末端质量作用下变长度轴向运动梁的振动特性

Vibration Characteristics of an Axially Moving Variable Length Beam With a Tip Mass

对带末端质量,变长度(或速度)轴向运动梁的振动特性采用两种精确方法求解.首先,对变长度轴向运动Euler(欧拉)梁横向自由振动方程进行化简,通过复模态分析得到本征方程,并在有末端质量的边界条件下得到频率方程,用数值方法求解固有频率和模态函数.然后,采用有限元方法建立运动梁自由振动的方程,求解矩阵方程得到复特征值和复特征向量,结合形函数得到复模态位移.最后,将两种方法的计算结果进行了分析和对比.数值算例的结果表明:不同的轴向运动速度和末端质量对变长度轴向运动梁的振动特性有显著影响,两种计算方法的结果接近且均有效.

A semi-analytical method and a numerical method were used to investigate the vibra- tion characteristics of an axially moving variable length （velocity） beam with a tip mass. First, the equation of transverse free vibration for the axially moving Euler beam was simplified. The eigenequation was derived with the complex modal analysis method. Moreover, the frequency e- quation was obtained under the boundary conditions with a tip mass. The numerical method was used to calculate the natural frequencies and modal shapes. Then, the equation of trans- verse free vibration was also derived with the finite element method （FEM）. The complex ei- genvalues and eigenvectors were obtained as solutions to the complex matrix equation, and the complex modal displacements were given through combination with the shape functions. Final- ly, the results from these 2 methods were comparatively analyzed. The numerical example illus- trates that different velocities and tip masses influence the beam vibration characteristics signifi- cantly. The calculated results from the 2 methods are close to each other and effective.