A universal phase condition in solving quasi-periodic responses with multiple self-excited fundamental frequencies

Abstract

Solving quasi-periodic (QP) responses of nonlinear dynamical systems, particularly when multiple self-excited fundamental frequencies have to be determining, has been a challenging task. The presence of unknown frequencies usually leads to an under-determined problem, where the number of unknowns exceeds that of equations, as obtained through harmonic balancing or difference techniques, for instance. Therefore, it is necessary to introduce supplementary equations to ensure the well-posedness of the solving process. To the best of our knowledge, some existing methods are only applicable to cases with a single unknown frequency or they require prior numerical results attained, for example, by time-marching integration techniques. In this paper, we propose a new phase condition that is unconditionally valid for QP solutions with multiple unknown frequencies. This condition reveals an inherent feature of QP motion in a multi-dimensional hyper-time domain and provides a theoretical basis for solving QP responses in such a domain. To demonstrate its effectiveness, we apply the phase condition to both the finite difference and the harmonic balance methods. Numerical examples illustrate that, by employing the phase condition, both methods can solve stable as well as unstable QP solutions accurately. As the phase condition holds universally, it is suitable to be incorporated into other solution techniques based on either frequency or time-domain analysis.