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

described below. Taking the logarithm of both sides of (4) twice yields that ln (ln(1/ p i ))= b ln(N i -N 0 )- b ln( λ ) (10) if set, Y i = ln(ln(1/ p i )), X i = ln(N i -N 0 ) (11) d =- b ln(λ), λ=exp(- d/b ) (12) So (10) could been become, Y i =bX i +d (13) This is a system of linear regression equations that can be derived by the least squares method with coefficients b and d . However, it is important to note that here X i is related not only to the data N already given, but also to the required safety lifetime N 0 of Weibull distribution. This problem can be solved by determining the extreme value of the absolute value of the relative coefficient r of the regression line to determine the corresponding N 0 , but the mathematical derivation of this method is complex and error-prone [9] . It is better to use a different idea to use Python to find the series of r about N 0 directly in the interval 0≤ N 0 < N min (here N min is taken as the minimum value of the given data). Then Python intelligently finds the N 0 of r with the largest correlation coefficient, and at the same time determines b and λ . This is known as the Z.T. Gao algorithm. It is abbreviated as the Z.T. Gao method [1], [5] or GZT method. Example 2: Now, using the data of Example 1, three parameters of Weibull distribution are determined by using GZT method, and the results are compared with Gaussian distribution. The results are as follows: Fig. 2: Schematic graph of Z.T. Gao method This figure graphically demonstrates how GZT method finds the corresponding safe lifetime that maximizes the correlation coefficient. Since it is clear at the beginning of the process that N 0 cannot be greater than the minimum lifetime of the data, it is not possible to have a situation where it is inconsistent. Again, if the data are fitted with a Gaussian distribution and the coefficient of determination of the Weibull distribution estimated by GZT method, respectively, fitted with the ideal reliability (9): Coefficient of determination obtained by fitting the Weibull distribution = 0.97999 Coefficient of determination obtained by fitting the Gaussian distribution = 0.95044 It can be seen that the fitted coefficient of determination of the Weibull distribution obtained by GZT method is greater than that of the Gaussian distribution. That is, in this sense the data are more realistically depicted by the Weibull distribution. The advantage of GZT method is that the physical meaning is very intuitive, and there is no problem of "inconsistent". This method is not only convenient for solving the problem of estimating the three parameters of the Weibull distribution, but also easy to determine whether the original data fits better with the Weibull distribution or with the Gaussian distribution. It is also easy to extend to solve similar problems, such as fitting fatigue performance curves with three parameters [1] , and the confidence intervals of these three parameters will be discussed in separate papers [10], [11] . V. P roblems of L ogarithmization of O riginal D ata Due to the complexity of the Weibull distribution, when the original data is not so consistent with the Gaussian distribution, often take its logarithmic, from a mathematical point of view is equivalent to do a spatial transformation, at this time because the data "compressed", it may be closer to the Gaussian distribution [4] . This has the advantage of making the PDF of the original data taken logarithmically will be fitted quite well by the Gaussian distribution, which will be more convenient for people to study and apply. However, this will lose the physical meaning of the safety lifetime, while making the original data density distribution is "distorted". This is illustrated in the following two examples. Example 3: Using the (large sample) 100 fatigue life data of a structure from Table 12-3 of [1] P253, the Python code gives: Also the following parameter table and histogram can be obtained. © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 3 Year 2023 ( ) I From Gaussian Distribution to Weibull Distribution Fatigue life (original data) N= [3.08, 3.26, 3.32, 3.48, 3.49, 3.56, 3.69, 3.7, 3.78, 3.79, 3.8, 3.87, 3.95, 4.07, 4.08, 4.1, 4.12, 4.2, 4.24, 4.25, 4.28, 4.31, 4.31, 4.36, 4.54, 4.58, 4.6, 4.62, 4.63, 4.65, 4.67, 4.67, 4.72, 4.73, 4.75, 4.77, 4.8, 4.82, 4.84, 4.9, 4.92, 4.93, 4.95, 4.96, 4.98, 4.99, 5.02, 5.03, 5.06, 5.08, 5.06, 5.1, 5.12, 5.15, 5.18, 5.2, 5.22, 5.38, 5.41, 5.46, 5.47, 5.53, 5.56, 5.6, 5.61, 5.63, 5.64, 5.65, 5.68, 5.69, 5.73, 5.82, 5.86, 5.91, 5.94, 5.95, 5.99, 6.04, 6.08, 6.13, 6.16, 6.19, 6.21, 6.26, 6.32, 6.33, 6.36, 6.41, 6.46, 6.81, 7.0, 7.35, 7.82, 7.88, 7.96, 8.31, 8.45, 8.47, 8.79, 9.87] (10^5cycle).