Statistics
Probability of Failure
Think about the time it takes for a particular type of part to fail for some particular reason. Let's assume that each type of part has only one failure mode, so given time all the parts of that type fail for that reason.
Here we have plotted the graphs for different types of parts, in different colours;
The proportion of parts that have failed at a given time is the same as the Probability of Failure (PF).
The graphs show that;
- Red parts - start to fail rapidly as soon as they come into service but most parts that get through the first 50,000 hours seem to last up to 400,000 hours and beyond.
- Green parts - fail uniformly through time.
- Blue parts - nearly all fail between 150,000 hours and 250,000 hours.
The formula that generated these probability graphs is;
p(t) = 1 - e-(t/l)k
t = Time in hours, k = Shape parameter and l = Time parameter
The shape parameter was 1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8, 16 colour run from red through green to blue. The Time parameter was fixed at 200,000 hours.
Throughout I wil talk about hours as the measure of life but it could easily be substituted for cycles.
Probability of Failure per Hour
From the Probability of Failure (PF) graph we can derive the Probability of Failure per Hour (PF/H) with Time graph.
The first thing to notice is that the y-axis has some very small values on it and that the curves all look very different this time. The formula for p(t) is above so to work out the probability of failure between t+1 and t, i.e. over the hour following time t, it is just p(t+1)-p(t)
Differentiation
The above method works but also the more mathematical among you my prefer to "differentiate" the formula;
p(t) = 1 - e-(t/l)k
to obtain the formula to generate the second graph set;
p(t) = (k/l)(t/l)k-1e-(t/l)k
The same values were used for k and l as in the first graph set.
One part three failure modes
Some more realistic graphs for a single part with three main failure modes follow...
The Time parameter value l and shape parameter k can very quite considerably, for any given failure mode with any given part type.
The real part represented in the two previous graphshas three main failure modes.
- Burn-In failures - Parts start to fail rapidly as soon as they are used but most parts that don't fail in the first 10,000 hours carry on. These failures could be as a result of manufacturing defects.
- Service Life failures - Parts fail uniformly through time. These failures could be as a result of the operating environment.
- Wear-Out failures - Parts fail from about 180,000 hours to 370,000 hours.
- All failures - The combined failures due to Burn-In, Service Life and Wear-Out.
The manufacturer burns-in each part at the factory for 10,000 hours before it is sold to the customer. The manufacturer has quoted a PF/H of 2e-6 and a service life of 20 years i.e. which is 175,315 hours.
The probability of something not having failed i.e. the probability of having survived to a particular time is just PS = 1 - PF
The probability of surviving to a given time i.e. not failing in any mode is:
PS =PSburnin x PSservice x PSwear
The probability of failure due to failing in one or more modes at any given time is:
PF = 1 - (1-PFburnin)x(1-PFservice)x(1-PFwear)
So that is how the black graph was calculated for each point in time.
Notice that the black PF/H curve looks rather like a bathtub. Hence the name the bathtub curve.
Naturally the assumptions are that there is only one dominant failure mode in each of the three categories. In simple parts this may be true but as things get more complicated more failure modes may need to be considered.
Having a bath!
The cyan vertical line is the MTTF you can see that it bears little relation to any Service Life and PF/H you may choose to quote for the part! It will move around a lot and often appear on the right of the hump. It is left of the hump when you start because there are an unrealistically high number of failures during Burn-In and Service Life, with the figures as they are.
Just type your own numbers and click on the submit button to create your own bathtub and if the system can it will calculate the MTTF to an accuracy of about 1% otherwise it does not draw it. Sometimes it can't be seen because it is off the scale to the right though.
The Black line is the bathtub curve. The values here are not that realistic but are rather set up so all the graphs are clear, so you do have to play around for a bit to see the real thing.
PF with Log Time
This is the same graph as "PF with Time" above but with a logarithmic Time axis.
PF/H with Log Time
This is the same graph as "PF/H with Time" above but with a logarithmic Time axis.
The main benefit of this is that the Burn-In time is far clearer to see.
PF Weibull scale with Log Time
The next graph is the same graph as "PF with Log Time" above but as well as the logarithmic Time axis the PF scale has been distorted in such a way that the PF graphs become lines. There is one exception, the black graph representing PF for all failure modes is not a line though it is clear that it is derived from the other three. It is the failure characteristic graph for the part.
A Weibull probability plot like this is very useful for plotting failure statistics in order to establish how many failure modes there may be and what the contribution of each failure mode may be to the final failure characteristic.