Insight into the dynamic condensation technique of non-classically damped models

The dynamic condensation technique fully depends on the definition and computational scheme of the dynamic condensation matrix. Four definitions for the dynamic condensation matrix in the single-mode-, m-mode-, response-dependent dynamic condensation and modal reduction of non-classically damped models are presented. They are, respectively, defined as the relations of the single eigenvector, m eigenvectors, P eigenvectors, and responses between the master and the slave degrees of freedom. Using the complex mode superposition technique, the response-dependent dynamic condensation matrix may be interpreted as any-mode-, including whole-mode, m-mode, P-mode and single-mode, dependent condensation matrix. Computational equations for the dynamic condensation matrix are derived for each of definitions. After the proper introduction of the assumptions for the single-mode and the response-dependent dynamic condensation, the same computational equation is obtained from the former three definitions. In the modal reduction, the dynamic condensation matrix is directly computed from the eigenvector matrix of full model. Because the eigenvector matrix of the non-classically damped models is generally complex, the complex numerical operations are required in the commonly used expression. An alternative expression is derived in which only the real numerical operations are necessary. Furthermore, it is proven that the dynamic condensation matrix and the reduced system matrices resulted from the modal reduction all have real values.