Mass minimization of an AFG Timoshenko cantilever beam with a large body placed eccentrically at the beam end

Abstract

Shape optimization of an AFG Timoshenko cantilever beam of a variable cross-sectional area, with a specified fundamental frequency, is considered. The cantilever beam has a finitedimensional body placed eccentrically at the right end. Optimization is performed in terms of beam mass minimization. Considerations involve the case of coupled axial and bending oscillations, where contour conditions are the cause of coupling, which exist at the place of junction between the cantilever beam and the body. The problem is solved applying Pontryagin’s maximum principle, with the beam cross-sectional area being taken for control. The two-point boundary value problem is obtained, and the shooting method is applied to solving it. The property of self-adjoint systems is employed, where all costate variables are expressed by state variables, which facilitates solving the appropriate differential equations. Also, the percent saving of the beam mass is determined, achieved by using the cantilever beam of an optimum variable square cross-section compared to the cantilever beam of a constant cross-section at specified value of the fundamental frequency.