Cumulative Damage Model for Step Stress Profiles - Revision history

Kate Racaza at 18:22, 28 September 2015

2015-09-28T18:22:55Z

← Older revision		Revision as of 18:22, 28 September 2015
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	{{Reference Example\|{{Banner ALTA Reference Examples}}}}		{{Reference Example\|{{Banner ALTA Reference Examples}}}}

	This example ~~compares~~ the results of a cumulative damage model for a step stress test.		This example validates the results for a cumulative damage model for a step stress test in an ALTA standard folio.

Kate Racaza at 21:43, 16 June 2014

2014-06-16T21:43:31Z

Kate Racaza at 20:50, 16 June 2014

2014-06-16T20:50:14Z

← Older revision		Revision as of 20:50, 16 June 2014
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	The log likelihood value calculated by hand in Excel is -115.0334, as given in the above table. It matches the value in ALTA. The slight difference is caused by the precision error. Therefore, the log likelihood value calculated in ALTA is correct.		The log likelihood value calculated by hand in Excel is -115.0334, as given in the above table. It matches the value in ALTA. The slight difference is caused by the precision error. Therefore, the log likelihood value calculated in ALTA is correct.

	In fact, the optimization tool ~~“Solver”~~ in Excel can also be used to find the ML solutions. Using ~~“Solver”~~ and the “GRG” nonlinear optimization, we get the same results in Excel as in ALTA.		In fact, the optimization tool in Excel can also be used to find the ML solutions. Using the “Solver Add-in” program in Excel and the “GRG” nonlinear optimization, we get the same results in Excel as in ALTA.

	The following reliability plot shows that the predicted values (the line) by the model can match the observed values (the points) very well.		The following reliability plot shows that the predicted values (the line) by the model can match the observed values (the points) very well.

	[[Image:Step Stress_plot.png\|center\|550 px]]		[[Image:Step Stress_plot.png\|center\|550 px]]

Kate Racaza at 20:16, 16 June 2014

2014-06-16T20:16:35Z

@@ Line 372: / Line 372: @@
 | 9||100||102||36||27||1.333333||0.287682072||0.031977014
 |-
-| ||||||||||||'''A'''=||0.132138165
+|colspan="7" style="text-align:right;"|'''A ='''||0.132138165
 |-
 |}
-The total cumulative damage at failure time of 102 is '''A''' in the above table.
+The total cumulative damage at a failure time of 102 is '''A''' in the above table.
 Since the reliability function is:
@@ Line 385: / Line 385: @@
-The probability density function (pdf) at time t is:
+The probability density function (pdf) at time ''t'' is:
 ::<math>\begin{align}
@@ Line 404: / Line 404: @@
-Take log of the above value, we get the log likelihood value for the first observation as  .
+Take the logarithm of the above value and we get the log likelihood value for the first observation as <math>log(0.016) = -4.1489\,\!</math>.
-Repeating the above procedure, we can get the log likelihood value for each observation as given in the following table.
+Repeating the above procedure, we can get the log likelihood value for each observation, as given in the following table.

Kate Racaza at 18:48, 16 June 2014

2014-06-16T18:48:25Z

← Older revision		Revision as of 18:48, 16 June 2014
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	The total ~~cumulated~~ damage at failure time of 102 is '''A''' in the above table.		The total cumulative damage at failure time of 102 is '''A''' in the above table.

	Since the reliability function is:		Since the reliability function is:

			::<math>R(t) = e^{-\left[\int^{t}_{0}e^{-(\alpha_{0}+\alpha_{1}ln(S(x)))} dx \right]^{\beta}} = e^{-A^{\beta}}\,\!</math>



	The probability density function (pdf) at time t is:		The probability density function (pdf) at time t is:

			::<math>\begin{align}
			f(t) =& R(t) \cdot \beta \cdot \left[\int^{t}_{0}e^{-(\alpha_{0}+\alpha_{1}ln(S(x)))} dx \right]^{\beta-1}\cdot e^{-(\alpha_{0}+\alpha_{1}ln(S(t)))} \\
			=& e^{-A^{\beta}}\cdot \beta \cdot A^{\beta-1}\cdot e^{-(\alpha_{0}+\alpha_{1}ln(S(t)))} \\
			\end{align}\,\!</math>


			If an observation is a failure, then the pdf will be used to calculate its log likelihood value. If an observation is a suspension, then the reliability will be used to calculate its log likelihood value.

	~~If an~~ observation is failure, ~~then~~ the pdf ~~will be~~ used ~~to calculate its log likelihood value~~. ~~If an observation is a suspension~~, ~~then~~ the ~~reliability will be used to calculate its log likelihood~~ value.		Since the 1st observation is a failure, the pdf is used. Using the model parameters and the cumulative damage A, the value of the pdf is:

	~~Since the 1st observation is a failure, so the pdf is used~~. ~~Using the model parameters and the cumulated damage A~~, ~~the value of the pdf is:~~		::<math> \begin{align}
			f(t) =& e^{-A^{\beta}} \cdot \beta \cdot A^{\beta - 1} \cdot e^{-(\alpha_{0}+\alpha_{1}ln(S(t)))} \\
			=& e^{-0.13214^{0.7906}} \times 0.7906 \times 0.13214^{0.7906-1} \times e^{-(8.7685 - 16.103 \times 0.2877)} \\
			=& 0.016
			\end{align}\,\!</math>



	Take log of the above value, we get the log likelihood value for the first observation as .		Take log of the above value, we get the log likelihood value for the first observation as .

	Repeating the above procedure, we can get the log likelihood value for each observation as given in the following table.		Repeating the above procedure, we can get the log likelihood value for each observation as given in the following table.

Kate Racaza at 18:22, 16 June 2014

2014-06-16T18:22:15Z

← Older revision		Revision as of 18:22, 16 June 2014
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	'''Validate the Likelihood Calculation in ALTA'''		'''Validate the Likelihood Calculation in ALTA'''

			For the ML solution given in the book, the parameters are: <math>\beta\,\!</math> = 0.75597,  <math>\alpha_{0}\,\!</math> = 9.573776, <math>\alpha_{1}\,\!</math> = -19.937, and the log likelihood value is -103.53. Entering these parameters in ALTA, we get a log likelihood value of -115.502692. We will validate the calculation for the log likelihood value in ALTA to make sure it is correct.

			Define the stress value at step ''i'' as <math>S_{i}\,\!</math>. If the test unit stayed in this stress level for a time period of <math>\Delta t_{i}\,\!</math>, then the cumulative damage at this step will be <math>\frac{1}{\eta(S_{i})}\times \Delta t_{i}\,\!</math> and

			::<math>A_{i} = \frac{1}{\eta(S_{i})} \times \Delta t_{i} = e^{-(\alpha_{0}+\alpha_{1}ln(S_{1}))} \times \Delta t_{i}\,\!</math>


			For example, for the first failure, the failure occurred at 102, which is at the 9th step of profile G1. The test unit accumulates damage from each step stress it experienced before it fails. The first step stress value is 0.185185 which is the voltage divided by the thickness (5 Kvolts/27 mils). The length of the first step is 10 mins. Therefore, the damage at the 1st step segment is:

			::<math>A_{1} = e^{-(\alpha_{0}+\alpha_{1}ln(S_{1}))} \times \Delta t_{i} = e^{-(8.757876+16.095189 \times 1.6864)}\times 10 = 2.56 \times 10^{-15}\,\!</math>


			We can use this method to calculate the damage at each stress segment. The total damage will be the sum of the damage at each segment. For the first failure at time 102 under stress profile G1, the damage at each stress segment is calculated as given below.

			{\| {{table}}
			! Step\|\|Profile G1\|\|Segment End\|\|Voltage\|\|Thickness\|\|Stress\|\|ln(S)\|\|Cum Damage
			\|-
			\| 1\|\|0\|\|10\|\|5\|\|27\|\|0.185185\|\|-1.68639895\|\|2.50025E-15
			\|-
			\| 2\|\|10\|\|20\|\|10\|\|27\|\|0.37037\|\|-0.99325177\|\|1.76011E-10
			\|-
			\| 3\|\|20\|\|30\|\|15\|\|27\|\|0.555556\|\|-0.58778666\|\|1.20553E-07
			\|-
			\| 4\|\|30\|\|40\|\|20\|\|27\|\|0.740741\|\|-0.30010459\|\|1.23907E-05
			\|-
			\| 5\|\|40\|\|55\|\|26\|\|27\|\|0.962963\|\|-0.03774033\|\|0.0012707
			\|-
			\| 6\|\|55\|\|70\|\|28.5\|\|27\|\|1.055556\|\|0.054067221\|\|0.005573181
			\|-
			\| 7\|\|70\|\|85\|\|31\|\|27\|\|1.148148\|\|0.138150338\|\|0.021584512
			\|-
			\| 8\|\|85\|\|100\|\|33.4\|\|27\|\|1.233333\|\|0.209720531\|\|0.071720247
			\|-
			\| 9\|\|100\|\|102\|\|36\|\|27\|\|1.333333\|\|0.287682072\|\|0.031977014
			\|-
			\| \|\|\|\|\|\|\|\|\|\|\|\|'''A'''=\|\|0.132138165
			\|-
			\|}


			The total cumulated damage at failure time of 102 is '''A''' in the above table.

			Since the reliability function is:


			The probability density function (pdf) at time t is:



			If an observation is failure, then the pdf will be used to calculate its log likelihood value. If an observation is a suspension, then the reliability will be used to calculate its log likelihood value.

			Since the 1st observation is a failure, so the pdf is used. Using the model parameters and the cumulated damage A, the value of the pdf is:



			Take log of the above value, we get the log likelihood value for the first observation as .

			Repeating the above procedure, we can get the log likelihood value for each observation as given in the following table.

Kate Racaza at 18:07, 16 June 2014

2014-06-16T18:07:30Z

← Older revision		Revision as of 18:07, 16 June 2014
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	The next step is to enter the failure data into an ALTA standard folio and ~~then~~ use the stress profiles to define the stress values, as shown next.		The next step is to enter the failure data into an ALTA standard folio and use the stress profiles to define the stress values, as shown next.

			[[Image: Step Stress_folio.png\|center]]


			The log transformation is used for the stress. To compare the parameter values used in the book with the results obtained by ALTA, we need do convert the values in the book in terms of the parameters used in ALTA. Therefore the parameters in the book are: <math>\beta\,\!</math> = 0.75597,   <math>\alpha_{0}\,\!</math> = 9.573776,   <math>\alpha_{1}\,\!</math> = -19.937. Comparing these values to the value shown in the picture above, we see that the results obtained in ALTA are close to the results shown in the book.

			The picture above also shows that the maximum log likelihood (LK Value) is -114.932651. This value is smaller than the value given in the book. To validate what the LK value would be if the results in the book were used in ALTA, we use the Alter Parameters tool, as shown next.

			[[Image: Step Stress_alter parameters.png\|center]]


			The resulting LK Value is -115.502692, as shown next.

			[[Image: Step Stress_alter parameters results.png\|center\|Altered Parameters]]
			<center>Altered Parameters</center>


			This value is different from the value of -13.53 given in the book. Later, we will validate the calculations for this result.

			Using the parameters values originally calculated in ALTA:

			* The 1% percentile point (B1 life) at 0.4 Kvolts/mil is 4.8342 X 10<sup>7</sup> hours, as shown next.

			[[Image: Step Stress qcp.png\|center\|550px]]


			* The 2-sided 95% confidence intervals for model parameters are shown next.

			[[Image: Step Stress bounds.png\|center]]


			'''Validate the Likelihood Calculation in ALTA'''

Kate Racaza at 16:35, 13 June 2014

2014-06-13T16:35:51Z

← Older revision		Revision as of 16:35, 13 June 2014
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	{{Reference Example\|~~ALTA_Reference_Examples_Banner.png\|ALTA_Reference_Examples~~}}		{{Reference Example\|{{Banner ALTA Reference Examples}}}}

	This example compares the results of a cumulative damage model for a step stress test.		This example compares the results of a cumulative damage model for a step stress test.

Kate Racaza at 23:00, 11 June 2014

2014-06-11T23:00:36Z

← Older revision		Revision as of 23:00, 11 June 2014
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	{{Reference_Example_Heading2}}		{{Reference_Example_Heading2}}

	A step-stress test is conducted for one type of cable insulation to estimate the insulation life at a constant design stress of 400 volts/mil. The test cables are of different thickness in unit of mils (0.001 inch). The stress is the applied step voltage divided by the thickness. A total of 6 step-stress profiles are used. These stress profiles are calculated based the applied voltage (Kilovolts) and the insulation thickness given on page 495 and Table 2.1. For all the profiles, the holding time is 10 mins for the first 4 steps. From step 5, different a holding time is applied at each step for each of the stress profiles. These profiles are given in the following tables.		A step-stress test is conducted for one type of cable insulation to estimate the insulation life at a constant design stress of 400 volts/mil. The test cables are of different thickness, in unit of mils (0.001 inch). The stress is the applied step voltage divided by the thickness.

			A total of 6 step-stress profiles are used. These stress profiles are calculated based on the applied voltage (Kilovolts) and the insulation thickness given on page 495 and Table 2.1. For all the stress profiles, the holding time for the first 4 steps is 10 mins. From step 5 onwards, a different a holding time is applied at each step for each of the stress profiles. These profiles are given in the following tables.

	Profile 1 (named "G1"): ~~Holding~~ time is 15 mins ~~after step 4~~ and the thickness is 27 mils.
			Profile 1 (named "G1"): The holding time after step 4 is 15 mins and the thickness is 27 mils.

	{\| {{table}}		{\| {{table}}
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	Profile 2 (named "G2"): ~~Holding~~ time is 60 mins ~~after step 4~~ and the thickness is 29.5 mils.		Profile 2 (named "G2"): The holding time after step 4 is 60 mins and the thickness is 29.5 mils.

	{\| {{table}}		{\| {{table}}
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	\|}		\|}

	Profile 3 (named "G3"): ~~Holding~~ time is 60 mins ~~after step 4~~ and the thickness is 28 mils.		Profile 3 (named "G3"): The holding time after step 4 is 60 mins and the thickness is 28 mils.

	{\| {{table}}		{\| {{table}}
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	\|}		\|}

	Profile 4 (named "G4"): ~~Holding~~ time is 240 mins ~~after step 4~~ and the thickness is 29 mils.		Profile 4 (named "G4"): The holding time after step 4 is 240 mins and the thickness is 29 mils.

	{\| {{table}}		{\| {{table}}
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	\|}		\|}

	Profile 5 (named "G5"): ~~Holding~~ time is 240 mins ~~after step 4~~ and the thickness is 30 mils.		Profile 5 (named "G5"): The holding time after step 4 is 240 mins and the thickness is 30 mils.

	{\| {{table}}		{\| {{table}}
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	\|}		\|}

	Profile 6 (named "G6"): ~~Holding~~ time is 960 mins ~~after step 4~~ and the thickness is 30 mils.		Profile 6 (named "G6"): The holding time after step 4 is 960 mins and the thickness is 30 mils.

	{\| {{table}}		{\| {{table}}
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	{{Reference_Example_Heading3}}		{{Reference_Example_Heading3}}

	The power law life stress relationship and the Weibull distribution is used ~~for~~ the data. At a constant stress ''V'', the <math>\eta\,\!</math> is:		The power law life stress relationship and the Weibull distribution is used to analyze the data. At a constant stress ''V'', the <math>\eta\,\!</math> is:

	::<math>\eta(V) = \left(\frac{V_{0}}{V} \right)^p\,\!</math>		::<math>\eta(V) = \left(\frac{V_{0}}{V} \right)^p\,\!</math>
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	The reliability function at time ''t'' is stress ''V'' is:		The reliability function at time ''t'' and stress ''V'' is:

	::<math>R(t,V) = e^{-\left(\frac{t}{\eta(V)} \right)^\beta}\,\!</math>		::<math>R(t,V) = e^{-\left(\frac{t}{\eta(V)} \right)^\beta}\,\!</math>

Kate Racaza at 22:26, 10 June 2014

2014-06-10T22:26:37Z

← Older revision		Revision as of 22:26, 10 June 2014
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	{{Reference_Example_Heading4\|ALTA}}		{{Reference_Example_Heading4\|ALTA}}

			First, we create each stress profile in ALTA. For example, the following picture shows the data for Profile G1.

			[[Image: Step Stress_G1.png\|center\|550px]]


			The following picture shows the plot for this stress profile.

			[[Image: Step Stress_G1 plot.png\|center\|550px]]


			The next step is to enter the failure data into an ALTA standard folio and then use the stress profiles to define the stress values, as shown next.

← Older revision		Revision as of 21:43, 16 June 2014
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	{{Reference_Example_Heading2}}		{{Reference_Example_Heading2}}

	A step-stress test is conducted for one type of cable insulation to estimate the insulation life at a constant design stress of 400 volts/mil. The test cables are of different thickness, in ~~unit~~ of mils (0.001 inch). The stress is the applied step voltage divided by the thickness.		A step-stress test is conducted for one type of cable insulation to estimate the insulation life at a constant design stress of 400 volts/mil. The test cables are of different thickness, in units of mils (0.001 inch). The stress is the applied step voltage divided by the thickness.

	A total of 6 step-stress profiles are used. These stress profiles are calculated based on the applied voltage (Kilovolts) and the insulation thickness given on page 495 and Table 2.1. For all the stress profiles, the holding time for the first 4 steps is 10 mins. From step 5 onwards, a different a holding time is applied at each step for each of the stress profiles. These profiles are given in the following tables.		A total of 6 step-stress profiles are used. These stress profiles are calculated based on the applied voltage (Kilovolts) and the insulation thickness given on page 495 and Table 2.1. For all the stress profiles, the holding time for the first 4 steps is 10 mins. From step 5 onwards, a different a holding time is applied at each step for each of the stress profiles. These profiles are given in the following tables.
Line 257:		Line 257:
	{{Reference_Example_Heading3}}		{{Reference_Example_Heading3}}

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

	::<math>\eta(V) = \left(\frac{V_{0}}{V} \right)^p\,\!</math>		::<math>\eta(V) = \left(\frac{V_{0}}{V} \right)^p\,\!</math>


	where <math>V_{0}\,\!</math> and <math>p\,\!</math> are the model parameters used in the book.		where <math>V_{0}\,\!</math> and <math>p\,\!</math> are the model parameters used in the book. The above equation can be rewritten as:


	The above equation can be rewritten as:

	::<math>\eta(V) = e^{\alpha_{0}+\alpha_{1}ln(V)}\,\!</math>		::<math>\eta(V) = e^{\alpha_{0}+\alpha_{1}ln(V)}\,\!</math>


	where <math>\alpha_{0} = ~~pln~~(V_{0})\,\!</math>  and  <math>\alpha_{1} = -p\,\!</math>		where <math>\alpha_{0} = p \cdot ln(V_{0})\,\!</math>  and  <math>\alpha_{1} = -p\,\!</math>


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	Using the parameters values originally calculated in ALTA:		Using the parameters values originally calculated in ALTA:

	* The 1% percentile point (B1 life) at 0.4 Kvolts/mil is 4.8342 X 10<sup>7</sup> hours, as shown next.		* The 1% percentile point (B1 life) at 0.4 Kvolts/mil is 4.8342 X 10<sup>7</sup> hours, as shown next. The value given in the book is 2.81 X 10<sup>9</sup>. The results are different because of the differences in the values of the parameters of the model.

	[[Image: Step Stress qcp.png\|center\|550px]]		[[Image: Step Stress qcp.png\|center\|550px]]


	* The 2-sided 95% confidence intervals for model parameters are shown next.		* The 2-sided 95% confidence intervals for the model parameters are shown next. These results are also different from the results in the book due to the estimated parameter values.

	[[Image: Step Stress bounds.png\|center]]		[[Image: Step Stress bounds.png\|center]]
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	Define the stress value at step ''i'' as <math>S_{i}\,\!</math>. If the test unit stayed in this stress level for a time period of <math>\Delta t_{i}\,\!</math>, then the cumulative damage at this step will be <math>\frac{1}{\eta(S_{i})}\times \Delta t_{i}\,\!</math> and		Define the stress value at step ''i'' as <math>S_{i}\,\!</math>. If the test unit stayed in this stress level for a time period of <math>\Delta t_{i}\,\!</math>, then the cumulative damage at this step will be <math>\frac{1}{\eta(S_{i})}\times \Delta t_{i}\,\!</math> and

	::<math>A_{i} = \frac{1}{\eta(S_{i})} \times \Delta t_{i} = e^{-(\alpha_{0}+\alpha_{1}ln(S_{1}))} \times \Delta t_{i}\,\!</math>		::<math>A_{i} = \frac{1}{\eta(S_{i})} \times \Delta t_{i} = e^{-(\alpha_{0}+\alpha_{1}ln(S_{i}))} \times \Delta t_{i}\,\!</math>


	For example, for the first failure, the failure occurred at 102, which is at the 9th step of profile G1. The test unit accumulates damage from each step stress it experienced before it fails. The first step stress value is 0.185185 which is the voltage divided by the thickness (5 Kvolts/27 mils). The length of the first step is 10 mins. Therefore, the damage at the 1st step segment is:		For example, for the first failure, the failure occurred at 102, which is at the 9th step of profile G1. The test unit accumulates damage from each step stress it experienced before it fails. The first step stress value is 0.185185, which is the voltage divided by the thickness (5 Kvolts/27 mils). The length of the first step is 10 mins. Therefore, the damage at the 1st step segment is:

	::<math>A_{1} = e^{-(\alpha_{0}+\alpha_{1}ln(S_{1}))} \times \Delta t_{i} = e^{-(8.757876+16.095189 \times 1.6864)}\times 10 = 2.56 \times 10^{-15}\,\!</math>		::<math>A_{1} = e^{-(\alpha_{0}+\alpha_{1}ln(S_{1}))} \times \Delta t_{i} = e^{-(8.757876+16.095189 \times 1.6864)}\times 10 = 2.56 \times 10^{-15}\,\!</math>


	We can use this method to calculate the damage at each stress segment. The total damage will be the sum of the damage at each segment. For the first failure at time 102 under stress profile G1, the damage at each stress segment is calculated as given below.		We can use this method to calculate the damage at each stress segment. The total damage will be the sum of the damage at each segment. For the first failure at time 102 under stress profile G1, the damage at each stress segment is calculated, as given below.

	{\| {{table}}		{\| {{table}}
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	The log likelihood value calculated by hand in Excel is -115.0334, as given in the above table. It matches the value in ALTA. The slight difference is caused by the precision error. Therefore, the log likelihood value calculated in ALTA is correct.		The log likelihood value calculated by hand in Microsoft Excel is -115.0334, as given in the above table. It matches the value in ALTA. The slight difference is caused by the precision error. Therefore, the log likelihood value calculated in ALTA is correct.

	In fact, the optimization tool in Excel can also be used to find the ML solutions. Using the “Solver Add-in” program in Excel and the “GRG” nonlinear optimization, we get the same results in Excel as in ALTA.		In fact, the optimization tool in Excel can also be used to find the ML solutions. Using the “Solver Add-in” program in Excel and the “GRG” nonlinear optimization, we get the same results in Excel as in ALTA.