A normal form approach for nonlinear normal modes

The definition of a non-linear normal mode (NNM) is considered through the framework of normal form theory. Following Shaw and Pierre, a NNM is defined as an invariant manifold which is tangent to its linear counterpart at the origin. It is shown that Poincaré and Poincaré-Dulac's theorems define a non-linear change of variables which permits to span the phase space with the coordinates linked to the non-linear invariant manifolds. Hence, the equations governing the geometry of the NNMs are contained within the coordinate transformation. Moreover, the attendant dynamics onto the manifolds are given by the normal form of the problem. General calculations for a conservative N-degrees of freedom system are provided. The relevance of the method for the study of non-linear vibrations of continuous damped structures is discussed