State Enumeration Tool MIL-STD-756 Example

The Reliability Analytics Toolkit System States tool provides the equivalent functionality as the Method 1002 procedure described in MIL-STD-756, Reliability Modeling and Prediction. While the approach described in MIL-STD-756 is very tedious, the System States tool makes the analysis process far easier. Continue reading

Reliability Modeling: Series Configuration

The reliability functions of some simple, well known structures will be derived. These functions are based upon the exponential distribution of time to failure.

Series Configurations

The simplest and perhaps most commonly occurring configuration in reliability mathematical modeling is the series configuration. The successful operation of the system depends on the proper functioning of all the system components. A component failure represents total system failure. A series reliability configuration is represented by the block diagram as shown below with n components.

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Exponential Distribution

This is probably the most important distribution in reliability work and is used almost exclusively for reliability prediction of electronic equipment. It describes the situation wherein the hazard rate is constant which can be shown to be generated by a Poisson process. This distribution is valuable if properly used. It has the advantages of:

  1. single, easily estimated parameter (λ)
  2. mathematically very tractable
  3. fairly wide applicability
  4. is additive  that is, the sum of a number of independent exponentially distributed variables is exponentially distributed.

Some particular applications of this model include:

  1. items whose failure rate does not change significantly with age.
  2. complex and repairable equipment without excessive amounts of redundancy.
  3. equipment for which the early failures or “infant mortalities” have been eliminated by “burning in” the equipment for some reasonable time period.

The failure density function is

for t > 0, where λ is the hazard (failure) rate, and the reliability function is

the mean life (θ) = 1/λ, and, for repairable equipment the MTBF = θ = 1/λ .

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Normal Distribution

There are two principal applications of the normal (or Gaussian) distribution to reliability. One application deals with the analysis of items which exhibit failure due to wear, such as mechanical devices. Frequently the wearout failure distribution is sufficiently close to normal that the use of this distribution for predicting or assessing reliability is valid.

The other application deals with the analysis of manufactured items and their ability to meet specifications. No two parts made to the same specification are exactly alike. The variability of parts leads to a variability in systems composed of those parts. The design must take this part variability into account, otherwise the system may not meet the specification requirement due to the combined effect of part variability. Another aspect of this application is in quality control procedures.

The basis for the use of normal distribution in this application is the central limit theorem which states that the sum of a large number of identically distributed random variables, each with finite mean and variance, is normally distributed. Thus, the variations in value of electronic component parts, for example, due to manufacturing are considered normally distributed.

The failure density function for the normal distribution is
Equ. 1

μ = the population mean
σ = the population standard deviation, which is the square root of
the variance.

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Reliability Theory

Most modern engineering disciplines are based on applied mathematics. An engineer or scientist observes a particular event and formulates a hypothesis (or conceptual model) which describes a relationship between the observed facts and the event being studied. In the physical sciences, conceptual models are, for the most part, mathematical in nature. Mathematical models represent an efficient, shorthand method of describing an event and the more significant factors which may cause, or affect, the occurrence of the event. Such models are useful to engineers since they provide the theoretical foundation for the development of an engineering discipline and a set of engineering design principles which can be applied to cause or prevent the occurrence of an event.

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