轴向运动悬臂梁横向振动理论及实验研究

Theoretical and Experimental Study on Transverse Vibration of Axially Moving Cantilever Beam

针对轴向运动悬臂梁横向振动理论研究中存在的不足,结合实验研究纠正文献[19]中的求解表达式。根据D’Alembert原理建立了轴向运动悬臂梁模型的横向振动方程,取悬臂梁的瞬时模态作为试函数,采用修正的Galerkin法对振动方程进行离散求解;通过轴向运动梁实验平台对模型的悬臂边界条件进行了修正,当梁长在0.662~1.107 m之间变化时,长度修正量为0.0475 m,修正量不随梁长变化而改变;采用对数衰减率法识别出了梁的20组衰减系数,发现衰减系数随梁长增加而逐渐减小;最后比较了梁以不同速度伸展和收缩运动时梁端加速度响应理论计算结果和试验数据,二者吻合较好,验证了理论计算模型的正确性,并且说明了模型修正和阻尼确定方法的有效性。

In view of the deficiency in theoretical research on transverse vibration of an axially moving cantilever beam,the solution expressions in literature[19]were corrected by combining theoretical and experimental research.The transverse vibration equation for the axially moving cantilever beam was derived by D’Alembert’s principle.The modified Galerkin’s method,which uses the instantaneous modal function of the cantilever beam as a trial function,was used to solve the vibration equation.The boundary condition of the axially moving cantilever beam was updated by experiments on a test platform,and it can be obtained that the correction value is 0.0475 m as the measured length of the beam varies from 0.662 m to 1.107 m.It is determined that the length correction value of the beam remains unchanged as the beam length increases.Then the logarithmic decay rate method was used to identify 20 groups of decay coefficients of the beam and it is found that the decay coefficient decreases with the increase of the beam length.The theoretical calculation results and experimental data of the acceleration response of the beam end were compared when the beam moves at different speed.The good agreement between the two results verifies the correctness of the theoretical calculation model and illustrates the effectiveness of the method of model modification and damping determination.