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

Abstract

The Aerodynamic force acting on compact structures is often modeled as a quadratic function of the wind velocity, which uctuates randomly due to the atmospheric turbulence. When a exible 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 dierential equation characterized by a quadratic feedback term. This class of dynamical systems has been deeply investigated (with dierent levels of simplications) applying several alternative mathematical approaches. In this paper we dene 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 Scale 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. IntroductionIn several engineering problems, the driving force is expressed as a nonlinear transformation of the input, the response and its time derivatives. In structural engineering, this is the case in vibrations of base-excited structures, where the internal forces depend on the (nonlinear transformation of) relative displacements and velocities between the structure and the base motion [Constantinou and Papageorgiou, 1990]. Other examples include the aerodynamic or hydrodynamic loads on exible structures, where the eective loads might be expressed, in a quasi-steady framework, as a memoryless nonlinear transformation of some relative velocities (e.g. Kareem, 1987). The common feature shared by these models with nonlinear feedback is the presence of random parametric excitation terms. The nonlinearity associated with these terms, as well as other more specic terms (like the quadratic structural velocity term considered in this paper) make the development of exact analytical solutions rather challenging. In this paper we consider the problem of a single degree-of-freedom linear oscillator, subject to a quasi-steady aerodynamic loading. The aerodynamic force is dened as the square of the wind-structure relative velocity and results from the sum of ve terms: a constant term, terms proportional to the turbulence uctuation and its square, terms proportional to the structure velocity and its square and a term proportional to the product of wind velocity uctuation and structure velocity. Another specicity of this problem concerns the stochasticity of the wind velocity uctuation, which is usually modeled as a zero-mean, stationary, Gaussian random process. The need to regard this problem from a probabilistic viewpoint makes it even more dicult to develop closed-form solutions. Of course Monte Carlo simulations are able to deal with the m...

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