On the existence of solutions to optimization problems with eigenvalue constraints
Quarterly of Applied Mathematics
The optimum tapering of Bernoulli-Euler beams, i.e. the shape for which a given total mass yields the highest possible value of the first fundamental frequency of harmonic transverse small oscillations, is determined. The question of the existence of a solution to the optimization problem is considered. It is shown that, irrespective of the relationship between the flexural rigidity and linear mass density of the cantilever beam, the necessary conditions for optimality lead to a contradiction.
... o a contradiction. This result is in partial disagreement with that obtained by earlier investigators. By imposing additional constraints on the optimization variable, a numerical solution for the case of the cantilever beam is obtained, using the formulation of the maximum principle of Pontryagin. Introduction. Optimization of elasto-mechanic systems in which the behavioral constraint is a deformation bound has become a standard design procedure. However, optimization with a frequency constraint is a more difficult problem. The object of optimization in such problems is to establish, from among all designs of given style and specified total mass, the one for which the lowest natural mode is a maximum. This design is at the same time the minimum-volume (or -mass) structure for a specified fundamental natural frequency. The proper vehicle for modelling the above optimization problem is the calculus of variations. The theory for the extremum problem of Bolza, which is the classical mathematical tool, was developed by Bliss, McShane, Hestenes and others. On the non-classical side two principles have evolved and since become popular, namely the Maximum Principle of Pontryagin  which will be used here and the Principle of Optimality of Bellman. These represent the necessary extremum conditions obtainable by the use of first derivatives. This paper is motivated by two publications, one by Brach  and the other by Karihaloo and Niordson  . Brach has raised the questions of the existence of the solution of the optimization problem for a cantilever beam (and free-free beams) for the restricted case where the flexural rigidity and the mass distribution along the span of the beam are linearly related. Although the distinction between the simply supported case and the cantilever/free-free cases is not very clearly brought out, it is clear that the existence * Received April 19, 1972.