
Re: Failure rate of population of components: Underdamped response to step function
Posted:
Nov 4, 2013 7:26 PM


On Mon, 4 Nov 2013 10:03:47 0800 (PST), paul.domaskis@gmail.com wrote:
>When I read reliability lecture notes and reliability data information, I've seen references to the apparent failure rate of a population of components. Plotted against time, this failure rate looks like the stepfunction response of an underdamped secondorder system. That is, from time zero (system is new), the curve accelerates upward from zero value and zero derivative until it peaks at t=MTTF, then is followed by decaying oscillations as parts are replaced and their ages become unsynchronized. At steadystate, the population failure rate is essentially constant (Poisson). >
What you describe as the result sounds like results you might get from simulation if you start out with an exponentially increasing function, or even better, with a typical mechanical failure curve that has a distinct "lifespan" (and, as you say, skips past the fastfailure phase). Bathtub curve, minus the early failure rate.
(Low, then exponential.) The human rateofdeath is drops off from a peak at birth, and after the age 30, it doubles for every 9 (or so) years of age. It might stop increasing in the 80s or 90s. What underlies this is still a matter of interest.
(More narrowly specified than human life span?) Automobile axle ballbearings that I read about 30 years ago had a more drastic curve of failure, with a sharp increase at 40,000 (?) miles. The study explained it in good detail. Micro pitting is "uniform" at the start. Eventually, pits start to overlap, and the size of new pit and the amount of damageperhit increase. When my left front wheel (under the engine) froze up, I thought the mechanic had been trying to scam me about changing both front wheels; until the other one froze up a few hundred miles later.
How are curves obtained? Industries probably observed the picture before they described it mathematically. Ballbearings. Lightbulbs. Lots of experience. Set in your numbers and simulate it.
>I'm trying to find a nonmathmetician's treatment of how this curve is obtained [ ideally online :) ]  say, for a person with postgraduate engineering background. Actually, I haven't even been able to find a very mathematical treatment that explains how it is arrived at. Intuitively, I understand that there is a peak at t=MTTF, and the failures abate as parts begin to be replaced in earnest (though failures are still replaced before this). But I can't square this off with the probability density function (PDF) for failure in a Poisson process. Since it is an exponential decay, shouldn't we see heavy failures at t=0? Note that I am treating the components as governed by Poisson failures and hence assuming that those parts that die due to infant mortality have been vetted away by a burnin stage before parts are released/sold for use. So the peaks in the population failure curve are due to synchronicity of component age, which diminish with time. > >I thought of using the CDF for a Poisson process rather than the PDF, since it grows with time, but I don't have an intuitive justification for that. Furthermore, the population failure rate accelerates from a zero derivative, which doesn't correspond to the Poisson CDF. > >I also thought of using the hazard function for a Poisson process, which constant. Not sure I that sheds light on the above "ringing" curve.
 Rich Ulrich

