A Mathematical Formulation for as Generalized Hertz Impact Problem

Abstract

The approximate solution developed by Hertz for the frictionless impact between two linearly elastic bodies with simple smooth contact surfaces has been extremely useful in engineering applications for more than one century [1,2]. A major simplification in Hertz’s solution is the following: only the inertia forces associated with the rigid body translational motion of approach and rebound of the two bodies is taken into account. As a consequence, the problem studied by Hertz decouples into two problems which can be solved sequentially. First, an unilateral contact problem in elastostatics is solved, the solution of which (also developed by Hertz) gives the pressure distribution on the contact surface as a function of the translational approach of the two bodies. Next, the linear momentum balance for the rigid body translational motion of the two bodies is established. When the resultant of the contact pressure is expressed (using the solution to the first problem) as a function of the translational approach of the bodies, the problem reduces to the solution of an ordinary differential equation.