In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius $\beta$ increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.

在这项研究中，基于所谓的光谱几何方法研究了材料特性和横截面沿轴方向任意变化的梁的自由振动。梁的横截面积和面积二阶矩都展开为傅立叶余弦级数，它们在数学上能够表示任何可变横截面。沿梁长度方向变化的杨氏模量、质量密度和剪切模量也展开为傅立叶余弦级数。通过加入一些多项式函数，将截面的平移位移和旋转表达为傅里叶级数，这些多项式函数用于处理弹性边界条件，精度更高，收敛速度更快。根据哈密顿原理，可以得到傅里叶级数的特征值和系数。给出了一些例子来验证该方法的准确性并研究参数对梁振动的影响。结果表明，前四个固有频率随着半径的系数逐渐减小$\beta$在夹紧 - 夹紧端部支撑下，随着梯度参数n 的增加而增加和减少。与均匀梁相比，任意截面的功能铁木辛柯梁的刚度是可变的，这使得梁的振幅有不同的变化。