风力发电机增速器齿轮修形技术研究
|
摘要
本论文在国家高技术研究发展计划项目(863计划)(No.2006AA04Z101)的资助下,结合实际工程项目背景,对大功率风力发电机增速器齿轮修形技术理论与方法进行了深入研究。
首先建立斜齿轮修形的数学模型。根据齿廓修形参数和齿向修形参数的几何定义,在啮合平面上建立修形曲面方程的统一表达式；以渐开线发生线滚动角和轴向截面的旋转角为参数,建立修形齿面切极坐标方程；根据连续相切接触条件,建立修形齿面啮合方程。为后续的几何模型和有限元分析模型奠定理论基础。
其次,给出计算齿轮箱整体变形的具体方法。应用有限元刚体耦合技术构造箱体柔度矩阵,计算箱体变形；根据赫兹接触理论,计算滚动轴承变形。提出了基于SQL Server数据库的斜齿轮有限元信息结构模型,开发了参数化轮齿接触分析有限元建模程序。该程序首先应用APDL生成原始直齿轮模型,将单元、节点信息导入SQL Server数据库,然后在SQL Server数据库中批量化变换、修改节点坐标,并导入ANSYS依据原单元与节点关系重建有限元网格,实现斜齿轮微观几何拓扑修形有限元分析的精确、高效建模。提出了一种基于分级剖分的网格细化方法,解决轮齿接触区网格细化难题,与以往的全齿面细化相比,可以有效地缩减节点规模,提高计算效率和精度。
然后,将轮齿齿面离散成若干点,以离散点对的接触代替实际齿面接触,根据轮齿啮合变形协调条件和力平衡关系,建立轮齿面-面接触分析数字化模型。以修形参数作为设计变量,以避免啮入、啮出冲击和载荷均布为目标函数,结合面-面接触分析模型,建立考虑增速器整体变形条件的齿轮修形优化设计模型。将齿轮箱体、轴承、轴、齿轮体的受载变形耦合至齿面各接触点上,对接触点的法向变形进行三次样条拟合,从而得到齿面变形函数,再基于“反变形”思想对齿面进行修形,分析修形后的接触应力、载荷分布、接触斑点等啮合性质,从而确定修形参数。
最后,通过分析考察修形参数对某风力发电机增速器输出级斜齿轮传动啮合性质的影响,确定最佳修形参数。在某企业1.5MW风力发电机增速器试验台上,采用DP1500振动信号采集分析系统,测试4种工况下增速器的振动信号。测试结果显示在计算工况下,啮合频率上的振动信号幅值最小,表明根据本文方法确定的齿轮修形参数是基本合理的,验证了本文理论和方法的正确性和有效性。
A new methodology for gear modification of high-power speed increasing gearboxes for wind turbine is presented. This research is supported by the National High Technology Research and Development Program of China (863 program) under Grant No. 2006AA04Z101.
Firstly, the mathematics model of modified helical gear is established. The unified expression of the modification surface equation on action plane is built according to the geometric definition of modification parameters, the tangent-polar coordinate equation of gear tooth is put forward with involute generation line roll angle and cross-section twist angle as surface parameters and the meshing equation of modified tooth flank is built based on the continuous tangent contact condition. These mentioned above are the theory foundation of the following geometric model and finite element model of modificated gears.
Secondly, the flexibility matrix of gearbox is generated by advantage of finite element rigid coupling method in order to calculate the deformation of gearbox housing and the deformation of rolling bearings is calculated by Hertz law. The information structure model of helical gear finite element analysis based on SQL Server data base is put forward. The parameterized modeling program for tooth contact finite element analysis is developed. The original spur gear finite element model is established by APDL, then, node and element is input in to SQL Server data base. The rotation transformation of node coordinates is made and the elements are built according to the correlation of original node and element in order to create quickly helical gear finite element model. A hierarchical subdivision algorithm is developed for mesh refinement of tooth contact zone instead of entire tooth flank. So, the number of nodes is reduced significantly. Accordingly, computation efficiency and computation precision is increased.
Then, the tooth flank is discreted to a set of points which are called reference points, and the tooth contact is replaced by the contact of these point pairs on the gears. The digitized surface to surface contact analysis model is established based on the deformation compatibility condition of teeth meshing and force equilibrium relationship. The optimum design model is put forward based on the considerations in the whole deformation of gearbox. In which, the modification parameters are regarded as design variables and Avoiding dynamic impact and geometric interference at both mesh-in position and mesh-out position as well as axial uniformly distributed load is regarded as goal function. The deformation including gearbox housing, rolling bearing, shaft and gear body is maped to the contact points on tooth flanks and the normal deformation of the contact points is fited by cubic spline. The gear is modified based on the thought of reverse deformation to determine the modification parameters by analyzing the tooth contact zone and von Mises stress etc.
Finally, an example of a Speed Increasing Gearboxes for Wind Turbines is presented to demonstrate the influences of the gear's modification parameters on meshing characteristics so as to determine the optimum modification parameters. Furthermore, the speed Increasing Gearbox's vibration signal under four work conditions is measured by DP1500 type vibration signal collector at a 1.5MW gear test rig. It is demonstrated that the amplitude of vibration under design work condition at gear meshing frequency is minimum. It proves that the modification parameters determined is correct and the theory and method proposed is available.
首先建立斜齿轮修形的数学模型。根据齿廓修形参数和齿向修形参数的几何定义,在啮合平面上建立修形曲面方程的统一表达式；以渐开线发生线滚动角和轴向截面的旋转角为参数,建立修形齿面切极坐标方程；根据连续相切接触条件,建立修形齿面啮合方程。为后续的几何模型和有限元分析模型奠定理论基础。
其次,给出计算齿轮箱整体变形的具体方法。应用有限元刚体耦合技术构造箱体柔度矩阵,计算箱体变形；根据赫兹接触理论,计算滚动轴承变形。提出了基于SQL Server数据库的斜齿轮有限元信息结构模型,开发了参数化轮齿接触分析有限元建模程序。该程序首先应用APDL生成原始直齿轮模型,将单元、节点信息导入SQL Server数据库,然后在SQL Server数据库中批量化变换、修改节点坐标,并导入ANSYS依据原单元与节点关系重建有限元网格,实现斜齿轮微观几何拓扑修形有限元分析的精确、高效建模。提出了一种基于分级剖分的网格细化方法,解决轮齿接触区网格细化难题,与以往的全齿面细化相比,可以有效地缩减节点规模,提高计算效率和精度。
然后,将轮齿齿面离散成若干点,以离散点对的接触代替实际齿面接触,根据轮齿啮合变形协调条件和力平衡关系,建立轮齿面-面接触分析数字化模型。以修形参数作为设计变量,以避免啮入、啮出冲击和载荷均布为目标函数,结合面-面接触分析模型,建立考虑增速器整体变形条件的齿轮修形优化设计模型。将齿轮箱体、轴承、轴、齿轮体的受载变形耦合至齿面各接触点上,对接触点的法向变形进行三次样条拟合,从而得到齿面变形函数,再基于“反变形”思想对齿面进行修形,分析修形后的接触应力、载荷分布、接触斑点等啮合性质,从而确定修形参数。
最后,通过分析考察修形参数对某风力发电机增速器输出级斜齿轮传动啮合性质的影响,确定最佳修形参数。在某企业1.5MW风力发电机增速器试验台上,采用DP1500振动信号采集分析系统,测试4种工况下增速器的振动信号。测试结果显示在计算工况下,啮合频率上的振动信号幅值最小,表明根据本文方法确定的齿轮修形参数是基本合理的,验证了本文理论和方法的正确性和有效性。
A new methodology for gear modification of high-power speed increasing gearboxes for wind turbine is presented. This research is supported by the National High Technology Research and Development Program of China (863 program) under Grant No. 2006AA04Z101.
Firstly, the mathematics model of modified helical gear is established. The unified expression of the modification surface equation on action plane is built according to the geometric definition of modification parameters, the tangent-polar coordinate equation of gear tooth is put forward with involute generation line roll angle and cross-section twist angle as surface parameters and the meshing equation of modified tooth flank is built based on the continuous tangent contact condition. These mentioned above are the theory foundation of the following geometric model and finite element model of modificated gears.
Secondly, the flexibility matrix of gearbox is generated by advantage of finite element rigid coupling method in order to calculate the deformation of gearbox housing and the deformation of rolling bearings is calculated by Hertz law. The information structure model of helical gear finite element analysis based on SQL Server data base is put forward. The parameterized modeling program for tooth contact finite element analysis is developed. The original spur gear finite element model is established by APDL, then, node and element is input in to SQL Server data base. The rotation transformation of node coordinates is made and the elements are built according to the correlation of original node and element in order to create quickly helical gear finite element model. A hierarchical subdivision algorithm is developed for mesh refinement of tooth contact zone instead of entire tooth flank. So, the number of nodes is reduced significantly. Accordingly, computation efficiency and computation precision is increased.
Then, the tooth flank is discreted to a set of points which are called reference points, and the tooth contact is replaced by the contact of these point pairs on the gears. The digitized surface to surface contact analysis model is established based on the deformation compatibility condition of teeth meshing and force equilibrium relationship. The optimum design model is put forward based on the considerations in the whole deformation of gearbox. In which, the modification parameters are regarded as design variables and Avoiding dynamic impact and geometric interference at both mesh-in position and mesh-out position as well as axial uniformly distributed load is regarded as goal function. The deformation including gearbox housing, rolling bearing, shaft and gear body is maped to the contact points on tooth flanks and the normal deformation of the contact points is fited by cubic spline. The gear is modified based on the thought of reverse deformation to determine the modification parameters by analyzing the tooth contact zone and von Mises stress etc.
Finally, an example of a Speed Increasing Gearboxes for Wind Turbines is presented to demonstrate the influences of the gear's modification parameters on meshing characteristics so as to determine the optimum modification parameters. Furthermore, the speed Increasing Gearbox's vibration signal under four work conditions is measured by DP1500 type vibration signal collector at a 1.5MW gear test rig. It is demonstrated that the amplitude of vibration under design work condition at gear meshing frequency is minimum. It proves that the modification parameters determined is correct and the theory and method proposed is available.
引文
[1]王梦.对我国开发利用风力发电的思考[J].中国国土资源经济,2005(2)：19-20.
[2]钟伟强.国内外风力发电的概况[J].风机技术,2005(5)：44-46.
[3]刘元杰.风电发展与齿轮商机[J].现代零部件,2007,6：28-29
[4]蒋莉萍.2005年我国风力发电情况综述[J].电力技术经济,2006,18(3)：专稿
[5]仙波正状.高强度齿轮设计[M].北京：机械工业出版社,1981
[6]李瑰贤,马亮,陶建国.舰船用齿轮传动啮合刚度及动态性能研究[J].船舶工程,2000,5：41-43
[7]常山,徐振忠,陈堪闻.船用传动齿轮齿面接触应力分布及修形研究[J].舰船科学与技术,1995,5：31-34
[8]齿轮手册—渐开线圆柱齿轮修形计算[M].北京：机械工业出版社2000.8
[9]杨廷力,王玉璞,叶新等.渐开线高速齿轮的齿向修形[J].齿轮,1982,24(4)：1-11
[10]杨廷力,王玉璞,叶新等.渐开线高速齿轮的齿高修形[J].齿轮,1982,24(4)：14-24
[11]王朝晋,丁玉成.关于齿廓修形的研究(一).齿轮,1986,10(2)
[12]王朝晋,丁玉成.关于齿廓修形的研究(二).齿轮,1986,10(3)
[13]Munro, R. G., N. Yildirim and D. M. Hall. Optimum Profile Relief and Transmission Error in Spur Gears[J]. Proceedings I. Mech. E. Gear Noise and Vibration, Cambridge,1990.
[14]Remmers, E. P. Analytical Gear Tooth Profile Design [J]. ASME Paper No.72-PTG-47,1972.
[15]Munro, R. G., N. Yildirim and D. M. Hall. Optimum Profi le Relief and Transmission Error in Spur Gears[J]. Proceedings I. Mech. E. Gear Noise and Vibration, Cambridge,1990.
[16]Palmer, D., and R. G. Munro. Measurements of Transmission Error, Vibration and Noise in Spur Gears[J]. Proceedings, I. Mech. E. Gear Dynamics Conference,1995.
[17]Hsiang Hsi Lin, Fred B. Oswald, Dennis P. Townsend. Dynamic Loading of Spur Gears With Linear or parabolic tooth profile modifications[J].Mechanism and Machine Theory,1994,29 (8):1115-1129.
[18]Tavakoli, M. S. and D. R. Houser. Optimum Profi le Modifi cations for the Minimization of Static Transmission Errors of Spur Gears[J]. Journal Mechanics Transmission, ASME,1986 (108):86-95
[19]Houser, D. R. "Transmission Error Modeling. " Seminar:Plastic Gear Fundamentals Design and Manufacturing, Rosemont, IL, April 26-27,2005.
[20]Walker H. Gear Tooth Defection and Profile Modification (Part1) [J]. The Engineer,1938, 166(4318):409-412
[21]Walker H. Gear Tooth Defection and Profile Modification(Part2) [J]. The Engineer,1938, 166(4319):434-436
[22]Yoshio, Treauchi, Kazuteru, et al.直齿圆柱齿轮轮齿挠曲计算及其齿廓修缘[J].冯澄宙译.齿轮,1983,7(3)：37-40
[24]李绍彬,李润方,林腾蛟.基于热弹变形的圆柱齿轮理想修形曲线[J].中国机械工程,2003,7(14)：1175-1179
[25]Welbourn, D. B. Fundamental Knowledge of Gear Noise—A Survey[J]. Proceedings Ⅰ. Mech. E. Conference on Noise and Vibrations of Engines and Transmissions, Cranfield Institute of Technology, July,1979.
[26]Band R V and Peterson R E. Load and Stress Cycle in Gear Teeth [J]. Mechanical Engineening, 1929,51(9):653-662
[27]Walker H. Gear Tooth Deflection and Profile Modification (Part 1) [J]. The Engineer, 1938,166 (4318):409-412
[28]Walker H. Gear Tooth Deflection and Profile Modification (Part 2) [J]. The Engineer, 1938,166 (4319):434-436
[29]Walker H. Gear Tooth Deflection and Profile Modification (Part 3) [J]. The Engineer, 1940,170 (4414):102-104
[30]Weber C. The Deformation of Loaded Gears and the Effect on Their Load-Carrying Capacity, Sponsored Research (Germany). British Scientific and Industrial Research,1949, London, Report No 3
[31]Terauchi Y and Nagamua K. Study on Deflection of Spur Gear Teeth[J]. (1st Report). Bull JSM E,1980,23(184):1682-1688
[32]Teranchi Y and Nagamura K. Study on Deflection of Spur Gear Teeth[J]. (2nd Report). Bull JSM E,1981,24 (188):447-452
[33]程乃土,刘温.用平面弹性理论的复变函数解法精确确定直齿轮的轮齿挠度[J].应用数学与力学,1985,6(7)
[34]魏任之等.关于直齿圆柱齿轮轮齿受载变形量的计算[J].齿轮,1980
[35]丁玉成,王建军,李润方.直齿轮接触有限元分析及轮齿热弹变形[J].重庆大学学报,1987,10(2)：1-9
[36]Su-Jin Parka, Wan-Suk Yoo. Deformation overlap in the design of spur and helical gear pair[J]. Finite Elements in Analysis and Design,2004 (40) 1361-1378
[37]Jiande Wang, Ian Howard. Finite Element Analysis of High Contact Ratio Spur Gears in Mesh [J]. Journal of Tribology,2005 (127):469-483
[38]詹东安,王树人,唐树为.高速齿轮齿部修形技术研究[J].机械设计,2000.8：8-10
[39]崔传勋.工程机械齿轮的修形设计与加工[J].机械传动,1996,3(20)：50-54
[40]Niemann G, Retting H. Dynamische Zahnkrafte. VDIZ,1957,99(89).131
[41]Niemann G, Retting H. Erfahrungen ber Statischen and Dynamischen Zahndehnungsmessungen. VDI Ber,1959,32.17
[42]Eiselt H. Quantitative Bestimmung der Profilrlrucknahme bei Evolventischen Zahmliden—mit Beruckschtigung von Verzahnungsfehlem. Maschinenbautech,1971,20.91
[43]Sigg H.Profile Deflection and Scuffing in Design of Highly Loaded Gears. Prod. En gg.,1947,18(2):141
[44]Hsiang Hsi Lin, Fred B. Oswald. Profile Modification to Minimize Spur Gear Dynamic Loading[J]. Prepared for the Design Engineering Technical Conference sponsored by the American Society of Mechanical Engineers Orlando, Florida, September 24-28,1988
[45]Wyeth, M. and W. Rouverol. Differential Crowning:A New Weapon Against Gear Noise and DynamicLoad[J].AGMA,1996.
[46]F. L. Litvin, J. Zhang, R. F. Handschuh. Crowned spur gears:methods for generation and tooth contact analysis-part I:basic concepts, generation of the pinion tooth surface by a plane[J]. Journal of Mechanisms, Transmissions, and Automation in Design, 1988(110):337-342
[47]F. L. Litvin, J. Zhang, R. F. Handschuh. Crowned spur gears:methods for generation and tooth contact analysis-part II:basic concepts, generation of the pinion tooth surface by a revolution[J].Journal of Mechanisms, Transmissions, and Automation in Design, 1988(110):343-347
[48]T. F. Conry, A. Seireg. A mathematical programming technique for the evaluation of load distribution and optimal modification for gear systems [J]. Journal of Engineering for Industry,1973,11:1115-1122
[49]Shuting LI. Effects of machining errors, assembly errors and tooth modifications on loading capacity, load-sharing ratio and transmission error of a pair of spur gears [J]. Mechanism and Machine Theory,2007 (42):698-726
[50]Shuting LI. Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear[J]. Journal of Mechanical Design,2002,124(9): 511-517
[51]Tobe, T., and K. Inoue. Longitudinal Load Distribution Factor of Helical Gears[J]. J Mech. Trans. and Auto. in Design,1985 (107)
[52]Wingate, D., and R. Walsh. Evaluating the Load Distribution Factor for Spur Gears [J]. SAE Paper 891930, Milwaukee,1989.
[53]Sainsot, V. and D. Berthe. A Static and Dynamic Analysis of Misaligned Gears with Partial Contact[J]. Proceedings of the 15th Leeds-Lyon Symposium on Tribology, Univ. of Leeds, UK,1988:167-172
[54]De Vaujany, Jean-Pierre. Axis Misalignments and Crowning for Helical cylindrical Gears[J].4th World Congress on Gearing and Power Transmission; Paris, France,1999.
[55]Chang, S., D.R. Houser and J.Harianto. Tooth Flank Corrections of Wide Face Width Helical Gears that Account for Shaft Deflections[J]. Proceedings of the 9th International Power Transmission and Gearing Conference, Chicago, IL,2003.
[56]Linke, H. and M. Senf. Longitudinal Load Distribution in Gearing—Calculation and Discussion[J].Maschinenbautechnik,1983 (32).437-444,.
[57]Hosel, T.H. Calculation of Contact Pattern and Longitudinal Flank Correction in Spur and Helical Involute Gears [J]. Proceedings 6th World Congress on the Theory of Machines and Mechanism, New Delhi,1983.
[58]Fujita, K. Gear-Tooth Stress Calculation Method for Heavily Crowned Gear[J]. JSME, 1974,1 (17):264-272
[59]Umezawa, K., T. Suzuki, H. Hou joh and K. Bagiasna. Influence of Misalignment on Vibration of Helical Gear [J]. Proceedings 2nd World Congress on Gearing, Paris,1986:615-626
[60]Guilbault,R. Effects of Helical Slope and Form Deviation on the Contact and Fillet Stresses of Helical Gears[J]. AGMA Technical Conference.
[61]Jaram illo T. Deflect ion and Moments Due to a Concent rated Load on a Cant ilever Plate of Infinite Length[J]. ASME Appl. Mech,1950, (17):67-72
[62]Umezawa K. The Meshing Test on Helical Gear Under Load Transmission (1st Report) [J]. Bull JSME,1972,15:1632-1639
[63]Umezawa K. The Meshing Test on Helical Gears Under Load Transmission (2nd Report) [J]. Bull JSME,1973,16 (97):1285-1293
[64]Umezawa K. The Meshing Test on Helical Gears Under Load Transmission (3rd Report) [J]. Bull JSME,1974,17 (112):1348-1355
[65]唐恒龄.斜齿轮的啮合刚度[J].昆明工学院学报,1989,10(14)：43-50
[66]Umezawa K and Ishikawa. Deflection Due to Contact Between Gear Teeth with Finite Width[J]. Bull. JSME,1973, Vo 114
[67]唐赠宝,钟毅芳,戴玉堂.斜齿圆柱齿轮传动的静态啮合刚度和动态啮合刚度[J].机械设计,1993(6Z)：10-13
[68]Vilmos Simon. Load and stress distribution in spur and helical gears[J]. Journal of Mechanisms, Transmissions, and Automation Design,1988(110):197-202
[69]Vilmos Simon. Load Distribution in Hypoid Gears[J]. Journal of Mechanical Design, 2000(122):529-535
[70]Vilmos Simon. Load Distribution in Spiral Bevel Gears [J]. Journal of Mechanical Design, 2007(129):201-209
[71]Vilmos Simon. Load Distribution in Cylindrical Worm Gears[J]. Journal of Mechanical Design,2003(125):356-364
[72]Vilmos Simon. Optimal Machine Tool Setting for Hypoid Gears Improving Load Distribution[J]. Journal of Mechanical Design,2001(123):577-582
[73]Vilmos Simon. The Influence of Gear Hobbing on Worm Gear Characteristics [J]. Journal of Mechanical Design,2001(123):577-582
[74]Vilmos Simon. Optimal Tooth Modifications in Hypoid Gears[J]. Journal of Manufacturing Science and Engineering,2007(129):919-925
[75]Vilmos Simon. The influence of misalignments on mesh performances of hypoid gears [J]. Mechanism and Machine Theory,1998 (33) 1277-1291
[76]Vilmos Simon. Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears [J]. Mechanism and Machine Theory,2006 (41) 707-724
[77]Vilmos Simon. Influence of Tooth Errors and Misalignments on Tooth Contact in Spiral Bevel Gears[J].12th IFToMM World Congress, Besancon (France), June18-21,2007
[78]Vilmos Simon. Computer Aided Loaded Tooth Contact Analysis in Cylindrical Worm Gears[J]. Journal of Manufacturing Science and Engineering,2005(127):973-981
[79]Vilmos Simon. Computer simulation of tooth contact analysis of mismatched spiral bevel gears [J]. Mechanism and Machine Theory,2007 (42) 365-381
[80]张涛,冯守卫,濮良贵.齿轮轮齿刚度与齿间载荷分配系数的再研究[J].西安公路交通大学学报,1999,10(19)：112-116
[81]方宗德.斜齿轮齿面柔度矩阵与修形的有限元计算[J].航空动力学,1994(9)：242-244
[82]方宗德.修形斜齿轮的承载接触分析[J].航空动力学报,1997(12)：251-254
[83]常山,徐振忠,霍肇波.斜齿圆柱齿轮瞬时啮合刚度及齿廓修形的研究[J].热能动力工程1997(4)：270-274
[84]常山,徐振忠,李威,陈堪闻.重载齿轮的最佳轮齿修形[J].热能动力工程,1995(5)：330-333
[85]Y. Zhang, Z. Fang Analysis of Transmission Errors Under Load of Helical Gears With Modified Tooth Surfaces[J]. Journal of Mechanical Design,1997,3 (119):121 -126
[86]Umeyama, M. Effects of Modified Tooth Surface of a Helical Gear Pair on the Transmission Error and its Optimal Design[J]. International Gearing Conference.University of Newcastle, Tyne, UK, p.377,1994
[87]Maki, H. A Study on Optimum Tooth Modifi cations of Helical Gears under Various Loads [J]. Proceedings of the Transmission and Driveline Systems Symposium, SAE,1999:257-261
[88]A. H. Elkholy. Load distribution on contact lines of helical gear teeth
[89]江荣贵,胡青春.斜齿圆柱齿轮啮合刚度的一种简化计算[J].机械,2005,12(32)：18-19
[90]杨生华.齿轮接触有限元分析[J].计算力学学报,2003(20)：138-143
[91]田涌涛,李从心,佟维.考虑传动轴变形影响的齿轮载荷分布[J].上海交通大学学报,2001,35(10)：1526-1530.
[92]田涌涛,李从心.基于子结构技术的复杂齿轮系统有限元三维接触分析[J].机械工程学报,2002(38)：133-137
[93]龙慧,张光辉,欧恒安.渐开线直齿轮啮合过程中载荷及应力的计算机模拟[J].机械工程学 报,1995,31(5)：39-45.
[94]李润方,陈大良.斜齿轮三维有摩擦接触应力分析及前后处理方法[J].齿轮,1990(14)：29-31
[95]Parag Wagaj Ahmet Kahraman. Influence of Tooth Profile Modification on Helical Gear Durability[J]. Journal of Mechanical Design,2002(124):501-510
[96]大野耕作,田中直行.采用三维有限元方法进行斜齿轮接触应力分析
[97]方宗德,张永才,蔺天存.斜齿轮的齿廓修形的实验研究[J].机械传动,1992(16)：27-30
[98]霍肇波,许德林,徐振忠.斜齿轮降噪修形实验研究[J].热能动力工程1996(11)：9-11
[99]刘更,蔺天存,李树庭等.三维修形对薄轮缘斜齿轮共振应力影响的实验研究[J].航空动力学报,1996,1(11)：41-44
[100]Donald R. Houser, Fred B. Oswald, Mark J. Valco. Comparison of Transmission Error Predictions With Noise Measurements for Several Spur and Helical Gears[J]. Prepared for the 30th Joint Propulsion Conference cosponsored by the AIAA, ASME, SAE, and ASEE Indianapolis, Indiana, June 27-29,1994
[101]Piermaria Davoli, Carlo Gorla, Francesco Rosa. Transmission Error and Noise Emission of Spur Gears [J]. GEARTECHNOLOGY,2007:34-38
[102]方宗德,赵宁.圆柱齿轮三维修形的计算与实验[J].制造技术与机床,1995,3：24-26
[103]常山,孟祥生,陈堪闻.圆柱渐开线斜齿轮啮合过程受载变形的实验研究[J].哈尔滨工业大学学报,1997(29)：37-39
[104]Faydor L Litvin, Alfonso Fuentes, Ignacio Gonzalez-Perez, et al. Modified involute helical gears:computerized design, simulation of meshing and stress analysis [J]. Comput. Methods Appl. Mech. Engrg.,2003,192:3619-3655
[105]王树人.齿轮啮合理论简明教程第1版[M]天津天津大学出版社2005.11 83-84
[106]魏巍MATLAB应用数学工具箱技术手册第1版[M]北京国防工业出版社2004.1 285-289
[107]万艳丽.基于ANSYS耦合箱体刚度的齿轮修形研究[D]大连理工大学硕士论文,2009.11
[108]戴曙.机床滚动轴承应用手册[M]机械工业出版社,1993.3
[109]博弈创作室.APDL参数化有限元分析技术及其应用实例[M]北京中国水利水电出版社2004.2
[110]丁能根.斜齿轮三维有限元网格和接触单元的自动生成[J].合肥工业大学学报(自然科学版),2003(26)：1094-1097
[111]顾守丰,连小珉,严磊.斜齿轮轮齿三维有限元网格自动生成及细化[J].清华大学学报(自然科学版),1996(36)：77-82
[112]赵海峰,蒋迪.ANSYS8.0工程结构实例分析[M] 北京中国铁道出版社2004.10
[113]方宗德,张永才.斜齿轮的齿廓修形[J].汽车工程,1993,15(1)：59-64
[114]霍肇波,徐振忠,常山.降低斜齿轮噪声的齿轮修形优化设计[J].热能动力工程,1995,10(9)：334-337
[115]吴序堂,王小椿.点啮合共扼齿面失配传动性能的预控[J].齿轮,1988(12)：1-7
[116]刘鹄然,王小椿.点啮合失配齿面传动性能及接触区形状的预控[J].机械设计,1995(1)：4-6
[117]王军,姜虹,王小椿等.差曲面的性质及其在弧齿锥齿轮中的应用[J].机械传动2003(27)：23-27
[2]钟伟强.国内外风力发电的概况[J].风机技术,2005(5)：44-46.
[3]刘元杰.风电发展与齿轮商机[J].现代零部件,2007,6：28-29
[4]蒋莉萍.2005年我国风力发电情况综述[J].电力技术经济,2006,18(3)：专稿
[5]仙波正状.高强度齿轮设计[M].北京：机械工业出版社,1981
[6]李瑰贤,马亮,陶建国.舰船用齿轮传动啮合刚度及动态性能研究[J].船舶工程,2000,5：41-43
[7]常山,徐振忠,陈堪闻.船用传动齿轮齿面接触应力分布及修形研究[J].舰船科学与技术,1995,5：31-34
[8]齿轮手册—渐开线圆柱齿轮修形计算[M].北京：机械工业出版社2000.8
[9]杨廷力,王玉璞,叶新等.渐开线高速齿轮的齿向修形[J].齿轮,1982,24(4)：1-11
[10]杨廷力,王玉璞,叶新等.渐开线高速齿轮的齿高修形[J].齿轮,1982,24(4)：14-24
[11]王朝晋,丁玉成.关于齿廓修形的研究(一).齿轮,1986,10(2)
[12]王朝晋,丁玉成.关于齿廓修形的研究(二).齿轮,1986,10(3)
[13]Munro, R. G., N. Yildirim and D. M. Hall. Optimum Profile Relief and Transmission Error in Spur Gears[J]. Proceedings I. Mech. E. Gear Noise and Vibration, Cambridge,1990.
[14]Remmers, E. P. Analytical Gear Tooth Profile Design [J]. ASME Paper No.72-PTG-47,1972.
[15]Munro, R. G., N. Yildirim and D. M. Hall. Optimum Profi le Relief and Transmission Error in Spur Gears[J]. Proceedings I. Mech. E. Gear Noise and Vibration, Cambridge,1990.
[16]Palmer, D., and R. G. Munro. Measurements of Transmission Error, Vibration and Noise in Spur Gears[J]. Proceedings, I. Mech. E. Gear Dynamics Conference,1995.
[17]Hsiang Hsi Lin, Fred B. Oswald, Dennis P. Townsend. Dynamic Loading of Spur Gears With Linear or parabolic tooth profile modifications[J].Mechanism and Machine Theory,1994,29 (8):1115-1129.
[18]Tavakoli, M. S. and D. R. Houser. Optimum Profi le Modifi cations for the Minimization of Static Transmission Errors of Spur Gears[J]. Journal Mechanics Transmission, ASME,1986 (108):86-95
[19]Houser, D. R. "Transmission Error Modeling. " Seminar:Plastic Gear Fundamentals Design and Manufacturing, Rosemont, IL, April 26-27,2005.
[20]Walker H. Gear Tooth Defection and Profile Modification (Part1) [J]. The Engineer,1938, 166(4318):409-412
[21]Walker H. Gear Tooth Defection and Profile Modification(Part2) [J]. The Engineer,1938, 166(4319):434-436
[22]Yoshio, Treauchi, Kazuteru, et al.直齿圆柱齿轮轮齿挠曲计算及其齿廓修缘[J].冯澄宙译.齿轮,1983,7(3)：37-40
[24]李绍彬,李润方,林腾蛟.基于热弹变形的圆柱齿轮理想修形曲线[J].中国机械工程,2003,7(14)：1175-1179
[25]Welbourn, D. B. Fundamental Knowledge of Gear Noise—A Survey[J]. Proceedings Ⅰ. Mech. E. Conference on Noise and Vibrations of Engines and Transmissions, Cranfield Institute of Technology, July,1979.
[26]Band R V and Peterson R E. Load and Stress Cycle in Gear Teeth [J]. Mechanical Engineening, 1929,51(9):653-662
[27]Walker H. Gear Tooth Deflection and Profile Modification (Part 1) [J]. The Engineer, 1938,166 (4318):409-412
[28]Walker H. Gear Tooth Deflection and Profile Modification (Part 2) [J]. The Engineer, 1938,166 (4319):434-436
[29]Walker H. Gear Tooth Deflection and Profile Modification (Part 3) [J]. The Engineer, 1940,170 (4414):102-104
[30]Weber C. The Deformation of Loaded Gears and the Effect on Their Load-Carrying Capacity, Sponsored Research (Germany). British Scientific and Industrial Research,1949, London, Report No 3
[31]Terauchi Y and Nagamua K. Study on Deflection of Spur Gear Teeth[J]. (1st Report). Bull JSM E,1980,23(184):1682-1688
[32]Teranchi Y and Nagamura K. Study on Deflection of Spur Gear Teeth[J]. (2nd Report). Bull JSM E,1981,24 (188):447-452
[33]程乃土,刘温.用平面弹性理论的复变函数解法精确确定直齿轮的轮齿挠度[J].应用数学与力学,1985,6(7)
[34]魏任之等.关于直齿圆柱齿轮轮齿受载变形量的计算[J].齿轮,1980
[35]丁玉成,王建军,李润方.直齿轮接触有限元分析及轮齿热弹变形[J].重庆大学学报,1987,10(2)：1-9
[36]Su-Jin Parka, Wan-Suk Yoo. Deformation overlap in the design of spur and helical gear pair[J]. Finite Elements in Analysis and Design,2004 (40) 1361-1378
[37]Jiande Wang, Ian Howard. Finite Element Analysis of High Contact Ratio Spur Gears in Mesh [J]. Journal of Tribology,2005 (127):469-483
[38]詹东安,王树人,唐树为.高速齿轮齿部修形技术研究[J].机械设计,2000.8：8-10
[39]崔传勋.工程机械齿轮的修形设计与加工[J].机械传动,1996,3(20)：50-54
[40]Niemann G, Retting H. Dynamische Zahnkrafte. VDIZ,1957,99(89).131
[41]Niemann G, Retting H. Erfahrungen ber Statischen and Dynamischen Zahndehnungsmessungen. VDI Ber,1959,32.17
[42]Eiselt H. Quantitative Bestimmung der Profilrlrucknahme bei Evolventischen Zahmliden—mit Beruckschtigung von Verzahnungsfehlem. Maschinenbautech,1971,20.91
[43]Sigg H.Profile Deflection and Scuffing in Design of Highly Loaded Gears. Prod. En gg.,1947,18(2):141
[44]Hsiang Hsi Lin, Fred B. Oswald. Profile Modification to Minimize Spur Gear Dynamic Loading[J]. Prepared for the Design Engineering Technical Conference sponsored by the American Society of Mechanical Engineers Orlando, Florida, September 24-28,1988
[45]Wyeth, M. and W. Rouverol. Differential Crowning:A New Weapon Against Gear Noise and DynamicLoad[J].AGMA,1996.
[46]F. L. Litvin, J. Zhang, R. F. Handschuh. Crowned spur gears:methods for generation and tooth contact analysis-part I:basic concepts, generation of the pinion tooth surface by a plane[J]. Journal of Mechanisms, Transmissions, and Automation in Design, 1988(110):337-342
[47]F. L. Litvin, J. Zhang, R. F. Handschuh. Crowned spur gears:methods for generation and tooth contact analysis-part II:basic concepts, generation of the pinion tooth surface by a revolution[J].Journal of Mechanisms, Transmissions, and Automation in Design, 1988(110):343-347
[48]T. F. Conry, A. Seireg. A mathematical programming technique for the evaluation of load distribution and optimal modification for gear systems [J]. Journal of Engineering for Industry,1973,11:1115-1122
[49]Shuting LI. Effects of machining errors, assembly errors and tooth modifications on loading capacity, load-sharing ratio and transmission error of a pair of spur gears [J]. Mechanism and Machine Theory,2007 (42):698-726
[50]Shuting LI. Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear[J]. Journal of Mechanical Design,2002,124(9): 511-517
[51]Tobe, T., and K. Inoue. Longitudinal Load Distribution Factor of Helical Gears[J]. J Mech. Trans. and Auto. in Design,1985 (107)
[52]Wingate, D., and R. Walsh. Evaluating the Load Distribution Factor for Spur Gears [J]. SAE Paper 891930, Milwaukee,1989.
[53]Sainsot, V. and D. Berthe. A Static and Dynamic Analysis of Misaligned Gears with Partial Contact[J]. Proceedings of the 15th Leeds-Lyon Symposium on Tribology, Univ. of Leeds, UK,1988:167-172
[54]De Vaujany, Jean-Pierre. Axis Misalignments and Crowning for Helical cylindrical Gears[J].4th World Congress on Gearing and Power Transmission; Paris, France,1999.
[55]Chang, S., D.R. Houser and J.Harianto. Tooth Flank Corrections of Wide Face Width Helical Gears that Account for Shaft Deflections[J]. Proceedings of the 9th International Power Transmission and Gearing Conference, Chicago, IL,2003.
[56]Linke, H. and M. Senf. Longitudinal Load Distribution in Gearing—Calculation and Discussion[J].Maschinenbautechnik,1983 (32).437-444,.
[57]Hosel, T.H. Calculation of Contact Pattern and Longitudinal Flank Correction in Spur and Helical Involute Gears [J]. Proceedings 6th World Congress on the Theory of Machines and Mechanism, New Delhi,1983.
[58]Fujita, K. Gear-Tooth Stress Calculation Method for Heavily Crowned Gear[J]. JSME, 1974,1 (17):264-272
[59]Umezawa, K., T. Suzuki, H. Hou joh and K. Bagiasna. Influence of Misalignment on Vibration of Helical Gear [J]. Proceedings 2nd World Congress on Gearing, Paris,1986:615-626
[60]Guilbault,R. Effects of Helical Slope and Form Deviation on the Contact and Fillet Stresses of Helical Gears[J]. AGMA Technical Conference.
[61]Jaram illo T. Deflect ion and Moments Due to a Concent rated Load on a Cant ilever Plate of Infinite Length[J]. ASME Appl. Mech,1950, (17):67-72
[62]Umezawa K. The Meshing Test on Helical Gear Under Load Transmission (1st Report) [J]. Bull JSME,1972,15:1632-1639
[63]Umezawa K. The Meshing Test on Helical Gears Under Load Transmission (2nd Report) [J]. Bull JSME,1973,16 (97):1285-1293
[64]Umezawa K. The Meshing Test on Helical Gears Under Load Transmission (3rd Report) [J]. Bull JSME,1974,17 (112):1348-1355
[65]唐恒龄.斜齿轮的啮合刚度[J].昆明工学院学报,1989,10(14)：43-50
[66]Umezawa K and Ishikawa. Deflection Due to Contact Between Gear Teeth with Finite Width[J]. Bull. JSME,1973, Vo 114
[67]唐赠宝,钟毅芳,戴玉堂.斜齿圆柱齿轮传动的静态啮合刚度和动态啮合刚度[J].机械设计,1993(6Z)：10-13
[68]Vilmos Simon. Load and stress distribution in spur and helical gears[J]. Journal of Mechanisms, Transmissions, and Automation Design,1988(110):197-202
[69]Vilmos Simon. Load Distribution in Hypoid Gears[J]. Journal of Mechanical Design, 2000(122):529-535
[70]Vilmos Simon. Load Distribution in Spiral Bevel Gears [J]. Journal of Mechanical Design, 2007(129):201-209
[71]Vilmos Simon. Load Distribution in Cylindrical Worm Gears[J]. Journal of Mechanical Design,2003(125):356-364
[72]Vilmos Simon. Optimal Machine Tool Setting for Hypoid Gears Improving Load Distribution[J]. Journal of Mechanical Design,2001(123):577-582
[73]Vilmos Simon. The Influence of Gear Hobbing on Worm Gear Characteristics [J]. Journal of Mechanical Design,2001(123):577-582
[74]Vilmos Simon. Optimal Tooth Modifications in Hypoid Gears[J]. Journal of Manufacturing Science and Engineering,2007(129):919-925
[75]Vilmos Simon. The influence of misalignments on mesh performances of hypoid gears [J]. Mechanism and Machine Theory,1998 (33) 1277-1291
[76]Vilmos Simon. Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears [J]. Mechanism and Machine Theory,2006 (41) 707-724
[77]Vilmos Simon. Influence of Tooth Errors and Misalignments on Tooth Contact in Spiral Bevel Gears[J].12th IFToMM World Congress, Besancon (France), June18-21,2007
[78]Vilmos Simon. Computer Aided Loaded Tooth Contact Analysis in Cylindrical Worm Gears[J]. Journal of Manufacturing Science and Engineering,2005(127):973-981
[79]Vilmos Simon. Computer simulation of tooth contact analysis of mismatched spiral bevel gears [J]. Mechanism and Machine Theory,2007 (42) 365-381
[80]张涛,冯守卫,濮良贵.齿轮轮齿刚度与齿间载荷分配系数的再研究[J].西安公路交通大学学报,1999,10(19)：112-116
[81]方宗德.斜齿轮齿面柔度矩阵与修形的有限元计算[J].航空动力学,1994(9)：242-244
[82]方宗德.修形斜齿轮的承载接触分析[J].航空动力学报,1997(12)：251-254
[83]常山,徐振忠,霍肇波.斜齿圆柱齿轮瞬时啮合刚度及齿廓修形的研究[J].热能动力工程1997(4)：270-274
[84]常山,徐振忠,李威,陈堪闻.重载齿轮的最佳轮齿修形[J].热能动力工程,1995(5)：330-333
[85]Y. Zhang, Z. Fang Analysis of Transmission Errors Under Load of Helical Gears With Modified Tooth Surfaces[J]. Journal of Mechanical Design,1997,3 (119):121 -126
[86]Umeyama, M. Effects of Modified Tooth Surface of a Helical Gear Pair on the Transmission Error and its Optimal Design[J]. International Gearing Conference.University of Newcastle, Tyne, UK, p.377,1994
[87]Maki, H. A Study on Optimum Tooth Modifi cations of Helical Gears under Various Loads [J]. Proceedings of the Transmission and Driveline Systems Symposium, SAE,1999:257-261
[88]A. H. Elkholy. Load distribution on contact lines of helical gear teeth
[89]江荣贵,胡青春.斜齿圆柱齿轮啮合刚度的一种简化计算[J].机械,2005,12(32)：18-19
[90]杨生华.齿轮接触有限元分析[J].计算力学学报,2003(20)：138-143
[91]田涌涛,李从心,佟维.考虑传动轴变形影响的齿轮载荷分布[J].上海交通大学学报,2001,35(10)：1526-1530.
[92]田涌涛,李从心.基于子结构技术的复杂齿轮系统有限元三维接触分析[J].机械工程学报,2002(38)：133-137
[93]龙慧,张光辉,欧恒安.渐开线直齿轮啮合过程中载荷及应力的计算机模拟[J].机械工程学 报,1995,31(5)：39-45.
[94]李润方,陈大良.斜齿轮三维有摩擦接触应力分析及前后处理方法[J].齿轮,1990(14)：29-31
[95]Parag Wagaj Ahmet Kahraman. Influence of Tooth Profile Modification on Helical Gear Durability[J]. Journal of Mechanical Design,2002(124):501-510
[96]大野耕作,田中直行.采用三维有限元方法进行斜齿轮接触应力分析
[97]方宗德,张永才,蔺天存.斜齿轮的齿廓修形的实验研究[J].机械传动,1992(16)：27-30
[98]霍肇波,许德林,徐振忠.斜齿轮降噪修形实验研究[J].热能动力工程1996(11)：9-11
[99]刘更,蔺天存,李树庭等.三维修形对薄轮缘斜齿轮共振应力影响的实验研究[J].航空动力学报,1996,1(11)：41-44
[100]Donald R. Houser, Fred B. Oswald, Mark J. Valco. Comparison of Transmission Error Predictions With Noise Measurements for Several Spur and Helical Gears[J]. Prepared for the 30th Joint Propulsion Conference cosponsored by the AIAA, ASME, SAE, and ASEE Indianapolis, Indiana, June 27-29,1994
[101]Piermaria Davoli, Carlo Gorla, Francesco Rosa. Transmission Error and Noise Emission of Spur Gears [J]. GEARTECHNOLOGY,2007:34-38
[102]方宗德,赵宁.圆柱齿轮三维修形的计算与实验[J].制造技术与机床,1995,3：24-26
[103]常山,孟祥生,陈堪闻.圆柱渐开线斜齿轮啮合过程受载变形的实验研究[J].哈尔滨工业大学学报,1997(29)：37-39
[104]Faydor L Litvin, Alfonso Fuentes, Ignacio Gonzalez-Perez, et al. Modified involute helical gears:computerized design, simulation of meshing and stress analysis [J]. Comput. Methods Appl. Mech. Engrg.,2003,192:3619-3655
[105]王树人.齿轮啮合理论简明教程第1版[M]天津天津大学出版社2005.11 83-84
[106]魏巍MATLAB应用数学工具箱技术手册第1版[M]北京国防工业出版社2004.1 285-289
[107]万艳丽.基于ANSYS耦合箱体刚度的齿轮修形研究[D]大连理工大学硕士论文,2009.11
[108]戴曙.机床滚动轴承应用手册[M]机械工业出版社,1993.3
[109]博弈创作室.APDL参数化有限元分析技术及其应用实例[M]北京中国水利水电出版社2004.2
[110]丁能根.斜齿轮三维有限元网格和接触单元的自动生成[J].合肥工业大学学报(自然科学版),2003(26)：1094-1097
[111]顾守丰,连小珉,严磊.斜齿轮轮齿三维有限元网格自动生成及细化[J].清华大学学报(自然科学版),1996(36)：77-82
[112]赵海峰,蒋迪.ANSYS8.0工程结构实例分析[M] 北京中国铁道出版社2004.10
[113]方宗德,张永才.斜齿轮的齿廓修形[J].汽车工程,1993,15(1)：59-64
[114]霍肇波,徐振忠,常山.降低斜齿轮噪声的齿轮修形优化设计[J].热能动力工程,1995,10(9)：334-337
[115]吴序堂,王小椿.点啮合共扼齿面失配传动性能的预控[J].齿轮,1988(12)：1-7
[116]刘鹄然,王小椿.点啮合失配齿面传动性能及接触区形状的预控[J].机械设计,1995(1)：4-6
[117]王军,姜虹,王小椿等.差曲面的性质及其在弧齿锥齿轮中的应用[J].机械传动2003(27)：23-27