An efficient way to perform stochastic dynamic analysis in fractional order systems

Several dynamic analysis in engineering problems involve structural elements that can be modeled as multi-degree-of-freedom (MDOF) systems embedded with fractional derivatives [1]. In this kind of systems the equations of motion are represented by a set of coupled fractional-order differential equations in terms of force-displacement relation. An efficient approach to obtain the exact solution of fractional MDOF for deterministic problem has been proposed by several authors [2, 3]. This approach is based on a proper state variable analysis in conjunction with a complex modal transformation that allows to decouple the equations of motion. The fractional state variable analysis has been also applied in stochastic mechanics problems [4, 5]. However, such kind of dynamic analysis has a numerical limitation when the dimension of the problem increases because the exact solution can become cumbersome. For this reason, an ap- proach to obtain an approximate solution with a proper complex modal truncation is proposed in this article. In this way, the fractional state variable analysis can be readily applied in Monte Carlo simulation, also in the case in which many degrees of freedom are involved.