Double scale analysis of periodic solutions of some non linear vibrating systems

AbstractWe consider small solutions of a vibrating system with smooth non-linearities for which we provide an approximate solution by using a doublescale analysis; a rigorous proof of convergence of a double scale expansionis included; for the forced response, a stability result is needed in orderto prove convergence in a neighbourhood of a primary resonance. Key-words: double scale analysis; periodic solutions; nonlinear vibrations,resonance MSC: 34e13, 34c25, 74h10, 74h45 1 Introduction In this work we look for an asymptotic expansion of small periodic solutionsof free vibrations of a discrete structure without damping and with local nonlinearity; then the same system with light damping and a periodic forcing withfrequency close to a frequency of the free system is analyzed (primary reso-nance). For a small solution, we recover a behavior with some similarity withthe linear case; in particular the amplitude of the forced response reaches alocal maximum at the frequency of the free response. On the other hand thefrequency of the free response is amplitude dependent and the superpositionprinciple does not apply. The work of Lyapunov [oL49] is often cited as a basisfor the existence of periodic solutions which tends towards linear normal modesas amplitudes tend to zero; the proof of this paper uses the hypothesis of ana-lycity of the non linearity involved in the di erential system. In [Rou11], weaddressed the case of a non linearity which is only lipschitzian and we proveexistence of periodic solutions with a constructive proof; in this case the resultof Lyapunov obviously may not be applied. Non-linearity of oscillations is a1