含碰摩故障的航空发动机整机振动建模与分析
|
摘要
现代航空发动机通过减小转静间隙来追求高推重比的同时,大大加剧了转子叶片与静子机匣间的碰撞摩擦事故发生。随着航空事业的发展,对航空发动机的结构完整性和可靠性的要求也越来越高,这就需要深入认识转静碰摩故障的振动特性,为有效地排除和预防碰摩故障提供理论依据。基于此,本文进行了如下研究工作:
(1)针对航空发动机转子系统,建立了含转子碰摩故障的单/双转子-滚动轴承-机匣耦合动力学模型。在模型中,考虑了由转静碰摩所导致的机匣运动,滚动轴承外圈与轴承座之间的弹性支承和挤压油膜阻尼,同时,充分考虑了滚动轴承间隙、滚动轴承滚珠与滚道的非线性赫兹接触力以及由滚动轴承支撑刚度变化而产生的VC(Varying compliance)振动。在双转子模型中还考虑了高压转子和低压转子的耦合振动。最后,运用四阶龙格库塔法获取了系统响应,并验证了模型的正确性。
(2)针对单转子系统,利用时域波形图、轴心轨迹图、频谱图、Poincaré图、分岔图等方法,讨论了转速对系统非线性的影响。利用频谱图和三维瀑布图分析了含碰摩故障的单转子系统频谱响应,比较了发生碰摩与不发生碰摩故障的频率响应的差异,发现了单转子碰摩故障诊断的故障特征频率;最后,利用航空发动机转子实验器进行了碰摩故障实验,并将实验结果与仿真结果进行了比较,结果取得了很好的一致性。
(3)在双转子模型中,不仅对滚动轴承进行了较为详细的建模,而且考虑了高低压转子通过中介轴承的耦合作用。根据双转子系统的仿真结果,首先,分析了不同转速比时,系统随碰摩刚度变化的非线性响应规律。然后,分析了双转子系统碰摩与不碰摩时频率响应的差异性,并为双转子的碰摩故障诊断提供了故障特征频率。最后,分析了转速对双转子系统的拍振响应和振信号强度的影响,并利用双转子实验器验证了双转子系统的拍振响应规律。
(4)研究了支持向量机分类器算法,利用遗传算法实现了SVM模型参数的自动选取。并将SVM运用与转子碰摩故障诊断。对于单转子系统,首先利用转子-滚动轴承-机匣动力学模型仿真出大量的碰摩故障样本对支持向量机进行训练,然后利用训练好的SVM模型,对碰摩故障实验样本进行诊断,结果达到了较高的识别率。
Along with the increasing of push weight rate of modern aero-engine, the probability of rubbing between rotor and stator has a great increase. With the higher need of integrity and reliability for the aero-engine structure, Vibration feature o the mechanism, vibration feature and diagnostic technology of rub-impact between rotor and stator in rotating machinery are required to research deeply. The main work of this paper is briefly summarized below:
(1) Aiming at the single/dual rotor-bearing system of practical aero-engine, a rotor-ball bearing-support-stator coupling system dynamic model is established. In the model, the stator motion is considered, and the flexible support and squeeze film damper (SDF) are established, and the nonlinear factors of ball bearing such as the clearance of bearing, nonlinear Hertzian contract force between balls and races, and the varying compliance vibration because of periodical variety of contact position between balls and races are modeled. In the dual rotor model, the coupling effect of inter-rotor bearing between the low pressure rotor and high pressure rotor is also considered. At last, the Fourth-Order Runge-Kutta method is employed to obtain system’s responses and the verification of the model is carried out.
(2) The time waveforms, the orbits, the frequency spectra, waterfall plot, the Poincarémaps and the bifurcation plots are used to analysis the spectral response and the non-linear response with the different rotate speeds for the single system. We compare the response of rub-impact fault with the no-impact fault, we find the characteristic frequency of the rub-impact fault. Finally, an aero-engine tester is used to carry out the rubbing fault experiment, and the result has had a high coherence between simulation and experiment.
(3) In the dual model, the roll bearing model is not only built detailedly, but also the coupling effect of inter-rotor bearing between the low pressure rotor and high pressure rotor is also considered. According to the simulation results, we analysis the spectral response and the non-linear response with the different impact-rub stiffness for the dual rotor system, then the characters of beat vibration are quantitatively analyzed and verified by numerical analysis and experiment. The results show that the beat is related to the speed difference of the dual rotor.
(4) Support Vector Machine (SVM) is studied, and the genetic algorithm (GA) are used to optimize the parameters of SVM. Then the SVM is trained by the simulation samples, which are obtained by rotor-ball bearing-support-stator coupling system dynamic model, and the trained classifier is used to diagnose the experiment rubbing fault samples. Finally, we have a high recognition from the SVM.
(1)针对航空发动机转子系统,建立了含转子碰摩故障的单/双转子-滚动轴承-机匣耦合动力学模型。在模型中,考虑了由转静碰摩所导致的机匣运动,滚动轴承外圈与轴承座之间的弹性支承和挤压油膜阻尼,同时,充分考虑了滚动轴承间隙、滚动轴承滚珠与滚道的非线性赫兹接触力以及由滚动轴承支撑刚度变化而产生的VC(Varying compliance)振动。在双转子模型中还考虑了高压转子和低压转子的耦合振动。最后,运用四阶龙格库塔法获取了系统响应,并验证了模型的正确性。
(2)针对单转子系统,利用时域波形图、轴心轨迹图、频谱图、Poincaré图、分岔图等方法,讨论了转速对系统非线性的影响。利用频谱图和三维瀑布图分析了含碰摩故障的单转子系统频谱响应,比较了发生碰摩与不发生碰摩故障的频率响应的差异,发现了单转子碰摩故障诊断的故障特征频率;最后,利用航空发动机转子实验器进行了碰摩故障实验,并将实验结果与仿真结果进行了比较,结果取得了很好的一致性。
(3)在双转子模型中,不仅对滚动轴承进行了较为详细的建模,而且考虑了高低压转子通过中介轴承的耦合作用。根据双转子系统的仿真结果,首先,分析了不同转速比时,系统随碰摩刚度变化的非线性响应规律。然后,分析了双转子系统碰摩与不碰摩时频率响应的差异性,并为双转子的碰摩故障诊断提供了故障特征频率。最后,分析了转速对双转子系统的拍振响应和振信号强度的影响,并利用双转子实验器验证了双转子系统的拍振响应规律。
(4)研究了支持向量机分类器算法,利用遗传算法实现了SVM模型参数的自动选取。并将SVM运用与转子碰摩故障诊断。对于单转子系统,首先利用转子-滚动轴承-机匣动力学模型仿真出大量的碰摩故障样本对支持向量机进行训练,然后利用训练好的SVM模型,对碰摩故障实验样本进行诊断,结果达到了较高的识别率。
Along with the increasing of push weight rate of modern aero-engine, the probability of rubbing between rotor and stator has a great increase. With the higher need of integrity and reliability for the aero-engine structure, Vibration feature o the mechanism, vibration feature and diagnostic technology of rub-impact between rotor and stator in rotating machinery are required to research deeply. The main work of this paper is briefly summarized below:
(1) Aiming at the single/dual rotor-bearing system of practical aero-engine, a rotor-ball bearing-support-stator coupling system dynamic model is established. In the model, the stator motion is considered, and the flexible support and squeeze film damper (SDF) are established, and the nonlinear factors of ball bearing such as the clearance of bearing, nonlinear Hertzian contract force between balls and races, and the varying compliance vibration because of periodical variety of contact position between balls and races are modeled. In the dual rotor model, the coupling effect of inter-rotor bearing between the low pressure rotor and high pressure rotor is also considered. At last, the Fourth-Order Runge-Kutta method is employed to obtain system’s responses and the verification of the model is carried out.
(2) The time waveforms, the orbits, the frequency spectra, waterfall plot, the Poincarémaps and the bifurcation plots are used to analysis the spectral response and the non-linear response with the different rotate speeds for the single system. We compare the response of rub-impact fault with the no-impact fault, we find the characteristic frequency of the rub-impact fault. Finally, an aero-engine tester is used to carry out the rubbing fault experiment, and the result has had a high coherence between simulation and experiment.
(3) In the dual model, the roll bearing model is not only built detailedly, but also the coupling effect of inter-rotor bearing between the low pressure rotor and high pressure rotor is also considered. According to the simulation results, we analysis the spectral response and the non-linear response with the different impact-rub stiffness for the dual rotor system, then the characters of beat vibration are quantitatively analyzed and verified by numerical analysis and experiment. The results show that the beat is related to the speed difference of the dual rotor.
(4) Support Vector Machine (SVM) is studied, and the genetic algorithm (GA) are used to optimize the parameters of SVM. Then the SVM is trained by the simulation samples, which are obtained by rotor-ball bearing-support-stator coupling system dynamic model, and the trained classifier is used to diagnose the experiment rubbing fault samples. Finally, we have a high recognition from the SVM.
引文
[1]宋兆泓.航空发动机典型故障分析[M].北京:北京航空航天大学出版社, 1993.
[2]王德友.发动机转静件碰摩振动特征的提取与理论研究[D].北京:北京航空航天大学, 1995.
[3] Muszynska A. Stability of whirl and whip in rotor bearing system .Journal of Sound and Vibration, 1988, 127(1): 49~64.
[4] Agnes Mxjszynska, Paul Goldman. Chaotic Responses of Unbalanced Rotor/Bearing/Stator Systems with Looseness or Rubs [J]. Chaos, Solitons & Fractals, 1995, 5(9):1683~1704.
[5] Agnes Mxjszynska, Paul Goldman. Rotor-to-Stator, Rub-related, Thermal/Mechanical Effects in Rotating Machinery [J]. Chaos, Solitons & Fractals, 1995, 5(9):1579~1601.
[6] Chu F L, Zhang Z S. Bifurcation and chaos in a rub impact Jeffcott rotor system [J]. Journal of Sound and Vibration, 1998, 210 (1): 1~18.
[7] Chu Fulei, Lu Wenxiu. Experimental observation of nonlinear vibrations in a rub-impact rotor system [J]. Journal of Sound and Vibration, 2005 (283): 621–643.
[8]诸福磊,冯冠平,张正松.碰摩转子系统中的阵发性及混沌现象[J].航空动力学报,1996, 11(3), 261~264.
[9]袁惠群,闻邦椿,王德友,刘叔伦.滚动轴承-转子-定子系统的碰摩故障分析[J].东北大学学报(自然科学版), 2003, 24(3):244~247.
[10] E1-Sacidv, F.M.A., Effect of tooth backlash and ball earing deadband clearance on vibration spectrum in spur earboxes. Journal of the Acoustical Socity of America, 1991, 89(6), 2766~2773.
[11] El-Sacidy, F.M.A., Effect of backlash on vibration spectrum in spur gearboxes incorporating ball bearings with radial clearance, Proceedings of the 8th International Modal Analysis Conference Kissmmee. FL, January 29-Feburary 1, 1158~1164.
[12] Kim, Y. B. and Noah, S. T., Steady-state analysis of a nonlinear rotor-housing system. ASME Journal o f Engineering for Gas Turbines and Power, 1991, 113, 550~556.
[13] Childs, n. w., The space shuttle main engine highpressure fuel turbopum protor dynamic instability problem .A SME Journal o f Engineering for Industry ,1978,100,4 8~57.
[14] Sankaravelu, A, Noah, S. T and Burger, C .P .Bifurcation and chaos in hall bearings. ASME Winter Annual Meeting, Symposiumon Stochastic and Nonlinear Dynamics , 1994 AMD一Vol.192, DE-Vol.7 8, Chicago, 313~325.
[15]陈果.带碰摩耦合故障的转子-滚动轴承-机匣耦合动力学模型[J].振动工程学报, 2007,20(4): 361~368.
[16]陈果.转子-滚动轴承-机匣耦合系统的不平衡-碰摩耦合故障非线性动力学响应分析[J].航空动力学报, 2007, 22(10): 1771~1778.
[17]赵凌燕.滚动轴承-转子系统的非线性动力学研究:[D].西安:西北工业大学, 2003.
[18]张利芹.具有非线性支承刚度的轴承-转子系统非线性动力学研究:[D].西安:西安建筑科技大学, 2006.
[19]K. D. Gupta, K. Gupta, K. Athre. Stability analysis of dual rotor system by extended transfer matrix method[J]. American Society of Mechanical Engineers, NewYork, 1989.
[20] K. D. Gupta, K. Gupta, K. Athre. Unbalance response of a dual rotor system: theory and experiment. Journal of Vibration and Acoustics. 1993, 115:427~435.
[21]邓四二,滕弘飞,周彦伟等.滚动轴承-双转子系统动态性能分析[J].轴承, 2006, (4): 1~4.
[22]唐六丁,邓四二,孟瑾,杜辉.双转子-轴承系统动态性能仿真分析[J].机械传动, 2006, 30(4): 55~58.
[23]李宗成.反向旋转的双转子系统振动响应特性分析及实验研究[D].南京:南京航空航天大学,2001.
[24]胡绚,罗贵火,高德平.反向旋转双转子稳态响应计算分析与试验.航空动力学报, 2007, 22(7): 1044～1049.
[25]胡绚,罗贵火,高德平.航空发动机中介轴承的特性分析[J].航空动力学报, 2007, 22(3): 439～443.
[26]庞辉,李育锡,王三民.基于Riccati变换的整体传递矩阵法计算双转子理解转速的研究[J].机械科学与技术, 2005, 24(7): 832～834.
[27]庞辉.多转子系统动力特性分析及应用研究[D].西安:西北工业大学, 2005.
[28] Chiang H W D, Hsu C N, Tu S H. Rotor-bearing analysis for turbo machinery single- and dual-rotor systems[J]. Journal of Propulsion and Power, 2004, 20(6):1096~1104.
[29] Chiang, H. W. D., Hsu, C. N., Jeng, W., Tu, S. H., and Li, W. C., Turbo machinery dual rotor-bearing system analysis,”American Society of Mechanical Engineers, Paper GT-2002-30315, June 2002.
[30]晏砺堂,王德友.航空双转子发动机动静件碰摩振动特征研究[J].航空动力学报, 1998, 13(2): 173～176.
[31]刘献栋.旋转机械转静件碰摩故障及其诊断技术的研究[D].北京:北京航空航天大学, 1999
[32]刘献栋,李其汉,王德友.具有转静件碰摩故障双转子系统的动力学模型及其小波变换特征.航空动力学报, 2000, 15(2): 187~19.
[33]孟越,李其汉.应用整体传递系数法分析复杂转子系统转静件碰摩振动特征[J].航空动力学报, 2003, 18(1): 146~150.
[34]单颖春,刘献栋,何田,李其汉.双转子系统碰摩有限元接触分析模型及故障诊断[J].航空动力学报, 2005, 20(5): 789~794.
[35]姜洪开,何正嘉,段晨东,陈雪峰.基于提升方法的小波构造及早期故障特征提取[J].西安交通大学学报,2005, 39(5):494~498.
[36]段晨东,张建丁.基于第二代小波变换的转子碰摩故障特征提取方法研究[J].机械科学与技术, 2006, 25(10):1229~1232.
[37]刘献栋,李其汉.小波变换在转子系统动静件早期碰摩故障诊断中的应用[J].航空学报, 1999, 20(3):220~223.
[38]宋友,柳重堪,李其汉.基于小波包分解的早期碰摩故障诊断研究[J].北京航空航天大学学报,2003, 29(1):87~90.
[39]宋友,柳重堪,李其汉.基于小波变换的转子动静件碰摩故障诊断研究[J].振动工程学报,2002, 15(3):319~322.
[40]万方义,许庆余,李松涛.谐波小波变换在转子—轴承系统多故障监测中的应用[J].工程力学, 2004, 21(1):102~106.
[41]赵玉成,张玉莲,张亚红等.时间序列关联维数在非线性系统运动性态识别中的应用[J].航空学报, 2003, 24(1):28~32.
[42]徐玉秀,侯荣涛,杨文平.广义分形维数在旋转机械故障诊断中的应用研究[J].中国机械工程, 2003, 14(21):1812~1814.
[43]任辉,裴承鸣,贺尔铭等.转子碰摩振动响应的非线性时间序列分析[J].航空动力学报, 2002, 17(4):437~441.
[44]宋晓萍,廖明夫.双转子航空发动机振动信号的分离测试技术[J].机械科学与技术, 2006, 25(4): 487~490.
[45]陈岳东,蒋林,屈梁生.机械故障信号的分离[[J],中国机械工程,1995,6(2): 4850.
[46]韩震宇,朱鲁闯,屈梁生.大型回转机械故障的神经网络分类器[[J],成都科技大学学报,1996,6: 1827.
[47]张小栋,苗晓燕,朱均.模糊神经网络诊断模型理论及应用研究[[J],航空动力学报,2000,15(2): 196200.
[48]任辉,裴承鸣.基于LVQ神经网络的混沌时间序列分类识别[J].机械科学与技术, 2001,20(6):916~917.
[49]杨世锡,胡小平.大型旋转机械状态监测与故障诊断系统的面向对象建模研究[[J],机电工程,1999,3: 44~47.
[50]吴峰崎,孟光.基于支持向量机的转子振动信号故障分类研究[J].振动工程学报, 2006, 19(2):238~241.
[51]田路,田干,张炜,李亮.基于支持向量机的涡轮泵故障诊断方法研究[J].控制工程, 2007, 14(S):138~140.
[52]张周锁,李凌均,何正嘉.基于支持向量机的机械故障诊断方法研究[J].西安交通大学学报, 2002, 36(12):1303~1306.
[53] GeM, Du R, Zhang C C, Xu Y S. Fault diagnosis using support vector machine with an application in metal stamping operations[ J ]. Mechanical Systems and Signal Processing, 2004, 18: 143.
[54] Sheng-Fa Yuan, Fu-Lei Chu. Support vector machines-based fault diagnosis for turbo-pump rotor [J]. Mechanical Systems and Signal Processing, 2006, 20:939~952.
[55] Wu Fengqi, G. Meng. Compound rub malfunctions feature extraction based on full-spectrum cascade analysis and SVM [J]. Mechanical Systems and Signal Processing, 2006, 20:2007~2021.
[56] Wu Fengqi, G. Meng. Fault diagnosis based on support vector machines with parameter optimisation by artificial immunisation algorithm [J]. Mechanical Systems and Signal Processing, 2007, 21:1318~1330.
[57] Sheng-Fa Yuan, Fu-Lei Chu. Fault diagnosis based on support vector machines with parameter optimization by artificial immunization algorithm [J]. Mechanical Systems and Signal Processing, 2007, 21:1318~1330.
[58] L. B. Jack, A. K. Nandi. Fault detection using support vector machine and artificial neural networks, augmented by genetic algorithms [J]. Mechanical Systems and Signal Processing, 2002, 16(2–3):373–390.
[59]陈果.基于遗传算法的支持向量机分类器模型参数优化[J].机械科学与技术, 2007, 26(3): 347~350.
[60]于德介,陈淼峰,程军圣,杨宇.一种基于经验模式分解与支持向量机的转子故障诊断方[J].中国电机工程学报,2006, 26(16):162~167.
[61]万长森.滚动轴承的分析方法[M].北京:机械工业出版社, 1987.
[62] Hertz H. The contact of elastic solids. J. Reine Angew. Math, 1881, (92): 156~171.
[63]马东升.数值计算方法[M].北京:机械工业出版社, 2002.
[64] Mevel B, Guyader J L. Routs to chaos in ball bearing [J]. Journal of Sound and Vibration, 1993, 162(3): 471~487.
[65] Fukata S, Gad E H, Kondou T etc. On the radial vibration of ball bearings (computer simulation) [J], Bulletin of the JSME, 1985, 28: 899~904.
[66]航空发动机设计手册总编委会.航空发动机设计手册(第19分册)转子动力学及整机振动.北京:航空工业出版社, 2000.
[67]闻邦椿,顾家柳,夏松波等.高等转子动力学[M].北京:机械工业出版社, 2000.
[68] NPR 8831.2D. Reliability centered building and equipment acceptance duide. Nation Aeronautics and Space Administration (NASA), 2004.
[69]韩军,高德平,胡绚,陈高杰.航空发动机双转子系统的拍振分析[J].航空学报, 2007, 28(6), 1369~1373.
[70] Buscarello R T. Practical solutions to machinery and maintenance vibration problems [M]. Fourth Revised Edition, Denver: Update International, 2002.
[71]贝达特J S,皮尔索A G.随机数据处理[M].北京:科学出版社, 1989:30~34.
[72]胡海岩.应用非线性动力学[M].北京:航空工业出版社, 2000.
[73] Nayfeh A H, Mook D T.Nonlinear Oscillations [M]. New York: John Wiley&Sons, 1979:183~184, 368~395.
[74]闻邦椿,李以农,徐培民等.工程非线性振动[M].北京:科学出版社, 2007, 8.
[75] Vapnik V. The nature of statistical learning theory [M]. Springer-Verlag, New York. 1995.
[76]褚蕾蕾,陈绥阳,周梦编著.计算智能的数学基础[M].北京:科学出版社, 2002.
[77]魏海坤.神经网络结构设计的理论与方法[M ].北京:国防工业出版社, 2005.
[78] Goldberg D. Genetic Algorithms in Search, Optimization and Machine Learning [M]. Addison-Wesley, Reading, MA, 1989.
[79]张希军. WP13发动机试车振动分析系统研究[D].西安:西北工业大学, 2005.
[80]唐云冰.航空发动机高速滚动轴承力学特性研究[D].南京:南京航空航天大学, 2005
[2]王德友.发动机转静件碰摩振动特征的提取与理论研究[D].北京:北京航空航天大学, 1995.
[3] Muszynska A. Stability of whirl and whip in rotor bearing system .Journal of Sound and Vibration, 1988, 127(1): 49~64.
[4] Agnes Mxjszynska, Paul Goldman. Chaotic Responses of Unbalanced Rotor/Bearing/Stator Systems with Looseness or Rubs [J]. Chaos, Solitons & Fractals, 1995, 5(9):1683~1704.
[5] Agnes Mxjszynska, Paul Goldman. Rotor-to-Stator, Rub-related, Thermal/Mechanical Effects in Rotating Machinery [J]. Chaos, Solitons & Fractals, 1995, 5(9):1579~1601.
[6] Chu F L, Zhang Z S. Bifurcation and chaos in a rub impact Jeffcott rotor system [J]. Journal of Sound and Vibration, 1998, 210 (1): 1~18.
[7] Chu Fulei, Lu Wenxiu. Experimental observation of nonlinear vibrations in a rub-impact rotor system [J]. Journal of Sound and Vibration, 2005 (283): 621–643.
[8]诸福磊,冯冠平,张正松.碰摩转子系统中的阵发性及混沌现象[J].航空动力学报,1996, 11(3), 261~264.
[9]袁惠群,闻邦椿,王德友,刘叔伦.滚动轴承-转子-定子系统的碰摩故障分析[J].东北大学学报(自然科学版), 2003, 24(3):244~247.
[10] E1-Sacidv, F.M.A., Effect of tooth backlash and ball earing deadband clearance on vibration spectrum in spur earboxes. Journal of the Acoustical Socity of America, 1991, 89(6), 2766~2773.
[11] El-Sacidy, F.M.A., Effect of backlash on vibration spectrum in spur gearboxes incorporating ball bearings with radial clearance, Proceedings of the 8th International Modal Analysis Conference Kissmmee. FL, January 29-Feburary 1, 1158~1164.
[12] Kim, Y. B. and Noah, S. T., Steady-state analysis of a nonlinear rotor-housing system. ASME Journal o f Engineering for Gas Turbines and Power, 1991, 113, 550~556.
[13] Childs, n. w., The space shuttle main engine highpressure fuel turbopum protor dynamic instability problem .A SME Journal o f Engineering for Industry ,1978,100,4 8~57.
[14] Sankaravelu, A, Noah, S. T and Burger, C .P .Bifurcation and chaos in hall bearings. ASME Winter Annual Meeting, Symposiumon Stochastic and Nonlinear Dynamics , 1994 AMD一Vol.192, DE-Vol.7 8, Chicago, 313~325.
[15]陈果.带碰摩耦合故障的转子-滚动轴承-机匣耦合动力学模型[J].振动工程学报, 2007,20(4): 361~368.
[16]陈果.转子-滚动轴承-机匣耦合系统的不平衡-碰摩耦合故障非线性动力学响应分析[J].航空动力学报, 2007, 22(10): 1771~1778.
[17]赵凌燕.滚动轴承-转子系统的非线性动力学研究:[D].西安:西北工业大学, 2003.
[18]张利芹.具有非线性支承刚度的轴承-转子系统非线性动力学研究:[D].西安:西安建筑科技大学, 2006.
[19]K. D. Gupta, K. Gupta, K. Athre. Stability analysis of dual rotor system by extended transfer matrix method[J]. American Society of Mechanical Engineers, NewYork, 1989.
[20] K. D. Gupta, K. Gupta, K. Athre. Unbalance response of a dual rotor system: theory and experiment. Journal of Vibration and Acoustics. 1993, 115:427~435.
[21]邓四二,滕弘飞,周彦伟等.滚动轴承-双转子系统动态性能分析[J].轴承, 2006, (4): 1~4.
[22]唐六丁,邓四二,孟瑾,杜辉.双转子-轴承系统动态性能仿真分析[J].机械传动, 2006, 30(4): 55~58.
[23]李宗成.反向旋转的双转子系统振动响应特性分析及实验研究[D].南京:南京航空航天大学,2001.
[24]胡绚,罗贵火,高德平.反向旋转双转子稳态响应计算分析与试验.航空动力学报, 2007, 22(7): 1044～1049.
[25]胡绚,罗贵火,高德平.航空发动机中介轴承的特性分析[J].航空动力学报, 2007, 22(3): 439～443.
[26]庞辉,李育锡,王三民.基于Riccati变换的整体传递矩阵法计算双转子理解转速的研究[J].机械科学与技术, 2005, 24(7): 832～834.
[27]庞辉.多转子系统动力特性分析及应用研究[D].西安:西北工业大学, 2005.
[28] Chiang H W D, Hsu C N, Tu S H. Rotor-bearing analysis for turbo machinery single- and dual-rotor systems[J]. Journal of Propulsion and Power, 2004, 20(6):1096~1104.
[29] Chiang, H. W. D., Hsu, C. N., Jeng, W., Tu, S. H., and Li, W. C., Turbo machinery dual rotor-bearing system analysis,”American Society of Mechanical Engineers, Paper GT-2002-30315, June 2002.
[30]晏砺堂,王德友.航空双转子发动机动静件碰摩振动特征研究[J].航空动力学报, 1998, 13(2): 173～176.
[31]刘献栋.旋转机械转静件碰摩故障及其诊断技术的研究[D].北京:北京航空航天大学, 1999
[32]刘献栋,李其汉,王德友.具有转静件碰摩故障双转子系统的动力学模型及其小波变换特征.航空动力学报, 2000, 15(2): 187~19.
[33]孟越,李其汉.应用整体传递系数法分析复杂转子系统转静件碰摩振动特征[J].航空动力学报, 2003, 18(1): 146~150.
[34]单颖春,刘献栋,何田,李其汉.双转子系统碰摩有限元接触分析模型及故障诊断[J].航空动力学报, 2005, 20(5): 789~794.
[35]姜洪开,何正嘉,段晨东,陈雪峰.基于提升方法的小波构造及早期故障特征提取[J].西安交通大学学报,2005, 39(5):494~498.
[36]段晨东,张建丁.基于第二代小波变换的转子碰摩故障特征提取方法研究[J].机械科学与技术, 2006, 25(10):1229~1232.
[37]刘献栋,李其汉.小波变换在转子系统动静件早期碰摩故障诊断中的应用[J].航空学报, 1999, 20(3):220~223.
[38]宋友,柳重堪,李其汉.基于小波包分解的早期碰摩故障诊断研究[J].北京航空航天大学学报,2003, 29(1):87~90.
[39]宋友,柳重堪,李其汉.基于小波变换的转子动静件碰摩故障诊断研究[J].振动工程学报,2002, 15(3):319~322.
[40]万方义,许庆余,李松涛.谐波小波变换在转子—轴承系统多故障监测中的应用[J].工程力学, 2004, 21(1):102~106.
[41]赵玉成,张玉莲,张亚红等.时间序列关联维数在非线性系统运动性态识别中的应用[J].航空学报, 2003, 24(1):28~32.
[42]徐玉秀,侯荣涛,杨文平.广义分形维数在旋转机械故障诊断中的应用研究[J].中国机械工程, 2003, 14(21):1812~1814.
[43]任辉,裴承鸣,贺尔铭等.转子碰摩振动响应的非线性时间序列分析[J].航空动力学报, 2002, 17(4):437~441.
[44]宋晓萍,廖明夫.双转子航空发动机振动信号的分离测试技术[J].机械科学与技术, 2006, 25(4): 487~490.
[45]陈岳东,蒋林,屈梁生.机械故障信号的分离[[J],中国机械工程,1995,6(2): 4850.
[46]韩震宇,朱鲁闯,屈梁生.大型回转机械故障的神经网络分类器[[J],成都科技大学学报,1996,6: 1827.
[47]张小栋,苗晓燕,朱均.模糊神经网络诊断模型理论及应用研究[[J],航空动力学报,2000,15(2): 196200.
[48]任辉,裴承鸣.基于LVQ神经网络的混沌时间序列分类识别[J].机械科学与技术, 2001,20(6):916~917.
[49]杨世锡,胡小平.大型旋转机械状态监测与故障诊断系统的面向对象建模研究[[J],机电工程,1999,3: 44~47.
[50]吴峰崎,孟光.基于支持向量机的转子振动信号故障分类研究[J].振动工程学报, 2006, 19(2):238~241.
[51]田路,田干,张炜,李亮.基于支持向量机的涡轮泵故障诊断方法研究[J].控制工程, 2007, 14(S):138~140.
[52]张周锁,李凌均,何正嘉.基于支持向量机的机械故障诊断方法研究[J].西安交通大学学报, 2002, 36(12):1303~1306.
[53] GeM, Du R, Zhang C C, Xu Y S. Fault diagnosis using support vector machine with an application in metal stamping operations[ J ]. Mechanical Systems and Signal Processing, 2004, 18: 143.
[54] Sheng-Fa Yuan, Fu-Lei Chu. Support vector machines-based fault diagnosis for turbo-pump rotor [J]. Mechanical Systems and Signal Processing, 2006, 20:939~952.
[55] Wu Fengqi, G. Meng. Compound rub malfunctions feature extraction based on full-spectrum cascade analysis and SVM [J]. Mechanical Systems and Signal Processing, 2006, 20:2007~2021.
[56] Wu Fengqi, G. Meng. Fault diagnosis based on support vector machines with parameter optimisation by artificial immunisation algorithm [J]. Mechanical Systems and Signal Processing, 2007, 21:1318~1330.
[57] Sheng-Fa Yuan, Fu-Lei Chu. Fault diagnosis based on support vector machines with parameter optimization by artificial immunization algorithm [J]. Mechanical Systems and Signal Processing, 2007, 21:1318~1330.
[58] L. B. Jack, A. K. Nandi. Fault detection using support vector machine and artificial neural networks, augmented by genetic algorithms [J]. Mechanical Systems and Signal Processing, 2002, 16(2–3):373–390.
[59]陈果.基于遗传算法的支持向量机分类器模型参数优化[J].机械科学与技术, 2007, 26(3): 347~350.
[60]于德介,陈淼峰,程军圣,杨宇.一种基于经验模式分解与支持向量机的转子故障诊断方[J].中国电机工程学报,2006, 26(16):162~167.
[61]万长森.滚动轴承的分析方法[M].北京:机械工业出版社, 1987.
[62] Hertz H. The contact of elastic solids. J. Reine Angew. Math, 1881, (92): 156~171.
[63]马东升.数值计算方法[M].北京:机械工业出版社, 2002.
[64] Mevel B, Guyader J L. Routs to chaos in ball bearing [J]. Journal of Sound and Vibration, 1993, 162(3): 471~487.
[65] Fukata S, Gad E H, Kondou T etc. On the radial vibration of ball bearings (computer simulation) [J], Bulletin of the JSME, 1985, 28: 899~904.
[66]航空发动机设计手册总编委会.航空发动机设计手册(第19分册)转子动力学及整机振动.北京:航空工业出版社, 2000.
[67]闻邦椿,顾家柳,夏松波等.高等转子动力学[M].北京:机械工业出版社, 2000.
[68] NPR 8831.2D. Reliability centered building and equipment acceptance duide. Nation Aeronautics and Space Administration (NASA), 2004.
[69]韩军,高德平,胡绚,陈高杰.航空发动机双转子系统的拍振分析[J].航空学报, 2007, 28(6), 1369~1373.
[70] Buscarello R T. Practical solutions to machinery and maintenance vibration problems [M]. Fourth Revised Edition, Denver: Update International, 2002.
[71]贝达特J S,皮尔索A G.随机数据处理[M].北京:科学出版社, 1989:30~34.
[72]胡海岩.应用非线性动力学[M].北京:航空工业出版社, 2000.
[73] Nayfeh A H, Mook D T.Nonlinear Oscillations [M]. New York: John Wiley&Sons, 1979:183~184, 368~395.
[74]闻邦椿,李以农,徐培民等.工程非线性振动[M].北京:科学出版社, 2007, 8.
[75] Vapnik V. The nature of statistical learning theory [M]. Springer-Verlag, New York. 1995.
[76]褚蕾蕾,陈绥阳,周梦编著.计算智能的数学基础[M].北京:科学出版社, 2002.
[77]魏海坤.神经网络结构设计的理论与方法[M ].北京:国防工业出版社, 2005.
[78] Goldberg D. Genetic Algorithms in Search, Optimization and Machine Learning [M]. Addison-Wesley, Reading, MA, 1989.
[79]张希军. WP13发动机试车振动分析系统研究[D].西安:西北工业大学, 2005.
[80]唐云冰.航空发动机高速滚动轴承力学特性研究[D].南京:南京航空航天大学, 2005