Cyclic Testing with Confidence and Reliability
Reliable products are tested for many reasons. It can be for durability, life expectancy, fatigue and a number of other reasons. In order to test confidently and reliably, statistics gets involved. How do you ensure that the cycle requirement that you set is met? There is a calculation for reliability and confidence. They are usually represented in a look up table based on the confidence, reliability and sample size. There is only so much that can be printed on these tables. If you are looking for granularity, you will not find it in these tables especially for larger sample sizes.
There is also the question of how do you account for failures? One failure does not mean you do not meet your reliability and confidence criteria. You can increase the number of cycles for the rest of the sample set to meet your criteria. This can account for a small percentage of premature failures in your manufacturing processes. Again, we can use look up tables but we come up with the same challenge as before, granularity.
There are equations that will tell you how many cycles to test your sample set including one to account for failures. It is not a trivial equation and that probably the reason why look up tables are provided instead of the equation. I derived the equation using standard statistical equations. For the paper on the derivation please click on this link on academia.edu:
Before we present the equations we need to know the variables. Here is a list of the variables we will be using:
T = number of cycles or amount of time tested
T_f = number of cycles or amount of time to test each sample to
γ = location parameter (number of cycles or amount of time that have zero chance of failure which is usually zero)
β = shape parameter (characterizes the chance of failure as the test is running: β > 1 means the risk of failure increases as it is being cycled or time is being passed)
η = scale parameter (characterizes the spread of the distribution)
r = number of failures
N = number of samples
k = index of iteration for the summation (increment from 0 to the number of failures)
R(T) = Reliability
C = Confidence
Q(T) = 1-R(T) = unreliability
P(k≤r) = 1-C = the probability that the results will be above the confidence level
The variable we are looking to solve for is T_f. That will tell us how many cycles to run our sample set to achieve our set criteria.
If we set r=0 (no failures) we can simplify and rearrange the equation to:
The full equation includes the term for failures. When there are no failures it simplifies to the equation above. The full equation is:
Notice that I mentioned that we needed to solve for T_f. If you look at the equation, doing so is not a simple task. This would have to be solved numerically (with a computer program). There are many ways to execute this. Unfortunately, I do not have a robust program to provide. I am currently working on one. Keep an eye out for it.
