Наближений розв’язок інтегрального рівняння удару тіл з сингулярною точкою на поверхні контакту

Abstract

The integral equation of the impact force of two elastic bodies of revolution, one of which has a singular point on the contact surface, where the curvature of the limiting surface is infinite, is compiled and reduced to a dimensionless form. In formulating the problem of dynamic compression of bodies the assumptions, introduced by G. Hertz whendeveloping his own theory of quasistatic impact ofsolids, and the well-known solution of the axisymmetrical static contact problem of the theoryof elasticity, constructed by I. Ya. Shtaerman, are used. By the method of successive approximations, after three iterations, an approximate solution to the integral equation of the impactforce is obtained and is presented in the form of a power series. This series is reduced to a closed form by the approximate Shanks method, which results in a compact analytical solution of the impact problem is obtained. It is convenient for engineering calculations and describes the change in time ofthe impact force and the processes of compressing and unclenching of bodies. Compact formulas are also obtained for calculating the maximumsof the impact force and the approach of the centers of mass of the bodies, as well as formulas for calculating the durations of the dynamic compression process and the entire impact. In order not to go beyond the limits of the elastic statement ofthe problem, it is recommended to use the theory presented at low speeds of collision of bodies (up to 5 m/s). The main emphasis in the work is on the formulation and solution of the integral equation by analytical methods rather than numerically. The high accuracy of the solution obtained is confirmed by small deviations of the results to which itleads from the results of numerical integration of the shock equation on a computer. The relative error does not exceed 0.5 %. It is shown that the obtained formulas can also be used to approximate periodic Ateb-functions through which the exact solution to this impact problem is expressed. Approximate solutions serve as a good approximation of these special functions in the first quarter of their period. Examples of calculations are given, witha discussion of the results obtained, and comparisons are made with the numerical data of other publications. The correspondence of numerical results obtained by different methods is established, which confirms the relevance of the developed model of elastic impact of bodies of revolution in the presence of a singular point on the surface of one of them.