Fatigue life evaluation of bogie frame based on kernel density extrapolation for stress spectrum
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摘要: 提出了一种基于多样本核密度应力谱外推的疲劳寿命评估方法，研究了核密度估计中的最优带宽与核函数的确定问题，采用灰色关联度分析方法对应力谱的外推优度进行量化评价与检验，并讨论了疲劳寿命评估的相对误差与外推倍数之间的关系；为了验证方法的正确性与可行性，以某转向架构架转臂定位安装座焊缝附近某测点为研究对象，选取了该测点在车轮分别处于镟轮初期、中期和末期时的3组动应力测试数据，进行多样本核密度应力谱外推与疲劳评估。研究结果表明：基于最小渐近均方积分误差的概率密度函数拟合优度良好，所研究的4种核函数类型中，基于Epanechekov核函数外推的相关性最好，相关系数为0.99，较其他3种核函数提高了0.01%~0.12%，基于Circular核函数外推的一致性最好，灰色关联度为0.592 0，较其他3种核函数提高了0.17%~0.32%；基于多样本核密度应力谱外推10倍后的疲劳评估寿命相比基于线性外推的评估寿命减少了1.15%；当应力谱外推至全寿命周期时，基于核密度外推所评估的安全运营里程减少了6.45%。可见，基于核密度应力谱外推的疲劳寿命评估更偏于安全，能保证车辆结构的安全服役。Abstract: A fatigue life evaluation method based on the multi-sample kernel density stress spectrum extrapolation was proposed. The determination of optimal bandwidth and kernel function in the kernel density estimation was studied. The grey correlation analysis method was proposed to quantitatively evaluate and verify the extrapolation optimization of stress spectrum. The relationship between the relative error of fatigue life evaluation and the extrapolation multiple was discussed. To verify the correctness and feasibility of the method, taking a measurement point near the weld of the positioning mounting seat of a bogie frame at the research object, three sets of dynamic stress test data were selected to conduct the multi-sample knernel density stress spectrum extrapolation and fatigue evaluation when the wheel was in the initial, middle, and final stages, respectively. Research results show that the probability density function based on the minimum asymptotic integral mean square error has a goodness of fit. Among the four types of studied kernel functions, the correlation based on the Epanechekov kernel function is the best, and the correlation coefficient is 0.99 and 0.01%-0.12% higher than the coefficients of the other three kernel functions. The consistency based on the Circular kernel function is the best, and the grey correlation degree is 0.592 0 and 0.17%-0.32% higher than the degrees of the other three kernel functions. The assessment fatigue life based on 10-time multi-sample kernel density stress spectrum extrapolation reduces by 1.15% compared with that based on the linear extrapolation. When the stress spectrum is extrapolated to the whole life cycle, the safe operation mileage evaluated based on the kernel density extrapolation reduces by 6.45%. Therefore, the fatigue life evaluation based on the extrapolation of kernel density stress spectrum is safer, and can ensure the safe service of the vehicle structure. 4 tabs, 12 figs, 31 refs.
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Key words:
- railway vehicle /
- bogie /
- fatigue life /
- kernel density /
- stress spectrum /
- extrapolation /
- kernel function /
- bandwidth /
- grey correlation degree
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图 2 应力起始值-应力终止值雨流矩阵
Figure 2. Rain flow matrix of starting and ending values for stress
图 8 应力起始值-应力终止值雨流矩阵
Figure 8. Rain flow matrices of starting and ending values of stress
图 9 综合应力起始值-应力终止值雨流矩阵
Figure 9. Rain flow matrices of starting and ending values for combined stress
图 10 综合应力起始值-应力终止值雨流矩阵(外推10倍后)
Figure 10. Rain flow matrices of starting and ending values for combined stress (after 10 times of extrapolation)
图 12 外推倍数与疲劳评估相对误差关系
Figure 12. Relation between extrapolation multiple and relative error of fatigue life evaluation
表 1 核函数表达式
Table 1. Expressions of kernel functions
序号 核函数类型 函数表达式 1 Circular $K_{1}(x)= \begin{cases}\left(1-x^{2}\right)^{\frac{1}{2}} & x \in[-1, 1] \\ 0 & \text { 其他 }\end{cases} $ 2 Rectangular $K_{2}(x)= \begin{cases}\frac{1}{2} & x \in[-1, 1] \\ 0 & \text { 其他 }\end{cases} $ 3 Triangular $K_{3}(x)= \begin{cases}1-|x| & x \in[-1, 1] \\ 0 & \text { 其他 }\end{cases} $ 4 Epanechekov $ K_{4}(x)= \begin{cases}\frac{2}{\pi}\left(1-x^{2}\right) & x \in[-1, 1] \\ 0 & \text { 其他 }\end{cases}$ 
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表 2 分析结果对比
Table 2. Comparison of analysis results
核函数类型 相关系数 疲劳寿命评估 灰色关联度 线性外推寿命/万公里 核密度外推寿命/万公里 相对误差/
%Circular 0.989 9 304.76 270.76 11.16 0.592 0 Rectangular 0.989 1 304.76 265.96 12.73 0.590 4 Triangular 0.988 8 304.76 264.80 13.11 0.590 1 Epanechekov 0.990 0 304.76 284.20 6.75 0.591 0
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表 3 基于4种应力谱外推方案的疲劳寿命评估结果对比
Table 3. Comparison of fatigue life evaluation results based on four extrapolation schemes of dynamic stress spectrum
方案 初期方案 中期方案 末期方案 组合方案 疲劳寿命/万公里 284.20 195.35 92.98 157.73 相对误差/% 78.10 22.42 -41.73 -1.15
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表 4 各外推倍数下的疲劳寿命评估结果(部分)
Table 4. Fatigue life evaluation results under different extrapolation multiples (partial)
外推倍数 疲劳损伤 疲劳寿命/万公里 相对误差/
%线性外推 核密度外推 线性外推 核密度外推 1 0.000 0.000 159.57 159.57 0.00 10 0.003 0.003 159.57 157.73 1.15 30 0.009 0.010 159.57 157.07 1.57 60 0.019 0.019 159.57 155.91 2.30 100 0.031 0.032 159.57 155.38 2.63 200 0.062 0.064 159.57 154.53 3.16 400 0.125 0.130 159.57 153.32 3.92 600 0.187 0.196 159.57 152.39 4.50 1 000 0.312 0.329 159.57 151.54 5.03 1 500 0.468 0.495 159.57 150.8 5.50 2 000 0.624 0.663 159.57 150.19 5.88 2 500 0.780 0.832 159.57 149.63 6.23 3 000 0.936 1.001 159.57 149.28 6.45
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[1] XIU Rui-xian, SPIRYAGIN M, WU Qing, et al. Fatigue life assessment methods for railway vehicle bogie frames[J]. Engineering Failure Analysis, 2020, 116: 104725. doi: 10.1016/j.engfailanal.2020.104725 [2] BAEK S H, CHO S S, JOO W S. Fatigue life prediction based on the rainflow cycle counting method for the end beam of a freight car bogie[J]. International Journal of Automotive Technology, 2008, 9(1): 95-101. doi: 10.1007/s12239-008-0012-y [3] WANG Qiu-shi, ZHOU Jin-song, GONG Dao, et al. The influence of the motor traction vibration on fatigue life of the bogie frame of the metro vehicle[J]. Shock and Vibration, 2020, 2020: 1-11. [4] YANG Guang-xue, WANG Meng, LI Qiang, et al. Methodology to evaluate fatigue damage of high-speed train welded bogie frames based on on-track dynamic stress test data[J]. Chinese Journal of Mechanical Engineering, 2019, 32(1): 1-8. doi: 10.1186/s10033-018-0313-7 [5] 张建成. 基于核函数的数控车床载荷谱编制关键技术研究[D]. 长春: 吉林大学, 2019.ZHANG Jian-cheng. Research on key technologies for compiling load spectrum of CNC lathe based on kernel function[D]. Changchun: Jilin University, 2019. (in Chinese) [6] 李研. 轮式装载机载荷谱编制中非参数外推方法研究[D]. 长春: 吉林大学, 2016.LI Yan. Research on non-parametric extrapolation method in compiling load spectra of wheel loader[D]. Changchun: Jilin University, 2016. (in Chinese) [7] 刘岩, 张喜逢, 王振雨, 等. 载荷谱外推方法的对比[J]. 现代制造工程, 2011(11): 8-11. doi: 10.3969/j.issn.1671-3133.2011.11.002LI Yan, ZHANG Xi-feng, WANG Zhen-yu, et al. Contrast of extrapolations in compiling load spectrum[J]. Modern Manufacturing Engineering, 2011(11): 8-11. (in Chinese) doi: 10.3969/j.issn.1671-3133.2011.11.002 [8] 杨子涵, 宋正河, 尹宜勇, 等. 基于POT模型的大功率拖拉机传动轴载荷时域外推方法[J]. 农业工程学报, 2019, 35(15): 40-47. doi: 10.11975/j.issn.1002-6819.2019.15.006YANG Zi-han, SONG Zheng-he, YIN Yi-yong, et al. Time domain extrapolation method for load of drive shaft of high-power tractor based on POT model[J]. Transactions of the Chinese Society of Agricultural Engineering, 2019, 35(15): 40-47. (in Chinese) doi: 10.11975/j.issn.1002-6819.2019.15.006 [9] WANG Qiu-shi, ZHOU Jin-song, GONG Dao, et al. Fatigue life assessment method of bogie frame with time-domain extrapolation for dynamic stress based on extreme value theory[J]. Mechanical Systems and Signal Processing, 2021, 159: 107829. doi: 10.1016/j.ymssp.2021.107829 [10] 李凡松, 邬平波, 曾京. 基于核密度估计法的载荷历程外推研究[J]. 铁道学报, 2017, 39(7): 25-31. doi: 10.3969/j.issn.1001-8360.2017.07.004LI Fan-song, WU Ping-bo, ZENG Jing. Extrapolation of load histories based on kernel density method[J]. Journal of the China Railway Society, 2017, 39(7): 25-31. (in Chinese) doi: 10.3969/j.issn.1001-8360.2017.07.004 [11] 熊峻江, 高镇同. 实测载荷谱数据处理系统[J]. 北京航空航天大学学报, 1996, 22(4): 438-441. https://www.cnki.com.cn/Article/CJFDTOTAL-BJHK604.009.htmXIONG Jun-jiang, GAO Zhen-tong. Actural load spectrum data treatment system[J]. Journal of Beijing University of Aeronautics and Astronautics, 1996, 22(4): 438-441. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-BJHK604.009.htm [12] NAGODE M, KLEMENC J, FAJDIGA M. Parametric modelling and scatter prediction of rainflow matrices[J]. International Journal of Fatigue, 2001, 23(6): 525-532. doi: 10.1016/S0142-1123(01)00007-X [13] NAGODE M, FAJDIGA M. An improvedalgorithm for parameter estimation suitable for mixed Weibull distributions[J]. International Journal of Fatigue, 2000, 22(1): 75-80. doi: 10.1016/S0142-1123(99)00112-7 [14] SLIVERMAN B W. On a Gaussian process related to multivariate probability density estimation[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1976, 80(1): 135-144. doi: 10.1017/S0305004100052762 [15] DRESSLER K, HACK M, KRUGER W. Stochastic reconstruction of loading histories from a rainflow matrix[J]. Applied Mathematics and Mechanics, 1997, 77(3): 217-226. [16] WAND M P, JONES M C. Comparison of smoothing parameterizations in bivariate kernel density estimation[J]. Journal of the American Statistical Association, 1993, 88(422): 520-528. doi: 10.1080/01621459.1993.10476303 [17] KHOSROVANEH A K, DOWLING, N E. Fatigue loading history reconstruction based on the rainflow technique[J]. International Journal of Fatigue, 1990, 12(2): 99-106. doi: 10.1016/0142-1123(90)90679-9 [18] RATHANRAJ K J, SRIVIDYA A, VERMA A K, et al. Rescaled range analysis of service load data[J]. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2010, 224(3): 361-367. doi: 10.1243/09544070JAUTO1247 [19] 栾世杰, 于佳伟, 郑松林, 等. 基于非参数核密度估计法的商用车服役载荷环境研究与应用[J]. 机械强度, 2020, 42(2): 443-452. https://www.cnki.com.cn/Article/CJFDTOTAL-JXQD202002028.htmLUAN Shi-jie, YU Jia-wei, ZHENG Song-lin, et al. Research and application of service-loads environment for commercial vehicle based on non-parametric kernel density estimation method[J]. Journal of Mechanical Strength, 2020, 42(2): 443-452. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JXQD202002028.htm [20] 宋清椿, 宋正河, 杜岳峰. 基于非参数雨流外推法的车架载荷谱与疲劳寿命分析[J]. 中国农业大学学报, 2019, 24(2): 154-160. https://www.cnki.com.cn/Article/CJFDTOTAL-NYDX201902017.htmSONG Qing-chun, SONG Zheng-he, DU Yue-feng. Load spectrum and fatigue life analysis of frame based on non-parametric rainflow extrapolation method[J]. Journal of China Agricultural University, 2019, 24(2): 154-160. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-NYDX201902017.htm [21] 陈道云, 孙守光, 李强. 一种新的高速列车动应力谱分布估计方法[J]. 机械工程学报, 2017, 53(8): 109-114. https://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201708016.htmCHEN Dao-yun, SUN Shou-guang, LI Qiang. A new dynamic stress spectrum distribution estimation method of high-speed train[J]. Journal of Mechanical Engineering, 2017, 53(8): 109-114. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201708016.htm [22] WANG Qiu-shi, ZHOU Jin-song, WANG Teng-fei, et al. Extrapolation of the dynamic stress spectrum of train bogie frame based on kernel density estimation method[J]. Fatigue and Fracture of Engineering Materials and Structures, 2021, 44(7): 1783-1798. doi: 10.1111/ffe.13459 [23] WANG Jian-bin, SONG Chun-yuan, WU Ping-bo, et al. Wheel reprofiling interval optimization based on dynamic behavior evolution for high speed trains[J]. Wear, 2016, 366/367: 316-324. doi: 10.1016/j.wear.2016.06.016 [24] SEDEHI O, PAPADIMITRIOU C, TEYMOURI D, et al. Sequential Bayesian estimation of state and input in dynamical systems using output-only measurements[J]. Mechanical Systems and Signal Processing, 2019, 131: 659-688. doi: 10.1016/j.ymssp.2019.06.007 [25] ZHANG Xi-bin, KING M L, HYNDMAN R J. A Bayesian approach to bandwidth selection for multivariate kernel density estimation[J]. Computational Statistics and Data Analysis, 2006, 50(11): 3009-3031. doi: 10.1016/j.csda.2005.06.019 [26] 谢国, 王竹欣, 黑新宏. 面向热轴故障的高速列车轴温阈值预测模型[J]. 交通运输工程学报, 2018, 18(3): 129-137. doi: 10.3969/j.issn.1671-1637.2018.03.014XIE Guo, WANG Zhu-xin, HEI Xin-hong, et al. Axle temperature threshold prediction model of high-speed train for hot axle fault[J]. Journal of Traffic and Transportation Engineering, 2018, 18(3): 129-137. (in Chinese) doi: 10.3969/j.issn.1671-1637.2018.03.014 [27] 张子璠, 李强, 丁然, 等. 转向架构架载荷谱频域校准与建立方法[J]. 交通运输工程学报, 2019, 19(5): 74-83. doi: 10.3969/j.issn.1671-1637.2019.05.009ZHANG Zi-fan, LI Qiang, DING Ran, et al. Frequency domain calibration and establishment method for load spectrum of bogie frame[J]. Journal of Traffic and Transportation Engineering, 2019, 19(5): 74-83. (in Chinese) doi: 10.3969/j.issn.1671-1637.2019.05.009 [28] CICERO S, GARCÍA T, ÁLVAREZ J A, et al. Fatigue behaviour and BS 7608 fatigue classes of steels with thermally cut holes[J]. Journal of Constructional Steel Research, 2017, 128: 74-83. doi: 10.1016/j.jcsr.2016.08.012 [29] 赵方伟, 谢基龙. 小应力循环对C70E型车体疲劳损伤的影响研究[J]. 机械工程学报, 2014, 50(10): 121-126. https://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201410018.htmZHAO Fang-wei, XIE Ji-long. Influence of small stress cycles on the fatigue damage of C70E car body[J]. Journal of Mechanical Engineering, 2014, 50(10): 121-126. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201410018.htm [30] 卢耀辉, 向鹏霖, 曾京, 等. 高速列车转向架构架动应力计算与疲劳全寿命预测[J]. 交通运输工程学报, 2017, 17(1): 62-70. doi: 10.3969/j.issn.1671-1637.2017.01.008LU Yao-hui, XIANG Peng-lin, ZENG Jing, et al. Dynamic stress calculation and fatigue whole life prediction of bogie frame for high-speed train[J]. Journal of Traffic and Transportation Engineering, 2017, 17(1): 62-70. (in Chinese) doi: 10.3969/j.issn.1671-1637.2017.01.008 [31] 李玉宁. 基于灰色关联分析的仿真模型验证及辅助工具研究[D]. 哈尔滨: 哈尔滨工业大学, 2015.LIU Yu-ning. Research on simulation model validation methods and assistant tool based on grey relational analysis[D]. Harbin: Harbin Institute of Technology, 2015. (in Chinese) -
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