Cumulative Damage Model for Progress Stress Profiles: Difference between revisions

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The data set is from Table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. Model of Eqn. (3.10) is used and the results are given in Table 3.2 on page 511.
The data set is from Table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. Model of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.




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The following life stress relationship and the Weibull distribution are used for the data. At a constant stress V, the   is:
The following life stress relationship and Weibull distribution are used for the data. At a constant stress ''V'', the <math>\eta\,\!</math> is:


where  is the voltage and  is the area.
::<math>ln(\eta(V, A)) = \alpha_{0}+\alpha_{1}ln(V)+\alpha_{2}ln(A)\,\!</math>


The reliability function at time t and stress V is:


When stress is varying with time, the reliability at time   is given as following:
where ''V'' is the voltage and ''A'' is the area.
 
The reliability function at time ''t'' and stress ''V'' is:
 
::<math>R(t,V,A) = e^{-\left(\frac{t}{\eta(V,A)} \right)^\beta}\,\!</math>
 
 
When stress is varying with time, the reliability at time ''t'' is given as following:
 
::<math>R(t,V,A) = e^{-\left(\int_{0}^{t}\frac{1}{\eta(x,A)}dx\right)^\beta}</math>
 


In the book, the following results are provided:
In the book, the following results are provided:
• ML solution for the parameters are given in section 3) of Table 3.2 on page 511:   ( ), , , and .
 
• The maximum log likelihood is -1035.4269.  
* The ML solution for the parameters are given in Section 3 of Table 3.2 on page 511: <math>1/\beta\,\!</math> = 0.07856677 (<math>\beta\,\!</math>=12.728027)&nbsp;,&nbsp; <math>\alpha_{0}\,\!</math> = 3.673202&nbsp;, &nbsp; <math>\alpha_{1} = \,\!</math>0.05843506&nbsp;, &nbsp;and&nbsp; <math>\alpha_{2}=\,\!</math>-0.058626.
* The maximum log likelihood is -1035.4269.  






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Revision as of 22:51, 9 June 2014

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ALTA_Reference_Examples

Compares the results for the cumulative damage model for ramp stress.


Reference Case

The data set is from Table on page 232 in the book Accelerated Testing: Statistical Models, Test Plans, and Data Analysis by Dr. Nelson, John Wiley & Sons, 1990. Model of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.


Data An accelerated test employed a pair of parallel disk electrodes immersed in insulating oil. Voltage V across the pair was increased linearly with time t at a specified rate, and the voltage at oil breakdown was recorded. Since V = Rt(R is the ramp rate), the time to breakdown can be recorded. The breakdown time is also affected by the two electrode areas A. Three voltage linear rates and two electrode areas are used. The six stress profiles are:


Voltage Ramp Rate (Volts/sec) Area (Sq. Inch)
10 1
100 1
1000 1
10 9
100 9
1000 9


The following table shows the failure data for an area of 1 square inch.

Time to Failure Voltage Ramp Rate Time to Failure Voltage Ramp Rate Time to Failure Voltage Ramp Rate
41 10 46 100 55 1000
43 10 50 100 57 1000
42 10 39 100 59 1000
43 10 36 100 57 1000
44 10 47 100 55 1000
40 10 55 100 60 1000
38 10 49 100 53 1000
47 10 58 100 51 1000
43 10 50 100 57 1000
45 10 48 100 54 1000
38 10 53 100 57 1000
44 10 54 100 64 1000
49 10 55 100 53 1000
42 10 37 100 63 1000
42 10 53 100 51 1000
51 10 52 100 62 1000
39 10 53 100 62 1000
34 10 50 100 56 1000
41 10 52 100 62 1000
41 10 50 100 57 1000
35 10 45 100 41 1000
44 10 48 100 41 1000
46 10 53 100 51 1000
39 10 50 100 58 1000
41 10 43 100 59 1000
40 10 50 100 60 1000
52 10 42 100 58 1000
40 10 45 100 55 1000
35 10 47 100 59 1000
40 10 34 100 63 1000
39 10 46 100 63 1000
46 10 42 100 53 1000
47 10 46 100 63 1000
44 10 46 100 61 1000
41 10 52 100 59 1000
46 10 47 100 53 1000
46 10 53 100 60 1000
42 10 52 100 58 1000
45 10 45 100 62 1000
42 10 47 100 56 1000
44 10 43 100 69 1000
41 10 45 100 65 1000
44 10 54 100 51 1000
38 10 51 100 56 1000
36 10 46 100 55 1000
44 10 55 100 57 1000
50 10 44 100 54 1000
47 10 49 100 63 1000
49 10 49 100 65 1000
46 10 53 100 65 1000
34 10 53 100 56 1000
47 10 54 100 54 1000
49 10 53 100 65 1000
43 10 53 100 60 1000
43 10 51 100 60 1000
48 10 48 100 64 1000
34 10 49 100 60 1000
38 10 52 100 54 1000
47 10 45 100 57 1000
35 10 49 100 61 1000


The following table shows the failure data for an area of 9 square inch.

Time to Failure Voltage Ramp Rate Time to Failure Voltage Ramp Rate Time to Failure Voltage Ramp Rate
33 10 43 100 50 1000
37 10 42 100 53 1000
38 10 45 100 50 1000
38 10 48 100 49 1000
38 10 38 100 53 1000
37 10 44 100 51 1000
27 10 37 100 47 1000
42 10 44 100 44 1000
39 10 43 100 53 1000
38 10 42 100 42 1000
42 10 43 100 49 1000
32 10 49 100 46 1000
42 10 44 100 50 1000
40 10 45 100 38 1000
32 10 50 100 48 1000
38 10 44 100 43 1000
36 10 44 100 52 1000
42 10 45 100 53 1000
20 10 41 100 52 1000
37 10 48 100 48 1000
43 10 45 100 45 1000
40 10 48 100 53 1000
38 10 43 100 52 1000
43 10 49 100 50 1000
39 10 50 100 55 1000
41 10 45 100 50 1000
35 10 45 100 43 1000
41 10 46 100 52 1000
40 10 47 100 50 1000
32 10 42 100 54 1000
38 10 47 100 51 1000
40 10 48 100 40 1000
37 10 47 100 52 1000
29 10 48 100 53 1000
31 10 39 100 47 1000
41 10 49 100 45 1000
38 10 44 100 53 1000
36 10 47 100 47 1000
35 10 34 100 54 1000
40 10 41 100 50 1000
37 10 45 100 32 1000
41 10 48 100 48 1000
36 10 44 100 53 1000
39 10 47 100 52 1000
43 10 45 100 45 1000
42 10 50 100 48 1000
43 10 40 100 48 1000
43 10 47 100 51 1000
41 10 47 100 53 1000
44 10 43 100 48 1000
37 10 49 100 54 1000
43 10 45 100 51 1000
38 10 45 100 50 1000
40 10 45 100 54 1000
40 10 47 100 35 1000
38 10 39 100 56 1000
33 10 44 100 51 1000
40 10 37 100 48 1000
35 10 47 100 48 1000
41 10 48 100 46 1000

Result

The following life stress relationship and Weibull distribution are used for the data. At a constant stress V, the [math]\displaystyle{ \eta\,\! }[/math] is:

[math]\displaystyle{ ln(\eta(V, A)) = \alpha_{0}+\alpha_{1}ln(V)+\alpha_{2}ln(A)\,\! }[/math]


where V is the voltage and A is the area.

The reliability function at time t and stress V is:

[math]\displaystyle{ R(t,V,A) = e^{-\left(\frac{t}{\eta(V,A)} \right)^\beta}\,\! }[/math]


When stress is varying with time, the reliability at time t is given as following:

[math]\displaystyle{ R(t,V,A) = e^{-\left(\int_{0}^{t}\frac{1}{\eta(x,A)}dx\right)^\beta} }[/math]


In the book, the following results are provided:

  • The ML solution for the parameters are given in Section 3 of Table 3.2 on page 511: [math]\displaystyle{ 1/\beta\,\! }[/math] = 0.07856677 ([math]\displaystyle{ \beta\,\! }[/math]=12.728027) ,  [math]\displaystyle{ \alpha_{0}\,\! }[/math] = 3.673202 ,   [math]\displaystyle{ \alpha_{1} = \,\! }[/math]0.05843506 ,  and  [math]\displaystyle{ \alpha_{2}=\,\! }[/math]-0.058626.
  • The maximum log likelihood is -1035.4269.


Results in ALTA

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