A high order purely frequential harmonic balance formulation for continuation of periodic solutions
Résumé
The combination of the harmonic balance method (HBM) with a continuation technique is a widely used approach to follow the periodic solutions of dynamical systems with respect to a bifurcation parameter. However, since the derivation of the algebraic system containing the Fourier coefficients can be very cumbersome, the classical HBM is often limited to simple nonlinearities (quadratic, cubic) and/or to a small number of harmonics. Several variations of the classical HBM have been introduced in the literature to overcome this shortcoming, for instance, the incremental HBM or the alternating frequency/time domain HBM. In this paper, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with regular equation. The main idea is to systematically recast the dynamical system under a quadratic form before applying the HBM. Once the equations are quadratic, the derivation of the algebraic system and its resolution by the so-called ANM continuation technique become obvious. Several classical examples are presented to illustrate this numerical technique.
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