﻿ 半球型动压气浮轴承陀螺仪的静态误差模型<sup>*</sup>
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Static error model of a gyroscope with gas-dynamic hemispherical bearings
LI Yan, DUAN Fuhai
School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China
Received: 2017-09-11; Accepted: 2017-11-03; Published online: 2017-12-05 10:47
Foundation item: Aeronautical Science Foundation of China (20150863003)
Corresponding author. DUAN Fuhai, E-mail: duanfh@dlut.edu.cn
Abstract: In order to investigate the influence of deformation of gas film in gas-dynamic hemispherical bearings on the output of three-floated gyroscopes in the platform initial navigation system subject to 3-DOF specific forces, a mathematical model is established to calculate the static error by solving Reynolds equation. Firstl, Reynolds equation is modified to describe gas flow in hemispherical bearings considering the effect of gas rarefaction. Secondl, it is solved by finite difference method to obtain the pressure distribution, and the relationship between load and rotor displacement is used to calculate the gyroscope error. Finally, by regression analysis, a static error model of the gyroscope with gas-dynamic hemispherical bearings is obtained. To simplify the ternary regression analysis to binary regression analysis, the circumferential angle between interference torque and specific force, and the radial interference torque are introduced as intermediate parameters. Numerical results show that the radial interference torque increases with the increase of axial specific force. With the increase of radial specific force, the radial interference torque increases when the radial specific force is small, and decreases when the interference torque is large. Interference torque is 1.35-1.55 rad ahead of specific force in radial direction. The proposed static error model can predict the gyroscope static error caused by rotor displacement with any specific force below 300 m/s2.
Keywords: platform inertial navigation     three-floated gyroscope     hemispherical bearing     gas lubrication     error model

1 半球型动压气浮轴承的静态特性

 图 1 半球型动压气浮轴承的结构 Fig. 1 Structure of gas-dynamic hemispherical bearing
1.1 气膜压力场控制方程

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 图 2 计算域的局部坐标系与网格划分 Fig. 2 Local coordinate system and meshing of calculation domain

1.2 控制方程的求解

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2 静态误差模型的建立 2.1 陀螺仪的静态误差

 1—传感器；2—壳体；3—浮筒；4—转子；5—力矩器；6—气浮轴承。 图 3 三浮陀螺仪的结构简图 Fig. 3 Sketch of three-floated gyroscope structure

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2.2 回归分析过程的简化

 图 4 主要变量的计算关系 Fig. 4 Calculation relationship among main variables

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f′引起的干扰力矩M′也等于Mz轴转过了β角。

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3 计算结果与分析

 参数 数值 轴承半径R/mm 6 轴承宽度b/mm 5 轴承间隙c/mm 2 两轴承间距d/mm 8 沟槽深度hg/μm 1 沟槽数量Ng 6 沟槽方向角βg/(°) 45 转子质量m/g 60 转子角动量Hr/(kg·m2·s-1) 0.016 7 气体黏度μ/(Pa·s) 1.79×10-5 转速n/(r·min-1) 30 000 环境压力Pa/Pa 1.013×105

3.1 气膜压力分布

 图 5 气膜压力分布 Fig. 5 Pressure distribution of gas film
3.2 陀螺仪误差模型回归分析

 图 6 径向干扰力矩与2个比力分量的关系 Fig. 6 Radial interference torque versus specific force in two directions

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 系数 数值 b00/(N·m) 2.48×10-6 b10/(N·s2) -4.88×10-7 b01/(N·s2) 1.60×10-6 b20/(N·s4·m-1) -4.41×10-7 b11/(N·s4·m-1) -8.13×10-8 b30/(N·s6·m-2) 4.56×10-7 b21/(N·s6·m-2) -4.54×10-7 b40/(N·s8·m-3) -1.11×10-7 b31/(N·s8·m-3) 1.59×10-7

 图 7 比力与干扰力矩的周向夹角与2个比力分量的关系 Fig. 7 Circumferential angle between specific force and interference torque versus specific force in two directions

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 系数 a00/rad a10/(rad·s2·m-1) a01/(rad·s2·m-1) 数值 1.482 -3.746×10-2 3.608×10-3

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4 结论

1) 通过引入2个中间参数——干扰力矩与比力的周向夹角和径向干扰力矩，成功地将陀螺仪静态误差模型的三元回归分析问题转化为2个二元回归分析问题。

2) 径向干扰力矩随着轴向比力的增大而增大，随着径向比力的增大呈现先增大后减小的趋势。当比力的径向或轴向分量为0时，转子只有轴向或径向的位移，不会产生陀螺仪静态误差。干扰力矩在周向上与比力相比较超前1.35~1.55 rad。

3) 建立的陀螺仪静态误差模型克服了润滑计算中必须以转子位移为自变量的不便，可直接预测300 m/s2以内的任意三自由度环境比力下由转子位移引起的陀螺仪静态误差。

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#### 文章信息

LI Yan, DUAN Fuhai

Static error model of a gyroscope with gas-dynamic hemispherical bearings

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(8): 1705-1711
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0570