Linear and nonlinear model reduction in structural dynamics with application to model updating

Developments of the state-of-art technology in the area of model reduction and model updating/damage detection for linear and nonlinear systems are addressed in this dissertation.;Direct model reduction (MR) defines transformation between master states which are to be retained in the reduced model and slave states which are to be eliminated. Direct model reduction for linear system defines exact transformation such that the reduced model preserves exact eigen-structure of the full model. For model reduction of nonlinear systems, a "linear based reduction" which utilizes the exact-for-the linear case master/slave transformation, is developed. This is compared to a "NNM-based reduction" which is based on the leading order calculation of the nonlinear master/slave state transformation. Numerical studies illustrate that the linear based reduction is much simpler to implement and provides better results for nonlinear systems. The linear based model reduction also captures essential nature of non-analytic nonlinearity such as deadband and discontinuous saturation, whereas these would be extremely difficult to characterize accurately using the NNM based reduction.;Ritz reduction for linear system is presented as an efficient model reduction technique to approximate eigen-properties of a large structure under influence of structural changes due to non-proportional damping or damage. This method enables the reduced model to retain the effect of structural changes as a means of correction and improves lower modes approximations. Numerical studies illustrate the reduced model obtained using this method can predict the complete system response with exceptional accuracy. The application of Ritz vector is extended to characterize damage of a finite element beam. Mode shape expansion based on Ritz vectors are formulated for damaged structural systems. It turns out to be a simple and accurate method to do mode shape expansion.;Two frequency domain model updating/damage detection methods are formulated for linear systems: (1) expanded mode shape based method and (2) reduced model based method. The two methods are iterative, and the optimization for each method is performed based on minimization of measurement errors embedded in the residual and reduced residual. For the method (1), the expanded mode shape and the changes in stiffness due to damage are updated. Method (2) involves updating of change in stiffness and calculations of reduced model at the perturbation stage. Numerical examples show that fast and accurate convergence is achieved for an spring-mass system and a finite element beam.;Two model updating/damage detection methods in the time domain, combined with the linear-based model reduction, are addressed: (1) Root Mean Square(RMS) of which the optimization is performed by minimizing the error between simulated and measured response. (2) Singular Value Decomposition (SVD) of which the optimization is performed by minimizing error between parameters of SVD's of simulated and measured response. Numerical examples show that both methods produce reasonably successful updating for linear and nonlinear (cubic nonlinearity) systems. The advantage of using RMS method over SVD is that it eliminates the computation of SVD's of the response histories of structural systems.