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Rigid-flexible coupling multibody model for the tethered satellite system based on recursive dynamics algorithm
ZHONG Rui
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:A multibody dynamics model of hinged flexible rods was established for the spatial tethered satellite system (TSS) based on the recursive dynamics algorithm. The system was consisted of two rigid end-satellites and a flexible tether connecting them. In the model, the tether was discretized into a series of flexible rods connected by spherical hinges and the non-uniform longitude deformations of the rods were considered, whereas the bending and torsion of the tether were simulated by the motion of the spherical hinges. Then, the equation of motion of the rigid-flexible coupling multibody model was derived based on the recursive dynamics algorithm. The simulation results prove the efficiency of the proposed model in simulation spatial motion of the TSS, including tether vibrations and oscillations in different directions. The introduction of the recursive dynamics algorithm effectively decreases the dimension of the discretized model and therefore reduces the computational efforts sharply.
Key words: tethered satellite system (TSS)     multi-body system     recursive dynamics algorithm     rigid-flexible coupling model     dynamics simulation

1 基本假设和坐标系定义

 Lj—j点到Oj点的矢量；ej,x、ej,y、ej,z—坐标系Sj的x、y、z轴单位向量； rj—地心到Ij点的矢量；SIN—地心惯性坐标系； eIN,x、eIN,y、eIN,z—坐标系SIN的x、y、z轴单位矢量. 图 1 铰接柔性杆绳系模型示意图 Fig. 1 Sketch map of hinged-flexible-rods tether model

1) 地心惯性坐标系SIN:坐标系原点位于地心,x轴指向春分点,z轴指向天球北极,与y轴构成右手坐标系.

2) 轨道坐标系So:坐标系原点位于地心,x轴指向绳系系统的质心;坐标系SoSIN的关系为,其中Rz(Ω)为绕着z轴转过Ω角度，Rx(i)为绕着x轴转过i角度，Rz(+υ)为绕着z轴转过+υ角度， Ω、iυ分别为轨道的升焦点赤经、轨道倾角、近地点幅角和真近点角.

3) 各体的本体坐标系S1~Sn+2:本体系Sj固连于体j,原点位于Ij,当系绳处于当地铅垂方向时,各本体系与轨道系方向一致.

2 刚柔混合模型的递推算法

2.1 单体动力学方程

2.2 运动学递推关系

1) 主星(j=n+2).

2) 柔性杆(j=3,4,…,n+1).

3) 柔性杆(j=2,内接刚体).

4) 子星(j=1).

2.3 动力学递推关系

2.4 算法修正

1) |uj(Lj)|≤δ,δ是一个较小的正参数.即此段系绳虽然松弛但变形不大,可以认为体j的惯量和质量参数不发生较大变化,即无需改变前文建立的单链式递推算法,只需要将体j的弹性模量E设为0即可.

2) |uj(Lj)|>δ.此时该段系绳发生较大的变形,无法用杆近似,惯量参数也有较大的变化,可以认为系绳断为两截,应分段采用前文建立的单链式递推算法,体j－1成为链1~j－1的端体,j+1则成为链j+1~n+2的根体.另外时时计算uj(Lj)的值,若减小至δ以下,则切换回单链递推算法.uj(Lj)可由以下公式计算:

3 数值仿真

 参量名 数值 参量名 数值 主星质量/kg 1 000 系绳密度/(g·cm-3) 1.44 主星尺寸/m 1×1×1 系绳弹性模量/GPa 131 子星质量/kg 100 系绳剪切模量/GPa 1.8 子星尺寸/m 0.4×0.4×0.4 纵向振动阻尼 0.005 系绳长度/km 5 弯曲扭转阻尼 0.005 系绳直径/mm 2 计算步长/s 0.01

 图 2 主星姿态角变化(z轴) Fig. 2 Attitude angle of main satellite (z axis)

 图 3 球铰1和2转角(z轴) Fig. 3 Rotation angles of spherical hinges 1 and 2 (z axis)

 图 4 球铰5和6转角(z轴) Fig. 4 Rotation angles of spherical hinges 5 and 6 (z axis)

 图 5 杆单元1和2外接点的纵向变形(x轴) Fig. 5 Longitude deformations at outer-point of elements 1 and 2 (x axis)

 图 6 最后时刻各杆单元的纵向变形(x轴) Fig. 6 Longitude deformations of elements at final moment (x axis)

 图 7 主星姿态角变化(y轴) Fig. 7 Attitude angle of main satellite (y axis)

 图 8 球铰1和2转角(y轴) Fig. 8 Rotation angles of spherical hinges 1 and 2 (y axis)

 图 9 主星姿态角变化 Fig. 9 Attitude angle of main satellite

 图 10 子星姿态角变化 Fig. 10 Attitude angle of sub satellite

 图 11 球铰1和2转角 Fig. 11 Rotation angles of spherical hinges 1 and 2

 图 12 球铰5转角 Fig. 12 Rotation angles of spherical hinge 5

4 结 论

1) 所提出离散模型能够用较少的单元数模拟空间系绳的纵向不均匀变形,同时模拟系绳的弯曲和扭转,适用范围较广.

2) 系绳阻尼作用下接近大惯量绳端星的柔绳段横向振动较靠近小惯量绳端星的柔绳段减小更快.

3) 采用递推算法易于编程实现,且计算量随离散单元数量线性增长,在同等精度下运算效率高.本文数值仿真中未考虑重力梯度外的空间干扰力,模型不能直接用于仿真系绳的释放和回收,系绳纵向模态的选取仍需考察,这些都是今后的研究要点.

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文章信息

ZHONG Rui

Rigid-flexible coupling multibody model for the tethered satellite system based on recursive dynamics algorithm

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(7): 1188-1195.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0525