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sa iL^i'-^' /7
HARVARD
COLLEGE
LIBRARY
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7
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Google
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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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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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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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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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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
Digitized by CjOOQIC
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.
Digitized by CjOOQI£
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
Digitized by CjOOQIC
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- •>••;■ ^^>
Digitized by VjOOQIC
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,
Digitized by VjOOQIC
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— />.,///. ^ ,
Digitized by VjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOO^C
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-
Digitized by CjO(§^IC
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
Digitized by LjOOQIC
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
Digitized by CjOOQIC
(•)
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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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so
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
Digitized
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
Digitized by CjOOQIC
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
Digitized by VjOOQ
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/.
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by VjOOQIC
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
Digitized by VjOOQIC
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
Digitized by VjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by VjOOQ
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by VjOOQIC
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
Digitized by CjOOQIC
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.)
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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. ^
Digitized by CjOOQIC
^ = 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
Digitized by CjOOQIC
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^,
Digitized by CjOOQIC
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. ■
Digitized by CjOOOJC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIc
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
Digitized by CjOOQIC
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
Digitized by VjOOyiC
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
* * Digitized by V^j
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
^_-^. . . ■ . • • •
Digitized by VjOOQIC
(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
Digitized by VjOOQ
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)
Digitized by CjOOQIC
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
Digitized by VjOOQIC
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 <.«)
Digitized by CjOOQIC
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.
Digitized by VjOOQ
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
Digitized by CjOOQIC
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
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•1063
•95
•2755
•1227
I -02
•2944 -14021
69
•1992
•0770
■74
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•0898
•80
2307
•1036
•85
•2465
•1184
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•67
•1938
•0814
•72
•2080
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•77
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•1071
•55
•1601
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•60
•1727
•0728
•64
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•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
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•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^^
Digitized by CjOOQIC
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).
Digitized by VjOOQ
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,
Digitized by VjOOQ
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
Digitized by CjOOQIC
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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Digitized by CjOOQIC
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.
Digitized by CjOOglC
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-
Digitized by CjOOQIC
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."
Digitized by CjOO^^
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.
Digitized by CjOOQIC
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.'
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by LjOOQIC
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
Digitized by CjOOQI®
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 "
Digitized by CjOOQIC
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
Digitized by VjOOQvI 2
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-
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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,
Digitized by CjOOQIC
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.
Digitized
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),
Digitized by CjOOQIC
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
Digitized by Vj N 2
i8o
Cycles* in General
CHAP. XV.
Fiu. 167
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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'
Digitized by VjOOQ
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.
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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.
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOO^
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
Digitized by CjOOQIC
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
Digitized by CjOOVIC
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
Digitized by CjOOQIC
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
Digitized by VjOOQ
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 ' ■ ^"^^
Digitized by CjOOQIC
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
Digitized by VjOOQ
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
Digitized by VjOOQ
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
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by LjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOO^C
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.
Digitized by CjOOQIC
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
Digitized by VjOO^^
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-
Digitized by VjOOQ
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
Digitized by LjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by VjOOQ
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)
Digitized by VjOOQIC
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).
Digitized by LjOOQIC
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.
Digitized by LjOOQIC
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
Digitized by V^jOOQ
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
Digitized by CjOOQIC
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.
Digitized by CjOOQI^
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
Digitized by V^jOOQ
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
Digitized by CjOO^ ^
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
Digitized by CjOOQIC
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
Digitized by VjOOQ
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
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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.
Digitized by CjOOQIC
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-
Digitized by VjOOQIC
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.
Digitized by CjOCglC
Digitized by CjOOQIC
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
Digitized by VjDQQIC
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
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by V^jOOQ
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.
Digitized by CjOOQIC
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
Digitized by CjOCWfe
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
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by VjOOQIC
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
Digitized by CjOOQIC
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
Digitized by VjDOQ IC
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.
Digitized by CjOOQIC
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
Digitized by VjOOQIC
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
Digitized by V^jOOQ
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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.
Digitized by VjOOQIC
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
Digitized by Vj y^ «
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.
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjiOOQ IC
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.
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by VjOOQ
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-
Digitized by CjOOQIC
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
Digitized by VjOOQ
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by VjOOQ
■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
Digitized by CjOOQIC
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
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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
Digitized by V^jOOQ
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.
Digitized by CjOOQIC
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,
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOglC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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'
Digitized by CjOOQIC
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).
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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-
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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
^,
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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
Digitized by V^jOOQ
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
Digitized by CjOOQIC
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.
Digitized by CjOQQIC
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,
Digitized by CjOOQIC
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.
Digitized by V^jOOQ
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.
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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)
Digitized by V^OOQlC
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
Digitized by CjOOQIC
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
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' 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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
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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
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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'
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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
Digitized by CjOOQIC
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
Digitized by V^jOOQ
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)
Digitized by CjOOQIC
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-
Digitized by CjOOqIc
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.
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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.
Digitized by CjOOgte
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.
Digitized by CjOOQIC
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
Digitized by CjOOQIC
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c:Acme
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300 Suinm«r Stritt
0*tton. Mau 03210
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7