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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 183
Stochastic Stability of Linear and Nonlinear Fluid Structure Coupled System with NonGaussian Multiplicative Noise F. Poirion^{1}, B. Puig^{2} and I. Zentner^{3}
^{1}French Aeronautics and Space Research Center, Chatillon, France
F. Poirion, B. Puig, I. Zentner, "Stochastic Stability of Linear and Nonlinear Fluid Structure Coupled System with NonGaussian Multiplicative Noise", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 183, 2006. doi:10.4203/ccp.84.183
Keywords: aeroelasticity, nonGaussian, random dynamical system, stability.
Summary
In a previous paper [2] the stability of fluidstructure coupled systems in turbulent flow has
been investigated under the assumption that turbulence was introduced as a stationnary
Gaussian random process. The coupled system behaviour will be described using the random
dynamical system context [3].
It is well known that the stability of linear random dynamical systems can be examined by
the means of Lyapounov exponents. This approach can be extended to the case of nonlinear systems
by linearising in the vicinity of fixed points. The use of Lyapounov exponents and more generally of
the Oseledets' theorem necessitates rather restricting assumptions for the random noise which must be
modelled as a regular diffusion. That explains why, in practice, only Gaussian multiplicative noises
are considered. Recently [1] a general method for constructing nonGaussian stochastic processes has
been proposed, based on memoryless transformation of stationnary Gaussian processes.
Moreover the Gaussian process can be considered as a diffusion, solution of a stochastic differential equation.
Assuming that the diffusion has the good properties, this permits the use of Oseledets' results
for studying the stability of a dynamical system with nonGaussian multiplicative noise.
The purpose of this paper is to highlight the incidence of the noise probability distribution on the stability
of an airfoil when the longitudinal turbulence component is introduced as a nonGaussian noise.
In a first stage, the state space model is derived for the airfoilaerodynamic coupled system.
Using rational approximation of the aerodynamic forces [5], the system motion equation can be
written:
where is a multiplicative noise wich can model for instance longitudinal turbulence. Using the method described in [1], the nonGaussian process is constructed using a truncated Hermite chaos polynomial expansion: where The Lyapounov exponent is finally constructed, using a simulated sample path of equation (93) solution: In this paper our goal is to consider the effect of the distribution on the overall stability. We will consider here two families of distribution, the centered exponential distribution, and the centered Rayleigh distribution, . The spectrum will be modeled by the von Karman model. Numerical results show , as it was noticed in [2,4] for Gaussian noise, that if multiplicative noise has an impact on the stability of the linear airfoil, so does the distribution of the multiplicative noise, the more nonGaussian the distribution is the more important the effect. In presence of freeplay, the role of the noise distribution is crucial since, for this particular application, a Gaussian noise does not change significally the stability behaviour of the airfoil. References
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