Maximizing the lowest eigenvalue of a constrained affine sum with application to the optimal design of structures and vibrating systems

Abstract

This article deals with the problem of maximizing the lowest eigenvalue of an affine sum of symmetric matrices subject to a constraint. It is shown that by the repeated use of eliminants, the problem may be reduced in a systematic manner to that of finding the roots of certain polynomials. However, the process of finding the analytical solution is tedious. Therefore, a Newton iterative method, which solves the problem numerically, is developed. To demonstrate the results, the Lagrange problem of determining the shape of the strongest column is formulated in the discrete model setting and solved by using the developed method. The design problem of finding the mass distribution in a vibratory system that optimizes its extreme natural frequencies is also given.