A Galerkin approach for power spectrum determination of nonlinear oscillators

A numerical method to estimate spectral properties of nonlinear oscillators with random input is presented. The stationary system response is expanded into a trigonometric Fourier series. A set of nonlinear algebraic equations, solved by Newton's method, leads to the determination of the unknown Fourier series coefficients of single samples of the response process. For cubic polynomial nonlinearities, closed-form expressions are used to find the nonlinear terms at each step of the solution scheme. Further, a simple procedure yields an approximation of an arbitrary nonlinearity by a cubic polynomial. Power spectral density estimates for the response process are constructed by averaging the square modulus of the computed Fourier coefficients over various samples or by means of well-established smoothing techniques of spectral analysis. Two applications are presented illustrating the effectiveness of the method as compared to statistical linearization and digital Monte-Carlo simulation.