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Using the total probability theorem, we propose a method to calculate the failure rate of a linear vibratory system with random parameters excited by stationary Gaussian processes. The response of such a system is non-stationary because of the randomness of the input parameters. A space-filling design, such as optimal symmetric Latin hypercube sampling or maximin, is first used to sample the input parameter space. For each design point, the output process is stationary and Gaussian. We present two approaches to calculate the corresponding conditional probability of failure. A Kriging metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure and the failure rate of the dynamic system. The proposed method is demonstrated using a vibratory system.

Design optimization often relies on computational models, which are subjected to a validation process to ensure their accuracy. Because validation of computer models in the entire design space can be costly, we have previously proposed an approach where design optimization and model validation, are concurrently performed using a sequential approach with variable-size local domains. We used test data and statistical bootstrap methods to size each local domain where the prediction model is considered validated and where design optimization is performed. The method proceeds iteratively until the optimum design is obtained. This method however, requires test data to be available in each local domain along the optimization path. In this paper, we refine our methodology by using polynomial regression to predict the size and shape of a local domain at some steps along the optimization process without using test data.

The system response of many engineering systems depends on time. A random process approach is therefore, needed to quantify variation or uncertainty. The system input may consist of a combination of random variables and random processes. In this case, a time-dependent reliability analysis must be performed to calculate the probability of failure within a specified time interval. This is known as cumulative probability of failure which is in general, different from the instantaneous probability of failure. Failure occurs if the limit state function becomes negative at least at one instance within a specified time interval. Time-dependent reliability problems appear if for example, the material properties deteriorate in time or if random loading is involved which is modeled by a random process. Existing methods to calculate the cumulative probability of failure provide an upper bound which may grossly overestimate the true value.