A new method for computation of eigenvector derivatives with distinct and repeated eigenvalues in structural dynamic analysis

Zhengguang Li, Siu Kai Lai, Baisheng Wu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

10 Citations (Scopus)


Determining eigenvector derivatives is a challenging task due to the singularity of the coefficient matrices of the governing equations, especially for those structural dynamic systems with repeated eigenvalues. An effective strategy is proposed to construct a non-singular coefficient matrix, which can be directly used to obtain the eigenvector derivatives with distinct and repeated eigenvalues. This approach also has an advantage that only requires eigenvalues and eigenvectors of interest, without solving the particular solutions of eigenvector derivatives. The Symmetric Quasi-Minimal Residual (SQMR) method is then adopted to solve the governing equations, only the existing factored (shifted) stiffness matrix from an iterative eigensolution such as the subspace iteration method or the Lanczos algorithm is utilized. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner. Three numerical examples are given to illustrate the accuracy and validity of the proposed algorithm. Highly accurate approximations to the eigenvector derivatives are obtained within a few iteration steps, making a significant reduction of the computational effort. This method can be incorporated into a coupled eigensolver/derivative software module. In particular, it is applicable for finite element models with large sparse matrices.

Original languageEnglish
Pages (from-to)78-92
Number of pages15
JournalMechanical Systems and Signal Processing
Publication statusPublished - 1 Jul 2018


  • Distinct and repeated eigenvalues
  • Eigenvector derivatives
  • Finite element model
  • Real symmetric eigensystems
  • Singularity

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Signal Processing
  • Civil and Structural Engineering
  • Aerospace Engineering
  • Mechanical Engineering
  • Computer Science Applications


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