A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions

Details A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions

2008-08-28

arxiv.org/abs/0808.3839v2

Journal of Sound and Vibration 324 (2009) 243-262

Mathematics

Dynamical Systems

PACS numbers: 02.30.Mv, 02.30.Nw, 02.30.Px, 02.60.-x, 02.70.-c

Scientific paper

10.1016/j.jsv.2009.01.054

Combinig the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called ANM continuation technique. Several classical examples are presented to illustrate the use of this numerical approach.

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