Arrhenius Lognormal

The pdf for the Arrhenius relationship and the lognormal distribution is given next.

The pdf of the lognormal distribution is given by:

(13)

$images\6_6_13.gif$

Where

$images\6_6_1.gif$

and

$images\tdash.gif$ = mean of the natural logarithms of the times-to-failure.

$images\ot2.gif$ = standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

(14)

$images\6_6_14.gif$

The Arrhenius-lognormal model pdf can be obtained first by setting $images\tu.gif$ = L(V) in Eqn. (1). Therefore,

$images\6_6_2.gif$

$images\6_6_3.gif$

Thus,

(15)

$images\6_6_15.gif$

Substituting Eqn. (15) into Eqn. (16) yields the Arrhenius-lognormal model pdf or,

(16)

$images\6_6_16.gif$

Note that in Eqn. (16), it was assumed that the standard deviation of the natural logarithms of the times-to-failure, $images\ot2.gif$ is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( $images\ot2.gif$ is the shape parameter of the lognormal distribution).

Arrhenius-Lognormal Statistical Properties Summary

The Mean

· The mean life of the Arrhenius-lognormal model (mean of the times-to-failure), $images\tline.gif$ , is given by:

(17)

$images\6_611_17.gif$

· The mean of the natural logarithms of the times-to-failure, $images\tdash.gif$ , in terms of $images\tline.gif$ and $images\ot.gif$ is given by:

$images\6_611_1.gif$

The Standard Deviation

· The standard deviation of the Arrhenius-lognormal model (standard deviation of the times-to-failure), $images\ot.gif$ , is given by:

(18)

$images\6_612_18.gif$

· The standard deviation of the natural logarithms of the times-to-failure, $images\ot2.gif$ , in terms of $images\tline.gif$ and $images\ot.gif$ is given by:

$images\6_612_1.gif$

The Mode

The mode of the Arrhenius-lognormal model is given by:

$images\6_613_1.gif$

Lognormal Reliability

The reliability for a mission of time T, starting at age 0, for the Arrhenius-lognormal model is determined by:

$images\6_614_1.gif$

or,

$images\6_614_2.gif$

There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods.

Reliable Life

For the Arrhenius-lognormal model, the reliable life, or the mission duration for a desired reliability goal, $images\tr2.gif$ is estimated by first solving the reliability equation with respect to time, as follows,

$images\6_615_1.gif$

where,

$images\6_615_2.gif$

and,

$images\6_615_3.gif$

Since $images\tdash2.gif$ = ln(T) the reliable life, $images\tr2.gif$ is given by:

$images\6_615_4.gif$

Lognormal Failure Rate

The Arrhenius-lognormal failure rate is given by:

$images\6_616_1.gif$

When Using The Lognormal Distribution in ALTA

The parameters returned for the Arrhenius-lognormal distribution are always $images\ot2.gif$ , C and B. The returned $images\ot2.gif$ is always the square root of the variance of the natural logarithms to failure. Also, if the Show Mean option is checked (under the Tools menu), the returned mean value is always the mean of the natural logarithms of the times-to-failure, given by Eqn. (15). Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as parameters of the distribution, and are thus the mean (a function of stress as it can be seen in Eqn. (15)) and standard deviation of the natural logarithms of the data. The mean life value of the times-to-failure, as well as the standard deviation of times-to-failure (not the parameter) can be obtained through the Quick Calculation Pad or the Function Wizard in ALTA.

Parameter Estimation

Maximum Likelihood Estimation Method

The lognormal log-likelihood function for the Arrhenius-lognormal model is composed of two summation portions,

$images\66_21_1.gif$

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.

· $images\ot2.gif$ is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).

· B is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

· C is the second Arrhenius 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 $images\otdash2.gif$ , $images\bhat.gif$ , $images\chat.gif$ so that $images\votline.gif$ = 0, $images\vb2.gif$ = 0 and $images\vc.gif$ = 0, where

$images\6_621_2.gif$

and,

$images\6_621_3.gif$