Asymptotic analysis of chattering oscillations for an impacting inverted pendulum

The chattering oscillations for an inverted pendulum impacting between lateral walls, a prototype of a class of impact dampers, are analysed. Attention is focused on the periodic chattering appearing when the rest positions cease to be attracting, and the aim is that of computing the time tau required by the micro-oscillations to come to rest. An algorithm is proposed to compute tau, and it is shown how this time depends on the excitation amplitude. When tau becomes equal to the excitation period, the considered chattering is observed to lose attractivity. This occurs for a certain excitation amplitude threshold, which is computed and whose (very weak) dependence on the excitation frequency is illustrated. An asymptotic estimate of the chattering time with respect to a small parameter proportional to the excitation amplitude is given. This is obtained by an appropriate expansion in Taylor series of the relevant quantities involved in the analysis. It is shown that tau is asymptotically proportional to the square root of the excitation amplitude.