770 MECHANICS [APPLIED MECHANICS. denote the unsteadiness of the motion of the fly-wheel; the denom inator 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 per formed, and which has consequently to be alternately stored by acceleration and restored by retardation of the fly-wheel. The value of this periodical excess is ...... (77), from which, dividing both sides by A 2 , we obtain the following equations : 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. 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 fly-wheel 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 fly-wheel in connexion with each of those points. For example, in a machine- work, the steam-engine, which is the prime mover of the various tools, has a fly-wheel 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, rivetting machines, and other tools has a fly-wheel 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. According to the computation of General Morin, the periodical excess e in steam-engines with single cranks is from ^th to nearly ^th of the energy exerted during one revolution of the crank. For a pair of steam-engines driving one shaft, with a pair of cranks at right angles to each other, the value of e is one-fourth of its value for a single cranked engine of the same kind, and of the same power with the two combined. The ordinary radius of gyration of a steam-engine fly-wheel is from three to rive times the length of the crank- arm. (For further particulars on this subject, see STEAM-ENWINE.) 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. 133. Brakes. A brake is an apparatus for stopping and diminish ing the velocity of a machine by friction, such as the friction-strap already referred to in sect. 112. 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 sect. 105, 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 ..... (79) - 134. 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 pro jectile ; and such as the propulsion of vessels, where the velocity of the water thrown backward by the paddle, screw, or other pro peller 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 propor tional 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 W 1( W 2 be the weights of the bodies ; P the effort exerted between them ; S the distance through which it acts ; Rj, R 2 the resistances opposed to the effort overcome by W 1; W 2 respectively ; E 1; E 2 the shares of the whole energy E exerted upon W^, W 2 respectively. Then F - <; F T^ . . Wo(P-R 1 ) + V 1 (P-R 3 l P- I*i Pj^Ra [ (80). ~ WjW 2 W a W 2 ) If E 1 = R 2 , 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 : EJ : E 2 ) C8T) W. w ( O1 J > >. >< i i that is to say, the energy is exerted on the bodies in shares inversely proportional to their weights ; and they receive accelerations in versely proportional to their weights, according to the principle of dynamics, already quoted in a note to sect. 321, 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, HI of the energy exerted by the powder in exploding will be employed in propelling the ball, and r | T 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. 135. 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 ..... (82) g t 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 da. be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body; then ^Ic?(a 2 ) = lada 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, sect. 104, the energy exerted by this couple in the interval dt is i&a.dt, which, being equated to the variation of energy, gi^es ,, ,da R 2 V da , OQ v -T7 dt g -rr dt Now (fig. 35), let the required deviation be a rotation of the body BB about an axis 0, not traversing the centre of gravity G, da being, as before, the deviation of angular velocity to be produced in the interval dt. A rotation with the angular velocity a about an axis may be considered as compounded of a rotation with the same angular velocity about an axis drawn through G parallel to and a translation with the velocity a. OG, OG being the perpendicular distance between the two ^^^ axes. Hence the required deviation may be regarded as compounded of a deviation / of translation dv = QG.da, to produce which there would be required, according to equation 82, a force applied at G per pendicular to the plane OG OG dt (84), and a deviation da of rotation about an axis drawn through G parallel to 0, to produce which there would be required a couple of the moment M given by equa tion 83. According to the principles of statics, the resultant of the force P, ap plied at G perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value is Thus is determined the position of the centre of percussion C, corresponding to the axis of rotation 0. It is obvious from this
equation that, for an axis of rotation parallel to traversing C, the