摘要：提出了一种计算变截面梁横向振动特性的半解析法。基于欧拉-伯努利梁理论给出的弯曲刚度、质量分布沿梁轴线连续或非连续变化的变截面梁横向振动方程；将该变截面梁等效为多段均匀梁，并基于相邻两段连接处的位移(位移、转角)和力(弯矩、剪力)的连续条件，建立了两相邻均匀段之间模态函数的关系；针对简支边界条件给出了计算变截面梁横向振动固有频率的特征方程和模态函数，并用Newton-Raphson方法计算其固有频率。通过与有限元法的数值结果比较说明了本文半解析解的高精度和有效性。

Abstract: A semi-analytical approach for bending vibration of variable cross-section beam is developed in this research. The variable cross-section beam model allowed parameters, such as bending stiffness and mass per unit, to vary in a manner of continuous and non-continuous which are related to axial coordinate. Based on Euler-Bernoulli beam theory, the transverse oscillation equation of variable cross-section beam is derived. Then, the beam is modeled a number of segments connected by continuity condition, and each segment of the continuous beam is assumed to obey characteristic of uniform cross section. The relation of modal functions of contiguous segments is derived by using the conditions for continuity of displacement, slope of deflection curve, moment, and shear force at the connecting point between contiguous segments. Furthermore, the modal functions are determined for simply supported boundary conditions, and the natural frequencies can be obtained using Newton-Raphson method. To verify the effect of this method, the first five natural frequencies and modal shape of beam are presented. Results show that the natural frequencies of variable cross-section beam approximately encouraging agreement those numerically obtained by the finite-element method.