A Galerkin multiharmonic procedure for nonlinear multidimensional random vibration

Abstract

The stationary vector-response of a multidimensional system of nonlinear dynamical equations with random input, is expanded into a finite trigonometric Fourier series on a time interval T (the period). The unknown Fourier coefficients are solutions of a system of nonlinear algebraic equations, obtained by harmonic balance. Let d be the dimension of the problem and M be the number of harmonics. The computational iterative algorithm consists of solving at each step, M-times a 2 d-dimensional linear system in place of a linear system with dimension 2 Md, via repetitive applications of the Fast Fourier Transform (F.F.T.) procedure to compute at each iteration the Fourier coefficients of the nonlinear terms. The finite Fourier series thus obtained, is called a Galerkin approximation of order M, on the time interval T, of the stationary response process. Good trajectory approximations will be obtained, provided M is large enough. Consistent P.S.D. estimators will be obtained by “smoothing” the square modulus of the computed Fourier coefficients, as in a real time spectral analyser, provided M and T are large enough. The method will be carried out on examples from vehicle dynamics, in Section 4. The results will be compared to the corresponding experimental ones. The mathematical justification of the existence of Galerkin's approximations is given in the Appendix.