轴承支承刚度对行星轮系统动态稳定性的影响
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摘要
分析了轴承支承刚度对行星轮传动机构动态稳定性的影响,建立了同时考虑各齿轮和行星架横向和扭转振动、时变啮合刚度、齿侧间隙、脱齿现象和齿轮偏心误差的非线性动力学模型。提出了新的模态分类方式,该分类方式同时考虑中心轮和行星轮的振动情况。借助多尺度法,依据该模态分类,分析了轴承支承刚度对行星轮系统各模态固有频率、各模态不稳定边界、各不稳定区域内振幅的影响。分析结果与数值模拟取得了一致的结论:轴承支承刚度对行星轮系统稳定性的影响主要取决于行星轮的振动类别,各模态不稳定振动的激励条件同样依据行星轮的振动类型而不同;分析轴承支承刚度对行星轮动态稳定性的影响,可以为行星轮系统轴承的选择提供指导。
The effects of bearing stiffness on planetary gear parameter instability were analytically investigated. A nonlinear model with multiple degrees of freedoms was built taking consideration of both translations and rotations of each component, variable mesh stiffness, backlash, loss of contact and eccentricity errors in the analysis. New classification of modal types was proposed depending on the motions of both center gears and planet gears. The effects of bearing stiffness on the natural frequencies, the instability boundaries and the response amplitudes of each modal type were analytically investigated through the method of multiple scales. And the results agree well with those of numerical integrations. The results show that the different specific influences mainly depend on the motions of planet gears. This study is expected useful for selections of bearings in the designing process of planetary gear trains.
The effects of bearing stiffness on planetary gear parameter instability were analytically investigated. A nonlinear model with multiple degrees of freedoms was built taking consideration of both translations and rotations of each component, variable mesh stiffness, backlash, loss of contact and eccentricity errors in the analysis. New classification of modal types was proposed depending on the motions of both center gears and planet gears. The effects of bearing stiffness on the natural frequencies, the instability boundaries and the response amplitudes of each modal type were analytically investigated through the method of multiple scales. And the results agree well with those of numerical integrations. The results show that the different specific influences mainly depend on the motions of planet gears. This study is expected useful for selections of bearings in the designing process of planetary gear trains.
引文
[1]LIN J,PARKER R G.Planetary gear parametric instability caused by mesh stiffness variation[J].Journal of Sound&Vibration,2002,249(1):129-145.
[2]COOLEY C G,PARKER R G.Mechanical stability of highspeed planetary gears[J].International Journal of Mechanical Sciences,2013,69(4):59-71.
[3]CANCHI S V,PARKER R G.Effect of ring-planet mesh phasing and contact ratio on the parametric instabilities of a planetary gear ring[J].Journal of Mechanical Design,2007,130(1):842-849.
[4]李同杰,朱如鹏,鲍和云,等.行星齿轮系扭转非线性振动建模与运动分岔特性研究[J].机械工程学报,2011,47(21):76-83.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Nonlinear torsional vibration modeling and bifurcation characteristic study of a planetary gear train[J].Journal of Mechanical Engineering,2011,47(21):76-83.
[5]李同杰,朱如鹏,鲍和云,等.行星齿轮传动系的周期运动及其稳定性[J].振动工程学报,2013,26(6):815-822.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Coexisting periodic solutions and their stability of a nonlinear planetary gear train[J].Zhendong Gongcheng Xuebao,2013,26(6):815-822.
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[7]李同杰,朱如鹏,鲍和云,等.行星齿轮非线性振动系统参数稳定域的计算方法[J].航空动力学报,2012,27(6):1416-1423.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Method of stability region determination for planetary gear train’s parameters based on nonlinear vibration model[J].Hangkong Dongli Xuebao,2012,27(6):1416-1423.
[8]李同杰,朱如鹏,鲍和云,等.满足稳定性要求的行星轮系统工作转速区间确定方法[J].振动与冲击,2012,31(22):183-187.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Determination method of working speed range for a planetary gear train based on stability requirement and a nonlinear vibration model[J].Journal of Vibration&Shock,2012,31(22):183-187.
[9]SINGH A.行星传动滚针轴承对单和双小齿轮行星齿轮传动性能的影响[J].传动技术,2009,23(3):28-39.SINGH A.Influence of planetary needle bearings on the performance of single and double pinion planetary systems[J].Journal of Mechanical Design,2009,23(3):28-39.
[10]BU Z,LIU G,WU L.Modal analyses of herringbone planetary gear train with journal bearings[J].Mechanism&Machine Theory,2012,54:99-115.
[11]GILL-JEONG C,PARKER R G.Influence of bearing stiffness on the static properties of a planetary gear system with manufacturing errors[J].Journal of Mechanical Science&Technology,2004,18(11):1978-1988.
[12]陈安华,罗善明,王文明,等.齿轮系统动态传递误差和振动稳定性的数值研究[J].机械工程学报,2004,40(4):21-25.CHEN Anhua,LUO Shanming,WANG Wenming,et al.Numerical investigations on dynamic transmission error and stability of a geared rotor-bearing system[J].Chinese Journal of Mechanical Engineering,2004,40(4):21-25.
[13]GUO Y,KELLER J,PARKER R G.Nonlinear dynamics and stability of wind turbine planetary gear sets under gravity effects[J].European Journal of Mechanics-A/Solids,2014,47:45-57.
[14]GUO Y,PARKER R G.Dynamic analysis of planetary gears with bearing clearance[J].Journal of Computational&Nonlinear Dynamics,2012,7(4):614-620.
[15]MASOUMI A,PELLICANO F,SAMANI F S,et al.Symmetry breaking and chaos-induced imbalance in planetary gears[J].Nonlinear Dynamics,2015,80(1/2):561-582.
[16]张锁怀,石守红,丘大谋.结构参数对齿轮耦合的二平行转子-轴承系统稳定性的影响[J].机械科学与技术,2000,17(7):13-15.ZHANG Suohuai,SHI Shouhong,QIU Damou.The influence of structural parameters on stability of gear meshed double parallel rotor-bearing system[J].Machine Design,2000,17(7):13-15.
[17]ASME M,PARKER R G.Mesh stiffness variation instabilities in two-stage gear systems[J].Journal of Vibration&Acoustics,2002,124(1):68-76.
[18]BAHK C J,PARKER R G,et al.Analytical study on nonlinear dynamics of planetary gears[J].Dissertations&Theses-Gradworks,2012.
[19]LIN J,PARKER R G,PARKER R G.Analytical characterization of the unique properties of planetary gear free vibration.Transaction of ASME[J].Journal of Vibration&Acoustics,1999,121(3):316-321.
[2]COOLEY C G,PARKER R G.Mechanical stability of highspeed planetary gears[J].International Journal of Mechanical Sciences,2013,69(4):59-71.
[3]CANCHI S V,PARKER R G.Effect of ring-planet mesh phasing and contact ratio on the parametric instabilities of a planetary gear ring[J].Journal of Mechanical Design,2007,130(1):842-849.
[4]李同杰,朱如鹏,鲍和云,等.行星齿轮系扭转非线性振动建模与运动分岔特性研究[J].机械工程学报,2011,47(21):76-83.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Nonlinear torsional vibration modeling and bifurcation characteristic study of a planetary gear train[J].Journal of Mechanical Engineering,2011,47(21):76-83.
[5]李同杰,朱如鹏,鲍和云,等.行星齿轮传动系的周期运动及其稳定性[J].振动工程学报,2013,26(6):815-822.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Coexisting periodic solutions and their stability of a nonlinear planetary gear train[J].Zhendong Gongcheng Xuebao,2013,26(6):815-822.
[6]PARKER R G,WU X,WU X.Parametric instability of planetary gears having elastic continuum ring gears[J].Journal of Vibration&Acoustics,2012,134(4):1002-1010.
[7]李同杰,朱如鹏,鲍和云,等.行星齿轮非线性振动系统参数稳定域的计算方法[J].航空动力学报,2012,27(6):1416-1423.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Method of stability region determination for planetary gear train’s parameters based on nonlinear vibration model[J].Hangkong Dongli Xuebao,2012,27(6):1416-1423.
[8]李同杰,朱如鹏,鲍和云,等.满足稳定性要求的行星轮系统工作转速区间确定方法[J].振动与冲击,2012,31(22):183-187.LI Tongjie,ZHU Rupeng,BAO Heyun,et al.Determination method of working speed range for a planetary gear train based on stability requirement and a nonlinear vibration model[J].Journal of Vibration&Shock,2012,31(22):183-187.
[9]SINGH A.行星传动滚针轴承对单和双小齿轮行星齿轮传动性能的影响[J].传动技术,2009,23(3):28-39.SINGH A.Influence of planetary needle bearings on the performance of single and double pinion planetary systems[J].Journal of Mechanical Design,2009,23(3):28-39.
[10]BU Z,LIU G,WU L.Modal analyses of herringbone planetary gear train with journal bearings[J].Mechanism&Machine Theory,2012,54:99-115.
[11]GILL-JEONG C,PARKER R G.Influence of bearing stiffness on the static properties of a planetary gear system with manufacturing errors[J].Journal of Mechanical Science&Technology,2004,18(11):1978-1988.
[12]陈安华,罗善明,王文明,等.齿轮系统动态传递误差和振动稳定性的数值研究[J].机械工程学报,2004,40(4):21-25.CHEN Anhua,LUO Shanming,WANG Wenming,et al.Numerical investigations on dynamic transmission error and stability of a geared rotor-bearing system[J].Chinese Journal of Mechanical Engineering,2004,40(4):21-25.
[13]GUO Y,KELLER J,PARKER R G.Nonlinear dynamics and stability of wind turbine planetary gear sets under gravity effects[J].European Journal of Mechanics-A/Solids,2014,47:45-57.
[14]GUO Y,PARKER R G.Dynamic analysis of planetary gears with bearing clearance[J].Journal of Computational&Nonlinear Dynamics,2012,7(4):614-620.
[15]MASOUMI A,PELLICANO F,SAMANI F S,et al.Symmetry breaking and chaos-induced imbalance in planetary gears[J].Nonlinear Dynamics,2015,80(1/2):561-582.
[16]张锁怀,石守红,丘大谋.结构参数对齿轮耦合的二平行转子-轴承系统稳定性的影响[J].机械科学与技术,2000,17(7):13-15.ZHANG Suohuai,SHI Shouhong,QIU Damou.The influence of structural parameters on stability of gear meshed double parallel rotor-bearing system[J].Machine Design,2000,17(7):13-15.
[17]ASME M,PARKER R G.Mesh stiffness variation instabilities in two-stage gear systems[J].Journal of Vibration&Acoustics,2002,124(1):68-76.
[18]BAHK C J,PARKER R G,et al.Analytical study on nonlinear dynamics of planetary gears[J].Dissertations&Theses-Gradworks,2012.
[19]LIN J,PARKER R G,PARKER R G.Analytical characterization of the unique properties of planetary gear free vibration.Transaction of ASME[J].Journal of Vibration&Acoustics,1999,121(3):316-321.