Harmonic Balance for quasi-periodic vibrations under nonlinear hysteresis
Nidish Narayanaa Balaji, Johann Groß, Malte Krack
Abstract
Harmonic Balance is probably the most popular method for computing both periodic and quasi-periodic vibrations. To treat hysteresis in the periodic case, a marching procedure is commonly used where the evolution law that governs the hysteresis is incremented in discrete time until the steady cycle is reached. In the present work, this marching procedure is generalized to the quasi-periodic case. The numerical examples demonstrate the superior computational performance of the proposed approach over the state-of-the-art. The latter requires the representation of internal variables defining the hysteresis state, which have discontinuous derivatives, whereas the remaining variables are two-times continuously differentiable. This leads to the Gibbs phenomenon and poor convergence, while the proposed method achieves higher accuracy and a typical computational speedup by 2 to 3 orders of magnitude. The proposed approach is illustrated for the case of the popular Jenkins element; the application to other models of mechanical hysteresis and non-mechanical hysteresis problems seems easily possible.