Dissipation in the Bouc–Wen model: Small amplitude, large amplitude and two-frequency forcing

We study energy dissipation in the famous Bouc–Wen model both analytically and numerically. We demonstrate power law behavior for small-amplitude, simple periodic forcing. We develop new analytical expressions for the leading order power law, as well as the first correction term, for arbitrary parameter values. We separately find the asymptotic linear dissipation formula for large amplitudes, using special functions and in a form not available before. We obtain the regimes of validity of the small- and large-amplitude approximations as well. We then undertake a numerical study of aperiodic two-frequency forcing. We observe that superposition of individual dissipation rates from the two frequency components does not give the combined nonlinear dissipation rate. We theoretically anticipate, and numerically verify, a simple scaling law in the small-amplitude two-frequency regime. Our results may encourage uses of the versatile Bouc–Wen model in studies of damping under multifrequency vibration of lightly damped engineering materials.