Response of an oscillator to a random quadratic velocity-feedback loading

The aerodynamic force acting on compact structures is often modeled as a quadratic function of the wind velocity, which fluctuates randomly due to the atmospheric turbulence. When a flexible structure is considered and the quasi-steady assumption is applied, the wind velocity is substituted by the wind-structure relative velocity and, even in case of linear structures, the composite aerodynamic-mechanical system is governed by a nonlinear differential equation characterized by a quadratic feedback term. This class of dynamical systems has been deeply investigated (with different levels of simplifications) applying several alternative mathematical approaches. In this paper we define a system approximation based on a 2nd-order Volterra series and obtain its statistical response in terms of cumulants. The response cumulants are calculated applying the Multiple Timescale Spectral Analysis leading to analytic or semi-analytic expressions. All the approximations are validated through Monte Carlo simulation within a wide parameter space. Then, the analytical structure of the obtained expressions is used to discuss, from a qualitative point of view, the behavior of the considered dynamical system.