Analytical solution for stochastic response of a fractionally damped beam

This paper presents a general analytical technique for stochastic analysis of a continuous beam whose damping characteristic is described using a fractional derivative model. In this formulation, the normal-mode approach is used to reduce the differential equation of a fractionally damped continuous beam into a set of infinite equations, each of which describes the dynamics of a fractionally damped spring-mass-damper system. A Laplace transform technique is used to obtain the fractional Green's function and a Duhamel integral-type expression for the system's response. The response expression 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 for two cases, and numerical results are presented for one of the cases. The approach can be extended to all those systems for which the existence of normal modes is guaranteed.