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[APPLIED DYNAMICS
MECHANICS


If the component along the direction of motion acts with the motion, it is called an ejfort; 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 ejort, or potential energy, is measured y 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 Ejort 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 As of the corresponding portion of the path of the point of application; the sum 2 . PAS (50)

is the whole energy exerted. If the eliort 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 fPds. (51)

Similar processes are applicable to the finding of the work performed in overcoming a varying resistance. The work done b a machine can be actually measured by means of a dynamo meter (ba).

§ 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 o posite. In other words, the sum of) the negative roducts 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 o 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 gi' en interval of time by ds'; then

E Pds=E . 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 off Illechanisms.-The principle stated in the preceding section, name y, 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 bein 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 dividin 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 == R1/', 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:- V

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 cen-tre

Obs, and resolve each °

load in the respective "

directions of motion of the

points x and y;-

X

C

»

i'

g/ 6”

/ P

b -

thus there are obtained, !

the components P cos ' ', ~-P-' P

0 and R cos qh. Let

the mechanism have a ¢l

small motion; then, for I

the instant, the link b l, ,, » d

is turning about its

instantaneous centre

OM, and, if co is its

instantaneous angular

velocity, the velocity »=

of the point x is wr, '7

and the velocity of the

point y is ws. Hence,

by the principle just

stated, P cos 0X wr-

cos ¢ X ws. 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. I 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 fi acting on the link s, to find the force f, 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 t us:-

(1) Find the three virtual centres Of., Ori, 0, ¢, which must be three points in a line.

(2) Resolve f, into two components, one of which, namely, fu, passes through O", and may be neglected, and the other fp passes through On.

(3) Find the point M, where fp joins the given direction of ft, and resolve fp into two components, of which one is in the direction MO, , and may be neglected because it passes through Off, and the other is in the given direction of ft and is therefore the force required.

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. Land. Math. Soc. (1878-1879), by the same author. ' fq

An example of the method of

solution stated above, and

taken from the Jllechanics 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 fs 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 Om Oat are at the respective axes of the wheels r and t, and the centre On 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 palper by Professor Fleeming ]enkin, “ On the application of graphic met ods to the determination of the efficiency of machinery " A

ll

g FIG. 126.

pf

fi

e o, ,

S .ft

if.

Flo. 127.