Multiple Impacts of a Ball Between Two Plates—Part 2: Mathematical Modelling

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Abstract
A mathematical model is developed to portray the vibroimpacts of a steel sphere which is trapped between two flat steel plates with clearance, while the plates are oscillated by an electromagnetic shaker. Data from a long series of experimental observations are reported in Part 1 of this paper [53]. The aim is to determine a law of motion by which a computer simulation can satisfactorily reproduce the major characteristics of the observed movement. During each impact the motion of the ball is taken to be a brief half wave, due to the highly nonlinear forces of surface compliance and surface damping. Modelling is by analog simulation. It was found first that linearization of the surface stiffness does not reproduce the observed phenomena. The mathematical model formulated is that the motion of the ball during contact is governed by the equation mx¨ + cx1.5 x˙ + kx1.5 = 0, where x is the penetration, c is a damping constant, and kx1.5 is the Hertzian force of compliance. The results of experiments can be corroborated only on the basis of a variable damping coefficient cxn with n = 3/2.

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