-
摘要:
为研究三自由度比力作用下半球型动压气浮轴承气膜变形对平台惯导中三浮陀螺仪输出的影响,提出了一种通过求解Reynolds方程来计算陀螺仪静态误差的数学模型。首先,在考虑气体稀薄效应条件下,针对三浮陀螺仪中的半球型动压气浮轴承给出对应的Reynolds润滑方程;然后,用有限差分法求解气膜压力场,并利用得到的载荷与转子位移计算陀螺仪静态误差;最终,通过回归分析,得到半球型动压气浮轴承陀螺仪的静态误差模型。为简化回归分析的过程,引入干扰力矩与比力的周向夹角和径向干扰力矩作为中间参数,将三元回归分析问题转化为二元回归分析问题。计算结果表明: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;干扰力矩在周向上超前比力1.35~1.55 rad。本文静态误差模型可预测300 m/s2以内任意方向比力作用下由转子位移所引起的陀螺仪静态误差。
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.
-
表 1 计算参数
Table 1. Parameters in calculation
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 表 2 式(10)各系数拟合结果
Table 2. Fitted results of each coefficient in Eq.(10)
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 表 3 式(11)各系数拟合结果
Table 3. Fitted values of coefficients in Eq.(11)
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 -
[1] 严恭敏, 李四海, 秦永元.惯性仪器测试与数据分析[M].北京:国防工业出版社, 2015:10-55.YAN G M, LI S H, QIN Y Y.Test and data analysis of inertial meter[M].Beijing:National Defense Industry Press, 2015:10-55(in Chinese). [2] DELLACORTE C, RADIL K C, BRUCKNER R J, et al.Design, fabrication, and performance of open source generation Ⅰ and Ⅱ compliant hydrodynamic gas foil bearings[J].Tribology Transactions, 2008, 51(3):254-264. doi: 10.1080/10402000701772579 [3] 秦冬黎. 一种球形气浮气动陀螺仪的设计方法及误差分析研究[D]. 哈尔滨: 哈尔滨工业大学, 2009: 11-27. http://cdmd.cnki.com.cn/Article/CDMD-10213-2010031070.htmQIN D L. Research on design method and error analysis of a spherical gas-floated and driven gyroscope[D]. Harbin: Harbin Institute of Technology, 2009: 11-27(in Chinese). http://cdmd.cnki.com.cn/Article/CDMD-10213-2010031070.htm [4] 陈广强, 杨云军, 雷娟棉, 等.锥台型气体润滑动压轴承动力学数值模拟研究[J].机械工程学报, 2016, 52(4):185-191. http://d.old.wanfangdata.com.cn/Periodical/jxgcxb201604027CHEN G Q, YANG Y J, LEI J M, et al.Numerical simulation research on cone self-acting gas lubrication bearing dynamics[J].Chinese Journal of Mechanical Engineering, 2016, 52(4):185-191(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/jxgcxb201604027 [5] ARMENISE M N, CIMINELLI C, DELL'OLIO F, et al.Advances in gyroscope technologies[M].New York:Springer Science & Business Media, 2010:52-133. [6] 武丽花, 凌林本.三浮陀螺仪漂移模型的建立及MATLAB实现[J].中国惯性技术学报, 2004, 12(6):77-80. http://d.old.wanfangdata.com.cn/Periodical/zggxjsxb200406018WU L H, LING L B.Model of gyro drift and realizing in MATLAB[J].Journal of Chinese Inertial Technology, 2004, 12(6):77-80(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/zggxjsxb200406018 [7] GOLIKOV A N, IGNATOVSKAYA A A.The statement of design and application questions for the gyroscope with a gas-dynamic suspension of ball rotor in the navigation support drilling system[J].Journal of Physics, 2016, 671:012020. http://adsabs.harvard.edu/abs/2016JPhCS.671a2020G [8] 刘晶石. 气浮陀螺仪干扰力矩影响因素研究[D]. 哈尔滨: 哈尔滨工业大学, 2011: 18-31. http://cdmd.cnki.com.cn/Article/CDMD-10213-1012000318.htmLIU J S. Research on influencing factors of interference torque of gas-floated gyroscope[D]. Harbin: Harbin Institute of Technology, 2011: 18-31(in Chinese). http://cdmd.cnki.com.cn/Article/CDMD-10213-1012000318.htm [9] YANG Q, ZHANG H, LIU Y.Improved modified Reynolds equation for thin-film gas lubrication from an extended slip velocity boundary condition[J].Microsystem Technologies, 2016, 22(12):2869-2875. doi: 10.1007/s00542-015-2667-4 [10] ZHANG W, MENG G, WEI X, et al.Slip flow and heat transfer in microbearings with fractal surface topographies[J].International Journal of Heat and Mass Transfer, 2012, 55(23-24):7223-7233. doi: 10.1016/j.ijheatmasstransfer.2012.07.045 [11] CHENG F, JI W.A new model of water-gas turbulent lubrication for analysis of the static and dynamic characteristics in a journal bearing[J].Proceedings of the Institution of Mechanical Engineers, Part J:Journal of Engineering Tribology, 2016, 230(12):1439-1451. doi: 10.1177/1350650116635927 [12] GERTZOS K P, NIKOLAKOPOULOS P G, PAPADOPOULOS C A.CFD analysis of journal bearing hydrodynamic lubrication by Bingham lubricant[J].Tribology International, 2008, 41(12):1190-1204. doi: 10.1016/j.triboint.2008.03.002 [13] BHORE S P, DARPE A K.Investigations on characteristics of micro/meso scale gas foil journal bearings for 100-200 W class micro power systems using first order slip velocity boundary conditions and the effective viscosity model[J].Microsystem Technologies, 2013, 19(4):509-523. doi: 10.1007/s00542-012-1639-1 [14] LIU R, WANG X L, ZHANG X Q.Effects of gas rarefaction on dynamic characteristics of micro spiral-grooved thrust bearing[J].Journal of Tribology, 2012, 134(2):222011-222017. http://europepmc.org/abstract/med/23904692 [15] GAD A M, KANEKO S.A new structural stiffness model for bump-type foil bearings:Application to generation Ⅱ gas lubricated foil thrust bearing[J].Journal of Tribology, 2014, 136:0417014. http://tribology.asmedigitalcollection.asme.org/article.aspx?articleid=1870280 [16] FENG K, LI W, XIE Y, et al.Theoretical analysis of the slip flow effect on gas-lubricated micro spherical spiral groove bearings for machinery gyroscope[J].Microsystem Technologies, 2016, 22(2):387-399. doi: 10.1007/s00542-015-2487-6 [17] DEHERI G M, PATEL S J.Combined effect of slip velocity and surface roughness on a magnetic squeeze film for a sphere in a spherical seat[J].Indian Journal of Materials Science, 2015, 1155(10):1-9. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=Doaj000003919282 [18] CUI D P, YAO Y X, QIN D L.Study on the dynamic characteristics of a new type externally pressurized spherical gas bearing with slot-orifice double restrictors[J].Tribology International, 2010, 43(4):822-830. doi: 10.1016/j.triboint.2009.11.009 [19] FUKUI S, KANEKO R.A database for interpolation of Poiseuille flow-rates for high Knudsen number lubrication problems[J].Journal of Tribology-Transactions of the ASME, 1990, 112(1):78-83. doi: 10.1115/1.2920234 [20] 黄平.润滑数值计算方法[M].北京:高等教育出版社, 2012:93-103.HUANG P.Lubrication numerical calculation methods[M].Beijing:Higher Education Press, 2012:93-103(in Chinese). [21] SAHU M, SARANGI M, MAJUMDAR B C.Thermo-hydrodynamic analysis of herringbone grooved journal bearings[J].Tribology International, 2006, 39(11):1395-1404. doi: 10.1016/j.triboint.2005.11.022