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

In this work, the spectral density of the following multi-DOF nonlinear damping model is investigated: Mx¨+D₀x˙+γD(x, x˙)+Kx=σn(t) where γ>0 is a small parameter. A formula for the spectral density is established with O(γ2) accuracy based upon the Fokker-Planck technique and perturbation. One of the features of the multi-DOF oscillation system is that x and x˙ 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_xx(0) and R_xy(0), where y=x˙, appear in the spectral density formula, it is desirable to have the explicit stationary probability density for the purpose of evaluating R_xx(0) and R_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 “energy”-type nonlinear damping model Mx¨+μ(E_D)Dx˙+Kx=σn(t) in which the “energy” E_D is defined as E_D=1/2(x ^TKDx+y^TMDy) 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