摘要
精密超精密机床是国防工业、航空航天工业和民用光学工业中精密超精密零件的关键加工装备,是事关国防安全的战略性装备。液体静压主轴是精密超精密机床的核心部件,其回转精度对于机床的加工精度具有决定性影响。虽然国内外已有液体静压主轴的成功应用案例,但迄今尚缺乏针对回转精度形成机理的专门研究成果,现有工作不能有效揭示主轴旋转形成回转误差的动态发展过程,不能定量解释静压轴承均化效应和揭示提升静压主轴回转精度的物理本质,难以为静压主轴回转精度的工程设计和预测提供系统深入的理论支持。
本文着眼于轴心运动轨迹的动态发展过程,建立了静压主轴回转精度的流固耦合动力学模型,提出了基于CFD动网格模型的求解新方法,实现了主轴形成回转精度时轴心运动轨迹的定量仿真,揭示了静压主轴回转误差的物理机理；系统研究了工作参数和结构参数以及静压轴承摩擦副形状误差对回转精度的影响规律；建立了液体静压主轴回转精度测试平台,对比研究了理论计算和实验测试的主轴误差轨迹和回转精度。本文主要研究工作包括：
(1)建立了基于N-S润滑方程和主轴动力学方程的静压主轴流固耦合动力学模型,提出了基于CFD动网格模型的实现回转精度动态过渡过程的求解新方法。该方法将主轴旋转和平动方程嵌入CFD并求解N-S方程,计算瞬态油膜压力,将轴颈旋旋转动边界转换为静边界,利用弹簧光顺模型更新主轴位移引起的油膜网格变形,界定了主轴在油膜中运动速度的取值范围。通过对比分析理论计算与实验测试典型静压主轴的四个油膜刚度,证实了静压主轴回转精度的流固耦合动力学模型及求解方法是可行和有效的。
(2)定量计算了主轴由轴颈与轴承同心位置开始过渡到稳态回转运动轨迹的动态发展过程,揭示了静压主轴形成回转精度的物理机理。采用全程动网格方法、平衡位置动网格方法和油膜刚度阻尼公式三种方式,定量计算和仿真了主轴由轴颈与轴承的同心初始位置到形成回转误差的运动轨迹,研究揭示了静压主轴形成平均回转中心和回转误差时主轴合力、轴承流量和轴心位置的动态发展规律以及扰动因素与主轴合力、轴承流量和轴心位置之间内在联系。通过对比三种方法计算的平均回转中心和回转精度,结果表明采用动网格与油膜刚度阻尼公式相结合方法计算主轴回转精度可兼顾计算效率和仿真精度。
(3)系统研究揭示了静压主轴结构参数和工作参数对回转精度的影响规律。采用动网格方法定量研究了工作参数(包括工作转速和供油压力)和结构参数(包括长径比、节流孔径和轴承间隙)对平均回转中心和油膜刚度阻尼的影响规律；采用简化公式计算了动不平衡激励下主轴的回转误差,研究揭示了工作参数和结构参数对主轴回转精度的影响规律。结果表明：增加供油压力和长径比,可提高主轴回转精度；增大节流孔径、工作转速和轴承间隙,将降低主轴回转精度。
(4)研究揭示了轴承形状误差对主轴回转精度的影响规律。定量计算并分析了轴承形状误差影响下主轴平均回转中心位置和油膜刚度阻尼的变化规律；通过对比分析误差轴承与理想轴承的平均回转中心位置和油膜刚度阻尼,发现了轴承形状误差是影响平均回转中心位置和油膜刚度阻尼的根本原因,研究揭示了轴承形状误差对静压主轴回转精度的影响规律。结果表明：光滑轴颈条件下,轴承本身的形状误差虽然不会使主轴发生位置变动而产生回转误差,但通过影响主轴平均回转中心位置及其油膜刚度阻尼,影响主轴的回转误差。
(5)研究揭示了轴颈形状误差对主轴回转精度的影响规律。定量计算并分析了轴颈圆度误差和圆柱度误差对主轴平衡位置和油膜刚度阻尼的影响规律,重点分析了主轴旋转时轴颈圆度误差峰谷交替变化对主轴轴心位置产生的影响,揭示了静压轴承对轴颈形状误差的均化机理。结果表明：光滑轴承条件下,轴颈圆度误差使主轴发生位置变动,形成回转误差,回转误差大小与轴颈圆度误差的幅值和频率有关；轴颈圆柱度误差虽然不会使主轴产生位置变动而产生回转误差,但通过影响主轴平均回转中心及其油膜刚度阻尼,影响主轴的回转误差。
(6)建立了液体静压主轴试验台,测试了静压主轴的误差轨迹和回转精度,验证了理论计算误差轨迹和回转精度的部分结果。测试了静压主轴的三维运动形态,分析了磨床主轴倾角运动简化为径向运动的科学依据；对比研究了理论计算与实验测试的误差轨迹和回转精度,结果表明：理论计算与实验测试的误差轨迹均呈椭圆形状,理论计算椭圆长轴约为实验值的78.8%；理论计算与实验测试的回转精度值之比约为75.6%。理论计算结果与实验测试结果一致性好。
本研究提出的静压主轴回转精度的流固耦合动力学模型及其动网格求解新方法,发展了回转精度的分析方法；研究揭示了静压主轴形成回转误差的物理机理；理论阐释了静压轴承对轴颈误差的均化效应；建立并发展了静压主轴回转精度的理论体系,研究成果为超精密液体静压主轴开发提供有力的理论指导和技术支撑。
The precision&ultra-precision machine becomes vital processing equipment for ultra-precision parts in national defense industry, aerospace, industrial and civil optical industry, also strategic equipment for national defense security. Machine spindle is the core part of ultra-precision machine, and the rotational accuracy of hydrostatic spindle plays a determinative influence on the machine's processing precision. Currently, there is successful application for ultra-precision hydrostatic spindle. However, specialized research on the mechanism of rotary accuracy is still lacking, the physical origin of rotation error caused by the spindle rotation is not effectively revealed, the transitional process of rotary accuracy is not quantitatively explained, and systematically guidance and prediction cannot be provided for engineering design.
Focused on the dynamic development process, the dynamic model of the fluid-structure interaction for hydrostatic spindle rotary accuracy is constructed in this paper, and its new solution based on CFD dynamic mesh model is proposed. The trajectory of the axis of spindle during spindle forming rotary accuracy is quantitatively simulated, meanwhile the physical origin of rotary error caused by the spindle rotation is revealed. On such basis, the effect of working parameter, structure parameter and manufacturing error of hydrostatic spindle friction pair on the rotary accuracy is systematically studied in this paper, and the hydrostatic spindle rotary accuracy test platform is constructed, the spindle error trajectory and rotary accuracy obtained by the theoretically calculated and experimentally tested are comparatively studied, which confirms the correctness and validity of the research. The research work and key result of this paper is listed as follows:
(1) The dynamic model of the fluid-structure interaction for hydrostatic spindle rotary accuracy based on Navier-stokes equation and spindle dynamics equation is constructed, and its new solution based on CFD moving mesh model is proposed. This solution inserts the spindle rotational and translational equation into CFD software and solves the N-S equation, calculates the transient oil film pressure, the spindle mesh rotational boundary is transformed into static boundary, the influence of speed disturbance and displacement disturbance on the oil film force is compared, which determined the numerical range of spindle speed effect. The theoretical calculation and experimental test on hydrostatic spindle's stiffness is compared and analysed, which ensures the validity and feasibility of the hydrostatic spindle rotary accuracy model and solving method.
(2) The trajectory of the axis of spindle is quantitatively simulated, and the physical mechanism of hydrostatic spindle forming rotary accuracy is revealed. Three types of method, which is full moving mesh, equilibrium position and moving mesh, and equilibrium position and simplified formula, are applied to calculate and simulate the trajectory of spindle from concentric position to form rotary error. While the hydrostatic spindle forming rotary centre and rotary error, the dynamic development of total force, flow and spindle position are comparatively analysed. Through comparatively studying the influence of these three calculation methods on the rotary centre position and rotary error of hydrostatic spindle, the physical reason of the difference of output among the three methods is found, which shows that combining the moving mesh method and simplified formula to calculate the rotary accuracy of hydrostatic spindle is feasible and valid.
(3) The effect of the spindle structure parameters and working parameters on the spindle rotary accuracy is revealed. The operating parameter (including operating rotation speed and oil supply pressure) and the structure parameter (including length to diameter ratio, orifice diameter and bearing clearance) on the rotary centre and oil film dynamic characteristics are quantitatively calculated, and the mechanical reasons of attitude angle, eccentricity, stiffness and damping is analysed, and then the working parameter and structure parameter's influence on the spindle rotational precision under the condition of the dynamic imbalance excitation is revealed. The result show that when increasing the oil supply pressure and the length to diameter ratio, the rotary accuracy will increase. When increasing the orifice diameter, rotational speed and bearing clearance, the rotary accuracy will decrease.
(4) The effect of the shape errors of the bearing on the spindle rotary accuracy is revealed. The influence of the roundness error and the cylindrical error on the rotary centre and oil stiffness&damping were quantitatively calculated. After compared and analysed mechanical root of the structure difference between the error bearing and smooth bearing, the reason of the rotary centre and oil stiffness&damping influenced by the shape errors of the bearing is shown. The effect of the roundness error and the cylindrical error on rotary accuracy of hydrostatic was discussed, the result shows that the shape error of the bearing doesn't form rotation error, but it affects rotational accuracy through the central position of spindle and the stiffness and damping of oil film.
(5) The effect of the shape errors of the journal on the spindle rotary accuracy is revealed. The influence of the roundness error and the cylindrical error on the equilibrium position and oil stiffness&damping were quantitatively calculated. The change of equilibrium position affected by switching between peak and valley of the roundness error was emphatically analysed. Through comparatively analysis the rotary centre position and rotary accuracy affected by the error journal and smoothing journal, the averaging effect of the hydrostatic bearing on the shape error of the journal was discovered finally. The result shows that, the roundness error of the journal forms the rotation error, and the accuracy has relation with the roundness error's amplitude and frequency. The cylindrical error of the journal doesn't forms rotation error, but affects rotational accuracy through the central position of spindle and the stiffness and damping of oil film.
(6) The trajectory and rotary accuracy of the spindle received by the theoretical calculation and experiments were comparatively studied. A hydrostatic spindle rotary accuracy experiment apparatus was developed, and three dimensional motion of the hydrostatic spindle was experimentally studied. For simplification inclination motion of grinder spindle as cylindrical motion, the scientific basis was found. The spindle trajectory and rotary accuracy obtained by theoretical calculation and experimental test were compared. The result shows that the trajectory of spindle of theoretical calculation and experimental test were ellipse, the theoretical calculated major axis of the ellipse is around78.8%of experimental result. The theoretical calculated rotary accuracy is around75.6%of the experimental result, with a high degree of consistency of theory and experiment.
The dynamic model of fluid-structure interaction for hydrostatic spindle rotary accuracy and its new moving mesh method is proposed in this thesis, develops the theoretically analytical methods of rotary accuracy of hydrostatic spindle is developed, the mechanical mechanism of hydrostatic spindle forming rotational error is revealed, the average effect of hydrostatic bearing on the shape error of the journal is theoretical interpreted, the analyze theory of hydrostatic spindle rotation accuracy preliminarily is established and developed, theoretical guidance and technical support for the development of ultra-precision hydrostatic spindle is provided.
引文
[1]Corbett J, Mckeown P A, Peggs G N, et al. Nanotechnology:International developments and emerging products. Annals of the CIRP,2000,49(2): 523-545.
[2]袁巨龙,王志伟,文东辉等.超精密加工现状综述.机械工程学报,2007,43(1)：35-48.
[3]袁哲俊,王先逵.精密和超精密加工技术.北京：机械工业出版社,2009,6-20.
[4]Lkawa N, Donaldson R R, Komanduri R, et al. Ultra-precision Metal Cutting-The Past, the Present and the Future. CIRP Annals-Manufacturing Technology, 1991,40(2):587-594.
[5]李圣怡,戴一帆.超精密加工机床新进展.机械工程学报,2003,39(8)：7-14.
[6]Ghosh M K, Majumdar B C. Stiffness and damping characteristics of hydrostatic multi-recess oil journal bearings. International Journal of Machine Tool Design and Research,1978,18(3):139-151.
[7]熊万里,阳雪冰,吕浪等.液体动静压电主轴关键技术综述.机械工程学报,2009,45(9)：1-18.
[8]CIRP. Unification Document Me. Axis of rotation Annals of CIRP,1976,25(2): 545-564.
[9]朱训生,薛秉源,贝季瑶.主轴回转误差的运动本质及测试计算方法.上海交通大学学报,1990,24(4)：49-57.
[10]黄仁贵.主轴径向平面运动的几何分析.机械工程学报,1994,30(2)：26-31.
[11]熊有伦.精密测量的数学方法.北京：中国计量出版社,1989.
[12]Reynolds O. On the theory of lubrication and Its application to Mr. Beauchamp Tower's experiments, Including an experimental determination of the viscosity of olive oil. Phil. Trans. Roy. Soc.1886,177:157-234
[13]Gumbel L. Das problem der lagerreibung, Bezirks Verein Deutscher Ingenieure, 1914,5:87-104,109-120.
[14]Gumbel L, EVERLING E, Reibung und Schmierung im Maschinenbau, Berlin: Krayn,1925.
[15]Falz E. Grundzuge der schmiertechnik. Berlin:Verlag von Julius Springer, 1926.
[16]Fuller D D. Theory and practice of lubrication for engineers, first edition. USA: John Wiley & Sons,1956.
[17]Mori H, Yabe H. A theoretical investigation on hydrostatic bearing. Bulletin of JSME,1963,6:354-363.
[18]Rowe W B, Xu S X, Chong F S, et al. Hybrid Journal Bearings with particular reference to hole-entry configurations. Tribology International,1982, 15(6):339-348
[19]Adams M L, Shapiro W. Squeeze film characteristics in flat hydrostatic bearings with incompressible flow. ASLE Transactions,1969,12(3):183-189.
[20]Majumdar B C, Saha A K. Temperature distribution in oil journal bearing. Wear, 1974,28(2):259-266.
[21]Christensen H, T(?)nder K. Stochastic models for hydrodynamic lubrication of rough surface. International Journal of Mechanical Engineers,1970,104: 1022-1033.
[22]Patir N, Cheng H S. Application of average flow model to lubrication between rough sliding surfaces. Journal of Lubrication Technology,1979,101:220-229.
[23]Guha S K, Ghosh M K, Majumdar B C. Rotor dynamic coefficients of multi-recess hybrid journal bearings part2:fluid inertia effect. Wear,1989,129: 261-272.
[24]San A L. Turbulent hybrid bearings with fluid inertia effects. Journal of Tribology,1990,112:699-707.
[25]孙恭寿,王中阳,芦伟健.动静压混合轴承的温度场研究.机械工程学报,1992,28(6)：74-81
[26]Yang Shuzi. A study of the static stiffness of machine tool spindles. International Journal of Machine Tool Design and Research.1981,21(1):23-40.
[27]Rsjan M, Rajan S D, Nelson H D, et al. Optimal placement of critical speeds in rotor-bearing systems. ASME Journal of Vibration Acoustics, Stress, and Reliability in Design,1987,109:152-157.
[28]金瑞琪.机床主轴轴承的合理配置.江苏大学学报,1980,1(1)：103-118.
[29]Chung I S, Tsutsumi M, Ito Y. Dynamic characteristic of spindle bearing system. Bullet of JSME,1985,28:1782-1788.
[30]Montusiewicz J, Osyczka A. Computer aided optimum design of machine tool spindle systems with hydrostatic bearings. Proceedings of the Institution of Mechanical Engineers Part B,1997,211:43-51.
[31]黎永明,瞿元尝,巫嘉耀.一种新型精密机床主轴箱.机床,1990,(6)：18-19.
[32]Tucker P G, Keogh P S. On the dynamic thermal state in a hydrodynamic bearing with a whirling journal using CFD techniques. Journal of Tribology, 1996,118(2):356-363.
[33]Huang B W, Kung H K. Variation of instability in a rotating spindle system with various bearing.2003,35:57-72.
[34]Braun M J, Dzodzo M B. Three-Dimensional Flow and Pressure Patterns in a Hydrostatic Journal Bearing Pocket. Journal of Tribology,1997,119(10): 711-719.
[35]Keogh P S, Gomiciaga R, Khonsari M M. CFD based design techniques for thermal prediction in a generic two axial groove hydrodynamic journal bearing. Journal of Tribology,1997(7),119:428-435.
[36]Sawicki J T, Rao T V V L N. A nonlinear model for prediction of dynamic coefficients in a hydrodynamic journal bearing. International Journal of Rotating Machinery,2004,10(6):507-513.
[37]Kim H S, Jeong K S, Lee D G. Design and manufacture of a three axis ultra precision CNC grinding machine. Journal of Materials Processing Technology, 1997,71(2):258-266.
[38]张耀满,王伟,刘永贤.数控车床主轴部件有限元分析及其验证.东北大学学报,2009,30(5)：720-723.
[39]Liu Huiping, Xu Hua, Ellison P J, et al. Application of computational fluid dynamic and fluid structure interaction method to the lubrication study of a rotor bearing system. Tribology Letter,2010,38:325-336.
[40]Li Hongqi, Shin Y C. Analysis of bearing configuration effects on high speed spindles using an integrated dynamic thermo-mechanical spindle model. International Journal of Machine Tools and Manufacture.2004,44(4):347-364
[41]熊万里,侯志泉,吕浪,等.气体悬浮电主轴动态特性研究进展.机械工程学报,2011,47(5)：40-58.
[42]Kashchenevsky L, Johson D. Theoretical analysis of rotational accuracy for thrust hydrostatic bearings. ASPE summer topical meeting,2007,41:7-9.
[43]Berthe D, Fantino B, Frene J, et al. Influence of shape defects and surface roughness on the hydrodynamics of lubricated systems. Journal of Mechanical Engineering Science,1974,16(3):156-159.
[44]孙方金.空气静压轴承径向回转精度的分析.仪器仪表学报,1991,12(3)：324-328.
[45]郑颖君,李东升.气体静压轴系径向回转误差均化机理的研究.工具技术,2002,36(2)：38-40.
[46]Kim K M, Kim K W. An analytical study on the rotational accuracy of externally pressurized air journal bearing. JSME International Journal,1992, 35(3):485-492.
[47]Yabe H, Yamamoto M. A study on the running accuracy of an externally pressurized gas thrust bearing (bearing stiffness and damping coefficient). JSME International Journal,1989,32(4):618-624
[48]Yabe H. A study on run out characteristics of externally pressurized gas journal bearing (modified DF method for point source solution). JSME International Journal,1989,37(2):362-368.
[49]Kashchenevsky L, Knapp B R. Predicting the rotational accuracy of hydrostatic spindle. ASPE summer topical meeting,2001,24:52-55.
[50]Akira K, Masakzu M. Effects of part accuracy on rotational accuracy in hydrostatic bearing. Journal of the Japan Society of Precision Engineering, 1979,45(10):1174-1176.
[51]景岗,张立平,张思矩,等.高精度主轴回转精度的测试与研究.制造技术与机床,1996,(6)：24-26.
[52]翁善惠.油阻尼对主轴回转精度的影响.振动与动态测试.1985,(1)：22-28.
[53]巫嘉耀.CG6510型高精度球面车床小间隙液体静压轴承主轴.装备机械,1987,(3)：32-35.
[54]Rowe W B, Donoghue J P. A review of hydrostatic bearing design. Proc. Conf. Externally Pressurized Bearings. London,1971,167-187.
[55]Rowe W B, Stout K J. Design data and a manufacture technique for spherical hydrostatic bearings in machine tool applications. International Journal of Machine Tool Design and Research.1971,11(4):293-307.
[56]黎永明.超高精度液体静压球轴系.计量学报,1986,7(3)：199-203.
[57]Leung P S, Craighead I A, Wikinson T S. An analysis of the steady and dynamic characteristics of a spherical hydrodynamic journal bearing. Journal of Tribology,1989,111:459-467.
[58]Gupta J, Deher G. Effect of roughness on the behaviour of squeeze in a spherical bearings. STLE Tribology Transaction,1996,39(1):99-102
[59]Yacout A W, Ismaeel A S, Kassab S Z. The combined effects of the centripetal inertia and the surface roughness on the hydrostatic thrust spherical bearings performance. Tribology International,2007,40(3):522-532.
[60]Kosmynin A V, Kabaldin Y G, Vinogradov V S, et al. Partly porous gas static bearings of high speed spindle units of metal working machines. European Journal of Natural History,2007, (2):105-109.
[61]Chattopadhay A K, Majumdar B C. Steady state solution of finite hydrostatic porous oil journal bearings with tangential velocity slip. Tribology International,1984,17:317-323.
[62]Kwan Y B P, Stephenson D J, Alcock J R. The dependence of pore size distribution on porosity in hot isostatically pressed porous alumina, Journal of Porous Material,2001,8:119-127.
[63]Durazo I S, Stephenson D J, Corbett J. Controlled porosity alumina structures for ultra precision hydrostatic journal bearings. Journal of the American Ceramic Society,2010,93(11):3671-3678.
[64]Slocum A H, Scagnetti P A, Kane N R, et al. Design of self-compensated water hydrostatic bearings. Precision Engineering,1995,17(3):173-185.
[65]Okuyama S, Yui A, Kumagai M, et al. Development of a linear motor driven table with hydrostatic water bearings. The Japan Society of Mechanical Engineers,2009,75(2):454-459.
[66]Zumgahr K H, Mathieu M, Brylka B. Friction control by surface engineering of ceramic sliding pairs in water. Wear,2007,263:920-929
[67]Feldeman Y, Kligerman Y, Etsion I. Stiffness and efficiency optimization of a hydrostatic laser surface textured gas seal. Journal of Tribology,2007, 129:407-410
[68]韩中领,汪家道,陈大融.凹坑表面形貌在面接触润滑状态下的减阻研究.摩擦学报,2009,29(1)：10-16.
[69]Durazo I S, Corbett J, Stephenson D J. The performance of a porous ceramic hydrostatic journal bearing. Proceedings of the Institution of Mechanical Engineers, Part J:Journal of Engineering Tribology,2010,224(1):81-89.
[70]Nishitani Y, Yoshimoto S, Somaya K. Numerical investigation of static and dynamic characteristics of water hydrostatic porous thrust bearings. International Journal of Automation Technology,2011,5(6):773-779.
[71]Geraghty P. Ultra-precision machine spindle using porous ceramic bearings. https://docs.google.com, USA:2010-11-23.
[72]Corbett J, Almond R J, Stephenson D J, et al. Porous ceramic water hydrostatic bearings for improved for accuracy performance. Annals of the CIRP,1998, 47(1):467-470.
[73]许尚贤.液体静压和动静压滑动轴承设计.南京：东南大学出版社,1989.
[74]Su C T and Lie K N. Approximate solutions for hydrostatic journal bearings with multi-array of hole-entry. STLE Lub Eng,2002; 4(58):19-28.
[75]Yoshimoto S, Rowe WB, and Ivesa D. A theoretical investigation of the effect of inlet pocket size on the performance of holes entry hybrid journal bearings employing capillary restrictors. Wear 1988; 3(127):307-318.
[76]Sharma S C, Kumar V, Jain S C, et al. Thermo-hydrostatic analysis of slot-entry hybrid journal bearing. Tribology International.2002, Vol.35: 561-577
[77]Sharma S C, Jain S C and Reddy N M M. A Study of non-recessed hybrid flexible journal bearing with different restrictors. Tribol Trans 2001; 2(44): 310-317
[78]丁振乾.流体静压支承设计.上海：上海科学技术出版社,1989,7-11.
[79]Singh N, Sharma S C, Jain S C. Performance of membrane compensated multi-recess hydrostatic/hybrid flexible journal bearing system considering various shapes. Tribology International,2004,37:11-24
[80]Kane N R, Sihler J, Slocum A H. A hydrostatic rotary bearing with angled surface self-compensation. Precision Engineering,2003,27:125-139.
[81]Santos L F, Watanabe F Y. Compensation of cross-coupling stiffness and increase of direct damping in multi-recess journal bearings using active hybrid lubrication, part I-theory. Transaction of the ASME, Journal of Tribology,2004, 126,146-155.
[82]刘学杰,王镛,余之泳.以可控液体静压轴承补偿主轴回转误差.包头钢铁学院学报,1990,9(1)：25-33.
[83]Mizumoto H, Arii S, Kami Y, et al. Active inherent restrictor for air bearing spindles. Precision Engineering,1996,19(2):141-147.
[84]Mizumoto H, Arii S, Yabuya M. Rotational accuracy and positioning resolution of an air bearing spindle with active inherent restrictors. Initiatives of precision engineering at the beginning of a millennium,2002, Ⅲ:709-713.
[85]王元勋,陈儿昌,师汉民,等.主动控制节流气体润滑轴承静动态特性研究.机械工程学报,1999,35(3)：12-15.
[86]伍良生,杨勇,周大帅.机床主轴径向回转误差的测试与研究.机械设计与制造,2009,(1)：107-109.
[87]王凯,韩大勇,韦增战.动不平衡对高速主轴回转运动的影响.南京理工大学学报,2008,28(4)：341-344.
[88]李增强,孙涛,夏广岚,等.高速空气静压轴承的精密动平衡.纳米技术与精密工程2006,4(4)：296-300.
[89]Rho B H, Kim K W. A study of the dynamic characteristics of synchronously controlled hydrodynamic journal bearings. Tribology International,2002,35(5): 339-345.
[90]Moon J D, Kim B S, Lee S H. Development of the active balancing device for high-speed spindle system using influence coefficient. International Journal of Machine Tools & Manufacture,2006,46:978-987.
[91]Adiletta G, Guido A R, Rossi C. Nonlinear dynamics of a rigid unbalanced rotor in journal bearings. Part Ⅰ:Theoretical analysis. Nonlinear Dynamics, 1997,14:57-87.
[92]Adiletta G, A Guido R, Rossi C. Nonlinear dynamics of a rigid unbalanced rotor in journal bearings. Part Ⅱ:Experimental analysis. Nonlinear Dynamics, 1997,14:157-189.
[93]安晨辉,许乔,张飞虎.空气静压主轴气膜的压强分布与动态特性分析.纳米技术与精密工程.2009,5(7)：459-468.
[94]李树森,刘暾.精密离心机静压气体轴承主轴系统的动力学特性分析.机械工程学报,2005,41(2)：28-32.
[95]Sahlin F, Glavatskih S B, Almquist, et al. Two dimensional CFD analyses of micro-patterned surfaces in hydrodynamic lubrication. Journal of Tribology, 2005,127(1):96-127.
[96]Han Jin, Fang Liang, Sun Jiapeng, et al. Hydrodynamic lubrication of micro-dimple textured surface using three-dimensional CFD. Tribology Transactions,2010,53(6):860-870.
[97]Gertzos K P, Nikolakopoulos P G, Papadopoulos C A. CFD analysis of journal bearing hydrodynamic lubrication by Bingham lubricant. Tribology International,2008,41(12):1190-1204.
[98]孙方金,陈世杰.精密轴系回转精度测试.哈尔滨：哈尔滨工业大学出版社,1997.
[99]苏波,袁行飞,聂国隽,等.流固耦合理论研究述评-流固耦合问题研究进展.第七届全国现代结构工程学术研讨会论文集.杭州：工业建筑(增刊),2007,703-709.
[100]FLUENT. FLUENT 6.3 User's Guide. Lebanon:New Hampshire (USA) Fluent Inc.,2006.
[101]Versteeg H K, Malalaekera W. An introduction to Computational Fluid Dynamics:The Finite Volume Method. England:Longman Group Ltd.1995.
[102]Gluck M, Breuer M, Durst F, et al. Computational of wind-induced vibrations of flexible shells and membranous structures. Journal of Fluids and Structures, 2003,17(5):739-765.
[103]Issa R I. Solution of the implicitly discretised fluid flow equations by operator splitting. Journal of Computational Physics,1985,62:40-65.
[104]Patankaer S V, Spalding D B. A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows. International Journal of Heat Mass Transfer,1972,15(10):1787-1806.
[105]王福军.计算流体动力分析-CFD软件原理与应用.北京：清华大学出版社,2004.
[106]李人宪.有限体积法基础(第二版).北京：国防工业出版社,2008.
[107]FLUENT. FLUENT 6.3 UDF Manual. Lebanon:New Hampshire (USA) Fluent Inc.,2006.
[108]许尚贤,陆永颐,万鹏.小孔式动静压轴承的特性分析与实验研究.机械工程学报,1993,29(5)：88-95.
[109]张直明.滑动轴承的流体动力润滑理论.北京：高等教育出版社,1986.
[110]吴石林,张玘.误差分析与数据处理.北京：清华大学出版社,2010.
[111]陶黎国,杨质苍,姚俊源等.小孔节流型空气静压径向轴承回转精度的定性分析.上海工业大学学报,1987,8(2)：207-211.
[112]陈明,徐建民.应用边界元法研究小孔节流空气静压径向轴承的运动精度.哈尔滨工业大学学报,1990,(2)：108-113.
[113]Tlusty J. System and methods of testing machine tools, Microtecnic,1959, 13(4):162-178.
[114]Donaldson R R. A simple method for separating spindle error from test ball roundness error. CIRP Annals,1972,21(1):125-136.
[115]Shino H, Mitsui K, Tatsue Y, et al. A new method for evaluating error motion of ultra precision spindle. CIRP Annals,1987,36(1):381-384
[116]Gao W, Kiyono S, Nomura T. A new multiprobe method of roundness measurements. Precision Engineering,1996,19(1):37-45
[117]张宇华,王晓林.三点法中测头角位置的精密测量方法.光学精密工程,1999,7(1)：120-124.
[118]Gao W, Kiyano S, Sugawara T. High accuracy roundness measurement by a new error separation method. Precision Engineering,1997,21(2):123-133.
[119]Marsh E R. Precision Spindle Metrology. USA:DEStech Publications, Inc., 2008.
[120]王建敏.超精密气体静压轴系部分关键技术研究：[国防科技大学博士论文]. 长沙：国防科技大学,2007.
[121]Grejda R, Marsh E, Vallance R. Tecniques for calibrating spindles with nanometer error motion. Precision Engineering,2005,29(1):113-123.
[122]Marsh E, Couey, J. Nanameter-level comparision of three spindle error motion separation tecniques. Journal of Manufacturing Science and Engineering,2006, 128(1),180-187.