Temperature-Nonthermal (TNT)-Weibull Model
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This example compares the results for the temperature-nonthermal relationship for Weibull distribution.
Reference Case
Data is from Table 7.10 on page 300 in book Life Cycle Reliability Engineering by Dr. Guangbin Yang, John Wiley & Sons, 2007.
Data
Temperature and switching rate are the two stresses used in the accelerated life test for a type of 18-V compact electromagnetic relays. The cycles to failure are provided next.
| Number in Group | State F/S | Time to State | Temperature (F) | Switching Rate | Subset ID | Number in Group | State F/S | Time to State | Temperature (F) | Switching Rate | Subset ID |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | F | 47154 | 337.15 | 10 | 1 | 1 | F | 29672 | 398.15 | 10 | 3 |
| 1 | F | 51307 | 337.15 | 10 | 1 | 1 | F | 38586 | 398.15 | 10 | 3 |
| 1 | F | 86149 | 337.15 | 10 | 1 | 1 | F | 47570 | 398.15 | 10 | 3 |
| 1 | F | 89702 | 337.15 | 10 | 1 | 1 | F | 56979 | 398.15 | 10 | 3 |
| 1 | F | 90044 | 337.15 | 10 | 1 | 6 | S | 57600 | 398.15 | 10 | 3 |
| 1 | F | 129795 | 337.15 | 10 | 1 | 1 | F | 7151 | 398.15 | 30 | 4 |
| 1 | F | 218384 | 337.15 | 10 | 1 | 1 | F | 11966 | 398.15 | 30 | 4 |
| 1 | F | 223994 | 337.15 | 10 | 1 | 1 | F | 16772 | 398.15 | 30 | 4 |
| 1 | F | 227383 | 337.15 | 10 | 1 | 1 | F | 17691 | 398.15 | 30 | 4 |
| 1 | F | 229354 | 337.15 | 10 | 1 | 1 | F | 18088 | 398.15 | 30 | 4 |
| 1 | F | 244685 | 337.15 | 10 | 1 | 1 | F | 18446 | 398.15 | 30 | 4 |
| 1 | F | 253690 | 337.15 | 10 | 1 | 1 | F | 19442 | 398.15 | 30 | 4 |
| 1 | F | 270150 | 337.15 | 10 | 1 | 1 | F | 25952 | 398.15 | 30 | 4 |
| 1 | F | 281499 | 337.15 | 10 | 1 | 1 | F | 29154 | 398.15 | 30 | 4 |
| 59 | S | 288000 | 337.15 | 10 | 1 | 1 | F | 30236 | 398.15 | 30 | 4 |
| 1 | F | 45663 | 337.15 | 30 | 2 | 1 | F | 33433 | 398.15 | 30 | 4 |
| 1 | F | 123237 | 337.15 | 30 | 2 | 1 | F | 33492 | 398.15 | 30 | 4 |
| 1 | F | 192073 | 337.15 | 30 | 2 | 1 | F | 39094 | 398.15 | 30 | 4 |
| 1 | F | 212696 | 337.15 | 30 | 2 | 1 | F | 51761 | 398.15 | 30 | 4 |
| 1 | F | 304669 | 337.15 | 30 | 2 | 1 | F | 53926 | 398.15 | 30 | 4 |
| 1 | F | 323332 | 337.15 | 30 | 2 | 1 | F | 57124 | 398.15 | 30 | 4 |
| 1 | F | 346814 | 337.15 | 30 | 2 | 1 | F | 61833 | 398.15 | 30 | 4 |
| 1 | F | 452855 | 337.15 | 30 | 2 | 1 | F | 67618 | 398.15 | 30 | 4 |
| 1 | F | 480915 | 337.15 | 30 | 2 | 1 | F | 70177 | 398.15 | 30 | 4 |
| 1 | F | 496672 | 337.15 | 30 | 2 | 1 | F | 71534 | 398.15 | 30 | 4 |
| 1 | F | 557136 | 337.15 | 30 | 2 | 1 | F | 79047 | 398.15 | 30 | 4 |
| 1 | F | 570003 | 337.15 | 30 | 2 | 1 | F | 91295 | 398.15 | 30 | 4 |
| 1 | F | 12019 | 398.15 | 10 | 3 | 1 | F | 92005 | 398.15 | 30 | 4 |
| 1 | F | 18590 | 398.15 | 10 | 3 |
Result
The following temperature non-thermal life stress relationship is used:
- [math]\displaystyle{ \,\!L\left ( f,T \right )=Af^{B}e^{\left ( \frac{E_{a}}{kT} \right )} }[/math]
where [math]\displaystyle{ \,\!f }[/math] is the switching rate, [math]\displaystyle{ \,\!T }[/math] is temperature. [math]\displaystyle{ \,\!L\left ( f,T \right ) }[/math] is the life characteristic affected by the two stresses. This relationship is called temperature non-thermal model in ALTA.
This relationship also can be expressed as the following:
- [math]\displaystyle{ \,\!ln\left ( L\left ( x_{1},x_{2} \right ) \right )=\alpha _{0}+\alpha _{1}x_{1}+\alpha _{2}x_{2} }[/math]
where [math]\displaystyle{ \,\!x_{1}=\frac{1}{T} }[/math] and [math]\displaystyle{ \,\!x_{2}=ln\left ( f \right ) }[/math] . This is the General log-linear model with the proper stress transformation in ALTA.
The failure time distribution is a Weibull distribution. The book has the following results:
- The maximum likelihood estimation (MLE) results for the parameters are: [math]\displaystyle{ \,\!\alpha _{0}=0.671 }[/math] , [math]\displaystyle{ \,\!\alpha _{1}=4640.1 }[/math] , [math]\displaystyle{ \,\!\alpha _{2}=-0.445 }[/math] and [math]\displaystyle{ \,\!\beta =1.805 }[/math].
- The eta parameter in the Weibull distribution at temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is estimated as [math]\displaystyle{ \,\!4.244\times 10^{6} }[/math].
- The estimated reliability at 200,000 cycles and temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is 0.996. Its one-sided lower 90% confidence bound is 0.992.
- The two-sided 90% confidence interval for parameter [math]\displaystyle{ \,\!\alpha _{2} }[/math] is [-0.751, -0.160].
Results in ALTA
- The maximum likelihood estimation (MLE) results for the parameters are:
- These results (especially [math]\displaystyle{ \,\!\alpha _{2} }[/math]) are slightly different from the one given in the book. If we use the results in the book, then the calculated log likelihood value is -710.354601 as given below.
- This likelihood value is slightly smaller than the value given in ALTA which is -710.268519. Therefore, the result in ALTA is better in terms of maximizing the log likelihood value.
- The η parameter in the Weibull distribution at temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is estimated as [math]\displaystyle{ \,\!4.172\times 10^{6} }[/math] .
- The estimated reliability at 200,000 cycles and temperature of 30°C (303.15 F) and switching rate of 5 cycles/minute is 0.996. Its one-sided lower 90% confidence bound is 0.992.
- The two-sided 90% confidence interval for parameter [math]\displaystyle{ \,\!\alpha _{2} }[/math] is [-0.751, -0.160].
If the temperature-Nonthermal relationship is used directly, the same results will be obtained. The following is the estimated model parameters for the temperature-nonthermal model in ALTA. By doing the right transformations for the stresses, a general log linear model can become a temperature-nonthermal model.
