Template:Inverse power law relationship - Revision history

Richard House: Redirected page to Inverse Power Law (IPL) Relationship

2012-08-15T22:55:34Z

Redirected page to Inverse Power Law (IPL) Relationship

← Older revision		Revision as of 22:55, 15 August 2012
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	~~=Inverse Power Law (IPL) Relationship=~~		#REDIRECT [[Inverse_Power_Law_(IPL)_Relationship]]
	~~<br>~~
	~~==Introduction==~~
	~~<br>~~
	~~The inverse power law (IPL) model (or relationship) is commonly used for non-thermal accelerated stresses and is given by:~~

	~~<br>~~
	~~::<math>L(V)=\frac{1}{K{{V}^{n}}}</math>~~

	~~<br>~~
	~~where:~~
	~~<br>~~
	~~• <math>L</math> represents a quantifiable life measure, such as mean life, characteristic life, median life, <math>B(x)</math> life, etc.~~

	~~• <math>V</math> represents the stress level.~~

	~~• <math>K</math> is one of the model parameters to be determined, <math>(K>0).</math>~~

	~~• <math>n</math> is another model parameter to be determined.~~
	~~<br>~~
	[[~~Image:ALTA8.1.png\|center\|400px\|The inverse power law relationship on linear scales at different life characteristics and with a Weibull life distribution.]]~~

	~~<br>~~
	~~The inverse power law appears as a straight line when plotted on a log-log paper. The equation of the line is given by:~~

	~~<br>~~
	~~::<math>\ln~~ (~~L)=-\ln (K)-n\ln (V)</math>~~
	~~<br>~~
	Plotting methods are widely used in estimating the parameters of the inverse power law relationship since obtaining <math>K</math> and <math>n</math> is as simple as finding the slope and the intercept in the above equation.

	~~<br>~~
	~~[[Image:ALTA8.2.png\|center\|400px\|Graphical look at the~~ IPL ~~relationship (log-log scale~~)]]
	~~<br>~~
	~~===A Look at the Parameter <math>n</math>===~~
	~~<br>~~
	The parameter <math>n</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n</math> increases, the greater the effect of the stress. Negative values of <math>n</math> indicate an increasing life with increasing stress. An absolute value of <math>n</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.</math>
	~~<br>~~
	~~[[Image:ALTA8.3.png\|center\|400px\|Life vs. Stress for different values of n.]]~~
	~~<br>~~

	~~{{ipl acceleration factor}}~~

	~~{{ipl exponential}}~~

	~~{{ipl weibull}}~~

	~~{{Example:IPL}}~~

	~~{{ipl lognormal}}~~

	~~==IPL and the Coffin-Manson Relationship==~~
	~~<br>~~
	In accelerated life testing analysis, thermal cycling is commonly treated as a low-cycle fatigue problem, using the inverse power law relationship. Coffin and Manson suggested that the number of cycles-to-failure of a metal subjected to thermal cycling is given by [[Reference Appendix D: References\|[28]]]:

	~~<br>~~
	~~::<math>N=\frac{C}{{{\left( \Delta T \right)}^{\gamma }}}</math>~~

	~~<br>~~
	~~where:~~
	~~<br>~~
	~~• <math>N</math> is the number of cycles to failure.~~
	~~<br>~~
	~~• <math>C</math> is a constant, characteristic of the metal.~~
	~~<br>~~
	~~• <math>\gamma </math> is another constant, also characteristic of the metal.~~
	~~<br>~~
	~~• <math>\Delta T</math> is the range of the thermal cycle.~~

	~~<br>~~
	This relationship is basically the inverse power law relationship, where the stress <math>V,</math> is substituted by the range <math>\Delta V</math>. This is an attempt to simplify the analysis of a time-varying stress test by using a constant stress model. It is a very commonly used methodology for thermal cycling and mechanical fatigue tests. However, by performing such a simplification, the following assumptions and shortcomings are inevitable. First, the acceleration effects due to the stress rate of change are ignored. In other words, it is assumed that the failures are accelerated by the stress difference and not by how rapidly this difference occurs. Secondly, the acceleration effects due to stress relaxation and creep are ignored.
	~~<br>~~
	~~<br>~~
	~~===Example===~~
	~~In this example the use of the Coffin-Manson relationship will be illustrated. This is a very simple example which can be repeated at any time. The reader is encouraged to perform this test.~~
	~~<br>~~


	~~'''Product:''' ACME Paper Clip Model 4456~~
	~~<br>~~
	~~'''Reliability Target:''' 99% at a 90% confidence after 30 cycles of 45º~~

	~~<br>~~
	After consulting with our paper-clip engineers, the acceleration stress was determined to be the angle to which the clips are bent. Two bend stresses of 90º and 180º were used. A sample of six paper clips was tested to failure at both 90º and 180º bends with the following data obtained.
	~~<br>~~
	~~<br>~~
	~~[[Image:chp8degrees2cyclesTbl.png\|center\|300px\|]]~~
	~~<br>~~
	~~<br>~~
	~~The test was performed as shown in the next figures (a side-view of the paper-clip is shown).~~

	~~<br>~~
	~~[[Image:90degrees.png\|center\|300px\|]]~~
	~~<br>~~
	~~Using the IPL-lognormal model, determine whether the reliability target was met.~~
	~~<br>~~
	~~<br>~~
	~~<br>~~
	~~====Solution====~~
	~~<br>~~
	By using the IPL relationship to analyze the data, we are actually using a constant stress model to analyze a cycling process. Caution must be exercised when performing the test. The rate of change in the angle must be constant and equal for both the 90º and 180º bends and constant and equal to the rate of change in the angle for the use life of 45º bend. Rate effects are influencing the life of the paper clip. By keeping the rate constant and equal at all stress levels, we can then eliminate these rate effects from our analysis. Otherwise the analysis will not be valid.

	~~<br>~~
	~~The data were entered and analyzed using ReliaSoft's ALTA.~~
	~~<br>~~
	~~[[Image:90-180time.png\|thumb\|center\|625px\|]]~~
	~~<br>~~

	~~The parameters of the IPL-lognormal model were estimated to be:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \sigma = & 0.198533 \\~~
	~~& K= & 0.000012 \\~~
	~~& n= & 1.856808~~
	~~\end{align}</math>~~

	~~<br>~~
	~~Using the QCP, the 90% lower 1-sided confidence bound on reliability after 30 cycles for a 45º bend was estimated to be <math>99.6%</math> , as shown below.~~

	~~<br>~~
	~~[[Image:stdprobqcp.png\|thumb\|center\|500px\|]]~~
	~~<br>~~
	~~This meets the target reliability of 99%.~~

	~~<br>~~

Chris Kahn: /* Solution */

2012-06-25T10:41:18Z

Solution

← Older revision		Revision as of 10:41, 25 June 2012
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	The data were entered and analyzed using ReliaSoft's ALTA.		The data were entered and analyzed using ReliaSoft's ALTA.
	<br>		<br>
	[[Image:90-180time.png\|thumb\|center\|~~500px~~\|]]		[[Image:90-180time.png\|thumb\|center\|625px\|]]
	<br>		<br>

Chris Kahn: /* Introduction */

2012-06-25T10:38:25Z

Introduction

← Older revision		Revision as of 10:38, 25 June 2012
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	<br>		<br>
	[[Image:ALTA8.2.png\|center\|~~200px~~\|Graphical look at the IPL relationship (log-log scale)]]		[[Image:ALTA8.2.png\|center\|400px\|Graphical look at the IPL relationship (log-log scale)]]
	<br>		<br>
	===A Look at the Parameter <math>n</math>===		===A Look at the Parameter <math>n</math>===
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	The parameter <math>n</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n</math> increases, the greater the effect of the stress. Negative values of <math>n</math> indicate an increasing life with increasing stress. An absolute value of <math>n</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.</math>		The parameter <math>n</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n</math> increases, the greater the effect of the stress. Negative values of <math>n</math> indicate an increasing life with increasing stress. An absolute value of <math>n</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.</math>
	<br>		<br>
	[[Image:ALTA8.3.png\|center\|~~200px~~\|Life vs. Stress for different values of n.]]		[[Image:ALTA8.3.png\|center\|400px\|Life vs. Stress for different values of n.]]
	<br>		<br>

Chris Kahn: /* Introduction */

2012-06-25T10:38:06Z

Introduction

← Older revision		Revision as of 10:38, 25 June 2012
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	• <math>n</math> is another model parameter to be determined.		• <math>n</math> is another model parameter to be determined.
	<br>		<br>
	[[Image:ALTA8.1.png\|center\|~~200px~~\|The inverse power law relationship on linear scales at different life characteristics and with a Weibull life distribution.]]		[[Image:ALTA8.1.png\|center\|400px\|The inverse power law relationship on linear scales at different life characteristics and with a Weibull life distribution.]]

	<br>		<br>

Chris Kahn: /* Example */

2012-06-22T15:17:10Z

Example

← Older revision		Revision as of 15:17, 22 June 2012
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	<br>		<br>
	<br>		<br>
	[[Image:chp8degrees2cyclesTbl.png\|center\|~~175px~~\|]]		[[Image:chp8degrees2cyclesTbl.png\|center\|300px\|]]
	<br>		<br>
	<br>		<br>
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	<br>		<br>
	[[Image:90degrees.png\|center\|~~200px~~\|]]		[[Image:90degrees.png\|center\|300px\|]]
	<br>		<br>
	Using the IPL-lognormal model, determine whether the reliability target was met.		Using the IPL-lognormal model, determine whether the reliability target was met.
Line 103:		Line 103:
	The data were entered and analyzed using ReliaSoft's ALTA.		The data were entered and analyzed using ReliaSoft's ALTA.
	<br>		<br>
	[[Image:90-180time.png\|thumb\|center\|~~200px~~\|]]		[[Image:90-180time.png\|thumb\|center\|500px\|]]
	<br>		<br>

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	<br>		<br>
	[[Image:stdprobqcp.png\|thumb\|center\|~~200px~~\|]]		[[Image:stdprobqcp.png\|thumb\|center\|500px\|]]
	<br>		<br>
	This meets the target reliability of 99%.		This meets the target reliability of 99%.

	<br>		<br>

Chuck Smith: /* Inverse Power Law (IPL) Relationship */

2012-04-26T15:48:48Z

Inverse Power Law (IPL) Relationship

← Older revision		Revision as of 15:48, 26 April 2012
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	• <math>n</math> is another model parameter to be determined.		• <math>n</math> is another model parameter to be determined.
	<br>		<br>
	[[Image:ALTA8.1.png\|center\|~~400px~~\|The inverse power law relationship on linear scales at different life characteristics and with a Weibull life distribution.]]		[[Image:ALTA8.1.png\|center\|200px\|The inverse power law relationship on linear scales at different life characteristics and with a Weibull life distribution.]]

	<br>		<br>
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	<br>		<br>
	[[Image:ALTA8.2.png\|center\|~~400px~~\|Graphical look at the IPL relationship (log-log scale)]]		[[Image:ALTA8.2.png\|center\|200px\|Graphical look at the IPL relationship (log-log scale)]]
	<br>		<br>
	===A Look at the Parameter <math>n</math>===		===A Look at the Parameter <math>n</math>===
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	The parameter <math>n</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n</math> increases, the greater the effect of the stress. Negative values of <math>n</math> indicate an increasing life with increasing stress. An absolute value of <math>n</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.</math>		The parameter <math>n</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n</math> increases, the greater the effect of the stress. Negative values of <math>n</math> indicate an increasing life with increasing stress. An absolute value of <math>n</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.</math>
	<br>		<br>
	[[Image:ALTA8.3.png\|center\|~~400px~~\|Life vs. Stress for different values of n.]]		[[Image:ALTA8.3.png\|center\|200px\|Life vs. Stress for different values of n.]]
	<br>		<br>

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	<br>		<br>
	<br>		<br>
	[[Image:chp8degrees2cyclesTbl.png\|center\|~~350px~~\|]]		[[Image:chp8degrees2cyclesTbl.png\|center\|175px\|]]
	<br>		<br>
	<br>		<br>
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	<br>		<br>
	[[Image:90degrees.png\|center\|~~400px~~\|]]		[[Image:90degrees.png\|center\|200px\|]]
	<br>		<br>
	Using the IPL-lognormal model, determine whether the reliability target was met.		Using the IPL-lognormal model, determine whether the reliability target was met.
Line 103:		Line 103:
	The data were entered and analyzed using ReliaSoft's ALTA.		The data were entered and analyzed using ReliaSoft's ALTA.
	<br>		<br>
	[[Image:90-180time.png\|thumb\|center\|~~400px~~\|]]		[[Image:90-180time.png\|thumb\|center\|200px\|]]
	<br>		<br>

Line 119:		Line 119:

	<br>		<br>
	[[Image:stdprobqcp.png\|thumb\|center\|~~400px~~\|]]		[[Image:stdprobqcp.png\|thumb\|center\|200px\|]]
	<br>		<br>
	This meets the target reliability of 99%.		This meets the target reliability of 99%.

	<br>		<br>

Nicolette Young: /* Example */

2012-04-03T16:58:47Z

Example

← Older revision		Revision as of 16:58, 3 April 2012
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	<br>		<br>
	<br>		<br>
	[[Image:chp8degrees2cyclesTbl.png\|center\|~~400px~~\|]]		[[Image:chp8degrees2cyclesTbl.png\|center\|350px\|]]
	<br>		<br>
	<br>		<br>

Nicolette Young: /* Example */

2012-03-20T17:53:28Z

Example

← Older revision		Revision as of 17:53, 20 March 2012
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	<br>		<br>
	[[Image:90degrees.~~gif\|thumb~~\|center\|~~300px~~\|]]		[[Image:90degrees.png\|center\|400px\|]]
	<br>		<br>
	Using the IPL-lognormal model, determine whether the reliability target was met.		Using the IPL-lognormal model, determine whether the reliability target was met.

Nicolette Young: /* Example */

2012-03-20T17:52:59Z

Example

← Older revision		Revision as of 17:52, 20 March 2012
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	<br>		<br>
	<br>		<br>
	[[Image:chp8degrees2cyclesTbl.~~gif\|thumb~~\|center\|~~300px~~\|]]		[[Image:chp8degrees2cyclesTbl.png\|center\|400px\|]]
	<br>		<br>
	<br>		<br>

Nicolette Young: /* Introduction */

2012-03-20T17:34:37Z

Introduction

← Older revision		Revision as of 17:34, 20 March 2012
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	• <math>n</math> is another model parameter to be determined.		• <math>n</math> is another model parameter to be determined.
	<br>		<br>
	[[Image:ALTA8.1.png~~\|thumb~~\|center\|~~300px~~\|The inverse power law relationship on linear scales at different life characteristics and with a Weibull life distribution.]]		[[Image:ALTA8.1.png\|center\|400px\|The inverse power law relationship on linear scales at different life characteristics and with a Weibull life distribution.]]

	<br>		<br>
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	<br>		<br>
	[[Image:ALTA8.2.png~~\|thumb~~\|center\|~~300px~~\|Graphical look at the IPL relationship (log-log scale)]]		[[Image:ALTA8.2.png\|center\|400px\|Graphical look at the IPL relationship (log-log scale)]]
	<br>		<br>
	===A Look at the Parameter <math>n</math>===		===A Look at the Parameter <math>n</math>===
Line 36:		Line 36:
	The parameter <math>n</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n</math> increases, the greater the effect of the stress. Negative values of <math>n</math> indicate an increasing life with increasing stress. An absolute value of <math>n</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.</math>		The parameter <math>n</math> in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of <math>n</math> increases, the greater the effect of the stress. Negative values of <math>n</math> indicate an increasing life with increasing stress. An absolute value of <math>n</math> approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when <math>n=0.</math>
	<br>		<br>
	[[Image:ALTA8.3.~~gif\|thumb~~\|center\|~~300px~~\|Life vs. Stress for different values of n.]]		[[Image:ALTA8.3.png\|center\|400px\|Life vs. Stress for different values of n.]]
	<br>		<br>