Weibull 6 Training Guide 9th Printing - ReliaSoft

Software Training Guide

ReliaSoft’s

Weibull++ 6

All the tools you will ever need for Life Data Analysis, and more...

No part of this training guide may be reproduced or transmitted in any form or by any means, for any purpose, without the expressed written consent of ReliaSoft Corporation, Tucson, AZ, USA. ReliaSoft and Weibull++ are trademarks of ReliaSoft Corporation.

©1997-2004 ReliaSoft Corporation. ALL RIGHTS RESERVED

ReliaSoft Corporation ReliaSoft Plaza 115 South Sherwood Village Drive Tucson, AZ 85710 USA Phone: +1.520.886.0366 Fax: +1.520.886.0399 Toll Free: 1.888.886.0410 (U.S. and Canada) [email protected] http://www.ReliaSoft.com

Printed in the United States of America 10 9

1 Weibull++ 6 Training Guide

1.1 About this Training Guide This training guide is intended to provide you with many examples of Weibull++ 6. This training guide begins with step-by-step examples and then proceeds into more advanced examples and questions. At any time during the training, please feel free to ask the instructor(s) any questions you might have.

1.2 Weibull++ 6 Documentation Like all of ReliaSoft's standard software products, Weibull++ 6 is shipped with detailed printed documentation on the product (Weibull++ 6 User's Guide) and the underlying principles and theory (ReliaSoft's Life Data Analysis Reference). This training guide is intended to be a supplement to those references.

Introduction

Weibull++ 6

1.3 Contacting ReliaSoft ReliaSoft can be reached at: ReliaSoft Corporation ReliaSoft Plaza 115 South Sherwood Village Drive Tucson, AZ 85710 USA Phone: +1.520.886.0366 Fax: +1.520.886.0399 E-mail: [email protected] For up-to-date product information, visit our Web site at: http://www.ReliaSoft.com

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2 Features Summary

The examples included in this training guide have been designed to introduce you to the features available in Weibull++ Version 6. This section presents a brief summary of these features. If you are already familiar with Weibull++'s features, you can proceed to Chapter 3, First Steps.

2.1 Intuitive Work Environment ReliaSoft's Weibull++ 6 builds upon the strength of its predecessors and takes a quantum leap forward in features, power and flexibility to deliver the new standard in life data analysis. Weibull++ 6's interface is an intelligent, flexible and completely integrated work center designed around the Data Folio, which is similar to Excel® Worksheets and Quattro Pro® Sheets. Within the Data Folio, you can enter, edit and manipulate data, create plots and perform analyses. Each Data Folio can contain up to 256 data sheets with over 16,000 data rows per sheet and can easily interface with other Microsoft Windows applications. Weibull++ includes all the tools and options you will need for life data analysis and provides support for all major life data types. You can easily analyze data from all censoring schemes (i.e. complete, right censored or suspended, left censored, interval censored and multiply censored data) and enter the data individually or in groups of observations.

2.2 Advanced Features Weibull++ 6 offers four new utilities that let you perform even more advanced analyses on your data. These utilities perform degradation analysis, warranty analysis, non-parametric analysis and analysis of variance. Each utility can be saved with the Data Folio so you can view and alter the data at any time and easily share your analysis with colleagues--all included in a single file.

Features Summary

Weibull++ 6

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The Degradation Analysis utility allows you to extrapolate the failure times of a product based on its performance over a period of time. You can analyze the extrapolated failure times as times-to-failure data and transfer the data into the current Data Folio. In addition, this utility also creates a plot of the analysis, which can help you visualize the results.

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The Warranty Analysis utility allows you to quickly and easily convert shipping and warranty return data for a product into times-to-failure and suspension data. This utility even gives you a summary of the generated forecast results, allowing you to create a report of your analysis.

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The Non-Parametric Analysis utility allows you to perform non-parametric data analysis using KaplanMeier, simple actuarial or standard actuarial techniques. You can calculate the reliability of a product at specific time periods and then generate non-parametric reliability vs. time plots. You can also transfer the data to the Weibull++ Data Folio for further analysis.

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The analysis of variance (ANOVA) utility gives you a method to identify the cause and source(s) of variability during a process. The utility also helps you determine whether a change in output is due to a change in input or due to random error.

2.3 Additional Enhancements In addition to many new utilities, Weibull++ 6 is also packed with enhancements to standard Weibull++ features that make analyzing data even easier.

2.3.1 Analysis Methods Weibull ++ 6 now offers an additional life distribution, generalized gamma, in addition to the Weibull, normal, lognormal, exponential and mixed Weibull distribution options. Also, you can now use the competing failure modes analysis option to assign a separate distribution to each failure mode identified in your data set. In addition to the rank regression on X, rank regression on Y and maximum likelihood estimation analysis methods, Weibull++ 6 offers Kaplan-Meier as an alternative ranking method. With the new version of Weibull++, you also have the ability to treat grouped data as individual data in rank regression.

2.3.2 Plots and Graphics Always a strength of Weibull++, the graphing and plotting capabilities in Version 6 are even better! New features include: •

Side-by-Side Plots tool, which allows you to view up to six different plots at a time in a single window.

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Likelihood Ratio Contour Plots for comparing different data sets.

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Option to represent suspension data points on plots as well as the number in a group for grouped data.

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Option to shade the upper and lower confidence bounds area on probability plots.

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Option to set and show the target reliability on plots.

ReliaSoft Draw, an integrated metafile graphics editor, allows you to take complete graphical control of all plots and graphics you create and to annotate and enhance any aspect of the plot. In Version 6, RS Draw has an improved and easier-to-use interface and offers a new custom arrow tool.

2.3.3 Custom Analyses and Reports ReliaSoft's Report Work Center allows you to streamline and automate report creation. You can generate report templates with fields that return results from linked data sheets. In Version 6, the Report Work Center has been redesigned and expanded and now provides the following:

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Features Summary

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Ability to reference fields from multiple data sets in a single report.

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Option to automatically insert plots into the reports.

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Ability to include tables with embedded fields.

Weibull++ 6

In addition to data sheets and plot sheets, Weibull++ Data Folios include general spreadsheets that can be used just like an Excel worksheet, with complete in-cell formula support, cell references and over 140 builtin functions. ReliaSoft's new advanced Function Wizard can be used with the General Spreadsheet and Report Work Center to generate a variety of calculated results. The Function Wizard includes multiple functions that return additional results from any referenced data set. The Function Wizard in Weibull++ 6 is now capable of nesting these functions, which allows you to combine multiple functions to create a composite function that meets your particular requirements. You can also use the Function Wizard to insert plot graphics into your reports and analyses.

2.3.4 Quick Analysis Tools The Quick Calculation Pad (QCP) provides a quick and easy method for you to obtain results for frequently asked reliability questions, including reliability or probability of failure for a specific mission time or specified time range. With Weibull++ 6, you can now keep the QCP open while you return to the current Data Folio to make modifications or change data sets and then continue to perform QCP calculations. Weibull++'s Design of Reliability Tests (DRT) tool, which is now easier to use and more automated, helps you to determine the appropriate sample size, test duration or any other variable related to a reliability demonstration test. In Version 6, the DRT includes carpet plots for better visualization of the results. The Non-Linear Equation Fit Solver allows you to quickly and easily solve user-defined equations using an iterative process. New and improved for Version 6, the solver fits data to any user-defined function and can automatically detect the parameters of the equation and provide plotted outputs of the results. Weibull++'s Quick Statistical Reference (QSR) tool, which now has an improved interface in Version 6, returns a variety of statistical values such as Median Ranks and Polynomial Interpolation Funtion values, so you don't have to spend time looking them up in tables.

2.3.5 User Customization In addition to these new utilities and enhancements, Weibull++ 6 also offers an array of customization options, which allow you to configure the application to meet your particular needs. You can hide or display any of the panels or toolbars. You can also change the shape and location of any of the toolbars and customize the icons displayed in each toolbar. In addition, you can set which tools and options you want displayed in the Data Folio's Control Panel. Not only can you set many options within Weibull++ yourself, but Version 6 also offers three pre-set configuration settings, depending on the level of data analysis you are performing and your familiarity with the application. These are the Simple, Intermediate and Expert settings. Each setting has an effect on your data analysis options and the tools that are displayed in the application.

2.4 Easy to Use Using the familiar Windows interface, as well as the given tips and wizards, you can be up and running from the moment Weibull++ 6 is installed. This allows you to be productive immediately, without the normal downtime you would expect when learning a new software application. For example, all you need to analyze a complete data set is to type your data into the Data Entry Spreadsheet and click a single button to get the parameters and a probability plot. It's that easy! In addition, ReliaSoft's Weibull++ comes with complete and detailed on-line help files as well as a multitude of example files and guides designed to get you up and

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Features Summary

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running quickly. Complete and detailed printed documentation and an accompanying Life Data Analysis reference book are also included.

2.5 System Requirements Weibull++ is compiled and designed for Windows 98, Windows Me, Windows NT, Windows 2000 and Windows XP and takes full advantage of the features available in these platforms. Minimum system requirements are a Pentium class processor with 32 MB RAM, SVGA display and at least 35 MB of hard disk space.

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3 First Steps

3.1 Starting Weibull++ 6 Weibull++ 6 has been designed to work with Windows 98, Windows Me, Windows NT, Windows 2000 and Windows XP. To start Weibull++, from Start select Programs, ReliaSoft Office, Weibull++ and then Weibull++ 6. The figures used in this manual are all from Windows 98. However, the Weibull++ internal screens and commands are identical regardless of which operating system you are using, and this manual is equally applicable.

First Steps

Weibull++ 6

3.2 Multiple Document Interface The Multiple Document Interface (MDI) is the main window and "manager" for Weibull++. The MDI serves as the container for all of the Data Folios you will use to analyze your data, manages the different active windows and keeps track of each data set. Data entry is performed within a Data Folio. The Data Folio, or Folio, is very much like an Excel Workbook. The Folio contains all the data, plots and spreadsheets associated with a particular file name. The Folio works very much like a folder within a filing cabinet where the MDI is the filing cabinet. Each Folio can contain up to 256 sheets with over 16,000 data rows per sheet. The MDI remains open until you close the program. Closing the MDI has the same effect as terminating the program. The appearance of the MDI will vary depending on the window(s) currently open and on the configuration settings that you have established. The next figure shows the MDI of Weibull++ Version 6 with the Expert configuration settings established.

3.3 Getting Help in the Weibull++ Environment ReliaSoft's Weibull++ includes complete on-line help documentation. This help can be obtained at any time by pressing F1 or by selecting Contents from the Help menu.

3.4 Weibull++ Configuration Settings Weibull++ 6 provides many powerful features and several customization options that allow you to manage Weibull++'s features in a way that most completely and efficiently meets your particular needs. To make customization as simple as possible, Weibull++ gives you the option of working with one of three different pre-set configurations for the application. Depending on the level of data analysis you are performing and your familiarity with the application, you may choose to configure the application with the Simple, Intermediate or Expert settings. Each setting has an effect on how you create a new Data Entry Spreadsheet, on your data analysis options and on the tools that are displayed on the MDI toolbars and on the Folio Control Panel. You will be prompted to choose a setting when you first run Weibull++ 6. However, you can always change or modify the setting within the application at any point from the User Setup. The Expert setting is used for the examples contained in this training manual. To verify that your application is configured with Expert settings or to change the settings selection to Expert, do the following:

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First Steps

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Select User Setup from the File menu to open Weibull++'s User Setup window.

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From the first page of the User Setup window, the General page, click the Modify Default Presets button. A window like the one shown next will appear.

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If the red Expert button on the right side of the window is selected, then the appropriate default configuration has already been set on your computer. Click OK to close the Default Settings window and OK to close the User Setup window. You are ready to proceed with the examples in this guide.

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If the Expert configuration is not currently selected for your application, click the Expert button in the Default Settings window and then click OK. A window will appear explaining that you need to re-start the application in order for the change to take effect. Click OK. In the User Setup window, click OK to return to the MDI.

•

Shut down the application by selecting Exit from the File menu. When you re-start the application, the tools that are displayed on the MDI toolbars and on the Folio Control Panel, as well as some analysis and other settings, will be those associated with the Expert configuration.


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First Steps

Weibull++ 6

3.5 Weibull++, A Familiarization 3.5.1 A Quick Overview Example This section presents you with a very simple example and guides you through the solution. The test case for this example: Six units were reliability tested and the following times-to-failure were observed: 64, 46, 83, 123, 105 and 150 hr. Do the following: •

Open a new Folio by clicking the New icon,

or by selecting New from the File menu. The Data Type Wizard window will appear like the one shown next, which helps you select the type of Data Entry Spreadsheet you want to create in your Folio.

The Data Type Wizard allows you to move step-by-step through a series of questions designed to determine the type of Data Entry Spreadsheet that will be appropriate for your data. If you already know what type of data you are dealing with, you can click the Data Type Expert button in the upper right corner of the window. The Data Type Expert allows you to use buttons to select whether you will enter Time-to-Failure and Suspensions data or Freeform data. If you are an experienced user and know the data type to be used, the Data Type Expert feature allows you to quickly and easily select the appropriate data type. If you are unsure of the data type to be entered, use the Data Sheet Wizard. For users who are familiar with the format of the Data Type Expert from Weibull++ 5.0, this window is also available. If you prefer to use this window to create Data Entry Spreadsheets, use the Data Folio page of the User Setup to indicate this preference. Weibull++ always remembers your preference. •

You will create a Data Entry Spreadsheet within the new Data Folio with times-to-failure data without suspensions. The data set is non-grouped and you have exact times-to-failure. Click NEXT > to proceed to the second window in the Data Type Wizard.

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The second window asks whether you will be entering times-to-failure/suspension data or freeform data. Select Yes, I am entering times-to-failure data and click NEXT >.

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The next window asks if you have any suspended units in your data set. Select No and click NEXT >.

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First Steps

Weibull++ 6

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The next window asks if you want to enter the data in groups. Select No, each entry will be unique and click NEXT >.

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The next window asks you if the data are given in intervals. In this case, the failure times are given as exact times. Therefore, select No, I have exact times and click NEXT >.

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The Data Type Wizard has now determined the type of data you will be entering in the Data Entry Spreadsheet. Click Finish to generate the Folio.

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Once the Folio has been generated, the Data Entry Spreadsheet will appear. The Data Entry Spreadsheet is where all the data are entered in the Folio. Enter the data in the first column of the Data Entry Spreadsheet under Time Failed, as shown next (leave the Subset ID column blank for now).

Note that the times-to-failure do not need to be sorted; the application will automatically do this for you. •

The next step is to select a distribution and parameter estimation method to calculate the parameters. We assume that you have not changed any of the default settings. The default distribution is the 2-parameter Weibull distribution. You can verify this on the Main page of the Data Folio Control Panel by noticing that the Weibull button is RED and is therefore selected. Also, notice that the 2-parameter option has been selected under Parameters/Type.

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The default parameter estimation method is Rank Regression on X (RRX) and the regression method will be standard regression (SRM). You can check this by looking at the Parameter Estimation Method and Regression Method boxes at the bottom of the Main page of the Control Panel, as shown next.


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First Steps

Weibull++ 6

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You can also verify this by clicking the Set Analysis page index tab and noticing that Rank Regression on X (RRX) is selected and Use RS Regression Method is not selected.1 Now return to the Main page of the Folio Control Panel by clicking the Main page index tab.

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Once the distribution, number of parameters and parameter estimation method have been selected, you can calculate the parameters by clicking the Calculate icon,

or by selecting Calculate Parameters from the Data menu. The Data Entry Spreadsheet with its parameters calculated is shown next. The results will appear in the Results Area, which is located just below the Parameters/Type options.

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Plot the data by clicking the Plot icon,

or by selecting Plot Probability from the Data menu. If you click the Plot icon before the parameters have been calculated, the application will automatically calculate the parameters and then plot the data.

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Besides choosing an option from the Set Analysis page, you can switch between parameter estimation methods (RRX, RRY or MLE) by clicking the Parameter Estimation Method Box and between regression methods by clicking the Regression Method Box.

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First Steps

Weibull++ 6

You will now have a window with the Weibull probability plot on your screen, as shown next. The plot appears within the Plot Sheet.

A legend that presents information about the data being plotted is displayed in the upper right corner of the plot. This includes the distribution, number of parameters, parameter estimation method, regression method, ranking method and the number of failures and suspensions. The values of the parameters for the data set are located in the bottom left corner of the plot. You can toggle the option to display the legend and results on the plot by selecting/de-selecting Show Results and Legends from the Plot Options menu. •

At this time, save the data file as 1stStep.rw6. To do this, click the Save icon,

or select Save from the File menu. In the Save Data to a Weibull Folio window, enter 1stStep for the filename and press OK, as shown next. You do not have to include .RW6 after the filename. Weibull++ will do this for you if you do not provide a file extension.

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After saving the file, close the Folio by clicking the close button (X) in the upper right corner of the Folio or by selecting Close from the File menu. You will now be looking at the MDI without any Folios open.


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4 Step-by-Step Examples

Please note that these examples use the default registry settings that are associated with the Expert setting. If you have changed any of your settings, either reset the default configuration setting to Expert or click the Reset Registry Settings button on the General page of the User Setup. Weibull++ configuration settings are presented in detail in Chapter 3.

4.1 General Examples 4.1.1 Example 1 Ten identical units were reliability tested at the same application and operation stress levels. Six of these units failed during the test after operating for the following times, T j : 16, 34, 53, 75, 93 and 120. Four other units were still operating, i.e. right censored or suspended, after 120 hr. Determine the parameters of the Weibull pdf that represents this data set and create the Probability and pdf plots. Solution •

Create a new Data Folio by selecting New from the File menu or by clicking the New icon.

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The Data Type Expert will appear. If the Data Type Wizard opens instead, click the Data Type Expert button in the upper right corner of the window to open the Data Type Expert.

Step-by-Step Examples

Weibull++ 6

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Select the Times to Failure and the with Right Censored Data (Suspensions) options, as shown next, then click OK to open a Data Folio with the appropriate Data Entry Spreadsheet..

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Enter the data in the Data Entry Spreadsheet. Enter the state of the unit when it was removed from the test, F for failure and S for suspension (right censored), in the State F or S column. You do not have to enter the state (F or S) capitalized; the application will automatically do this for you. Enter the times-tofailure in the Time to F or S column.1

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The 2-parameter Weibull distribution and Rank Regression on X (RRX) will be used to calculate the parameters. Make sure that Weibull is selected. Make sure that RRX is displayed in the Parameter

1.

Weibull++ provides a shortcut for entering right censored data. If you enter a negative time value, the state is automatically assigned an S for suspended. If you enter a positive time value, the state is automatically assigned an F for a failure.

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Estimation Method box. If RRX is not displayed, switch to the Set Analysis page and select Rank Regression on X (RRX).

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If you switched to the Set Analysis page of the Folio Control Panel to change the parameter estimation method, return to the Main page by clicking the Main page index tab.

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Calculate the parameters by clicking the Calculate icon,

or by selecting Calculate Parameters from the Data menu. The results will appear in the Results area, as shown next.



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or by selecting Plot Probability from the Data menu. The Weibull Probability plot for this data set is shown next.

If the straight line does indeed represent the points, as in this case, then the location parameter, γ , is zero. Note that the plotted line represents the unreliability, Q(T), which is defined as:

Q(T) = 1 – R(T) where R(T) is the reliability. If desired, the 2-parameter Weibull pdf representing this data set can be written as:

β T f ( T ) = ---  --- η  η

 T β – 1 –  --η-

β

e

or 1.2097 T  - 0.2097 –  -------------------- 144.3631 

1.2097 T f ( T ) = ----------------------  ---------------------- 144.3631 144.3631 •

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e

You can also plot the Weibull pdf by selecting Pdf Plot from the Special Plot Type drop-down menu located on the Plot Sheet Control Panel.



Weibull++ 6

The pdf plot is shown next.

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Now save this Folio as Example1.rw6 and then close the Folio.

4.1.2 Example 2 Determine the Median Rank value for the sixth failure from a sample size of 10. This will demonstrate two of the built-in functions contained in the Quick Statistical Reference. The Quick Statistical Reference is presented in detail in Chapter 18 of the User's Guide. Solution • From the theoretical rank equation, with Q = 0.50, where Q is the probability of failure:

1 MR = -----------------------------------------------------N–j+1 1 +  --------------------- F Q ;n1 ;n2   j For this example: N = 10 (sample size) j = 6 (failure)

n 1 = 2 ( 10 – 6 + 1 ) = 10 And:

n 2 = 2 × 6 = 12 ( n 1, n 2 are degrees of freedom)



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Therefore:

1 MR = -----------------------------------------5 1 +  --- F 0.50 ;10 ;12  6 •

The value of F needs to be determined. This can be done using the Quick Statistical Reference (QSR). Open the Quick Statistical Reference by clicking its icon located on the General Tools toolbar,

or by selecting More Tools then Quick Statistical Reference from the Tools menu. The value of F can be calculated using the Inverse F-Distribution function. Select Inverse F-Distribution Values as the Function Option. Input the values for n 1 , n 2 and Q in the appropriate text boxes and then click Calculate. The results are shown next.

The value of F was found to be:

F 0.50 ;10 ;12 = 0.9886 Consequently:

1 MR = ----------------------------------- = 0.5483 5 1 + --- × 0.9886 6 Therefore:

MR( % ) = 54.83 %

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Another method to calculate the Median Ranks is to use the Median Ranks function directly. Select Median Ranks as the Function Option and input the appropriate values for the sample size and order number (j). Click Calculate to obtain the result, as shown next.

The value of MR was found to be:

MR( % ) = 54.8306 % •

Close the Quick Statistical Reference by clicking the Close button.

4.1.3 Example 3 What is the unreliability of the units in Example 1 for a mission duration of 226 hr, starting the mission at T = 0? Solution There are several methods of solution for this problem. The first and more laborious method is to extract the information directly from the plot. It is possible to extract the information from a plot within Weibull++ or from a printed copy. For this example, let's obtain the information from within the application. This can be done using ReliaSoft Draw (RS Draw). •

Open Example1.rw6 by clicking the Open icon,

or by selecting Open from the File menu.



Weibull++ 6

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Select Example1.rw6, as shown next.

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Click Open to open the file.

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The Data Folio you created in Example 1 will appear like the one shown next.

•

Click the Plot of Data 1 page index tab to open the plot of the data. The pdf plot will be displayed; however, the probability plot is needed to answer this question. First, select the Auto Refresh option, located on the Plot Sheet Control Panel. Next, select Probability-Weibull from the Special Plot Type drop-down menu. While viewing the probability plot, click the RS Draw icon,

or select ReliaSoft Draw from the Tools menu. RS Draw is presented in detail in Chapter 12 of the User's Guide. RS Draw can automatically track the position of the mouse cursor and translate the coordinates for you. Obtain the unreliability at T = 226 hr by placing the cursor at the intersection of the plotted line and T = 226. The position of the cursor is indicated by the Position Indicator located in the lower right corner of the RS Draw window. The x-coordinate (time) is displayed on the left and the y-coordinate (unreliability) is displayed on the right. When the x-coordinate reads approximately 226, read off the value of the y-coordinate. You may not be able to obtain the value of the unreliability at exactly 226 hr.

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This is one of the disadvantages of reading the value off the plot. The position of the cursor at the intersection of the plotted line and T = 226 is shown below.

The value of the y-coordinate from the Position Indicator, the unreliability Q(T), is such that Q ( T ) ≅ 82 %. Then, a good estimate of the probability of failure (unreliability) at 226 hr is 82%. The reliability is then:

R ( T ) = 1 – Q ( T ) = 1 – 0.82 = 0.18 or 18 % •

Close the RS Draw window by clicking the close button and return to the Data Entry Spreadsheet by clicking the Data 1 page index tab.

•

The second method involves the use of the Quick Calculation Pad (QCP). Open the Quick Calculation Pad by clicking its icon located on the Folio Control Panel's Main page,

or by selecting Quick Calculation Pad from the Tools menu. The Quick Calculation Pad is presented in detail in Chapter 22 of the User's Guide. When the QCP appears, do the following: • On the Basic Calculations page, select Std. Prob. Calculations under Options for Calculations. • Under Results Option, select Results as Probability of Failure.



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• Under Required Input From User, type 226 for the Mission End Time. Click Calculate to obtain the results shown below.

The QCP returns a result of Q ( T ) = 0.8209 ≅ 82 %. This agrees with the result found using the plot. However, using the QCP obviously is more accurate and easier to use. •

Close the QCP by clicking Close but leave Example1.rw6 open.

4.1.4 Example 4 Using Example1.rw6, what is the reliability for a mission duration of t = 30 hr, starting the mission at T = 30 hr? Also, what is the warranty time for a reliability of 85%? Solution The functions used to find the reliability are shown below.

R(T + t) R ( T, t ) = -------------------R(T) Rˆ ( 30hr + 30hr ) Rˆ ( 60hr ) Rˆ ( 30hr, 30hr ) = --------------------------------------- = -------------------Rˆ ( 30hr ) Rˆ ( 30hr ) •

Open the Quick Calculation Pad by clicking its icon located on the Folio Control Panel's Main page,

or by selecting Quick Calculation Pad from the Tools menu. Do the following: • On the Basic Calculations page, select Std. Prob. Calculations. • Under Results Option, select Results as Reliability.

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• Enter a Mission End Time = 60. Click Calculate to obtain the results shown below.

Rˆ ( 60 – hr ) = 0.7077 •

Again using the QCP, enter a Mission End Time = 30 and click Calculate to obtain the reliability at t =30 hr.

Rˆ ( 30hr ) = 0.8612



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Divide the reliability at 60 hr by the reliability at 30 hr and you will obtain the reliability for this example.

0.7077 Rˆ ( 30hr, 30hr ) = ---------------- = 0.8218 0.8612 Rˆ ( 30hr, 30hr ) = 82.18 % The reliability is 82.18%. However, the Quick Calculation Pad can provide this result directly and more efficiently. The Std. Prob. Calculations are based on a starting mission time, T, equal to zero. This example had a starting mission time equal to T = 30 hr. A starting mission time greater than zero can be accounted for using Conditional Calculations. •

Do the following: • Select Conditional Calculations under Options for Calculations. • Enter a Mission Start Time = 30 and the Mission Additional Time = 30. • Click Calculate to obtain the result. as shown next.

The reliability is equal to 82.18%. This is the same result found previously. The second part of this example involves determining the warranty time for a reliability of 85%. •

Do the following: • On the Basic Calculations page, select Warranty (Time) Information. • Enter a Required Reliability = 0.85.

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• Click Calculate to obtain the results.

The time required for a reliability of 85% is equal to 32.1482. •

Close the QCP by clicking Close. Close the Folio but do not save your changes.

4.2 3-Parameter Weibull Analysis The 3-parameter option calculates the correct γ and adjusts the points by this value, such that they fall on a straight line, and then plots both the adjusted and the unadjusted points.

4.2.1 Example 5 Ten identical units, N = 10, were reliability tested at the same application and operation stress levels. Six of these units failed during the test after operating for the following times, T j : 46, 64, 83, 105, 123 and 150. The four other units were suspended after operating for 150 hr. Using the 3-parameter Weibull distribution and Rank Regression on X (RRX), find the parameters of the Weibull pdf that represents these data points. Solution •

Open a new Folio by clicking the New icon,

or by selecting New from the File menu. Use the Data Type Expert, Data Type Wizard or Quick Data Type Set window to create a Data Entry Spreadsheet that can be used to analyze the data for this example.



Weibull++ 6

•

Enter the data as you did previously in Example 1. The Data Entry Spreadsheet with the entered data is shown next.

•

Select the Weibull distribution and the 3-parameter option under Parameters/Type, as shown next.

Check the Parameter Estimation Method box to see if RRX is displayed. If RRX is not displayed, go to the Set Analysis page of the Data Folio Control Panel and select Rank Regression on X (RRX) or click the Parameter Estimation Method box until RRX is displayed. Return to the Data Folio Control Panel Main page if you are not already there. •

Calculate the parameters by clicking the Calculate icon,

or by selecting Calculate Parameters from the Data menu.

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The calculated parameters are shown next.

•


or by selecting Plot Probability from the Data menu.



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The plot of the data will appear, as shown next.

Note that the original data points, plotted along the curved line, were adjusted by γ = 30.92 hr to yield the straight line.2 Also note, that the x-axis for the plot is T - γ . Therefore, to obtain the time (T) required for a given unreliability level using the straight line, you must add the value of γ to the x-axis value found at (T - γ ). •

Save the Folio as Example5.rw6 and then close the Folio.

4.3 Grouped Data Analysis To use Weibull++ 6 to analyze grouped data, simply open a new Folio by clicking the New icon,

or by selecting New from the File menu. Use the Data Type Expert, Data Type Wizard or Quick Data Type Set window to create a Data Entry Spreadsheet that can be used to analyze grouped data with times-tofailure (no censoring).

2.

If two lines do not appear on the plot by default, one adjusted for Gamma and one not adjusted for Gamma, select to display both the adjusted for Gamma and unadjusted for Gamma Weibull 3P points and lines from the Show Points and Show Plot Line submenus under the Plot Options menu.

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4.3.1 Entering Grouped Data Grouped data are entered similarly to individual data; however, you need to specify the number of units in each group within the Number in State column.

4.3.2 Analyzing Grouped Data The results you get with grouped data will vary depending on the analysis method you use. The available parameter estimation methods as they apply to grouped data are discussed next. 4.3.2.1 Using Rank Regression When using Rank Regression (Least Squares), Weibull++ plots the data point corresponding to the highest rank position in each group. In other words, given three groups of ten units, each failing at 100, 200 and 300 hr respectively, the three plotted points will be the end point of each group, or the 10th rank position out of 30, the 20th rank position out of 30 and the 30th rank position out of 30. (This procedure is identical to standard procedures for using grouped data in rank regression and is equivalent to saying what is the probability of the 10th failure out of a sample size of 30, the 20th and the 30th, respectively.) 4.3.2.2 Using Maximum Likelihood Estimation When using Maximum Likelihood Estimation (MLE), each individual time is explicitly used in the calculation of the parameters, thus there is no difference in the entry of a group of ten units failing at 100 hr and 10 individual entries of 100 hr. However, if there is uncertainty as to the exact times at which the units failed, it is recommended that you enter the data as interval data, i.e. ten units failed by 100 hr, another ten units failed between 100 and 200 hr and another ten failed between 200 hr and 300 hr.

4.3.3 Example 6 Seventeen units, N = 17, were put through a 1,000 hr reliability test. Every 200 hr, the number of units that failed were counted. One failed at 200 hr, 2 failed at 400 hr, 4 failed at 600 hr, 5 failed at 800 hr and 5 failed at 1,000 hr. Find the parameters of the 2-parameter Weibull pdf representing this data set using Rank Regression on X and Maximum Likelihood Estimation.



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Solution •

Using the previously entered grouped data, select Rank Regression on X (RRX) as the parameter estimation method.3 RRX will be displayed within the Parameter Estimation Method box. Click the Calculate icon to calculate the parameters. The next figure displays the results.

•

Now calculate the parameters using Maximum Likelihood Estimation (MLE). Switch to MLE by clicking the Parameter Estimation Method Box twice. The Parameter Estimation Method Box will toggle from RRX to RRY and then to MLE. You can also go to the Set Analysis page and select Maximum Likelihood (MLE). Return to the Main page.

•

Click the Calculate icon to calculate the parameters. The results are shown next.

As you can see, the results found using RRX and MLE are very different. 3.

For this example, the Use all data if grouped option should not be selected on the Set Analysis page of the Data Folio Control Panel.

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•

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Save the Folio as Example6.rw6 and then close it.

4.4 Looking At Other Distributions and Functions 4.4.1 Example 7 Twenty units, N = 20, were reliability tested with the following times-to-failure: seven units failed at 100 hr, five units failed at 200 hr, three units failed at 300 hr, two units failed at 400 hr, one unit failed at 500 hr and two units failed at 600 hr. Do the following: • Enter the data. Use the 2-parameter exponential distribution and Rank Regression on X as the parameter estimation method to calculate the parameters. • Obtain the Exponential Probability plot. • Obtain the Reliability vs. Time plot. • Obtain the pdf plot. • Obtain the Failure Rate vs. Time plot. Solution •

Open a new Folio. Enter the data into the appropriate Data Entry Spreadsheet and select Exponential as the distribution to be used, as shown next.



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•

Select the 2-parameter option and Rank Regression on X as the parameter estimation method. Calculate the parameters by clicking the Calculate icon. The calculated parameters are shown next.

•

Plot the data. The Exponential Probability plot is shown below.

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•

Obtain the Reliability vs. Time plot by selecting Reliability vs Time from the Special Plot Type dropdown menu. The Reliability vs. Time plot is shown next.

•

Select Pdf Plot from the Special Plot Type menu to obtain the pdf plot, as shown next.



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•

Select Failure Rate vs Time from the Special Plot Type menu to obtain the Failure Rate vs. Time plot. The Failure Rate vs. Time plot is shown next.

•

You can change the scaling of the plots using the X and Y scaling boxes located on the Plot Sheet Control Panel.

•

First, de-select the Automatic Scaling option below the scaling boxes to disable the automatic scaling. This will activate the X and Y scaling boxes. Next, enter the new value within the appropriate input box. After typing the new value, press Enter if the Auto Refresh option is selected. If the Auto Refresh option is not selected, click the Refresh icon.

•

To re-activate the automatic scaling, select the Automatic Scaling option and refresh the plot. The plot will be redrawn to the new scale settings.

•

Save the Folio as Example7.rw6 and close the Folio.

4.4.2 Example 8 Six units were tested to failure and the following times-to-failure were observed: 11,260; 12,080; 12,125; 12,825; 13,550 and 14,670 hr. Assume the data are normally distributed. Do the following: • Determine the parameters for the data using the normal distribution and Rank Regression on X as the parameter estimation method.

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• Obtain the Normal Probability plot for the data with 90%, 2-sided Time (Type 1) confidence bounds. • Obtain the pdf plot. Solution •

Open a new Folio. Enter the data into the appropriate Data Entry Spreadsheet. Select Normal as the distribution and set RRX as the parameter estimation method. Calculate the parameters. The next figure displays the results.

•

Create the Normal Probability plot by clicking the Plot icon. The Probability plot is shown next.

•

Next, plot the confidence bounds by selecting Confidence Bounds and then Show Confidence Bounds from the Plot Options menu. The Confidence Bounds window will appear, as shown next. Under Sides,



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select Two Sided and under Type, select Time (Type I). The Confidence Level set at (in%) will be set at 90%. If it is not, simply type 90 in the text box. Finally, click OK.

The Probability plot with confidence bounds is shown next.

The plot legend now contains information about the confidence bounds that have been plotted.

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•

Plot the pdf plot. The pdf plot is shown next.

•

Save the Folio as Example8.rw6 and return to the Data Entry Spreadsheet by clicking the Data 1 page index tab. This Folio will be used for the next example.

4.4.3 Example 9 Using the data and results from the previous example, do the following: • Determine the reliability for a mission of 11,000 hr, as well as the two-sided 90% confidence bounds on this reliability. • Determine the MTBF, as well as the two-sided 90% confidence bounds on this MTBF. Solution •

You can easily obtain these results using the Quick Calculation Pad. Open the QCP by clicking its icon located on the Folio Control Panel's Main page,

or by selecting Quick Calculation Pad from the Tools menu. •

You can use the Std. Prob. Calculations to solve the first part of this question. Select Std. Prob. Calculations and Results as Reliability, since reliability is the answer required.



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•

Since confidence bounds are also required, click the Confidence Bounds tab. Select Show Confidence Bounds and more options will become available. Select Two Sided and type .90 for the Confidence Level, as shown next.

•

Return to the Basic Calculations page of the QCP. Enter 11000 for the Mission End Time. Finally, click Calculate to obtain the results as shown below.

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•

You can solve the second part of this problem using the Mean Life calculation option. Select Mean Life. The Confidence Level is already set at 0.90. Click Calculate. The results are shown next.

•

Exit the QCP by clicking Close, but leave the Folio open.

4.4.4 Example 10 Using the data from Example 8, obtain tabulated values for the failure rate for ten different mission end times. The mission end times are 1,000 to 10,000 hr, incremented by 1,000 hr. Solution •

You can easily accomplish this via the use of the Function Wizard within the General Spreadsheet. The General Spreadsheet is very similar to an Excel Spreadsheet. You can input formulas and edit the cells in the same manner.



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•

To insert a General Spreadsheet, select Insert General Spreadsheet from the Folio menu. The Folio will look like the figure below.

•

Next, open the Function Wizard by clicking its icon located on the Spreadsheet Tools toolbar,

or by selecting Function Wizard from the Tools menu. The Function Wizard is presented in detail in Chapter 25 of the User's Guide. Scroll down through the list of functions and select Failure Rates for range of times, as shown next. Note that the functions are in alphabetical order.

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•

In the available input boxes, enter 1000 for the Iter. Start T, 10000 for the Iter. End T and 1000 for the Iter. Incr., as shown next. The CL will remain at 0.9.

•

Click OK to place the table into the General Spreadsheet, as shown next.

•

Save the Folio by clicking the Save icon,

or by selecting Save from the File menu. Then close the Folio.



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4.4.5 Example 11 Consider the interval data given below. The data represent the inspection times of 167 identical parts in a machine and the number of them found cracked at the end of each inspection time. Number in State 5 16 12 18 18 2 6 17 73

Last Inspected 0 6.12 19.92 29.64 35.40 39.72 45.24 52.32 63.48

State F F F F F F F F S

State End Time 6.12 19.92 29.64 35.40 39.72 45.24 52.32 63.48 63.48

Determine the parameters of the 2-parameter Weibull distribution using Maximum Likelihood Estimation and obtain the 3D Log-Likelihood Function surface plot. Solution •

Enter the data in the appropriate Data Entry Spreadsheet. For example, within the Data Type Expert, select the Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations buttons. The next figure shows the data entered and the parameters calculated.

•

The Log-Likelihood Function surface plot can be obtained by clicking the 3D Plot icon located on the Folio Control Panel's Main page,

or by selecting ReliaSoft 3D (LK Function) from the Tools menu.

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The next figure shows the 3D Log-Likelihood Function surface plot.

•

To rotate the plot, first click the plot. Then, while pressing the CTRL key, hold down the left mouse button and move the mouse around to rotate the plot. Note that the peak of the surface is at the center of the beta-eta plane.

•

Close the 3D Plot by clicking the Close icon located on the toolbar.

•

Save the Folio as Example11.rw6 and then close the Folio.



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4.4.6 Example 12 A new product was reliability tested. Since the life of this product under operating conditions is expected to be more than 15,000 hr, testing under these conditions is not time-wise feasible. For this reason, it was decided to run an accelerated test for this product. The operating temperature for this product is 323K (50°C) and temperature is the only acceleration variable. A table of the data obtained from the test for three different operating stress levels is given below. Stress Level, K

Times-to-failure, hr

393K 3850 4340 4760 5320 5740 6160 6580 7140 7980 8960

408K 3300 3720 4080 4560 4920 5280 5640 6120 6840 7680

423K 2750 3100 3400 3800 4100 4400 4700 5100 5700 6400

Do the following: • Determine the parameters of the 2-parameter Weibull distribution at each stress level using Rank Regression on X. • Estimate the parameters for the Eyring model. • What is the reliability of the unit for a mission duration of 9,000 hr, starting the mission at T = 0 and at the operating temperature (323K)? Solution •

46

Enter the data into the appropriate data sheet. In the Subset ID column, enter the corresponding temperatures for each time-to-failure. The calculated parameters for the combined data are shown next.



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Be sure to enter all of the data shown in the preceding table. Not all of the data that are applicable for this example appear in the next figure, as there are more than twenty rows of data.

You can change the heading of the Subset ID column, as shown above, by double-clicking the heading. A window that allows you to change the heading will appear. Type Temperature in K (Use is 323K) and click OK. You can edit any of the available column headings, row headings and page index tabs in the same manner. •

Use the Batch Auto Run feature to break down the original data into three different data sets based on the different operating temperatures. The Batch Auto Run uses the Subset IDs to extract the data. Open the Batch Auto Run by clicking its icon,

or by selecting Batch Auto Run from the Data menu. The following window will appear. Batch Auto Run is presented in detail in Chapter 24 of the User's Guide.



•

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The three operating stress levels are listed under Available Subset IDs. The Batch Auto Run process will be applied to all three Subset IDs, so click Select All Available> or double-click each Subset ID individually. The three stress levels now appear under Selected Subset IDs.

The Set Action Preferences for Processing page, shown next, allows you to customize the Batch Auto Run process. The Calculate Parameters for Selected Subsets option will already be selected. This indicates that the parameters will automatically be calculated when extracted into the Data Entry Spreadsheets. You will not have to initiate the calculation of the parameters yourself.

•

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Click OK to proceed through the Batch Auto Run process.



•

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When the Batch Auto Run is complete, the Data Folio will look like the one shown next.

You will notice that three new page index tabs labeled 393K1, 408K1 and 423K1 will appear in the Data Folio. Each new page index tab represents a particular Subset ID and the number of times the Subset ID has been through the Batch Auto Run process. See the figure below for details.

You will be on the 423K1 page. Notice that the parameters have already been calculated for each Subset ID. Click the appropriate page index tabs to view the other Data Entry Spreadsheets. •

Insert a General Spreadsheet into the Folio. Enter the parameters corresponding to each stress level into the General Spreadsheet, as shown next.

•

The parameters of the Eyring model can be estimated using the Non-Linear Equation Fit Solver. Highlight the cells containing the stress and eta values (it is assumed that beta remains constant at each stress level, which is the case for this data set). Copy the data to the Clipboard. Open the Non-Linear



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Equation Fit Solver by selecting Non-Linear Equation Fit Solver from the More Tools submenu under the Tools menu. The Non-Linear Equation Fit Solver is presented in detail in Chapter 32 of the User's Guide.

•

Place the cursor into the first cell in the X column on the right side of the window and paste the data that you copied from the General Spreadsheet.

•

Create an equation for the Eyring model by typing Eyring Acceleration Model in the Equation Name box and typing the following equation in the y(x)= box:

( 1 ⁄ X )∗ exp ( – ( ( A – ( B ⁄ X ) ) ) )

50



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•

Click Add to Template so that the equation for the Eyring acceleration model that you just created can be added to the Equation Template. The equations in the Equation Template can be re-used at a later time in other analyses. The window will look like the figure shown next.

•

Enter the values for the Initial guess, Is less than… and Is greater than… columns for the parameters of the Eyring model, as shown in the next figure. The values A and B are the parameters to be solved for.



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•

Click Calculate. The results are shown next.

•

The values for A and B are given within the Value column. The standard deviation for each parameter appears within the SDEV column. Using the solution of the parameters for the Eyring model, you can calculate the value of eta ( η ) for any temperature (stress level), particularly in this case for the operating temperature of 323K. Save the file as a ReliaSoft NLF (*.nlf) file and then copy the solutions for the parameters of the Eyring model. You can then paste the data into the General Spreadsheet, as in the figure shown next.

•

To aid in the process of calculating eta at different stress levels, define the parameters, A and B, as variables. In other words, define the contents of the cells that contain the values of A and B as named variables. This allows you to input these values into a formula without having to input the actual value. To do this, click the cell containing the solution of A, in this case cell E11, and select Define Name from the Edit menu. A window like the one shown next will appear. Type a name for the cell being

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defined in the Variable Name box. Type A for parameter A. You can refer to the defined cell by its Variable Name. Just type the Variable Name within the formula.

•

After typing the name, click Add. The contents of the cell have now been defined. Follow a similar process to define a name for the cell containing the solution of B. Type B for the name.

•

Using the Eyring model and the defined parameters, A and B, create a table like the one in the figure below to calculate the value of eta for a range of temperatures (323K-443K). For example, for cell C20, enter the following formula: =(1/B20)*exp(-(A-(B/B20))) and press Enter. Type the formula as it applies to the current structure of your General Spreadsheet. You can carry this formula down to C32 by selecting cell C20 and moving the cursor over the bottom right corner of the cell. The cursor will become a small crosshair. Click and drag the cursor down to cell C32. Release the mouse button after cell C32 has become highlighted.

The value of eta at a stress level of 323K is 17933.8499. The final part of this example involves estimating the reliability of the product at the operating stress level (323K) for 9000 hr of operation. The Weibull distribution was used to calculate the values of the parameters eta and beta.



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The Weibull reliability function is given by:

R(T) = e •

T –  ---  η

Create a table of reliabilities for a range of different end times (1000-10000 hr) within the General Spreadsheet using the Weibull reliability function. When referencing the cells containing the values of beta and eta within the formula to calculate the reliability, you must refer to these cells as an absolute reference. This involves inserting a $ in front of the column letter and the row number. In the next figure, the values of beta and eta are located within cells B36 and B37, respectively. When typing the formula for the Weibull reliability function, these two cells must be referenced using the following format: $B$36 and $B$37. In cell B41, enter the following formula: =exp(-((A41/$B$37)^$B$36)). Carry the formula down to B50. The reliability has now been calculated for all of the end times in question.

The reliability at 9,000 hr is 94.6979%. •

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4.4.7 Example 13 ACME company manufactures widgets. There are three failure modes for a particular widget. A table of the times-to-failure for each mode is given below. Mode 1 Times-to-Failure, hr 2.3 3.4 15 16 20 35 50 53 75 80 106 147 157 175 183 233 262 358.3

Mode 2 Times-to-Failure, hr 10 12 25 56 80 106 108 147 150 170 180 181

Mode 3 Times-to-Failure, hr 2 3 12 15 16 16.5 20

Determine the parameters using the 2-parameter Weibull distribution and Rank Regression on X. Conduct a simple criticality analysis (FMECA) for this widget, given that the probability of total loss due to each mode is 100%, 60% and 35%, respectively, and the mission time is 10 hr. Solution •

Open a new Folio to enter the data given. Enter the data for each mode into a different Data Entry Spreadsheet. Insert additional data sheets into the existing Folio by selecting Insert Data Sheet from



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the Folio menu. Name each data sheet as Mode 1, Mode 2 and Mode 3, respectively. The Folio with the calculated parameters is shown next.

•

Insert a General Spreadsheet into the Folio and construct a table as shown below. Format your table the best you can. The specific formatting shown below is not as important as the procedures involved in solving the example. The formatting is there to show what is possible with the General Spreadsheet.

•

Complete the table as follows: • Use the Function Wizard to insert the number of failures for each mode into the Number of Failures column. For example, place the cursor in the Number of Failures for Mode 1 cell (B4 in the example shown) and click the Function Wizard icon,

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or select Function Wizard from the Tools menu to open the Function Wizard. Select Total Number of Failures from the list of available functions and verify that Mode 1 is selected in the Data Source box, as shown next.

Click OK to insert the value into the General Spreadsheet at the location of the cursor. Use a similar procedure to insert the total number of failures into the Mode 2 and Mode 3 rows, changing the Data Source selection to the appropriate Data Entry Spreadsheet. • In the Totals row, use a formula to compute the sum of the values in the Number of Failures column. For example: =sum(B4+B6+B8) • The Failure Mode Frequency Ratio is equal to the total number of failures for each mode divided by the total number of failures for all three modes combined. Use a formula to compute this for each cell. For example: =b4/b10 • The Probability of Total Loss Due to this Mode was stated earlier for each mode: Mode 1 = 100%, Mode 2 = 60% and Mode 3 = 35%. • Use the Function Wizard to insert the Unreliability (also called Probability of Failure) at T = 10 hr for each mode into the Unreliability column. For example, place the cursor into the Unreliability for Mode 1 cell and open the Function Wizard. Select Probability of Failure from the list of available functions, type 10 in the Mission End T input box and verify that Mode 1 is



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selected in the Data Source box and None is selected in the Show Confidence box, as shown next.

•

Click OK to insert the value into the General Spreadsheet at the location of the cursor. Use a similar procedure to insert the Unreliability into the Mode 2 and Mode 3 rows, changing the Data Source selection to the appropriate Data Entry Spreadsheet. • The Criticality for each mode is equal to the Failure Mode Frequency Ratio times the Probability of Total Loss Due to this Mode times the Unreliability. Again, use a formula in the spreadsheet to calculate this value. For example: =c4*d4*e4

•

The results of the analysis are shown below.

•


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4.5 Examples Using Additional Reliability Analysis Utilities 4.5.1 Example 14 Five turbine blades were tested for crack propagation. The test units were cyclically stressed and inspected every 100,000 cycles for crack length. Failure is defined as a crack of length 30mm or greater. Following is a table of the test results: Cycles (x1000) 100 200 300 400 500

Unit A 15mm 20mm 22mm 26mm 29mm

Unit B 10mm 15mm 20mm 25mm 30mm

Unit C 17mm 25mm 26mm 27mm 33mm

Unit D 12mm 16mm 17mm 20mm 26mm

Unit E 10mm 15mm 20mm 26mm 33mm

Using Weibull++ 6's Degradation Analysis utility and Quick Calculation Pad, determine the B10 life for the blades using degradation analysis with an exponential model for the extrapolation. Solution •

Create a new Data Folio.

•

Activate the Degradation Analysis utility by clicking the Degradation Analysis icon,

or by selecting Degradation Analysis from the Tools menu. •

When you first activate Degradation Analysis, the Open Degradation Analysis window will appear. For this example, select Create a new Degradation base data set for this sheet and click OK. When you create a new Degradation base data set, it will automatically attach itself to the current Data Entry Spreadsheet so you can access it at any time. The attachment of this data set will be indicated by a red tab that will appear in the Utility Attachment area of the Data Folio Control Panel's Main Page. To open the existing Degradation base data set file, you can either click the tab or right-click and select Open Attachment from the menu. The Degradation Analysis window will open with a Data sheet and a Control Panel. The Control Panel, located on the right side of the window, allows you to select a degradation model, suspension time and the critical degradation for the analysis. In the Control Panel, select Exponential for the Degradation Model and type 30 for the Critical Degradation. There is no suspension time for this example.



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•

On the Data sheet, enter the data from the test results. The Data sheet will look similar to the one shown next.

•

Once you have entered the data in the Data sheet and made the specifications in the Control Panel, click the Calculate icon on the toolbar,

or select Calculate from the Tools menu. •

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When you click Calculate, the utility will create two new sheets. The Output Parameters sheet, which can be accessed by clicking its tab, shows the parameters for the degradation model for each of the units. For this example, the Output Parameters sheet contains the parameter values for each unit, as shown next.



•

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You can plot the data from the degradation analysis by clicking the Plot icon on the toolbar,

or by selecting Plot from the Tools menu. •

The results from the degradation analysis will be shown graphically on the Plot sheet. On the Control Panel located on the right side of the Plot sheet, select to show all the units on the plot by clicking Select All.

•

Next, refresh the plot by clicking the Refresh Plot icon.



•

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Your plot will look similar to the one shown next. Degradation vs Time 40

30

20

10

0 0

87.5

175

262.5

350

437.5

525

612.5

700

Time

Data a

Exponential Fit b

Data d

Exponential Fit e

Exponential Fit a

Data c

Exponential Fit d

Critical Degradation

Data b

Exponential Fit c

Data e

•

The Output Raw Data sheet, which returns the time-to-failure data for each of the units, was also created when you calculated the data set. Go to the Output Raw Data sheet by clicking its tab.

•

Next, transfer the data from this sheet to the Data Folio by clicking the icon on the toolbar,

or by selecting Transfer Weibull Data to Folio from the Tools menu.

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•

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The data from Degradation Analysis will be transferred to the Data Folio. To analyze the data in the Data Folio, select a two-parameter Weibull distribution and MLE estimation method and click Calculate. The Data Folio will look similar to the one shown next.

The calculated results are beta = 8.0552 and eta = 519.5559. •

Activate the Quick Calculation Pad. Perform a calculation using BX information. Enter 10 for the BX% information. Click Calculate. The results are shown next.

Using the Degradation Analysis utility and the QCP, the B10 life is calculated to be 392,918 cycles.



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4.5.2 Example 15 A company keeps track of its shipments and warranty returns on a month-by-month basis. Following is a table for shipments in June, July and August and the warranty returns through September:

Months June July Aug.

Shipments 100 140 150

July 3 -

Returns Aug. 3 2 -

Sept. 5 4 4

Do the following: •

Using Weibull++ 6's Warranty Analysis utility, convert this information to life data and determine the parameters for a 2-parameter Weibull distribution.

•

Predict the number of products from each of the three shipment periods that will be returned under warranty in October.

Solution •

Create a Data Entry Spreadsheet for times-to-failure data. In the Folio Control Panel, select the 2parameter Weibull distribution with MLE and median ranks using the standard regression method.

•

Activate Warranty Analysis by clicking the Warranty Analysis icon on the toolbar,

or by selecting Warranty Analysis from the Tools menu. •

When you first activate Warranty Analysis, the Open Warranty Analysis window will appear. In this window, select Create a new Warranty base data set for this sheet. When you create a new Warranty base data set, it will use the analysis settings of the linked Folio and automatically attach itself to the current Data Entry Spreadsheet so you can access it at any time. The attachment of this data set will be indicated by a yellow tab that will appear in the Utility Attachment area of the Data Folio Control Panel's Main Page.

•

The Warranty Analysis window will open with a Data sheet and the Control Panel. The Control Panel, located on the right side of the window, allows you to select the number of sales periods you want to analyze. For this example, type 3 for the number of periods and click the Update icon on the Control Panel.

Notice that the Warranty Data Input sheet now reflects the specified number of sales periods.

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•

In the Warranty Data Input sheet, type the data from the shipments and returns table, as shown next. Type the number of shipments in the Sales column and the number of returns in the Returns columns. The Returns columns are the columns located to the right of the Sales column. You only need to type data in the white cells.

•

To convert this data set to times-to-failure data, click the Create Weibull Data icon on the toolbar,

or select Create Weibull Data from the Tools menu. The data will be placed in the Weibull++ Data sheet, indicated by the corresponding tab, as shown next.



•

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From the Weibull++ Data sheet, you can predict the total expected returns from all shipments for the next month, October, by clicking the Generate Forecast icon on the toolbar,

or by selecting Generate Forecast from the Tools menu. •

Depending on the type of Data Entry Spreadsheet the Warranty Analysis data set is attached to, a window may appear warning you that when you generate the forecast, the current Data Entry Spreadsheet will be modified to correspond with the Weibull++ Data Sheet and prompting you to indicate whether you want to proceed. For this example, click Yes.

Another window will appear, allowing you to enter the number of time periods you want to forecast for. Type 1 since you are forecasting data for one month (October) and click OK.

Note that the calculation options selected in the linked Data Entry Spreadsheet are used to generate forecasts in the Warranty Analysis utility. For your reference, these are displayed in the Analysis area of the Warranty Analysis utility's Control Panel. •

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Two new sheets will be added to the Warranty Analysis window, the Forecast Output sheet and the Summary Output sheet, indicated by their corresponding tabs. View the Forecast Output sheet, which shows the expected returns for October. October is indicated as period 4. The expected returns for each shipment period are indicated by the purple (shaded) cells, as shown next.



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•

As shown previously, the predicted number of products that will be returned under warranty in October are 12 from the June shipment, 11 from the July shipment and 6 from the August shipment. You can also view the summarized forecast results in the Summary Output sheet by clicking its corresponding tab.

•

As shown on this sheet, the total expected returns from all three shipments for Period 4 (October) is 29 units.

•

Next, go back to the Weibull++ Data sheet by clicking its tab. Transfer the data on this sheet to the current Data Folio by clicking the Transfer Weibull Data to Folio icon on the toolbar,

or by selecting Transfer Weibull Data to Folio from the Tools menu.



•

Weibull++ 6

The data from Warranty Analysis will be transferred to the Data Folio. To estimate the parameters in the Data Folio, click Calculate. The Data Folio will look similar to the one shown next.

The calculated results are beta = 2.4928 and eta = 6.6951. These results also appear in the Warranty Analysis Summary Output sheet.

4.5.3 Example 16 This example has been abstracted from Example 15.6 from the Meeker and Escobar textbook Statistical Methods for Reliability Data, published by John Wiley and Sons. An electrical component has two failure modes. One failure mode is due to random voltage spikes, which cause failure by overloading the system. This failure mode is denoted by a V in the table. The other failure mode is due to wear-out failures, which usually happen only after the system has run for many cycles. This failure mode is denoted by a W in the table. Time-to-failure data for each mode and suspension data are shown in the following table: Number in State 1 1 1 2 1 1 1 1 1 1 1

Failure Time* 2 10 13 23 28 30 65 80 88 106 143

Failure Mode V V V V V V V V V V V

*Failure times are given in thousands of cycles.

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Number in State 1 1 1 1 1 1 1 1 1 1 8

Failure Time* 147 173 181 212 245 247 261 266 275 293 300

Failure Mode W V W W W V V W W W suspended


Weibull++ 6

Do the following: •

Determine the overall reliability for the component at 100,000 cycles.

•

Plot the competing failure modes.

Solution •

Create a new Data Entry Spreadsheet for times-to-failure data with suspensions and grouped data.

•

Enter the data from the failures table into the Data Entry Spreadsheet. Identify the failure mode in the Subset ID column.

•

Select a Weibull distribution with competing failure modes, MLE and median ranks using the standard regression method. Your Folio will look similar to the one shown next.

•

To calculate the parameters, click the Calculate icon,

or select Calculate Parameters from the Data menu.



•

Weibull++ 6

In the Competing Failure Modes window that appears, define the V and W failure modes. Place the V failure mode into the Mode 1 column and place the W failure mode into the Mode 2 column, as shown next. To do this, click a subset ID (mode) from the Available panel and select a Mode panel to place the subset ID into. Click (+) to add the selected Subset ID to the Mode panel. Click (X) to remove the selected Subset ID from the Mode panel. You can also double-click or drag and drop a Subset ID to add or remove it from a panel.

Please note that you must have distinct Subset IDs in the Data Entry Spreadsheet to use Competing Failure Modes. •

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After you have separated the failure modes, click OK. The parameters will be estimated for both failure modes. You can compare the two failure modes in the Control Panel by selecting which mode you want to view from the Subpopulation drop-down box. The calculated results will correspond to the selected failure mode in the Results Area. The picture below shows the parameters for failure mode 1, or the V failure mode.



Weibull++ 6

•

The estimated parameters for Failure Mode 1 (the random voltage spikes failures) are beta = .6711 and eta = 449.4272. The estimated parameters for Failure Mode 2 (the wear-out failures) are beta = 4.3373 and eta = 340.3842.

•

Next, activate the Quick Calculation Pad and determine the overall reliability of the system at 100,000 cycles. To calculate this, use standard probability calculations and select to Show Results as Reliability. Enter a mission end time of 100 (since the failure times were estimated in thousands of cycles). The results are shown next.

Using the Competing Failure Modes and QCP, the overall reliability for the component at 100,000 cycles is 69.1%. •

Close the QCP.

•

Next, plot the competing failure modes by clicking the Plot icon,

or by selecting Plot Probability from the Data menu. Your plot can contain the combined mode line as well as the individual mode lines. This option can be accessed by selecting Plot Setup from the Plot Options menu. On the Lines page of the Plot Setup window, click Set Up CFM Mode Colors. In the window that appears, select Show Individual Mode Lines and click OK.



•

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Weibull++ 6

Click OK in the Plot Setup window. Your plot will look similar to the one shown next. (Note that the labels in the figure have been added to show the different modes.)


5 Practice Questions

This section contains some additional practice questions. Chapter 6 presents solutions to these examples.

5.1 Practice Question 1 Six units were reliability tested and the following times-to-failure were observed: 46, 64, 83, 105, 123 and 150 hr. Do the following: 1.

Determine how you would classify this data set, i.e. individual, grouped, suspended, censored, uncensored, etc.

2.

Open a new Folio and enter the data.

3.

Select Rank Regression on X as the parameter estimation method.

4.

Determine the parameters for this data set using the following distributions and plot the data for each distribution. From the plot, note how well you think each distribution tracks the data, i.e. how well does the fitted line track the plotted points? i.

2-parameter Weibull

ii.

3-parameter Weibull

iii. Normal iv. Lognormal v.

1-parameter Exponential

vi. 2-parameter Exponential 5.

Use the Distribution Wizard to determine the distribution that best fits your data. Use Begin Auto Run to conduct the analysis. Compare with what you have observed.

6.

Save your file as QUEST1.RW6 and close the Folio.

Practice Questions

Weibull++ 6

5.2 Practice Question 2 ACME manufacturing implemented a reliability tracking program for a recent product. A total of 96 units were released in a small test market. Ten units failed sometime between 0-68 hr, 10 units failed between 6890 hr, ten failed between 90-120 hr and seven failed between 120-130 hr. 59 units were still in the field after successfully operating for 153 hr. Do the following: 1.

Determine the parameters of the 2-parameter Weibull distribution using Rank Regression on X.

2.

Obtain the probability plot for this data set.

3.

Plot the 90%, 2-sided confidence bounds (Type 2). Open RS Draw and save your plot as MyPlot1.wmf.

4.

Experiment with annotating the plot while in RS Draw.

5.

From the plot, determine: i.

The reliability of these units for a mission of 50 hr, R(50hr).

ii.

The 90%, 2-sided confidence limits on the reliability for a mission of 50 hr (Type 2).

iii. The mission duration for these units if ACME Inc. requires a 90% reliability at the 50% confidence level. iv. The mission range if ACME Inc. requires a 90% reliability at the 90% confidence level (Type 1). 6.

Using the Quick Calculation Pad, determine: i.

The reliability of these units for a mission of 50 hr, R(50hr).

ii.

The 90%, 2-sided confidence limits on the reliability for a mission of 50 hr.

iii. The mission duration for these units if ACME Inc. requires a 90% reliability at the 50% confidence level. iv. The mission range if ACME Inc. requires a 90% reliability at the 90% confidence level. v.

The 2-sided confidence limits on the parameters.

7.

Obtain the Reliability vs. Time plot for these units.

8.

Obtain the pdf plot for these units.

9.

Obtain the Failure Rate vs. Time plot for these units. From the plot, what is the failure rate of these units at 100 hr?

10. Save your file as QUEST2.RW6. 11. Close the Folio.

5.3 Practice Question 3 Open the file you previously saved as QUEST1.RW6 (Practice Question 1). Do the following: 1.

Open a new Folio for non-grouped data entry for times-to-failure (no censoring). You will now have an empty Folio, as well as the one you saved from Question 1.

2.

Copy the data from Question 1 and paste it into the empty Data Entry Spreadsheet. Use the 2-parameter Weibull distribution and Maximum Likelihood Estimation to calculate the parameters.

3.

Insert a new Data Entry Spreadsheet into this Folio. Copy the data set from Question 1 and paste it into the new Data Entry Spreadsheet.

4.

Switch the parameter estimation method in the new Data Entry Spreadsheet to Rank Regression on X and calculate the parameters.

5.

Plot both data sets individually.

6.

Customize both plots to your liking.

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Practice Questions

Weibull++ 6

7.

Insert a Multiple Plot Sheet into the Folio and select both data sets to be displayed. (Note that both are made up of the same data. Do you see a difference? Why?)

8.

Close all Folios when you are finished.

5.4 Practice Question 4 ACME Manufacturing is preparing to purchase a new component for one of its products. Two competing manufacturers presented ACME with the following test data. Each manufacturer tested eight units to failure with the following results: Manufacturer A Time to failure, hr 1900 2600 1400 1300 800 2650 1850 2400

Manufacturer B Time to failure, hr 2000 4250 1300 600 3000 6600 9000 12000

1.

Determine the parameters of the 2-parameter Weibull distribution using Rank Regression on X.

2.

Use the Tests of Comparison utility to calculate the probability that Design A is better than Design B. Which design would you choose?

5.5 Practice Question 5 As you may have noticed, plotting the data points and seeing how well the line runs through the points is an indication of how well the calculated parameters fit the data (or how correct the calculation is). However, when you use Maximum Likelihood Estimation, the data points do not, in general, track the plotted line. The reasons lie in the way the parameters are estimated. One way to view this convergence, since MLE maximizes the Likelihood Function, is to view the 3D plot of the Log-Likelihood Function. Weibull++ lets you do just that. 1.

Open QUEST1.RW6.

2.

Calculate the parameters using the 2-parameter Weibull distribution and MLE as the parameter estimation method.

3.

Plot the Log-Likelihood Function using 3D Plot.

4.

Press Ctrl and click the plot. Move the mouse simultaneously in the direction you would like to rotate the plot.

5.

Rotate the plot and view the maximum beta.

6.

Rotate the plot again and view the maximum eta.

7.

Close 3D Plot and insert a General Spreadsheet in the Folio.

8.

Activate the Function Wizard and choose to generate a Table of Reliabilities given a time range. Enter time increments from T = 10 to 100 incrementing by 5, at the 90% 2-sided confidence level.

9.

When finished, save the Folio and leave it open.


Practice Questions

Weibull++ 6

5.6 Practice Question 6 Once you have generated the data, created your plots and calculated what you wanted to calculate, the next step is to generate a report. Weibull++ 6 allows you to do that very easily. 1.

QUEST1.RW6 will already be open.

2.

If the parameters are not calculated, select a distribution and a calculation method to calculate them.

3.

Open the Report Work Center by clicking its icon on the Data Tools toolbar, or by selecting Report Work Center from the Tools menu. By default, the Template page will be open.

4.

The first time the Template page is activated, it shows you a default template with some instructions. Read the default template and follow the instructions.

5.

Click the Create Report or Editor button to create the report based on the default template. When the Create Report from Template window appears, select Update by Replacing Editor's existing report with a new report and click Yes. You will now be in the Editor page. Do you notice any differences?

6.

Go back to the Template page by clicking the Template button and select New from the File menu.

7.

Create your own report by doing the following: i.

Insert a title and a few lines of text.

ii.

Use the Function Wizard by clicking Insert Field to insert special commands. These commands will return results based on the linked data set.

iii. Create a table of Reliability vs. Time, starting at 10 hr and ending at 100 hr with an increment of 10 hr. iv. Create a table of Failure Rate vs. Time, starting at 10 hr and ending at 100 hr with an increment of 10 hr. v.

Let's include some graphics. Create a Probability plot based on the linked data set. a. Click Insert Field to open the Function Wizard. Select Plot Probability vs. Time from the function list and click OK.

vi. Save your Template as MyTMPL.RLT. vii. Click the Editor button to compile the template and finalize your report. viii. Save your report as MyREPORT.RLW. NOTE: Files saved within the Template page have the extension *.rlt. File saved within the Editor page have the extension *.rlw.

5.7 Practice Question 7 The Monte Carlo Data generation is a useful tool when dealing with simulations. You can use it to generate values for any distribution or function. This example will take you through such a scenario. 1.

Open the Monte Carlo Data Simulation by clicking its icon located on the General Tools toolbar, or by selecting Generate Monte Carlo Data... from the Data menu.

2.

Generate a data set using the Weibull distribution with beta = 2 and eta = 1500 hr.

3.

Calculate the parameters.

4.

Are the parameters equal to the ones used in the Monte Carlo tool? Comments?

5.

Generate another Monte Carlo data set using the same distribution and parameters.

6.

Compare the two sets. Are they the same?

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Practice Questions

Weibull++ 6

5.8 Practice Question 8 Your new manager has to be convinced that product X has an MTBF of 1000 hr with a 95% confidence level. You know from past experience that the shape parameter for this product is 1.5. Due to time constraints, you are allowed to test the units for no more than 500 hr. Using ReliaSoft's Design of Reliability Tests (DRT) find the number of units that have to be tested if: 1.

No failures are allowed.

2.

One failure is allowed.

3.

Experiment with the other cases using the DRT.

5.9 Practice Question 9 The next table presents a data set corresponding to the times-to-failure for two samples of the same product, samples A and B. Both samples are assumed to be identical, so the times-to-failure were entered together into Weibull++, utilizing the Subset ID column to identify each sample. Time-to-failure, hr 49 71 80 85 93 100 16 18 30 34 50 201

Subset ID A A A A A A B B B B B B

Using the Batch Auto Run feature, extract two different data sets. Do the following: 1.

Calculate the parameters for each data set using the 2-parameter Weibull distribution with Rank Regression on X.

2.

Compare the two sets using the Tests of Comparison utility.

3.

Create a Multiple Plot and plot the two data sets together.

5.10 Practice Question 10 For this example, design a test to demonstrate a reliability of 90% at t = 100 hr, with a 95% confidence. Assume a Weibull distribution with a shape parameter beta = 1.5. No failures will be allowed on this test, or f = 0. Using the DRT (Design of Reliability Testing) utility, determine the following: 1.

The number of units to test for 48 hr (in integers).

2.

The test time for 20 units.


6 Answers to Practice Questions

This section provides answers to the Practice Questions in Chapter 5. Due to the structure and process required, answers are not provided for Questions 6 and 7.

6.1 Practice Question 1 1.

Individual

4.

Parameter values for each distribution: i.

β = 2.4381, η = 108.4277

ii.

β = 1.9469, η = 92.3005, γ = 15.2800

iii. µ = 95.1666, σ = 43.4372 iv. µ = 4.4805, σ = 0.4901 v.

λ = 0.0119

vi. λ = 0.0191, γ = 46 5.

The Distribution Wizard suggests the 3-parameter Weibull distribution.


β = 1.6284, η = 195.4625

5.

From the plot: i.

R(50 hr) @ 89%

ii.

Upper CL @ 93.6%, Lower CL @ 83%

iii. Mission Duration @ 50 hr

Answers to Practice Questions

Weibull++ 6

iv. Requires a mission range of approximately 37-68 hr 6.

From the QCP: i.

R(50 hr) = 89.71%

ii.

Upper CL = 93.58%, Lower CL = 83.71%

iii. Mission Duration = 49.0802 hr iv. Requires a mission range of 36.1231-66.6849 hr v. 9.

For β : 1.3039, 2.0338

For η : 171.3291, 222.9953 Failure Rate @ 5.5E-03


For MLE: β = 3.0139, η = 106.9455

4.

For RRX: β = 2.4381, η = 108.4277


Manufacturer A: β = 2.7017, η = 2107.2287

2.

Manufacturer B: β = 1.0796, η = 5322.1346 Manufacturer B is better with a 73% probability


β = 3.0139, η = 106.9455

8.

Reliability for time increments from 10 to 100 with a 2-sided confidence level at 90%: Time 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

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Lower CL 0.9525 0.9171 0.8776 0.8352 0.7908 0.7451 0.6985 0.6515 0.6045 0.5577 0.5113 0.4656 0.4207 0.3767 0.3338 0.2921 0.2519 0.2135 0.1772

Reliability 0.9992 0.9973 0.9936 0.9876 0.9785 0.9661 0.9497 0.9290 0.9038 0.8739 0.8393 0.8001 0.7567 0.7095 0.6591 0.6062 0.5518 0.4967 0.4419

Upper CL 1 0.9999 0.9997 0.9991 0.9980 0.9960 0.9926 0.9874 0.9799 0.9694 0.9553 0.9370 0.9142 0.8864 0.8535 0.8158 0.7738 0.7282 0.6801

Answers to Practice Questions

Weibull++ 6


10 units

2.

17 units


Product A: β = 4.2669, η = 87.3813 Product B: β = 1.3121, η = 55.4934


86 units

2.

126 hr
