— Figure 546 is 
a sketch showing part of the crank- 
axle /z, the crank and pedal-pin, the 
latter acted on by the force P, Intro- 
duce two equal and opposite forces P^ 
and P4 at the bearing -4, and two equal 
and opposite forces P, and P^ at a 
point B on the axis of the crank-axle, 
the forces Py 7>„ and P^ lying in a plane parallel to thexrank and 
at right angles to the crank-axle. The forces P and P^ constitute a 

Digitized by CjOOQIC 




CHAP. XXX. 



Pedals^ Cranks^ and Bottom Brackets 513 



twisting couple of magnitude /'/acting on the axle. This twist- 
ing-moment is constant on the portion of the axle between the 
crank and the chain-wheel. The forces P^ and P^ constitute a 
bending couple of magnitude P/2 at the point Ay l^ being the dis- 
tance from the bearing to the middle of the pedal, measured 
parallel to the axis. The force P^ produces a pressure on the 
bearing at A, 

Example, — If l^ be 3 ins., the other dimensions being as in the 
previous examples, il/'= 150x3 = 450 inch-lbs., 7^= 150 x6| 
= 975 inch-lbs., 7^ = 1524 inch-lbs., J^ = 762 inch-lbs. The dia- 
meter d of the axle will be obtained by substituting in formula 
(15), chap, xii., thus : ^ 

— X 20,000= 1524 

d^ = -381, //= 725, say I in. 

If the axle be tubular, Z = ' = 0381. 

20,000 

From Table IV., p. 112, a tube \ in. external diameter, 13 W. G., 
will be sufficient. 

Comparing the hollow and solid axles, their sectional areas are 
•226 and '442 square inches respectively ; thus by increasing the 
external diameter \ inch and hollowing out the axle its weight 
may be reduced by one half ; while, if the external diameter be 
increased to i in., from Table IV., p. 112, a tube 16 W. G. will 
be sufficient, the sectional area being -188 square inches ; less than 
43 per cent, of that of the solid axle. 

In riding ahead the maximum stresses on the axle, crank, and 
pedal-pins vary from zero, during the up-stroke of the pedal, to the 
maximum value / If back-pedalling be indulged in, the range 
of stresses will be from -f / to — / The dimensions of the axle 
and crank above obtained by taking /= 20,000 lbs. per sq. in. 
are a little greater than those obtaining in ordinary practice. A 
total range of stress of 40,000 lbs. per sq. in. is very high, and 
cranks or axles subjected to it may be expected to break after a few 
years' working, unless they are made of steel of very good quality. 
It may be pointed out here that a pedal thrust of 150 lbs. will not 

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SH 



Details 



be exerted continuously even in hard riding, though it may be 
exceeded in mounting by, and dismounting from, the pedal. 




Fig. 547. 



348. Crank-brackets.— The bracket and bearings for support- 
ing the crank-axle form a kinematic inversion of the bearing shown 
in figure 404 ; the outer portion ^ forming the bracket is fastened 




Fig. 548. 



to the frame of the machine, while the spindle S becomes the 
crank-axle, to the ends of which the cranks are fastened. In the 

Digitized by CjOOQIC 



CHAP. XXX. Pedals^ Cranks^ and Bottotn Brackets 515 



earlier patterns of crank-brackets, hard steel cups D were forced 
into the ends of the bracket, and cones C were screwed on the 
axle, the adjusting cone being fixed in position by a lock nut. 

The barrel bottom-bracket is now more generally adopted ; 
being oil-retaining and more nearly dust-proof, it is to be preferred 
to the older pattern. The axle ball-races are fixed, and the adjust- 
able ball-race can be moved along the bracket. In the * Centaur ' 
crank-bracket (fig. 547) the bearing discs or cups are screwed to 
the bracket, and secured by lock nuts. In the * R. F. Hall ' bracket 
(fig. 548) one cup is fastened to the bracket by a pin, and the other 
is adjusted by means of a stud screwed to the cup and working 
in a diagonal slot cut in the bracket. The pitch of this slot is so 
coarse that the adjustment is performed by pushing the stud for- 
ward as far as it will go, it being impossible to adjust too tightly. 
The cup is then clamped in place by the external screwed pin. 

349. The Frestnre on Crank-axle bearings is the resultant of 
the thrust on the pedals and the pull of the chain. 

Example, — Taking the rows of balls 3^ ins. apart, and the 
rest of the data as in the example of section 238, and considering 
first the vertical components due to the pressure P on the pedals, 
the condition of affairs is represented by figure 549. Taking 
moments about by we get 

z\Px = 3^3, .-. Pz = ^^ X 150 = 1607 lbs. 

35 

In the same way, taking moments about r, we find P^^ = 3107 lbs. 



zi 



^j i* W--"-> 



-Ji 



P, 



-j|-»j 



3i ^^i 



Fig. 549. Fig. 550. 

Consider now the horizontal forces. Fig. 550 represents the 
condition of affairs ; /^„ F^y F^ being respectively the horizontal 



5i6 



Details 



CHAP. XZX. 



components of the pull of the chain and of the pressure on the 
bearings. Taking moments about b^ we get 

IF, = z\F^, therefore, F^ = *^75 ^340 = 36-4 lbs. 

In the same way, taking the moments about r, we find 
^2 = 376-4 lbs. 



>S 




Fig. 551. 



4^ 



Fig. 55a. 



The resultant pressures Rx, and R^ on the bearings b and c can 
be found graphically as shown in figures 551 and 552, or by cal- 
culation, thus : 



R^^^P^^F^^ ^3112+376^ = 488 lbs. 



J?,= >/:^^5q:?i«= v/i6i^ + 36> =i65lbs. 



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517 



Y//a////////u/u^////a^ 



CHAPTER XXXI 

SPRINGS AND SADDLES 

350. Springs under the Action of suddenly Applied Load. — 

We have already seen (sec. 82) that when a load is applied at the 
end of a long bar, the bar is stretched, and a definite amount of 
work is done. If the load be not too great, such a solid bar of 
iron or steel forms a perfect spring. If a greater ■ extension be 
required for a given load, instead of a cylindrical bar a spiral 
spring is used. The relation between the steady load and the 
extension of a spiral spring is expressed by an equation similar to 
(2), chap. X., and the stress-strain curve is, as in figure 74, a 
straight line inclined to the axis of the spring. 

Let a spiral spring be fixed at one end with its axis vertical 
(fig. 553), and let A^ be the position of its free end when support- 
ing no load. Let ^, be the position of 
the free end when supporting a load W^ 
the ordinate A^ P^ being equal to W^ to 
a convenient scale. Let A^ Py P^ P^ be 
the stress-strain curve of the spring. 
When this spring is supporting steadily 
the load IVy tet an extra load w be sud- 
denly applied. The end of the spring 
when supporting the load W -{■ w will be 
in the position A^- The work done by 
the loads in descending from A^ to A.^ 
is (IV -}- w) Xy and is graphically repre- 
sented by the area of the rectangle 
Ay A^P^py, The work done in stretch- ^'c- 553. 

ing the spring is Wx ■\- \w x^ and is represented by the area 
Ax A2 P2 Pi' 

Digitized by CjOOQIC 




5 1 8 Details chap. xm. 

The difference of the quantities of work done by the falling 
weight and in stretching the bar is \wxj and is graphically repre- 
sented by the triangle P\px P^. In the position A^oi the end of 
the spring, this exists as kinetic energy, so that in this position the 
load must be still descending with appreciable speed. The spring 
continues to stretch until its end reaches a point A^ where it 
comes to rest and then begins to contract. At the position of 
rest -^3, the work done by the loads in falling the distance Ay A^ 
must be equal to the work done in stretching the spring, since no 
kinetic energy exists in the position A^. Therefore, area 
^1 ^zPzPx == area A^ A^P^ P^. 

It is easily seen that this is equivalent to saying that the 
triangles P^p\P\ and P^p^ P^ are equal, and therefore j^ = x ; 
i,e, a load suddenly applied to a spring will stretch it twice as 
much as the same load applied gradually. 

In the position ^j, the tension on the spring is greater than 
the load supported, and therefore the spring begins to contract 
and raise the load. If the spring had no internal friction it would 
contract as far as the original f)osition ^„ and continue vibrating 
with simple harmonic motion between Ax and A^ ; but owing to 
internal friction of the molecules (or hysteresis) the spring will 
ultimately come to rest in the position of equilibrium A^^ and 
therefore the work lost internally is 

PxPxP^^\wx (i) 

For a stiff spring the slope of A^^Px PiP^ is great, i.e. the 
extension x corresponding to a load w is small, and therefore the 
work lost is also small. For a weak spring the slope A^p\% small, 
and for a given load w the extension x^ and therefore work lost, 
is large. But for i given extension x the work lost with a stiff 
spring is greater than with a weak spring. 

351. Spring Supporting Wheel. — The function of a spring 
supporting the frame of a vehicle from the axle of a rolling 
wheel is to allow the frame to move along in a horizontal line 
without partaking of any vertical motion due to the inequalities 
of the road. This ideal motion would be attained if the stress- 
strain curve of the spring were a straight line parallel to its 
axis, and distant from it IV ; IV being the steady load to be 

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CHAP. ZZZl. 



Springs and Saddles 



519 



supported. The wheel centre would then remain indifferently at 
any distance (within certain limits) from the frame of the vehicle, 
and since the pressure of the spring in all positions would be just 
equal to the weight supported, no vertical motion would be 
communicated to the frame. With this ideal spring the motion 
would be perfect until the spring got to one end or other of 
its stops, when a shock would be communicated to the frame. 
A better practical form of spring would be one having a stress- 
strain curve with a portion distant W from, and nearly parallel to, 
the axis ; the slope increasing at lower and higher loads, practi- 
cally as shown in figure 554. 





Fig. 554- 



Fig. 555- 



Fig. 556. 



Let a cycle wheel running along a level road be supported by 
a spring under compression, the steady load on the latter being Wy 
Ax and B^ (fig. 555) being the steady positions of the ends of the 
spring at the wheel axle and frame respectively. Let the wheel 
suddenly move over an obstacle so that its centre is raised the 
distance A^A^ and the spring is further compressed. The frame 
end B of the spring may be considered fixed, while the wheel- 
centre is being raised. The work AxP\P'iA<^'\% expended in 
compressing the spring. The end A^ may now be considered 

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520 Details chap. zzn. 

fixed, and as the pressure on the spring is greater than the 
load supported, the end B will rise and lift the frame. The 
work B^ B^ P%P\ (fig- 556) is expended in raising the frame 
from B^ to -B^ where static equilibrium takes place. If the 
wheel-centre remain at the level A^ the difference of energy 
P\p\P<i'^\wx'\& dissipated, the frame end of the spring vibrat- 
ing between positions B^ and B^, If the wheel return quickly to 
its former level A^^ little or no energy may be lost The quantity 
of lost energy is smaller the more nearly the stress-strain curve P 
is parallel to the axis of the spring ; therefore a spring for a spring- 
frame or wheel should be long, or the equivalent. An ideal spring 
would have to be very carefully adjusted, as a small deviation from 
the load it was designed for would send it to one end or other of 
its stops. 

352. Saddle Springs. — With a rigid frame cycle, the saddle 
spring should perform the function above described, so that no 
vertical motion due to the inequality of the road be communicated 
to the rider ; practically, the vertical springs of saddles are 
arranged so as to make as comfortable a seat as possible. It has 
been shown (Chap. XIX.) that in riding over uneven roads, the 
horizontal motion of the saddle is compounded of that of the 
mass-centre of the machine, and a horizontal pitching due to the 
inequalities of the road. If the saddle springs cannot yield 
horizontally, the rider will slip slightly on his saddle. 

A saddle, as in figure 557, with three vertical spiral springs 
interposed between the upper and lower fi^mes will yield hori- 
zontally more than one in which the frame and spring are merged 
into one structure (fig. 560). 

353. Cylindrical Spiral Spring8.-~Let d be the diameter of 
the round wire from which the spring is made; D the mean 
diameter, and n the number, of the coils ; C the modulus of trans- 
verse elasticity ; 8 the deflection, and q the maximum torsional 
shear, produced by a load W. Then 

'-'-^^ (■) 

^O <3) 

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OMAP. xxxi. Springs and Saddles 521 

Mr. Hartnell says that a safe value for q for |-inch to J-inch 
vrire, as used in safety-valve springs, is 60,000 to 70,000 lbs. per 
sq. in. Probably cycle springs have not such a large margin of 
strength as safety-valve springs. If q be taken slightly under 
80,000 lbs. per sq. in., the greatest safe load, W, is given by the 
equation 3 

JF= 30,000^ (4) 

lbs. and inches being the units. 

The value of C is between 12 and 14,000,000 lbs. per 
sq. in. If we take C= 12,800,000 the deflection is given by the 
equation 

i^»J?^^ (S) 

1,600,000 d* 

Example. — A spiral spring i^-inch mean diameter, made 
from ^-inch steel wire, will carry safely a load 

JF= 33_<^^-^,i = 40 lbs. nearly. 
1*5 X 8* 

The deflection per coil with this load will be 

5 1*5^ X 8^ X 40 ^ • u 

I = ^ !L_ = '345 inch. 

1,600,000 XI 

Round wire is more economical than wire of any other section 
for cylindrical spiral springs. 

354. Flat Springs. — The deflection of a beam of uniform 
section of span /, supported at its two ends and carrying a load 
Win the middle, is given by the formula 

;5=J^ (6) 

48^/ ^^ 

E being the modulus of elasticity of the material, and / the 
moment of inertia of the section. For steel wire, tempered, E = 
13,000 to 15,000 tons per sq. in. If -£ be taken 33,600,000 lbs. 
per sq. in., substituting for / its value for a circular section 

^ d\ we get 

64 . ^ Wl^ .V 

80,000,000//^ 

lbs. and inches being the units. ogtzed by Google 



522 



Details 



CHAP. ZZXL 



In many saddles the springs are made of round wire, and are 
subjected both to bending and direct compression. The deflection 
due to stress along the axis of the wire is very small in comparison 
with that due to bending, and may be neglected. 

355. Saddles. — The seat of a cycle is almost invariably made 
of a strip of leather supported hammock fashion at the two ends, 




Fig. 557- 

the sides being left free. In the early days of the. * Ordinary ' 
bicycle the seat was carried by a rigid iron frame, to which the 
peak and back of the leather were riveted. After being in use 
for some time such a seat sagged considerably, and the necessity 
for providing a tension adjustment soon became apparent This 
tension adjustment is found on all modern saddles. The iron 




frame was itself bolted direct either to the backbone or to a flat 
spring, the saddle and spring were considered to a certain extent 
as independent parts, and were often supplied by different manu- 
facturers. In modern saddles the seat, frame, and springs are so 
intimately connected that it is impossible to treat them separately. 
One of the most comfortable types of saddles consists of the 
leather seat, the top-frame with the tension adjustment, an under- 
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Cttil^ ZZXI. 



springs and Saddles 



523 



,^ ^^^imil^^^ ^ 



Fig. 559. 



frame with clip to fasten to the L-pin of the bicycle, and three 
vertical spiral springs between the top- and under-frames. 




Fig. 560. 



In the * Brampton' saddle (fig. 557) the under- frame forms 
practically a double-trussed beam made of two wires. In 




Fig. 561. 



Lamplugh's saddle (fig. 558) the under-frame is made of two thin 
plates. 

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524 



Details 



CHAT, ZZZI. 



A simple hammock saddle with the seat supported by springs 
(fig. 559), made by Messrs. Birt & Co., consists of leather seat, 
tubular frame, and three spiral springs subjected to tension, no 
top-frame being necessary. 

The springs, top- and under-frames, are often merged into 
one structure, as in the saddle shown in figure 560, made by 




Fig. 56?. 

Mr. Wm. Middlemore, and that shown in figure 561, made by 
Messrs. Brampton & Co. In the former two wires, in the latter 
six wires, are used for the combined springs and frames. 

All saddle-clips should be of such a form that the rider can 
adjust the tilt of the saddle so as to get the most comfortable 




Fig. 563. 

position. In the 'Automatic Cycle Saddle (fig. 562) the rider 
can alter the tilt while riding. 

It may be noticed that the leather seats of the saddles illus- 
trated above are slit longitudinally, the object being to avoid 
injurious pressure on the perineum. 

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OHAP. XXXI. Springs and Saddles 525 

The ' Sar ' saddle (fig. 565), of the Cameo Cycle Company, is 
provided with a longitudinal depression, for the same purpose. 

356. Pneumatic Saddles. — A number of pneumatic saddles 
have been made, in which the resilience is provided by com- 




Fic. 564. 



pressed air instead of steel springs. The * Guthrie-Hall ' saddle 
(fig. 563) is one of the most successful. The. * Henson Anatomic ' 
saddle (fig. 564) is made without a peak, and consists of two 
air pads, each with a depression in which the ischial tuberosities 




rest, the whole design of the saddle being to avoid perineal 
pressure. The ' Sar * saddle (fig. 565) is also provided with two 
depressions for the same reasons. 



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526 Details 



OHAV. ZXXII. 



CHAPTER XXXII 

BRAKES 

357. Brake SesiBtanoe on the Level.— Let W be the total 
weight of machine and rider, W^, the load supported by the wheel 
to which the brake is applied, and \ig the coefficient of friction 
between the ground and the tyre. If the brake be powerful 
enough, it may actually prevent the wheel from rotating, in which 
case the tyre will rub along the ground while the machine is being 
brought to a standstill. Then R^ the greatest possible brake 
resistance, would be /i^ ^. The pressure applied at the brake 
handle should be, and usually is, less than that necessary to make 
the tyre rub on the ground ; this rubbing might have disastrous 
results. Let v be the speed in feet per second, V in miles per 
hour, and / the distance in feet which must be travelled when 
pulling up under the greatest brake resistance. Then, since the 
kinetic energy of the machine and rider is expended in overcoming 
the brake resistance, 

^^^' = .0334 ivy^ = ^, W,l, 

or 

•0334 wv^ 

nW, ^'^ 

Example. — Taking the data of the example in section 228, 
with the weight of the machine, 30 lbs., equally divided between 
the two wheels, speed 20 miles per hour, fi = 0*4, and the brake 
applied to the front wheel, we have ^= 180 lbs., W^, = 54-3 lbs., 
^ = 0*4 X 54'3 = 217 lbs., and substituting in (i), 

/= 0334 X 180 X4OO ^ J J J fj 

o*4 X 54-3 

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CHAP, xxxii. Brakes 527 

If the brake be applied to the rear wheel, W^^ 1257 lbs., and 

/ = -0334 X 180 X 400 _ g ^^ 
04 X 1257 

It should be noticed that the load W^ should be taken as that 
actually on the wheel while the brake is applied (see sec. 164). 

358. Brake Besistanoe Down-hill.— If the machine be on a 
gradient of x part vertical to i on the slope, the force parallel to 
the road surface necessary to keep it from running downhill is 
X W (see fig. 58). The brake resistance is fi^ W^ cos ^ = 
fa ^bN/^— ^^ ^ being the angle of inclination to the horizontal. 
For all but very steep gradients, n/ T ^^0^ does not differ much 
from I, and therefore the brake resistance is approximately \i^ \i\^ 
as on the level. Thus, if the brake be fully applied, the resultant 
maximum retarding force is \i^ W^ \/i — Jir* — x IV, and there- 
fore, as in section 357, the distance which must be travelled before 
being pulled up is given by the equation 



•0334 W^F2=(^, iV,y/T^^^-^x W)I , . (2) 
or 

If 

X IV^fi, W, (4) 

the machine cannot be pulled up by the brake, however powerful ; 
while if ^ Wis greater than ^g IV^ the speed will increase, and 
the machine run away. 

Example /. — With the data of the example of section 357, 
brake on the front wheel, running down a gradient of i in 10, 
^ = o*i ; substituting in (3), 

/^ -03 34x18 0x400 ^ 6 ft, 
o'4 X 54'3 — o-i X 180 

Example IL — With the same data except as to gradient, find 
the steepest gradient that can be safely ridden down, with the 
brake. 

Substituting in (4), o-i x 180=04 x 54*3; or ^ = -121. 
That is, no brake, however powerful, can stop the machine on a 
gradient of 121 in 1,000, about i in 8. 

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528 Details 



CilAP. ZZXXI. 



If the brake be applied to the back wheel, the corresponding 
gradient is 

o*4 X 1257 

/>. about I in 4. 

359. Tyre and Bim Brakes.— The brake is usually applied to 
the tyre of the front wheel, not because this is the best position, 
but on account of the simplicity of the necessary brake gear. In 
the early days of the 'Ordinary* a roller or spoon brake was 
sometimes applied to the rear wheel, a cord communicating with 
the handle-bar (fig. 338). The ordinary spoon brake (fig. 131) 
at the top of the front wheel fork is depressed by a rod or plunger 
operated by the brake -lever on the handle-bar, the leverage being 
about 2^ or 3 to I. If r be this leverage, and ^, the coefficient of 
friction between the brake-spoon and the tyre, the pressure P on 
the brake-handle necessary to produce the maximum effect is 
given by the equation it^r F=^ fjg W^^ or 

P^f^^J^ (5) 

^'^ 
Example, — With the data of the example of section 357, /•=3, 
and /I, = o*2 ; substituting in (5), we get 

/>=?:4_X.5_4= 61bs. 
0-2x3 

In the pneumatic brake the movement of the brake block on 
to the tyre is produced by means of compressed air, pumped by a 
rubber collapsible ball placed on the handle-bar, and led through 
a small india-rubber tube to an air chamber, which can be fastened 
to any convenient part of the frame. With this simple apparatus 
the brake can be as easily applied to the rear as to the front wheel. 

360. Band Brakes are applied to the hubs of both the front and 
rear wheels, and have been occasionally applied at the crank-axle. 
The spoon brake, rubbing on the tyre, may possibly injure it ; the 
band brake is not open to this objection. Since a small drum 
fixed to the hub has, relative to the frame, a less linear speed than 
the rim of the wheel, to produce a certain effect the brake resist- 
ance must be correspondingly larger. One end of the band is 
fastened to the frame, the other can be tightened by means of the 

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CHAP. xxxn. Brakes 529 

brake gear. The gear should be arranged so that when the brake 
is applied the tension on the fixed end of the band is the greater. 
If /, and /2 be the tensions on the ends of the band, the resistance 
at the drum is t^ — f^^ and, as in section 251, 

log. ^-^=-4343/^^ (6) 

If Z> and d be respectively the diameters of the wheel and the 
brake drum, to actually make the wheel stop revolving we must 
have 

(ii-f2)^=l^,fV, (7) 

Example L — Let the band have an arc of contact of three 

right angles with the drum, />. ^ = ^^ '^ = 471, let /i = '15, Z? = 

2 

28 in., ^ = sJ in., and the rest of the data as in section 357, 

then, substituting in (6) 

log- 7^= "4343 X 0*15 X 471 = '3068. 
Consulting a table of logarithms, 

-» = 2-027 ; 
and /| — /2 = I '02 7 /j- Substituting in (7), 
1027/2 X ^^1= -4 X 54, 

or 

. '4 X 54 X 28 ,, 

/a = ^ - - -^-? = 112 lbs. 

5i X ro27 

Example J I. — If a band brake of the same diameter as in last 
example be applied at the crank-axle, the necessary tension /a will 

be -^ times as great, N^ and N^ being the numbers of teeth in 

the chain- wheels on the driving-hub and crank-axle respectively. 
With Ny = 8, 7V^2 = 18, 

. 18 X 112 ^ ,t 
/2 = g =252 lbs. 



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530 Detaik 



CHAT. mil. 



This example shows the ineffectiveness of a crank-axle band 
brake, since the elasticity of the gear is such that the brake lever 
would be close up against the handle-bar long before the required 
pull was exerted on the band. 

If oil gets in between the band and its drum, the coefficient 
of friction will be much less, and a much greater pull vnll be 
required, than in the above examples. 



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INDEX 



{TAi figures indicate page numbers,) 



ABI 
* A BINGDON.' nipples, 353 ; chain, 398, \ 

Absolute, motion, 17: unit of force, 12 ^ | 

Acceleration, 12 ; addition and resolution, ^ 

17 ; angular, 66 ; radial, Z2 ; tangential, 



Action and reaction, 85 

Adams, J. H., 24 hours' ride, 153 

Addendum*circle, chain-wheel, 403, 404 ; 

toothed-wheel, 440 
Addition, of accelerations^ 2: ; rotations, 31, 

39 ; vectOTs, 17 ; velocities, 15 
' Adjosuble ' Safety, Hawkin's, 186 
Air-tube, 491 
Alloys of copper, 137 
Aluminium, 137 ; bronze, 137 ; frames, 287 ; 

hubs^ 361 
' American Star ' bicycle, 189 
Angle of friction, of repose, 79 
Angular, acceleration, 66 ; momentum, 66 ; 



knkie ac ' 



' Ariel bicycle, 342 
Ash, strenrai of, 139 
Auto Machinery Co., balls. 



393 



Ankle action, 271 

Arc of, approach and recess, 445 ; contact, 

449 
Areas, sectional, of round bars, 109 ; spokes, 

346; tubes, X12 
' Ariel ' bicycle, 342 



O ACK-PEDALLING, 216 
'-' Balance gear, Starley's, 240 
Balancing, on bicycle, 196 ; on ' Otto ' dicycle, 

aoo 
Ball-bearings, 370; thrust-block, 3741,391; 

adjustable, 370 ; Sharp's ideal, 381 ; single, | 

388 ; double, 390 ; spherical, 394 ; univer- 1 

sal. ^95 ; with cages, 386 
Rand^beld t3rres, 498 
Banking of racing tracks, 204, 206 , 

* Bantam ' front-driver, 158 ; motion oi — , 

over a stone, 244 

* Bantamette,' lady's front-driver,, 276 | 
Bars, sectional areas and moduli of round, 

112 ; torbion of, 126 



CHA 

Bauschinger, on repetition of stress, X43 , 

Beam, 93 ; shearing-force on, 93 ; bending- 
moment on, 94 ; examples. 97 ; nature of 
bending stresses on, 102 ; neutral axis of, 
10^ ; of uniform stren^h, X09 

Bearing pressure on cliam rivet, ^21 

Bearings 366 ; ball {See Ball-beanngs) ; coni- 
cal, 368 : roller, 369, 371 ; dust-proof, 361, 
392 ; oil-retaining, 361, 393 ; thrust, 371, 
391 ; Meneely's tubular, 387 

* Keeston ' tyre, 500 

Bending, 86, 93 (See Beam) j — moment, 94 ; 
tension and, 120 ; and twisting. 130 

Birmingham Small Arms Co., Safety, 278 

Birt's hammock saddle, 524 

' Boneshaker ' bicycle, 149 

Boothroyd, single-tube tyre, 495 

Bottom-bracket {See Crank-bracket) 

Boudard driving gear, 462 

Bown's ball-be^ngs, 358, 388 ; pedals, 507, 
508 

Brake, 216, 218, 526 

Brampton's self-lubricating chain, 399 ; 
saddle, 523 

' British Star ' spring-frame bicycle, 296 

Bronze, aluminium,, 137 

Brooke's, tandem bicycle, 291 



(BARTER'S gear case, i6r, 430 

^^ Cast iron, 136 

' Centaur,' hub, 361, 393 ; pedal, 508 ; crank- 
bracket, 5x2 

Centre of gravity, 50 

* Centric 'front-driving gear, 457 

Centrifugal f(m:e, 55 

Chain, 396 ; adjustment, 3x8, 342 : Bramp- 
ton's self-lubricating, 399 ; comparison of 
different forms, 431 ; early — , 397 ; fric- 
tion, 429 ; H umber, 398 ; influence on 
frame, 318; Perry's, 390;. pivot, 401; 
rivets, 420 ; roller, 399 ; ruDbing ana wear, 
409; side-plates, 398,, 415 ; Simpson lever, 
^9, 263, 404 ; single-link, 400, ^19 ; stretch- 
ing, 286 ; struts, J22 ; variation of speed 

Digitized by V^OOQlC 



532 



Index 



CHA 

Chain-wheel, 401 ; design of, 411 ; elliptical, 

438 ; faults in design, 411 ; Humber, 406, 

415 ; expandini^, 470; pitch circle,^ 405; 

roller, 401 ; section of blanks, 416 ; size of, 

427 ; spring, 4*6 
Circular, motion, 21, 66 ; wheel-teeth, 449 ^ 
Classification, of cycles, 183 ; of pneumatic 

tyres, 494 
' Claviger ' bicycles, 477, 480 ; pedal and 

knee-joint speeds, 476 
Clearance, in wheel-teeth, 440; in chain 

searing, 406 

* Clincher ' tyre, 495, 497 
•Club 'tricycle, 168 

Clutch gear for tricycle axle, 238 

* Cob,' •Rover,' isp 

Coefficient of fricuon, 79 ; apparent reduc- 
tion of. 210 

* Collier two-speed gear, 465 
Collision, 72 

Columns, 221 

Component, forces, 4^ ; velocities, 16 

Compression, 86 ; and bending, lao ; spokes, 

Concurrent forces, 45 

Conical bearings, 368 

Conservation of energy, 60 

Contact, arc of, ^2 ; path of, 443, 445 i mo- 
tion of bodies in, 34, 41 

Convertible tricvcle, 179 

Copper, 137 : alloys o\, 137 

Coventry Machinists Co., tricycle, 168 ; tan- 
dem bicycle, 290 

' Coventry Rotary ' tricycle. 166 1 

Crank, 507, 509; Southard, i;j9, 140, 512; | 
variable leverage, 265 ; ^ort diagram, 267 ; 
and levers, 264 

Crank-axle, 512 

Crank-bracket, ^14 

' Cremome ' spnng frame, 296 

* Cripper • tricycle, 177 : steering, 223 
Crowns, 297, 334 

Crushing pressure on balls, 393 

' Crypto ' front-driving gear, 439, 458 

Curved tubes, 316 

Cycle Components Mfg. Co., steering head, 

298 ; driving gears, 462 ; pedals, 508 ; 

crank, 511 ; crank-bracket, 514 
Cycle Gear Co., two-speed gear, 467 
Cyclo^aph, Scott's, 269 
Cycloid, 24 ; hypo-, 26 ; epi-, 26 ; -al wheel ' 

teeth, 443 I 



P)ALZELL'S early bicycle, 148 1 

*^ * Dandy-horse,' 147 
' Dayton ' handle-bar, 299 

* Deburgo ' spring-wheel, 365 

* Decourdemanche ' tyre, 497 

* Delta' metal, 137 

Development of cycles, the bicycle, 145 ; 
tandem bicycles, 162 ; tricycles, 165 ; tan- , 
dem tric)rcles, 176 ; quadncycles, x8i 

Devoirs driving gear, 461 

Diamond-frame, 156, 307; Humber, 310; 
open, 312 I 



FRI 

Dicycle, ' Otto,' 171 ; balaxKing oo, aoo ; 
steering of. 237 

Differential driving gear, 169, 339 

Disapation of energy, 63 

Double-driving tricycles, x68, 191, 338 

Drais' dandy-horse, 147 

* Drayton ' tyre, 502 

Driving gears, front, 455 ; rear, 461 ; com- 
pound, 461 

D tubes, X15 ; for chain-struts, 335 

' Dublin,' quadricycle, 246, i8x ; tricyck, 
166 

' Dunlop ' tyre, x6o, 495. 499, 500 

Dust-proof beanngs, 392 

Dynamics, i : of a particle, 65 ; rigid body, 
70 ; system of bodies, 77 



P ITE and Todd's two-speed gear, 466 
■'-' Elasticity, 87 ; index of, 73 ; transverse, 

135 ; modulus of, 88, 134 
Elastic limit, 1^3 ; raising of, 138 
Ellipse of inertia, 207 
Elliptical chain-wheel, 427 ; tubes, 212 
Elm, strength of, 139 
' Elswick ' hub, 360, 393 
Ener^, kinetic, 60; potential, 61 ; coiBer- 

vation of, 62 ; dissipation of, 63 ; loss of, 

Epicyclic tram, 437, 457 

Epicycloid, 26 

Epitrochoid, 27 

Equilibrium, stable, unstable, and neutral, 

54. 183 
Everett's spnng-wheel, 364 
Expenditure of power, 250 



* P ACILE,' bicycle, 151 ; speed of knce- 
•'■ joint, 472 ; geared, 155 ; tricycle, 173 
Factor of safety, 132 
Fairbank's wood nm, 358 
' Falcon ' steering-head, 399 
Falling bodies, 66 

Fichtel and Sach's ball-bearing, 394 
Fir. strength of, 139 
' Fleuss ' tubeless tyre, 498 
Force, 12: co-planar, 46; -diagram, 91; 
parallelogram, 43 ; -polygon, 45 : -triangle. 

Fork, sides, 333 ; duplex, 300 ; back, 3« 
' Referee,' 300 

Frame, 275 ; aluminium, 287 ; bambocs 386 . 
cross-, 313 ; diagram, 91 ; diamoDd, 256, 
307, 310; front, 397, ^32 ; front-drivers, 
^7S% 303 ; generail considerations, 275, 335 
lady's Safety, 287, 3x5; open diamcmd-; 
278, 3x2 'j pyramia-, 386; rear-drivers, 277, 
307 ; spring-, 395 j -d structures, 89 ; tan 
dem, 280,^ 327 ; tncycle, 392, 330 

' French ' bicycle, 249 

Friction, 78 ; chain gearing, 438 ; -gearing, 
434 ; journal, 79, 368 j pivot, 368 ; rolling, 
78 ; rubbing, 78 ; spmning, 8a ; toothed- 
wheels, 447 ; wheel and ground, 203 



Digitized by VjOOQIC 



Index 



533 



FRO 

Front-drivini;, bicycle, 140, 187 ; gears, 454 ; 

Safety, 158 ; tricycle, 165, 169, 193 
Front-frame, 297^ 332 
Front-steering, Dicycle, 185 ; tricycle, 191 
Frost, F. D., loo-mile race, 254 
' Furore,* tandem bicycle, 300 



MUL 



' Grappler ' tyre, 498 

Gravity, centre of, 50 ; work done 

Gnffin, ' Bicycles of the Year,' 156 
Gun metal, 137 

Guthrie-Hall pneumatic saddle, 525 
Gyroscope, 75 ; -ic action, 207, 231 



against, 



H-^Haai^si- 



corrugated chain, 401 ; 



flEAR, 257, 396 ; -case, 160, 430 ; chain, 

^^ ^ ^ ; compound driving, 461 ; front- 
driving, 456 ; lever-and-crank, 471 ; rear- 
driving, 460 ; two-speed, 465 ; variable- 
speed, 262, ^64 

Geared 'Claviger,' 476, 481 ; ' Facile,' 155, 
'73. 456. 476 ; * Ordinary,' 158 

'German tncyde, 165 

* Giraffe* bicycle, 159 

Gordon's formula for columns, 123 

Graphic r^resentation of, force, 43 ; velocity, 
15 



Handfe-bar, 299, ^34 

Hand-power, mechanism, 272 ; tricycle, 271 

Hart's driving gear, 461 

Hawkin's 'Adjustable' Safety, 186 

Headers, 216 

Healy's driving gear, 462 

Heat, 63 ; medianical equivalent, 64 

Helical tube. 141 

Henson saddle. 525 

Hillman, Herbert & Cooper's 'Kangaroo,' 

Ho^rt, Bird & Co.'s chain adjustment, 433 

Hodograph, 21 

Hook tyres, 497 

Horse-power, 60 

Hubs, 358 ; geared, ^62 

H umber, 'Ordinary, 149; chain, 398, 424; 

• Safety,' 155, 157 ; frame, 280, 286, 310 ; 

spring-frame, 297 ; tricycle, 170 ; steering 

of -tncyde, 235 ; tyre, 498 
Hypocycloid, 26 
Hyi>otrochoid, 1 

IMPACT, 72 

* Impulse, 13 

Index of elastidty, 7^ 

Inertia, moment of, 06, 68, 71, 105 

Inflator, 504 

Instantaneous centre, 19, 24, 38 

' Invindble ' Safety, 280 ; rims, 357 ; tricyde, 

177 
Involute, 27 ; -teeth, 4^2 
Iron, cast, 136 ; wrought, 135 
Ivcl ' Safety, 279 



' T & R,'_two-speed gear, 466 



Jointless rims, 357 
dting. 2^6 
ournal fnction, 79, 368 



I 



• l^ ANGAROO • bicycle, 152 
**• Kauri, strength of, 139 
Kinematics, i, 4, 15 
Kinetics, i; energy, 60 
Knee-joint speed, when pedalling crank, 29, 

265 : with ' Facile' gear, 472 ; ' Claviger,' 

476 : • Xtraordinary, 473, 479 



T ACED tyres, 496 

*^ Lady's ' Safety ' frame, 287, 315 

Lamina, rotation of, 67 

Lamplueh's saddle, ^23 

Lawson s ' Safety ' bicycle, 153 

Larch, stren^h of, 139 

Laws of motion, 56 

Lever, chain, 59, 263, 404 : and crank gear, 

471 ; tension driving-wheel, 341 
Linear speed, 4 

Link, mechanbro, 27 ; -polygon, 47 
Linley & Biggs' expanding chain-wheel, 

,.433 

Lisle s early^ tricycle, 165 

Lloyd's semi-tangent hubs, 359 

LooUised vector, 43 

Loss of energy, 247 ; by vibration, 251 



A/fACHINE, 250; efficiency, 258 
■•^'^ \ Macintosh tyre, 503 
Macmillan's early bicycle, 148 
Macready & Stoney, 'Art of Cycling," 14B, 

271 
Manumotive cydes, 271 
Marriott _& Cooper's driving gear, 461 : 

' Olympia' tricycle, 176. 235 
Marston's 'Sunbeam ' cycles, 271, 288 
Mass, 3 : -centre, 50 
Matter, 3 

Mechanical, equivalent of heat, 64 ; treat- 
ment of metals, 141 
Meneely tubular bearing, 387 
' Merlin ' bicycle, 188 
Metric system, 2 
* Michelin * tyre, 501 
Middlemore'.s saddle, 524 
Mild steel, i^ts 
Modulus, of bending resistance, 108 ; of 

round bars, 100 ; tubes. 112; dastidty, 

87 ; resilience, SiS 
' Mohawk,' Safety, 285 ; tandem, 164 
Moment, of a force, 14 ; bending-, 94 ; of 

inertia, 66, 68, 71 : of momentum, 14 ; 

twisting-, 125 
Momentum, 15 ; angular, 66 
Monocycles, 184 
Morj^an & Wright tyre, 496, 503 
MultKyclcs, 196 



Digitized by CjOOQIC 



534 



Index 



NEU 

TaEUTRAL, axis, loa, 104; equilibrium. 

New Howe Co., tandem bic>-clc, 291 
Nipples, 35a 
Non-slipping covers, 504 
Nottingham, Machinists' hollow rim, 357 ; 
Sociable, 179 



SCA 

Pressure, crushing-, onbalb. 393 : on pedals 
368 ; rivet-pins of chain, 421 ; working-, 
on toothed wheels, 454 

' Prcston-Davics ' tyre, 498, 501, 503 

Pump, 504 

Punctures, prevention of, 503 

Pyramid frame. a86 



/^AK, strength of, 139 

^^ ' Oarsman * tncycle, 273 

• Olympia ' tricycle, 176 ; steering of, 235 

• Orainary,' 149 ; frame, 276, 303 ; motion 

over a stone, 343 

• Ormonde * Safety, 285 

Oscillation, of bicycle, 199 ; * Otto * dicycle, 
200 

' Otto ' dicycle, 170; balancing, aoo; steer- 
ing, 237 

Outer-cover, 491 

Oval tubes, iii 



pAlRS. higher and lower, 257 

* • Palmer ' tyre, 493, 497 
Palmer. R., loo-mile race, 254 
Parallel, forces, 49 : shafts, 443 
ParallelojKram of, forces, 43 ; roUtions, 39 ; 

velocities, 17 

Path, of contact, 442 ; of pedals, 474 ; point, 
24, 27 

Pedial, 507 ; clutch mechanism, 266 ; pres- 
sure, 268 ; influence on frame, 320 ; speed 
with lever-and-crank gears, 473, 48^ : work 
done per stroke of -, 60 ; and side-slipping, 
209 ; -pins, 509 

Pedalling, 270 : speed of knee-joint when — , 
39* 265, 4731.484 

Perpetual mouon, 263 

Perry, chain, 399 ; front-driving gear, 456 

•Persil' spring wheel, 364 

' Phantom ' tncycle, 175 

Phillips, 'Construction of Cycles,' 162 

Pine, strength of, 139 

* Pioneer ' Safety, 277 ^ 

Pitch, circular ana diametral, 435 ; -num- 
ber j 436 ; -line of chain-wheels, 401, 405 
Pitching, 246 
Pivot, 368 ; friction, 81 

* Platnauer ' geared hub, 463 
' Plymouth ' wood rim, 358 
Pneumatic saddle, 525 

Pneumatic tyres, 91, 159, 488 ; classification, 

4^ ; side-slipping, 210 : interlocking, 496 ; 

single tube, 495 ; wire-held, 499 
Point-path, 23, 24 
PolyRon, offerees, 45 ; link, 47 
Potential energ>', 6t 
Poundal, 13 
Power, 60, 259 : brake and indicated, 250 ; 

expenditure, 250 ; of a cyclist, 262 ; horse-, 

69 ; transmission, 396, 434 
'Premier,' ball-beanngs, ^85; cranks, 511; 

helical lube, 141 ; tandem bicjcle, 186 ; 

tricycle, I7J, 29J 



•/QUADRANT,' bicycle, 283 ; tricycle, 171 
Vc^ Quadricycles, 146, 181 ; ' Rodge 
triplets, 182 



O ACE, sUrting in, 72 

^^ Racins; tracks, iMUiking of, 004 

Radial acceleration, 13 

Radian, 6 

' Rapid ' tangent spokes, 346 

Ravenshaw, resistance of cycles, 356 

Rear-driving, bicycles, 148, 153, 188, 377, 307 
gears, 460 : tricjrdes, 166, 770. 191 
I Rear-steering, 185 : tricycles, 168, 175, 192 
I Rectangular tubes, 117 
I 'Referee' back-fork, 336: Safety, 3S4; 

steering-head, too 
I ' Regent ' tandem tricycle, 179 
, Relative motion, 16 : of cluun and wheel, 
i ^02 ; of toothed-wheels. 441 ; of two bodies 
in contact, 34, 41 ; of balls and bearii^ 

Renold, chain, 420 

Resilience, modulus of, 88 

Resistance, air, 252 ; of cycles, 250 : on com- 
mon roads, 256 ; rollinjg, 351 : total, 254 

Resolution of, accelerations, 31 : forces, 47 : 
velocities, 30 

Resultant, of co-planar forces, 46 : of non- 
planar forces, 53 : plane motion, 31 ; of two 
routions, 31, 40 ; velocity, 16 

Reynolds, rolling friction, 82 

Rin»*i >38, 353 ; hollow, 357 ; wood, 138, 

Rivets of chain, ^18 

' RoadscuUer ' tncycle, 273 

Roller, bearings, 369, 371 ; brake, 528 ; 

chain, 399, 424 
Rolling, 35 ; friction, 82 ; of balls in bearing, 

379. 

Rotadon, 5 ; resultant of, 31, 39 ; parallelo- 
gram of, 39 

'Rover* Safety, 153, 278, 283; lady's, 287; 
' Cob,' 159 

'Royal Crescent' tricycle, 17a; steering, 
834 

Rubbing, 35 ; of balls in bearing, 384 

Rucker tandem bicycle, 161, 163 

Rudge, ' Coventry Rotary,' x6i, 236 : 'Royal 
Crescent,' 172, 234; qtiadricycle, iSi, 335 



C ADDLE, 522 : position, 245 ; influence 

'^ on frame, 316 

Safety, factor of, 133 

Safety bicycle i^See various sub-headings) 

Sar saddle, 535 

Scalar, y 

Digitized by CjOOQIC 



I'hdex 



535 



SCO 

Scott, cyclograph, 269 : motion of bic^le 
over a stone, 344 ; pedal clutch mechanism, 
267 

• Scottish * t;rre, yn 
Screw, motion, 41 ; pair, 357 
Seat-lng, 318 

* Seddon ' tyre, 501 
Self-healine air-chamber^ 503 
Shaft, bending and twisting, 130 

Sharp, ball-bearings, 374, 381, 395 ; circular 
wheel'teeth, 449; seat-lug, 318: tandem 
frame, 393, 303 ; tangent wheel, 346. ^61 ; 
two-speed gear, 468 ; frame for fiont-dnver, 
376, 306 ; valve fitting, 506 ; chain-wheel, 

43a 

Shearing, 86 ; -force, 93 ; -stress, 124 

'ShellaiS'Safetyji87 

Side-plates of chain, 398 ; design, 415 

Side-slipping, 209; speed and, 3ti ; pedal 
eflfbrtand, 214 

' Silvertown * tyre, 495, 503 

Simple harmonic motiwi, 30 

Simpson lever chain, 59, 363, 404 

Singer & Co., ' Xtraordinaxy , 150,^ 473; 
ball-bearing, 390, Safety, 284 ; tricycle, 
293 ; * Velodman,* 271 

Single-driving tricycles, 166, 168, 174, 191, 
238 

Single-link chain, 400, 419 

Sinsle-tube tyres, 495 

Sliding, 35 

Snuth, loo-mile road race, 153 ; * Balloon ' 
tyre, 496 

Sociable, monocycle, 185 ; tricjrdes, 179 

Southard, crank, 139, 240, 512 

Space. I 

' Sparkbrook ' Safety, 280 ^ 

Speed. 4 \ angular, 5 ; linear, 4 ; in link 
mecnanisms, 28 ; -ratio variation in chain 
gearing, 421; do. with drcolar wheel- 
teeth, 449 ; of pedals and knee-joint, 99, 

• Speed tyre, 501 

Spindle, hub, 362 

Spinning, 35, 42 j of balls in bearing, 379 

Spoke, 337; direct-, 340, 344; tangent-, 

342, 344 ; Sharp's, 346 ; * Rapid,' 346 ; 

spread of, 351 ; weight of, 346 
Spoon brake, 538 
Spring. 517: spiral, 590; flat, 521; -frame, 

295 ; wheel, 364 ; chain-wheel, 426 
Sprocket wheel {See Chain-wheel) 
Square tubes, 117 
Stability of bicycles, 198, 203 ; quadricycle, 

107 ; tricycle^ 197, 208 
Stable equilibnum, ^4, 183 
Starley, diflerentiaf gear, 169; 'Rover/ 

153 > 'Cob,' 159 ; tricycle axle, 294 ; tri- 
cycle frame, 331 
Starting in race, 72 
Statics, I. 43 
Steel, mild, 135 ; tool*, 136 ; spokes, 346 ; 

tubes, 112 
Steering, 185, 221 : weight on -wheel, 223; 

without hands, 227 ; -head, 297, 333 
.Step, 363 _ . 

Strain, 86 ; straining action, 85, 93 



VEL 

Strength of materials, 132 ; woods, 139 ; 
wheel-teethj 45a 

Stress, breaking and working, 133; com- 
pound, xio ; repeated, 142 ; -strain dia- 
gram, 133, X39 

Sturmey, '^Girafle,' 159; pneonuttic tyres, 
494 

* Sun-and-Planet ' gear, 455 

' Sunbeam ' bicycle, a7x ; lady's, 288 

Swerving of tricycles, 223 

' Swift ' Safety, 279 ; tandem bicycle, 990 

* Swiftsure * tyre, ^ 
Swinging back-fork, 319 



TTANDEM, bicycles, 162; frames, 289, 

•■• 327 ; tricycles, 176 

Tangent spokes, 349, 346 

TangentiaJ, acceleration, 12 ; effort on crank, 
267 

Teeth of wheels, 449 ; circular, 4^9 ; cycloi- 
dal^ 444 ; involute, 4^2 ; strength, 453 

Tension, 86 ; and bending, 120 

Tetmajer's * value figure, 135 

Thrust bearing, 371, 391 

Time, 3 

Toothed-wheels, 444 ; friction, 447 

Torsion, 86, 125 

Tracks, banking of racing, 204 

Train of wheels, 437; epicjrclic, 438 

Translation, 31, 39 ; and rotation, 34, 41 
I Transmission of power, 396, 433 
I Triangle, of forces. 44 ; of rotations, 39 ; of 
' velociues, 17 ; masb-centre of, 51 
) Tubes, areas, and moduli of, tx9 ; circular, 
no; cunred, 316; D, 115, 395; elliptical, 
in; helical, 141 ; internal pressure on, 91 ; 
oval, in; square and rectangular, 117: 
torsion of, 127 

Twisting, 86 ; -moment, 195 ; and bending, 
'30 

Two-speed gears, 465 

Tyres, 22$, 249. 484 ; cushion, 356 ; iron, 
486 ; inurlockiiu;, 496 ; pneumatic (.9^^ 
Pneumatic tyres); rubber, 487 ; single- 
tube, 495 ; wire-held, 499 



T JNEVEN road, motion over, 247 
^ Uniform, motion, 4 ; speed, 4 
Unstable equilibrium, 54, 183 
Unwin, strength of materials, 133, 136. 143 
bearing pressures, 421 ; toothed- wheels 
I 436 

VALVE, 504 

* Variable, acceleration, 12; leverage 
' cranks, 265 ; speed, 8 ; -speed gears, 264, 
' 46s . 

I Variation of speed-ratio, with chain gearing, 
' 432 ; with circular wheel-teeth, 450 
I Vector^ 9 ; addition, 17 

' Veloaman,' hand-power tricycle, 271 
I Velocity, 9, 15 ; parallelogram, 16 ; resultant 
I 16; triangle, 17 



Digitized by CjOOQIC 



536 Index 

WAR YOS 

AlirARWICK & Stockton Co., hub, 361 ; ' Wohler, repetition of stress, 143 
' ' pedal, 508 I Wood, rims, 358 ; weight aud strength, 358 

Weight, 3 ; steel spokes, 346 ; steel tubes, Woodley t>Te, 49^ 

112; woods, 139 
Westwood rims, 352, 357 ' 

'*S^;n1?,iuri4ir.|«rf„"'^'l^lj:i ! •XTRAORDINARYJbicycle,,^: pedal 

ShSp-s1!S?gentft^6;«lodtyS>5nt%' ! and knec-joim speeds, «4, 480 

22 ; sire, 232, 245 ; toothed-, 433 ; chain-, 
405 
'Whippet* spring-frame, 297 Y^^^^'P^'NT, 135 

Wire-held tyres, 499 ^ • Yost' hub, 361 ; steering-head, 290 



PRINTED BY 
SPOTTISWOOUK AND CO., NEW-STREET SQUARE 



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c:Acme 

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6ool(bimii;i£ Co.. Inc. 

300 Suinm«r Stritt 

0*tton. Mau 03210 



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