Global Journal of Researches in Engineering, I: Numerical Methods, Volume 23 Issue 1

Fig. 6: Histogram after logarithm of the original data (small sample) and fitting diagram of Gaussian and Weibull distribution As seen in Fig. 5 and 6, similar to the case of the large sample, the original data are also left-skewed and appear symmetric after taking the logarithm. However, if the Weibull distribution is fitted, there is no need to take the logarithm of the original data. Even if the logarithm is taken, the data looks more symmetric, but the Weibull distribution does not fit worse than the Gaussian distribution. So in this sense, even for symmetric data, fitting with the Weibull distribution is possible. However, the difficulty in fitting the Weibull distribution is that it is more difficult to estimate the three parameters, but now there is no problem with GZT method. VI. C onclusion 1. The three-parameter Weibull distribution is a more general full state distribution than the Gaussian distribution. In the field of reliability, the physical meaning of its position parameter is particularly important, that is, the safe life under 100% reliability. 2. Based on the complexity of the three-parameter Weibull distribution, the previous methods to determine its three parameters by test data are complicated. The graphical method is more error- prone and inconvenient to use; while the analytical method may be inconsistent; and the GZT method makes full use of the advantages of Python, which solves this problem better. 3. In the past, the fatigue life data that were not so well fitted with Gaussian distribution were taken logarithmically so that they might be more consistent with Gaussian distribution, but the result of doing so made the 100% reliability of the safe life no longer exist. The fact is that the data itself is more consistent with Weibull distribution. Since Weibull distribution is a full state distribution, it is generally not necessary to take the fatigue life as logarithm in the future and directly fit the fatigue life data with the three-parameter Weibull distribution to get a better fit. 4. The two parameters of Gaussian distribution (mean and variance) are not very significant for asymmetric data, while for asymmetric data like structural fatigue life the three parameters of Weibull distribution (safety life, shape and scale parameters) will be much more significant, and in a sense these three parameters "contain" the two parameters of Gaussian distribution. This is probably the reason why the Weibull distribution can "contain" the Gaussian distribution. 5. Finally, it can be concluded that for asymmetric fatigue life, it is not necessary to take logarithms to fit with Gaussian distribution, but can be directly fitted with three-parameter Weibull distribution. Further even for the more symmetric data, it is better to fit directly with the three-parameter Weibull distribution. A cknowledgments We thank Mr. Wan Weihao for his support to this paper and related research work. R eferences R éférences R eferencias 1. Xu J. J.(2021), Gao Zhentong Method in the Fatigue Statistics Intelligence, Journal of Beijing University of Aeronautics and Astronautics, Vol. 47(10), 2024- 2033, Doi: 10.13700 /j.bh. 1001-5965.2020.0373 (in Chinese). 2. Weibull WALODDI. (1951), A statistical distribution function of wide applicability Journal of Applied Mechanic Reliab., 1951, Vol. 28(4): 613-617. 3. Hallinan A J. (1993), A review of the Weibull distribution (1993), Journal of Quality Technology, 1993, 25(2): 85-93. 4. Gao Z T. Fatigue applied statistics (1986), Beijing: National Defense Industry Press, (in Chinese). 5. Gao Z.T., XU J,J. (2022), Intelligent Fatigue Statistics Beijing: Beihang publishing house, (in Chinese). 6. Fan Yang, Hu Ren, and Zhili Hu, (2019), Maximum Likelihood Estimation for Three-Parameter Weibull Distribution Using Evolutionary Strategy Mathematical Problems in Engineering, Volume, Article ID 6281781, 8 pages, https://doi.org/10. 1155/2019/6281781. 7. Mahdi Teimouri, d Arjun K. Gupta, (2013), On the Three-Parameter Weibull Distribution Shape Parameter Estimation, Journal of Data Science 11, 403-414. 8. Trivedi K.S (2015), Probability and Statistics with Reliability, Queuing, and Computer Science Applications, Beijing: Electronic Industry Press, (in Chinese). 9. Fu H M, Gao Z T. (1990), An optimization method of correlation coefficient for determining A three- parameters Weibull distribution, Acta Aeronautica et © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 5 Year 2023 ( ) I From Gaussian Distribution to Weibull Distribution