Analytical solution of the generalized problem of elastic impact of solids

Abstract

The analytical solution of the generalized problem of mechanical shock of two elastic bodies of revolution in the formulation of H. Hertz is expressed through the periodic Ateb-sine and its degrees. The formulas for calculating the time variation of the convergence of the centers of mass of bodies, the force of the shock interaction, the radius of the contact area and the pressure at its center are derived. Compact formulas for the maxima of these quantities are obtained, which are achieved at the end of the process of dynamic compression of bodies. The formula for the duration of the impact in time is also derived. It is noted that the duration depends on the order of the boundary surfaces of the bodies subjected to impact. To account for regional informational deformations of bodies in the zone of their interaction, a generalized solution of the contact problem of the theory of elasticity, constructed by I.Ya. Shtaermann for the case when solids are bounded by surfaces that have an order greater than the second. It is shown that from the obtained theoretical results, which relate to the dense contact of bodies subject to impact, as a special case, the well-known classical results obtained by H. Hertz follow. The formula of a definite integral of the degree of Ateb-sine, which expresses the shock pulse, is obtained. An example of elastic impact of bodies, one of which is limited to a fourth-order surface, is considered. To calculate the values of the Ateb-sine, it is recommended to use its approximation by elementary functions. It is shown that the numerical results obtained using an analytical solution, using this approximation, agree well with the results of numerical integration of the impact equation on a computer. The influence of the geometric characteristics of the boundary surfaces on the design parameters of the impact, which occurs with a small initial velocity, is investigated. Taking into account the symmetry of the characteristics of elastic impact in time, it is recommended to use analytical solutions built for the stage of compression of bodies to calculate the process of dynamic unclamping of bodies. The theory presented deals with exceptional elastic impact, when dynamic compression does not lead to plastic deformations. This imposes a significant limitation on the initial impact velocity.