The integral equation of the impact of the elastic cone on obstacle and its approximate solution

Abstract

An approximate analytical solution of the integral equation of impact with a low velocity of two elastic bodies, one of which is a cone of revolution, is constructed. For this, the method of successive approximations and the Shanks’ formula are used, which in a closed form approximately expresses the sum of a power series. The solutions obtained provide a time sweep of the impact process and the possibility of calculating the maximum compression force, the maximum convergence of the centers of mass of bodies and the impact duration. The described method for constructing approximate solutions of the integral equations of the impact force of unsecured bodies is quite universal. It can be used for the theoretical study of the impact of bodies bounded by high-order surfaces describing tight contact, as well as surfaces with singular points, for which there are analytical solutions of the contact problem of the theory of elasticity. In compiling the integral equation of the impact force, the theory of H. Hertz and the solution of the axisymmetric contact problem of the theory of elasticity obtained by I.Ya. Shtaerman. It is shown that the constructed approximate solutions can also be used to approximate those periodic Ateb-functions, in the first quarter of their period, through which the exact analytical solution of this impact problem is expressed. The errors of approximate solutions are less than 0,5 %. This is established by comparing the numerical results to which they lead to the results of integrating the differential equation of impact on a computer. Examples of calculations are given, confirming the reliability of the analytical solutions obtained. The stated theory concerns only low impact velocities, when a large area of plastic deformations does not arise during the dynamic interaction of bodies. The appearance of a small region of such deformations is inevitable in the vicinity of the apex of the cone and at low velocities of collisions of bodies, but we do not take it into account in this work.