Efficient spectral response of locally uncertain linear systems

It is difficult to model any real dynamical system with fully deterministic characteristics and yet, capture its behavior reasonably. Randomness arises from many sources, such as uncertain material properties, assumptions involved in structural modeling, and the stochastic nature of input forces. Thus, the random vibration analysis of systems with uncertain parameters is a crucial component of structural design and optimization procedures. In this paper, a new method is presented for fast spectral analysis of locally uncertain systems subjected to random inputs based on the response of one such system (called the nominal system). Unlike other methods, such as modal expansion methods, the proposed method is applicable to general uncertainties in the damping and stiffness matrices with the sole restriction that the system remains stable with probability one. Moreover, the proposed method yields exact responses for the perturbed systems and its accuracy is not affected by the size or magnitude of the uncertainties. However, the degree of locality of the uncertainty dictates the observed gains in computational efficiency when using the proposed method. When the uncertainty is extremely localized, one can expect gains in computational efficiency of two to three orders of magnitude while only modest gains of 2–3 times are observed when half the model is uncertain. Two numerical examples are presented to illustrate the accuracy and gains in computational efficiency of the proposed method.