Analysis of multi-degree of freedom strongly non-linear mechanical systems with random input: Part II: equivalent linear system with random matrices and power spectral density matrix

We present a method for estimating the (power spectral density) PSD matrix of the stationary response of lightly damped randomly excited multi-degree of fredom mechanical systems with strong non-linear asymmetrical restoring forces. The PSD matrix is defined as the mean value of the PSD matrix response of an equivalent linear system (ELS) whose damping and stiffness matrices depend on non-linear vibration modes of the associated conservative system, the frequencies and modes shapes being amplitude dependent. Based on a generalized van der Pol transformation and using a stochastic averaging principle, as developed in a companion paper, a stationary probability density function for the amplitude process is derived to characterize the ELS fully. Some possible simplifications of the method, such as modal reduction and/or local linearization, are also discussed. The results obtained are in good agreement with those of direct numerical simulations taking two typical examples.