H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. 75 



of impact, the quantities of motion produced by the finite elastic forces do not appear as, that 

 period being indefinitely small, those quantities are indefinitely small also and disappear in 

 the limit. 



In order to effect the summation of the products of the quantities of motion of the several 

 particles by their respective virtual velocities, it is necessary to ascertain the form which the 

 beam assumes at the instant after impact. We set out with the hypothesis, that the side of 

 the beam, which is struck, begins to take the form of a curve concave in every part. The 

 ordinates of this curve are the distances which the several parts of the beam described in an 

 indefinitely short time. Consequently the ordinates are proportional to the initial velocities of 

 the several particles, and by substituting their values from the equation to the curve, the 

 summation required is reduced to the summation of a function of a single variable. The 

 curve in question is here assumed to be the elastic curve, as determined by Poisson and Prof. 

 Moseley to be the form assumed by a uniform beam deflected by a pressure applied perpen- 

 dicularly at its centre. The accuracy of the computation does not, however, depend essen- 

 tially on the selection of this particular curve, for the quantity to be computed from it is 

 involved in such a manner, that if the curve had been assumed to be a portion of a circle 

 or parabola, the final results would not be very widely different. 



Poisson has shewn (Traite de Mecanique, Tom. i. p. 641) the equation to the elastic 



curve to be 



f 

 y = — 3 (3a?x -4>a?), 



where a is the whole length of the beam, and x the ordinate measured from one end along 

 the beam when undeflected. This equation is the same, mutatis mutandis, with that arrived 

 at by different methods by Professor Moseley, and given in his Principles of Engineering. 

 Squaring both sides of the equation, and integrating between limits x = ^a, and x = 0, 



I 



17 



, 7(T m 



For the whole length of the beam the integral will have double the above value, 



if 



*tfdm = ^ffxa (2), 



o So 



and therefore equation (l) becomes 



17 



f- /ma 

 35 •* 



-*/ 



Or, dividing by /, putting v for the velocity at the centre of the beam, and /xa the mass of 

 the beam = M, 



P=- Mv (3). 



35 



Now by the ordinary principles of impact, if u be the velocity of the ball before impact, and 

 m its mass, the blow is equal to the quantity of motion lost by the ball, or since v is the 

 same for both beam and ball after impact, 



P = m (u — v). 



10—2 



