Study For Some Problems On Structural Dynamics Model Updating

The structural dynamics model updating has become increasingly important in the structural dynamics. This dissertation studies some model updating problems on undamped structural systems, damped structural systems and gyroscopic structural systems by means of the theory and method of the algebraic inverse eigenvalue problems. It provides the mathematical theory and efficient numerical methods for structural dynamics modification. The main contribution is as follows:For the undamped structural systems, the problem of updating mass matrix is firstly considered. The optimal corrected mass matrix complied with the required orthogonality condition is found by using the QR-decomposition. Secondly, a new method for updating stiffness matrix based on the orthogonality relations of the eigenvectors of the finite element model is established. The corrected stiffness matrix obtained by the method is symmetric positive semidefinite. Moreover, the measured frequencies and mode shapes are embedded into the corrected system and the remaining modal data remain unchanged. Next, in view of the magnitude of the stiffness terms being far greater than that of the mass terms, an efficient numerical method with a weighted factor is addressed. Using techniques of matrix decomposition and the optimization method, the explicit representations of updated matrices are given, and a method for choosing the weighted factor is provided. Finally, a new method for updating the inaccurate elements in finite element dynamics model is presented by concentrating the accurate elements in a region, and using the orthogonality condition. The modified model does not exert any influence on the accurate elements. Numerical examples show that the presented method is efficient.For the damped structural systems, the problem of updating the viscous damping matrix using the complex measured modal data is firstly considered. By applying the inverse eigenvalue technique, the optimal corrected positive semidefinite damping matrix complied with the required eigenvalue equation is found. Secondly, a model identification problem is firstly addressed under the assumption that M a is completely accurate and M a> 0, i.e., how many eigenpairs should be needed to determine uniquely the damping and stiffness matrices? Necessary and sufficient conditions for the existence of solution of the problem are derived using the matrix decomposition, and the expression of the solution is provided. Finally, an efficient numerical method for updating the damping and stiffness matrices simultaneously using a few of complex measured modal data is developed. By applying the singular value decomposition, the optimal corrected damping and stiffness matrices that satisfy the equation of motion are found in a weighted Frobenius norm. This model updating method is direct and the updating process is very simple. Numerical results show that the presented method can be used to update the finite element model to get better agreement between analytical and experimental modal parameters.For the gyroscopic systems, the problem of updating the gyroscopic matrix from incomplete complex experimental modal data is firstly considered. Based on the eigenequation and the skew-symmetry of the gyroscopic matrix, the corrected matrix that is closest to the finite element gyroscopic matrix is found by applying Lagrange multiplier method and the matrix decomposition. Secondly, using the technique of linearization, a problem of reconstructing an undamped gyroscopic system is considered when the complete spectral data are given. The solvability conditions and a solution to the problem are presented, and a method for model updating is developed using the obtained results. Using the method, the measured modal data are embedded into the corrected system and the remainder spectral data are the same as those of the original system. Finally, an efficient numerical method for the finite element model updating of damped gyroscopic system based on incomplete complex measured modal data is proposed. Applying the Lagrange multiplier method, the optimal corrected damping and gyroscopic matrices complied with the required eigenvalue equation are found. Numerical example shows that the agreement between the experimental and analytical modal data is remarkably improved.