On the study of random oscillations in non-autonomous mechanical systems using the fokker-planck-kolmogorov equations

Abstract

A method of integrating the Fokker-Plank-Kolmogorov equations (FPKE) used in the theory of random oscillations /1–4/ is proposed. The Duffing equation is first studied as an example. The method is then used, together with the method of averaging, to study random oscillations of non-autonomous mechanical systems with one degree of freedom when the eigenfrequency varies in a random manner. The Van-der-Pol equation is considered for the case of a randomly varying eigenfrequency and periodic parametric excitation. When the function sought is replaced, the FPKE transform into another equation whose trivial solutions have the corresponding particular solutions of the FPKE. The condition of integrability of the FPKE is obtained as the direct consequence of the change in question.