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2 Computer Assisted Mechanics and Engineering Sciences

2 1999

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Efficient free vibration analysis of large structures with close or multiple natural frequencies. P. 1 : Undamped structures

100%

Lee I. W. , Jung H. J. , Kim M. C.

Computer Assisted Mechanics and Engineering Sciences

1999

tom Vol. 6, No. 3-4

403-414

An efficient numerical method which can calculate the eigenproblem for the large structural system with multiple or close natural frequencies is presented. The method is formulated by the accelerated Newton-Raphson method to the transformed problem. The method can calculate the natural frequencies and mode shapes without any numerical instability which may be encountered in the well-known methods such as the subspace iteration method or the determinant search method which has been widely used for solving eigenvalue problem. The efficiency of the method verified by comparing convergence and solution time for numerical examples with those of the subspace iteration method and the determinant search method.

Efficient free vibration analysis of large structures with close or multiple natural frequencies. P. 2 : Damped structures

100%

Lee I. W. , Jung H. J. , Kim M. C.

Computer Assisted Mechanics and Engineering Sciences

1999

tom Vol. 6, No. 3-4

415-424

An efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of nonproportionally damped structural systems with close or multiple eigenvalues. The proposed method is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linear eigenproblem format through matrix augmentation of the quadratic eigenvalue problem. In the iteration methods such as the inverse iteration method and the subspace iteration method, singularity may be occurred during the factorizing process when the shift value is close to an eigenvalue of the system. However, even if the shift value is an eigenvalue of the system, the proposed method guarantees nonsingularity, which is analytically proved. The initial values of the proposed method can be taken as the intermediate results of iteration methods or results of approximate methods. Two numerical examples are also presented to demonstrate the effectiveness of the proposed method and the results are compared with those of the well-known subspace iteration method and the Lanczos method.