Spectral Modification Problems In Structural Dynamics

The spectral modification of vibration systems has become increasingly important in thestructural dynamics. In this dissertation, we consider some spectral modification problems in poleassignment and model updating by means of the theory and method of spectral decomposition ofvibration systems and algebraic inverse eigenvalue problems. The main contribution is as follows:According to the GLR theory, the spectral decomposition theories for various vibration systemsare given, including undamped system, viscous damped system, undamped and damped gyroscopicsystem. Moreover, the orthogonality relations for the eigenvectors of the associating vibration systemsare established, which provide the mathematical theory for spectral modification problems.For pole assignment problems, firstly, the partial pole assignment with delayed state feedback ofa first-order system is considered. A constructive method based on the orthogonality relations of theeigenvectors is provided. The explicit solution is given for the single-input case, and the parametricsolution is given for the multi-input case. Secondly, under the uncertain small time delay, the partialpole assignment with time-delay robustness for the first-order system is investigated. A time-delayrobustness measure is constructed by analyzing the sensitivity of the assigned eigenvalues withrespect to time-delay. The problem is formulated into an unconstrained minimization problem. Theformulation of the gradient vector for the objective function is provided. Then an algorithm forsolving this problem is presented. Next, the partial pole assignment with delayed state feedback of asecond-order system is considered. A constructive method based on the orthogonality relations of theeigenvectors is provided. The explicit solution for the single-input case and the parametric solution forthe multi-input case are given. Finally, the partial pole assignment problem for high-order system isaddressed. The new orthogonality relations for the eigenvectors of matrix polynomial are introducedto solve the problem, such that the unwanted eigenvalues are moved to the prescribed values and theremaining eigenvalues remain unchanged. The solvability condition and the parametric expression ofthe solution are derived. An optimization-based algorithm for solving the minimum norm solution isproposed.For model updating problems, firstly, the model updaing for undamped structural system isconsidered. Based on the spectral decomposition of the undamped system, a new method for updatingmass and stiffness matrices is presented. Using the method, the corrected mass matrix is symmetricpositive definite and the corrected stiffness matrix is symmetric positive semidefinite. Moreover, the measured frequencies and mode shapes are embedded into the corrected system and the remainingmodal data remains unchanged (no spill-over). We give the solvability condition and the expression ofthe solution, and propose an algorithm for computing the minimum modification solution. Secondly,we present a new model updating method for damped structural systems. The method can deal withthe spatially incomplete measured modal data without using any standard modal expansion orreduction techniques. Moreover, the updating is structure-preserving and no spill-over which meansthat the measured modal data is embedded into the corrected system and the remaining spectral data isthe same as those of the original system. The solvability condition of the problem is presented.Numerical example shows that the presented method is efficient. Finally, the model updaing forundamped gyroscopic systems is investigated. Using the spectral decomposition and the orthogonalityrelation, we give two structure-preserving eigenvalue embedding methods. The parametric solutionsare given by analyzing the freedom of the eigenvectors. Furthermore, the minimum modificationproblem is considered and an algorithm for solving this problem is presented.