Stochastic Dynamics of a Noisy Two Degree-of-Freedom Nonlinear Aeroelastic System

Abstract

In this paper we study numerically the stochastic phenomenological bifurcations of a two-degree-of-freedom noisy aeroelastic system oscillating in pitch and plunge, with a cubic non-linearity in pitch. We consider a mathematical model expressed by a non-linear system of Stratonovich stochastic differential equations, and we separately study the systems with additive or multiplicative Gaussian white noise. For both types of noise a local solution exists and is unique for any initial conditions and any values of the parameters. Moreover, for the system with additive noise we prove that a global (non-explosive) solution exists for any initial conditions, if the cubic non-linearity is represented by a hard spring. However, for the system with additive noise and a cubic non-linearity represented by a soft spring, we prove that the explosion time can be finite, and consequently a global solution does not always exist. Without the stochastic perturbation, the deterministic system has a Hopf-bifurcation. Here we analyze the corresponding phenomenological bifurcation of the stochastic model. The study of the phenomenological bifurcations concerns the qualitative changes of the density of the stationary distribution associated with the system, i.e of the time independent solution of the corresponding Fokker-Planck equation. We use a numerical algorithm to estimate the shape of the density of the stationary distribution, and we study the structural changes of this density. A stochastic analysis in this case is useful for validating the mathematical model associated with the aeroelastic system, and for studying the uncertainties in the limit cycle oscillations.