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

a good approximation. In fact [4] , the data of various fatigue lives are often not fit Gaussian distribution but better fit Weibull distribution, and sometimes the fatigue life is logarithmically distributed, but it is only an approximation. Because of this, Weibull distribution needs to be introduced and studied in more depth. III. B rief I ntroduction of W eibull D istribution There are various expressions for the Weibull distribution, and a more general form is taken here [1] , with a probability density function: f( x )=( b/λ )[( x-x 0 )/λ )] b -1 exp{-[ (x-x 0 )/ λ ] b } (3) where b is the shape parameter, λ is the scale or proportional parameter, and x 0 is called the position parameter. In the field of fatigue it is customary to use the fatigue life N instead of x, N 0 instead of x 0 , and call it the safe life. In a non-strict sense [1] , "when 0 < b < 1 resembles a power-law function, while 1 < b < 3 is a left-skewed distribution, 3 < b < 4 approximates a Gaussian distribution, and b > 4 is a right-skewed distribution". This is the reason why the Weibull distribution is called the "full state distribution". As shown in the following fig.1 [5] : Fig. 1: PDF of various Three-Parameter Weibull distributions when x 0 =0.5 It is easy to prove that the life is x i and the corresponding reliability [1] is, p i = exp{-[ (x i -x 0 )/λ ] b } (4) It can be seen that when x=x 0 , p 0 =100%. This is the origin of 100% reliability safety life. If p 50 =50%, it means that the corresponding X is called the median value x m of X, that is, there are, 50%=exp{-[ (x m -x 0 )/λ ] b } (5) It is not difficult to get the expectation and variance of Weibull distribution with three parameters according to the definition [4] , E( X )= x 0 +λΓ(1+1/ b ) (6) Var( X )=λ 2 [Γ(1+2/ b )-Γ 2 (1+1/ b )] (7) In this way, the fatigue life data are given and the three parameters of Weibull distribution can be derived by (5), (6) and (7), which is the analytical metho [4] . In addition to the analytical method, the maximum likelihood method and some methods derived from it [6], [7] have been used more recently, but they have problems such as cumbersome derivation and inconvenient calculation, so we will not discuss them in depth here. IV. O rigin of Z.T. GAO M ethod and F itting S tandard Theoretically if a set of fatigue life data N is given, then using the median ( N m ), mean ( N av ) and mean squared deviation (s) of this array, then using the three equations (5), (6) and (7) is possible to solve for the estimated values of the three parameters of the Weibull distribution. However, for convenience (5), (6) and (7) can be reduced to a transcendental equation [1] with respect to b: ( N av -N m )[Γ(1+2/b)-Γ 2 (1+1/ b )]+s[D 1/b - Γ(1+1/ b )] 1/2 =0 (8) It is not difficult to find that N 0 (=127) derived from the analytical method is greater than the minimum value of 124 for this group of fatigue lives. And this is in contradiction with the definition of safe life N 0 . That is, the problem of inconsistent occurs. Another question is what happens if we fit this set of data with a Gaussian distribution? That is, which is the more appropriate distribution to fit? The second problem can be judged by the magnitude of the determination coefficient [8] R 2 fitting the ideal reliability based on the so-called "average rank" [4] . The so-called ideal reliability means that the following formula is independent of the specific distribution, p i =1- i /(n+1) (9) where i is the order of the data from smallest to largest, and n is the number of data. And the first problem is solved by the Z. T. Gao method [1] . The basic idea of the method is briefly Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 2 Year 2023 © 2023 Global Journals ( ) I From Gaussian Distribution to Weibull Distribution where D = ln2. This equation is solvable by Newton's method, and after obtaining b, then λ and N 0 can be found by (7),(6). Example 1: The data in Table 8-2 in [4] are used to find the three parameters of the Weibull distribution by analytical method. Table 1: A set of fatigue life data (10 3 c) You can get it through Python code, Parameter estimation: b=1.221,N 0 =127, λ =22.46