Eyring Weibull

The pdf for the Eyring relationship and the Weibull distribution is given next.

The pdf for 2-parameter Weibull distribution is given by:

(6)

$images\7_4_6.gif$

The scale parameter (or characteristic life) of the Weibull distribution is $images\etan.gif$ . The Eyring-Weibull model pdf can then be obtained by setting $images\etan.gif$ = L(V) in Eqn. (1),

$images\7_4_1.gif$

or,

$images\7_4_2.gif$

Substituting for $images\etan.gif$ into Eqn. (6),

$images\7_4_3.gif$

Eyring Weibull Statistical Properties Summary

Mean or MTTF

The mean, $images\tline.gif$ , or mean time to failure (MTTF) for the Eyring-Weibull relationship is given by:

$images\7_411_1.gif$

where $images\gamma.gif$ is the gamma function evaluated at the value of $images\1b.gif$ .

Median

The median, $images\tu.gif$ for the Eyring-Weibull relationship is given by:

(7)

$images\7_412_7.gif$

Mode

The mode, $images\twave.gif$ for the Eyring-Weibull relationship is given by:

(8)

$images\7_413_8.gif$

Standard Deviation

The standard deviation, $images\ot.gif$ for the Eyring-Weibull relationship is given by:

$images\7_414_1.gif$

Eyring-Weibull Reliability Function

The Eyring-Weibull reliability function is given by:

$images\7_415_1.gif$

Conditional Reliability Function

The Eyring-Weibull conditional reliability function at a specified stress level is given by:

$images\7_416_1.gif$

or,

$images\7_416_2.gif$

Reliable Life

For the Eyring-Weibull relationship, the reliable life, $images\tr3.gif$ , of a unit for a specified reliability and starting the mission at age zero is given by:

(9)

$images\7_417_9.gif$

Eyring-Weibull Failure Rate Function

The Eyring-Weibull failure rate function, $images\lambdal.gif$ (T), is given by:

$images\7_418_1.gif$

Parameter Estimation

Maximum Likelihood Estimation Method

The Eyring-Weibull log-likelihood function is composed of two summation portions,

$images\7_421_1.gif$

where:

· $images\fe.gif$ is the number of groups of exact times-to-failure data points.

· $images\ni2.gif$ is the number of times-to-failure data points in the $images\ith.gif$ time-to-failure data group.

· $images\betab.gif$ 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).

· $images\vi.gif$ is the stress level of the $images\ith.gif$ group.

· $images\tsubi.gif$ is the exact failure time of the $images\ith.gif$ group.

· S is the number of groups of suspension data points.

· $images\nlinei.gif$ is the number of suspensions in the $images\ith.gif$ group of suspension data points.

· $images\tlinei.gif$ is the running time of the $images\ith.gif$ suspension data group.

The solution (parameter estimates) will be found by solving for the parameters $images\betab.gif$ , A and B so that $images\vbeta.gif$ = 0, $images\va.gif$ = 0 and $images\vb2.gif$ = 0 where,

$images\7_421_2.gif$

Example

Consider the following times-to-failure data at three different stress levels.

$images\7_5ex.gif$

The data set was analyzed jointly and with a complete MLE solution over the entire data set using ReliaSoft's ALTA yielding,

$images\betahat.gif$ = 4.29186497,

$images\ahat.gif$ = -11.08784624,

$images\bhat.gif$ = 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,

$images\7_5_1.gif$

or,

$images\7_5_2.gif$