The relation between the effort and the resistance in a machine to include the effect of friction at the joints has been investigated in a paper by Professor Fleeming Jenkin, “On the application of graphic methods to the determination of the efficiency of machinery” (Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented by a frame of links the stresses in which are identical with the pressures at the joints of the mechanism. This self-strained frame is called the dynamic frame of the machine. The driving and resisting efforts are represented by elastic links in the dynamic frame, and when the frame with its elastic links is drawn the stresses in the several members of it may be determined by means of reciprocal figures. Incidentally the method gives the pressures at every joint of the mechanism.
§ 91. Efficiency.—The efficiency of a machine is the ratio of the useful work to the total work—that is, to the energy exerted—and is represented by
| Σ · Ruds′ | = | Σ · Ru ds′ | = | Σ · Ru ds′ | = | U | . |
| Σ · R ds′ | Σ · Ru ds′ + Σ · Rp ds′ | Σ · P ds | E |
(53)
Ru being taken to represent useful and Rp prejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine.
§ 92. Power and Effect.—The power of a machine is the energy exerted, and the effect the useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.
The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.
§ 93. Modulus of a Machine.—In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances arc generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of the lost work, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine. The general form of the modulus may be expressed thus—
E = U + φ (U, A) + ψ (A),
(54)