Reliability of uncertain linear and nonlinear systems

The reliability of uncertain single degree-of-freedom linear and nonlinear systems subjected to broadband random excitation is examined. A robust Petrov-Galerkin finite element solution for the reliability of deterministic single degree-of-freedom systems is used in conjunction with the theorem of total probability and a numerical integration scheme to obtain the sought reliabilities. Particular examples are given for the linear system in which the stiffness is modeled as a discrete random variable or as a continuous random variable. Use of the mean stiffness in an analysis is shown to be generally unconservative in the region of design interest, whereas using the Poisson approximation in conjunction with the moments of first passage time indicates that the results should be conservative. A particular example is given in which the error introduced by using the mean system parameters in the analysis is increased by increasing the failure bound width. A final example is given for the Duffing oscillator.