Cosserat弹性杆动力学普遍定理的守恒量问题

薛纭; 王鹏

doi:10.7498/aps.60.114501

摘要

根据Cosserat弹性杆的动力学普遍定理,讨论其守恒量问题. 因弹性杆的动力学方程是以截面为对象,并且是以弧坐标和时间为双自变量,其守恒量必定是以积分的形式给出,分别存在关于弧坐标或时间守恒的问题. 根据弹性杆的动量和动量矩方程,导出其动量守恒和动量矩守恒的存在条件及其表达,并讨论了关于沿中心线弧坐标的守恒问题;再分别根据弹性杆关于时间和弧坐标的能量方程导出了各自的关于时间和弧坐标的守恒量存在条件及其表达, 结果包括了弹性杆的机械能守恒以及平衡时的应变能积分;守恒问题给出了例子. 积分形式的守恒量对于弹性杆动力学的理论分析和数值计算都具有实际意义.

关键词:

Abstract

Conserved quantities of the Cosserat elastic rod dynamics are studied according to the general theorems of dynamics. The rod dynamical equation takes the cross section of the rod as its objective of study and is expressed by two independent variables, the arc coordinate of the rod and the time, so the conserved quantities are written in the integral forms and there exist the arc coordinate conservation and the time conservation. The existence conditions and the formulas of conservations of momentum and moment of momentum are derived from the theorem of momentum and the theorem of moment of momentum respectively, which contain two cases of conserved quanties, one is the time and the other is arc coordinate. Also existence conditions and formulas of conservations of energy about time and are coordinate, which contain mechanical energy conservation, are derived from energy equations about the time and arc coordinate of the rod respectively. All of conservative motions of the rod are explained by examples. The conserved quantities in the integral form are of practical significance in both theoretical and numerical analysis for the Cosserat elastic rod dynamics.

Keywords:

作者及机构信息

薛纭¹,
王鹏^2,3

1.
上海应用技术学院,机械工程学院,上海 201418;

2.
上海大学,上海市应用数学和力学研究所,上海 200072;

3.
新疆师范大学物理系, 乌鲁木齐 830054

基金项目: 国家自然科学基金(批准号:10972143)资助的课题.

Authors and contacts

Xue Yun¹,
Wang Peng^2,3

1.
School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China;

2.
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University,Shanghai 200072, China;

3.
Department of Physics, Xinjiang Normal University, Urumqi 830054, China

参考文献

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[5]	Liu Y Z, Xue Y, Chen L Q 2004 Acta Phys. Sin. 53 2424(in Chinese)[刘延柱、薛纭、陈立群 2004 53 2424]
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[8]	Xue Y,Weng D W 2009 Acta Phys. Sin. 58 34(in Chinese)[薛纭、翁德玮 2009 58 34]
[9]	Yaoming S, Hearst J E 1994 J. Chem. Phys. 101 5186
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施引文献

[1]	Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod:Theoritical Basis of Mechanical Model of DNA (Beijing: Tsinghua University Press,Springer) p14,32 (in Chinese)[刘延柱 2006 弹性细杆的非线性力学:DNA力学模型的理论基础(北京:清华大学出版社, Springer) 第14,32页)]
[2]	Liu Y Z 2003 Mech.Eng. 25(1) 1 (in Chinese) [刘延柱 2003 力学与实践 25(1) 1]
[3]	Ou Yang Z C 2003 Phys. 32 728 (in Chinese) [欧阳钟灿 2003 物理 32 728]
[4]	Xue Y, Liu Y Z 2009 Acta Phys. Sin. 58 6737(in Chinese)[薛纭、刘延柱 2009 58 6737 ]
[5]	Liu Y Z, Xue Y, Chen L Q 2004 Acta Phys. Sin. 53 2424(in Chinese)[刘延柱、薛纭、陈立群 2004 53 2424]
[6]	Xue Y, Liu Y Z, Chen L Q 2005 Chin. J. Theor. Appl. Mech. 37 485 (in Chinese) [薛纭、刘延柱、陈立群 2005 力学学报 37 485]
[7]	Xue Y,Liu Y Z 2006 Acta Phys. Sin. 55 3845 (in Chinese)[薛纭、刘延柱 2006 55 3845]
[8]	Xue Y,Weng D W 2009 Acta Phys. Sin. 58 34(in Chinese)[薛纭、翁德玮 2009 58 34]
[9]	Yaoming S, Hearst J E 1994 J. Chem. Phys. 101 5186
[10]	Xue Y, Liu Y Z, Chen L Q 2004 Chin. Phys. 13 794
[11]	Liu Y Z 2009 Chin. Phys. B 18 1
[12]	Cao D Q, Tucker R W 2008 International J. Solids and Structure 45 460
[13]	Maddocks J H, Dichmann D J 1994 J. Elasticity 34 83
[14]	Zhao W J,Weng Y Q, Fu J L 2007 Chin. Phys. Lett. 24 2773
[15]	Wei P J 2007 Integral Equation and Its Numerical Caculation (Beijing: Metallurgical Industry Press) p8(in Chinese)[魏培君 2007 积分方程及其数值计算(北京:冶金工业出版社) 第8页]
[16]	Xue Y, Weng D W, Chen L Q 2009 Chin. Quart. Mech. 30(1) 119(in Chinese)[薛纭、翁德玮、陈立群 2009 力学季刊 30(1)116]
[17]	Mei F X, Shang M 2000 Acta Phys. Sin. 49 1901(in Chinese)[梅凤翔、尚玫 2000 49 1901]
[18]	Zhang Y, Ge W K 2009 Acta Phys. Sin. 58 7447 (in Chinese)[张毅、葛伟宽 2009 58 7447]

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