(6)The pdf for the Eyring relationship and the Weibull distribution is given next.
The pdf for 2-parameter Weibull distribution is given by:
(6)
The scale parameter (or characteristic life) of the Weibull distribution is η. The Eyring-Weibull model pdf can then be obtained by setting η = L(V) in Eqn. ( 1):
or:
Substituting for η into Eqn. (6):
Eyring Weibull Statistical Properties Summary
Mean or MTTF
The mean, 
, or mean time
to failure (MTTF) for the Eyring-Weibull model is given by:
where 
is the gamma function
evaluated at the value of 
.
Median
The median, 
for the Eyring-Weibull
model is given by:
(7)
Mode
The mode, 
for the Eyring-Weibull
model is given by:
(8)
Standard Deviation
The standard deviation, 
for
the Eyring-Weibull model is given by:
Eyring-Weibull Reliability Function
The Eyring-Weibull reliability function is given by:
Conditional Reliability Function
The Eyring-Weibull conditional reliability function at a specified stress level is given by:
or:
Reliable Life
For the Eyring-Weibull model , the reliable life, TR, of a unit for a specified reliability and starting the mission at age zero is given by:
(9)
Eyring-Weibull Failure Rate Function
The Eyring-Weibull failure rate function, λ(T), is given by:
Parameter Estimation
Maximum Likelihood Estimation Method
The Eyring-Weibull log-likelihood function is composed of two summation portions:
where:
Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure data points in the ith time-to-failure data group.
β is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
A is the Eyring parameter (unknown, the second of three parameters to be estimated).
B is the second Eyring parameter (unknown, the third of three parameters to be estimated).
Vi is the stress level of the ith group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is the number of suspensions in the ith
group of suspension data points.
is the running time of the ith
suspension data group.
FIis the number of interval data groups.
is the number of intervals in the ith
group of data intervals.
is the beginning of the ith
interval.
is the ending of the ith
interval.
The solution (parameter estimates) will be found by solving for the
parameters β, A
and B so that 
= 0, 
= 0 and 
= 0 where:
Example
Consider the following times-to-failure data at three different stress levels.
The data set was analyzed jointly and with a complete MLE solution over the entire data set using ReliaSoft's ALTA yielding:
= 4.29186497
= -11.08784624
= 1454.08635742
Once the parameters of the model are defined, other life measures can be directly obtained using the appropriate equations. For example, the MTTF can be obtained for the use stress level of 323K using:
or:
See Also:
Eyring Relationship
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