Applications of dual MRM for determining the natural frequencies and natural modes of a rod using the singular value decomposition method

In this article, the dual multiple reciprocity method is employed to solve the natural frequencies and natural modes for a rod. The conventional appro ach using dual MRM is not qualified as a systematic method because of the f ollowing two reasons: (1) it needs to distinguish the spurious eigenvalue o nly after the corresponding eigenmode is obtained; (2) the possible indeter minancy of eigenvector may be encountered when the constraint equations cho sen are highly dependent such that the rank of the leading coefficient matr ix is insufficient. To construct a systematic way, we propose to consider a ll constraint equations together instead of using the singular or hypersing ular equation alone as the conventional MRM uses. The singular value decomp osition method is, then, used to solve the eigenproblem after combining the singular and hypersingular equations. This method can avoid the spurious e igenvalue problem and the possible indeterminancy of boundary eigenvectors at the same time. Three numerical examples are given to verify the validity of the present method. (C) 1999 Elsevier Science Ltd. All rights reserved.