Shape optimization of an axially functionally graded (AFG) Timoshenko beams of a variable cross-sectional area, with a specified fundamental frequency, is considered. Optimization is performed in terms of beam mass minimization. Considerations involve the case of coupled axial and bending vibrations, where contour conditions are the cause of coupling. 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 adjoint 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 beam of an optimum variable square cross-section compared to the beam of a constant cross-section at specified value of the fundamental frequency. The procedure described can be used when the cross-sectional area is limited. The lower limit may be defined based on beam strength, whereas the upper limit may correspond to validity limits of the Timoshenko beam theory. The above procedure can be also applied to different case of contour conditions at the beam ends, including bodies eccentrically positioned at both ends, different types of supports at beam ends, as well as clamping of the bodies with different springs. The present paper is a bidirectional generalization of paper [1]. Here, instead of a homogeneous material we consider the case of AFG material and instead of Euler-Bernoulli beams we have a more complex case of Timoshenko beams.
References
[1] Obradović A, Šalinić S, Grbović A., Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency, Engineering Structures, Vol. 228, 111538, 2021.