Parametric Excitation in the Self-excited Vibration System with Dry Friction : 1st Report, Parametric Resonance

Abstract

An application of solutions of linear Mathieu equation by a new method gives approximate solutions in a parametrically excited self-excitation system with dry friction. Steady solutions and the stability in the regions of parametric resonance of first and second orders are determined. Solutions outside the regions of parametric resonances are approximated by two limit cycles. One corresponds to a stable solution and the other corresponds to an unstable solution. Beyond a certain value of dry friction self-excited vibrations are completely suppressed and a parametric resonance of second order soon disappears, so that only a parametric resonance of first order remains within a certain interval of frequency. Numerical results by the present analysis almost agree with those by Runge-Kutta method and it is seen that the proposed approximate methods have high accuracy.