## 留言板

 引用本文: 易壮鹏, 张勇, 王连华. 弹性约束浅拱的非线性动力响应与分岔分析[J]. 应用数学和力学, 2013, 34(11): 1182-1196.
YI Zhuang-peng, ZHANG Yong, WANG Lian-hua. Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch[J]. Applied Mathematics and Mechanics, 2013, 34(11): 1182-1196. doi: 10.3879/j.issn.1000-0887.2013.11.008
 Citation: YI Zhuang-peng, ZHANG Yong, WANG Lian-hua. Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch[J]. Applied Mathematics and Mechanics, 2013, 34(11): 1182-1196.

## 弹性约束浅拱的非线性动力响应与分岔分析

##### doi: 10.3879/j.issn.1000-0887.2013.11.008

###### 作者简介:易壮鹏(1979—),男,湖南长沙人,副教授,博士,硕士生导师(通讯作者. E-mail: yizhuangpeng@163.com).
• 中图分类号: U311.2; O302

## Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch

Funds: The National Natural Science Foundation of China(11002030; 11032004); The Program for New Century Excellent Talents in University of China(NCET-09-0335)
• 摘要: 浅拱采用竖向、转动方向弹性约束时,自振频率和模态与理想的铰支/固结边界存在差异,不同约束刚度将改变外激励下的非线性响应及各种分岔产生的参数域．由浅拱基本假定建立无量纲动力学方程, 采用在频率和模态中考虑约束刚度大小的方法,通过Galerkin全离散和多尺度摄动分析导出极坐标、直角坐标形式的平均方程, 其中方程系数与约束刚度一一对应．用数值方法分析了周期激励下竖向弹性约束系统最低两阶模态之间1∶2内共振时的动力行为, 所得结果与有限元的对比以及平均方程系数的收敛性证明了所采用方法是可行的．随着激励幅值、频率的变化存在若干分岔点,分岔发生时的参数分布与约束刚度值有关,在由分岔点连接的不稳定区或共振区附近,存在一系列稳态解、周期解、准周期解和混沌解窗口,且随参数的变化可观测到倍周期分岔．
•  [1] Malhotra N, Namachchivaya N S. Chaotic dynamics of shallow arch structures under 1∶2 resonance[J]. Journal of Engineering Mechanics,1997, 123(6): 612-619. [2] Malhotra N, Namachchivaya N S. Chaotic motion of shallow arch structures under 1∶1 internal resonance[J]. Journal of Engineering Mechanics,1997, 123(6): 620-627. [3] Bi Q, Dai H H. Analysis of non-linear dynamics and bifurcations of a shallow arch subjected to periodic excitation with internal resonance[J]. Journal of Sound and Vibration,2000, 233(4): 557-571. [4] 王钟羡, 江波, 孙保昌．周期激励浅拱的全局分岔[J]. 江苏大学学报（自然科学版）, 2004, 25(1): 85-88.(WANG Zhong-xian, JIANG Bo, SUN Bao-chang. Global bifurcation of shallow arch with periodic excitation[J]. Journal of Jiangsu University (Natural Science Edition), 2004, 25(1): 85-88.(in Chinese)) [5] Lacarbonara W, Arafat H N, Nayfeh A H. Non-linear interactions in imperfect beams at veering[J]. International Journal of NonLinear Mechanics,2005, 40(7): 987-1003. [6]Zhou L Q, Chen Y S, Chen F Q. Global bifurcation analysis and chaos of an arch structure with parametric and forced excitation[J]. Mechanics Research Communications,2010, 37(1): 67-71. [6] 刘习军, 陈予恕, 侯书军. 拱型结构在参、强激励下的非线性振动分析[J]. 力学学报, 2000, 32(1): 99-102.(LIU Xi-jun, CHEN Yu-shu, HOU Shu-jun. Analysis of nonlinear vibration of the arch structures under the parametric and forced exciting[J].Acta Mechanica Sinica,2000, 32(1): 99-102.(in Chinese)) [7] Chen J S, Li Y T. Effects of elastic foundation on the snapthrough buckling of a shallow arch under a moving point load[J]. International Journal of Solids and Structures,2006, 43(14/15): 4220-4237. [8] Xu J X, Huang H, Zhang P Z, Zhou J Q. Dynamic stability of shallow arch with elastic supports-application in the dynamic stability analysis of inner winding of transformer during short circuit[J].International Journal of NonLinear Mechanics,2002, 37(4/5): 909-920. [9] Nayfeh A H, Mook D T. Nonlinear Oscillations [M]. New York: John Wiley & Sons Inc, 1979: 320-385. [10] Nayfeh A H, Balachandran B. Applied Nonlinear Dynamics [M]. New York: WileyInterscience, 1995: 423-460. [11] Ermentrout B. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students [M]. Philadelphia: Siam, 2002: 161-193. [12] Lacarbonara W, Rega G, Nayfeh A H. Resonant non-linear normal modes—part I: analytical treatment for structural one-dimenensional systems[J]. International Journal of nNon-Linear Mechanics,2003, 38(6): 851-872. [13] Lacarbonara W, Rega G. Resonant nonlinear normal modes—part II: activation orthogonality conditions for shallow structural systems[J]. International Journal of Non-Linear Mechanics,2003, 38(6): 873-887.
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##### 出版历程
• 收稿日期:  2013-07-05
• 刊出日期:  2013-11-15

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