Tailored nonlinear stiffness and geometric damping: Applied to a bistable vibration absorber

Abstract

A novel device is proposed to obtain arbitrary restoring force characteristics and nonlinear geometric damping. It consists of linear springs and dampers that are compressed along a track. The obtained nonlinear damping and stiffness law depend on the track’s shape. The device is then applied to obtain a nonlinear energy sink (NES) to damp the vibrations of a host system. Both the case of transient vibrations, induced by shock loads, and sustained vibrations, induced by harmonic loads, are studied. For arbitrary stiffness and damping, slow flow dynamics and slow invariant manifolds (SIMs) are derived by applying harmonic balancing and multiple timescales techniques. Under transient vibrations, two performance measures are derived from the SIM, the relative residual energy and the pumping time, and are found for generic nonlinear spring force and nonlinear geometric damping. For sustained vibrations, a load-dependent frequency response (FR) is derived from the SIMs. This FR predicts the occurrence of the efficient strongly modulated responses but also the occurrence of the unfavorable detached responses called isolas, where the NES fails to mitigate and even amplifies vibrations.This research investigates the performance of a conventional cubic NES and a bistable NES (BNES) with nonlinear damping obtained from the proposed device. Especially the BNES with nonlinear damping shows attractive properties, with a high degree of robustness over a wide energy range and a low vibration threshold under transient loading. Under harmonic loads, the nonlinear damping helps reduce the effect of isolas. A novel tuning methodology is proposed to avoid isolas for a range of load magnitudes. The new device opens up a whole range of possibilities regarding energy dissipation through nonlinear damping.