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Article(id=1228654097906205359, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228654089437901468, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.12.014, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1668873600000, receivedDateStr=2022-11-20, revisedDate=1677600000000, revisedDateStr=2023-03-01, acceptedDate=null, acceptedDateStr=null, onlineDate=1770863564230, onlineDateStr=2026-02-12, pubDate=1735315200000, pubDateStr=2024-12-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770863564230, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770863564230, creator=13701087609, updateTime=1770863564230, updator=13701087609, issue=Issue{id=1228654089437901468, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='12', pageStart='1993', pageEnd='2167', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770863562211, creator=13701087609, updateTime=1770863940325, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228655675413299456, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228654089437901468, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228655675413299457, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228654089437901468, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=2124, endPage=2131, ext={EN=ArticleExt(id=1228654098204000951, articleId=1228654097906205359, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Double Gamma distribution model for the probability density of rainflow-range of broadband random vibration stress, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

A double Gamma distribution model to determine the probability density function （PDF） of the time domain rainflow-range corresponding to the broadband random stress power spectral density （PSD） is proposed，and a neural network method is used to implement the parameter prediction of the model. A series of stress PSDs are given，and the corresponding stress time histories are generated using the time-domain randomization method. The number of rainflow-range is counted for the stress time histories using the rainflow counting method，and the stress rainflow-range probability density values are calculated. Based on the calculation results of each stress PSD mentioned above，the proposed stress rainflow-range probability density double Gamma distribution model is parametrically fitted to obtain a set of corresponding model parameters. The results of the double Gamma distribution model are compared with the Dirlik method and fatigue life prediction is carried out，and the results show that the proposed double Gamma distribution model is more accurate for determining the broadband random stress rainflow-range PDF.

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提出了一种确定宽带随机振动应力谱密度（power spectral density，PSD）对应的时域雨流变程概率密度函数（probability density function，PDF）的双Gamma分布模型，并采用神经网络方法实现了模型的参数预测。给定一系列应力PSD，利用时域随机化方法生成对应的应力时间历程。运用雨流计数法统计应力时间历程的雨流变程数，计算出应力变程概率密度值。根据上述每一个应力PSD的计算结果，对所提出的应力雨流变程概率密度双Gamma分布模型进行参数拟合，得到一组对应的模型参数。利用所得数据进行神经网络模型训练，实现由给定的应力PSD直接预测出所对应的时域雨流变程PDF。将双Gamma分布模型结果与Dirlik方法结果进行了对比，并进行了疲劳寿命预测，结果表明，提出的双Gamma分布模型对宽带随机振动应力雨流变程PDF的确定更为准确。

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陈怀海（1965―），男，博士，教授。E-mail： 。
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王杰（1996—），男，博士研究生。E-mail： 。

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王杰（1996—），男，博士研究生。E-mail： 。

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王杰（1996—），男，博士研究生。E-mail： 。

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articleId=1228654097906205359, language=EN, label=Tab.1, caption=

Parameter fitting results of double Gamma distribution model

, figureFileSmall=null, figureFileBig=null, tableContent=
α1λ1α2λ2c
f1=6 Hz，f2=89 Hz0.68321.08574.85755.55510.4418
f1=27 Hz，f2=194 Hz1.08531.59874.40418.69290.7121
f1=49 Hz，f2=138 Hz1.03552.79208.05265.14630.6793
), ArticleFig(id=1228654128591733550, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=CN, label=表1, caption=

双Gamma分布模型参数拟合结果

, figureFileSmall=null, figureFileBig=null, tableContent=
α1λ1α2λ2c
f1=6 Hz，f2=89 Hz0.68321.08574.85755.55510.4418
f1=27 Hz，f2=194 Hz1.08531.59874.40418.69290.7121
f1=49 Hz，f2=138 Hz1.03552.79208.05265.14630.6793
), ArticleFig(id=1228654128671425330, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=EN, label=Tab.2, caption=

Spectrum moments of PSD

, figureFileSmall=null, figureFileBig=null, tableContent=
m0m1m2m3m4m5m6
0.499831.51762.7951×1033.2802×1054.6661×1077.5509×1091.3512×1012
), ArticleFig(id=1228654128755311413, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=CN, label=表2, caption=

PSD谱矩

, figureFileSmall=null, figureFileBig=null, tableContent=
m0m1m2m3m4m5m6
0.499831.51762.7951×1033.2802×1054.6661×1077.5509×1091.3512×1012
), ArticleFig(id=1228654128843391802, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=EN, label=Tab.3, caption=

Spectrum parameters of PSD

, figureFileSmall=null, figureFileBig=null, tableContent=
m1/m0m2/m1m3/m2m4/m3m5/m4m6/m5
63.065888.6852117.3547142.2500161.8240178.9395
), ArticleFig(id=1228654128939860798, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=CN, label=表3, caption=

PSD谱参数

, figureFileSmall=null, figureFileBig=null, tableContent=
m1/m0m2/m1m3/m2m4/m3m5/m4m6/m5
63.065888.6852117.3547142.2500161.8240178.9395
), ArticleFig(id=1228654129023746883, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=EN, label=Tab.4, caption=

Stress PSD parameters of the test groups

, figureFileSmall=null, figureFileBig=null, tableContent=
组别f1/Hzf2/Hz组别f1/Hzf2/Hz
11682517162
29178618196
314192735160
427152818164
), ArticleFig(id=1228654129103438661, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=CN, label=表4, caption=

检验组的应力PSD参数

, figureFileSmall=null, figureFileBig=null, tableContent=
组别f1/Hzf2/Hz组别f1/Hzf2/Hz
11682517162
29178618196
314192735160
427152818164
), ArticleFig(id=1228654129178936138, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=EN, label=Tab.5, caption=

S-N curve parameters of spring steel

, figureFileSmall=null, figureFileBig=null, tableContent=
Ck
1.413×103711.7
), ArticleFig(id=1228654129258627915, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=CN, label=表5, caption=

弹簧钢的S-N曲线参数

, figureFileSmall=null, figureFileBig=null, tableContent=
Ck
1.413×103711.7
), ArticleFig(id=1228654129338319696, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=EN, label=Tab.6, caption=

Calculation results of fatigue life

, figureFileSmall=null, figureFileBig=null, tableContent=
组别雨流计数法Dirlik方法本文方法
寿命L/s寿命L/s相对误差/%寿命L/s相对误差/%
11521940.438.17104231.49
2847.6594.829.82787.17.13
3366.1248.332.17351.04.12
4131.797.8425.70102.622.09
5315.1147.153.31275.712.50
6211.6123.741.54153.827.31
792.3267.6726.7082.9110.19
8166.3119.528.14134.619.06
), ArticleFig(id=1228654129422205779, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228654097906205359, language=CN, label=表6, caption=

疲劳寿命计算结果

, figureFileSmall=null, figureFileBig=null, tableContent=
组别雨流计数法Dirlik方法本文方法
寿命L/s寿命L/s相对误差/%寿命L/s相对误差/%
11521940.438.17104231.49
2847.6594.829.82787.17.13
3366.1248.332.17351.04.12
4131.797.8425.70102.622.09
5315.1147.153.31275.712.50
6211.6123.741.54153.827.31
792.3267.6726.7082.9110.19
8166.3119.528.14134.619.06
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宽带随机振动应力雨流变程概率密度双Gamma分布模型
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王杰 , 陈怀海 , 郑荣慧
振动工程学报 | 2024,37(12): 2124-2131
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振动工程学报 | 2024, 37(12): 2124-2131
宽带随机振动应力雨流变程概率密度双Gamma分布模型
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王杰 , 陈怀海 , 郑荣慧
作者信息
  • 南京航空航天大学航空航天结构力学及控制全国重点实验室，江苏　南京　210016

    • 王杰（1996—），男，博士研究生。E-mail： 。

通讯作者:

陈怀海（1965―），男，博士，教授。E-mail： 。
Double Gamma distribution model for the probability density of rainflow-range of broadband random vibration stress
Jie WANG , Huai-hai CHEN , Rong-hui ZHENG
Affiliations
  • State Key Laboratory of Mechanics and Control for Aerospace Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，China
出版时间: 2024-12-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.12.014
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摘要
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提出了一种确定宽带随机振动应力谱密度（power spectral density，PSD）对应的时域雨流变程概率密度函数（probability density function，PDF）的双Gamma分布模型，并采用神经网络方法实现了模型的参数预测。给定一系列应力PSD，利用时域随机化方法生成对应的应力时间历程。运用雨流计数法统计应力时间历程的雨流变程数，计算出应力变程概率密度值。根据上述每一个应力PSD的计算结果，对所提出的应力雨流变程概率密度双Gamma分布模型进行参数拟合，得到一组对应的模型参数。利用所得数据进行神经网络模型训练，实现由给定的应力PSD直接预测出所对应的时域雨流变程PDF。将双Gamma分布模型结果与Dirlik方法结果进行了对比，并进行了疲劳寿命预测，结果表明，提出的双Gamma分布模型对宽带随机振动应力雨流变程PDF的确定更为准确。

随机振动  /  疲劳寿命  /  应力概率密度  /  雨流变程  /  神经网络

A double Gamma distribution model to determine the probability density function （PDF） of the time domain rainflow-range corresponding to the broadband random stress power spectral density （PSD） is proposed，and a neural network method is used to implement the parameter prediction of the model. A series of stress PSDs are given，and the corresponding stress time histories are generated using the time-domain randomization method. The number of rainflow-range is counted for the stress time histories using the rainflow counting method，and the stress rainflow-range probability density values are calculated. Based on the calculation results of each stress PSD mentioned above，the proposed stress rainflow-range probability density double Gamma distribution model is parametrically fitted to obtain a set of corresponding model parameters. The results of the double Gamma distribution model are compared with the Dirlik method and fatigue life prediction is carried out，and the results show that the proposed double Gamma distribution model is more accurate for determining the broadband random stress rainflow-range PDF.

random vibration  /  fatigue life  /  stress probability density  /  rainflow-range  /  neural network
王杰, 陈怀海, 郑荣慧. 宽带随机振动应力雨流变程概率密度双Gamma分布模型. 振动工程学报, 2024 , 37 (12) : 2124 -2131 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.12.014
Jie WANG, Huai-hai CHEN, Rong-hui ZHENG. Double Gamma distribution model for the probability density of rainflow-range of broadband random vibration stress[J]. Journal of Vibration Engineering, 2024 , 37 (12) : 2124 -2131 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.12.014
正文
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引言
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在航空航天、车辆工程、船舶工业等领域内，机载结构设备在实际运行中都会承受来自发动机自身和外部环境等产生的振动载荷，其中有很大一部分为宽带随机振动［1⁃2］。宽带随机振动会使结构在某些位置产生疲劳损伤，这是导致结构破坏失效的主要原因，关系到设备的安全性、耐久性和经济性等［3⁃4］。因此，研究结构在宽带随机振动应力下的振动疲劳问题，并进行损伤计算和寿命预估，具有非常重要的意义［5⁃6］。
疲劳分析方法有基于应力、应变、断裂力学、能量等的多种方法，本文主要研究适用于高周疲劳分析的基于应力的方法。基于应力的方法又可分为时域和频域两种，它们分别基于应力时间历程进行计数统计和基于功率谱密度函数进行计算。时域方法通过直接对结构危险点处的应力时间历程数据采用各种循环计数方法［7］（如雨流计数法、幅值穿越法、区间计数等），统计每个应力变程下的循环次数，计算其对应的损伤，之后采用损伤累积方法计算结构总的疲劳损伤，预估疲劳寿命。频域方法则是直接根据结构危险点处的应力响应PSD函数，得到应力的雨流幅值概率密度函数，再进行损伤计算和疲劳寿命预测。
在工程结构设计中，为描述结构在复杂振动环境下的随机振动应力，采用频域表达更为方便和直接。频域信号在理论上是对所有可能的时域信号的统计，能反映信号的总体特征。另外，采用频域分析方法相较于采用时域分析方法，有数据量少、计算速度快等优点，因此频域分析方法已经成为结构振动疲劳寿命分析的主流方法之一［8⁃12］。
采用频域方法进行结构疲劳损伤计算的关键是如何根据给定的应力PSD确定应力雨流幅值PDF。在窄带情形下，应力雨流幅值的PDF与应力幅值的PDF是相同的，BENDAT等［13］从理论上推导得到了窄带随机振动应力的应力幅值PDF为瑞利分布函数。然而实际结构所处的随机振动环境大多是宽带的，因此WIRSCHING等［14］、BENASCIUTTI等［15］、ORITZ等［16］、ZHAO等［17］进一步提出了多种能适用于宽带的修正计算方法，但这些方法在估算寿命时大多是偏向保守的［18⁃19］。DIRLIK［20］通过大量的模拟和计算，提出了一个描述宽带随机振动应力雨流变程PDF的封闭经验公式。BISHOP［21］进一步验证了Dirlik方法的正确性。如今，Dirlik方法凭借其对寿命计算的有效性，成为了频域中基于PSD计算疲劳损伤和寿命的首选算法，被各类商业计算软件广泛应用。
随着计算机技术的发展，计算能力不断提高，有研究者提出使用神经网络方法建立宽带疲劳的损伤计算与寿命预测模型。LI等［22］研究了海洋平台的悬链线式系泊系统的宽带疲劳问题，提出了基于神经网络的宽带损伤预测模型，发现谱矩法是预测宽带疲劳损伤的更合适的方法。SUN等［23］将多种宽带PSD谱型的谱参数和对应的损伤系数作为神经网络的输入和输出，训练得到了一种预测宽带随机疲劳寿命的人工神经网络模型。
本文通过研究分析时域内宽带应力随机过程的雨流变程PDF统计结果，结合Gamma分布特征，提出了一个双Gamma分布PDF模型，并给出了确定该模型参数的拟合方法。然后通过双Gamma分布模型参数和相应PSD谱参数之间的对应关系，使用神经网络进行训练学习，建立了一个由频域PSD到其对应的幅值域PDF模型的预测网络。这样就可以根据给定的宽带随机振动应力PSD，直接预测得到进行损伤计算和寿命预测所需要的应力雨流变程PDF函数。最后通过算例验证本文提出的双Gamma分布模型的准确性。
1　频域疲劳寿命分析方法
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本文研究的随机振动应力仅限于平稳高斯随机过程，将其用表示。在频域中，随机过程的PSD表示为，为单边谱。的第阶谱矩定义为［9］：
其中，频率的单位为Hz。
随机过程的方差可表示为。谱宽参数表示为［9］：
峰值频率和正向穿零频率分别定义为［12］：
随机过程的不规则因子定义为［12］：
当时，表示该随机过程为窄带随机过程，而宽带随机过程的不规则因子。
为了方便计算结构的随机疲劳寿命，频域方法主要关注单位时间内结构产生的疲劳损伤。设材料的S⁃N曲线为：
式中　 k和C为材料参数；S为应力幅值；N为S对应的循环次数即寿命。
由于变幅值应力载荷可以用一系列的载荷块模拟，根据Miner损伤累积准则，疲劳损伤可定义为：
式中　 为应力幅值的循环次数；为应力幅值对应的寿命，可由式（5）中的S⁃N曲线计算得到。
假设随机过程中峰值出现的总次数等于总的循环次数，那么，应力幅值的概率为［9］：
因此，应力幅值的PDF为。那么，式（6）中的疲劳损伤可写为：
结合式（5）中S⁃N曲线的表达，疲劳损伤可写为：
在单位时间内，峰值出现次数的含义为峰值频率，即。如果用一个连续函数表示应力幅值的PDF，那么单位时间内的疲劳损伤可写为［9］：
式中　 为应力雨流幅值PDF，可见的确定是频域随机疲劳分析的关键所在。对于窄带随机过程，峰所对应的循环数与雨流计数法提取的循环数是相同的，前人已研究证实此时为一瑞利函数［13］；而对于宽带过程，应力在低频波形上可能会叠加高频波动，此时应力雨流变程数将多于峰数，峰所对应的循环数与雨流计数法提取的循环数是不同的，这种情况下则必须采用应力雨流幅值进行损伤计算，但到目前为止尚不能从理论上推导得到的解析表达式。
2　宽带随机振动应力PSD模拟
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本节是本文所提方法使用过程的示例，实际应用时需根据具体情况对相关参数进行适当调整，以生成相应的数据集。
结构的宽带随机振动应力响应PSD通常情况下是具有多个峰值的，因为当结构的某几阶模态的固有频率位于随机激励的频带范围内时，结构在危险点处的应力响应主要是由这几阶模态的应力响应叠加而成的，因此在仿真模拟中不能简单用几个矩形谱来表示结构的宽带应力PSD。由于应力属于位移型量，在工程实际中，结构的高阶模态对结构的应力响应贡献不大，总的应力响应主要是由结构的低阶模态贡献的。通常，工程上结构应力的带宽在200 Hz左右，因此本文将分析频带范围设置为。
为模拟宽带应力，本文给出一种双峰傅氏幅值谱：
式中　 ，；，，其中和分别为两个峰值所在的位置，设，；为阻尼比，。
运用周期图法，将其转化为应力PSD：
式中　 的单位为MPa，；为频率分辨率，设。
公式（11）中的阻尼比是按照工程实际给定的具有代表性的值，具体使用时要根据实际情况设定。对于式（11）和（12）描述的应力PSD，采用式（4）可计算得到对应的不规则因子在0.2026~0.6194之间，也即宽带随机过程。改变式（11）中和的值，便可得到多个应力PSD。图1所示为三个不同参数的应力PSD。
3　宽带随机振动应力雨流变程PDF模型
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3.1　Dirlik模型
DIRLIK［20］用一个指数分布和两个瑞利分布来近似宽带随机振动应力的雨流变程PDF（变程是幅值的2倍），他提出的宽带随机过程雨流变程PDF的公式为：
式中　 为规范化的雨流变程，其余参数分别为：
其中：
3.2　双Gamma分布模型
当变量X服从参数的Gamma分布时，可表示为：
其概率密度函数为：
其中，为Gamma函数，其表达式为：
式中　 和分别为Gamma分布的形状参数和逆尺度参数。控制Gamma分布的PDF形状的变化趋势，只有当时，PDF为单峰函数，否则为递减函数；控制Gamma分布的PDF的峰值大小。当时，Gamma分布为参数为的指数分布，；当，时，Gamma分布为自由度为的分布，。图2所示为不同参数的Gamma分布的PDF曲线。
为了研究宽带随机过程的应力雨流变程PDF，将图1中的应力PSD通过时域随机化方法［24］转化为时域应力数据，并对其进行雨流计数得到应力雨流变程PDF，如图3所示。
分析式（13）可知，Dirlik模型是指数分布与瑞利分布的拼接，Gamma分布则可以通过参数的调整模拟这两种分布。结合图3的曲线变化趋势以及图2中Gamma分布的概率密度受参数影响的变化趋势，且为了适应谱参数的大范围变化，考虑用两个具有不同参数Gamma分布的概率密度函数拟合宽带随机过程的雨流变程PDF，表示如下：
式中　 为该双Gamma分布模型的5个参数，可通过对应力雨流变程PDF结果进行参数拟合得到，本文用到的优化表达式如下：
式中　 为雨流计数得到的PDF序列值；为式（17）计算得到的序列值。
通过对赋初值，使用非线性规划算法使得式（17）取最小值，得到的5个参数的最优解即为双Gamma分布模型的参数，代入式（17）即可确定应力PSD的雨流变程PDF。
例如，对图1中三个应力PSD分别进行双Gamma分布模型参数拟合，得到结果如表1所示。图4中给出了本文双Gamma分布模型参数拟合与雨流计数法及Dirlik方法的结果对比。
进一步，将图1所示的应力PSD分别使用雨流计数法、Dirlik方法计算以及双Gamma分布模型参数拟合，得到应力雨流变程PDF，如图4所示。
3.3　对比分析
由图4的三种PDF结果可知，本文提出的双Gamma分布模型可对应力雨流变程PDF进行准确的拟合，比Dirlik方法与雨流计数结果更吻合。并且从图4中可以看出，Dirlik方法的结果不太稳定，在概率密度峰值附近常大于雨流计数结果，这会使其计算的损伤偏大，对宽带疲劳寿命的预测偏于保守。
4　神经网络模型的建立
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对于3.2节中提出的双Gamma分布模型，在已知应力PSD的情况下，确定该模型的参数以及建立谱参数与双Gamma分布模型参数的关系流程如图5所示。对于一系列给定的应力PSD，一方面计算其各阶谱矩，得到共6个谱参数，另一方面可以通过对其雨流变程PDF进行拟合，得到5个模型参数。分别将这两组参数作为神经网络训练的输入项和输出项。各阶谱矩数值的数量级相差较大（表2所示为， Hz时的应力PSD谱矩），由于输入数据之间的数量级差别过大可能会导致神经网络不收敛，以及基于量纲一致考虑，在实际建立神经网络时，本文提出将各阶相邻谱矩的比值作为输入参数（表3所示为对应的PSD谱参数），这样处理之后可使得各谱参数保持在相当的数量级上，便于训练神经网络模型。
本文神经网络模型使用的是贝叶斯正则化神经网络［25］，如图6所示，其中为神经网络的权重函数。
根据式（11）和（12），改变和的值，可以产生多组应力PSD。例如，令从5 Hz开始，每次增大1 Hz，直到55 Hz，共51组值；类似地，可从60 Hz增大到200 Hz，共141组值。通过和的不同随机组合，共可得到7191个应力PSD。在这些应力PSD中随机挑选出8组不参与神经网络训练，而作为检验组来验证最终的网络训练结果，将其余应力PSD按照图5中的流程处理，生成输入数据和输出数据，训练神经网络，在一台普通PC上训练总共耗时13 s，训练结束之后便得到了可预测宽带随机振动应力雨流变程PDF的神经网络模型。
5　算例
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5.1　问题描述
第4节中在训练神经网络时随机挑选出来的检验组的应力PSD参数如表4所示，对这8组宽带应力PSD用本文训练得到的神经网络模型预测其应力雨流变程PDF，并预估弹簧钢结构（S-N曲线参数如表5所示）在该宽带应力下的寿命。
5.2　结果
以第8组为例，其PSD参数为，时，雨流计数PDF结果与本文神经网络预测PDF结果以及使用Dirlik方法确定的PDF结果的对比如图7所示。
为了研究神经网络预测PDF结果与雨流计数PDF结果的一致程度，定义相关系数为：
式中　 X和Y分别为两种不同方法得到的PDF结果；和分别表示均值和方差。
分别计算每组PSD使用本文方法和雨流计数结果的相关系数，以及使用Dirlik方法和雨流计数结果的相关系数，结果如图8所示。
使用雨流计数法计算的参考寿命、使用Dirlik方法预测的寿命以及使用本文方法预测的寿命结果如表6所示，对比结果如图9所示。总体来看，Dirlik方法和本文方法得到的寿命预测结果都是偏保守的。但相较于使用Dirlik方法计算得到的寿命，使用本文提出的双Gamma分布模型计算得到的寿命更接近于雨流计数得到的参考寿命。Dirlik方法寿命计算结果的相对误差最高为53.31%，最低为25.70%；而双Gamma分布模型的寿命计算结果的相对误差最高为31.49%，最低为4.12%。
6　结论
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频域疲劳分析方法是当前广泛使用的一种结构疲劳寿命分析方法，尤其在结构抗疲劳设计过程中具有极为重要的使用价值。本文针对宽带随机振动应力下结构的疲劳寿命计算模型进行研究，重点分析了宽带随机振动应力雨流变程PDF模型的建立，提出了双Gamma分布模型，并且通过训练神经网络建立了从谱参数到双Gamma分布模型参数的预测神经网络模型。所得结论如下：
（1）提出的双Gamma分布参数预测的神经网络模型对宽带随机振动应力的雨流变程PDF预测结果的精度高于Dirlik方法，并且在对不同参数的应力PSD进行预测时的结果稳定性更强，与时域雨流统计结果的相关系数均稳定保持在0.9~1之间。
（2）使用本文方法预测得到的PDF用来进行频域寿命预测时，寿命与雨流计数法结果之间的相对误差最低只有4.12%，相较于Dirlik方法，使用本文提出的方法进行寿命预测的结果更接近雨流计数结果。因此，本文提出的预测模型在进行宽带随机振动应力疲劳寿命预测时可得到更为准确的结果。
（3）Dirlik方法在实际应用中较为方便，但该方法只能用于宽带高斯随机过程。本文提出的双Gamma分布神经网络模型在实际应用中相对复杂一些，但由于本文模型是根据PSD的前六阶矩建立的，因此本文的方法或可以进一步应用到非高斯、非平稳等更为复杂的随机疲劳分析中。
（4）用本文7000多个样本数据训练神经网络时的耗时较短，因此本文的神经网络模型有望拓展至更大级别数据量的训练中。
基金
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  • 国家自然科学基金资助项目(12202187)
  • 江苏省卓越博士后项目
文献
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2024年第37卷第12期
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文章信息
doi: 10.16385/j.cnki.issn.1004-4523.2024.12.014
  • 接收时间：2022-11-20
  • 首发时间：2026-02-12
  • 出版时间：2024-12-28
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  • 收稿日期：2022-11-20
  • 修回日期：2023-03-01
基金
国家自然科学基金资助项目(12202187)
江苏省卓越博士后项目
作者信息
    南京航空航天大学航空航天结构力学及控制全国重点实验室，江苏　南京　210016

通讯作者:

陈怀海（1965―），男，博士，教授。E-mail： 。
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