Nonlinear Mechanical Systems, Sinusoidal Excitation Signal and Modal Analysis: Theory

Abstract

In this study, we shall show that this normal transformation is naturally linked to the notion of nonlinear modes.The nonlinear modes appear as an extansion of the linear modes but, in opposition to the latter, they are associated with complex scheduling laws with periods dependant on the amplitudes. Rosenberg in [16], [17] was the first author to have studied them in the case of a system with two degrees of freedom. Moreover, in the context of a general study on the forced response of systems with several degrees of freedom, Szemplinska-Stupnicka [19], [20] proposed the generalization of the Ritz-Galerkin method, which consists of utilizing a natural normal mode dependant on the movement’s amplitude. The author clearly showed that it allowed one to find with precision the resonance amplitudes. The mode shapes utilized corresponded to an approximation of the non-linear modes defined by Rosenberg calculated with the help of the harmonic balance principle. However, it has been shown in [19], [20] that it was not possible to use a superposition principle for the normal non-linear modes analogous to that often used in a linear context. This paper deals with the generalization of superposition technique to the nonlinear case. We consider multidegree-of-freedom mechanical systems with analytical nonlinearities. First, we introduce normal form theory: we recall what is the Jordan normal form of an analytical system of ordinary differential equations and how to compute this normal form and the associated normal transform. Then we show how to compute nonlinear normal modes in the case of forced vibrations of the system from the nonlinear modes obtained in the case of free vibrations. Here we study primary resonance as an example. By using the previous computations, the level of decoupling, the true physical orders of responses and forcings and by slightly improving the former theoretical presentation, we are able to build blocks of general masses, general nonlinear frequencies, and nonlinear modes.The amplitudes of the response look like a generalized superposition type. The numerical part of this work can be realized because of an algebraic nonlinear resonant equation derived when we look for approximated periodical solutions of the normalized system, and the normal transform achieved from the computations of the free case. In order to achieve the computations, we could use Maple computer algebra.