变截面梁横向振动特性的半解析法

提出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半解析解的高精度和有效性。

Semi-analytical method for calculating vibration characteristics of variable cross-section beam

A semi-analytical approach for analysing bending vibration of variable cross-section beam was developed.The variable cross-section beam model allows parameters,such as bending stiffness and mass per unit,to vary in a manner of continuity or non-continuity,related to axial coordinate.Based on Euler-Bernoulli beam theory,the transverse oscillation equation of a variable cross-section beam was derived.Then,the beam was modeled as a number of segments connected according to continuity condition,and each segment of the continuous beam was assumed to obey the characteristics of uniform cross section beam.The relation between modal functions of contiguous segments was derived by using the continuity conditions of displacement,slope of deflection curve,moment,and shear force at the connected point between contiguous segments.Furthermore,the modal functions were determined for simply supported boundary conditions,and the natural frequencies were obtained using Newton-Raphson method.To verify the effectiveness of the method,first five natural frequencies and modal shapes of the beam were presented.The results show that the natural frequencies of variable cross-section beam are approximately in agreement with those numerically obtained by the finite-element method.