﻿ 结构偏差对二维连续地月载荷转移系统动力学影响
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Two dimensional dynamics of continuous cislunar payload transfer system considering structural deviation effect
QI Naiming, YANG Yong, HUANG Panxing, SUN Kang, HAN Bing
School of Astronautics, Harbin Institute of Technology, Harbin 150006, China
Abstract: Based on the motorized momentum exchange tether (MMET), with the principle of momentum exchange and by considering the structural deviation (including the tether length deviation and payload mass deviation), the two dimensional error dynamics of continuous cislunar payloads transferring system (CCPTS) was built by Lagrange method and its numerical simulated solution was solved by Mathematica software. Under the conditions of existing the tether length deviation and payload mass deviation, the dynamic simulations of the CCPTS were presented. It is shown that tether length deviation and payload mass deviation have similar influence on CCPTS's dynamics (including generalized coordinates and generalized velocity). With the increment of tether length deviation or payload mass deviation, the maximum of the CCPTS's generalized coordinates' deviation and generalized velocities' deviation increases linearly. However, the maximum of the CCPTS's generalized accelerations have no evident linear relation with the structure deviation of the CCPTS. Lastly, the external torque can weaken the influence degree of the structure deviation of the CCPTS on the dynamic parameters.
Key words: tether satellite     momentum exchange     cislunar transfer     payload     dynamics     structural deviation

1 CCPTS概念

CCPTS包括两套长期驻轨的子系统:一套子系统称为驻地载荷转移系统(Earth Motorized Momentum Exchange Tether,EMMET),运行于地球某椭圆轨道上;另一套子系统运行于月球某椭圆轨道上,称为驻月载荷转移系统(Lunar Motorized Momentum Exchange Tether,LMMET).两套系统工作原理相似,MMET示意图如图 1所示.

 图 1 MMET示意图 Fig. 1 Schematic of MMET

 图 2 CCPTS进行载荷转移过程示意图 Fig. 2 Process schematic of transferring payloads by CCPTS

2 CCPTS二维刚性动力学模型 2.1 坐标系定义

1) 惯性坐标系OEXYZ.

2) 运动坐标系OMxoyozo.

 Ω—赤经;i—轨道倾角;ω—近地点幅角; θ—真近角;RM—母星质心矢径. 图 3 惯性坐标系与运动坐标系 Fig. 3 Inertial frame and motion coordinate

3) 轨道面坐标系OEXoYoZo.

4) 体轴坐标系OMxbybzb.

OMxb由低轨载荷(lower payload)指向高轨载荷(upper payload),OMyb位于轨道面内,垂直于OMxb,OMzb轴与OMxbOMyb构成右手坐标系.3个轴方向上的单位矢量分别为$\bar i$、$\bar j$和$\bar k$;轨道面坐标系与体轴坐标系的几何关系如图 4所示.

 mP1,mP2—载荷P1、P2的质量;l1,l2—系绳T1和T2的长度;x—系绳质量单元与母星的距离;mM—母星质量;Ψ—系绳展长方向与当地重力梯度方向夹角;dm—质量微元. 图 4 轨道面坐标系与体轴坐标系 Fig. 4 Orbital plane coordinate and body coordinate

2.2 建立动力学模型

3 仿真分析

 参数 数值 μ/(m3·s-2) 3.9877848×1014 l1/m 10000 mM/kg 5000 mP1/kg 1000 ρ/(kg·m-3) 970 A/m2 6.283×10-5 rM/m 0.5 rP/m 0.5 rT/m 0.00447207

 图 5 CCPTS质心矢径随时间变化关系 Fig. 5 Relation of radius of CCPTS’s center-of-mass

 图 6 系绳长度偏差对CCPTS质心矢径的影响 Fig. 6 Influence of tether length deviation on radius of CCPTS’s COM

 图 7 CCPTS质心径向速度随时间变化关系 Fig. 7 Relation of radial velocity of CCPTS’s COM changing with time

 图 8 系绳长度偏差对CCPTS质心径向速度的影响 Fig. 8 Influence of tether length deviation on radial velocity of CCPTS’s COM

 图 9 系绳与重力方向夹角随时间变化关系 Fig. 9 Relation of angle between tether-span and local gravity gradient changing with time

 图 10 系绳长度偏差对系绳与重力方向之间夹角的影响 Fig. 10 Influence of tether length deviation on angle between tether-span and local gravity gradient

 图 11 系绳长度偏差对系绳旋转角速度的影响 Fig. 11 Influence of tether length deviation on angular velocity of tether

 图 12 系绳长度偏差对系绳角加速度的影响 Fig. 12 Influence of tether length deviation on angular acceleration of tether

 图 13 系绳长度偏差对真近角的影响 Fig. 13 Influence of tether length deviation on true anomaly angular

 图 14 系绳长度偏差对CCPTS真近角速度的影响 Fig. 14 Influence of tether length deviation on true anomaly angular velocity of CCPTS

 轨道参数 偏差 外力矩/(kN·m) Δl=0.01m Δl=0.1m Δl=1m Δl=10m (RM,θ) RM绝对偏差/m τ=0 (7.6,7.0×10-6) (75.7,6.6×10-5) (757.4,6.67×10-4) (7572,6.67×10-3) θ绝对偏差/rad τ=250 (7.6,7.0×10-6) (75.7,6.6×10-5) (757.1,6.67×10-4) (7574,6.67×10-3) RM相对偏差/% τ=0 (9.36×10-5,1.59×10-5) (0.00093,0.00015) (0.0093,0.00152) (0.093,0.015) θ相对偏差/% τ=250 (9.36×10-5,1.59×10-5) (0.00093,0.00015) (0.00933,0.001516) (0.0933,0.015) (${\dot R_{\rm M}},\dot \theta$) ${\dot R_{\rm M}}$绝对偏差/(m·s-1) τ=0 (9.78×10-3,1.9×10-9) (9.78×10-2,1.96×10-8) (0.976,1.95×10-7) (9.769,1.96×10-6) $\dot \theta$绝对偏差/(rad·s-1) τ=250 (0.0101,2.0×10-9) (0.101,1.96×10-8) (1.014,1.956×10-7) (9.772,1.996×10-6) ${\dot R_{\rm M}}$相对偏差/% τ=0 (0.057,0.000188) (0.571,0.00193) (5.698,0.0193) (57.03,0.194) $\dot \theta$相对偏差/% τ=250 (0.0857,0.000185) (0.8567,0.001808) (8.601,0.01804) (2.808,0.1974) (${\ddot R_{\rm{M}}},\ddot \theta$) ${\ddot R_{\rm{M}}}$绝对偏差/(m·s-2) τ=0 (2.2×10-5,-5.63×10-12) (2.2×10-4,-5.69×10-11) (2.1×10-3,-5.94×10-10) (0.022,-5.79×10-9) $\ddot \theta$绝对偏差/(rad·s-2) τ=250 (2.7×10-5,-7.02×10-12) (2.8×10-4,-6.98×10-11) (2.7×10-3,-6.29×10-10) (0.028,-7.55×10-9) ${\ddot R_{\rm{M}}}$相对偏差/% τ=0 (0.00687,0.0659) (0.06869,0.666) (0.6556,6.952) (6.8686,67.767) $\ddot \theta$相对偏差/% τ=250 (0.00357,0.00495) (0.036,0.0492) (0.3339,0.443) (4.0603,63.127)

 姿态参数 偏差 外力矩/(kN·m) Δl=0.01m Δl=0.1m Δl=1m Δl=10m ψ 绝对偏差/rad τ=0 0.0072 0.0715 0.715 7.13 τ=250 0.0043 0.043 0.430 4.285 相对偏差/% τ=0 0.0001365 0.001356 0.01356 0.1352 τ=250 0.0000695 0.000695 0.00695 0.0693 $\dot \psi$ 绝对偏差/rad τ=0 2.120×10-7 2.124×10-6 2.123×10-5 1.944×10-4 τ=250 1.1×10-7 1.06×10-6 1.021×10-5 1.011×10-4 相对偏差/% τ=0 0.00024 0.00242 0.02411 0.2204 τ=250 0.0000934 0.00090 0.00867 0.0856 $\ddot \psi$ 绝对偏差/rad τ=0 1.103×10-8 1.123×10-7 1.046×10-6 1.692×10-6 τ=250 6.427×10-9 6.267×10-8 4.828×10-7 1.288×10-6 相对偏差/% τ=0 3.4718 222.4203 106.7783 135.2518 τ=250 0.5465 5.0869 42.203 96.5517

 轨道参数 偏差 外力矩/(kN·m) Δm=0.01kg Δm=0.1kg Δm=1kg Δm=10kg (RM,θ) RM绝对偏差/m τ=0 (47.0,4.2×10-5) (470.5,4.15×10-4) (4704,4.101×10-3) (46930,0.0413) θ绝对偏差/rad τ=250 (47,4.1×10-5) (470.4,4.09×10-4) (4705,0.0041) (46430,0.0413) RM相对偏差/% τ=0 (5.76×10-4,9.55×10-5) (0.0058,0.000943) (0.0577,0.00938) (0.555,0.0939) θ相对偏差/% τ=250 (5.79×10-4,9.268×10-5) (0.0058,0.000925) (0.058,0.00937) (0.569,0.0977) (${\dot R_{\rm M}},\dot \theta$) ${\dot R_{\rm M}}$绝对偏差/(m·s-1) τ=0 (0.0608,-1.17×10-8) (0.594,-1.18×10-7) (6.268,-1.17×10-6) (61.77,-1.19×10-5) $\dot \theta$绝对偏差/(rad·s-1) τ=250 (0.0627,-1.17×10-8) (0.6298,-1.20×10-7) (6.044,-1.20×10-6) (62.85,-1.19×10-5) ${\dot R_{\rm M}}$相对偏差/% τ=0 (0.763,0.00116) (0.164,0.0117) (45.987,0.108) (89.353,1.113) $\dot \theta$相对偏差/% τ=250 (0.0505,0.00115) (5.071,0.011) (1.718,0.112) (84.204,1.112)

 姿态参数 偏差 外力矩/(kN·m) Δm=0.01kg Δm=0.1kg Δm=1kg Δm=10kg ψ 绝对偏差/rad τ=0 0.0444 0.4444 4.443 44.51 τ=250 0.0426 0.4278 4.277 42.43 相对偏差/% τ=0 0.000844 0.00844 0.0843 0.8382 τ=250 0.000689 0.00692 0.0691 0.6817 $\dot \psi$ 绝对偏差/(rad·s-1) τ=0 1.319×10-6 1.328×10-5 1.229×10-4 1.207×10-3 τ=250 1.2×10-6 1.159×10-5 1.153×10-4 1.142×10-3 相对偏差/% τ=0 0.1499 1.508 13.944 135.268 τ=250 0.1016 0.981 9.766 95.805

4 结 论

1) 在考虑母星两端系绳长度不等、两载荷质量不同的前提下,本文建立了相应的误差动力学模型.

2) 仿真结果表明,系绳长度偏差以及载荷质量偏差对CCPTS广义坐标、广义速度均产生了相似的影响.随着系绳长度偏差以及载荷质量偏差的增加,反映系统轨道运行以及姿态运动的变量(如ΔRM、Δ${\dot R_{\rm M}}$、Δθ、Δ$\dot \theta$、Δψ及Δ$\dot \psi$)的幅值最大值均随之呈线性增加,而对于广义加速度而言,则没有明显的线性关系.

3) 结构偏差对CCPTS轨道参数的影响要小于对姿态参数的影响.随着CCPTS广义坐标对时间求导阶次的增加,由于结构偏差所导致的相应变量的偏差明显增加.此外,外力矩的存在能够削弱结构偏差对CCPTS动力学的影响程度.

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#### 文章信息

QI Naiming, YANG Yong, HUANG Panxing, SUN Kang, HAN Bing

Two dimensional dynamics of continuous cislunar payload transfer system considering structural deviation effect

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(11): 2000-2009.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0730