摆线钢球行星传动动力学性能研究
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摘要
摆线钢球行星传动作为一种精密行星传动机构,其工作过程中所产生的振动,影响了机械设备的精度、生产效率和使用寿命,使得人们对其动态性能提出了更高的要求。研究摆线钢球行星传动系统的动力学问题,对其动态性能优化具有重要的意义,同时对工程中应用的其它行星传动系统也具有普遍的科学意义。
首先,利用超静定方法建立了摆线钢球啮合副的非线性力学模型,通过求解能量平衡方程得到啮合副的载荷关系式。利用赫兹公式计算出啮合副接触区中心变形和中心压强等参数,并求出传动过程中各参数极值对应的啮合位置。利用叠加法积分求出半椭球状分布载荷作用下啮合副的接触区变形挠曲面方程、接触区表面以下不同深度处的应力分布,以及最大切应力的深度位置。建立了摆线钢球啮合副的有限元模型,根据啮合副的载荷分布状态对有限元模型进行加载,通过对啮合过程的有限元仿真,得到啮合副的应力分布状态,并对啮合过程中承载接触对的交替变化规律和啮合点的应力变化规律进行了分析。
其次,综合考虑时变啮合刚度、轴承支承刚度和陀螺效应等影响因素,建立了摆线钢球行星传动系统在偏心轴随动坐标系下的多自由度平移—扭转耦合动力学模型,并推导出系统的运动微分方程和动力学方程,以揭示其固有特性。对系统的啮合刚度激励进行时域分析和频域分析,得到其频谱成分,并分析了不同系统参数对啮合刚度激励的影响。通过求解系统的特征值和特征向量问题,得到系统的各阶固有频率和主振型。对系统的固有频率进行灵敏度分析,进而探讨了系统参数和传动结构对各阶固有频率的影响。
然后,通过对摆线钢球行星传动系统动力学方程的变换和解耦,构建了系统的参数振动分析模型。利用多尺度法计算出系统的组合共振频率,并分析了系统的动力稳定性。利用林滋泰德-庞加莱摄动方法推导出内共振状态下系统的各特征函数,得到系统的稳定区边界曲线,绘制出稳定图,并分析了系统的广义质量和啮合刚度对动力稳定性的影响。在系统的动力学方程内增加线性阻尼项,分析了系统阻尼对动力稳定性的影响。利用摄动法求解了系统的参数振动的稳态响应。
最后,利用三维设计软件Pro/E对摆线钢球行星传动机构进行参数化造型,并进行虚拟装配。根据摆线钢球啮合副的正确啮合条件,研究了摆线槽的截面齿形、槽形角和槽深的设计准则,并推导出钢球密排结构对应的摆线短幅系数设计公式。利用Pro/E的Pro/NC模块设计了摆线槽的数控加工工序,通过轨迹加工方法模拟摆线槽的加工过程,并编制出摆线槽的数控铣削程序。总结了摆线钢球行星传动机构的设计步骤,并成功的加工制造出样机。
The cycloid ball planetary transmission is a precise planetary transmission mechanism, and the vibration caused in working process will influence the precision, the production efficiency and the operating life of mechanical devices, so the higher requirements are present to improve the dynamic characteristics. By investigating the dynamic problems of the cycloid ball planetary transmission system, it has an important significance to the dynamic characteristics optimization, and offers a general scientific reference to the other planetary transmission systems in practical engineering applications.
Firstly, the nonlinear mechanical model of the cycloid ball engagement pair is established using the hyperstatic method. The load of the engagement pair is present by solving the energy equilibrium equation. The central deformation and the central pressure of the engagement pair in contact zone are calculated by the Hertz contact formula, and the engagement positions of the parameter extremes in transmission process are obtained. The deformed deflection surface equation of contact zone, the stress distribution under contact surface, and the depth of the maximum shear stress are presented using the superposition method. The finite element models of the engagement pair are established, and the loads are applied to the models according to the load distribution of the engagement pair. The stress distribution states are present by the finite element simulation of the engagement process, and the alternate variation regularity of the loading contact pairs as well as the stress variation regularity of the engagement points is analyzed.
Secondly, the translational-torsional coupling dynamic model of the cycloid ball planetary transmission system is established in the following coordinate system of the eccentric shaft, and the model includes some key factors such as the time-variant engagement rigidity, the bearing rigidity and the gyroscopic effect. The governing differential equations and the dynamic equations of system are derived to investigate the natural characteristics. The engagement rigidity excitation is analyzed in time-domain and frequency-domain. The frequency spectrum components of the engagement rigidity excitation are present, and the influences of the different system parameters to the engagement rigidity excitation are analyzed. The natural frequencies and the principal modes of system are present by solving the eigenvalues and the eigenvectors. The sensitivities of the natural frequencies are analyzed, and the influence of the system parameters as well as the transmission structures to the natural frequencies is discussed.
Thirdly, the parametric vibration analytic model of the cycloid ball planetary transmission system is established by transformation and uncoupling of the dynamic equations. The combination resonance frequencies of system are calculated using the multi-scale method, and the dynamic stability of system is analyzed. The characteristic functions of system of the internal resonance are derived using the Lindstedt-Poincaréperturbation method, the boundary curves of the stability zones are obtained, the stability diagrams are drawn, and the influence of the generalized mass and the engagement rigidity to the dynamic stability is analyzed. The linear damping is added in the dynamic equations, and the influence of the damping to the dynamic stability is analyzed. The steady-state response of the parametric vibration of system is solved using the perturbation method.
Finally, the parametric models and the simulated assembly of the cycloid ball planetary transmission mechanism are completed using the three-dimensional design software Pro/E. According to the accurate engagement conditions of the cycloid ball engagement pair, the design criterions for the sectional profile, the angle and the depth of the cycloid groove are present, and the design formula for the cycloid curtate coefficient of the compact arrangement structure of balls is also present. The numerical control manufacture process is designed using the module of Pro/NC. The manufacture process of the cycloid groove is simulated by the locus manufacture method, and the procedure of the numerical control milling is programmed. The design process of the cycloid ball planetary transmission mechanism is summarized, and the prototype is manufactured successfully.
首先,利用超静定方法建立了摆线钢球啮合副的非线性力学模型,通过求解能量平衡方程得到啮合副的载荷关系式。利用赫兹公式计算出啮合副接触区中心变形和中心压强等参数,并求出传动过程中各参数极值对应的啮合位置。利用叠加法积分求出半椭球状分布载荷作用下啮合副的接触区变形挠曲面方程、接触区表面以下不同深度处的应力分布,以及最大切应力的深度位置。建立了摆线钢球啮合副的有限元模型,根据啮合副的载荷分布状态对有限元模型进行加载,通过对啮合过程的有限元仿真,得到啮合副的应力分布状态,并对啮合过程中承载接触对的交替变化规律和啮合点的应力变化规律进行了分析。
其次,综合考虑时变啮合刚度、轴承支承刚度和陀螺效应等影响因素,建立了摆线钢球行星传动系统在偏心轴随动坐标系下的多自由度平移—扭转耦合动力学模型,并推导出系统的运动微分方程和动力学方程,以揭示其固有特性。对系统的啮合刚度激励进行时域分析和频域分析,得到其频谱成分,并分析了不同系统参数对啮合刚度激励的影响。通过求解系统的特征值和特征向量问题,得到系统的各阶固有频率和主振型。对系统的固有频率进行灵敏度分析,进而探讨了系统参数和传动结构对各阶固有频率的影响。
然后,通过对摆线钢球行星传动系统动力学方程的变换和解耦,构建了系统的参数振动分析模型。利用多尺度法计算出系统的组合共振频率,并分析了系统的动力稳定性。利用林滋泰德-庞加莱摄动方法推导出内共振状态下系统的各特征函数,得到系统的稳定区边界曲线,绘制出稳定图,并分析了系统的广义质量和啮合刚度对动力稳定性的影响。在系统的动力学方程内增加线性阻尼项,分析了系统阻尼对动力稳定性的影响。利用摄动法求解了系统的参数振动的稳态响应。
最后,利用三维设计软件Pro/E对摆线钢球行星传动机构进行参数化造型,并进行虚拟装配。根据摆线钢球啮合副的正确啮合条件,研究了摆线槽的截面齿形、槽形角和槽深的设计准则,并推导出钢球密排结构对应的摆线短幅系数设计公式。利用Pro/E的Pro/NC模块设计了摆线槽的数控加工工序,通过轨迹加工方法模拟摆线槽的加工过程,并编制出摆线槽的数控铣削程序。总结了摆线钢球行星传动机构的设计步骤,并成功的加工制造出样机。
The cycloid ball planetary transmission is a precise planetary transmission mechanism, and the vibration caused in working process will influence the precision, the production efficiency and the operating life of mechanical devices, so the higher requirements are present to improve the dynamic characteristics. By investigating the dynamic problems of the cycloid ball planetary transmission system, it has an important significance to the dynamic characteristics optimization, and offers a general scientific reference to the other planetary transmission systems in practical engineering applications.
Firstly, the nonlinear mechanical model of the cycloid ball engagement pair is established using the hyperstatic method. The load of the engagement pair is present by solving the energy equilibrium equation. The central deformation and the central pressure of the engagement pair in contact zone are calculated by the Hertz contact formula, and the engagement positions of the parameter extremes in transmission process are obtained. The deformed deflection surface equation of contact zone, the stress distribution under contact surface, and the depth of the maximum shear stress are presented using the superposition method. The finite element models of the engagement pair are established, and the loads are applied to the models according to the load distribution of the engagement pair. The stress distribution states are present by the finite element simulation of the engagement process, and the alternate variation regularity of the loading contact pairs as well as the stress variation regularity of the engagement points is analyzed.
Secondly, the translational-torsional coupling dynamic model of the cycloid ball planetary transmission system is established in the following coordinate system of the eccentric shaft, and the model includes some key factors such as the time-variant engagement rigidity, the bearing rigidity and the gyroscopic effect. The governing differential equations and the dynamic equations of system are derived to investigate the natural characteristics. The engagement rigidity excitation is analyzed in time-domain and frequency-domain. The frequency spectrum components of the engagement rigidity excitation are present, and the influences of the different system parameters to the engagement rigidity excitation are analyzed. The natural frequencies and the principal modes of system are present by solving the eigenvalues and the eigenvectors. The sensitivities of the natural frequencies are analyzed, and the influence of the system parameters as well as the transmission structures to the natural frequencies is discussed.
Thirdly, the parametric vibration analytic model of the cycloid ball planetary transmission system is established by transformation and uncoupling of the dynamic equations. The combination resonance frequencies of system are calculated using the multi-scale method, and the dynamic stability of system is analyzed. The characteristic functions of system of the internal resonance are derived using the Lindstedt-Poincaréperturbation method, the boundary curves of the stability zones are obtained, the stability diagrams are drawn, and the influence of the generalized mass and the engagement rigidity to the dynamic stability is analyzed. The linear damping is added in the dynamic equations, and the influence of the damping to the dynamic stability is analyzed. The steady-state response of the parametric vibration of system is solved using the perturbation method.
Finally, the parametric models and the simulated assembly of the cycloid ball planetary transmission mechanism are completed using the three-dimensional design software Pro/E. According to the accurate engagement conditions of the cycloid ball engagement pair, the design criterions for the sectional profile, the angle and the depth of the cycloid groove are present, and the design formula for the cycloid curtate coefficient of the compact arrangement structure of balls is also present. The numerical control manufacture process is designed using the module of Pro/NC. The manufacture process of the cycloid groove is simulated by the locus manufacture method, and the procedure of the numerical control milling is programmed. The design process of the cycloid ball planetary transmission mechanism is summarized, and the prototype is manufactured successfully.
引文
1张国瑞,张展.行星传动技术.上海:上海交通大学出版社,1989:4-6
2 Jian Lin, Parker R G. Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration. Journal of Vibration and Acoustics, Transaction of the ASME, 1999, 116:316-322
3杨建明,张策,林忠钦,等.行星齿轮传动动力学特性研究进展.航空动力学报,2003, 18(2):299-304
4 Vijaya K A. Nonlinear Dynamics of Planetary Gears Using Analytical and Finite Element Models. Journal of Sound and Vibration, 2006, 302(3):577-595
5李润方,王建军.齿轮系统动力学——振动、冲击、噪声.北京:科学出版社,1997:259-267
6 August A, Kasuba R. Torsional Vibration and Dynamic Loads in a Basic Planetary Gear System. Journal of Acoustics Stress and Reliability in Design, Transaction of the ASME, 1986, 108:348-253
7 Parker R G. A Physical Explanation for the Effectiveness of Planet Phasing to Suppress Planetary Gear Vibration. Journal of Sound and Vibration, 2000, 236(4):561-573
8 Kahraman A. Load Sharing Characteristics of Planetary Transmission. Mech. Mach. Theory, 1994, 29(8):1151-1165
9 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (1st Report: Load Distribution in Planetary Gear). Bulletin of the JSME, 1976, 132(19):690-698
10 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (2nd Report: Displacement of Sun and Ring Gear). Bulletin of the JSME, 1976, 138(19):1563-1570
11 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (3rd Report: Displacement of Ring Gear in Direction of Line of Aetion). Bulletin of the JSME, 1977, 150(20):1663-1672
12 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (4th Report: Influence of the Transmitted Tooth Load). Bulletin of the JSME, 1979, 167(22):877-884
13 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (5th Report: Dynamic Increment of Torque). Bulletin of the JSME, 1979, 169(22):1017-1025
14 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (6th Report: Influence ofMeshing-Phase). Bulletin of the JSME, 1979, 169(22):1026-1033
15 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (7th Report: Influence of the Thickness of the Ring Gear). Bulletin of the JSME, 1979, 170(22):1142-1149
16宋轶民,许伟东,张策,等.2K-H行星传动的修正扭转模型建立与固有特性分析.机械工程学报,2006,42(5):16-21
17王世宇,张策,宋轶民,等.行星传动固有特性分析.中国机械工程,2005,16(16):1461-1464
18王世宇,张策,宋轶民,等.行星齿轮传动固有频率的统计特性分析.机械科学与技术,2005,24(6):705-709
19王世宇,宋轶民,沈兆光,等.行星传动系统固有特性及模态跃迁研究.振动工程学报,2005,18(4):417-421
20 Comparin R J, Singh R. Frequency Response Characteristics of a Multi-degree-of-freedom System with Clearances. Journal of Sound and Vibration, 1990,142(1):101-124
21 Saada A, Velex P. An Extended Model for the Analysis of the Dynamic Behavior of Planetary Trains. Journal of Mechanical Design, Transaction of the ASME, 1995, 117:241-247
22 Lin J, Parker R G. Planetary Gear Parametric Instability Caused by Mesh Stiffness Vibration. Journal of Sound and Vibration, 2002, 249(1):129-145
23 Velex P, Flamand L. Dynamic Response of Planetary Trains to Mesh Parametric Excitation. Journal of Mechanical Design, Transaction of the ASME, 1996, 118:7-14
24 Jian Lin, Parker R G. Parametric Resonance in Two Stage Gears Form Fluctuating Mesh Stiffness. Proceedings of the International Conference on Mechanical Transmission, Chongqing, China, 2001, 4:233-238
25孙涛,沈允文,孙智民,等.行星齿轮传动非线性动力学模型与方程.机械工程学报,2002,38(3):6-9
26孙智民,沈允文,李素有.封闭行星齿轮传动系统的动态特性研究.机械工程学报,2002,38(2):44-48
27孙智民,季林红,沈允文.2K-H行星齿轮传动非线性动力学.清华大学学报(自然科学版),2003,43(5):636-639
28孙涛,沈允文,孙智民,等.行星齿轮传动非线性动力学方程求解与动态特性分析.机械工程学报,2002,38(3):11-15
29 R. W. Cornell. Compliance and Stress Sensitivity of Spur Gear Teeth. ASME Mech. Des., 1981,103(2):477-459
30 Y. Terauchi, K. Nagamura. Study on Deflection of Spur Gear Teeth. Bull. JSME, 1981, 24(188):447-452
31宋轶民,刘欣,张策,等.高速行星传动内齿圈的有限元分析.中国机械工程,2007, 18(9):1016-1019
32朱才朝,陈兵奎,洪沙.平行动轴少齿差传动多齿接触问题研究.农业机械学报,2002,33(6):96-99
33黄慧春,沈永鹤.三环传动中内齿轮环板的刚度分析.机械设计与研究,2004,20(5):87-89
34单丽君,何卫东,关天民.环板式针摆行星传动非线性动态啮合特性分析.机械设计,2008,25(3):46-48
35方宗德,沈允文等.2K-H行星减速器的动态特性.西北工业大学学报,1990,8(4):361-371
36 Bodas A. Influence of Carrier and Gear Manufacturing Errors on Static Load Sharing Behavior of Planetary Gear Sets. JSME, 2004, 43(3):908-915
37 Singh A. Application of a System Level Model to Study the Planetary Load Sharing Behavior. Journal of Mechanical Design, 2005, 127:474-568
38姚文席,魏任之.渐开线直齿轮的啮合冲击响应.振动、测试与诊断,1992,12(2):27-30
39王玉芳,童忠钫.齿轮加速度噪声的研究.振动与冲击,1991,1:42-48
40 Kahraman A. Planetary Gear Train Dynamics. Journal of Mechanical Design, Transaction of the A SME, 1994, 116:713-720
41 Kahraman A. Natural Modes of Planetary Gear Trans. Journal of Sound and Vibration, 1994, 173(1):125-130
42 Kahraman A. Free Torsional Vibration Characteristics of Compound Planetary Gear Sets. Mech. Mach. Theory, 2001, 36:953-971
43赵玉香,孙首群,朱卫光.行星齿轮传动机构动力学分析.机械传动,2008,32(4):69-72
44孙智民,沈允文,王三民,等.星型齿轮传动非线性动力学建模与动载荷研究.航空动力学报,2001,16(4):402-407
45孙智民,沈允文,王三民,等.星型齿轮传动非线性动力学分析.西北工业大学学报,2002,20(2):222-226
46王三民,沈允文,董海军.含间隙和时变啮合刚度的弧齿锥齿轮传动系统非线性振动特性研究.机械工程学报,2003,39(2):28-32
47杨先勇,周晓军,胡宏伟,等.螺旋锥齿轮非线性振动特性及参数影响.浙江大学学报(工学版),2009,43(3):505-510
48 Lin Liu, et al. The Effect of Ring Gear Flexibility on Planetary Gear Train Dynamics. Proceedings of the International Conference on Mechanical Transmissions, Chongqing, China, 2001:687-690
49宋轶民,张俊,张君,等.3K-Ⅱ型直齿行星齿轮传动的固有特性.机械工程学报,2009,7(45):23-28
50梁尚明,张均富,徐礼钜,等.摆动活齿传动系统振动的动力学模型.振动工程学报,2003,16(3):285-289
51 Rajiv Dama, et al. Modeling and Analysis of a High-Speed Transmission with Multi-Path Elastometric Load-Share Gears. Mech. Mach. Theory, 1997, 32(3):295-312
52 Parker R G, Agashe V, Vijayakar S M. Dynamic Response of a Planetary Gear System Using a Finite Element Contact Mechanics Model. Journal of Mechanical Design, Transaction of the ASME, 2000, 122(3):305-311
53梁尚明,罗伟,徐俊光,等.摆动活齿减速器的有限元模态分析.机械设计,2004,21(7):14-16
54裘建新,叶桦,姜少杰.ADAMS对长幅外摆线行星传动的动力学仿真.机械设计,2004, 21(12):48-50
55梁尚明,张均富,徐礼钜,等.摆动活齿减速器箱体的有限元模态分析.机械设计,2003,20(1): 26-28
56 Ambarisha V K, Parker R. G. Suppression of Planet Mode Response in Planetary Gear Dynamics through Mesh Phasing. Journal of Vibration and Acoustics, 2006, 128(4):133-142
57杨建明,张策.行星齿轮传动的固有频率对设计参数的灵敏度分析.机械设计,2001,18(4):41-43
58 Lin J, Parker R G. Sensitivity of Planetary Gear Natural Frequencies and Vibration Modes to Model Parameters. Journal of Sound and Vibration, 1999, 228(1):109-128
59郝秀红,许立忠.机电集成超环面传动啮合参数激励下的稳定性分析.中国机械工程,2007,18(24):2947-2950
60郝秀红,许立忠.机电集成超环面传动随惯量波动的稳定性分析.机械工程学报,2008,44(4):69-73
61 Velex P, et al. Some Numerical Methods for the Simulation of Geared Transmission Dynamic Behavior Formulation and Assessment. Journal of Mechanical Design, Transaction of the ASME, 1997, 119:292-298
62 Kahraman A, et al. Interactions between Commensurate Parametric and Forcing Excitations in System with Clearance. Journal of Sound and Vibration, 1996, 194(3):317-336
63杨绍普,申永军,刘献栋.基于增量谐波平衡法的齿轮系统非线性动力学.振动与冲击,2005,24(3):40-42
64杨建明,张策.三环减速机的弹性动力学分析.机械工程学报,2000,36 (10):54-58
65杨玉虎.高速凸轮分度传动系统动力学理论与实验研究.[天津大学博士学位论文].2000:6-12
66曾强.空间凸轮活齿传动力学特性研究.[重庆大学硕士学位论文].2003:5-9
67 J. W. David, et al. Using Transfer Matrices for Parametric System Forced Response. ASME Vib. Acous. Stress Reli. Des., 1987, 109(2):356-360
68 L. D. Mitchell. A New Branching Technique for the Static and Dynamic Analysis of Gears System. Second Inter. Conf. Vib. in Rotating Machinery, Cambridge, England, 1980:437-441
69安子军,曲志刚.摆线钢球传动啮合副间隙及其调整机构.机械设计,1996,12:33-35
70 Qu Jifang, An Zijun. Research on Zero Clearance Cycloid Ball Transmission. Chinese Journal of Mechanical Engineering, 1994, 7(1):17-23
71 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (1st Report). JSPE, 1988, 54(11):2101-2106
72 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (2nd Report). JSPE, 1990, 56(4):751-756
73 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (3rd Report). JSPE, 1995, 61(12):1075-1079
74 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (4th Report). JSPE, 1997, 63(6):834-838
75 Roy S. McFarland. Employs Trochoidal Motion and Bearing Balls. Mechanical Engineering, 1982:1-60
76安子军,曲志刚,张荣贤.摆线钢球传动齿形综合研究.机械工程学报, 1996,32(5):41-46
77 Yang Zuomei, An Zijun, Zhang Peng. Manufacturing Mathematic Models and Meshing Analysis for the Cycloid Grooves of the Cycloid Ball Planetary Drive. Proceedings of International Conference on Measuring Technology and Mechatronics Automation, Zhangjiajie, China, 2009, 4:89-92
78杨作梅,安子军,张鹏.基于空间啮合理论的摆线钢球行星传动根切研究.农业机械学报,2009,40(10):216-222
79安子军,曲继方.摆线钢球传动齿形及其压力角分析.机械设计,1997,1:24-26
80 An Zijun, Yi Yali. Force Analysis and Stress Calculation of Non-clearance Cycloid Ball Transmissions. Proceedings of IEEE International Conference on Mechatronics and Automation, Japan, 2008, 8:1089-1093
81安子军.端面啮合摆线钢球行星传动啮合副力及强度分析.机械传动,2003,27(4):29-31
82安子军,胡瑞雪.摆线钢球行星传动齿廓误差的参数分析.机械传动,2007,31(2):63-65
83安子军,曲志刚,王广欣.基于模糊理论的摆线钢球行星传动接触疲劳强度可靠性研究.中国机械工程,2002,13(23):2010-2012
84安子军,王广欣.摆线钢球行星传动的模糊可靠性优化设计.机械传动,2004,28(5):17-19
85宜亚丽,安子军.摆线钢球传动中等速输出机构的研究设计.机械电子,2004,11:22-23
86宋敬稳,安子军.浮动式无回差钢球等速输出机构动平衡研究与创新设计.机械传动,2008,32(5):6-9
87安子军,宜亚丽.精密钢球等速输出机构的力学性能研究.机械传动,2000,24(4):1-3
88宋敬稳,安子军,胡瑞雪.精密钢球环槽等速传动机构运动精度及可靠性分析.燕山大学学报,2009,33(2):95-98
89 Zhou Jianjun, Chen Zichen. Study and Performance Test of Full Complement Cycloidal Ball Reducer with Ceramic Balls as Its Gear Teeth. Chinese Journal of Aeronautics, 2001, 14(4): 245-251
90 Zhou J. J., Hu J. X., Huang M., Geometric Design of Cycloidal Drives with Ceramic Ball Meshing Elements. Key Engineering Materials, 2007, 336(3):2484-2486
91李思益,王博,张彩丽.摆线钢球行星传动减速器振动特性分析.现代制造工程,2004,3:70-71
92刘锡峰,李思益.双摆线钢球行星传动减速器振动分析.西安工业学院学报,2003,23(3):218-222
93李思益,王海燕,张彩丽.两齿差摆线钢球行星传动减速器的试验研究.现代制造工程,2002,10:81-82
94姚素芬,高东强,李进.基于UG的双摆线钢球减速器的三维建模技术.陕西科技大学学报,2008,26(6):138-141
95高东强,姚素芬,李进.基于PRO/E的双摆线钢球减速器的参数化设计技术的研究.机械设计与制造,2009,3:222-224
96吴勤保,冉朝,辛晓峰.双摆线钢球减速器摆线槽的设计及数控加工.机械设计,2007,24(10): 68-70
97西庆坤,朱俊平,赵曙光,等.摆线钢球行星减速器误差的蒙特卡洛分析,西北农林科技大学学报(自然科学版),2008,36(7):224-228
98万长森.滚动轴承的分析方法.北京:机械工业出版社,1987:259-263
99李有法.数值计算方法.北京:高等教育出版社,1996:68-75
100徐芝纶.弹性力学(上册).北京:人民教育出版社,1982:299-302
101 K. L. Johnson.徐秉业,罗学富,刘信声,等译.接触力学.北京:高等教育出版社,1992:51-77
102李润方,陈晓南.弹塑性接触有限元混合法及其在齿轮传动中的应用.工程力学,1989, 6(2):87-97
103李润方,龚剑霞.接触问题数值方法及其在机械设计中的应用.重庆:重庆大学出版社,1989:1-20
104周宁.ANSYS机械工程应用实例.北京:中国水利水电出版社,2006:94-117
105刘文芝,张乃仁,张春林,赵永忠.谐波齿轮传动中杯形柔轮的有限元计算与分析.机械工程学报,2006,42(4):52-57
106付军锋,董海军,沈允文.谐波齿轮传动中柔轮应力的有限元分析.中国机械工程,2007,18(18):2210-2214
107安子军.摆线钢球行星传动研究.[燕山大学博士学位论文].2000:64-80
108安子军,曲继方.双摆线无隙钢球传动间隙调节力矩分析.机械传动,1995,19(4):1-3
109哈尔滨工业大学理论力学教研组.理论力学(下册).北京:高等教育出版社,1997:19-27
110扈英超.线性振动.北京:高等教育出版社,1981:8-13
111张荣贤,曲继方,安子军.摆线钢球传动的传动比计算.太原重型机械学院学报,1995,16(1):14-19
112刘延柱,陈立群.非线性振动.北京:高等教育出版,2001:57-95
113徐韫知.综合数学手册.北京:商务印书馆,1958:125-126
114郝秀红.机电集成超环面传动机电耦合动力学研究.[燕山大学硕士学位论文].2006:20-22
115宜亚丽.端面啮合摆线钢球行星传动的动力性能研究.[燕山大学硕士学位论文].2001:67-76
116过永德.机械振动基础.北京:兵器工业出版社,1992:112-114
117王广欣.摆线钢球行星传动模糊可靠性优化与建模仿真研究.[燕山大学硕士学位论文].2003:59-69
118王广欣,朱莉莉,关天民.基于Pro/E的摆线钢球行星传动建模仿真研究.机械传动,2005, 9(3):27-29
119孙江宏,黄小龙. Pro/ENGINEER Wildfire(野火版)数控加工教程.北京:清华大学出版社,2005:2-8
120吴勤保.双摆线钢球减速器齿廓曲线参数的选择.机械设计,2007,24(7):64-66
2 Jian Lin, Parker R G. Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration. Journal of Vibration and Acoustics, Transaction of the ASME, 1999, 116:316-322
3杨建明,张策,林忠钦,等.行星齿轮传动动力学特性研究进展.航空动力学报,2003, 18(2):299-304
4 Vijaya K A. Nonlinear Dynamics of Planetary Gears Using Analytical and Finite Element Models. Journal of Sound and Vibration, 2006, 302(3):577-595
5李润方,王建军.齿轮系统动力学——振动、冲击、噪声.北京:科学出版社,1997:259-267
6 August A, Kasuba R. Torsional Vibration and Dynamic Loads in a Basic Planetary Gear System. Journal of Acoustics Stress and Reliability in Design, Transaction of the ASME, 1986, 108:348-253
7 Parker R G. A Physical Explanation for the Effectiveness of Planet Phasing to Suppress Planetary Gear Vibration. Journal of Sound and Vibration, 2000, 236(4):561-573
8 Kahraman A. Load Sharing Characteristics of Planetary Transmission. Mech. Mach. Theory, 1994, 29(8):1151-1165
9 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (1st Report: Load Distribution in Planetary Gear). Bulletin of the JSME, 1976, 132(19):690-698
10 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (2nd Report: Displacement of Sun and Ring Gear). Bulletin of the JSME, 1976, 138(19):1563-1570
11 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (3rd Report: Displacement of Ring Gear in Direction of Line of Aetion). Bulletin of the JSME, 1977, 150(20):1663-1672
12 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (4th Report: Influence of the Transmitted Tooth Load). Bulletin of the JSME, 1979, 167(22):877-884
13 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (5th Report: Dynamic Increment of Torque). Bulletin of the JSME, 1979, 169(22):1017-1025
14 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (6th Report: Influence ofMeshing-Phase). Bulletin of the JSME, 1979, 169(22):1026-1033
15 Teruaki Hidaka, Yoshio Terauchi. Dynamic Behavior of Planetary Gear (7th Report: Influence of the Thickness of the Ring Gear). Bulletin of the JSME, 1979, 170(22):1142-1149
16宋轶民,许伟东,张策,等.2K-H行星传动的修正扭转模型建立与固有特性分析.机械工程学报,2006,42(5):16-21
17王世宇,张策,宋轶民,等.行星传动固有特性分析.中国机械工程,2005,16(16):1461-1464
18王世宇,张策,宋轶民,等.行星齿轮传动固有频率的统计特性分析.机械科学与技术,2005,24(6):705-709
19王世宇,宋轶民,沈兆光,等.行星传动系统固有特性及模态跃迁研究.振动工程学报,2005,18(4):417-421
20 Comparin R J, Singh R. Frequency Response Characteristics of a Multi-degree-of-freedom System with Clearances. Journal of Sound and Vibration, 1990,142(1):101-124
21 Saada A, Velex P. An Extended Model for the Analysis of the Dynamic Behavior of Planetary Trains. Journal of Mechanical Design, Transaction of the ASME, 1995, 117:241-247
22 Lin J, Parker R G. Planetary Gear Parametric Instability Caused by Mesh Stiffness Vibration. Journal of Sound and Vibration, 2002, 249(1):129-145
23 Velex P, Flamand L. Dynamic Response of Planetary Trains to Mesh Parametric Excitation. Journal of Mechanical Design, Transaction of the ASME, 1996, 118:7-14
24 Jian Lin, Parker R G. Parametric Resonance in Two Stage Gears Form Fluctuating Mesh Stiffness. Proceedings of the International Conference on Mechanical Transmission, Chongqing, China, 2001, 4:233-238
25孙涛,沈允文,孙智民,等.行星齿轮传动非线性动力学模型与方程.机械工程学报,2002,38(3):6-9
26孙智民,沈允文,李素有.封闭行星齿轮传动系统的动态特性研究.机械工程学报,2002,38(2):44-48
27孙智民,季林红,沈允文.2K-H行星齿轮传动非线性动力学.清华大学学报(自然科学版),2003,43(5):636-639
28孙涛,沈允文,孙智民,等.行星齿轮传动非线性动力学方程求解与动态特性分析.机械工程学报,2002,38(3):11-15
29 R. W. Cornell. Compliance and Stress Sensitivity of Spur Gear Teeth. ASME Mech. Des., 1981,103(2):477-459
30 Y. Terauchi, K. Nagamura. Study on Deflection of Spur Gear Teeth. Bull. JSME, 1981, 24(188):447-452
31宋轶民,刘欣,张策,等.高速行星传动内齿圈的有限元分析.中国机械工程,2007, 18(9):1016-1019
32朱才朝,陈兵奎,洪沙.平行动轴少齿差传动多齿接触问题研究.农业机械学报,2002,33(6):96-99
33黄慧春,沈永鹤.三环传动中内齿轮环板的刚度分析.机械设计与研究,2004,20(5):87-89
34单丽君,何卫东,关天民.环板式针摆行星传动非线性动态啮合特性分析.机械设计,2008,25(3):46-48
35方宗德,沈允文等.2K-H行星减速器的动态特性.西北工业大学学报,1990,8(4):361-371
36 Bodas A. Influence of Carrier and Gear Manufacturing Errors on Static Load Sharing Behavior of Planetary Gear Sets. JSME, 2004, 43(3):908-915
37 Singh A. Application of a System Level Model to Study the Planetary Load Sharing Behavior. Journal of Mechanical Design, 2005, 127:474-568
38姚文席,魏任之.渐开线直齿轮的啮合冲击响应.振动、测试与诊断,1992,12(2):27-30
39王玉芳,童忠钫.齿轮加速度噪声的研究.振动与冲击,1991,1:42-48
40 Kahraman A. Planetary Gear Train Dynamics. Journal of Mechanical Design, Transaction of the A SME, 1994, 116:713-720
41 Kahraman A. Natural Modes of Planetary Gear Trans. Journal of Sound and Vibration, 1994, 173(1):125-130
42 Kahraman A. Free Torsional Vibration Characteristics of Compound Planetary Gear Sets. Mech. Mach. Theory, 2001, 36:953-971
43赵玉香,孙首群,朱卫光.行星齿轮传动机构动力学分析.机械传动,2008,32(4):69-72
44孙智民,沈允文,王三民,等.星型齿轮传动非线性动力学建模与动载荷研究.航空动力学报,2001,16(4):402-407
45孙智民,沈允文,王三民,等.星型齿轮传动非线性动力学分析.西北工业大学学报,2002,20(2):222-226
46王三民,沈允文,董海军.含间隙和时变啮合刚度的弧齿锥齿轮传动系统非线性振动特性研究.机械工程学报,2003,39(2):28-32
47杨先勇,周晓军,胡宏伟,等.螺旋锥齿轮非线性振动特性及参数影响.浙江大学学报(工学版),2009,43(3):505-510
48 Lin Liu, et al. The Effect of Ring Gear Flexibility on Planetary Gear Train Dynamics. Proceedings of the International Conference on Mechanical Transmissions, Chongqing, China, 2001:687-690
49宋轶民,张俊,张君,等.3K-Ⅱ型直齿行星齿轮传动的固有特性.机械工程学报,2009,7(45):23-28
50梁尚明,张均富,徐礼钜,等.摆动活齿传动系统振动的动力学模型.振动工程学报,2003,16(3):285-289
51 Rajiv Dama, et al. Modeling and Analysis of a High-Speed Transmission with Multi-Path Elastometric Load-Share Gears. Mech. Mach. Theory, 1997, 32(3):295-312
52 Parker R G, Agashe V, Vijayakar S M. Dynamic Response of a Planetary Gear System Using a Finite Element Contact Mechanics Model. Journal of Mechanical Design, Transaction of the ASME, 2000, 122(3):305-311
53梁尚明,罗伟,徐俊光,等.摆动活齿减速器的有限元模态分析.机械设计,2004,21(7):14-16
54裘建新,叶桦,姜少杰.ADAMS对长幅外摆线行星传动的动力学仿真.机械设计,2004, 21(12):48-50
55梁尚明,张均富,徐礼钜,等.摆动活齿减速器箱体的有限元模态分析.机械设计,2003,20(1): 26-28
56 Ambarisha V K, Parker R. G. Suppression of Planet Mode Response in Planetary Gear Dynamics through Mesh Phasing. Journal of Vibration and Acoustics, 2006, 128(4):133-142
57杨建明,张策.行星齿轮传动的固有频率对设计参数的灵敏度分析.机械设计,2001,18(4):41-43
58 Lin J, Parker R G. Sensitivity of Planetary Gear Natural Frequencies and Vibration Modes to Model Parameters. Journal of Sound and Vibration, 1999, 228(1):109-128
59郝秀红,许立忠.机电集成超环面传动啮合参数激励下的稳定性分析.中国机械工程,2007,18(24):2947-2950
60郝秀红,许立忠.机电集成超环面传动随惯量波动的稳定性分析.机械工程学报,2008,44(4):69-73
61 Velex P, et al. Some Numerical Methods for the Simulation of Geared Transmission Dynamic Behavior Formulation and Assessment. Journal of Mechanical Design, Transaction of the ASME, 1997, 119:292-298
62 Kahraman A, et al. Interactions between Commensurate Parametric and Forcing Excitations in System with Clearance. Journal of Sound and Vibration, 1996, 194(3):317-336
63杨绍普,申永军,刘献栋.基于增量谐波平衡法的齿轮系统非线性动力学.振动与冲击,2005,24(3):40-42
64杨建明,张策.三环减速机的弹性动力学分析.机械工程学报,2000,36 (10):54-58
65杨玉虎.高速凸轮分度传动系统动力学理论与实验研究.[天津大学博士学位论文].2000:6-12
66曾强.空间凸轮活齿传动力学特性研究.[重庆大学硕士学位论文].2003:5-9
67 J. W. David, et al. Using Transfer Matrices for Parametric System Forced Response. ASME Vib. Acous. Stress Reli. Des., 1987, 109(2):356-360
68 L. D. Mitchell. A New Branching Technique for the Static and Dynamic Analysis of Gears System. Second Inter. Conf. Vib. in Rotating Machinery, Cambridge, England, 1980:437-441
69安子军,曲志刚.摆线钢球传动啮合副间隙及其调整机构.机械设计,1996,12:33-35
70 Qu Jifang, An Zijun. Research on Zero Clearance Cycloid Ball Transmission. Chinese Journal of Mechanical Engineering, 1994, 7(1):17-23
71 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (1st Report). JSPE, 1988, 54(11):2101-2106
72 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (2nd Report). JSPE, 1990, 56(4):751-756
73 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (3rd Report). JSPE, 1995, 61(12):1075-1079
74 Hidetsugu Terada, Hiroshi Makino, Kenji Imase. Fundamental Analysis of Cycloid Ball Reducer (4th Report). JSPE, 1997, 63(6):834-838
75 Roy S. McFarland. Employs Trochoidal Motion and Bearing Balls. Mechanical Engineering, 1982:1-60
76安子军,曲志刚,张荣贤.摆线钢球传动齿形综合研究.机械工程学报, 1996,32(5):41-46
77 Yang Zuomei, An Zijun, Zhang Peng. Manufacturing Mathematic Models and Meshing Analysis for the Cycloid Grooves of the Cycloid Ball Planetary Drive. Proceedings of International Conference on Measuring Technology and Mechatronics Automation, Zhangjiajie, China, 2009, 4:89-92
78杨作梅,安子军,张鹏.基于空间啮合理论的摆线钢球行星传动根切研究.农业机械学报,2009,40(10):216-222
79安子军,曲继方.摆线钢球传动齿形及其压力角分析.机械设计,1997,1:24-26
80 An Zijun, Yi Yali. Force Analysis and Stress Calculation of Non-clearance Cycloid Ball Transmissions. Proceedings of IEEE International Conference on Mechatronics and Automation, Japan, 2008, 8:1089-1093
81安子军.端面啮合摆线钢球行星传动啮合副力及强度分析.机械传动,2003,27(4):29-31
82安子军,胡瑞雪.摆线钢球行星传动齿廓误差的参数分析.机械传动,2007,31(2):63-65
83安子军,曲志刚,王广欣.基于模糊理论的摆线钢球行星传动接触疲劳强度可靠性研究.中国机械工程,2002,13(23):2010-2012
84安子军,王广欣.摆线钢球行星传动的模糊可靠性优化设计.机械传动,2004,28(5):17-19
85宜亚丽,安子军.摆线钢球传动中等速输出机构的研究设计.机械电子,2004,11:22-23
86宋敬稳,安子军.浮动式无回差钢球等速输出机构动平衡研究与创新设计.机械传动,2008,32(5):6-9
87安子军,宜亚丽.精密钢球等速输出机构的力学性能研究.机械传动,2000,24(4):1-3
88宋敬稳,安子军,胡瑞雪.精密钢球环槽等速传动机构运动精度及可靠性分析.燕山大学学报,2009,33(2):95-98
89 Zhou Jianjun, Chen Zichen. Study and Performance Test of Full Complement Cycloidal Ball Reducer with Ceramic Balls as Its Gear Teeth. Chinese Journal of Aeronautics, 2001, 14(4): 245-251
90 Zhou J. J., Hu J. X., Huang M., Geometric Design of Cycloidal Drives with Ceramic Ball Meshing Elements. Key Engineering Materials, 2007, 336(3):2484-2486
91李思益,王博,张彩丽.摆线钢球行星传动减速器振动特性分析.现代制造工程,2004,3:70-71
92刘锡峰,李思益.双摆线钢球行星传动减速器振动分析.西安工业学院学报,2003,23(3):218-222
93李思益,王海燕,张彩丽.两齿差摆线钢球行星传动减速器的试验研究.现代制造工程,2002,10:81-82
94姚素芬,高东强,李进.基于UG的双摆线钢球减速器的三维建模技术.陕西科技大学学报,2008,26(6):138-141
95高东强,姚素芬,李进.基于PRO/E的双摆线钢球减速器的参数化设计技术的研究.机械设计与制造,2009,3:222-224
96吴勤保,冉朝,辛晓峰.双摆线钢球减速器摆线槽的设计及数控加工.机械设计,2007,24(10): 68-70
97西庆坤,朱俊平,赵曙光,等.摆线钢球行星减速器误差的蒙特卡洛分析,西北农林科技大学学报(自然科学版),2008,36(7):224-228
98万长森.滚动轴承的分析方法.北京:机械工业出版社,1987:259-263
99李有法.数值计算方法.北京:高等教育出版社,1996:68-75
100徐芝纶.弹性力学(上册).北京:人民教育出版社,1982:299-302
101 K. L. Johnson.徐秉业,罗学富,刘信声,等译.接触力学.北京:高等教育出版社,1992:51-77
102李润方,陈晓南.弹塑性接触有限元混合法及其在齿轮传动中的应用.工程力学,1989, 6(2):87-97
103李润方,龚剑霞.接触问题数值方法及其在机械设计中的应用.重庆:重庆大学出版社,1989:1-20
104周宁.ANSYS机械工程应用实例.北京:中国水利水电出版社,2006:94-117
105刘文芝,张乃仁,张春林,赵永忠.谐波齿轮传动中杯形柔轮的有限元计算与分析.机械工程学报,2006,42(4):52-57
106付军锋,董海军,沈允文.谐波齿轮传动中柔轮应力的有限元分析.中国机械工程,2007,18(18):2210-2214
107安子军.摆线钢球行星传动研究.[燕山大学博士学位论文].2000:64-80
108安子军,曲继方.双摆线无隙钢球传动间隙调节力矩分析.机械传动,1995,19(4):1-3
109哈尔滨工业大学理论力学教研组.理论力学(下册).北京:高等教育出版社,1997:19-27
110扈英超.线性振动.北京:高等教育出版社,1981:8-13
111张荣贤,曲继方,安子军.摆线钢球传动的传动比计算.太原重型机械学院学报,1995,16(1):14-19
112刘延柱,陈立群.非线性振动.北京:高等教育出版,2001:57-95
113徐韫知.综合数学手册.北京:商务印书馆,1958:125-126
114郝秀红.机电集成超环面传动机电耦合动力学研究.[燕山大学硕士学位论文].2006:20-22
115宜亚丽.端面啮合摆线钢球行星传动的动力性能研究.[燕山大学硕士学位论文].2001:67-76
116过永德.机械振动基础.北京:兵器工业出版社,1992:112-114
117王广欣.摆线钢球行星传动模糊可靠性优化与建模仿真研究.[燕山大学硕士学位论文].2003:59-69
118王广欣,朱莉莉,关天民.基于Pro/E的摆线钢球行星传动建模仿真研究.机械传动,2005, 9(3):27-29
119孙江宏,黄小龙. Pro/ENGINEER Wildfire(野火版)数控加工教程.北京:清华大学出版社,2005:2-8
120吴勤保.双摆线钢球减速器齿廓曲线参数的选择.机械设计,2007,24(7):64-66