SOLUTION OF THE EQUATION OF IMPACT THE ELASTIC BODY, ONE OF WHICH CONE

Abstract

The impact interaction of elastic bodies with a small initial velocity is considered, when one of them is limited in the contact zone with another conical surface of revolution. Using the well-known solution of the static contact problem of the theory of elasticity, which found I. Shtaermann, and the assumptions of G. Hertz, which he made when creating his own theory of quasistatic impact of solids, compiled a nonlinear differential equation of impact force as a function of time. His closed analytical solution, which describes the process of dynamic interaction of bodies in time, is expressed through the periodic Ateb-sine. To simplify the use of the obtained analytical solution in the calculations, a separate table of the specified special function has been compiled and its approximation with elementary functions has been proposed, the relative error of which is less than one percent. In order to confirm the reliability of the constructed solutions, the integration of the equation of impact force on the computer was carried out in parallel. A good agreement is established between the results obtained by the constructed analytical solution and the numerical integration of the nonlinear Cauchy problem on a computer for a second-order differential equation. Compact formulas for the maxima of the impact force and the magnitude of the compression of bodies, as well as the formula for the duration of the impact process are derived. It is noted that the obtained results can be used in determining the dynamic loads acting on the rubber-lined rolls of the vibration classifier when pieces of solid raw materials fall on them. Examples of calculations are given and a comparative analysis of the results is carried out.