The nature of epistemic uncertainty in linear dynamical systems

In the modeling of complex dynamical systems, high-resolution finite element models are routinely adopted to reduce the discretization error. This is often implemented by exploiting cost-effective computing hardware through parallel processing to solve the resulting large scale linear systems. Such an approach fails to enhance confidence in simulation-based predictions when the dynamical systems exhibit significant variability in their data and model leading to so-called data and modeling uncertainty. When substantial statistical information is available, data uncertainty can be tackled using probabilistic methods by modeling the parameters of the data as random variables or stochastic processes. Model uncertainty poses significant challenges as no parameter is available a priori as opposed to the case of data uncertainty. In modelling complex systems such marine (e.g. ships, submarines) and aerospace systems (e.g. helicopters and space shuttles) modeling uncertainty arises naturally due to the lack of complete knowledge and even presence of many subsystems attached to the main structural components. In the low frequency regions, effect of such substructure may be modeled by rigid masses attached to the primary structures. In higher frequency, the mechanics of energy flow among the primary and secondary systems may not be captured by these rigid masses alone as dynamics of the subsystems becomes more important. The additional degree-of-freedom arising from the subsystems should be incorporated to model the entire system. A sprung-mass models are adopted in the current study to investigate the effect of such subsystem on the vibration of a thin steel plate. The location of the attachments of these sprung-mass systems and their natural frequencies are assumed to be uncertain while the constitutive and geometric properties of the steel plate (e.g. the primary structure) are known. In contrast to the case of data uncertainty (traditionally modeled in the framework of stochastic finite element method), the model uncertainty arising from the sprung-masses (attached randomly to the plate) gives rise to entirely different variety of dynamical system for each sample. This can be observed from the variation in sparsity structure of the mass, stiffness and damping matrices of the total system from sample to sample. Clearly such change in sample-wise sparsity pattern can not be modeled by data uncertainty alone. In the case of data uncertainty, the actual configuration of dynamical system remains unchanged, just its local parameters change from sample to sample and therefore, sparsity structure of the system matrices for each sample remains the same. In this study, we investigate the feasibility of adopting a global probabilistic model to represent such entire ensemble of different dynamical systems derived from perturbing the model of a baseline system. In the current study, the baseline dynamical system is just the thin plate (without any sprung-mass attachment). A range of dynamical systems is then generated from the random attachment topology of the sprung-masses with the thin plate. As mentioned before, each of these dynamical systems possesses mass, stiffness and damping matrices for which sparsity pattern differ from sample to sample. The objective of this investigation is to represent uncertainty arising from model perturbation. We explore the possibility of stochastic representation of this entire variety derived from the baseline system with model perturbation (in contrast to data perturbation). More specifically, in the framework of random matrix theory, we fit the parameters of Wishart random matrices to model uncertainty in the mass, s