轴向功能梯度变截面Timoshenko梁自由振动的研究

Free vibration analysis of axially functionally Timoshenko beams with a non-uniform cross-section

功能梯度材料可以提高结构的强度、改善质量分布和保证工程结构的完整性,因此轴向功能梯度变截面梁已广泛应用于土木、机械和航空工程。提出了用插值矩阵法计算轴向功能梯度Timoshenko梁自由振动固有频率;基于Timoshenko梁理论,将轴向功能梯度Timoshenko梁自由振动固有频率的计算转化为一组非线性变系数常微分方程特征值问题;运用插值矩阵法可一次性地计算出轴向功能梯度变截面梁各阶振动固有频率,并可同时获取相应的振型函数。该方法对于材料梯度函数和截面几何轮廓的具体形式无任何限制条件,计算结果与现有结果对比,发现吻合良好,表明了该方法的有效性。

Non-uniform beams with varying axially material properties are widely used in civil,mechanical and aeronautical engineering,due to the fact that they can improve distribution of strength and weight,and guarantee structural integrity. In this paper,an interpolating matrix method（ IMM） for determining the natural frequencies of free transverse vibration of axially functionally graded Timoshenko beams was proposed. Firstly,based on the Timoshenko beam theory,the governing equations of free vibration analysis of an axially functionally graded Timoshenko beam were transformed into a set of nonlinear characteristic ordinary differential equations with variable coefficients. Then,the interpolating matrix method（ IMM） was adopted to solve the established equations. All the natural frequencies of free transverse vibration companying with the corresponding vibration mode functions of the axially functionally graded beam were calculated at a time. Furthermore,the present methods do not pose any restrictions on both the type of material gradation and the variation of the cross section profile. By comparing with the existing results of numerical examples,the validity of the present method was confirmed.