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1016
[APPLIED DYNAMICS
MECHANICS

The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the motion of the body, is

(v2v1) V = gE / V2w.
(74)

§ 118. Actual Energy of a Shifting Body.—The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by overcoming resistance while being retarded from the same velocity of translation v to a state of rest, is

wv2 / 2g.
(75)

This is called the actual energy of the motion of the body, and is half the quantity which in some treatises is called vis viva.

The energy stored or restored, as the case may be, by the deviations of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished.

§ 119. Principle of the Conservation of Energy in Machines.—The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in § 89 for machines of uniform speed.

In any given interval during the working of a machine, the energy exerted added to the energy restored is equal to the energy stored added to the work performed.

§ 120. Actual Energy of Circular Translation—Moment of Inertia.—Let a small body of the weight w undergo translation in a circular path of the radius ρ, with the angular velocity of deflexion α, so that the common linear velocity of all its particles is v = αρ. Then the actual energy of that body is

wv2 / 2g = wa2ρ2 / 2g.
(76)

By comparing this with the expression for the centrifugal force (wa2ρ/g), it appears that the actual energy of a revolving body is equal to the potential energy Fρ/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.

The product wρ2/g, by which the half-square of the angular velocity is multiplied, is called the moment of inertia of the revolving body.

§ 121. Flywheels.—A flywheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its angular velocity.

The principles according to which variations of angular velocity store and restore energy are the same as those of § 117, only substituting moment of inertia for mass, and angular for linear velocity.

Let W be the weight of a flywheel, R its radius of gyration, a2 its maximum, a1 its minimum, and A = ½ (α2 + α1) its mean angular velocity. Let

I/S = (α2α2) / A

denote the unsteadiness of the motion of the flywheel; the denominator S of this fraction is called the steadiness. Let e denote the quantity by which the energy exerted in each cycle of the working of the machine alternately exceeds and falls short of the work performed, and which has consequently to be alternately stored by acceleration and restored by retardation of the flywheel. The value of this periodical excess is—

e = R2W (α22α12), 2g,
(77)

from which, dividing both sides by A2, we obtain the following equations:—

e / A2 = R2W / gS

R2WA2 / 2g = Se / 2.

(78)

The latter of these equations may be thus expressed in words: The actual energy due to the rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical excess of energy multiplied by the steadiness.

In ordinary machinery S = about 32; in machinery for fine purposes S = from 50 to 60; and when great steadiness is required S = from 100 to 150.

The periodical excess e may arise either from variations in the effort exerted by the prime mover, or from variations in the resistance of the work, or from both these causes combined. When but one flywheel is used, it should be placed in as direct connexion as possible with that part of the mechanism where the greatest amount of the periodical excess originates; but when it originates at two or more points, it is best to have a flywheel in connexion with each of these points. For example, in a machine-work, the steam-engine, which is the prime mover of the various tools, has a flywheel on the crank-shaft to store and restore the periodical excess of energy arising from the variations in the effort exerted by the connecting-rod upon the crank; and each of the slotting machines, punching machines, riveting machines, and other tools has a flywheel of its own to store and restore energy, so as to enable the very different resistances opposed to those tools at different times to be overcome without too great unsteadiness of motion. For tools performing useful work at intervals, and having only their own friction to overcome during the intermediate intervals, e should be assumed equal to the whole work performed at each separate operation.

§ 122. Brakes.—A brake is an apparatus for stopping and diminishing the velocity of a machine by friction, such as the friction-strap already referred to in § 103. To find the distance s through which a brake, exerting the friction F, must rub in order to stop a machine having the total actual energy E at the moment when the brake begins to act, reduce, by the principles of § 96, the various efforts and other resistances of the machine which act at the same time with the friction of the brake to the rubbing surface of the brake, and let R be their resultant—positive if resistance, negative if effort preponderates. Then

s = E / (F + R).
(79)

§ 123. Energy distributed between two Bodies: Projection and Propulsion.—Hitherto the effort by which a machine is moved has been treated as a force exerted between a movable body and a fixed body, so that the whole energy exerted by it is employed upon the movable body, and none upon the fixed body. This conception is sensibly realized in practice when one of the two bodies between which the effort acts is either so heavy as compared with the other, or has so great a resistance opposed to its motion, that it may, without sensible error, be treated as fixed. But there are cases in which the motions of both bodies are appreciable, and must be taken into account—such as the projection of projectiles, where the velocity of the recoil or backward motion of the gun bears an appreciable proportion to the forward motion of the projectile; and such as the propulsion of vessels, where the velocity of the water thrown backward by the paddle, screw or other propeller bears a very considerable proportion to the velocity of the water moved forwards and sideways by the ship. In cases of this kind the energy exerted by the effort is distributed between the two bodies between which the effort is exerted in shares proportional to the velocities of the two bodies during the action of the effort; and those velocities are to each other directly as the portions of the effort unbalanced by resistance on the respective bodies, and inversely as the weights of the bodies.

To express this symbolically, let W1, W2 be the weights of the bodies; P the effort exerted between them; S the distance through which it acts; R1, R2 the resistances opposed to the effort overcome by W1, W2 respectively; E1, E2 the shares of the whole energy E exerted upon W1, W2 respectively. Then

        E : E1 : E2  
:: W2 (P − R1) + W1 (P − R2) : P − R1 : P − R2 .
W1W2 W1 W2
(80)

If R1 = R2, which is the case when the resistance, as well as the effort, arises from the mutual actions of the two bodies, the above becomes,

  E  : E1 : E2
W1 + W2 : W2 : W1,
(81)

that is to say, the energy is exerted on the bodies in shares inversely proportional to their weights; and they receive accelerations inversely proportional to their weights, according to the principle of dynamics, already quoted in a note to § 110, that the mutual actions of a system of bodies do not affect the motion of their common centre of gravity.

For example, if the weight of a gun be 160 times that of its ball 160/161 of the energy exerted by the powder in exploding will be employed in propelling the ball, and 1/161 in producing the recoil of the gun, provided the gun up to the instant of the ball’s quitting the muzzle meets with no resistance to its recoil except the friction of the ball.

§ 124. Centre of Percussion.—It is obviously desirable that the deviations or changes of motion of oscillating pieces in machinery should, as far as possible, be effected by forces applied at their centres of percussion.

If the deviation be a translation—that is, an equal change of motion of all the particles of the body—the centre of percussion is obviously the centre of gravity itself; and, according to the second law of motion, if dv be the deviation of velocity to be produced in the interval dt, and W the weight of the body, then

P = W · dv
g dt
(82)

is the unbalanced effort required.

If the deviation be a rotation about an axis traversing the centre of gravity, there is no centre of percussion; for such a deviation can only be produced by a couple of forces, and not by any single force. Let dα be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body about an axis through its centre of gravity; then ½Id(α2) = Iα dα is the variation of the body’s actual energy. Let M be the moment of the unbalanced couple required to produce the deviation; then by equation 57, § 104, the energy exerted by this couple in the interval dt is Mα dt, which, being equated to the variation of energy, gives

M = I dα = R2W · dα .
dt g dt
(83)

R is called the radius of gyration of the body with regard to an axis through its centre of gravity.

Fig. 133.

Now (fig. 133) let the required deviation be a rotation of the body BB about an axis O, not traversing the centre of gravity G, dα