# Reliability by Using Weibull Distribution Based on Vibration Fatigue Damage
## 5 Discussion  
The paper presents an alternative to fit a probabilistic vibration fatigue analysis based on the use of the damage accumulated for two parameters provided by the Weibull distribution model. The main contribution of the proposed method is the probabilistic approach that the Weibull distribution function has in the mechanical industrial field. A considerable number of industrial standards and guidelines employ the Weibull model for their fatigue analysis; thus, reliability engineers are especially familiar with this model. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(n_{i}\) & \(Y_{i}\) & \(\mu_{y}\) & \(R(t)\) & \(t_{oi}\) & \(\sigma_{2i}\) & \(\sigma_{1i}\) & \(F(t)\) \\  & **Equation (21)** & **Equation (14)** & **Equation (20)** & **Equation (15)** & **Equation (16)** & **Equation (17)** & **Equation (7)** \\ \hline
25 & 0.6062 & 0.0209 & 0.1599 & 2.2349 & 156.0158 & 31.2347 & 0.8401 \\ \hline
26 & 0.7288 & 0.0251 & 0.1259 & 2.6298 & 183.5827 & 26.5445 & 0.8741 \\ \hline
27 & 0.8703 & 0.0300 & 0.0918 & 3.1730 & 221.4969 & 22.0008 & 0.9082 \\ \hline
28 & 1.0474 & 0.0361 & 0.0578 & 4.0133 & 280.1599 & 17.3940 & 0.9422 \\ \hline  & **1.1109** & **1.4738** & **0.0480** & **4.3656** & **304.7500** & **15.9900** & **0.9520** \\ \hline
29 & 1.3185 & 0.0455 & 0.0238 & 5.7498 & 401.3795 & 12.1409 & 0.9762 \\ \hline \hline \end{tabular}
\begin{tabular}{c} BOLD: The principal vibration stresses above and below the \(\eta\) parameter, the \(R(t)\) index of 95%, and the \(R(t)\) index of the \(S_{y}\) 430 MPa. \\ \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{l c c} \hline \hline
**Feature** & **Proposed Method** & **Median Rank Method** \\ \hline Weibull Shape Parameter & & \\ Equation (9) & \(\beta=0.9102\) & \(\beta=0.7538\) \\ \hline Weibull Scale Parameter & & \\ Equation (10) & \(\eta=69.8077\) & \(\eta=69.8077\) \\ \hline Principal Stresses, & & \\ Equations (17) and (16) & & \\ \(\sigma_{1}=304.7600\) MPa & \(R(t)=0.8260\) & \(R(t)=0.7757\) \\ \(\sigma_{2}=15.9900\) MPa & & \\ Material strength, & & \\ \(S_{y}=430\) MPa & & \\ \hline \(R(t)=0.95\) & \(\sigma_{1}=1824.3190\) MPa & \(\sigma_{1}=3591.0886\) MPa \\ Equation (8) & \(\sigma_{2}=2.6712\) MPa & \(\sigma_{2}=1.3570\) MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The proposed method and median rank method comparison results. This paper presents a probabilistic aspect that involves fatigue experiments and the proper use of vibration fatigue damage during the investigation and design of mechanical components. The use of vibration fatigue damage as the cumulative failure percentile in the process allows the transfer to obtain an estimation of the reliability and a prediction of failure, in spite of the significant variability involved in the vibration fatigue. Regarding the results obtained for the probabilistic prediction of reliability and cumulated failure, based on the principal vibration stress values \(\sigma_{1}=304.76\) and \(\sigma_{2}=15.99\), the proposed method predicts a reliability of \(R(t)=0.826\) and a cumulative failure of \(F(t)=0.174\). Here, it is important to notice that if we use the standard median rank approach, then \(R(t)=0.776\) and \(F(t)=0.224\). By their comparison, we have an \(R(t)\) variation of 5%. Moreover, it can be observed that because the used cumulated damage depends, among other factors, on the vibration load distribution applied, the material's strength, and the geometry of the tested mechanical component, then because each analysis is different, the proposed method is completely dynamic, and can be used in any application where the accumulated damage can be measured. Here, we highlight that the only restriction to use the cumulated damage is that the failure must be defined when the damage is equal to one.