轴向运动梁的横纵耦合非线性振动Galerkin截断研究

Nonlinear transverse-longitudinal coupled vibration of axially moving beam using Galerkin method and truncation technique

考虑径向变化的张力、外部黏性阻尼系数和支撑刚度,构建了一个描述轴向变速运动黏弹性梁的横纵耦合数学模型,并进行了数值分析以研究在轴向变张力条件下该梁的稳态响应。应用Galerkin方法将连续模型转化为一系列理论上可以无限截断的常微分方程组,并给出了精确的一般表达式,同时纠正了相关文献中的错误项。进一步,我们比较了不同截断阶数对最终结果的影响,并分析了计算时间随截断阶数的变化。基于M=N=8阶Galerkin截断方法和四阶Runge-Kutta方法,数值求解了系统的稳态响应。然后对比分析了文献中的近似解析法和不同的数值方法下,耦合模型与简化模型的振动幅值结果。通过时间历程图、相图和频谱分析三方面,揭示了次谐波参数共振下运动梁的非线性振动特性。从纵向振动和横向振动的频谱分析中均可检测到系统存在3∶1内共振。研究表明,在特定条件下,为了模型简化而忽略纵向位移的高阶项是可行的。

Here,considering longitudinal variation of tension,external viscous damping coefficient and support stiffness,a transverse-longitudinal coupled mathematical model was constructed to describe a viscoelastic beam with axial variable speed.Numerical analysis was performed to study steady-state response of the beam under axial variable tension conditions.Galerkin method was used to convert the continuous model into a series of ordinary differential equation systems which were theoretically infinite but could be truncated.Their accurate general expressions were given,and wrong terms in relevant literature were corrected.Furthermore,effects of different truncation orders on the final results were compared,and variations of computation time with truncation orders varying were analyzed.Based on 8th order Galerkin expansions truncated and 4th order Runge-Kutta numerical integration method,the steady-state response of the beam system was numerically solved.Then,vibration amplitude results of the coupled model and the simplified model were analyzed contrastively under the approximate analytical method and different numerical methods in literature.Nonlinear vibration characteristics of the moving beam under subharmonic parametric resonance were revealed in 3 aspects of time history diagram,phase diagram and frequency spectral analysis.The presence of 3∶1 internal resonance in the beam system could be detected from frequency spectral analyses of both longitudinal and transverse vibrations.The study showed that under specific conditions,it is feasible to ignore high-order terms of longitudinal displacement for model simplification.