The Kinematics of Machinery

by Franz Reuleaux

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		v
Introduction		1
OUTLINES OF THE THEORY OF MACHINES.
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		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		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		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		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		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		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		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		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		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		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		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		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		585
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."