计及摩擦的摆线轮齿与针齿的分形接触模型
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摘要
为综合体现摆线轮齿与针齿的宏观特征和微观特征对接触特性的影响,应用Weierstrass-Mandelbrot函数和矢量函数构建了摆线轮齿与针齿的表面形貌模型,应用MATLAB绘制了各向同性的粗糙针齿以及单个摆线轮齿的二维截面图,提出了摆线轮齿与针齿的接触比例系数,该接触比例系数始终小于1,且随啮合点的变化而变化,针齿与摆线轮齿的内凹部分接触时的接触比例系数远大于针齿与摆线轮齿外凸部分接触时的值。计及摩擦因素的影响,构建了单对摆线轮齿与针齿的分形接触模型,分析了摩擦因子、结合面的微观特征和宏观特征对接触特性的影响。研究结果表明,相同载荷下,接触面积随摩擦因子的增大而增大,随结合面粗糙度的增大先增大后减小,随针齿半径的增大而减小,随中心距的增大而减小,随针轮中心圆半径的增大而增大,随针轮齿数的增加而减小。
In order to comprehensively reflect the influences of macroscopic and microscopic characteristics of cycloid teeth and pin teeth, the surface topography model of cycloid teeth and pin teeth was established by Weierstrass-Mandelbrot function and vector function. The section plots of pin tooth and cycloid tooth with obvious fractal characteristics were described using MATLAB. In order to confirm the contact area between rough surfaces, the parameter called contact proportionality coefficient was proposed and the relations among the coefficient and fractal dimension, characteristic scale, center distance, radius of pin tooth, number of pin wheel were analyzed. The analyses indicate that the coefficient is less than 1 in any case and the values of inner contacts are larger than that of outer contacts. The fractal contact model with friction was set up. The model shows that the relationship between the contact load and area is affected by gross characteristics, and by the microscopic characteristics of the contact surfaces. It may be indicated by the analyses of the contact characteristics of the model, that the contact area in the same load increases with the increases of friction factors, increases at first and then decrease with the creases of surface roughness, decreases with the decreases of radius of pin teeth, decreases with the decreases of center distance, increases with the decreases of radius of pin wheel and decreases with the decreases of number of pin teeth.
In order to comprehensively reflect the influences of macroscopic and microscopic characteristics of cycloid teeth and pin teeth, the surface topography model of cycloid teeth and pin teeth was established by Weierstrass-Mandelbrot function and vector function. The section plots of pin tooth and cycloid tooth with obvious fractal characteristics were described using MATLAB. In order to confirm the contact area between rough surfaces, the parameter called contact proportionality coefficient was proposed and the relations among the coefficient and fractal dimension, characteristic scale, center distance, radius of pin tooth, number of pin wheel were analyzed. The analyses indicate that the coefficient is less than 1 in any case and the values of inner contacts are larger than that of outer contacts. The fractal contact model with friction was set up. The model shows that the relationship between the contact load and area is affected by gross characteristics, and by the microscopic characteristics of the contact surfaces. It may be indicated by the analyses of the contact characteristics of the model, that the contact area in the same load increases with the increases of friction factors, increases at first and then decrease with the creases of surface roughness, decreases with the decreases of radius of pin teeth, decreases with the decreases of center distance, increases with the decreases of radius of pin wheel and decreases with the decreases of number of pin teeth.
引文
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[12] WEI Bo,WANG Jiaxu,ZHOU Guangwu.Mixed Lubrication Analysis of Modified Cycloidal Gear Used in the RV Reducer[J].Journal of Engineer Tribolody,2016,230(2):121-134.
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[14] BROWN C A,SAVARY G.Describing Ground Surface Texture Using Contact Profilometry and Fractal Analysis[J].Wear,1991,141(2):211-226.
[15] MAJUMDAR A,BHUSHAN B.Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surfaces[J].Journal of Tribology,1990,112:205-216.
[16] MAJUMDAR A,BHUSHAN B.Fractal Model of Elastic-Plastic Contact between Rough Surfaces[J].Journal of Tribology,1991,113:1-11.
[17] MORAG Y,ETSION I.Resolving the Contradiction of Asperities Plastic to Elastic Mode Transition in Current Contact Models of Fractal Rough Surfaces[J].Wear,2007,262(5):624-629.
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[19] 陈奇,黄康,张彦,等.基于分形接触模型的齿轮接触强度影响参数分析[J].中国机械工程,2013,16(16):2208-2211.CHEN Qi,HUANG Kang,ZHANG Yan,et al.Analysis of Influence Parameters on Gear’s Contact Strength Based on Fractal Contact Model[J].China Mechanical Engineering,2013,16(16):2208-2211.
[20] CHEN Qi,FAN Xu,PENG Liu,et al.Research on Fractal Model of Normal Contact Stiffness between Two Spheroidal Joint Surfaces Considering Friction Factor[J].Tribology International,2016,97:256-264.
[21] HUANG Kang,XIONG Yangshou,WANG Tao.Research on the Dynamic Response of High-contact-ratio Spur Gears Influenced by Surface Roughness under EHL Condition[J].Applied Surface Science,2017,392:8-18.
[22] BERRY M V,LEWIS Z V.On the Weierstrass-Mandelbrot Fractal Function[J].Proceedings of the Royal Society,1980,370:459-484.
[23] CHEN Q,ZHAO Y,MA Y B,et al.Testing Experiment of Characteristic-scale Coefficient about Cylinder Surface[J].Journal of Mechanical Strength,2014,36(5):687-697.
[24] 葛世荣,朱华.摩擦学的分形[M].北京:机械工业出版社,2005:122.GE Shirong,ZHU Hua,Fractal in Tribology[M].Beijing:China Machine Press,2005:122.
[25] 田红亮,陈甜敏,郑金华,等.平行轴圆柱副接触分析[J].西安交通大学学报,2016,50(1):1-8.TIAN Hongliang,CHEN Tianmin,ZHENG Jinhua,et al.Contact Analysis of Cylindrical Pair with Parallel Axes[J].Journal of Xi’an Jiao Tong University,2016,50(1):1-8.
[26] 韩炬,李威,王志军.采用分形理论与微分几何的摆线轮形貌模型[J].西安交通大学学报,2017,51(11):97-105.HAN Ju,LI Wei,WANG Zhijun.Surface Morphology Models of Cycloid and Pin Teeth Based on Fractal Theory and Differential Geometry[J].Journal of Xi’an Jiao Tong University,2017,51(11):97-105.
[27] 李小彭,郭浩,刘井年,等.考虑摩擦的结合面法向刚度分形模型及仿真[J].振动、测试与诊断,2013,33(2):210-213.LI Xiaopeng,GUO Hao,LIU Jingnian,et al.Fractal Model and Simulation of Normal Contact Stiffness Considering the Friction between Joint Surfaces[J].Journal of Vibration,Measurement & Diagnosis,2013,33(2):210-213.
[28] LIU Peng,ZHAO Han,HUANG Kang,et al.Research on Normal Contact Stiffness of Rough Surface Considering Friction Based on Fractal Theory[J].Applied Surface Science,2015,349:43-48.
[29] DORIANA H H,GAN Yixiang,EINAN I.Static Friction at Fractal Interfaces[J].Tribology International,2016,93:229-238.
[30] KOGUT L,ETSION I.Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat[J].ASME Journal of Applied Mechanics,2002,69:657-662.
[31] 成雨,原园,甘立,等.尺度相关的分形粗糙表面弹塑性接触力学模型[J].西北工业大学学报,2016,34(3):485-492.CHENG Yu,YUAN Yuan,GAN Li,et al.The Elastic-Plastic Contact Mechanics Model Related Scale of Rough Surface[J].Journal of Northwestern Polytechnical University,2016,34(3):485-492.
[32] 李小彭,鞠行,赵光辉,等.考虑摩擦因素的结合面切向接触刚度分形预估模型及其仿真分析[J].摩擦学学报,2013,33(5):463-468.LI Xiaopeng,JU Xing,ZHAO Guanghui,et al.Fractal Prediction Model for Tangential Contact Stiffness of Joint Surface Considering Friction Factors and Its Simulation Analysis[J].Tribology,2013,33(5):463-468.
[33] 陈奇,黄守武,张振,等.考虑摩擦因素的两圆柱体表面接触承载能力的分形模型研究[J].机械工程学报,2016,52(7):114-121.CHEN Qi,HUANG Shouwu,ZHANG Zhen,et al.Research on Fractal Contact Model for Contact Carrying Capacity of Two Cylinders’ Surfaces Considering Friction Factors[J].Journal of Mechanical Engineering,2016,52(7):114-121.
[34] 丁雪兴,严如奇,贾永磊.基于基底长度的粗糙表面分形接触模型的构建与分析[J].摩擦学学报,2014,34(4):341-347.DING Xuexing,YAN Ruqi,JIA Yonglei.Construction and Analysis of Fractal Contact Mechanics Model for Rough Surface Based on Base Length[J].Tribology,2014,34(4):341-347.
[2] 王嘉宁,顾京君,言勇华.摆线轮基本齿形参数与RV 减速器啮合刚度的关系[J].机械设计与研究,2017,33(4):63-67.WANG Jianing,GU Jingjun,YAN Yonghua.Research on the Relationship between the Basic Tooth Profile Parameters of Cycloid Wheel and the Meshing Stiffness of RV Reducer[J].Machine Design and Research,2017,33(4):63-67.
[3] WANG Jianing,GU Jingjun,YAN Yonghua.Study on the Relationship between the Stiffness of RV Reducer and the Profile Modification Method of Cycloid-pin Wheel[C]//Intelligent Robotics and Applications-ICIRA 2016.Berlin:Springer Verlag,2016:721-735.
[4] YU Hongliu,YI Jinhua,HU Xin,et al.Study on Teeth Profile Modification of Cycloid Reducer Based on Non-Hertz Elastic[J].Mechanics Research Communications,2013,48:87-92.
[5] HSIEH C F.The Effect on Dynamics of Using a New Transmission Design for Eccentric Speed Reducers[J].Mechanism and Machine Theory,2014,80:1-16.
[6] LI Shuting.Design and Strength Analysis Methods of the Trochoidal Gear Reducers[J].Mechanism and Machine Theory,2014,81:140-154.
[7] HSIEH C F.Traditional Versus Improved Designs for Cycloidal Speed Reducers with a Small Tooth Difference:the Effect on Dynamics[J].Mechanism and Machine Theory,2015,86:15-35.
[8] DO T P,ZIEGLER P,EBERHARD P.Review on Contact Simulation of Beveloid and Cycloid Gears and Application of a Modern Approach to Treat Deformations[J].Mathematical and Computer Modelling of Dynamical Systems,2015,21(4):359-388.
[9] 许立新,杨玉虎.一种摆线针轮传动多齿啮合接触特性分析方法[J].中国机械工程,2016,27(10):1382-1388.XU Lixin,YANG Yuhu.A General Method for Multi-tooth Contact Analysis of Cycloid-pin Gear Transmission[J].China Mechanical Engineering,2016,27(10):1382-1388.
[10] XU Lixin,YANG Yuhu.Dynamic Modeling and Contact Analysis of a Cycloid-pin Gear Mechanism with a Turning Arm Cylindrical Roller Bearing [J].Mechanism and Machine Theory,2016,104:327-349.
[11] TSAI S J,CHANG L C,HUANG C H.Design of Cycloid Planetary Gear Drives with Tooth Number Difference of Two:a Comparative Study on Contact Characteristics and Load Analysis[J].Forsch Ingenieurwes,2017,81:325-336.
[12] WEI Bo,WANG Jiaxu,ZHOU Guangwu.Mixed Lubrication Analysis of Modified Cycloidal Gear Used in the RV Reducer[J].Journal of Engineer Tribolody,2016,230(2):121-134.
[13] MANDELBROT B B.The Fractal Geometry of Nature[M].New York:W.H.Freeman and Company,1982:35-45.
[14] BROWN C A,SAVARY G.Describing Ground Surface Texture Using Contact Profilometry and Fractal Analysis[J].Wear,1991,141(2):211-226.
[15] MAJUMDAR A,BHUSHAN B.Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surfaces[J].Journal of Tribology,1990,112:205-216.
[16] MAJUMDAR A,BHUSHAN B.Fractal Model of Elastic-Plastic Contact between Rough Surfaces[J].Journal of Tribology,1991,113:1-11.
[17] MORAG Y,ETSION I.Resolving the Contradiction of Asperities Plastic to Elastic Mode Transition in Current Contact Models of Fractal Rough Surfaces[J].Wear,2007,262(5):624-629.
[18] MIAO Xiaomei,HUANG Xiaodiao.A Complete Contact Model of a Fractal Rough Surface[J].Wear,2014,309(1):146-151.
[19] 陈奇,黄康,张彦,等.基于分形接触模型的齿轮接触强度影响参数分析[J].中国机械工程,2013,16(16):2208-2211.CHEN Qi,HUANG Kang,ZHANG Yan,et al.Analysis of Influence Parameters on Gear’s Contact Strength Based on Fractal Contact Model[J].China Mechanical Engineering,2013,16(16):2208-2211.
[20] CHEN Qi,FAN Xu,PENG Liu,et al.Research on Fractal Model of Normal Contact Stiffness between Two Spheroidal Joint Surfaces Considering Friction Factor[J].Tribology International,2016,97:256-264.
[21] HUANG Kang,XIONG Yangshou,WANG Tao.Research on the Dynamic Response of High-contact-ratio Spur Gears Influenced by Surface Roughness under EHL Condition[J].Applied Surface Science,2017,392:8-18.
[22] BERRY M V,LEWIS Z V.On the Weierstrass-Mandelbrot Fractal Function[J].Proceedings of the Royal Society,1980,370:459-484.
[23] CHEN Q,ZHAO Y,MA Y B,et al.Testing Experiment of Characteristic-scale Coefficient about Cylinder Surface[J].Journal of Mechanical Strength,2014,36(5):687-697.
[24] 葛世荣,朱华.摩擦学的分形[M].北京:机械工业出版社,2005:122.GE Shirong,ZHU Hua,Fractal in Tribology[M].Beijing:China Machine Press,2005:122.
[25] 田红亮,陈甜敏,郑金华,等.平行轴圆柱副接触分析[J].西安交通大学学报,2016,50(1):1-8.TIAN Hongliang,CHEN Tianmin,ZHENG Jinhua,et al.Contact Analysis of Cylindrical Pair with Parallel Axes[J].Journal of Xi’an Jiao Tong University,2016,50(1):1-8.
[26] 韩炬,李威,王志军.采用分形理论与微分几何的摆线轮形貌模型[J].西安交通大学学报,2017,51(11):97-105.HAN Ju,LI Wei,WANG Zhijun.Surface Morphology Models of Cycloid and Pin Teeth Based on Fractal Theory and Differential Geometry[J].Journal of Xi’an Jiao Tong University,2017,51(11):97-105.
[27] 李小彭,郭浩,刘井年,等.考虑摩擦的结合面法向刚度分形模型及仿真[J].振动、测试与诊断,2013,33(2):210-213.LI Xiaopeng,GUO Hao,LIU Jingnian,et al.Fractal Model and Simulation of Normal Contact Stiffness Considering the Friction between Joint Surfaces[J].Journal of Vibration,Measurement & Diagnosis,2013,33(2):210-213.
[28] LIU Peng,ZHAO Han,HUANG Kang,et al.Research on Normal Contact Stiffness of Rough Surface Considering Friction Based on Fractal Theory[J].Applied Surface Science,2015,349:43-48.
[29] DORIANA H H,GAN Yixiang,EINAN I.Static Friction at Fractal Interfaces[J].Tribology International,2016,93:229-238.
[30] KOGUT L,ETSION I.Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat[J].ASME Journal of Applied Mechanics,2002,69:657-662.
[31] 成雨,原园,甘立,等.尺度相关的分形粗糙表面弹塑性接触力学模型[J].西北工业大学学报,2016,34(3):485-492.CHENG Yu,YUAN Yuan,GAN Li,et al.The Elastic-Plastic Contact Mechanics Model Related Scale of Rough Surface[J].Journal of Northwestern Polytechnical University,2016,34(3):485-492.
[32] 李小彭,鞠行,赵光辉,等.考虑摩擦因素的结合面切向接触刚度分形预估模型及其仿真分析[J].摩擦学学报,2013,33(5):463-468.LI Xiaopeng,JU Xing,ZHAO Guanghui,et al.Fractal Prediction Model for Tangential Contact Stiffness of Joint Surface Considering Friction Factors and Its Simulation Analysis[J].Tribology,2013,33(5):463-468.
[33] 陈奇,黄守武,张振,等.考虑摩擦因素的两圆柱体表面接触承载能力的分形模型研究[J].机械工程学报,2016,52(7):114-121.CHEN Qi,HUANG Shouwu,ZHANG Zhen,et al.Research on Fractal Contact Model for Contact Carrying Capacity of Two Cylinders’ Surfaces Considering Friction Factors[J].Journal of Mechanical Engineering,2016,52(7):114-121.
[34] 丁雪兴,严如奇,贾永磊.基于基底长度的粗糙表面分形接触模型的构建与分析[J].摩擦学学报,2014,34(4):341-347.DING Xuexing,YAN Ruqi,JIA Yonglei.Construction and Analysis of Fractal Contact Mechanics Model for Rough Surface Based on Base Length[J].Tribology,2014,34(4):341-347.