7 Forced Response of Nonlinear Systems Under Combined Harmonic and Random Excitation 73 x2 x1 x1 1) 2) 3) 4) 5) −5 −0.4 −0.2 0 0.2 0.4 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 5 10 20 30 40 50 0 0 p(1) (x 1) Fig. 7.2 Galerkin-approximation of p(1,2) (x1,x2) [left] and p(1) x1, t = 2π with (1) MCS-results, (2) ˆ x1,det, (3) ˆp (1) 0 (x1), (4) p (1) Galerkin x1, t = 2π and (5) ˆ p (1) Galerkin (x1) [right] the Galerkin solution. One can see a small difference between the course of the maxima of the weighting function and the Galerkin solution. This means that the weighting function could be chosen even better or that there is a deviation from a purely Gaussian distribution due to the non-linearity. There is also a small difference between the PDF from the MCS and the Galerkin-type method. This can be primarily explained by different time discretization, but also partly by the general inaccuracy of the MCS. Other than that, the results of the MCS and the Galerkin-type method also agree very well in the individual time steps. The calculation took around 10 s with low approximation orders and around 2 min for the final high quality results. An MCS of comparable quality would take much longer. 7.5.2 Duffing-van der Pol-Oscillator The Duffing-van der Pol-oscillator is a combination of the Duffing-oscillator and the van der Pol-oscillator. Its differential equation is m¨q +c˙q +kq +λq 3 +νq 2 ˙ q =fex (7.40) with the cubic stiffness λ and the van der Pol-coefficient ν. The external force fex is identical with the one used before, so that the corresponding stochastic differential equation is as follows dx = x2 −c m x2 − k m x1 − λ m x 3 1 − ν m x 2 1x2 + ˆfex m cos( t) - ./ 0 f dt + 0 0 0 σ m - ./ 0 G dw (7.41) with the drift vector f and diffusion matrix B= 1 2 · 0 0 0 σ 2 m2 (7.42) resulting from the matrix G. The system is characterized by its parameters m=0.2, c =0.425, k =40, λ =150, ν =120. The external force is given byσ =0.075, ˆfex =2.5 and =14. The harmonic order used is Nh,μ =5 aswell as Nh,σ =5 and the orders of the final correction functions are N (1) ϕ =6, N (2) ϕ =6 andN (t) ϕ =10. In order to improve the quality of the results, the calculation is also carried out several times in this case, whereby the approach is always adapted to the intermediate results Figures 7.3 and 7.4 show the results for the Duffing-van der Pol-oscillator, for which there is also a large similarity between the results of MCS and the Galerkin-type method. Due to the strong damping of the Duffing-van der Pol-oscillator, the Galerkin-type method is able to calculate the temporal course of the PDF for any excitation frequency using Gaussian ansatz functions. If the solutions calculated for adjacent
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