弹性介质中任意形状细长曲梁的Cosserat模型

刘延柱

doi:10.6052/j.issn.1000-4750.2013.02.0141

弹性介质中任意形状细长曲梁的Cosserat模型

刘延柱

上海交通大学工程力学系,上海 200240

基金项目: 国家自然科学基金项目(11372195)

详细信息

作者简介:
刘延柱(1936―),男,江苏南京人,教授,博导,中国力学学会名誉理事,从事动力学与控制研究(Email: liuyzhc@gmail.com).

计量
- 文章访问数: 378
- HTML全文浏览量: 15
- PDF下载量: 172
出版历程
- 收稿日期: 2013-02-21
- 修回日期: 2013-11-20
- 刊出日期: 2014-08-24

COSSERAT MODELING OF THIN-LONG CURVE BEAMS WITH ARBITRARY SHAPE IN ELASTIC MEDIUM

LIU Yan-zhu

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

摘要

摘要: Kirchhoff动力学比拟理论使动力学的概念和方法进入弹性杆力学的研究领域。Cosserat弹性杆模型考虑Kirchhoff模型所忽略的截面剪切变形、中心线伸缩变形和分布载荷等因素,更适合工程中大变形细长梁的动力学建模。该文以弹性介质中任意形状中心线的圆截面细长曲梁为对象,基于Cosserat模型建立以截面的姿态角和挠度为未知变量的精确动力学方程。其直梁小变形特例为弹性介质中的Timoshenko梁。将Lyapunov运动稳定性理论的时间变量置换为空间变量,可用于判断梁的平衡稳定性。以弹性介质中轴向受压Timoshenko梁为例,讨论梁平衡状态的Lyapunov稳定性与欧拉失稳传统概念之间的区别和相互联系。导致梁屈曲的欧拉载荷可利用满足Lyapunov稳定性梁的受扰挠性线和端部约束条件导出。在一次近似条件下证明空间域内的Lyapunov 稳定性和欧拉稳定性是时间域内的Lyapunov稳定性的必要条件。
- 弹性介质中的曲梁 /
- Kirchhoff动力学比拟 /
- Cosserat弹性杆模型 /
- Lyapunov稳定性 /
- 欧拉载荷
Abstract: According to Kirchhoff’s kinetic analogy, the concepts and approaches of dynamics can be applied to mechanics of elastic rods. The Cosserat model of elastic rods considers factors neglected by the Kirchhoff model, such as the cross-sectional shear deformation, the tensile deformation of rod centerline and loadings, and therefore is more suitable to the modeling of thin-long beams with large deformation. In this study, the Cosserat model is applied to establish the exact dynamic equations of a thin-long curve beam with circular cross sections and an arbitrary shape of centerline. The attitude angles of rod cross sections and deflections of the centerline are selected as unknown variables. The Timoshenko beam in elastic medium represents the special case with beam of a straight centerline and small deformation. The Lyapunov’s stability theory is applied to analyze the equilibrium status of the beam when temporal variables are replaced by spatial variables. The stability of a Timoshenko beam in elastic medium subject to axial compression is analyzed to identify the difference and relationship between Lypunov’s stability and the traditional Euler’s buckling theory. The Euler’s load is obtained by using the perturbed centerline of the beam satisfying constraint conditions on beam ends when the Lyapunov’s stability conditions are satisfied. It is proven that the Lyapunov’s and Euler’s stability conditions in the spatial domain are the necessary conditions of Lyapunov’s stability in the time domain in the sense of the first-order approximation.
- curve beam in elastic medium /
- Kirchhoff&rsquo /
- s kinetic analogy /
- Cosserat model of elastic rod /
- Lyapunov&rsquo

HTML全文

参考文献(18)

[1]	A E H. A treatise on mathematical theory of elasticity [M]. 4th ed. New York: Dover, 1927.
[2]	弹性细杆的非线性力学: DNA力学模型的理论基础[M]. 北京: 清华大学清华Springer出版社, 2006[2] Liu Yanzhu. Nonlinear mechanics of thin elastic rod [M]. Beijing: Tsing Hua University Press / Springer, 2006. (in Chinese)
[3]	Y Z, Zu J W. Stability and bifurcation of helical equilibrium of a thin elastic rod [J]. Acta Mechanica, 2004, 167(1/2): 29―39.
[4]	Y Z, Sheng L W. Stability and vibration of a helical rod with circular cross section in a viscous medium [J]. Chinese Physics, 2007, 16(4): 891―896.
[5]	Y Z, Sheng L W. Stability and vibration of a helical rod constrained by a cylinder [J]. Acta Mechanica Sinica, 2007, 23(2): 215―219.
[6]	潘家祯. 拱形跨越管线边值问题的数值解[J]. 工程力学, 2008, 25(3): 126―131.[6] Zhang Huimin, Pan Jiazhen. Numerical analysis of circular arch-span pipelines [J]. Engineering Mechanics, 2008, 25(3): 126―131. (in Chinese)
[7]	悬垂弹性细杆的几何形态[J]. 力学季刊, 2011, 32(3): 295―299.[7] Liu Yanzhu. The shape of a hanging thin elastic rod [J]. Chinese Quarterly of Mechanics. 2011, 32(3): 295―299 (in Chinese)
[8]	薛纭. 受圆柱面约束螺旋杆伸展为直杆的动力学分析[J]. 力学学报, 2011, 43(6): 1151―1156.[8] Liu Yanzhu, Xue Yun. Dynamical analysis of stretching process of helical rod to straight rod under constraint of cylinder [J], Acta Mechanica Sinica, 2011, 43(6): 1151―1156. (in Chinese)
[9]	S S. Nonlinear problems of elasticity [M]. New York: Springer, 1995.
[10]	R W, Wang C. Torsional vibration control and Cosserat dynamics of a drill-rig assembly [J]. Meccanica 2003, 38(1): 143―159.
[11]	D Q, Liu D S, Wang C H T. Nonlinear dynamic modeling for MEMS components via the Cosserat rod element approach [J]. Journal of Micromechanics and Microengineering, 2005, 15(6): 1334―1343.
[12]	D Q, Liu D, Wang C H T. Three dimensional nonlinear dynamics of slender structures: Cosserat rod element approach [J]. International Journal of Solids and Structures, 2006, 43(3/4): 760―783.
[13]	D Q, Tucker R W. Nonlinear dynamics of elastic rods using the Cosserat theory: Modelling and simulation [J]. Intern J Solids and Structures, 2008, 45(2): 460―477.
[14]	Yanzhu. On dynamics of elastic rod based on exact Cosserat model [J]. Chinese Physics B, 2009, 18(1): 1―8.
[15]	D A, Tucker R W. Practical applications of simple Cosserat methods [C]. Mechanics of Continua. Advances in Mechanics and Mathematics Berlin, Springer, 2010, 21: 87―98.
[16]	Yanzhu, Xue Yun. Stability analysis of a helical rod based on exact Cosserat model [J]. Applied Mathemaics and Mechanics, 2011, 32(5): 603―612.
[17]	宋敉淘. 预应力曲杆的Cosserat动力学模型[J]. 哈尔滨工业大学学报, 2012, 44(11): 1―6.[17] Cao Dengqing, Song Mitao. Dynamic modeling of prestressed curved Cosserat rods [J]. J Harbin Institute Technology, 2012, 44(11): 1―6 (in Chinese)
[18]	大变形轴向运动梁的精确动力学模型 [J]. 力学学报, 2012, 44(5): 832―838.[18] Liu Yanzhu. Exact dynamical model of axially moving beam with large deformation [J]. Acta Mechanica Sinica, 2012, 44(5): 832―838. (in Chinese)

施引文献

资源附件(0)

计量

文章访问数: 378
HTML全文浏览量: 15
PDF下载量: 172
被引次数: 0

弹性介质中任意形状细长曲梁的Cosserat模型

作者简介: 刘延柱(1936―),男,江苏南京人,教授,博导,中国力学学会名誉理事,从事动力学与控制研究(Email: liuyzhc@gmail.com).

计量

出版历程

COSSERAT MODELING OF THIN-LONG CURVE BEAMS WITH ARBITRARY SHAPE IN ELASTIC MEDIUM

计量

出版历程

目录

作者简介:
刘延柱(1936―),男,江苏南京人,教授,博导,中国力学学会名誉理事,从事动力学与控制研究(Email: liuyzhc@gmail.com).