Arrhenius Weibull

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

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

(8)

$images\6_4_8.gif$

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

$images\6_4_1.gif$

and substituting for $images\etan.gif$ in Eqn. (8),

$images\6_4_2.gif$

An illustration of the pdf for different stresses is shown in Figure 6. As expected, the pdf at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3 in Figure 6). This behavior is observed when the parameter B of the Arrhenius model is positive.

$images\2d_and_3d_pdf.gif$

Fig. 6: Behavior of the probability density function at different stresses and with the parameters held constant.

The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution was presented in greater detail in the Life Distributions chapter.

Arrhenius Weibull Statistical Properties Summary

Mean or MTTF

The mean, $images\tline.gif$ (also called MTTF by some authors), of the Arrhenius-Weibull relationship is given by:

$images\6_411_1.gif$

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

Median

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

(9)

$images\6_412_9.gif$

Mode

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

(10)

$images\6_413_10.gif$

Standard Deviation

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

$images\6_414_1.gif$

Arrhenius-Weibull Reliability Function

The Arrhenius-Weibull reliability function is given by:

$images\6_415_1.gif$

If the parameter B is positive, then the reliability increases as stress decreases.

$images\2d_and_3d_reliability.gif$

Fig. 7: Behavior of the reliability function at different stresses and constant parameter values.

The behavior of the reliability function of the Weibull distribution for different values of $images\betab.gif$ was illustrated in the Life Distributions chapter. In the case of the Arrhenius-Weibull model however, the reliability is a function of stress also. A 3D plot such as in Figure 8 is now needed to illustrate the effects of both the stress and $images\betab.gif$ .

$images\3dreliab.gif$

Fig. 8: Reliability function for $images\betab.gif$ < 1, $images\betab.gif$ = 1, and $images\betab.gif$ > 1.

Conditional Reliability Function

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

(11)

$images\6_416_11.gif$

$images\6_416_1.gif$

Reliable Life

For the Arrhenius-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:

(12)

$images\6_417_12.gif$

This is the life for which the unit will function successfully with a reliability of $images\rtr.gif$ . If $images\rtr.gif$ = 0.50 then $images\tr3.gif$ = $images\tu.gif$ , the median life, or the life by which half of the units will survive.

Arrhenius-Weibull Failure Rate Function

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

$images\6_418_1.gif$

$images\failrate.gif$

Fig. 9: Failure rate function for $images\betab.gif$ < 1, $images\betab.gif$ = 1, and $images\betab.gif$ > 1.

Parameter Estimation

Maximum Likelihood Estimation Method

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

$images\6_421_1.gif$

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).

· 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 $images\betahat.gif$ , $images\bhat.gif$ , $images\chat.gif$ so that $images\vbeta.gif$ = 0, $images\vb2.gif$ = 0 and $images\vc.gif$ =0, where,

$images\6_421_2.gif$

Example

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

$images\6_5ex.gif$

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

$images\6_5_1.gif$

Once the parameters of the model are estimated, extrapolation and other life measures can be directly obtained using the appropriate equations. Using the MLE method, confidence bounds for all estimates can be obtained. Note in Figure 10 below that the more distant the accelerated stress from the operating stress, the greater the uncertainty of the extrapolation. The degree of uncertainty is reflected in the confidence bounds. (General theory and calculations which underly confidence intervals are presented in Appendix A: Brief Statistical Background. Specific calculations for confidence bounds on the Arrhenius model are presented in Arrhenius Confidence Bounds.)

$images\f0ryvr0g.gif$

Fig. 10: Comparison of the confidence bounds for different use stress levels.