Humpage's Gear. This compound epicyclic bevil train is used by Messrs. Humpage, Jacques, and Pedersen, of Bristol, as a variable-speed gear for machine tools (see The Engineer, December 30, 1898).

=

The number of teeth in the wheels are: A 46, B = 40, B1 = 16, C 12, E = 34. The wheels A and C are loose on the shaft F, but E is keyed. The wheel A is rigidly attached to the frame of the machine, and C is driven by a stepped pulley; the arm d rotates on the shaft F; the two wheels B and B, are fixed together. Let d make one complete clockwise revolution; then the other wheels will make

= −0·541 + 1 = 0'459

Whence for one revolution of E, C makes revolutions.

The sign in the expressions for the speed of C and E is

on account of these wheels of the epicyclic train rotating in the same sense as that of the arm d.

As stated above, the nth wheel in a bevil train rotates in the same sense as the arm when n is odd, and in the opposite sense when is even; hence the sign is + for odd axes, and even axes, always counting the first as "one."

- for

It may help some readers to grasp the solution of this problem more clearly if we work it out by another method. Let A be free, and let d be prevented from rotating; turn A through one revolution; then

Revs.) product of teeth in drivers

=

==

of E product of teeth in driven wheels TXT.

Hence, when A is fixed by clamping the split bearing G, and dis rotated, the train becomes epicyclic, and since C and E are on odd axes of bevil trains, they rotate in the same sense as the arm; consequently, for reasons already given, we have

Particular attention must be paid to the sense of rotation. Bevil gears are more troublesome to follow than plain gears; hence it is well to put an arrow on the drawing, showing the direction in which the observer is supposed to be looking.

CHAPTER VI.

DYNAMICS OF THE STEAM-ENGINE.

Reciprocating Parts.-On p. 133 we gave the construction for a diagram to show the velocity of the piston at each part of the stroke when the velocity of the crank-pin was assumed to be constant. We there showed that, for an infinitely long connecting-rod or a slotted cross-head (see Fig. 159), such a diagram is a semicircle when the ordinates represent the velocity of the piston, and the abscissæ the distance it has moved through. The radius of the semicircle represents the constant velocity of the crank-pin. We see from such a diagram that the velocity of the reciprocating parts is zero at each end of the stroke, and is a maximum at the middle; hence during the first half of the stroke the velocity is increased, or the reciprocating parts are accelerated, for which purpose energy has to be expended; and during the second half of the stroke the velocity is decreased, or the reciprocating parts are retarded, and the energy expended during the first half of the stroke is given back. This alternate expenditure and paying back of energy very materially affects the smoothness of running of high-speed engines, unless some means are adopted for counteracting these disturbing effects.

We will first consider the case of an infinitely long connecting-rod, and see how to calculate the pressure at any part of the stroke required to accelerate and retard the reciprocating parts.

<-

X2

FIG. 173

The velocity diagram for this case is given in Fig. 173 (see p. 133). Let V = the linear velocity of the crank-pin, assumed constant; then the ordinates

V1, V2 represent to the same scale the velocity of the piston when it is at the positions A, A, respectively, and V1 = V sin 0.

Let the total weight of the reciprocating parts = W.

This energy must have come from the steam or other motive fluid in the cylinder.

Let P = the pressure on the piston required to accelerate the moving parts.

Work done on the piston in accelerating the

moving parts during the interval

the}

where x is the mean distance *+* of the piston from the

2

M

middle of the stroke; and when x R at the beginning and end of the stroke, we have

We shall term P the "acceleration pressure." Thus with an infinitely long connecting-rod the pressure at the end of the stroke required to accelerate or retard the reciprocating parts is equal to the centrifugal force (see p. 19), assuming the parts to be concentrated at the crank-pin, and at any other part of the stroke distant x from the middle the pressure is less in the ratio

x

R'

Another simple way of arriving at the result given above is as follows: If the connecting-rod be infinitely long, then it

C

FIG. 174.

always remains parallel to the centre line of the engine; hence the action is the same as if the connecting-rod were rigidly attached to the cross-head and piston, and the whole

rotated together as one solid body, then each point in the body would describe the arc of a circle, and would be subjected to WV2 the centrifugal force C = but we are only concerned with R the component along the centre line of the piston, marked P in the diagram. It will be seen that P vanishes in the middle of the stroke, and increases directly as the distance from the middle, becoming equal to C at the ends of the stroke. When the piston is travelling towards the middle of the

<

R

FIG. 175.

stroke the pressure P is positive, and when travelling away from the middle it is negative. Thus, in constructing a diagram to show the pressure exerted at all parts of the stroke, we put the p_wv2 first half above, and the second 9R half below the base-line. We show such a diagram in Fig. 175. The height of any point in the

sloping line ab above the base-line represents the pressure