A Harmonic Balance-Based Tracking Procedure for Amplitude Resonances

In a number of applications, predicting the maximum displacement, velocity, or acceleration amplitude that can be undergone by a forced nonlinear system is of crucial importance. Existing resonance tracking methods rely on phase resonance or approximate the response via a single-harmonic Fourier series, which constitutes an approximation in both cases. This chapter addresses this problem and proposes a harmonic balance tracking procedure to follow the amplitude extrema of nonlinear frequency responses. Means to compute the amplitude of multi-harmonic Fourier series and their time derivatives are first outlined. A set of equations describing the local extrema of the amplitude of a nonlinear frequency response is then derived. The associated terms and their derivatives can be computed via an alternating frequency–time procedure without resorting to finite differences. The whole method can be embedded in an efficient predictor–corrector continuation framework to track the evolution of amplitude resonances with a changing parameter such as the external forcing amplitude. The proposed approach is illustrated on two examples: a Helmholtz–Duffing oscillator and a doubly clamped von Kàrmàn beam with a nonlinear tuned vibration absorber.