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Weibull Distribution

Background:

The Weibull distribution can be used to model many different failure distributions. Given a shape parameter, β, and characteristic life, η, the reliability can be determined at a specific point in time (x). The two-parameter Weibull distribution probability density function, reliability function and hazard rate are given by:

Weibull Distribution PDF Equation Probability Density Function

Weibull Distribution Reliability Function Reliability Function

Weibull Distribution Hazard Rate Hazard Rate

The characteristic life, η, is the point where 63.2% percent of the population will have failed. This is true regardless of the shape parameter (β) chosen. The following shape parameter characteristics are noted:

β = 1.0 : Exponential distribution, constant failure rate
β = 3.5 : Normal distribution (approximation)
β < 1.0 : Decreasing failure (hazard) rate
β > 1.0 : Increasing failure (hazard) rate

The calculation inputs below show units of "hours," but any life metric (cycles, years, etc.) can be used as long as there is consistency among the three inputs. Because the minimum time step for chart generation is 1.0, if small single digit numbers are entered for η, the charts will appear choppy (regardless of the "time step division" input selected). If charts are desired, say for η = 2 years, values entered should be it terms of hours, not years, in order to generate smooth charts. Note, inputs of 2*8760 to represent two years time, in terms of hours, is a valid input for items #2 and #3 below.


Calculation Inputs:

1. Shape parameter (β):
2. Chacteristic life (η, hours):
3. Time period of interest (x, hours):
4. Decimal places:
5. Options:

Calculate mean life

Charts:
f(t)
R(t)
h(t)

Chart size:
Font size:
Time step division:
Title:

* Larger values of time step division result in smoother plots, but require more processing time. Auto font size is based on chart size.




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References:

  1. MIL-HDBK-338, Electronic Reliability Design Handbook.
  2. Bazovsky, Igor, Reliability Theory and Practice.
  3. O'Connor, Patrick, D. T., Practical Reliability Engineering.
  4. http://en.wikipedia.org/wiki/Weibull_distribution
  5. Weibull Distribution, NIST Engineering Statistics Handbook .
  6. Barringer, Paul, Typical beta (β) values: http://www.barringer1.com/wdbase.htm.