Study For Some Problems On Model Updating

The structural dynamics model updating has become increasingly important in the structuraldynamics. This dissertation mainly studies some problems in finite element model updating ofundamped structures. The main contributions are listed as follows:Under the assumption that the measured data is exact, the problems of updating analyticalstiffness matrix and updating analytical mass matrix and stiffness matrix simultaneously arediscussed. Assume that the analytical mass matrix is exact, the least change adjustment to theanalytical stiffness matrix is found in Frobenius norm sense constrained by eigenequation,positive semi-definiteness and sparsity requirement. Conditions to ensure the feasible region ofthe minimization problem is nonempty are drawn and the KKT conditions are also deduced forthe optimal solutions. The alternating projection method, the dual approach and the matrix linearvariational inequality technique are used to solve the minimization problem. Numerical testsshow that the proposed methods work well. The simultaneous updating of analytical mass matrixand stiffness matrix is reformulated as an optimal matrix pencil approximation problem witheigenequation, orthogonal condition of measured modes, positive semi-definiteness and sparsityrequirement imposed as side constraints. Conditions to ensure the feasible region of the optimalproblem is nonempty are drawn. The alternating projection method is applied to solve theminimization problem and the convergence of the algorithm is proved. Numerical examplesillustrate the efficiency of the proposed method.When the measured data is noised, the problems of updating analytical stiffness matrix andupdating analytical mass matrix and stiffness matrix simultaneously are studied. Assume that theanalytical mass matrix is exact, the least change adjustment to the analytical stiffness matrix isfound in Frobenius norm sense constrained by positive semi-definiteness, sparsity requirementand the minimal residual of the eigenequation. Conditions to ensure the feasible region of theminimization problem is nonempty are drawn. The alternating projection method is used to solvethe minimization problem. The simultaneous updating of analytical mass matrix and stiffnessmatrix is reformulated as an optimal matrix pencil approximation problem with minimal residualof eigenequation, minimal residual of orthogonal condition of measured modes, positivesemi-definiteness and sparsity requirement imposed as side constraints. Conditions to ensure thefeasible region of the optimal problem is nonempty are drawn. The alternating projection methodis applied to solve the minimization problem and numerical examples show that the proposed method works well.Problems of local updating of mass matrix and stiffness matrix using inexact measured dataare reformulated as a minimization problem with submatrix constraint. The least changeadjustment to the analytical stiffness matrix is found in Frobenius norm sense constrained bypositive semi-definiteness, sparsity requirement and the minimal residual of the eigenequation.Similarly, the least change adjustment to the analytical mass matrix is also found in Frobeniusnorm sense with positive semi-definiteness, sparsity requirement and the minimal residual of theorthogonal condition of measured modes imposed as constraints. The alternating projectionmethod is proposed and numerical examples illustrate the efficiency of the proposed method.Problem of updating the mass matrix and the stiffness matrix of gyroscopic systemsimultaneously from incomplete complex modes is considered. Assume that the analytical massmatrix of gyroscopic system is exact, the model updating problem is transformed to be a bestmatrix pencil approximation problem constrained by eigenequation, skew symmetry ofgyroscopic matrix and the symmetry of stiffness matrix. Numerical algorithm based on QRdecomposition is proposed.