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研究了黏弹性轴向运动梁在外部激励和参数激励共同作用下横向振动的混沌非线性动力学行为. 引入有限支撑刚度, 并考虑黏弹性本构关系取物质导数, 同时计入由梁轴向加速度引起的沿径向变化的轴力, 建立轴向运动黏弹性梁横向非线性振动的偏微分-积分模型. 通过Galerkin截断方法研究了外部激励的频率和因速度简谐脉动引起的参数激励的频率在不可通约关系时轴向运动连续体的非线性动力学行为, 并对不同截断阶数的数值预测进行了对比. 基于对控制方程的Galerkin截断, 得到离散化的常微分方程组, 使用四阶Runge-Kutta方法求解. 基于此数值解, 运用非线性动力学时间序列分析方法, 通过Poincaré 映射, 观察到轴向运动梁随扰动速度幅值的倍周期分岔现象, 并比较了有无外部激励对倍周期分岔的影响. 分别在低速以及近临界高速运动状态下, 从相平面图、Poincaré 映射以及频谱分析的角度识别了系统中存在的准周期运动形态.In this paper, the chaotic behaviour in the transverse vibration of an axially moving viscoelastic tensioned beam under the external harmonic excitation is studied. The parametric excitation comes from harmonic fluctuations of the moving speed. A nonlinear integro-partial-differential governing equation is established to include the material derivative in the viscoelastic constitution relation and the finite axial support rigidity. Moreover, the longitudinally varying tension due to the axial acceleration is also considered. The nonlinear dynamics of axially moving beam is investigated under incommensurable relationships between the forcing frequency and the parametric frequency. Based on the Galerkin truncation and the Runge-Kutta time discretization, the numerical solutions of the nonlinear governing equation are obtained. The time history of the center of the axially moving viscoelastic beam is chosen to represent the motion of the beam. Based on the time history of the axially moving beam, the Poincaré map is constructed by sampling the displacement and the velocity of the center. The bifurcation diagram of the axially moving beam is used to show the influence of the external excitation. Furthermore, quasi-periodic motions are identified using different methods including the Poincaré map, the phase-plane portrait, and the fast Fourier transforms.
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Keywords:
- axially moving beam /
- nonlinearity /
- chaotic /
- bifurcation
[1] Xue Y, Liu Y Z, Chen L Q 2006 Acta Phys. Sin. 55 3845 (in Chinese) [薛纭, 刘延柱, 陈立群 2006 物理学报 55 3845]
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[6] Liu D, Xu W, Xu Y 2012 J. Sound Vib. 331 4045
[7] Li Q H, Yan Y L, Wei L M, Qin Z Y 2013 Acta Phys. Sin. 62 120505 (in Chinese) [李群宏, 闫玉龙, 韦丽梅, 秦志英 2013 物理学报 62 120505]
[8] Ravindra B, Zhu W D 1998 Arch. Appl. Mech. 68 195
[9] Yang X D, Chen L Q 2005 Chaos Soliton. Fract. 23 249
[10] Ding H, Chen L Q 2009 Acta Mech. Solid. Sin. 22 267
[11] Ghayesh M H 2012 J. Sound Vib. 331 5107
[12] Yao M H, Zhang W, Zu J W 2012 J. Sound Vib. 331 2624
[13] Chen L Q, Tang Y Q 2011 J. Sound Vib. 330 5598
[14] Chen L Q, Tang Y Q 2012 ASME J. Vib. Acoust. 13 011008
[15] Yang T Z, Fang B, Chen Y, Zhen Y X 2009 Int. J. Non-Lin. Mech. 44 230
[16] Ghayesh M H, Kafiabad H A, Reid T 2012 Int. J. Solids Struct. 49 227
[17] Zhang W, Li S B 2010 Nonlinear Dynam. 62 673
[18] Gholizadeh H, Hassannia A, Azarfar A 2013 Chin. Phys. B 22 010503
[19] Pan W Z, Song X J, Yu J 2010 Chin. Phys. B 19 030203
[20] Chen L Q, Liu Y Z 1996 Physics 25 278 (in Chinese) [陈立群, 刘延柱 1996 物理 25 278]
[21] Chai Y, L L, Chen L Q 2012 Chin. Phys. B 21 030506
[22] Zhao D M, Zhang Q C 2010 Chin. Phys. B 19 030518
[23] Ding H, Zu W J 2013 Int. J. Appl. Mech. 5 1350019
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[1] Xue Y, Liu Y Z, Chen L Q 2006 Acta Phys. Sin. 55 3845 (in Chinese) [薛纭, 刘延柱, 陈立群 2006 物理学报 55 3845]
[2] Shen H J, Wen J H, Yu D L, Wen X S 2009 Acta Phys. Sin. 58 8357 (in Chinese) [沈惠杰, 温激鸿, 郁殿龙, 温熙森 2009 物理学报 58 8357]
[3] Wang L H, Hu Z D, Zhong Z, Ju J W 2009 Acta Mech. 206 149
[4] Li Y H, L H W, Li Z H, Li L 2012 J. Chongqing Univ. Technol. (Natural Science) 26 16 (in Chinese) [李映辉, 吕海炜, 李中华, 李亮 2012 重庆理工大学学报(自然科学) 26 16]
[5] Chen S H, Huang J L, Sze K Y 2007 J. Sound Vib. 306 1
[6] Liu D, Xu W, Xu Y 2012 J. Sound Vib. 331 4045
[7] Li Q H, Yan Y L, Wei L M, Qin Z Y 2013 Acta Phys. Sin. 62 120505 (in Chinese) [李群宏, 闫玉龙, 韦丽梅, 秦志英 2013 物理学报 62 120505]
[8] Ravindra B, Zhu W D 1998 Arch. Appl. Mech. 68 195
[9] Yang X D, Chen L Q 2005 Chaos Soliton. Fract. 23 249
[10] Ding H, Chen L Q 2009 Acta Mech. Solid. Sin. 22 267
[11] Ghayesh M H 2012 J. Sound Vib. 331 5107
[12] Yao M H, Zhang W, Zu J W 2012 J. Sound Vib. 331 2624
[13] Chen L Q, Tang Y Q 2011 J. Sound Vib. 330 5598
[14] Chen L Q, Tang Y Q 2012 ASME J. Vib. Acoust. 13 011008
[15] Yang T Z, Fang B, Chen Y, Zhen Y X 2009 Int. J. Non-Lin. Mech. 44 230
[16] Ghayesh M H, Kafiabad H A, Reid T 2012 Int. J. Solids Struct. 49 227
[17] Zhang W, Li S B 2010 Nonlinear Dynam. 62 673
[18] Gholizadeh H, Hassannia A, Azarfar A 2013 Chin. Phys. B 22 010503
[19] Pan W Z, Song X J, Yu J 2010 Chin. Phys. B 19 030203
[20] Chen L Q, Liu Y Z 1996 Physics 25 278 (in Chinese) [陈立群, 刘延柱 1996 物理 25 278]
[21] Chai Y, L L, Chen L Q 2012 Chin. Phys. B 21 030506
[22] Zhao D M, Zhang Q C 2010 Chin. Phys. B 19 030518
[23] Ding H, Zu W J 2013 Int. J. Appl. Mech. 5 1350019
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