An efficient matrix tridiagonalization method for 3D finite element analysis of free vibration

In order to avoid mechanical resonance, a vibrating structure needs to be designed such that its working frequency interval is sufficiently far from its natural frequencies. Natural frequencies of a mechanical system are obtained from free vibration analysis commonly done by the finite element method. One important step in the analysis is matrix tridiagonalization commonly performed by the block Lanczos method. However, the classical block Lanczos method suffers numerical instability due to the loss of matrix orthogonality. In this paper, we demonstrate the implementation of the block Lanczos method with an orthogonality fixing scheme for 3D free vibration problems. The solution accuracy and computational time of this method are compared with those of the classical block Lanczos method and the Householder method. The results show that the block Lanczos method with the orthogonality fixing scheme employed in this work can effectively avoid the numerical instability due to the loss of matrix orthogonality. Furthermore, the block Lanczos method with the orthogonality fixing scheme provides solution accuracy as good as the Householder method while using significantly less computational time.