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提出了一种考虑剪切效应的三结点弹性轴段单元,并将其用于轴系振动的分析。给出了三结点单元的插值函数,推导了弯曲刚度矩阵和剪切刚度矩阵,得到了单元的运动方程。研制了考虑剪切效应的有限元计算程序。按七种不同弯曲刚度计算了单圆盘转子(算例1)和某水轮发电机组轴系(算例2)的模态频率和临界转速,得出了模态频率和临界转速的变化规律和不同方法的相对误差。计算结果表明,剪切变形对于轴系振动的模态频率和临界转速均有一定的影响。算例1的结果显示,当轴的弯曲刚度较大(k=EI/l3>106N·m-1)时,必须考虑剪切效应;当轴的刚度较小(k≤106N·m-1)时,可以忽略剪切变形的影响,由此所带来的计算误差不超过2%。算例2的计算结果表明,考虑剪切变形作用时,一阶模态频率和临界转速的结果比不考虑剪切时大1.5%以上,且随着轴的刚度的增大,相对误差也逐渐增大。
A three-node elastic axis segment considering shear effect is proposed and used to analyze the vibration of shaft system. The interpolation function of three-node unit is given. The bending stiffness matrix and the shear stiffness matrix are deduced. The equations of motion of the unit are obtained. A finite element calculation program considering shear effect was developed. The modal frequencies and critical speeds of the single disc rotor (Example 1) and the shaft of a hydroelectric generator (Example 2) were calculated according to seven different bending stiffness. The variation of the modal frequency and the critical speed were obtained And the relative error of different methods. The calculation results show that the shear deformation has a certain influence on the modal frequency and critical speed of shaft vibration. The results of Example 1 show that the shearing effect must be considered when the bending stiffness of the shaft is large (k = EI / l3> 106 N · m-1); when the rigidity of the shaft is small (k≤106 N · m -1) , You can ignore the impact of shear deformation, the resulting calculation error does not exceed 2%. The calculation results of Example 2 show that the first-order modal frequencies and the critical speed result in more than 1.5% of the shear without considering the effect of shear deformation. As the stiffness of the shaft increases, the relative error gradually increases Increase.