Nonlinear responses of an SDOF structure with a light, whirling, driven, untuned pendulum

Abstract

We study a spring-block system, as a one-mode model of a structure, on which is mounted a light, lightly damped, and lightly driven pendulum. This system differs from classical tuned vibration absorbers in that the pendulum is untuned, and can act as either a stabilizer or a destabilizer. We first study negative damping, wherein sub-synchronous whirl can stabilize the structure. We obtain a new analytical stability criterion for such stabilization. When the stability criterion is not met, vibration amplitudes may still remain bounded. The pendulum’s whirl speed then evolves, cyclically and possibly aperiodically, through multiple bifurcations while the vibration amplitude oscillates slowly. A 1:1 resonance between whirl and vibration plays a key role in this surprising finite-amplitude stabilization of the structure. For larger vibration amplitudes the whirling pendulum fails to stabilize the negatively damped spring-block system. We next study destabilization by the pendulum in the presence of positive structural damping in the spring-block system, and relatively larger driving torques. In this regime the pendulum can cause large vibration amplitudes through two differing mechanisms: a driving torque that starts small and grows slowly, and a driving torque that starts large and stays constant. In the first case, initially sub-synchronous whirl is captured in a 1:1 resonance and vibration amplitudes then grow; later, the pendulum escapes resonance and whirls fast, and vibrations decay. In the second case the pendulum initially whirls fast as vibrations grow. The pendulum is then captured into resonance, and vibration amplitudes grow further towards saturation, possibly mediated by period doubling bifurcations in whirl motions. Both stabilization and destabilization dynamics studied here include novel aspects as above, brought out using a combination of multiple scales analysis, detailed numerical parameter studies, and harmonic balance approximations.