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 哈尔滨工业大学学报  2019, Vol. 51 Issue (7): 128-134  DOI: 10.11918/j.issn.0367-6234.201806041 0

### 引用本文

HUANG Xiaokai, LIU Shouwen, HUANG Shouqing, YAO Zemin. Modeling and verification technology of bearing multi-stress acceleration model based on response surface method[J]. Journal of Harbin Institute of Technology, 2019, 51(7): 128-134. DOI: 10.11918/j.issn.0367-6234.201806041.

### 文章历史

Modeling and verification technology of bearing multi-stress acceleration model based on response surface method
HUANG Xiaokai, LIU Shouwen, HUANG Shouqing, YAO Zemin
Beijing Institute of Spacecraft Environment Engineering, Beijing 100094, China
Abstract: This paper addresses the scientific problems in bearing accelerated life test, such as lack of multi-stress acceleration model, and not well understanding of failure law. Firstly, the interface micro failure mode and mechanism of fluid lubricated bearing under environment of vacuum, temperature, preload, speed and microgravity are analyzed. Secondly, the micro contact uniform lubrication model considering multi-stress comprehensive working mechanism is derivated and established, and the numerical solution method based on improved Newton Raphson method is proposed. Thirdly, the orthogonal simulation scheme with 5 factors and 5 levels orthogonal table is designed, the bearing failure law in the work of vacuum, temperature, preload, speed and microgravity is simulated and analyzed, and the failure response value of micro contact surface film thickness, pressure peak, friction factor, maximum undersurface stress which changes with the stress level and contact region dimension is obtained. Finally, the multi-stress acceleration model about vacuum, temperature, preload, speed and microgravity based on response surface method has been established, which is verified by a test of actual engineering example.
Keywords: space bearing     multi-stress acceleration model     response surface method     accelerated life test

1 空间轴承多应力失效机理分析 1.1 真空失效模式及机理

1.2 温度失效模式及机理

1.3 预紧力失效模式及机理

1.4 转速失效模式及机理

1.5 微重力失效模式及机理

2 统一润滑机理模型及数值求解 2.1 失效机理模型

 $\eta=\eta_{0} \mathrm{e}^{\alpha P-\beta\left(T-T_{0}\right)},$
 $\rho / \rho_{0}=\left(1+\frac{0.6 \times 10^{-9} P}{1+1.7 \times 10^{-9} P}\right) \times\left(1-\alpha_{T}\left(T-T_{0}\right)\right).$

 $u=\frac{1}{2 \eta} \times \frac{\partial P}{\partial x}\left(z^{2}-z h\right)+\left(V-V_{0}\right) \frac{z}{h}+\left.V_{0}\right|_{z=h, u=V}.$

 $W=\iint p \mathrm{d} x \mathrm{d} y.$

2.2 微观接触统一润滑模型

 $\frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{12 \eta} \frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\rho h^{3}}{12 \eta} \frac{\partial p}{\partial y}\right)=u \frac{\partial(\rho h)}{\partial x}+\frac{\partial(\rho h)}{\partial t}.$ (1)

 $\begin{array}{c}{h=h_{0}(t)+\frac{x^{2}}{2 R_{x}}+\frac{y^{2}}{2 R_{y}}+\nu_{e}(x, y, t)+} \\ {\quad \delta_{1}(x, y, t)+\delta_{2}(x, y, t)}.\end{array}$ (2)

 图 1 接触界面表面粗糙度 Fig. 1 Surface roughness of contact area
2.3 数值解算方法及流程

 图 2 数值解算流程 Fig. 2 Numerical calculation algorithm

 $\operatorname{err}=\frac{\sum\limits_{i=0}^{400}\left|\overline{P}_{i}^{m+1}-\overline{P}_{i}^{m}\right|}{\sum\limits_{i=0}^{400}\left|\overline{P}_{i}^{m+1}\right|}<0.000 \;1.$
3 失效规律仿真与分析

1) 真空(Pa):10-3、102、103、104、105.

2) 温度(℃):-20、20、40、60、80.

3) 预紧力(轴向N):35、60、80、100、120.

4) 转速(r/min):1、10、102、103、6×103.

5) 微重力(入口膜厚μm):1、2、3、4、5.

 $R_{x}=\max \left\{K_{x}^{y}\right\}-\min \left\{K_{x}^{y}\right\}, x=1,2, \ldots, 5 ; y=1,2, \ldots, 5.$

 图 3 各应力类型对失效响应值的敏感程度 Fig. 3 Sensitive degree of failure response value with each stress type

4 多应力加速模型建模与试验验证 4.1 二阶响应面建模

 $\left\{ \begin{array}{l} {y_1} = - 0.1025 + 3.55 \times {10^{ - 6}}{x_1} - 1.7 \times {10^{ - 3}}{x_2} + 5.7 \times {10^{ - 4}}{x_3} + 2.1 \times {10^{ - 4}}{x_4}\\ \qquad + 0.09{x_5} - 2.0 \times {10^{ - 7}}{x_1}{x_5} + 1.241 \times {10^{ - 5}}{x_2}{x_3} - 2.72 \times {10^{ - 6}}{x_2}{x_4}\\ \qquad - 2.8424 \times {10^{ - 4}}{x_2}{x_5} - 2.3 \times {10^{ - 7}}{x_3}{x_4} - 3.9248 \times {10^{ - 4}}{x_3}{x_5} + 2.632 \times \\ \qquad {10^{ - 5}}{x_4}{x_5} + 2.344 \times {10^{ - 5}}x_2^2 - 8.4 \times {10^{ - 7}}x_3^2 - 9.647 \times {10^{ - 3}}x_5^2,\\ {y_2} = 716.16 - 1.65 \times {10^{ - 2}}{x_1} + 1.466{x_2} + 9.5465{x_3} + 3.97 \times {10^{ - 2}}{x_4} + \\ \qquad 68.0258{x_5} + 8.49 \times {10^{ - 5}}{x_1}{x_2} + 4.086 \times {10^{ - 5}}{x_1}{x_3} + 8.8 \times {10^{ - 4}}{x_1}{x_5} - \\ \qquad 5.036 \times {10^{ - 2}}{x_2}{x_3} - 1.085 \times {10^{ - 3}}{x_2}{x_4} + 0.262274{x_2}{x_5} + 1.625 \times {10^{ - 4}}\\ \qquad {x_3}{x_4} - 7.147 \times {10^{ - 2}}{x_3}{x_5} + 1.4178 \times {10^{ - 2}}{x_4}{x_5} + 2.55374 \times {10^{ - 2}}x_2^2 - \\ \qquad 8.9 \times {10^{ - 3}}x_3^2 - 4.58 \times {10^{ - 6}}x_4^2 - 16.5324x_5^2,\\ {y_3} = - 0.05942 - 8.25 \times {10^{ - 6}}{x_1} - 4.9692 \times {10^{ - 4}}{x_2} + 2.187 \times {10^{ - 4}}{x_3} - \\ \qquad 1.79 \times {10^{ - 5}}{x_4} + 0.0715{x_5} - 1.1 \times {10^{ - 7}}{x_1}{x_3} - 3.2 \times {10^{ - 7}}{x_1}{x_5} + 1.622\\ \qquad \times {10^{ - 5}}{x_2}{x_3} - 2.12 \times {10^{ - 4}}{x_2}{x_5} - 5.8 \times {10^{ - 7}}{x_3}{x_4} - 3.69 \times {10^{ - 4}}{x_3}{x_5} + \\ \qquad 3.53 \times {10^{ - 6}}{x_4}{x_5} + 3.94 \times {10^{ - 6}}x_2^2 + 9.6 \times {10^{ - 6}}x_3^2 - 7.62 \times {10^{ - 3}}x_5^2,\\ {y_4} = 51.203 - 1.39 \times {10^{ - 2}}{x_1} + 0.46127{x_2} + 14.588{x_3} - 5.34 \times {10^{ - 3}}{x_4} + \\ \qquad 166.765{x_5} + 6.7 \times {10^{ - 5}}{x_1}{x_2} + 1.542 \times {10^{ - 5}}{x_1}{x_3} + 2.7 \times {10^{ - 7}}{x_1}{x_4} + 7.39\\ \qquad \times {10^{ - 4}}{x_1}{x_5} - 0.04277{x_3}{x_4} + 8.76 \times {10^{ - 6}}{x_2}{x_4} + 0.4902{x_2}{x_5} + 2.67\\ \qquad \times {10^{ - 4}}{x_3}{x_4} - 0.45249{x_3}{x_5} - 4.56 \times {10^{ - 3}}{x_4}{x_5} + 0.0261x_2^2 - 0.01089x_3^2\\ \qquad - 6.03 \times {10^{ - 6}}x_4^2 - 28.837x_5^2. \end{array} \right.$ (3)

4.2 响应面方程的显著性检验

 $F_{R}=\frac{S_{R}^{2} / f_{R}}{S_{E}^{2} / f_{E}}.$

FRFa(fR, fE)时认为响应面方程显著，各参数计算公式如下：

 $S_{R}^{2}=S_{T}^{2}-S_{E}^{2},$
 $S_{T}^{2}=\sum\limits_{i=1}^{N}\left(y_{i}-\overline{y}\right)^{2},$
 $S_{E}^{2}=\sum\limits_{i=1}^{N}\left(\hat{y}_{i}-y_{i}\right)^{2},$
 $f_{R}=2 p+\frac{p(p-1)}{2},$
 $f_{E}=f_{T}-f_{R},$
 $f_{T}=N-1.$

4.3 回归系数的显著性检验

 $S^{2}=S_{E}^{2} / f_{E},$
 $Q_{j}=\frac{\beta_{j}^{2}}{e^{-1}}, j=1, 2, \cdots, p,$
 $e=m_{c}+2 r^{2}.$

 $F_{j}=Q_{j} / S^{2}, j=1, 2, \cdots, p.$

4.4 试验验证案例

 图 4 轴承加速试验装置 Fig. 4 Bearing acceleration test device

 图 5 正常条件下可靠度函数 Fig. 5 Reliability function under normal condition

5 总结与分析

1) 深入分析了真空、温度、预紧力、转速、微重力作用下的失效机理，并将其集成到统一润滑模型中，获得了多机理竞争失效的加速模型数值解，奠定了理论基础.

2) 设计了正交仿真方案，得到了各应力作用下的疲劳和磨损失效规律，预紧力和转速是最为敏感应力，为加速试验方案设计的敏感应力类型及水平选取，提供了机理支撑.

3) 分别建立了膜厚、压力峰值、摩擦系数、最大下表面应力的多应力加速模型，通过试验验证了模型的有效性和准确性，弥补了多应力加速模型的空白.

 [1] 张森, 石军, 王九龙. 卫星在轨失效统计分析[J]. 航天器工程, 2010, 19(4): 41. ZHANG Sen, SHI Jun, WANG Jiulong. Satelliteon-board failure statistics and analysis[J]. Spacecraft Engineering, 2010, 19(4): 41. DOI:10.3969/j.issn.1673-8748.2010.04.007 [2] 黄敦新, 白越, 黎海文, 等. 飞轮轴系润滑剂损失及寿命分析[J]. 润滑与密封, 2009, 34(9): 20. HUANG Dunxin, BAI Yue, LI Haiwen, et al. Analysis of molecular diffusion of lubricants and lubrication life of flywheel shafting[J]. Lubrication Engineering, 2009, 34(9): 20. DOI:10.3969/j.issn.0254-0150.2009.09.005 [3] 陈磊, 梁波, 田林涛. 真空环境中轴承油润滑失效原因分析[J]. 轴承, 2002(8): 31. CHEN Lei, LIANG Bo, TIAN Lintao. Failure analysis on bearing lubricated by oil in vacuum[J]. Bearing, 2002(8): 31. DOI:10.3969/j.issn.1000-3762.2002.08.011 [4] 王福亚.卫星天线双轴驱动机构健康评估与寿命预测方法研究[D].哈尔滨: 哈尔滨工业大学, 2013 WANG Fuya. Health evaluation and life rediction method study of the dual-axis drive mechanism for satellite antenna[D]. Harbin: Harbin Institute of Technology, 2013 http://cdmd.cnki.com.cn/Article/CDMD-10213-1014002951.htm [5] 宁峰平, 姚建涛, 孙锟, 等. 多因素耦合对空间轴承热学特性的影响[J]. 浙江大学学报, 2016, 50(1): 129. NING Fengping, YAO Jiantao, SUN Kun, et al. Effect of multi-factor coupling on thermal properties of space bearing[J]. Journal of Zhejiang University, 2016, 50(1): 129. DOI:10.3785/j.issn.1008-973X.2016.01.019 [6] 赵慧.固体润滑滚动轴承加速寿命试验方法研究[D].重庆: 重庆大学机械工程学院, 2013 ZHAO Hui. Accelerated life testing method research for solid lubrication rolling bearing[D].Chongqing: College of mechanical engineering of Chongqing University, 2013 http://www.wanfangdata.com.cn/details/detail.do?_type=degree&id=D356095 [7] 李俊阳.空间润滑谐波减速器失效机理及其加速寿命试验方法研究[D].重庆: 重庆大学机械工程学院, 2012 LI Junyang. Failure mechanism theory and accelerated life testing method research for space lubrication harmonic drive[D].Chongqing: College of Mechanical Engineering of Chongqing University, 2012 http://cdmd.cnki.com.cn/Article/CDMD-10611-1013007911.htm [8] 陈仁详, 陈思杨, 杨黎霞, 等. 基于振动敏感时频特征的航天轴承寿命状态识别方法[J]. 振动与冲击, 2016, 35(17): 135. CHEN Renxiang, CHEN Siyang, YANG Lixia, et al. Life state recognition method for space bearings based on sensitive time-frequency features of vibration[J]. Journal of Vibration and Shock, 2016, 35(17): 135. [9] 阙子俊, 金晓航, 孙毅. 基于UKF的轴承剩余寿命预测方法研究[J]. 仪器仪表学报, 2016, 37(9): 2037. QUE Zijun, JIN Xiaohang, SUN Yi. Remaining useful life prediction for bearings with the unscented Kalman filter-based approach[J]. Chinese Journal of Scientific Instrument, 2016, 37(9): 2037. [10] 黄志洋, 杨鹤, 彭茜, 等. 空间液体润滑剂的研究进展[J]. 石油商技, 2014(1): 20. HUANG Zhiyang, YANF He, PENG Qian, et al. Research progress of space lubricant[J]. Petroleum Business Technology, 2014(1): 20. DOI:10.3969/j.issn.1006-1479.2014.01.004 [11] JONES W R. Lubrication for Space Applications[J]. Acad.r.p.romne Stud.cerc.mat, 2005, 91. [12] BROWN J R, FORSTER N H. Operating temperatures in the mist lubricated rolling element bearing for gas turbines[R]. AIAA-2000-3027: 1268 https://ieeexplore.ieee.org/document/870940 [13] SATHYAN K, HSU H Y, LEE S H, et al. Long-term lubrication of momentum wheels used in spacecrafts-An overview[J]. Tribology International, 2010, 43: 259. DOI:10.1016/j.triboint.2009.05.033 [14] DAMIENS B, VENNER C H, CANN P M E, et al. Starved lubrication of elliptical EHD contacts[J]. Journal of Tribology, 2004, 126(1): 105. DOI:10.1115/1.1631020 [15] CANN P M E, DAMIENS B, LUBRECHT A A. The transition between fully flooded and starved regimes in EHL[J]. Tribology International, 2004, 37(10): 859. DOI:10.1016/j.triboint.2004.05.005