Stochastic response of chain-like hysteretic MDOF systems endowed with fractional derivative elements and subjected to combined stochastic and periodic excitation

A statistical linearization method is proposed for determining the response of the chain-like multi-degree-of-freedom (MDOF) hysteretic system endowed with fractional derivative elements and subjected to combined periodic and stochastic excitation. The method is developed based on representing the system response into a combination as a deterministic and a zero-mean stochastic component. Specifically, first, the equation of motion can be transformed into two sets of coupled nonlinear differential equations for the deterministic and the stochastic response components, respectively. Next, the deterministic and the stochastic differential equations are treated by the harmonic balance method and the statistical linearization method, respectively, yielding two sets of coupled nonlinear algebraic equations, which can be solved simultaneously via standard numerical methods for the deterministic response time history and the stochastic response second moment. Pertinent numerical examples demonstrate the applicability of the proposed method.