Maximizing Higher Order Eigenfrequencies of Beams with Constraints on the Design Geometry

Abstract

With the cross-sectional area as the design variable, we seek the optimal design of a transversely vibrating beam of given volume that maximizes a natural frequency ω _n of specified order, n. The beams considered are thin, elastic, and of geometrically similar cross-sections, and a minimum constraint is prescribed for the cross-sectional area. In addition, the length and the boundary conditions are assumed to be given, and the beams may carry specified nonstructural masses. Variational analysis is used for deriving the governing equations, which are strongly coupled and nonlinear. These equations are solved numerically by a successive iteration procedure based on a finite difference discretization. A number of optimal solutions are obtained, and significant features of the spectra of their eigenfrequencies are discussed.