Determining the parameters of the transverse impact of a ball on an elastic beam in the Timoshenko problem

Abstract

Timoshenko's integral equation describing an impact of a ball on a beam is replaced by a system of nonlinear differential equations in dimensionless form. The effect of the parameters of the ball-beam system on the number of impacts and the maximum dimensionless impact force, the deflection of the beam, and the duration of impact is analyzed Keywords: elastic beam, ball, system of nonlinear differential equations, impact force, beam deflection, duration of impact, number of impacts Introduction. The impact of a weight on an elastic system has long attracted the attention of researchers. Saint-Venant was the first to find a solution describing a transverse impact of a weight on an elastic beam. He supposed that the weight is concentrated and is in contact with the beam when its deflection becomes maximum during impact. He also neglected the local strains in the contact area. It was later discovered that Saint-Venant's solution cannot be used to calculate the dynamic stresses in cross-sections of the beam because the series for transverse forces diverge. Timoshenko made significant corrections in the solution by taking into account both global and local strains. The relationship between the impact force and the midspan deflection of the beam was derived by expanding the solution into series of natural vibration modes. Moreover, it was shown that the shear strains and the rotational inertia of elements of the beam may be neglected in analyzing the impact process. Allowing for these factors would unjustifiably complicate the solution of the problem.Despite some computational difficulties, many researchers continued to return to the Timoshenko problem. Theoretical and experimental studies confirmed some results obtained by Timoshenko. One of them is repeated collisions of the beam and the weight. In this connection, it is necessary to identify the major factors affecting the number of collisions. There was no doubt that it was expedient to develop the Timoshenko method and to generalize the obtained results instead of considering separate examples.In this paper, we attempt to do such a generalization. To analyze the impact process, it seems convenient to pass from Timoshenko's integral equation to a system of nonlinear differential equations in dimensionless form, which would allow us to use standard algorithms.1. Problem Formulation. The Timoshenko solution describing a transverse impact of a ball on an elastic beam [5] was a major contribution to impact theory. Timoshenko proposed the following integral equation [4,5] to determine the force P dependent on time t and generated upon transverse impact of a ball of mass M on the middle of a hinged-hinged beam: