Response of Van Der Pol ' s Oscillator to Random Excitation
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T h i s paper considers the response of V a n der Pol's oscillator to random excitation I t i s shown that the output of the oscillator consists of a periodic term, plzrs a narrow band noise term centered at the 7zatural frequency of the oscallator. The root-mean-square amfilitude of this noise term i s shown to be probortional to the square root of the spectral density of the excitation, and inr~evsel~ proportional to the amplitz~rle of self-oscillation IN mcrh-r yews considerabk interest has been aroused in the response of nonlinear systems to various t ~ p e s of excitation. Most of the recent work has been in the field of servomechar~isms and feedback systems, where the equivalent linearization technique of nonlinear mechanics has been used in various guises and under various names. With the exception of Garstens' paper,I the analytical studies have been on the response of systems to either periodic excitation alone or to random excitation alone. I n this paper an analysis will be presented on a self-excited oscillator acted upon by an external random excita-tion. Because of the self-excitation, the solution will contain periodic terms and, owing to the random excitation, the solution also will contain random terms. The results of this study show that the noise component of the solution decreases with increasing amplitude of self-excited oscillation; however, the band width of the noise increases with increasing amplitude of self-oscillation To verify the results of the analysis, the problem was simulated on an analog computer. The results of this study are in good agret2ment with the theorv. The equation of a Van der Pol oscillator acted upon by noise is where N (t) is a small Gaussian noise force, assumed to have a white power spectrum of density 4D/cycle. Let where V p (t) = A sin wut is the periodic part of the solution and V N (t) is the randomly varying part of the solution. If equation (1) was linear, the noise component V N (t) would be Gaussian since N (t) is Gaussian; further, if the damping is small the noise component TrN(t) is confined to a narrow band of frequencies close to wo. I n a nonlinear system the noise component in general will not be Gaussian; however, if the nonlinear-will be accepted until one month after final publication of the paper itself in the JOURNAL OF APPLIED …