Overall Load Extrapolation Method Based on Frequency and Extreme Value Extrapolation

doi:10.12382/bgxb.2023.0072

Abstract

Abstract:

Load extrapolation, as an important technical means for compiling load spectra, is inconvenient to generate the high-precision load spectra to support equipment performance design due to lacking of the comprehensive review of overall load extrapolation method, the insufficient adaptability of solution method for Markov steady-state distribution, and the comparison and selection principles for different nonparametric frequency extrapolation methods. Focusing on the compilation of tank’s load spectrum under high mobility and extreme conditions, the nonparametric extrapolation method based on rain flow matrix and kernel density estimation, the signal reconstruction based on Markov Chain Monte Carlo (MCMC) method, and the Metropolis-Hastings (MH) direct sampling method are used to perform the frequency extrapolation according to the barrel displacement dataset of a tank in motion. Besides, an improved Markov steady-state distribution solution method is proposed for the signal reconstruction method of MCMC. Finally, the proposed overall frequency-extreme load extrapolation method is used to expand the frequency and predict the extreme values of the dataset. The accuracy of overall method is verified based on experimental results. It is shown that the improved Markov steady-state distribution solution method is effective. The MH direct sampling method can be used as a new frequency extrapolation method when the sample length is sufficient and the extrapolation accuracy is not very high. The accuracy of overall frequency-extreme load extrapolation method is relatively high. The principles for selecting frequency extrapolation methods have certain guiding significance for the selection of methods in the process of compiling the load spectra. The research work provides a mature technical route and reference for the high-quality equipment load spectra compiling.

Key words: load extrapolation, kernel density estimation, Markov Chain Monte Carlo method, Markov steady-state distribution, Metropolis-Hastings sampling

CLC Number:

TJ303

GAO Hua, SHAN Chunlai, LIU Jun, ZHANG Fanfan, LIU Pengke. Overall Load Extrapolation Method Based on Frequency and Extreme Value Extrapolation[J]. Acta Armamentarii, 2024, 45(6): 1942-1953.

Figures/Tables 22

Fig.1 Overall load extrapolation method

Fig.2 Schematic diagram of MH sampling method

Fig.3 The code of 2D MH sampling

Fig.4 Operation process of signal reconstruction based on MCMC

Fig.5 Displacement-time graph of a tank barrel after pretreatment (Level C road, 25km/h)

Fig.6 Barrel displacement From-To matrix (Level C road, 25km/h)

Fig.7 Barrel displacement From-To matrix (3D)

Fig.8 Kernel density estimation results of rain flow matrix

Fig.9 Nonparametric frequency extrapolation of time-series signals based on rain flow matrix and kernel density estimation (partial)

Fig.10 Time-series signal based on the signal reconstruction method of MCMC

Fig.11 Time-series signal after MH direct sampling (partial)

Table 1 Statistical comparison before and after frequency extrapolation by 3 methodsmm

类别	均值	标准差	前20个极大值			前20个极小值
类别	均值	标准差	最大值	均值	中位数	最小值	均值	中位数
外推前	10.0884	0.0111	10.1288	10.1227	10.1214	10.0533	10.0574	10.0574
方法1	10.0888	0.0111	10.1267	10.1202	10.1193	10.0504	10.0585	10.0594
方法2	10.0884	0.011	10.1277	10.1268	10.1277	10.0546	10.055	10.0546
方法3	10.0886	0.011	10.1285	10.126	10.126	10.0495	10.055	10.0555

Fig.12 Comparison of probability densities before and after frequency extrapolation by 3 methods

Fig.13 Probability densities before and after extrapolation under different data volumes (Method 3)

Table 2 Probability density difference statistics before and after extrapolation under different data volumes (Method 3)

n	最大值	均值	标准差
1000	3.0768	-0.1356	2.3830
3000	1.6686	0.2764	2.1118
5000	-0.81	0.1148	1.2565

Fig.14 Comparison of power spectra before and after frequency extrapolation by 3 methods

Table 3 Comparison of key factors and advantage/disadvantage of three extrapolation methods

外推方法	关键因素	优点	缺点
方法1	核函数的选择,最优带宽确定	与样本的分布类型无关,任意形状载荷谱的有效估计	需要大量样本数据,核函数和带宽选择是有影响的
方法2	马尔可夫分布初始状态,状态转移矩阵	基本不需要调参,可兼顾载荷与时间间隔^[7]	需要判断是否马氏收敛,求解马尔可夫初始状态繁琐
方法3	最小样本长度	操作简单、外推方便	样本长度需要校验

Fig.15 MEF curve of signal after frequency extrapolation

Fig.16 Comparison between empirical distribution and theoretical distribution of over threshold value

Fig.17 Displacement-time signal after extremum extrapolation

Fig.18 Time-series signals of verification test

Fig.19 Probability density comparison of verification test and load extrapolation

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