Analysis of non-linear dynamical systems by the normal form theory
Résumé
A method is proposed for calculating the periodic solutions of non-linear mechanical systems with analytical non-linearities. The Jordan normalization procedure for the case of non-linear autonomous systems is described and generalized to dampened harmonically excited oscillators. Non-linear modes for Hamiltonian systems are introduced; normal forms simplify the analysis of bifurcation. It is shown how to extend, the modal synthesis procedure: the proposed non-linear modes obtained from free vibrations are used to construct a superposition technique to describe the forced response of harmonically excited systems. These results are tested for one- and two-degrees-of-freedom systems with cubic non-linearities. The results are compared with expressions obtained by classical analytical methods (averaging or multiple scales methods) or Runge-Kutta numerical methods.