基于四元数表示的一种改进的刚体动力学保辛积分

徐小明; 钟万勰

doi:10.3879/j.issn.1000-0887.2014.11.001

基于四元数表示的一种改进的刚体动力学保辛积分

doi: 10.3879/j.issn.1000-0887.2014.11.001

大连理工大学工程力学系;工业装备结构分析国家重点实验室(大连理工大学), 辽宁大连 116024

基金项目: 国家自然科学基金（面上项目）(11472067)

详细信息

作者简介:
徐小明（1986—），男，辽宁东港人，博士生（通讯作者. E-mail: xxm@mail.dlut.edu.cn）;钟万勰（1934—），男，浙江德清人，教授，中科院院士（E-mail: zwoffice@dlut.edu.cn）.

中图分类号: TP391.9;O313.3
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出版历程
- 收稿日期: 2014-06-26
- 修回日期: 2014-09-11
- 刊出日期: 2014-11-18

An Improved Symplectic Integration for Rigid Body Dynamics in Terms of Unit Quaternions

State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology); Department of Engineering Mechanics,Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

Funds: The National Natural Science Foundation of China(General Program)(11472067)

摘要

摘要: 根据四元数刚体动力学基本理论，将四元数时间导数与角速度之间的恒等变换引入动能项，由此可以直接得到非奇异的四元数质量矩阵.将其与分析结构力学结合，可以得到4种形式的保辛积分算法.该算法以离散系统作用量变分原理代替四元数微分方程，单位长度约束以代数约束的方式在积分格点处满足.数值仿真结果表明该方法不仅避免了陀螺稳态进动数值仿真中严重的章动误差，并且对于一般情况也展现出很大的精度改善.
- 分析结构力学 /
- 单位四元数 /
- 刚体动力学 /
- 保辛积分 /
- 恒等变换 /
- 重陀螺
Abstract: An identity transformation between the time derivative of quaternions and angular velocity was introduced into the kinetic energy term, according to the theory of quaternionbased rigid body dynamics. This proposed approach yielded a nonsingular mass matrix. Combined with the analytical structural mechanics, a new symplectic integration scheme with 4 formulations, was proposed. In practice, the discrete variational principle of the action function was employed to replace the relevant quaternion differential equations for the proposed method. Correspondingly, the unit length constraint was met explicitly by means of the algebraic constraint at the integration grid points. The numerical results show that the new scheme avoids the severe periodical nutation errors for the special cases of steady precession of a gyro top, which is a puzzling phenomenon in recent researches. In addition, the new scheme presents an impressive improvement of accuracy for the general cases as well.
- analytical structural mechanics /
- unit quaternion /
- rigid body dynamics /
- symplectic integration /
- identity transformation /
- heavy top

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参考文献(14)

[1]	Goldstein H, Poole Jr C P, Safko J L. Classical Mechanics [M]. 3rd ed. Boston: Addison Wesley, 2002.
[2]	Mclachlan R I, Scovel C. Equivariant constrained symplectic integration[J]. Journal of Nonlinear Science,1995,5(3): 233-256.
[3]	Hairer E, Lubich C, Wanner G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations [M]. Springer, 2006.
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[5]	Simo J C, Wong K K. Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum[J]. International Journal for Numerical Methods in Engineering,1991,31(1): 19-52.
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[8]	Betsch P, Siebert R. Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration[J]. International Journal for Numerical Methods in Engineering,2009,79(4): 444-473.
[9]	Nielsen M B, Krenk S. Conservative integration of rigid body motion by quaternion parameters with implicit constraints[J]. International Journal for Numerical Methods in Engineering,2012,92(8): 734-752.
[10]	Krenk S, Nielsen M B. Conservative rigid body dynamics by convected base vectors with implicit constraints[J]. Computer Methods in Applied Mechanics and Engineering,2014,269: 437-453.
[11]	徐小明, 钟万勰. 刚体动力学的四元数表示及保辛积分[J]. 应用数学和力学, 2014,35(1): 1-11.(XU Xiao-ming, ZHONG Wan-xie. Symplectic integration of rigid body motion by quaternion parameters[J]. Applied Mathematics and Mechanics,2014,35(1): 1-11.(in Chinese))
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[13]	钟万勰, 高强, 彭海军. 经典力学——辛讲[M]. 大连: 大连理工大学出版社, 2013.(ZHONG Wan-xie, GAO Qiang, PENG Hai-jun. Classical Mechanics—Its Symplectic Description [M]. Dalian: Dalian University of Technology Press, 2013. (in Chinese))
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基于四元数表示的一种改进的刚体动力学保辛积分

doi: 10.3879/j.issn.1000-0887.2014.11.001

作者简介:
徐小明（1986—），男，辽宁东港人，博士生（通讯作者. E-mail: xxm@mail.dlut.edu.cn）;钟万勰（1934—），男，浙江德清人，教授，中科院院士（E-mail: zwoffice@dlut.edu.cn）.

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An Improved Symplectic Integration for Rigid Body Dynamics in Terms of Unit Quaternions

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留言板

基于四元数表示的一种改进的刚体动力学保辛积分

doi: 10.3879/j.issn.1000-0887.2014.11.001

作者简介: 徐小明（1986—），男，辽宁东港人，博士生（通讯作者. E-mail: xxm@mail.dlut.edu.cn）;钟万勰（1934—），男，浙江德清人，教授，中科院院士（E-mail: zwoffice@dlut.edu.cn）.

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出版历程

An Improved Symplectic Integration for Rigid Body Dynamics in Terms of Unit Quaternions

计量

出版历程

目录

作者简介:
徐小明（1986—），男，辽宁东港人，博士生（通讯作者. E-mail: xxm@mail.dlut.edu.cn）;钟万勰（1934—），男，浙江德清人，教授，中科院院士（E-mail: zwoffice@dlut.edu.cn）.