航空发动机高速滚动轴承及转子系统的动态性能研究
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摘要
滚动轴承作为航空发动机转子系统广泛采用的一种支承,其动态力学特性对转子系统的动力特性有重要的影响,随着航空发动机的性能不断提高,这种影响更为显著。因此,为了优化和提升现有发动机的动态性能,满足新一代先进航空发动机的研制需要,进行高速滚动轴承及其转子系统动态性能的研究十分必要。本文围绕航空发动机高速滚动轴承及其支承的转子系统,建立了滚动轴承的拟动力学分析模型、滚动轴承-转子系统的动特性和非线性振动模型,分析了不同工况参数和轴承结构参数对滚动轴承及转子系统动特性的影响,开发了分析程序,在现有的高速轴承台架上进行了高速轴承转子系统的验证实验。本文的研究为高速滚动轴承及其转子系统的分析提供理论基础,为航空发动机的可靠性分析提供技术储备。
论文首先对高速滚动轴承的动态性能进行了研究,通过建立滚动轴承的拟动力学分析模型对高速球轴承和高速滚子轴承进行分析,工况和轴承结构参数对轴承动态性能有较大影响:球轴承的径向载荷和弯矩的存在会使滚动体的旋滚比、微区PV值、保持架滑动率等动态性能在数值上和变化规律上都有较大变化,且弯矩的影响更为明显。较小的接触角、较大的沟曲率对于降低球的PV值、降低保持架滑动率都有明显的影响。滚子轴承在高速轻载时,较小的弯矩就使滚子接触载荷峰值和PV值将变得很大,外圈存在椭圆度、具有较小径向游隙的滚子轴承对于降低滚子PV值、降低保持架滑动率都有明显的影响。
研究了轴向预载荷对球轴承打滑的影响规律,给出了防止球轴承打滑的临界轴向载荷的定义:承受联合载荷作用或有弯矩存在的球轴承,当承载滚动体不打滑且保持架转速明显趋于平稳时对应的轴向载荷为临界轴向载荷;仅受轴向载荷作用的球轴承,滚动体打滑率为零时对应的轴向载荷为临界轴向载荷。并提供了在给定保持架滑动率下的轴承轴向预载荷。
考虑轴承的动刚度用Riccati传递矩阵法分析了转子系统的动特性。与定刚度值计算相比,考虑动刚度对转子系统的临界转速、稳定性、失稳门槛值有较大的影响。转子系统的瞬态响应会影响支承轴承的动态性能,建立了滚动轴承与转子整体计算的模型,用Newmark-β积分法计算并分析了转子的响应和轴承的动态性能之间的耦合规律,发现当转子载荷变化速率较大、突加不平衡力较大时会引起转子的振幅较大,同时引起支承轴承保持架的滑动率也较大。
为了揭示滚动轴承支承的转子系统的复杂振动特性,推导了滚动轴承的非线性轴承力,建立了滚动轴承非线性振动的运动微分方程,使用Runge-Kutta积分法求解。研究表明,滚动轴承的非线性轴承力会诱发变刚度振动,使系统响应中存在周期振动和拟周期、混沌非周期振动。选取适当的参数,轴承的接触角、游隙、预紧力、阻尼等,可以改善非线性轴承力引起的非周期振动行为。转子系统中存在不平衡力时,将使非周期振动区域明显增多,产生变刚度振动频率和不平衡力振动频率的组合振动。研究给出了滚动轴承外圈波纹度、内圈波纹度、滚动体波纹度引起的振动频率与轴承结构参数和波纹度阶数的关系。考虑转子的非线性振动发现在非周期振动范围内,滚动轴承的打滑率、刚度有较大变化。
编制了滚动轴承设计与分析软件,使用软件集成技术实现了滚动轴承参数化设计、滚动轴承-转子系统性能预测、滚动轴承温度场分析和滚动轴承寿命预测的集成,为滚动轴承的设计与分析提供依据。
使用经过实验验证的滚动轴承动力学算例对本文建立的滚动轴承拟动力学模型进行了验证,两者计算结果较为接近,对轴承稳态性能预测较准确,对于轴承瞬态运动学参数预测的误差在5%以内,表明本文建立的拟动力学模型可以较为准确的预测滚动轴承的动态性能。
以现有的高速滚动轴承试验台架为基础,进行了高速滚动轴承转子系统动态性能的实验验证,测试了滚动轴承转子系统的临界转速、振动位移、轴心轨迹,与软件计算结果对比:理论计算结果与实验结果比较吻合,并揭示了转子系统中存在的复杂轴心轨迹,表明本文提出和建立的滚动轴承转子系统动态性能分析模型和方法可以较为精确和可靠的预测轴承和转子的动态性能。
Rolling bearing is widely used as support in aero-engine rotor system, and its dynamic mechanical performances have important influence on the dynamic performances of the rotor system. The influence becomes more and more prominent with the enhancement of the performance of aero-engine. Therefore, in order to optimize and upgrade the existing aero-engine dynamic performance to meet the development needs of new generation aero-engine, it is necessary to study the dynamic performances of high-speed rolling bearings and rotor system. The objects of the research are high-speed rolling bearings and rotor system of aero-engine. The quasi-dynamic models of rolling bearing, dynamic property models of rolling bearing and rotor system, nonlinear vibration models of rolling bearing and rotor system are established. The influence of operating parameters and structure parameters of rolling bearing to the dynamic performances of rolling bearings and rotor system are analyzed. Computer programs are developed. High-speed rolling bearing and rotor system experiments are performed to validate the theory on an existing small high-speed rolling bearing bench. The research provides theoretical foundation for analysis of high-speed rolling bearings and rotor system, and also provides technical reserves for the reliability of aero-engine.
The dynamic performances of high-speed rolling bearing are investigated. Through the analysis of high-speed ball bearing and roller bearing using the quasi-dynamic models, it is found that the operating parameters and structure parameters of rolling bearing have evident influence on the performances of the bearing. For the ball bearing with radial load and bending moment, spin to roll ratio, PV value of balls and sliding ratio of cage are changed greatly in value and rule when compared to axial loaded or radial loaded bearing, and the influences of bending moment are more evident. PV value of balls and sliding ratio of cage are decreased evidently by small contact angle or large groove curvature. For the roller bearing with bending moment, peak of contact load and PV value become large. Bending moments have significant influence on the dynamic performances of high-speed light-loaded roller bearing. PV value of rollers and sliding ratio of cage are decreased evidently by non-circular ring or small radial clearance.
The influence rules of the axial pre-load to the skid of ball bearing are studied. The definition of the critical axial load to prevent skid of ball bearing is given. For the ball bearing with axial load and radial load or bending moment, the axial load is the critical axial load when there aren’t skidding in loaded balls and rotational speed of cage becomes stable. For the axial loaded ball bearing, the axial load is the critical axial load when loaded balls don’t skid. The axial load of ball bearing also can be provided when skid ratio of cage is given.
The dynamic characteristics of the rotor system are analyzed considering dynamic stiffness of bearing using Riccati transfer matrix method. Compared to certain stiffness computation method, it is found that there are great influences on critical speed, stability, instability threshold of rotor system when dynamic stiffness of bearing are considered. For instantaneous response of rotor system can influence dynamic performances of rolling bearing, the whole computation model of bearing rotor system is established. The couple rules of rotor response and dynamic property of bearing are analyzed using Newmark-βmethod, it is found that amplitude of the rotor and cage sliding rate will becomes great larger when change ratio of load or sudden unbalance force is large.
In order to reveal the complex vibration behavior of rotor system supported by rolling bearing, a nonlinear vibration analysis model is established and solved by Runge-Kutta method. The results show that the nonlinear variable stiffness vibration will be induced by nonlinear bearing force, which resulting in periodic vibration and quasi-periodic, chaos vibration in the system. Select appropriate parameters, such as, contact angle, clearance, preload of bearing and damping, etc., can improve the nonlinear vibration behavior caused by nonlinear bearing force. When unbalance force exist in the rotor system, non-periodic vibration regions will increase and combined vibration constitute of variable stiffness vibration frequency and unbalance vibration frequency will occur. The relationship between vibration frequency caused by outer waviness, inner waviness, rolling element waviness of bearing and bearing structure parameters, waviness orders is given. Considering nonlinear vibration of rotor system, it is found that skid ratio and stiffness of bearing are changed greatly.
A design and analysis software for rolling bearing has been developed using software integration technology. The software integrates parametric design of bearing, performance prediction of bearing rotor system, temperature field analysis and life prediction of bearing, which provide method for design and analysis of rolling bearing.
The quasi-dynamic model of rolling bearing is validated by dynamic example which has been validated by test. The results in this dissertation are very close to the example and the difference is less than 5%, which shows that the model and method put forward in the dissertation to forecast the bearing characteristics is precise.
At last, dynamic characteristics test researches of high-speed rolling bearing rotor system have been finished using the rolling bearing bench test. Critical speed, vibration displacement, axis orbit are obtained. The theoretical results in this dissertation are very close to test data and disclose the complex axis orbit of the rotor system, which shows that the model and method put forward in the dissertation to forecast dynamic property of rolling bearing rotor system is precise and credible.
论文首先对高速滚动轴承的动态性能进行了研究,通过建立滚动轴承的拟动力学分析模型对高速球轴承和高速滚子轴承进行分析,工况和轴承结构参数对轴承动态性能有较大影响:球轴承的径向载荷和弯矩的存在会使滚动体的旋滚比、微区PV值、保持架滑动率等动态性能在数值上和变化规律上都有较大变化,且弯矩的影响更为明显。较小的接触角、较大的沟曲率对于降低球的PV值、降低保持架滑动率都有明显的影响。滚子轴承在高速轻载时,较小的弯矩就使滚子接触载荷峰值和PV值将变得很大,外圈存在椭圆度、具有较小径向游隙的滚子轴承对于降低滚子PV值、降低保持架滑动率都有明显的影响。
研究了轴向预载荷对球轴承打滑的影响规律,给出了防止球轴承打滑的临界轴向载荷的定义:承受联合载荷作用或有弯矩存在的球轴承,当承载滚动体不打滑且保持架转速明显趋于平稳时对应的轴向载荷为临界轴向载荷;仅受轴向载荷作用的球轴承,滚动体打滑率为零时对应的轴向载荷为临界轴向载荷。并提供了在给定保持架滑动率下的轴承轴向预载荷。
考虑轴承的动刚度用Riccati传递矩阵法分析了转子系统的动特性。与定刚度值计算相比,考虑动刚度对转子系统的临界转速、稳定性、失稳门槛值有较大的影响。转子系统的瞬态响应会影响支承轴承的动态性能,建立了滚动轴承与转子整体计算的模型,用Newmark-β积分法计算并分析了转子的响应和轴承的动态性能之间的耦合规律,发现当转子载荷变化速率较大、突加不平衡力较大时会引起转子的振幅较大,同时引起支承轴承保持架的滑动率也较大。
为了揭示滚动轴承支承的转子系统的复杂振动特性,推导了滚动轴承的非线性轴承力,建立了滚动轴承非线性振动的运动微分方程,使用Runge-Kutta积分法求解。研究表明,滚动轴承的非线性轴承力会诱发变刚度振动,使系统响应中存在周期振动和拟周期、混沌非周期振动。选取适当的参数,轴承的接触角、游隙、预紧力、阻尼等,可以改善非线性轴承力引起的非周期振动行为。转子系统中存在不平衡力时,将使非周期振动区域明显增多,产生变刚度振动频率和不平衡力振动频率的组合振动。研究给出了滚动轴承外圈波纹度、内圈波纹度、滚动体波纹度引起的振动频率与轴承结构参数和波纹度阶数的关系。考虑转子的非线性振动发现在非周期振动范围内,滚动轴承的打滑率、刚度有较大变化。
编制了滚动轴承设计与分析软件,使用软件集成技术实现了滚动轴承参数化设计、滚动轴承-转子系统性能预测、滚动轴承温度场分析和滚动轴承寿命预测的集成,为滚动轴承的设计与分析提供依据。
使用经过实验验证的滚动轴承动力学算例对本文建立的滚动轴承拟动力学模型进行了验证,两者计算结果较为接近,对轴承稳态性能预测较准确,对于轴承瞬态运动学参数预测的误差在5%以内,表明本文建立的拟动力学模型可以较为准确的预测滚动轴承的动态性能。
以现有的高速滚动轴承试验台架为基础,进行了高速滚动轴承转子系统动态性能的实验验证,测试了滚动轴承转子系统的临界转速、振动位移、轴心轨迹,与软件计算结果对比:理论计算结果与实验结果比较吻合,并揭示了转子系统中存在的复杂轴心轨迹,表明本文提出和建立的滚动轴承转子系统动态性能分析模型和方法可以较为精确和可靠的预测轴承和转子的动态性能。
Rolling bearing is widely used as support in aero-engine rotor system, and its dynamic mechanical performances have important influence on the dynamic performances of the rotor system. The influence becomes more and more prominent with the enhancement of the performance of aero-engine. Therefore, in order to optimize and upgrade the existing aero-engine dynamic performance to meet the development needs of new generation aero-engine, it is necessary to study the dynamic performances of high-speed rolling bearings and rotor system. The objects of the research are high-speed rolling bearings and rotor system of aero-engine. The quasi-dynamic models of rolling bearing, dynamic property models of rolling bearing and rotor system, nonlinear vibration models of rolling bearing and rotor system are established. The influence of operating parameters and structure parameters of rolling bearing to the dynamic performances of rolling bearings and rotor system are analyzed. Computer programs are developed. High-speed rolling bearing and rotor system experiments are performed to validate the theory on an existing small high-speed rolling bearing bench. The research provides theoretical foundation for analysis of high-speed rolling bearings and rotor system, and also provides technical reserves for the reliability of aero-engine.
The dynamic performances of high-speed rolling bearing are investigated. Through the analysis of high-speed ball bearing and roller bearing using the quasi-dynamic models, it is found that the operating parameters and structure parameters of rolling bearing have evident influence on the performances of the bearing. For the ball bearing with radial load and bending moment, spin to roll ratio, PV value of balls and sliding ratio of cage are changed greatly in value and rule when compared to axial loaded or radial loaded bearing, and the influences of bending moment are more evident. PV value of balls and sliding ratio of cage are decreased evidently by small contact angle or large groove curvature. For the roller bearing with bending moment, peak of contact load and PV value become large. Bending moments have significant influence on the dynamic performances of high-speed light-loaded roller bearing. PV value of rollers and sliding ratio of cage are decreased evidently by non-circular ring or small radial clearance.
The influence rules of the axial pre-load to the skid of ball bearing are studied. The definition of the critical axial load to prevent skid of ball bearing is given. For the ball bearing with axial load and radial load or bending moment, the axial load is the critical axial load when there aren’t skidding in loaded balls and rotational speed of cage becomes stable. For the axial loaded ball bearing, the axial load is the critical axial load when loaded balls don’t skid. The axial load of ball bearing also can be provided when skid ratio of cage is given.
The dynamic characteristics of the rotor system are analyzed considering dynamic stiffness of bearing using Riccati transfer matrix method. Compared to certain stiffness computation method, it is found that there are great influences on critical speed, stability, instability threshold of rotor system when dynamic stiffness of bearing are considered. For instantaneous response of rotor system can influence dynamic performances of rolling bearing, the whole computation model of bearing rotor system is established. The couple rules of rotor response and dynamic property of bearing are analyzed using Newmark-βmethod, it is found that amplitude of the rotor and cage sliding rate will becomes great larger when change ratio of load or sudden unbalance force is large.
In order to reveal the complex vibration behavior of rotor system supported by rolling bearing, a nonlinear vibration analysis model is established and solved by Runge-Kutta method. The results show that the nonlinear variable stiffness vibration will be induced by nonlinear bearing force, which resulting in periodic vibration and quasi-periodic, chaos vibration in the system. Select appropriate parameters, such as, contact angle, clearance, preload of bearing and damping, etc., can improve the nonlinear vibration behavior caused by nonlinear bearing force. When unbalance force exist in the rotor system, non-periodic vibration regions will increase and combined vibration constitute of variable stiffness vibration frequency and unbalance vibration frequency will occur. The relationship between vibration frequency caused by outer waviness, inner waviness, rolling element waviness of bearing and bearing structure parameters, waviness orders is given. Considering nonlinear vibration of rotor system, it is found that skid ratio and stiffness of bearing are changed greatly.
A design and analysis software for rolling bearing has been developed using software integration technology. The software integrates parametric design of bearing, performance prediction of bearing rotor system, temperature field analysis and life prediction of bearing, which provide method for design and analysis of rolling bearing.
The quasi-dynamic model of rolling bearing is validated by dynamic example which has been validated by test. The results in this dissertation are very close to the example and the difference is less than 5%, which shows that the model and method put forward in the dissertation to forecast the bearing characteristics is precise.
At last, dynamic characteristics test researches of high-speed rolling bearing rotor system have been finished using the rolling bearing bench test. Critical speed, vibration displacement, axis orbit are obtained. The theoretical results in this dissertation are very close to test data and disclose the complex axis orbit of the rotor system, which shows that the model and method put forward in the dissertation to forecast dynamic property of rolling bearing rotor system is precise and credible.
引文
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35 Lars-Erik Stacke, Dag Fritzson. Bearing Simulation Get Real. Machine Design. 2002
36 Tomoya Sakaguchi, Kazuyoshi Harada. Dynamic Analysis of Cage Behavior in a Tapered Roller Bearing. Trans of ASME. 2006,128: 604~611
37 El-Sayed H R. Stiffness of deep-groove ball bearings. Wear, 1980, 63(1): 89~94
38 Hernot X, Sartor M, Guillot J. Calculation of the stiffness matrix of angular contact ball bearings by using the analytical approach. Journal of Mechanical Design, Transactions of the ASME, 2000, 122(1): 83~90
39桃野达信(日).滚动轴承的刚性.国外轴承技术. 2002, (3): 56~64
40 Bugra H Ertas, John M Vance. The effect of static and dynamic misalignment on ball bearing radial stiffness. AIAA-2002-4160, 2002
41 Walford T L H, Stone B J. Some damping and stiffness characteristics of angular contact bearings under oscillation radial load. Institution of Mechanical Engineers, Conference Publications, 1980: 157~162
42 Stone B J. State of the art in the measurement of the stiffness and damping of rolling element bearings. CIRPAnnals, 1982, 31(2): 529~538
43罗祝三,孙心德,吴林丰.滚动轴承在任意方向的支承刚度.南京航空学院学报. 1992, 24(3): 248~256
44刘卫群,罗继伟,吴长春等.滚动轴承刚度分析程序.计算力学学报. 2001, 18(3): 375~378
45黄浩,张鹏顺,温建民.航空发动机角接触球轴承刚度的一种实用分析方法.南京航空航天大学学报. 2000, 32(4): 422~427
46王刚,郭茂林.航空航天滚动轴承刚度.哈尔滨工业大学学报,2001, 33(5): 644~650
47王硕桂,夏源明.从随机振动响应中提取滚动轴承的刚度参数.振动工程学报. 2004, 17(4): 388~392
48 Kraus J, Blech J J, Braun S G. In situ determination of rolling bearing stiffness and damping by model analysis. Journal of Vibration, Acoustic, Stress, and Reliability in design, 1987, 109(3): 235~240
49 Goodwin M J. Experimental techniques for bearing impedance measurement. Journal of Engineering for Industry, Transactions of the ASME, 1991, 113(3): 335~342
50 Rajiv Tiwari, Nalinaksh S. Vyas. Stiffness estimation from random response in multi-mass rotor bearing systems. Prob. Engng. Mech. 1998,13(4): 255~268
51 Raffa F. A, Vatta V. The dynamic stiffness method for linear rotor-bearing system. ASME Journal of Vibration and Acoustics, 1996,118(7): 332~339
52 M. A. Prohl, Ageneral method for calculating critical speed of flexible rotors, Journal of Appl, Mech.Trans.ASME,1945,12(8): 142~148
53 17 G. C. Horner and W. D. Pilkey, The Riccati Transfer matrix method, J. of Mech. Des. Trans ASME, 1978,100: 297~302
54 Gu Z, Zhi X, Meng G,et al. Transient response analysis of large-scale rotor-bearing system with strong non-linear elements by a transfer matrix—Newmark formulation integration method[J].Journal of Sound and Vibration, 2003,259(3): 559~570
55 Hua J, Liu Z S, Xu Q Y. A new high precision direct integration scheme for nonlinear rotor-seal system. International Conference on Scientific & Engineering Computation. Singapore: National University of Singapore, 2002, 552~556
56蒋书运,陈照波,须根法.用整体传递矩阵法计算航空发动机整机临界转速特性.哈尔滨工业大学学报. 1998, 30(1): 32~35
57柴山,刚宪约.整体传递矩阵法的Riccati变换.上海理工大学学报, 2002, 24(2): 98~100
58李永强,朱亮,刘宇等.用整体传递矩阵法计算旋转机械整机固有频率特性.东北大学学报(自然科学版). 2005, 26(12): 1178~1180
59闻邦椿,顾家柳,夏松波.高等转子动力学.北京,机械工业出版社. 2000
60 Tiwari M. Effect of radial internal clearance of a ball bearing in the dynamics of a balanced horizontal rotor. Journal Sound and Vibration. 2000, 238(5): 723~756
61 Tiwari M. Dynamics response of a unbalance spindle supported on ball bearings. Journal Sound and Vibration. 2000, 238(5): 757~779
62 Panda K C. Optimum support characteristics for rotor-shaft system with preloaded rolling element bearing. Journal Sound and Vibration. 2003, 260: 731~755
63 Mohammed A Alfares, Abdallah A. Elsharkawy. Effects of axial preloading of angular contact ball bearing on the dynamics of a grinding machine spindle system. Journal of Materials Processing Technology. 2003, 136: 48~59
64邓四二,滕弘飞,周彦伟.滚动轴承-双转子系统动态性能分析.轴承. 2006, (4): 1~4
65李松生.高速电主轴轴系转子动力学特性分析.轴承. 2002, (2): 15~17
66朱爱斌,彭祥多,谢友柏.滚动轴承刚度及转子系统动力学网络计算系统的设计和实现.润滑与密封. 2005, 9(5): 126~132
67 Dong-Soo Lee, Dong-Hoon Choi. Reduced Weight Design of a Flexible Rotor with Ball Bearing Stiffness Characteristics Varying with Rotational Speed and Load. Journal of Vibration and Acoustics. 2000, 122(7): 203~208
68 David P. Fleming. Unbalance Response Prediction for Rotors on Ball Bearings Using Speed and Load Dependent Nonlinear Bearing Stiffness. NASA/TM. 2003-212527
69 Hoon-Voon Liewa, Teik C. Lim. Analysis of time-varying rolling element bearing characteristics. Journal of Sound and Vibration. 2005,283: 1163~1179
70黄太平,罗贵火.转子动力学优化设计.航空动力学报. 1994, 9(2): 113~116
71 K. C. Panda, J. K. Dutt. Design of optimum support parameters for minimum rotor response and maximum stability limit. Journal of Sound and Vibration. 1999, 223(1): 1~21
72 C. S. SUNNERSJO. Varying compliance vibrations of rolling bearings. Journal of Sound and vibration, 1978, 58: 363~373.
73 Fukata S, Gad EH, TamuraH. On the radial vibration of ball bearings (computer simulation)[J].Bulletin of the JSME, 1985,28:899~904
74 B. MEVEL and J. L. GUYADER. Routes to chaos in ball bearings. Journal of Sound and vibration. 1993, 162: 471~487
75 Sankaravelu A, Noah ST, Burger C P. Bifurcation and chaos in ball bearings.Nonlinear and Stochastic Dynamics, 1994,192:313~325
76 D. W. CHILDS. Fractional frequency rotor motion due to clearance effects. Journal of Engineering Power. 1982,104: 533~536
77 S. SAITO. Calculation of nonlinear unbalance response of horizontal Jeffcott rotors supported by ball bearings with radial clearances. Journal of vibration, Acoustics, Stress, and Reliability in Design. 1985,107: 416~420
78 Y. B. KIM and S. T. NOAH. Bifurcation analysis for a modified Jeffcott rotor with bearing clearances. Nonlinear Dynamics. 1990, 1: 221~241
79 Tiwari M, Gupta K. Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor. Journal of Sound and Vibration. 2000, 238(5): 723~756
80 Tiwari M, Gupta K. Dynamic Response of an unbalanced rotor supported on ball beaings. Journal of Sound and Vibration. 2000, 238(5):757~779
81 S.P. Harsha, K. Sandeep, R. Prakash. The effect of speed of balanced rotor on nonlinear vibrations associated with ball bearings. Mechanical Sciences. 2003, 45: 725~740
82 A.N. Lioulios, I.A. Antoniadis. Effect of rotational speed fluctuations on the dynamic behaviour of rolling element bearings with radial clearances. International Journal of Mechanical Sciences. 2006, 48: 809~829
83 N. Aktu.rk. The Effect of Waviness on Vibrations Associated with Ball Bearings. Journal of Tribology. 1999, 121: 667~677
84 S.P. Harsha, K. Sandeep, R. Prakash. Non-linear Dynamic Behaviors of Rolling Element Bearings Due to Surface Waviness. Journal of Sound and Vibration. 2004, 272: 557~580
85 BAI Chang-qing, XU Qing-yu, ZHANG Xiao-long. Nonlinear Stability of Balanced Rotor Due to the Effect of Ball Bearing Internal Clearance. Applied Mathematics and Mechanics. 2006, 294: 23~48
86唐云冰,高德平,罗贵火.滚动轴承非线性轴承力及其对轴承系统振动特性的影响.航空动力学报. 2006, 21(2): 366~373
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88 G.H. Jang, S.W. Jeong. Nonlinear Excitation Model of Ball Bearing Waviness in a Rigid Rotor Supported by Two or More Ball Bearings Considering Five Degrees of Freedom. ASME Journal of Tribology. 2002, 124: 82~90
89 BAI Chang-qing, XU Qing-yu. Dynamic Model of Ball Bearings with Internal Clearance and Waviness. Journal of sound and vibration, 2006, 294: 23~48
90 S.P. Harsha. Nonlinear Dynamic Analysis of an Unbalanced Rotor Supported by Roller Bearing. Chaos, Solitons and Fractals 2005, 26: 47~66
91 S.P. Harsha. Nonlinear Dynamic Response of a Balanced Rotor Supported by Rolling Element Bearings Due to Radial Internal Clearance Effect. Mechanism and Machine Theory, 2006, 41: 688~706
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93 A.Tauqir. Causes of Fatigue Failure in the main bearing of an Aero-engine. Engineering Failure Analysis. 2000, 7: 127~143
94 Boness R. J. Load Requirement for the Prevention of Skidding in High Speed Thrust Loaded Ball Bearings. Trans of ASME, Journal of Lubrication Technology. 1981, 103(1): 35~39
95 Neng Tung Liao, Jen Fin lin. Ball bearing skidding under radial and axial loads. Mechanism and Machine Theory. 2002, 37: 91~113
96蒋兴奇,马家驹,顾小兵.主轴轴承最小预负荷的计算.轴承. 2001, (3): 1~4
97 Michael Flouros. Correlations for heat generation and outer ring temperature of high speed and highly loaded ball bearings in an aero-engine. Aerospace Science and Technology. 2006,10: 611~617
98 J. R. Brown, N. H. Forster. Operating Temperatures in Lubricated Rolling Element Bearings for Gas Turbines. AIAA-2000-3027. 2000: 1268~1275
99 K. Mizuta, T. Inoue, Y. Takahashi, S. Huang, K. Ueda, H. Omokawa. Heat Transfer Characteristics Inner and Outer Rings of an Angular Ball Bearing. Heat Transfer Asia Research. 2003, 32(1):42~57
100蒋兴奇.主轴轴承热特性及对速度和动力学性能影响的研究.杭州,浙江大学. 2001
101宁练,周孑民.滚动轴承内部温度状态监测技术.轴承. 2007, (2): 25~27
102 M. O. T. Cole, P. S. Keogh, C. R. Burrows. The Dynamics Behavior of a Rolling Element Auxiliary Bearing Following Rotor Impact. ASME. 2002,124:406~413
103 Y. S. Wang, B. Y. Yang, L. Q. Wang. Investigation into the Traction Coefficient in Elastohydro -dynamic Lubrication. Tribotest. 2004, 11(2): 113~124
104王燕霜.航空润滑油流变特性及其对润滑性能影响的研究.哈尔滨工业大学博士学位论文. 2006: 63
105李斌才.航空润滑油与润滑脂.烃加工出版社.第1版. 1987:156
106林基恕.航空燃气涡轮发动机机械系统设计.航空工业出版社. 2005
107 Z. Gu, X. Zhi, G. Meng, et al. Transient Response Analysis of Large-scale Rotor-bearing System with Strong Non-linear Elements by a Transfer Matrix-newmark Formulation Itegration Method. Jounal of Sound and Vibration. 2003, (259): 559~570
108 S. C. Hsieh, J. H. Chen, A. C. Lee. A Modified Transfer Matrix Method for the Coupling Lateral and Torsional Vibrations of Symmetricrotor-bearing Systems. Jounal of Sound and Vibration. 2006, (289): 294~333
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112陈观慈.航空发动机主轴高速滚动轴承热分析.哈尔滨工业大学博士学位论文. 2008: 88
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16 P. K. Gupta. Some Dynamic Effects in High-speed Solid-Lubricated Ball Bearings. ASLE Transactions. 1983, 26(3): 393~400
17 C. R. Meeks. The Dynamics of Ball Separators In Ball Bearings-Part I: Analysis. Trans of ASLE. 1985, 28(3): 21~29
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27杜辉,周彦伟,邓四二.高速圆柱滚子轴承零件间相互作用力瞬态动力学分析.轴承. 2005, (9): 12~15
28马国华,胡桂兰.滚动轴承弹性接触问题的数值计算.轴承. 2005, (1): 1~3
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31 R. J. Kleckner, J. Pirvics. High Speed Cylindrical Roller Bearing Analysis, SKFComputer Program CYBEAN. NASA-CR-159460~159461. 1979
32 Hirotoshi Aramaki. Rolling Bearing Analysis Program Package“BRAIN”. Motion&Control. 1997, (3): 15~24
33李媛媛,徐浩.高速机床主轴用陶瓷轴承动态性能研究.现代零部件. 2004, (8): 54~57
34唐云冰,高德平,罗贵火.航空发动机高速滚珠轴承力学特性分析与研究.航空动力学报. 2006, 21(2): 354~360
35 Lars-Erik Stacke, Dag Fritzson. Bearing Simulation Get Real. Machine Design. 2002
36 Tomoya Sakaguchi, Kazuyoshi Harada. Dynamic Analysis of Cage Behavior in a Tapered Roller Bearing. Trans of ASME. 2006,128: 604~611
37 El-Sayed H R. Stiffness of deep-groove ball bearings. Wear, 1980, 63(1): 89~94
38 Hernot X, Sartor M, Guillot J. Calculation of the stiffness matrix of angular contact ball bearings by using the analytical approach. Journal of Mechanical Design, Transactions of the ASME, 2000, 122(1): 83~90
39桃野达信(日).滚动轴承的刚性.国外轴承技术. 2002, (3): 56~64
40 Bugra H Ertas, John M Vance. The effect of static and dynamic misalignment on ball bearing radial stiffness. AIAA-2002-4160, 2002
41 Walford T L H, Stone B J. Some damping and stiffness characteristics of angular contact bearings under oscillation radial load. Institution of Mechanical Engineers, Conference Publications, 1980: 157~162
42 Stone B J. State of the art in the measurement of the stiffness and damping of rolling element bearings. CIRPAnnals, 1982, 31(2): 529~538
43罗祝三,孙心德,吴林丰.滚动轴承在任意方向的支承刚度.南京航空学院学报. 1992, 24(3): 248~256
44刘卫群,罗继伟,吴长春等.滚动轴承刚度分析程序.计算力学学报. 2001, 18(3): 375~378
45黄浩,张鹏顺,温建民.航空发动机角接触球轴承刚度的一种实用分析方法.南京航空航天大学学报. 2000, 32(4): 422~427
46王刚,郭茂林.航空航天滚动轴承刚度.哈尔滨工业大学学报,2001, 33(5): 644~650
47王硕桂,夏源明.从随机振动响应中提取滚动轴承的刚度参数.振动工程学报. 2004, 17(4): 388~392
48 Kraus J, Blech J J, Braun S G. In situ determination of rolling bearing stiffness and damping by model analysis. Journal of Vibration, Acoustic, Stress, and Reliability in design, 1987, 109(3): 235~240
49 Goodwin M J. Experimental techniques for bearing impedance measurement. Journal of Engineering for Industry, Transactions of the ASME, 1991, 113(3): 335~342
50 Rajiv Tiwari, Nalinaksh S. Vyas. Stiffness estimation from random response in multi-mass rotor bearing systems. Prob. Engng. Mech. 1998,13(4): 255~268
51 Raffa F. A, Vatta V. The dynamic stiffness method for linear rotor-bearing system. ASME Journal of Vibration and Acoustics, 1996,118(7): 332~339
52 M. A. Prohl, Ageneral method for calculating critical speed of flexible rotors, Journal of Appl, Mech.Trans.ASME,1945,12(8): 142~148
53 17 G. C. Horner and W. D. Pilkey, The Riccati Transfer matrix method, J. of Mech. Des. Trans ASME, 1978,100: 297~302
54 Gu Z, Zhi X, Meng G,et al. Transient response analysis of large-scale rotor-bearing system with strong non-linear elements by a transfer matrix—Newmark formulation integration method[J].Journal of Sound and Vibration, 2003,259(3): 559~570
55 Hua J, Liu Z S, Xu Q Y. A new high precision direct integration scheme for nonlinear rotor-seal system. International Conference on Scientific & Engineering Computation. Singapore: National University of Singapore, 2002, 552~556
56蒋书运,陈照波,须根法.用整体传递矩阵法计算航空发动机整机临界转速特性.哈尔滨工业大学学报. 1998, 30(1): 32~35
57柴山,刚宪约.整体传递矩阵法的Riccati变换.上海理工大学学报, 2002, 24(2): 98~100
58李永强,朱亮,刘宇等.用整体传递矩阵法计算旋转机械整机固有频率特性.东北大学学报(自然科学版). 2005, 26(12): 1178~1180
59闻邦椿,顾家柳,夏松波.高等转子动力学.北京,机械工业出版社. 2000
60 Tiwari M. Effect of radial internal clearance of a ball bearing in the dynamics of a balanced horizontal rotor. Journal Sound and Vibration. 2000, 238(5): 723~756
61 Tiwari M. Dynamics response of a unbalance spindle supported on ball bearings. Journal Sound and Vibration. 2000, 238(5): 757~779
62 Panda K C. Optimum support characteristics for rotor-shaft system with preloaded rolling element bearing. Journal Sound and Vibration. 2003, 260: 731~755
63 Mohammed A Alfares, Abdallah A. Elsharkawy. Effects of axial preloading of angular contact ball bearing on the dynamics of a grinding machine spindle system. Journal of Materials Processing Technology. 2003, 136: 48~59
64邓四二,滕弘飞,周彦伟.滚动轴承-双转子系统动态性能分析.轴承. 2006, (4): 1~4
65李松生.高速电主轴轴系转子动力学特性分析.轴承. 2002, (2): 15~17
66朱爱斌,彭祥多,谢友柏.滚动轴承刚度及转子系统动力学网络计算系统的设计和实现.润滑与密封. 2005, 9(5): 126~132
67 Dong-Soo Lee, Dong-Hoon Choi. Reduced Weight Design of a Flexible Rotor with Ball Bearing Stiffness Characteristics Varying with Rotational Speed and Load. Journal of Vibration and Acoustics. 2000, 122(7): 203~208
68 David P. Fleming. Unbalance Response Prediction for Rotors on Ball Bearings Using Speed and Load Dependent Nonlinear Bearing Stiffness. NASA/TM. 2003-212527
69 Hoon-Voon Liewa, Teik C. Lim. Analysis of time-varying rolling element bearing characteristics. Journal of Sound and Vibration. 2005,283: 1163~1179
70黄太平,罗贵火.转子动力学优化设计.航空动力学报. 1994, 9(2): 113~116
71 K. C. Panda, J. K. Dutt. Design of optimum support parameters for minimum rotor response and maximum stability limit. Journal of Sound and Vibration. 1999, 223(1): 1~21
72 C. S. SUNNERSJO. Varying compliance vibrations of rolling bearings. Journal of Sound and vibration, 1978, 58: 363~373.
73 Fukata S, Gad EH, TamuraH. On the radial vibration of ball bearings (computer simulation)[J].Bulletin of the JSME, 1985,28:899~904
74 B. MEVEL and J. L. GUYADER. Routes to chaos in ball bearings. Journal of Sound and vibration. 1993, 162: 471~487
75 Sankaravelu A, Noah ST, Burger C P. Bifurcation and chaos in ball bearings.Nonlinear and Stochastic Dynamics, 1994,192:313~325
76 D. W. CHILDS. Fractional frequency rotor motion due to clearance effects. Journal of Engineering Power. 1982,104: 533~536
77 S. SAITO. Calculation of nonlinear unbalance response of horizontal Jeffcott rotors supported by ball bearings with radial clearances. Journal of vibration, Acoustics, Stress, and Reliability in Design. 1985,107: 416~420
78 Y. B. KIM and S. T. NOAH. Bifurcation analysis for a modified Jeffcott rotor with bearing clearances. Nonlinear Dynamics. 1990, 1: 221~241
79 Tiwari M, Gupta K. Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor. Journal of Sound and Vibration. 2000, 238(5): 723~756
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