臺灣博碩士論文加值系統

摘要
本研究探討由氣靜壓軸承所支撐的剛性主軸之靜態和動態特性及穩定性。其氣體通過雙排小孔或環面節流之軸承間隙且為絕熱過程。軸承間隙內的氣體薄膜遵循雷諾方程式，並耦合由主軸旋轉所形成的邊界效應及由軸頸振動所造成的擠壓效應。使用有限差分法求解雷諾方程式，然後用數值積分的方法求出氣體薄膜的靜態和動態特性。轉子軸承系統的運動方程式由微擾法及特徵解法得到臨界慣性力及臨界漩振比的穩定性門檻。增加氣腔的體積可以增加軸承承載能力及剛性，但所伴隨的問題是系統漩振不穩定的現象發生。最後，分析氣腔的體積比對軸承動態性能的影響，以作為氣體軸承設計之參考。

Abstract
This paper studies static and dynamic characteristics of a rigid spindle supported by aerostatic bearings. The air passes through the double-row hole entries, which consist of orifices restrictors, into the bearing clearance. The process is adiabatic. The air film in this bearing clearance is governed by Reynolds equation, coupling both wedge effect due to spindle rotation and squeezed effect due to journal oscillation. The -method is used to analyze Reynolds equation, solved by the finite difference method and numerical integration, and then yields static and dynamic characteristics of air film. The motion equation of the rotor-bearing system is obtained by perturbation method. The eigensolution method is used to determine the stability threshold including critical inertial force and critical whirl ratio. Various eccentricities, rotor speeds, and restriction parameters are taken into considerations. To increase volume ratio of air recess can get higher loading capacity and stiffness, but it will occur the instability of pneumatic hammer. The research gives the volume effect of air recess in stability of rotor-aerostatic bearing system.

目錄
摘要……………………………………………………………………………………I
Abstract……………………………………………………………………………….II
誌謝………………………………………………………………………………….III
目錄………………………………………………………………………………….IV
圖目錄………………………………………………………………………………VII
表目錄……………………………………………………………………………….IX
符號說明……………………………………………………………………………..X
第一章 緒論………………………………………………………………………….1
1.1 研究背景……………………………………………………………………1
1.2 文獻回顧……………………………………………………………………2
1.3 研究內容……………………………………………………………………7
第二章 理論推導…………………………………………………………………….8
2.1空氣靜壓軸承潤滑方程式的推導…………………………………………..8
2.2本研究所使用的氣體軸承及節流器之型式………………………………10
2.2.1氣體頸軸承型式…………………………………………………….10
2.2.2 節流器之型式………………………………………………………12
2.3流量平衡方程式……………………………………………………………12
2.3.1孔口節流器的流量平衡方程式……………………………………..12
2.3.2氣腔的流量平衡方程式……………………………………………..16
2.4微擾化分析…………………………………………………………………20
2.4.1雷諾方程式的微擾化………………………………………………..20
2.4.2孔口節流器的流量平衡方程式的微擾化…………………………..22
2.4.3氣腔流量平衡方程式的微擾化……………………………………..23
第三章 空氣靜壓軸承系統的性能分析…………………………………………..29
3.1靜態承載能力及靜態偏位角………………………………………………29
3.2剛度係數及阻尼係數………………………………………………………30
3.3轉子之運動方程式…………………………………………………………32
3.4穩定性分析…………………………………………………………………33
3.5漩振頻率……………………………………………………………………34
第四章 介紹數值方法……………………………………………………………...37
4.1求解方法……………………………………………………………………37
4.2雷諾方程式之離散化………………………………………………………40
4.3氣腔流量平衡方程式之離散化……………………………………………41
4.4 S.O.R. 方法………………………………………………………………..52
第五章 結果與討論………………………………………………………………...53
第六章 結論與未來展望…………………………………………………………...67
6.1 研究結論…………………………………………………………………..67
6.2 未來研究方向……………………………………………………………..67
參考文獻…………………………………………………………………………….69
個人資料…………………………………………………………………………….78
論文發表…………………………………………………………………………….79

圖目錄
圖2-1 平面軸頸與軸承之橫截面示意圖…………………………………………...9
圖2-2 氣體軸承示意圖(a)結構圖 (b)主軸中心擾動軌跡………………………..11
圖2-3 孔口節流器………………………………………………………………….12
圖2-4 從氣腔流出之流量 的積分路徑………………………………………….18
圖3-1 轉子-軸承系統圖……………………………………………………………33
圖4-1 數值計算區域……………………………………………………………….37
圖4-2 數值計算流程圖…………………………………………………………….39
圖5-1 六個供氣孔軸承型式的構造……………………………………………….53
圖5-2 空氣軸承沿周向表面通過孔口節流器在 及 時的壓力分布(左欄為 ，右欄為 ）……………………………………………...55
圖5-3 軸承的承載能力( )與其相關的偏位角( )在 及 時隨節流參數( )變化之分布………………………………………………………….56
圖5-4 軸承的承載能力( )與其相關的偏位角( )在 及 時隨偏心率( )變化之分布……………………………………………………………..56
圖5-5軸承的剛度係數( )及阻尼係數( )隨漩振比( )變化之分布， ， ， 及 ……………………………………………………...58
圖5-6 軸承的(a)剛度係數( ) (b)阻尼係數( ) (c)臨界慣性力( )及漩振比的平方( )隨漩振比( )變化之分布， , , 及 ………60
圖5-7 軸承的臨界慣性力( )及漩振比( )在 及 時隨節流參數( )變化之分布………………………………………………………………….61
圖5-8 軸承的臨界慣性力( )及漩振比( )隨偏心率( )變化之分布(a) (b) (c) …………………………………………………………..61
圖5-9 節流參數( )與供壓比( )的相對關係…………………………………..63
圖5-10臨界慣性力( )在 ， 及 時與速度參數( )的關係………………………………………………………………………….…64
圖5-11臨界慣性力( )在 ， 及 時與供壓比( )的關係……….64
圖5-12無因次質量( )在 及 時與承載能力( )的關係……………..65
圖5-13無因次質量( )在 及 時與供壓比( )的關係…………….….66
附圖一 流體的動力黏度……………………………………...……………………74
附圖二 流體的運動黏度…………………………………………………………...75


表目錄
附表一 在標準海平面大氣壓下空氣的物理特性………………………………...76
附表二 標準大氣壓的性質………………………………………………………...76
附表三 在標準海平面大氣壓下普通液體的物理特性…………………………...77
附表四 在標準海平面大氣壓下普通氣體的物理特性…………………………...77

參考文獻
[1]Gross W. A. and Zachmanaglou E C., “Perturbation solutions for gas-lubricating films,” Journal of Basic Engineering, Transactions of the ASME, Vol. 83, pp. 139-144, 1961.
[2] Lund J.W., “The hydrostatic gas journal bearing with journal rotation and vibration,” Journal of Basic Engineering, Transactions of the ASME, Vol. 87, No.2, pp. 328-336, 1964.
[3] Lund J. W., “A theoretical analysis of whirl instability and pneumatic hammer for a rigid rotor in pressurized gas journal bearing,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 89, No.2, pp. 154-165, 1967.
[4] Lund J. W., “Calculation of stiffness and Damping properties of gas journal bearing,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 90, No. 4, pp. 793-803, 1968.
[5]Majumdar B. C., “On the general solution of externally pressurized gas journal bearings,” Journal of Lubrication Technology, Transactions of the ASME, pp. 291-296, 1972.
[6] Leonard R. and Rowe W. B., “Dynamic force coefficient and the mechanism of instability in hydrostatic journal bearings,” Wear, Vol. 23, No. 3, pp. 277-282, 1973.
[7]Majumdar B. C. “Analysis of externally pressurized porous wall gas journal bearings-I,” Wear, Vol. 33, No. 3, pp. 25-35, 1975.
[8]Majumdar B. C. “Dynamic charactertics of externally pressurized rectangular porous gas thrust bearings.” Journal of Lubrication Technology, Transactions of the ASME, Vol. 98, pp. 181-186, 1976.
[9]Boffey D. A., “A study of the stability of an externally-pressurized gas-lubricated thrust bearing with a flexible damped support.” Journal of Lubrication Technology, Transactions of the ASME, Vol. 100, pp. 364-368, 1978.
[10]Pink E. G. and Stout K. J., “Design procedures for orifice compensated gas journal bearings based on experimental data,” Tribology International, Vol. 11, No. 1, pp. 63-75, 1978.
[11] Kazimierski Z. and Trojnarski J., “Investigations of Externally Pressurized Gas Bearings with Different Feeding System,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 102, No. 1, pp. 59-64, 1980.
[12]Blondeel E., Snoeys R. and Devrieze L., “Dynamic Stability of Externally Pressurized Gas Bearings,” Journal of Lubrication Technology, Transactions of the ASME, Vol. 102, pp. 511-519, 1980.
[13]Gross W. A., “Fluid film lubrication,” Wiley, New York, 1980.
[14]Wadhwa S. S., Sinhasan R. and Singh D. V., “Analysis of orifice compensated externally pressurized gas bearing,” Tribology International, Vol. 16, No. 4, pp.203-211, 1983.
[15]Pal D. K. and Majumdar B. C., “Stability analysis of externally-pressurized gas-lubricated porous bearings with journal rotation part 2: conical whirl,” Tribology International Vol. 17, No. 2, pp. 92-98, 1984.
[16]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. 165-168, 1985.
[17]Stout K. J., “Design of aerostatic flat pad bearings using annular orifice restrictors,” Tribology International, Vol. 18, No. 4, pp. 209-214, 1985.
[18]Gero L. R. and Ettles C. M. McC., “An evaluation of finite defference and finite element methods for the solution of the Reynolds equation,” ASLE Transaction Vol. 29, No.2, pp. 166-172, 1986.
[19]Lin G., Aoyama T., and Inasaki I., “A computer simulation method for dynamic and stability analyses of air bearings,” Wear, Vol. 126, Issue 3, pp. 307-319, 1988.
[20]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. 333-345, 1990.
[21]Han D. C., Park S. S., Kim W. J. and Kim J. W., “A study on the characteristics of externally pressurized air bearings,” Precision Engineering, Vol. 16, No. 3, pp. 164-173, 1994.
[22]Zhang R. Q. and Chang H. S., “A new type of hydrostatic/hydrodynamic gas journal bearing and its optimization for maximum stability,” Tribology Transactions, Vol. 38, No. 3, pp. 589-594, 1995.
[23]Fourka M., Tian Y., and Bonis M., “Prediction of the stability of air thrust bearings by numerical analytical and experimental methods,” Wear, Vol. 198, No. 1-2, pp. 1-6, 1996.
[24]Fourka M., and Bonis M., “Comparison between externally pressurized gas thrust bearings with different orifice and porous feeding system.” Wear, Vol. 210, No. 1-2, pp. 311-317, 1997.
[25]Kassab S. Z., Noureldeen E. M. and Shawky M. A., “Effects of operating conditions and supply hole diameter on the performance of a rectangular aerostatic bearing,” Tribology International, Vol. 30, No. 7, pp.533-545, 1997.
[26]Kwan Y. B. and Post J. B., “A tolerancing procedure for inherently compensated, rectangular aerostatic thrust bearings,” Tribology International, Vol. 33, No. 8, pp. 581-585, 2000.
[27]Talukder H. M. and Stowell T. B., “Pneumatic hammer in an externally pressurized orifice compensated air journal bearing,” Tribology International, Vol. 36, No. 8, pp. 585-591, 2003.
[28]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. 4, pp.309-315, 2004.
[29]Stout K. J. and Barrans S. M., “The design of Aerostatic Bearings for application to nanometer resolution manufacturing machine systems,” Tribology International, Vol. 33, No. 12, pp. 803-809, 2000.
[30]Lo C. Y., Wang C. C. and Lee Y. H., “Performance analysis of high-speed spindle aerostatic bearings,” Tribology International, Vol. 38, No. 1, pp. 5-14, 2005.
[31]Belforte G., Raparelli T., Viktorov V., Trivella A. and Colombo F., “An experimental study of high-speed rotor supported by air bearings: test RIG and first experimental results,” Tribology International, Vol. 42, No. 4, pp. 839-845, 2006.
[32]Su J. C. T. and Lie K. N., “Rotor dynamic instability analysis on hybrid air journal bearings,” Tribology International, Vol. 39, No. 1, pp. 238-248, 2006.
[33]Anton V. B. and Ostayen R. A. J. V., “The design of partially grooved externally pressurized bearings,” Tribology International, Vol. 39, pp. 833-838.2006.
[34]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.
[35]Yang D. W., “The studies of static/dynamic characteristics and stability in rotor-aerostatic bearing system,” Ph.D. Dissertation, Chung Yuan Christian University, Taiwan, 2009.
[36]Yang D. W., Chen C. H., Kang Y., Hwang R. M. and Shyr S. S., “Influence of orifices on stability of rotor-aerostatic bearing system”, Tribology International, Vol. 42, No. 8, pp. 1206-1219, 2009.
[37]Chen C. H., Kang Y., Yang D. W., Hwang R. M. and Shyr S. S., “Influence of the number of feeding holes on the performances of aerostatic bearings,” Industrial lubrication and Tribology, Vol. 62, Issue 3, 2010.
[38]Wang C. C. and Yau H. T., “Theoretical analysis of high speed spindle air bearings by a hybrid numerical method,” Applied Mathematics and Computation, Vol. 217, pp. 2084-2096, 2010.
[39] Wei L. and Gang B., “Theoretical study of effects on static characteristics of orifice compensation aerostatic bearings with different structures of pockets,” 2nd International Symposium on Systems and Control in Aerospace and Astronautics, ISSCAA 2008, art. No. 4776398.
[40]Li Y. and Ding H., “Influences of the geometrical parameters of aerostatic thrust bearing with pocketed orifice -type restrictor on its performance,” Tribology International, Vol. 40, pp. 1120-1126, 2007.
[41]Boffey D. A. and Desai D. M., “An Experimental investigation into the Rubber-Stabilization of an Externally-Pressurized Air-Lubricated thrust bearing,” Journal of Lubrication Technology, Vol. 102, pp. 65-70, 1980.
[42] FausettL. V., “Applied Numerical Analysis Using Matlab®,” Prentice-Hall Inc. 1999.
[43] Wilcock D. F., “The MTI Gas Bearing Design Manual.” New York, Mechanical Technology incorporation, 1971.
[44] 王雲飛，”氣體潤滑理論與氣體軸承設計” 機械工業出版社，1999。
[45] 楊丁文，” 空氣靜壓軸承主軸之靜動態特性及穩定性研究” 中原大學博士論文，2009。