考虑不确定性因素的齿轮系统动力学研究综述

doi:10.3901/JME.2016.01.001

摘要/Abstract

摘要： 齿轮系统应用广泛，并在风力发电、航空航天、汽车和盾构机等机械设备中发挥关键作用。其动力学特性的优劣将直接影响设备整机的工作性能。然而齿轮传动的结构形式多样、内外部激励和非线性因素丰富，同时工作环境复杂多变，这使其动力学分析相比较于其他机械系统更加复杂。另外，制造、加工、装配等的误差、磨损、润滑和运行环境等因素将导致齿轮系统的内外部激励和系统参数具有不确定性。齿轮系统动力学分析需要考虑这些不确定性因素。目前，针对齿轮系统动力学特性的不确定性分析，国内外学者已开展了广泛的研究工作。从不确定性因素的描述方式、动力学方程的求解方法、动力学特性分析、可靠性与优化设计、以及不确定性分析的试验研究等方面系统地评述国内外学者对齿轮传动系统不确定性动力学特性的研究现状，并给出需要进一步研究的问题。

关键词: 不确定性, 齿轮传动, 动力学, 随机误差, 统计分析

Abstract: Geared system plays a critical role in wind power, aerospace, car, shield tunneling machine and other mechanism equipment. The dynamic characteristics of system will directly affect the performance of mechanism equipment. However, due to complicated structures, richness of internal and external excitations and nonlinear factors, and complex work environment, the dynamic analysis of gear train is more complex than that of other mechanical systems. In addition, owing to errors of manufacturing, processing, and assembling, wear, lubrication, operating environment and other factors, the internal and external excitations and system parameters are always uncertain. Those uncertain factors should be considered in the dynamic analysis of geared system. Many researchers have carried out extensive and in-depth research on the uncertain dynamic analysis of geared systems. The advances of research are reviewed from the description of uncertain factors, method for solving dynamic equations, analysis of dynamic characteristics, reliability analysis and optimization design, and experiments of uncertain analysis. Some critical issues for further study are also summarized.

Key words: dynamics, gear train, random error, statistical analysis, uncertainty

魏莎, 韩勤锴, 褚福磊. 考虑不确定性因素的齿轮系统动力学研究综述[J]. 机械工程学报, doi: 10.3901/JME.2016.01.001.

WEI Sha, HAN Qinkai, CHU Fulei. Review on Research of Uncertain Dynamics for Gear System[J]. Journal of Mechanical Engineering, doi: 10.3901/JME.2016.01.001.

参考文献

[1] LI X，WANG Y. Analysis of mixed model in gear transmission based on ADAMS[J]. Chinese Journal of Mechanical Engineering，2012，25(5)：968-973.
[2] 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)：47-50，54.SUN Yuehai，ZHANG Ce，PAN Fengzhang，et al. Dynamic model of a spur gear transmission system vibration[J]. Chinese Journal of Mechanical Engineering，2000，36(8)：47-50，54.
[3] 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. SUN Tao，SHEN Yunwen，SUN Zhimin，et al. Study on nonlinear dynamic behavior of planetary gear train dynamic model and governing equations[J]. Chinese Journal of Mechanical Engineering，2002，38(3)：6-10.
[4] AMBARISHA V K，PARKER R G. Nonlinear dynamics of planetary gears using analytical and finite element models[J]. Journal of Sound and Vibration，2007，302(3)：577-595.
[5] 孙涛，胡海岩. 基于离散傅里叶变换与谐波平衡法的行星齿轮系统非线性动力学分析[J]. 机械工程学报，2002，38(11)：58-61. SUN Tao，HU Haiyan. Nonlinear dynamics of planetary gear transmission by harmonic balance method based on DFT[J]. Chinese Journal of Mechanical Engineering，2002，38(11)：58-61.
[6] 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求解与动态特性分析[J]. 机械工程学报，2002，38(3)：11-15.SUN Tao，SHEN Yunwen，SUN Zhimin，et al. Study on nonlinear dynamic behavior of planetary gear train solution and dynamic behavior analysis[J]. Chinese Journal of Mechanical Engineering，2002，38(3)：11-15.
[7] AL-SHYYAB A，KAHRAMAN A. Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method：period-one motions[J]. Journal of Sound and Vibration，2005，284(1)：151-172.
[8] AL-SHYYAB A，KAHRAMAN A. Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method：sub-harmonic motions[J]. Journal of Sound and Vibration，2005，279(1)：417-451.
[9] LIN J，PARKER R G. Analytical characterization of the unique properties of planetary gear free vibration[J]. Journal of Vibration and Acoustics，1999，121：316-321.
[10] 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，2005，41(7)：50-55.YANG Tongqiang，SONG Yimin，ZHANG Ce，et al. Property analysis of free vibration of helical planetary gear trains[J]. Chinese Journal of Mechanical Engineering，2005，41(7)：50-55.
[11] 常乐浩，刘更，吴立言. 齿轮综合啮合误差计算方法及对系统振动的影响[J]. 机械工程学报，2015，51(1)：123-130. CHANG Lehao，LIU Geng，WU Liyan. Determination of composite meshing errors and its Influence on the vibration of gear system[J]. Journal of Mechanical Engineering，2015，51(1)：123-130.
[12] 任朝晖，谢吉祥，周世华，等. 斜齿轮-转子-轴承弯扭轴耦合振动特性分析[J]. 机械工程学报，2015，51(15)：75-89. REN Chaohui，XIE Jixiang，ZHOU Shihua，et al. Vibration characteristic analysis of helical gear-rotor-bearing system with coupled lateral-torsional-axial[J]. Journal of Mechanical Engineering，2015,51(15)：75-89.
[13] 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. CHEN Siyu，TANG Jinyuan. Effect of backlash on dynamics of spur gear pair system with friction and time-varying stiffness[J]. Journal of Mechanical Engineering，2009，45(8)：119-124.
[14] LIU C，QIN D，LIM T C，et al. Dynamic characteristics of the herringbone planetary gear set during the variable speed process[J]. Journal of Sound and Vibration，2014，333(24)：6498-6515.
[15] 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，LU Shenming，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.
[16] AMABILI M，RIVOLA A. Dynamic analysis of spur gear pairs：Steady-state response and stability of the SDOF model with time-varying meshing damping[J]. Mechanical Systems and Signal Processing，1997，11(3)：375-390.
[17] LIN J，PARKER R G. Mesh stiffness variation instabilities in two-stage gear systems[J]. Journal of Vibration and Acoustics，2002，124(1)：68-76.
[18] 郜志英，沈允文，李素有，等. 间隙非线性齿轮系统周期解结构及其稳定性研究[J]. 机械工程学报，2004，40(5)：17-22.GAO Zhiying，SHEN Yunwen，LI Suyou，et al. Research on the periodic solution structure and its stability of nonlinear gear system with clearance[J]. Chinese Journal of Mechanical Engineering，2004，40(5)：17-22.
[19] ABUID B A，AMEEN Y M. Procedure for optimum design of a two-stage spur gear system[J]. JSME International Journal. Series C，Mechanical Systems，Machine Elements and Manufacturing，2003，46(4)：1582.
[20] 唐增宝，钟毅芳，刘伟忠. 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)：66-75.TANG Zengbao，ZHONG Yifang，LIU Weizhong. Many stage helical gear train system optimal design which takes the dynamic properties as an objective function[J]. Chinese Journal of Mechanical Engineering，1994，30(5)：66-75.
[21] 秦大同，邢子坤，王建宏. 基于动力学和可靠性的风力发电齿轮传动系统参数优化设计[J]. 机械工程学报，2008，44(7)：24-31.QIN Datong，XING Zikun，WANG Jianhong. Optimization design of system parameters of the gear transmission of wind turbine based on dynamics and reliability[J]. Chinese Journal of Mechanical Engineering，2008，44(7)：24-31.
[22] WEI J，LÜ C，SUN W，et al. A study on optimum design method of gear transmission system for wind turbine[J]. International Journal of Precision Engineering and Manufacturing，2013，14(5)：767-778.
[23] 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.LI Runfang，WANG Jianjun. Dynamics of gear systems： Vibration，shock，and noise[M]. Beijing：Science Press，1997.
[24] 国家电监会. 风电安全监管报告2011年(一)[J]. 太阳能，2011，24：37-40. State Electricity Regulatory Commission. Wind power safety regulation report 2011 (a)[J]. Solar Energy，2011，24：37-40.
[25] TOBE T，SATO K. Statistical analysis of dynamic loads on spur gear teeth[J]. Transactions of the Japan Society of Mechanical Engineers，1976，20(146)：3652 - 3661.
[26] TOBE T，SATO K，TAKATSU N. Statistical analysis of dynamic loads on spur gear teeth ：Experimental study[J]. Transactions of the Japan Society of Mechanical Engineers，1977，20(148)：746-752.
[27] KUMAR A S，OSMAN M O M，SANKAR T S. On statistical analysis of gear dynamic loads[J]. Journal of Vibration，Acoustics，Stress，and Reliability in Design，1986，108：362-368.
[28] NERIYA S V，BHAT R B，SANKAR T S. On the dynamic response of a helical geared system subjected to a static transmission error in the form of deterministic and filtered white noise inputs[J]. Journal of Vibration，Acoustics，Stress，and Reliability in Design，1988，110(4)：501-506.
[29] PAPOULIS A. Probability，random variables，and stochastic processes[M]. New York：McGraw-Hill，1991.
[30] ALEFELD G，HERZBERGER J. Introduction to interval computations[M]. New York：Academic Press，1983.
[31] 李润方，王建军. 国外齿轮系统动态特性的统计分析[J]. 机械传动，1995，19(4)：45-47.LI Runfang，WANG Jianjun. Statistical analysis of characteristics for gear system at overseas[J]. Journal of Mechanical Transmission，1995，19(4)：45-47.
[32] 刘延柱，陈文良，陈立群. 振动力学[M]. 高等教育出版社，1998. LIU Yanzhu，CHEN Wenliang，CHEN Liqun. Vibration mechanics[M]. Beijing：Higher Education Press，1998.
[33] SATO K，YAMAMOTO S，KAMADA O. Jump phenomena in gear system to random excitation[J]. Bulletin of JSME，1985，28(240)：1271-1278.
[34] GELMAN L，GIURGIUTIU V，BAYOUMI A. Statistical analysis of the dynamic mean excitation for a spur gear[J]. Journal of Vibration and Acoustics，2005，127(2)：204.
[35] FENG Q，PFEIFFER F. Stochastic model on a rattling system[J]. Journal of Sound and Vibration，1998，215(3)：439-453.
[36] 陈会涛，吴晓铃，秦大同，等. 随机内外激励对齿轮系统动态特性的影响分析[J]. 中国机械工程，2013(04)：533-537.CHEN Huitao，WU Xiaoling，QIN Datong，et al. Dynamic characteristics of gear transmission system subjected to random internal and external excitation[J]. China Mechanical Engineering，2013(04)：533-537.
[37] WANG Y，ZHANG W. Stochastic vibration model of gear transmission systems considering speed-dependent random errors[J]. Nonlinear Dynamics，1998，17(2)：187 - 203.
[38] 王勇，章文俊. 齿轮传动系统非白噪声随机误差激振的建模[J]. 科学通报，1998(11)：1116-1118. WANG Yong，ZHANG Wenjun. Modelling of gear transmission error excitation with non-white noise[J]. Chinese Science Bulletin，1998(11)：1116-1118.
[39] WANG Y，ZHANG W. Modelling of gear stochastic vibration considering non-white noise errors[J]. Chinese Science Bulletin，1999，44(4)：375-378.
[40] 王勇，许蕴梅. 随机振动最小为目标的齿轮系统优化[J]. 山东大学学报，2003(3)：257-260.WANG Yong，XU Yunmei. Optimization for gear system with a minimum stochastic vibration[J]. Journal of Shandong University，2003(3)：257-260.
[41] DRIOT N，PERRET-LIAUDET J. Variability of modal behavior in terms of critical speeds of a gear pair due to manufacturing errors and shaft misalignments[J]. Journal of Sound and Vibration，2006，292(3-5)：824-843.
[42] BONORI G，PELLICANO F. Non-smooth dynamics of spur gears with manufacturing errors[J]. Journal of Sound and Vibration，2007，306(1-2)：271-283.
[43] 邓绪山，杨兵，刘增民. 基于轮齿随机误差的齿轮系统动力学分析[J]. 机械传动，2011(10)：31-34.DENG Chushan，YANG Bing，LIU Zengmin. Dynamic analysis of gear system based on the random gear errors[J]. Journal of Mechanical Transmission，2011(10)：31-34.
[44] PURI M L，RALESCU D A. Fuzzy random variables[J]. Journal of Mathematical Analysis and Applications，1986，114(2)：409-422.
[45] 马亮，李瑰贤，吴俊飞. 含有模糊参数的齿轮副动态响应分析[J]. 机械科学与技术，2001(2)：203-204，152.MA Liang，LI Huaixian，WU Junfei. Dynamic response of meshing gears including fuzzy parameters[J]. Mechanical Science and Technology，2001(2)：203-204，152.
[46] 李瑰贤，马亮，林少芬，等. 基于模糊误差激励的齿轮副单自由度扭转振动分析[J]. 重型机械，2000(5)：35-36，43. LI Huaixian，MA Liang，LIN Shaofen，et al. Torsional vibration analysis of single degree of freedom gear pairs under fuzzy error excitation[J]. Heavy Machinery，2000(5)：35-36，43.
[47] 王靖岳，郭立新，张亮，等. 齿轮传动系统随机振动模型与仿真[J]. 东北大学学报，2013(4)：578-582.WANG Jingyue，GUO Lixin，ZHANG Liang，et al. Stochastic vibration model and simulation of gear transmission system[J]. Journal of Northeasten University，2013(4)：578-582.
[48] 廖映华，秦大同，刘长钊. 考虑内部激励随机性的两级分流式人字齿轮传动动力学特性[J]. 振动与冲击，2015(1)：206-212.LIAO Yinghua，QIN Datong，LIU Changzhao. Dynamic characteristics of two-stage split herringbone gear trains considering the randomness of internal excitations[J]. Journal of Vibration and Shock，2015(1)：206-212.
[49] PFEIFFER F，KUNERT A. Rattling models from deterministic to stochastic processes[J]. Nonlinear Dynamics，1990，1(1)：63-74.
[50] NAESS A，KOLNES F E，MO E. Stochastic spur gear dynamics by numerical path integration[J]. Journal of Sound and Vibration，2007，302(4-5)：936-950.
[51] MO E，NAESS A. Nonsmooth dynamics by path integration：An example of stochastic and chaotic response of a meshing gear pair[J]. Journal of Computational and Nonlinear Dynamics，2009，4(3)：034501.
[52] YANG J. Vibration analysis on multi-mesh gear-trains under combined deterministic and random excitations[J]. Mechanism and Machine Theory，2013，59：20-33.
[53] YANG J，YANG P. Random vibration analysis of planetary gear trains[J]. Journal of Vibration and Acoustics，2013，135(2)：021005.
[54] WEN Y，YANG J，WANG S. Random dynamics of a nonlinear spur gear pair in probabilistic domain[J]. Journal of Sound and Vibration，2014，333(20)：5030-5041.
[55] 秦大同，李超，唐飞熊. 变风载下变速风力发电机传动系统动力学特性研究[J]. 太阳能学报，2013，34(2)：186-195. QIN Datong，LI Chao，TANG Feixiong. Study on dynamic characteristics of gear transmission system of vibriable speed wind generator under varying wind load[J]. Acta Energiae Solaris Sinica，2013，34(2)：186-195.
[56] 秦大同，张壮飞，周海波. 随机风速下风力发电机传动系统动态外载荷计算[J]. 可再生能源，2012，30(9)：21-26.QIN Datong，ZHANG Zhuangfei，ZHOU Haibo. Dynamic loads calculation of wind turbines transmission system under random wind speeds[J]. Renewable Energy Resources，2012，30(9)：21-26.
[57] 陈会涛，秦大同，吴晓铃，等. 考虑随机制造误差的风力机行星齿轮系统动力学特性[J]. 机械工程学报，2012，48(21)：77-83. CHEN Huitao，QIN Datong，WU Xiaoling，et al. Dynamic characteristics of planetary gear transmission system of wind turbine with random manufacturing error[J]. Journal of Mechanical Engineering，2012，48(21)：77-83.
[58] SUN W，CHEN T，WEI J. Dynamic reliability of gears in a wind turbine gearbox under the conditions of variable wind speed and small samples[J]. Proceedings of the Institution of Mechanical Engineers，Part C：Journal of Mechanical Engineering Science，2012，226(12)：3032-3042.
[59] 张建忠，程明. 风力发电随机风速时间序列生成方法分析与评价[J]. 风能，2012(1)：58-61.ZHANG Jianzhong，CHENG Ming. Analysis and evaluation of generating methods of random wind speed time series[J]. Wind Energy，2012(1)：58-61.
[60] 王雷，楚峥，姚兴佳. 风轮机输出特性模拟系统的设计[J]. 可再生能源，2007(1)：62-65.WANG Lei，CHU Zheng，YAO Xingjia. Design of simulating system of wind turbine output characteristic[J]. Renewable Energy Resources，2007(1)：62-65.
[61] 杨军，秦大同，陈会涛，等. 风力发电机传动系统随机风速下的载荷特性研究[J]. 中国机械工程，2011(15)：1837-1842.YANG Jun，QIN Datong，CHEN Huitao，et al. Research on dynamic load characteristics of wind power generation transmission system under random wind[J]. China Mechanical Engineering，2011(15)：1837-1842.
[62] 李东东，陈陈. 风力发电系统动态仿真的风速模型[J]. 中国电机工程学报，2005(21)：44-47.LI Dongdong，CHEN Chen. Wind speed model for dynamic simulation of wind power generation system[J]. Proceedings of the CSEE，2005(21)：44-47.
[63] 秦大同，田苗苗，杨军. 变风载下风力发电机齿轮传动系统动力学特性研究[J]. 太阳能学报，2012(2)：190-196.QIN Datong，TIAN Miaomiao，YANG Jun. Study on dynamic characteristics of gear transmission system of wind generator under varying wind load[J]. Acta Energiae Solaris Sinica，2012(2)：190-196.
[64] 秦大同，杨军，周志刚，等. 变载荷激励下风电行星齿轮系统动力学特性[J]. 中国机械工程，2013(3)：295-301.QIN Datong，YANG Jun，ZHOU Zhigang，et al. Dynamics characteristic of planetary gear system of wind turbines under varying load[J]. China Mechanical Engineering，2013(3)：295-301.
[65] 周云山，金红光. 多种动载荷激励下变速器箱体的动态响应分析[J]. 机械传动，2015(8)：122-125.ZHOU Yunshan，JIN Hongguang. Dynamic response analysis of gearbox housing under the multi-source dynamic excitation[J]. Journal of Mechanical Transmission，2015(8)：122-125.
[66] OGNJANOVIC M，MILUTINOVIC M. Design for reliability based methodology for automotive gearbox load capacity identification[J]. Strojniški vestnik – Journal of Mechanical Engineering，2013，59(5)：311-322.
[67] 李东兵，赵宇. 汽车直齿锥齿轮系统内外部激励数值模拟[J]. 机械传动，2014(8)：130-133.LI Dongbing，ZHAO Yu. Numerical simulation of the internal and external excitation of automotive differential straight bevel gear system[J]. Journal of Mechanical Transmission，2014(8)：130-133.
[68] 刘梦军，沈允文，董海军. 含随机间隙的齿轮非线性系统的全局分析[J]. 西北工业大学学报，2004，22(4)：482-486.LIU Mengjun，SHEN Yunwen，DONG Haijun. A global analysis for nonlinear gear system with stochastic backlash[J]. Journal of Northwestern Polytechnical University，2004，22(4)：482-486.
[69] LU J，CHEN H，ZENG F，et al. Influence of system parameters on dynamic behavior of gear pair with stochastic backlash[J]. Meccanica，2014，49(2)：429-440.
[70] GUERINE A，EL HAMI A，WALHA L，et al. A perturbation approach for the dynamic analysis of one stage gear system with uncertain parameters[J]. Mechanism and Machine Theory，2015，92：113-126.
[71] GUERINE A，DRISS Y，BEYAOUI M，et al. A polynomial chaos method for the analysis of uncertain spur gear system[M]. Mechatronic Systems：Theory and Applications，Springer International Publishing，2014.
[72] 王世宇，张策，宋轶民，等. 行星齿轮传动固有频率的统计特性分析[J]. 机械科学与技术，2005，24(6)：705-709.WANG Shiyu，ZHANG Ce，SONG Yimin，et al. Statistical analysis of the eigenvalue of planetary gears[J]. Mechanical Science and Technology，2005，24(6)：705-709.
[73] PERRET-LIAUDET J，DRIOT N. A modified Taguchi's method applied to parametric and nonlinear primary resonances of gears under parameter uncertainties[C]//12th IFToMM World Congress. Besancon，France，2007：1-6.
[74] 魏永祥，陈建军，王敏娟. 随机参数齿轮-转子系统的扭转振动分析[J]. 航空动力学报，2010(11)：2637-2642.WEI Yongxiang，CHEN Jianjun，WANG Minjuan. Dynamic response of torsional vibration of gear-rotor with random parameters[J]. Journal of Aerospace Power，2010(11)：2637-2642.
[75] LU J，ZENG F，XIN J，et al. Influences of stochastic perturbation of parameters on dynamic behavior of gear system[J]. Journal of Mechanical Science and Technology，2011，25(7)：1667-1673.
[76] 王靖岳，王浩天，郭立新. 随机外干扰下齿轮传动系统的混沌振动分析[J]. 振动•测试与诊断，2014(6)：1099-1104，1172.WANG Jingyue，WANG Haotian，GUO Lixin. Chaotic vibration analysis of gear transmission system with stochastic disturbance[J]. Journal of Vibration，Measurement & Diagnosis，2014(6)：1099-1104，1172.
[77] MOORE R E. Interval analysis[M]. Englewood Cliffs，New Jersey：Prentice-Hall，1966.
[78] LIN Y，STADTHERR M A. Validated solution of initial value problems for ODEs with interval parameters[C]// NSF Workshop Proceeding on Reliable Engineering Computing. Savannah GA，2006：1-14.
[79] LIN Y，ENSZER J A，STADTHERR M A. Enclosing all solutions of two-point boundary value problems for ODEs[J]. Computers & Chemical Engineering，2008，32(8)：1714-1725.
[80] HU J，QIU Z. Non-probabilistic convex models and interval analysis method for dynamic response of a beam with bounded uncertainty[J]. Applied Mathematical Modelling，2010，34(3)：725-734.
[81] RAMA R M V，POWNUK A，VANDEWALLE S，et al. Transient response of structures with uncertain structural parameters[J]. Structural Safety，2010，32(6)：449-460.
[82] DONG R，SUN W，XU H. Robust design optimization of gear parameters for a wind turbine using interval analysis[J]. Proceedings of the Institution of Mechanical Engineers，Part C：Journal of Mechanical Engineering Science，2010，224(10)：2235-2245.
[83] 段文峰. 基于响应面法的齿轮断裂非概率可靠性分析[J]. 煤矿机械，2014(5)：109-111.DUAN Wenfeng. Non-probabilistic fracture reliability analysis of gears based on response surface method[J]. Coal Mine Machinery，2014(5)：109-111.
[84] BAHK C，PARKER R G. Analytical solution for the nonlinear dynamics of planetary gears[J]. Journal of Computational and Nonlinear Dynamics，2011，6(2)：021007-1-15.
[85] 张义民. 机械振动[M]. 北京：清华大学出版社，2007.
ZHANG Yimin. Machinery vibration[M]. Beijing：Tsinghua University Press，2007.
[86] ROBERTS J B，SPANOS P D. Random vibration and statistical linearization[M]. Chichester，England：Wiley，1990.
[87] ROBERTS J B，SPANOS P D. Stochastic averaging：An approximate method of solving random vibration problems[J]. International Journal of Non-Linear Mechanics，1986，21(2)：111-134.
[88] STRATONOVICH R L. Topics in the theory of random noise[M]. New York：Gordon and Breach，1967.
[89] KHAS MINSKII R Z. Principle of averaging for parabolic and elliptic differential equations and for Markov processes with small diffusion[J]. Theory of Probability & Its Applications，1963，8(1)：1-21.
[90] 朱位秋. 随机平均法及其应用[J]. 力学进展，1987(3)：342-352. ZHU Weiqiu. Stochastic averaging methods and their applications[J]. Advances in Mechanics，1987(3)：342-352.
[91] SATO K，YAMAMOTO S，KAMADA O，et al. Approximate solution of a gear system subjected to random excitation[J]. Bulletin of JSME，1986，29(251)：1586-1589.
[92] BERNARD P，FLEURY G. Stochastic Newmark method[C]// Third International Conference on Computational Stochastic Mechanics. Rotterdam，1999，365-373.
[93] BERNARD P，FLEURY G. Stochastic newmark scheme[J]. Probabilistic Engineering Mechanics，2002，17(1)：45-61.
[94] ROY D. A weak form of stochastic Newmark method with applications to engineering dynamical systems[J]. Applied Mathematical Modelling，2003，27(6)：421-436.
[95] TO C W S. A stochastic version of the Newmark family of algorithms for discretized dynamic systems[J]. Computers & Structures，1992，44(3)：667-673.
[96] ZHANG L，ZU J W，ZHENG Z. The stochastic Newmark algorithm for random analysis of multi-degree-of-freedom nonlinear systems[J]. Computers & Structures，1999，70(5)：557-568.
[97] WEHNER M F，WOLFER W G. Numerical evaluation of path-integral solutions to Fokker-Planck equations[J]. Physical Review A，1983，27(5)：2663--2670.
[98] WEHNER M F，WOLFER W G. Numerical evaluation of path-integral solutions to Fokker-Planck equations. III. Time and functionally dependent coefficients[J]. Physical Review A，1987，35(4)：1795-1801.
[99] WEHNER M F，WOLFER W G. Numerical evaluation of path-integral solutions to Fokker-Planck equations. II. Restricted stochastic processes[J]. Physical Review A，1983，28(5)：3003-3011.
[100] SOBOL I M. A primer for the Monte Carlo method[M]. Boca Raton：CRC Press，1994.
[101] FISHMAN G S. Monte Carlo：Concepts，algorithms，and applications[M]. New York：Springer-Verlag，1996.
[102] 卢剑伟，刘梦军，陈磊，等. 随机参数下齿轮非线性动力学行为[J]. 中国机械工程，2009，10(3)：330-333.LU Jianwei，LIU Mengjun，CHEN Lei，et al. Nonlinear dynamics behavior of gear system with stochastic parameters[J]. China Mechanical Engineering，2009，10(3)：330-333.
[103] ZHANG J，GUO F. Statistical modification analysis of helical planetary gears based on response surface method and Monte Carlo simulation[J]. Chinese Journal of Mechanical Engineering，2015.
[104] STOKER K，CHAUDHURI A，KIM N H. Safety of spur gear design under non-ideal conditions with uncertainty[C]// Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. Montreal，Quebec，Canada，2010.
[105] ELISHAKOFF I，REN Y J，SHINOZUKA M. Improved finite element method for stochastic problems[J]. Chaos，Solitons and Fractals，1995，5(5)：833-846.
[106] NAYFEH A H. Perturbation methods[M]. New York：John Wiley & Sons，1973.
[107] 胡鹏，路金昌，张义民. 基于随机摄动法的齿轮系统动态响应及灵敏度分析[J]. 东北大学学报，2014(2)：257-262.HU Peng，LU Jinchang，ZHANG Yimin. Analysis of dynamic response and sensitivity of gear systems based on stochastic perturbation theory[J]. Journal of Northeastern University，2014(2)：257-262.
[108] TAGUCHI G. Performance analysis design[J]. The International Journal of Production Research，1978，16(6)：521-530.
[109] D'ERRICO J R，ZAINO N A. Statistical tolerancing using a modification of Taguchi’s method[J]. Technometrics，1988，30(4)：397-405.
[110] CHEN J J，CHEN J W，SUN H A，et al. Probabilistic dynamic analysis of truss structures[J]. Structural Engineering and Mechanics，2002，12(2)：231-239.
[111] 魏永祥，陈建军，王敏娟. 随机参数齿轮-转子系统扭转振动固有频率分析[J]. 工程力学，2011(4)：172-177.WEI Yongxiang，CHEN Jianjun，WANG Minjuan. Natural frequency analysis for torsional vibration of a gear-rotor system with random parameters[J]. Engineering Mechanics，2011(4)：172-177.
[112] 魏永祥，陈建军，王敏娟，等. 平稳随机激励下随机参数齿轮副的动力可靠性分析[J]. 机械强度，2011(1)：6-10.WEI Yongxiang，CHEN Jianjun，WANG Minjuan，et al. Dynamic reliability analysis of gear with stochastic parameters under stationary random excitation[J]. Journal of Mechanical Strength，2011(1)：6-10.
[113] 魏永祥，陈建军，马洪波. 随机参数齿轮系统的非线性动力响应分析[J]. 工程力学，2012(11)：319-324.WEI Yongxiang，CHEN Jianjun，MA Hongbo. Analysis of nonlinear dynamic response of gear-rotor with random parameters[J]. Engineering Mechanics，2012(11)：319-324.
[114] 邱志平，陈塑寰，刘中生. 区间参数结构振动问题的矩阵摄动法[J]. 应用数学和力学，1994(6)：519-527.QIU Zhiping，CHEN Suhuan，LIU Zhongsheng. Matrix perturbation method for the vibration problem of structures with interval parameters[J]. Applied Mathematics and Mechanics，1994(6)：519-527.
[115] MCWILLIAM S. Anti-optimization of uncertain structures using interval analysis[J]. Computers and Structures，2001，79(4)：421-430.
[116] MUSCOLINO G，SOFI A. Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis[J]. Probabilistic Engineering Mechanics，2012，28：152-163.
[117] MA Y，LIANG Z，CHEN M，et al. Interval analysis of rotor dynamic response with uncertain parameters[J]. Journal of Sound and Vibration，2013，332(16)：3869-3880.
[118] DIMAROGONAS A D. Interval analysis of vibrating systems[J]. Journal of Sound and Vibration，1995，183(4)：739-749.
[119] ZADEH L A. Fuzzy sets[J]. Information and Control，1965，8：338-353.
[120] BEZDEK J C. Fuzzy models - what are they，and why?[J]. Transactions on Fuzzy Systems，1993，1：1-6.
[121] KAUFMANN A，GUPTA M M. Introduction to fuzzy arithmetic：theory and applications[M]. New York：Van Nostrand Reinhold Company，1985.
[122] WOOD K L，OTTO K N. Engineering design calculations with fuzzy parameters[J]. Fuzzy Sets and Systems，1992，52(1)：1-20.
[123] QUIN S，WIDERA G E O. Application of fuzzy relational modeling to industrial product quality control[J]. Journal of Pressure Vessel Technology-Transactions of the ASME，1996，118：121-124.
[124] KRASLAWSKI A，KOIRANEN T，NYSTROM L. Concurrent engineering：Robust design in fuzzy environment[J]. Computers & Chemical Engineering，1993，17：S447-S452.
[125] 王宇宁，孙志礼，杨强，等. 基于热分析的齿轮模态及共振可靠性灵敏度研究[J]. 东北大学学报，2013(3)：408-412. WANG Yuning，SUN Zhili，YANG Qiang，et al. Study on the gear mode and resonance reliability sensitivity based on thermal analysis[J]. Journal of Northeastern University，2013(3)：408-412.
[126] 卢剑伟，曾凡灵，杨汉生，等. 随机装配侧隙对齿轮系统动力学特性的影响分析[J]. 机械工程学报，2010，46(21)：82-86.LU Jianwei，ZENG Fanling，YANG Hansheng，et al. Influence of stochastic assembling backlash on nonlinear dynamic behavior of transmission gear pair[J]. Journal of Mechanical Engineering，2010，46(21)：82-86.
[127] SRIKANTH P，SEKHAR A. Dynamic analysis of wind turbine drive train subjected to nonstationary wind load excitation[J]. Proceedings of the Institution of Mechanical Engineers，Part C：Journal of Mechanical Engineering Science，2014，229(3)：429-446.
[128] GUERINE A，DRISS Y，BEYAOUI M，et al. A polynomial chaos method for the analysis of uncertain spur gear system[M]. Mechatronic Systems：Theory and Applications，Springer International Publishing，2014，89-97.
[129] 鲍春梅，吕士宝，张丽. 齿轮时变系统在随机激励下的响应分析[J]. 重庆理工大学学报，2012(7)：34-37.BAO Chunmei，LÜ Shibao，ZHANG Li. Dynamic response of gear system with time-varying under random excitation[J]. Journal of Chongqing University of Technology，2012(7)：34-37.
[130] STOKER K，CHAUDHURI A，KIM N H. Safety of spur gear design under non-ideal conditions with uncertainty[C]// Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. Montreal，Quebec，Canada，2010.
[131] 程五一，李季. 系统可靠性理论及其应用[M]. 北京：北京航空航天大学出版社，2012.CHENG Wuyi，LI Ji. System reliability theory and its application[M]. Beijing：Beijing University of Aeronautics and Astronautics Press，2012.
[132] 孙志礼，袁哲. 齿轮随机参数系统非线性振动响应可靠性分析[J]. 东北大学学报，2011(6)：838-842.SUN Zhili，YUAN Zhe. Reliability analysis of nonlinear vibration response for gear system with stochastic parameters[J]. Journal of Northeastern University，2011(6)：838-842.
[133] 王倩倩，张义民，张振先，等. 齿轮系统随机振动及其传动误差的可靠性及灵敏度分析[J]. 东北大学学报，2011(12)：1741-1744.WANG Qianqian，ZHANG Yimin，ZHANG Zhenxian，et al. The reliability and sensitivity analysis of the gear random vibration system and transmission error[J]. Journal of Northeastern University，2011(12)：1741-1744.
[134] ALEMAYEHU F M，EKWARO-OSIRE S. Uncertainty considerations in the dynamic loading and failure of spur gear pairs[J]. Journal of Mechanical Design，2013，135(8)：084501.
[135] 陈会涛，秦大同，吴晓铃，等. 考虑载荷和参数随机性的风电齿轮传动系统动力可靠性研究[J]. 太阳能学报，2014(10)：1936-1943.CHEN Huitao，QIN Datong，WU Xiaoling，et al. Dynamic reliability analysis of gear transmission system of wind turbine with considering randomness of loading and parameters[J]. Acta Energiae Solaris Sinica，2014(10)：1936-1943.
[136] ALEMAYEHU F M，EKWARO-OSIRE S. Loading and design parameter uncertainty in the dynamics and performance of high-speed-parallel-helical-stage of a wind turbine gearbox[J]. Journal of Mechanical Design，2014，136(9)：091002.
[137] ALEMAYEHU F M，EKWARO-OSIRE S. Probabilistic performance of helical compound planetary system in wind turbine[J]. Journal of Computational and Nonlinear Dynamics，2014，10(4)：041003-1-12.
[138] 苏春. 航空齿轮传动的模糊可靠性优化设计[J]. 机械，1995(6)：5-8. SU Chun. Fuzzy reliability optimization design of aerospace gear transmission[J]. Machine，1995(6)：5-8.
[139] 张干清，龚宪生，王欢欢，等. 基于可靠灰色粒子群算法的盾构机行星减速器轮系的多目标优化设计[J]. 机械工程学报，2010，46(23)：135-145. ZHANG Ganqing，GONG Xiansheng，WANG Huanhuan，et al. Multi-objective optimization design on gear train of planetary reducer in shield tunneling machine based on reliably grey particle swarm optimization[J]. Journal of Mechanical Engineering，2010，46(23)：135-145.
[140] HANDSCHUH M J，KAHRAMAN A，MILLIREN M R. Impact of tooth spacing errors on the root stresses of spur gear pairs[J]. Journal of Mechanical Design，2014，136(6)：061010-1-10.
[141] INALPOLAT M，HANDSCHUH M，KAHRAMAN A. Influence of indexing errors on dynamic response of spur gear pairs[J]. Mechanical Systems and Signal Processing，2015，60-61：391-405.
[142] WOJNAROWSKI J，ONISHCHENKO V. Tooth wear effects on spur gear dynamics[J]. Mechanism and Machine Theory，2003，38(2)：161-178.
[143] DING H，KAHRAMAN A. Interactions between nonlinear spur gear dynamics and surface wear[J]. Journal of Sound and Vibration，2007，307(3-5)：662-679.
[144] SHON S，KAHRAMAN A，LABERGE K，et al. Influence of surface roughness on traction and scuffing performance of lubricated contacts for aerospace and automotive gearing[C]// Proceedings of the ASME/ STLE 2012 International Joint Tribology Conference. Denver，Colorado，USA，2012.
[145] MOENS D，VANDEPITTE D. Interval sensitivity theory and its application to frequency response envelope analysis of uncertain structures[J]. Computer Methods in Applied Mechanics and Engineering，2007，196(21-24)：2486-2496.