n-dimensional Harmonic Balance Method extended to non-explicit nonlinearities

Abstract

The harmonic balance method is widely used for the analysis of strongly nonlinear problems under periodic excitation. The concept of hypertime allows for the generalization of the usual formulation to multi-tone excitations. In this article, the method is applied to a system involving a nonlinearity which cannot be explicitly expressed in the multifrequency domain in terms of harmonic coefficients. The direct and inverse Discrete Fast Fourier Transforms are then necessary to alternate between time and frequency domains in order to take into account this nonlinearity. The results show the efficiency and the precision of the method. Keywords: harmonic balance method, hypertime domain, unilateral contact RésuméLa méthode de l'équilibrage harmonique permet l'analyse de problèmes fortement non linéaires sous excitation périodique. En utilisant le concept d'hyper-temps, il est possible de généraliser la formulation usuelle à des excitations apériodiques. Dans cet article, cette méthode est appliquée à un système impliquant une non-linéarité qui ne s'exprime pas de façon explicite dans l'espace multi-fréquentiel des coefficients harmoniques. Les transformées de Fourier rapides directe et inverse sont alors nécessaires pour alterner domaines temporel et fréquentiel afin de prendre en compte cette non-linéarité. Les résultats montrent l'efficacité et la précision de la méthode.

Computation of quasi-periodic localised vibrations in nonlinear cyclic and symmetric structures using harmonic balance methods

“…is not directly computed since displacement dependent nonlinear forces are, most of the times, defined in the time domain. Therefore, an alternating frequency-time (AFT) domain technique based on the hyper-time concept [27,28,30], as depicted in Fig. 1, is implemented.…”

Quasi-periodic harmonic balance method for rubbing self-induced vibrations in rotor–stator dynamics

“…Lau [15] studied the quasi-periodic free response of a clamped beam with large displacements. Legrand [16,17] used the quasi-periodic HBM for the computation of limit cycles for autonomous systems with application to aircraft turbomachinery with rotor-stator interaction. Coudeyras [18] used a similar approach applied to the study of break squeal.…”

Forced vibrations of a turbine blade undergoing regularized unilateral contact conditions through the wavelet balance method

“…It has been used to study steady‐state response of turbine engine blades with friction dampers using a multiterm approximation . This approach was extended to unilateral contact and friction conditions through an alternating frequency/time domain strategy proposed by Cameron et al and Pilipchuk . It is worthy to note that the HBM formulations of Equations – are identical.…”