基于薄壁圆环理论的机器人用柔性轴承变形特征快速求解
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摘要
谐波减速器内部柔性轴承是机器人关节的重要传动部件,因在工作中会产生较大的预变形而与普通轴承不同,导致传统轴承理论不适用,所以建立新的研究方法对其各项性能进行分析非常必要。该文通过建立柔性轴承的理论计算模型,求解计算柔性轴承工作时内部应力与变形特征,具体步骤包括:1)建立变形协调方程,并通过莫尔积分定理求解方程,得到变形过程中柔性轴承外圈的弯曲力矩;2)根据薄壁圆环理论,求得外圈的变形特征与内圈弯曲力矩;3)建立三弯矩方程,计算外载荷作用下柔性轴承外圈所承受的最大弯曲力矩。最后,建立柔性轴承有限元仿真模型对比验证理论模型,两者最大误差为7%,其中理论求解时间为5～8 min,有限元计算需4～5 h,通过理论模型可以快速求得柔性轴承内部变性特征与受力情况。计算结果表明,对于CSF-25-80型柔性轴承,外圈厚度设计在1.3～1.6 mm,宽度设计在9 mm左右,都可以有效改善外圈的应力状况。该研究可为对柔性轴承的设计和优化提供理论参考。
Agricultural robots are an important part of the modern agriculture,and its function execution is mainly accomplished by robotic arm.Flexible bearing of the harmonic driver is a key part of the joint of the robots' arm.At present,there is little research on the design of flexible bearings.Different from most rolling bearings which are used as supporting components in ordinary,the flexible bearings are used as transmission components.A non-circular cam which is called wave generator is assembled into the inner ring of the bearing before working and causes a greatly pre-deformation of the flexible bearing.Previous theories of rolling bearing will not applicable because of this deformation.Meanwhile,a pair of radial force in the opposite direction is applied to both ends of the long axis of deformation of the flexible bearing in the transmission process.Therefore,more complicated shape state and bending stresses are generated.Those unusual working conditions will lead the flexible bearing more prone to damage than ordinary bearing.Although FEA simulation can get relatively accurate results,it usually takes several hours to solve finite element model,and the calculation process is difficult to converge.Therefore,it is necessary to establish a new calculation method to obtain the performance of flexible bearings.In this paper,the stress and deformation characteristics of the inner and outer rings of the flexible bearing were solved separately by the following methods:1) First,the outer ring was equivalent to a statically indeterminate structure.The deformation coordination equation of the outer ring was established according to the equivalent model and solved by Mohr's integral theory.Overall bending moment of the outer ring of the flexible bearing formed by deformation was obtained.2) Combined the theory of thin-walled ring with the bending moment equation which was obtained above,the radial and circumferential deformation characteristics of the outer ring of the flexible bearing were obtained.3) According to the theory of multi span beam which was introduced in mechanics of materials,the loading model of the outer ring of the flexible bearing was built into three moment equations.The maximum stress was obtained by combining the three moment equation and the loading formula which was summarized by Ivanov's experiment.Above theoretical equations were compiled by MATLAB programs.Finally,a finite element simulation model of flexible bearing was established by ANSYS Workbench.The time consumed by simulation was about 4-5 hours while the calculation of theoretical equations only needed 5-8 min.By comparison,the maximum error between simulation value and theoretical value was only within 7%,it proved the correctness of the theoretical model calculation.Through analysis of bending stress and deformation of the flexible bearing,conclusions can be drawn as the following:1) the force state of the outer ring was different from the inner one during rotation of flexible bearings in harmonic drive,cyclic deformation of the outer ring caused a large alternating bending stress and prone to fatigue failure;2) stress of the outer ring formed by deformation increased sharply with the thickness while stress caused by external load would decline.And width only affected stress caused by external load.Increase of thickness was beneficial to carrying capacity of flexible bearing.However,increase of thickness lead to increasing of bending stress formed by deformation.The total bending stress had a minimum value in optimal thickness.3) Width was an important parameter which had a greater effect on the carrying capacity of the bearing,but it was constrained by external structural and cannot be too large.Calculation results would provide a theoretical reference for the design and optimization of flexible bearings.
Agricultural robots are an important part of the modern agriculture,and its function execution is mainly accomplished by robotic arm.Flexible bearing of the harmonic driver is a key part of the joint of the robots' arm.At present,there is little research on the design of flexible bearings.Different from most rolling bearings which are used as supporting components in ordinary,the flexible bearings are used as transmission components.A non-circular cam which is called wave generator is assembled into the inner ring of the bearing before working and causes a greatly pre-deformation of the flexible bearing.Previous theories of rolling bearing will not applicable because of this deformation.Meanwhile,a pair of radial force in the opposite direction is applied to both ends of the long axis of deformation of the flexible bearing in the transmission process.Therefore,more complicated shape state and bending stresses are generated.Those unusual working conditions will lead the flexible bearing more prone to damage than ordinary bearing.Although FEA simulation can get relatively accurate results,it usually takes several hours to solve finite element model,and the calculation process is difficult to converge.Therefore,it is necessary to establish a new calculation method to obtain the performance of flexible bearings.In this paper,the stress and deformation characteristics of the inner and outer rings of the flexible bearing were solved separately by the following methods:1) First,the outer ring was equivalent to a statically indeterminate structure.The deformation coordination equation of the outer ring was established according to the equivalent model and solved by Mohr's integral theory.Overall bending moment of the outer ring of the flexible bearing formed by deformation was obtained.2) Combined the theory of thin-walled ring with the bending moment equation which was obtained above,the radial and circumferential deformation characteristics of the outer ring of the flexible bearing were obtained.3) According to the theory of multi span beam which was introduced in mechanics of materials,the loading model of the outer ring of the flexible bearing was built into three moment equations.The maximum stress was obtained by combining the three moment equation and the loading formula which was summarized by Ivanov's experiment.Above theoretical equations were compiled by MATLAB programs.Finally,a finite element simulation model of flexible bearing was established by ANSYS Workbench.The time consumed by simulation was about 4-5 hours while the calculation of theoretical equations only needed 5-8 min.By comparison,the maximum error between simulation value and theoretical value was only within 7%,it proved the correctness of the theoretical model calculation.Through analysis of bending stress and deformation of the flexible bearing,conclusions can be drawn as the following:1) the force state of the outer ring was different from the inner one during rotation of flexible bearings in harmonic drive,cyclic deformation of the outer ring caused a large alternating bending stress and prone to fatigue failure;2) stress of the outer ring formed by deformation increased sharply with the thickness while stress caused by external load would decline.And width only affected stress caused by external load.Increase of thickness was beneficial to carrying capacity of flexible bearing.However,increase of thickness lead to increasing of bending stress formed by deformation.The total bending stress had a minimum value in optimal thickness.3) Width was an important parameter which had a greater effect on the carrying capacity of the bearing,but it was constrained by external structural and cannot be too large.Calculation results would provide a theoretical reference for the design and optimization of flexible bearings.
引文
[1]王儒敬,孙丙宇.农业机器人的发展现状及展望[J].中国科学院院刊,2015(6):803-809.Wang Rujing,Sun Bingyu.Development status and expectation of agricultural robot[J].Bulletin of Chinese Academy of Sciences,2015(6):803-809.(in Chinese with English abstract)
[2]李玉林,崔振德,张园,等.中国农业机器人的应用及发展现状[J].热带农业工程,2014,38(4):30-33.Li Yulin,Cui Zhende,Zhang Yuan,et al.Application and development status of China's agricultural robot[J].Tropical Agricultural Engineering,2014,38(4):30-33.(in Chinese with English abstract)
[3]夏田,杨世勇,何乃如,等.工业机器人用谐波减速器传动性能正交试验分析[J].科学技术与工程,2017(19):138-141.Xia Tian,Yang Shiyong,He Nairu,et al.Orthogonal experiment analysis on transmission performance of harmonic drive for industrial robots[J].Science Technology and Engineering,2017(19):138-141.(in Chinese with English abstract)
[4]Taghirad H D,Belanger P R.Modeling and parameter identification of harmonic drive systems[J].Journal of Dynamic Systems Measurement&Control,1997,120(4):439-444.
[5]Timothy D,Tuttle T,Seering W P,et al.A nonlinear model of a harmonic drive gear transmission[J].IEEE Transactions on Robotics and Automation,1996,12(3):368-374.
[6]Shah D B,Patel K M,Trivedi R D.Analyzing hertzian contact stress developed in a double row spherical roller bearing and its effect on fatigue life[J].Industrial Lubrication&Tribology,2016,68(3):361-368.
[7]Gunia D,Smolnicki T.The analysis of the stress distribution in contact pairs ball-wire and wire-ring in wire raceway slewing bearing[C]//International Conference on Renewable Energy Sources-Research&Business.Springer,Cham,2016.
[8]You Bindi,Wen Jianmin.Load distribution calculation of fexible ball bearing with elliptical cam wave generator[C]//International Conference on Mechanics,2016.
[9]张立勇,刘新猛,王长路,等.径向变形量对谐波减速器啮合特性及柔轮应力的影响分析[J].机械传动,2017(9):172-175.Zhang Liyong,Liu Xinmeng,Wang Changlu,et al.Influence of radial deformation on stress of flexspline and meshing characteristic of harmonic reducer[J].Journal of Mechanical Transmission,2017(9):172-175.(in Chinese with English abstract)
[10]关崇复.柔性轴承动载荷下的接触应力计算[J].燕山大学学报,1994(3):220-226.Guan Chongfu.Calculation for the contact pressure of flexible bearings under dynamic load[J].Journal of Yanshan University,1994(3):220-226.(in Chinese with English abstract)
[11]刘鸿文.材料力学[M].北京:高等教育出版社,1985.
[12]陈晓霞,刘玉生,邢静忠,等.谐波齿轮中柔轮中性层的伸缩变形规律[J].机械工程学报,2014,50(21):189-196.Cheng Xiaoxia,Liu Yusheng,Xing Jingzhong,et al.Neutral line stretch of flexspline in harmonic driver[J].Journal of Mechnical Engineering,2014,50(21):189-196.(in Chinese with English abstract)
[13]Harris T A,Kotzalas M N.Advanced Concepts of Bearing Technology:Rolling Bearing Analysis[M].Boca Raton:CRCPress,2007.
[14]王静静.某谐波减速机波发生器柔性轴承疲劳寿命研究与结构优化[D].长沙:湖南大学,2016.Wang Jingjing.The Research on Fatigue Life and Structure Optimization of the Flexible Bearing in a Harmonic Rreducer Wave Generator[D].Changsha:Hunan University,2016.(in Chinese with English abstract)
[15]伊万诺夫,沈允文,李克美.谐波齿轮传动[M].北京:国防工业出版社,1987.
[16]柏德恩,全齐全,李贺,等.伺服加载的谐波减速器启动力矩测试系统[J].吉林大学学报:工学版,2017(6):141-147.Bai Deen,Quan Qiquan,Li He,et al.Starting torque test system for harmonic driver based on servo loading[J].Journal of Jilin University:Engineering and Technology Edition,2017(6):141-147.(in Chinese with English abstract)
[17]Kayabasi O,Erzincanli F.Shape optimization of tooth profile of a flexspline for a harmonic drive by finite element modelling[J].Materials&Design,2007,28(2):441-447.
[18]唐小欢,王湘江,冯栋彦.谐波减速器输入轴迟滞曲线的特性研究[J].机械传动,2016(8):29-32.Tang Xiaohuan,Wang Xiangjiang,Feng Dongyan.Study on hysteresis curve characteristic of input shaft of harmonic reducer[J].Journal of Mechanical Transmission,2016(8):29-32.(in Chinese with English abstract)
[19]Tian Lin,Jiang Yi,Wang Yazhen,et al.Load distribution research on flexible bearing in harmonic drive[C]//International Conference on Mechanical Design.Springer,Singapore,2017:165-176.
[20]刘正宁.谐波齿轮传动柔性轴承受力分析[J].大连大学学报,1995(4):512-517.Liu Zhengning.Analysing stress of fIexible bearing under harmonic gear drive[J].Journal of Dalian University,1995(4):512-517.(in Chinese with English abstract)
[21]Ostapski W.Analysis of the stress state in the harmonic drive generator-flexspline system in relation to selected structural parameters and manufacturing deviations[J].Bulletin of the Polish Academy of Sciences:Technical Sciences,2010,58(4):683-698.
[22]Ostapski W,Mukha I.Stress state analysis of harmonic drive elements by FEM[J].Bulletin of the Polish Academy of Sciences:Technical Sciences,2007,55(1):115-123.
[23]Pacana J,Witkowski W,Mucha J.FEM analysis of stress distribution in the hermetic harmonic drive flexspline[J].Strength of Materials,2017,49(1):1-11.
[24]Li Junyang,Wang Jiaxun,Fan Kaijie,et al.Accelerated life model for harmonic drive under adhesive wear[J].Tribology,2016(3):297-303.
[25]Liang Yong,Fan Yuanxun.Analysis of tooth deflection of flexspline and its influence on harmonic drive[J].Journal of Mechanical Transmission,2018(2):31-35.
[26]Gravagno F,Mucino V H,Pennestrì,et al.Influence of wave generator profile on the pure kinematic error and centrodes of harmonic drive[J].Mechanism and Machine Theory,2016,104:100-117.
[27]Dennis León,Arzola N,Andrés Tovar.Statistical analysis of the influence of tooth geometry in the performance of a harmonic drive[J].Journal of the Brazilian Society of Mechanical Sciences&Engineering,2015,37(2):723-735.
[28]沙晓晨,范元勋.谐波减速器传动误差的研究[J].机械制造与自动化,2015(5):50-54.Sha Xiaochen,Fan Yuanxun.Study of transmission error of harmonic drive reducer[J].Machine Building&Automation,2015(5):50-54.(in Chinese with English abstract)
[29]Ye Zhenhuan,Liu Zhansheng,Wang Liqin.Optimization analysis of structure parameter of angular-contact ball bearings based on fatigue life[J].Journal of Mechanical Transmission,2016(1):59-63.
[30]Kim S W,Kang K,Yoon K,et al.Design optimization of an angular contact ball bearing for the main shaft of a grinder[J].Mechanism&Machine Theory,2016,104:287-302.
[2]李玉林,崔振德,张园,等.中国农业机器人的应用及发展现状[J].热带农业工程,2014,38(4):30-33.Li Yulin,Cui Zhende,Zhang Yuan,et al.Application and development status of China's agricultural robot[J].Tropical Agricultural Engineering,2014,38(4):30-33.(in Chinese with English abstract)
[3]夏田,杨世勇,何乃如,等.工业机器人用谐波减速器传动性能正交试验分析[J].科学技术与工程,2017(19):138-141.Xia Tian,Yang Shiyong,He Nairu,et al.Orthogonal experiment analysis on transmission performance of harmonic drive for industrial robots[J].Science Technology and Engineering,2017(19):138-141.(in Chinese with English abstract)
[4]Taghirad H D,Belanger P R.Modeling and parameter identification of harmonic drive systems[J].Journal of Dynamic Systems Measurement&Control,1997,120(4):439-444.
[5]Timothy D,Tuttle T,Seering W P,et al.A nonlinear model of a harmonic drive gear transmission[J].IEEE Transactions on Robotics and Automation,1996,12(3):368-374.
[6]Shah D B,Patel K M,Trivedi R D.Analyzing hertzian contact stress developed in a double row spherical roller bearing and its effect on fatigue life[J].Industrial Lubrication&Tribology,2016,68(3):361-368.
[7]Gunia D,Smolnicki T.The analysis of the stress distribution in contact pairs ball-wire and wire-ring in wire raceway slewing bearing[C]//International Conference on Renewable Energy Sources-Research&Business.Springer,Cham,2016.
[8]You Bindi,Wen Jianmin.Load distribution calculation of fexible ball bearing with elliptical cam wave generator[C]//International Conference on Mechanics,2016.
[9]张立勇,刘新猛,王长路,等.径向变形量对谐波减速器啮合特性及柔轮应力的影响分析[J].机械传动,2017(9):172-175.Zhang Liyong,Liu Xinmeng,Wang Changlu,et al.Influence of radial deformation on stress of flexspline and meshing characteristic of harmonic reducer[J].Journal of Mechanical Transmission,2017(9):172-175.(in Chinese with English abstract)
[10]关崇复.柔性轴承动载荷下的接触应力计算[J].燕山大学学报,1994(3):220-226.Guan Chongfu.Calculation for the contact pressure of flexible bearings under dynamic load[J].Journal of Yanshan University,1994(3):220-226.(in Chinese with English abstract)
[11]刘鸿文.材料力学[M].北京:高等教育出版社,1985.
[12]陈晓霞,刘玉生,邢静忠,等.谐波齿轮中柔轮中性层的伸缩变形规律[J].机械工程学报,2014,50(21):189-196.Cheng Xiaoxia,Liu Yusheng,Xing Jingzhong,et al.Neutral line stretch of flexspline in harmonic driver[J].Journal of Mechnical Engineering,2014,50(21):189-196.(in Chinese with English abstract)
[13]Harris T A,Kotzalas M N.Advanced Concepts of Bearing Technology:Rolling Bearing Analysis[M].Boca Raton:CRCPress,2007.
[14]王静静.某谐波减速机波发生器柔性轴承疲劳寿命研究与结构优化[D].长沙:湖南大学,2016.Wang Jingjing.The Research on Fatigue Life and Structure Optimization of the Flexible Bearing in a Harmonic Rreducer Wave Generator[D].Changsha:Hunan University,2016.(in Chinese with English abstract)
[15]伊万诺夫,沈允文,李克美.谐波齿轮传动[M].北京:国防工业出版社,1987.
[16]柏德恩,全齐全,李贺,等.伺服加载的谐波减速器启动力矩测试系统[J].吉林大学学报:工学版,2017(6):141-147.Bai Deen,Quan Qiquan,Li He,et al.Starting torque test system for harmonic driver based on servo loading[J].Journal of Jilin University:Engineering and Technology Edition,2017(6):141-147.(in Chinese with English abstract)
[17]Kayabasi O,Erzincanli F.Shape optimization of tooth profile of a flexspline for a harmonic drive by finite element modelling[J].Materials&Design,2007,28(2):441-447.
[18]唐小欢,王湘江,冯栋彦.谐波减速器输入轴迟滞曲线的特性研究[J].机械传动,2016(8):29-32.Tang Xiaohuan,Wang Xiangjiang,Feng Dongyan.Study on hysteresis curve characteristic of input shaft of harmonic reducer[J].Journal of Mechanical Transmission,2016(8):29-32.(in Chinese with English abstract)
[19]Tian Lin,Jiang Yi,Wang Yazhen,et al.Load distribution research on flexible bearing in harmonic drive[C]//International Conference on Mechanical Design.Springer,Singapore,2017:165-176.
[20]刘正宁.谐波齿轮传动柔性轴承受力分析[J].大连大学学报,1995(4):512-517.Liu Zhengning.Analysing stress of fIexible bearing under harmonic gear drive[J].Journal of Dalian University,1995(4):512-517.(in Chinese with English abstract)
[21]Ostapski W.Analysis of the stress state in the harmonic drive generator-flexspline system in relation to selected structural parameters and manufacturing deviations[J].Bulletin of the Polish Academy of Sciences:Technical Sciences,2010,58(4):683-698.
[22]Ostapski W,Mukha I.Stress state analysis of harmonic drive elements by FEM[J].Bulletin of the Polish Academy of Sciences:Technical Sciences,2007,55(1):115-123.
[23]Pacana J,Witkowski W,Mucha J.FEM analysis of stress distribution in the hermetic harmonic drive flexspline[J].Strength of Materials,2017,49(1):1-11.
[24]Li Junyang,Wang Jiaxun,Fan Kaijie,et al.Accelerated life model for harmonic drive under adhesive wear[J].Tribology,2016(3):297-303.
[25]Liang Yong,Fan Yuanxun.Analysis of tooth deflection of flexspline and its influence on harmonic drive[J].Journal of Mechanical Transmission,2018(2):31-35.
[26]Gravagno F,Mucino V H,Pennestrì,et al.Influence of wave generator profile on the pure kinematic error and centrodes of harmonic drive[J].Mechanism and Machine Theory,2016,104:100-117.
[27]Dennis León,Arzola N,Andrés Tovar.Statistical analysis of the influence of tooth geometry in the performance of a harmonic drive[J].Journal of the Brazilian Society of Mechanical Sciences&Engineering,2015,37(2):723-735.
[28]沙晓晨,范元勋.谐波减速器传动误差的研究[J].机械制造与自动化,2015(5):50-54.Sha Xiaochen,Fan Yuanxun.Study of transmission error of harmonic drive reducer[J].Machine Building&Automation,2015(5):50-54.(in Chinese with English abstract)
[29]Ye Zhenhuan,Liu Zhansheng,Wang Liqin.Optimization analysis of structure parameter of angular-contact ball bearings based on fatigue life[J].Journal of Mechanical Transmission,2016(1):59-63.
[30]Kim S W,Kang K,Yoon K,et al.Design optimization of an angular contact ball bearing for the main shaft of a grinder[J].Mechanism&Machine Theory,2016,104:287-302.