Keywords: stochastic dynamical systems, uncertainty propagation, asymptotic expansion.

High-resolution finite element models are routinely adopted as predictive tools for complex dynamical systems such as aerospace, marine and automotive vehicles. Recent availability of cost-effective high performance computing platforms offers practical means to solve such large-scale linear systems using parallel processing. Although such high-resolution numerical models can reduce discretisation errors, in numerous cases, experimental results and numerical predictions exhibit significant variabilities stemming from the random scatter in model parameters and imperfect models. When substantial statistical information is available, the scatter in the model parameters can be represented using probabilistic methods. In this case one can fit probability density functions corresponding to available data and consequently the model parameters can be expressed as random variables or random processes. In this paper a new approach to obtain the first two statistical moments of the dynamic response of a linear system with uncertain parameters in the frequency domain has been proposed.

Over the past three decades, the stochastic finite element methods have been developed to analyze uncertain dynamical systems. There are two broad approaches, namely (a) the direct inversion of the complex stochastic dynamic stiffness matrix, and (b) dynamic response using the random eigensolutions. In this paper the second approach is taken. Explicit analytical expressions of the mean and variance of the amplitude of the frequency response function have been derived using the Laplace's method of asymptotic integration. Firstly, single-degree-of-freedom-system is considered and later the results are extended to general multiple-degrees-of-freedom dynamic systems. The eigenvalues of the system are assumed to be independent and identically distributed random variables with different mean and variance. Several probability density functions, including gama, normal, uniform and lognormal distributions have been considered. The results have been validated against the Monte Carlo simulation results. It is shown that the closed-form expressions derived in the paper offer computational advantage without introducing significant error.

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