不同因素激励下齿轮传动动力学仿真及实验研究
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摘要
课题来源于国家“十一五”科技支撑计划“高速、重载、精密机械传动系统共性关键技术研究”(2006BAF01B07-01)。齿轮传动是极其重要的机械传动形式，广泛应用于制造装备业的各个领域，其动态特性和振动噪声水平是重要的性能评价指标，直接影响相关产品品质。齿轮传动的动态特征一方面由系统所固有的动力学性能决定，另一方面则是由不同的激励因素下形成的不同动态响应表现出来。因此，从不同激励因素角度出发，探讨齿轮传动的动态激励-系统特性-响应特征关系才具有实际意义，这对于把握齿轮传动系统动力学设计本质，指导低振动低噪声的齿轮传动产品开发具有重要的理论和工程实用价值，对提升我国制造装备业的整体水平具有积极的促进作用。
本文主要从齿轮传动激励因素入手，以实际工程产品为研究载体，基于摩擦学、机械动力学，以理论分析、数值仿真、虚拟样机、实验验证为手段，对齿轮传动系统中相关的动态特性和减振降噪方法开展研究。研究内容主要包括：
①从齿面摩擦因素激励出发，基于不同齿轮摩擦模型获得时变摩擦系数分布特征，建立计及齿面时变摩擦力影响，同时考虑啮合刚度时变性的六自由度精确动力学模型，利用Runge-Kutta法进行数值仿真，分析了不同齿面微观特征下齿面摩擦因素对系统的动态激励及其对啮合线方向动传动误差的影响，并通过实验对结论进行了验证，明确了齿面摩擦激励对齿轮传动的动态特征的影响规律。
②从啮合线方向传动误差因素激励出发，讨论了齿轮传动误差计算的薄片理论并基于有限元方法进行了验证，以某机械式变速箱为研究载体，建立耦合动力学仿真模型，计算了固有模态频率和振型，分析了二档档位齿轮和主减齿轮传动误差激励下的模态柔度，获得了各轴承轴承力的动态响应以及箱体表面加速度响应；以齿轮传动误差为目标，应用遗传算法对档位齿轮的齿廓修形方案进行优化，并通过敏感性分析验证了优化后齿廓的传动误差对制造公差的稳健性，最终获得了从传动误差激励入手的减振降噪解决方案。
③从齿轮非线性拍击激励出发，研究齿轮传动由齿轮侧隙形成的拍击振动的动力学特性，先从单齿对拍击研究入手，利用集中参数法建立考虑齿轮侧隙、离合器多级刚度、拖曳力矩和载荷波动影响的非线性扭转振动动力学模型，利用数值仿真方法考察了多参数影响下的拍击动态特征，分析了拍击强度和拍击门槛值；最后建立了某五档机械式变速箱的拍击动力学仿真模型，评价了驱动工况下各档位的拍击强度，基于台架实验对仿真结果进行了验证，为控制齿轮传动中的拍击振动提供有益参考。
④从多个齿轮啮合激励信号的相位调制现象出发，研究了齿轮行星轮系中存在的非对称边带频特征和形成机理，建立了行星轮系调制边带频分析模型，探讨了行星轮系的齿数、啮合相位等基本参数以及激励阶次对边带频特征的影响，获得可以根据已知参数预估行星轮系边带频特征的通用判定方法，并通过这种方法来改善由非对称边带频引起的噪声问题，最后在实际工程设计中，利用实验验证了该方法的有效性。
⑤针对某开发中的手动变速箱开展动力学实验研究，分别搭建了传动误差和动态特性测试平台，比较档位齿轮传动误差的仿真和实验结果较为吻合，获得变速箱不同工况下的固有特性和动态响应，基于阶次跟踪方法分析了主要的振动成分，对比了仿真与实验以及传动误差优化前后的结果，验证了从传动误差激励入手减小振动噪声的有效性。
This research is supported by “11thFive-Year National Key Technology R&DProgram”, named “Key technology for high-speed, heavy-duty, precision mechanicaltransmission system”(2006BAF01B07-01). The gear transmission is an extremelyimportant form of mechanical transmission, which is widely used in various fields ofmanufacturing equipment industry. Its dynamic characteristics and vibration noiselevels are important indicators for the performance evaluation, which directly impact onthe quality of related products. The gear dynamics, on the one hand, decided that theinherent characteristics of the system, on the other hand, in the complex dynamicresponse under different motivational factors. Focusing on different motivationalfactors, invesitgating relationship of dynamic excitation, system characteristics andresponse characteristics, finding out the essence of dynamics design for geartransmission system, are vital important to guide vibration and noise reduction for gearproduct development theoretically and practically. To conduct relavant research has apositive role in promoting the overall level of manufacturing equipment industry.
The article emphasising on various motivating factor, applying in the actualengineering products, based on the tribological and mechanical dynamics theory and bymeans of theoretical analysis, numerical simulation, virtual prototyping andexperimental validation, carry out research for dynamic characteristics and noisereduction in the gear transmission. The research mainly include:
①With consideration of gear tooth surface excitation factor, first to investigatevarying friction coefficient distribution based on the different gear friction model.Established the accurate6DOF dynamic model including effects of friction and timevarying mesh stiffness. Using Runge-Kutta method for the numerical simulation, effectof gear surface friction, under various tooth surface microscopic features, on systemdynamic excitation and transmission errror in line of action has been studied. Resultwas verified by experiment, which clearly shows the characteristics of gear dynamicsconsidering the effect from gear friction.
②Focusing on the exciation from the transmission error in line of action,disscussed the “thin slice” theory to calculate transmission error (TE) which verified byfinite element method. Taking a manual transmission as research example, establishedcoupling dynamics model, found out inherent modal frequency and modal shape. Modal flexibility, as well as bearing dynamic load and acceleration response on housing, wereinvestigated under TE excitation for2ndspeed; Setting gear transmission error as target,optimized the gear profile modification by genetic algorithm method, and conductedsensitivity analysis of the the tooth profile with consideration of manufacturingtolerances. Fanally obtained a effective way to reduce noise and vibration from thetransmission error optimization.
③Aiming at excitation from gear impact, studied dynamic characteristics of gearrattle formed by the gear backlash. Starting with single gear pair rattle, establishednonlinear torsional vibration dynamics model using lumped parameter method,considering the gear backlash, clutch multi-stiffness, drag torque and load fluctuations.Investigated rattle dynamic characteristics under the influence of multi-parameter, rattleintensity and rattle threshold value; Finally established a rattle simulation analysismodel for five-speed manual transmission. Evaluated the rattle intensity for all speedsand verified the result by experimental studies. This provides a useful reference forrattling control in gear tansmission.
④Starting from the phase modulation of the multiple gear mesh excitation signal,the asymmetric sideband frequency characteristics and formation mechanism inplanetary gear train were investigated. Established a analysis model for modulationsideband frequency in automatic transmission with planetary gear train. Investigated theeffect of basic parameters, such as the number of teeth of the planetary gear train, themeshing phase, and meshing order on sideband frequency characteristics. Finallyobtained an universal method to estimate characteristics of sideband frequency inplanetary gear train. By carefully selecting the known parameters of planetary gear train,it is possible to reduce the asymmetric sideband frequency noise in design stage. Thecorrectness of the method was experimentally verify in the end.
⑤Dynamic experimental study carried out for a manual gearbox. Test rig fortransmission error and dynamic characteristics measurement were setted up respectively.Transmission error result from simulation and test were compared to show the goodagreement. The inherent dynamic characteristics and response of the gearbox underdifferent conditions were achieved. Based on order tracking method, the maincomponents of the vibration were analyzed. Comparing the results of simulation andexperiment, as well as before and after optimization of transmission error, verified theeffectiveness to reduce the noise and vibration from the transmission error excitaion.
本文主要从齿轮传动激励因素入手，以实际工程产品为研究载体，基于摩擦学、机械动力学，以理论分析、数值仿真、虚拟样机、实验验证为手段，对齿轮传动系统中相关的动态特性和减振降噪方法开展研究。研究内容主要包括：
①从齿面摩擦因素激励出发，基于不同齿轮摩擦模型获得时变摩擦系数分布特征，建立计及齿面时变摩擦力影响，同时考虑啮合刚度时变性的六自由度精确动力学模型，利用Runge-Kutta法进行数值仿真，分析了不同齿面微观特征下齿面摩擦因素对系统的动态激励及其对啮合线方向动传动误差的影响，并通过实验对结论进行了验证，明确了齿面摩擦激励对齿轮传动的动态特征的影响规律。
②从啮合线方向传动误差因素激励出发，讨论了齿轮传动误差计算的薄片理论并基于有限元方法进行了验证，以某机械式变速箱为研究载体，建立耦合动力学仿真模型，计算了固有模态频率和振型，分析了二档档位齿轮和主减齿轮传动误差激励下的模态柔度，获得了各轴承轴承力的动态响应以及箱体表面加速度响应；以齿轮传动误差为目标，应用遗传算法对档位齿轮的齿廓修形方案进行优化，并通过敏感性分析验证了优化后齿廓的传动误差对制造公差的稳健性，最终获得了从传动误差激励入手的减振降噪解决方案。
③从齿轮非线性拍击激励出发，研究齿轮传动由齿轮侧隙形成的拍击振动的动力学特性，先从单齿对拍击研究入手，利用集中参数法建立考虑齿轮侧隙、离合器多级刚度、拖曳力矩和载荷波动影响的非线性扭转振动动力学模型，利用数值仿真方法考察了多参数影响下的拍击动态特征，分析了拍击强度和拍击门槛值；最后建立了某五档机械式变速箱的拍击动力学仿真模型，评价了驱动工况下各档位的拍击强度，基于台架实验对仿真结果进行了验证，为控制齿轮传动中的拍击振动提供有益参考。
④从多个齿轮啮合激励信号的相位调制现象出发，研究了齿轮行星轮系中存在的非对称边带频特征和形成机理，建立了行星轮系调制边带频分析模型，探讨了行星轮系的齿数、啮合相位等基本参数以及激励阶次对边带频特征的影响，获得可以根据已知参数预估行星轮系边带频特征的通用判定方法，并通过这种方法来改善由非对称边带频引起的噪声问题，最后在实际工程设计中，利用实验验证了该方法的有效性。
⑤针对某开发中的手动变速箱开展动力学实验研究，分别搭建了传动误差和动态特性测试平台，比较档位齿轮传动误差的仿真和实验结果较为吻合，获得变速箱不同工况下的固有特性和动态响应，基于阶次跟踪方法分析了主要的振动成分，对比了仿真与实验以及传动误差优化前后的结果，验证了从传动误差激励入手减小振动噪声的有效性。
This research is supported by “11thFive-Year National Key Technology R&DProgram”, named “Key technology for high-speed, heavy-duty, precision mechanicaltransmission system”(2006BAF01B07-01). The gear transmission is an extremelyimportant form of mechanical transmission, which is widely used in various fields ofmanufacturing equipment industry. Its dynamic characteristics and vibration noiselevels are important indicators for the performance evaluation, which directly impact onthe quality of related products. The gear dynamics, on the one hand, decided that theinherent characteristics of the system, on the other hand, in the complex dynamicresponse under different motivational factors. Focusing on different motivationalfactors, invesitgating relationship of dynamic excitation, system characteristics andresponse characteristics, finding out the essence of dynamics design for geartransmission system, are vital important to guide vibration and noise reduction for gearproduct development theoretically and practically. To conduct relavant research has apositive role in promoting the overall level of manufacturing equipment industry.
The article emphasising on various motivating factor, applying in the actualengineering products, based on the tribological and mechanical dynamics theory and bymeans of theoretical analysis, numerical simulation, virtual prototyping andexperimental validation, carry out research for dynamic characteristics and noisereduction in the gear transmission. The research mainly include:
①With consideration of gear tooth surface excitation factor, first to investigatevarying friction coefficient distribution based on the different gear friction model.Established the accurate6DOF dynamic model including effects of friction and timevarying mesh stiffness. Using Runge-Kutta method for the numerical simulation, effectof gear surface friction, under various tooth surface microscopic features, on systemdynamic excitation and transmission errror in line of action has been studied. Resultwas verified by experiment, which clearly shows the characteristics of gear dynamicsconsidering the effect from gear friction.
②Focusing on the exciation from the transmission error in line of action,disscussed the “thin slice” theory to calculate transmission error (TE) which verified byfinite element method. Taking a manual transmission as research example, establishedcoupling dynamics model, found out inherent modal frequency and modal shape. Modal flexibility, as well as bearing dynamic load and acceleration response on housing, wereinvestigated under TE excitation for2ndspeed; Setting gear transmission error as target,optimized the gear profile modification by genetic algorithm method, and conductedsensitivity analysis of the the tooth profile with consideration of manufacturingtolerances. Fanally obtained a effective way to reduce noise and vibration from thetransmission error optimization.
③Aiming at excitation from gear impact, studied dynamic characteristics of gearrattle formed by the gear backlash. Starting with single gear pair rattle, establishednonlinear torsional vibration dynamics model using lumped parameter method,considering the gear backlash, clutch multi-stiffness, drag torque and load fluctuations.Investigated rattle dynamic characteristics under the influence of multi-parameter, rattleintensity and rattle threshold value; Finally established a rattle simulation analysismodel for five-speed manual transmission. Evaluated the rattle intensity for all speedsand verified the result by experimental studies. This provides a useful reference forrattling control in gear tansmission.
④Starting from the phase modulation of the multiple gear mesh excitation signal,the asymmetric sideband frequency characteristics and formation mechanism inplanetary gear train were investigated. Established a analysis model for modulationsideband frequency in automatic transmission with planetary gear train. Investigated theeffect of basic parameters, such as the number of teeth of the planetary gear train, themeshing phase, and meshing order on sideband frequency characteristics. Finallyobtained an universal method to estimate characteristics of sideband frequency inplanetary gear train. By carefully selecting the known parameters of planetary gear train,it is possible to reduce the asymmetric sideband frequency noise in design stage. Thecorrectness of the method was experimentally verify in the end.
⑤Dynamic experimental study carried out for a manual gearbox. Test rig fortransmission error and dynamic characteristics measurement were setted up respectively.Transmission error result from simulation and test were compared to show the goodagreement. The inherent dynamic characteristics and response of the gearbox underdifferent conditions were achieved. Based on order tracking method, the maincomponents of the vibration were analyzed. Comparing the results of simulation andexperiment, as well as before and after optimization of transmission error, verified theeffectiveness to reduce the noise and vibration from the transmission error excitaion.
引文
[1] Smith J D. Gear noise and vibration [M]. New York: Marcel Dekker, Inc.2003.
[2]李润方,王建军.齿轮系统动力学-振动、冲击、噪声[M].北京:科学出版社,1997.
[3] Nelson H D, McVaugh J M. The dynamics of rotor-bearing systems using finite elements [J].ASME Journal of Engineering for Industry,1985,2:593~600.
[4] Umezawa K. Vibration of power transmission helical gear with narrow face width [J].ASME Paper,1984:84-DET-159.
[5] Umezawa K, Houjoh H. The effect of shaft stiffness on the gear vibration [J]. Transaction ofJSME,1988, C54:699~705.
[6] Ozguven H N. Assessment of some recently developed mathematical models in geardynamics [J]. Proceedings of the8thWorld Congress on the Theory of Machines andMechanisms, Prague,1991:605~608.
[7] OZguzen H N, Houser D R. Dynamic analysis of high speed gears by using loaded statictransmission error [J]. Journal of Sound and Vibration,1998,125(1):71~83.
[8] Tsuta T. Excitation force analysis of helical gear pair tolerance in their tooth shape and pitchmounted on f1exible shaft [C]. Proceedings of International Conference on Motion andPower Transmission, Hiroshima,1991:72~77.
[9] Neriya S V, Bhat R B, Sankar T S. On the dynamic response of a helical geared systemsubjected to a static transmission error in the form of deterministic and filtered white noiseinputs [J]. Journal of Vibration, Acoustics, Stress, and Reliability in Design,1988,110(4):501~506.
[10] Kahraman A. Effect of axial vibration on the dynamic of a helical gear pair [J]. Journal ofVibration and Acoustics,1993,115(1):33~39.
[11] Kahraman A, Ozguven H N, Houser D R. Dynamic analysis of geared rotors by finiteelements [J]. Journal of Mechanical Design,1992,114(3):507~515.
[12] Velex P, Maatar M. A mathematical model for analyzing the influence of shape deviationsand mounting errors on gear dynamic behavior [J]. Journal of Sound and Vibration,1996,191(5):629~660.
[13] Iwatsubo N, Arii S, Kawai R. Coupled lateral-torsional vibration of rotor system trained bygears (Part I: Analysis by transfer matrix method)[J]. Bulletin of JSME,1984,27:271~277.
[14] Neriya S V, Bhat R B, Sanker T S. Effect of coupled torsional-flexural vibration of a gearedshaft system on the dynamic tooth loads [J].Shock and Vibration Bulletin,1984,54:67~76.
[15] Umezawa K. The developed simulator and the performance diagrams for the vibration ofhelical gear [C]. Proceeding of International Conference On Gears. Zhengzhou, China,1988:659~662.
[16] Tordion G V, Gouvin R. Dynamic stability of a two-stage gear train under the influence ofvariable meshing stiffness [J]. ASME Journal of Engineering for Industry, l997,99:785~791.
[17] Velex P, Sanda A. A Model for the Dynamic Behavior of Multi-Stage Geared System [J].Proceedings of the8thInternational Federation for the Theory of Machines and MechanismsWorld Congress, Prague,1991:621~625.
[18]刘更,沈允文等.齿轮参数对斜齿轮传动振动特性的影响[J].西北工业大学学报,1992,10(1):67-73.
[19]唐增宝,钟毅芳.直齿圆柱齿轮传动系统的振动分析[J].机械工程学报,1992,28(4).
[20]唐增宝.提高多级齿轮传动系统动态性能的优化设计[J].机械工程学报,1994,30(5):65~75.
[21]陶泽光,李润方,林腾蛟.齿轮系统有限元模态分析[J].机械设计与研究,2000(3):45~46.
[22]李润方,韩西,林腾蛟等.齿轮系统耦合振动的理论分析与实验研究[J].机械工程学报,2000,30(6):79~81.
[23]林腾蛟,李润方,陶泽光.齿轮传动三维间隙非线性冲击─动力接触特性数值仿真[J].机械工程学报,2000,30(6):55~58.
[24]郑光泽.齿轮传动系统动态性能优化分析研究[D].重庆:重庆大学,2004.
[25]李润方.齿轮传动的刚度分析和修形方法[M].重庆:重庆大学,2004
[26] Gregory R W, Harris S L, Munro R G. Dynamic behavior of spur gears [J]. Proc Inst MechEng,1963,178(8):207~226.
[27] Harris S L. Dynamic loads on the teeth of spur gears [J]. Proceeding Institution MechanicalEngineer,1958,172:87~112.
[28] Honda S. Rotational vibration of a helical gear pair with modified tooth surfaces-modifiedtooth surface and its equivalent tooth profile [J]. JSME International Journal, Series3:Vibration Control Engineering for Industry,1993,36:125~134.
[29] Velex P, Ajmi M. On the modeling of excitations in geared systems by transmission errors[J]. Journal of Sound and Vibration,2006,290(3):882~909.
[30] Tavakoli M S, Houser D R. Optimum profile modifications for the minimization of statictransmission errors of spur gears [J]. ASME Journal of Mechanisms, Transmissions andAutomation in Design,1996,108(3):86~90.
[31] Chang S L, Tsay C B, Tseng C H. Kinematic optimization of a modified helical simple gearTrain [J]. ASME Journal of Mechanical Design,1997,119(2):307~314.
[32] Sundaresan S, Ishii K, Houser D R. Design optimization for robustness using performancesimulation programs [J]. ASME Journal of Design Automation,1991,32:249~256.
[33] Taguchi G. Taguchi methods: design of experiments [M]. Dearborn, MI: ASI Press,1993.
[34] Fonseca D J, Shishoo S, Lim T C. A genetic algorithm approach to minimize transmissionerror for automotive spur gear sets [J]. Applied Artificial Intelligence,2005,19(2):153~179.
[35] Bonori G, Barbieri M, Pellicano F. Optimum profile modifications of spur gears by meansof genetic algorithms [J]. Journal of Sound and Vibration,2008,313(3):603~616.
[36] Beghini M, Presicce F, Santus C. A method to define profile modification of spur gear andminimize the transmission error [J]. AGMA Technical Paper,04FTM3,2004.
[37]唐增宝,钟毅芳,陈久荣.修形齿轮的最佳修形量和修形长度的确定[J].华中理工大学学报,1995,23(2):125~128.
[38]朱传敏,秦慧芳,宋孔杰等.齿轮传动动态系统辨识及修形研究[J].机械工程学报,1999,35(4):60~63.
[39]张树生,刘增文,庞守美.齿轮传动动态性能的优化设计-最优修形曲线的确定[J].中国机械工程,1999,10(3):245~248.
[40]石照耀,康焱,林家春.基于齿轮啮合副整体误差的直齿轮动力学模型及其动态特性[J].机械工程学报.2010,46(17):55~61.
[41] Ozguven H N, Houser D R. Mathematical models used in gear dynamics-a review [J].Journal of Sound and Vibration,1988,121(3):383~411.
[42] Vaishya M, Houser D R. Modeling and measurement of sliding friction for gear analysis [J].AGMA Technical Paper,1999,99FTMS1:1~12.
[43] Houser D R, Vaishya M, Sorenson J D. Vibro-acoustic effects of friction in gears: anexperimental investigation [J]. SAE Paper,2001,#2001-01-1516.
[44] Borner J, Houser D R. Friction and bending moments as gear noise excitations [J]. SAEPaper,1996,#961816.
[45] Kim S, Singh R. Gear surface roughness induced noise prediction based on a lineartime-varying model with sliding friction [J]. Journal of Vibration and Control.2007,13(7):1045~1063.
[46] Iida H, Tamura A, Yamada Y. Vibrational characteristics of friction between gear teeth [J].Bulletin of JSME,1985,241(28):1512~1519.
[47] Hochmann D. Friction force excitations in spur and helical involute parallel axis gearing [D].PhD Dissertation, The Ohio State University,1997.
[48] Vaishya M, Singh R. Strategies for modeling friction in gear dynamics [J]. ASME Journal ofMechanical Design,2003,125:383~393.
[49] Vaishya M, Singh R. Analysis of periodically varying gear mesh systems with Coulombfriction using Floquet theory [J]. Journal of Sound and Vibration,2001,243(3):525~545.
[50] Vaishya M, Singh R. Sliding friction-induced non-linearity and parametric effects in geardynamics [J]. Journal of Sound and Vibration,2001,248(4):671~694.
[51] Velex P, Cahouet V. Experimental and numerical investigations on the influence of toothfriction in spur and helical gear dynamics [J]. ASME Journal of Mechanical Design,2000,122(4):515~522.
[52] Velex P, Sainsot P. An analytical study of tooth friction excitations in spur and helical gears[J]. Mechanism and Machine Theory,2002,37(7):641~648.
[53] Lundvall O, Str mberg N, Klarbring A. A flexible multi-body approach for frictional contactin spur gears [J]. Journal of Sound and Vibration,2004,278(3):479~499.
[54] Liu G, Park R G. Impact of tooth friction and its bending effect on gear dynamics [J].Journal of Sound and Vibration,2009,320(4):1039~1063.
[55] He S, Gunda R, Singh R. Effect of sliding friction on the dynamics of spur gear pair withrealistic time-varying stiffness [J]. Journal of Sound and Vibration,2007,301(3):927~949.
[56] He S, Gunda R, Singh R. Inclusion of sliding friction in contact dynamics model for helicalgears [J]. ASME Journal of Mechanical Design,2007,129(1):48~57.
[57] He S, Rook T, Singh R. Construction of semi-analytical solutions to spur gear dynamicsgiven periodic mesh stiffness and sliding friction functions [J]. ASME Journal ofMechanical Design,2008,130:1226011~1226019.
[58] He S, Cho S, Singh R. Prediction of dynamic friction forces in spur gears using alternatesliding friction formulations [J]. Journal of Sound and Vibration,2008,309(3):843~851.
[59]王三民，沈允文，董海军.含摩擦和间隙直齿轮副的混沌与分叉研究[J].机械工程学报,2002,38(9):8~11.
[60]孙月海，张策，潘凤章等.直齿圆柱齿轮传动系统振动的动力学模型[J].机械工程学报,2000,36(8):47~50.
[61]陈思雨，唐进元.间隙对含摩擦和时变刚度的齿轮系统动力学响应的影响[J].机械工程学报,2009,45(8):119~124.
[62]冯治恒，王时龙，雷松.时变摩擦因数对弧齿锥齿轮动力学特性的影响[J].中国机械工程,2011,22(2):148~152.
[63] Benedict G H, Kelley B W. Instantaneous coefficients of gear tooth friction [J]. Transactionsof the American Society of Lubrication Engineers,1961,4:59~70.
[64] Xu H, Kahraman A, Anderson N E, Maddock D G. Prediction of mechanical efficiency ofparallel-axis gear pairs [J]. ASME Journal of Mechanical Design,2007,129(1):58~68.
[65] Xu H. Development of a generalized mechanical efficiency prediction methodology [D].PhD dissertation, The Ohio State University,2005.
[66] Seireg A A. Friction and Lubrication in Mechanical Design [M]. Marcel Dekker Inc,1998,New York.
[67] Baranov V M, Kudryavtsev E M, Sarychev G A. Modeling of the parameters of acousticemission under sliding friction of solids [J]. Wear,1997,202:125~133.
[68] Drozdov Y N, Gavrikov Y A. Friction and scoring under the conditions of simultaneousrolling and sliding of bodies [J]. Wear,1968,11:291~302.
[69] Duan C, Singh R. Super-Harmonics in a torsional system with dry friction path subject toharmonic excitation under a mean torque [J]. Journal of Sound and Vibration,2005,285:803~834.
[70] Hamrock B J, Dowson D. Isothermal elasto-hydrodynamic lubrication of point contacts,Part III–Fully flooded results [J]. Journal of Lubrication Technology,1977,99(2):264~276.
[71]王建军,陆明万,张雄等.汽车变速系统的拍击动力学基本理论[J].机械科学与技术,1997,16(3):392~297.
[72]王建军,李润方.齿轮系统间隙非线性振动研究综述[J].非线性动力学学报,1995,2(3):214~221.
[73]王建军,李其汉,李润方.齿轮系统非线性振动研究进展[J].力学进展,2005,35(1):37~51.
[74] Ozguven H N, Houser D R. Mathematical models used in gear dynamics--a review [J].Journal of Sound and Vibration,1988,121:383~411.
[75] Comparin R J, Singh R. Non-linear frequency response characteristics of an impact pair [J].Journal of Sound and Vibration,1989,134:259~290.
[76] Comparin R J, Singh R. Frequency response characteristics of a multi-degree-of freedomsystem with clearances [J]. Journal of Sound and Vibration,1990,142(1):101~124.
[77] Comparin R J. A study of the frequency response of impact pairs with application toautomotive gear rattle [D]. PhD Dissertation, Ohio State University,1988.
[78] Comparin R J, Singh R. An analytical study of automobile neutral gear rattle [J]. ASMEMechanical Design,1990,112:237~245.
[79] Karaman A, Singh R. Non-linear dynamics of a spur gear pair [J]. Journal of Sound andVibration,1990,142(1):49~75.
[80] Karaman A, Singh R. Non-linear dynamics of a geared rotor-bearing system with multipleclearances [J]. Journal of Sound and Vibration,1991,144(3):469~506.
[81] Padmanabhan C, Singh R. Spectral coupling issues in a two degree of freedom system withclearance non-linearity [J]. Journal of Sound and Vibration,1992,155(2):209~230.
[82]冯奇.随机调幅Rattling系统建模[J].应用数学和力学,1999,20(1):85~92.
[83]冯奇. Rattling系统的一种随机模型[J].同济大学学报,1998,26(4):396~400.
[84]温建明,冯奇.随机调幅Rattling振动系统的二级传动模型[J].振动工程学报,2000,13(3):353~360.
[85] Feng Q, Pfeiffer F. Stochastic model on a rattling system [J]. Journal of Sound and Vibration,1998,215(3):439~453.
[86]董海军,沈允文,刘梦军.齿轮系统Rattling动力学行为研究[J].机械工程学报,2004,40(1):136~141.
[87]董海军,沈允文,刘梦军.转速激励齿轮系统拍击振动的分岔特性Rattling动力学行为研究[J].机械工程学报,2004,40(1):136~141.
[88]董海军,沈允文,刘梦军等.齿轮系统的拍击振动分析模型[J].中国机械工程,2003,14(16):1416~1419.
[89]张锁怀,李忆平,丘大谋.齿轮耦合的转子-轴承系统非线性动力特性的研究[J].机械工程学报,2001,37(9):53～57.
[90]张锁怀,沈允文,董海军等.齿轮时变系统对扭矩激励的响应[J].航空动力学报,2003,18(6):737~743.
[91]张锁怀,沈允文,董海军等.单级齿轮系统排击特性[J].机械工程学报,2003,29(3):28~31.
[92] Sakai T, Doi Y, Yamamoto K, Ogasawara T, etc. Theoretical and experimental analysis ofrattling noise of automotive gearbox[J], SAE Technical Paper,1981,810773
[93] Singh R. Analysis of automotive gear rattles [J]. Journal of Sound and Vibration,1989,131(2):177~196.
[94] Padmanabhan C. Modeling of automotive gear rattle phenomenon: state of the art [J]. SAETechnical Series,1995,951316:669~679.
[95] Pfeiffer F, Prestl W. Decoupling measures for rattling noises in gear boxes[J]. IME PaperC404/025, Institute of Mechanical Engineers,1990:129~134.
[96] Sebulke A. The two mass flywheel-A torsional vibration damper for the power train ofpassenger cars-state of the art and further technical development [C]. SAE Technical PaperSeries, International Congress and Exposition, Detroit, MI,1987.
[97] Drexl H J. Torsional dampers and alternative systems to reduce drive-line vibrations [C].SAE Technical Paper Series, International Congress and Exposition, Detroit, MI,1987.
[98] Padmanabhan C, Singh R. Influence of clutch design on the reduction and perception ofautomotive transmission rattle noise [C]. Noise Conference, Williamsburg, VA,1993.
[99] Robinette D, Beikmann R S, Piorkowski P. Characterizing the onset of manual transmissiongear rattles Part I: experimental results [J]. SAE International Journal of Passenger Cars–Mechanical System,2009(2):1352~1364.
[100] Robinette D, Beikmann R S, Piorkowski P. Characterizing the onset of manual transmissiongear rattle Part II: analytical results [J]. SAE International Journal of Passenger Cars–Mechanical System,2009,2:1367~1376.
[101] Kadmiri Y, Rigaud E, Perret-Liaudet J, Vary L. Experimental and numerical analysis ofautomotive gearbox rattle noise [J]. Journal of Sound and Vibration.2012,331(13):3144~3157.
[102] Barthod M, Hayne B, Tebec J L, Pin J C. Experimental study of gear rattle excited by amulti-harmonic excitation [J]. Applied Acoustics,2007,68(9):1003~1025.
[103] Wang M Y, Manoj R, Zhao W. Gear rattle modeling and analysis for automotive manualtransmissions [J]. Proceedings of the Institution of Mechanical Engineers, Part D: Journal ofAutomobile Engineering,2001:215~241.
[104] Cruz M D, Theodossiades S, Rahnejat H. An investigation of manual transmission driverattle [J]. Proceeding of Institution of Mechanical Engineers, Part K: J. Multi-bodyDynamics,2009,224:167~181.
[105] Rocca E, Russo R. Theoretical and experimental investigation into the influence of theperiodic backlash fluctuations on the gear rattle [J]. Journal of Sound and Vibration,2011,330(20):4738~4752.
[106] Russo R, Brancati R. Rocca E. Experimental investigations about the influence of oillubricant between teeth on the gear rattle phenomenon [J]. Journal of Sound and Vibration,2009,321(3):647~661.
[107] Dogan S N. Loose part vibration in vehicle transmissions-gear rattle [J]. Journal ofEngineering and Environmental Science,1999,23:439~454.
[108] Tangasawi O, Theodossiades S, Rahnejat H, Kelly P. Non-linear vibro-impact phenomenonbelying transmission idle rattle [J]. Proceeding of Institution of Mechanical Engineers, PartC: Journal of Mechanical Engineering Science,2008,222:1909~1923.
[109] Tangasawi O, Theodossiades S, Rahnejat H. Lightly loaded lubricated impacts: idle gearsrattle [J]. Journal of Sound and Vibration,2007.308(3):418~430.
[110] Theodossiades S, Tangasawi O, Rahnejat H. Gear teeth impacts in hydrodynamicconjunctions promoting idle gear rattle [J]. Journal of Sound and Vibration,2007,303(3):632~658.
[111] Ottewill J R, Neild S A, Wilson R E. An investigation into the effect of tooth profile errorson gear rattles [J]. Journal of Sound and Vibration,2010,329(3):3495~3506.
[112] Han B K, Cho M K, KIM C. Prediction of vibration forces on meshing gear for a gear rattleusing a new multi-body dynamic model [J]. International Journal of Automotive Technology,2009,10(4):469~474.
[113] Yakoub R Y. Prediction of System-level gear rattles using multi-body and vibro-acoustictechniques [J]. SAE paper,2004.
[114] Inalpolat M, Kahraman A. Dynamic modelling of planetary gears of automatictransmissions [J]. Proceedings of the Institution of Mechanical Engineers, Part K: Journal ofMulti-body Dynamics,2008,222:229~242.
[115] Hidaka T, Terauchi Y, Ishioka K. Dynamic behavior of planetary gear (2ndreport:displacement of sun gear and ring gear)[J], Bulletin of the JSME,1976,19(138):1563~1570.
[116] Kahraman A. Planetary gear train dynamics [J]. ASME Journal of Mechanical Design,1994,116:713~720.
[117] Singh A, Kahraman A, Ligata H. Internal gear strains and load sharing in planetarytransmissions-model and experiments [J]. ASME Journal of Mechanical Design,2008,130(7):072602-1~072602-10.
[118] Kahraman A. Load sharing characteristics of planetary transmissions [J]. Mechanism andMachine Theory,1994,29(8):1151~1165.
[119] Lin J, Park R G. Planetary gear parametric instability caused by mesh stiffness variation [J].Journal of Sound and Vibration,2002,249(1):129~145.
[120]杨通强,宋轶民,张策等.斜齿行星齿轮系统自由振动特性分析[J].机械工程学报,2005,7:50~55.
[121]杨富春,周晓军,郑津洋.复式行星齿轮传动系统综合动力学模型及振动特性研究[J].振动与冲击,2011,8:144~148.
[122]孙涛,沈允文,孙智民等.行星齿轮传动非线性动力学模型与方程[J].机械工程学报，2002,38(3):6~10.
[123]孙涛,沈允文,孙智民等.行星齿轮传动非线性动力学方程求解与动态特性[J].机械工程学报,2002,38(3):11~15.
[124]李同杰,朱如鹏,鲍如云等.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,2011,21:76~83.
[125] Randall R B. A new method of modeling gear faults [J]. ASME Journal of MechanicalDesign,1982,104:259~267.
[126]丁康.齿轮及齿轮箱故障诊断实用技术[M].机械工业出版社,北京,2009.
[127] McFadden P D, Smith J D. An explanation for the asymmetry of the modulation sidebandsabout the tooth meshing frequency in epicyclic gear vibration [J]. Proceedings of theInstitution of Mechanical Engineers,1985,199(1):65~70.
[128] McFadden P D. A technique for calculating the time domain averages of the vibration of theindividual planet gears and the sun gear in an epicyclic gearbox [J]. Journal of Sound andVibration,1991,144(1):163~172.
[129] McNames J. Fourier series analysis of epicyclic gearbox vibration [J]. ASME Journal ofVibration and Acoustics,2002,124(1):150~152.
[130] Schlegel R G, Mard K C. Transmission noise control-approaches in helicopter design [C].ASME Design Engineering Conference, New York,1967,67-DE-58.
[131] Seager D L. Conditions for the neutralization of excitation by the teeth in epicyclic gearing[J]. Journal of Mechanical Engineering Science,1975,17(5):293~299.
[132] Yu S, Kaatz S. Asymmetric gear noise sidebands and application to planetary gear noisereduction [C]. SAE Noise and Vibration Conference, Michigan,2005,01:2462
[133] Lin J, Park R G. Analytical characterization of the unique properties of planetary gear freevibration [J]. Journal of Sound and Vibration,1999,121:316~321.
[134] Parker R G. A physical explanation for the effectiveness of planet phasing to suppressplanetary gear vibration [J]. Journal of Sound and Vibration,2000,236(4):561~573.
[135] Parker R G, Lin J. Mesh phasing relationship in planetary and epicyclic gears [J]. Journal ofMechanical Design,2004,126:365~370.
[136] Inalpolat M, Kahraman A. A dynamic model to predict modulation sidebands of a planetarygear set having manufacturing errors [J]. Journal of Sound and Vibration,2010,329(4):371~393.
[137] Inalpolat M, Kahraman A. A theoretical and experimental investigation of modulationsidebands of planetary gear sets [J]. Journal of Sound and Vibration,2009,323:677~696.
[138] Chen Y, Ishibashi A. Investigation of the noise and vibration of planetary gear drives [C].ASME Proceeding of Design Engineering Technical Conferences,2003,4:507~512.
[139] Wang S, Huo M, Zhang C, et al. Effect of mesh phase on wave vibration of spur planetaryring gear [J]. European Journal of Mechanics A/Solids,2011,30:820~827.
[140]秦大同，肖正明，王建宏.基于啮合相位分析的盾构机减速器多级行星齿轮传动动力学特性[J].机械工程学报,2011,23:20~29.
[141]王世宇.基于相位调谐的直齿行星齿轮传动动力学理论与实验研究[D].天津大学,2005.
[142]雷英杰,张善文等. Matlab遗传算法工具箱及应用[M].西安:西安电子科大出版社,2005.
[143] Seaman R L, Johnson C E, Hamilton. Component inertial effects on transmission design [J].SAE Technical Paper,00399048,1984.
[144] Szadkowski, A. Mathematical model and computer simulation of idle gear rattle [J]. SAETechnical Paper,960641,1991.
[145] Fyfe K R, Munck E D S. Analysis of computed order tracking [J]. Mechanical Systems andSignal Processing,1997,11(2):187~205.
[146] Bossley K M, Mckendrick R J, Harris C J, et al. Hybrid computed order tracking [J].Mechanical Systems and Signal Processing,1999,13(4):627~641.
[147]傅志方,华宏星.模态分析理论与应用[M].上海:上海交通大学出版社,2000.
[2]李润方,王建军.齿轮系统动力学-振动、冲击、噪声[M].北京:科学出版社,1997.
[3] Nelson H D, McVaugh J M. The dynamics of rotor-bearing systems using finite elements [J].ASME Journal of Engineering for Industry,1985,2:593~600.
[4] Umezawa K. Vibration of power transmission helical gear with narrow face width [J].ASME Paper,1984:84-DET-159.
[5] Umezawa K, Houjoh H. The effect of shaft stiffness on the gear vibration [J]. Transaction ofJSME,1988, C54:699~705.
[6] Ozguven H N. Assessment of some recently developed mathematical models in geardynamics [J]. Proceedings of the8thWorld Congress on the Theory of Machines andMechanisms, Prague,1991:605~608.
[7] OZguzen H N, Houser D R. Dynamic analysis of high speed gears by using loaded statictransmission error [J]. Journal of Sound and Vibration,1998,125(1):71~83.
[8] Tsuta T. Excitation force analysis of helical gear pair tolerance in their tooth shape and pitchmounted on f1exible shaft [C]. Proceedings of International Conference on Motion andPower Transmission, Hiroshima,1991:72~77.
[9] Neriya S V, Bhat R B, Sankar T S. On the dynamic response of a helical geared systemsubjected to a static transmission error in the form of deterministic and filtered white noiseinputs [J]. Journal of Vibration, Acoustics, Stress, and Reliability in Design,1988,110(4):501~506.
[10] Kahraman A. Effect of axial vibration on the dynamic of a helical gear pair [J]. Journal ofVibration and Acoustics,1993,115(1):33~39.
[11] Kahraman A, Ozguven H N, Houser D R. Dynamic analysis of geared rotors by finiteelements [J]. Journal of Mechanical Design,1992,114(3):507~515.
[12] Velex P, Maatar M. A mathematical model for analyzing the influence of shape deviationsand mounting errors on gear dynamic behavior [J]. Journal of Sound and Vibration,1996,191(5):629~660.
[13] Iwatsubo N, Arii S, Kawai R. Coupled lateral-torsional vibration of rotor system trained bygears (Part I: Analysis by transfer matrix method)[J]. Bulletin of JSME,1984,27:271~277.
[14] Neriya S V, Bhat R B, Sanker T S. Effect of coupled torsional-flexural vibration of a gearedshaft system on the dynamic tooth loads [J].Shock and Vibration Bulletin,1984,54:67~76.
[15] Umezawa K. The developed simulator and the performance diagrams for the vibration ofhelical gear [C]. Proceeding of International Conference On Gears. Zhengzhou, China,1988:659~662.
[16] Tordion G V, Gouvin R. Dynamic stability of a two-stage gear train under the influence ofvariable meshing stiffness [J]. ASME Journal of Engineering for Industry, l997,99:785~791.
[17] Velex P, Sanda A. A Model for the Dynamic Behavior of Multi-Stage Geared System [J].Proceedings of the8thInternational Federation for the Theory of Machines and MechanismsWorld Congress, Prague,1991:621~625.
[18]刘更,沈允文等.齿轮参数对斜齿轮传动振动特性的影响[J].西北工业大学学报,1992,10(1):67-73.
[19]唐增宝,钟毅芳.直齿圆柱齿轮传动系统的振动分析[J].机械工程学报,1992,28(4).
[20]唐增宝.提高多级齿轮传动系统动态性能的优化设计[J].机械工程学报,1994,30(5):65~75.
[21]陶泽光,李润方,林腾蛟.齿轮系统有限元模态分析[J].机械设计与研究,2000(3):45~46.
[22]李润方,韩西,林腾蛟等.齿轮系统耦合振动的理论分析与实验研究[J].机械工程学报,2000,30(6):79~81.
[23]林腾蛟,李润方,陶泽光.齿轮传动三维间隙非线性冲击─动力接触特性数值仿真[J].机械工程学报,2000,30(6):55~58.
[24]郑光泽.齿轮传动系统动态性能优化分析研究[D].重庆:重庆大学,2004.
[25]李润方.齿轮传动的刚度分析和修形方法[M].重庆:重庆大学,2004
[26] Gregory R W, Harris S L, Munro R G. Dynamic behavior of spur gears [J]. Proc Inst MechEng,1963,178(8):207~226.
[27] Harris S L. Dynamic loads on the teeth of spur gears [J]. Proceeding Institution MechanicalEngineer,1958,172:87~112.
[28] Honda S. Rotational vibration of a helical gear pair with modified tooth surfaces-modifiedtooth surface and its equivalent tooth profile [J]. JSME International Journal, Series3:Vibration Control Engineering for Industry,1993,36:125~134.
[29] Velex P, Ajmi M. On the modeling of excitations in geared systems by transmission errors[J]. Journal of Sound and Vibration,2006,290(3):882~909.
[30] Tavakoli M S, Houser D R. Optimum profile modifications for the minimization of statictransmission errors of spur gears [J]. ASME Journal of Mechanisms, Transmissions andAutomation in Design,1996,108(3):86~90.
[31] Chang S L, Tsay C B, Tseng C H. Kinematic optimization of a modified helical simple gearTrain [J]. ASME Journal of Mechanical Design,1997,119(2):307~314.
[32] Sundaresan S, Ishii K, Houser D R. Design optimization for robustness using performancesimulation programs [J]. ASME Journal of Design Automation,1991,32:249~256.
[33] Taguchi G. Taguchi methods: design of experiments [M]. Dearborn, MI: ASI Press,1993.
[34] Fonseca D J, Shishoo S, Lim T C. A genetic algorithm approach to minimize transmissionerror for automotive spur gear sets [J]. Applied Artificial Intelligence,2005,19(2):153~179.
[35] Bonori G, Barbieri M, Pellicano F. Optimum profile modifications of spur gears by meansof genetic algorithms [J]. Journal of Sound and Vibration,2008,313(3):603~616.
[36] Beghini M, Presicce F, Santus C. A method to define profile modification of spur gear andminimize the transmission error [J]. AGMA Technical Paper,04FTM3,2004.
[37]唐增宝,钟毅芳,陈久荣.修形齿轮的最佳修形量和修形长度的确定[J].华中理工大学学报,1995,23(2):125~128.
[38]朱传敏,秦慧芳,宋孔杰等.齿轮传动动态系统辨识及修形研究[J].机械工程学报,1999,35(4):60~63.
[39]张树生,刘增文,庞守美.齿轮传动动态性能的优化设计-最优修形曲线的确定[J].中国机械工程,1999,10(3):245~248.
[40]石照耀,康焱,林家春.基于齿轮啮合副整体误差的直齿轮动力学模型及其动态特性[J].机械工程学报.2010,46(17):55~61.
[41] Ozguven H N, Houser D R. Mathematical models used in gear dynamics-a review [J].Journal of Sound and Vibration,1988,121(3):383~411.
[42] Vaishya M, Houser D R. Modeling and measurement of sliding friction for gear analysis [J].AGMA Technical Paper,1999,99FTMS1:1~12.
[43] Houser D R, Vaishya M, Sorenson J D. Vibro-acoustic effects of friction in gears: anexperimental investigation [J]. SAE Paper,2001,#2001-01-1516.
[44] Borner J, Houser D R. Friction and bending moments as gear noise excitations [J]. SAEPaper,1996,#961816.
[45] Kim S, Singh R. Gear surface roughness induced noise prediction based on a lineartime-varying model with sliding friction [J]. Journal of Vibration and Control.2007,13(7):1045~1063.
[46] Iida H, Tamura A, Yamada Y. Vibrational characteristics of friction between gear teeth [J].Bulletin of JSME,1985,241(28):1512~1519.
[47] Hochmann D. Friction force excitations in spur and helical involute parallel axis gearing [D].PhD Dissertation, The Ohio State University,1997.
[48] Vaishya M, Singh R. Strategies for modeling friction in gear dynamics [J]. ASME Journal ofMechanical Design,2003,125:383~393.
[49] Vaishya M, Singh R. Analysis of periodically varying gear mesh systems with Coulombfriction using Floquet theory [J]. Journal of Sound and Vibration,2001,243(3):525~545.
[50] Vaishya M, Singh R. Sliding friction-induced non-linearity and parametric effects in geardynamics [J]. Journal of Sound and Vibration,2001,248(4):671~694.
[51] Velex P, Cahouet V. Experimental and numerical investigations on the influence of toothfriction in spur and helical gear dynamics [J]. ASME Journal of Mechanical Design,2000,122(4):515~522.
[52] Velex P, Sainsot P. An analytical study of tooth friction excitations in spur and helical gears[J]. Mechanism and Machine Theory,2002,37(7):641~648.
[53] Lundvall O, Str mberg N, Klarbring A. A flexible multi-body approach for frictional contactin spur gears [J]. Journal of Sound and Vibration,2004,278(3):479~499.
[54] Liu G, Park R G. Impact of tooth friction and its bending effect on gear dynamics [J].Journal of Sound and Vibration,2009,320(4):1039~1063.
[55] He S, Gunda R, Singh R. Effect of sliding friction on the dynamics of spur gear pair withrealistic time-varying stiffness [J]. Journal of Sound and Vibration,2007,301(3):927~949.
[56] He S, Gunda R, Singh R. Inclusion of sliding friction in contact dynamics model for helicalgears [J]. ASME Journal of Mechanical Design,2007,129(1):48~57.
[57] He S, Rook T, Singh R. Construction of semi-analytical solutions to spur gear dynamicsgiven periodic mesh stiffness and sliding friction functions [J]. ASME Journal ofMechanical Design,2008,130:1226011~1226019.
[58] He S, Cho S, Singh R. Prediction of dynamic friction forces in spur gears using alternatesliding friction formulations [J]. Journal of Sound and Vibration,2008,309(3):843~851.
[59]王三民，沈允文，董海军.含摩擦和间隙直齿轮副的混沌与分叉研究[J].机械工程学报,2002,38(9):8~11.
[60]孙月海，张策，潘凤章等.直齿圆柱齿轮传动系统振动的动力学模型[J].机械工程学报,2000,36(8):47~50.
[61]陈思雨，唐进元.间隙对含摩擦和时变刚度的齿轮系统动力学响应的影响[J].机械工程学报,2009,45(8):119~124.
[62]冯治恒，王时龙，雷松.时变摩擦因数对弧齿锥齿轮动力学特性的影响[J].中国机械工程,2011,22(2):148~152.
[63] Benedict G H, Kelley B W. Instantaneous coefficients of gear tooth friction [J]. Transactionsof the American Society of Lubrication Engineers,1961,4:59~70.
[64] Xu H, Kahraman A, Anderson N E, Maddock D G. Prediction of mechanical efficiency ofparallel-axis gear pairs [J]. ASME Journal of Mechanical Design,2007,129(1):58~68.
[65] Xu H. Development of a generalized mechanical efficiency prediction methodology [D].PhD dissertation, The Ohio State University,2005.
[66] Seireg A A. Friction and Lubrication in Mechanical Design [M]. Marcel Dekker Inc,1998,New York.
[67] Baranov V M, Kudryavtsev E M, Sarychev G A. Modeling of the parameters of acousticemission under sliding friction of solids [J]. Wear,1997,202:125~133.
[68] Drozdov Y N, Gavrikov Y A. Friction and scoring under the conditions of simultaneousrolling and sliding of bodies [J]. Wear,1968,11:291~302.
[69] Duan C, Singh R. Super-Harmonics in a torsional system with dry friction path subject toharmonic excitation under a mean torque [J]. Journal of Sound and Vibration,2005,285:803~834.
[70] Hamrock B J, Dowson D. Isothermal elasto-hydrodynamic lubrication of point contacts,Part III–Fully flooded results [J]. Journal of Lubrication Technology,1977,99(2):264~276.
[71]王建军,陆明万,张雄等.汽车变速系统的拍击动力学基本理论[J].机械科学与技术,1997,16(3):392~297.
[72]王建军,李润方.齿轮系统间隙非线性振动研究综述[J].非线性动力学学报,1995,2(3):214~221.
[73]王建军,李其汉,李润方.齿轮系统非线性振动研究进展[J].力学进展,2005,35(1):37~51.
[74] Ozguven H N, Houser D R. Mathematical models used in gear dynamics--a review [J].Journal of Sound and Vibration,1988,121:383~411.
[75] Comparin R J, Singh R. Non-linear frequency response characteristics of an impact pair [J].Journal of Sound and Vibration,1989,134:259~290.
[76] Comparin R J, Singh R. Frequency response characteristics of a multi-degree-of freedomsystem with clearances [J]. Journal of Sound and Vibration,1990,142(1):101~124.
[77] Comparin R J. A study of the frequency response of impact pairs with application toautomotive gear rattle [D]. PhD Dissertation, Ohio State University,1988.
[78] Comparin R J, Singh R. An analytical study of automobile neutral gear rattle [J]. ASMEMechanical Design,1990,112:237~245.
[79] Karaman A, Singh R. Non-linear dynamics of a spur gear pair [J]. Journal of Sound andVibration,1990,142(1):49~75.
[80] Karaman A, Singh R. Non-linear dynamics of a geared rotor-bearing system with multipleclearances [J]. Journal of Sound and Vibration,1991,144(3):469~506.
[81] Padmanabhan C, Singh R. Spectral coupling issues in a two degree of freedom system withclearance non-linearity [J]. Journal of Sound and Vibration,1992,155(2):209~230.
[82]冯奇.随机调幅Rattling系统建模[J].应用数学和力学,1999,20(1):85~92.
[83]冯奇. Rattling系统的一种随机模型[J].同济大学学报,1998,26(4):396~400.
[84]温建明,冯奇.随机调幅Rattling振动系统的二级传动模型[J].振动工程学报,2000,13(3):353~360.
[85] Feng Q, Pfeiffer F. Stochastic model on a rattling system [J]. Journal of Sound and Vibration,1998,215(3):439~453.
[86]董海军,沈允文,刘梦军.齿轮系统Rattling动力学行为研究[J].机械工程学报,2004,40(1):136~141.
[87]董海军,沈允文,刘梦军.转速激励齿轮系统拍击振动的分岔特性Rattling动力学行为研究[J].机械工程学报,2004,40(1):136~141.
[88]董海军,沈允文,刘梦军等.齿轮系统的拍击振动分析模型[J].中国机械工程,2003,14(16):1416~1419.
[89]张锁怀,李忆平,丘大谋.齿轮耦合的转子-轴承系统非线性动力特性的研究[J].机械工程学报,2001,37(9):53～57.
[90]张锁怀,沈允文,董海军等.齿轮时变系统对扭矩激励的响应[J].航空动力学报,2003,18(6):737~743.
[91]张锁怀,沈允文,董海军等.单级齿轮系统排击特性[J].机械工程学报,2003,29(3):28~31.
[92] Sakai T, Doi Y, Yamamoto K, Ogasawara T, etc. Theoretical and experimental analysis ofrattling noise of automotive gearbox[J], SAE Technical Paper,1981,810773
[93] Singh R. Analysis of automotive gear rattles [J]. Journal of Sound and Vibration,1989,131(2):177~196.
[94] Padmanabhan C. Modeling of automotive gear rattle phenomenon: state of the art [J]. SAETechnical Series,1995,951316:669~679.
[95] Pfeiffer F, Prestl W. Decoupling measures for rattling noises in gear boxes[J]. IME PaperC404/025, Institute of Mechanical Engineers,1990:129~134.
[96] Sebulke A. The two mass flywheel-A torsional vibration damper for the power train ofpassenger cars-state of the art and further technical development [C]. SAE Technical PaperSeries, International Congress and Exposition, Detroit, MI,1987.
[97] Drexl H J. Torsional dampers and alternative systems to reduce drive-line vibrations [C].SAE Technical Paper Series, International Congress and Exposition, Detroit, MI,1987.
[98] Padmanabhan C, Singh R. Influence of clutch design on the reduction and perception ofautomotive transmission rattle noise [C]. Noise Conference, Williamsburg, VA,1993.
[99] Robinette D, Beikmann R S, Piorkowski P. Characterizing the onset of manual transmissiongear rattles Part I: experimental results [J]. SAE International Journal of Passenger Cars–Mechanical System,2009(2):1352~1364.
[100] Robinette D, Beikmann R S, Piorkowski P. Characterizing the onset of manual transmissiongear rattle Part II: analytical results [J]. SAE International Journal of Passenger Cars–Mechanical System,2009,2:1367~1376.
[101] Kadmiri Y, Rigaud E, Perret-Liaudet J, Vary L. Experimental and numerical analysis ofautomotive gearbox rattle noise [J]. Journal of Sound and Vibration.2012,331(13):3144~3157.
[102] Barthod M, Hayne B, Tebec J L, Pin J C. Experimental study of gear rattle excited by amulti-harmonic excitation [J]. Applied Acoustics,2007,68(9):1003~1025.
[103] Wang M Y, Manoj R, Zhao W. Gear rattle modeling and analysis for automotive manualtransmissions [J]. Proceedings of the Institution of Mechanical Engineers, Part D: Journal ofAutomobile Engineering,2001:215~241.
[104] Cruz M D, Theodossiades S, Rahnejat H. An investigation of manual transmission driverattle [J]. Proceeding of Institution of Mechanical Engineers, Part K: J. Multi-bodyDynamics,2009,224:167~181.
[105] Rocca E, Russo R. Theoretical and experimental investigation into the influence of theperiodic backlash fluctuations on the gear rattle [J]. Journal of Sound and Vibration,2011,330(20):4738~4752.
[106] Russo R, Brancati R. Rocca E. Experimental investigations about the influence of oillubricant between teeth on the gear rattle phenomenon [J]. Journal of Sound and Vibration,2009,321(3):647~661.
[107] Dogan S N. Loose part vibration in vehicle transmissions-gear rattle [J]. Journal ofEngineering and Environmental Science,1999,23:439~454.
[108] Tangasawi O, Theodossiades S, Rahnejat H, Kelly P. Non-linear vibro-impact phenomenonbelying transmission idle rattle [J]. Proceeding of Institution of Mechanical Engineers, PartC: Journal of Mechanical Engineering Science,2008,222:1909~1923.
[109] Tangasawi O, Theodossiades S, Rahnejat H. Lightly loaded lubricated impacts: idle gearsrattle [J]. Journal of Sound and Vibration,2007.308(3):418~430.
[110] Theodossiades S, Tangasawi O, Rahnejat H. Gear teeth impacts in hydrodynamicconjunctions promoting idle gear rattle [J]. Journal of Sound and Vibration,2007,303(3):632~658.
[111] Ottewill J R, Neild S A, Wilson R E. An investigation into the effect of tooth profile errorson gear rattles [J]. Journal of Sound and Vibration,2010,329(3):3495~3506.
[112] Han B K, Cho M K, KIM C. Prediction of vibration forces on meshing gear for a gear rattleusing a new multi-body dynamic model [J]. International Journal of Automotive Technology,2009,10(4):469~474.
[113] Yakoub R Y. Prediction of System-level gear rattles using multi-body and vibro-acoustictechniques [J]. SAE paper,2004.
[114] Inalpolat M, Kahraman A. Dynamic modelling of planetary gears of automatictransmissions [J]. Proceedings of the Institution of Mechanical Engineers, Part K: Journal ofMulti-body Dynamics,2008,222:229~242.
[115] Hidaka T, Terauchi Y, Ishioka K. Dynamic behavior of planetary gear (2ndreport:displacement of sun gear and ring gear)[J], Bulletin of the JSME,1976,19(138):1563~1570.
[116] Kahraman A. Planetary gear train dynamics [J]. ASME Journal of Mechanical Design,1994,116:713~720.
[117] Singh A, Kahraman A, Ligata H. Internal gear strains and load sharing in planetarytransmissions-model and experiments [J]. ASME Journal of Mechanical Design,2008,130(7):072602-1~072602-10.
[118] Kahraman A. Load sharing characteristics of planetary transmissions [J]. Mechanism andMachine Theory,1994,29(8):1151~1165.
[119] Lin J, Park R G. Planetary gear parametric instability caused by mesh stiffness variation [J].Journal of Sound and Vibration,2002,249(1):129~145.
[120]杨通强,宋轶民,张策等.斜齿行星齿轮系统自由振动特性分析[J].机械工程学报,2005,7:50~55.
[121]杨富春,周晓军,郑津洋.复式行星齿轮传动系统综合动力学模型及振动特性研究[J].振动与冲击,2011,8:144~148.
[122]孙涛,沈允文,孙智民等.行星齿轮传动非线性动力学模型与方程[J].机械工程学报，2002,38(3):6~10.
[123]孙涛,沈允文,孙智民等.行星齿轮传动非线性动力学方程求解与动态特性[J].机械工程学报,2002,38(3):11~15.
[124]李同杰,朱如鹏,鲍如云等.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,2011,21:76~83.
[125] Randall R B. A new method of modeling gear faults [J]. ASME Journal of MechanicalDesign,1982,104:259~267.
[126]丁康.齿轮及齿轮箱故障诊断实用技术[M].机械工业出版社,北京,2009.
[127] McFadden P D, Smith J D. An explanation for the asymmetry of the modulation sidebandsabout the tooth meshing frequency in epicyclic gear vibration [J]. Proceedings of theInstitution of Mechanical Engineers,1985,199(1):65~70.
[128] McFadden P D. A technique for calculating the time domain averages of the vibration of theindividual planet gears and the sun gear in an epicyclic gearbox [J]. Journal of Sound andVibration,1991,144(1):163~172.
[129] McNames J. Fourier series analysis of epicyclic gearbox vibration [J]. ASME Journal ofVibration and Acoustics,2002,124(1):150~152.
[130] Schlegel R G, Mard K C. Transmission noise control-approaches in helicopter design [C].ASME Design Engineering Conference, New York,1967,67-DE-58.
[131] Seager D L. Conditions for the neutralization of excitation by the teeth in epicyclic gearing[J]. Journal of Mechanical Engineering Science,1975,17(5):293~299.
[132] Yu S, Kaatz S. Asymmetric gear noise sidebands and application to planetary gear noisereduction [C]. SAE Noise and Vibration Conference, Michigan,2005,01:2462
[133] Lin J, Park R G. Analytical characterization of the unique properties of planetary gear freevibration [J]. Journal of Sound and Vibration,1999,121:316~321.
[134] Parker R G. A physical explanation for the effectiveness of planet phasing to suppressplanetary gear vibration [J]. Journal of Sound and Vibration,2000,236(4):561~573.
[135] Parker R G, Lin J. Mesh phasing relationship in planetary and epicyclic gears [J]. Journal ofMechanical Design,2004,126:365~370.
[136] Inalpolat M, Kahraman A. A dynamic model to predict modulation sidebands of a planetarygear set having manufacturing errors [J]. Journal of Sound and Vibration,2010,329(4):371~393.
[137] Inalpolat M, Kahraman A. A theoretical and experimental investigation of modulationsidebands of planetary gear sets [J]. Journal of Sound and Vibration,2009,323:677~696.
[138] Chen Y, Ishibashi A. Investigation of the noise and vibration of planetary gear drives [C].ASME Proceeding of Design Engineering Technical Conferences,2003,4:507~512.
[139] Wang S, Huo M, Zhang C, et al. Effect of mesh phase on wave vibration of spur planetaryring gear [J]. European Journal of Mechanics A/Solids,2011,30:820~827.
[140]秦大同，肖正明，王建宏.基于啮合相位分析的盾构机减速器多级行星齿轮传动动力学特性[J].机械工程学报,2011,23:20~29.
[141]王世宇.基于相位调谐的直齿行星齿轮传动动力学理论与实验研究[D].天津大学,2005.
[142]雷英杰,张善文等. Matlab遗传算法工具箱及应用[M].西安:西安电子科大出版社,2005.
[143] Seaman R L, Johnson C E, Hamilton. Component inertial effects on transmission design [J].SAE Technical Paper,00399048,1984.
[144] Szadkowski, A. Mathematical model and computer simulation of idle gear rattle [J]. SAETechnical Paper,960641,1991.
[145] Fyfe K R, Munck E D S. Analysis of computed order tracking [J]. Mechanical Systems andSignal Processing,1997,11(2):187~205.
[146] Bossley K M, Mckendrick R J, Harris C J, et al. Hybrid computed order tracking [J].Mechanical Systems and Signal Processing,1999,13(4):627~641.
[147]傅志方,华宏星.模态分析理论与应用[M].上海:上海交通大学出版社,2000.
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