Random Vibration of limit cycle systems and stochastic strings.

Abstract

The behaviour of the Van der Pol oscillator under periodic excitation is well known. The effect of adding noise to the excitation has been studied in this chapter. Three different solutions are obtained based on the Gaussian closure technique, the method of equivalent linearization, and the combined averaging and nonlinearization technique. These solutions display qualitatively identical behaviour and compare well with numerical simulations. From the results, it is seen that at resonance, the periodic term controls the response, leading to higher mean amplitudes and lower variance levels. However, as the external frequency is changed, the noise effects become more significant: the mean decreases and the variance increases. For a fixed value of harmonic excitation level, an increase in noise level reduces the mean response but increases the response variance. The existence of multiple steady states is typical of nonlinear systems under deterministic excitation. Here, the Gaussian closure solution and the equivalent linearization solution-which are identical in stationary state-show multivalued moments. These results have been interpreted using a stochastic stability analysis for plausible steady states. This stability analysis also serves to delineate regions in parameter space where the assumed form of the solution is acceptable. Although approximate, the predictions of the stability analysis compare well with numerical simulations. Some limitations of the linearization solution developed here include: • the method yields Gaussian estimates for the response, and • the solution is not valid in the limit P0P \to 0P0. These limitations are overcome in the solution based on the combined averaging and nonlinearization method. The response moment estimates obtained using this method compare well with digital simulation results and demonstrate the usefulness of this approach. The power spectrum of a combination of a periodic function and white noise has: • a discrete component at the periodic frequency, and • a constant part corresponding to the noise. When the noise level is low, the power is essentially concentrated at a single frequency-similar to narrow band random processes. Thus, in data analysis and random process measurements, these two types of processes may be confused with each other. However, unlike the noise plus periodic term, a narrow band process is a zero mean stationary random process. It is therefore natural to question how the Van der Pol oscillator responds to a narrow band process. An attempt to partially answer this question is made in the next chapter.