Transient probability density of nonlinear multi-degree-of-freedom system with time delay

The transient probability densities of nonlinear multi-degree-of-freedom systems with time delay are investigated. The system is firstly approximated by the corresponding non-time-delay system through appropriate relations between the current states and the delay states. Stochastic averaging is adopted to reduce the dimension of the equivalent system, and the corresponding Fokker–Planck–Kolmogorov equation with regard to transient probability density is obtained. This equation is solved by expressing the transient probability density as multiple series in terms of a set of properly state-dependent orthogonal basis functions with time-dependent coefficients. Two examples are given to illustrate the accuracy and efficacy of the proposed procedure.