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Keywords:

  • structural dynamics;
  • eigenvalue analysis;
  • nonproportional damping;
  • Arnoldi method

Abstract

This paper presents and discusses an Arnoldi-based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices. This modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems.

The eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non-linear eigenvalue problem at given values of the parameter. Numerical examples illustrate that this method, used in conjunction with a secant interpolation, accurately evaluates the complex natural frequencies and modes of the quadratic non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models.