Template:Repairable systems analysis rga

Repairable Systems Analysis

Background

Most complex systems, such as automobiles, communication systems, aircraft, printers, medical diagnostics systems, helicopters, etc., are repaired and not replaced when they fail. When these systems are fielded or subjected to a customer use environment, it is often of considerable interest to determine the reliability and other performance characteristics under these conditions. Areas of interest may include assessing the expected number of failures during the warranty period, maintaining a minimum mission reliability, evaluating the rate of wearout, determining when to replace or overhaul a system and minimizing life cycle costs. In general, a lifetime distribution, such as the Weibull distribution, cannot be used to address these issues. In order to address the reliability characteristics of complex repairable systems, a process is often used instead of a distribution. The most popular process model is the Power Law model. This model is popular for several reasons. One is that it has a very practical foundation in terms of minimal repair. This is the situation when the repair of a failed system is just enough to get the system operational again. Second, if the time to first failure follows the Weibull distribution, then each succeeding failure is governed by the Power Law model in the case of minimal repair. From this point of view, the Power Law model is an extension of the Weibull distribution.

Sometimes, the Crow Extended model , which was introduced in Chapter 9 for the developmental data, is also applied for fielded repairable systems. Applying the Crow Extended model on repairable system data allows analysts to project the system MTBF after reliability-related issues are addressed during the field operation. Projections are calculated based on the mode classifications (A, BC and BD). The calculation procedure is the same as the one for the developmental data.and is not repeated in this chapter.

Distribution Example

Visualize a socket into which a component is inserted at time [math]\displaystyle{ 0 }[/math] . When the component fails, it is replaced immediately with a new one of the same kind. After each failure, the socket is put back into an as good as new condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a renewal process. In the illustration below, the component life is [math]\displaystyle{ {{X}_{j}} }[/math] and [math]\displaystyle{ {{t}_{j}} }[/math] is the system time to the [math]\displaystyle{ {{j}^{th}} }[/math] failure.

Each component life [math]\displaystyle{ {{X}_{j}} }[/math] in the socket is governed by the same distribution [math]\displaystyle{ F(x) }[/math] .
A distribution, such as the Weibull, governs a single lifetime. There is only one event associated with a distribution. The distribution [math]\displaystyle{ F(x) }[/math] is the probability that the life of the component in the socket is less than [math]\displaystyle{ x }[/math] . In the illustration above, [math]\displaystyle{ {{X}_{1}} }[/math] is the life of the first component in the socket. [math]\displaystyle{ F(x) }[/math] is the probability that the first component in the socket fails in time [math]\displaystyle{ x }[/math] . When the first component fails, it is replaced in the socket with a new component of the same type. The probability that the life of the second component is less than [math]\displaystyle{ x }[/math] is given by the same distribution function, [math]\displaystyle{ F(x) }[/math] . For the Weibull distribution:

[math]\displaystyle{ F(x)=1-{{e}^{-\lambda {{x}^{\beta }}}} }[/math]

A distribution is also characterized by its density function, such that:

[math]\displaystyle{ f(x)=\frac{d}{dx}F(x) }[/math]

The density function for the Weibull distribution is:

[math]\displaystyle{ f(x)=\lambda \beta {{x}^{\beta -1}}\cdot {{e}^{-\lambda \beta x}} }[/math]

In addition, an important reliability property of a distribution function is the failure rate, which is given by:

[math]\displaystyle{ r(x)=\frac{f(x)}{1-F(x)} }[/math]

The interpretation of the failure rate is that for a small interval of time [math]\displaystyle{ \Delta x }[/math] , [math]\displaystyle{ r(x)\Delta x }[/math] is approximately the probability that a component in the socket will fail between time [math]\displaystyle{ x }[/math] and time [math]\displaystyle{ x+\Delta x }[/math] , given that the component has not failed by time [math]\displaystyle{ x }[/math] . For the Weibull distribution, the failure rate is given by:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

It is important to note the condition that the component has not failed by time [math]\displaystyle{ x }[/math] . Again, a distribution deals with one lifetime of a component and does not allow for more than one failure. The socket has many failures and each failure time is individually governed by the same distribution. In other words, the failure times are independent of each other. If the failure rate is increasing, then this is indicative of component wearout. If the failure rate is decreasing, then this is indicative of infant mortality. If the failure rate is constant, then the component failures follow an exponential distribution. For the Weibull distribution, the failure rate is increasing for [math]\displaystyle{ \beta \gt 1 }[/math] , decreasing for [math]\displaystyle{ \beta }[/math] [math]\displaystyle{ \lt 1 }[/math] and constant for [math]\displaystyle{ \beta =1 }[/math] . Each time a component in the socket is replaced, the failure rate of the new component converts back to the value at time [math]\displaystyle{ 0 }[/math] . This means that the socket is as good as new after each failure and the subsequent replacement by a new component. This process is continued for the operation of the socket.

Process Example

Now suppose that a system consists of many components with each component in a socket. A failure in any socket constitutes a failure of the system. Each component in a socket is a renewal process governed by its respective distribution function. When the system fails due to a failure in a socket, the component is replaced and the socket is again as good as new. The system has been repaired. Because there are many other components still operating with various ages, the system is not typically put back into a like new condition after the replacement of a single component. For example, a car is not as good as new after the replacement of a failed water pump. Therefore, distribution theory does not apply to the failures of a complex system, such as a car. In general, the intervals between failures for a complex repairable system do not follow the same distribution. Distributions apply to the components that are replaced in the sockets but not at the system level. At the system level, a distribution applies to the very first failure. There is one failure associated with a distribution. For example, the very first system failure may follow a Weibull distribution.

For many systems in a real world environment, a repair is only enough to get the system operational again. If the water pump fails on the car, the repair consists only of installing a new water pump. If a seal leaks, the seal is replaced but no additional maintenance is done, etc. This is the concept of minimal repair. For a system with many failure modes, the repair of a single failure mode does not greatly improve the system reliability from what it was just before the failure. Under minimal repair for a complex system with many failure modes, the system reliability after a repair is the same as it was just before the failure. In this case, the sequence of failure at the system level follows a non-homogeneous Poisson process (NHPP). The system age when the system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the NHPP, the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

This is the Power Law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the Power Law model governs each succeeding system failure. If the system has a constant failure intensity [math]\displaystyle{ u(t) }[/math] = [math]\displaystyle{ \lambda }[/math] , then the intervals between system failures follow an exponential distribution with failure rate [math]\displaystyle{ \lambda }[/math] . If the system operates for time [math]\displaystyle{ T }[/math] , then the random number of failures [math]\displaystyle{ N(T) }[/math] over [math]\displaystyle{ 0 }[/math] to [math]\displaystyle{ T }[/math] is given by the Power Law mean value function.

[math]\displaystyle{ E[N(T)]=\lambda {{T}^{\beta }} }[/math]

Therefore, the probability [math]\displaystyle{ N(T)=n }[/math] is given by the Poisson probability.

[math]\displaystyle{ \frac{{{\left( \lambda T \right)}^{n}}{{e}^{-\lambda T}}}{n!};\text{ }n=0,1,2\ldots }[/math]

This is referred to as a homogeneous Poisson process because there is no change in the intensity function. This is a special case of the Power Law model for [math]\displaystyle{ \beta =1 }[/math] . The Power Law model is a generalization of the homogeneous Poisson process and allows for change in the intensity function as the repairable system ages. For the Power Law model, the failure intensity is increasing for [math]\displaystyle{ \beta \gt 1 }[/math] (wearout), decreasing for [math]\displaystyle{ \beta \lt 1 }[/math] (infant morality) and constant for [math]\displaystyle{ \beta =1 }[/math] (useful life).

Using the Power Law Model to Analyze Complex Repairable Systems

The Power Law model is often used to analyze the reliability for complex repairable systems in the field. A system of interest may be the total system, such as a helicopter, or it may be subsystems, such as the helicopter transmission or rotator blades. When these systems are new and first put into operation, the start time is [math]\displaystyle{ 0 }[/math] . As these systems are operated, they accumulate age, e.g. miles on automobiles, number of pages on copiers, hours on helicopters. When these systems fail, they are repaired and put back into service.

Some system types may be overhauled and some may not, depending on the maintenance policy. For example, an automobile may not be overhauled but helicopter transmissions may be overhauled after a period of time. In practice, an overhaul may not convert the system reliability back to where it was when the system was new. However, an overhaul will generally make the system more reliable. Appropriate data for the Power Law model is over cycles. If a system is not overhauled, then there is only one cycle and the zero time is when the system is first put into operation. If a system is overhauled, then the same serial number system may generate many cycles. Each cycle will start a new zero time, the beginning of the cycle. The age of the system is from the beginning of the cycle. For systems that are not overhauled, there is only one cycle and the reliability characteristics of a system as the system ages during its life is of interest. For systems that are overhauled, you are interested in the reliability characteristics of the system as it ages during its cycle.

For the Power Law model, a data set for a system will consist of a starting time [math]\displaystyle{ S }[/math] , an ending time [math]\displaystyle{ T }[/math] and the accumulated ages of the system during the cycle when it had failures. Assume the data exists from the beginning of a cycle (i.e. the starting time is 0), although non-zero starting times are possible with the Power Law model. For example, suppose data has been collected for a system with 2000 hours of operation during a cycle. The starting time is [math]\displaystyle{ S=0 }[/math] and the ending time is [math]\displaystyle{ T=2000 }[/math] . Over this period, failures occurred at system ages of 50.6, 840.7, 1060.5, 1186.5, 1613.6 and 1843.4 hours. These are the accumulated operating times within the cycle and there were no failures between 1843.4 and 2000 hours. It may be of interest to determine how the systems perform as part of a fleet. For a fleet, it must be verified that the systems have the same configuration, same maintenance policy and same operational environment. In this case, a random sample must be gathered from the fleet. Each item in the sample will have a cycle starting time [math]\displaystyle{ S=0 }[/math] , an ending time [math]\displaystyle{ T }[/math] for the data period and the accumulated operating times during this period when the system failed.

There are many ways to generate a random sample of [math]\displaystyle{ K }[/math] systems. One way is to generate [math]\displaystyle{ K }[/math] random serial numbers from the fleet. Then go to the records corresponding to the randomly selected systems. If the systems are not overhauled, then record when each system was first put into service. For example, the system may have been put into service when the odometer mileage equaled zero. Each system may have a different amount of total usage, so the ending times, [math]\displaystyle{ T }[/math] , may be different. If the systems are overhauled, then the records for the last completed cycle will be needed. The starting and ending times and the accumulated times to failure for the [math]\displaystyle{ K }[/math] systems constitute the random sample from the fleet. There is a useful and efficient method for generating a random sample for systems that are overhauled. If the overhauled systems have been in service for a considerable period of time, then each serial number system in the fleet would go through many overhaul cycles. In this case, the systems coming in for overhaul actually represent a random sample from the fleet. As [math]\displaystyle{ K }[/math] systems come in for overhaul, the data for the current completed cycles would be a random sample of size [math]\displaystyle{ K }[/math] .

In addition, the warranty period may be of particular interest. In this case, randomly choose [math]\displaystyle{ K }[/math] serial numbers for systems that have been in customer use for a period longer than the warranty period. Then check the warranty records. For each of the [math]\displaystyle{ K }[/math] systems that had warranty work, the ages corresponding to this service are the failure times. If a system did not have warranty work, then the number of failures recorded for that system is zero. The starting times are all equal to zero and the ending time for each of the [math]\displaystyle{ K }[/math] systems is equal to the warranty operating usage time, e.g. hours, copies, miles. In addition to the intensity function [math]\displaystyle{ u(t) }[/math] given by Eqn. (intensity) and the mean value function given by Eqn. (expected failures), other relationships based on the Power Law are often of practical interest. For example, the probability that the system will survive to age [math]\displaystyle{ t+d }[/math] without failure is given by:

[math]\displaystyle{ R(t)={{e}^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}} }[/math]

This is the mission reliability for a system of age [math]\displaystyle{ t }[/math] and mission length [math]\displaystyle{ d }[/math] .