Highly Accurate Modal Method for Calculating Eigenvector Derivatives in Viscous Damping Systems

A highly accurate modal superposition method for computing complex eigenvector derivatives in viscous damp- ing systems has been developed. When higher modes are truncated, the conventional modal superposition method cannot give an accurate solution, and the errors may become significant. In this paper, calculating the derivatives is regarded as calculating the structural response to harmonic exciting. Using multiple modal accelerations and shifted-poles, highly accurate results would be obtained when only few modes are used. All of the available modal superposition methods would be directly obtained from the proposed method. Numerical examples show that it achieves better calculation efficiency than all of the available modal methods and Nelson's method when more than one eigenvector derivative is of interest. Moreover, the presented method can be used to improve response calculations and substructure syntheses. IGENVECTOR derivatives with respect to system parameters are widely used in structural design, structural stability anal- ysis, finite element model updating, and structural control. Since the earlier work of Fox and Kapoor,1 many methods have been developed.2'3 All of the available methods could be categorized into three groups: modal method,1'3'4 direct method,1'4 and iter- ation method.5'6 Most of the iteration methods suffer from slow convergence rates and are less efficient. Recently, Ting6 improved the effectiveness of the method. Because the direct methods give out the exact solutions, they are widely used. However, when the derivatives of a large number of eigenvectors are demanded, the cal- culation is very time consuming. Here, the modal methods are more suitable. The modal methods employ a modal superposition idea; there- fore, the accuracy is dependent on the number of modes used in calculation. When the complete set of eigenvectors is used, the ex- act solution could be obtained. To guarantee the accuracy, the clas- sical modal method1 needs higher eigenvectors. Recently, Wang4 presented a modified modal method, in which a modal acceleration type approach was used to approximate the contribution of trun- cated higher modes. Therefore, the modified modal method con- verges faster than the classical method. Ma and Hagiwara7). im- proved the modified method with an appropriate shift value, better convergence than that of the modified method is achieved, and sys- tems having zero eigenvalues can be directly analyzed. To further improve the convergence, the improved modified modal method8'10 and accurate modal superposition method9 have been proposed by several researchers. The basic original idea of the aforementioned improved modal methods was found in the Hu's book.11 The idea was used to improve the convergency of dynamic flexibility in series form for no damping systems. All of the aforementioned methods were used for real eigen- vectors. For the cases of complex eigenvectors in viscous damp- ing systems, the three groups of methods were presented.12'13 The classical modal method for the complex eigenvectors, which was similar to the method presented in Ref. 1 for real eigenvectors, suffers from heavy modal truncation errors. Recently, Akgun14 presented a new family of modal methods for the calculation of the derivatives in non-self-adjo int systems. The modal accelera- tion approach was used to improve the convergence. In this paper, calculating the derivatives in viscous damping systems is regarded as calculating a structural response to a harmonic exciting, using multiple modal accelerations and shifted-poles; a highly accurate modal method for calculating the derivatives has been developed. There are clear mathematical and physical meanings in conduct- ing the proposed method, from which all of the available meth- ods would be directly obtained. Furthermore, fewer eigenvectors (even only the specific eigenvector that is being considered) are re- quired for the predetermined accuracy. As a result, the method is more efficient in computation than Akgun's method and Nelson's method when more than one eigenvector derivative is of inter- est. Numerical examples show that the proposed method is cor- rect and drastically improves solution accuracy. The method can directly be applied in systems with zero eigenvalues. Although the method is developed to calculate eigenvectors derivatives, it can be used to improve response calculations and substructure syntheses.