Vibration Reanalysis

It has been noted that in dynamic analysis by mode superposition, the main computational effort is spent in the solution of the eigenproblem. Since exact solution of the problem can be prohibitively expensive, approximate solution techniques have been developed, primarily to calculate the lowest eigenvalues and corresponding eigenvectors [1–3]. Several studies have been published on eigenvalue reanalysis, where the object is to calculate only the modified eigenvalues [e.g. 4–6]. Other studies deal with eigenproblem reanalysis, or vibration reanalysis, where the object is to calculate both the modified eigenvalues and eigenvectors [e.g. 7, 8]. Vibration reanalysis is needed in various problems of structural analysis, design and optimization. Different reanalysis methods, which have been used for linear static analysis, are usually not suitable for vibration reanalysis.

In this chapter effective procedures for vibration reanalysis, based on the CA approach, are developed. Using these procedures, significant improvements in the accuracy of the results and the efficiency of the calculations can be achieved [9–12].

In Sect. 6.1 vibration reanalysis by the CA approach is introduced. The approximate reduced eigenproblem is formulated and procedures for determining the basis vectors are developed. Various means intended to improve the accuracy of the results are developed in Sect. 6.2. The procedures presented, which improve the quality of the basis vectors, include Gram-Schmidt orthogonalizations of the approximate modes, shifts of the basis vectors and Gram-Schmidt orthogonalizations of the basis vectors. A general solution procedure for vibration reanalysis is presented in Sect. 6.3 and various numerical examples are demonstrated in Sect. 6.4. In Sect. 6.5 it is shown how the CA approach can be used to improve the solution efficiency when various common iterative procedures for eigenproblem analysis are considered. These iterative procedures, formulated as reanalysis problems in Sect. 4.3.2, include inverse iteration, inverse iteration with shifts and subspace iteration.