ing is A.-A; regard being had to the sign of A, taking the direction of A, as positive.

Ans. Applying a velocity V. to the whole engine at right angles to OP reduces the centre of the bearing to rest. O moves with velocity Vo, so the crank pin rotates round P with angular velocity A.; and C, having a velocity compounded of V, and V., it will be found is rotating round P with angular velocity A (page 177). Since pin and bearing both rotate round the same centre, the relative velocity of rotation is evidently the difference of A, and A if they are in the same direction, and the sum if in opposite directions. The relative velocity is then A.-A.

This is one case of a very important principle, viz. that the relative angular velocity of two bodies as above is the difference of their angular velocities, irrespective of what their centres of motion may be. The general principle can be proved in the above

manner.

9. Find the velocity of rubbing of the crank pin of No. 5 in its bearing when on either dead centre.

Diameter 14 ins.
Ans. 8.6; 5.33 f.s.

CHAPTER IX

THE DIRECT ACTOR, CONTINUED FORCES—CRANK EFFORTS

HAVING investigated the relative motions of, we next proceed to consider the forces which act between, the pieces of the engine.

The effort driving the engine is the total steam pressure on the piston. The resistance is a moment applied in some way to the crank shaft; it may be due to an actual moment, as the resistance of the water to the turning of a screw propeller; or it may be caused by some linear resistance, such as the resistance to a cutting tool of a planing machine, this being transmitted back along the mechanism of the machine to the shop shaft, becoming a resisting moment. In all cases, no matter what the original source be, the resistance finally shows itself as a moment applied to the shaft.

We proceed to investigate the action under Balanced Forces and neglecting friction. Let

M=resisting moment. P=total piston pressure.

Then we have

Energy exerted = work done.

Now, as we have before stated (page 137), we must always define clearly what period of time we are going to consider. This at once leads to the question, Are P and M constant or variable? For without this knowledge we cannot tell for any period what the energy or

work will be. Actually we know that P is rarely, if ever, constant; and, as we shall soon see, if P were so, M could not be. In all investigations, however, we commence with simple cases, and so we will suppose that P is a constant pressure during the whole stroke.

Again, for balanced forces, there must either be no change at any instant in the motion of the parts of the machine, or else we must select such a period of time that there is no change on the whole (compare page 96). If we kept strictly to this we should be only able to consider a whole revolution in which to compare energy

and work.

For at the commencement of the stroke,

drawing Fig. 131 with the usual lettering

OP is revolving clockwise round O,

Now CD begins

and CD (the piston and rod) is still. moving, and is not still again till the end of the forward stroke, and then OP is revolving clockwise as before, but CP is now also revolving clockwise (Fig. 132). So

D

Fig. 132.

there has been an effect produced on the mechanism (page 60); and we cannot reach a stage of no effect produced until we reach the position of Fig. 131 again.

We are then debarred from comparing together any

thing more than the mean values of P and M during a whole revolution. For this we have, P being constant,

But we want if possible to examine more closely than this, even if we can only obtain approximate results, and this we do by agreeing for the present to consider the pieces as weightless. Then if they have no weight it does not matter whether their motion be changed or no,

Fig. 133.

because no energy can be expended on them. then that we neglect the weight of the parts.

We say

We are now at liberty to apply the principle of Balanced Forces to any period we please, and the period we will select is the indefinitely short one, while the engine is in the position of Fig. 133.

Since the effect of a moment is independent of the forces of which it consists (page 65), we are at liberty to suppose the moment M applied to the shaft by means of a force R at right angles to OP, acting at P, and such that

Ra= M.

Then R exactly replaces the moment, so far as resistance to turning is concerned.

The Principle of Work, we have seen, may be written

Whence for the small period considered, i.e. at the instant,

R

=

velocity of piston

P velocity of point where R is applied

But the latter is the velocity ratio of piston to crank, and equals OT/OP (page 173). Hence

R: P=OT: OP.

[And we may notice that this holds irrespective of the constancy or otherwise of P.]

Crank Effort. Since the forces are balanced it follows that the useful effect of P must be to produce, at the end of OP, a force exactly equal and opposite to R. This force is called the Crank Effort.

It must be particularly noticed that we say the useful effect of P ; this is because the turning of the arm against R is the useful effect required. The crank effort is not the total effect of P on the shaft, as we shall see a page or two farther on.

The numerical value R then, found above, represents either the resistance, or the crank effort, as we take it in one or the other direction.

Looking back to page 173 we found there

Vp: Vo=OT: OP,

and we proceeded to draw a curve of piston velocity, OP representing the constant V..

Plainly then, since we now have

R: P=OT: OP,

if we take OP to represent the constant P, OT represents R; and the curve of piston velocity becomes a