A Reduction Method for Large Scale Unsymmetric Eigenvalue Problem in Structural Dynamics

The discussion begins with the classification of eigenvalue problems arising from conservative and non-conservative structural systems. The conservative type includes undamped structural eigenvalue problems and undamped gyroscopic eigenvalue problems. The non-conservative type includes damped structural eigenvalue problems, damped gyroscopic eigenvalue problems and constrainedly damped eigenvalue problems. The methods for solving large scale unsymmetric eigenvalue problems are briefly reviewed. The advantages and properties of Arnoldi's method have also been discussed. Arnoldi's reduction method has been generalized and the partial solution of large scale unsymmetric-definite eigenvalue problems in structural dynamics is presented in detail. A very simple reduction algorithm is obtained by simplifying the proposed method for undamped gyroscopic eigenvalue problems. To make the proposed reduction method feasible for engineering problems, a restart technique is introduced to work with Arnoldi's reduction method for checking and computing missing eigenvalues. Numerical examples are also presented to demonstrate the effectiveness of the proposed reduction method.