If the component along the direction of motion acts with the motion, it is called an effort; if against the motion, a resistance. The component across the direction of motion is a lateral pressure; the unbalanced lateral pressure on any piece, or part of a piece, is deflecting force. A lateral pressure may increase resistance by causing friction; the friction so caused acts against the motion, and is a resistance, but the lateral pressure causing it is not a resistance. Resistances are distinguished into useful and prejudicial, according as they arise from the useful effect produced by the machine or from other causes.
§ 86. Work.—Work consists in moving against resistance. The work is said to be performed, and the resistance overcome. Work is measured by the product of the resistance into the distance through which its point of application is moved. The unit of work commonly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a foot-pound.
Work is distinguished into useful work and prejudicial or lost work, according as it is performed in producing the useful effect of the machine, or in overcoming prejudicial resistance.
§ 87. Energy: Potential Energy.—Energy means capacity for performing work. The energy of an effort, or potential energy, is measured by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work.
When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved.
§ 88. Variable Effort and Resistance.—If an effort has different magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length Δs of the corresponding portion of the path of the point of application; the sum
| Σ · PΔs | (50) |
is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum approaches continually as the divisions of the path are made smaller and more numerous, and is expressed by
| Pds. | (51) |
Similar processes are applicable to the finding of the work performed in overcoming a varying resistance.
The work done by a machine can be actually measured by means of a dynamometer (q.v.).
§ 89. Principle of the Equality of Energy and Work.—From the first law of motion it follows that in a machine whose pieces move with uniform velocities the efforts and resistances must balance each other. Now from the laws of statics it is known that, in order that a system of forces applied to a system of connected points may be in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero,—products being considered positive or negative according as the direction of the forces and the possible motions of their points of application are the same or opposite.
In other words, the sum of the negative products is equal to the sum of the positive products. This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time the energy exerted is equal to the work performed.
The symbolical expression of this law is as follows: let efforts be applied to one or any number of points of a machine; let any one of these efforts be represented by P, and the distance traversed by its point of application in a given interval of time by ds; let resistances be overcome at one or any number of points of the same machine; let any one of these resistances be denoted by R, and the distance traversed by its point of application in the given interval of time by ds′; then
| Σ · Pds = Σ · Rds′. | (52) |
The lengths ds, ds′ are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.
§ 90. Static Equilibrium of Mechanisms.—The principle stated in the preceding section, namely, that the energy exerted is equal to the work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds′ were made, it becomes Pds/dt = Rds′/dt, that is, Pv = Rv′, which shows that the force ratio is the inverse of the velocity ratio. It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points. The relation between the effort and the resistance may be found by means of this principle for all kinds of mechanisms, when the friction produced by the components of the forces across the direction of motion of the two points is neglected. Consider the following example:—
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| Fig. 126. |
A four-bar chain having the configuration shown in fig. 126 supports a load P at the point x. What load is required at the point y to maintain the configuration shown, both loads being supposed to act vertically? Find the instantaneous centre Obd, and resolve each load in the respective directions of motion of the points x and y; thus there are obtained the components P cos θ and R cos φ. Let the mechanism have a small motion; then, for the instant, the link b is turning about its instantaneous centre Obd, and, if ω is its instantaneous angular velocity, the velocity of the point x is ωr, and the velocity of the point y is ωs. Hence, by the principle just stated, P cos θ × ωr = R cos φ × ωs. But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to Pp = Rq, which shows that the ratio of the two forces may be found by taking moments about the instantaneous centre of the link on which they act.
The forces P and R may, however, act on different links. The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force ƒs, acting on the link s, to find the force ƒt acting in a given direction on the link t, which will keep the mechanism in static equilibrium. The graphic solution of this problem may be effected thus:—
(1) Find the three virtual centres Ors, Ort, Ost, which must be three points in a line.
(2) Resolve ƒs into two components, one of which, namely, ƒq, passes through Ors and may be neglected, and the other ƒp passes through Ost.
(3) Find the point M, where ƒp joins the given direction of ƒt, and resolve ƒp into two components, of which one is in the direction MOrt, and may be neglected because it passes through Ort, and the other is in the given direction of ƒt and is therefore the force required.
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| Fig. 127. |
This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is given in ch. 8 of his Mechanics of Machinery. Another general solution of the problem is given in the Proc. Lond. Math. Soc. (1878–1879), by the same author. An example of the method of solution stated above, and taken from the Mechanics of Machinery, is illustrated by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r fixed. Let it be required to find the vertical force which must act at the pitch radius of the last wheel t to balance exactly a force ƒs acting vertically downwards on the arm at the point indicated in the figure. The two links concerned are the last wheel t and the arm s, the wheel r being the fixed link of the mechanism. The virtual centres Ors, Ost are at the respective axes of the wheels r and t, and the centre Ort divides the line through these two points externally in the ratio of the train of wheels. The figure sufficiently indicates the various steps of the solution.
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”
