Analytical solution of the elastic impact problem for a cone over a half-space

Abstract

Using the main provisions of the theory of H. Hertz on the mechanical impact of solids, the dynamic interaction of an elastic cone with an elastic half-space bounded by a flat surface is considered. The case is investigated when the axis of the cone of rotation is perpendicular to the boundary of the half-space, and the initial point of contact of the bodies is the vertex of the cone. To describe the local deformations of bodies in the zone of their interaction, the well-known solution of the axisymmetric static contact problem of the theory of elasticity, constructed by I. Ya. Shtaermann, is used. The problem of the collision of bodies is reduced to a second-order differential equation with quadratic nonlinearity. Two forms of analytical solution of this nonlinear Cauchy problem are obtained. The first uses the Ateb-sine, and the second uses the elliptical cosine. The equivalence of the obtained forms of solutions is established, that is, the possibility of replacing one form with another. To calculate the values of the Ateb-sine using the linear interpolation method, a special table is presented, and an analytical approximation by its elementary functions is proposed. The consistency of the results to which these two methods of approximate calculation of the Ateb-sine values are shown is shown. An approximate formula for calculating the values of the elliptic cosine is also derived and its reliability is confirmed. As a result of solving the impact problem, formulas were obtained that describe the change over time: the convergence of the centers of mass of the bodies, the forces of impact interaction, the radius of the circular contact area and the contact pressure. It is noted that the pressure is infinite in the center of the site, where the top of the cone is in contact with the half-space. The results are compared, to which two analytical forms of the solution result and numerical computer integration of the differential equation of compression of bodies subjected to impact. Established a good agreement of numerical results obtained in different ways. The effect of the taper angle on the main parameters of the dynamic interaction of bodies is investigated. It is shown that an increase in the angle of a solution of a conical body, which strikes, leads to a decrease in the maximum dynamic compression of bodies and the duration of their interaction, as well as to an increase in the maximum impact force, with a constant value of its momentum. A numerical example of calculation is given, where steel is taken as the material of a conical body, and rubber is used as the material of a fixed half-space. Problems of this type arise when calculating the parameters of impact of a piece of mineral raw materials on rubber-lined rolls of the vibration classifier.