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1. 装备学院 航天装备系, 北京 101416;
2. 北京科技大学 机械工程学院, 北京 100083

Dynamic modeling and key error sources analysis of magnetically suspended sensitive gyroscopes
XIN Chaojun1 , CAI Yuanwen1 , REN Yuan1 , MIAO Cunxiao2 , ZHANG Liyuan2
1. Department of Space Equipment, Equipment Academy, Beijing 101416, China ;
2. School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
Received: 2015-10-08; Accepted: 2015-11-06; Published online: 2016-01-08 15:17
Foundation item: National Natural Science Foundation of China (51475472)
Corresponding author. REN Yuan Tel.: 010-66364384 E-mail: renyuan823@aliyun.com
Abstract: The mathematical model of a new magnetically suspended sensitive gyroscope (MSSG) with rotor of double spherical envelope surfaces is established based on rotor dynamics, and the key error sources of the gyro in sensing angular rate are analyzed theoretically. First, structural features and sensing principles of the MSSG are described. Second, the theoretical models of electromagnetic forces and moments acting on the rotor are built respectively. The influencing mechanisms of rotor translation and titling on its mechanical state are analyzed. Simulation results by finite-element method software ANSYS are basically in agreement with the calculated results. Finally, two key error sources of rotor asphericity and Lorentz force magnetic bearing process errors are analyzed, and the analytical expression of the disturbing torques has been constructed. Calculation results show that the rotor aspheric factors and the inhomogeneous magnetic field in Lorentz force magnetic bearing are the major factors inducing disturbance torques. The model provides a valuable theoretical basis for further research on optimization design and analysis of the MSSG.
Key words: magnetically suspended sensitive gyroscope (MSSG)     double spherical envelope surfaces     Lorentz force     aspheric error     error sources analysis

1 MSSG结构与工作原理 1.1 MSSG结构

 图 1 MSSG结构示意图 Fig. 1 Schematic diagram of configuration of MSSG

1.2 MSSG角速率敏感原理

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2 数学模型

2.1 电磁力模型

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 图 2 MSSG坐标系统 Fig. 2 Coordinate systems of MSSG

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r0在定子坐标系中可表示为

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MSSG转子发生轴向和径向偏移时，其受到轴向磁轴承的电磁力随转子偏移量的变化如图 3所示，计算过程中的MSSG各项参数如表 1所示。

 图 3 轴向磁轴承电磁力与转子偏移量关系 Fig. 3 Relations between electromagnetic forces of axial bearing and rotor translations

 参数 数值 磁极间气隙δ/mm 0.35 轴向磁轴承定子磁极张角φ0/(°) 32.6 真空磁导率μ0 4π×10-7 转子径向磁极包络面半径r01/mm 73.65 转子轴向磁极包络面半径r0/mm 41 定子轴向磁极包络面半径R0/mm 41.35 定子径向磁极包络面半径R1/mm 74 磁轴承保护间隙d0/mm 0.2 洛伦兹力磁轴承线圈半张角φ0/(°) 37 洛伦兹力磁轴承线圈匝数n 200 轴向磁轴承线圈匝数N 100 轴向磁轴承线圈电流I/A 0.14 洛伦兹力磁轴承线圈电流ib/A 1 洛伦兹力磁轴承磁场磁密B/T 0.4 洛伦兹力磁轴承支撑架半径Lr/mm 57.5

 图 4 MSSG仿真结构与计算结果比较 Fig. 4 Comparison between MSSG simulation structure and computed results

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2.2 电磁力矩模型

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 图 5 洛伦兹力磁轴承结构原理 Fig. 5 Structure principle of Lorentz force bearing

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 图 6 转子偏转时洛伦兹力方向变化示意图 Fig. 6 Schematic diagram of direction change of Lorentz force when spinning axis tilts

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 图 7 洛伦兹力磁轴承电磁力、力矩与激励电流及转子偏转角的关系 Fig. 7 Relations among electromagnetic force and moment of Lorentz force bearing, exciting current and rotor deflection angles

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3 敏感精度影响因素分析

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3.1 转子非球形误差

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 图 8 干扰力矩与转子非球形系数关系 Fig. 8 Relations of disturbance torque and rotor aspheric coefficients
3.2 洛伦兹力磁轴承误差

 图 9 洛伦兹力磁轴承磁场与磁力线分布仿真 Fig. 9 Simulation of magnetic field and magnetic lines distribution in Lorentz force bearing

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2种误差耦合导致的干扰力矩变化如图 10所示。可知，干扰力矩随着磁密误差和加工误差的增大而增大，在小范围内近似线性，干扰力矩随磁密误差变化的斜率更大，说明磁场分布的不均匀性对陀螺的检测精度影响更大。

 图 10 磁密分布及支撑架加工误差与干扰力矩关系 Fig. 10 Relations among magnetic density distribution, frame processing errors and disturbing torque
4 结论

1)在对该MSSG结构和测量原理分析的基础上，构建了动力学模型，得到了径向、轴向磁轴承电磁力以及偏转磁轴承电磁力矩的解析表达式。

2)仿真计算表明，当转子包络面为理想球形时，转子的平移运动仅会导致定子磁极对转子电磁力的作用，而不会对转子产生力矩作用，平移运动在垂直方向上的耦合电磁力比运动方向上的主电磁力小2个数量级，理论计算与有限元仿真结果基本吻合。

3)洛伦兹力磁轴承单独控制的偏转力矩对转子偏转角度不敏感，但磁轴承对转子的转动控制会在转子轴向上产生耦合电磁力，其值小于1N。

4)基于MSSG的动力学解析模型，得到了转子非球形条件下干扰力矩解析表达式，分析了洛伦兹力偏转磁轴承加工误差对MSSG敏感精度的影响。初步分析表明，转子非球形和洛伦兹力陀螺磁场分布不均是产生干扰力矩的2个关键因素，转子包络面非球形系数中，奇次谐波系数对干扰力矩的生成贡献更大，当转子位于平衡位置时，由于转子形状的对称性，偶次谐波系数不会导致干扰力矩的生成。

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#### 文章信息

XIN Chaojun, CAI Yuanwen, REN Yuan, MIAO Cunxiao, ZHANG Liyuan

Dynamic modeling and key error sources analysis of magnetically suspended sensitive gyroscopes

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(10): 2048-2058
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0650