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| {{Reference_Example_Heading4|ALTA}} | | {{Reference_Example_Heading4|ALTA}} |
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| In ALTA, the General log linear model can be used. To have the same life-stress relation as the one in the book, the following transformation should be used for each stress:
| | We will first perform the analysis using the general log-linear (GLL) life-stress relationship, and then compare its results with the temperature-nonthermal model (TNT) life-stress relationship. |
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| [[image:Two Stress GLL Weibull_Stress Transform.png|center]] | | [[image:Two Stress GLL Weibull_Stress Transform.png|center]] |
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| *The maximum likelihood estimation (MLE) results for the parameters are: | | *Based on this model, the maximum likelihood estimation (MLE) results for the parameters are: |
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| [[image:Two Stress GLL Weibull_Analysis Summary GLL.png|center]] | | [[image:Two Stress GLL Weibull_Analysis Summary GLL.png|center]] |
| | Result in ALTA |
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| :These results (especially <math>\,\!\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 -710.268628 as given below. | | :These results are slightly different from the results given in the book (especially for <math>\,\!\alpha _{2}</math>). To see what the log likelihood value (LK Value) would be if we used the parameter values in the book, we use the Alter Parameters tool, as shown next. |
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| | [[image:Two Stress GLL Weibull_Alter Parameters.png|center]] |
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| | The resulting LK Value for the altered parameters is -710.356064, as shown next. |
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| [[image:Two Stress GLL Weibull_Analysis Summary GLL new alpha.png|center]] | | [[image:Two Stress GLL Weibull_Analysis Summary GLL new alpha.png|center]] |
| | Altered Parameters |
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| | :This likelihood value is slightly smaller than the value that was originally calculated in ALTA, which was -710.268519. Therefore, the result in ALTA is better in terms of maximizing the log likelihood value. |
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| :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.
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| *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>\,\!4.172\times 10^{6}</math> .
| | The resulting LK Value for the altered parameters is -710.356064, as shown next. |
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| | [[image:Two Stress GLL Weibull_Analysis Summary GLL new alpha.png|center]] |
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| [[image:Two Stress GLL Weibull_Eta.png|center]]
| | Using the parameters originally calculated in ALTA: |
| | *The <math>\,\!\eta</math> parameter in the Weibull distribution at temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is estimated as . |
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Temperature-Nonthermal (TNT)-Weibull Model
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This example compares the results for the temperature-nonthermal life-stress relationship with a 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 relay. 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. In ALTA, this life-stress relationship is called the "temperature non-thermal" model.
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] .
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
We will first perform the analysis using the general log-linear (GLL) life-stress relationship, and then compare its results with the temperature-nonthermal model (TNT) life-stress relationship.
- Based on this model, the maximum likelihood estimation (MLE) results for the parameters are:
Result in ALTA
- These results are slightly different from the results given in the book (especially for [math]\displaystyle{ \,\!\alpha _{2} }[/math]). To see what the log likelihood value (LK Value) would be if we used the parameter values in the book, we use the Alter Parameters tool, as shown next.
The resulting LK Value for the altered parameters is -710.356064, as shown next.
Altered Parameters
- This likelihood value is slightly smaller than the value that was originally calculated in ALTA, which was -710.268519. Therefore, the result in ALTA is better in terms of maximizing the log likelihood value.
The resulting LK Value for the altered parameters is -710.356064, as shown next.
Using the parameters originally calculated in ALTA:
- The [math]\displaystyle{ \,\!\eta }[/math] parameter in the Weibull distribution at temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is estimated as .
- 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.
