An Analytical Scheme for Stochastic Dynamic Systems Containing Fractional Derivatives

This paper presents an analytical technique for the analysis of a stochastic dynamic system whose damping behavior is described by a fractional derivative of order 1/2. In this approach, an eigenvector expansion method proposed by Suarez and Shokooh is used to obtain the response of the system. The properties of Laplace transforms of convolution integrals are used to write a set of general Duhamel integral type expressions. The general response contains two parts, namely zero state and zero input. For a stochastic analysis the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed form stochastic response expressions are obtained for white noise. Numerical results are presented to show the stochastic response of a fractionally damped system subjected to white noise. INTRODUCTION All structural and flexible multibody systems exhibit some degree of internal damping. Accurate modeling of these systems requires accurate modeling of damping. It has been shown that fractional derivative models describe the frequency dependence of the structural damping very well (Bagley and Torvik; 1983a, 1983b, 1985). Koeller (1984) considered a fractional calculus model to obtain expressions for creep and relaxation functions for viscoelastic materials. Makris and Constantinou (1991) presented a fractional-derivative Maxwell model for viscous dampers and validated their model using experimental results. They also presented some analytical results for a fractionally damped single-degree-of-freedom system. Mainardi (1994) presented the thermoelastic coupling in anelastic solids to account for a temperature fractional relaxation due to diffusion. Fractional derivative based techniques to model damping behavior of materials and systems have been considered by Shen and Soong (1995), Pritz (1996), and Papoulia and Kelly (1997) also. Makris and Constantinou (1992) and Lee and Tsai (1994) have used fractional derivatives to model seismic and vibration isolation. Fractional derivative models have also been used to model stability and control of viscoelastic structures. Skaar, Michel and Miller (1988) and Makroglou, Miller and Skaar (1994) presented root locus analyses for one-dimensional controlled distributed structures whose damping is modeled with fractional order derivatives. Bagley and Calico (1991) presented a fractional order state equations for the control of viscoelastically damped structures. Their study showed that the feedback of fractional order time derivatives of structural displacements improves the system control performance. Mbodje et al. (1994) presented a linear-quadratic optimal control of a rod whose damping mechanism were described in terms of fractional derivatives. Makris, Dargush and Constantinou (1993) presented a general boundary-element