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

Abstract

In this article, the dual multiple reciprocity method is employed to solve the natural frequencies and natural modes for a rod. The conventional approach using dual MRM is not qualified as a systematic method because of the following two reasons: (1) it needs to distinguish the spurious eigenvalue only after the corresponding eigenmode is obtained; (2) the possible indeterminancy of eigenvector may be encountered when the constraint equations chosen are highly dependent such that the rank of the leading coefficient matrix is insufficient. To construct a systematic way, we propose to consider all constraint equations together instead of using the singular or hypersingular equation alone as the conventional MRM uses. The singular value decomposition method is, then, used to solve the eigenproblem after combining the singular and hypersingular equations. This method can avoid the spurious eigenvalue 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.