Nonstationary random vibration analysis of fractionally-damped nonlinear systems by equivalent linearization based on explicit time-domain method

Fractionally-damped nonlinear systems can be classified into two categories based on whether the nonlinearity originates from the fractional derivative term. The first category consists of classical nonlinear systems augmented by a linear fractional derivative term, while the second category comprises systems where the nonlinearity arises from the fractional derivative term itself. In recent years, random vibration analysis of fractionally-damped nonlinear systems has received increasing attention, but it has been limited to the first-category systems. This study presents an equivalent linearization method (ELM) to solve the random vibration problems of general fractionally-damped nonlinear systems under nonstationary random excitations. The recently developed explicit time-domain method (ETDM) is integrated into the ELM as an efficient solver for the repetitive nonstationary random vibration analyses of a series of fractionally-damped linearized systems involved in the iteration process. Five numerical examples are investigated to demonstrate the effectiveness of the present ETDM-based ELM, including a Duffing oscillator with a fractional derivative term, a Duffing-like oscillator with nonlinear fractional damping, a van der Pol-like oscillator with nonlinear fractional damping, a 5-degree-of-freedom (5-DOF) structure with nonlinear viscous dampers and fractional viscoelastic dampers as well as a 5-DOF structure with magnetorheological dampers described by nonlinear fractional Bingham models.