Energy transmission by impact in a system of two discrete oscillators

This paper studies the transfer of energy accompanying the impact of two otherwise linear, viscously damped, single-degree-of-freedom oscillators, one of which is driven by harmonic base motion. A semi-analytical solution of the oscillator equations of motion, in which the times of impact are determined by bisection, is used to simulate the responses while the parameters of the system are varied. The existence of a stable 1:1 resonance, which means the response is stable and both oscillators vibrate with the period of the excitation, is investigated using Floquet theory, and predictions are made about the ranges of excitation frequency and amplitude leading to such a response. It is found that the assumption of a coefficient of restitution, defined as the ratio of the relative velocity of the two oscillators after their collision to their relative velocity before collision, smaller than 1 (i.e., modeling energy loss during impact) increases the range of stable solutions. A transmission coefficient, the ratio of transmitted energy to input energy, is defined to quantify the efficiency of energy transfer between the oscillators, and it is found that the energy transmission efficiency from the higher-frequency oscillator to the lower-frequency one is much greater than the transmission efficiency for energy flow from the lower-frequency to the higher-frequency oscillator. There is an optimal excitation amplitude for maximum energy transfer in higher-to-lower-frequency transmission, while such an optimum does not exist for transmission in the opposite direction. The numerical results are validated by experiments.