Research On Condensation Methods For Dynamic Models Of Non-classically Damped Structures

This dissertation is devoted to constructing low-dimensional and high accuracy dynamic models of non-classically damped structures based on the view of condensation of degrees of freedoms (DOFs). Some research work is made to improve the computational efficiency and reduce the required computer resources. Several efficient model condensation algorithms are proposed. Based on the proposed methods, the low-dimensional models corresponding to the high-dimensional dynamic models can be obtained rapidly and precisely, which facilitate dynamic analysis, design, optimization and control of structures. The numerical examples show that the model condensation methods proposed in this dissertation improve the efficiency of the computation, and decrease the computer resources requirement compared with existing model condensation methods.Because model condensation of undamped structures is the starting point to investigate the model condensation approaches, the first part of this dissertation makes some research for model condensation methods of undamped structures. The main work is listed as follows:(1) Based on the equivalent coefficients of master DOFs of original and reduced system models, an equivalent condensation algorithm is given by constructing the iterative form of transformation matrix. This condensation procedure not only has the same precision of numerical results with the iterated improved reduced system method, but also has the merits of simply derivation and explicit meaning.(2) An equivalent-substructuring condensation algorithm and an iterative substructuring condensation algorithm are obtained by combining the equivalent condensation procedure and the iterative condensation procedure with the substructuring scheme, respectively. As compared with the original equivalent condensation method and the iterative condensation method, the new condensation procedures divide the whole structure into several small regions, and condense the system model for each substructure respectively. This leads to dramatic dimension decreasing of operation matrices. Meanwhile, the unrelated slave DOFs are discarded between the substructures. Thus the computational efficiency is improved, which implies that it is convenient to accomplish the model-condensation problems of whole structures with limited computer resources. The results of numerical examples demonstrate the validity of the proposed condensation algorithms.Based on the condensation approaches derived from complex eigenvalue equation, the second part of this dissertation investigates the model condensation methods for the non-classically damped structures with complex eigenvalue equation in the state space. The main work is listed as follows:(1) An improved iterative model condensation algorithm is obtained by extending the iterative model condensation method from displacement space to state space with complex eigenvalue equations. This condensation procedure can fully take into account the effect of non-classical damping on the transformation matrix, so that the convergence of iterations is guaranteed. As compared with the other state-space condensation procedures, the improved iterative condensation method can reduce the computational time and decrease the required computer resources. The results of numerical examples show that the proposed algorithm not only guarantees the precision of complex eigenvalues but also has a high computational efficiency.(2) An improved iterative sub structuring algorithm is obtained by combining the above improved iterative scheme and the sub structuring scheme. As compared with the improved iterative condensation technique, the new algorithm discards the unrelated slave DOFs and operates the matrices calculation in the substructures. This leads to the low-dimensional operation matrices and the improvement of computational efficiency. It also has advantages in the accomplishment of model condensation of whole high-dimensional systems with limited computer resources. The numerical examples show that this method not only can guarantee the precision of the reduced systems but also has a high computational efficiency.To overcome the limitation of too much computer resources requirement of state-sapce model condensation approaches, the third part of this dissertation researches the corresponding model condensation methods for non-classically damped structures with the equivalent matrices in displacement space. The main work is listed as follows:(1) An equivalent condensation procedure is proposed by utilizing the equivalent equations for master DOFs of original and reduced systems. The transformation matrix is constructed by including the stiffness and non-classical damping. As compared with the state-space condensation methods, because the transformation matrix defined in the displacement space, the size of system matrices will not be double, implying that it is efficient to construct the condensation matrix with less computer resources.(2) A more efficient equivalent substructuring condensation approache is obtained by combining the above equivalent method with the substructuring scheme. Because the unrelated slave DOFs in each substructure is removed and the corresponding matrix is calculated in each substructure, the computational efficiency is improved and the required computer resources are reduced.