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Statistical analysis of fastener vibration life tests

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Title:
Statistical analysis of fastener vibration life tests
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English
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Cheatham, Christopher
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University of South Florida
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Subjects

Subjects / Keywords:
Preload loss
Loosening
Population predictions
Threaded insert
Bolt
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
ABSTRACT: This thesis presents methods to statistically quantify data from fastener vibration life tests. Data from fastener vibration life tests with secondary locking features of threaded inserts is used. Threaded inserts in three different configurations are examined: no locking feature, prevailing torque locking feature, and adhesive locking feature. Useful composite plots were developed by extracting minimum preloads versus cycles from test data. Minimum preloads were extracted due to the overlapping of varying test data and because the minimum preload is of most interest in such tests. In addition to composite plots, descriptive statistics of the samples were determined including mean, median, quartiles, and extents. These descriptive statistics were plotted to illustrate variability within a sample as well as variability between samples.These plots also reveal that characteristics of loosening for a sample, such as preload loss and rates of preload loss, are preserved when summarizing such tests. Usually fastener vibration life tests are presented and compared with one test sample, which is why statistically quantifying them is needed and important. Methods to predict the sample population have been created as well. To predict populations, tests to determine the distribution of the sample, such as probability plots and probability plot correlation coefficient, have been conducted. Once samples were determined to be normal, confidence intervals were created for test samples, which provides a range of where the population mean should lie. It has been shown that characteristics of loosening are preserved in the confidence intervals. Populations of fastener vibration life tests have never before been presented or created.The evaluation of loosening has been conducted for fastener vibration life tests in the past with plots of one test sample; however, in this work statistically quantified results of multiple tests were used. This is important because evaluating loosening with more than one test sample can determine variation between tests. It has been found that secondary locking features do help reduce the loss of preload. The prevailing torque secondary locking feature is found to be more effective as preload is lost. The best secondary locking feature has been found to be the adhesive.
Thesis:
Thesis (M.S.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
Statement of Responsibility:
by Christopher Cheatham.
General Note:
Title from PDF of title page.
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Document formatted into pages; contains 193 pages.

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oclc - 225865156
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ABSTRACT: This thesis presents methods to statistically quantify data from fastener vibration life tests. Data from fastener vibration life tests with secondary locking features of threaded inserts is used. Threaded inserts in three different configurations are examined: no locking feature, prevailing torque locking feature, and adhesive locking feature. Useful composite plots were developed by extracting minimum preloads versus cycles from test data. Minimum preloads were extracted due to the overlapping of varying test data and because the minimum preload is of most interest in such tests. In addition to composite plots, descriptive statistics of the samples were determined including mean, median, quartiles, and extents. These descriptive statistics were plotted to illustrate variability within a sample as well as variability between samples.These plots also reveal that characteristics of loosening for a sample, such as preload loss and rates of preload loss, are preserved when summarizing such tests. Usually fastener vibration life tests are presented and compared with one test sample, which is why statistically quantifying them is needed and important. Methods to predict the sample population have been created as well. To predict populations, tests to determine the distribution of the sample, such as probability plots and probability plot correlation coefficient, have been conducted. Once samples were determined to be normal, confidence intervals were created for test samples, which provides a range of where the population mean should lie. It has been shown that characteristics of loosening are preserved in the confidence intervals. Populations of fastener vibration life tests have never before been presented or created.The evaluation of loosening has been conducted for fastener vibration life tests in the past with plots of one test sample; however, in this work statistically quantified results of multiple tests were used. This is important because evaluating loosening with more than one test sample can determine variation between tests. It has been found that secondary locking features do help reduce the loss of preload. The prevailing torque secondary locking feature is found to be more effective as preload is lost. The best secondary locking feature has been found to be the adhesive.
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Statistical Analysis of Fast ener Vibration Life Tests by Christopher Cheatham A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Daniel P. Hess, Ph.D. Craig Lusk, Ph.D. Nathan Crane, Ph.D. Date of Approval: November 1, 2007 Keywords: preload loss, loosening, populat ion predictions, threaded insert, bolt Copyright 2007, Christopher Cheatham

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Dedication I would like to dedicate this work to my late grandfather, Russ Woods. He is an immense and inspiring influence in my life, having always dedicated his time and effort for others, demonstrating his unselfishness and endless generosity. I believe that he lived by the motto “you do what you have to”, and in that sprit he help ed and supported his family, friends, and even strangers. Like all great teachers, my grandfather’s greatest lessons were imparted to me through his actions. Regardless of what I was involved in, he was always there to cheer me on and offer his support. Through his work ethic he taught me the value of hard work, efficientcy, and thoroughness. He taught me to never quit until the job is done, and done right. In life he taught me how to be gene rous, helpful, caring, punctual, brave, and obedient. Most importantly, he has taught me to enjoy life no matter how long or short it may be. Thank you Old Man, this one is for you.

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Acknowledgment I would like to express my appreciati on for Dr. Daniel Hess for all of the guidance, time, and support he has given me throughout the completion of this work. I would like to thank Dr. Craig Lusk and Dr Nathan Crane for giving their time and energy to be on my committee. I am grateful to my mother Shelley Dougherty who has gone above and beyond of what a mother needs to do. She is one of th e main reasons I am where I am today. I am equally as grateful to my father Jeff Ch eatham, who has supported me through my life's journey. He has helped make me the person I am today.

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i Table of Contents List of Tables ................................................................................................................ ..... iii List of Figures ............................................................................................................... ..... iv Abstract ...................................................................................................................... ....... xii Chapter 1 Introduction ........................................................................................................ .1 1.1 Introduction ........................................................................................................1 1.2 Background ........................................................................................................2 1.3 Overview ............................................................................................................5 Chapter 2 Test Data ........................................................................................................... ..7 2.1 Introduction ........................................................................................................7 2.2 Test apparatus ....................................................................................................7 2.3 Test specimens ...................................................................................................8 2.4 Test setup ...........................................................................................................9 2.5 Experiment design ...........................................................................................10 2.6 Test data ...........................................................................................................12 Chapter 3 Descriptive Analysis .........................................................................................31 3.1 Introduction ......................................................................................................31 3.2 Minimum preload extraction ............................................................................31 3.3 Sample statistics ...............................................................................................40 Chapter 4 Test for Normality .............................................................................................50 4.1 Introduction ......................................................................................................50 4.2 Probability plots ...............................................................................................51 4.3 Probability plot correlation coefficient test for normality ...............................53 Chapter 5 Population Predictions .......................................................................................58 5.1 Introduction ......................................................................................................58 5.2 Confidence interval ..........................................................................................58 5.3 Prediction intervals a nd tolerance intervals .....................................................64 Chapter 6 Interpretation of Results ....................................................................................66 Chapter 7 Conclusions .......................................................................................................80 References .................................................................................................................... ......82

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ii Bibliography .................................................................................................................. ....84 Appendices .................................................................................................................... .....85 Appendix A: M-files for mini mum preload composite plots.................................86 Appendix B: M-files for sample statistic plots ....................................................106 Appendix C: Normal plots ...................................................................................128 Appendix D: M-files for PPCC tests for normality .............................................141 Appendix E: M-files for confidence intervals .....................................................168

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iii List of Tables Table 1.2 Experiment run numbers for three locking levels. .....................................11 Table 1.2 Randomized test sequence. ........................................................................11 Table 4.1 Correlation coefficients pR. ......................................................................55 Table 4.2 Empirical percentage points for correlation coefficient test based on Blom’s plotting position. ............................................................56 Table 4.3 Correlation coefficients pRfor “Locking Heli-Coil with Braycote” with sample size 11. ..................................................................57

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iv List of Figures Figure 2.1 Schematic of test machine. ..........................................................................8 Figure 2.2 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 1. .............................................................................................12 Figure 2.3 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 2. .............................................................................................13 Figure 2.4 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 3. .............................................................................................13 Figure 2.5 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 4. .............................................................................................14 Figure 2.6 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 5. .............................................................................................14 Figure 2.7 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 6. .............................................................................................15 Figure 2.8 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 7. .............................................................................................15 Figure 2.9 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 8. .............................................................................................16 Figure 2.10 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 9. .............................................................................................16 Figure 2.11 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 10. ...........................................................................................17 Figure 2.12 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 11. ...........................................................................................17 Figure 2.13 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 12. ...........................................................................................18

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v Figure 2.14 Preload versus cycles fo r “Locking Heli-Coil with Braycote” run number 13. ...........................................................................................18 Figure 2.15 Preload versus cycles for “Locking Heli-Coil with Braycote” run number 14. ...........................................................................................19 Figure 2.16 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 15. ...........................................................................................19 Figure 2.17 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 16. ...........................................................................................20 Figure 2.18 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 17. ...........................................................................................20 Figure 2.19 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 18. ...........................................................................................21 Figure 2.20 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 19. ...........................................................................................21 Figure 2.21 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 20. ...........................................................................................22 Figure 2.22 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 21. ...........................................................................................22 Figure 2.23 Preload versus cycles for “Locking Heli-Coil with Braycote” run number 22. ...........................................................................................23 Figure 2.24 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 23. ...........................................................................................23 Figure 2.25 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 24. ...........................................................................................24 Figure 2.26 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 25. ...........................................................................................24 Figure 2.27 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 26 ............................................................................................25 Figure 2.28 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 27. ...........................................................................................25

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vi Figure 2.29 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 28 ............................................................................................26 Figure 2.30 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 29 ............................................................................................26 Figure 2.31 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 30. ...........................................................................................27 Figure 2.32 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 31. ...........................................................................................27 Figure 2.33 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 32. ...........................................................................................28 Figure 2.34 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 33. ...........................................................................................28 Figure 2.35 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 34. ...........................................................................................29 Figure 2.36 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 35. ...........................................................................................29 Figure 2.37 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 36. ...........................................................................................30 Figure 3.1 Plot of all data for “Stand ard Heli-Coil with Braycote” runs. ...................32 Figure 3.2 Plot of all data for “Loc king Heli-Coil with Braycote” runs. ....................33 Figure 3.3 Plot of all data for “Sta ndard Heli-Coil with Loctite” runs. ......................33 Figure 3.4 Composite plot for “Standar d Heli-Coil with Braycote” runs. ..................34 Figure 3.5 Composite plot for “Locki ng Heli-Coil with Braycote” runs. ...................34 Figure 3.6 Composite plot for “Stand ard Heli-Coil with Loctite” runs. .....................35 Figure 3.7 Comparison plot of the actu al test data and extracted data for run number 7. .............................................................................................37 Figure 3.8 Comparison plot of the ac tual test data and extracted data for run number 7 zoomed in. ..........................................................................37

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vii Figure 3.9 Comparison plot of the actu al test data and extracted data for run number13. ............................................................................................38 Figure 3.10 Comparison plot of the actu al test data and extracted data for run 13 zoomed in. ......................................................................................38 Figure 3.11 Comparison plot of the actu al test data and extracted data for run number 30. ...........................................................................................39 Figure 3.12 Comparison plot of the ac tual test data and extracted data for run number 30 zoomed in. ........................................................................39 Figure 3.13 Sample mean for “Standard Heli-Coil with Braycote” runs. .....................41 Figure 3.14 Sample mean for “Locking Heli-Coil with Braycote” runs. ......................41 Figure 3.15 Sample mean for “Standard Heli-Coil with Loctite” runs. ........................42 Figure 3.16 Sample median for “Standar d Heli-Coil with Braycote” runs. ..................42 Figure 3.17 Sample median for “Locki ng Heli-Coil with Braycote” runs. ...................43 Figure 3.18 Sample median for “Stand ard Heli-Coil with Loctite” runs. .....................43 Figure 3.19 Sample upper and lower quartile s for “Standard Heli-Coil with Braycote” runs. ..........................................................................................44 Figure 3.20 Sample upper and lower quart iles for “Locking Heli-Coil with Braycote” runs. ..........................................................................................44 Figure 3.21 Sample upper and lower quartile s for “Standard Heli-Coil with Loctite” runs...............................................................................................45 Figure 3.22 Upper and lower extent s for “Standard Heli-Coil with Braycote”runs. ...........................................................................................45 Figure 3.23 Upper and lower extents for “Locking Heli-Coil with Braycote” runs. ..........................................................................................46 Figure 3.24 Upper and lower extent s for “Standard Heli-Coil with Loctite” runs..............................................................................................46 Figure 3.25 Mean, Median, upper and lower quartile, and extent curves for “Standard Heli-Coil w ith Braycote” runs. .................................................47

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viii Figure 3.26 Mean, Median, upper and lower quartile, and extent curves for “Locking Heli-Coil w ith Braycote” runs. ..................................................48 Figure 3.27 Mean, Median, upper and lower quartile, and extent curves for “Standard Heli-Coil with Loctite” runs. ....................................................48 Figure 4.1 Normal plot for “Standard Heli-Coil with Braycote” minimum preloads at 250 cycles. ...............................................................................51 Figure 4.2 Normal plot for “Locking Heli-Coil with Braycote” minimum preloads at 250 cycles. ...............................................................................52 Figure 4.3 Normal plot for “Standard Heli-Coil with Loctite” minimum preloads at 250 cycles. ...............................................................................52 Figure 5.1 95% Confidence Intervals for the population mean of “Standard Heli-Coil with Braycote” runs. ..................................................................61 Figure 5.2 95% Confidence Intervals for the population mean of “Locking Heli-Coil with Braycote” runs. ..................................................................61 Figure 5.3 95% Confidence Intervals for the population mean of “Standard Heli-Coil with Loctite” runs. .....................................................................62 Figure 5.4 99% Confidence Intervals for the population mean of “Standard Heli-Coil with Braycote” runs. ..................................................................62 Figure 5.5 99% Confidence Intervals for the population mean of “Locking Heli-Coil with Braycote” runs. ..................................................................63 Figure 5.6 99% Confidence Intervals for the population mean of “Standard Heli-Coil with Loctite” runs. .....................................................................63 Figure 6.1 Composite Plot (a), De scriptive Statistics Plot (b), 95% Confidence Interval Plot (c), and 99% Confidence Interval Plot (d) for the“Standard HeliCoil with Braycote” runs. ..........................67 Figure 6.2 Composite Pl ot (a), Descriptive Stat istics Plot (b), 95% Confidence Interval Plot (c), and 99% Confidence Interval Plot (d) for the “Locking He li-Coil with Braycote” runs. .........................68 Figure 6.3 Composite Plot (a), De scriptive Statistics Plot (b), 95% Confidence Interval Plot (c), and 99% Confidence Interval Plot (d) for the “Standard He li-Coil with Loctite” runs. ............................69 Figure 6.4 Loosening regions for “Standa rd Heli-Coil with Braycote” plots. ............70

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ix Figure 6.5 Loosening regions for “Loc king Heli-Coil with Braycote” plots. .............72 Figure 6.6 Grip Coil on “Locking Heli-Coil with Braycote” provided by [13]. .......................................................................................................73 Figure 6.7 Free body diagram of fast ener with prevailing torque. ..............................73 Figure 6.8 Locking minimum preloa d minus Std minimum preload versus loss of Locking preload. .............................................................................75 Figure 6.9 Loosening regions for “Sta ndard Heli-Coil with Loctite” plots. ...............76 Figure 6.10 Loctite minimum preloa d minus Std minimum preload versus loss of Loctite preload. ...............................................................................78 Figure 6.11 95 percent Confidence Intervals for al l three locking levels. ....................79 Figure 6.12 99 percent Confidence Intervals for al l three locking levels. ....................79 Figure C.1 Normal plot for “Standard He li-Coil with Braycote” preloads at 10 cycles...................................................................................................128 Figure C.2 Normal plot for “Standard He li-Coil with Braycote” preloads at 250 cycles.................................................................................................128 Figure C.3 Normal plot for “Standard He li-Coil with Braycote” preloads at 500 cycles.................................................................................................129 Figure C.4 Normal plot for “Standard He li-Coil with Braycote” preloads at 750 cycles.................................................................................................129 Figure C.5 Normal plot for “Standard He li-Coil with Braycote” preloads at 1000 cycles...............................................................................................130 Figure C.6 Normal plot for “Standard He li-Coil with Braycote” preloads at 1250 cycles...............................................................................................130 Figure C.7 Normal plot for “Locking He li-Coil with Braycote” preloads at 10 cycles...................................................................................................131 Figure C.8 Normal plot for “Locking He li-Coil with Braycote” preloads at 250 cycles.................................................................................................131 Figure C.9 Normal plot for “Locking He li-Coil with Braycote” preloads at 500 cycles.................................................................................................132

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x Figure C.10 Normal plot for “Locking He li-Coil with Brayco te” preloads at 750 cycles.................................................................................................132 Figure C.11 Normal plot for “Locking He li-Coil with Braycote” preloads at 1000 cycles...............................................................................................133 Figure C.12 Normal plot for “Locking He li-Coil with Brayco te” preloads at 1250 cycles...............................................................................................133 Figure C.13 Normal plot for “Locking He li-Coil with Brayco te” preloads at 1500 cycles...............................................................................................134 Figure C.14 Normal plot for “Locking He li-Coil with Braycote” preloads at 1750 cycles...............................................................................................134 Figure C.15 Normal plot for “Locking He li-Coil with Brayco te” preloads at 2000cycles................................................................................................135 Figure C.16 Normal plot for “Locking He li-Coil with Brayco te” preloads at 2250 cycles...............................................................................................135 Figure C.17 Normal plot for “Standard He li-Coil with Loctite” preloads at 10 cycles...................................................................................................136 Figure C.18 Normal plot for “Standard He li-Coil with Loctite” preloads at 250 cycles.................................................................................................136 Figure C.19 Normal plot for “Standard He li-Coil with Loctite” preloads at 500 cycles.................................................................................................137 Figure C.20 Normal plot for “Standard He li-Coil with Loctite” preloads at 750 cycles.................................................................................................137 Figure C.21 Normal plot for “Standard He li-Coil with Loctite” preloads at 1000 cycles...............................................................................................138 Figure C.22 Normal plot for “Standard He li-Coil with Loctite” preloads at 1250 cycles...............................................................................................138 Figure C.23 Normal plot for “Standard He li-Coil with Loctite” preloads at 1500 cycles...............................................................................................139 Figure C.24 Normal plot for “Standard He li-Coil with Loctite” preloads at 1750 cycles...............................................................................................139

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xi Figure C.25 Normal plot for “Standard He li-Coil with Loctite” preloads at 2000 cycles...............................................................................................140 Figure C.26 Normal plot for “Standard He li-Coil with Loctite” preloads at 2250 cycles...............................................................................................140

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xii Statistical Analysis of Fa stener Vibration Life Tests Christopher Cheatham ABSTRACT This thesis presents methods to statisti cally quantify data from fastener vibration life tests. Data from fastener vibration life tests with se condary locking features of threaded inserts is used. Threaded inserts in three different configurations are examined: no locking feature, prevailing torque locki ng feature, and adhesive locking feature. Useful composite plots were developed by ex tracting minimum preloads versus cycles from test data. Minimum preloads were extrac ted due to the overlapping of varying test data and because the minimum preload is of most interest in such tests. In addition to composite plots, desc riptive statistics of the samples were determined including mean, median, quartiles, and extents. These descriptive statistics were plotted to illustrate va riability within a sample as well as variability between samples. These plots also reveal that charac teristics of loosening for a sample, such as preload loss and rates of preload loss, are preserved when summarizing such tests. Usually fastener vibration life tests are pres ented and compared with one test sample, which is why statistically quantifying them is needed and important. Methods to predict the sample population have been created as well. To predict populations, tests to determine the distribution of the sample such as probability plots and probability plot correlation coefficien t, have been conducted. Once samples were determined to be normal, confidence interv als were created for test samples, which

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xiii provides a range of where the population mean should lie. It has been shown that characteristics of loosening are preserved in the confidence intervals. Populations of fastener vibration life test s have never before been presented or created. The evaluation of loosening has been conduc ted for fastener vibration life tests in the past with plots of one test sample; however, in this work statistically quantified results of multiple tests were used. This is important because evaluating loosening with more than one test sample can determine vari ation between tests. It has been found that secondary locking features do help reduce the loss of prel oad. The prevailing torque secondary locking feature is found to be more effective as preload is lost. The best secondary locking feature has been found to be the adhesive.

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1 Chapter 1 Introduction 1.1 Introduction On August 20, 2007, China Airlines flight CI120 landed at Naha airport in Japan after a flight from Taiwan. Shortly after the 737-800 pulled into it’s parking spot the ground crew informed the captain of the plane that there was a fuel leak. The 165 occupants of the plane immediately evacuat ed. Captured on video was the horrific explosion which occurred at the instant th at the pilot jumped from the cockpit. Thankfully, there were no deaths and no inju ries cause by the expl osion. Later officials explained that a loose bolt had come off of th e slat on one of the main wings and pierced through the fuel tank, causing the fuel to leak out. This is a prime example of how loose bolts cause costly problems and sometimes even catastrophic failures. Though loosening is still a problem, thr eaded fasteners are used in almost all modern applications. Th e advantages of thread ed fasteners outweigh the disadvantages. One of the most useful advant ages of a threaded fastener is that it has the ability to be disassembled. It has been understood for several decade s [1] that threaded fasteners become loose from dynamic loads of vibration and shock. Enduring and resisting these dynamic loads is key for fasteners to retain their clamping force. Usually the friction at the head and threads is the primary for ce that retains the clamping forc e (preload). In addition to the friction at the head and thr eads, some fasteners use seconda ry locking features to help

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2maintain preload. Some of the secondary lock ing features include: lo ck wire, adhesives, pins, and lock washers. However, even with these secondary locki ng features loosening can occur. Typically vibration life tests of fasten ers produce loosening curves or preload versus cycle data, from which plots are create d. In the literature, there is limited data for these fastener life tests, and there are no current methods that quantify the results of such tests. Since these tests can and have been run, there is a need for a formal test that will quantify their results. To cont ribute to this area, this th esis develops methods that statistically quantify and predict the pr eload loss over cycles for a fastener. 1.2 Background For more than half of a century [1] re searchers have studied the loosening of threaded fasteners resulting from vibration. One researcher in the 1960’s named Gerhard Junker, showed that transverse vibration (she ar loading) was the most severe condition for self-loosening of thread fasteners. With the use of hysteresis curves, he [2] determined that loosening results from gross slip (friction loss) at the head and thread interfaces. More recently, Pai and Hess [3, 4] extende d the work of Junker and showed that there are four possible conditions that allow l oosening to exist: loca lized head slip and localized thread slip; localized head slip and complete thread slip; complete head slip and localized thread slip; complete head slip and complete thread slip. Junker showed that the there was complete slip at the head and the threads, but Pai and Hess were able to show that there is also localized slip as well as complete slip in the head and the threads when

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3loosening is occurring. Pai and Hess were ab le to show this th rough finite element models and used hysteresis plots as well as preload verses cycles plots. Also with the use of the hysteresis plots, Pai and Hess were able to show in the preloads versus cycles plots where localized slip and complete slip occurs for a particular test run. These preloads versus cycles plots helped to present how the loss of preload coincides with loosening of a threaded fastener. In other work, Finkelston [5] measured preloads versus cycles data from transverse vibration machines to study and compare preload loss for different threaded fastener conditions. He presents plots that ha ve one test run with another test run of a different configuration. Finkelsto n compared the preload loss of threaded fasteners with different initial preloads fasteners with different thread pitch and the effects of prevailing torque locknuts. Aided by the preload-versus -cycle plots he provides reasons of why the preload loss occurs differently in the different configurations. In another study, Sanclemente and Hess [6] use preload versus time data from a transverse vibration test to determine prel oad loss for threaded fasteners that have multiple high level and low level factors. With the preload loss information, they created a statistical model that determined which of the level factors had the most influence in loosening. Nord-Lock [7] presents test results of tr ansverse vibration tests in the form of preloads versus time to show that their product retains most of the initial preload compared to other available products. Like Finkelston, these plot s present comparison runs between the company’s product against other available products. Similarly, Faroni [8] provides reasons for self-loosening of thr eaded fasteners and presents a plot that

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4contains the range of preload loss over the durat ion of a vibration test for different quality fasteners (i.e., best-Aircraft fasten ers, poorest-Commercial fasteners). These methods of studying preload loss in th e form of preloads versus cycles are beneficial in assisting in th e understanding the self-loosening of threaded fasteners. The use of preloads versus cycles plots are useful when comparing the loosening between two different threaded fastener configurations. Most of these works use pr eloads versus cycles plots, but only use the prel oad loss for individual test samples to determine which configuration loosens less. Finkelston [5] pr ovided explanations of loosening using the preload versus cycles plots, but he described the results from one test run. Pai and Hess [3] also describes how a fastener loosens with the preloads ve rsus cycles plots in a plot that shows one test run; howev er Pai and Hess used more than one test run to develop their conclusions. Methods that wi ll quantify test data of fast eners that undergo transverse vibration tests are needed in order to quan tify the loosening when using preload-versuscycle plots. In a technical paper produced by the So ciety of Manufacturing Engineers [9], it was shown how statistics can be used to repor t, design, or even pred ict the behavior of bolted joints. By using the test sample data one can statistically describe the population. In testing bolted joints, the technical paper suggests that the populat ion could consist of: all of the bolts in a given joint and all of the bolts in all such joints in a given assembly. This technical paper provides ways to statis tically evaluate the a pproximate torque for a given preload, nut factors, rela xation, and friction; however it never provides methods of statically quantifying a vibration life test of threaded fasteners.

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5 Similar to vibration life tests of fasteners, stress life tests or fatigue tests present fatigue data in plots of stress versus cycles to failure. Th ese plots, also known as S-N curves are created by fitting the observed data which was collected from tests that have sample sizes greater than one [10]. Collins [10] explains that because of the scatter of fatigue life data, a statistical description of fatigue failure data should be used to make appropriate S-N curves. Using st atistics, curves of constant probability of failure can be created and are seen in S-N-P curves. S-N-P curves show many curves that present different probabilities of failure at a percent of reliability. This s hows that statistically quantifying the variation of vibration life te sts of fasteners can be done. The variation between fastener life tests should be quantifie d statistically to help explain the loosening occurring in the fastener vibration life test. All of this work [2-9] uses preload versus cycles data from vibration life tests. Most of these works use data from individua l test runs to develop conclusions. Because individual test runs will vary from one run to the next, a method to quantify and predict the loss of preload that incor porates that variation is need ed. Such quantified results can be used to help develop more meaningful conclusions to descri be a population with a sample size greater than one. This thesis contri butes to this void in the literature. This thesis provides methods that statistically qua ntify the preload versus cycles data for a reasonable sample size and predicts the loos ening of threaded fasteners for a population. 1.3 Overview This thesis presents methods that statis tically quantify the loos ening curve life test data. Chapter 2 presents how fast ener vibration life test data was obtain in this thesis so

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6that methods to quantify the data from such test can be created. To quantify the test results, Chapter 3 explains the techniques used to plot all of the minimum preloads versus cycles for a sample. This chapter also presents how plots of the sample statistics were created. Before statistical quantifications can be made, the distribution from which the data belongs must be identified. Chapter 4 disc usses the tests of normality that were used to determine if the sample data was normally distributed. After the data was identified to be normally distributed, predictions of th e population are created in Chapter 5. The predictions of the population are in the form of confidence intervals, which provided an interval in which the population’s mean will lie at a percent confidence. To demonstrate how these quantified results can describe the underlying mechanisms of fastener loosening, Chapter 6 uses the results develope d in this thesis to describe the various mechanisms of loosening. Finally, in Chapter 7 the conclusions are stated.

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7 Chapter 2 Test Data 2.1 Introduction In this thesis, test data is used to pr edict populations that describe loosening for three different locking features. This chapter pres ents the data used in this work as well as information on how the data was acquired. Even though this test data was not measured as part of this thesis, the te st apparatus, test specimens, and how the tests were run are described. 2.2 Test apparatus The test apparatus typically used to st udy loosening is shown in Figure 2.1. This machine applied a cyclic shear load by means of an adjustable eccentric that was driven by a five HP motor. The adjustable eccentric was connected to the top plate by an arm, and the top plate was fastened to a fixed base through a threaded insert using a test screw. The test screw clamped the top plate to the fi xed base through a wash er, a cone, and load cell fixture as shown in Figure 2.1. The load cell fixture is se t in a preload a load cell and the cone was set in the top plate. Roller bear ings were located between the top plate and the fixed base; which allowed the top plate to move when a shear load was applied through the arm from the eccentric. Measurements of shear force on the top plate as well as preload of the screw were measured by load cells. A Linear Variable Differential

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8Transformer, or LVDT for short, was used to measure the transverse displacement of the top plate. Figure 2.1 Schematic of test machine. 2.3 Test specimens The test data used in this work was obtai ned from the apparatus described above with NAS 1004 -28 UNJF-3A hex head screws [11] with: 1. Standard free-running Heli-Coil inserts w ith Braycote 601 EF high vacuum grease 2. Locking Heli-Coil inserts with Bray cote 601 EF high vacuum grease, and 3. Standard free-running Heli-Coil inse rts with Loctite 242 threadlocker.

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9These three different locking le vels were tested twelve times each, providing data in sets of twelve. The specifications for the screw, Heli-Coil inserts, and washers used in the tests are as follows: 1. Thirty-six NAS 1004 -28 UNJF-3A, 2.356 inch long, hex head screws, made of A286 Stainless steel [11] 2. Thirty-six NAS 1149-C0463R washers for inch screw made of corrosion resistant steel with a passivated finish [12] 3. Twenty-four MS124696, 0.375 inch long, standard, free-running Heli-Coil inserts, made of stainless steel [13] 4. Twelve MS21209-F4-15, 0.375 inch long, locking Heli-Coil inserts, made of 304 stainless steel [14] The thirty-six cones and load cell fixtures were made of 15-5 stainless steel and heat treated to RC35. The cones had a thru-hole for the test screw to run through, and the load cell fixtures had tapped holes ready fo r the Heli-Coils to be installed. 2.4 Test setup The tests were set up to provide signifi cant loosening for the “Standard Heli-Coil with Braycote” configuration over a finite number of cycles without causing screws to break for any of the locking levels. Before the tests were run, the thirty-six cones and thirty-six load cell fixtures were thoroughly pre-cleaned in an ultrasonic cleaner with MEK (methyl ethyl kentone). After the pre-cl eaning the Heli-Coil inserts were installed

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10into the load cell fixture. All parts, including the test screw, washer, cone, and load cell with the Heli-Coil installed, were cleaned ag ain in an ultrasonic cleaner with MEK. When the parts are installed in the test apparatus, Braycote grease was applied under the screw head and washer for all the tests. Tests for “Sta ndard Heli-Coil with Braycote” runs and “Locking Heli-Coil with Braycote” have Braycote applied to their threads, and then are tightened to 2,400 lbs of preload. The tests for “Standard Heli-Coil with Loctite” runs have Loctite applied to their threads, and then are tightened to 2,400 lbs of preload. After test specimens were fully assembled, the test machine was run at 15 Hz with a 0.12 inch (3 mm) eccentric. The data obtaine d from each test was collected at 51.2 samples/second for a total of 8,192 data points for each measured variable (displacement, preload, and shear force). 2.5 Experiment design The tests were designed for replicati on and randomization for a single-factor experiment. The single-factor in the tests was the secondary lo cking feature. There were three levels of this factor: “Standard Heli-Co il with Braycote”, “Locking Heli-Coil with Braycote”, and “Standard Heli-Coil with Loctite”. Replication was achieved by using twelve replicas for each of the locking levels for a total of thirty-six runs. Run numbers for each level are presented in Table 2.1. Ra ndomization of the tests ensured that the observations were independently distributed random variables and it also averaged out the effects of extraneous factors [15]. Th e test sequence was randomized using the Excel “rand” function and sort t ool. Table 2.2 lists the te st sequence for the runs.

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11Table 1.1 Experiment run num bers for three locking levels. Locking LevelExperiment Run Number Std Heli-Coil w/ Braycote123456789101112 Locking Heli-Coil w/ Braycote131415161718192021222324 Std Heli-Coil w/ Loctite252627282930313233343536 Table 1.2 Randomized test sequence. Test sequenceRun Numbe r Locking level 17Std Heli-coil w/ Brycote 213Locking Heli-coil w/ Brycote 31Std Heli-coil w/ Brycote 410Std Heli-coil w/ Brycote 530Std Heli-coil w/ Loctite 627Std Heli-coil w/ Loctite 76Std Heli-coil w/ Brycote 83Std Heli-coil w/ Brycote 916Locking Heli-coil w/ Brycote 1022Locking Heli-coil w/ Brycote 115Std Heli-coil w/ Brycote 1228Std Heli-coil w/ Loctite 1326Std Heli-coil w/ Loctite 1421Locking Heli-coil w/ Brycote 1512Std Heli-coil w/ Brycote 1614Locking Heli-coil w/ Brycote 1731Std Heli-coil w/ Loctite 1824Locking Heli-coil w/ Brycote 1929Std Heli-coil w/ Loctite 2036Std Heli-coil w/ Loctite 212Std Heli-coil w/ Brycote 2234Std Heli-coil w/ Loctite 2325Std Heli-coil w/ Loctite 2419Locking Heli-coil w/ Brycote 254Std Heli-coil w/ Brycote 268Std Heli-coil w/ Brycote 2723Locking Heli-coil w/ Brycote 289Std Heli-coil w/ Brycote 2932Std Heli-coil w/ Loctite 3017Locking Heli-coil w/ Brycote 3133Std Heli-coil w/ Loctite 3215Locking Heli-coil w/ Brycote 3311Std Heli-coil w/ Brycote 3435Std Heli-coil w/ Loctite 3518Locking Heli-coil w/ Brycote 3620Std Heli-coil w/ Loctite

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122.6 Test data The test data obtained from these test are in the form of plots that show preload versus cycles. There are 8192 data points fo r each test that was run. The plots allow individual test runs to be viewed, and allows preload loss over a finite number of cycles to be determined for each test. Comparisons of preload loss can be made from one plot to the next. From the test data that was acquired, comparisons can be made for each of the thirty-six tests. Figures 2.2 -2.37 show the data that was obtained. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.2 Preload versus cycles for “Sta ndard Heli-Coil with Braycote” run number 1.

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13 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.3 Preload versus cycles for “Sta ndard Heli-Coil with Braycote” run number 2. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.4 Preload versus cycles for “Sta ndard Heli-Coil with Braycote” run number 3.

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14 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.5 Preload versus cycles for “Sta ndard Heli-Coil with Braycote” run number 4. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.6 Preload versus cycles for “Sta ndard Heli-Coil with Braycote” run number 5.

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15 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.7 Preload versus cycles for “Sta ndard Heli-Coil with Braycote” run number 6. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.8 Preload versus cycles for “Sta ndard Heli-Coil with Braycote” run number 7.

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16 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.9 Preload versus cycles for “S tandard Heli-Coil with Braycote” run number 8. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.10 Preload versus cycles for “S tandard Heli-Coil with Braycote” run number 9.

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17 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.11 Preload versus cycles for “S tandard Heli-Coil with Braycote” run number 10. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.12 Preload versus cycles for “Standard Heli-Coil with Braycote” run number 11.

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18 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.13 Preload versus cycles for “S tandard Heli-Coil with Braycote” run number 12. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.14 Preload versus cycles for “Locking Heli-Coil with Braycote” run number 13.

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19 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.15 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 14. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.16 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 15.

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20 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.17 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 16. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.18 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 17.

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21 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.19 Preload versus cycles for “Loc king Heli-Coil with Braycote” run number 18. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.20 Preload versus cycles for “Loc king Heli-Coil with Braycote” run number 19.

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22 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.21 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 20. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.22 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 21.

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23 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.23 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 22. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.24 Preload versus cycles for “Locking Heli-Coil with Braycote” run number 23.

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24 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.25 Preload versus cycles for “L ocking Heli-Coil with Braycote” run number 24. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.26 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 25.

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25 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.27 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 26. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.28 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 27.

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26 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.29 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 28. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.30 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 29.

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27 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.31 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 30. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.32 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 31.

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28 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.33 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 32. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.34 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 33.

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29 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.35 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 34. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.36 Preload versus cycles for “S tandard Heli-Coil with Loctite” run number 35.

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30 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 CyclesPreload (lb) Figure 2.37 Preload versus cycles for “Sta ndard Heli-Coil with Loctite” run number 36.

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31 Chapter 3 Descriptive Analysis 3.1 Introduction Ordinarily, test data are st udied on an individual basi s. The minimum preload is found at a predetermined number of cycles fo r each of the test runs and a statistical analysis is conducted on these minimum preloads. The goal of this thesis is to provide a way to quantify the complete preload vers us cycles results of these tests both descriptively and sta tistically so predicti ons of how the population is loosening for a given configuration can be cr eated. This chapter describe s how methods of condensing the test data were created in order to pr ovide useful quantitative results. 3.2 Minimum preload extraction Comparing the minimum preloads at a predetermined number of cycles statistically is very useful, but it does not fully explain wh at is occurring during the full run of these tests. A plot th at would show all samples from the same locking level would illustrate the level of repeatability and variab ility of the loosening in each configuration. Plotting all of the test runs for the same locking level on one plot seems like an obvious solution. However in Figures 3.1-3.3 it can be seen that plots of this kind present results with limited information. The test data over laps one another with the variation of preload during a cycle, which makes it hard to obser ve minimum preloads and loosening trends between runs especially with little differences between te sts. Since we are generally

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32interested in the minimum prel oads, it would be useful to extract the minimum preloads from each run and plot them against cycles for each run of each locking level. Figures 3.4-3.6 show these plots. M-files for MatLab ha ve been written to create these composite plots, and can be found in Appendix A. Th ese composite plots present the minimum preload versus cycles for the test data for all three locking levels. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.1 Plot of all data for “S tandard Heli-Coil with Braycote” runs.

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33 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.2 Plot of all data for “L ocking Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.3 Plot of all data for “S tandard Heli-Coil with Loctite” runs.

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34 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.4 Composite plot for “Sta ndard Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.5 Composite plot for “Loc king Heli-Coil with Braycote” runs.

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35 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.6 Composite plot for “Sta ndard Heli-Coil with Loctite” runs. The m-file minstdlines2.m creates the composite plot for the “Standard Heli-Coils with Braycote” test runs in Figure 3.4, m-file minlockinglines2.m creates the composite plot for the “Locking Heli-Coils with Bray cote” test runs in Figure 3.5, and m-file minloctitelines2.m creates the composite plot for the “Standard Heli-Coil with Loctite” test runs in Figure 3.6. The obtai ned test data was recorded in to data files which can be read by MatLab. These files recorded pr eload and time at a sample rate of 51.2 samples/second for 160 seconds creating 8,192 data points for preload and time. In the mfiles the data files are loaded as arrays. Each m-file loads th e data obtained from runs of the same locking level. The minimum preloads for every 18.75 cycles are extracted using “for” loops. Sixty-four data points are grouped into an arra y 128 times for a total of 8,192 data points for both preload and cycles. The f unction “min” is used to sort the preload arrays from least to greatest. The first elem ent in the array, which is the minimum preload

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36for the sixty-four data points, is then put into another array ) (j Ci for a particular test where i=1-12 and j=1-128. The array ) (j Cicontains the 128 minimum preloads for every 64 recorded preloads; which is also the minimum preload for every 18.75 cycles. The cycle arrays do not need to be sorted and the median of each array is assigned to the array ) (j Difor a particular test. That is to sa y every 18.75 cycles makes up the arrays ) (j Di. Then all of the preload arrays ) (j Cifor a locking level are plotted against the cycle’s arrays ) (j Di. Even though the minimum preloads are bei ng plotted against the median of cycles per sixty-four data points, the composite plot s show either a very accurate representation or a slightly conservative representation. Figures 3.7-3.12 pr esent comparisons of the actual data and the minimum preload extrac ted data. Figure 3.7 and Figure 3.8 represent the most conservative minimum preload ex traction, Figure 3.9 and Figure 3.10 represent the typical minimum preload extractions, and Figure 3.11 and Figure 3.12 represent one of the best minimum preload extractions. When rates of preload per-cycles loss are larger, the minimum preload extraction plots are less accurate as seen in Figure 3.7 and Figure 3.8. This may be adjusted in the m-file by changing dpcm to be equal to dpc. This would change the median of cycles to equa l the end point of every sixty-four cycles. However, in this study we found using the median of cycles acceptable as the conservative representations were only slightly conservative.

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37 0 500 1000 1500 2000 -500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Extracted Data Actual Test Data Figure 3.7 Comparison plot of the actual test data and ex tracted data for run number 7. 0 100 200 300 400 0 500 1000 1500 2000 Preload (lb)Cycles Extracted Data Actual Test Data Figure 3.8 Comparison plot of the actual test data and ex tracted data for run number 7 zoomed in.

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38 0 500 1000 1500 2000 0 500 1000 1500 2000 2500 Preload (lb)Cycles Extracted Data Actual Test Data Figure 3.9 Comparison plot of the actual test data and ex tracted data for run number 13 1050 1100 1150 1200 1250 500 600 700 800 900 Preload (lb)Cycles Extracted Data Actual Test Data Figure 3.10 Comparison plot of the actual te st data and extracted data for run number 13 zoomed in.

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39 0 500 1000 1500 2000 1700 1800 1900 2000 2100 2200 2300 2400 Preload (lb)Cycles Extracted Data Actual Test Data Figure 3.11 Comparison plot of the actual test data and ex tracted data for run number 30. 1200 1400 1600 1800 2000 2200 1780 1800 1820 1840 1860 1880 Preload (lb)Cycles Extracted Data Actual Test Data Figure 3.12 Comparison plot of the actual test data and extracted data for run number 30 zoomed in. Compared to viewing the test data one sample at a time or all composite plots with preload cycle variation, the composite plots of minimum prel oads versus cycles provide many advantages. It allows the all sample s of a test to be vi ewed in one plot to

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40see if the loosening is simila r for each related sample. In th e case of the obtained test data, these composite plots reduced thirty-s ix plots down to three. Comparison of the locking performance is then easily evaluate d by comparing the thre e composite plots. Patterns of loosening emerge out of the co mposite plot, and it sheds insight on the loosening process as a whole. These composite plot m-files also contribute to a technique that will allow the predictions of the populations to be created. 3.3 Sample statistics In addition to the composite plots, presenti ng sample statistics in a plot that would summarize test samples by preload versus cy cles would help to describe what is happening with the samples as a whole. This will compliment the composite plots and provide a tool to summarize sample test results. The sample statistics that will be evaluated are the sample mean, sample median, sample quartiles, and sample extents. Individual plots of these samp le statistics were made in Ma tLab and are shown in Figures 3.13-3.24.

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41 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.13 Sample mean for “Stand ard Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.14 Sample mean for “Loc king Heli-Coil with Braycote” runs.

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42 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.15 Sample mean for “Stand ard Heli-Coil with Loctite” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.16 Sample median for “Sta ndard Heli-Coil with Braycote” runs.

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43 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.17 Sample median for “Loc king Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.18 Sample median for “Sta ndard Heli-Coil with Loctite” runs.

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44 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.19 Sample upper and lower quartiles for “Standard Heli-Co il with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.20 Sample upper and lower quartil es for “Locking Heli-Coil with Braycote” runs.

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45 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.21 Sample upper and lower quartiles for “Standard Heli-Co il with Loctite” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.22 Upper and lower extents for “Standard Heli-Coil with Braycote” runs.

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46 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.23 Upper and lower extents fo r “Locking Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.24 Upper and lower extents for “Standard Heli-Coil with Loctite” runs.

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47Though these sample statistics plots are very descriptive, it is still difficult to evaluate a sample as a whole when trying to compare four figures per sample. To create a tool that would allow all sample statistics to be presented in one pl ot, another type of mfile has been created and can be viewed in Appendix B. These m-files create preload versus cycle plots of sample means, samp le medians, sample quartiles, and upper and lower extents of the tests for each locki ng level in one plot. Figures 3.25 -3.27 shows these descriptive statistic plots. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.25 Mean, Median, upper and lower qu artile, and extent curves for “Standard Heli-Coil with Braycote” runs.

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48 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.26 Mean, Median, upper and lower quar tile, and extent curves for “Locking Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 3.27 Mean, Median, upper and lower qu artile, and extent curves for “Standard Heli-Coil with Loctite” runs.

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49M-files quanstd2.m, quanlocking2.m, and quanloctite2.m create the descriptive statistical plots of minimum preloads versus cycles for the “Standard Heli-Coil with Braycote”, “Locking Heli-Coil with Braycot e”, and “Standard HeliCoil with Loctite” tests data respectively. In the m-files, the minimum preloads and median of cycles are found by the same method used in m-files fo r the composite plots. That method creates arrays that contain 128 minimum preloads from every 18.75 cycles. Then minimum preloads for all runs are put into matrix td in a manner in which each row is the minimum preload for all runs at a partic ular cycle. A “for” loop then se lects each row of the matrix td and extracts the upper and lower extents, medians, 25% quartile, and 75% quartiles from each row and puts that extracted data into arrays. The medians and quartiles are found using the MatLab function “quantile” a nd the extents are found using the MatLab function “min” and “max”. The means of the samples are calculated by adding each minimum preload array together and then dividing them by the number of samples. This creates on array containing 128 m eans of all the minimum prel oads. Then the arrays of means, medians, quartiles, and extent s are plotted against a cycle array) (j Di. These m-files provide a tool to aid evalua tion of the sample data in a clear and concise manner. At a glance these plots pr esent where 100 percent of the data lies (between upper and lower extents), where 50 pe rcent of the data lies (between the upper and lower quartile), the sample mean, and th e sample median. Sample test results then can be easily evaluated.

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50 Chapter 4 Tests for Normality 4.1 Introduction This work is focused on developing popul ation predictions that quantify the loosening of preloaded bolted joints. In pa rticular this worked develops population predictions for three different locking levels of threaded inserts, but could be used for most tests from a transverse vibration machin e. In a perfect scenario the results would have shown a single curve for preload versus cy cles (i.e., no variation from test to test) for each of the three different locking levels However, like most experiments, the test data obtained for this work showed varia tion within individual locking levels. To describe the statistical model of loosening, a graphical model to describe the population is desired. To do this the distribution that the samples belongs to must be determined, and tests for normality will be conducted. Using the minimum preloads at approximately 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles, ten sets of sample size twelve are created for the “Locking Heli-Coil with Braycote” and “Standard Heli-Coil with Loctite” locking levels. The “Standard HeliCoil with Braycote” locking level only has six sets of sample size twelve using the minimum preloads at 10, 250, 500, 750, 1000, 1250 cycles. This is because only twenty-five percent of the “Standard Heli-Coil with Braycote” minimum preload data has a value greater than zero af ter 1250 cycles. These sample subsets will be tested for normalit y, to determine whether or not the samples belong to the Normal Distribution.

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514.2 Probability plots To determine whether the data collected fr om the three different locking levels are normally distributed, normal probability plots ha ve been developed. If the sample sets are normally distributed then populat ion predictions that describe the loosening of the three different locking levels can be developed usi ng equations for normally distributed sample sets. The normal probability plot is a graphi cal technique that allows for a quick visual test to determine whether or not the samp le data is from the Normal Distribution. Representative normal probability plots for mi nimum preloads at 250 cycles for each of the three locking levels are shown in Figures 4.1-4.3. Figures 4.1-4.3 show a good linear relationship on the minimum preloads at 250 cy cles. The normal probability plots for all twenty-six sample sets are provided in Appendix C. 1300 1400 1500 1600 1700 1800 1900 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbab ility Figure 4.1 Normal plot for “Standard Heli-Coil with Bray cote” minimum preloads at 250 cycles.

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52 1550 1600 1650 1700 1750 1800 1850 1900 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure 4.2 Normal plot of “Locking Heli-Coil with Brayco te” minimum preloads at 250 cycles. 1750 1800 1850 1900 1950 2000 2050 2100 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure 4.3 Normal plot of “Standard Heli-Coil with Loctite ” minimum preloads at 250 cycles.

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53 Normal probability plots were created in MatLab using the command “probplot”. The “probplot” function in MatLab uses one of the most commonly used plotting positions to create Normal Plots. The pl otting position is defined by the following equation: n i pi) 5 0 ( (4.1) were ip is the plotting position, n is the sample, and i= 1.. n. When these plots are approximately linear it can be said that the samples belong to the Normal Distribution. Most of the plots appear to have a strong linear relations hip, but to see how linearly related they are, a further check for normality is pursued. 4.3 Probability plot correlation co efficient test for normality A probability plot correlation coefficient (PPCC) test for normality was used to complement the results of the probability plots. One of the most useful features of the PPCC is that it produces a single number that quantifies normality. PPCC tests describe how linear the probability plot s are. M-files for MatLab we re created to calculate PPCC for the same sample sets as the Proba bility Plots. In Appendix D, m-file normstd2.m calculates the correlation coefficients for “S tandard Heli-Coil with Braycote” test data, m-file normlocking2.m calculates correlation coefficients for the “Locking Heli-Coil with Braycote” test data, and the m-file normloctite2.m calculates the correlation coefficient for “Standard Heli-Coil with Loctite” test da ta. These m-files calculate a total of twentysix correlation coefficients, one for each of the normal plots. Thus the same minimum

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54preloads that were used to cr eate the normal plots were also used in these m-files to run PPCC. To calculate the correlation coefficien t the minimum preload data must be arranged in order from smallest to largest, 12 2 1.... x x x, where 1xis the smallest and 12xis the largest for each sample set. To do this, th e same method used in the sample statistic m-files is implemented and the minimum preload arrays are put into a matrix td. Rows 5, 18, 31, 44, 58, 71, 84, 98, 111, 124 of matrix td contain minimum pr eloads at 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles respectively. Which are the minimum preloads used for creating the nor mal plots. Then using “for” loop these preloads are put into arrays ) ( j mi, where i=1..10 and j=1..12. These arrays are sorted by the function “sort” from least to greatest. Th e means of these arrays are calculated by the function “mean”, which will be needed later to find the correlation coefficient. Next the uniform order statistic median is calculated from the Blom plotting position equation (4.2). ) 25 0 /( ) 375 0 ( n i pi (4.2) Here i is 1, 2...12 and n is twelve for a sample size of twelve. The Blom plotting position equation (4.2) is used because it has been found to provide a more powerful correlation coefficient test than the Shanpiro-Francia, Filliben tests, and tests using the plotting position equation (4.1) [16]. Then, the normal order statistic median was approximated by equation (4.3). ) ( 001308 0 183269 0 432788 1 1 ) 010328 0 82853 0 515517 2 (3 2 2p e t t t t t t bi (4.3) where,

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55 ip t 1 ln (4.4) 410 5 4 ) (p e (4.5) Finally the correlation coefficient pR is calculated with equation (4.6). 2 2) ( ) ( ) ( ) ( b b x x b b x x Ri i i i p (4.6) The summations of the correlation coefficient are calculated using “for” loops. The actual correlation coefficients are the output of the m-files and are seen in an array. Elements one through ten represent the correlation coeffi cients in order for minimum preloads at cycles 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250. Results of the Correlation Coefficients for all thirty samples can be seen in Table 4.1. When the correlation coefficient has a hi gher value than the critical value the sample set can be said to belong to the Normal Dist ribution. The critical values for the Blom plotting position are found in tables provi de by Looney and Gulledge [16]. Table 4.2 gives a few of these critical values for data with different sample sizes and significant levels. Table 4.1 Correlation coefficients pR. Cycles Standard Heli-Coil w/ Braycote Locking Heli-Coil w/ Braycote Standard Heli-Coil w/ Loctite 10 0.992 0.950 0.993 250 0.977 0.959 0.969 500 0.938 0.920 0.944 750 0.881 0.981 0.931 1000 0.704 0.954 0.904 1250 0.603 0.835 0.875 1500 NA 0.781 0.866 1750 NA 0.778 0.855 2000 NA 0.783 0.854 2250 NA 0.799 0.876

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56Table 4.2 Empirical percentage points fo r correlation coefficient test based on Blom’s plotting position. Sample Size Significant Level 0 0.005 0.01 0.025 0.05 0.1 10 0.578 0.862 0.879 0.901 0.918 0.934 11 0.560 0.870 0.886 0.907 0.923 0.938 12 0.544 0.876 0.892 0.912 0.928 0.942 13 0.529 0.885 0.899 0.918 0.932 0.945 The sets of samples are normally distribut ed with a significance level of 0.005 for all the “Standard Heli-Coils w ith Braycote” except for the minimum preloads at 1000 and 1250 cycles. This is due to the fact that more th an half of the test runs have a preload of zero at 1000 and 1250 cycles, as other runs still record a preload above zero. Samples for the “Standard Heli-Coils with Loctite” are normally distributed at a significance level 0.005 or highe r except for minimum preloads sample sets at cycles 1500 to 2000. This is mainly due to the lager difference in test runs, and small sample size. However, with more than half of the samples coming from the Normal Distribution, a useful model can still be deve loped for this locking level. The sets of samples for the “Locking Heli-Coil with Braycote” are normally distributed for minimum prel oads at cycles 10 to 1000 at a significance level above 0.025; however, minimum preloads at cycles 1250 to 2250 are not norma lly distributed at all. Looking at both the normal plots and com posite plots for the Locking Heli-Coil with Braycote, it is apparent that R un 20 is a large outlier. Outliers tend to appear at the tails of probability plots where Run 20 is seen in all of the normal plots. Looking at the composite plot in Figure 3.5 it is very noticea ble that Run 20 is a sample that is very different from the other eleven samples. So Run 20 was taken out of the samples sets, and

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57a PPCC test was conducted again for the Locki ng Heli-Coil with Braycote with a sample size of 11. These new correlation coeffi cients are present in Table 4.3. Table 4.3 Correlation coefficients pR for “Locking Heli-Coil with Braycote” with sample size 11. Cycles Locking Heli-Coil w/ Braycote 100.935 2500.945 5000.887 7500.965 10000.975 12500.986 15000.987 17500.986 20000.979 22500.981 After removing Run 20 from the “Locking He li-Coil with Braycote” samples, all samples are normally distribute d at a significance level of 0. 01 or higher. Now that the most of the data is known to be normally di stributed predictions of the population can be created.

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58 Chapter 5 Population Predictions 5.1 Introduction Now that the sample data has been determined to be normally distributed, as defined in Chapter 4, predictions of the populat ion that describe the loosening of the three different locking levels can be created. Th ese population predictions will use the sample data to produce graphical models that accura tely describe the population of the samples. This is important because if the population is identified, conclusions about the loosening process for each of the three locking levels can be made. Th is chapter presents methods used to describe loosening of the populations themselves. 5.2 Confidence intervals Confidence intervals are used to provide an interval that will include the true population parameter at a specified probability This is useful when describing the population. Using the sample means and standard deviations, it is possi ble to describe the population’s mean with a predetermined conf idence level. These confidence intervals will provide a useful graphical model of loosening for the three different locking levels. Confidence intervals are created for the mini mum preloads at cycles 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, and 2250 if the samples were determined to have come from the Normal Distribution for each of the three different locking levels. A sample size of twelve was used for the Standard Heli-Coil with Braycote and the Standard Heli-Coil

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59with Loctite runs. A sample size of eleven was used for the Locking Heli-Coil with Braycote runs due to the remova l of Run 20 from the samples. M-files for MatLab in Appendix E have b een written to calculate the confidence intervals and plot them with their sample means of preload versus cycles for 95% and 99% confidence levels for all three locki ng levels. The 95% confidence intervals for minimum preloads for “Standard Heli-Coil with Braycote”, “Locking Heli-Coil with Braycote”, and “Standard Heli-Coil with Locite” test data ar e found using m-files CI95std2.m CI95locking2.m and CI95locite2.m respectively, and the 99% confidence intervals for minimum preloads for “Standard Heli-Coil with Braycote”, “Locking HeliCoil with Braycote”, and “Standard Heli-Coil with Locite” test data are found using mfiles CI99std2.m CI99locking2.m and CI99locite2.m respectively. To find the minimum preloads at 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, and 2250 cycles, these m-files that create the confidence interval plots use the same technique as the m-files that found the correlation coeffici ent. See Chapter Four to reference that technique. The sample means and standard deviation of each array of minimum preloads were calculated using the MatLab functions “mean” and “std”. The ten means and ten standard devi ations are put into arrays mm and sdm. Then “for” loops calculate the upper and lower confidence inte rval using the doubl e sided Student’s tdistribution confidence interval equation ) ( ) (1 2 / 1 2 /n s t x n s t xn n (5.1) where x is the sample mean, s is the sample sta ndard deviation, n is the sample size, is the mean of the population, is the level of confidence, and 1 2 / nt is the Student’s t-

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60distribution’s critical value found in a Student’s t Distribu tion table as a function of degrees of freedom and confidence level. Critic al values from the St udent’s t-Distribution are: 2.20 for a sample size of twelve and 95% confidence level, 2.23 for a sample size of eleven and 95% confidence level, 3.11 fo r a sample size twelve and 99% confidence level, and 3.17 for a sample size eleven a nd 99% confidence level. The upper and lower confidence intervals are then paired with ma tching cycles and plotted with the sample mean. The double sided Student’s t-Distribution confidence interval equation was used because the samples were normally distribut ed and the sample sizes were less than twenty. The Student’s t-distribution is appr oximately equal to th e Normal Distribution when sample sizes are twenty or larger; howev er with sample sizes less than twenty, the Student’s t-distribution’s area und er the bell curve increases at the tails. This is because for small sample sizes, the sample standard deviation, s is not a good estimate of the population’s standard deviation and so the Student’s t-dist ribution provides different critical values for different sample sizes ; whereas the Normal Distribution uses one critical value for all sample sizes. Figures 5.1-5.6 present these plots of the confidence intervals. The sample mean is the dashed line, and the confidence interval of minimum preload at cycles 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, and 2250 are solid lines.

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61 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 5.1 95% Confidence Intervals for the population mean of “Standard Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 5.2 95% Confidence Intervals fo r the population mean of “Locking Heli-Coil with Braycote” runs.

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62 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 5.3 95% Confidence Intervals for th e population mean of “Standard Heli-Coil with Loctite” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 5.4 99% Confidence Intervals for the population mean of “Standard Heli-Coil with Braycote” runs.

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63 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 5.5 99% Confidence Intervals fo r the population mean of “Locking Heli-Coil with Braycote” runs. 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Preload (lb)Cycles Figure 5.6 99% Confidence Intervals for th e population mean of “Standard Heli-Coil with Loctite” runs.

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64The 95 percent confidence interval defines the population mean range with 95 percent confidence, and the 99 percent confidence in terval defines the population mean range with 99 percent confidence. 5.3 Prediction intervals and tolerance intervals Prediction intervals, sometimes know as fo recast intervals, are able to provide an interval that contains a future run within th e sample; which is determined from previous runs. In the case of this study, a prediction in terval would predict a thirteenth sample for sample sets of twelve and woul d predict twelfth sample for sample sets of eleven. Using a normal two sided prediction interval, interval s were made for the same minimum preload sample sets as the normal plots and confidence intervals. However, due to larger sample standard deviations and small sample sizes, pr ediction intervals predicted that in a future sample the preload could increa se to larger than the initial preload as cycles increase. This would indicate a possibility of tightening. Due to the fact that all test results show loosening, the results for prediction interv als have been left out of this study. Tolerance intervals are intervals that contain a specified proportion of the population with a confidence level. This is to say that tolerance inte rvals contain a certain percentage of the population with a degree of confidence. These intervals are created with sample means, standard deviations, si zes, and critical values from the Normal Distribution and the Chi-Square d distributions. Tolerance inte rvals were also created for the same minimum preload sample set as the normal plots and confidence intervals. But just like the prediction intervals, small sa mple sizes and large standard deviations calculated toleran ce intervals that sugges t that the population may have an increase in

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65preload instead of decrease when cycles incr ease. All test results show no evidence of this occurring, instead all results show a loss of preload as cycles increase, therefore the tolerance interval results have been left out of this study.

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66 Chapter 6 Interpretation of Results Fastener life test data has been statisti cally quantified in this thesis. In this chapter, the underlying mechanisms of fasten er loosening for the tests are examined. Previous analysis and the preload versus cy cles data are used to identify the various mechanisms of loosening. The statistical treatments are examin ed to see if the characteristics inherent of the underlying mech anisms of loosening are preserved or lost. These characteristics include initial drop and changes in rate of preload loss. This work goes beyond one test sample w ith twelve test samples and predicts preload means versus cycles for the sample ’s population. This not only allows one to predict population preload means versus cycles but it also allows one to predict the characteristics of these curves. First, comparisons of the confidence interv al plots to those of actual sample runs will be made to determine if the confidence intervals accurately represent loosening for a particular locking level. In Figure 6.1, all of the “Standard Heli-Coil with Braycote” plots can be seen. In this figure it is apparent that the 95 percent confiden ce intervals, (c), and 99 percent confidence intervals, (d), are an excellent estimation for loosening. These confidence intervals display intervals that th e population’s mean lies within. On average (c) and (d) of Figure 6.1 shows the characteri stics inherent the underl ying mechanisms of loosening are preserved.

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67 Figure 6.1 Composite Plot (a), Descri ptive Statistics Plot (b), 95% Confidence Interval Plot (c), and 99% Confidence Interval Plot (d) for the “Standard Heli-Coil with Braycote” runs. Similarly to the “Standard Heli-Coil with Braycote” plots, the confidence intervals for “Locking Heli-Coil with Brayco te” preserve characteristics of loosening when comparing the individual sample runs from the composite plot, which can be observed in section (a) of Figure 6.2. Apparent in sections (a) and (b) in Figure 6.2, Run 20 lies separate from the rest of the other eleven test runs. The characteristics of loosening for Run 20 are different. With this understood, Run 20 was determined to be an outlier and was left out when cal culating the confidence intervals.

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68 Figure 6.2 Composite Plot (a), Descriptive Statistics Plot (b), 95% Confidence Interval Plot (c), and 99% Confidence Interval Plot (d) for the “Locking Heli-Coil with Braycote” runs. Figure 6.3 presents similar plots created for the “Standard Heli-Coil with Loctite” runs. When comparing the individual test runs to the confidence interval plots, it is apparent that the confidence intervals repr esent the loosening of individual samples. Three of the test runs from the “Standard Heli -Coil with Loctite” samples are seen to lose preload faster than the other nine test runs. The confiden ce intervals account for those different samples and suggest the population’s characteristic s are a combination of all samples.

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69 Figure 6.3 Composite Plot (a), Descriptive Statistics Plot (b), 95% Confidence Interval Plot (c), and 99% Confidence Interval Plot (d) for the “Standard Heli-Coil with Loctite” runs. Now that the confidence intervals have b een shown to capture the characteristics of the three different locking levels, estimati ons of what is occurring during the loosening process can be made. The Composite Plots and Descriptive Statistic Plots will be examined in regions to describe how looseni ng is occurring for each of the locking levels. This information will also provided insight on which secondary lo cking feature helps prevent the loss of preload the best. To help explain how the “Standard He li-Coil with Braycote” locking level loosens; the plots for the “Standard Heli-Co il with Braycote” will be divided into three regions. Looking at Region I of the “Standard Heli-Coil with Braycote” plots in Figure

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706.4 it can be seen that there is 5.1 to 8.8 percent drop of initial preload when the test begins. Figure 6.4 Loosening regions for “Sta ndard Heli-Coil with Braycote” plots. Pai and Hess provided a possible explanation to preload loss at the beginning of vibration tests. Pai and Hess [3] explained that when a tightening torque is applied to the screw head, a portion of that torque is retained by th e friction forces at the head and the threads of the screw. However this stored torque also provides a loosening moment at the head of the screw when loosening begins. Region I is at the beginning of th e vibration test and the seven percent preload loss occurs in the fi rst few cycles. This moment of loosening at the head created by the applied torque helps in the loss of friction at the head and threads of the bolt resulting in complete head slip wi th complete or localized thread slip of the bolt at Region I. For such large rates of prel oad loss to occur in such a short time, the threaded fastener would have to be completely turning. In Region II of Figure 6.4, a lower rate of preload loss occurs, compared to the loss of preload in Region I. Lower rates of pr eload loss would indicate that the friction at the head and threads interact more after the initial drop of preload. The preload loss in Region III is seen to be very significant and mo st of the preload is completely lost well

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71before 1000 cycles. Without a secondary lock ing feature the “Standard Heli-Coil with Braycote” relies on the friction at the head and threads of the screw. In Region II of Figure 6.4, the friction is overcome and slip occu rs at the head and the threads. As seen in Pai and Hess’ work [3] the rates of prel oad loss in Region II would be caused by localized slip at the head with complete or lo calized slip at the thre ads. With larger rates of preload loss in Region III the slip at the head and threads is more than Region II, and results in complete loss of preload. So with more slip, the localized slip at the head in Region II, changes to complete slip in Region III, while the threads experience complete or localize slip. The “Locking Heli-Coil with Braycote” plot s have been divided into four regions and are presented in Figure 6.5. When co mparing Figure 6.4 of the “Standard Heli-Coil with Braycote” plots with Figure 6.5 of th e “Locking Heli-Coil with Braycote” plots there is a noticeable similarity in the Regi ons I, II, and III. This would indicate that loosening for “Locking Heli-Coil with Brayco te” is similar to the loosening of the “Standard Heli-Coil with Braycote” for the first three regions of loosening. The difference between the two comes with the emergence of a transition region and Region IV in Figure 6.5. The “Locking Heli-Coil w ith Braycote” locking level has a transition region after Region III and before a new region, defined as Region IV, where preload loss is significantly changing.

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72 Figure 6.5 Loosening regions for “L ocking Heli-Coil with Braycote” plots. Unlike the “Standard Heli-Coil with Bray cote”, the “Locking Heli-Coil with Braycote” has a secondary locking feature that creates a prevailing torque. This prevailing torque is created from a series of straight segmen ts or “chords” in one of the insert coils [13]. When the bolt is inserted in to the “grip” coil, seen in Figure 6.6, the chord segments push outward on the bolt and creates pressure on the bolt. Thus the prevailing torque comes from this pressure exerted on the bolt threads. It has been documented by Finkelston [5] that prevailing to rque not only reduces the rate of preload loss but it can also stop the loosening proce ss completely when the prevailing torque counteracts with the loosening torque. This is what is occurring in the transition region and Region IV in Figure 6.5. The prevailing to rque counteracts the loosening torque and the loss of preload is reduced and stopped. In Figure 6.5 the transition region as well as Region IV shows when the prevailing torque counteracts with the loosening torque. In the transition region the preload loss is reduced significan tly which is the result of th e prevailing torque. To help explain this, Figure 6.7 presen ts a free body diagram shows the moments of a preloaded fastener that

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73 Figure 6.6 Grip Coil on “Locking He li-Coil with Braycote” provided by [13]. Figure 6.7 Free body diagram of fastener with prevailing torque. has a prevailing torque. It can be obser ved that the prevailing torque moment, pT acts in the same direction as the moment s created by friction at the head, hM and threads, tM pM is a moment acting agiansts the other mo ments and is created from the pitch. For this reason it is understood that the prevailing torque works against loosening. The prevailing torque help s reduce the loss of preloa d throughout the fastener vibrations life test, but seems to be more effective near th e end of the test. Reasons for this can be developed with a simplified ve rsion of the long-form torque equation. P n n t t P offT r r p F T ) cos 2 ( (6.1)

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74Here offT is the net torque tending to lessen the fastener,pF is the preload, p is the thread pitch, t is coefficient of friction in the thread, n is coefficient of friction in the head, tr is effective radii of the thread, nr is effective radii of the head, pT is the prevailing torque, and is the half-angle of thread tooth. Th is equation shows th e prevailing torque acts against the removal torque Pai and Hess [3] use this eq uation to show the condition for maintaining preload in the absence of external loads. p n n p t t p pT r F r F p F ) cos( 2 (6.2) By dividing equation (6.2) through with the pr eload it can be observed that the prevailing torque’s effectiveness is proportional to the preload. p p n n t tF T r r p ) cos( 2 (6.3) Equation (6.3) shows the condition when loosen ing occurs in the absence of external loads and how the prevailing torque is aff ected by the preload. Though this equation is only valid in absence of exte rnal loads, it demonstrates how preload can affect the performance of the prevailing torque. It can be seen in equation (6.3) that at high preloads, pF the prevailing torque, pT contribute less, and at lower preloads, the prevailing torque will cont ribute more and help slow or stop loosening. Looking at Figure 6.8, this is more appa rent. Figure 6.8 shows the difference in preload of the “Locking Heli-Coil with Br aycote” and the “Standard Heli-Coil with Braycote” mean and median versus the lo ss of preload for “Locking Heli-Coil with Braycote”. This figure shows that the prev ailing torque locking feature is only more effective, than no secondary locking feat ure, after significan t preload is lost.

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75 0 500 1000 1500 2000 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 P Locking minus P Std (lbs)Preload Locking (lbs) Mean Median 0 500 1000 1500 2000 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 P Locking minus P Std (lbs)Preload Locking (lbs) Mean Median Figure 6.8 Locking minimum preload minus Std minimum preload versus loss of Locking preload. In the transition region, the preload r eaches a low enough value for the prevailing torque to be more effective, and helps slow the loss of preload. As the loosening moment counteracts the prevailing torque, complete slip seen in Region III, becomes partial slip in the transition region. In Region IV the prel oad loss in some individual runs has stopped completely, but in others the preload is still lo st at very slow rates. When it is apparent that preload has reached a steady value, there is neither slip at the head or threads. For the test runs that still lose preload, there is localized slip at the head and threads of the fastener. When analyzing the “Standard Heli-Coil with Loctite” plots the loosening process was divided into three sections, which can be observed in Figure 6.9. Just like the other two locking levels, the “Standard Heli-Coil with Loctite” has an initia l loss in preload in

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76Region I. For the “Standard Heli-Coil with Braycote” plots and the “Locking Heli-Coil with Braycote” plots, Region I of Figure 6.9 showed a loss of initial preload to be 4.6 to 9.2 percent, but the loss of initial preload of the “Standard Heli-Coil with Loctite” plots is higher at 7.9 to 14.4 percent. This gross loss of preload in Region I for all three locking levels can be contributed to complete slip at the head and complete or localized slip at the threads. Figure 6.9 Loosening regions for “S tandard Heli-Coil with Loctite” plots. In Region II in Figure 6.9, it can be obs erved that rates of preload loss are slow, which would indicate that the complete slip in Region I has become localized slip. For most of the test runs in this sample, locali zed slip at the heads and threads occurs from Region I through Region II. An increase in preloa d loss rate occurs for three of the twelve tests as identified by Region III in Figure 6.9. For these three tests a transition to complete head and/or thread slip occurs. Some of the reasons that these three tests have different characteristics may be due to: curing issues of the adhesive, cleaning issues, and assembly issues. The secondary locking feature for the “S tandard Heli-Coil with Loctite” locking level is in the form of a thread locking adhesi ve. This adhesive “Locti te” is applied to the

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77threads during assembly of each sample of the “Standard Heli-Coil with Loctite” samples. Advertised by Henkel Corporation [18], Loctite threadlocker is an anaerobic liquid that cures to a hard thermoset plasti c that locks the thre ads together. In the “Standard Heli-Coil with Loctite” test runs all runs loosen to some degree, which suggest that this thr eadlocker does not comple tely lock the thread s together. When the threadlocker cures into a solid, the voids a nd gaps where air would usually be present without threadlocker, are now f illed with a solid. Just like tighter thread tolerances, the voids and gaps are filled at the threads, which improves the re sistance to vibrationinduced loosening [19]. This explains th e low rate of preload loss in Region II. Figure 6.9, similar to Figure 6.8, shows the difference in preload of the “Standard Heli-Coil with Loctite” and the “Standard He li-Coil with Braycote” mean and median versus the loss of preload for “Standard Heli-Coil with Loctite”. In this figure it appearant that the adhesive secondary locking feature help s prevent the loss of preload near intial preload when compaired to no locking feature or the prevailing torque secondary locking feature. Aft er the intial preload drop the adhesive maintains more preload which can be seen in Figure 6.9. As the difference in preload increase the loss of preload for “Standard Heli-Coil with Loctite” samples remains relatively slow.

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78 0 500 1000 1500 2000 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 P Loctite minus P Std (lbs)Preload Loctite (lbs) Mean Median Figure 6.10 Loctite minimum preload mi nus Std minimum preload versus loss of Loctite preload. The overall assessment of the three locki ng levels can be developed by looking at the Confidence Interval plot s. Figure 6.9 and Figure 6. 10 respectively show the 95 percent Confidence Intervals and the 99 percen t Confidence Intervals for all three locking levels. These figures indicate that the “Standard Heli-Coil with Loctite” locking level is the best when trying to retain initial preload. The next best locking level is the “Locking Heli-Coil with Braycote” followed by the “Standard Heli-Coil with Braycote” locking level. These figures also show how the progr ams created in this thesis can aid in the comparison of different threaded fasteners.

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79 Figure 6.11 95 percent Confidence In tervals for all three locking levels. Figure 6.12 99 percent Confidence In tervals for all three locking levels.

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80 Chapter 7 Conclusion The methods developed in this work provide quantified a nd statistical test results for tests of threaded fasteners subjected to vi bration life tests. The tools in this thesis allow one to quantify multiple test samples from vibration life tests. In previous work life tests are persented individually. In this work, fastener vibration life test data of threaded inserts with different secondary locking f eatures was obtained to help develop these methods of quantification. First, methods to summarize the test data of vibration life te st were developed. This was done by plotting the minimum preloads versus cycles for all runs in the same sample in one plot. The minimum preloads ar e usually of most interest in fastener vibration life test, which is the main r eason the minimum preload was extracted. In addition to sample composite plots, sample statistics include sample mean, sample median, sample quartiles, and sample extents were determined. Predictions for sample population were es timated. To do this the distribution of the sample data was assessed. Using normal plots and PPCC test it was found that the data is approximately normal. This allowed predictions of the sample population to be predicted. Using a double sided Student’s t-di stribution confidence in terval, intervals for which the population's mean lies have been cr eated for the minimum preload versus cycle data. It has been shown that these confiden ce intervals do preserve the characteristics

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81inherent of the underlying mechanisms of l oosening when comparing them to individual test runs. Finally in Chapter 6, the shapes of th e loosening curves are examined. It was found that the plots generated in this work do preserve the characteri stics inherent of the underlying mechanisms of loosening. These plot s were then used to aid in the evaluation of the loosening characteristics for the three locking levels. It was also found that having a secondary locking feature does help reta in preload loss. The prevailing torque secondary locking feature for the “Locking Heli-C oil with Braycote” test runs is found to be more effective as the prel oad is lost. The best secondary locking feature was found to be the Loctite Threadlocker.

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82 References 1. Hess, D. P., “Vibrationand ShockInduced Loosening,” Chapter 40 in Handbook of Bolts and Bolted Joints Marcel Dekker Inc., New York, pp. 757824, 1998a. 2. Junker, G. H., “New Criteria for Self-L oosening of Fasteners Under Vibration,” Society of Automotive Engineers Transactions Vol. 78, pp. 314-335, 1969. 3. Pai, N.P. and Hess, D.P., “Experiment al Study of Loosening of Threaded Fasteners Due to Dynamic Shear Load,” Journal of Sound and Vibration Vol. 253, pp. 585-692, 002. 4. Pai, N.P. and Hess, D.P., “Three-Dimensiona l Finite Element analysis of threaded fastener loosening due to dynamic shear load,” Engineering Failure Analysis Vol. 9, pp. 383-402, 2002. 5. Finkelston, R. F., “How Much Sh ake Can Bolted Joints Take,” Machine Design pp. 122–125, 1972. 6. Sanclemente, J.A. and Hess, D.P., “Parametric Study of Threaded Fastener Loosening Due to Cyclic Transverse Load,” Engineering Failure Analysis Vol. 14, pp. 239-249, 2007. 7. “Nord-Lock Bolt securing system”, http://www.nord-lock.com Accessed 2007. 8. Faroni, C. C., “Maintaining Tightness of Threaded Fasteners”, Amerace Corporation, ESNA Divi sion, Union, NJ, 1967. 9. “A Statistical Look at Bolt Preload, Friction and Relaxation”, SME Technical Paper, 1979. 10. Collins, J.A., Mechanical Design of Machine Elem ents and Machines: A Failure Prevention Perspective, New York: John Wiley & Sons, Inc., 2003. 11. NAS 1003 thru 1020, National Aerospace Standard, pp. 1-3, 1991. 12. NAS 1149, National Aerospace Standard, pp. 1-6, 1994. 13. Emhart Teknologies Heli-Coil Bulletin, http://www.emhart.com 2003

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8314. MIL-STD-1312-7A, 1984, Fasten er Test Methods Met hod 7: Vibration, U.S. Department of Defens e, Washington D.C. 15. Montgomery, D. C., Design and Analysis of Experiments 6th ed., New Jersey: John Wiley & Sons, Inc., 2005. 16. Looney S. W. and Gulledge Jr. T. R., “U se of the Correlation Coefficient with Normal Probability Plots”, The American Statistician Vol. 39, No. 1, pp. 75-79, 1985. 17. Bickford, J.H., An Introduction to the Design and Behavior of Bolted Joints 3rd ed., Marcel Dekker, 1995. 18. The Adhesive Sourcebook Vol. 7, Henkel Technologies-Loctite, 2007. 19. Hess, D. P., “Mechanism of Vibration-Induced Loosening,” Fastener Technology International Vol. XXII, 1999, pp. 86-87.

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84 Bibliography 1. Abramoxitz, M. and Stegun, I. A., National Bureau of Standards Applied Mathematics Series, 55: Handbook of Mathematical Functions, Washington, DC: US Government Printing Office, 1964. 2. Daily, J. W., Riley, W. F., and McConnell, K. G., Instrumentation for Engineering Measurements, 2nd ed. New Jersey: John Wiley & Sons, Inc, 1993. 3. Filliben, J. J., “The Probabiltiy Plot Co rrelation Coefficient Test for Normality,” Technometerics Vol. 17, No. 1, pp. 111-117, 1975. 4. Krishnamoorthy, K., Handbook of Statistical Dist ributions with Applications Florida: Chapman & Hall/CRC Press, 2006. 5. Patel, J. K. and Read, C. B., Handbook of the Normal Distribution 2nd ed. New York: Marcel Dekker, Inc, 1996.

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85 Appendices

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86Appendix A: M-Files for Composite Plots The M-files minstdlines2.m minlockinglines2.m and minloctitelines2.m create a composite plot of all runs for each of the lo cking levels. These composite plots show the minimum preload versus cycles for the runs. Th e raw data from the runs are recorded into files. These M-files load the raw data files a nd assigns time, preload, and cycles to proper calibrated values. With a sampling rate of 51.2 samples/second ther e are a total of 8192 data points for time, preload, and cycles. The minimum preloads for every 18.75 cycles are found using “for” loops. Every 64 data poi nts are grouped into a rrays 128 times for a total of 8192 data points for both preload and cy cles. The function “min” is used to sort the preload array from least to greatest. The first element in the array is then put into another array) ( j Ci. The cycle arrays do not need to be sorted and the median of the cycles array is assigned to the array) ( j Di. Then the preload arrays ) ( j Ciare plotted against the cycle’s arrays ) ( j Di. M-File minstdlines2.m begins here. % Minimum Preloads of Test data from Junker machine % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to beplotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2;

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87Appendix A: (Continued) % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test1.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test3.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ;

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88Appendix A: (Continued) load test4.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test7.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test8.vna -mat ;

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89Appendix A: (Continued) time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test11.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test15.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas;

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90Appendix A: (Continued) cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test21.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test25.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas;

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91Appendix A: (Continued) cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test26.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test28.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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92Appendix A: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test33.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; % Plots of all Runs are plotted together. Plots show the minimum % preload versus cycles. axes( 'FontSize' ,16); plot(D1,C1,D2,C2,D3,C3,D4,C4,D5,C5,D6,C6,D7,C7,D8,C8,D9,C9,D10,C1 0,D11,C11,D12,C12, 'LineWidth' ,2); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18);

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93Appendix A: (Continued) xlabel( 'Cycles' 'FontSize' ,18); legend( 'Test 1' 'Test 3' 'Test 4' 'Test 7' 'Test 8' 'Test 11' 'Test 15' 'Test 21' 'Test 25' 'Test 26' 'Test 28' 'Test 33' ); grid; M-File minstdlines2.m ends here. M-File minlockinglines2.m begins here. % Minimum Preloads of Test data from Junker machine % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" be change to be the end lowest cycles as well by making % it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test2.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1);

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94Appendix A: (Continued) d=cycles(a:b); D1(i)=d(dpcm); end ; load test9.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test10.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ;

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95Appendix A: (Continued) load test14.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test16.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ;

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96Appendix A: (Continued) load test18.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test24.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ;

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97Appendix A: (Continued) load test27.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test30.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test32.vna -mat ;

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98Appendix A: (Continued) time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test35.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test36.vna -mat ; time = SLm.tdxvec;

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99Appendix A: (Continued) preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; % Plots of all Runs are plotted together. Plots show the minimum % preload versus cycles. axes( 'FontSize' ,16); plot(D1,C1,D2,C2,D3,C3,D4,C4,D5,C5,D6,C6,D7,C7,D8,C8,D9,C9,D10,C1 0,D11,C11,D12,C12, 'LineWidth' ,2); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); legend( 'Test 2' 'Test 9' 'Test 10' 'Test 14' 'Test 16' 'Test 18' 'Test 24' 'Test 27' 'Test 30' 'Test 32' 'Test 35' 'Test 36' ); grid; M-File minlockinglines2.m ends here. M-File minloctitelines2.m begins here. % Minimum Preloads of Test data from Junker machine % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all

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100Appendix A: (Continued) mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test5.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test6.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc;

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101Appendix A: (Continued) end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test12.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test13.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p);

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102Appendix A: (Continued) C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test17.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test19.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ;

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103Appendix A: (Continued) load test20.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test22.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test23.vna -mat ;

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104Appendix A: (Continued) time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test29.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test31.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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105Appendix A: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test34.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; % Plots of all Runs are plotted together. Plots show the minimum % preload versus cycles. axes( 'FontSize' ,16); plot(D1,C1,D2,C2,D3,C3,D4,C4,D5,C5,D6,C6,D7,C7,D8,C8,D9,C9,D10,C1 0,D11,C11,D12,C12, 'LineWidth' ,2); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); legend( 'Test 5' 'Test 6' 'Test 12' 'Test 13' 'Test 17' 'Test 19' 'Test 20' 'Test 22' 'Test 23' 'Test 29' 'Test 31' 'Test 34' ); grid; M-File minloctitelines2.m ends here.

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106Appendix B: M-files for Sample Statistic Plots M-files quanstd2.m quanlocking2.m and quanloctite2.m plot sample means, medians, 25% quartiles, 75% quartiles, and ex tents of minimum prel oads versus cycles for all three locking levels. The data for each run is loaded, and minimum preloads for every 18.75 cycles are found using “for” l oops. Every 64 data points are grouped into arrays 128 times for a total of 8192 data point s for both preload and cycles. The function “min” is used to sort the preload array from least to greatest. The first element in the array is then put in to another array ) ( j CiThe cycle arrays do not need to be sorted and the median of the cycle’s a rray is assigned to the array D# ( i ). Then minimum preloads for all runs are put into matrix td in a manner in which each row is the minimum preload for all runs at a particular cycle. Then a nother for loop selects each row of the matrix td and extracts the extents, medians, 25% quart ile, and 75% quartiles from each row and puts them into arrays. The medians and qua rtiles are found using the MatLab function “quantile” and the extents are found using th e MatLab function “min” and “max”. Then the arrays are plotted against a cycle’s array D ( i ). M-file quanstd2.m begins here. % Mean, median, quartiles, and extents of minimum preloads. % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128;

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107Appendix B: (Continued) dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test1.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test3.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1);

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108Appendix B: (Continued) d=cycles(a:b); D2(i)=d(dpcm); end ; load test4.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test7.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm);

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109Appendix B: (Continued) end ; load test8.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test11.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ;

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110Appendix B: (Continued) load test15.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test21.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ;

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111Appendix B: (Continued) load test25.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test26.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test28.vna -mat ;

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112Appendix B: (Continued) time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test33.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12]; td=transpose(td); sum=C1+C2+C3+C4+C5+C6+C7+C8+C9+C10+C11+C12; mean1=sum/12;

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113Appendix B: (Continued) for i=1:mdp m=[td(i,1);td(i,2);td(i,3);td(i,4);td(i,5);td(i,6);td(i,7);td(i,8 );td(i,9);td(i,10);td(i,11);td(i,12)]; med = quantile(m, 0.5); upq = quantile(m, 0.75); loq = quantile(m, 0.25); mp=min(m); mp1=max(m); medd(i)=med; upqd(i)=upq; loqd(i)=loq; mind(i)=mp(1); maxd(i)=mp1(1); end ; medd = transpose(medd); upqd = transpose(upqd); lowq = transpose(loqd); mind=transpose(mind); maxd=transpose(maxd); axes( 'FontSize' ,16); plot(D1,mind, '--k' ,D1,upqd, ':c' ,D1,medd, '-r' ,D1,mean1, '-g' ,D1, loqd, ':c' ,D1,maxd, '--k' 'LineWidth' ,2.5); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); %legend('Extent Curves','Upper and Lower Quartiles','Median','Mean'); grid; M-file quanstd2.m ends here. M-file quanlocking2.m begins here. % Mean, median, quartiles, and extents of minimum preloads. % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all

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114Appendix B: (Continued) mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test2.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test9.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ;

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115Appendix B: (Continued) p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test10.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test14.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b);

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116Appendix B: (Continued) p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test16.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test18.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1);

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117Appendix B: (Continued) d=cycles(a:b); D6(i)=d(dpcm); end ; load test24.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test27.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm);

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118Appendix B: (Continued) end ; load test30.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test32.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ;

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119Appendix B: (Continued) load test35.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test36.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12]; td=transpose(td);

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120Appendix B: (Continued) sum=C1+C2+C3+C4+C5+C6+C7+C8+C9+C10+C11+C12; mean1=sum/12; for i=1:mdp m=[td(i,1);td(i,2);td(i,3);td(i,4);td(i,5);td(i,6);td(i,7);td(i,8 );td(i,9);td(i,10);td(i,11);td(i,12)]; med = quantile(m, 0.5); upq = quantile(m, 0.75); loq = quantile(m, 0.25); mp=min(m); mp1=max(m); medd(i)=med; upqd(i)=upq; loqd(i)=loq; mind(i)=mp(1); maxd(i)=mp1(1); end ; medd = transpose(medd); upqd = transpose(upqd); lowq = transpose(loqd); mind=transpose(mind); maxd=transpose(maxd); axes( 'FontSize' ,16); plot(D1,mind, '--k' ,D1,upqd, ':c' ,D1,medd, '-r' ,D1,mean1, '-g' ,D1, loqd, ':c' ,D1,maxd, '--k' 'LineWidth' ,2.5); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); %legend('Extent Curves','Upper and Lower Quartiles','Median','Mean'); grid; M-file quanlocking2.m ends here. M-file quanloctite2.m begins here. % Mean, median, quartiles, and extents of minimum preloads. % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles

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121Appendix B: (Continued) % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test5.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test6.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1;

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122Appendix B: (Continued) b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test12.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test13.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else

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123Appendix B: (Continued) a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test17.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test19.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ;

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124Appendix B: (Continued) p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test20.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test22.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p);

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125Appendix B: (Continued) C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test23.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test29.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b);

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126Appendix B: (Continued) D10(i)=d(dpcm); end ; load test31.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test34.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ;

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127Appendix B: (Continued) td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12]; td=transpose(td); sum=C1+C2+C3+C4+C5+C6+C7+C8+C9+C10+C11+C12; mean1=sum/12; for i=1:mdp m=[td(i,1);td(i,2);td(i,3);td(i,4);td(i,5);td(i,6);td(i,7);td(i,8 );td(i,9);td(i,10);td(i,11);td(i,12)]; med = quantile(m, 0.5); upq = quantile(m, 0.75); loq = quantile(m, 0.25); mp=min(m); mp1=max(m); medd(i)=med; upqd(i)=upq; loqd(i)=loq; mind(i)=mp(1); maxd(i)=mp1(1); end ; medd = transpose(medd); upqd = transpose(upqd); lowq = transpose(loqd); mind=transpose(mind); maxd=transpose(maxd); axes( 'FontSize' ,16); plot(D1,mind, '--k' ,D1,upqd, ':c' ,D1,medd, '-r' ,D1,mean1, '-g' ,D1, loqd, ':c' ,D1,maxd, '--k' 'LineWidth' ,2.5); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); %legend('Extent Curves','Upper and Lower Quartiles','Median','Mean'); grid; M-file quanloctite2.m ends here.

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128Appendix C: Normal Plots 2070 2080 2090 2100 2110 2120 2130 2140 2150 2160 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbab ility Figure C.1 Normal plot for “Standard He li-Coil with Braycote” preloads at 10 cycles. 1300 1400 1500 1600 1700 1800 1900 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbab ility Figure C.2 Normal plot for “Standard He li-Coil with Braycote” pr eloads at 250 cycles.

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129Appendix C: (Continued) -500 0 500 1000 1500 2000 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbab ility Figure C.3 Normal plot for “Standard He li-Coil with Braycote” pr eloads at 500 cycles. -500 0 500 1000 1500 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbab ility Figure C.4 Normal plot for “Standard He li-Coil with Braycote” pr eloads at 750 cycles.

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130Appendix C: (Continued) -200 0 200 400 600 800 1000 1200 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbab ility Figure C.5 Normal plot for “Standard Heli-Coil with Braycot e” preloads at 1000 cycles. -200 0 200 400 600 800 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbab ility Figure C.6 Normal plot for “Standard Heli-Coil with Braycot e” preloads at 1250 cycles.

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131Appendix C: (Continued) 2060 2080 2100 2120 2140 2160 2180 2200 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.7 Normal plot for “Locking HeliCoil with Braycote” pr eloads at 10 cycles. 1550 1600 1650 1700 1750 1800 1850 1900 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.8 Normal plot for “Locking HeliCoil with Braycote” pr eloads at 250 cycles.

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132Appendix C: (Continued) 600 800 1000 1200 1400 1600 1800 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.9 Normal plot for “Locking HeliCoil with Braycote” pr eloads at 500 cycles. 0 500 1000 1500 2000 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.10 Normal plot for “Locking Heli -Coil with Braycote” preloads at 750 cycles.

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133Appendix C: (Continued) 0 200 400 600 800 1000 1200 1400 1600 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.11 Normal plot for “Locking He li-Coil with Braycote” preloads at 1000 cycles. 0 500 1000 1500 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.12 Normal plot for “Locking He li-Coil with Braycote” preloads at 1250 cycles.

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134Appendix C: (Continued) 0 200 400 600 800 1000 1200 1400 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.13 Normal plot for “Locking He li-Coil with Braycote” preloads at 1500 cycles. 0 200 400 600 800 1000 1200 1400 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.14 Normal plot for “Locking He li-Coil with Braycote” preloads at 1750 cycles.

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135Appendix C: (Continued) 0 200 400 600 800 1000 1200 1400 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.15 Normal plot for “Locking He li-Coil with Braycote” preloads at 2000 cycles. 0 200 400 600 800 1000 1200 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.16 Normal plot for “Locking He li-Coil with Braycote” preloads at 2250 cycles.

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136Appendix C: (Continued) 1950 2000 2050 2100 2150 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.17 Normal plot for “Standard He li-Coil with Loctite” preloads at 10 cycles. 1750 1800 1850 1900 1950 2000 2050 2100 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.18 Normal plot for “Standard He li-Coil with Loctite” preloads at 250 cycles.

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137Appendix C: (Continued) 1500 1600 1700 1800 1900 2000 2100 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.19 Normal plot for “Standard He li-Coil with Loctite” preloads at 500 cycles. 1300 1400 1500 1600 1700 1800 1900 2000 2100 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.20 Normal plot for “Standard He li-Coil with Loctite” preloads at 750 cycles.

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138Appendix C: (Continued) 1000 1200 1400 1600 1800 2000 2200 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.21 Normal plot for “Standard He li-Coil with Loctite” pr eloads at 1000 cycles. 800 1000 1200 1400 1600 1800 2000 2200 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.22 Normal plot for “Standard He li-Coil with Loctite” pr eloads at 1250 cycles.

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139Appendix C: (Continued) 500 1000 1500 2000 2500 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.23 Normal plot for “Standard He li-Coil with Loctite” pr eloads at 1500 cycles. 0 500 1000 1500 2000 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.24 Normal plot for “Standard He li-Coil with Loctite” pr eloads at 1750 cycles.

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140Appendix C: (Continued) 0 500 1000 1500 2000 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.25 Normal plot for “Standard He li-Coil with Loctite” pr eloads at 2000 cycles. 0 500 1000 1500 2000 0.05 0.1 0.25 0.5 0.75 0.9 0.95 DataProbability Figure C.26 Normal plot for “Standard He li-Coil with Loctite” pr eloads at 2250 cycles.

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141Appendix D: M-files for PPCC Tests for Normality M-files normstd2.m normlocking2.m and normloctite2.m have been written to conduct probability plot correla tion coefficient tests for normality (PPCC) for minimum preloads at 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles. These M-files output correlation coefficients in an array for each of the three locking levels. The data for each run is loaded, and mini mum preloads for every 18.75 cycles are found using “for” loops. Every 64 data point s are grouped into a rrays 128 times for a total of 8192 data points for both preload and cy cles. The function “min” is used to sort the preload array from least to greatest. The first element in the array is then put into another array) ( j Ci. The cycle arrays do not need to be sorted and the median of the cycles array is assigned to the array ) ( j Di. Then minimum preloads for all runs are put into matrix td in a manner in which each row is th e minimum preload for all runs at a particular cycle. Rows 5, 18, 31, 44, 58, 71, 84, 98, 111, 124 of matrix td contain minimum preloads at 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles respectively. Using “for” loop these preloads are put into arrays, which are sorted by the function “sort” from least to greatest. The means of these arrays are calculated by the function “mean”. “For” loops also calcul ate the uniform order statistic median p ( i ), normal order statistic median b ( i ), and summations of the correlation coefficient. The correlation coefficient is then calculated and is put into an array. M-file normstd2.m begins here. % Probability Plot Correlation Coefficient Test for normality % (PPCC) for minimum preloads of test data from Junker machine % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points

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142Appendix D: (Continued) % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test1.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test3.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp

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143Appendix D: (Continued) if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test4.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test7.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1;

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144Appendix D: (Continued) b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test8.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test11.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else

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145Appendix D: (Continued) a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test15.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test21.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc;

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146Appendix D: (Continued) end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test25.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test26.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b);

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147Appendix D: (Continued) p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test28.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test33.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1);

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148Appendix D: (Continued) d=cycles(a:b); D12(i)=d(dpcm); end ; % Minimum preloads for all runs for this locking level in a % matrix. td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12]; td=transpose(td); % n is the number of test runs n=12; % The for loop selects minimum preloads at 10, 250, 500, % 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles. % The uniform order statistic median is calculated from the Blom % plotting position equation p(i). for i=1:n; m1(i)=td(5,i); m2(i)=td(18,i); m3(i)=td(31,i); m4(i)=td(44,i); m5(i)=td(58,i); m6(i)=td(71,i); m7(i)=td(84,i); m8(i)=td(98,i); m9(i)=td(111,i); m10(i)=td(124,i); p(i)=(i-(3/8))/(n+(1/4)); end ; % Here the minimum preloads are being sorted from least to % greatest using "sort". m1=sort(m1); m2=sort(m2); m3=sort(m3); m4=sort(m4); m5=sort(m5); m6=sort(m6); m7=sort(m7); m8=sort(m8); m9=sort(m9); m10=sort(m10);

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149Appendix D: (Continued) % The mean of the minimum preloads are found using "mean" mm1=mean(m1); mm2=mean(m2); mm3=mean(m3); mm4=mean(m4); mm5=mean(m5); mm6=mean(m6); mm7=mean(m7); mm8=mean(m8); mm9=mean(m9); mm10=mean(m10); % The normal order statistic median b(i) for i=1:n; b(i)=(2.515517+.802853*(log(1/p(i)^2))^.5+.010328*((log(1/p(i)^2) )^.5)^2)/(1+1.432788*(log(1/p(i)^2))^.5+0.189269*((log(1/p(i)^2)) ^.5)^2+.0013088*(log(1/p(i)^2)^.5)^3)(log(1/p(i)^2))^0.5+0.00045; end ; bm=mean(b); h1=0;h2=0;h3=0;h4=0;h5=0;h6=0;h7=0;h8=0;h9=0;h10=0; hb1=0;hb2=0;hb3=0;hb4=0;hb5=0;hb6=0;hb7=0;hb8=0;hb9=0;hb10=0; b2=0; %Correlation Coefficient sum's for i=1:n; h1=h1+(m1(i)-mm1)^2; h2=h2+(m2(i)-mm2)^2; h3=h3+(m3(i)-mm3)^2; h4=h4+(m4(i)-mm4)^2; h5=h5+(m5(i)-mm5)^2; h6=h6+(m6(i)-mm6)^2; h7=h7+(m7(i)-mm7)^2; h8=h8+(m8(i)-mm8)^2; h9=h9+(m9(i)-mm9)^2; h10=h10+(m10(i)-mm10)^2; hb1=hb1+(m1(i)-mm1)*(b(i)-bm); hb2=hb2+(m2(i)-mm2)*(b(i)-bm); hb3=hb3+(m3(i)-mm3)*(b(i)-bm);

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150Appendix D: (Continued) hb4=hb4+(m4(i)-mm4)*(b(i)-bm); hb5=hb5+(m5(i)-mm5)*(b(i)-bm); hb6=hb6+(m6(i)-mm6)*(b(i)-bm); hb7=hb7+(m7(i)-mm7)*(b(i)-bm); hb8=hb8+(m8(i)-mm8)*(b(i)-bm); hb9=hb9+(m9(i)-mm9)*(b(i)-bm); hb10=hb10+(m10(i)-mm10)*(b(i)-bm); b2=b2+(b(i)-bm)^2; end ; % Correlation Coefficient is calculated. Rp1=hb1/((h1*b2)^0.5); Rp2=hb2/((h2*b2)^0.5); Rp3=hb3/((h3*b2)^0.5); Rp4=hb4/((h4*b2)^0.5); Rp5=hb5/((h5*b2)^0.5); Rp6=hb6/((h6*b2)^0.5); Rp7=hb7/((h7*b2)^0.5); Rp8=hb8/((h8*b2)^0.5); Rp9=hb9/((h9*b2)^0.5); Rp10=hb10/((h10*b2)^0.5); %Correlation Coefficient the output as an array. Rp=[Rp1,Rp2,Rp3,Rp4,Rp5,Rp6,Rp7,Rp8,Rp9,Rp10] M-file normstd2.m ends here. M-file normlocking2.m begins here. % Probability Plot Correlation Coefficient Test for normality % (PPCC) for minimum preloads of test data from Junker machine % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2;

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151Appendix D: (Continued) .% C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test2.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test9.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ;

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152Appendix D: (Continued) load test10.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test14.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test16.vna -mat ; time = SLm.tdxvec;

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153Appendix D: (Continued) preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test18.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test24.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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154Appendix D: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test27.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test30.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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155Appendix D: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test32.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test35.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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156Appendix D: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test36.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; % Minimum preloads for all runs for this locking level in a % matrix. % C12 is test 36 or run 20 which is a larger outlier, here it is % taken out an n is changed to 11 td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11]; td=transpose(td);

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157Appendix D: (Continued) % n is the number of test run n=11; % The for loop selects minimum preloads at 10, 250, 500, % 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles. % The uniform order statistic median is calculated from the Blom % plotting position equation p(i). for i=1:n; m1(i)=td(5,i); m2(i)=td(18,i); m3(i)=td(31,i); m4(i)=td(44,i); m5(i)=td(58,i); m6(i)=td(71,i); m7(i)=td(84,i); m8(i)=td(98,i); m9(i)=td(111,i); m10(i)=td(124,i); p(i)=(i-(3/8))/(n+(1/4)); end ; % Here the minimum preloads are being sorted from least to % greatest using "sort". m1=sort(m1); m2=sort(m2); m3=sort(m3); m4=sort(m4); m5=sort(m5); m6=sort(m6); m7=sort(m7); m8=sort(m8); m9=sort(m9); m10=sort(m10); % The mean of the minimum preloads are found using "mean" mm1=mean(m1); mm2=mean(m2); mm3=mean(m3); mm4=mean(m4); mm5=mean(m5); mm6=mean(m6); mm7=mean(m7); mm8=mean(m8);

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158Appendix D: (Continued) mm9=mean(m9); mm10=mean(m10); % The normal order statistic median b(i) for i=1:n; b(i)=(2.515517+.802853*(log(1/p(i)^2))^.5+.010328*((log(1/p(i)^2) )^.5)^2)/(1+1.432788*(log(1/p(i)^2))^.5+0.189269*((log(1/p(i)^2)) ^.5)^2+.0013088*(log(1/p(i)^2)^.5)^3)(log(1/p(i)^2))^0.5+0.00045; end ; bm=mean(b); h1=0;h2=0;h3=0;h4=0;h5=0;h6=0;h7=0;h8=0;h9=0;h10=0; hb1=0;hb2=0;hb3=0;hb4=0;hb5=0;hb6=0;hb7=0;hb8=0;hb9=0;hb10=0; b2=0; %Correlation Coefficient sum's for i=1:n; h1=h1+(m1(i)-mm1)^2; h2=h2+(m2(i)-mm2)^2; h3=h3+(m3(i)-mm3)^2; h4=h4+(m4(i)-mm4)^2; h5=h5+(m5(i)-mm5)^2; h6=h6+(m6(i)-mm6)^2; h7=h7+(m7(i)-mm7)^2; h8=h8+(m8(i)-mm8)^2; h9=h9+(m9(i)-mm9)^2; h10=h10+(m10(i)-mm10)^2; hb1=hb1+(m1(i)-mm1)*(b(i)-bm); hb2=hb2+(m2(i)-mm2)*(b(i)-bm); hb3=hb3+(m3(i)-mm3)*(b(i)-bm); hb4=hb4+(m4(i)-mm4)*(b(i)-bm); hb5=hb5+(m5(i)-mm5)*(b(i)-bm); hb6=hb6+(m6(i)-mm6)*(b(i)-bm); hb7=hb7+(m7(i)-mm7)*(b(i)-bm); hb8=hb8+(m8(i)-mm8)*(b(i)-bm); hb9=hb9+(m9(i)-mm9)*(b(i)-bm); hb10=hb10+(m10(i)-mm10)*(b(i)-bm); b2=b2+(b(i)-bm)^2; end ;

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159Appendix D: (Continued) % Correlation Coefficient is calculated. Rp1=hb1/((h1*b2)^0.5); Rp2=hb2/((h2*b2)^0.5); Rp3=hb3/((h3*b2)^0.5); Rp4=hb4/((h4*b2)^0.5); Rp5=hb5/((h5*b2)^0.5); Rp6=hb6/((h6*b2)^0.5); Rp7=hb7/((h7*b2)^0.5); Rp8=hb8/((h8*b2)^0.5); Rp9=hb9/((h9*b2)^0.5); Rp10=hb10/((h10*b2)^0.5); %Correlation Coefficient the output as an array. Rp=[Rp1,Rp2,Rp3,Rp4,Rp5,Rp6,Rp7,Rp8,Rp9,Rp10] M-File normlocking2.m ends here. M-File normloctite2.m begins here. % Probability Plot Correlation Coefficient Test for normality % (PPCC) for minimum preloads of test data from Junker machine % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test5.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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160Appendix D: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test6.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test12.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp

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161Appendix D: (Continued) if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test13.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test17.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc;

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162Appendix D: (Continued) else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test19.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test20.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc;

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163Appendix D: (Continued) end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test22.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test23.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b);

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164Appendix D: (Continued) p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test29.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test31.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b);

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165Appendix D: (Continued) D11(i)=d(dpcm); end ; load test34.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; % Minimum preloads for all runs for this locking level in a % matrix. td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12]; td=transpose(td); % n is the number of test run n=12; % The for loop selects minimum preloads at 10, 250, 500, % 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles. % The uniform order statistic median is calculated from the Blom % plotting position equation p(i). for i=1:n; m1(i)=td(5,i); m2(i)=td(18,i); m3(i)=td(31,i); m4(i)=td(44,i); m5(i)=td(58,i); m6(i)=td(71,i);

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166Appendix D: (Continued) m7(i)=td(84,i); m8(i)=td(98,i); m9(i)=td(111,i); m10(i)=td(124,i); p(i)=(i-(3/8))/(n+(1/4)); end ; % Here the minimum preloads are being sorted from least to % leatest using "sort". m1=sort(m1); m2=sort(m2); m3=sort(m3); m4=sort(m4); m5=sort(m5); m6=sort(m6); m7=sort(m7); m8=sort(m8); m9=sort(m9); m10=sort(m10); % The mean of the minimum preloads are found using "mean" mm1=mean(m1); mm2=mean(m2); mm3=mean(m3); mm4=mean(m4); mm5=mean(m5); mm6=mean(m6); mm7=mean(m7); mm8=mean(m8); mm9=mean(m9); mm10=mean(m10); % The normal order statistic median b(i) for i=1:n; b(i)=(2.515517+.802853*(log(1/p(i)^2))^.5+.010328*((log(1/p(i)^2) )^.5)^2)/(1+1.432788*(log(1/p(i)^2))^.5+0.189269*((log(1/p(i)^2)) ^.5)^2+.0013088*(log(1/p(i)^2)^.5)^3)(log(1/p(i)^2))^0.5+0.00045; end ; bm=mean(b); h1=0;h2=0;h3=0;h4=0;h5=0;h6=0;h7=0;h8=0;h9=0;h10=0;

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167Appendix D: (Continued) hb1=0;hb2=0;hb3=0;hb4=0;hb5=0;hb6=0;hb7=0;hb8=0;hb9=0;hb10=0; b2=0; %Correlation Coefficient sum's for i=1:n; h1=h1+(m1(i)-mm1)^2; h2=h2+(m2(i)-mm2)^2; h3=h3+(m3(i)-mm3)^2; h4=h4+(m4(i)-mm4)^2; h5=h5+(m5(i)-mm5)^2; h6=h6+(m6(i)-mm6)^2; h7=h7+(m7(i)-mm7)^2; h8=h8+(m8(i)-mm8)^2; h9=h9+(m9(i)-mm9)^2; h10=h10+(m10(i)-mm10)^2; hb1=hb1+(m1(i)-mm1)*(b(i)-bm); hb2=hb2+(m2(i)-mm2)*(b(i)-bm); hb3=hb3+(m3(i)-mm3)*(b(i)-bm); hb4=hb4+(m4(i)-mm4)*(b(i)-bm); hb5=hb5+(m5(i)-mm5)*(b(i)-bm); hb6=hb6+(m6(i)-mm6)*(b(i)-bm); hb7=hb7+(m7(i)-mm7)*(b(i)-bm); hb8=hb8+(m8(i)-mm8)*(b(i)-bm); hb9=hb9+(m9(i)-mm9)*(b(i)-bm); hb10=hb10+(m10(i)-mm10)*(b(i)-bm); b2=b2+(b(i)-bm)^2; end ; % Correlation Coefficient is calculated. Rp1=hb1/((h1*b2)^0.5); Rp2=hb2/((h2*b2)^0.5); Rp3=hb3/((h3*b2)^0.5); Rp4=hb4/((h4*b2)^0.5); Rp5=hb5/((h5*b2)^0.5); Rp6=hb6/((h6*b2)^0.5); Rp7=hb7/((h7*b2)^0.5); Rp8=hb8/((h8*b2)^0.5); Rp9=hb9/((h9*b2)^0.5); Rp10=hb10/((h10*b2)^0.5); %Correlation Coefficient the output as an array. Rp=[Rp1,Rp2,Rp3,Rp4,Rp5,Rp6,Rp7,Rp8,Rp9,Rp10] M-file normloctite2.m ends here.

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168Appendix E: M-files for Confidence Intervals M-files CI95std2.m CI95locking2.m and CI95loctite2.m create plots that show 95% confidence intervals for the population me ans of minimum preloads at 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250 cycles. M-files CI99std2.m, CI99locking2.m, and CI99loctite2.m create plot s that show 99% confidence intervals for the population means for the same minimum preloads as the 95% confidence interval s. All M-files plot the confidence intervals with the sample mean by preload versus cycles. The data for each run is loaded, and mini mum preloads for every 18.75 cycles are found using for loops. Every 64 data points ar e grouped into arrays 128 times for a total of 8192 data points for both preload and cycles The function “min” is used to sort the preload array from least to grea test. The first element in the a rray is then put into another array ) ( j Ci. The cycle arrays do not need to be sort ed and the median of the cycle arrays are assigned to the array ) ( j Di. Then minimum preloads for all runs are put into matrix td in a manner in which each row is the minimum pr eload for all runs at a particular cycle. Rows 5, 18, 31, 44, 58, 71, 84, 98, 111, 124 of matrix td contain minimum preloads at 10, 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250 cy cles respectively. These preloads were put into arrays for each row selected The sample means and standard deviation were calculated using the MatLab functions “m ean” and “std”. Critical values from the Student’s t-Distribution are: 2.20 for a sample size of twelve and 95% confidence level, 2.23 for a sample size of eleven and 95% c onfidence level, 3.11 for sample size twelve and 99% confidence level, and 3.17 for sample sizes eleven and 99% confidence level. With the sample means, standard deviati ons, critical values, and sample sizes the confidence interval is calculated a nd plotted with the sample mean.

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169Appendix E: (Continued) M-file 95CIstd2.m begins here. % 95% Confidence Intervals for Minimum Preloads at cycles 10, % 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250. % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test1.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test3.vna -mat ;

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170Appendix E: (Continued) time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test4.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test7.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas;

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171Appendix E: (Continued) cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test8.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test11.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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172Appendix E: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test15.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test21.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1

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173Appendix E: (Continued) a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test25.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test26.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1;

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174Appendix E: (Continued) b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test28.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test33.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else

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175Appendix E: (Continued) a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12]; td=transpose(td); sum=C1+C2+C3+C4+C5+C6+C7+C8+C9+C10+C11+C12; mean1=sum/12; % n is the number of test run n=12; for i=1:n; m1(i)=td(5,i); m2(i)=td(18,i); m3(i)=td(31,i); m4(i)=td(44,i); m5(i)=td(58,i); m6(i)=td(71,i); m7(i)=td(84,i); m8(i)=td(98,i); m9(i)=td(111,i); m10(i)=td(124,i); end ; mm1=mean(m1); mm2=mean(m2); mm3=mean(m3); mm4=mean(m4); mm5=mean(m5); mm6=mean(m6); mm7=mean(m7); mm8=mean(m8); mm9=mean(m9); mm10=mean(m10); mm=[mm1,mm2,mm3,mm4,mm5,mm6,mm7,mm8,mm9,mm10]; mm=transpose(mm);

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176Appendix E: (Continued) sdm1=std(m1); sdm2=std(m2); sdm3=std(m3); sdm4=std(m4); sdm5=std(m5); sdm6=std(m6); sdm7=std(m7); sdm8=std(m8); sdm9=std(m9); sdm10=std(m10); sdm=[sdm1,sdm2,sdm3,sdm4,sdm5,sdm6,sdm7,sdm8,sdm9,sdm10]; sdm=transpose(sdm); % 2.20 is the upper critical value for the CI which comes from % Percentage Points of the t-Distribution table for % t(alpha/2,N-1). Thus we have t(0.025,11) for a CI of 95 percent % and 12 samples ta=2.20; for i=1:10; ciu(i)=mm(i)+ta*sdm(i)/(12)^0.5; cil(i)=mm(i)-ta*sdm(i)/(12)^0.5; end ; ci10=[ciu(1),cil(1)]; d10=[D1(5),D1(5)]; ci250=[ciu(2),cil(2)]; d250=[D1(18),D1(18)]; ci500=[ciu(3),cil(3)]; d500=[D1(31),D1(31)]; ci750=[ciu(4),cil(4)]; d750=[D1(44),D1(44)]; ci1000=[ciu(5),cil(5)]; d1000=[D1(58),D1(58)]; ci1250=[ciu(6),cil(6)]; d1250=[D1(71),D1(71)]; ci1500=[ciu(7),cil(7)]; d1500=[D1(84),D1(84)]; ci1750=[ciu(8),cil(8)]; d1750=[D1(98),D1(98)]; ci2000=[ciu(9),cil(9)]; d2000=[D1(111),D1(111)]; ci2250=[ciu(10),cil(10)]; d2250=[D1(124),D1(124)]; axes( 'FontSize' ,16);

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177Appendix E: (Continued) plot(d10,ci10, '-k' ,d250,ci250, '-k' ,d500,ci500, '-k' ,d750,ci750, 'k' ,D1,mean1, '--k' 'LineWidth' ,2); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); %legend(); grid; M-File CI95std2.m ends here. M-file CI95locking2.m begins here. % 95% Confidence Intervals for Minimum Preloads at cycles 10, % 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250. % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data points. % D arrays are the medians of cycles for each range of data points. load test2.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b);

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178Appendix E: (Continued) p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test9.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test10.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b);

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179Appendix E: (Continued) D3(i)=d(dpcm); end ; load test14.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test16.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ;

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180Appendix E: (Continued) load test18.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test24.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test27.vna -mat ;

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181Appendix E: (Continued) time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test30.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test32.vna -mat ; time = SLm.tdxvec;

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182Appendix E: (Continued) preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test35.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test36.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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183Appendix E: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; % C12 is test 36 or run 20 which is a larger outlier, here it is % taken out an n is changed to 11 td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11]; td=transpose(td); sum=C1+C2+C3+C4+C5+C6+C7+C8+C9+C10+C11; mean1=sum/11; % n is the number of test run n=11; for i=1:n; m1(i)=td(5,i); m2(i)=td(18,i); m3(i)=td(31,i); m4(i)=td(44,i); m5(i)=td(58,i); m6(i)=td(71,i); m7(i)=td(84,i); m8(i)=td(98,i); m9(i)=td(111,i); m10(i)=td(124,i); end ; mm1=mean(m1); mm2=mean(m2); mm3=mean(m3); mm4=mean(m4);

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184Appendix E: (Continued) mm5=mean(m5); mm6=mean(m6); mm7=mean(m7); mm8=mean(m8); mm9=mean(m9); mm10=mean(m10); mm=[mm1,mm2,mm3,mm4,mm5,mm6,mm7,mm8,mm9,mm10]; mm=transpose(mm); sdm1=std(m1); sdm2=std(m2); sdm3=std(m3); sdm4=std(m4); sdm5=std(m5); sdm6=std(m6); sdm7=std(m7); sdm8=std(m8); sdm9=std(m9); sdm10=std(m10); sdm=[sdm1,sdm2,sdm3,sdm4,sdm5,sdm6,sdm7,sdm8,sdm9,sdm10]; sdm=transpose(sdm); % 2.23 is the upper critical value for the CI which comes from % Percentage Points of the t Distribution table for % t(alpha/2,N-1). Thus we have t(0.025,10) for a CI of 95 percent % and 11 samples ta=2.23; for i=1:10; ciu(i)=mm(i)+ta*sdm(i)/(11)^0.5; cil(i)=mm(i)-ta*sdm(i)/(11)^0.5; end ; ci10=[ciu(1),cil(1)]; d10=[D1(5),D1(5)]; ci250=[ciu(2),cil(2)]; d250=[D1(18),D1(18)]; ci500=[ciu(3),cil(3)]; d500=[D1(31),D1(31)]; ci750=[ciu(4),cil(4)]; d750=[D1(44),D1(44)]; ci1000=[ciu(5),cil(5)]; d1000=[D1(58),D1(58)]; ci1250=[ciu(6),cil(6)]; d1250=[D1(71),D1(71)];

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185Appendix E: (Continued) ci1500=[ciu(7),cil(7)]; d1500=[D1(84),D1(84)]; ci1750=[ciu(8),cil(8)]; d1750=[D1(98),D1(98)]; ci2000=[ciu(9),cil(9)]; d2000=[D1(111),D1(111)]; ci2250=[ciu(10),cil(10)]; d2250=[D1(124),D1(124)]; axes( 'FontSize' ,16); plot(d10,ci10, '-k' ,d250,ci250, '-k' ,d500,ci500, '-k' ,d750,ci750, 'k' ,d1000,ci1000, '-k' ,d1250,ci1250, '-k' ,d1500,ci1500, 'k' ,d1750,ci1750, '-k' ,d2000,ci2000, '-k' ,d2250,ci2250, 'k' ,D1,mean1, '--k' 'LineWidth' ,2); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); %legend(); grid; M-file CI95locking2.m ends here. M-file CI95loctite2.m begins here. % 95% Confidence Intervals for Minimum Preloads at cycles 10, % 250, 500, 750, 1000, 1250, 1500, 1750, 2000, 2250. % "mdp" is the total number of data points to be plotted. "dpc" % is the data point count. It controls the range of data points % to be selected to choose a specific data point. "dpcm" is the % data point count median. It is use to select the median cycles % to be plotted against the lowest data point from each group. % "dpcm" could be change to be the end lowest cycles as well by % making it equal to "dpc". clear all mdp=128; dpc=8192/mdp; dpcm=dpc/2; % C arrays are the minimum preloads for each range of data % points. % D arrays are the medians of cycles for each range of data % points. load test5.vna -mat ;

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186Appendix E: (Continued) time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C1(i)=p(1); d=cycles(a:b); D1(i)=d(dpcm); end ; load test6.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C2(i)=p(1); d=cycles(a:b); D2(i)=d(dpcm); end ; load test12.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas;

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187Appendix E: (Continued) cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C3(i)=p(1); d=cycles(a:b); D3(i)=d(dpcm); end ; load test13.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C4(i)=p(1); d=cycles(a:b); D4(i)=d(dpcm); end ; load test17.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time;

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188Appendix E: (Continued) for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C5(i)=p(1); d=cycles(a:b); D5(i)=d(dpcm); end ; load test19.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C6(i)=p(1); d=cycles(a:b); D6(i)=d(dpcm); end ; load test20.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc;

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189Appendix E: (Continued) else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C7(i)=p(1); d=cycles(a:b); D7(i)=d(dpcm); end ; load test22.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C8(i)=p(1); d=cycles(a:b); D8(i)=d(dpcm); end ; load test23.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc;

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190Appendix E: (Continued) end ; p=preload(a:b); p=min(p); C9(i)=p(1); d=cycles(a:b); D9(i)=d(dpcm); end ; load test29.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C10(i)=p(1); d=cycles(a:b); D10(i)=d(dpcm); end ; load test31.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p);

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191Appendix E: (Continued) C11(i)=p(1); d=cycles(a:b); D11(i)=d(dpcm); end ; load test34.vna -mat ; time = SLm.tdxvec; preload = 6000*SLm.scmeas(2).tdmeas; cycles = 15*time; for i=1:mdp if i==1 a=1; b=dpc; else a=i*dpc-dpc; b=(i+1)*dpc-dpc; end ; p=preload(a:b); p=min(p); C12(i)=p(1); d=cycles(a:b); D12(i)=d(dpcm); end ; td=[C1;C2;C3;C4;C5;C6;C7;C8;C9;C10;C11;C12]; td=transpose(td); sum=C1+C2+C3+C4+C5+C6+C7+C8+C9+C10+C11+C12; mean1=sum/12; % n is the number of test run n=12; for i=1:n; m1(i)=td(5,i); m2(i)=td(18,i); m3(i)=td(31,i); m4(i)=td(44,i); m5(i)=td(58,i); m6(i)=td(71,i); m7(i)=td(84,i); m8(i)=td(98,i); m9(i)=td(111,i);

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192Appendix E: (Continued) m10(i)=td(124,i); end ; mm1=mean(m1); mm2=mean(m2); mm3=mean(m3); mm4=mean(m4); mm5=mean(m5); mm6=mean(m6); mm7=mean(m7); mm8=mean(m8); mm9=mean(m9); mm10=mean(m10); mm=[mm1,mm2,mm3,mm4,mm5,mm6,mm7,mm8,mm9,mm10]; mm=transpose(mm); sdm1=std(m1); sdm2=std(m2); sdm3=std(m3); sdm4=std(m4); sdm5=std(m5); sdm6=std(m6); sdm7=std(m7); sdm8=std(m8); sdm9=std(m9); sdm10=std(m10); sdm=[sdm1,sdm2,sdm3,sdm4,sdm5,sdm6,sdm7,sdm8,sdm9,sdm10]; sdm=transpose(sdm); % 2.20 is the upper critical value for the CI which comes from % Percentage Points of the t Distribution table for % t(alpha/2,N-1). Thus we have t(0.025,11) for a CI of 95 percent % and 12 samples ta=2.20; for i=1:10; ciu(i)=mm(i)+ta*sdm(i)/(12)^0.5; cil(i)=mm(i)-ta*sdm(i)/(12)^0.5; end ; ci10=[ciu(1),cil(1)]; d10=[D1(5),D1(5)]; ci250=[ciu(2),cil(2)];

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193Appendix E: (Continued) d250=[D1(18),D1(18)]; ci500=[ciu(3),cil(3)]; d500=[D1(31),D1(31)]; ci750=[ciu(4),cil(4)]; d750=[D1(44),D1(44)]; ci1000=[ciu(5),cil(5)]; d1000=[D1(58),D1(58)]; ci1250=[ciu(6),cil(6)]; d1250=[D1(71),D1(71)]; ci1500=[ciu(7),cil(7)]; d1500=[D1(84),D1(84)]; ci1750=[ciu(8),cil(8)]; d1750=[D1(98),D1(98)]; ci2000=[ciu(9),cil(9)]; d2000=[D1(111),D1(111)]; ci2250=[ciu(10),cil(10)]; d2250=[D1(124),D1(124)]; axes( 'FontSize' ,16); plot(d10,ci10, '-k' ,d250,ci250, '-k' ,d500,ci500, '-k' ,d750,ci750, 'k' ,d1000,ci1000, '-k' ,d1250,ci1250, '-k' ,d2250,ci2250, 'k' ,D1,mean1, '--k' 'LineWidth' ,2); axis([-50 2700 0 2500]); ylabel( 'Preload (lb)' 'FontSize' ,18); xlabel( 'Cycles' 'FontSize' ,18); %legend(); grid; M-file CI95loctite2.m ends here. M-file CI99std2.m is identical to CI95std2.m except ta =3.11. M-file CI99lcoking2.m is identical to CI95locking2.m except ta =3.17. M-file CI99loctite2.m is identical to CI95loctite2.m except ta =3.11.