# Reliability by Using Weibull Distribution Based on Vibration Fatigue Damage
## 3 Numerical Application
### Fatigue Damage Accumulation
\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
1 & \(-3.7256\) & \(-0.1285\) & \(0.9762\) & \(0.0071\) & \(0.4981\) & \(9783.4030\) & \(0.0238\) \\ \hline  & \(-2.9702\) & \(-0.1024\) & **0.9500** & **0.0194** & **1.3570** & **3591.0886** & **0.0500** \\ \hline
2 & \(-2.8207\) & \(-0.0973\) & \(0.9422\) & \(0.0237\) & \(1.6546\) & \(2945.1760\) & \(0.0578\) \\ \hline
3 & \(-2.3400\) & \(-0.0807\) & \(0.9082\) & \(0.0449\) & \(3.1311\) & \(1556.3556\) & \(0.0918\) \\ \hline
4 & \(-2.0062\) & \(-0.0692\) & \(0.8741\) & \(0.0698\) & \(4.8755\) & \(999.5045\) & \(0.1259\) \\ \hline
5 & \(-1.7476\) & \(-0.0603\) & \(0.8401\) & \(0.0984\) & \(6.8706\) & \(709.2668\) & \(0.1599\) \\ \hline
6 & \(-1.5347\) & \(-0.0529\) & \(0.8061\) & \(0.1305\) & \(9.1130\) & \(534.7438\) & \(0.1939\) \\ \hline  & \(-1.3704\) & \(-1.8180\) & **0.7757** & **0.1623** & **11.3328** & **430.0000** & **0.2243** \\ \hline
7 & \(-1.3524\) & \(-0.0466\) & \(0.7721\) & \(0.1663\) & \(11.6070\) & \(419.8408\) & \(0.2279\) \\ \hline
8 & \(-1.1918\) & \(-0.0411\) & \(0.7381\) & \(0.2058\) & \(14.3630\) & \(339.2823\) & \(0.2619\) \\ \hline  & \(-1.1109\) & \(-1.4738\) & **0.7194** & **0.2291** & **15.9900** & **304.7500** & **0.2806** \\ \hline
9 & \(-1.0474\) & \(-0.0361\) & \(0.7041\) & \(0.2492\) & \(17.3959\) & \(280.1293\) & \(0.2959\) \\ \hline
10 & \(-0.9154\) & \(-0.0316\) & \(0.6701\) & \(0.2969\) & \(20.7257\) & \(235.1238\) & \(0.3299\) \\ \hline
11 & \(-0.7930\) & \(-0.0273\) & \(0.6361\) & \(0.3492\) & \(24.3773\) & \(199.9034\) & \(0.3639\) \\ \hline
12 & \(-0.6784\) & \(-0.0234\) & \(0.6020\) & \(0.4066\) & \(28.3815\) & \(171.7005\) & \(0.3980\) \\ \hline
13 & \(-0.5699\) & \(-0.0197\) & \(0.5680\) & \(0.4695\) & \(32.7756\) & \(148.6811\) & \(0.4320\) \\ \hline
14 & \(-0.4663\) & \(-0.0161\) & \(0.5340\) & \(0.5387\) & \(37.6055\) & \(129.5852\) & \(0.4660\) \\ \hline
15 & \(-0.3665\) & \(-0.0126\) & \(0.5000\) & \(0.6149\) & \(42.9271\) & \(113.5207\) & \(0.5000\) \\ \hline
16 & \(-0.2697\) & \(-0.0093\) & \(0.4660\) & \(0.6992\) & \(48.8095\) & \(99.8395\) & \(0.5340\) \\ \hline
17 & \(-0.1751\) & \(-0.0060\) & \(0.4320\) & \(0.7927\) & \(55.3389\) & \(88.0595\) & \(0.5680\) \\ \hline
18 & \(-0.0819\) & \(-0.0028\) & \(0.3980\) & \(0.8971\) & \(62.6241\) & \(77.8153\) & \(0.6020\) \\ \hline  & **0.0000** & **0.0000** & **0.3679** & **1.0000** & **69.8065** & **69.8065** & **0.6321** \\ \hline
19 & \(0.0107\) & \(0.0004\) & \(0.3639\) & \(1.0143\) & \(70.8051\) & \(68.8243\) & \(0.6361\) \\ \hline
20 & \(0.1033\) & \(0.0036\) & \(0.3299\) & \(1.1469\) & \(80.0653\) & \(60.8642\) & \(0.6701\) \\ \hline
21 & \(0.1969\) & \(0.0068\) & \(0.2959\) & \(1.2986\) & \(90.6512\) & \(53.7567\) & \(0.7041\) \\ \hline
22 & \(0.2925\) & \(0.0101\) & \(0.2619\) & \(1.4741\) & \(102.9040\) & \(47.3559\) & \(0.7381\) \\ \hline
23 & \(0.3913\) & \(0.0135\) & \(0.2279\) & \(1.6805\) & \(117.3143\) & \(41.5390\) & \(0.7721\) \\ \hline
24 & \(0.4950\) & \(0.0171\) & \(0.1939\) & \(1.9285\) & \(134.6219\) & \(36.1985\) & \(0.8061\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Weibull vibration fatigue damage statistics by using median rank method. A comparison of the principal results between both the proposed method and the median rank method by following the formulation given in Section 2 is shown in Table 5.  
In Table 5, notice that although we are using the same \(\sigma_{1}=304.76\), \(\sigma_{2}=15.99\) and \(S_{y}\) values in both methods, the component's reliability \(R(t)\) is different. This difference occurs because in both methods, the Weibull \(\beta\) parameter shape have different values. This is because the arithmetic mean of both methods is different.