On Optimal Design of Vibrating Structures

Optimal design problems of linearly elastic vibrating structural members have been formulated in two ways. One is to minimize the total mass holding the frequency fixed; the other is to maximize the fundamental frequency holding the total mass fixed. Generally, these two formulations are equivalent and lead to the same solution. It is shown in this work that the equivalence is lost when the design variable (the specific stiffness) appears linearly in Rayleigh's quotient and when there is no nonstructural mass. The maximum-frequency formulation then is a normal Lagrange problem, whereas the minimum-mass problem is abnormal. The lack of recognition of this can lead to incorrect conclusions, particularly concerning existence of solutions. It is shown that existence depends directly on the boundary conditions and, when a sandwich beam has a free end, a solution to the maximum-frequency problem does not exist.