Vibration of mechanical systems with geometric nonlinearities: Solving Harmonic Balance Equations with Groebner basis and continuations methods

Aurélien Grolet (1) , Fabrice Thouverez (1)

1 LTDS - Laboratoire de Tribologie et Dynamique des Systèmes

This paper is devoted to the study of vibration of mechanicals systems with geometric nonlinearities. After applying the harmonic balance method, one has to solve a system of multivariate polynomial equations whose solutions give the frequency component of the possible steady states. Computing solutions of HBM (harmonic balance method) equations for a particular frequency is usually done iteratively with Newton-Raphson methods resulting in only one solution depending on the first iterate. Here we intend to compute all solutions of HBM equations by using methods based on Groebner basis computation. This approach allows to reduce the complete system to an unique polynomial equation in one variable driving all solution of the problème. This way the procedure avoid the computation of multiple paths, as in homotopie techniques, which can be quickly to much time consuming. In addition continuation methods are used to extend the solution for multiple values of the frequency parameter. We apply those methods to a simple forced nonlinear dynamic system and give a representation of the multiple states possible versus frequency.

Mots clés: nonlinear vibration; harmonic balance method; groebner basis; numeric continuation

Domaines: Mécanique [physics]

Aurélien Grolet, Fabrice Thouverez. Vibration of mechanical systems with geometric nonlinearities: Solving Harmonic Balance Equations with Groebner basis and continuations methods. 11e colloque national en calcul des structures, CSMA, May 2013, Giens, France. ⟨hal-01722052⟩