Direct Quadrature Method of Moments Solution of Fokker-Planck Equations in Aeroelasticity

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker-Planck equation. The theoretical details of the solution procedure are first presented. The method is then used to solve the Fokker-Planck equations for both one- and two-dimensional processes that possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared with analytic (one-dimensional process) and Monte Carlo (two-dimensional process) solutions. Trends in the standard deviation and coefficient of kurtosis with respect to the additive noise level and bifurcation parameter are reported for what appears to be the first time for the saddle-node/subcritical Hopf bifucation problem. The deterministic form of this problem exhibits hysteresis, which is often a phenomenon present in more-complex nonlinear aeroelastic systems. Finally, statistical results are shown for a typical section airfoil with nonlinear stiffness subjected to random aerodynamic loading. The constants for the nonlinear stiffness are chosen such that the deterministic behavior exhibits both a saddle-node and subcoritcal bifurcation. Standard deviation results computed using the direct quadrature method of moments for a reduced frequency below the deterministic saddle-node bifurcation point compare well with Monte Carlo results.