# The elements of mechanism - Goodeve

THE ELEMENTS OF MECHANISM

Designed for students of applied mechanics

BY T. M. GOODEVE

LONDON, LONGMAN AND GREEN, 1912

The elements of mechanism

INTRODUCTION.

A machine may be defined to be an assemblage of moving parts, constructed for the purpose of transmitting motion or force, and of modifying, in various ways, the motion or force  transmitted.

The parts of a machine are set in motion by some moving power, which may be derived from any convenient source, and the machine itself must be constructed with reference to the character of the power from which its motion is derived.

The introduction of the steam-engine has greatly simplified the art of constructive mechanism, by rendering the source of power uniform and undeviating; the steam-engine is always employed to give rotation to a piece of shafting, and the mechanic is required to derive from the smooth and steady rotation of a shaft every movement which the nature of the work may demand. Thus the starting point for steam-machinery is everywhere the same, and the problem of making a machine resolves itself into a question of the resolution or transfer of circular motion in every variety of manner, and subject to every possible modification. Accordingly we propose to commence with a discussion of the methods adopted for the conversion of circular into reciprocating motion; then to proceed to the inverse problem, or to examine the conversion of reciprocating into circular motion; to pass on to an enquiry into the results deduced from combinations of wheelwork in trains, giving the theory which has led to an accurate shaping of the teeth of wheels; to consider, further, some arrangements by which a moving piece may be made the recipient of two or more independent motions; and finally, to conclude by collecting and analyzing certain miscellaneous contrivances which produce results of a specific and very noticeable character.

It will be necessary to premise a few general remarks and definitions.

In the transfer of motion or force from one axis to another, wheels furnished with teeth are commonly employed. The various calculations connected with the forms of teeth which are suitable for this purpose will be given hereafter; at the present time we may remark, that the most simple case of the transfer of motion from one axis to another occurs when a circular disc or plate moves another by
rolling contact.

In such a case the uniform motion of the axis, a, conveys a perfectly even and uniform motion to the other axis, B. (Fig. 1.)

If A and B were circular plates with smooth flat edges, and very accurately adjusted, we might expect a to move B by friction alone, without any slipping of the surfaces in contact, but we could never expect a to overcome any great resistance to motion in b ; or, in other words, we could not convey force by the action of one disc upon the other.

The transmission of force being an essential condition in machinery, the discs a and b are provided with teeth as in the annexed figure, and the mechanist endeavours so to form and shape the teeth that the motion  shall be exactly the same as if one circle rolled upon another.

Herein consists the perfection of wheel-work, a perfectly uniform motion of the axis a is to be conveyed by means of teeth to the axis b j and the motion of b, when tested with microscopic accuracy, is to be no less even and uniform than that of A.

Since, then, it appears that the motions of a and b are exactly the same as those of two circles rolling upon each other, such imaginary circles may always be conceived to exist, and are called the pitch circles of the wheels in question. They are represented by the dotted lines in the annexed sketch.

So much of the tooth lies within the pitch circle is called its root or flank, and the portion beyond the pitch circle is called the point or addendum.

The pitch of a tooth is the space a c upon the pitch circle cut off by the corresponding edges o (Fig 3.)

Spur wheels are represented in Fig. 2, and are those in which the teeth stand out radially along the circumference. In a face wheel the cogs or pins are placed perpendicularly to the plane of the wheel. (Fig. 4.) A crown wheel is provided with teeth upon the edge of its rim (Fig. 5.) In annular wheels the teeth are cut upon the inside of an annulus or ring. A straight bar, provided with teeth, is called a rack, and a wheel with a small number of teeth is termed a pinion. The wheels a and b are suited to convey motion only between parallel axes ; it often happens, however, that the axes concerned in any movement are not parallel, and as a consequence they may, or may not, meet in a point In the latter case we proceed by successive steps and continually introduce intermediate intersecting axes, and thus we are led to the use of inclined wheels whose axes meet each other, and which are known as Bevil wheels, (Fig. 6.) It is a well known fact in Geometry that two right cones which have a common vertex will roll upon each other, and the same would be true of the frustra of two cones as c r p b, D Q p B, in Fig. 7. The rolling of the cones will allow us to consider any pair of circles in contact as the pitch circles of the cones, and teeth may be shaped upon the frustra, so as to produce the same even motion as that which exists in the case of spur wheels.

Equal bevil wheels whose axes are at right angles are termed mitre wheels.

It is sometimes convenient that the axes of the bevil wheels should pass close to each other without intersecting; the teeth have then a peculiar form, and the wheels are known as skew bevils.

If a horizontal line A p, Fig. 8, revolve uniformly in one direction, and at the same time be made to ascend or descend with a uniform velocity, it will trace out a screw surface APRB, The points of intersection of this generating line with any circular cylinder whose axis coincides with A b, will form a screw thread, p R, upon the surface of such cylinder.

The screw thread used in machinery is a projecting rim of a certain definite form running round the cylinder and obeying the same geometrical law as the ideal thread just described.

The pitch of a screw is the space along A b, through which the generating line moves in completing one entire revolution, but in practice the pitch of a screw bolt is usually estimated by observing the number of ridges which occur in an inch of its length ; thus we speak of the screw of one-eighth of an inch pitch as being a screw with eight threads to the inch.

INTRODUCTION
General Statement. Definitions. Spur, Crown, and Bevil Wheels. Screw Surface. Screw Thread. Pitch of a Screw. Worm Wheel. Belts or Bands. Shafting. Arbor. Driver. Follower. Gearing or Gear

ON THE CONVERSION OF CIRCULAR INTO RECIPROCATING MOTION.
Art 1 - 3. Elementary Considerations. 4. The Crank and Connecting Rod. 5. The Eccentric Circle. 6. The Swash Plate. 7. The Eccentric. 8 - 10. Intermittent Motion. 11 - 14. Escapements. 15 - 29. Cams of various kinds: the Heart Wheel, the Worm Barrel, the Expansion Cam, Double Cams. 30 - 35. Mangle Wheels and Racks. 36 - 43. Circular in Reciprocating Motion by Wheelwork: Collier's Planing Machine, Whitworth*s Reversing Motions, Screwing Machines. 44, 45. Crossed and Open Bands. 46- 50. Reciprocating Motion with a Quick Return : Whitworth's Shaping Machine. 51. Stanhope Levers. 52, 53. Reciprocation by Linkwork

ON THE CONVERSION OF RECIPROCATING INTO CIRCULAR MOTION.
Art. 54 - 56. General Principles. 57. Ratchet Wheels. 5S. Practical Subdivision of the Teeth. 59. Detent. 60. Equivalent for a Ratchet Wheel. 61, 62. Feed in a Planing Machine. 63. Escapement. 64. Levers of Lagarousse. 65. Screw Barrel

ON THE TEETH OF WHEELS.
Art. 66 - 68. Statement of the Problem. 69. Epicycloid and Hypocycloid. 70-72. Solution of the Problem. 73-75. Teeth with Radial Flanks. 76-78. Pin Wheels. 79,80. Involute Teeth. 81, 82. Rucks and Pinions. 83-87. General Considerations. 88. Bevil Wheels

ON THE USE OF WHEELS IN TRAINS.
Art. 89-91. Wheels in Trains: Value of the Train, Idle Wheels, Marlborough Wheel 92. Circular and Diametral Pitch. 93. Eight day Clock. 94. The Screw-cutting Lathe. 95. Reducing Motion. 96, 97. Connection of Axes

ON AGGREGATE MOTION.
Art 98 - 100. Examples: Lazy Tongs, Differential Screw, Chinese Windlass. 101, 102. Epicyclic Trains. 103. Ferguson’s Paradox. 104. Sun and Planet Wheels. 105 - 107. Houldsworth’s Bobbin Motion. 108. Phases of the Moon. 109. Slow Motion. 110. Comparison of Fixed and Epicyclic Trains. 111. Equation Clocks. 112 - 116. Parallel Motion. 117. Deviation from the Vertical. 118. Similar Curves. 119. The Pantograph. 120, 121. Parallel Motions of Beam Engines. 122, 123. Parallel Motion of the “Gorgon” Engines. 124. Parallel Motion for rolling Steel Plates. 125 - 128. Drilling and Boring Machinery. 130. Wheel Boiling. 131. Watt's Indicator. 132, 133. The oval Chuck. 134. Siemen’s Chronometric Governor

CHAPTER VI - ON MISCELLANEOUS CONTRIVANCES.
Art. 135 - 142. The Fusee and its Applications : Formation of a "Cop," Boberts's Winding-on Motion, Screw of varying Pitch, The Snail, The Disc and Boiler. 143. Conical Pulleys. 144. Speed Pulleys. 145-147. Boiling Curves. 148. The Double Eccentric 149. Bell Crank Levers. 150. The Geneva Stop. 151. The Star Wheel. 152. Counting Wheels 2 The Differential Worm Wheel. 153. The Toggle Joint. 154. Saxtons Differential Pulleys. 155. Step Wheels. 156. Hooke's Joint