Relability of van der Pol-Duffing oscillator with fractional derivative under wide-band noise excitations

To investigate the first-passage time of van der Pol-Duffing oscillator with fractional derivative under wide-band noise excitations, the restoring force described by a fractional derivative was firstly separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique, which yields an equivalent nonlinear stochastic system without fractional derivative. Then, the equivalent nonlinear stochastic system was approximated as one-dimensional diffusive process by using the stochastic averaging method, and the backward Kolmogorov equation associated with the averaged equation was then established and solved to yield the conditional reliability function and mean first-passage time of system. Finally, the numerical simulation demonstrates that the analytical results agree well with those derived by the Monte Carlo simulation; the system reliability improves with the fractional order; and the restoring force described by the fractional derivative can not be regarded as a special damping force.