CHAPTER VII.

MECHANICAL ILLUSTRATIONS OF SIMPLE HARMONIC

MOTION.

83. In this chapter we propose to describe instances of simple harinonic motion not depending on vibrations of pendulums, or on forces of elasticity, but on arrangements for the transmission and transformation of motion, and especially on arrangements for transforming circular into rectilinear motion.

84. Uniform circular motion can be converted into simple harmonic motion by the arrangement shown in Fig. 29. A B is a piece which by means of the guides GG

A

G

FIG. 29.

is constrained to travel in a straight line; and in one part of it there is a slot at right angles to the line of travel. A crank revolving uniformly in a circle (both indicated by dotted lines) passes through this slot, which it fits so accurately that there is no shake, and at the same time

no pinching. The slot and the piece A B in which it is cut will evidently be compelled to take a simple harmonic motion.

This plan involves a large amount of friction; and it would be difficult to preserve a good fit, as the parts would wear loose. Instead of allowing the crank to rub against the slot, it is better, as suggested by, the Rev. F. Bashforth, in 1845, to make the crank work in a circular hole in the centre of a sliding-piece which travels in the slot. Fig. 30, which is copied from Mr. Bashforth's

FIG. 30.

drawing, shows the slot and sliding-piece with the hole. in its centre.

85. A motion approximately simple harmonic can be obtained by the arrangement shown in Fig. 31.

A B is a

B

C

FIG. 31.

crank revolving uniformly round the centre A.

CD is a

rod guided to move in a straight line passing through a,

and B C is a connecting-rod jointed at B and C to the other two pieces. CD will have a reciprocating motion, which will be more nearly simple harmonic as the ratio of the length of the connecting-rod to that of the crank is greater.1 The arrangement here described is met with in many of the commonest forms of steam-engine, CD being the piston-rod and A B the crank, which it drives with a nearly uniform velocity of rotation; the uniformity being maintained by means of a fly-wheel, or in the locomotive by the inertia of the engine and train. A similar arrangement is also generally employed for working the slide-valves, the crank AB being usually replaced by an eccentric mounted on the axle of the fly-wheel. If в be the centre of this eccentric and a the fixed point round which в revolves, the motion of the connecting-rod BC and of the valve-rod C D will be precisely the same as if A B were a crank.

86. In steam-engines which admit of being reversed, the apparatus for reversing consists usually of a combination of two eccentrics, each having its own connectingrod for giving an approximately simple harmonic motion to the slide-valves. The principle of its action is illustrated by Fig. 32, where A and B are the two eccentrics, their centres revolving in one and the same circle (the

1 If the projection of BC upon the straight line A C D can be regarded as of constant length, the motion of c is the same as the motion of the projection of B, and is therefore simple harmonic. In order that the motion of c may be sensibly simple harmonic, the difference between the greatest and least projections of BC (the former being BC itself) must be negligible in comparison with the amplitude A B.

dotted circle in the figure) round a fixed centre. The line joining the centres of the two eccentrics is always a diameter of this circle, so that when one is in the extreme position to the right the other is in the extreme position to the left. CD is the valve-rod, which is constrained to travel in what is very nearly a straight line, passing through the centre of the dotted circle. The ends, E F, of the two connecting-rods are joined by a piece in which

a slot is cut for the purpose of receiving a button G, which moves with the valve-rod. In the figure the button is represented as midway along the slot. In this position, the opposite motions of the points EF combine to leave the button and valve-rod nearly at rest. This accordingly is the position for stopping the engine.

By means of the bar FII, which contains three holes.

for fixing it by a pin L, the slot can be made to travel along the button. When the hole a is brought down to L, the upper end of the slot will be brought down to the button, and the movement of the button will be governed by the eccentric A. This is the position for driving the engine in one direction-say forwards. Then, to reverse the engine, the hole 6 is brought to L, and the lower end of the slot is thus brought to the button, which will now be driven by the eccentric в instead of by A, and the

phases of motion for the valve-rod (and consequently fo the piston) will be reversed..

87. New method of obtaining S.H. motion.

A rigorous simple harmonic motion can be obtained without the friction of guides by employing a pantagraph (as described in § 22) to give the arithmetical mean of two equal and opposite uniform circular motions (see § 26), as