The spectral density of a nonlinear damping model: multi-DOF case

Abstract

In this work, the spectral density of the following multi-DOF nonlinear damping model is investigated: Mx/spl uml/+D/sub 0/x/spl dot/+/spl gamma/D(x, x/spl dot/)+Kx=/spl sigma/n(t) where /spl gamma/>0 is a small parameter. A formula for the spectral density is established with O(/spl gamma/2) accuracy based upon the Fokker-Planck technique and perturbation. One of the features of the multi-DOF oscillation system is that x and x/spl dot/ are generally correlated in stationary state. This is true even for linear systems. Necessary and sufficient conditions for uncorrelatedness are given for linear systems. Since the first-order statistics R/sub xx/(0) and R/sub xy/(0), where y=x/spl dot/, appear in the spectral density formula, it is desirable to have the explicit stationary probability density for the purpose of evaluating R/sub xx/(0) and R/sub xy/(0). However, in general, as in the single DOF case, an expression for the stationary density is not available. This note gives the explicit stationary density of an nonlinear damping model Mx/spl uml/+/spl mu/(E/sub D/)Dx/spl dot/+Kx=/spl sigma/n(t) in which the energy E/sub D/ is defined as E/sub D/=1/2(x/sup T/KDx+y/sup T/MDy) where D>0 is assumed to commute with K and M. In the end, an energy-type nonlinear damping model is worked out completely as an illustration. >