Reliability of quasi-integrable Hamiltonian systems with coupled viscoelastic dampings driven by wideband noises

Viscoelasticity is the time-dependent anelastic behavior of materials, which means the response of a viscoelastic system to a stimulus depends on the instantaneous state and its history. For quasi-integrable Hamiltonian systems with coupled viscoelastic dampings driven by wideband noises, the key to estimate their reliability is how to handle the coupled viscoelastic dampings between different degrees of freedom. In the present paper, the con-tributions of the coupled viscoelastic dampings are decoupled into elastic restoring forces and viscous dampings by using the generalized harmonic balance technique, and the equivalent quasi-integrable Hamiltonian systems without coupled viscoelastic dampings are obtained. Then by using the stochastic averaging method, the aver-aged equations for all slowly varying processes of the equivalent systems in both non-resonant and internally resonant cases are derived, which retain the main dynamical characteristics of the equivalent systems. The lower-dimensional backward Kolmogorov (BK) equation for reliability and the Pontryagin equation for the mean first passage time of the equivalent systems are derived. Two examples are worked out in detail to illustrate the application of the proposed procedure. Comparisons of the results obtained by using the proposed method with those from the Monte Carlo simulations of original systems show that the proposed method is well applicable.