Stochastic analysis of a 1-D system with fractional damping of order 1/2

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 eigen-vector expansion method 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 for the response of the system. 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, namely the variance and covariance responses of the system. Closed form stochastic response expressions are obtained for white noise. Numerical results art, presented to show the stochastic response of a fractionally damped system subjected to white noise. Results show that stochastic response of the fractionally damped system oscillates even when the damping ratio is greater than its critical value.