There are two major quality topics that concern the element of time. They’re statistical process control (SPC), which is how the process performs over time; and reliability, which is how long the product lasts. Reliability is the probability of mission success over a defined period of time, under a defined operating environment. Today it’s possible to substitute the word "cycles" in place of the word "time." But before defining reliability, it’s important to define failure. In the light bulb example, it’s simple: it either lights or it doesn’t. For more complex systems, the definition of failure isn’t as straightforward. What would be the definition of failure for human beings, for example?
The rate at which products fail, λ, is constant within the time period defined as useful life. Early failures generally occur during a period called “infant mortality” and happen because of poor quality workmanship or defective raw materials. The infant-mortality period generally represents a small fraction of the product that wasn’t detected during a burn-in phase. It’s the product within the steady state or constant failure rate that is defined as the useful life of the product.
A model that’s frequently used to predict the reliability of a product while it’s in its useful life is the exponential model:
Where R = the reliability at time t
θ = the mean time before failure (MTBF)
Because the failure rate is constant, place a number of units in a testing program and test them until there are failures. Testing can be done until there’s a predetermined amount of time (cycles) of the units or until there’s a predetermined number of failed units. For this discussion, assume that all the units have failed (complete data). By averaging all the times to failure, it’s possible to determine the MTBF.
Consider the following example:
A motor is run to 1,800 rpms under a constant load for one hour. It’s then allowed to cool to 20° C and then run for another hour. When the current draw exceeds a specified number of amps, the motor is defined as having failed.
The number of cycles to failure are given:
53,717 | 52,900 | 53,500 | 50,648 | 53,195 |
49,600 | 51,280 | 52,293 | 54,495 | 53,280 |
53,100 | 54,277 | 54,080 | 49,928 | 53,307 |
The MTBF is given by:
But there’s one major problem. It took more than six years to complete the study. There are other ways to accelerate the test and effectively compress the time to failure. One method is elevating the test temperature, known as the high temperature operating life acceleration model. The devices are subjected to an elevated temperature for the life of the test. It’s assumed that the failure mechanism will follow an Arrhenius relationship. Many failure modes follow this relationship as do chemical processes, diffusion, rates of oxidation, etc.
For life testing where elevated temperatures to accelerate failure are used, the Arrhenius model can be expressed as:
T = the temperature in degrees Kelvin ( 273.16 +° C)
= Boltzmann’s constant, 8.6173 x 10 -5 eV/ ° K
The degree to which an increased temperature will increase the MTBF can be expressed using a temperature acceleration factor,
The life of the product under normal operating temperature is usually much longer than it would be under the higher operating temperature. An acceleration factor of 50 indicates that one hour in the accelerated or stressed condition would be equivalent to 50 hours in the normal use condition.
The MTBF is assumed to be inversely proportional to the temperature. If tests at two different temperatures are done , the times to failure are related to the respective temperatures as follows:
Combining these three equations gives the temperature acceleration factor:
Solving for the activation energy, :
See the following example of determination of the activation energy:
The MTBF at 90° C for the previous motor example is 7,000 cycles and the MTBF at 120° C is 635 hours. The activation energy would be:
Note: the temperature is expressed as degrees Kelvin. Add 273.16 to the degreed Centigrade.
Both the 90° C and 120° C represent a stressed or elevated condition relative to the normal ambient operating temperature of 20° C.
Knowing the activation energy, it’s possible to determine the acceleration factor:
= Boltzmann’s constant, 8.6173 x 10-5 eV/ ° K
TUSE = 20° C ( 293.16 ° K)
TSTRESS = 90° C ( 363.16 ° K)
One cycle at 90° C is equivalent to 7.52 cycles at 20° C.
The MTBF at 20° C = (MTBF @ 90° C)
MTBF @ 20° C = (7.52)(7,000) = 52,640
The same conclusion as with the six-year study was reached but in only 9.6 months.
In all these cases, the MTBF is determined by testing units until failure. Because sample data is used, we are subject to statistical error—as with all statistics generated from samples. MTBFs are estimates of the truth. The following equation is used to calculate the two sided confidence interval for the MTBF, upper > True MTBF > MTBF, lower:
MTBF, upper =
MTBF, lower =
Confidence = 1 – α and α = risk
= total accumulated test cycles or time.
r = total number of failures
Example of the 90% confidence interval for the original motor test data:
Total number of cycles for all units, ta = 789,600
Total number of failed units, r = 15
MTBF, upper =
MTBF, lower =
MTBF, point estimate = 52,640
In spite of not knowing the true MTBF, there’s a 90% confidence that it’s between 36,077 and 85,394 cycles.
There are two ways to reduce this range:
- Reduce the level of confidence
- Test more units to increase the total number of failures in the test program.
Of these two, the latter is the better choice.
- Reliability Improvement with Design of Experiments (Marcel Dekker, 2001) by Lloyd W. Condra
- Quality and Reliability Series #41 (Rochester Institute of Technology) ISBN 0-8247-888-5
- The Desk Reference of Statistical Quality Methods (ASQ Quality Press, 2000) by Mark L. Crossley
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