变截面铁木辛柯梁振动特性快速计算方法

提出了一种快速计算变截面铁木辛柯梁横向振动特性的方法.基于铁木辛柯梁理论建立的变截面梁的横向振动方程,其梁的截面参数如有效剪切面积、密度、弯曲刚度、转动惯量等沿梁轴线连续或非连续变化；首先将变截面梁等效为多段均匀阶梯梁；然后基于相邻两段连接处的位移(位移、转角)和力(弯矩、剪力)连续条件,建立相邻两段模态函数间相互关系,并递推出首段段与末段模态函数相互关系,利用边界条件得到相应特征方程,使用Newton-Raphson方法计算其固有频率；最后针对梁常见边界条件,得到计算变截面铁木辛柯梁横向振动固有频率特征

A solution for vibration characteristic of Timoshenko beam with variable cross-section

A method for bending vibration of variable cross-section beam was developed. Based on Timoshenko beam theory, the transverse oscillation equation of variable cross-section beam was derived, which allows the parameters, such as effective shear area, density, bending stiffness and moment of inertia, to vary in a manner of continuous and non-continuous related to axial coordinate. Then, the beam was modeled as a number of segments connected by continuity condition, and each segment was assumed to obey characteristic of uniform cross section. The relation of modal functions of contiguous segments was 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 were 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 third natural frequencies of a model were presented. The first three-order frequencies of a variable cross-section beam with the common boundary conditions were calculated by applying the new method introduced. The finite element results were adopted to verify the proposed method. Through the comparison with the results for Euler-Bernoulli beam model, it shows that the proposed method to solve the natural frequency of the beam with larges slenderness ratio has a better applicability.