Response Power Spectrum of Multi-Degree-of-Freedom Nonlinear Systems by a Galerkin Technique

Abstract

This paper deals with the estimation of spectral properties of randomly excited multi-degree-of-freedom (MDOF) nonlinear vibrating systems. Each component of the vector of the stationary system response is expanded into a trigonometric Fourier series over an adequately long interval T. The unknown Fourier coefficients of individual samples of the response process are treated by harmonic balance, which leads to a set of nonlinear equations that are solved by Newton’s method. For polynomial nonlinearities of cubic order, exact solutions are developed to compute the Fourier coefficients of the nonlinear terms, including those involved in the Jacobian matrix associated with the implementation of Newton’s method. The proposed technique is also applicable for arbitrary nonlinearities via a cubicization procedure over the interval T. Upon determining the Fourier coefficients, estimates of the response power spectral density matrix are constructed by averaging their squared moduli over the samples ensemble. Examples of application prove the reliability of the technique by comparison with digital simulation data.