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| A general matrix perturbation method for complex modes using singular value decomposition |
| Revised:July 15, 1999 |
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| DOI：10.7511/jslx20014089 |
| KeyWord:complex modes,matrix perturbation,singular value decomposition |
| LIU Ji-ke 1 XU Wei-hua 2 YANG Hong-yan 2 |
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| Abstract: |
| A good understanding of the dynamic behavior of a structure under vibration and the ability to perform dynamic analysis is of vital importance for a designer nowadays. To satisfy a variety of requirements of structural dynamics, quite often in the dynamic design of structural systems, a so-called iterative design or reanalysis is generally carried out until a satisfactory outcome is achieved. A frequently encountered scenario in the dynamic reanalysis is determining the changes of the dynamic characteristics as a result of the changes in the system parameters. For general damped gyroscopic systems, the dynamic analysis is significantly difficult and most complicated. In such systems, there frequently co-exist distinct, repeated and closely spaced complex eigenvalues. In the existing techniques dealing with the three cases of eigenvalues, complete modal expansion is usually used. For large complicated systems, however, quite often it is difficult to obtain all modes. Therefore, using complete modal expansion probably gives rise to errors. In some cases these errors maybe too large to be acceptable.\;To this end, in this paper a simple and general matrix perturbation method is proposed for the dynamic reanalysis of systems with complex modes. This method is developed by performing a sub-space condensation and by using the singular value decomposition of complex matrix. The first and second perturbation formulas are derived. The present method can be universally applicable to systems with all the three cases of complex eigenvalues: distinct, repeated and closely spaced complex eigenvalues but complete modal expansion is not used. Illustrative examples are given to demonstrate the validity of this proposed technique. Three examples corresponding to the three cases of complex eigenvalues have shown that the present method is very effective. |
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