Random Vibration Analysis of Linear/Nonlinear Systems with Fractional Derivatives Subjected to Colored Noise

In this paper, the stochastic responses of multi-degree-of-freedom (MDOF) linear/nonlinear systems with rational-order fractional derivatives excited by colored noise are investigated. A closed-form expression for the PSDs of MDOF linear systems with fractional derivatives under exponential colored noise is derived by combining the eigenvector expansion method (EEM) and the pole-residue representation of the transfer function. The equivalent linearization method (ELM) is developed to determine the stochastic responses of fractional-order nonlinear systems. This is achieved by replacing the original system with an equivalent fractional-order linear system whose parameters are determined by minimizing the mean-square difference. The EEM is incorporated into the linearization framework to solve the linear fractional-order equations during the iterative process for the equivalent coefficients. Two widely adopted linearization criteria, the energy-based criterion and the equation-based criterion, are examined in detail. Numerical examples, including two linear systems and three nonlinear fractional-order systems with various types of nonlinearity under colored noise, are presented to verify the accuracy and applicability of the proposed method. Compared with Monte Carlo (MC) simulations, both criteria provide satisfactory accuracy. In addition, the results from the energy-based linearization criterion are more accurate than those from the equation-based criterion for single-degree-of-freedom (SDOF) systems, whereas for MDOF systems, the equation-based criterion yields slightly more accurate results than the energy-based criterion. The effects of nonlinear intensity and the order of the fractional derivative on the response statistics are also investigated.