非均勻 Timoshenko 曲樑的平面外振動分析

Abstract

The three coupled governing differential equations for the out-of-plane vibrations of a curved non-uniform Timoshenko beam are derived via the Hamilton’s principle. Two physical parameters are introduced to simplify the analysis. By eliminating all the terms with the flexural displacement parameter, then reducing the order of differential operator acting on the angle parameter of the rotation due to bending, one uncouples the three governing characteristic differential equations with variable coefficients and reduces them into a sixth-order ordinary differential equation with variable coefficients in terms of the torsional angle for the first time. The explicit relations between the flexural displacement and the angle of the rotation due to bending and the torsional angle are also revealed. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the out-of-plane vibrations of a curved non-uniform Timoshenko beam can be obtained. Several limiting cases are studied. Finally, the influence of the boundary conditions, the taper ratio, the slender ratio, and the arc angle parameter on the first three natural frequencies of the curved beams is explored.