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



The subject of the present investigation may be stated as follows : 



An elastic beam of uniform density and section throughout its length, abuts at each 

 extremity against a fixed vertical prop, and is impinged upon at its centre by a ball moving 

 horizontally, with an assigned velocity in a direction perpendicular to the length of the beam 

 before collision, and subsequently moving in contact with the beam throughout its deflection. 

 It is required to determine the deflection of the beam produced by the impact. 



The dynamical circumstances of the problem may be divided into two stages. The first 

 consists in the sudden alteration of the velocity of the ball, at the instant of collision ; the 

 second, the effect of the elastic forces developed in the beam by deflection, in destroying the 

 vis viva which the system has immediately after collision. 



(1) In order to determine the first part of the problem, it will be assumed for the present 

 that the ends of the beam remain in contact with the fixed abutments, and that the side of 

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

 case in which the ends of the beam recoil from their bearings after impact, requires a different 

 method of investigation. Now in the first case referred to, the curve will not, while the 

 deflection is small, differ considerably from the elastic curve of a beam deflected by statical 

 pressure at its centre. 



As the beam begins to take a curvilinear form, it begins to move in different parts with 

 different velocities. But while the deflection is indefinitely small, the velocities are parallel to 

 the direction of impact and proportional to the spaces described. 



D'Alemberfs Principle holds for simultaneous percussions as well as for finite forces and, 

 as it reduces every dynamical problem to a statical form, may here be combined with the 

 principle of virtual velocities. The legitimacy of combining the two principles is specially 

 shewn by Poisson in the ninth chapter of his Traite de Mecanique, numero 535. 



Let it be supposed that the arbitrary displacement is that which actually occurs during 

 motion. To construct the equation of virtual velocities, this displacement of each particle 

 must be multiplied by its quantity of motion ; i. e. by its initial velocity multiplied by its 

 mass. The sum of the products so formed must be put equal to the external impulsive force 

 of the ball multiplied by its virtual velocity. 



If d,v be an element of the length of the beam and /udx its mass, initially at rest, and y 

 the small distance through which it moves in the indefinitely small time t, parallel to the 



direction of impact, - is the velocity, fxd.v.- is the "quantity of motion," and is the 



* ' t 



product of the quantity of motion by the virtual velocity. Also let P be the force of impact, 

 and / the value of y at the centre of the beam. Pf is the product of the blow and its virtual 

 velocity. Hence combining the principles above referred to, we have the equation 



^fi-jy (i). 



where the integral includes the product of the quantity of motion of every particle of the 

 beam by its virtual velocity. 



In forming the equation of the equilibrium of the quantities of motion and the force 



