Quantification of non-proportionality of damping in discrete vibratory systems

Quantification of the non-proportionality of damping is very important in modal testing, model correlation, and model modifications. Three new indices measuring the damping non-proportionality are proposed in this study. The proposed indices are based on the complex modes of a generally damped system. The mathematical development is built on a common assumption that the modal matrix can be represented by a product of a real matrix and complex matrix. The complex matrix represents modal coupling due to the non-proportionality. The relation between the complex matrix and the modal damping matrix is derived and used as a base for the proposed indices. The first index measures the correlation between the real and the imaginary parts of a complex mode. The second index measures the magnitude of the imaginary parts of a complex mode that is scaled in a way that maximizes its real parts. The third index quantifies the degree of modal coupling using the magnitude of the imaginary part of the entire scaled modal matrix. The applicability of the proposed indices is demonstrated by numerical simulations. The performance of the proposed indices is compared with three existing indices. The example also tests the robustness of the proposed indices, when the complex modes are corrupted by a measurement noise.