ROTMAC and subspace MAC for the correlation of systems with complex modes: Theory and practical application

Modal correlation techniques are widely used to compare the dynamic behaviour of different structural models. However, classical mode-by-mode indicators such as the Modal Assurance Criterion (MAC) may become misleading when repeated or closely spaced modes are present, especially in systems with complex mode shapes. In these cases, small changes in mass, stiffness, damping, modelling assumptions, or experimental conditions, may produce large rotations of the modal basis within an almost invariant subspace, without necessarily indicating a relevant change in the underlying dynamic behaviour. This paper addresses this limitation by extending the concept of rotated MAC (ROTMAC) to systems with complex modes and by introducing subspace-based correlation indicators for complex modal bases. Structural dynamic modification theory is used to express the perturbed complex modes as linear combinations of the unperturbed modes, providing a direct interpretation of modal rotations and modal mixing. Then, Euclidean and mass-weighted ROTMAC formulations are developed by estimating an optimal unitary transformation that aligns the modal bases before mode-by-mode comparison. In addition, Euclidean and mass-weighted subspace MAC formulations are proposed based on orthonormal bases and principal angles between modal subspaces. Practical guidelines are provided to clarify the use of MAC, ROTMAC, or subspace MAC. The proposed methodology is assessed using an analytical four-degree-of-freedom system with repeated modes and an experimental symmetric steel cantilever beam. The results show that ROTMAC improves mode-by-mode correlation when discrepancies are mainly caused by modal-basis rotations, whereas subspace MAC provides a more robust and physically meaningful assessment when modal uniqueness is lost.