Semi-analytical calculation of stiffness and damping in gas foil bearings
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摘要:
为克服传统雷诺方程法在模型精度与普适性上的不足，提出了一种基于流体力学基本方程组的半解析方法，用于精确计算箔片气体动压轴承的刚度阻尼系数。首先建立气膜与箔片之间的流固耦合模型，然后不断改变偏心率以找到轴颈在确定转速下的静平衡位置，再取一次偏心扰动进行稳态计算得到四个刚度系数，最后对轴颈施加两次简谐振动通过瞬态计算获得4个阻尼系数。结果表明：随着转速增大，轴颈静平衡位置逐渐靠近轴承中心，主刚度系数逐渐下降，交叉刚度系数逐渐上升，4个阻尼系数皆逐渐下降。计算得到的刚度阻尼系数可用于后续转子动力学计算。搭建实验平台测试了箔片轴承的刚度系数，实验结果验证了所提计算方法的正确性。
Abstract:To address the limitations of the traditional Reynolds equation method in terms of model accuracy and generalizability, a semi-analytical method based on the fundamental equations of fluid mechanics was proposed for accurate calculation of the stiffness and damping coefficients of gas foil bearings. First, the fluid-structure interaction between the gas film and the foil structure was established. Then, the eccentricity ratio was continuously varied in order to locate the static equilibrium position of the journal at a determined rotational speed. Subsequently, a first-order eccentricity perturbation was applied to obtain the four stiffness coefficients via steady-state calculations. Finally, the four damping coefficients were determined through transient calculations by imposing two simple harmonic vibrations on the journal. The results showed that as the rotational speed increased, the static equilibrium position of the journal gradually moved closer to the bearing center, the direct stiffness coefficients decreased, the cross-coupled stiffness coefficients increased, and all four damping coefficients decreased. The calculated stiffness and damping coefficients can be used in subsequent rotor dynamics analyses. The stiffness coefficients of the foil bearing were tested experimentally, and the experimental results demonstrated the feasibility of the proposed calculation method.
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图 1 静平衡位置时轴颈受力示意图
Figure 1. Force schematic diagram of the journal at the static equilibrium position
图 2 水平方向静载荷不变时轴颈受力示意图
Figure 2. Schematic diagram of force on the journal shaft when the horizontal static load remains unchanged
图 3 竖直方向静载荷不变时轴颈受力示意图
Figure 3. Schematic diagram of force on the journal shaft when the vertical static load remains unchanged
图 5 箔片轴承刚度阻尼系数半解析计算流程图
Figure 5. Flowchart for calculating stiffness and damping coefficients of foil bearings
图 10 转速为20 000 r/min、偏心距为19.6 $ \text{μm} $时的气膜压力云图
Figure 10. Gas film pressure contour plot at 20 000 r/min and19.6 μm eccentricity
图 11 转速为20 000 r/min、偏心距为19.6 μm时的箔片变形云图
Figure 11. Foil deformation contour plot at 20 000 r/min and19.6 μm eccentricity
图 12 不同转速下轴颈的静平衡位置
Figure 12. Static equilibrium position of the journal at different rotational speeds
图 13 转速为20 000 r/min的刚度系数与扰动位移的关系
Figure 13. Relationship between stiffness coefficients and disturbance displacement at rotational speed is20 000 r/min
图 14 刚度系数随转速变化曲线图
Figure 14. Curve of stiffness coefficient variation with rotational speed
图 15 不同时间步数下的气膜力波形（转速为20 000 r/min）
Figure 15. Waveforms of gas film force at different time steps （rotational speed is 20 000 r/min）
图 16 x方向位移正弦激励引起的气膜力变化图（转速为20 000 r/min）
Figure 16. Variation of aerodynamic film force caused by harmonic excitation of displacement in the x-direction （rotational speed is 20 000 r/min）
图 17 阻尼系数随转速变化曲线图
Figure 17. Curve of damping coefficient variation with rotational speed
图 18 转速为20 000 r/min的阻尼系数与时间步数的关系
Figure 18. Relationship between damping coefficient and time steps at rotational speed is 20 000 r/min
图 19 转速为20 000 r/min的阻尼系数与扰动速度的关系
Figure 19. Relationship between damping coefficient and perturbation velocity at rotational speed is 20 000 r/min
图 21 箔片轴承刚度测试实验装置
Figure 21. Experimental setup for foil bearing stiffness measurement
图 22 实验测得的气膜力增量与位移增量关系图
Figure 22. Measured relationship curve of air film force increment versus displacement increment.
表 1 箔片轴承结构参数
Table 1. Structural Parameters of the foil bearings
结构参数 数值 轴颈半径R/mm 11 半径间隙C/mm 0.027 平箔片厚度$ {t}_{\text{t}} $/mm 0.2 波箔片厚度$ {t}_{\text{b}} $/mm 0.3 波箔片高度$ {h}_{\text{b}} $/mm 0.7 波箔片半弦长l/mm 1.6 波箔片节距s/mm 3.6 静荷载W/N 4.96 波箔片波瓣轴向宽度$ {L}_{1} $,$ {L}_{2} $,$ {L}_{3} $/mm 5.4, 6.8, 5.4 波箔片波瓣轴向间隔$ {L}_{4} $/mm 1.2 轴承轴向长度L/mm 20 箔片材料密度$ \rho $/（kg/m3） 8000 箔片材料弹性模量E/GPa 214 箔片材料泊松比$ \mu $ 0.3 
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表 2 刚度系数的实验拟合值与误差
Table 2. Experimental fitted values and errors of stiffness coefficients
参数 半解析计算值 实验拟合值 误差/% $ {k}_{xx} $/（$ \text{kN/m} $） 30.97 31.1 0.42 $ {k}_{yy} $/（$ \text{kN/m} $） 103.53 109.3 5.57
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[1] 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. Yu Lie, Qi Shemiao, Geng Haipeng. Compressible gas lubrication and elastic foil gas bearing technology[M]. Beijing: Science Press, 2011. (in ChineseYu Lie, Qi Shemiao, Geng Haipeng. Compressible gas lubrication and elastic foil gas bearing technology[M]. Beijing: Science Press, 2011. (in Chinese) [2] 王云飞. 气体润滑理论与气体轴承设计[M]. 北京: 机械工业出版社, 1999. Wang Yunfei. Gas lubricated theory and design manual of gas bearings[M]. Beijing: China Machine Press, 1999. (in ChineseWang Yunfei. Gas lubricated theory and design manual of gas bearings[M]. Beijing: China Machine Press, 1999. (in Chinese) [3] 熊万里, 汪剑, 陈振宇, 等. 箔片气体动压轴承研究进展综述[J]. 机械工程学报, 2022, 58(21): 92-113. Xiong Wanli, Wang Jian, Chen Zhenyu, et al. Review of research status and development of foil air bearings[J]. Journal of Mechanical Engineering, 2022, 58(21): 92-113. (in ChineseXiong Wanli, Wang Jian, Chen Zhenyu, et al. Review of research status and development of foil air bearings[J]. Journal of Mechanical Engineering, 2022, 58(21): 92-113. (in Chinese) [4] 冯凯. 先进箔片气体动压轴承技术及其工程应用[M]. 北京: 科学出版社, 2022. Feng Kai. Advanced foil gas dynamic bearing technology and its engineering application[M]. Beijing: Science Press, 2022. (in ChineseFeng Kai. Advanced foil gas dynamic bearing technology and its engineering application[M]. Beijing: Science Press, 2022. (in Chinese) [5] 刘佳琦, 毕春晓, 韩东江, 等. 弹性箔片轴承的研究进展[J]. 润滑与密封, 2022, 47(9): 166-178. Liu Jiaqi, Bi Chunxiao, Han Dongjiang, et al. Research progress of elastic foil bearings[J]. Lubrication Engineering, 2022, 47(9): 166-178. (in ChineseLiu Jiaqi, Bi Chunxiao, Han Dongjiang, et al. Research progress of elastic foil bearings[J]. Lubrication Engineering, 2022, 47(9): 166-178. (in Chinese) [6] 钟一谔. 转子动力学[M]. 北京: 清华大学出版社, 1987. Zhong Yi’e. Rotor dynamics[M]. Beijing: Tsinghua University Press, 1987. (in ChineseZhong Yi’e. Rotor dynamics[M]. Beijing: Tsinghua University Press, 1987. (in Chinese) [7] Lund J W. The stability of an elastic rotor in journal bearings with flexible, damped supports[J]. Journal of Applied Mechanics, 1965, 32(4): 911-920. doi: 10.1016/0043-1648(67)90152-4 [8] 虞烈, 戚社苗, 耿海鹏. 弹性箔片空气动压轴承的完全气弹润滑解[J]. 中国科学: E辑, 2005(7): 746-760. Yu Lie, Qi Shemiao, Geng Haipeng. Complete aeroelastic lubrication solution of elastic foil aerodynamic bearing[J]. Science in China: Series E, 2005(7): 746-760. (in ChineseYu Lie, Qi Shemiao, Geng Haipeng. Complete aeroelastic lubrication solution of elastic foil aerodynamic bearing[J]. Science in China: Series E, 2005(7): 746-760. (in Chinese) [9] Xu Haojie, Yang Jiapeng, Gao Lei, et al. The influences of bump foil structure parameters on the static and dynamic characteristics of bump-type gas foil bearings[J]. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 2020, 234(10): 1642-1657. doi: 10.1177/1350650120912609 [10] 张镜洋, 赵晓荣, 常海萍, 等. 边界滑移对波箔型动压气体轴承静特性的影响[J]. 推进技术, 2018, 39(2): 388-395. Zhang Jingyang, Zhao Xiaorong, Chang Haiping, et al. Effects of sliding boundary on static characteristics of aerodynamic compliant foil bearing[J]. Journal of Propulsion Technology, 2018, 39(2): 388-395. (in ChineseZhang Jingyang, Zhao Xiaorong, Chang Haiping, et al. Effects of sliding boundary on static characteristics of aerodynamic compliant foil bearing[J]. Journal of Propulsion Technology, 2018, 39(2): 388-395. (in Chinese) [11] 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 bearing[J]. Journal of Vibration Engineering, 2019, 32(5): 908-917. (in ChineseWu Yao, Yang Lihua, Xu Tengfei, et al. Analysis on dynamic characteristics of the ultra-thin gas film lubrication in cylindrical gas journal bearing[J]. Journal of Vibration Engineering, 2019, 32(5): 908-917. (in Chinese) [12] Miyanaga N, Tomioka J. Effect of support stiffness and damping on stability characteristics of herringbone-grooved aerodynamic journal bearings mounted on viscoelastic supports[J]. Tribology International, 2016, 100: 195-203. doi: 10.1016/j.triboint.2016.01.019 [13] Guo Zenglin, Hirano T, Kirk R G. Application of CFD analysis for rotating machinery: part I: hydrodynamic, hydrostatic bearings and squeeze film damper[J]. Journal of Engineering for Gas Turbines and Power, 2005, 127(2): 445-451. doi: 10.1115/1.1807415 [14] Heshmat H, Walowit J A, Pinkus O. Analysis of gas-lubricated foil journal bearings[J]. Journal of Lubrication Technology, 1983, 105(4): 647-655. [15] Heshmat H, Walowit J A, Pinkus O. Analysis of gas lubricated compliant thrust bearings[J]. Journal of Lubrication Technology, 1983, 105(4): 638-646. doi: 10.1115/1.3254696 [16] Iordanoff I. Analysis of an aerodynamic compliant foil thrust bearing: method for a rapid design[J]. Journal of Tribology, 1999, 121(4): 816-822. doi: 10.1115/1.2834140 [17] Iordanoff I, Bou Said B, Mezianne A, et al. Effect of internal friction in the dynamic behavior of aerodynamic foil bearings[J]. Tribology International, 2008, 41(5): 387-395. doi: 10.1016/j.triboint.2007.09.010 [18] Feng Kai, Kaneko S. Analytical model of bump-type foil bearings using a link-spring structure and a finite-element shell model[J]. Journal of Tribology, 2010, 132(2): 021706. doi: 10.1115/1.4001169 [19] Feng Kai, Guo Zhiyang. Prediction of dynamic characteristics of a bump-type foil bearing structure with consideration of dynamic friction[J]. Tribology Transactions, 2014, 57(2): 230-241. doi: 10.1080/10402004.2013.864790 [20] 李强, 许伟伟, 赵月, 等. 基于瞬态流场计算的滑动轴承静平衡位置求解[J]. 中国石油大学学报(自然科学版), 2015, 39(2): 105-110. Li Qiang, Xu Weiwei, Zhao Yue, et al. Solution of static equilibrium position of journal bearing based on transient flow calculation[J]. Journal of China University of Petroleum (Edition of Natural Science), 2015, 39(2): 105-110. (in ChineseLi Qiang, Xu Weiwei, Zhao Yue, et al. Solution of static equilibrium position of journal bearing based on transient flow calculation[J]. Journal of China University of Petroleum (Edition of Natural Science), 2015, 39(2): 105-110. (in Chinese) [21] 熊万里, 侯志泉, 吕浪, 等. 基于动网格模型的液体动静压轴承刚度阻尼计算方法[J]. 机械工程学报, 2012, 48(23): 118-126. Xiong Wanli, Hou Zhiquan, (lü/lv/lu/lyu) Lang, et al. Method for calculating stiffness and damping coefficients of hybrid bearings based on dynamic mesh model[J]. Journal of Mechanical Engineering, 2012, 48(23): 118-126. (in ChineseXiong Wanli, Hou Zhiquan, (lü/lv/lu/lyu) Lang, et al. Method for calculating stiffness and damping coefficients of hybrid bearings based on dynamic mesh model[J]. Journal of Mechanical Engineering, 2012, 48(23): 118-126. (in Chinese) [22] Li Qiang, Liu Shulian, Pan Xiaohong, et al. A new method for studying the 3D transient flow of misaligned journal bearings in flexible rotor-bearing systems[J]. Journal of Zhejiang University: SCIENCE A, 2012, 13(4): 293-310. doi: 10.1631/jzus.A1100228 [23] Wang Bin, Ding Qian, Yang Tianzhi. Soft rotor and gas bearing system: Two-way coupled fluid-structure interaction[J]. Journal of Sound and Vibration, 2019, 445: 29-43. doi: 10.1016/j.jsv.2019.01.007 [24] Wang Bin, Sun Yongtao, Ding Qian. Dynamic characteristics of the herringbone groove gas journal bearings: numerical simulations[J]. Shock and Vibration, 2016, 2016: 8743016. [25] 贾晨辉, 刘书明, 刘恒, 等. 基于流固耦合的箔片气体轴承动态特性分析[J]. 振动工程学报, 2024, 37(3): 394-401. Jia Chenhui, Liu Shuming, Liu Heng, et al. Dynamic characteristic analysis of foil gas bearings based on fluid-structure coupling[J]. Journal of Vibration Engineering, 2024, 37(3): 394-401. (in ChineseJia Chenhui, Liu Shuming, Liu Heng, et al. Dynamic characteristic analysis of foil gas bearings based on fluid-structure coupling[J]. Journal of Vibration Engineering, 2024, 37(3): 394-401. (in Chinese) [26] 池长青. 流体力学润滑[M]. 北京: 国防工业出版社, 1998. Chi Changqing. Hydromechanical lubrication[M]. Beijing: National Defense Industry Press, 1998. (in ChineseChi Changqing. Hydromechanical lubrication[M]. Beijing: National Defense Industry Press, 1998. (in Chinese) -
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