Temperature-Nonthermal (TNT)-Weibull Model: Difference between revisions

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{{Reference Example|ALTA_Reference_Examples_Banner.png|ALTA_Reference_Examples}}
{{Reference Example|{{Banner ALTA Reference Examples}}}}


This example validates the calculation of Temperature-nonthermal relationship for Weibull distribution.
This example validates the calculations for the temperature-nonthermal life-stress relationship with a Weibull distribution in ALTA standard folios.  




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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.
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.


{| {{table}}
{| {{table}}
| align="center" style="background:#f0f0f0;"|'''Number in Group'''
!Number in Group
| align="center" style="background:#f0f0f0;"|'''State F/S'''
!State F/S
| align="center" style="background:#f0f0f0;"|'''Time to State'''
!Time to State
| align="center" style="background:#f0f0f0;"|'''Temperature (F)'''
!Temperature (F)
| align="center" style="background:#f0f0f0;"|'''Switching Rate'''
!Switching Rate
| align="center" style="background:#f0f0f0;"|'''Subset ID'''
!Subset ID
| align="center" style="background:#f0f0f0;"|'''Number in Group'''
!Number in Group
| align="center" style="background:#f0f0f0;"|'''State F/S'''
!State F/S
| align="center" style="background:#f0f0f0;"|'''Time to State'''
| align="center" style="background:#f0f0f0;"|'''Time to State'''
| align="center" style="background:#f0f0f0;"|'''Temperature (F)'''
| align="center" style="background:#f0f0f0;"|'''Temperature (F)'''
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The following temperature non-thermal life stress relationship is used:
::<math>\,\!L\left ( f,T \right )=Af^{B}e^{\left ( \frac{E_{a}}{kT} \right )}</math>
where <math>\,\!f</math> is the switching rate, <math>\,\!T</math> is temperature. <math>\,\!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>\,\!ln\left ( L\left ( x_{1},x_{2} \right ) \right )=\alpha _{0}+\alpha _{1}x_{1}+\alpha _{2}x_{2}</math>
where <math>\,\!x_{1}=\frac{1}{T}</math> and <math>\,\!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>\,\!\alpha _{0}=0.671</math> , <math>\,\!\alpha _{1}=4640.1</math> , <math>\,\!\alpha _{2}=-0.445</math> and <math>\,\!\beta =1.805</math>.
*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 <math>\,\!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>\,\!\alpha _{2}</math> is [-0.751, -0.160].




{{Reference_Example_Heading4|ALTA}}
{{Reference_Example_Heading4|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 (TNT) life-stress relationship.
'''General Log-Linear (GLL)-Weibull Model'''
To use the GLL-Weibull model with the same life-stress relationship as the one in the book, the following stress transformations should be used:
[[image:Two Stress GLL Weibull_Stress Transform.png|center]]
Based on this model, the maximum likelihood estimation (MLE) results for the parameters are:
[[image:Two Stress GLL Weibull_Analysis Summary GLL.png|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Results in ALTA'''</div>
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.
[[image:Two Stress GLL Weibull_Alter Parameters.png|center]]
The resulting LK Value for the altered parameters is -710.356064, as shown next.
[[image:Two Stress GLL Weibull_Analysis Summary GLL new alpha.png|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Altered Parameters'''</div>
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.
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 <math>\,\!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, as shown next.
[[image:Two Stress GLL Weibull_QPC Reliability.png|center]]
*The two-sided 90% confidence interval for parameter <math>\,\!\alpha _{2}</math> is [-0.751, -0.160], as shown next.
[[image:Two Stress GLL Weibull_Parameter Bounds.png|center]]
'''Temperature-Nonthermal (TNT)-Weibull Model'''
If we use the temperature-nonthermal life-stress relationship to analyze the data, the same results would be obtained, as shown in the following picture. Therefore, by selecting the appropriate stress transformations, a general log-linear model can become a temperature-nonthermal model.
[[image:Two Stress GLL Weibull_Analysis Summary TNT.png|center]]

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Temperature-Nonthermal (TNT)-Weibull Model

This example validates the calculations for the temperature-nonthermal life-stress relationship with a Weibull distribution in ALTA standard folios.


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 [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 [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 (TNT) life-stress relationship.


General Log-Linear (GLL)-Weibull Model

To use the GLL-Weibull model with the same life-stress relationship as the one in the book, the following stress transformations should be used:

Two Stress GLL Weibull Stress Transform.png


Based on this model, the maximum likelihood estimation (MLE) results for the parameters are:

Two Stress GLL Weibull Analysis Summary GLL.png
Results 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.

Two Stress GLL Weibull Alter Parameters.png


The resulting LK Value for the altered parameters is -710.356064, as shown next.

Two Stress GLL Weibull Analysis Summary GLL new alpha.png
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.

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 [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, as shown next.
Two Stress GLL Weibull QPC Reliability.png


  • The two-sided 90% confidence interval for parameter [math]\displaystyle{ \,\!\alpha _{2} }[/math] is [-0.751, -0.160], as shown next.
Two Stress GLL Weibull Parameter Bounds.png


Temperature-Nonthermal (TNT)-Weibull Model

If we use the temperature-nonthermal life-stress relationship to analyze the data, the same results would be obtained, as shown in the following picture. Therefore, by selecting the appropriate stress transformations, a general log-linear model can become a temperature-nonthermal model.

Two Stress GLL Weibull Analysis Summary TNT.png
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