may be arranged in various relative positions, but the one now practically universal is to have the cranks lying at 120° apart. In this case (Fig. 141) it is evidently

A

Fig. 141.

impossible to call either the leading crank, and so they must be distinguished in some other way. Usually they will belong to a triple expansion engine, and we distinguish them as high, intermediate, and low. It will be noticed, however, that in construction it is unnecessary to specify along which crank the combined effort is set off. For looking back to Fig. 138, when one is at OA, one is at 08, and one at O16, we add these three and set them off along OA say. Then we add 01, 09, O17, and set off along O1, and so on; till 07, 015, O23 are set off along 07. Now we come again to 08, O23, OA to set off now along 08, and so we get the same sets of three over again starting from 08 instead of OA. And we shall come to the same again at 016. So whether we start by setting off along OA, or 08, or 016, exactly the same curve will result.

The actual construction of this case we will set as an example, and it will be found that the ratio of maximum to mean crank effort is still further reduced. We append in tabular form results showing the effect of number of cylinders, and of connecting rod length, on the regularity of the turning effort. We assume equal pressures in the cylinders, and that the pressure is uniform throughout the stroke. For the method of dealing with the question, when the pressures are unequal and varying, we must refer to the larger treatise. It may be noticed, however, that equality of total pressure in the two cylinders of a compound, or the three of a triple expansion engine, is one of the conditions the designer aims at. So the assumption of such an equality is not at variance with the practical facts.

1. The curve of question 5, p. 178, is also a curve of crank effort. If the diameter of the cylinder be 68 inches, and the constant effective steam pressure 18 lbs. per square inch, find the scale of the curve; and construct carefully combined diagrams, both polar and linear-Ist, for two such cylinders on cranks at right angles; 2d, for three cylinders on cranks at 120°. The cylinders are not of the same diameter, but the total pressure is taken to be the same (page 194).

Give numerical results for the maximum, minimum, and mean crank efforts in each case.

Ans. Scale I inch to 16.68 tons

2. Draw a combined curve for the second case of the above when two of the cranks are at right angles, and the third makes equal angles with the other two. Compare the regularity of the effort in the two cases.

Ans. Maximum to mean 1.28 as against 1.07.

Minimum to mean .6 as against .77.

The value of the graphic method depends not only on the facility with which single results can be obtained, but also on the obtaining and preservation of a continuous set of results. Hence the curves above should be carefully and accurately drawn, and inked in for future reference.

CHAPTER X

UNBALANCED FORCES

IN the present chapter we shall investigate the manner of dealing with cases in which the energy applied to a body or machine does not leave it in the same state, at the end of the time considered, as it found it in at the commencement-or in other words, where the forces are not balanced.

Taking the simplest case, Fig. 142, A is a sliding piece, moved by an effort P, resisted by a resistance R. P and R not being equal, what kind of motion will ensue?

Take a movement of the slider. Then,

P

A

R

Fig. 142.

If then P>R, an amount of energy (P – R)r remains to be accounted for. If on the other hand P<R, an amount of work (RP)r has been done at the expense of some source other than the effort P. Now we know from experience that, combined with the effects just stated, in the first case A will be moving faster at the end of the period considered than it was at the commencement, and in the second case slower at the end than at the commencement.

We are then inevitably led to the conclusion that— in the first case, the body has in some way stored up in

itself the amount of energy (P–R)x, this stored up energy showing its presence by the increased velocity; while in the second case the body is in some way capable of exerting energy to an amount (R − P)x, by having its velocity decreased. Taking the two cases together they lead to this one conclusion that—a body in motion possesses, by virtue of its motion, a store of energy which can be added to or be drawn on.

Kinetic Energy.—The amount of energy thus stored up in a body in motion is called the Kinetic Energy of the body.

We can now see what form the principle of work takes when applied to the cases just considered. For

In the first case the body had, at the commencement, a certain amount of kinetic energy, this amount we call its Initial K. E. (K. E. being a common abbreviation for kinetic energy). But during the motion an amount (PR) is added to the store of energy, therefore

or

Final K. E. = Initial K. E.+(P – R)x.,

Px Rx+Final K. E. - Initial K. E.,

and it can be easily seen that the second case gives exactly the same equation. Putting this equation in words, it is

Energy exerted = work done + change of K. E. of moving

body,

which is the new form of the principle of work.

It must be noticed that change here means Final – Initial, which may be either positive or negative. We have given the form above since it is the one usually given, but the student will find it advisable, certainly on a first study, and probably even in all cases, to use the more extended form

Energy exerted = work done + Final K. E. - Initial K. E.