On the extremal fundamental frequencies of vibrating beams

Abstract

Bernoulli-Euler Beams with variable cross section are optimized with respect to their fundamental frequency of transverse oscillation. The cross section is allowed to vary in a manner such that the second area moment is linearly related to the area. Using calculus of variations, the fundamental frequency is made stationary. The closed-form solution is found for all sets of homogeneous boundary conditions. In most cases, the resulting beam is uniform, however, the frequency in some cases is a minimum, others a maximum. For cantilever and free-free beams no maximum fundamental frequency exists for this type of cross section variation.