Exact statistics of systems with uncertainties: An analytical theory of rank-one stochastic dynamic systems

This paper concerns the response of uncertain vibro-acoustic and structural dynamic systems. Here, exact expressions are presented for the statistics of systems with a random rank-one component. The expressions are derived using the Sherman–Morrison update formula that gives the exact expression of the disturbed response for any magnitude of the disturbance. It is shown that the probability density function (pdf) of any transfer function is a simple function of the pdf of the disturbance magnitude of the random component. The expressions for the mean, variance, and covariance of any transfer function, and at any frequency, of a random system necessitate non-trivial integrals. Exact, including closed-form, expressions of these integrals are derived in the particular cases of a real or complex normal disturbance magnitude, and qualitative differences between these two cases are highlighted. The theoretical and practical advantages of the theory are discussed and applied to a model of a bladed disk subjected to random damage. The comparison with Monte-Carlo simulations demonstrates that the statistics can be evaluated efficiently and precisely. The theory, derived formally in the context of discretised systems, is directly applicable to continuous systems.