The Power Spectral Density Of Response For A Strongly Non-linear Random Oscillator

Abstract

An oscillator with a non-linear restoring force and a small linear damping under wide-band random excitation is considered. A modified Van Der Pol transformation with a suitable amplitude dependent frequency, is used to transform the original system into a first order vector system to which the stochastic averaging method (SAM) and its higher approximations apply, despite the presence of a large non-linear term. The averaged equations are derived in the Appendix by using a two-timescale approach after performing a random time change which depends on the solution. An efficient equivalent linear system with random coefficients is then proposed from which the power spectral density (PSD) is deduced. The use of corrective terms yields to an approximation of the PSD in the region of the third harmonic resonant frequency (as well as at higher harmonics). The analytical results are in excellent agreement with numerical simulations.