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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]
[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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[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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