Consolidated expansions for estimating the response of a randomly driven nonlinear oscillator

Abstract

We consider a nonlinear oscillator driven by random, Gaussian “noise.” The oscillator, which is damped and has linear and cubic terms in the restoring force, is often called the “Duffing Equation.” The Fourier transform of the response is expanded in a series in the coefficient of the nonlinear term. This series is then squared and averaged, and each term in the resulting response spectrum series is expressed in terms of the response spectrum of the linearized harmonic oscillator (i.e., without the cubic term). Since the forcing function is Gaussian, the linear solution is Gaussian. The terms in the series for the response spectrum are then regrouped so that common quantities can be factored out. This process leads to “consolidated equations” for the response spectrum and the “common factors.” These consolidated equations are truncated in various ways, and the corresponding solutions are compared with an analog computer experiment. This technique was proposed for turbulent flow by Kraichnan and followed up by Wyld, and has yielded some good results. The numerical results indicate that the truncated consolidated equations can provide a substantial improvement over some other methods used to solve this type of problem. The methods compared with it are (1) the traditional truncated parametric expansion, (2) statistical linearization, and (3) use of the joint-normal hypothesis to express the fourth and sixth moments in terms of the second.