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 [
]. 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 [
]. We use the exact energy-conserving Simo scheme [
] 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 [
] for this purpose, since it is particularly efficient for such difficult problems with quite complex bifurcation diagrams.