Periodic response and stability of rigid mass resting on friction-damped SDOF oscillator

This paper presents closed-form periodic solutions, accompanying stability analyses, and an analytically generated response spectra for a passive isolation system subjected to a harmonic base motion. This isolation system, a rigid mass resting on a single degree of freedom (SDOF) oscillator, is a piecewise linear problem that has been historically studied using numerical techniques. By carefully expressing initial periodicity conditions as a function of an excitation phase angle, both the initiation times for stick and slip behaviors and the symmetric steady-state slip-slip and slip-stick responses are analytically obtained. The stability analysis, based on error-propagation techniques, shows that the steady-state solutions are stable and are realized after the transient motion decays in a beating-type manner.