On Physical Realizability for Inverse Structural Designs: Bounding the Least Eigenvalue of an Unknown Mass Matrix

In the field of structural engineering analysis, a common requirement is to calculate the modal frequencies of a structure that has undergone an update, either naturally (such as from material degradation), or due to man-made influences (by placing point masses along a structure). In addition to this requirement, it is common to only have access to truncated modal testing results. In this paper, we derive analytical bounds for the first eigenvalue of a completely unobserved mass matrix for linear elastic systems. Doing so allows engineers to proceed with modifying linear elastic systems, without requiring direct access to the mass matrix. This is because it is often difficult to know exactly what negative mass perturbations are allowable, given that the full mass matrix is an unknown quantity. Ultimately, the analysis in this paper will proceed by assuming only access to the left and right eigenvectors of the underlying system - which are both possible to obtain via physical experiments, so that the bounds are not only physically realizable, but also practically realizable.