Semi-analytical stochastic analysis of the generalized van der Pol system

Abstract

The paper is motivated by a series of wind tunnel experiments, which deal with aeroelastic Single Degree of Freedom (SDOF) and Two Degrees of Freedom (TDOF) section models. Most of them can be mathematically expressed by van der Pol-Duffing type equations or their combination. Excitation due to aeroelastic forces consists mostly of a deterministic periodic part and random components, both of them are applied as additive processes. The lock-in state represents an auto-synchronization of the vortex shedding and basic eigen-frequency of the system. This problem seems to be very polymorphous and, therefore, several isolated regimes have been outlined together with their characterization. Parameter setting with solely random excitation is further investigated in the paper. The strategy of stochastic averaging is then employed to formulate normal form of stochastic system for partial amplitudes of harmonic approximates of the response. The random part of excitation is considered as a Gaussian process with significantly variable spectral density. Hence, a conventional way of investigation based on an idea of white noise excitation is no more applicable. Therefore, the general formulation of diffuse and drift coefficients should be used to construct the relevant Fokker-Planck equation (FPE). Semi-analytical solution of FPE is deduced in the exponential form by means of a probability potential. It is later used for stochastic stability investigation together with consideration about the stationary probability distribution existence. Open problems and further research steps are outlined.