Analytical Solutions of Contact Impact Problems

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Abstract
In this article, analytical solutions of the normal and tangential impact of half-spaces at low velocities are formulated. After a brief introduction, a classification of contact problems is proposed and some publications about impact are mentioned. The basic contact laws for monotonously increasing normal, tangential, and torsional contact have been found decades ago, while the general analytical solutions for tangential and torsional load histories, which are necessary for impact calculations, have only been found recently. Insertion of these contact laws into the equations of motion yields a system of nonlinear differential equations, which are uncoupled in the case of the oblique central impact. The normal solution of the uncoupled equations for Hertzian and some axisymmetric surfaces can be written as a hypergeometric function, which generalizes earlier solutions. Solutions in tangential direction for the compression phase and the restitution phase can be found for the case of complete adhesion. Finally, the kinematic coefficient of restitution is shown. This solutions may help to understand some open problems of the collision of rigid bodies, which are in contact on a small elastic domain, where a part of the kinetic energy is transformed into elastic energy and reconverted into kinetic energy in normal and tangential directions.

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