The Kinematics Of Machinery: Outlines Of A Theory Of Machines - Franz Reuleaux
Macmillan (1876)
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The greater part of the Theoretische Kinematik of Prof. Reuleaux, which I have now the pleasure of presenting to English and American leaders, was originally published in chapters in the Berliner Verhandlungen, under the title of Kinematische Mittheilungen, These papers, revised and enlarged, and with the addition of a chapter on Kinematic Synthesis, were published collectively in 1874-5 in the work of which the present is a translation. They have attracted considerable attention in Germany, and the principles laid down in them have already made their way into Polytechnic School instruction, not only in that country but also in Russia and Italy.

The book addresses itself to somewhat different classes of readers, or rather to readers who have had very different training, on the Continent and here. Its readers there are to a great extent the past or present students of the Polytechnic Schools, or at least those who are acquainted with polytechnic teaching. They are familiar with a regularly systematised system of machine instruction and its somewhat extended literature. Here, on the other hand, neither systematised instruction nor extended literature exists. The book addresses itself greatly to practical engineers and mechanicians, men who have often enough worked out their knowledge of the subject for themselves to a far greater extent than they have acquired it from books or lectures. To these readers some sections of the book may appear unnecessary, as referring to opinions or combating conclusions of which they have scarcely heard, and the erroneousness of which they are perfectly ready to admit. No doubt had the work originally been written for its English readers these passages might have been omitted or changed; as it is, I must merely remind those readers of the fact I have just mentioned. Here and there I have made small alterations in the text on this account, otherwise the sections referred to remain as in the original. The conclusions arrived at in them are not the less interesting that they might have been reached here, sometimes, in a more direct manner.

It may be well for me to mention here some of the leading characteristics of Prof. Reuleaux's treatment of his subject, and to point out in what respects it differs from that of his predecessors. In the oldest books upon machinery each machine was taken up as a whole, to be described and treated by itself from beginning to end. Gradually it became recognised that similar parts occurred again and again in different machines, and these parts received the name of mechanisms. They sometimes appear in a more or less abstract form in text-books of Elementary Mechanics, and have received more complete treatment in separate works. With the growth of clear ideas in physical science it became possible to separate the ideas of force, time and motion, and to consider the latter merely for its own sake without reference to the other two. Prof. Willis adopted this treatment unreservedly in his Principles of Mechanism - a work too well known to need any characterisation here - calling the study thus marked out the "Science of Pure Mechanism." Here, however, the matter stopped, later writers have been content to follow upon Willis's lines, not carrying the analytic process further, and con- tenting themselves with the examination of mechanisms as a whole in the forms in which they are presented to us by tradition or invention, without attempting to analyse them, or to investigate their mode of formation.

It is unquestionably true that by the aid of mathematics this treatment of mechanisms has given us many most valuable results, but it is equally true that the method itself is defective, and was only used for want of a better. This better method Prof. Reuleaux has attempted, and I think with great success, to indicate. Starting with the idea of motion as change of position only - and limiting himself to cases where such changes are absolutely determinate at every instant, - as always in the machine - he points out that they are conditioned simply by the geometric form of the moving bodies. Two bodies, such for instance as a screw and nut, having such forms that at any instant there is only one possible motion for each relatively to the other, form the simplest combination available for machinal purposes - such bodies he calls a pair of elements. Two or more elements from as many different pairs can be combined into a link, and such links united into kinematic chains, and it is by fixing, that is, preventing the motion of, someone link of such a chain that a mechanism is obtained. Stated thus in a few words the analysis is simple and obvious enough; like many other simple things, however, it leads to most important consequences. As one illustration merely of this, I may point to the collection of “rotary” engines and pumps examined in Chap. IX. Here will be found, among others, over thirty forms of "rotary" engines of which the kinematic chain used in the driving mechanism is absolutely identical with that of the common direct-acting engine! Their constructive forms differ most widely, and have of course too often misled their inventors, but the application of what Prof. Reuleaux calls “kinematic analysis” shows at once both their identity as kinematic chains and their relation as mechanisms. In Fig. 3, PI. XX., for instance, is shown a rotary engine which has been patented every few years since 1805 in one or another form, and in which no doubt some of my readers will recognise an old friend "schemed" in the days of their apprenticeship. Its driving mechanism is absolutely the same as that of the direct-acting engine, but with the crank fixed and the frame allowed to move round it.

In order to utilise the kinematic analysis Prof. Reuleaux has devised and elaborated the notation which is explained in Chap. VII. and used in the later part of the book. That this notation is both exceedingly simple and of practical use will be admitted by all readers of Chap. IX., but its full advantages will only be realised by those who use it for themselves. The way in which it aids the resolution of apparently complex mechanisms into quite familiar forms is often most remarkable. Use will no doubt suggest modifications and improvements in its details, but Prof. Reuleaux is very anxious that its essential features, and especially the symbols for the elements (which have been so chosen as to be as suitable as possible for the principal European languages) should not be altered.

I may mention here only one other feature in Reuleaux's work, namely, his treatment of fluids when they occur in mechanisms or machines (Chap. IV. &c.). It has long been customary, of course, to treat cords, chains, belts &c., as organs which could legitimately form part of machines, but fluids have been universally (so far as I know) excluded from consideration in this way. Reuleaux points out that fluids - "pressure-organs" - are simply contrapositives of the "tension-organs" just mentioned, and that if one be included in the study of "pure mechanism" there can be no reason for excluding the other. He gives also many instances of the way in which engineers use the one or the other as the column of fluid or the cord best suits their purpose. In examining mechanisms we consider the motions of each body as a whole, ignoring altogether its molecular condition, or more strictly assuming that it is so arranged that its molecular stability is not disturbed during the motion. This pre-supposition is made tacitly in the case of "rigid" bodies, where molecular stability is independent of the application of external force. It is made also in the case of ropes, belts, &c., for when these occur in machines it is always assumed that they are kept in tension by some force external to themselves, in any other case their motions would be quite indeterminate. With fluids it is not necessary to make any other assumption than this, but the external force must be a pressure instead of a pull, and must be supplied in directions other than that in which motion takes place

My own work in connection with Prof. Reuleanx's book has been chiefly, of course, that of translation; but a comparison of this edition with the German one will show several not un- important improvements. Some of these have been suggested by the author ; in all cases where they involved more than the changing of a few words they have been submitted to him. I may take this opportunity of acknowledging the assistance I have received from him and the interest he has taken in the progress of this English edition of his work, (which has been already published in Italian, and is now being translated into French. I have also great pleasure in acknowledging the help I have received on many occasions from my friend and colleague Prof. Henrici, F.R.S.

The references given in footnotes are mostly those of the original edition; in Chapters IX and X I have added English references where I was able to do so. The longer footnotes I am responsible for, except in cases where I have placed "R" after them. Of the notes at the end of the book I have added those which are placed in square brackets. A few of the notes in the original, which referred to matters with which English readers would probably be unacquainted, or to passages which have been altered in the text, have been shortened or omitted.

The names which Prof. Reuleaux gives to the various mechanisms have in most cases been invented by himself, and in several other instances he has had to coin words to express ideas to which individual distinctness has now first been given. Such names and words I have not tried to translate, but only to replace by equally good English ones, with what success I must leave my readers to judge. I shall be happy to receive suggestions for improvements in this matter. The names have, however, been very carefully considered, and so arranged as to fit in with each other - I venture to hope, therefore, that those who use them for instruction will not alter them without good reason. For the word “centroid,” for which I anticipate great usefulness, I have to thank Prof. Clifford.



CHAPTER I. General Outlines
1. Nature of the Machine-Problem
2. The Science of Machines
3. General Solution of the Machine-Problem

Chapter II. Phoronomic Propositions
4. Preliminary Remarks
5. Relative Motion in a Plane
6. Temporary Centre; the Central Polygon
7. Centroids; Cylindric Rolling
8. The Determination of Centroids
9. Reduction of Centroids
10. Rotation about a Point
11. Conic Rolling
12. Most general Form of the Relative Motion of Rigid Bodies
13. Twisting and Rolling of Ruled Surfaces

Chapter III. Pairs of Elements
14. Different Forms of Pairs of Elements
15. The Determination of Closed Pairs
16. Motion in Closed Pairs
17. The necessary and sufficient Restraint of Elements
18. Restraint against Sliding
19. Restraint against Turning
20. Simultaneous Restraint of Sliding and Turning
21. The Higher Pairs of Elements
22. Higher Pairs - Duangle and Triangle
23. Point-paths of the Duangle relatively to the Equilateral Triangle
24. Point-paths of the Triangle relatively to the Duangle
25. Figures of Constant Breadth
26. Higher Pairs of Elements. - Equilateral Curve-triangle and Rhombus
27. Paths of Points of the Curve-triangle relatively to the Square
28. Paths described by Points of the Square relatively to the Curve-triangle
29. Higher Pairs of Elements: other Curved Figures of Constant Breadth
30. General Determination of Profiles of Elements for a given Motion
31. First Method. - Determination of the Profile of one Element, that of the other being arbitrarily assumed
32. Second Method. - Auxiliary Centroids
33. Third Method. - Profiles described by Secondary Centroids
34. Fourtli Method. - Point-paths of Elements used as Profiles
35. Fifth Method. - Parallels or Equidistants to the Roulettes as Profiles
36. Sixth Method. - Approximations to Curved Profiles by Circular Arcs. Willis's Method
37. Seventh Method. - The Centroids themselves as Profiles of Elements
38. Generalisation of the foregoing Methods

Chapter IV. Incomplete Pairs of Elements
39. Closure of Pairs of Elements by Sensible Forces
40. Force-Closure in the Rolling of Axoids
41. Flectional Kinematic Elements
42. Springs
43. Closure of a Pair of Elements by a Kinematic Chain
44. Complete Kinematic Closure of the Flectional Elements

Chapter V. Incomplete Kinematic Chains
45. Dead Points in Mechanism, - their Passage by Means of Sensible Forces
46. Passage of the Dead Points by Chain-Closure
47. Closure of Kinematic Chains by Pairs of Elements

Chapter VI. Sketch of the History of Machine Development
48. The Origin and Early Growth of Machines
49. The Development of the Machine from a Kinematic point of view
50. The Growth of Modem Machinery
51. The Present Tendency of Machine Development

Chapter VII. Kinematic Notation
52. Necessity for a Kinematic Notation
53. Former Attempts
54. Nature of the Symbols required
55. Class or Name-Symbols
56. Form-Symbols
57. Symbols of Relation
58. Formulae for simple Kinematic Chains and Mechanisms
59. Contracted Formulae
60. Formulae for Compound Chains
61. Formulae for Chains containing Pressure-organs
62. Contracted Formulae for Single Mechanisms

Chapter VIII. Kinematic Analysis
63. The Problems of Kinematic Analysis 274
64. The “Mechanical Powers” or “Simple Machines”
65. The Quadric (Cylindric) Crank Chain
66. Parallel Cranks
67. Anti-parallel Cranks
68. The Isosceles Crank-train
69. The Cylindric Slider-crank Chain
70. The Isosceles Slider-crank Chain
71. Expansion of Elements in the Slider-crank Chain
72. The Normal Double Slider-crank Cham
73. The Crossed Slider-crank Chain
74. Recapitulation of the Cylindric Crank Trains
75. The Conic Quadric Crank Chain
76. Reduction of a Kinematic Chain
77. Augmentation of Kinematic Chains

Chapter IX. Analysis of Chamber-crank Trains
78. Chaining of Crank Mechanisms with Pressure-Organs
79. Chamber-crank Trains from the Turning Slider-crank
80. Chamber-crank Trains from the Isosceles Turning Slider-crank
81. Chamber-crank Trains from the Swinging-block
82. Chamber-crank Trains from the Turning-block
83. Chamber-crank Trains from the Swinging Slider-crank
84. Chamber- crank Trains from the Turning Double Slider-crank
85. Chamber-crank Trains from the Turning Cross-block
86. Chamber-crank Trains from the Lever-crank
87. Chamber-crank Trains from the Double-crank
88. Chamber Trains from Conic Crank Mechanisms
89. Chamber-gear from the Conic Turning Double-slider
90. Chamber-gear from the Conic Swinging Cross-block
91. Chamber-gear from the Conic Turning Cross-block
92. Review of the preceding Results

Chapter X. Analysis op Chamber-wheel Trains
93. Chaining of Spur-Gearing with Pressure-Organs
94. The Pappenheim Chamber-wheels
95. Fabry's Ventilator
96. Root's Blower
97. Payton's Water Meter
98. Evrard's Chamber- wheel Gear
99. Repsold's Pump
100. Dart's or Behrens' Chamber-wheel Gear
101. Eve's Chamber- wheel Gear
102. Revillion's Chamber-wheel Gear
103. Other Simple Chamber-wheel Trains
104. Compound Chamber- wheel Gear
105. Epicyclic Chamber-wheel Gear

Chapter XI. Analysis of the Constructive Elements op Machinery
106. The Machine as a Combination of Constructive Elements
107. Screws and Screwed Joints
108. Keys, Cutters, &c., and Keyed Joints
109. Rivets and Riveting, Forced or Strained Joints
110. Pins, Axles, Shafts, Spindles
111. Couplings
112. Plummer Blocks, Bedplates, Brackets and Framing
113. Ropes, Belts, and Chains
114. Friction- wheels; Belt and Rope-gearing
115. Toothed-wheels, Chain-wheels
116. Fly-wheels
117. Levers, Cranks, Connecting-rods
118. Crossheads and Guides
119. Click-wheels and Gear
120. Reversed Motion in Free Click-trains
121. Katchet-trains
122. Brakes and Brake-gear
123. Engaging and Disengaging Gear
124. Recapitulation of the Methods used for Stopping and Setting in Motion
125. Pipes, Steam and Pump-cylinders, Pistons and Stuffing-boxes
126. Valves
127. Springs as Constructive Elements
128. General Conclusions from the Foregoing Analysis

Chapter XII. The Analysis of Complete Machines
129. Existing Methods and Treatment
130. The Tool
131. Kinematic Nature of the Tool
132. The Receptor
133. Kinematic Nature of the Complete Machine
134. Prime-movers and Direct-actors
135. The Principal Subdivisions of Complete Machines. Descriptive Analysis
136. Examples of the Descriptive Anal3'sis of Complete Machines
137. The Relation of Machinery to Social Life

Chapter XIII. Kinematic Synthesis
138. General Nature of Kinematic Synthesis
139. Direct Kinematic Synthesis
140. Indirect Kinematic Synthesis
141. Diagram of the Synthetic Processes
142. Synthesis of the Lower Pairs of Elements
143. The Simpler Higher Pairs
144. Synthesis of Toothed- wheel Pairs
145. Cam Pairs
146. Recapitulation of the Pairs of Rigid Elements
147. Pairs of Elements containing Tension-Organs
148. Pairs of Elements containing Pressure-Organs
149. Recapitulation of the Pairs containing Flectional Elements
150. Determination of the Simple Chains
151. The Screw Chain
152. Cylinder-Chains
153. Prism Chains
154. The Crossed and Skew Screw Chains
156. Substitution of Higher Pairs for Pairs of Re volutes
156. The Simple Wheel-chains
157. The Slider-cam Trains
158. Pulley Chains
159. Chains with Pressure-Organs
160. Compound Chains
161. Examples of Compound Chains
162. Closing Remarks

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