Mass minimization of an AFG Timoshenko beam with a coupled axial and bending vibrations

Abstract

The paper considers mass minimization of an axially functionally graded (AFG) Timoshenko beam of a variable cross-sectional area, with a specified fundamental frequency. The analyzed case of coupled axial and bending vibrations involves contour conditions as the cause of coupling. The problem is solved applying Pontryagin’s maximum principle, where the beam cross-sectional area is taken for control. The two-point boundary value problem is obtained, and the shooting method is used to solving it. The property of self-adjoint systems is deployed. The percent saving of the beam mass is determined, achieved by using the beam of an optimum variable square cross-section as compared to the beam of a constant cross-section. The procedure developed by the author in his earlier papers is extended herein to the case of a limited cross-sectional area. The second generalization relates to the general case of contour conditions at the beam ends.