Weibull Plots Predict System Reliability
Authored by John Berner
Applications Research Inc.
Golden Valley, Minn.
and James McLinn
Edited by Paul Dvorak
• Weibull plots fit a curve through data, not data to a predetermined distribution’s curve.
• After evaluating subsystems, their Weibull data is combined to
represent the complete system.
Applications Research Inc.,
Engineering Statistics Handbook,
Weibull plots are statistical tools that give engineers a good look at the reliability of their designs while in prototype stages. For relatively simple products, such as a lawn edger, a test team would put several prototypes through their paces and track when they fail. Putting that hour-to- failure data into Weibull software lets it predict the reliability of a much larger number of devices. The team can then decide whether a redesign is necessary or not.
The Weibull plot differs from a normal distribution in that the normal distribution fits data to a bell curve. The Weibull method fits a curve to the data. Even with just a dozen or so prototypes it makes good predictions. The technique works for anything from heart pumps to battle tanks.
For more complex products, Weibull software is only slightly more involved. A system’s overall reliability is the product of the reliabilities of its components. Take a hybrid auto, for example. Its component groups include a control system, engine, generators, cooling, batteries, transaxle, structural members, and safety components. Each of these is likely to have multiple subcomponents. And each would be tested to failure, meaning it no longer performs according to a predetermined definition of its function. Reliability is, by definition, 1.0 minus the probability of failure within a specified period.
Test data can be processed by software such as Weibull- Ease from Applications Research Inc., software from other companies, or by applying equations available in text books.
To show how Weibull plots can help predict the reliability of a system of subsystems, consider the hybrid vehicle. Start with the control system. Assume we’ve identified the most critical stresses that affect reliability and correlated those stresses to a measurable variable that can apply to all components in the system. Kilo-miles (1,000-mile increments) is logical because most failure modes relate directly to operating miles or cycles. Furthermore, a pretest investigation can determine a stresscycles- to-miles conversion factors. Once done, tests will correlate to kilo- miles traveled under agreed upon stress levels. All final units will be in kilo-miles. How the design team correlates kilo-miles to stress levels must be well documented.
Processing initial data
Data, in the form of kilo-miles to failure, is generated by tests under controlled conditions. It is then typed into the standard Weibull analysis routines in the order of its occurrence. Sometimes an individual test must be suspended. In this case, applying a system called Median Ranks gives partial credit for the length of time a part has been tested even though it did not fail. Most standard routines include a Median Rank calculation. The process is described in various texts.
Next, using the data entered into the table Times to failure in kilomiles, a distribution routine fits the Weibull equation to the data using to a least-squares regression routine. In the process, it calculates three figures. A shape parameter, β, represents the slope of the time-to-failure Weibull graph. It governs the curve shape. The scale factor, Θ, has units of time or kilo-miles. It represents the point where 63.2% of the units will have failed. The third, an “offset” parameter, is a single value subtracted from each data point to improve the correlation coefficient of the least-squares-regression fit of the data. The observable effect is a general straightening of the data points on the Weibull plot. Its final value is determined by an iterative process that finds the closest local maxima of the correlation coefficient with respect to the offset. This value is then added back to correct life values as they are entered into the matrix.
The fraction failed F, also defined by 1-R, comes from data in an accompanying table using: F=1-e-[t-x]ß/θ
where Θ = 151.9 kilo-miles
X = 39.98 kilo-miles
β = 1.983, and
t = kilo-miles.
The engineering team repeats the process for all components (or modes) critical to the system. In some cases, this is done in steps, as other components may be composed of subcomponents that must be dealt with separately.
Equivalent Weibull or capability distribution
A next step consolidates data generated from all critical components into one useful relationship. This requires entering data for each mode or subsystem into the software starting with Mode 1, the control system.
After entering data from all modes, the software takes data from the consolidation calculation and refits it to a single new set of Weibull distribution parameters. Now we can represent the complete machine or system with a single equation.
Confidence plots associated with this total system distribution are the result of sweeping the full graphic range in about 250 steps or increments of reliability and evaluating the Upper Control Limit (UCL) and Lower Control Limit (LCL). The overall confidence level is based on the system’s least reliable component or mode. The UCL and the LCL% values correspond to the times within which the indicated % failure of a component or system fails.
The percents generated by the Multiple Mode Reliability Model show the chance or probability of a single disabling problem occurring that requires a dealer visit anywhere within the completed system, and within the indicated time or, in this case, kilo-miles.
Demand or use distribution
A next step compares this “capability” to “expected use.” Guidance should come from an organization’s market research group. These people generally have responsibility for gathering expected-use figures.
The software lets users choose between entering customer-use data (% users versus demand) by quartiles, or by way of a normaldistribution description of the population. Quartile entries, for example, might read: 25% of users will drive 100 hr/month, 25% will drive 300 hr/month, and so on. If normal distribution parameters attempt to overlap the origin, the shape of the distribution morphs to something between normal and log normal. This prevents a situation in which some parts would appear to fail before a zero-use level. The resulting curve looks much the same for trimmers or lawn mowers. Many use such products for a few hours, while others use them for many hours.
It’s now possible to calculate an expected reliability based on customer use. This is done by a horizontal chart sweep comparing incremental strips of “Use” and “Capability” distributions. This generates marketbased failure rates.
When an increment of units exceeds its capability, it is counted as failed. For example, if an individual unit or component is capable of 86 kilo-miles, it will instantly fail when the use figure exceeds that value.
We now total all contributions to failure represented by the use and capability distributions. This gives the product’s extended cumulative failures based upon expected use.