 Reliability HotWire

Issue 81, November 2007

Hot Topics

Life Data Analysis of Load Sharing Units

Typically in life data analysis, in a population of similar units, it is assumed that unit failures are independent from each other. In other words, one unit's failure does not make the other unit(s) less or more likely to fail. This independence assumption is not always valid, however, as there are cases where a unit's failure affects the failure likelihood of other unit(s). A common example for this type of dependence is the load sharing redundant configuration. Examples of units in such a configuration include pumps, routers and motors. In this article we present an approach for analyzing data obtained from units that are in a load sharing relationship using ALTA 7. The obtained model can also be used in an RBD analyzed in BlockSim 7.

Example
In this article, we use the example of two motors, A and B, that are in a load sharing redundant configuration. It is assumed that the load is shared equally between the two motors during operation and that when one motor fails, the other motor carries the whole load by itself.

The following table shows the failure data for 15 systems that contain the two redundant motors.

Table 1 - Load Sharing Motors Failure Data

 System 1 Time to First Motor Failure Failed Motor Time to Second Motor Failure Failed Motor System 1 65 B 102 A System 2 84 A 148 B System 3 88 A 202 B System 4 121 B 156 A System 5 123 B 148 A System 6 156 B 245 A System 7 172 B 235 A System 8 192 B 220 A System 9 207 A 214 B System 10 212 B 250 A System 11 220 A 275 B System 12 243 A 300 B System 13 248 B 300 A System 14 257 A 330 B System 15 263 A 350 B

Analysis Procedure
To analyze the above data set, we need to distinguish between two "environments" that a motor operates in:

• The two motors are working. In this case each motor supports 50% of the load (i.e. the stress is 50%).

• When a motor fails, the remaining motor supports 100% of the load (i.e. the stress is 100%).

Therefore, we can model the stress experienced by a motor with one of the following stress profiles:

• Profile 1: The stress is at 50% all the way until t1 (as seen in Fig. 1). This profile is used by the motors that failed first.

• Profile 2: The stress is at 50% until t1 and then increases to 100% until t2 (as seen in Fig. 2). This profile is used by the motors that failed second.

Where:

• t1 is the failure time of the first failed motor.

• t2 is the failure time of the second failed motor. Fig. 1 -
Stress Profile for Motors That Fail First Fig. 2 -
Stress Profile for Motors That Fail Second

To bring the effect of stress level into the analysis, a life-stress relationship is needed. Because the stress is of a mechanical load nature, the power law relationship can be used to express the expected life as a function of the stress, x. The stress is expressed as time-dependent, x(t), to accommodate the type of stress profile shown in Fig. 2. Specifically: where n and a are model parameters.

In this example, the motors are assumed to exhibit the same failure behavior (i.e. come from the same population) and therefore we can use the same distribution to model them.

The Weibull distribution is assumed for the life of the motors. Therefore, the life-stress distribution model can be expressed as follows (where the shape parameter, β, is assumed to be independent of stress, while the characteristic life, η, is assumed to be stress-dependent): Because some motors (those that failed second) would experience a step stress (time-varying stress), the cumulative damage or cumulative exposure of the motors to the applied load needs to be taken into account. The cumulative damage model can be used for this purpose.

The reliability function of the unit is given by: where: Therefore, the pdf  is: Note: In ALTA 7, the power relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the power law relationship: (Eqn. 1)

The parameters (β, α0 , α1) in the pdf model can be estimated based on the data using Maximum Likelihood Estimation (MLE).

Using ALTA 7 for Data Analysis
We use ALTA 7 to perform the analysis. First, for each of the motors in the 15 systems, we create the stress profile that it experienced during its life. As an example, the following two figures show the profiles for the two motors in System 1 as created in ALTA 7. Fig. 3 - Stress Profile for Motor B in System 1 Fig. 4
- Stress Profile for Motor A in System 1

The profiles are constructed in a similar way for all the motors in the data set.

Note: In this example, a total of 30 profiles will be needed for the analysis. However, the number of profiles can be reduced by using a common stress profile for all the motors that failed first. This profile (called C50 in Fig. 3) is constant from 0 to infinity. Using the common stress profile, C50, the total number of profiles will be 16 instead of 30.

The following figure shows the data set (failures and corresponding stress profiles) as entered in ALTA 7 after selecting the Cumulative Damage model. The inverse power law life-stress relationship is set in the stress transformation window by using the logarithmic transformation for the stress, as shown next. The data set is then analyzed. The obtained parameters using MLE are:

β = 2.6784
α0
= 9.7373
α1
= -1.064

The following plot shows the reliability model for the motor, assuming it operated only under the 50% load. The following plot shows the reliability model for the motor, assuming it operated only under the 100% load. Using the Cumulative Damage Model in System RBD Modeling
The above analysis can then be used to facilitate system modeling of the load sharing redundant configuration in a system's reliability block diagram (RBD). For more information about the Load Sharing configuration in BlockSim 7, click here. The next figure shows the RBD of the load sharing redundant configuration. The load sharing container properties are shown next. The failure properties of the block require a life-stress relationship and a distribution model. The inverse power law relationship is expressed in terms of n and k (the standard parameterization of the inverse-power law) as follows: where V is the stress.

However, in the case of the power cumulative damage model, ALTA 7 uses the general log-linear parameterization as expressed in Eqn. 1. The n and k parameters can be easily obtained as follows. So the inverse power law relationship parameters for the motors are:

n = -1.064
k
=
5.90397e-5

They are entered in BlockSim 7 for each of the two blocks (along with the β and the weight proportionality factors) as shown next. The following plot shows the reliability plot for the load sharing redundant motors configuration. Conclusion:
This article presented an approach to analyzing life data of units that are dependent and are in a load sharing redundant configuration. It utilized the cumulative damage concept used in life data analysis of quantitative accelerated tests with time-varying stresses. The obtained life-stress distributions model was then used in an RBD model to model two load sharing redundant motors. 