Template:Example: Weibull Distribution Example-Demonstrate MTTF: Difference between revisions

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'''Weibull Distribution Example - Demonstrate MTTF'''
#REDIRECT [[Reliability_Test_Design]]
 
In this example, we will design a test to demonstrate  <math>MTTF=75</math>  hours, with a 95% confidence.  We will once again assume a Weibull distribution with a shape parameter  <math>\beta =1.5</math>.  No failures will be allowed on this test, or  <math>f=0</math>. We want to determine the number of units to test for  <math>{{t}_{TEST}}=60</math>  hours to demonstrate this goal.
 
The first step in this case involves determining the value of the scale parameter  <math>\eta </math>  from the  <math>MTTF</math>  equation. The equation for the  <math>MTTF</math>  for the Weibull distribution is:
 
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<center><math>MTTF=\eta \cdot \Gamma (1+\frac{1}{\beta })</math></center>
 
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where  <math>\Gamma (x)</math>  is the gamma function of  <math>x</math>. This can be rearranged in terms of <math>\eta</math>:
 
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<center><math>\eta =\frac{MTTF}{\Gamma (1+\tfrac{1}{\beta })}</math></center>
 
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Since  <math>MTTF</math>  and  <math>\beta </math>  have been specified, it is a relatively simple matter to calculate  <math>\eta =83.1</math>. From this point on, the procedure is the same as the reliability demonstration example. Next, the value of  <math>{{R}_{TEST}}</math>  is calculated as:
 
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<center><math>{{R}_{TEST}}={{e}^{-{{({{t}_{TEST}}/\eta )}^{\beta }}}}={{e}^{-{{(60/83.1)}^{1.5}}}}=0.541=54.1%</math></center>
 
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The last step is to substitute the appropriate values into the cumulative binomial equation. The values of  <math>CL</math>,  <math>{{t}_{TEST}}</math>,  <math>\beta </math>,  <math>f</math>  and  <math>\eta </math>  have already been calculated or specified, so it merely remains to solve the binomial equation for  <math>n</math>.  The value is calculated as  <math>n=4.8811,</math>  or  <math>n=5</math>  units, since the fractional value must be rounded up to the next integer value.  This example solved in Weibull++ is shown next.
 
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[[Image:RDT Weibull Demonstrate MTTF.png|thumb|center|300px| ]]  
 
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The procedure for determining the required test time proceeds in the same manner, determining  <math>\eta </math>  from the  <math>MTTF</math>  equation, and following the previously described methodology to determine  <math>{{t}_{TEST}}</math>  from the binomial equation with Weibull distribution.

Latest revision as of 08:58, 9 August 2012

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