Optimization of Vibrating Beams with Respect to Higher Order Natural Frequencies

Abstract

We seek the design of a transversely vibrating beam that yields a maximum value of a higher natural frequency ω_n of specified order n. The beams considered are thin, elastic and of geometrically similar cross sections. The volume, length, and boundary conditions of the beams are assumed to be given. The governing equations are derived by the calculus of variations and solved numerically by successive iterations for low values of n. A simple method of obtaining optimal solutions corresponding to any higher value of n by aid of scaled optimal beam elements is developed. The paper presents the optimal solutions for beams with any combination of free, simply supported and clamped ends for any given value of n.