A Direct Method for Quantifying Limit-Cycle Oscillation Response Characteristics in the Presence of Uncertainties

A cyclic method was developed to compute directly the stochastic response of limitcycle oscillations in an aeroelastic system subject to parametric uncertainties. The system is a typical-section airfoil in incompressible flow with nonlinear behavior modeled in the torsional structural coupling. The imposed nonlinearities cause the formation of a subcritical Hopf bifurcation, with the associated presence of a cyclic fold at reduced velocities below the Hopf point, and large-amplitude limit cycles at reduced velocities above the Hopf point. The cubic component of the pitch stiness is treated as a Gaussian random variable, while other parameters are specified deterministically. The method used to evaluate the stochastic response is based on a highly implicit, iterative technique that strongly converges to the complete time-discretized, limit-cycle oscillation without the need for time integration. The method is cast in both intrusive and non-intrusive forms, each relying on polynomial chaos expansions to obtain spectral representations of stochastic system responses. Each form is observed to well characterize stochastic responses at dierent reduced velocities with polynomial chaos expansions containing only two terms, a great improvement over previous time-domain methods. The nonintrusive method is found to be highly ecient, requiring few function evaluations to compute deterministically the limit cycle, and few deterministic samples to compute the expansion coecients.