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UBC Theses and Dissertations

An evaluation of component mode synthesis for modal analysis of finite element models Smith, Malcolm J.


Component mode synthesis (CMS) is a condensation method for vibration analysis which preserves the low frequency vibrational characteristics of a structure. In this method, the structure is treated as an assemblage of components whose displacements are described in terms of component modes. These modes may be some combination of static response, free vibration, or rigid body displacements of a component. In this thesis, the component mode sets used by other researchers are reviewed with a view to establishing which is most suitable for large-order finite element models. Two component mode sets are identified as ideally satisfying the basic requirements for inter-component compatibility, high convergence rate, linear independence and completeness. Fixed-interface and free-interface CMS formulations in the form of matrix eigen value equations are derived from these mode sets and describe approximately the low-frequency free vibration modes of the structure. They are improvements over previous formulations in that they can be systematically and efficiently applied to linear, undamped, discrete systems of an arbitrarily complex geometry. The free-interface formulation is derived both with and without an approximation of the high-frequency component inertia, and this results in two different structural mass matrices. Two new developments of the free-interface formulation are presented: (1) a method for calculating upper and lower bounds to the exact natural frequencies is given, providing a measure of the absolute accuracy of the structural frequencies; (2) the convergence and interlacing properties of the free-interface method are explored through the analysis of a two-component vibrating rod. Both the fixed- and free-interface methods have been implemented in the general-purpose finite element program VAST. Three finite element models are analyzed and a comprehensive comparison of the frequency and mode shape results obtained with CMS, direct finite element analysis, and Guyan reduction is presented. The complexity of the test cases is sufficient to infer general performance characteristics of the CMS methods. It is shown that with CMS, accuracy equal to a direct analysis is readily obtained in the low frequency modes, and that by using a frequency cutoff criterion to select dynamic modes, the natural frequencies converge in a fairly uniform manner. It is also shown that in terms of computational cost and order-reduction, the relative advantages of using CMS increase with the size of the model and with the stringency of the accuracy requirements. The free-interface method with second-order mass approximation gives the best overall performance because of its high convergence rate and superior condensation in complex two and three dimensional models. Application of CMS to structural dynamic modification and inverse modification is also studied. These techniques use a baseline modal analysis as a reference point for the modified system dynamics. A generalized CMS formulation for the baseline system is used to derive a linear-equivalent perturbation equation from which modified modes can be efficiently determined without recalculating the component modes. Also, two new methods are presented for predicting design changes which satisfy prescribed frequency constraints. An iterative scheme is proposed in which the energy-balance perturbation equations are solved with a full account of the nonlinear coupling terms; and a Newton's method algorithm using inverse iteration eigenvector updating is applied to the linear-equivalent equation. Numerical results using a finite element model are presented which show that for large structural changes, the two new methods give similar or better results than an established method.

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