Rotation of mode shapes in structural dynamics due to mass and stiffness perturbations

According to the structural dynamic modification theory, the perturbed mode shapes can be expressed as a linear combination of the unperturbed mode shapes through a transformation matrix T. This matrix is proposed in this paper as a powerful technique to determine whether the discrepancies between two models can be attributed to differences in stiffness, in mass, or both. It is demonstrated that matrix T becomes a rotation matrix when there are no mass discrepancies. In the case of mass discrepancies, or a combination of mass and stiffness differences, matrix T can be decomposed into a product of a rotation matrix and a matrix containing information about the changes in scaling and in shear. The angle of rotation depends on the closeness of the modes, and large rotations can be obtained when the system presents closely spaced or repeated modes. The polar and the QR decompositions are used in this paper to factorize matrix T as a product of two matrices, one of them being a rotation matrix. A new version of the modal assurance criterion (MAC), denoted in this paper as rotated MAC or ROTMAC, is proposed to detect mass discrepancies between two models. The equations and the conclusions derived in this paper have been validated through numerical simulations on a 2-degrees-of-freedom system and by correlating a numerical model and an experimental model of a square laminated glass plate.