Analysis of multi-degree of freedom strongly non-linear mechanical systems with random input: Part I: non-linear modes and stochastic averaging

The displacement and velocity vector responses of multi-degree of freedom non-linear systems are expanded as a series of non-linear modes, in which the mean term, frequencies and mode shapes depend on the modal amplitudes. The non-linear modes, defined in the first harmonic sense, are obtained by solving a generalized non-linear eigenvalue problem for each fixed value of the amplitude vector. Based on a generalized van der Pol transformation and a stochastic averaging principle, adapted for multiperiodic systems with coupled fast and slow movements, an averaged Itô differential system governing the amplitude vector process is deduced. Then, an approximate probability density function for the amplitude vector is derived. It is shown on a two-dimensional example with cubic non-linearities that, for different system parameters, the first two moments of the displacement and velocity vector responses calculated analytically with the method are in agreement with those calculated with the Gaussian linearization procedure and also with the Monte Carlo simulation. Nevertheless, the efficiency of the method will be clearly demonstrated in Part II of the paper, where an equivalent linear system with random matrices is proposed, greatly improving upon the usual linearization with constant matrices in terms of the predicted PSD matrix response.