Hamiltonian Structures and Stability Analysis for Rigid-Liquid Coupled Spacecraft Systems
-
摘要: 该文采用3D刚体摆来等效推进剂的非线性晃动行为. 由此研究了该刚-液耦合航天器系统的Hamilton结构，介绍了系统的$ \mathbb{R}^3$约化(对应系统的平移不变性或总线动量不变性)以及So(3)约化(对应系统的旋转不变性或总角动量不变性)，并推导了系统在约化空间$ \mathfrak{s}_0^*(3) \times \mathfrak{s}_0^*(3) \times {S_0}(3)$上的约化Poisson括号. 接着研究了刚-液耦合航天器系统的自旋稳定性特征，先根据对称临界原理推导了刚-液耦合航天器系统的相对平衡态，由此根据能量-动量方法与分块对角化技术，推导了系统的自旋稳定性条件和Arnold形式的稳定性边界. 最后根据具体模型参数，给出了以图形方式展现的自旋稳定域.
-
关键词:
- 液体非线性晃动 /
- 等效力学模型 /
- Hamilton结构 /
- 稳定性分析 /
- 能量-动量方法
Abstract: For the dynamics problems of rigid-liquid coupling spacecraft systems with liquid propellant, a 3D rigid pendulum model was used to simulate the nonlinear sloshing behavior of the propellant. On this basis, the Hamiltonian structure of the rigid-liquid coupling spacecraft system was studied, the $ \mathbb{R}^3$ reduction (corresponding to the translation invariance or the bus momentum invariance of the system) and the So(3) reduction (corresponding to the rotation invariance or the total angular momentum invariance of the system) of the system were introduced, with the reduced Poisson brackets of the system in reduced space $ \mathfrak{s}_0^*(3) \times \mathfrak{s}_0^*(3) \times {S_0}(3)$ derived. Then, the spin stability characteristics of the rigid-liquid coupled spacecraft system were studied. Firstly, the relative equilibrium of the rigid-liquid coupled spacecraft system was derived under the principle of symmetric criticality. Based on the energy-momentum method and the block diagonalization technology, the spin stability conditions and the Arnold form stability boundaries of the system were derived. Finally, the spin stability domains illustrated in the form of graph were given according to the specific model parameters. -
图 1 刚-液耦合航天器系统的等效力学模型
Figure 1. The equivalent mechanical model for the rigid-liquid coupled spacecraft system
-
[1] DODGE F T. The new "dynamic behavior of liquids in moving containers"[R]. San Antonio, TX, USA: Southwest Research Institute, 2000. [2] IBRAHIM R A. Liquid Sloshing Dynamics: Theory and Applications[M]. Cambridge: Cambridge University Press, 2005. [3] KANA D D. Validated spherical pendulum model for rotary liquid slosh[J]. Journal of Spacecraft and Rockets, 1989, 26(3): 188-195. doi: 10.2514/3.26052 [4] KANG J Y, LEE S. Attitude acquisition of a satellite with a partially filled liquid tank[J]. Journal of Guidance, Control, and Dynamics, 2008, 31(3): 790-793. doi: 10.2514/1.31865 [5] MIAO N, LI J F, WANG T S. Equivalent mechanical model of large-amplitude liquid sloshing under time-dependent lateral excitations in low-gravity conditions[J]. Journal of Sound and Vibration, 2017, 386: 421-432. doi: 10.1016/j.jsv.2016.08.029 [6] YUE B Z. Study on the chaotic dynamics in attitude maneuver of liquid-filled flexible spacecraft[J]. AIAA Journal, 2011, 49(10): 2090-2099. doi: 10.2514/1.J050144 [7] 邓明乐. 液体大幅晃动等效力学模型及航天器刚-液-柔-控耦合动力学研究[D]. 博士学位论文. 北京: 北京理工大学, 2017.DENG Mingle. Studies on the equivalent mechanical model of large amplitude liquid slosh and rigid-liquid-flex-control coupling dynamics of spacecraft[D]. PhD Thesis. Beijing: Beijing Institute of Technology, 2017. (in Chinese) [8] TANG Y, YUE B Z. Simulation of large-amplitude three-dimensional liquid sloshing in spherical tanks[J]. AIAA Journal, 2017, 55(6): 2052-2059. doi: 10.2514/1.J055798 [9] 刘峰. 液体非线性晃动类柔性航天器大范围运动动力学与姿态控制研究[D]. 博士学位论文. 北京: 北京理工大学, 2020.LIU Feng. Studies on large motion dynamics and attitude control of flexible spacecraft with nonlinear liquid slosh[D]. PhD Thesis. Beijing: Beijing Institute of Technology, 2020. (in Chinese) [10] LIU F, YUE B Z, TANG Y, et al. 3DOF-rigid-pendulum analogy for nonlinear liquid slosh in spherical propellant tanks[J]. Journal of Sound and Vibration, 2019, 460: 114907. doi: 10.1016/j.jsv.2019.114907 [11] ABRAHAM R, MARSDEN J E. Foundations of Mechanics[M]. New York: Addison-Wesley, 1978. [12] MARSDEN J E, RATIU T S. Introduction to Mechanics and Symmetry: a Basic Exposition of Classical Mechanical Systems[M]. New York: Springer, 1998. [13] HOLM D D, MARSDEN J E, RATIU T, et al. Nonlinear stability of fluid and plasma equilibria[J]. Physics Reports, 1985, 123(1/2): 1-116. http://www.sciencedirect.com/science/article/pii/0370157385900286 [14] KRISHNAPRASAD P S, MARSDEN J E. Hamiltonian structures and stability for rigid bodies with flexible attachments[J]. Archive for Rational Mechanics and Analysis, 1987, 98(1): 71-93. doi: 10.1007/BF00279963 [15] OZKAZANC Y. Dynamics and stability of spacecraft with fluid-filled containers[D]. PhD Thesis. MD, USA: University of Maryland, 1994. [16] ARDAKANI H A, BRIDGES T J, GAY-BALMAZ F, et al. A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion[J]. Proceedings of the Royal Society A, 2019, 475(2224): 20180642. doi: 10.1098/rspa.2018.0642 [17] GASBARRI P, SABATINI M, PISCULLI A. Dynamic modelling and stability parametric analysis of a flexible spacecraft with fuel slosh[J]. Acta Astronautica, 2016, 127: 141-159. doi: 10.1016/j.actaastro.2016.05.018 [18] SALMAN A, YUE B Z. Bifurcation and stability analysis of the Hamiltonian-Casimir model of liquid sloshing[J]. Chinese Physics Letters, 2012, 29(6): 060501. doi: 10.1088/0256-307X/29/6/060501 [19] 闫玉龙. 航天器刚液柔耦合动力学及姿态稳定性研究[D]. 博士学位论文. 北京: 北京理工大学, 2017.YAN Yulong. Study on dynamics and attitude stability of the rigid-liquid-flex coupling spacecraft system[D]. PhD Thesis. Beijing: Beijing Institute of Technology, 2017. (in Chinese) [20] SIMO J C, POSBERGH T A, MARSDEN J E. Stability of coupled rigid body and geometrically exact rods: block diagonalization and the energy-momentum method[J]. Physics Reports, 1990, 193(6): 279-360. doi: 10.1016/0370-1573(90)90125-L [21] SIMO J C, LEWIS D, MARSDEN J E. Stability of relative equilibria, part Ⅰ: the reduced energy-momentum method[J]. Archive for Rational Mechanics and Analysis, 1991, 115(1): 15-59. doi: 10.1007/BF01881678 [22] SIMO J C, POSBERGH T A, MARSDEN J E. Stability of relative equilibria, part Ⅱ: application to nonlinear elasticity[J]. Archive for Rational Mechanics and Analysis, 1991, 115(1): 61-100. doi: 10.1007/BF01881679 [23] YI Z G, YUE B Z. Study on the dynamics, relative equilibria, and stability for liquid-filled spacecraft with flexible appendage[J]. Acta Mechanica, 2022, 233(9): 3557-3578. doi: 10.1007/s00707-022-03269-5 [24] ABRAMSON H N. The dynamic behavior of liquids in moving containers, with applications to space vehicle technology: NASA-SP-106[R]. San Antonio, TX, USA: Southwest Research Institute, 1966. [25] GROSSMAN R, KRISHNAPRASAD P S, MARSDEN J E. The dynamics of two coupled rigid bodies[J]. Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits, 1987, 1988: 373-378. http://drum.lib.umd.edu/bitstream/handle/1903/4531/TR_87-22.pdf?sequence=1&isAllowed=y [26] KRISHNAPRASAD P. Lie-Poisson structures, dual-spin spacecraft and asymptotic stability[J]. Nonlinear Analysis: Theory, Methods & Applications, 1985, 9(10): 1011-1035. http://www.sciencedirect.com/science/article/pii/0362546X85900835/part/first-page-pdf [27] OH Y G, SREENATH N, KRISHNAPRASAD P, et al. The dynamics of coupled planar rigid bodies Ⅱ: bifurcations, periodic solutions, and chaos[J]. Journal of Dynamics and Differential Equations, 1989, 1(3): 269-298. doi: 10.1007/BF01053929 [28] SREENATH N, OH Y G, KRISHNAPRASAD P, et al. The dynamics of coupled planar rigid bodies, part Ⅰ: reduction, equilibria and stability[J]. Dynamics and Stability of Systems, 1988, 3(1/2): 25-49. http://doc.paperpass.com/foreign/rgArti1988139257856.html [29] WANG LISHENG, KRISHNAPRASAD P S. Gyroscopic control and stabilization[J]. Journal of Nonlinear Science, 1992, 2(4): 367-415. doi: 10.1007/BF01209527 [30] GE X S, YI Z G, CHEN L Q. Optimal control of attitude for coupled-rigid-body spacecraft via Chebyshev-Gauss pseudospectral method[J]. Applied Mathematics and Mechanics, 2017, 38(9): 1257-1272. doi: 10.1007/s10483-017-2236-8 [31] MARSDEN J E, WEINSTEIN A. Reduction of symplectic manifolds with symmetry[J]. Reports on Mathematical Physics, 1974, 5(1): 121-130. doi: 10.1016/0034-4877(74)90021-4 [32] MARSDEN J E, RATIU T. Reduction of Poisson manifolds[J]. Letters in Mathematical Physics, 1986, 11(2): 161-169. doi: 10.1007/BF00398428 [33] 易中贵. 几何力学建模和谱方法离散及航天工程应用[D]. 博士学位论文. 北京: 北京理工大学, 2022.YI Zhonggui. Geometric mechanics modeling and spectral methods discretization and aerospace engineering applications[D]. PhD Thesis. Beijing: Beijing Institute of Technology, 2022. (in Chinese) [34] SHI D H, ZENKOV D V, BLOCH A M. Hamel's formalism for classical field theories[J]. Journal of Nonlinear Science, 2020, 30(1): 1307-1353. http://d.wanfangdata.com.cn/periodical/9b0df1df64c9df5bcf813e0802b2e532 [35] SIMO J C, MARSDEN J E, KRISHNAPRASAD P. The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates[J]. Archive for Rational Mechanics and Analysis, 1988, 104(2): 125-183. doi: 10.1007/BF00251673 [36] POSBERGH T A, KRISHNAPRASAD P S, MARSDEN J E. Stability analysis of a rigid body with a flexible attachment using the energy-Casimir method: SRC-TR-87-23[R]. MD, USA: University of Maryland, 1987. [37] YI Z G, YUE B Z, DENG M. Hamilton-Pontryagin spectral-collocation methods for the orbit propagation[J]. Acta Mechanica Sinica, 2021, 37(11): 1698-1715. http://www.sciengine.com/doi/pdf/88730759c9fe4886b0a3881299112762 [38] 冯康, 秦孟兆. 哈密尔顿系统的辛几何算法[M]. 杭州: 浙江科学技术出版社, 2003.FENG Kang, QIN Mengzhao. Symplectic Geometric Algorithms for Hamiltonian Systems[M]. Hangzhou: Zhejiang Science and Technology Press, 2003. (in Chinese) [39] 高山, 史东华, 郭永新. Hamel框架下几何精确梁的离散动量守恒律[J]. 力学学报, 2021, 53(6): 1712-1719. https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB202106017.htmGAO Shan, SHI Donghua, GUO Yongxin. Discrete momentum conservation law of geometrically exact beam in Hamel's framework[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(6): 1712-1719. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB202106017.htm [40] 满淑敏, 高强, 钟万勰. 非完整约束Hamilton动力系统保结构算法[J]. 应用数学和力学, 2020, 41(6): 581-590. doi: 10.21656/1000-0887.400375MAN Shumin, GAO Qiang, ZHONG Wanxie. A structure-preserving algorithm for Hamiltonian systems with nonholonomic constraints[J]. Applied Mathematics and Mechanics, 2020, 41(6): 581-590. (in Chinese) doi: 10.21656/1000-0887.400375 [41] 刘晓梅, 周钢, 朱帅. Hamilton系统下基于相位误差的精细辛算法[J]. 应用数学和力学, 2019, 40(6): 595-608. doi: 10.21656/1000-0887.390249LIU Xiaomei, ZHOU Gang, ZHU Shuai. A highly precise symplectic direct integration method based on phase errors for Hamiltonian systems[J]. Applied Mathematics and Mechanics, 2019, 40(6): 595-608. (in Chinese) doi: 10.21656/1000-0887.390249 [42] 张素英, 邓子辰. Poisson流形上广义Hamilton系统的保结构算法[J]. 西北工业大学学报, 2002, 20(4): 625-628. https://www.cnki.com.cn/Article/CJFDTOTAL-XBGD200204030.htmZHANG Suying, DENG Zichen. An algorithm for preserving the generalized Poisson bracket structure of generalized Hamiltonian system[J]. Journal of Northwestern Polytechnical University, 2002, 20(4): 625-628. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-XBGD200204030.htm -
下载:
点击查看大图
图(2)
计量
- 文章访问数: 261
- HTML全文浏览量: 106
- PDF下载量: 58
- 被引次数: 0
