# The Kinematics of Machinery

London : Macmillan (1876). Scanned in July 2007 by the University of Toronto for the Internet Archive.
THE
KINEMATICS OF MACHINERY.
OUTLINES
OF A
THEORY OF MACHINES.
BY
F. REULEAUX,
Director of and Professor in the Königlichen Gewerbe-Akademie in Berlin. Member of the Königl. technischen Deputation für Gewerbe.
TRANSLATED AND EDITED BY
ALEX. B. W. KENNEDY, C.E.,
Professor of Civil and Mechanical Engineering in University College, London.
WITH NUMEROUS ILLUSTRATIONS.
London:
MACMILLAN AND CO.
1876.
[The Right of Translation and Reproduction is Reserved.]
LONDON: R. CLAY, SONS, AND TAYLOR, PRINTERS, BREAD STREET HILL, QUEEN VICTORIA STREET

## CONTENTS.

| | | Page |
| Preface | Preface | v |
| Introduction | Introduction | 1 |
| OUTLINES OF THE THEORY OF MACHINES. |||
| Chapter I. General Outlines | Chapter I. General Outlines | 29 |
| § 1. | Nature of the Machine-Problem | 29 |
| 2. | The Science of Machines | 36 |
| 3. | General Solution of the Machine-Problem | 41 |
| Chapter II. Phoronomic Propositions | Chapter II. Phoronomic Propositions | 56 |
| § 4. | Preliminary Remarks | 56 |
| 5. | Relative Motion in a Plane | 57 |
| 6. | Temporary Centre; the Central Polygon | 60 |
| 7. | Centroids; Cylindric Rolling | 63 |
| 8. | The Determination of Centroids | 65 |
| 9. | Reduction of Centroids | 70 |
| 10. | Rotation about a Point | 76 |
| 11. | Conic Rolling | 77 |
| 12. | Most general Form of the Relative Motion of Rigid Bodies | 78 |
| 13. | Twisting and Rolling of Ruled Surfaces | 79 |
| Chapter III. Pairs of Elements | Chapter III. Pairs of Elements | 86 |
| § 14. | Different Forms of Pairs of Elements | 86 |
| 15. | The Determination of Closed Pairs | 87 |
| 16. | Motion in Closed Pairs | 92 |
| 17. | The necessary and sufficient Restraint of Elements | 96 |
| 18. | Restraint against Sliding | 98 |
| 19. | Restraint against Turning | 103 ​ |
| § 20. | Simultaneous Restraint of Sliding and Turning | 112 |
| 21. | The Higher Pairs of Elements | 115 |
| 22. | Higher Pairs.—Duangle and Triangle | 116 |
| 23. | Point-paths of the Duangle relatively to the Equilateral Triangle | 121 |
| 24. | Point-paths of the Triangle relatively to the Duangle | 125 |
| 25. | Figures of Constant Breadth | 129 |
| 26. | Higher Pairs of Elements.—Equilateral Curve-triangle and Rhombus | 131 |
| 27. | Paths of Points of the Curve-triangle relatively to the Square | 133 |
| 28. | Paths described by Points of the Square relatively to the Curve-triangle | 139 |
| 29. | Higher Pairs of Elements:—other Curved Figures of Constant Breadth | 139 |
| 30. | General Determination of Profiles of Elements for a given Motion | 146 |
| 31. | First Method.—Determination of the Profile of one Element, that of the other being arbitrarily assumed | 148 |
| 32. | Second Method.—Auxiliary Centroids | 152 |
| 33. | Third Method.—Profiles described by Secondary Centroids | 155 |
| 34. | Fourth Method.—Point-paths of Elements used as Profiles | 156 |
| 35. | Fifth Method.—Parallels or Equidistants to the Roulettes as Profiles | 157 |
| 36. | Sixth Method.—Approximations to Curved Profiles by Circular Arcs. Willis's Method | 160 |
| 37. | Seventh Method. The Centroids themselves as Profiles of Elements | 163 |
| 38. | Generalisation of the foregoing Methods | 164 |
| Chapter IV. Incomplete Pairs of Elements | Chapter IV. Incomplete Pairs of Elements | 169 |
| § 39. | Closure of Pairs of Elements by Sensible Forces | 169 |
| 40. | Force-Closure in the Rolling of Axoids | 171 |
| 41. | Flectional Kinematic Elements | 173 |
| 42. | Springs | 176 |
| 43. | Closure of a Pair of Elements by a Kinematic Chain | 178 |
| 44. | Complete Kinematic Closure of the Flectional Elements | 183 |
| Chapter V. Incomplete Kinematic Chains | Chapter V. Incomplete Kinematic Chains | 186 |
| § 45. | Dead Points in Mechanism,—their Passage by Means of Sensible Forces | 186 |
| 46. | Passage of the Dead Points by Chain-Closure | 188 |
| 47. | Closure of Kinematic Chains by Pairs of Elements | 191 ​ |
| Chapter VI. Sketch of the History of Machine Development | Chapter VI. Sketch of the History of Machine Development | 201 |
| § 48. | The Origin and Early Growth of Machines | 201 |
| 49. | The Development of the Machine from a Kinematic point of view | 226 |
| 50. | The Growth of Modern Machinery | 232 |
| 51. | The Present Tendency of Machine Development | 242 |
| Chapter VII. Kinematic Notation | Chapter VII. Kinematic Notation | 247 |
| § 52. | Necessity for a Kinematic Notation | 247 |
| 53. | Former Attempts | 248 |
| 54. | Nature of the Symbols required | 251 |
| 55. | Class or Name-Symbols | 252 |
| 56. | Form-Symbols | 253 |
| 57. | Symbols of Relation | 255 |
| 58. | Formulæ for simple Kinematic Chains and Mechanisms | 258 |
| 59. | Contracted Formulæ | 263 |
| 60. | Formulæ for Compound Chains | 264 |
| 61. | Formulæ for Chains containing Pressure-organs | 268 |
| 62. | Contracted Formluæ for Single Mechanisms | 270 |
| Chapter VIII. Kinematic Analysis | Chapter VIII. Kinematic Analysis | 274 |
| § 63. | The Problems of Kinematic Analysis | 274 |
| 64. | The "Mechanical Powers" or "Simple Machines" | 275 |
| 65. | The Quadric (Cylindric) Crank Chain | 283 |
| 66. | Parallel Cranks | 287 |
| 67. | Anti-parallel Cranks | 290 |
| 68. | The Isosceles Crank-train | 292 |
| 69. | The Cylindric Slider-crank Chain | 294 |
| 70. | The Isosceles Slider-crank Chain | 302 |
| 71. | Expansion of Elements in the Slider-crank Chain | 304 |
| 72. | The Normal Double Slider-crank Chain | 313 |
| 73. | The Crossed Slider-crank Chain | 318 |
| 74. | Recapitulation of the Cylindric Crank Trains | 323 |
| 75. | The Conic Quadric Crank Chain | 327 |
| 76. | Reduction of a Kinematic Chain | 333 |
| 77. | Augmentation of Kinematic Chains | 341 |
| Chapter IX. Analysis of Chamber-crank Trains | Chapter IX. Analysis of Chamber-crank Trains | 342 |
| § 78. | Chaining of Crank Mechanisms with Pressure-Organs | 342 |
| 79. | Chamber-crank Trains from the Turning Slider-crank | 344 |
| 80. | Chamber-crank Trains from the Isosceles Turning Slider-crank | 355 ​ |
| § 81. | Chamber-crank Trains from the Swinging-block | 356 |
| 82. | Chamber-crank Trains from the Turning-block | 360 |
| 83. | Chamber-crank Trains from the Swinging Slider-crank | 371 |
| 84. | Chamber-crank Trains from the Turning Double Slider-crank | 374 |
| 85. | Chamber-crank Trains from the Turning Cross-block | 375 |
| 86. | Chamber-crank Trams from the Lever-crank | 378 |
| 87. | Chamber-crank Trains from the Double-crank | 382 |
| 88. | Chamber Trains from Conic Crank Mechanisms | 384 |
| 89. | Chamber-gear from the Conic Turning Double-slider | 386 |
| 90. | Chamber-gear from the Conic Swinging Cross-block | 391 |
| 91. | Chamber-gear from the Conic Turning Cross-block | 393 |
| 92. | Review of the preceding Results | 400 |
| Chapter X. Analysis of Chamber-wheel Trains | Chapter X. Analysis of Chamber-wheel Trains | 402 |
| § 93. | Chaining of Spur-Gearing with Pressure-Organs | 402 |
| 94. | The Pappenheim Chamber-wheels | 403 |
| 95. | Fabry's Ventilator | 409 |
| 96. | Root's Blower | 411 |
| 97. | Payton's Water Meter | 414 |
| 98. | Evrard's Chamber-wheel Gear | 416 |
| 99. | Repsold's Pump | 417 |
| 100. | Dart's or Behrens' Chamber-wheel Gear | 420 |
| 101. | Eve's Chamber-wheel Gear | 422 |
| 102. | Révillion's Chamber-wheel Gear | 422 |
| 103. | Other Simple Chamber-wheel Trains | 424 |
| 104. | Compound Chamber-wheel Gear | 425 |
| 105. | Epicyclic Chamber-wheel Gear | 427 |
| Chapter XI. Analysis of the Constructive Elements of Machinery | Chapter XI. Analysis of the Constructive Elements of Machinery | 436 |
| § 106. | The Machine as a Combination of Constructive Elements | 436 |
| 107. | Screws and Screwed Joints | 438 |
| 108. | Keys, Cutters, &c., and Keyed Joints | 441 |
| 109. | Rivets and Riveting, Forced or Strained Joints | 443 |
| 110. | Pins, Axles, Shafts, Spindles | 444 |
| 111. | Couplings | 445 |
| 112. | Plummer Blocks, Bedplates, Brackets and Framing | 447 |
| 113. | Ropes, Belts, and Chains | 451 |
| 114. | Friction-wheels; Belt and Rope-gearing | 452 |
| 115. | Toothed-wheels, Chain-wheels | 453 |
| 116. | Fly-wheels | 453 |
| 117. | Levers, Cranks, Connecting-rods | 454 ​ |
| § 118. | Crossheads and Guides | 454 |
| 119. | Click-wheels and Gear | 455 |
| 120. | Reversed Motion in Free Click-trains | 459 |
| 121. | Ratchet-trains | 461 |
| 122. | Brakes and Brake-gear | 467 |
| 123. | Engaging and Disengaging Gear | 468 |
| 124. | Recapitulation of the Methods used for Stopping and Setting in Motion | 472 |
| 125. | Pipes, Steam and Pump-cylinders, Pistons and Stuffing-boxes | 473 |
| 126. | Valves | 473 |
| 127. | Springs as Constructive Elements | 480 |
| 128. | General Conclusions from the Foregoing Analysis | 480 |
| Chapter XII. The Analysis of Complete Machines | Chapter XII. The Analysis of Complete Machines | 486 |
| § 129. | Existing Methods and Treatment | 486 |
| 130. | The Tool | 490 |
| 131. | Kinematic Nature of the Tool | 493 |
| 132. | The Receptor | 497 |
| 133. | Kinematic Nature of the Complete Machine | 502 |
| 134. | Prime-movers and Direct-actors | 505 |
| 135. | The Principal Subdivisions of Complete Machines. Descriptive Analysis | 510 |
| 136. | Examples of the Descriptive Analysis of Complete Machines | 516 |
| 137. | The Relation of Machinery to Social Life | 522 |
| Chapter XIII. Kinematic Synthesis | Chapter XIII. Kinematic Synthesis | 527 |
| § 138. | General Nature of Kinematic Synthesis | 527 |
| 139. | Direct Kinematic Synthesis | 528 |
| 140. | Indirect Kinematic Synthesis | 529 |
| 141. | Diagram of the Synthetic Processes | 531 |
| 142. | Synthesis of the Lower Pairs of Elements | 532 |
| 143. | The Simpler Higher Pairs | 533 |
| 144. | Synthesis of Toothed-wheel Pairs | 535 |
| 145. | Cam Pairs | 537 |
| 146. | Recapitulation of the Pairs of Rigid Elements | 538 |
| 147. | Pairs of Elements containing Tension-Organs | 539 |
| 148. | Pairs of Elements containing Pressure-Organs | 542 |
| 149. | Recapitulation of the Pairs containing Flectional Elements | 544 |
| 150. | Determination of the Simple Chains | 545 |
| 151. | The Screw Chain | 546 |
| 152. | Cylinder-Chains | 549 ​ |
| 153. | Prism Chains | 553 |
| 154. | The Crossed and Skew Screw Chains | 555 |
| 155. | Substitution of Higher Pairs for Pairs of Revolutes | 559 |
| 156. | The Simple Wheel-chains | 562 |
| 157. | The Slider-cam Trains | 563 |
| 158. | Pulley Chains | 565 |
| 159. | Chains with Pressure-Organs | 567 |
| 160. | Compound Chains | 569 |
| 161. | Examples of Compound Chains | 572 |
| 162. | Closing Remarks | 580 |
| Notes | Notes | 585 |
| Alphabetical Index | Alphabetical Index | 615 |

## ERRATA.

Page 47, line 8, after "mechanism" add "or train."
Page 66, line 5 from bottom, p. 67, line 13 from bottom, p. 68, line 2 from bottom, p. 77, line 13 from top, for "pole" read "instantaneous centre."
Page 77, lines 24 and 25 from top, and p. 78, line 2, for "polar" read "central."
Page 121, reference number at end of top line should be $^{15}$.
Page 145, line 7 from bottom, for "XII. 1" read "XII. 2."
Page 146, line 7 from top, for "XII. 2" read "XII. 1."
Page 205, after lower footnote add "R."
Page 216, Nos. 1 and 2, fig. 168, should be inverted.
Page 269, line 4 from bottom, in formula, for "P+" read "P+."
Page 291, line 2 from bottom, insert the sign Z in formula.
Page 294, line 4 from top, in formula, for "C″$_4$" read "C″$_2$."
Page 426, bottom line, in formula, for "±" read "+."
Page 429, line 10 from bottom, in formula, for "C′$_3$" read " C″$_3$."