# Accelerated life testing

This is used for obtaining quick information on item's life distributions, reliabilities, failure rates, etc., by subjecting the test items to conditions such that the failures occur earlier. Thus, the accelerated life testing is a quite useful tool for making long-term reliability prediction within a short time span. The following two methods are used for performing an accelerated life test [7, 9-11]:

- •
**Method I.**This is concerned with carrying out the test at very high stress (e.g., voltage, temperature, and humidity) levels so that malfunctions can be induced in a very short time interval. Typical examples of items for application under this method are communication satellites, air traffic control monitors, and components of a power-generating unit. Generally, accelerated failure time testing over the accelerated stress testing is preferred because there is no need for making assumptions concerning the relationship of time to failure distributions at both normal and accelerated conditions. Nonetheless, the results of the accelerated-stress testing are related to the normal conditions by using various mathematical models. One such model will be presented subsequently in this section. - •
**Method II.**This method is concerned with accelerating the test by using the item under consideration more intensively than its general usage. Normally, the items such as a high bulb of a telephone set and a crank shaft of a car used discretely or non-discretely can be tested by employing this method. However, it is to be noted that it is not possible to use this method for an item such as a mainframe computer in constant use. Under such a scenario, method I can be used.

## Relationship between the accelerated and normal conditions

This section presents relationships between the accelerated and normal conditions for the following four items [7,12]:

**• Probability density function**

The normal operating condition failure probability density function is defined by

= (7-8)

where

a is the acceleration factor.

t is time.

*f„* (f) is the normal operating condition failure probability density function.

Al “I *^{S t}^^{ie stress}^^{u}' °P^{eratin}§ ^{cor,}dition failure probability density function.

**Cumulative distribution function**

The normal operating condition cumulative distribution function is expressed by

F„(t) = F_{si}f — 1

(7.9)

where

F_{s(}^—j is the stressful operating condition cumulative distribution function.

**• Hazard rate**

The normal operating condition hazard rate is expressed by

M0 =

(7.10)

By substituting Equations (7.8) and (7.9) into Equation (7.10), we get

(7.11)

Thus, from Equation (7.11), we have

*h„(t) = —h _{sl }a*

*t_*

*a*

(7.12)

where is the stressful operating condition hazard rate.

**• Time to failure**

The time to failure at normal operating condition is expressed by

(7.13)

where

t„ is the time to failure at normal operating condition.

t_{s}, is the time to failure at stressful operating condition.

## Acceleration model

For an exponentially distributed time to failure at an accelerated stress, s, the cumulative distribution function is expressed by

F_{sf}(t) = l-e^" (7.14)

where

X_{s}, is the constant failure rate at the stressful level.

Thus, from Equations (7.9) and (7.14), we obtain

F„ *(t ) = F _{s}, ( -* Ï = 1 -

*e-*(7.15)

^{la}'^{/a}*aj*

Similarly, using Equation (7.12), we have

*K = —* (7.16)

*a*

where

is the constant failure rate at the normal operating conditions.

For both non-censored and censored data, the failure rate at the stressful level can be calculated by using the following two equations, respectively [7,13]:

**• Non-censored data**

**X _{s(} = **

*tn /*

(7.17)

**i=i**

where

t, is the ith failure time; i = 1,2,3,.. ,,m.

m is the total number of items under test at a certain stress.

**• Censored data**

(7.18)

where

n is the number of failed items at the accelerated stress. *t-* is the ith censoring time.

**Example 7.2**

Assume that a sample of 30 identical electronic items was accelerated life tested at 120°C, and their times to failure were exponentially distributed with a mean value of 6000 hours. If the value of the acceleration factor is 20 and the electronic items' normal operating temperature is 30°C, calculate the electronic items, operating at the normal conditions, failure rate, mean time to failure, and reliability for a 4000-hour mission.

In this case, the electronic items' failure rate at the accelerated temperature is expressed by

X_{s(} = 1/(electronic items' mean life under accelerated testing)

= —-— = 0.000166 failures/hour 6000

By substituting the above result and the given data into Equation (7.16), we obtain

_ 0.000166

20

= 8.3333 failures/hour

Thus, the electronic items' mean time to failure *(MTTF„)* at the normal operating condition is

*MTTF„= —*

= 120,000 hours

The electronic items' reliability for a 4000-hour mission at the normal operating condition is

R(4,O0O) = c-|^{8M33x,0}’^{6})^{(4}'^{0O0)}

= 0.9672

Thus, the electronic items' failure rate, mean time to failure, and reliability at the normal operation are 8.3333 x 10“^{6} failure/hour, 120,000 hours, and 0.9672, respectively.