International audienceThe aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped non-linear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms ''periodic orbits'' means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems
This paper concerns the computation of nonlinear modes of elastic structures under large displacements. We present a numerical method that we have implemented in a general purpose finite element code. Bifurcation of modes will be also addressed.We begin by introducing a general simple quadratic framework that is suitable for most elastic models (beam, plate, shell) and most classical finite elements. We define the non linear modes as two dimensional invariants of the phase space which are tangent to the eigenspaces of the associated linear system [1]. Theses invariant subsets are determined by making continuation of one dimensional families of periodic orbits. The periodic solutions are computed using the periodic orbit approach [2]. We use the exact energy-conserving Simo scheme [3] to time-discretise the periodic orbits. We do not use the classical shooting method to compute the periodic orbits but another one which consist to write the governing equation at each time step in a whole system. This lead to a large system of algebraic equations containing the displacement of the degres of freedom at each time step. The nonlinear modes (or their approximations) are obtained by making the continuation with respect to adequate parameters. We use the asymptotic-numerical method [4] for this purpose, since it is particularly efficient for such difficult problems with quite complex bifurcation diagrams.
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