行星齿轮传动动态特性的研究
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摘要
行星齿轮被广泛应用于船舶、飞机、汽车、重型机械等许多领域,它的振动和噪音一直以来都是普遍关注的问题。为了减小其振动和噪音,动力学分析是必不可少的。本文分析了行星齿轮动力学当中的一些关键性问题,提高了对于行星齿轮传动动态特性的理解。
为了对行星齿轮传动进行动力学分析,首先建立了两个分析模型,利用这两个模型回顾了具有回转对称性结构的行星齿轮固有频率和振动模态特点,为后续研究奠定了基础。
利用行星齿轮的振动特点,深入研究了行星齿轮固有特性对于设计参数中啮合刚度、支撑刚度、扭转刚度、质量和转动惯量变化的敏感度,得到了与模态动能和模态应变能有关的固有频率敏感度关系式。应用泰勒公式和特征值敏感度公式,通过理论推导,得到了特征值与参数变化曲线之间的分叉和交插规律。结果表明太阳轮、内齿圈和行星架的支撑刚度、扭转刚度、质量和转动惯量中,每一个参数的变化只对某一种类型振动模态具有影响,并且对同一种类型振动模态中各阶振动模态的影响程度也不同。行星轮的设计参数对振动的影响程度最大,几乎对所有振动模态都有影响。啮合刚度的变化对两个旋转模态、两组位移模态和两组行星模态影响较大。固有频率与参数变化曲线中相同振动模态的固有频率与参数变化曲线相互分叉,不同振动模态固有频率与参数变化曲线相互交插。
啮合刚度的变化产生了参数不稳定性,利用行星齿轮振动模态特点,应用多自由度系统稳定性分析的多尺度法,得到了稳定区域和不稳定区域分界线公式,探讨了重合度和啮合相位对于参数不稳定的影响。结果表明:通过调整重合度和啮合相位可以减小或者消除一些特定的不稳定区域。
Planetary gear noise and vibration are primary concerns in theirapplications in the transmissions of marine vessels, aircrafts, automobiles, andheavy machinery.Dynamic analysis is essential to the noise and vibrationreduction.This work analytically investigates some critical issues and advancesthe understanding of planetary gear dynamics.
Two parameter models are built for the dynamic analysis of planetary gear.The unique properties of the natural frequency spectra and vibration model aredisserted.The special vibration properties are useful for subsequent research.
Taking advantage of the disserted modal properties, the natural frequencyand vibration mode sensitivities to design parameters are closely investigated.The key parameters include mesh stiffness, support stiffness, twiststiffness,component mass,and moments of inertia.The eigensensitivities areexpressed in simple formulae associated with modal strain and kineticenergies.The results indicate each parameter of sun ,ring and carrier ,includingsupport stiffness, twist stiffness,component mass,and moments of inertia,affects one type natural frequency and the infuential extent is disparate todifferent order natural frequency of the same type. Planet design parametersare the most affect parameter and affect most natural frequencies.Meshstiffness affects six different frequencies associated with two rotational modes,two pair of translational modes and two groups of planet modes. Takingadvantage of Taylor formulae and eigensensitivites formulae, the rules ofveering and intersecting are derived to identify changes of natural frequencyunder parameter variations. These results indicate the plots of two same modesveer away and the plots of two different modes intersect with each other in the
figure of natural frequencies versus varying parameter curve. Parametric instabilities are excited by mesh stiffness varity.Using thewell-defined modal properties of planetary gear and the multi-scale method ofinstability analyses of multi-degree of freedom system, the boundary ofstability and instability are derived. The effect of contact ratio and mesh phaseon parametric instability is analytically approached. The results indicateadjusting contact ratio and mesh phasing can suppress or abate some particularinstability.
为了对行星齿轮传动进行动力学分析,首先建立了两个分析模型,利用这两个模型回顾了具有回转对称性结构的行星齿轮固有频率和振动模态特点,为后续研究奠定了基础。
利用行星齿轮的振动特点,深入研究了行星齿轮固有特性对于设计参数中啮合刚度、支撑刚度、扭转刚度、质量和转动惯量变化的敏感度,得到了与模态动能和模态应变能有关的固有频率敏感度关系式。应用泰勒公式和特征值敏感度公式,通过理论推导,得到了特征值与参数变化曲线之间的分叉和交插规律。结果表明太阳轮、内齿圈和行星架的支撑刚度、扭转刚度、质量和转动惯量中,每一个参数的变化只对某一种类型振动模态具有影响,并且对同一种类型振动模态中各阶振动模态的影响程度也不同。行星轮的设计参数对振动的影响程度最大,几乎对所有振动模态都有影响。啮合刚度的变化对两个旋转模态、两组位移模态和两组行星模态影响较大。固有频率与参数变化曲线中相同振动模态的固有频率与参数变化曲线相互分叉,不同振动模态固有频率与参数变化曲线相互交插。
啮合刚度的变化产生了参数不稳定性,利用行星齿轮振动模态特点,应用多自由度系统稳定性分析的多尺度法,得到了稳定区域和不稳定区域分界线公式,探讨了重合度和啮合相位对于参数不稳定的影响。结果表明:通过调整重合度和啮合相位可以减小或者消除一些特定的不稳定区域。
Planetary gear noise and vibration are primary concerns in theirapplications in the transmissions of marine vessels, aircrafts, automobiles, andheavy machinery.Dynamic analysis is essential to the noise and vibrationreduction.This work analytically investigates some critical issues and advancesthe understanding of planetary gear dynamics.
Two parameter models are built for the dynamic analysis of planetary gear.The unique properties of the natural frequency spectra and vibration model aredisserted.The special vibration properties are useful for subsequent research.
Taking advantage of the disserted modal properties, the natural frequencyand vibration mode sensitivities to design parameters are closely investigated.The key parameters include mesh stiffness, support stiffness, twiststiffness,component mass,and moments of inertia.The eigensensitivities areexpressed in simple formulae associated with modal strain and kineticenergies.The results indicate each parameter of sun ,ring and carrier ,includingsupport stiffness, twist stiffness,component mass,and moments of inertia,affects one type natural frequency and the infuential extent is disparate todifferent order natural frequency of the same type. Planet design parametersare the most affect parameter and affect most natural frequencies.Meshstiffness affects six different frequencies associated with two rotational modes,two pair of translational modes and two groups of planet modes. Takingadvantage of Taylor formulae and eigensensitivites formulae, the rules ofveering and intersecting are derived to identify changes of natural frequencyunder parameter variations. These results indicate the plots of two same modesveer away and the plots of two different modes intersect with each other in the
figure of natural frequencies versus varying parameter curve. Parametric instabilities are excited by mesh stiffness varity.Using thewell-defined modal properties of planetary gear and the multi-scale method ofinstability analyses of multi-degree of freedom system, the boundary ofstability and instability are derived. The effect of contact ratio and mesh phaseon parametric instability is analytically approached. The results indicateadjusting contact ratio and mesh phasing can suppress or abate some particularinstability.
引文
[1] 卜炎.机械传动装置设计手册(上册). 机械工业出版社.1988:446-453
[2] 马从谦.渐开线行星齿轮传动.机械工业出版社.1987:11-78
[3] 李建军.齿轮系统动力学—振动·冲击·噪声.科学出版社.1997:69-247
[4] R.L.Platt,R.D.Leipold. A Study on Helical Gear Planetary Phasing Effects on Transmission noise. VDI Beriche.1996,123:793-807
[5] 屈维得,唐恒龄.机械振动手册.机械工业出版设.2000:30-77
[6] T.Chiang,R.H.Badgley. Reduction of Vibration and Noise Generated by Planetary Ring Gears in Helicopter Aircraft Transmission. Journal of Engineering for Industry. 1973,93:1149-1158
[7] F.Cunliffe.J.D.Smith.D.B.Welbourn.Dynamic Tooth Loads in Epicyclic Gears. Journal of Engineering for Industry. 1974,94:578-584
[8] M.Botman.Epicyclic Gear Vibration.Journal of Engineering for Industry. 1976,96:811-815
[9] J.L.Frater.R.August,F.B.Oswald. Vibration in Planetary Gear System withUnequal Planet Stiffness. NASATechnical Memorandum. 1983,8:28-34
[10] Mark, J.Valco. Planetary Gear Train Ring Gear and Support Structure Investigation. Cleveland State University.1992:86-254
[11] P.Velex, L.Flamand.Dynamic Response of Planetary Trains to Mesh Parameter Excitations.ASME Journal of Mechanical Design.1996,118:7-14
[12] J.Lin,R.G.Parker.AnalyticalCharacterizationoftheUniquePropertiesofPlanetary Gear Free Vibration. Journal of Vibration andAcoustics. 1999,121:316-321
[13] J.Lin,R.G.Parker. Structured Vibration Characteristics of Planetary Gears with Unequally Spaced Planets. Journal of Sound and Vibration. 2000,233(5):921-928
[14] 杨建明. 行星齿轮机构弹性动力学建模. 桂林电子工业学院学报.2000,20(2):48-52
[15] Ahmet,Kahraman.Free Torsional Vibration Characteristics of Compound Planetary Gear Sets. Mechanism and Machine Theory. 2001,36:953-971
[16] A.W.Leissa. On a Curve Veering Aberration .Journal of Applied Mathematics and Physics.1974,106:451-463
[17] N.C.Perkins,C.D.Mote. Comments on Curve Veering in Eigenvalue Problems. Journal of Sound Vibration .1986,106: 451-463
[18] A.Kahraman. Planetary Gear Train Dynamics. ASME Journal of Mechanical Design.1994:116(3):713-720
[19] J.Lin,R.G.Parker. Sensitivity of Planetary Gear Natural Frequencies and Vibration Modes to Model Parameter. Journal of Sound and Vibration. 1999,228(1):109-128
[20] 朱才朝,秦大同,李润方,陈宗源.内齿行星齿轮传动动态特性的研究.重庆 工业大学学报.1997,20(5):20-32
[21] G.Bollinger,R.J.Harker. Instability Potential of High Speed Gear . Journal oof the ndustrial Mathematics. 1967,17:39-55
[22] M.Amabili, A.Rivola. Dynamic Analysis of Spur of Gear Pairs: Steady State Response and Stability of the SDOF Model with Time-Varying Meshing Damping. Mechanical Systems and Signal Processing. 1997, 11(3):357-390
[23] M.Benton,A.Seireg. Factors Influencing Instability and Resonances in Geared Systems.ASME Journal of Mechanical Design. 1981,103(2): 372-378
[24] A.Kahraman,G.W.Blankenship. Experiments on Nonlinear Dynamic Behavior of an Oscillator with Clearance and Periodically Time-Varying Parameters. Journal ofApplied Mechanics.1997,64: 217-226
[25] C.Nataraj,A.M.Whitman.Parameter Excitation Effects in Gear Dynamics. ASME Design Engineering .Technical Conferences. 1997: DETC97/VB-4018, Sacrame- nto,CA.
[26] C.Nataraj, N.K.Arakere .Dynamic Response and Stability of a Spur GearPair. ASME Design Engineering Technical Conferences. 1999:DETC99/VIB-8110. Las Vegas
[27] M.Benton,A.Seireg.ApplicationoftheRitzAveragingMethodtoDeterminingthe Response of Systems with Time Varying Stiffness to Harmonic Excitation . ASME Journal of Mechanical Design. 1980,102(2):384-390
[28] M.Botman. Vibration Measurements on Planetary Gears of Aircraft Engines. AIAAJournal.1980,17:351-357
[29] T.Hidaka,Y.Terauchi,K.Nagamura. Dynamic Behavior of PlanetaryGear-1st Report: Load Distribution. Bull.Jsme.1976,19(132): 690-698
[30] R.Kasuba,R.August. Gear Mesh Stiffness and Load Sharing in Planetary Gearing.1984:ASME Paper 84-DET-229
[31] P.Ma,M.Botman. Load Sharing in a Planetary Gear Stage in the Presence of Gear Errors and Misalignment. Journal of Mechanisms. Transmissions and Automation in Design.1984,104:1-7
[32] R.August, R.Kasuba. Torsional Vibrations and Dynamic Loads in a Basic Planetary Gear System. Journal of Vibration, Acoustics, Stress and Reliability in Design .1986,108(3):348-353
[33] P.Velex, L.Flamand. Dynamic Response of Planetary Trains to Mesh Parametric Excitations ,ASME Journal of Mechanical Design.1996,118:7-14
[34] A.Kahraman. Dynamic Analysis of a Multi-Mesh Helical Gear Trains. ASME Journal of Mechanical Design.1994,116(3):706-712
[35] A.Kahraman, G.W. Blankenship. Planet Mesh Phasing in Epicyclic Gear Sets. Proc. of International Gearing Conference.Newcastle,UK. 1994: 99-104
[36] A.Kahraman. Natural Modes of Planetary Gear Trains. Journal of Sound and Vibration .1994,173(1):125-130
[37] 张建云,阮忠唐,丘大谋.2K-H 型行星齿轮减速器的动态特性分析.西安工业 大学学报.1997,17(1):83-86
[38] Shu Xiao-Long. Determination of Load Sharing Factor for Planetary Gearing with Small Tooth Number Difference. Mech.Mach. Theoey . 1997,10(2): 313 -321
[39] A.Kahraman.StaticLoadSharingCharacteristicsofTransmissionPlanetaryGear Sets: Model and Experiment. Transmission and Driveline System Symposium SAE Paper .1999-01-1050
[40] V.Agashe. Computational Analysis of the Dynamic Response ofa Planetary Gear System. Master Thesis, Ohio StateUniversity.1998
[41] R.G.Parker. A Physical Explanation for the Effectiveness of Planet Phasing to Suppress Planetary Gear Vibration. Journal of Sound and Vibration. 2000,236(4):561-573
[42] G.Robert,A.Parker.Physical Explanation for the Effectiveness of Planetary Phasing to Suppress Planetary Gear Vibration .Journal of Sound and Vibration .2000, 236(4):561-573
[43] 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):44-52
[44] 孙涛,沈允文,孙智民,刘继岩.行星齿轮传动非线性动力学建模与方程. 机 械工程学报.2002,38(3):6-10
[45] 孙涛,沈允文,孙智民,刘继岩. 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
[46] A.Toda, M.Botman. Planet Indexing In Planetary Gears for Minimum Vibration. ASME Design Engineering Technical Conference,St.Louis. 1979:35-78
[47] R.G Parker, V.Agashe, S.M.Vijayakar. Dynamic Response of a Planetary Gear System Using a Finite Element/Contact Mechanics Model. Journal of Sound and Vibration.2000,122(3):305-311
[48] D.l.Seager. Condations for Neutralization by the Teeth in Epicyclic Gearing.Journal of Mechanical Engineering Science. 1975,17:293-298
[49] 谢志江,朱才朝.带运动副间隙的内齿行星齿轮传动动力学研究.机械设计 与制造.1999,6:44-46
[50] J.Lin,R.G.Parker. Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration. Journal of Vibration andAcoustics. 1999, 121:316-321
[51] 孙涛,胡海岩.基于离散傅立叶变换与谐波平衡法的行星齿轮系统非线性动 力学分析. 机械工程学报.2002,38(11):58-61
[52] A.Bier,L.Demkowicz. Dynamic Contact/Impact Problems, Energy Conservation, and Planetary Gear Trains. Computer method in Applied Mechanics and Engineering.2002,191:4159-4194
[53] A.Kahraman, G.W. Blankenship. Effect of Involute Contact Ratio on Spur Gear Dynamics. ASME Journal of Mechanical Design. 1999,121(1):112-118
[54] M.I.Friswell.The Derivatives of Repeated Eigenvalues and their Eigensensitivity with Repeated Roots. Journal of Vibration and Acoustics.1996,118:390-397
[55] Gyula,Mester,Szilvester,Pletl,Gizella,Pajior,Zoltan,Jeges.Flexible Planetary Gear Drives in Robotics. IEEE.1992: 0-7803-0582-5
[56] M.Botman. Epicyclic Gear Vibrations. Journal of Engineering for Industry. 1976:75-DET-14
[2] 马从谦.渐开线行星齿轮传动.机械工业出版社.1987:11-78
[3] 李建军.齿轮系统动力学—振动·冲击·噪声.科学出版社.1997:69-247
[4] R.L.Platt,R.D.Leipold. A Study on Helical Gear Planetary Phasing Effects on Transmission noise. VDI Beriche.1996,123:793-807
[5] 屈维得,唐恒龄.机械振动手册.机械工业出版设.2000:30-77
[6] T.Chiang,R.H.Badgley. Reduction of Vibration and Noise Generated by Planetary Ring Gears in Helicopter Aircraft Transmission. Journal of Engineering for Industry. 1973,93:1149-1158
[7] F.Cunliffe.J.D.Smith.D.B.Welbourn.Dynamic Tooth Loads in Epicyclic Gears. Journal of Engineering for Industry. 1974,94:578-584
[8] M.Botman.Epicyclic Gear Vibration.Journal of Engineering for Industry. 1976,96:811-815
[9] J.L.Frater.R.August,F.B.Oswald. Vibration in Planetary Gear System withUnequal Planet Stiffness. NASATechnical Memorandum. 1983,8:28-34
[10] Mark, J.Valco. Planetary Gear Train Ring Gear and Support Structure Investigation. Cleveland State University.1992:86-254
[11] P.Velex, L.Flamand.Dynamic Response of Planetary Trains to Mesh Parameter Excitations.ASME Journal of Mechanical Design.1996,118:7-14
[12] J.Lin,R.G.Parker.AnalyticalCharacterizationoftheUniquePropertiesofPlanetary Gear Free Vibration. Journal of Vibration andAcoustics. 1999,121:316-321
[13] J.Lin,R.G.Parker. Structured Vibration Characteristics of Planetary Gears with Unequally Spaced Planets. Journal of Sound and Vibration. 2000,233(5):921-928
[14] 杨建明. 行星齿轮机构弹性动力学建模. 桂林电子工业学院学报.2000,20(2):48-52
[15] Ahmet,Kahraman.Free Torsional Vibration Characteristics of Compound Planetary Gear Sets. Mechanism and Machine Theory. 2001,36:953-971
[16] A.W.Leissa. On a Curve Veering Aberration .Journal of Applied Mathematics and Physics.1974,106:451-463
[17] N.C.Perkins,C.D.Mote. Comments on Curve Veering in Eigenvalue Problems. Journal of Sound Vibration .1986,106: 451-463
[18] A.Kahraman. Planetary Gear Train Dynamics. ASME Journal of Mechanical Design.1994:116(3):713-720
[19] J.Lin,R.G.Parker. Sensitivity of Planetary Gear Natural Frequencies and Vibration Modes to Model Parameter. Journal of Sound and Vibration. 1999,228(1):109-128
[20] 朱才朝,秦大同,李润方,陈宗源.内齿行星齿轮传动动态特性的研究.重庆 工业大学学报.1997,20(5):20-32
[21] G.Bollinger,R.J.Harker. Instability Potential of High Speed Gear . Journal oof the ndustrial Mathematics. 1967,17:39-55
[22] M.Amabili, A.Rivola. Dynamic Analysis of Spur of Gear Pairs: Steady State Response and Stability of the SDOF Model with Time-Varying Meshing Damping. Mechanical Systems and Signal Processing. 1997, 11(3):357-390
[23] M.Benton,A.Seireg. Factors Influencing Instability and Resonances in Geared Systems.ASME Journal of Mechanical Design. 1981,103(2): 372-378
[24] A.Kahraman,G.W.Blankenship. Experiments on Nonlinear Dynamic Behavior of an Oscillator with Clearance and Periodically Time-Varying Parameters. Journal ofApplied Mechanics.1997,64: 217-226
[25] C.Nataraj,A.M.Whitman.Parameter Excitation Effects in Gear Dynamics. ASME Design Engineering .Technical Conferences. 1997: DETC97/VB-4018, Sacrame- nto,CA.
[26] C.Nataraj, N.K.Arakere .Dynamic Response and Stability of a Spur GearPair. ASME Design Engineering Technical Conferences. 1999:DETC99/VIB-8110. Las Vegas
[27] M.Benton,A.Seireg.ApplicationoftheRitzAveragingMethodtoDeterminingthe Response of Systems with Time Varying Stiffness to Harmonic Excitation . ASME Journal of Mechanical Design. 1980,102(2):384-390
[28] M.Botman. Vibration Measurements on Planetary Gears of Aircraft Engines. AIAAJournal.1980,17:351-357
[29] T.Hidaka,Y.Terauchi,K.Nagamura. Dynamic Behavior of PlanetaryGear-1st Report: Load Distribution. Bull.Jsme.1976,19(132): 690-698
[30] R.Kasuba,R.August. Gear Mesh Stiffness and Load Sharing in Planetary Gearing.1984:ASME Paper 84-DET-229
[31] P.Ma,M.Botman. Load Sharing in a Planetary Gear Stage in the Presence of Gear Errors and Misalignment. Journal of Mechanisms. Transmissions and Automation in Design.1984,104:1-7
[32] R.August, R.Kasuba. Torsional Vibrations and Dynamic Loads in a Basic Planetary Gear System. Journal of Vibration, Acoustics, Stress and Reliability in Design .1986,108(3):348-353
[33] P.Velex, L.Flamand. Dynamic Response of Planetary Trains to Mesh Parametric Excitations ,ASME Journal of Mechanical Design.1996,118:7-14
[34] A.Kahraman. Dynamic Analysis of a Multi-Mesh Helical Gear Trains. ASME Journal of Mechanical Design.1994,116(3):706-712
[35] A.Kahraman, G.W. Blankenship. Planet Mesh Phasing in Epicyclic Gear Sets. Proc. of International Gearing Conference.Newcastle,UK. 1994: 99-104
[36] A.Kahraman. Natural Modes of Planetary Gear Trains. Journal of Sound and Vibration .1994,173(1):125-130
[37] 张建云,阮忠唐,丘大谋.2K-H 型行星齿轮减速器的动态特性分析.西安工业 大学学报.1997,17(1):83-86
[38] Shu Xiao-Long. Determination of Load Sharing Factor for Planetary Gearing with Small Tooth Number Difference. Mech.Mach. Theoey . 1997,10(2): 313 -321
[39] A.Kahraman.StaticLoadSharingCharacteristicsofTransmissionPlanetaryGear Sets: Model and Experiment. Transmission and Driveline System Symposium SAE Paper .1999-01-1050
[40] V.Agashe. Computational Analysis of the Dynamic Response ofa Planetary Gear System. Master Thesis, Ohio StateUniversity.1998
[41] R.G.Parker. A Physical Explanation for the Effectiveness of Planet Phasing to Suppress Planetary Gear Vibration. Journal of Sound and Vibration. 2000,236(4):561-573
[42] G.Robert,A.Parker.Physical Explanation for the Effectiveness of Planetary Phasing to Suppress Planetary Gear Vibration .Journal of Sound and Vibration .2000, 236(4):561-573
[43] 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):44-52
[44] 孙涛,沈允文,孙智民,刘继岩.行星齿轮传动非线性动力学建模与方程. 机 械工程学报.2002,38(3):6-10
[45] 孙涛,沈允文,孙智民,刘继岩. 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
[46] A.Toda, M.Botman. Planet Indexing In Planetary Gears for Minimum Vibration. ASME Design Engineering Technical Conference,St.Louis. 1979:35-78
[47] R.G Parker, V.Agashe, S.M.Vijayakar. Dynamic Response of a Planetary Gear System Using a Finite Element/Contact Mechanics Model. Journal of Sound and Vibration.2000,122(3):305-311
[48] D.l.Seager. Condations for Neutralization by the Teeth in Epicyclic Gearing.Journal of Mechanical Engineering Science. 1975,17:293-298
[49] 谢志江,朱才朝.带运动副间隙的内齿行星齿轮传动动力学研究.机械设计 与制造.1999,6:44-46
[50] J.Lin,R.G.Parker. Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration. Journal of Vibration andAcoustics. 1999, 121:316-321
[51] 孙涛,胡海岩.基于离散傅立叶变换与谐波平衡法的行星齿轮系统非线性动 力学分析. 机械工程学报.2002,38(11):58-61
[52] A.Bier,L.Demkowicz. Dynamic Contact/Impact Problems, Energy Conservation, and Planetary Gear Trains. Computer method in Applied Mechanics and Engineering.2002,191:4159-4194
[53] A.Kahraman, G.W. Blankenship. Effect of Involute Contact Ratio on Spur Gear Dynamics. ASME Journal of Mechanical Design. 1999,121(1):112-118
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