An approximate solution for the spectral response of Duffing's oscillator with random input

An approximate method is presented for estimating the power spectral density of the response of Duffing's oscillator when it is driven with Gaussian white noise. The approach involves calculating the expected value of the spectral response of an equivalent linear system where the equivalent natural frequency is assumed to be a random variable. Comparisons of the response power spectral density obtained by the present method with results obtained by numerical simulations show remarkably close agreement, even for large non-linearities. It is found that as the amplitude of the random excitation is increased, the resonant response peak in the response spectrum tends to broaden. This effect results from the non-linear stiffness which causes the effective natural frequency to be non-deterministic when the system is subjected to random excitation.