THE POWER SPECTRAL DENSITY OF RESPONSE FOR A STRONGLY NONLINEAR RANDOM OSCILLATOR

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