Example R-2 - Reliability Considering Both Failure and Usage Distributions
Problem Statement:
Consider the case of a washing machine that can have varied loads. Assume the times to failure of the washing machine follow a Weibull distribution. When it is being used under a 10 lb load, the washing machine fails according to a Weibull distribution with beta = 3 and eta = 2,500 hr. When it is being used under a 5 lb load, the washing machine fails according to a Weibull distribution with beta = 3 and eta = 10,000 hr.
Assume that customer loads vary and are described by a normal distribution with mean = 5 and std = 1.
- Determine the Reliability of the machine for 2,000 hrs of operation.
- Obtain a plot of Reliability vs Time from 1,000 hrs to 10,000 hrs (using 1,000 hr increments).
- Generate a data set of times to failure for subsequent life data analysis.
RENO Solution:
Define a Random Variable to describe the failure distribution for the washing machine and an Equation Variable to describe the life-stress relationship, which is used to determine the scale parameter, eta, for each load. The equation refers to a second Random Variable called "Usage Load," which describes the distribution for customer loads.
In the "LSR" Equation Variable, an inverse power life-stress relationship is assumed to describe the effect of loading on life.
(See http://www.weibull.com/AccelTestWeb/acceltestweb.htm for more information.)
The Constants "k" and "n" are the variables of that relationship obtained by solving this model for the given conditions, or for L(V=5)=10,000 and L(V=10)=2,500.
Determine the Reliability at 2,000 Hours
To estimate the reliability at 2,000 hours, construct the flowchart as follows:
The Block obtains the time-to-failure from the Random Variable called "FailureTime."
The Conditional Block checks whether the failure time is less than or equal to 2,000 Hours (defined with the Constant called "Life"). If true, then a 100 is passed to the Result Storage construct in the TRUE path, which stores an average of the failures (across all simulations), and a 0 is passed to the FALSE path. If false, then a 100 is passed to the Result Storage construct in the FALSE path, which stores an average of the successes, and a 0 is passed to the TRUE path.
After performing 2,000 simulations (with a seed of 1 for repeatability), the results are displayed in the Simulation Results Explorer and in the flowchart, as shown next.
Create a Reliability vs. Time Plot
You can also use the above flowchart to create a plot of Reliability vs. Time. To do this, go to the Sensitivity Analysis tab of the Simulation Console and select to vary the "Life" Constant from 1,000 to 10,000 with increments of 1,000, as shown next.
After running the simulation with the new settings, you can generate the following plot.
Generate Data Set for Life Data Analysis
To generate a set of times-to-failure that can be used for subsequent life data analysis, construct the flowchart as follows:
The "Time to Failure" Block is defined as described above for the previous flowchart. The Result Storage construct stores an array of the failure times obtained by the Block.
After performing 1,000 simulations (with a seed of 1 for repeatability), the table of results is displayed in the Simulation Results Explorer, as shown next. Note that the Transpose Results command in the Tools menu may be required to place each result into a separate row (rather than a separate column) and that only the first five results are displayed in the picture.
A RENO project with the solution for this example (called "Failure and Usage Distribution.rnp") is shipped with the software and stored in the Examples\Reliability folder in the application directory (e.g. C:\Program Files\ReliaSoft\RENO\Examples\Reliability\Failure and Usage Distribution.rnp).
