Dynamic Flexibility Method with Hybrid Shifting Frequency for Eigenvector Derivatives

Abstract

When concentrated roots exist, the precision of the dynamic flexibility method proposed previously by the authors is changed severely to be poor because a power series in the method closes in divergency. For this reason, a dynamic flexibility method with hybrid shifting frequency is developed. This new dynamic flexibility method applies a hybrid shifting frequency technique, that is, two shifting frequency values, Δω 1 and Δω 2 , are put up. So a hybrid shifting frequency system is obtained. In this system Δω 1 , is used to guarantee that the stiffness matrix of the system is always nonsingular, and Δω 2 is employed to accelerate the convergence of the power series contained in the present system. Thus the dynamic flexibility method with hybrid shifting frequency is suitable to the concentrated root condition. However, the eigenvector derivative of this hybrid shifting frequency system is not the same as that of the original system. To give the eigenvector derivatives of the original system, the relationship between dynamic flexibility matrices of the original and the hybrid shifting frequency system is first established. Then the eigenvector derivatives of the original system can be found from the eigenvector derivatives of the hybrid shifting frequency system. In words, this method is powerful and suitable for any (constrained and free) structures with concentrated roots.