本論文的研究目的在探討與分析軸承系統在高速運轉下的運動和動態特性。並針對高速主軸所要求的運動與動態性能，對其軸承系統的設計提出初步簡單的設計方向。文中結合軸承運動學(Kinematics of Bearing Systems)
、轉子動力學(Rotordynamics)與彈液動潤滑理論(EHL Theory)來探討高速軸承系統的運動和動態特性，其中包括極限轉速、疲勞壽命和剛性等運動特性及穩定度、振動和軸承系統外傳力等動態特性。在了解這些特性與軸承系統的設計參數、動態參數和操作參數間的互動和果因關係後，於文中提出初步簡單之設計方向以達到所要求的運動與動態性能要求。

The aim of this thesis is to analyze the kinematic and dynamic characteristics of a high-speed bearing system. The examined kinematic characteristics will include the maximal speed, fatigue life and stiffness of a bearing system. The stability and transmission force of a bearing system as well as the vibration of its shaft were the studied dynamic characteristics. These analyses were based on the kinemitics of bearing systems, rotor dynamics and elastohydrodynamic lubrication theory. According to these analyses, the interaction and consequence among these characteristics and the parameters of a bearing system will be explained. After these analyses, preliminary and simple design trends about a high-speed bearing system are going to be proposed.

謝誌………………………………………………………………………Ⅰ
中文摘要…………………………………………………………………Ⅱ
英文摘要…………………………………………………………………Ⅲ
目錄………………………………………………………………………Ⅳ
圖索引……………………………………………………………………Ⅵ

第一章 緒論………………………………………………………1
1.1 前言……………………………………………………………1
1.2 研究動機………………………………………………………1
1.3 文獻回顧………………………………………………………2
1.4 本文內容………………………………………………………4

第二章 軸承系統運動學與參數設計……………………………5
2.1 軸承系統之運動行為模式…………………………………………5
2.1.1 單軸承系統運動學…………………………………………………5
2.1.2 軸承之設計與操作參數……………………………………………11
2.1.3 設計參數對單軸承系統運動行為之影響…………………………12
2.1.4 操作參數對單軸承系統運動行為之影響…………………………14
2.2 多軸承系統之運動行為模式………………………………………16
2.2.1 多軸承系統運動學…………………………………………………17
2.2.2 多軸承系統之主軸參數設計………………………………………25
2.2.3 主軸參數對主軸運動行為之影響…………………………………25
2.2.4 主軸參數對軸承運動行為之影響…………………………………26
2.3 總結…………………………………………………………………28

第三章 軸承系統運動特性分析……………………………………29
3.1 軸承系統之極限轉速………………………………………………29
3.1.1 設計參數對極限轉速的影響………………………………………29
3.1.2 操作參數對極限轉速的影響………………………………………30
3.1.3 多軸承系統之極限轉速……………………………………………32
3.1.4 總結…………………………………………………………………32
3.2 軸承系統之疲勞壽命………………………………………………33
3.2.1 疲勞破獲機制………………………………………………………33
3.2.2 疲勞壽命理論………………………………………………………33
3.2.3 設計參數對疲勞壽命的影響………………………………………36
3.2.4 操作參數對疲勞壽命的影響………………………………………37
3.2.5 多軸承系統之疲勞壽命……………………………………………37
3.2.6 總結…………………………………………………………………38
3.3 軸承系統之剛性……………………………………………………38
3.3.1 軸承油膜剛性與介面剛性…………………………………………39
3.3.2 軸向剛性與徑向剛性………………………………………………41
3.3.3 設計參數對軸承剛性之影響………………………………………42
3.3.4 操作參數對軸承剛性之影響………………………………………44
3.3.5 多軸承系統之剛性…………………………………………………45
3.3.6 總結…………………………………………………………………48

第四章 軸承系統之穩定度分析……………………………………49
4.1 軸承系統動力學……………………………………………………49
4.1.1 廣義的軸承系統模型………………………………………………49
4.1.2 軸承系統之動態參數………………………………………………55
4.2 軸承系統之動態穩定度……………………………………………56
4.2.1 穩定度理論分析……………………………………………………57
4.2.2 動態參數對穩定度的影響…………………………………………59
4.3 總結…………………………………………………………………64

第五章 軸承系統之動態特性分析……………………………………65
5.1 軸承系統之主軸振動分析…………………………………………65
5.1.1 主軸振動與軸承振動………………………………………………65
5.1.2 動態參數對主軸振動的影響………………………………………67
5.1.3 總結…………………………………………………………………71
5.2 軸承系統之外傳力分析……………………………………………71
5.2.1 軸承系統之外傳力機制……………………………………………72
5.2.2 動態參數對外傳力的影響…………………………………………72
5.2.3 總結…………………………………………………………………74

第六章 結論…………………………………………………………75

參考文獻 ……………………………………………………………………78

【1】 A. B. Jones, “Ball Motion and Sliding Friction in Ball Bearings,” Journal of Basic Engineering, Transactions of the ASME, Vol. 81, pp. 1~12, 1959.
【2】 A. B. Jones, “A General Theory for Elastically Constrained Ball and Radial Roller Bearings Under Arbitrary Load and Speed Conditions,” Journal of Basic Engineering, Transactions of the ASME, Vol. 82, pp. 309~320, 1960.
【3】 T. A. Harris, “Ball Motion in Thrust-Loaded, Angular Contact Bearings With Coulomb Friction,” Journal of Lubrication Technology, Transactions of the ASME, pp. 32~38, January 1971.
【4】 T. A. Harris, “An Analytical Method to Predict Skidding in Thrust-Loaded, Angular-Contact Ball Bearings,” Journal of Lubrication Technology, Transactions of the ASME, pp. 17~23, January 1971.
【5】 B. J. Hamrock, W. J. Anderson, “Analysis of an Arched Outer-Race Ball Bearing Considering Centrifugal Forces,” Journal of Lubrication Technology, Transactions of the ASME, pp. 265~275, July 1973.
【6】 B. J. Hamrock, “Ball Motion and Sliding Friction in an Arched Outer Race Ball Bearing,” Journal of Lubrication Technology, Transactions of the ASME, pp. 202~211, April 1975.
【7】 C. R. Gentle, R. J. Boness, “Prediction of Ball Motion in High-Speed Thrust-Loaded Ball Bearings,” Journal of Lubrication Technology, Transactions of the ASME, pp. 463~471, July 1976.
【8】 C. T. Walters, ”The Dynamics of Ball Bearings,” Journal of Lubrication Technology, Transactions of the ASME, pp. 1~9, January 1971.
【9】 P. K. Gupta, “Transient Ball Motion and Skid in Ball Bearings,” Journal of Lubrication Technology, Transactions of the ASME, pp. 261~269, April 1975.
【10】 P. K. Gupta, “Dynamics of Rolling-Element Bearings, Part I：Cylindrical Roller Bearing Analysis,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 101, pp. 293~304, July 1979.
【11】 P. K. Gupta, “Dynamics of Rolling-Element Bearings, Part II：Cylindrical Roller Bearing Results,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 101, pp. 305~311, July 1979.
【12】 P. K. Gupta, “Dynamics of Rolling-Element Bearings, Part III：Ball Bearing Analysis,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 101, pp. 312~318, July 1979.
【13】 P. K. Gupta, “Dynamics of Rolling-Element Bearings, Part IV：Ball Bearing Results,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 101, pp. 319~326, July 1979.
【14】 T. L. H. Walford, B. J. Stone, “The Measurement of the Radial Stiffness of Rolling Element Bearings Under Oscillating Conditions,” Journal of Mechanical Engineering Science, Vol. 22, No. 4, pp. 175~181, 1980.
【15】 Katsuya Touma, Hisashi Kawamura, Kazuaki Kawakita, “Ball Motion in High-Speed Angular Contact Ball Bearings,” Proceedings of the JSLE International Tribology Conference, July 8~10, 1985, Tokio, Japan.
【16】 W. Weibull, “A Statistical Representation of Fatigue Failure in Solid,” Acta Polytech. Mech. Eng. Ser. 1, R.S.A.E.E., No. 9, 49, 1949.
【17】 T. Tallian, “Weibull Distribution of Rolling Contact Fatigue Life and Deviations Therefrom,” ASLE Trans. 5(1), April 1962.
【18】 G. Lundberg and A. Palmgren, “Dynamic Capacity of Rolling Bearings,” Acta Polytech. Mech. Eng. Ser. 1, R.S.A.E.E., No. 3, 7, 1947.
【19】 G. Lundberg and A. Palmgren, “Dynamic Capacity of Rolling Bearings,” Acta Polytech. Mech. Eng. Ser. 2, R.S.A.E.E., No. 4, 96, 1952.
【20】 A. B .Jones, “The Life of High-Speed Ball Bearings,” Transactions of the ASME, pp.695~703, July 1952.
【21】 T. A. Harris, “Optimizing the Fatigue Life of Flexibly-Mounted Rolling Bearings,” ASLE Trans. 11, pp. 290~294, 1968.
【22】 T. A. Harris, “How to Compute the Effects of Preloaded Bearings,” Product Engineering, pp. 84~93, July 1965.
【23】 C. F. Smith, “Fatigue Failure Distributions for Ball Bearings,” Journal of Lubrication Technology, Transactions of the ASME, pp. 123~131, April 1973.
【24】 H. Hertz, “The Contact of Elastic Solids,” J. Reine Angew. Math., Vol. 92, pp. 156~171, 1881.
【25】 H. S. Cheng, “A Numerical Solution of the Elastohydrodynamic Film Thickness in an Elliptical Contact,” Journal of Lubrication Technology, Transactions of the ASME, pp. 155~162, January 1970.
【26】 D. E. Brewe, B. J. Hamrock, “Simplified Solution for Elliptical Contact Deformation Between Two Elastic Solids,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 99, No. 4, pp. 485~487, 1977.
【27】 D. E. Brewe, B. J. Hamrock, “Simplified Solution for Stresses and Deformations,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 105, No. 2, pp. 171~177, 1983.
【28】 B. J. Hamrock, D. Dowson, “Ball Bearing Lubrication-The Elastohydrodynamics of Elliptical Contacts,” Wiley-Interscience, 1981, New York.
【29】 H. H. Coe, R. J. Parker, “Evaluation of Electron-Beam Welded Hollow Balls for High-Speed Ball Bearings,” Journal of Lubrication Technology, pp. 47~59, 1971.
【30】 H. M. Dalal, Y. P. Chiu, E. Rabinowicz, “Evaluation of Hot Pressed Silicon Nitride as a Rolling Bearing Material,” ASLE Transactions, Vol. 18, 3, pp.211~221.
【31】 G. W. Hosang, “Experimental and Computed Performance Characteristics of High-Speed Silicon Nitride Hybrid Ball Bearings,” Journal of Engineering for Gas Turbines and Power, Transactions of the ASME, Vol. 113, pp.635~642, October 1991.
【32】 A. Igartua, J. Laucirica, A. Aranzabe, T. Leyendecker, O. Lemmer, G. Erkens, M. Weck, G.Hanrath, “Application of Low Temperature PVD Coatings in Rolling Bearings：Tribological Tests and Experiences with Spindle Bearing Systems,” Surface and Coatings Technology, No. 86~87, pp.460~466, 1996.
【33】 Ming Jiang, R. Komanduri, “On the Finishing of Si3N4 Balls for Bearing Applications,” Wear, 215, pp. 267~278, 1998.
【34】 L. Wang, R. W. Snidle, L. Gu, “Rolling Contact Silicon Nitride Bearing Technology：A Review of Recent Research,” Wear, 246, pp.159~173,2000.
【35】 W. J. M. Rankine, “On the Centrifugal Force of Rotating Shafts,” The Engineer, 27, April 9 1869.
【36】 A. D. Dimarogonas, Vibration for Engineers, Prentice Hall, p. 526, 1992.
【37】 H. H. Jeffcott, “The Lateral Vibration of Loaded Shafts in the Neighborhood of a Whirling Speed,” Phil. Mag., ser. 6 Vol. 7, pp. 304~314, 1919.
【38】 B. J. Newkirk, “Shaft Whipping,” General Electric Rev. 27, pp. 169~178, 1924.
【39】 A. T. Kimball, “Internal Friction as a Cause of Shaft Whirling,” Philosophical Magazine, series 6, Vol. 37, pp.724~727, 1925.
【40】 D. Robertson, “The Vibrations of Revolving Shafts, Philosophical magazine,” series 7, Vol. 20, No. 86, pp. 962~965, 1932.
【41】 D. Robertson, “Transient Whirling of a Rotor,” Philosophical Magazine, series 7, Vol. 20, No. 135, pp. 793~825, 1935.
【42】 Toshio Yamamoto, “On Critical Speeds of a Shaft Supported by a Ball Bearing,” Journal of Applied Mechanics, pp. 199~204, 1959.
【43】 I. C. Begg, “Friction Induced Rotor Whirl-A Study in Stability,” Journal of Engineering for Industry, Transactions of the ASME, pp. 450~454, May 1974.
【44】 J. M. Vance, “Stability of High Speed Rotors with Internal Friction, Journal of Engineering for Industry,” pp. 960~968, 1974.
【45】 F. F. Ehrich, “Shaft Whirl Induced by Rotor Internal Damping,” Journal of Applied Mechanics, Transactions of the ASME, pp.279~282, June 1964.
【46】 A. Tondl, “Some Problems of Rotor Dynamics,” Chapman and Hall Limited, 1965.
【47】 E. J. Gunter, “Dynamic Stability of Rotor-Bearing Systems,” NASA, 1966.
【48】 F. C. Pruvot, “High Speed Bearings for Machine Tool Spindles,” Annals of the CIRP, Vol. 29, No. 1, pp. 293~297, 1980.
【49】 M. S. Hundal, “Response of a Base Excited System with Coulomb and Viscous Friction,” Journal of Sound and Vibration, Vol. 64, No.3, pp. 371~378, 1979.
【50】 R. Aini, H. Rahnejat, R. Gohar, “A Five Degrees of Freedom Analysis of Vibrations in Precision Spindles,” International Journal of Machine Tools and Manufacture, Vol. 30, No. 1, pp. 1~18, 1990.
【51】 Y. C. Shin, “Bearing Nonlinearity and Stability Analysis in High Speed Machining,” Journal of Engineering for Industry, Transactions of the ASME, Vol. 114, pp. 23~30, February 1992.
【52】 C. H. Chen, K. W. Wang, Y. C. Shin, “An Integrated Approach Toward the Dynamic Analysis of High Speed Spindles, Part I: System Model,” Journal of Vibration and Acoustics, Vol. 116, pp. 506~513, 1994.
【53】 C. H. Chen, K. W. Wang, Y. C. Shin, “An Integrated Approach Toward the Dynamic Analysis of High Speed Spindles, Part II: Dynamics Under Moving End Load,” Journal of Vibration and Acoustics, Vol. 116, pp. 514~522, 1994.
【54】 C. H. Chen, K. W. Wang, Y. C. Shin, “Dynamic Analysis and Modeling of a High Speed Spindle System,” Transactions of NAMR/SME, pp. 298~304, 1990.
【55】 A. W. Lees, M. I. Friswell, “The Evaluation of Rotor Imbalance in Flexibly Mounted Machines,” Journal of Sound and Vibration, Vol. 208, No. 5, pp. 671~683, 1997.
【56】 Y.-S. Lee, C.-W. Lee, “Modelling and Vibration Analysis of Misaligned Rotor-Ball Bearing Systems,” Journal of Sound and Vibration, Vol. 224, No. 1, pp. 17~32, 1999.
【57】 S.-W. Hong, J.-H. Park, ”Dynamic Analysis of Multi-Stepped, Distributed Parameter Rotor-Bearing Systems,” Journal of Sound and Vibration, Vol. 227, No. 4, pp. 769~785, 1999.
【58】 N. Lynagh, H. Rahnejat, M. Ebrahimi, R. Aini, ”Bearing Induced Vibration in Precision High Speed Routing Spindles,” International Journal of Machine Tools and Manufacture, Vol. 40, pp.561~577, 2000.
【59】 G.Genta, C. Delprete, E. Busa, “Some Considerations on the Basic Assumptions in Rotordynamics,” Journal of Sound and Vibration, Vol. 227, No. 3, pp. 611~645, 1999.
【60】 M. Tiwari, K. Gupta, “Dynamic Response of an Unbalanced Rotor Supported on Ball Bearings,” Journal of Sound and Vibration, Vol. 238, No. 5, pp. 757~779, 2000.
【61】 Y. Kang, Y.-P. Chang, J.-W. Tsai, L.-H. Mu, Y.-F. Chang, “An Investigation in Stiffness Effects on Dynamics of Rotor-Bearing-Foundation Systems,” Journal of Sound and Vibration, Vol. 231, No. 2, pp. 343~374, 2000.
【62】 Y. B. Kim, S.-K. Choi, “A Multiple Harmonic Balance Method for the Internal Resonant Vibration of a Non-Linear Jeffcott Rotor,” Journal of Sound and Vibration, Vol. 208, No. 5, pp. 745~761, 1997.
【63】 G. T. Flowers, D. B. Margithu, G. Szasz, “The Application of Floquet Methods in the Analyses of Rotordynamic Systems,” Journal of Sound and Vibration, Vol. 218, No. 2, pp. 350~360, 1998.
【64】 M. Aleyaasin, M.Ebrahimi, R. Whalley, ”Multivariable Hybrid Models for Rotor-Bearing Systems,” Journal of Sound and Vibration, Vol. 233, No. 5, pp. 835~856, 2000.
【65】 R. Tiwari, N. S. Vyas, “Parameter Estimation in Imbalanced Non-Linear Rotor-Bearing Systems from Random Response, ournal of Sound and Vibration, Vol.208, No. 1, pp.1~14, 1997.
【66】 E. J. Routh, “Advanced Rigid Dynamics,” Macmillan, 1905, New York.
【67】 D. W. Nicholson, D. J. Inman, “Stable Response of Damped Linear Systems,” Shock and Vibration Digest, Vol. 15, pp. 19~25, November 1983.
【68】 林孟璋, “高速斜角滾珠軸承減振系統之動態分析”, 國立中山大學機械工程研究所碩士論文, 1999, 高雄.
【69】 許明雄, “斜角滾珠軸承之精度與疲勞壽命分析”, 國立中山大學機械工程研究所碩士論文, 1999, 高雄.
【70】 蘇耀藤, 傅光華, “陶瓷滾珠軸承之裝配與檢驗,” 國防科技學術合作協調小組研究成果期末報告, NSC 87-2623-D019-003, 1998.