基于精确Cosserat模型的螺旋杆稳定性分析

刘延柱; 薛纭

doi:10.3879/j.issn.1000-0887.2011.05.007

基于精确Cosserat模型的螺旋杆稳定性分析

doi: 10.3879/j.issn.1000-0887.2011.05.007

刘延柱¹,
薛纭²

1.
上海交通大学工程力学系,上海 200240;2.上海应用技术学院机械与自动化工程学院, 上海 200235

基金项目: 国家自然科学基金资助项目(10972143)

详细信息

作者简介:
刘延柱(1936- ),男,南京人,教授,博士生导师(联系人.E-mail:liuyzhc@online.sh.cn).

中图分类号: O317;O343
计量
- 文章访问数: 1595
- HTML全文浏览量: 121
- PDF下载量: 898
- 被引次数: 0
出版历程
- 收稿日期: 2011-02-21
- 修回日期: 2011-03-23
- 刊出日期: 2011-05-15

Stability Analysis of a Helical Rod Based on Exact Cosserat’s Model

LIU Yan-zhu¹,
XUE Yun²

1.
Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China;

摘要

摘要: 弹性杆的螺旋线平衡问题在DNA、纤维、海底电缆和输油管线等方面具有应用背景．Kirchhoff动力学比拟是分析弹性细杆平衡稳定性的有效方法．Kirchhoff模型中包括中心线无拉伸变形和截面无剪切变形的基本假定与生物大分子等软物质的实际状况有较大差异．基于精确Cosserat模型，考虑中心线的拉伸压缩变形和截面剪切变形，以及剪切变形引起杆中心线转动导致切线轴相对截面法线轴的偏离，以Euler角表达截面姿态，建立圆截面弹性杆的动力学普遍方程．在静力学范畴内讨论螺旋线平衡状态的Liapunov稳定性和Euler稳定性问题，导出稳定性条件及轴向力和扭矩的Euler临界值．证明螺旋杆平衡的静态Liapunov稳定性和Euler稳定性条件是动态Liapunov稳定性的必要条件．
- 精确Cosserat模型 /
- Kirchhoff杆 /
- Liapunov稳定性 /
- Euler载荷
Abstract: The helical equilibrium of a thin elastic rod has a practical background such as DNA, fiber, sub-ocean cable and oil-well dill string. The Kirchhoff's kinetic analogy is an effective approach in stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff's theory including no extension of the centerline and no shear deformation of the cross section are not adaptable to the real soft materials of biological fibers. The dynamic equations of a rod with circular cross section were established on the basis of exact Cosserat's model considering tension and shear deformations. The Euler's angles were applied as the attitude representation of the cross section. The deviation of the normal axis of cross section from the tangent of the centerline was considered as the result of shear deformation. The Liapunov's stability of helical equilibrium was discussed in static category and the Euler's critical values of axial force and torque were obtained. The Liapunov's and Euler's stability conditions in space domain are the necessary conditions of Liapunov's stability of the helical rod in time domain.
- exact Cosserat’s model /
- Kirchhoff’s rod /
- Liapunov’s stability /
- Euler’s load

HTML全文

参考文献(12)

[1]	Love A E H. A Treatise on Mathematical Theory of Elasticity[M]. 4th ed. New York: Dover, 1927.
[2]	刘延柱. 弹性细杆的非线性力学[M].北京: 清华大学出版社, Springer, 2006.(LIU Yan-zhu.Nonlinear Mechanics of Thin Elastic Rod[M]. Beijing: Tsinghua University & Springer, 2006.(in Chinese))
[3]	刘延柱, 薛纭. 弹性细杆螺旋线平衡的动态稳定性[J]. 力学季刊，2005, 26(1): 1-7.(LIU Yan-zhu, XUE Yun. Dynamical stability of helical equilibrium of a thin elastic rod[J]. Chinese Quarterly of Mechanics, 2005, 26(1): 1-7.(in Chinese))
[4]	刘延柱，盛立伟. 轴向受压螺旋杆的动态稳定性与振动[J]. 力学季刊，2006, 27(2): 190-195.(LIU Yan-zhu, SHENG Li-wei. Dynamical stability and vibration of an helical rod under axial compression[J]. Chinese Quarterly of Mechanics, 2006, 27(2): 190-195.(in Chinese))
[5]	刘延柱，盛立伟. 圆截面弹性螺旋杆的稳定性与振动[J]. 物理学报，2007, 56(4):2305-2310.(LIU Yan-zhu, SHENG Li-wei. Stability and vibration of an elastic helical rod with circular cross section[J]. Acta Physica Sinica，2007, 56(4): 2305-2310.(in Chinese))
[6]	Antman S S. Nonlinear Problems of Elasticity[M]. New York: Springer, 1995.
[7]	Bishop T C, Cortez R, Zhmudsky O O. Investigation of bend and Shear waves in a geometrically exact elastic rod model[J]. J of Comput Physics, 2004, 193: 642-665. doi: 10.1016/j.jcp.2003.08.028
[8]	Schuricht F. Regularity for shearable nonlinearly elastic rods in obstacle problems[J]. Arch Rational Mech Anal, 1998, 145: 23-49. doi: 10.1007/s002050050123
[9]	Tucker R W, Wang C. Torsional vibration control and Cosserat dynamics of a drill-rig assembly[J]. Meccanica, 2003, 38: 143-159.
[10]	Cao D Q, Liu D, Wang C H T. Nonlinear dynamic modelling of MEMS components via the Cosserat rod element approach[J]. J Micromechanics and Microengineering, 2005, 15: 1334-1343. doi: 10.1088/0960-1317/15/6/027
[11]	Cao D Q, Tucker RW. Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation [J]. Intern J Solids and Structures, 2008, 45: 460-477. doi: 10.1016/j.ijsolstr.2007.08.016
[12]	LIU Yan-zhu. On dynamics of elastic rod based on exact Cosserat model[J]. Chinese Physics B, 2009, 18(1): 1-8. doi: 10.1088/1674-1056/18/1/001