轴向运动梁的横纵耦合非线性振动Galerkin截断研究

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振动与冲击 ›› 2024, Vol. 43 ›› Issue (21) : 237-244.

论文

轴向运动梁的横纵耦合非线性振动Galerkin截断研究

唐有绮1，周星宇1，陈玲2，谭霞1，毛永恒1

作者信息 +

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

TANG Youqi1, ZHOU Xingyu1, CHEN Ling2, TAN Xia1, MAO Yongheng1

Author information +

文章历史 +

摘要

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。

Abstract

A transverse and longitudinal coupling mathematical model for an axially accelerating viscoelastic beam is constructed, considering the effects of the longitudinally varying tension, the external viscous damping coefficient, and the support stiffness coefficient. The steady-state response of the beam under the axial tension variation is numerically analyzed. The Galerkin method is used to transform the continuous model into a series of ordinary differential equations which can be truncated indefinitely in theory. The errors in the relevant literature are corrected. The running time of different truncation orders and their effects on the final result are compared. The numerical solution of the steady-state response is obtained based on the 8th-order Galerkin truncation method and the 4th-order Runge-Kutta method. Then, the vibration amplitude results of the coupled model and the simplified model are compared and analyzed under different numerical methods and the approximate analytic methods in the literature. Through the time history, the phase portrait, and the frequency spectrum analyses, the nonlinear vibration characteristics of the beam under subharmonic parameter resonance are revealed. From the spectrum analysis of longitudinal vibration and transverse vibration, it can be detected that the system has 3:1 internal resonance. The results show that it is feasible to ignore the higher order term of longitudinal displacement in order to simplify the model under certain conditions.

导出引用

唐有绮1, 周星宇1, 陈玲2, 谭霞1, 毛永恒1. 轴向运动梁的横纵耦合非线性振动Galerkin截断研究[J]. 振动与冲击, 2024, 43(21): 237-244

TANG Youqi1, ZHOU Xingyu1, CHEN Ling2, TAN Xia1, MAO Yongheng1. Nonlinear transverse-longitudinal coupled vibration of axially moving beam using Galerkin method and truncation technique[J]. Journal of Vibration and Shock, 2024, 43(21): 237-244

参考文献

[1] Li H, Lang T, Liu, Y, et al. Nonlinear Vibrations and Stability of an Axially Moving Plate Immersed in Fluid [J]. Acta Mechanica Solida Sinica. 2019, 32: 737-753.
[2] Mehdi Arazm, Hamidreza Eipakchi, Mehdi Ghannad. Vibrational behavior investigation of axially functionally graded cylindrical shells under moving pressure [J]. Acta Mechanical. 2019, 230: 3221-3234.
[3] Yang X D, Chen L Q. Bifurcation and chaos of an axially accelerating viscoelastic beam [J]. Chaos Solitons & Fractals. 2005, 23: 249-258.
[4] Sbaiti M, Hamrouni K, Aloui A, et al. Analytical and Numerical Approach for the Dynamic Behavior of a Beam Under a Moving Load [C]. International Conference Design & Modeling of Mechanical Systems. Springer, Cham. 2018, 1: 127-134.
[5] Sahoo B, Panda L N, Pohit G. Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity [J]. Modelling & Simulation in Engineering. 2013, 1-18.
[6] Sahoo B, Panda L N, Pohit G. Combination, principle parametric and internal resonances of an accelerating beam under two frequency parametric excitation [J]. International Journal of Non-linear Mechanics. 2016, 78: 35–44.
[7] Ding H, Jean W Z. Periodic and chaotic responses of an axially accelerating viscoelastic beam under two-frequency excitations [J]. International Journal of Applied Mechanics. 2013, 5: 1-39.
[8] Marynowski K, Kapitaniak T. Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension [J]. International Journal of Nonlinear Mechanical. 2007, 42: 118-131.
[9] Marynowski K. Dynamic analysis of an axially moving sandwich beam with viscoelastic core [J]. Composite Structures. 2012, 94: 2931-2936.
[10] Wang L, Hu Z, Zhong Z. Non-linear dynamical analysis for an axially moving beam with finite deformation [J]. International Journal Non-Linear Mechanics.2013, 54: 5-21.
[11] Bağdatli S M, Uslu B. Free vibration analysis of axially moving beam under non-ideal conditions [J]. Structural Engineering & Mechanics. 2015, 54: 597-605.
[12] Seddighi H, Eipakchi H. Natural frequency and critical speed determination of an axially moving viscoelastic beam [J]. Mechanics of Time-Dependent Materials. 2013, 17: 529–541.
[13] Pakdemirli M, Oz H R. Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations [J]. Journal of Sound and Vibration. 2008, 311: 1052–1074.
[14] YAN Hao,NI Qiao,ZHOU Kun,et al.YAN Hao,NI Qiao,ZHOU Kun,et al.Nonlinear Vibration and Stability Analysis of Axially Accelerating Beam in Axial Flow[J].Transactions of Nanjing University of Aeronautics and Astronautics,2022, 39: 12-22.
[15] Yang Q, Li D, Chu Y, Li X. Existence and Generation Mechanisms of Strange Nonchaotic Attractors in Axially Accelerating Beam Systems [J]. International Journal of Bifurcation and Chaos. 2022, 32: 2250212.
[16] 朱灿, 李梦瑶. 时滞控制下轴向运动纳米梁横向振动的稳定性研究[J]. 贵州大学学报（自然科学版）. 2022, 39: 22-30.
ZHU Can,LI Mengyao.Stability analysis and time delay feedback of axially moving nano beams[J].Journal of Guizhou University(Natural Sciences),2022, 39: 22-30.
[17] 刘金建, 谢锋, 姚林泉, 李成. 基于非局部理论的轴向运动黏弹性纳米板的参数振动及其稳定性[J]. 振动与冲击. 2017, 36(19): 13-20.
LIU Jinjian,XIE Feng,YAO Linquan,et al.Parametric vibration and stability of an axially moving viscoelastic nanoplate based on the nonlocal theory[J].Journal of Vibration and Shock,2017, 36(19): 13-20.
[18] Chen L Q, Tang Y Q. Combination and principal parametric resonances of axially accelerating viscoelastic beams: Recognition of longitudinally varying tensions [J]. Journal of Sound and Vibration. 2011, 330: 5598-5614.
[19] Chen L Q, Tang Y Q. Parametric stability of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions [J]. Journal of Vibration and Acoustics. 2012, 134: 1-11.
[20] Ding H, Yan Q Y, Zu J W. Chaotic dynamics of an axially accelerating viscoelastic beam in the supercritical regime [J]. International Journal of Bifurcation and Chaos. 2014, 24: 1-19.
[21] Tang Y Q, Zhang D B, Gao, J. M. Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions [J]. Nonlinear Dynamics. 2016, 83: 401-418.
[22] Tang Y Q, Zhang Y Y, Yang X D. On Parametric Instability Boundaries of Axially Moving Beams with Internal Resonance [J]. Acta Mechanica Solida Sinica. 2018, 31: 470-483.
[23] Tang Y Q, Ma Z G. Nonlinear vibration of axially moving beams with internal resonance, speed-dependent tension, and tension-dependent speed [J]. Nonlinear Dynamics. 2019, 98: 2475-2490.
[24] Tang Y Q, Zhou Y, Liu S, Jiang S Y. Complex stability boundaries of axially moving beams with interdependent speed and tension [J]. Applied Mathematical Modelling. 2021, 89: 208-224.