The Response Spectrum of a Nonlinear Oscillator

The response of a nonlinear oscillator excited by white noise is considered. A truncated Hermite polynomial series is used as an approximation to the probability density function. While this approach has been used before by many authors to obtain statistics such as the time-dependent mean or mean-square values, it has not been noted before that the approach can be extended to obtain the correlation function and spectrum. This series when substituted into the Fokker-Planck equation yields a set of time-dependent moment equations, which can be solved numerically for the correlation functions, or, after a Fourier transform, a set of complex algebraic equations which can be solved for the spectrum. Examples of spectra for the Duffing and van der Pol oscillators are shown.