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 work 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 Kármán beam with a nonlinear tuned vibration absorber.