臺灣博碩士論文加值系統

摘 要
本論文研究空氣靜壓軸承-轉子系統的靜態、動態特性及穩定性，探討主軸及其軸承之承載負荷、偏位角、動態剛度，抗振阻尼、臨界動態力及臨界旋振比，以數值分析方法模擬空氣靜壓軸承主軸運轉特性及工作性能之設計參數的影響。
　　本研究建立由雙排進氣孔之空氣靜壓軸承支撐的轉子模型，以有限差分法求解雷諾方程式，求得軸承之靜態壓力分佈、承載能力、偏位角及動態剛度及阻尼係數的靜動態特性，並以特徵值求解方法求得轉子-軸承系統之穩定性門檻。在分析結果中探討偏心率、轉速及節流參數等參數在不同的軸承進氣孔位置及孔數、不同的節流器的設計條件下，對轉子-空氣軸承系統不穩定性的影響。
　　經由運動方程式的特徵解，可得到兩個穩定性門檻：臨界慣性力、臨界旋振比，由此兩個穩定性門檻，可判別轉子的穩定性。

關鍵字：空氣靜壓軸承、軸承剛度、阻尼比、穩定性門檻、臨界慣性力、臨界旋振比

Abstract
This project has dealt with the integrated research of the spindles with aerostatic bearings. The contents include studies in load capacity, stiffness, damping ratio, critical inertial force, critical whirl ratio for aerostatic-bearing spindle, analysis for simulating the influences of design parameters in operation performances and characteristics and optimal design for aerobearing-spindle system.
The Reynolds equation is solved by the finite difference method and numerical integration to yield static and dynamic characteristics of air film. The motion equation of the rotor-bearing system is obtained by using the perturbation method and the eigensolution method is used to determine the stability threshold. The eccentricity, spindle speed, and restriction parameter are considered in the analysis of the whirl instability of the rotor-aerostatic bearing system for the different comparisons between various designs in the number and location of entry holes of aerostatic bearings.

目錄
中文摘要……………………………………………………………………I
英文摘要……………………………………………………………………II
誌謝………………………………………………………………III
目錄………………………………………………………………IV
圖目錄……………………………………………………………………VI
符號說明…………………………………………………………….....IX
第一章 導論……………………………………………………………………1
1.1 研究背景…………………………………………………………………1
1.2 文獻回顧…………………………………………………………………2
1.3 研究內容………………………………………………………………4
第二章 理論分析………………………………………………………………6
2.1氣體軸承潤滑方程式……………………………………………………6
2.2節流器流量連續方程式…………………………………………………8
2.3微擾化分析………………………………………………………………10
2.3.1雷諾方程式的微擾化………………………………………………10
2.3.2孔口節流器的微擾化……………………………………………11
2.3.3環面節流器的微擾化…………………………………………… 13
2.3.4孔口節流器與環面節流器的串聯分析…………………………... 14
2.4承載能力…………………………………………………………………20
2.5剛度及阻尼係數……………………………………………22
2.6 穩定性分析………………………………………………………………24
第三章 數值分析……………………………………………28
3.1求解方法…………………………………………………28
3.2雷諾方程式之離散化……………………………………34
第四章 結果與討論………………………………………………………………37
4.1孔口節流器…………………………………………………38
4.2環面節流器…………………………………………………52
4.3孔口節流器與環面節流器的串聯…………………………60
第五章 結論與未來展望………………………………………64
5.1 研究結論………………………………………………………………64
5.2 未來研究方向…………………………………………………………65
參考文獻…………………………………………………………………66
附錄A 空氣靜壓軸承Reynolds equation推導………………69
附錄B雷諾方程之微擾化………………………………………77
附錄C節流器流量之微擾化…………………………………..85
附錄D雷諾方程式離散化………………………………………….90
個人資料……………………………………………………………98
論文發表……………………………………………………………99
圖目錄
圖2-1 氣體軸承示意圖(a)結構圖 (b)主軸中心擾動軌跡....7
圖2-2 節流器 (a)孔口 (b) 環面...................8
圖2-3 (a) 節流器串聯示意圖 (b) 進氣孔之壓力圖.....15
圖2-4 轉子－軸承系統圖..............24
圖3-1 數值計算區域.....................28
圖3-2 數值計算流程圖.....................32
圖3-3 三個供氣孔type 1型式軸承、環面節流器 的
容許誤差比較圖(a)靜態特性 (b)動態特性.........33
圖3-4 數值分析之離散格子圖..............34
圖4-1軸承型式 (a)六個供氣孔 (b)第一種型式(type 1)
(c)第二種(type2)…………………………37
圖4-2空氣軸承沿周向表面通過孔口節流器在 及 時的
壓力分布 (a)三個供氣孔type 1型式 (b)三個供氣孔type 2型式
(c)六個供氣孔型式（左欄為 ，右欄為 ）…………39
圖4-3空氣軸承沿周向表面通過孔口節流器在 及 時
的壓力分布(a) 三個供氣孔type 1型式(b) 三個供氣孔type 2型式
(c) 六個供氣孔型式（左欄為 ，右欄為 ）…………40
圖4-4軸承的承載能力( )與其相關的偏位角隨節流參數
變化之分布(a) 三個供氣孔type 1型式(b) 三個供氣孔type 2型式(c) 六個供氣孔型式………………………42
圖4-5軸承的承載能力與其相關的偏位角隨偏心率
變化之分布(a) (b) ………………………………………….43
圖4-6軸承的剛度及阻尼係數隨旋振比( )變化之分布………46
圖4-7三個供氣孔type 1型式軸承的 (a)剛度係數 (b)阻尼係數(c)臨界動態力 及旋振比的平方隨旋振比變化之分布 ……47
圖4-8軸承的臨界動態力 隨節流參數變化之分布
(a)三個供氣孔type 1型式(b) 三個供氣孔type 2型式
(c) 六個供氣孔型式………………….………..48
圖4-9軸承的旋振比的平方隨節流參數變化之分布
(a)三個供氣孔type 1型式(b) 三個供氣孔type 2型式
(c) 六個供氣孔型式…………………….…….…….50
圖4-10軸承的臨界動態力 及旋振比的平方隨偏心率變化之分布(a) (b) …………………………………………………….…51
圖4-11軸承的承載能力與其相關的偏位角隨節流參數變化
之分布，於三個供氣孔type 1型式(a) (b) ………….…53
圖4-12軸承的承載能力( )與其相關的偏位角隨節流參數
變化之分布，於三個供氣孔type 2型式 (a) (b) …54

圖4-13軸承的承載能力與其相關的偏位角隨節流參數
變化之分布，於六個供氣孔型式 (a) (b) ……………....55
圖4-14軸承的臨界動態力與旋振比的平方隨節流參數
變化之分布，於三個供氣孔type 1型式 (a) (b) ……...57
圖4-15軸承的臨界動態力與旋振比的平方隨節流參數
變化之分布，於三個供氣孔type 2型式 (a) (b) ……..…58
圖4-16軸承的臨界動態力與旋振比的平方隨節流參數
變化之分布，於六個供氣孔型式 (a) (b) ……………….59
圖4-17軸承的承載能力與其相關的偏位角隨節流參數
變化之分布(a) 三個供氣孔type 1型式
(b) 三個供氣孔type 2型式(c) 六個供氣孔型式……………….…..……61
圖4-18軸承的臨界動態力 隨節流參數變化之分布
(a)三個供氣孔type 1型式 (b)三個供氣孔type 2型式
(c) 六個供氣孔型式……………………………………62
圖4-19軸承的旋振比的平方隨節流參數變化之分布
(a)三個供氣孔type 1型式(b) 三個供氣孔type 2型式
(c) 六個供氣孔型式……………………………………63

參考文獻
[1]Majumdar﹐B. C., “On the General Solution of Externally Pressurized Gas Journal Bearings,” Journal of Lubrication Technology, pp. 291-296, 1972.
[2]Kazimierski, Z. and Trojnarski. J., “Investigations of Externally Pressurized Gas Bearings with Different Feeding System,” ASME Journal of Lubrication Technology, Vol. 102, 1980.
[3]Stout, K. J. and Barrans, S. M., “The Design of Aerostatic Bearings for Application to Nanometer Resolution Manufacturing Machine Systems,” Tribology International, Vol. 33, pp. 803-809, 2005.
[4]Lo, C. Y., Wang, C. C. and Lee, Y. H., “Performance Analysis of High-Speed Spindle Aerostatic Bearings,” Tribology International, Vol. 38, pp. 5-14, 2005.
[5]Blondeel, E., Snoeys, R. and Devrieze, L., “Dynamic Stability of Externally Pressurized Gas Bearings,” Journal Lubrication Technology , Vol. 102 , pp. 511-519, 1980.
[6]Renn, J. C. and Hsiao, C. H., “Experimental and CFD Study on the Mass Flow-Rate Characteristic of Gas Through Orifice-Type Restrictor in Aerostatic Bearing,” Tribology International, Vol. 37, No. 5, pp. 309, 2004.
[7]Anton, V. B., Ron, A. J. and Ostayen,. V., “The Design of Partially Grooved Externally Pressurized Bearings,” Tribology International, Vol. 39, pp. 833-838.2006
[8]Boffey, D. A., Barrow, A. A. and Dearden, J. K., “Experimental Investigation Into the Performance of an Aerostatic Industrial Thrust Bearing,” Tribology International, Vol. 18, No. 3, pp. 165, 1985.
[9]Stout, K. J., “Design of Aerostatic Flat Pad Bearings Using Annular Orifice Restrictors,” Tribology International, Vol. 18, No. 4, pp. 209, 1985.
[10]Fourka, M. and Bonis, M., “Comparison Between Externally Pressurised Gas Thrust Bearings with Different Orifice and Porous Feeding System,” Wear, Vol. 210, No. 1-2, pp. 311, 1997.
[11]Roblee, J. W. and Mote Jr., C. D., “Design of Externally Pressurised Gas Bearings for Stiffness and Damping,” Tribology International, Vol. 23, No. 5, pp. 333, 1990.
[12]Yao, Y. X., Qin, D. U. and Zhang, H. B., “Static Performance Analysis of Orifice Compensated Externally Pressurized Gas Spherical Bearing based on FEM,” Key Engineering Materials, Vol. 315, pp. 860, 2006.
[13]Gross, W. A., Fluid Film Lubrication, Wiley, New York, 1980.
[14]Gross, W. A. and Zachmanaglou, E C., “Perturbation Solutions for Gas-lubricating Films,” Trans ASME. J Basic Eng 83, pp. 139-144, 1961.
[15]Majumdar, B. C. “Analysis of Externally Pressurized Porous Wall Gas Journal Bearings-I,” Wear 33, pp. 25-35, 1975.
[16]Majumdar, B. C. “Dynamic Charactertics of Externally Pressurized Rectangular Porous Gas Thrust Bearings.” Trans ASME , J. Lubrication Technology 98, pp. 181-186, 1976
[17]Gero, L. R., Ettles and C. M. M. cC. , “An evaluation of finite dufference and finite element methods for the solution of the Reynolds equation.” ASLE Trans. 29 (2), pp. 166-172 1985.
[18]Boffey, D. A., “A study of the stability of an externally-pressurized gas-lubricated thrust bearing with a flexible damped support.” Trans. ASME, J. Lubrication Technology 100, pp. 364-368, 1978.
[19]Pal, D. K., Majumdar and B. C., “Stability Analysis of Externally-Pressurized Gas-Lubricated Porous Bearings with Journal Rotation Part 1 Cylindrical whirl,” Tribology International Vol. 17, pp. 83-91, 1984.
[20]Pal, D. K., Majumdar and B. C., “Stability Analysis of Externally-Pressurized Gas-Lubricated Porous Bearings with Journal Rotation Part 2 Conical whirl,” Tribology International Vol. 17, pp. 92-98, 1984.
[21]Fourka, Mohamed, Tian and Yong, Bonis, Marc, “Prediction of the stability of air thrust bearings by numerical , analytical and experimental methods.” Wear 198, pp. 1-6, 1996.
[22]Fourka, Mohamed, Bonis. and Marc, “Comparison between externally pressurized gas thurst bearings with different orifice and porous feeding system.” Wear 210, pp. 311-317, 1997.
[23]Lin, Gang, Aoyama, Tojiro, and Inasaki, Ichiro, “A computer simulation method for dynamatic and stability analyses of air bearings,” Wear 126, pp. 307-319, 1988.
[24]Kazimierski, Z. and Trojnarski. J., “Investigations of Externally Pressurized Gas Bearings with Different Feeding System,” ASME Journal of Lubrication Technology, Vol. 102, 1980.
[25]Laurene V. Fausett, “Applied Numerical Analysis Using Matlab®, Prentice-Hall,” Inc. (1999)