PhysEx: A novel procedure to estimate full-rank physical matrices of a structure from an incomplete modal model

Abstract

This work is devoted to presenting and applying a novel methodology that estimates a linear physical model, represented by full-rank mass, stiffness and damping matrices, capable of replicating the dynamic behaviour of an incomplete modal model. No hypothesis is made with respect to the damping matrix, so the procedure is applicable assuming any linear viscous damping model. It is shown that, if the number of considered modes satisfies a certain restriction with respect to the number of degrees of freedom, the problem has infinite solutions that differ in their exogenous eigenvalues. By controlling their effect on the dynamic behaviour of the model in the frequency range of interest, a proper final solution is obtained. The methodology is applied to a three-degrees-of-freedom discrete model and a discretized cantilever beam. Comparisons are made in both frequency and time domains which reveal that the methodology provides physical models that accurately replicate the dynamic behaviour of the incomplete modal model. Even for complete models, where the solvability condition is not met, the methodology manages to provide meaningful results in some situations. To prove the usefulness of the estimated models, a mass is added on one DOF of both examples to show that they reproduce the same effects as the original modified ones. Finally, in the interests of reproducibility, all models and algorithms presented in this article have been made publicly available.