Uncertain systems: Modelling and updating

Abstract

In this article a new method, which relies on the concept of density matrix, for modelling un- certain vibrating structures is presented. The density matrix is defined as the product of the normalized to unity mode shape with its transpose. The resulting density matrices, one for each mode shape, together with a nominal model constitute the model of the uncertainty. Con- sequently, the modal properties of both certain and uncertain structures are defined as expected values of the system properties with respect to the density matrices. Furthermore, by taking kronecker products of the original mode shapes new density matrices are constructed, which can be used to demonstrate that the variances of the modal properties are also the expected values with respect to the covariance matrices of the system properties. Results of updating the mean and variance values of the uncertain system parameters are also discussed. Copyright © (2011) by the International Institute of Acoustics & Vibration.