Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases

Aurelien Grolet (1) , Fabrice Thouverez (1)

1 LTDS - Laboratoire de Tribologie et Dynamique des Systèmes

This paper is devoted to the study of vibration of mechanical systems with geometric nonlinearities. The harmonic balance method is used to derive systems of polynomial equations whose solutions give the frequency component of the possible steady states. Groebner basis methods are used for computing all solutions of polynomial systems. This approach allows to reduce the complete system to an unique polynomial equation in one variable driving all solutions of the problem. In addition, in order to decrease the number of variables, we propose to first work on the undamped system, and recover solution of the damped system using a continuation on the damping parameter. The search for multiple solutions is illustrated on a simple system, where the influence of the retained number of harmonic is studied. Finally, the procedure is applied on a simple cyclic system and we give a representation of the multiple states versus frequency.

Mots clés: Nonlinear dynamics; Multiple solutions; Polynomial nonlinearities; Harmonic balance method; Groebner basis; Continuation methods

Domaines: Vibrations [physics.class-ph]

Aurelien Grolet, Fabrice Thouverez. Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases. Mechanical Systems and Signal Processing, 2015, 52-53, pp.529-547. ⟨10.1016/j.ymssp.2014.07.015⟩. ⟨hal-02121532⟩