Consistent Method for PDF Solutions of Random Oscillators

The probability density function (PDF) of the responses of random oscillators with both external and parametric excitations is formulated as an exponential function of polynomials in state variables. The PDF is assumed to be governed by the Fokker-Planck-Kolmogorov equation. Special measure is taken to satisfy the Fokker-Planck-Kolmogorov equation in the average sense of integration with the assumed function, and quadratic algebraic equations are obtained for determining the unknown PDF. Examples are given to show the application of the method to the random oscillators with both external and parametric excitations. The PDFs obtained with the proposed method are compared with the obtainable exact ones. Numerical results showed that the PDF solutions obtained with the proposed method approach the exact solutions.