工程车辆车桥位移谱统计分布建模及分步参数识别
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摘要
针对非公路用车的车桥实测位移谱统计分布建模中模型选择、参数识别的初值选取主观性大和计算效率低等难题,该文以实测的车桥位移信号为研究对象,分别进行时域分析、频域功率谱分析,对信号进行分组,统计频数,获得统计直方图和累计概率分布曲线。分别采用正态分布、双峰正态分布、威布尔分布和双峰威布尔分布模型对位移谱进行建模,提出分步参数识别方法。引入灰色关联度目标函数,以人工鱼群算法获得的参数作为模型参数的初始值,采用迭代非线性最小二乘法levenberg-marquardt (LM)算法进行精确参数识别,使用相关系数和kolmogorov-smirnov(KS)检验对各模型的拟合优度进行比较。结果表明,混合威布尔分布与统计直方图的相关系数为(0.9800,0.9908,0.9867,0.9665),混合正态分布为(0.9793,0.9904,0.9783,0.9661),威布尔模型为(0.8613,0.9113,0.8618,0.8854),正态模型为(0.8611,0.9127,0.8624,0.8869),混合威布尔模型可以对车桥位移谱进行高精度拟合,而所提出的分步参数识别法可以高效、准确地进行模型的参数识别。研究结果可为车辆疲劳载荷谱的编制和台架试验提供参考。
The study of the statistical distribution is the basis for further loading spectrum and fatigue reliability platform test. Normal distribution and weibull distribution are 2 kinds of probability statistical distribution models widely used in reliability engineering. The idea of weighted superposition is used to approximate the actual distribution by so-called mixed model, and it has a strong practical application value, so it has been paid increasing attention by many scholars. The introduction of mixed distribution model brings many challenges in model parameter identification. Finding a simple, efficient and accurate mixed distribution model parameter estimation method has become a focus in the field of reliability research. The traditional reliability model parameter identification methods include graphic method, nonlinear least square method, maximum likelihood estimation and bias estimation, and so on. The main disadvantages of these algorithms are as follows:(1) The calculation efficiency needs to be improved, and the traditional algorithms mostly rely on iterative solution. Requirement to improve the accuracy of parameter estimation distinct increases the time cost.(2) The selection of parameter identification and optimization targets is improper. Most of the existing studies have defined the objective function of parameter identification as the square sum of the model and the measured data, which inevitably ignores the simulation error of the transverse coordinates between the sample points and the simulation points, that only considers the simulation error of the ordinate.(3) The empirical dependence of parameter identification is high, and the initial value of parameter identification has a great influence on the results. However, the intelligent algorithm shows great potential in the problem of parameter identification of the model with multidimensional nonlinearity and uneasy initial value. In view of this, the measured vehicle bridge displacement signal was taken as the research object in this paper, the time domain analysis and frequency domain power spectrum analysis were carried out respectively. In order to further study the statistical law of the displacement signals, the signal was grouped and the frequency was counted, the statistical histogram and the cumulative probability distribution curve were obtained. The normal distribution, mixed normal distribution, weibull distribution and mixed weibull distribution were employed respectively. A novel two-step parameter identification method was proposed, and the grey correlation degree objective function was introduced. The grey correlation coefficient objective function could ensure the maximum geometric similarity between the fitting curve and the original curve. By doing this, the inherent malpractice of the optimization process with the square sum of error as the fitness was overcome to some extent. The proposed parameter estimation method's tep was as following: Firstly, the parameters obtained by the artificial fish swarm algorithm were applied as the initial values of the model parameters. Secondly, the iterative nonlinear least square method, namely, levenberg-marquardt(LM) algorithm was used to identify the parameters accurately. Thirdly, the goodness of fit for each model were calculated by using the kolmogorov-smirnov test index and correlation coefficient. The result showed that the mixed weibull model could be used to describe the tested displacement signal best. The correlation coefficient between the mixed Weibull distribution and the statistical histogram was(0.9800, 0.9908,0.9867,0.9665), whereas, the mixed normal distribution was(0.9793,0.9904,0.9783,0.9661), the weibull model was(0.8613,0.9113,0.8618,0.8854), and the normal model was(0.8611,0.9127,0.8624,0.8869). The proposed two-step parameter identification method combined the advantages of the artificial fish swarm optimization algorithm and the traditional iterative algorithm, and used the artificial fish swarm optimization result as the initial value of the LM algorithm. It solved the problem of the difficulty in selecting the initial value of the nonlinear least square method and improved the efficiency of the parameter identification. This study can provide reference for the fatigue load spectrum and the bench test of off-road vehicles.
The study of the statistical distribution is the basis for further loading spectrum and fatigue reliability platform test. Normal distribution and weibull distribution are 2 kinds of probability statistical distribution models widely used in reliability engineering. The idea of weighted superposition is used to approximate the actual distribution by so-called mixed model, and it has a strong practical application value, so it has been paid increasing attention by many scholars. The introduction of mixed distribution model brings many challenges in model parameter identification. Finding a simple, efficient and accurate mixed distribution model parameter estimation method has become a focus in the field of reliability research. The traditional reliability model parameter identification methods include graphic method, nonlinear least square method, maximum likelihood estimation and bias estimation, and so on. The main disadvantages of these algorithms are as follows:(1) The calculation efficiency needs to be improved, and the traditional algorithms mostly rely on iterative solution. Requirement to improve the accuracy of parameter estimation distinct increases the time cost.(2) The selection of parameter identification and optimization targets is improper. Most of the existing studies have defined the objective function of parameter identification as the square sum of the model and the measured data, which inevitably ignores the simulation error of the transverse coordinates between the sample points and the simulation points, that only considers the simulation error of the ordinate.(3) The empirical dependence of parameter identification is high, and the initial value of parameter identification has a great influence on the results. However, the intelligent algorithm shows great potential in the problem of parameter identification of the model with multidimensional nonlinearity and uneasy initial value. In view of this, the measured vehicle bridge displacement signal was taken as the research object in this paper, the time domain analysis and frequency domain power spectrum analysis were carried out respectively. In order to further study the statistical law of the displacement signals, the signal was grouped and the frequency was counted, the statistical histogram and the cumulative probability distribution curve were obtained. The normal distribution, mixed normal distribution, weibull distribution and mixed weibull distribution were employed respectively. A novel two-step parameter identification method was proposed, and the grey correlation degree objective function was introduced. The grey correlation coefficient objective function could ensure the maximum geometric similarity between the fitting curve and the original curve. By doing this, the inherent malpractice of the optimization process with the square sum of error as the fitness was overcome to some extent. The proposed parameter estimation method's tep was as following: Firstly, the parameters obtained by the artificial fish swarm algorithm were applied as the initial values of the model parameters. Secondly, the iterative nonlinear least square method, namely, levenberg-marquardt(LM) algorithm was used to identify the parameters accurately. Thirdly, the goodness of fit for each model were calculated by using the kolmogorov-smirnov test index and correlation coefficient. The result showed that the mixed weibull model could be used to describe the tested displacement signal best. The correlation coefficient between the mixed Weibull distribution and the statistical histogram was(0.9800, 0.9908,0.9867,0.9665), whereas, the mixed normal distribution was(0.9793,0.9904,0.9783,0.9661), the weibull model was(0.8613,0.9113,0.8618,0.8854), and the normal model was(0.8611,0.9127,0.8624,0.8869). The proposed two-step parameter identification method combined the advantages of the artificial fish swarm optimization algorithm and the traditional iterative algorithm, and used the artificial fish swarm optimization result as the initial value of the LM algorithm. It solved the problem of the difficulty in selecting the initial value of the nonlinear least square method and improved the efficiency of the parameter identification. This study can provide reference for the fatigue load spectrum and the bench test of off-road vehicles.
引文
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[8]郑锐.三参数威布尔分布参数估计及在可靠性分析中的应用[J].振动与冲击,2015(5):78-81.Zheng Rui.Parameter estimation of three-parameter Weibull distribution and its application in reliability analysis[J].Vibration and shock,2015(5):78-81.(in Chinese with English abstract)
[9]丛楠,陈俊达,任焱晞,等.一种指定功率谱密度与峭度值的对称威布尔分布道路谱重构方法[J].振动与冲击,2018(2):1-5.Cong Nan,Chen Junda,Ren Yanxi,et al.A road spectral reconstruction method for symmetric Weibull distribution with specified power spectral density and kurtosis[J].Vibration and Shock,2018(2):1-5.(in Chinese with English abstract)
[10]Krifa M.A mixed Weibull model for size reduction of particulate and fibrous materials[J].Powder Technology,2009,194(3):233-238.
[11]凌丹,黄洪钟,张小玲,等.混合威布尔分布参数估计的L-M算法[J].电子科技大学学报,2008,37(4):634-636.Ling Dan,Huang Hongzhong,Zhang Xiaoling,et al.L-Malgorithm for parameter estimation of mixed Weibull distribution[J].Journal of the University of Electronic Science and Technology,2008,37(4):634-636.(in Chinese with English abstract)
[12]Nagode M,Fajdiga M.An improved algorithm for parameter estimation suitable for mixed Weibull distributions[J].International Journal of Fatigue,2000,22(1):75-80.
[13]王继利,杨兆军,李国发,等.基于改进EM算法的多重威布尔可靠性建模[J].吉林大学学报:工学版,2014,44(4):1010-1015.Wang Jili,Yang Zhaojun,Li Guofa,et al.Multiple Weibull reliability modeling based on improved EM algorithm[J].Journal of Jilin University:Engineering Edition,2014,44(4):1010-1015.(in Chinese with English abstract)
[14]王晓峰,张英芝,申桂香,等.基于ITLS和DE的加工中心三参数威布尔分布[J].华南理工大学学报:自然科学版,2015,43(6):84-88.Wang Xiaofeng,Zhang Yingzhi,Shen Guixiang,et al.Three parameter Weibull distribution of machining center based on ITLS and DE[J].Journal of South China University of Technology:Natural Science Edition,2015,43(6):84-88.(in Chinese with English abstract)
[15]张根保,李冬英,刘杰,等.面向不完全维修的数控机床可靠性评估[J].机械工程学报,2013,49(23):136-141.Li Genbao,Li Dongying,Liu Jie,et al.Reliability Evaluation of NC Machine tool for incomplete maintenance[J].Journal of Mechanical Engineering,2013,49(23):136-141.(in Chinese with English abstract)
[16]杨兆军,杨川贵,陈菲,等.基于PSO算法和SVR模型的加工中心可靠性模型参数估计[J].吉林大学学报:工学版,2015,45(3):829-836.Yang Zhaojun,Yang Chuangui,Chen Fei,et al.Parameter estimation of Machining Center Reliability Model based on PSO algorithm and SVR Model[J].Journal of Jilin University:Engineering Edition,2015,45(3):829-836.(in Chinese with English abstract)
[17]Datsiou K C,Overend M.Weibull parameter estimation and goodness-of-fit for glass strength data[J].Structural Safety,2018,73:29-41.
[18]魏艳华,王丙参,邢永忠.指数-威布尔分布参数贝叶斯估计的混合Gibbs算法[J].统计与决策,2017(16):70-73.Wei Yanhua,Wang Bingcan,Xing Yongzhong.Mixed Gibbs algorithm for Bayesian estimation of exponent-Weibull Distribution parameters[J].Statistics and Decision Making,2017(16):70-73.(in Chinese with English abstract)
[19]Ignacio M,Chubynsky M V,Slater G W.Interpreting the Weibull fitting parameters for diffusion-controlled release data[J].Physica A Statistical Mechanics&Its Applications,2017,486:1-24.
[20]吴龙涛,王铁宁,杨帆.基于贝叶斯法和蒙特卡洛仿真的威布尔型装备器材需求预测[J].兵工学报,2017,38(12):2447-2454.Wu Longtao,Wang Tiening,Yang Fan.Prediction of Weibull equipment demand based on Bayesian method and Monte Carlo simulation[J].Journal of Military Engineering,2017,38(12):2447-2454.(in Chinese with English abstract)
[21]Lan C,Bai N,Yang H,et al.Weibull modeling of the fatigue life for steel rebar considering corrosion effects[J].International Journal of Fatigue,2018,111:134-143.
[22]Voronov S,Frisk E,Krysander M.Data-driven battery lifetime prediction and confidence estimation for heavy-duty trucks[J].IEEE Transactions on Reliability,2018,67(2):1-17.
[23]刘大维,蒋荣超,朱龙龙,等.基于遗传算法的路面有理函数功率谱密度参数识别[J].农业工程学报,2012,28(8):128-133.Liu Dawei,Jiang Rongchao,Zhu Longlong,et al.Identification of pavement rational function power spectrum density parameters based on genetic algorithms[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2012,28(8):128-133.(in Chinese with English abstract)
[24]?rkcüH H,?zsoy V S,Aksoy E,et al.Estimating the parameters of 3-p Weibull distribution using particle swarm optimization:A comprehensive experimental comparison[J].Applied Mathematics&Computation,2015,268(9):201-226.
[25]Abbasi B,Niaki S T A,Khalife M A,et al.A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution[J].Expert Systems with Applications,2011,38(1):700-708.
[26]Abbasi B,Hosseinifard S Z,Coit D W.A neural network applied to estimate Burr XII distribution parameters[J].Reliability Engineering&System Safety,2010,95(6):647-654.
[27]Xu M,Droguett E L,Lins I D,et al.On the q-Weibull distribution for reliability applications:An adaptive hybrid artificial bee colony algorithm for parameter estimation[J].Reliability Engineering&System Safety,2017,158:93-105.
[28]Cai Y.Artificial fish school algorithm applied in a combinatorial optimization problem[J].International Journal of Intelligent Systems&Applications,2010,2(1):37-43.
[29]李晓磊.一种新型的智能优化方法-人工鱼群算法[D].杭州:浙江大学,2003.Li Xiaolei.A New Intelligent Optimization method-artificial Fish Swarm algorithm[D].Hangzhou:Zhejiang University,2003.(in Chinese with English abstract)
[30]刘志君,高亚奎,章卫国,等.混合人工鱼群算法在约束非线性优化中的应用[J].哈尔滨工业大学学报,2014,46(9):55-60.Liu Zhijun,Gao Yakui,Zhang Weiguo,et al.Application of hybrid artificial fish swarm algorithm in constrained nonlinear optimization[J].Journal of Harbin University of Technology,2014,46(9):55-60.(in Chinese with English abstract)
[31]刘思峰,杨英杰,吴利丰.灰色系统理论及其应用.[M].第7版北京:科学出版社,2014.
[32]Kayacan E,Ulutas B,Kaynak O.Grey system theory-based models in time series prediction[J].Expert Systems with Applications,2010,37(2):1784-1789.
[2]翟新婷,张晓晨,江柱锦,等.基于混合分布的轮式装载机半轴载荷谱编制[J].农业工程学报,2018,34(8):78-84.Zhai Xinting,Zhang Xiaochen,Jiang Zhujin,et al.Compilation of half-axle load spectrum of wheel loader based on mixed distribution[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2018,34(8):78-84.(in Chinese with English abstract)
[3]T Bu?ar,M Nagode,M Fajdiga.Reliability approximation using finite Weibull mixture distributions[J].Reliability Engineering&System Safety,2004,84(3):241-251.
[4]Hagey T J,Puthoff J B,Crandell K E,et al.Modeling observed animal performance using the Weibull distribution[J].Journal of Experimental Biology,2016,219(11):1603-1607.
[5]Davies I J.Unbiased estimation of the Weibull scale parameter using linear least squares analysis[J].Journal of the European Ceramic Society,2017,37(8):2973-2981.
[6]Wais P.Two and three-parameter Weibull distribution in available wind power analysis[J].Renewable Energy,2016,103:15-29.
[7]许伟,程刚,黄林,等.基于混沌模拟退火PSO算法的威布尔分布参数估计应用研究[J].振动与冲击,2017,36(12):134-139.Xu Wei,Cheng Gang,Huang Lin,et al.Research on Weibull distribution parameter estimation and application research based on chaotic simulated annealing PSOalgorithm[J].Vibration and Shock,2017,36(12):134-139.(in Chinese with English abstract)
[8]郑锐.三参数威布尔分布参数估计及在可靠性分析中的应用[J].振动与冲击,2015(5):78-81.Zheng Rui.Parameter estimation of three-parameter Weibull distribution and its application in reliability analysis[J].Vibration and shock,2015(5):78-81.(in Chinese with English abstract)
[9]丛楠,陈俊达,任焱晞,等.一种指定功率谱密度与峭度值的对称威布尔分布道路谱重构方法[J].振动与冲击,2018(2):1-5.Cong Nan,Chen Junda,Ren Yanxi,et al.A road spectral reconstruction method for symmetric Weibull distribution with specified power spectral density and kurtosis[J].Vibration and Shock,2018(2):1-5.(in Chinese with English abstract)
[10]Krifa M.A mixed Weibull model for size reduction of particulate and fibrous materials[J].Powder Technology,2009,194(3):233-238.
[11]凌丹,黄洪钟,张小玲,等.混合威布尔分布参数估计的L-M算法[J].电子科技大学学报,2008,37(4):634-636.Ling Dan,Huang Hongzhong,Zhang Xiaoling,et al.L-Malgorithm for parameter estimation of mixed Weibull distribution[J].Journal of the University of Electronic Science and Technology,2008,37(4):634-636.(in Chinese with English abstract)
[12]Nagode M,Fajdiga M.An improved algorithm for parameter estimation suitable for mixed Weibull distributions[J].International Journal of Fatigue,2000,22(1):75-80.
[13]王继利,杨兆军,李国发,等.基于改进EM算法的多重威布尔可靠性建模[J].吉林大学学报:工学版,2014,44(4):1010-1015.Wang Jili,Yang Zhaojun,Li Guofa,et al.Multiple Weibull reliability modeling based on improved EM algorithm[J].Journal of Jilin University:Engineering Edition,2014,44(4):1010-1015.(in Chinese with English abstract)
[14]王晓峰,张英芝,申桂香,等.基于ITLS和DE的加工中心三参数威布尔分布[J].华南理工大学学报:自然科学版,2015,43(6):84-88.Wang Xiaofeng,Zhang Yingzhi,Shen Guixiang,et al.Three parameter Weibull distribution of machining center based on ITLS and DE[J].Journal of South China University of Technology:Natural Science Edition,2015,43(6):84-88.(in Chinese with English abstract)
[15]张根保,李冬英,刘杰,等.面向不完全维修的数控机床可靠性评估[J].机械工程学报,2013,49(23):136-141.Li Genbao,Li Dongying,Liu Jie,et al.Reliability Evaluation of NC Machine tool for incomplete maintenance[J].Journal of Mechanical Engineering,2013,49(23):136-141.(in Chinese with English abstract)
[16]杨兆军,杨川贵,陈菲,等.基于PSO算法和SVR模型的加工中心可靠性模型参数估计[J].吉林大学学报:工学版,2015,45(3):829-836.Yang Zhaojun,Yang Chuangui,Chen Fei,et al.Parameter estimation of Machining Center Reliability Model based on PSO algorithm and SVR Model[J].Journal of Jilin University:Engineering Edition,2015,45(3):829-836.(in Chinese with English abstract)
[17]Datsiou K C,Overend M.Weibull parameter estimation and goodness-of-fit for glass strength data[J].Structural Safety,2018,73:29-41.
[18]魏艳华,王丙参,邢永忠.指数-威布尔分布参数贝叶斯估计的混合Gibbs算法[J].统计与决策,2017(16):70-73.Wei Yanhua,Wang Bingcan,Xing Yongzhong.Mixed Gibbs algorithm for Bayesian estimation of exponent-Weibull Distribution parameters[J].Statistics and Decision Making,2017(16):70-73.(in Chinese with English abstract)
[19]Ignacio M,Chubynsky M V,Slater G W.Interpreting the Weibull fitting parameters for diffusion-controlled release data[J].Physica A Statistical Mechanics&Its Applications,2017,486:1-24.
[20]吴龙涛,王铁宁,杨帆.基于贝叶斯法和蒙特卡洛仿真的威布尔型装备器材需求预测[J].兵工学报,2017,38(12):2447-2454.Wu Longtao,Wang Tiening,Yang Fan.Prediction of Weibull equipment demand based on Bayesian method and Monte Carlo simulation[J].Journal of Military Engineering,2017,38(12):2447-2454.(in Chinese with English abstract)
[21]Lan C,Bai N,Yang H,et al.Weibull modeling of the fatigue life for steel rebar considering corrosion effects[J].International Journal of Fatigue,2018,111:134-143.
[22]Voronov S,Frisk E,Krysander M.Data-driven battery lifetime prediction and confidence estimation for heavy-duty trucks[J].IEEE Transactions on Reliability,2018,67(2):1-17.
[23]刘大维,蒋荣超,朱龙龙,等.基于遗传算法的路面有理函数功率谱密度参数识别[J].农业工程学报,2012,28(8):128-133.Liu Dawei,Jiang Rongchao,Zhu Longlong,et al.Identification of pavement rational function power spectrum density parameters based on genetic algorithms[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2012,28(8):128-133.(in Chinese with English abstract)
[24]?rkcüH H,?zsoy V S,Aksoy E,et al.Estimating the parameters of 3-p Weibull distribution using particle swarm optimization:A comprehensive experimental comparison[J].Applied Mathematics&Computation,2015,268(9):201-226.
[25]Abbasi B,Niaki S T A,Khalife M A,et al.A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution[J].Expert Systems with Applications,2011,38(1):700-708.
[26]Abbasi B,Hosseinifard S Z,Coit D W.A neural network applied to estimate Burr XII distribution parameters[J].Reliability Engineering&System Safety,2010,95(6):647-654.
[27]Xu M,Droguett E L,Lins I D,et al.On the q-Weibull distribution for reliability applications:An adaptive hybrid artificial bee colony algorithm for parameter estimation[J].Reliability Engineering&System Safety,2017,158:93-105.
[28]Cai Y.Artificial fish school algorithm applied in a combinatorial optimization problem[J].International Journal of Intelligent Systems&Applications,2010,2(1):37-43.
[29]李晓磊.一种新型的智能优化方法-人工鱼群算法[D].杭州:浙江大学,2003.Li Xiaolei.A New Intelligent Optimization method-artificial Fish Swarm algorithm[D].Hangzhou:Zhejiang University,2003.(in Chinese with English abstract)
[30]刘志君,高亚奎,章卫国,等.混合人工鱼群算法在约束非线性优化中的应用[J].哈尔滨工业大学学报,2014,46(9):55-60.Liu Zhijun,Gao Yakui,Zhang Weiguo,et al.Application of hybrid artificial fish swarm algorithm in constrained nonlinear optimization[J].Journal of Harbin University of Technology,2014,46(9):55-60.(in Chinese with English abstract)
[31]刘思峰,杨英杰,吴利丰.灰色系统理论及其应用.[M].第7版北京:科学出版社,2014.
[32]Kayacan E,Ulutas B,Kaynak O.Grey system theory-based models in time series prediction[J].Expert Systems with Applications,2010,37(2):1784-1789.