Updating of a Nonlinear Oscillator

The first application is concerned with the nonlinear Duffing oscillator of known mass M = 1.0 kg subjected to zero-mean stationary Gaussian white noise / with spectral intensity Sfo.  

This oscillator has a nonlinear restoring force -Kix(t) - K xit). A stationary displacement response time history V was generated with parameter vector 0 = [C, K3, 5 q, 

There is no closed-form solution for the auto-correlation function or the power spectral density function of this nonlinear system. Therefore, an equivalent linear system is utilized to obtain the approximated mean of the spectral density estimator. Multiplying Equation (3.76) with x(t - t) and taking expectation yields  

It is a second-order linear ordinary differential equation with constant coefficients, and its analytical solution is given by  

By solving Equations (3.81)-(3.83), the response variance of the Duffing oscillator can be approximated by  

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