A partial derivative method for frequency dependent dynamic coefficients embracing complete variables perturbation

PDF(1727 KB)

Journal of Vibration and Shock ›› 2022, Vol. 41 ›› Issue (20) : 68-77.

BI Chunxiao1,2，HAN Dongjiang1,2，YANG Jinfu1,2

Author information +

History +

Abstract

The process fluid lubricated bearings can avoid oil pollution completely and simplify the seals and rotor dynamics of high-speed rotating machineries. However, fluid with special parameters has complicated thermodynamic properties, which brings many additional effects, such as turbulence, real gas compressible flow and so on. Based on compressible turbulent lubrication Reynolds equation, a partial derivative method embracing complete variable perturbation was presented by establishing dynamic mapping, which is a general method for static and dynamic characteristics and is independent of lubricating fluid and thermodynamic model. The frequency effect of dynamic coefficients caused by both structural perturbation and compressible lubrication can be involved together by this method, which is suitable for calculating frequency dependent dynamic characteristics of bearings and dry gas seals with additional effect. Compared to the finite increment method, the accuracy of the complete variables perturbation method is verified. A method presented in this research was used in supercritical carbon dioxide and oil lubricated tilting pad bearings as example. The results show that neglecting the additional perturbed variables besides pressure and film thickness will overestimate dynamic coefficients. The maximum deviations of stiffness and damping coefficients in the examples of this paper are 88% and 93% respectively.
Key words: lubricant; real gas effect; perturbation frequency; complete variable perturbation; turbulent lubrication

Key words

lubricant / real gas effect / perturbation frequency / complete variable perturbation / turbulent lubrication

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

BI Chunxiao1,2，HAN Dongjiang1,2，YANG Jinfu1,2. A partial derivative method for frequency dependent dynamic coefficients embracing complete variables perturbation[J]. Journal of Vibration and Shock, 2022, 41(20): 68-77

References

[1] 韩东江, 郝龙, 毕春晓, 等. 燃气轮机转子系统典型振动特性试验研究[J]. 振动与冲击, 2021, 40(4): 87-93.
HAN Dongjiang, HAO Long, BI Chunxiao, et al. An experimental study on typical vibration characteristics of a gas turbine rotor system[J]. Journal of Vibration and Shock, 2021, 40(4): 87-93.
[2] 陈昌婷. 高速气体轴承结构性能分析与实验研究[D]. 北京：中国科学院研究生院, 2014.
[3] QIN K, LI D J, HUANG C, et al. Numerical investigation on heat transfer characteristics of Taylor Couette flows operating with CO2[J]. Applied Thermal Engineering, 2020, 165: 114570.
[4] 顾延东, 成立, BOHLE Martin, 等. 多孔质静压径向轴承的理论建模与数值计算[J]. 振动与冲击, 2021, 40(20): 16-24.
GU Yandong, CHENG Li, BOHLE Martin, et al. Theoretical modeling and numerical solution of hydrostatic radial bearings with porous restrictor[J]. Journal of Vibration and Shock, 2021, 40(20): 16-24.
[5] 王昱, 候予, 赵琪, 等. 液氢自润滑动压轴承空化特性的数值研究[J]. 低温工程, 2020 (1): 50-54.
WANG Yu, HOU Yu, ZHAO Qi, et al. Numerical study on cavitation characteristics of liquid hydrogen self-lubricated hydrodynamic bearings[J]. Cryogenics, 2020 (1): 50-54.
[6] BOUCHEHIT B, BOU-SAID B, GARCIA M. Static and dynamic performances of refrigerant-lubricated bearings[J]. Tribology International, 2016, 96: 326-348.
[7] SAVILLE M, GU A, NUNEZ S. Foil-bearing LOx/LN2 turbopump performance during rapid transients and low flow conditions[C]// 31st Joint Propulsion Conference and Exhibit.[S.l.]：AIAA,1995.
[8] 丁增杰. 氢氧涡轮泵用箔片轴承的试验研究[J]. 导弹与航天运载技术, 1999 (5): 22-28.
DING Zengjie. Experiment and research of foil bearing for LH2/LOx turbo-pump[J]. Missiles and Space Vehicles, 1999(5) : 22-28.
[9] GALLO B M, EL-GENK M S. Brayton rotating units for space reactor power systems[J]. Energy Conversion and Management, 2009, 50(9): 2210-2232.
[10] MA Y, MOROZYUK T, LIU M, et al. Optimal integration of recompression supercritical CO2 Brayton cycle with main compression intercooling in solar power tower system based on exergoeconomic approach[J]. Applied Energy, 2019, 242: 1134-1154.
[11] 严新平, 王佳伟, 孙玉伟, 等. 船舶余热利用sCO2布雷顿循环发电技术综述[J]. 中国机械工程, 2019, 30(8): 939-946.
YAN Xinping, WANG Jiawei, SUN Yuwei, et al. Review on sCO2 brayton cycle power generation technology based on ship waste heat recovery utilization[J]. China Mechanical Engineering, 2019, 30(8): 939-946.
[12] 姜培学, 胥蕊娜, 祝银海, 等. 高超声速飞行器发动机热量回收发电系统及其控制方法：CN 201610832405.0. [P].2016-09-19.
[13] KIMBALL K J, CLEMENTONI E M. Supercritical carbon dioxide Brayton power cycle development overview[C]//Proceedings of the ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. Copenhagen：, ASME ，2012.
[14] CHO J, SHIN H, RA H S, et al. Development of the supercritical carbon dioxide power cycle experimental loop in KIER[C]//Proceedings of ASME Turbo Expo 2016: Turbomachinery Technical Conference and Exposition. Seoul：ASME, 2016.
[15] ERTAS B. Compliant hybrid gas bearing using integral hermetically sealed squeeze film dampers [C]//Proceedings of ASME Turbo Expo 2019: Turbine Technical Conference and Exposition. Phoenix, AZ：ASME; 2019.
[16] 程文杰, 邓志凯, 肖玲, 等. 三瓦双向箔片轴承—高速电机转子系统动力学研究[J]. 振动与冲击, 2021, 40(1): 218-225.
CHENG WenJie, DENG Zhikai, XIAO Ling, et al. Dynamic characteristics of a rotor system with 3-pad bidirectional GFB and high speed motor[J]. Journal of Vibration and Shock, 2021, 40(1): 218-225.
[17] HESHMAT H, WALTON II J F, CORDOVA J L. Technology readiness of 5th and 6th generation compliant foil bearing for 10 MWE S-CO2 turbomachinery systems[C]//Proceedings of the 6th International Supercritical CO2 Power Cycles Symposium. Pittsburgh：ISPCS, 2018.
[18] CONBOY T M. Real-gas effects in foil thrust bearings operating in the turbulent regime[J]. Journal of Tribology, 2013, 135(3): 031703.
[19] KIM D. Design space of foil bearings for closed-loop supercritical CO2 power cycles based on three-dimensional thermohydrodynamic analyses[J]. Journal of Engineering for Gas Turbines and Power, 2016, 138: 032504.
[20] QIN K, JAHN I H, JACOBS P A. Effect of operating conditions on the elastohydrodynamic performance of foil thrust bearings for supercritical CO2 cycles[J]. Journal of Engineering for Gas Turbines and Power, 2017, 139(4): 042505.
[21] XU F, KIM D. Three-dimensional turbulent thermo-elastohydrodynamic analyses of hybrid thrust foil bearings using real gas model[C]//Proceedings of ASME Turbo Expo 2016: Turbomachinery Technical Conference and Exposition.Seoul：ASME, 2016.
[22] 张国渊, 袁小阳. 基于混合均质模型的气液两相流润滑动静压轴承性能分析[J]. 低温工程, 2010 (2): 8-23.
ZHANG Guoyuan, YUAN Xiaoyang. Performance analysis of hybrid journal bearings in two phase flow based on liquid-vapor mixture model[J]. Cryogenics, 2010 (2): 8-23.
[23] 李智. 空间反应堆动态能量转换系统特性研究[D]. 北京：清华大学. 2017.
[24] 吴垚, 杨利花, 徐腾飞, 等. 气体动压径向轴承超薄气膜润滑动特性分析[J]. 振动工程学报, 2019, 32(5): 908-917.
WU Yao, YANG Lihua, XU Tengfei, et al. Analysis on dynamic characteristics of the ultra-thin gas film lubrication in cylindrical gas journal bearings[J]. Journal of Vibration Engineering, 2019, 32(5): 908-917.
[25] ZHANG Y F, HEI D, LIU C, et al. An approximate solution of oil film forces of turbulent finite length journal bearing[J]. Tribology International, 2014, 74: 110-120.
[26] LUND J W. Spring and damping coefficients for the tilting-pad journal bearing[J]. Tribology Transactions, 1964, 7(4): 342-352.
[27] CHEN W J, GUNTER E J. Introduction of dynamics of rotor-bearing systems[M]. Victoria: Trafford, 2005.
[28] WILKES J C, CHILDS D W. Tilting pad journal bearings—a discussion on stability calculation, frequency dependence, and pad and pivot[J]. Journal of Engineering for Gas Turbines and Power, 2012, 134: 122508.
[29] DELGADO A. Experimental identification of dynamic force coefficients for a 110 MM compliantly damped hybrid gas bearing[J]. Journal of Engineering for Gas Turbines and Power, 2015, 137: 072502.
[30] SIM K, LEE Y, SONG J, et al. Identification of the dynamic performance of a gas foil journal bearing operating at high temperatures[J]. Journal of Mechanical Science and Technology, 2014, 28(1): 43-51.
[31] 虞烈, 戚社苗, 耿海鹏. 可压缩气体润滑与弹性箔片气体轴承技术[M]. 北京: 科学出版社, 2011.
[32] SAN ANDRES L, TAO Y. The role of pivot stiffness on the dynamic force coefficients of tilting pad journal bearings[J]. Journal of Engineering for Gas Turbines and Power, 2013, 135: 112505.
[33] KIM D. Parametric studies on static and dynamic performance of air foil bearings with different top foil geometries and bump stiffness distributions[J]. Journal of Tribology, 2007, 129: 354-364.
[34] YANG L, SUN Y, YU L. Active control of unbalance response of rotor systems supported by tilting-pad gas bearings[J]. Proc. IMechE, Part J: Journal of Engineering Tribology, 2012, 226: 87-98.
[35] 李长林. 多滑动梁径向与层叠式止推箔片气体轴承静动特性研究[D]. 哈尔滨:哈尔滨工业大学, 2019.
[36] RAMAN S, KIM T, KIM H. A novel algorithm to estimate the CO2 flows across the critical point with real gas effects[J]. International Journal of Heat and Mass Transfer, 2019, 142: 118363.
[37] 温建全. 超临界二氧化碳介质箔片轴承弹流耦合研究[D]. 哈尔滨:哈尔滨工业大学. 2017.
[38] BI C X, HAN D J, YANG J F. The frequency perturbation method for predicting dynamic coefficients of supercritical carbon dioxide lubricated bearings[J]. Tribology International, 2020, 146: 106256.
[39] 江锦波, 滕黎明, 孟祥铠, 等. 基于多变量摄动的超临界CO2干气密封动态特性[J]. 化工学报, 2021, 72(4): 2190-2202.
JIANG Jinbo, TENG Liming, MENG Xiangkai, et al. Dynamic characteristics of supercritical CO2 dry gas seal based on multi variables perturbation[J]. CIESC Journal, 2021, 72(4): 2190-2202.
[40] 严如奇, 丁雪兴, 徐洁, 等. 基于湍流模型的S-CO2干气密封流场与稳态性能分析[J]. 化工学报, 2021,72(8):4292-4303.
YAN Ruqi, DING Xuexing, XU Jie, et al. Flow field and steady performance of supercritical carbon dioxide dry gas seal based on turbulence model[J]. CIESC Journal, 2021,72(8):4292-4303.
[41] FRENE J, NICOLAS D, DEGUEURCE B, et al. Hydrodynamic lubrication: bearings and thrust bearings[M]. Amsterdam: Elsevier, 1997.
[42] 闻邦椿, 顾家柳, 夏松波, 等. 高等转子动力学—理论、技术及应用[M]. 北京: 机械工业出版社, 1999.
[43] 杨期江. 柔性阻尼支承可倾瓦轴承油膜动力及减振特性研究[D].广州：华南理工大学，2016.
[44] SOMEYA T. Journal-bearing databook[M]. Berlin: Springer, 1989.