Extreme Value Inference of Axle Stress Spectrum for Intercity EMU
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摘要:为评估车轴的可靠性或扩展车轴的应力谱，需要预判车轴全寿命期内的最大应力. 首先，综合考虑推断精度和统计效率等因素，基于Sturges公式选取0.5 MPa为组距对实测车轴应力-时间历程进行雨流计数，得到车轴应力的经验累积分布函数. 其次，针对不同的门槛限值计算应力的经验均值超限函数，通过确定经验均值超限函数的拐点确定合理门槛限值. 然后，基于Pickands-Balkema-de Haan定理，利用广义帕累托分布拟合车轴应力的超限分布函数，并通过拟合分布外推车轴全寿命期内的最大应力. 最后，利用实测车轴应力数据进行了验证. 研究结果表明：为评估车轴寿命期内的最大应力，大约需要不少于 3000 km 的线路测试；基于13.5 t轴重实测数据推断的应力极值比基于14.1 t轴重计算的校核应力高出13%，因此完全按照设计轴重满载运营，需格外谨慎.Abstract:In order to evaluate the reliability and extend stress spectra of axles, the maximum stress during its full life need to be predicted. First, to obtain the empirical cumulative distribution function of axle stress, the bin width of 0.5 MPa is chosen, which is based on the Sturges formula as a tradeoff between the inferred accuracy and statistical efficiency, and the rain flow counting is performed against the measured axial stress-time history. Secondly, the empirical mean excess function is calculated according to different threshold values and the reasonable threshold is determined by identifying the turning point of the mean excess function. Then, based on Pickands-Balkema-de Haan theorem, the excess distribution function of the axle stress is fitted with the generalized Pareto distribution, and the maximum stress during the full life is inferred with the fitted distribution. Finally, this method is verified with the measured data. The results show that in order to assess the maximum axle stress during its full life, the tests of 3000 km are required at least. The inferred maximum stress with the measured data of 13.5 t wheel load is about 13% higher than that with the one of 14.1 t wheel load. Therefore it should be particularly cautious to operate at axle design weight.
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Key words:
- axle/
- load spectrum analysis/
- extreme value inference/
- reliability theory
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图 3不同分布概率密度市函数与累积分布函数的拟合效果
Figure 3.Fitted probability density function and cumulative distribution function with different distributions
表 1不同经验公式的组距计算结果
Table 1.Calculated results by different empirical formulas
公式名 表达式 h/MPa Sturges $k = {\rm{ent} }({\log _2}{n_{\rm{t}}}) + 1$ 0.50 square-root $k = {\rm{ent} }({\sqrt n _{\rm{t}}})$ 0.30 Scott $h = 3.5{\sigma _S} n_{\rm{t} }^{ - 1/3}$ 0.04 Freedman-Diaconis $h = 2{Q_S} n_{\rm{t} }^{ - 1/3}$ 0.03 
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表 2截面B的分布拟合结果
Table 2.Fitted results of sectionB
分布 位置参数 尺度参数 形状参数 $\chi _{{\rm{LN}}}^2/{\chi ^2} $ smax/MPa 正态 36.9 1.10 0.98 42.2 对数正态 3.6 0.03 1.00 42.6 两参数威布尔 37.20 36.3 0.52 39.9 三参数威布尔 30.0 7.20 6.9 0.66 40.5 下载:
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表 3各截面的实测最大值与应力的极值推断结果
Table 3.Measured maximum values and extreme value inferences of each section
截面 实测最大值/MPa $s_{\max }^*$/MPa $\hat \xi $ A 44 52 0.086 B 49 53 0.023 C 75 98 0.002 D 63 67 −0.350 E 75 78 −0.460 F 48 52 −0.140 G 43 47 −0.110 下载:
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表 4
$s_{\max }^*$ 与校核应力的对比Table 4.Comparison between
$s_{\max }^*$ and allowable stressesMPa 截面 $s_{\max }^* $ 轴重/t 17.0 14.1 13.5 A 52 60 50 48 B 53 56 47 45 C 98 103 86 83 D 67 94 80 77 E 78 103 86 83 F 52 56 47 45 G 47 60 50 48 下载:
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