The First Passage Failure of SDOF Strongly Nonlinear Stochastic System with Fractional Derivative Damping

The first passage failure of single-degree-of-freedom (SDOF) strongly nonlinear stochastic system with fractional derivative damping is studied. The stochastic averaging method for SDOF strongly nonlinear stochastic system with fractional derivative damping under white noise excitations using the generalized harmonic function is introduced. The averaged It equation for Hamiltonian is obtained by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the mean of the first passage time are established. The conditional reliability function, and the conditional probability density and mean first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions. Finally, two examples are worked out in detail and the solutions are confirmed by using Monte Carlo simulation of original systems.