Existence and Stability of Periodic Motions of a Harmonically Forced Impacting System

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Abstract
A simple idealization of a vibratory plow, which is impacting against an immovable relatively rigid obstruction, is analyzed to determine possible periodic motions and the stability of these motions. The plow is idealized as a translating rigid body with an applied harmonic force and viscous damping. It is pushed against a wall by an applied bias force, with the resulting impacts characterized by a coefficient of restitution. Necessary conditions are found for the existence of periodic motions with one impact per motion cycle. The infinitesimal stability of these periodic motions is investigated, and it is found that either a motion with the highest number of crank revolutions per cycle is stable or none are stable. Some cases are investigated numerically, and it is found that the infinitesimally stable solutions are approached even for large departures from the periodic initial conditions. For unstable cases, the two dependent variables of greatest interest, average number of force cycles per impact and average impact velocity, can be approximated by using the formulas developed for the one‐impact‐per‐cycle motionsanalyzed.

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