齿轮传动动态弹流润滑的数值计算模型研究
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摘要
渐开线齿轮在啮合传动过程中，其轮齿刚度、接触载荷、齿面速度和曲率半径等沿啮合线在始终变化着；同时，由于齿轮制造、安装误差等原因不可避免地会产生动载荷。所以说，齿轮啮合属于典型的动态过程。而现行的齿轮接触疲劳强度设计仅仅适用于静态干接触Hertz理论。因此，进行齿轮动态润滑研究对完善齿轮强度设计理论具有一定的理论意义。
本文的研究是建立在齿轮传动系统动力学及弹性流体动力润滑理论基础之上。首先采用系统分析法分析了轮齿接触动载荷沿齿轮传动啮合线的分布；接着提出了与动载荷相适应的弹性流体动力润滑理论；最后形成了齿轮传动动态弹流润滑的数值计算模型。该计算模型的建立，无疑为后续研究奠定了坚实的基础。分析、总结本文的主要工作，可以归
太原理工大学硕士研究生学位论文
纳如下几点:
1.采用系统分析法建立了齿轮传动系统的扭转振动、
横向振动及其祸合振动的等效动力学模型和相应的动力学
方程;计算了轮齿静刚度、动态啮合刚度;分析了轮齿的齿
形误差、基节误差对啮合刚度的影响;给出了轮齿刚度及齿
轮动载沿传动啮合线的分布。尤其是运用Matlab中的
Sim川ink求解齿轮传动系统动力学方程是本文的特色之一。
2.采用当量齿形法详细地分析了轮齿的总变形，其中
包括当量齿形中矩形和梯形两部分的弯曲、剪切变形以及由
轮齿基体弹性倾斜所引起的轮齿变形和轮齿接触变形等，并
给出了相应的变形曲线图。
3.将齿轮动载计算结果与润滑理论进行祸合从而形
成了齿轮动态弹流润滑计算公式。这些公式中不仅考虑了润
滑剂的热效应和非牛顿效应，还考虑了载荷、速度、曲率半
径等随时间和空间的双重变化。应用这些公式，可以算出轮
齿在啮合线上任意位置啮合时的齿面压力分布、油膜厚度、
温度分布及摩擦系数等。
4.提出了齿轮传动动态弹流润滑数值计算方法。由于
多重网格法数值稳定性好、收敛速度快，近年引起了各国弹
2
太原理工大学硕士研究生学位论文
流工作者的青睐111，鉴此，本文提出了采用多重网格法进行
计算的求解思路和数值方法。但由于时间所限，未能获得数
值计算结果，这也正是本文的缺憾之处。
During the transmission of the involute gear in mesh, the gear engagement stiffness, the load, the sliding velocity and the curvature are varying with the meshing line of the gear teeth. At the same time , because there are some manufacturing and install error and the like ,so the dynamic load inevitably produced .So we can say the transmission of a gear is a representative dynamic process, but the contact strength design which are used now are only applicable to the steady condition of the contact Hertz theory. Therefore, preceding the research of the elastohydro-dynamic lubrication of wheel gear for a perfect strength designing theories has the certain theories meaning. This study is based on a comprehensive method which
applies dynamics and tribology to the analyses of the gear transmission system. The rotation and traverse vibration of six parts in the gear transmission system are studied in this paper. And then the author bring forward the elstohydro-dynamic lubrication theory match with the dynamic load of the gear, and at the last, developing a calculating model for the elastohydro-dynamic lubrication of the transmission of the gear in mesh. The establishment of the model is doubtless a stability base for the continuing research. Analysis, summarizing the textual main work, can induce as follows:
1. The equivalent kinetics model and the dynamic equations of the rotation and traverse vibration of six parts in the gear transmission system are established in this paper. Calculates the static and dynamic gear engagement stiffness, analysis the influence of the tooth profile and pitch errors of gear to the gear engagement stiffness; and give out the distribution of the gear engagement stiffness and the dynamic loads of the gear along the meshing line of the gear teeth;
2. Especially using the Simulink tool in the Matlab
deduces the systemic kinetic equations are features of the thesis. In the paper , adoption equivalent gear teeth form method analyzed the round flection of the gear teeth, in it conclude the bending and cutting distortion of rectangle and trapezoid two parts , the gear teeth and contact distortion come frome the slants of the gear teeth body, and so on .moreover give out the correspond distortion graph.
3. Coupling the calculation results of the gear dynamic loads and the elastohydro-dynamic lubrication can form the gear wheel unsteady elastohydro-dynamic lubrication calculating formula. In this formula not only consider the heat effect of the lubricant but also the non-Newtonian effect, and also the changing of the loads, velocity and curvature with the time and the space. Applying this formula can calculate the stress distribution of gear teeth in meshing, the thickness of the film, distribution of the temperature and the coefficient of the friction, etc.
4. The author put forward a calculating method for unsteady elastohydro-dynamic lubrication in gear transmission
system. Because the multi-grid method have a good numerical value stability and have a quick convergence, so these years many researchers interests on it, and so in this paper the author put forward the idea how to using the method for the calculating and the numerical value method. Because the restrict of time, I cannot get the numerical value results, this is just a pity of this paper. Carries out the numerical simulations of tooth profile and pitch errors of gear. This paper demonstrates that gear engagement stiffness has a primary influence; tooth errors mentioned above have a secondary influence on the vibration of the gear transmission system.
本文的研究是建立在齿轮传动系统动力学及弹性流体动力润滑理论基础之上。首先采用系统分析法分析了轮齿接触动载荷沿齿轮传动啮合线的分布；接着提出了与动载荷相适应的弹性流体动力润滑理论；最后形成了齿轮传动动态弹流润滑的数值计算模型。该计算模型的建立，无疑为后续研究奠定了坚实的基础。分析、总结本文的主要工作，可以归
太原理工大学硕士研究生学位论文
纳如下几点:
1.采用系统分析法建立了齿轮传动系统的扭转振动、
横向振动及其祸合振动的等效动力学模型和相应的动力学
方程;计算了轮齿静刚度、动态啮合刚度;分析了轮齿的齿
形误差、基节误差对啮合刚度的影响;给出了轮齿刚度及齿
轮动载沿传动啮合线的分布。尤其是运用Matlab中的
Sim川ink求解齿轮传动系统动力学方程是本文的特色之一。
2.采用当量齿形法详细地分析了轮齿的总变形，其中
包括当量齿形中矩形和梯形两部分的弯曲、剪切变形以及由
轮齿基体弹性倾斜所引起的轮齿变形和轮齿接触变形等，并
给出了相应的变形曲线图。
3.将齿轮动载计算结果与润滑理论进行祸合从而形
成了齿轮动态弹流润滑计算公式。这些公式中不仅考虑了润
滑剂的热效应和非牛顿效应，还考虑了载荷、速度、曲率半
径等随时间和空间的双重变化。应用这些公式，可以算出轮
齿在啮合线上任意位置啮合时的齿面压力分布、油膜厚度、
温度分布及摩擦系数等。
4.提出了齿轮传动动态弹流润滑数值计算方法。由于
多重网格法数值稳定性好、收敛速度快，近年引起了各国弹
2
太原理工大学硕士研究生学位论文
流工作者的青睐111，鉴此，本文提出了采用多重网格法进行
计算的求解思路和数值方法。但由于时间所限，未能获得数
值计算结果，这也正是本文的缺憾之处。
During the transmission of the involute gear in mesh, the gear engagement stiffness, the load, the sliding velocity and the curvature are varying with the meshing line of the gear teeth. At the same time , because there are some manufacturing and install error and the like ,so the dynamic load inevitably produced .So we can say the transmission of a gear is a representative dynamic process, but the contact strength design which are used now are only applicable to the steady condition of the contact Hertz theory. Therefore, preceding the research of the elastohydro-dynamic lubrication of wheel gear for a perfect strength designing theories has the certain theories meaning. This study is based on a comprehensive method which
applies dynamics and tribology to the analyses of the gear transmission system. The rotation and traverse vibration of six parts in the gear transmission system are studied in this paper. And then the author bring forward the elstohydro-dynamic lubrication theory match with the dynamic load of the gear, and at the last, developing a calculating model for the elastohydro-dynamic lubrication of the transmission of the gear in mesh. The establishment of the model is doubtless a stability base for the continuing research. Analysis, summarizing the textual main work, can induce as follows:
1. The equivalent kinetics model and the dynamic equations of the rotation and traverse vibration of six parts in the gear transmission system are established in this paper. Calculates the static and dynamic gear engagement stiffness, analysis the influence of the tooth profile and pitch errors of gear to the gear engagement stiffness; and give out the distribution of the gear engagement stiffness and the dynamic loads of the gear along the meshing line of the gear teeth;
2. Especially using the Simulink tool in the Matlab
deduces the systemic kinetic equations are features of the thesis. In the paper , adoption equivalent gear teeth form method analyzed the round flection of the gear teeth, in it conclude the bending and cutting distortion of rectangle and trapezoid two parts , the gear teeth and contact distortion come frome the slants of the gear teeth body, and so on .moreover give out the correspond distortion graph.
3. Coupling the calculation results of the gear dynamic loads and the elastohydro-dynamic lubrication can form the gear wheel unsteady elastohydro-dynamic lubrication calculating formula. In this formula not only consider the heat effect of the lubricant but also the non-Newtonian effect, and also the changing of the loads, velocity and curvature with the time and the space. Applying this formula can calculate the stress distribution of gear teeth in meshing, the thickness of the film, distribution of the temperature and the coefficient of the friction, etc.
4. The author put forward a calculating method for unsteady elastohydro-dynamic lubrication in gear transmission
system. Because the multi-grid method have a good numerical value stability and have a quick convergence, so these years many researchers interests on it, and so in this paper the author put forward the idea how to using the method for the calculating and the numerical value method. Because the restrict of time, I cannot get the numerical value results, this is just a pity of this paper. Carries out the numerical simulations of tooth profile and pitch errors of gear. This paper demonstrates that gear engagement stiffness has a primary influence; tooth errors mentioned above have a secondary influence on the vibration of the gear transmission system.
引文
[1] 王静，杨沛然．时变等温线接触弹流润滑问题求解的多重网格技术．润滑与密封，2000(3)，9～11．
[2] 濮良贵，纪名刚．机械设计[M]．高等教育出版社．1996
[3] Martin H M. Lubrication of Gear Teeth. Engineering. London, 1916, V. 102.199.
[4] 温诗铸．摩擦学原理[M]．北京：清华大学出版社．1990
[5] Mc Eween E. The Effect of Variation of Viscosity with Pressure on the Load Carrying Capacity of Oil Film between Gear Teeth. Journal Institute of Petroleum. 1952, v.38, 655～665
[6] Adkins R W, Radzimovsky E I. Lubrication Phenomena in Spur Gears, Transaction ASME Journal Basic Eng. 1965, v.87,655～656.
[7] Dowson, Higginson, Theory of Involute Gear Lubrication, Inst. Pet. Gear Lubrication Symposium
[8] Gu A. Elatohydrodynamic Lubrication of Involute Gears. Transactions of the ASME Journal of Engineering for Industry. 1973, NOVEMBER, 1164～1170
[9] Vichard J P. Transient Effects in the Lubrication of Hertzian Contacts. Journal of Mechanical Engineering Science. 1971, Vol. 13, 173～180
[10] Wang K L, Cheng H S. A Numerical Solution to the Dynamic Load, Film
Thickness and Surface Temperature in Spur Gears. Transactions of Journal of Mechanical. Design, 1981, V.103,177～195
[11] Petrousevitch A I. The Investigation of Oil Film Thickness in Lubricated Ball-Race Rolling Contact. Wear, 1972, V. 19, 369～389
[12] Wada S, Tsukijihara M. Elastohydrodynamic Squeeze Problem of Two Rotating Cylinders. Bulletin JSME. 1981, V. 24, No. 190, 192, 937～942, 1072～1077
[13] 严升明，“非稳态弹流润滑理论及其应用的研究”，中国矿业学研硕士学位论文，1984
[14] Oh, K.P., "The Numerical Solution of Dynamically Loaded Elastohydrodynamic Contact as a Non-Linear Complementarity Problem, "ASME Journal of Tribology, Vol. 106, 1984, pp.88-95
[15] 任宁、温诗铸，“非稳态线接触弹流润滑的直接迭代法”，第四届全国摩擦学学术会议论文集，清华大学出版社，1987(5)，27～32．
[16] 艾晓岚、俞海清，“线接触非稳态弹流润滑完全数值解及其实际应运”，第四届全国摩擦学学术会议论文集，清华大学出版社，1987(4)，21～26．
[17] 杨沛然．非稳态热弹流体动力润滑研究．清华大学博士论文．1989．
[18] 华东耕，系统弹流动力学—系统动力学瞬态弹性流体动力润滑，上海工业大学应用数学和力学研究所博士学位论文，1990．3．
[19] Venner, C.H., ten Napel, W.E., Bosma, R. Advanced Multilevel Solution of the EHL Line Contact Problem[J].Transactions of the ASME, 1990,112(7): 426-431.
[20]张和豪，施孟浩，华东耘．上海工业大学报，1989，10(1)，16．
[21]卢立新，蔡莹，张和豪．齿轮几何参数对齿轮传动弹流润滑性能的影响．机械传动，1998，22(1)：24-27
[22]卢立新，蔡莹，张和豪．渐开线直齿圆柱齿轮传动瞬态弹流润滑研究．润滑与密封，1999，5，5-7，10．
[23]章易程．直齿轮啮合弹性流体动力润滑的非稳态效应的研究．机械工程学报，2000，36(1)：32～35．
[24]梅雪松，谢友柏．一种快速求解非稳态弹流润滑问题的直接迭代计算法．应用力学报，1991，8(3)：19～28．
[25]卢立新，张和豪．齿轮传动系统动载荷非稳态弹流润滑研究[J]．润滑与密封，2002(1)：381-384
[26]Roland L., Transient non-Newtonian elastohydrodynamic lubrication analysis of an involute spur gear, Wear, 1997,207: 67～73.
[27]R W Snidle, H P Evans, M P Alanou. Gears Elastohydrodynamie Lubrication and Durability. Proc Instn Mech Engrs, Vol.214 Part C, 2000, 39～49
[28]李廉枫．基于非牛顿流体的齿轮热弹流润滑，太原理工大学硕士学位文，2003，5
[29]章易程．齿轮传动系统的动力学与弹性流体动力润滑分析，西安交通大学硕士学位论文．1995
[30]朱孝录，鄂中凯，《齿轮承载能力分析》．高等教育出版社，1992．8
[31]Lin, H-H. Huston. R. L.: On Dynamic Loads in Parallel Shaft Automations,Journal of Mechanisms, Transmissions, and Automation in Design,
Vol. 110, 221-229,1988.
[32] D.R.Houser, 1988, Gear noise state of the art, Proc. of Inter-Noise 88, 601-606.
[33] Hahn, W. F. Study on Instantaneous Load to Which Gear Teeth are Subjected, Ph. D. Dissertation, University of Illinois,98-135,1969.
[34] 刘鸿文，《材料力学》，高等教育出版社，上册，40～187，下册，1～68，1983
[35] R. Kasuba and J. W. Evans, 1981, An extended model for determining dynamic loads in spur gearing, ASME Mech. Des., 103(2), 398～409.
[36] K. L. Wang and H. S. Cheng, 1981, A numerical solution to the dynamic load,film thickness and surface temperature in spur gear, ASME Mech. Des.,103(1), 177-187.
[37] 仙波正庄，张范孚译，《齿轮的误差与强度》，机械工业出版社，176-207，1982．
[38] Michalec，GW，楠波，成电译，《精密齿轮传动装置—理论与实践》，国防工业出版社，1～178，1978
[39] 苏金明．MATLAB6．1实用指南[M]．电子工业出版社．2002
[40] Yang Peiran, Wen Shizhu, "A generalized Reynolds equation for non-Newtonian thermal elastohydro-dynamic lubrication", Journal of Tribology, Trans. ASME, V. 112,N.4, 1990
[41] Yang Peiran, Wen Shizhu, "A generalized Reynolds equation based on non-Newtonian flow in lubrication mechanics", Acta Mechanica
Sinica, V.6,N.4,1990
[42]温诗铸，杨沛然．弹性流体动力润滑[M]．北京：清华大学出版社，1992
[43]杨沛然．流体润滑数值分析[M]．北京：国防工业出版社．1998
[44]李庆扬．数值分析[M]．北京：清华大学出版社，施普林格出版社．2001
[45]黄平，温诗铸．多重网格法求解线接触弹流问题．清华大学学报(自然科学版)，1992，32(32)：26～34
[2] 濮良贵，纪名刚．机械设计[M]．高等教育出版社．1996
[3] Martin H M. Lubrication of Gear Teeth. Engineering. London, 1916, V. 102.199.
[4] 温诗铸．摩擦学原理[M]．北京：清华大学出版社．1990
[5] Mc Eween E. The Effect of Variation of Viscosity with Pressure on the Load Carrying Capacity of Oil Film between Gear Teeth. Journal Institute of Petroleum. 1952, v.38, 655～665
[6] Adkins R W, Radzimovsky E I. Lubrication Phenomena in Spur Gears, Transaction ASME Journal Basic Eng. 1965, v.87,655～656.
[7] Dowson, Higginson, Theory of Involute Gear Lubrication, Inst. Pet. Gear Lubrication Symposium
[8] Gu A. Elatohydrodynamic Lubrication of Involute Gears. Transactions of the ASME Journal of Engineering for Industry. 1973, NOVEMBER, 1164～1170
[9] Vichard J P. Transient Effects in the Lubrication of Hertzian Contacts. Journal of Mechanical Engineering Science. 1971, Vol. 13, 173～180
[10] Wang K L, Cheng H S. A Numerical Solution to the Dynamic Load, Film
Thickness and Surface Temperature in Spur Gears. Transactions of Journal of Mechanical. Design, 1981, V.103,177～195
[11] Petrousevitch A I. The Investigation of Oil Film Thickness in Lubricated Ball-Race Rolling Contact. Wear, 1972, V. 19, 369～389
[12] Wada S, Tsukijihara M. Elastohydrodynamic Squeeze Problem of Two Rotating Cylinders. Bulletin JSME. 1981, V. 24, No. 190, 192, 937～942, 1072～1077
[13] 严升明，“非稳态弹流润滑理论及其应用的研究”，中国矿业学研硕士学位论文，1984
[14] Oh, K.P., "The Numerical Solution of Dynamically Loaded Elastohydrodynamic Contact as a Non-Linear Complementarity Problem, "ASME Journal of Tribology, Vol. 106, 1984, pp.88-95
[15] 任宁、温诗铸，“非稳态线接触弹流润滑的直接迭代法”，第四届全国摩擦学学术会议论文集，清华大学出版社，1987(5)，27～32．
[16] 艾晓岚、俞海清，“线接触非稳态弹流润滑完全数值解及其实际应运”，第四届全国摩擦学学术会议论文集，清华大学出版社，1987(4)，21～26．
[17] 杨沛然．非稳态热弹流体动力润滑研究．清华大学博士论文．1989．
[18] 华东耕，系统弹流动力学—系统动力学瞬态弹性流体动力润滑，上海工业大学应用数学和力学研究所博士学位论文，1990．3．
[19] Venner, C.H., ten Napel, W.E., Bosma, R. Advanced Multilevel Solution of the EHL Line Contact Problem[J].Transactions of the ASME, 1990,112(7): 426-431.
[20]张和豪，施孟浩，华东耘．上海工业大学报，1989，10(1)，16．
[21]卢立新，蔡莹，张和豪．齿轮几何参数对齿轮传动弹流润滑性能的影响．机械传动，1998，22(1)：24-27
[22]卢立新，蔡莹，张和豪．渐开线直齿圆柱齿轮传动瞬态弹流润滑研究．润滑与密封，1999，5，5-7，10．
[23]章易程．直齿轮啮合弹性流体动力润滑的非稳态效应的研究．机械工程学报，2000，36(1)：32～35．
[24]梅雪松，谢友柏．一种快速求解非稳态弹流润滑问题的直接迭代计算法．应用力学报，1991，8(3)：19～28．
[25]卢立新，张和豪．齿轮传动系统动载荷非稳态弹流润滑研究[J]．润滑与密封，2002(1)：381-384
[26]Roland L., Transient non-Newtonian elastohydrodynamic lubrication analysis of an involute spur gear, Wear, 1997,207: 67～73.
[27]R W Snidle, H P Evans, M P Alanou. Gears Elastohydrodynamie Lubrication and Durability. Proc Instn Mech Engrs, Vol.214 Part C, 2000, 39～49
[28]李廉枫．基于非牛顿流体的齿轮热弹流润滑，太原理工大学硕士学位文，2003，5
[29]章易程．齿轮传动系统的动力学与弹性流体动力润滑分析，西安交通大学硕士学位论文．1995
[30]朱孝录，鄂中凯，《齿轮承载能力分析》．高等教育出版社，1992．8
[31]Lin, H-H. Huston. R. L.: On Dynamic Loads in Parallel Shaft Automations,Journal of Mechanisms, Transmissions, and Automation in Design,
Vol. 110, 221-229,1988.
[32] D.R.Houser, 1988, Gear noise state of the art, Proc. of Inter-Noise 88, 601-606.
[33] Hahn, W. F. Study on Instantaneous Load to Which Gear Teeth are Subjected, Ph. D. Dissertation, University of Illinois,98-135,1969.
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