The first-passage failure of quasi-integrable Hamiltonian systems (multidegree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is investigated. The motion equations of such a system are first reduced to a set of averaged Ito^ stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamitonian systems. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving these equations with suitable initial and boundary conditions. Two examples are given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the procedure.

*Elements of Markov Processes and Their Applications*, McGraw-Hill, New York.

*The Theory of Stochastic Processes*, Chapman and Hall, New York.

*Proc. 47th Session of International Statistical Institute*,

**LIII**(Invited Papers), Book 3, pp. 517–531.

*Stochastic Structural Dynamics*, B. F. Spencer, Jr., and E. A. Johnson, eds., Balkema, Rotterdam, pp. 141–148.