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

##
See other formats

This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project to make the world's books discoverable online. It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover. Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book's long journey from the publisher to a library and finally to you. Usage guidelines Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we have taken steps to prevent abuse by commercial parties, including placing technical restrictions on automated querying. We also ask that you: + Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for personal, non-commercial purposes. + Refrain from automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the use of public domain materials for these purposes and may be able to help. + Maintain attribution The Google "watermark" you see on each file is essential for informing people about this project and helping them find additional materials through Google Book Search. Please do not remove it. + Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner anywhere in the world. Copyright infringement liability can be quite severe. About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ sa iL^i'-^' /7 HARVARD COLLEGE LIBRARY Digitized by CjOOQIC 7 Digitized by Google Digitized by CjOOQIC Digitized by CjOOQIC Digitized by CjOOQIC BICYCLES AND TRICYCLES Digitized by CjOOQIC Digitized by CjOOQIC 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 Digitized by CjOOQIC All rights reserved ^ ^ C^ \'oV. tr-'^"^ ^ HARVARD COLLEGE UBRARY 6IFT0F tSyVARD WORTHINGTON SARGENT Digitized by CjOOQIC 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, Digitized by CjOOQIC 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 Digitized by CjOOQIC 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. Digitized by CjOOQIC Digitized by CjOOQIC 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 Digitized by VjOOQlC 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 Digitized by CjOOQIC xii Bicycles and Tricycles CHAPTER XIII STRENGTH OF MATERIALS PAGE 112. Stress r Breaking and Working. — 113. Elastic Limit. — 114. Stress-strain Diagram. — 115. Mild Steel and Wrought Iron. — 116. Tool Steel. — 117. Cast Iron. — 118. Copper. — 119. Alloys of Copper. — 120. Aluminium, — 121. Wood. — 122. Raising of the Elastic Limit — 123. Complete Stress-strain Diagram. — 124. Work done in breaking a Bar. — 125. Me- chanical Treatment of Metals. — 126. Repeated Stress . . 132-143 PART 11 CYCLES lAT GENERAL CHAPTER XIV DEVELOPMENT OF CYCLES : THE BICYCLE 127. Introduction. — 128. The Dandyhorse. — 129. Early Bic>des. — 130. The * Ordinary.'— 131. The * Xtraordinary. ' — 132. The ♦Facile/ — 133. The 'Kangaroo.*— 134. The Rear-driving Safety.— 135. The 'Geared Facile.'— 136. The Diamond- frame Rear-driving Safety. — 137. The * Rational Ordinary.* — 138. The 'Geared Ordinary * and Front -driving Safety. — 139, The * Giraffe* and 'Rover Cob.* — 140. Pneumatic Tyres. — 141, Gear-cases% — 142.. Tandem Bicycles « « . , 145-164 CHAPTER XV DEVELOPMENT OF CYCLES : THE TRICYCLE 143. Early Tricycles. — 144. Tricycles with Differential Gear. — 145. Modem Single driving Tricycles. — 146. Tandem Tri- cycles. — 147. Sociables. — 148^ Convertible Tricycles. — 149. Quadricycles . r . . • r . . . 165-182 CHAPTER XVJ CLASSIFICATION OF CYCLES 150. Stable and Unstable Equilibrium. — -151. Method of Steering. — 152. Bicycles : Front-drivers. — 153. Bicycles; Rear-drivers. — Digitized by CjOOQIC Contents xiit 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 Digitized by V^jOO^ xiv Bicycles and Tricycles CHAPTER XXI GEARING IN GENERAL PACK 20 1. Machine. — 202. Higher and Lower Pairs. — 203. Classification of Gearing. — 204. Efficiency of a Machine. — 205. Power. — 206. Variable-speed Gear. — 207. Perpetual Motion. — 208. Downward Pressure. — 209. Cranks and Levers. — 210. Variable Leverage Cranks. — 211. Speed of Knee-joint during Pedalling. — 212. Pedal-clutch Mechanism. — 213. Diagrams of Crank Eflfort. — 214. Actual Pressure on Pedals.— 215. Pedalling. — 216. Manumotive Cycles. — 217. Auxiliary Hand- Power Mechanisms 257-273 PART III DETAILS CHAPTER XXII THE FRAME : DESCRIPTIVE 2 1 8. Frames in General. — 219. Frames of Front-drivers. — 22a Frames of Rear-drivers. — 221. Frames of Ladies' Safeties. — 222. Tandem Frames. — 223. Tricycle Frames. — 224. Spring- frames. — 225. The Front-frame ..•,.. 275-302 CHAPTER XXIII THE FRAME : STRESSES 226. Frames of Front-drivers. — 227. Rear-driving Safety Frame. — 228. Ideal Braced Safety Frame. — 229. Humber Diamond Frame. — 230. Diamond-frame with no Bending on Frame Tubes. — 231. Open Diamond -frame. — 232. Cross-frame. — 233. Frame of Ladies' Safety. — 234. Curved Tubes. — 235. Influence of Saddle Adjustment. — 236. Influence of Chain Ad- justment. — 237. Influence of Pedal Pressure. — 238. Influence of Pull of Chain on Chain-struts. — 239. Tandem Bicycle Frames. — 240. Stresses on Tricycle Frames. — 241. The Front-frame. — 242. General Considerations Relating to Design of Frame . 303-336 Digitized by CjOOQIC Contents xv CHAPTER XXIV WHEELS rAGB 243. Introductory. — 244. Compression-spoke Wheels. — 245. Ten- sion-spoke Wheels. — 246. Initial Compression on Rim. — 247. Direct-spoke Driving-Wheel. — 248. Tangent-spoke Wheel. — 249. Direct-spokes. — 250. Tangent -spokes. — 251. Sharp's Tangent Wheel. — 252. Spread of Spokes. — 253. Disc Wheels. — 254. Nipples. — 255. Rims. — 256. Hubs. — 257. Spindles. — 258. Spring Wheels 337-36$ CHAPTER XXV BEARINGS 259. Definition of Bearings. — 260. Journals, Pivot and Collar Bearings. — 261. Conical Bearings. — 262. Roller-bearings. — 263. Ball-bearings. — 264. Thrust Bearings with Rollers. — 265. Adjustable Ball-bearing for Cycles. — 266. Motion of Ball in Bearing. — 267. Magnitudes of the Rolling and Spinning of the Balls on their Paths. — 268. Ideal Ball-bearing. — 269. Mutual Rubbing of Balls in the Bearing. — 27a * Meneely ' Tubular Bearing. — 271. Ball-bearingfor Tricycle Axle. — 272. Ordinary Ball Thrust Bearing. — 273. Dust-proof Bearings. — 274. Oil-retaining Bearings. — 275. Crushing Pressure on Balls. — 276. Wear of Ball-bearings. — 277. Spherical Ball-races. — 278. Universal Ball-bearing 366-395 CHAPTER XXVI CHAINS AND CHAIN GEARING 279. Transmission of Power by Flexible Bands. — 280. Early Tri- cycle Chain. — 281. Humber Chain. — 282. Roller Chain. — 283. Pivot Chain. — 284. Roller Chain- wheel. — 285. Huml^er Chain- wheel. — 286. Side-clearance, and Stretching of Chain. — 287. Rubbing and Wear of Chain and Teeth. — 288. Common Faults in Design of Chain- wheels. — 289. Summary of Conditions determining the Proper Form of Chain-wheels. — 290. Form of Section of Wheel Blanks — 291. Design of Side-plates of Chain. — 292. Rivets. — 293. Width of Chain, and Bearing Pressure on Rivets. — 294. Speed-ratio of Two Shafts connected by Chain Gearing. — 295. Size of Chain-wheels. — 296. Spring Chain- wheel. — 297. Elliptical Chain-wheel. — 298. Friction of Chain Gearing. — 299. Gear-case. — 300. Comparison of Different Forms of Chain. — 301. Chain-tightening Gear . . . 396-433 Digitized by VjOOQ ^ 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 Digitized by LjOOQIC Contents xvii CHAPTER XXXI SPRINGS AND SADDLES PAGB 50. Spring under the Action of suddenly applied Load. — 351. Spring Supporting Wheel. — 352. Saddle Springs. — 353. Cylindrical Spiral Springs. ~ 354. Flat Springs. — 355. Saddles. — 356. Pneumatic Saddles 517-52$ CHAPTER XXXn BRAKES 57. Brake Resistance on the Level. - 358. Brake Resistance Down- hill — 359. Tyre and Rim Brakes. — 360. Band Brakes . 526-530 INDEX 531 LIST OF TABLES TABLE L Work done in Foot-lbs. per Stroke of Pedal in raising 100 lbs. Weight against Gravity 60 n. Work done in Foot-lbs. per Minute in pushing 100 lbs. Weight Up-hill 61 HL Sectional Areas and Moduli of Binding Resistance of Round Bars 109 IV. Sectional Areas, Weights per Foot run, and Moduli of Bend- ing Resistance of Steel Tubes 1 12-3 V. Ultimate and Elastic Strengths of Materials, and Coefficients of Elasticity 134 VI. Specific Gravity and Strength of Woods '139 VII. Tensile Strength of Helical and Solid-drawn Tubes . 142 VIIL Banking of Racing Tracks 205 IX. Banking of Racing Tracks 206 X, Air Resistance to Safety Bicycle and Rider , , , '253 Digitized by CjOOQIC 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 Digitized by CjOOQIC ^ 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 Digitized by CjOOQiC 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 Digitized by CjOOQIC (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 Digitized by CjOOQIC 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. Digitized by CjOOQIC 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 Digitized by CjOOQIC 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 Digitized by VjOOQIC 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). Digitized by CjOOQIC 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 Digitized by CjOOviC 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 Digitized by CjOOQIC 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^ Digitized by VjOOQIC 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 Digitized by V^jOOQ 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. Digitized by CjOOQIC 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. Digitized by CjOOQIC 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. Digitized by CjOOQIC 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 ; Digitized by VjOOQ 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 Digitized by LjOOQIC 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 Digitized by CjOOQIC 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. Digitized by CjOOQIC 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 Digitized by CjOOQIC 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. Digitized by Google 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 Digitized by CjOOQIC 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 Digitized by CjOOQIC 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) Digitized by CjOOglC (^ 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) Digitized by CjOOQIC 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 Digitized by CjOOQIC 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,. Digitized by CjOOQIC 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. Digitized by CjOOQIC 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 Digitized by CjOOQIC 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 Digitized by V3 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 Digitized by CjOOQIC 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 Digitized by CjOOQIC 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 Digitized by CjOOQIC 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 Digitized by V^jOOQ 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 Digitized by VjOOQ 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^ . , , Digitized by CjOOQIC 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 Digitized by CjOOQIC 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. Digitized by CjOOQIC 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 Digitized by CjOOQIC 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 •2566 •1063 •95 •2755 •1227 I -02 •2944 -14021 69 •1992 •0770 ■74 •2150 •0898 •80 2307 •1036 •85 •2465 •1184 •62 •1797 •0698 •67 •1938 •0814 •72 •2080 •0938 •77 •2221 •1071 •55 •1601 •0625 •60 •1727 •0728 •64 •I8S3 •0839 •68 •1978 •0958 49 •>405 •0551 •S3 •1515 •0642 •56 •1625 •0739 •60 •1735 •0843 •42 •1207 •0476 •45 •1301 •0554 •48 ■1396 •0638 •51 •1490 •0727 •38 •1108 •0438 •41 •119* 05101 •44 •I 28 1 -0586 '47 ' •1367 0669 ' Digitized byGoogk 114 Principles of Mechanics chap. xi. Therefore the moment of inertia of the area included between the ellipses is /=^-|(2>43+ (ibh^)t -{■ {12 h} ^ 12 bh)^ 4-(24^ + 8^)/ + i6/^| (31) If/ is small in comparison with b and h^ the second, third, and fourth terms in the expression for / are smaller and smaller com- pared with the first, and may be neglected. Therefore, the moment of inertia of the figure is approximately 7=J^>4V(>4 + 3^) (32) The modulus of bending resistance is approximately Z='^^ht(h^Zb) (33) \Vhen a tube of circular section is flattened to form an oval tube, its thickness will be nearly uniform throughout, but in the oval tube section, above discussed, the thickness is not constant throughout, but is a little less than / except at the ends of the major and minor axes. The strength of an oval tube of uniform thickness will therefore be under-estimated if the formula (33) be used, so that the error is on the safe side. The area of the ellipse is -«^; and in the same way as above 4 it can be shown that the area included between the two ellipses is A^'^-(b-^h)t (34) 2 Therefore, (32) and (33) may be respectively written, ^-tlfit^" w An oval tube of uniform thickness will be stronger than indi- cated by formula (36). This is clearly shown by figure 96, which represents a quarter section ; the modulus giveii in (36) is that of Digitized by VjOOQ CH^P. XI. Straining Action : Bending "5 the tube whose inner surface is represented by the dotted line. The inner continuous line represents a tube of the same sectional area A, and of uniform thickness. It has the area b in excess of the dotted tube, and the areas a and c deficient, a + r = ^. It is evident from the figure that the moment of inertia of b about the minor axis is greater than that of a and Cy and therefore the tube of uniform thickness is slightly stronger than the section above discussed. 98. D Tubes.— Tubes of D section have been recently introduced for the lower back fork of a bicycle ; it will be instructive to in- vestigate their bending resistances here. We will assume that the outline of the D tube is made up of a semicircle and its diameter (fig- 97)- Let r be the radius and h the dia- meter of the semicircle, / the thickness, which we will consider very small in comparison with r, and A the sectional area of the D tube. First, consider the moment of inertia about the axis a a 2X right angles to the flat side of the tube. The moment of inertia of the rectangle of depth h and width / is 12 h^ /, the moment of inertia of the semicircle is ^^/, therefore the moment of inertia of 16 the D tube about the axis a a\s '={L'-:y^- ■ ■ <37) 1 1 ^ 1 n at ic X, -Nt- y The modulus of the section, about the same axis, is Fig. 97. =(^s)^" W Consider now the moment of inertia about the axis b b^ coin- ciding with the flat side of the D tube. The moment of inertia of Digitized by CjOOQ I 2 Ii6 Principles of Mechanics chap. xi. the flat side \% ~ h /', that of Jhe semicircle is ^ ^^ / ; therefore 12 lO the moment of inertia of the section of the D tube is 12 10 If / be small in comparison with ^, the first term in this expres- sion may be neglected in comparison with the second, and there- fore, / = — i4^ /, = ^ t^ t^ approximately . . . (39) 16 2 'But the /just found is not about an axis through the centre of figure ; this we now proceed to find. Let G be the centre of figure ; the distance O G can be found as follows : The moment of the semicircle about the axis ^ i^ is 2 r* / (see sec. 50), that of the straight side about the same axis is zero, the total moment of the D tube about the axis ^ ^ is therefore 2 r* /. But the total moment is also equal to the total area multiplied by the distance O G ', therefore 2 r» /= (2 r + Trr) / x OG, ^"'i^^ = (.l^=air^'- = -3«9- . . . (40) Let /o be the moment of inertia about an axis gg passing through G parallel to ^^ ; then by section 92 Therefore./. = ;H/-^^=(;-4^y,. . (4.) But A = {2 + Tt)rf; therefore we may write Z, the modulus of bending resistance about the axis gg is equal to " A" being the extremity of the radius through O G, Now, GX=:OX'-'OG=r— ^- r= "" r. 2 -f TT 2 + TT 4 (2 + ^={;'.(2V.)} ^^=-^5^4^>; . (43) Digitized by CjOOQIC CHAP. XI. Straining Action : Bending 117 Let d be the diameter of a round tube equal in perimeter to the P tube. Then ir ^ = (2 4- ^y, /. r =-^ d=: '6iiod. 2 + TT Substituting this value of r in (43), we get, The Z of the original round tube is approximately A d, so 4 the strengths of a round tube and the D tube into which it can be pressed are in the ratio of 2500 to 1542, i.e. 1000 to 617. But since a D tube is used when the space O X is limited, it would seem fairer to compare it with a round tube of equal weight and of diameter O X, The Z of a round tube of diameter O X '\s '2^Ar. Comparing this value with that in (43), it is seen that the strength of the D 1 ^ tube is slightly greater than that of a round r ^ tube of equal weight, and of diameter equal to the smallest diameter of the D tube, the ratio being "252 to '2500, a difference of less than one percent, in favour of the Dtube. 99. Square and Eeotangolar Tabes.— Con- '''*' ^' sider the / of a square tube of section A B C D (fig. 98), about an axis a a parallel to the side A B. The / of each of the sides BC2Jv6.DA\% --, that of each of the sides ^^ and CZ> is 12 ht.\ therefore, for the whole section 4 I^2.'^A^2.htM^'-h^t (45) 12 4 3 The total sectional area is 4 ^ /, therefore J^\Ah^ (46) also, Z^i^^ Ah (47) ^ 3 Digitized by CjOOQIC 1 1 8 Principles of Mechanics chap. xi. Let d be the diameter of a round tube of the same perimeter as the square tube ; then 4 ^ = tt </ .'. h^'^ d=z 7854 </, and 4 Z =- Ad=z'26i2>Ad (48) 12 hence, comparing with (30), the moduH of bending resistance of the square tube and of the original round tube are in the ratio of — to -, or of TT to 3, i,e, 1047 to 1000, in favour of the square 12 4 tube. Compared with a round tube of equal sectional area, but of the same diameter as the side of the square tube, the ratio is Ad Ad I I - to — , i,e. - to , or i33'3 to 100 : i,e. the square tube is 33*3 3 4 3 4 per cent, stronger than the round tube of equal area and diameter. Rectangular Tubes. — If a round tube be drawn into a rectan- gular tube of the same thickness, perimeter, and sectional area, it can be shown that the strength of the latter will be greatest when its depth h is three times its width b. * For any rectangular section, approximately Z = m(^+^) (50) For the strongest rectangular tube, (49) becomes /=9/^=^=I^>i' (51) ^"^» Z^^Ah (52) 4 Comparing (33) and (50), it is seen that a thin rectangular tube is stronger than an elliptical tube of the same depth, width, and thickness in the ratio 16 : 3 tt. Now the ratio of the peri- meters, and therefore the weights, is never greater than 4 : ir ; this being the value when theeUipse and rectangle become a circle Digitized by CjOOQIC CBAP. XI. Straining Action : Bending 119 and square respectively. Weight for weight, then, the rectangular has at least ^ times the strength of the elliptical tube. That the rectangular is stronger than the elliptical tube of equal depth, width, and sectional area, can be easily shown from first principles, as follows : Figure 99 shows quadrants of rectangular and elliptical tubes of equal sectional area. Since the perimeter of the ellipse is less than that of the rectangle, its thickness is greater. Let a portion a of the ellipse be marked off equal in width to the corresponding part of the rectangle, so that the moments of inertia about the axis OXzxt equal. The part b is common to both ellipse and rect- angle, and there remain only the parts c. That belonging to the rectangle is at a much greater distance from the axis O X than that belonging to the ellipse ; its moment of inertia is therefore greater, and the rectangular is stronger than the elliptical tube to resist bending. Fig. 99. .Digitized by CjOOQIC 1 20 Principles of Mechanics chap. xrt. CHAPTER XII SHEARING, TORSION, AND COMPOUND STRAINING ACTION loo. Compression. — The laws relating to simple compressive stress are exactly the same as those of simple tension, the formula (i), (2), (3), and (4), of chapter X. will apply, / being in this case the compressive stress, e the compression per unit of length, and E the modulus of elasticity for compression. For a homo- geneous material with perfect elasticity, as above defined, E would be the same for tension and compression. On a bar which is short in comparison to its diameter, if the compressive stress be increased above the elastic limit of compres- sion, the bar gives way ultimately by lateral yielding. If the material be hard, the bar may actually split up into several pieces. If of a soft, ductile material it will bulge gradually in the middle while being shortened in length. loi. Compression or Tension combined with Bending. — If a bar be simultaneously subjected to bending, and a pull or thrust parallel to its axis, the maximum stress on the section is the sum of the separate stresses due to the separate straining actions. If the bar be subjected to a pull Fy and a bending-moment M, A being the area and Z the modulus of the section, the maximum tensile stress is ^-A^-2 • . (0 and the minimum tensile stress is /'=3-f <•) For circular tubes of small thickness, substituting the value of Z from (30), section 96, f^A^-A-^ ^- • .• • (3) A Ad Digitized by CjOOQIC CHAP. ZII. Shearing, Torsion^ drc 121 The bending-moment may be produced by applying the pull /'at a distance x from the neutral axis of the section (fig. loo). In this case M =^ Fx and (3) may be written ^'^'^ Ad (4) >P-. Fig. 101. If the bar be subjected to a compression P and a bending- moment My equations (i), (3), and (4) give the maximum compres- sive stress on the section, equation (2) the minimum compressive stress. 102. Columns. — If a long bar be subjected to tension, any slight devia- tion from straightness (fig. 100) will, under the action of the forces, tend to get less. If, on the other hand, the bar be subjected to compression, the deviation from straightness will tend to get greater, and the bar will give way by bending (fig. 10 1). The stresses on a straight short column supporting a load, placed eccentrically, are given by formulae (i) and (2). Example, — A bicycle tube i in. diameter, 16 W.G., is subjected to a compressive force, the axis of which is in. from the axis of 4 the tube. Find the breaking load, the breaking stress of the material being 30 tons per sq. in. From Table IV., A = . 1882 sq. in., Z = '0414 in.^, also M -^^ P inch-lbs. /= 30 x 2240 lbs. per 4 sq. in. ; substituting in (i) P ^ P 30x2240= QQ -h , ^ -1882 4 X -0414 from which, /> = 592 1 lbs. If the load were placed exactly co-axial with the tube, it would reach the value given by, — --. = 30 X 2240 •1882 ^ ^ />., P = 1 2650 lbs. Digitized by GoOglC 122 Principles of Mechanics CHA1». Xlt. 103. Limiting Load on Long Colomns.— If, under the action of the load, the deviation x becomes greater, the bending-moment also becomes greater without any addition being made to the load ; thus the deviation once started, may rapidly increase until fracture of the column takes place. Let the section of the column be such that, under the action of the load, its neutral axis bends into a circular arc AC B (fig. Id) of radius R, Let A D B he the chord, CZ> the greatest deviation, and C E2l diameter of the circle. Then, by the well- known proposition in elementary geometry, CD xDE^AD X DB. i,e. neglecting the difference between CE and DE^ 2 Rx^s- - approximately. EI But R = -^Tj^, from (17), chap, xi., and M^Fx. Substituting, M P^ ZEI (5) If the load be less than that given by (5), no deviation will take place. If the column be of constant section throughout its length its neutral axis bends into a curve of sines, and it can be shown that the limiting load is P^ 'EI (6) Fig. 102. If the middle section of the column be prevented from deviating laterally, it will bend into the form shown in figure 102. In this case the length of the segment of the curve corresponding to figure loi is half the total length, and the corresponding load will be r^ jP = (7) Again, if the ends of the column be held in such a manner that the directions of the axis at the end are always the same, it will give way by bending as shown in figure lot^ The segment Digitized by VjOOQ CBAP. ZII. Shearings Torsion^ &c. 123 bd in this case is of the same shape as the curve in figure loi, while the portions a b and e d are of the same form as ^ ^ and dc. In this case, therefore, the length of the segment ^ ^ is , and the corre- 2 sponding limiting load is given by the formula P^ ^ic^EI (8) If the column be fixed at one end (fig. 104), held laterally but free to turn at the other, /2 F = (9) ^- tn: it- 1 1 1 ' ••V 1/ iLl_ Fig. 103. Fig. 104. Fig. X05. If the column be fixed at one end and quite free at the other end (fig. 105), ^= -4/2- ('°) These are known as Euler's formulae, and are only applicable to bars or columns in which the length / is great as compared with ihe least transverse dimension. / is the length before bending ; though in the figures, in which the bending is greatly exaggerated, it is marked as after bending. 104. Gordon's Formula for Coliiiim8.--The pieces of tube used in bicycle building are too long to have the simple com- pression formula applied to them, and too short for the application Digitized by V^j 1 24 Principles of Mechanics chap. xn. of Euler's formula. A great many experiments on columns, principally cast iron, have been made by Hodgkinson, and Gordon has suggested an empirical formula which agrees very closely with his experiments. For thin tubes, Gordon's formula becomes W^ / A , .1/^ (") '^ cd' /and c being constants depending on the material. Actual experiments on the compressive strengths of weldless steel tubes are wanting, but taking / = 30 tons per sq. in., and c = 32,000, Gordon's formula becomes W 67200 A "■ 7^ r^ .... (12) 32000 (P Example, — A tube is i in. diameter. No. 16 W.G., 20 in. long ; required the crushing load by Gordon's formula. From Table IV., page 113, ^ = -1882 ; W 67200 67200 •1882 400 1*0125 32000 X I from which, JF= 12490 lbs., slightly less than for a short length of the same tube (sec. 102). ' 105. Shearing.— Let A B C D (fig. 106) be a small square prism of unit width perpendicular to the paper, subjected to shearing stress on the planes ^ ^ and C D. If the planes A B and C Z> be very close to each other, the shearing I A stress will be the same on /\q both. If q be the shearing ^ < > ' ' stress per unit of area, the downward force acting 2X A B Fic. ,06. F,c. X07. and the upward force at C Z> will each he q x A B. But since the portion A B C Diszi rest, the couple formed by the forces at -^4 ^ and C Z} must be Digitized by VjOOQIC U^ CHIP. xir. Shearing, Torsion, &c. 125 balanced by an equal and opposite couple, formed by forces acting at -^ Z> and B C, since no force acts normally at the surfaces A B and C D, Thus the shearing^ stress on the sides A D and B C'v^ equal to that on A B and Z> C ; or the shearing stress on a plane is always accompanied by an equal shearing stress on a plane at right angles to the former, and to the direction of the shearing stress on the former plane. Transverse Elasticity, — Under the action of the shearing forces the square A B C D (fig. 106) will be distorted into a rhombus, A^ B^ CD, the angle of distortion A D A^ being proportional to the shearing stress. Let ^ be this angle and q the shearing stress producing it ; then • q^ Ci^ (13) C being the modulus of transverse elasticity, or the coefficient of rigidity of the material. Shearing Stress equivalent to Simultaneous Tension and Com- pressive Stresses, —Dmw a diagonal B D (fig. 106) ; the triangular prism A D B 'y& m equilibrium under the action of the three forces, / gy and h, acting on its sides, which can therefore be repre- sented by the sides of a triangle {^g, 107). /and g being equal, the force h is evidently at right angles to the side B D, The triangles A B D and fgh^Lie similar ; that is, the forces/ g and A are proportional to the lengths of the sides on which they act ; the stress per unit area must therefore be the same for the three sides A B, B D, and D A, Thus, the stress on the plane B D'\s 2i compressive stress of the same intensity as the shearing stress on the planes A B and A D, In the same way it may be shown that a tensile stress of equal magnitude exists on the plane A C, Thus, in any body a pair of shearing stresses on two planes at right angles are equivalent to a pair of compressive and tensile stresses respectively on two planes mutually at right angles, and inclined 45° to the former planes. 106. Torrion. — If a long bar be subjected to two equal and opposite couples acting at its ends, the axes of the couples being parallel to the axis of the bar (fig. 108), it is said to be subjected to torsion. The moment of the couple applied is called the tivisting-moment on the bar. If one end be rigidly fixed, the Digitized by CjOOQIC 126 Principles of Mechanics other end will, under the action of the twisting-moment, be dis- placed through a small angle, and any straight line on the surface of the bar originally parallel to the axis will be twisted into a spiral curve a a. If the twisting-moment be increased indefinitely, the bar will ultimately break, the total angle of twist before break- ing depending on the nature of the material. Let figure 109 be the longitudinal elevation of a thin tube of mean radius r and thickness /, subjected to a twisting -moment 3: Fig. 108. Fig. 109. T' foot-lbs. A square, abcd^ drawn on the surface of the tube becomes distorted while in a strained condition into the rhombus abd cH, Thus, every transverse section of the tube is subjected to a shearing stress. If the tube be of uniform diameter and thickness, this shearing stress, ^, will be the same throughout, provided the thickness is very small in comparison with the diameter. The sectional area of the tube is nrtr \ and since q is the shear on unit area, the total shear on the section is nrgtr. The shearing-force on each element of the section acts at a distance r from the centre of the tube ; the moment of the total shearing-force is therefore 2irqtr^, This must be equal to the twisting-moment T', applied to the end ; therefore T.^iirqtr' (14) Thus the twisting-moment which can be transmitted by a thin tube of circular section is proportional to the square of its radius or diameter and to its thickness. 107. Torsion of a Solid Bar. — In a solid cylinder of radius r,, imagine the square abcd(^%, 109) drawn on a concentric cylin- drical surface of radius r \ it is easily seen that the angle of distortion of the fibres, <t>, or da d\ is proportip^ial to/*. If ^i be Digitized b )rtip|ial to r. d by Google CHAP. XH. Shearing, Torsion^ &c, 127 the angle of distortion for a square drawn on the surface of the cylindrical rod, q and qx the shearing stresses at radii r and r, respectively, then evidently and therefore *=*•- If now the solid rod be considered to be divided into a number of thin concentric tubes, all of the same thickness, /, the area of the tube of radius r is 2 ir /r, the total shear on this tube is and the twisting-moment resisted is The sum of the moments of all the concentric tubes into which the rod is divided is easily found, by one of ^}ie simplest examples in the integral calculus, to be or, r=^^^^V. =^^^approx. . (15) 108. Torsion of Thick Tubes.— If r^ and r^ be the external and internal radii of a hollow tube, the sum of the twisting-moments (45) o^ ^^^ v^T ^^*" concentric tubes into which it may be divided — and, therefore, the twisting- moment such a tube can resist— is or T^-^ WjnMfi .... (16) 16 d^ The quantity — "T — ^ depends simply on the dimensions Digitized by CjOOQIC 128 Principles of Mechanics chap. xu. of the section of the tube, and may be called the modulus of resistance to torsion ; it may be denoted by the symbdl Z^, Then Comparing Zp with Z, chapter xi., it will be seen that the modulus of resistance of a circular tube or solid bar to torsion is twice its modulus of resistance to bending. The strength of any tube to resist bending can therefore be obtained by multiplying the modulus from Table IV., page 112, by twice the maximum shear ^|. 109. Lines of Direct Tension and Compression on a Bar subject to Torsion. — From what has been said in section 105, there will be a compressive stress on the plane a r, and a tensile stress on the plane b d (fig. 109). This holds for every point on the surface of the tube. Now if the tube be split up into a number of narrow strips by the spiral lines / /, inclined 45** to the axis (fig. 10), the tensile stresses can be transmitted just as before. The spiral lines / / are said to be tension lines, and the spiral lines c cdX right angles compression lines. If the twisting-moment be in the opposite direction, however, the pressure and tension spiral lines will be interchanged, and the split tube will not be able to transmit the twisting-moment. no. Compound Stress. — If the straining actions on any part of a structure be all parallel to one plane, the stress on any plane section, at right angles to the plane of the straining actions, can be resolved into a normal stress, tension or compression — and a tangential stress, shearing. It can be shown that any system of stress in two dimensions is equivalent to a pair of normal stresses on two planes mutually at right angles, and that the stress on one of these planes is greater than, that on the other plane less than, on any other plane section of the ma- terial. On any other plane the stress will have a tangential com- ^'^- "°- ponent An important case of compound stress is that of a shaft sub- jected to bending and torsion ; a section at right angles to the CHAP. XII. Shearings Torsion, &c, 1 29 axis of the shaft is subjected to a nonnal stress, / and simul- taneously to a torsional shearing stress, q. Consider a small portion of a material (fig. no) subjected to stresses parallel to the plane of the paper. Let A B Che a, small prism, of unit depth at right angles to the paper, the face B C being subjected to a normal stress, / and a tangential stress, q. From section 106 we know that an equal shearing stress, ^, must exist on the face A B. Let us find the magnitude of the stress p on the face A C, on which the stress shall be wholly normal. Considering the equilibrium of the prism ABC, and resolv- ing the forces on the three faces parallel to the side A B, we have p.AC.sinB-q.AB-'f.BC^o or {p^f)tane=q (17) Similarly resolving the forces parallel to B C, we get, p,AC,cose-q,BC=o or p — qtanB (t8) Multiplying (17) and (18) together, we get from which ^ = i{/± >/7^TT^«} (19) the two values of / in (19) are the maximum and minimum normal stresses on the material. That is, the tension / and the shear q, on the face B C, produce on some plane A C the maximum tensile stress ^ {/ + s/P + 4 q^] , and on another plane the minimum tensile stress \{f — 'JP + 4 ^^} ; the latter plane being at right angles to the former. If the stresses on two planes at right angles be wholly normal and of equal intensity, it can easily be shown that the stress on any other plane is wholly normal and of the same intensity. If the normal stress be compression, the whole system of stress is of the nature of fluid pressure. If there be a tensile stress on one plane and an equal compressive stress on the plane at right angles, Digitized by V^jOOQK 130 Principles of Mechanics chap. xn. it has already been shown that this is equivalent to shearing stresses of the same intensity on two planes at angles of 45° with the planes of the normal stresses. This pair of shearing stresses tends to distort the body but not to alter its volume, whereas fluid pressure tends to alter the volume but not the shape of the body. Any set of stresses in two dimensions can be expressed as the sum of a fluid stress and a shearing stress. Let two planes, A and B^ at right angles be subjected to normal tensile stresses of in- tensity, p and ^, respectively. Then this state of stress is equivalent to the sum of two states of stress, the first being a tensile stress -^ ^ on both planes A and B^ the second a tensile stress - ^""^ 2 2 on A and an equal compressive stress on the plane B. For / =/_+.^. ^P-^lJ, and q =^^-±-? - t^.J, This principle will 2 2 2 2 be made use of when discussing the outer cover of a pneumatic tyre. III. Bending and Twisting of a Shaft— -In a circular shaft of diameter, //, subjected to a bending-moment, J/, and a twisting- moment, Ty the normal stress due to the bending-moment is 32 and the shearing-stress due to the twisting-moment is T q= -. '^ (P 16 Substituting these values in (19), 16 if the shaft be subjected to a twisting- moments 7*^, which would produce the same stress, p^ T 16 Digitized by CjOOQIC CHAP. XII. Shearings Torsion^ &c. 1 3 1 and 7i is said to be the twisting-moment equivalent to the given bending-moment and twisting-moment acting simultaneously. Comparing the two expressions for /, we get T,^M ^ A/J/mr» (20) Similarly, the equivalent bending-moment is M,^\T,^\{M^s/-AP^rT^^, . . (21) Digitized by CjOOglS 132 Principles of Mechanics chap. xht. CHAPTER XIII STRENGTH OF MATERIALS. 112. Stress, Breaking and Working.— Each part of a machine must be capable of resisting the greatest straining actions that may come on it. This condition fixes, as a rule, the smallest possible section of the part below which it is not permissible to go. In ordinary machines, where mere mass is sometimes re- quisite, the section actually used may often with advantage be considerably greater than the minimum ; but in bicycles, since * lightness' is always sought after, though it should always be secondary to * strength,' the actual section used must not be very much greater than the minimum consistent with safety. The magnitude of the stress on any piece depends on the general con- figuration of the machine and of the arrangement of the external forces acting on it. The strength of the various parts depends on the physical qualities of the materials of which they are made, as well as on their section ; this we will now proceed to discuss. Breaking Stress, — If a load be applied at the end of a bar and be gradually increased, the bar will ultimately break under it. If the bar be of unit section—one square inch— the load on it at the instant of breaking is called the breaking tensile strength of the material. A great number of experiments have been made from time to time on the strength of materials, and the values of the breaking tensile strength for all materials used in construction are fairly accurately known. Factor of Safety, — One method of designing parts of a machine or structure is to fix arbitrarily on a working stress which shall not be exceeded. This working stress is got by dividing the breaking stress of the material, as determined by experiment, by an arbitrary Digitized by VjOOQ CHAP. xin. Strength of Materials 133 number called 2: factor of safety. This factor of safety varies with the nature of the material used, and with the conditions to which the structure is subjected. Professor Unwin, in * Elements of Machine Design/ gives a table of factors of safety, the factor vary- ing from 3 for wrought iron and steel supporting a dead load, to 30 for brickwork and masonry subjected to a varying load. The factor of safety should be large for a material that can be easily broken by impact, and may be low for a material that undergoes considerable deformation before fracture actually takes place. 113. Elastic Limit — We have already seen (sec. 81) that the application of a load to a bar of what might be popularly called a rigid material produces an elongation, and that this elongation is proportional to the load applied up to a certain limit. If not loaded beyond this limit, on removing the load the bar returns to its original length, and no permanent alteration has been made. If, however, the load applied be greater than the above limit, the elongation produced by it becomes greater proportionally, and on the load being removed the bar is found to be permanently increased in length. The stress beyond which the elongation is no longer proportional to the load, is called the elastic limit. Since the elongation is in most metals proportional to the load applied up to this point, it has also been called the proportional limit (German, * Proporiionalitatsgrenze '). In a few metals — cast iron, brass—there is no well-defined proportional limit. The total elongation of a bar loaded up to a stress just inside the elastic limit is a very small fraction of its original length. On increasing the load beyond the elastic limit and up to the break- ing point, the elongation before fracture occurs, in the case of most materials, is a very much greater fraction of the original length. Table V. gives the breaking and elastic strengths and coefficients of elasticity of most of the materials used in cycle making ; the figures are taken from Professor Unwinds * Elements of Machine Design.* 114. Stress-strain Diagram. — The relation between the elongation and the load producing it can be conveniently exhi- bited in the form of a diagram. Let the stress be represented by an ordinate O y drawn vertically (not shown on the diagram), and Digitized by V^j 134 Principles of Mechanics CHAP. XIII. 1 I \8 fOO "^ o OOO*'* "^ ooo i ro W CO ^ »^ <» CS il I I ! I I I u D O 3 M ri i§§§ § §1 f^ ' o »^ o »^o o I I I I I III *n u^OD 0^5vrJ000000O I 000*^000000 . C5 •3 •c •I ti •a iu = o c y > Ac/) a 1 o Googk 3 -8 Digitized by CHAP. XIII. Strength of Materials 1 3 5 the corresponding extension be a lineJ^'/ drawn horizontally from y. The locus of the point / will be the stress-strain curve of the material. Stress-strain curves for a number of different materials subjected to tension are shown in figure iii. It has been proposed to represent the comparative values of materials for constructive purposes by figures derived from their stress-strain curves. The work done in breaking a test piece, reckoned per cubic inch of volume, may be used. This is pro- portional to the area included between the base and the stress- strain curve. Tetmajer's * value- figure' for a material is the product of the maximum stress and the elongation per unit length. It is the area of the rectangle formed by drawing from the final point of the stress-strain curve lines parallel to the axes. Of the materials represented in figure 1 1 1, * Delta ' metal and aluminium bronze have the highest * value-figures.' 115. Hild Steel.— Figure m shows the stress-strain curve for mild steel, such as the material from which weldless steel tubes are made. The straight portion O a represents the action within the elastic limit. If the load be increased beyond that represented by a, the extension takes place at a more rapid rate, as shown by the slightly curved portion a b. At a point, b^ somewhat above the elastic limit, a, a sudden lengthening of the bar takes place without any increase of load, this being represented by the portion b c oi the curve. The stress at which this occurs is called the yield-point of the material. On further increasing the load, exten- sion again takes place, at first comparatively slowly, but afterwards more rapidly, until the maximum stress at the point d is reached. Under this stress the bar elongates until it breaks. If, however, the stress be partially removed after the maximum stress, d^ is reached, as can be done in a testing machine, the curve falls gradually, as at d e, then more rapidly until fracture occurs at / The elongation represented by the curve up to E takes place uniformly over the whole length of the bar, that represented by efonXy on a small portion in the neighbourhood of the fracture. In wrought iron, the yield-point is not so distinctly marked as in mild steel ; the stresses at the elastic limit and at breaking are less, the elongation before fracture is also less. The specific gravity of wrought iron and mild steel is, on an average, 77. .oogle 136 Principles of Mechanics CHAP. xin. 1 1 6. Tool SteeL — For a tool steel of good quality, containing about one per cent, carbon, the maximum stress may be much higher ; the stress-strain curve takes the form shown in figure in, PERCENTAGE ELONGATION Fig. zxx. the extension being smaller, though the tenacity is very much greater, than that of mild steel. 117. Cart Iron has no well-defined elastic limit ; in fact, the stress-strain curve is not straight for any part of its length, so that for cast iron the term * elastic limit,' though often used, has no definite meaning. 118. Pore Copper varies greatly in tensile strength, according to the mechanical treatment to which it has been subjected. Rolling and wire-drawing both increase its tenacity. The stress- strain curve for rolled copper (fig. iii) is from Professor Unwinds * The Testing of Materials of Construction.' 119. The Alloys of Copper with other metals form a most Digitized by CjOOQIC CHAP. XIII. Strength of Materials 137 important series. Their mechanical properties are most fully discussed in Professor Thurston's 'Brasses, Bronzes, and other Alloys.' Brass contains 66-70 per cent, copper, and 34-30 per cent, zinc ; sometimes a little lead. The stress-strain diagram (fig. iii) shows that the stress at the elastic limit is very low in comparison with the ultimate breaking stress. Gun-metal is an alloy of copper and tin. The stress-strain diagram (fig. iii) is from a metal containing 98 per cent, copper, 2 per cent. tin. Ternary alloys of copper, zinc, and tin have been exhaustively investigated by Professor Thurston. He finds the best proportion, when toughness as well as tenacity is important, is copper 55, tin 0-5, zinc 44-5. Aluminium Bronze, — Copper and aluminium form a most useful series of alloys. The stress-strain curve (fig. iii) is from an alloy containing about 10 percent, aluminium ; it shows clearly the great strength and ductility of the material. Alloys containing a much larger proportion of aluminium are valuable where lightness is the first consideration, but since they possess little strength and ductility, they can only be sparingly used in structural work. Delta metal is a copper-zinc-iron alloy, which can be cast and worked hot or cold. It possesses great strength and ductility, as is shown by the stress-diagram (fig. iii) from a bar 79 sq. in. sectional area, tested by Mr. A. S. E. Ackermann at the Central Technical College. 1 20. AluminiiiTn. — A specimen of squirted aluminium, con- taining 98 per cent, of the pure metal, was tested at the Central Technical College by Mr. Ackermann ; the tenacity was 6*32 tons per sq. in. ; the elongation in 8" was i*i2", of which '53" was in the immediate neighbourhood of the fracture ; the general elonga- tion may, therefore, be taken as 10 per cent. For comparison this result is plotted in figure 1 1 1 . Pure aluminium has not sufficient strength and toughness to be of much value as a structural material, though its lightness as compared with other metals is a desirable quality. Some alloys, containing a small percentage of aluminium, possess great strength, 138 Principles of Mechanics chap, xm but they are, of course, heavy. It remains to be seen whether a strong alloy, containing a large percentage of aluminium, and therefore light, can be discovered. Such an alloy may possibly be of value in cycle making. The specific gravity of sheet aluminium is 2*67, of mild steel 77- 121. Wood is not so homogeneous as most metals ; it is, as a rule, much stronger along than across the grain. The fact that wood joints are generally of low efficiency is against its use in tension members of a frame. For compression members, where there is no loss of strength at the joints, it may be used with advantage in some cases, its compressive strength (see Table VI.) being not much inferior, weight for weight, to that of the metals. In beams of short span subjected to bending, it is, in some im- portant cases, immensely superior, weight for weight, to metal. The strength of a rectangular beam is proportional to its width, the square of its depth, and the strength of the material from which it is made (sec. 94), i,e, proportional to b z^f. If beams of equal weight be made from wood and steel, the width b being the same in both, the depth d of the wood beam will be greater than that of the steel beam ; and the product z^fmW be much greater for the wood than the steel beam. The rim of a bicycle wheel is subjected to compression and bending (sec. 255). Since its width must be made to suit the tyre, a wood rim will be much stronger than a solid steel rim of the same weight ; or, for equal strengths, the wood rim will be the lighter. A holloiv steel rim will possibly be stronger than a wood rim of equal weight. Table VI., taken from Laslett's * Timber and Timber Trees, gives the weights and strengths of a few woods. 122. Eaising of the Elastic Limit. — Let a bar be subjected to a stress — represented by the point k (fig. in) — consider- ably above its elastic limit. If the load be removed and the bar be again tested, it wfU be found that it is elastic up to a stress as high as that indicated by k. Thus the elastic limit in tension of a material like mild steel can be raised by simply applying an initial stress a little above the limit required. An important application of this principle occurs in the case Digitized by V^jOOQ CHAP. XIII. Strength of Materials 139 Table VI. Specific Gravity and Strength of Woods. Name of wood 1 Specific ^ water being taken I'ooo Transverse load on pieces 2"X2"X72" Tensile stress on pieces 2"xa"x3o" Vertical sire^ on pieces 2" X 2" X 2" Ash, English . ] ,, American Elm, English . ,, Canadian 1 -480 1 .558 1 748 lbs. 862 638 393 920 lbs. per sq. in. 3,780 5,495 5.460 9,182 lbs. per sq. in. 5U94 5.78| 7,418 Fir, Dantzic . „ Spruce, Canada Kauri, New Zealand Larch, Russian 1 -484 •646 877 670 816 626 3,231 3,934 4,543 4,203 7,104 5,985 Oak, English . „ French . ,y White, American Pine, Yellow . „ Pitch, American '735 •976 ' -983 1 659 776 878 804 505 1,049 7,571 8,102 7,021 2,027 4,666 7,640 7,942 6,964 4,172 6,462 of Southard's twisted cranks. Here the cranks are given a con- siderable initial twist in the direction in which they are strained while driving ahead ; their strength is considerably increased thereby. A twist (sec. 109) is equivalent to a direct pull along certain fibres, and a direct compression along other fibres at right angles. The initial twist in Southard's crank is, therefore, equi- valent to raising the elastic limit of tension of the fibres under tensile stress, and the elastic limit of compression of the fibres under compressive stress. 123. Complete Stress-strain Diagram. — ^The complete stress- strain diagram of a material should extend below the axis O X ; in other words, it should give the contractions of the bar under compressive stresses, as well as elongations under tensile stresses. Figure 112 represents such a curve, the point a denoting the elastic limit in tension, and b the elastic Hmit in compression. If the bar has had its elastic limit in tension raised artificially to the point k (fig. Ill), it is found experimentally that the elastic limit in compression has been lowered, and thus the new stress-strain curve would be somewhat as represented in figure 1 13^^ 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. Digitized by CjOOQIC 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. Digitized by CjOOQIC 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 Digitized by V^jOOQ 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 Digitized by CjOOQIC [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. Digitized by CjOOQIC 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 Digitized by CjOOQIC 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. Digitized by CjOOQIC 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 Digitized by Google 344 Details chap. iL.Tin. Comparing this example with that of section 247, the superiority of the tangent wheel in tangential stiffness is apparent. In this example it should be noted that the initial pull on the spokes does not enter into the calculation. Consequently, the initial pull on tangent- spokes may with advantage be less than that on direct-spokes. 249. Direct-spokes.— The spokes of a direct-spoke wheel are usually of the form shown in figure 340, the conical head at the , , . end engaging in the rim, and the \rm\ ' ' _ -L-j * ' ' ' — d other end being screwed into the Pj^j hub. For the sake of preserving the spoke of equal strength through- out, its end is often butted before being screwed (fig. 341), the section at the bottom of the thread in this case being ^^^ at least as great as at the middle HH 1 i ^ '3 of the spoke. Fig. 3n. -^^ ^^"^^ 339 the spoke is shown making an acute angle with the hub. As a matter of fact, under the action of a driving effort the spokes near the hub will be bent, as shown exaggerated in figure 342. The continual flexure under the driving effort weakens and ultimately causes breakage of direct spokes, unless made of greater sectional area than would be necessary if they could be connected to the hub by some form of pin-joint. The conical head lies loosely in the rim, and being quite free to Fig. 342. adjust itself to any alteration of direction, the spoke near the rim is not subjected to such severe strain- ing actions as at the hub. 250. Tangent-spokes. — Tangent-spokes cannot be con- veniently screwed into the hub, but are threaded through holes in a flange of the hub, the end of the 4 — 1-^— -^ ^ spoke being made as indicated in figure " . •i'' ^jEEaS-Hj, 343. This sharp bend of the spoke p,^, ^^^ . seriously affects its strength. Let P be the pull on the spoke,, and d its diameter. On the section of the sp6ke at a there will be a bending- moment P x^ x being the distance between the middle of the 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 CO p CM JO b b 8 Ximbiiqo uinuifx^j^ t.?? 8^.?.8.8' b b b b o _b b b_b. ■|-i b b b o S;^^^2?8 8 b b o b o b b b o § SI, ^'i^.^ M b b b b b b M^S».'n»?.8.8 *H 'o b 6 b II <5 c c td 1 D O t-H X 1 i5 t *Xiinl P P CO P P io sdnit ijiqo 1 umumrepi ° V » •• o« »>• 1 000.0 = ^ 1 b K o o o or;: Iffflff III m . „ •^ k boo o p_p_ _i:^o P o_<LP_ O VwVO NO *«« V V Vo "rr^'fftl 8 M b b b b b M b b b b b 0. rv lo N "(S .0 8 « M b b 0_0 _o_o_o_o_ _M ob_b 0. N (*>v3 « « *M 'o b b b b V) b b b « O r*.** •- p c : .- b be c c . Xiinbiiqo uin'uiixi!}^ b p p p c c .: .8i§ ^m **••»• o : : :^ .^«.2 opTi o o : 3 --Iv" M M S A f« « N p p p p p .: c ^-'k .11 i 1111 Xiinbiiqo uinuiprej^ *- •« « b 5 C 3 O b •>?!k- o o o coca p 1 b ^"Ik y\ p p 5- fT p 8 V^ « M b b b b m 1111 M M » o o : « b b b o ' Digiti :edbY *i air's 8jj,£»8 2?^ 8S^i s b b b b b b L.OOQ b b b c -•• CHAP. XX71I. Toothed'ivheel Gearing 453 be negative. In (12), (13), (14), and (15) substitute R^ ^R \ they become respectively, * - ^j'j,- (■«) 3 ' + 3 /?»(/? -if zK" " • • ^^^> 6-4is(ff_-j)(ff-__=^)«' , . (^:)'= Table XIX., with m = -2, is calculated from these equations. The values of ^\ ""', in Tables XVIII. and XIX. change so slowly, that their values corresponding to any value of R and ;/ not found in the tables can easily be found by interpolation. 314. Strength of Wheel-Teeth. — The mutual pressure i^ be- tween a pair of wheels is sometimes distributed over two or more teeth of each wheel \ but when one of the pair has a small number of teeth it is impossible to have an arc of contact equal to twice the pitch, and the whole pressure will be borne at times by a single tooth ; each tooth must therefore be designed as a cantilever fixed to the rim of the wheel and supporting a transverse load F at its point Let/ be the pitch of the teeth, b the width, h the thickness of a tooth at the root, and / the perpendicular distance from the middle of the root of the tooth to the line of action of F. Then the section at the root is subjected to a bending- moment F /, while the moment of resistance of the section is — - •^. Therefore, ^^= "6 (20) The width of the teeth is usually made some multiple of the pitch ; let b ^=i kp. The height of the tooth may also be expressed as a multiple of/ ; it is often as much as 7 /, but since long teeth are necessarily weak, the teeth should be made as short as possible consistent with the arc of contact being at least equal to the pitch. ** Digiffzed by Vj * 454 Details CHAP. ZXflL If the height be equal to -6/, the length / may be assumed equal to s /. If there be no side-clearance the thickness at the pitch- line will be '5 p^ and with a strong tooth form the thickness at the root will be greater. Even with side-clearance, we may assume h •=. '^p. Substituting in (20) we have ^=-o8333/«^/ (21) or, writing p :=z p, P being the diametral pitch-number, 7?*= -82246^ (22) The value that can be taken for / the safe working stress of the material, depends in a great measure on the conditions to which the wheels are subjected. If the teeth be accurately cut and run smoothly, they will be subjected to comparatively little shock. For steel wheels with machine-cut teeth, 20,000 lbs. per sq. in. seems a fairly low value for/ the safe working stress. Table XX. is calculated on the assumptions made above. Table XX. — Safe Working Pressure on Toothed Wheels. Calculated from equaiion (22). Pitch- Lbs. Pressure when k b number 548 14 2 24 3 5 822 1097 1371 1645 6 381 571 761 952 1 142 7 282 421 561 1 703 845 8 215 323 430 1 538 64s 9 170 266 341 1 427 5" 10 137 206 ^74 ; 343 1 411 II "4 171 228 i ^75 31^ 12 96 144 192 239 287 13 81 122 162 203 , 244 14 70 106 141 1 176 211 15 61 91 122 1 152 ! 183 16 54 80 107 134 161 18 43 64 85 107 128 20 35 51 69 86 1 103 22 28 42 55 ! 71 1 ^5 24 24 36 48 ; ^ 72 Digitized by CjOOQIC CHAP, xxvii. Toothed-wheel Gearing 455 The arc of contact is sometimes made equal to two or three times the pitch, with the idea of distributing the total pressure over two or three teeth. But in this case, although the pressure on each tooth may be less than the total, they must be made longer in order to obtain the necessary arc of contact. It is therefore possible that when the pressure is distributed, the teeth may be actually weaker than if made shorter and the pressure concentrated on one. In cycloidal teeth, for a given thickness at the pitch-line, the thickness at the root is greater the smaller the rolling-circle ; where strength is of primary importance, therefore, a small rolling- circle should be adopted. In involute teeth, the angle of obliquity influences the thickness at the root in the same manner; the greater the angle of obliquity the greater the root thickness. In circular teeth, the greater m be taken, the thicker will be the teeth at the root. 315. Choice of Tooth Form. — It has already been remarked that involute toothed-wheels possess the valuable property that their centres may be slightly displaced without injury to the motion. Involute tooth outlines are simpler than cycloidal outlines, the latter having a point of inflection at the pitch-circle ; involute teeth cutters are therefore much easier to make to the required shape than cycloidal. With involute teeth the direction of the line of action is always the same, but with cycloidal teeth it con- tinually changes, and therefore the pressure of the wheel on its bearing is continually changing. Taking everything into con- sideration, involute teeth seem to be preferable to cycloidal. The old millwrights and engineers invariably used cycloidal, but the opinion of engineers is slowly but surely coming round to the side of involute teeth. 316. Front-driving Gears.— Toothed-wheel gearing has been more extensively used for front-driving bicycles than for rear- drivers. A few special forms may be briefly noticed. ^ Sun-and'Plamt^ Gear, — In the * Sun-and- Planet ' Safety (fig. 479), the pedal-pins are not fixed direct to the ends of the main cranks, but to the ends of secondary links, hung from the crank-pin. A small pinion is fastened to each pedal-link and gears with a toothed-wheel fixed to the hub. This ^is^^ simple ^ Digitized by VjUOQ ^ 4S6 Details COAT. XZTIL form of epicyclic train, and can be treated as in section 306. If JVi and Jv^ be the numbers of teeth on the hub and p)edal-liiik respectively, it can be shown (sec. 306) that the speed-ratio of the driving-wheel and main crank is — l_ i — 1, If the driving-wheel be 40 in. diameter, and the pinion and hub have 10 and 30 teeth respectively, the bicyde is geared to ^ — — - X 40 in. = 53*3 inches. It should be noticed that the pedal-link will ^^^" ^'^' not hang vertically, owing to the pressure on the pinion. During the down-stroke the pedal will be behind the crank-pin ; while on the up- stroke, if pressure be applied to the pedal, it will be in front of the crank-pin. The pedal path is therefore an oval curve with its longer axis vertical. If the pressure on the pedal be always applied vertically, the pedal path will be an ellipse, with its minor axis equal to the diameter of the toothed-wheel on the driving-hub. This simple gear might repay a little consideration on the part of those who prefer an up-and-down to a circular motion for the pedals. T^e * Geared Facile ' is a combination of the * Facile ' and * Sun-and-Planet ' gears, the lower end of the pinion-link of the latter being jointed to the pedal-lever of the former. In figure 124, the planet-pinion is 2 in. diameter, the hub-wheel 4 in. diameter, and the driving-wheel 40 in. diameter ; the bicycle is 2) (4 + therefore geared to — 4 X 40 = 60 in. Perry's Front-driving Gear is similar in arrangement to the back gear of a lathe. The crank-axle (fig. 480) passes through the hub and is carried by it on ball-bearings. A toothed-wheel Digitized by V^jOOQ CHAP. XXVII. Toothed-wheel Gearinsr 457 fixed to the crank-axle gears with a wheel on a short intermediate spindle, to which is also fastened a wheel gearing in turn with one fastened to the hub of the driving-wheel ; the whole arrangement being the same as diagrammatically shown in figure 470. Fig. 460. The mutual pressure between the wheels D and E (fig. 470) is equal to the tangential effort on the pedal multiplied by the ratio of the crank length to the radius of wheel D, Example,— li the pedal pressure be 150 lbs., the crank length 6^ in., and the radius of wheel D \\ in., the pressure on the teeth will be "^ X 150 = 780 lbs. The * Centric ' Front-driving Gear affords an ingenious example of the application of internal contact. A large annular wheel is fixed to the crank-axle and drives a pinion fixed to the hub of the driving-wheel, the arrangement being diagrammatically shown in figure 481 ; a and b being the centres of the crank-axle and the driving-wheel hub respectively. As the crank-axle has to pass right through the hub, the latter must be large enough to encircle the former, as shown in section (fig. 482). The hub ball-races are of correspondingly large diameter, the inner race being a disc set Digitized by CjOOQ IC 458 Details CHAP. invn. eccentrically to the crank-axle centre. The central part of the hub must be large enough to enclose the toothed-wheel on the Fig. 481. Fig. 482. crank-axle. Instead of being made continuous and enclosing the toothed-wheel completely, the hub is divided in the middle, and the end| portions are united by a triangu- lar frame. From figure 482 it is evident that the * Centric' gear can only be used for speed-ratios of hub and crank-axle less than 2. The 'Crypto' Front-driving Gear is an epicyclic train, similar in principle to that shown in figure 472. Figure 484 is a longitudinal section of the gear ; figure 483 an end view, showing the toothed-wheels ; and figure 485 an outside view of the Fig. 4S3. Digitized by CjOOQIC CBAP. XXYII. Toothed-wheel Gearing 459 hub, bearings and cranks complete. The arm C (fig. 472) in this case takes the form of a disc fastened to the crank-axle A^ and carrying four wheels E^ which engage with the annular wheel G, forming part of the hub of the driving-wheel, and with the small wheel 2?, rigidly fastened to the fork. The crank-axle is carried on ball-bearings attached to the fork, the hub runs on ball-bearings on Fig. 484. the crank-axle, while the small wheels E run on cylindrical pins B riveted to the disc C. The pressures on the teeth of the wheels are found as follows : Considering the equilibrium of the rigid body formed by the pedal- pin, crank, crank-axle A^ disc C, and pins B, the moment — about the centre of the crank-axle — of the tangential pressure P, on the pedal-pin is equal to that of the pressures of the wheels E on the pins B. Let / be the length of the crank, and r the distance of the centre of the pin B from the crank-axle ; the pressure of each wheel E on its pin will be r 4r (23) Digitized by CjOOQIC 460 Details CHAP. XIVII This pressure of the pin on the wheel E is resisted by the pressures of the wheels D and G ; each of these pressures must, therefore, be equal to IP , . S-r <^^^ If iVi, N^i ^^^ ^3 t>e the numbers of teeth on the hub- wheel G, fork-wheel Z>, and intermediate wheels JE, respectively, the speeds of the wheels and arm C, relative to the latter, are respectively proportional to I and • (25) while, relative to the fork, the speeds (o ,;, CD 2), 0) £, and o> c 'ire propor- tional to iVa'^'^i )■ and Also JV,= JV,-JV, .(26) (27) l'"rom (26) the speed-ratio of the hub and crank, relative to the fork, is I ^i Fig. 485- From (28) it is evident that when the hub speed is to be more than twice that of the crank, JVy must be less than -A^, ; that is, the annular wheel must be fixed to the fork. From (25) and (27) the speed-ratio of wheels £ and Z>, rela- tive to the disc C, is _ ^2 _ 2M, AT. N^ - iV, = 2 (^-2) (»9) But the wheel Z> makes — i turn relative to the crank while the latter makes i turn relative to the fork. Therefore, for every Digitized byVjOOgle CHAF. xxTn. Toothed-wheel Gearing 461 turn of the crank, the wheels E make 2 ; "" ( turns in their bearings. Since these are plain cylindrical bearings, and the pressure on them is large, their frictional resistance will be the largest item in the total resistance of the gear. Example, — If /= 6i, r = i^in., and /*== 150 lbs. ; from (23) the pressure on the teeth is -^ 5_ = 97-5 lbs., and of the wheels E on their pins 195 lbs. Also if -/?=2*5, as in gearing a 28-inch driving-wheel to 70-inch, the wheels E each make ^~2J> = 6 turns on their pins to one turn of the crank. •5 317. Toothed-wheel Bear-driving Oears. — A number of gears have been designed from time to time with the object of replac- ing the chain, but none of them have attained any considerable degree of success. The ^ Burton^ Gear was a spur-wheel train, consisting of a spur-wheel on the crank-axle, a small pinion on the hub, and an intermediate wheel, gearing with both the former and running on an intermediate spindle on the lower fork. The intermediate wheel did not in any way modify the speed-ratio, so that the gearing up of the cycle depended only on the numbers of teeth of the wheels on the crank-axle and hub respectively. If r was the radius of the spur-wheel on the crank -axle and / the length of the crank, the upward pressure on the teeth of the IP intermediate wheel was — , and therefore the upward pressure of IP the intermediate wheel on its spindle was 2 . This upward pressure was so great, that an extra bracing member was required to resist it. Example.— If /*= 150 lbs., /= 6| in., ^ = 4, the pressure on the intermediate spindle = -^ ^^^ '5o ^ ^g^.^ ibg. 4 The Fearnhead Gear was a bevel-wheel gear, bevel- wheels being fixed on the crank-axle and hub respectively and geared together by a shaft enclosed in the lower frame tube. If bevel-wheels could be accurately and cheaply cut by machinery, it is possible that Digitized by CjOOQIC 462 Details CBA.V. zxm. gears of this description might supplant, to a considerable extent, the chain-driving gear ; but the fact that the teeth of bevel-wheels cannot be accurately milled is a serious obstacle to their practical success. 318. Componnd Driving Oears. — For front-driving, Messrs, Marriott and Cooper used an epicyclic train (fig. 486), formed from a pair of spur-wheels and a pair of chain-wheels. Two spur-wheels, I) and JS, rotate on spindles fixed to the crank C Rigidly fixed to -^ is a chain- wheel G, connected by a chain to a chain-wheel jF, fixed to the fork. If the arm C be fixed and the pinion D be rotated, the chain-wheel jF will be driven in the opposite direction. Let — « be the speed-ratio of the wheels 7^ and £> rela- tive to the arm C (in figure 486, — « = — i), then theangular speeds of Fy C, and D are respectively proportional to «, o, and - r. If a rotation + 1 be given to the whole system, their speeds will be proportional to (« + i), i and o respectively. The wheel JD is fixed to the fork, the wheel jF to the hub of the driving-wheel, and C is the crank. The driving-wheel, therefore, makes (n + i) turns to one turn of the crank. With this gear, any speed-ratio of driving-wheel and crank can be conveniently obtained. A number of compound rear- driving gears have been made, some of which have been designed with the object of avoiding the use of a chain. In * Hart's ' gear, a toothed-wheel was fixed on the crank-axle and drove through an intermediate wheel a small pinion ; a crank fixed on this pinion was connected by a coupling-rod to a similar crank on the back hub. In this gear, there was a dead-centre when the hub crank was horizontal, and when going up-hill at a slow pace the machine might stop. In * Devoirs ' gear the secondary axle was carried through to the other side of the driving-wheel, two coupling-rods and pairs of cranks were used, and the dead-centre avoided. T/ie ^Boudard' Gear (fig. 487) was the first of a number of compound driving gears in which the chain is retained. An annular wheel is fixed near one end of the crank-axle and gears with a pinion on a secondary axle ; at the other end of the Digitized by CjOOQIC CHAP. XXVII. Toothed-wheel Gearing 463 secondary axle a chain-wheel is fixed and is connected by a chain in the usual way to a chain-wheel on the back hub. A great deal of discussion has taken place on the merits and demerits of this gear ; probably its promoters at first made extravagant claims, and its opponents have overlooked some points that may be advanced in its favour. Of course, the mere introduction of an additional axle and a pair of spur-wheels is rather a disadvantage on account of the extra friction. In the chapter on Chain Gearing it has been shown that it is advantageous to make the chain run at a high speed ; this can be done with the ordinary chain gearing by making both chain-wheels with large numbers of teeth, but if the back hub chain- wheel be large, say with twelve teeth, that on the crank-axle must be so large as to interfere with the arrangement of the lower fork. The * Boudard ' gear is a convenient means of using a high gear with a large chain-wheel on the back hub. The * Healy ' Gear (fig. 488) is an epicyclic bevel gear having a speed-ratio of 2 to i, which has the advantage of being more compact than the * Boudard ' gear, but has the disadvantage which applies to all bevel-wheels, viz. the fact that they cannot be cheaply and accurately cut. Geared Hubs. — Compound chain gears have been used in which the toothed-wheel gearing is placed at the hub of the driving- wheel. In the * Platnauer ' gear (fig. 489) the small pinion is fixed to the hub and gears with a large annular wheel which runs on a disc set eccentrically to the hub spindle, a row of^balls being Digitized by V^jOOQ Fig. 487. 464 Details CHAP. xxm. introduced. The outer part of this wheel has projecting teeth to gear with the chain. Fig. 488. These hub gears, as far as we can see, have none of the advantages of the crank-axle gears to recommend them, since the speed of the chain cannot be increased unless a very large crank- FiG. 489. axle chain-wheel be used, and they possess the disadvantages of additional frictional resistance of the extra gear Digitized by CjOOQIC CHAP. XXVII. Toothed-wheel Gearing 465 319. Variable Speed Oears. — It has been shown, in Chapter XXI., that it is theoretically desirable to lower the gear of the cycle while riding up-hill. In the * Collier ' Two-Speed Gear, of which figure 490 is a section, and figure 491 a general sectional view, a stud-wheel D (that is, a wheel with pin teeth) fixed on the crank-axle gears with a toothed-pinion P attached to the chain-wheel C The crank- axle A is carried on a hollow axle B^ the axes of the two axles Kk;. 490. being placed eccentrically. The chain-wheel 6', and with it the toothed-pinion J\ revolves on a ball-bearing at the end of the hollow axle B, There are twelve and fifteen teeth respectively on the pinion and stud-wheel, so that the ratio of the high and low gears is 5 : 4. When the low gear is in use, the two axles are locked together by means of a slide bolt S in the hollow axle which engages with a hole in the stud-wheel />, the whole revolving together on ball-bearings in the bottom- bracket F. When the high gear is used, the bolt in the hollow axle is with- Digitized by LjO^QtC 466 Details CHAP. ZXTII. drawn from the hole in the stud-wheel and fits in a notch in the operating lever. The toothed-pinion, and with it the chain-wheel C, is then driven at a higher speed than the crank-axle. Fig. 491. The arrangement of the two axles is shown diagrammatically in figure 492. When the high gear is in use the centre b of the crank-axle is locked in position vertically about the centre a of ^--^ the hollow axle. If the cranks are exactly in line at high gear, the virtual cranks a Cy and a c^ will be slightly out of line at *^" • 49^ low gear. The pedals, however, describe practically equal circles with either gear in use. The * Eite and Todd ' Two-Speed Gear (fig. 493) consists of a double-barrelled bracket carrying the crank-axle — on which is keyed a toothed-wheel — and a secondary axle, to which is fixed two small pinions at one end, and the chain-wheel at the other. The pinions on the secondary axle are in gear with intermediate Digitized by CjOOQ IC CHAP. XXTII. Toothed-wheel Gearing 467 pinions running on balls on adjustable studs attached to an arm which can swing round the secondary axle. One or other of the intermediate pinions can be thrown into gear with the spur- wheel on the crank-axle, by the shifting mechanism under the control of the rider, by means of a lever placed close to the handle-bar. The Cycle Gear Company's Two-Speed Gear has an epicyclic train somewhat similar in principle to that of the * Crypto ' front- FiG. 493. driving gear. When high speed is required, the whole of the gear rotates as one rigid body ; but when low speed is required the small central wheel is fixed and the chain-wheel driven by an epicyclic train. The same Company also make a two-speed gear, the change of gearing being effected at the hub of the driving-wheel. The *y. and R,^ Tivo-Speed Gear (fig. 494) consists of an epicyclic gear in the back hub ; the central pinion of the gear is fixed to the driving-wheel spindle when the low gear is used, the wheel hub then rotating at a slower speed than the H i^ 468 Details CHAP. XXTll. chain-wheel. When the high gear is used, the epicyclic gear— and with it, of course, the chain-wheel— is locked to the driving-wheel hub. C, is the main portion of the driving-wheel hub ; to this is fastened the end portion Ca, on which are formed a ball-race for the chain-wheel G, and an annular wheel D^ in which the Fic. 494. central pinion D can be locked. The intermediate pinions E^ four in number, revolve on pins fastened to the hub C, and C,. The annular wheel G^^ which gears with the intermediate pinions, is made in one piece with the chain-wheel G, AVhen the low gear is in use the central pinion D is held by the axle-clutch />, Fig. 495. fastened to the spindle A, To change the gear, the central pinion D is shifted longitudinally out of gear with the axle clutch and into gear with the annular wheel D^. This shifting is done by means of a rack r and pinion / ; the latter is supported in a shifter-case S fixed to the driving-wheel spindle, and is operated by the rider at pleasure. Digitized by CjOOQIC CHAP. XXVII. Toothed-wheel Gearing 469 Figure 495 shows an outside view of the hub with the spindle and shifter-case partially removed. The * Sharp * Two-Speed Gear (fig. 496) is an adaptation of the * Boudard' driving gear. On the crank-axle A the disc D^ carries a drum D.^ on which are formed two annular wheels w^ and w^ which can gear with pinions p^ and p^ fastened to the secondary axle. The secondary axle is in two parts \ the chain-wheel ^is Fig. 496. fixed to one part ^,, the pinions/, and /a to the other part aj. The ball-bearing near the end of a^ is carried by a secondary bracket b^ which can be moved longitudinally in the main bracket B^ so that the pinion /j may be moved into gear with the wheel ?«;,, or pinion p^ into gear with wheel ^e'^ 'i while in the inter- mediate position, shown in the figure, the crank-axle may remain stationary while the machine runs down hill. A hexagonal surface on the portion a.i fits easily in a hollow hexagonal surface on the portion a^ of the secondary axle, so that the one cannot rotate without the other, although there is freedom of longitudinal movement. The longitudinal movement is provided by a stud j. 470 Details chap. xxm. which passes through a small spiral slot in the main bracket B^ and is screwed to the inner movable bracket b. The end of the stud can be raised or lowered, and the sliding bracket simulta- neously moved longitudinally, by the rider, by means of suitable mechanism, and the gear changed from high to low, or vice versa. The drum D^ is wider than that on the ordinary * Boudard ' gear, the corresponding crank, C^, may therefore be fastened to the outside of the drum instead of to the end of the crank-axle. The other crank, Cj, is fastened in the usual way to the crank- axle. IJnley and Biggs^ Expanding Chain-wheel (fig. 465) provides for three or four different gearings, and though there is no toothed-wheel gear about it, it may be mentioned here, since it has the same function as the two- speed gears above described. The rim of the chain-wheel on the crank-axle can be expanded and contracted by an ingenious series of latches and bolts, so as to contain different numbers of cogs. When pedalling ahead the driving effort is transmitted direct from the crank -axle to the chain-wheel; but if the chain- wheel be allowed to overrun the crank-axle the series of changes is effected in the former. The right pedal being above, below, before, or behind the crank-axle, corresponds to one particular size of the chain-wheel ; if pedalling ahead be begun from one of these positions, the chain-wheel will remain unaltered. The length of chain is altered by the changes, therefore a loose pulley at the end of a light lever, controlled by a spring (fig. 465), is used to keep it always tight. Back-pedalling is impossible with this expanding chain -wheel, so a very powerful brake is used in conjunction with it. A two-speed gear, with the gearing-down done at the hub, will be better than one with the gearing-down done at the crank- bracket, in so far that when driving with the low gear the speed of the chain will be greater, and therefore the pull on it will be less, presuming that the number of teeth on the back-hub chain- wheel is the same in both cases. The frictional resistance of an epicyclic two-speed gear is probably much greater than that of an annular toothed-wheel gear, such as the * Collier ' or * Sharp,' on account of the inter- mediate pinions revolving on plain cylindricaL^bearings under Digitized by V^jOOQ CHAP. XXVII. Tootlted'wheel Gearing 4^1 considerable pressure. The crank-axle of the former gear runs on plain qrlindrical bearings when the gear is in action. The * Sharp ' and the * Eite and Todd ' two-speed gears have the dis- advantage, compared with the others, that the additional gear and its consequent increased frictional resistance is always in action ; in this respect the former is exactly on a level with the ordinary * Boudard ' gear. 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 Digitized by CjOOQIC CHAP. XXX. Pedals^ Cranks^ and Bottom Brackets 513 twisting couple of magnitude /'/acting on the axle. This twist- ing-moment is constant on the portion of the axle between the crank and the chain-wheel. The forces P^ and P^ constitute a bending couple of magnitude P/2 at the point Ay l^ being the dis- tance from the bearing to the middle of the pedal, measured parallel to the axis. The force P^ produces a pressure on the bearing at A, Example, — If l^ be 3 ins., the other dimensions being as in the previous examples, il/'= 150x3 = 450 inch-lbs., 7^= 150 x6| = 975 inch-lbs., 7^ = 1524 inch-lbs., J^ = 762 inch-lbs. The dia- meter d of the axle will be obtained by substituting in formula (15), chap, xii., thus : ^ — X 20,000= 1524 d^ = -381, //= 725, say I in. If the axle be tubular, Z = ' = 0381. 20,000 From Table IV., p. 112, a tube \ in. external diameter, 13 W. G., will be sufficient. Comparing the hollow and solid axles, their sectional areas are •226 and '442 square inches respectively ; thus by increasing the external diameter \ inch and hollowing out the axle its weight may be reduced by one half ; while, if the external diameter be increased to i in., from Table IV., p. 112, a tube 16 W. G. will be sufficient, the sectional area being -188 square inches ; less than 43 per cent, of that of the solid axle. In riding ahead the maximum stresses on the axle, crank, and pedal-pins vary from zero, during the up-stroke of the pedal, to the maximum value / If back-pedalling be indulged in, the range of stresses will be from -f / to — / The dimensions of the axle and crank above obtained by taking /= 20,000 lbs. per sq. in. are a little greater than those obtaining in ordinary practice. A total range of stress of 40,000 lbs. per sq. in. is very high, and cranks or axles subjected to it may be expected to break after a few years' working, unless they are made of steel of very good quality. It may be pointed out here that a pedal thrust of 150 lbs. will not Digitized by KjO^S^gS^ SH Details be exerted continuously even in hard riding, though it may be exceeded in mounting by, and dismounting from, the pedal. Fig. 547. 348. Crank-brackets.— The bracket and bearings for support- ing the crank-axle form a kinematic inversion of the bearing shown in figure 404 ; the outer portion ^ forming the bracket is fastened Fig. 548. to the frame of the machine, while the spindle S becomes the crank-axle, to the ends of which the cranks are fastened. In the Digitized by CjOOQIC CHAP. XXX. Pedals^ Cranks^ and Bottotn Brackets 515 earlier patterns of crank-brackets, hard steel cups D were forced into the ends of the bracket, and cones C were screwed on the axle, the adjusting cone being fixed in position by a lock nut. The barrel bottom-bracket is now more generally adopted ; being oil-retaining and more nearly dust-proof, it is to be preferred to the older pattern. The axle ball-races are fixed, and the adjust- able ball-race can be moved along the bracket. In the * Centaur ' crank-bracket (fig. 547) the bearing discs or cups are screwed to the bracket, and secured by lock nuts. In the * R. F. Hall ' bracket (fig. 548) one cup is fastened to the bracket by a pin, and the other is adjusted by means of a stud screwed to the cup and working in a diagonal slot cut in the bracket. The pitch of this slot is so coarse that the adjustment is performed by pushing the stud for- ward as far as it will go, it being impossible to adjust too tightly. The cup is then clamped in place by the external screwed pin. 349. The Frestnre on Crank-axle bearings is the resultant of the thrust on the pedals and the pull of the chain. Example, — Taking the rows of balls 3^ ins. apart, and the rest of the data as in the example of section 238, and considering first the vertical components due to the pressure P on the pedals, the condition of affairs is represented by figure 549. Taking moments about by we get z\Px = 3^3, .-. Pz = ^^ X 150 = 1607 lbs. 35 In the same way, taking moments about r, we find P^^ = 3107 lbs. zi ^j i* W--"-> -Ji P, -j|-»j 3i ^^i Fig. 549. Fig. 550. Consider now the horizontal forces. Fig. 550 represents the condition of affairs ; /^„ F^y F^ being respectively the horizontal 5i6 Details CHAP. XZX. components of the pull of the chain and of the pressure on the bearings. Taking moments about b^ we get IF, = z\F^, therefore, F^ = *^75 ^340 = 36-4 lbs. In the same way, taking the moments about r, we find ^2 = 376-4 lbs. >S Fig. 551. 4^ Fig. 55a. The resultant pressures Rx, and R^ on the bearings b and c can be found graphically as shown in figures 551 and 552, or by cal- culation, thus : R^^^P^^F^^ ^3112+376^ = 488 lbs. J?,= >/:^^5q:?i«= v/i6i^ + 36> =i65lbs. Digitized by CjOOQIC 517 Y//a////////u/u^////a^ CHAPTER XXXI SPRINGS AND SADDLES 350. Springs under the Action of suddenly Applied Load. — We have already seen (sec. 82) that when a load is applied at the end of a long bar, the bar is stretched, and a definite amount of work is done. If the load be not too great, such a solid bar of iron or steel forms a perfect spring. If a greater ■ extension be required for a given load, instead of a cylindrical bar a spiral spring is used. The relation between the steady load and the extension of a spiral spring is expressed by an equation similar to (2), chap. X., and the stress-strain curve is, as in figure 74, a straight line inclined to the axis of the spring. Let a spiral spring be fixed at one end with its axis vertical (fig. 553), and let A^ be the position of its free end when support- ing no load. Let ^, be the position of the free end when supporting a load W^ the ordinate A^ P^ being equal to W^ to a convenient scale. Let A^ Py P^ P^ be the stress-strain curve of the spring. When this spring is supporting steadily the load IVy tet an extra load w be sud- denly applied. The end of the spring when supporting the load W -{■ w will be in the position A^- The work done by the loads in descending from A^ to A.^ is (IV -}- w) Xy and is graphically repre- sented by the area of the rectangle Ay A^P^py, The work done in stretch- ^'c- 553. ing the spring is Wx ■\- \w x^ and is represented by the area Ax A2 P2 Pi' Digitized by CjOOQIC 5 1 8 Details chap. xm. The difference of the quantities of work done by the falling weight and in stretching the bar is \wxj and is graphically repre- sented by the triangle P\px P^. In the position A^oi the end of the spring, this exists as kinetic energy, so that in this position the load must be still descending with appreciable speed. The spring continues to stretch until its end reaches a point A^ where it comes to rest and then begins to contract. At the position of rest -^3, the work done by the loads in falling the distance Ay A^ must be equal to the work done in stretching the spring, since no kinetic energy exists in the position A^. Therefore, area ^1 ^zPzPx == area A^ A^P^ P^. It is easily seen that this is equivalent to saying that the triangles P^p\P\ and P^p^ P^ are equal, and therefore j^ = x ; i,e, a load suddenly applied to a spring will stretch it twice as much as the same load applied gradually. In the position ^j, the tension on the spring is greater than the load supported, and therefore the spring begins to contract and raise the load. If the spring had no internal friction it would contract as far as the original f)osition ^„ and continue vibrating with simple harmonic motion between Ax and A^ ; but owing to internal friction of the molecules (or hysteresis) the spring will ultimately come to rest in the position of equilibrium A^^ and therefore the work lost internally is PxPxP^^\wx (i) For a stiff spring the slope of A^^Px PiP^ is great, i.e. the extension x corresponding to a load w is small, and therefore the work lost is also small. For a weak spring the slope A^p\% small, and for a given load w the extension x^ and therefore work lost, is large. But for i given extension x the work lost with a stiff spring is greater than with a weak spring. 351. Spring Supporting Wheel. — The function of a spring supporting the frame of a vehicle from the axle of a rolling wheel is to allow the frame to move along in a horizontal line without partaking of any vertical motion due to the inequalities of the road. This ideal motion would be attained if the stress- strain curve of the spring were a straight line parallel to its axis, and distant from it IV ; IV being the steady load to be 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 Digitized by CjOOQIC Digitized by CjOOQIC Digitized by CjOOQIC c:Acme IS Co.. m 03; Digitized by V^UOyiC 6ool(bimii;i£ Co.. Inc. 300 Suinm«r Stritt 0*tton. Mau 03210 THE BORROWER WILL BE CHARGED AN OVERDUE FEE IF THIS BOOK !S NOT RETURNED TO THE LrBRARY ON OR BEFORE THE LAST DATE STAMPED BELOW. NON^RECEIPT OF OVERDUE NOTICES DOES NOT EXEMPT THE BORROWER FROM OVERDUE FEES. 7