Stochastic Modal Models of Slender Uncertain Curved Beams Preloaded Through Clamping

This paper addresses the stochastic modeling of the stiffness matrix of slender uncertain curved beams that are forced fit into a clamped–clamped fixture designed for straight beams. Because of the misfit with the clamps, the final shape of the clamped–clamped beams is not straight and they are subjected to an axial preload. Both of these features are uncertain given the uncertainty on the initial, undeformed shape of the beams and affect significantly the stiffness matrix associated with small motions around the clamped–clamped configuration. A modal model using linear modes of the straight clamped–clamped beam with a randomized stiffness matrix is employed to characterize the linear dynamic behavior of the uncertain beams. This stiffness matrix is modeled using a mixed nonparametric–parametric stochastic model in which the nonparametric (maximum entropy) component is used to model the uncertainty in final shape while the preload is explicitly, parametrically included in the stiffness matrix representation.

Finally, a maximum likelihood framework is proposed for the identification of the parameters associated with the uncertainty level and the mean model, or part thereof, using either natural frequencies only or natural frequencies and mode shape information of the beams around their final clamped–clamped state. To validate these concepts, three simulated, computational experiments were conducted within Nastran to produce populations of natural frequencies and mode shapes of uncertain slender curved beams after clamping. The three experiments differed from each other by the nature of the clamping condition in the in-plane direction. One experiment assumed a no-slip condition (zero in-plane displacement), another a perfect slip (no in-plane force), while the third one invoked friction. The first two experiments gave distributions of frequencies with similar features while the latter one yielded a strong deterministic dependence of the frequencies on each other, a situation observed and explained recently and thus not considered further here. Then, the application of the stochastic modeling concepts to the no-slip simulated data was carried out and led to a good matching of the probability density functions of the natural frequencies and the modal components, even though this information was not used in the identification process. These results strongly suggest the applicability of the proposed stochastic model.