defining a model for tool consumption rate on asphalt

Brigham Young University Brigham Young University

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Theses and Dissertations

2006-11-30

Defining a Model for Tool Consumption Rate on Asphalt Defining a Model for Tool Consumption Rate on Asphalt

Reclamation Machines Reclamation Machines

Matthew H. Taylor Brigham Young University - Provo

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BYU ScholarsArchive Citation BYU ScholarsArchive Citation Taylor, Matthew H., "Defining a Model for Tool Consumption Rate on Asphalt Reclamation Machines" (2006). Theses and Dissertations. 1293. https://scholarsarchive.byu.edu/etd/1293

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DEVELOPING A MODEL FOR TOOL CONSUMPTION RATE ON

ASPHALT RECLAMATION MACHINES

by

Matthew H. Taylor

A thesis submitted to the faculty of

Brigham Young University

in partial fulfillment of the requirements for the degree of

Master of Science

Department of Mechanical Engineering

Brigham Young University

December 2006

BRIGHAM YOUNG UNIVERSITY

GRADUATE COMMITTEE APPROVAL

of a thesis submitted by

Matthew H. Taylor

This thesis has been read by each member of the following graduate committee andby majority vote has been found to be satisfactory.

Date Kenneth W. Chase, Chair

Date Craig C. Smith

Date Carl D. Sorensen

BRIGHAM YOUNG UNIVERSITY

As chair of the candidate’s graduate committee, I have read the thesis of MatthewH. Taylor in its final form and have found that (1) its format, citations, and bibli-ographical style are consistent and acceptable and fulfill university and departmentstyle requirements; (2) its illustrative materials including figures, tables, and chartsare in place; and (3) the final manuscript is satisfactory to the graduate committeeand is ready for submission to the university library.

Date Kenneth W. ChaseChair, Graduate Committee

Accepted for the Department

Matthew R. JonesGraduate Coordinator

Accepted for the College

Alan R. ParkinsonDean, Ira A. Fulton College ofEngineering and Technology

ABSTRACT

DEVELOPING A MODEL FOR TOOL CONSUMPTION RATE ON

ASPHALT RECLAMATION MACHINES

Matthew H. Taylor

Department of Mechanical Engineering

Master of Science

Asphalt and concrete reclamation machines are used to cut roadways when

a repair is required. The performance of these machines can affect the quality of

road repairs, and cost/profitability for both contractors and governments. We believe

that several performance characteristics in reclamation machines are governed by

the placement and pattern of cutting picks on the cutter head. Previous studies,

focused on mining and excavation applications, have shown strong correlation between

placement and wear.

The following study employs a screening experiment (observational study)

to find significant contributors to tool wear, in applications of asphalt milling or

reclamation. We have found that picks fail by two primary modes: tip breakage,

and body abrasive wear. Results indicate that the circumferential spacing of a bit,

relative to neighboring bits, has the strongest effect on tip breakage. We have also

shown that bit skew angle has a large positive effect on body abrasive wear.

ACKNOWLEDGMENTS

This research was mainly supported by Asphalt Zipper, Inc. Asphalt Zipper

has made significant contributions of both equipment and labor to make the study

possible, and they are gratefully acknowledged for their help.

I would like to acknowledge the long patience of Dr. Ken Chase. He has given

many hours of help, and been a great source of advice and ideas. His impact on me

extends beyond the engineering classroom.

I would also like to acknowledge the help of the Mechanical Engineering Faculty

and staff. They have been an immense help to me and have always been there.

Special thanks to Dr. Carl Sorensen for help and support on many difficult problems

encountered in this work. And, to Dr. Craig Smith for his thoughtful input.

Most of all, I would like to thank my wonderful wife. She is truly a pillar

of strength in my life. I am thankful for her patience with long, odd hours, for her

genuine interest in my research, her playful teasing, and her efforts to motivate me

through to the end.

Contents

Abstract v

Acknowledgments vi

List of Tables xii

List of Figures xv

List of Listings xvii

1 Introduction 1

1.1 Problem Statement and Motivation . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Features of Small Reclamation Machines . . . . . . . . . . . 4

1.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Volume Models . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Pick Position and Orientation . . . . . . . . . . . . . . . . . 10

1.3.3 Design Optimization . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Document Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Pick Position and Performance 15

2.1 Chevron vs. Scattered Patterns . . . . . . . . . . . . . . . . . . . . 15

2.2 Observations on Vibration . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Observations on Wear . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Pick Failure Mechanisms . . . . . . . . . . . . . . . . . . . . 18

vii

2.3.2 Pick Orientation Effects . . . . . . . . . . . . . . . . . . . . 20

2.3.3 Pick Position Effects . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Effects of Assembly Tolerance . . . . . . . . . . . . . . . . . . . . . 26

3 Modeling Per-Pick Volume Removal 27

3.1 Pick Groove Shape and Material Removal Zone . . . . . . . . . . . 28

3.2 Relative Depth and Cut Cross-Section . . . . . . . . . . . . . . . . 29

3.3 Cross-Sectional Area and Volume . . . . . . . . . . . . . . . . . . . 35

3.4 Verification of Model Results . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Summary of Chapter Variables . . . . . . . . . . . . . . . . . . . . 38

4 Developing an Experiment 39

4.1 Designed Experiment vs. Observational Study . . . . . . . . . . . . 39

4.1.1 Independent Variables and Manufacturing Variation . . . . . 40

4.2 Characterization of Experiment Variables . . . . . . . . . . . . . . . 42

4.2.1 Expanded and Condensed Factor Models . . . . . . . . . . . 45

4.3 Overall Average Pick Consumption Rates . . . . . . . . . . . . . . . 48

4.4 Independent Sampling and Bias . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Cut Overlap Bias . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Conducting the Experiment 51

5.1 Measuring Actual Pick Position and Orientation . . . . . . . . . . . 51

5.2 Data Collection Methods . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Controlling Machine Operating Parameters . . . . . . . . . . . . . . 54

6 Analysis of Results 57

6.1 Note on Regression Model Closeness . . . . . . . . . . . . . . . . . 57

6.2 Pick Tip Failure Model . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2.1 Condensed Factors . . . . . . . . . . . . . . . . . . . . . . . 58

6.2.2 Expanded Factors . . . . . . . . . . . . . . . . . . . . . . . . 59

6.3 Pick Body Failure Model . . . . . . . . . . . . . . . . . . . . . . . . 63

viii

6.3.1 Condensed Factors . . . . . . . . . . . . . . . . . . . . . . . 63

6.3.2 Expanded Factors . . . . . . . . . . . . . . . . . . . . . . . . 66

6.4 Contributors to Tip Radius Variation . . . . . . . . . . . . . . . . . 67

7 Conclusions and Recommendations 71

7.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.2.1 Multiple Failure Modes . . . . . . . . . . . . . . . . . . . . . 72

7.2.2 Skew Angle Effect . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2.3 Pick Failures, a Poisson Process . . . . . . . . . . . . . . . . 73

7.2.4 Manufacturing Variation . . . . . . . . . . . . . . . . . . . . 73

7.2.5 Observational Studies . . . . . . . . . . . . . . . . . . . . . . 74

7.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8 Future Work 77

8.1 Dataset Size and Randomization . . . . . . . . . . . . . . . . . . . 77

8.2 Material Flow Between Picks . . . . . . . . . . . . . . . . . . . . . 77

8.3 Predictive Models for Pick Consumption Rate . . . . . . . . . . . . 79

8.4 Design Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.5 Variations on Volume Calculation . . . . . . . . . . . . . . . . . . . 81

A Cutter Head Pattern Definitions 83

A.1 Specifications for 30 inch Cutter Head . . . . . . . . . . . . . . . . 83

A.2 Specifications for 48 inch Cutter Head . . . . . . . . . . . . . . . . 85

B Analysis Code Listings 93

B.1 MATLAB/Octave Per-Pick Volume Calculation . . . . . . . . . . . 93

B.2 SolidWorks/VBA Per-Pick Volume Calculation . . . . . . . . . . . . 105

B.3 Poisson Regression and Plotting Code . . . . . . . . . . . . . . . . . 110

B.4 Linear Regression and Plotting Code . . . . . . . . . . . . . . . . . 115

C Regression Procedure 137

ix

C.1 Regression Model Trimming . . . . . . . . . . . . . . . . . . . . . . 138

C.2 Factor Scaling and Similarity . . . . . . . . . . . . . . . . . . . . . 138

C.3 Regression Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C.3.1 Pick Tip Failures (Condensed Factors) . . . . . . . . . . . . 140

C.3.2 Pick Tip Failures (Expanded Factors) . . . . . . . . . . . . . 140

C.3.3 Pick Body Failures (Condensed Factors) . . . . . . . . . . . 142

C.3.4 Pick Body Failures (Expanded Factors) . . . . . . . . . . . . 144

C.4 Model Assumptions, Closeness, and Goodness of Fit . . . . . . . . . 145

C.4.1 Tip Failures, Expanded Factors . . . . . . . . . . . . . . . . 147

C.4.2 Body Failures, Expanded Factors . . . . . . . . . . . . . . . 149

C.4.3 Body Failures, Condensed Factors . . . . . . . . . . . . . . . 151

C.5 Example Backward-Stepwise Regression . . . . . . . . . . . . . . . 153

C.6 Regression Model Variable Values . . . . . . . . . . . . . . . . . . . 155

D Cutter Head Measurement 159

D.1 Variables Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

D.2 Sample Part-Program Report . . . . . . . . . . . . . . . . . . . . . 168

Bibliography 172

x

List of Tables

3.1 Description of labels for cross-section boundary points from Figure 3.4 31

3.2 Coordinates for pick tip locations shown in Figure 3.4 . . . . . . . . 32

3.3 Coordinates for intersection points shown in Figures 3.4 and 3.5 . . 32

4.1 Dependent response – directly observed output variables . . . . . . 42

4.2 Primary independent variables . . . . . . . . . . . . . . . . . . . . . 43

4.3 Global variables, applying to all picks collectively . . . . . . . . . . 43

4.4 Randomly varying experiment variables, adding noise to the results 44

4.5 Key definitions relating to proximate distance . . . . . . . . . . . . 45

4.6 Additional experiment variables potentially correlated to bit failures 47

5.1 Descriptive statistics for bit placement measurements (no edge bits) 51

5.2 Descriptions of Testing Applications . . . . . . . . . . . . . . . . . . 52

5.3 Number of failures by failure mode and application . . . . . . . . . 53

5.4 Descriptive statistics for bit failures on bits considered in the analysis 54

6.1 Tip failure regression results, expanded factors and first order interac-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Body failure regression results for condensed factors . . . . . . . . . 64

6.3 Body failure regression results for expanded factors . . . . . . . . . 66

A.1 Cutter head lacing pattern specification, 30 inch cutter head . . . . 84

A.2 Cutter head lacing pattern specification, 48 inch . . . . . . . . . . . 85

A.3 Actual lacing pattern measurements, 48 inch . . . . . . . . . . . . . 89

B.1 Function call structure for volume calculation . . . . . . . . . . . . 93

B.2 Function call structure for backward stepwise regression . . . . . . . 115

C.1 Descriptive statistics for un-scaled regression variables . . . . . . . . 139

C.2 Descriptive statistics for scaled regression variables . . . . . . . . . 140

C.3 Regression results for tip failures for expanded factor model . . . . 141

xi

C.4 Regression results for body failures for expanded factor model . . . 142

C.5 Regression results for body failures for expanded factor model . . . 144

C.6 Comparison of coefficients for pick body outlier regression model . . 151

C.7 Unscaled regression model variable values . . . . . . . . . . . . . . . 155

xii

List of Figures

1.1 Asphalt Zipper model AZ-480S mounted on a wheeled loader . . . . 5

1.2 Cutter head used on asphalt reclamation machines . . . . . . . . . . 6

1.3 Flattened pick lacing pattern from the cutter head of an asphalt recla-mation machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Diagram of asphalt reclamation process . . . . . . . . . . . . . . . . 7

1.5 Cutter head used on a continuous mining machine . . . . . . . . . . 9

1.6 The effect of pick spacing on specific energy . . . . . . . . . . . . . 11

1.7 Theory of rock tool interaction effects . . . . . . . . . . . . . . . . . 11

1.8 Conical bit, viewed from cutting direction, showing negative and pos-itive skew angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Chevron cutter head pick pattern . . . . . . . . . . . . . . . . . . . 16

2.2 The profile of processed material for an inverted chevron pattern (A)and a scattered chevron pattern (B) . . . . . . . . . . . . . . . . . . 17

2.3 Components of a carbide-tipped attack pick . . . . . . . . . . . . . 18

2.4 Failure paths associated with rock excavation using attack picks[1] . 20

2.5 Definitions of pick orientation angles . . . . . . . . . . . . . . . . . 21

2.6 Flattened pick lacing pattern in a 30 inch wide scattered chevron pat-tern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Map of impact locations for each attack pick in a 30 inch wide scatteredchevron pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Illustration of pattern lacing effect on cutting cross-section for picks 23and 50 of Figure 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Pick body has been found to leave a smooth track along the inside faceof the pick groove . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 A simplified pick cutting profile angle of 75 degrees was used to calcu-late per-pick material volume . . . . . . . . . . . . . . . . . . . . . 29

3.3 Section view of pick cut paths in an asphalt slab . . . . . . . . . . . 30

xiii

3.4 A representation of material cross-section for a single pick’s cuttingpath, from Detail B of Figure 3.3 . . . . . . . . . . . . . . . . . . . 31

3.5 Dimensioned sketch (in inches) of a pick’s material cross-section . . 32

3.6 Geometry for effective pick radius Re within a sectioning plane at angleθ and advance distance A . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Comparison between CAD and MATLAB models of per-pick volumeremoval rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.8 Complete CAD model describing cut simulation used in model valida-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Expected variation in per-pick volume removal, based on cutter headassembly tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Diagram showing definitions and sample values of tip and body sideproximate distance for a selected pick and its axially adjacent neighbors 46

5.1 Image of asphalt in ‘alligatored’ condition . . . . . . . . . . . . . . 53

6.1 Plot of main effects in pick tip failure regression model . . . . . . . 60

6.2 Predicted pick tip failures for extreme combinations of the rT ipPD×aBodyPD interaction effect . . . . . . . . . . . . . . . . . . . . . . 62

6.3 Predicted pick tip failures for extreme combinations of the rT ipPD×aT ipPD interaction effect . . . . . . . . . . . . . . . . . . . . . . . 63

6.4 Plot of main effects on pick body failure (condensed-factor model) . 65

6.5 Predicted pick body failures for extreme combinations of the attack-angle × per-pick-volume interaction effect . . . . . . . . . . . . . . 66

6.6 Plot of main effects on pick body failure (expanded-factor model) . 67

6.7 Plot of bit tip height and bit axial location, versus angular position 68

6.8 Drawing illustrating the definition of the row slope variable . . . . . 69

A.1 Flattened plot of designed bit positions, 30 inch cutter head . . . . 83

A.2 Flattened plot of designed bit positions, 48 inch cutter head . . . . 85

C.1 Plot of main effects on pick tip failure for expanded factor model . . 141

C.2 Plot of interaction effects on pick tip failure for expanded-factor model 142

C.3 Plot of main effects on pick body failure for condensed factor model 143

C.4 Plot of interaction effects on pick body failure, condensed factor model 143

C.5 Plot of main effects on pick body failure for expanded factor model 144

xiv

C.6 Normal probability of residuals for expanded-factor regression on picktip failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

C.7 Influential observations among pick tip failures . . . . . . . . . . . . 148

C.8 Normal probability of residuals for expanded-factor regression on pickbody failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

C.9 Influential observations among pick body failures (expanded factors,including outlier) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C.10 Influential observations among pick body failures (expanded factors) 150

C.11 Normal probability of residuals for condensed-factor regression on pickbody failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

C.12 Influential observations among pick body failures (condensed factors) 152

D.1 CMM measurement apparatus and setup . . . . . . . . . . . . . . . 160

D.2 Coordinate system created in the CMM part program . . . . . . . . 161

D.3 Angular construction used to find individual pick tip radius R . . . 163

D.4 Angular construction used to find each pick’s attack angle φa, and clockangle adjustment caa . . . . . . . . . . . . . . . . . . . . . . . . . . 164

D.5 Illustration of variables for transforming measured side angle (sa) intoskew angle φs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xv

Listings

B.1 CutterApp.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B.2 MainVolume.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.3 BitVolume.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B.4 BitArea.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.5 Dominance.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.6 SectPt.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.7 sw automation.bas . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.8 poisson regress body short sample.R . . . . . . . . . . . . . . . . . 110

B.9 WearAnalysis.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B.10 FactGen.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

B.11 EdgeTrim.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B.12 BackStepRegres.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B.13 LatexTabMed.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

B.14 PlotData.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

B.15 PlotInteract.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

xvii

Chapter 1

Introduction

1.1 Problem Statement and Motivation

Asphalt and concrete reclamation machines are used to cut roadways when

a repair is required. The performance of these machines can affect the quality of

road repairs, and cost/profitability for both contractors and governments. We believe

that several performance characteristics in reclamation machines are governed by the

placement and pattern of cutting picks on the cutter head. Following is a list of

important performance measures for reclamation machines.

• Amplitude and frequency of vibration

• Magnitude of dynamic loading on pick, drivetrain, and frame components

• Pick consumption rate

• Torque loading on drivetrain components

• Production rate (power consumption)

• Aggregate size of processed material

• Surface roughness of asphalt after milling

• Profile of reclaimed material after trenching or milling (where material is left in

the trench)

Our specific interest for this study is the pick consumption rate. The cost of

replacement picks, relative to other operating costs, can be quite high. We are aware

1

of a case in which a contractor’s cost for replacement picks turned out higher than his

bid price on the entire road repair project. The rate of pick consumption is difficult

to predict in practice. In normal use, the operator of a small asphalt reclamation

machine is required to stop frequently and inspect each pick.

Studies attempting to define comprehensive models for pick consumption rate

have been conducted, for mining and trenching applications, with limited success.

Other studies in these areas, focusing on the relative pick consumption rate among

picks in a given machine, have been more successful. Although there has been signif-

icant related work in the areas of mining, drilling, and trenching, there is still a lack

of knowledge specific to applications in asphalt.

Based on field observations, we have found that particular pick locations have

higher pick consumption rates than other locations on a given cutter head. This

research focuses on defining a model that will allow us to predict relative pick con-

sumption rate. The objective is to identify and characterize design and operational

parameters of asphalt reclamation machines that can be modified to make the pick

consumption rate more uniform between locations on a cutter head. Any changes

made to machine parameters, in an effort to improve pick consumption rate, are con-

strained by the other performance characteristics listed above. Following is a complete

list of factors believed to affect pick wear.

• Operational parameters

– Cutter head rotational speed

– Machine advance rate (or feed rate)

– Machine cutting depth

• Material flow

– Volume of reworked material

– Clearance for material flow between picks

– Pick tip height above cutter head skin (or cutter drum)

2

• Asphalt properties

– Aggregate size

– Matrix adhesive strength

– Asphalt temperature

– Asphalt and base material moisture content

• Pick position and orientation

– Skew angle

– Attack angle

– Pick tip radius from cutter axis of rotation

– Pattern of placement, relative to adjacent picks (or lacing pattern)

• Pick manufacturing characteristics

– Material strength

– Pick component bond strength

– Dimensional variation

• Pick design characteristics

– Tip diameter

– Cone angle

– Carbide profile

Pick positioning, or lacing pattern design, has been a strong area of focus for

improving cutter head performance. However, previous efforts have taken a casual

approach to controlling uniformity of wear. Ultimately, we would like to develop a set

of design tools capable of optimizing cutter head performance along the performance

measures listed above. The present study aims to lay the ground work for methods

of explicitly controlling pick wear. Particularly, we have developed a method for

3

measuring the volume removed by each pick, and have attempted to relate pick volume

to wear performance.

Dimensional tolerances in manufacturing are another obvious source of non-

uniform wear. Without the benefit of a model relating tolerances to pick wear, ap-

propriate tolerances are difficult to obtain. Since manufacturing methods and costs

are closely tied to manufacturing tolerances, identifying appropriate tolerances can

have a large effect on profitability.

The specific purpose of this study was to identify the primary contributing

factors to uneven bit wear, and rank them by size of effect. This information will

allow manufactures to make a focused improvement effort on production processes

and design of asphalt reclamation cutter heads.

By making the pick consumption rate more uniform, the overall average pick

consumption rate will decrease moderately. Additionally, with a uniform consumption

rate, the machine operator will be required to make fewer inspections, which will

significantly improve productivity.

1.2 Background

1.2.1 Features of Small Reclamation Machines

The small reclamation machines, on which this study will focus, have several

noteworthy features. The following list describes some of these distinguishing features.

• Cutter head powered by a dedicated diesel engine

• Cutter head belt driven, with a gear reduction

• Designed to work as an attachment on the bucket of a loader

• Cutter head rotates in a direction such that asphalt is cut in an upward motion

• Forward motion provided by the host vehicle

• Trench dimensions up to 48 inches wide and 12 inches deep

4

Figure 1.1: Asphalt Zipper model AZ-480S mounted on a wheeled loader

The major components of a small reclamation machine include the cutter head,

engine, frame, and drive system. Figure 1.1 shows an Asphalt Zipper model AZ-480S

mounted on a wheeled loader, with the cutter head displayed prominently. The engine

can be seen on top of the frame. A simplified view of the cutter head used on the

AZ-480S is illustrated in Figure 1.2. Figure 1.3 shows a flattened pattern of pick

locations on this same cutter head.

5

Figure 1.2: Cutter head used on asphalt reclamation machines

Figure 1.3: Flattened pick lacing pattern from the cutter head of an asphalt recla-mation machine

6

1.2.2 Applications

There are several applications in which small asphalt reclamation machines

are appropriate. However, the most common uses are in trenching, patching, and full

depth reclamation. Small asphalt reclamation machines are suited to most small to

medium sized road repair jobs. The class of machines that are the focus of this study

are capable of making cuts up to 48 inches wide. These narrow cuts work well for

most utility trenches and patches. Small reclamation machines weigh between 4,000

and 10,000 pounds, and can be towed by a full sized pickup truck. Transportation

of a large milling machine requires special equipment and permits, and can be very

costly.

Figure 1.4: Diagram of asphalt reclamation process

The rate of deterioration of an asphalt roadway depends, in part, on the

strength of the base that supports the road. Full depth reclamation is a way of

stabilizing the base of a roadway. A reclamation machine grinds the existing asphalt

into fine tailings (1 inch minus), and mixes it into the road base below. The com-

paction and adhesion properties of ground asphalt are significantly better than the

materials typically used for road base. After grading and compacting the new base,

fresh asphalt is laid on the surface.

7

For roadways that have a sufficiently strong base, repairs of surface deteriora-

tion are performed by milling only a thin layer from the top of the roadway surface.

Small reclamation machines are not well suited to this type of repair work. Rather,

large milling machines are used because of their stability, and ability to make fine

adjustments in depth and angle of cut.

The class of small reclamation machines in this study do not have material

load-out capability, meaning that the material is left in the trench (see Figure 1.4).

Although this can be a problem in a few situations, it is beneficial in most. By leaving

the material in the trench, the cost of adding road base to a road is eliminated. An-

other benefit is the ability of passenger vehicles to drive over the trench immediately

after being cut. This allows a road crew to work on a road, while not disrupting local

traffic.

1.3 Literature Review

Over the last 20 years there has been an increasing amount of effort devoted

to finding optimal parameters for mining equipment. Oil and gas drilling tools have

also received intense study over this period. The published literature includes research

from business, government, and academic sources. Much of this research is focused on

working in hard rock. Published research in the area of asphalt reclamation has been

sparse. However, the equipment and analysis methods used in mining applications

have significant similarities to working in asphalt.

Some of the similarities between mining equipment and asphalt reclamation

equipment are illustrated in Figures 1.2 and 1.5. Figure 1.2 is a drawing of the cutter

head from the reclamation machine shown in Figure 1.1. Figure 1.5 is an image of the

cutter head from a coal shearer. Both pieces of equipment use conical attack picks of

similar size. Also note that the picks are arranged and oriented in a similar pattern.

8

Figure 1.5: Cutter head used on a continuous mining machine

1.3.1 Volume Models

A study of the geometry of oil well drilling heads using diamond-faced cutters

was conducted by Ken Chase [2] in 1978. In conjunction with this study, a computer

program (named Stratapax) was developed, that attempts to optimize dynamic loads

and tool wear characteristics by altering the radial and angular spacing of cutters.

The objective of the optimization was to equalize wear for all of the cutters. Stratapax

calculates for each cutter: the volume of material removed, the cross-sectional area

and radius of gyration of the cut, the torque load, the contact arc, and the wear

surface. The optimal solution criteria of the Stratapax program were chosen based

on expert knowledge rather than analysis of physical data. Dr. Chase believes that

an effort to conduct in situ verification of the results was begun in 1978, but we have

been unable to find any documentation of this effort.

There are noteworthy differences between Chase’s study of drilling heads, and

asphalt reclamation machines. The characteristics of the material being cut are some-

what different. Drilling heads are typically designed to work in hard rock which may

have some similarities to concrete, but which we would expect to behave quite differ-

ently from asphalt. Also, the drilling head studied by Chase makes continuous cuts,

while the asphalt reclamation machine makes interrupted cuts. The effect of this

9

difference is that picks in the asphalt machine experience widely interrupted impact

loads.

An article published in the International Journal of Approximate Reasoning[3]

describes a practical approach for predicting performance of trenching machines.

There appear to be some similarities between trenching in soil and trenching in as-

phalt. However, the failure mechanisms for picks applied in a soil-rock mixture are

much more complex than those of trenching in a homogeneous material, like asphalt.

Parameters of the study included machine advance rate, excavation rate, wear of the

picks, and breakage of the picks, while trench dimensions and total excavated material

volume were fixed. The authors of this study chose to use an approximate reason-

ing method because “Not enough data were available to perform reliable statistical

correlation studies. Therefore, use was made of a Fuzzy Expert System to construct

models that predict production and tool consumption.” The authors report that the

results of their effort were encouraging, but fell short of the stated objective.

1.3.2 Pick Position and Orientation

Research conducted by Sandvik Rock Tools, Inc. relates to the proposed

problem. In a research document titled “Rock Tool Interaction”, Sandvik shows that

the spacing of cutting picks can have a nonlinear effect on the effort required to

remove material. Figure 1.6 shows a plot of specific energy for varying pick spacing.

These results relate to the present study in that changes in specific energy suggest

correlated changes in pick loads, based on pick spacing.

Gary Fuller, the engineer who supervised the Sandvik research, believes that

the effort required to cut asphalt (as opposed to hard rock) is probably linearly related

to pick spacing. The theory used to describe tool interaction in hard rock is illustrated

in Figure 1.7. Mr. Fuller feels that cutting picks will not interact through fracturing

when used in asphalt. He says that Sandvik’s experience has shown a spacing of 5/8

inch to be most effective, but they have not collected supporting data for asphalt.

10

Figure 1.6: The effect of pick spacing on specific energy

Figure 1.7: Theory of rock tool interaction effects

11

The US Bureau of Mines has conducted extensive research into the improve-

ment of mining processes. A pertinent technical report was published in 1985 under

the title ‘Conical Bit Rotation as a Function of Selected Cutting Parameters’[4]. The

study consisted of a central composite experiment relating attack angle, skew angle,

cut depth, and axial spacing to the amount of rotation per length of cut. As stated

in the article, “. . . conical tools are intended to rotate freely so they will wear sym-

metrically . . . ”. Results of the study showed that skew angle and attack angle were

the largest contributing factors to bit rotation.

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Figure 1.8: Conical bit, viewed from cutting direction, showing negative and positiveskew angles

The study conducted by the USBOM provides definitions for attack angle and

skew angle based on the geometry of that experiment. We have defined similar angles

for the present study, discussed in Section 2.3.2. Figure 1.8 shows the definition of

skew angle used in the USBOM report. Here, the sign of the skew angle is positive

when the pick is skewed away from the uncut material, and negative when skewed

toward the uncut material.

12

1.3.3 Design Optimization

The study of oil well drilling heads, conducted by Chase, applied an optimiza-

tion method used in minimum error linkage design. A detailed description of this

method can be found in another paper by Chase titled ‘Computer-Aided Design of

Precision Control Linkages in the Classroom’[5]. This optimization approach is based

on sensitivity analysis .

Optimization is accomplished for the drilling head, by defining parametric

intervals for bit positions. The magnitude of bit volume error (deviation from the

mean volume for all bits) is used to alter bit positions until volume magnitude is equal

for all bits. Applying sensitivity analysis to this problem proved very computationally

efficient, compared to other optimization techniques.

A study performed by Stephen Rogers and Brian Roberts[1] provides impor-

tant insights into the wear mechanics of cutting picks. They summarize pick wear

mechanics as “complex and almost certainly a composite mechanism”. An important

outcome of this research was a structured view of the possible wear paths for carbide

insert cutting picks. Rogers and Roberts conclude that “the cutting process is an op-

eration where productivity (namely cutting speed and depth), tool wear and product

size all need to be optimized”. Figure 2.4 shows some of the results of this study.

1.4 Document Overview

In the following chapters, we develop theories to explain asphalt reclamation

machine performance, and present research to support the theories.

Chapter 2 provides an introduction to pick failure mechanisms, and the sur-

rounding theory. This chapter also covers some of the constraints on cutter head

design.

Chapter 3 details methods for characterizing cutter head design and manu-

facturing in terms of volume removed by each pass of each individual pick. This

characterization helps to simplify our analysis of experimental data.

We provide a detailed description of our experimental design in Chapter 4.

And, Chapter 5 provides details on how the experiment was conducted.

13

Analysis of experimental results is contained in Chapter 6. The bulk of this

chapter is focused on statistical calculations of effects and their significance.

Chapter 8 outlines some possibilities for useful applications of the knowledge

gained in the present study. Also contained in Chapter 8 are some needed extensions

to the present study, and related projects.

The conclusions and recommendations, resulting from this body of work, are

detailed in Chapter 7.

14

Chapter 2

Pick Position and Performance

Most cutter heads do not wear evenly; i.e. certain locations on the cutter head

require more pick replacements than others. In some cases, casual observation will

reveal patterns in individual pick wear, that seem to be related to geometric condi-

tions. We discuss here some of the performance characteristics relating to geometric

parameters of cutter heads. The topics on geometric characteristics of a cutter head

can be categorized as either variational or designed.

2.1 Chevron vs. Scattered Patterns

A common method of laying out a pick pattern is to use a multiple start

chevron pattern. A typical chevron pattern is shown in Figure 2.1. For large milling

machines, this type of pattern has some significant advantages. The main advantage

is that the chevron pattern has the tendency to move material to the center of the

machine. This allows a load-out system to more efficiently move material out of the

cut. Another advantage, is the uniform volume removal rate of the picks. We will

deal with this characteristic of chevron patterns in later sections.

Chevron patterns tend to excite a frequency close to the rate of revolution of

the cutter head. Each time the end of the ‘V’ makes a cutting cycle, most of the

material that has been processed by other picks is reprocessed all at once by a few

picks in the center of the ‘V’. This phenomenon causes a cyclical imbalance in cutter

head loading, which excites a very low resonant frequency mode in the reclamation

machine and host vehicle.

15

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Figure 2.1: Chevron cutter head pick pattern

A significant wear effect can be observed in chevron patterns. Large asphalt

milling machines typically use a load-out system, which requires that processed ma-

terial be pushed toward the center of the cut. Because the material is moved to

the center of the machine, picks located at the center of the cutter head experience

excessive abrasive wear on the body of the picks.

Material processing characteristics of chevron patterns can also be problematic

in some applications. When the end of the ‘V’ engages the material, a few of the

center-most picks have the tendency to act as one large pick, and break out large pieces

of asphalt. In situations where the processed material is to be reused as roadbase,

this is not acceptable. Also, these large pieces of material can cause damage to other

picks, and other machine components.

One solution to these problems is to use an inverted chevron pattern. This

reduces the problems of center pick wear, but has other unwanted effects. Pushing

material to the outer edge of a cut causes the processed material to pile up on the outer

edges. The uneven profile of the processed material is problematic on city streets,

where residents may need to drive over the processed material. Figure 2.2 shows a

sketch of the resulting material profile when using an inverted chevron pattern.

16

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Figure 2.2: The profile of processed material for an inverted chevron pattern (A) anda scattered chevron pattern (B)

Because small reclamation machines generally do not require the load-out ca-

pability provided by a chevron pick lacing pattern, the designs of these machines have

shifted away from chevron patterns. An alternative to chevron patterns is the scat-

tered chevron pattern. Figure 2.6 shows a scattered chevron pattern. This pattern is

based on the chevron, but the chevron lines have been broken into several lines, with

much more distance between picks. This allows processed material to flow between

the picks, rather than being pushed along in front of a row of adjacent picks.

2.2 Observations on Vibration

Small asphalt reclamation machines are plagued by extreme vibration prob-

lems. They have low weight and rigidity relative to the magnitude of impact forces

originating at the cutter head. Typical cutter head patterns have the picks placed

in regular spacing around the circumference of the drum. This arrangement causes

the primary vibration mode to have a very narrow frequency band. In the particular

machine that is the focus of this study, the pick impact frequency has a mean of

approximately 33 cycles per second, with a range of 3 cycles per second.

Our effort to improve pick life is constrained by vibration in the entire system.

In altering the lacing pattern on the cutter head, we must not affect any significant

17

increase in vibration amplitude. Amplitude of vibration may be worsened by moving

to non-uniform circumferential pick spacing. This is because the resonant frequencies

of the reclamation machine and host vehicle will be similar, but not exactly the

same as the pick impact frequency. By randomizing the pick impact frequency, the

bandwidth of the impact frequency will be widened. This widened frequency band has

the effect of moving a portion of the input frequency closer to the resonant frequencies

of the system. Cutter heads are typically designed so that most picks have uniform

axial spacing, thereby condensing vibration excitation into less disruptive regions of

the system’s frequency response spectrum.

2.3 Observations on Wear

2.3.1 Pick Failure Mechanisms

In studies on mining, attack picks generally fail by a single mode, although

there are several paths that lead to failure. Components described here are illustrated

in Figure 2.3. Standard bits consist of a carbide insert at the tip, a conical body, a

brazed joint, and a support post. The conical body serves the purpose of directing

material away from the pick support block. The cylindrical support post allows the

pick to rotate in the holder.

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Figure 2.3: Components of a carbide-tipped attack pick

18

As each pick passes through the material (asphalt in this case), impact forces

and abrasive mechanisms cause a wear flat to form on the carbide insert. Once this

wear flat forms, the forces required to process material are increased. The increase in

force will, in some cases, immediately remove the carbide insert. When the carbide

insert is removed, the body of the pick is rapidly destroyed.

If the carbide insert is not immediately removed by an increase in cutting

forces, the excess heat generated on the pick introduces two main mechanisms: heat-

ing of the insert leads to thermal fatigue and cracking, and heating of the insert may

cause the brazed joint to fail.

Picks used in asphalt milling machines experience failure paths that are similar

to those of rock excavation picks. However, we have found in our study of asphalt

applications that an additional failure mode is present. Under conditions of high body

wear, a pick will require replacement before the carbide insert is destroyed. In these

cases, although the pick is essentially intact, it must be replaced to prevent abrasive

damage to the pick holder. The exact mechanism by which this happens is at present

unknown, but the result is easily observable.

Figure 2.4, from a study conducted by Stephen Rogers in 1991 [1], shows

typical paths to failure for picks employed in rock excavation.

19

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Figure 2.4: Failure paths associated with rock excavation using attack picks[1]

2.3.2 Pick Orientation Effects

For the purposes of our study, we define two primary angles for the orientation

of the attack pick: attack angle and skew angle. These angles have been shown

to directly affect pick performance. Pick manufacturers have issued research-based

20

recommended values for these angles, to be used in designing cutter heads. These

values vary by application.

Figure 2.5: Definitions of pick orientation angles

Attack angle is defined as the angle from a line tangent to the path of the tip

of a pick, to the axis of the pick body, measured in the plane defined by the circular

path traveled by the pick tip. Skew angle is the angle from the plane defined by the

circular path traveled by the pick tip, to the axis of the pick body, measured in a

plane tangent to the circular path traveled by the pick tip. The sketch of Figure 2.5

illustrates the angle definitions. These angles are described in greater detail in the

referenced literature, and in Appendix D.

So far, we have discussed two different forms of the pick skew angle: that

defined in the USBOM study[4], and the definition of Figure 2.5. In future sections,

we will also make use of the absolute value of the skew angle. It should be noted

21

that while we are using multiple forms of skew angle, all three have identical magni-

tude. For the USBOM study, the sign of the skew angle depends on the surrounding

material. As defined here, the sign of the skew angle is relative to the cutter head.

Other forms of the skew angle used here are the absolute skew angle, and the

standard skew angle. Absolute skew angle is simply the absolute value of the angle.

Standard skew angle is the angle found in cutter head measurements (Appendix D).

For asphalt reclamation machines, the attack angle, as shown in Figure 2.5, is

approximately constrained by the shape of the pick holder (hereafter called a block).

The standard block has constraining features that control its placement on a cutter

head. These features are designed to constrain the attack angle, and are fairly effec-

tive. Pick manufacturers have recommended a skew angle of between 4 and 6 degrees

for the present application. But, skew angle is not constrained by the shape of the

block, and is known to vary significantly in manufacturing.

In general, we have observed that pick orientation has a large effect on wear,

and the skew angle seems to have a larger effect than attack angle. Studies have shown

that, by introducing a small material-negative skew angle, forces are generated that

cause the pick to rotate in its holder. This rotation ensures uniform wear around the

pick. When this rotation is not present, a flat spot is generated on the pick, which

causes rapid failure. The US Bureau of Mines study[4], referenced above, showed that

a skew angle of between 5 and 15 degrees was most effective in inducing rotation in

carbide attack picks. For angles approaching 15 degrees, pick manufacturers find that

significant bending stress is introduced into the pick body. But, no recommendation

is given on the tolerance for skew angle variation.

22

2.3.3 Pick Position Effects

Simple experiments in altering relative pick position suggest that pick posi-

tioning has a large impact on pick wear. We have found that, when inserting a new

pick into a cutter head with worn picks, the new pick will wear rapidly to match the

worn state of the neighboring picks. We hypothesize that this phenomenon can be

quantified in terms of volume removed by the new pick.

When a new pick is inserted between worn picks, its relative height may be

as much as 1/8 of an inch greater than the neighboring picks. This additional height

causes the new pick to remove a greater volume of material in each cutting cycle.

When the new pick has worn to match the neighboring picks, the volumes equalize,

and the wear rate between picks equalizes.

To illustrate the effects of pick position on pick volume removal, we make a

rough estimate of the cross-sectional area of the material removed by each pick. The

volume removed by each pick will be approximately proportional to the estimated

cross-sectional area. This illustration focuses on pick numbers 23 and 50, numbered

in order of impact. While these particular picks represent an extreme case, they serve

well to illustrate the potential effect of pattern layout on volume removal.

The cutter head pattern used for this illustration is shown in Figure 2.6, with

each dot representing the tip of an attack pick. This pattern is a 30 inch wide scattered

chevron pattern, and is typical of most cutter head patterns. The location of pick

numbers 23 and 50 are also noted in figure 2.6. Horizontal lines have been drawn

through picks 23 and 50 so that the relative locations of neighboring picks can be

easily identified.

Dimensions defining the location of picks in a particular pattern are defined

in terms of axial position referenced from the edge of the cutter head drum, and in

terms of circumferential spacing around the drum based on an arbitrary starting line

(sometimes a seam in the tubing from which the drum is constructed).

23

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Figure 2.6: Flattened pick lacing pattern in a 30 inch wide scattered chevron pattern

A map of the impact location of each pick in the pattern is shown in Figure 2.7.

This map is based on a cutter head rotational speed of 150 revolutions per minute,

and a machine advance rate of 20 feet per minute. A simple explanation of this map

is obtained by imagining that the map is laid on the ground and then run over by the

machine. In this case, as the machine advances forward, the tip of each pick would

impact at the center of its corresponding circle on the map.

The circles in Figure 2.7 represent an impact-affected zone for each pick, or a

zone in which the pick is removing material from the cut. Overlapping circles indicate

that material has been removed by preceding picks. The size and shape of the impact-

affected zone are chosen arbitrarily for the purposes of this illustration. Figure 2.8

diagrams the difference in cross-sectional area of material removed by pick numbers

23 and 50. From Figure 2.8 we can conclude that Pick 50 will remove a larger volume

of material than pick 23.

24

Figure 2.7: Map of impact locations for each attack pick in a 30 inch wide scatteredchevron pattern

This model is similar to the model used by Chase [2] in his analysis of oil

well drilling heads. However, a simple study of the behavior of asphalt in milling

conditions, as described in later sections, shows that the shape of the material removal

zone is quite different from the shape found in Chase’s study. Later discussions detail

methods for finding a more appropriate model for the actual volume removed by each

pick.

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Figure 2.8: Illustration of pattern lacing effect on cutting cross-section for picks 23and 50 of Figure 2.7

25

2.4 Effects of Assembly Tolerance

As we have demonstrated in a previous section, per-pick volume removal is

determined by the axial spacing and angular placement of attack picks. Our earlier

discussion assumes that a cutter head is built exactly to engineering specifications.

Current manufacturing processes allow for variation not only in axial and angular

position, but also in radius. The effects of manufacturing variation, along dimensions

that define pick position, are similar to the effects of lacing pattern design.

The radial height of a pick is measured from the axis of rotation of the cutter

head, to the tip of the pick, measured in the plane defined by the circular path traveled

by the pick tip. Attack angle and skew angle are described in a previous section.

Road milling applications require tighter tolerances on pick position because

of the requirement for uniform depth in shallow cuts. When making shallow cuts in

asphalt, the effect of variation in pick position will be much larger as a percentage of

total depth. Reclamation machines are typically not used for making shallow cuts,

therefore the manufacturing requirements have been relaxed in an effort to reduce

cost.

The cutter heads used in this study have a specified manufacturing tolerance

for pick location of plus or minus 1/8 inch, in any direction (arc-wise, axial or radial).

Typical positional tolerances for the industry range from 1/16 inch to 1/8 inch. How-

ever, we have not been able to locate any studies that would help to define appropriate

tolerances for pick position in asphalt reclamation machines. Part of our objective in

this study is to determine which dimensional characteristics have significant effects

on pick wear, and therefore, where to focus manufacturing improvements.

26

Chapter 3

Modeling Per-Pick Volume Removal

In order to investigate the relationship between per-pick-volume and pick wear,

we first needed to define a model relating per-pick-volume to measurable parameters

of a cutter head. In this section, we show that a close approximation of per-pick

volume can be calculated from geometric, and operational parameters. In simple

terms, we calculate the cross-sectional area of a pick’s cut, at several angles through

the cutting cycle, and then integrate those areas to find volume. A summary of this

process follows.

1. Select a primary pick for which to calculate cut-volume, based on impact order

2. Select a subset of picks that interact with the primary pick

3. Divide the primary pick’s cutting path into angularly incremented sectioning

planes

4. For each sectioning plane:

(a) Calculate an effective radius for each interacting pick, lying in the current

sectioning plane

(b) Find intersection points between cutting profiles for the primary pick, its

own previous cutting cycle, and all interacting picks

(c) Calculate the area enclosed by the intersection points

5. Numerically integrate each of the pick’s cutting section area along its cutting

path

27

3.1 Pick Groove Shape and Material Removal Zone

By studying the shape of the grooves formed by picks as they pass through

asphalt, we have developed a model for cut profile. Figure 3.1 shows an area where

a reclamation machine began a cut in asphalt. The feature noted in this image is

a smooth track caused by the body of a pick contacting unprocessed material. The

hardened steel body of the pick continues to make occasional contact with the solid

portions of asphalt throughout the picks effective life. The observation of pick body

contact suggests that asphalt, not in the direct path of the pick (material between

picks), remains intact after a pick has made a near-by pass.

Figure 3.1: Pick body has been found to leave a smooth track along the inside faceof the pick groove

Based on the above observations, we have chosen to use a “V” shaped cutting

profile, at an angle of 75 degrees. In order to verify the geometry of the material

removal zone, we collected a sample of asphalt that had been cut by a single pick.

28

Figure 3.2 shows a cut-away view of the asphalt sample. A true scale sketch of a pick

has been superimposed on the image. A 75 degree “V” has been added to the sketch

to represent the approximated cutting profile.

Figure 3.2: A simplified pick cutting profile angle of 75 degrees was used to calculateper-pick material volume

3.2 Relative Depth and Cut Cross-Section

Most of the effort required to find volume is centered in finding the cross-

sectional area along a pick’s cutting path. In Section 2.3.3, we demonstrate the

relationship between pick position geometry, machine forward speed, cutter head

rotational speed, and pick cutting cross-section. For a more complete model, we also

consider the overall cutting depth of the machine.

Cross-sectional area for an individual pick is defined in a series of sectioning

planes, like the plane illustrated in Figures 3.3. Sectioning planes are defined by

the cutter head axis of rotation and the tip of the current pick. We describe a

material cross-section using a set of points lying in the sectioning plane. These points

are coordinates of the intersections between the “V” shaped cutting profile for all

neighboring picks. Coordinates for these intersections are based on the axis of rotation

of the cutter head, and the edge of the cutter head skin, in the directions shown in

29

Figure 3.4. The dimensioned sketch in Figure 3.5 shows a set of intersection points,

relative to the tip of the current pick.

Boundary points for the included figures are described in Table 3.1. As dis-

played in Figure 3.4, Lines BF and FD represent material remaining immediately

prior to the current picks cutting cycle. Lines EA and AG represent the new ma-

terial boundary, after the current pick completes its cutting cycle. The shaded area

represents the cross-section of material removed in this cutting cycle.

Figure 3.3: Section view of pick cut paths in an asphalt slab

30

Figure 3.4: A representation of material cross-section for a single pick’s cutting path,from Detail B of Figure 3.3

Table 3.1: Description of labels for cross-section boundary points from Figure 3.4

Label DescriptionA Tip location for current pickB Tip location for a previous pass of an adjacent pickC Tip location for previous pass of current pickD Tip location for a previous pass of an adjacent pickE Intersection point between cut profile of picks A and BF Intersection point between cut profile of picks B and DG Intersection point between cut profile of picks A and D

31

Figure 3.5: Dimensioned sketch (in inches) of a pick’s material cross-section

Table 3.2: Coordinates for pick tip locations shown in Figure 3.4

Label xp yp

A 9.00 18.00B 8.50 17.86C 9.00 16.99D 9.50 17.64

Table 3.3: Coordinates for intersection points shown in Figures 3.4 and 3.5

Label xs ys

A 9.00 18.00E 8.70 17.60F 9.09 17.10G 9.39 17.49

32

The coordinates for the intersection point between any two pick profiles, in

the sectioning plane, are given by Equation 3.1. This equation requires x and y

coordinates for the tip location (xp, yp) of two picks in the sectioning plane. With

the equation in this form, Pick 2 must have x coordinate greater than that of pick 1

(xp2 > xp1). Example pick tip and intersection point coordinates are shown in Tables

3.2 and 3.3 respectively.

xs =(yp2 + m · xp2)− (yp1 −m · xp1)

2m

ys = m · xs + (yp1 −m · xp1); (3.1)

Where:

m = Slope of the line defined by a single side of a pick cutting

profile, lying in the sectioning plane – In this case, m =

tan(90◦ − 75◦

2)

xp = Axial coordinate of a pick tip location, relative to refer-

ence point on edge of cutter head skin: a specific instance

of Da

yp = Radial coordinate of a pick tip location: a specific in-

stance of the variable Re

xs = Axial coordinate of a pick cut profile intersection point,

relative to tip location of current pick

ys = Radial coordinate of a pick cut profile intersection point,

relative to tip location of current pick

Pick tip effective depth within a particular sectioning plane (yp in Table 3.2),

can be found using Equation 3.2. This equation finds an effective radius Re for the

previous pass of a given pick, or the effective depth of its neighboring picks. Variable

definitions for the effective radius equation are listed below, and are illustrated in the

sketch of Figure 3.6.

33

Re = R

sin[π − arcsin

(AR

sin (θ))− θ

]sin (θ)

(3.2)

Where:

Re = Effective pick tip radius for a particular sectioning plane

R = Individual pick tip radius, from rotational axis of cutter

head

A = Machine advance distance between pick impact events

θ = Cutter head rotation angle, measured from negative of

machine advance direction to an individual pick tip

Figure 3.6: Geometry for effective pick radius Re within a sectioning plane at angleθ and advance distance A

34

3.3 Cross-Sectional Area and Volume

A pick’s cutting cross-sectional area can be found using a variation of Green’s

Theorem[6]. Our variation, shown in Equation 3.3, takes a list of ordered intersection

points (calculated in Section 3.2) and returns an enclosed area. The intersection

points for a typical pick are shown in Figure 3.5. Figures 3.3 and 3.4 further illustrate

the context of the plane and geometry in which Green’s theorem is applied.

Ab =1

2

n∑i=1

(xs,iys,i+1 − xs,i+1ys,i) (3.3)

θmin =π

2

θmax = arccos(

Rn −Dc

Rn

)+

π

2

θstep =θmax − θmin

nstep − 1

sstep = Rn · θstep

Vb =nstep∑i=1

(sstep

Ab,i + Ab,i+1

2

)(3.4)

The swept volume of removed material is calculated by numerically integrating

cross-sectional area at intervals along the pick’s path. Equation 3.4 shows the method

for numerically integrating for swept volume. This equation essentially implements

a trapezoidal numerical integration between the angles bmin and bmax, in angular

increments of bstep. The variable sstep is the arc distance along a single angular step,

and nstep is the number of angular steps in the integration.

3.4 Verification of Model Results

In order to confirm the results of the MATLAB calculations, we modeled the

expected volume removed by each pick using a CAD-based solid modeler. Solid vol-

ume calculations returned by this software are quite accurate, however, the processing

35

time for these calculations is excessive. Code used to implement volume calculations,

using an Application Program Interface (API) to the CAD package, is included in

Appendix B. The simplified CAD model, used for volume calculations, is also shown

in Appendix B. Figure 3.8 shows a rendering of the complete CAD model.

Per-Pick Volume Model Comparison

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30

Axial Position (in)

Pick Volume (in3)

CAD Model

MATLAB Model

Figure 3.7: Comparison between CAD and MATLAB models of per-pick volumeremoval rate

The comparison illustrated in Figure 3.7 is a plot of the per-pick volume for

each pick, ordered by axial position on the cutter head. Values found from the

MATLAB model lie almost directly over the CAD values, causing the plot to appear

as one line. These results are based on a MATLAB volume calculation using 30

integration steps (nstep = 30). Because processing time with 30 steps is relatively slow,

we used nstep = 10 for tasks requiring iterative model calls (Monte Carlo Simulations).

The processing time is dramatically reduced, with very little loss of accuracy. Overall,

the MATLAB model performs very well.

36

Figure 3.8: Complete CAD model describing cut simulation used in model validation

It should be noted that both methods described here, for finding per-pick

volume, ignore the forward movement of the machine during a pick’s cutting cycle.

We believe this to be a minimal source of error, considering that a given pick’s cutting

cycle is short compared to the forward speed of the machine. The advance distance

between impacts of a single pick is 0.8 inches, while the advance during cutting is only

about 0.1 inches. This approximation is made for all picks, therefore the net effect

is a change in shape for the volume element removed by a pick, but no significant

change in the magnitude of the volume.

37

3.5 Summary of Chapter Variables

m = Slope of the line defined by a single side of a pick cutting profile, lying

in the sectioning plane – In this case, m = tan(90◦ − 75◦

2)

xp = Axial coordinate of a pick tip location, relative to tip location of current

pick

yp = Radial coordinate of a pick tip location: a specific instance of the vari-

able Re

xs = Axial coordinate of a pick cut profile intersection point, relative to tip

location of current pick

ys = Radial coordinate of a pick cut profile intersection point, relative to tip

location of current pick

R = Individual pick tip radius, from rotational axis of cutter head

Rn = Nominal pick tip radius, specified by engineering drawings

A = Machine advance distance between pick impact events

Re = Effective pick tip radius for a particular sectioning plane

θ = Individual pick tip rotational angle, measured from negative of machine

advance direction

D = Pick tip depth relative to another pick (R−Re)

Ab = Pick cut cross-sectional area, at a particular integration angle

θmin = Rotational angle at which picks begin a cutting cycle (enter the asphalt)

θmax = Rotational angle at which picks end a cutting cycle (exit the asphalt)

θstep = Step size (in rotational angular units) used for volume integration

sstep = Step size (in arc distance units) used for volume integration

Vb = Volume of asphalt removed by a single pick in a single pass

38

Chapter 4

Developing an Experiment

The primary focus of this experiment is to determine the strength of the rela-

tionship between pick position/orientation, and pick failure rate. In previous sections,

we have defined three main variables that will be our primary experiment factors: ab-

solute skew angle, attack angle, and per-pick volume removal. The response variable

for our experimentation is the pick failure rate.

4.1 Designed Experiment vs. Observational Study

Using a designed experiment requires precise control over the factors of inter-

est, in our case pick position and orientation. This study is motivated by the fact

that these factors are difficult to control. Because of this difficulty, we have chosen

to obtain data under an observational study framework.

There are certain trade-offs when choosing between an observational study and

a designed experiment. Typically, a designed experiment would have to be conducted

in a laboratory in order to explicitly control all of the variables. And, some operating

conditions would be impossible to simulate in the lab.

On the other hand, observational studies have certain difficulties. Measure-

ments in the field are expected to be more difficult, and less accurate. In the present

study, we simply collect observations of performance under conditions of natural vari-

ation in all experiment variables. Under these conditions, it is impossible to detect

the presence of lurking variables from statistical analysis of the results. This means

there is a strong potential for confounded results, or results that cannot be verified

by statistical methods. In order to validate our results, we have shown by expert

39

knowledge of the systems involved, that all potential variables have been sufficiently

accounted for.

Designed experiments are setup in such a way that data are collected at the

boundaries of the model space. Since in an observational study we are unable to

choose factor values, the boundaries are not fully explored. This can lead to mislead-

ing results, as some regions of the predictive model actually use extrapolations of the

data.

The specific objective of the experiment was to relate the pick failure rate to

the volume removal and to the orientation of each pick location. Table 4.2 describes

the three primary explanatory variables that make up our experiment. Section 2.3.2

describes in detail how skew and attack angles affect individual pick wear.

4.1.1 Independent Variables and Manufacturing Variation

The attractiveness of an observational study for this research is derived from

the large manufacturing variation observed in the construction of cutter heads. There

are 96 picks in the main pattern of a 48 inch wide cutter head. We assume that each

pick is defined by a set of independent random variables. This situation provides us

with a large amount of data from a single cutter head.

Using the methods described in Chapter 3, we have calculated expected vari-

ation in the volume of material removed by each pick. These calculations are based

on the dimensional tolerance from engineering assembly drawings. Figure 4.1 shows

nominal volume for each pick location on a particular cutter head. Also shown, are

expected variation found from two different methods of calculation. The Monte Carlo

method is simple to implement, but very resource intensive. The Direct Linearization

Method (DLM) returns accurate results, with very few calculations. A comparison

of the two methods can be found in a study by Gao, Chase, and Magleby[7].

40

Volume Tolerance Analysis

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30

Axial Position (in)

Volume (in3)

MonteCarlo Tolerance

DLM Tolerance

Nominal Volume

Figure 4.1: Expected variation in per-pick volume removal, based on cutter headassembly tolerance

Both the DLM, and the Monte Carlo method produce a root-sum-squares

(RSS) assembly tolerance for per-pick material volume. The error bars on the nominal

volume represent a plus-or-minus three sigma (3σ) variation in per-pick volume. A

separate plot of the one-sided 3σ variation for each of the analysis methods is also

included in Figure 4.1.

It can be seen from the DLM and Monte Carlo tolerance plots, of Figure

4.1, that there is very close agreement between the two methods. There are three

main issues effecting the accuracy of these methods. First, Monte Carlo accuracy

depends on making a large number of model calls, which can be very time intensive

(we performed only 15,000 iterations). Second, DLM performs a linearization on the

input model, which can introduce errors where tolerances are large relative to nominal

dimensions. Third, the DLM assumes a Normal distribution for both manufacturing

tolerances and volume variation, but non-linearity in manufacturing tolerances can

cause the actual distribution of volume variation to be skewed. The DLM method

has been shown to have accuracy equivalent to a Monte Carlo analysis with 30,000

model calls[7].

41

4.2 Characterization of Experiment Variables

The response variable for our experimentation is the locational pick failure

rate. This rate is defined as the number of picks replaced in a particular location on

an existing cutter head, throughout all testing. Note that we have not defined this

rate in terms of time, but rather in terms of count-per-experiment (see Table 4.1).

Table 4.1: Dependent response – directly observed output variables

Variable Name Factor Description

tipFails Individual pick tip failure count, bycutter head location, for the presentexperiment

bodyFails Individual pick body failure count, bycutter head location, for the presentexperiment

Two main characteristics of the experimental design allow us to ignore time

between failures. First, defining per-pick volume makes each pick location on a cutter

head an independent statistical sample. Second, each pick location will experience

the same amount of run time. Under this scenario, the knowledge gained from the

experiment will only allow us to identify significant factors for predicting failures; we

will not be able to predict time-to-failure for a particular pick location.

The main predictive variables that are the focus of this study are pick absolute

skew angle, attack angle, and per-pick volume removal. These three factors can either

be measured directly, or calculated from direct measurements of a cutter head. The

measurement of the factors of interest is fairly simple, with appropriate equipment

(refer to Section 5.1).

42

Table 4.2: Primary independent variables

Factor Description Factor Handling

Absolute Skew angle Can be directly measured

Attack angle Can be directly measured

Per-pick volume removal Can be calculated from measurableparameters

Pick failure rate is affected by many different conditions present in asphalt

reclamation. For the current experiment, we are able to directly observe only a few

key variables. Several variables, known to contribute to pick consumption rate, are

very difficult to observe. However, the nature of the experiment causes these non-

observed variables to be effectively randomized. On average, all picks will experience

identical levels of these randomized factors. Observable and non-observable factors

are listed in Tables 4.3 and 4.4, respectively. The referenced tables show each of the

factors that we have been able to identify, and the approach we have taken to account

for them.

Table 4.3: Global variables, applying to all picks collectively


Machine depth Although each pick has a different cut-ting radius, all picks will experiencethe same machine depth

Cutter head RPM’s All picks will experience the same cut-ter head RPM

Amount of material being re-worked

In a scattered pattern, all picks willprocess approximately the same vol-ume of rework material

43

Table 4.4: Randomly varying experiment variables, adding noise to the results


Grain size of asphalt On average, all picks will experi-ence the same asphalt grain sizethroughout testing

Hardness of asphalt On average, all picks will experiencethe same asphalt hardness through-out testing

Temperature of asphalt On average, all picks will experi-ence the same temperature of as-phalt throughout testing

Moisture content of asphalt On average, all picks will experiencethe same moisture content, withwater cooling system disabled

Pick manufacturing characteristics The effects of pick manufacturingcharacteristics will be randomizedas picks are destroyed and replaced

Material flow characteristics This should be highly correlatedwith other factors that have alreadybeen accounted for

One potential factor that was not directly handled in this experiment is the

material flow characteristics between picks. When operating in deep applications

(i.e. trenching), a large volume of processed material flows between the picks. This

material flow could interfere with bit rotation, cause side loads on bits, and/or increase

abrasive wear to pick bodies. But, the limited resources of the project, and the

difficulty involved in measuring such parameters, lead us to ignore the effects of this

potential factor. We feel this is a safe decision, based on observations suggesting that

material flow will be highly correlated with other factors in the analysis (see Section

4.2.1).

44

4.2.1 Expanded and Condensed Factor Models

In order to help us develop a more detailed understanding of pick failure, we de-

fine and analyze a few additional factors (Table 4.6). The primary dependent factors,

defined previously, will hereafter be called condensed factors. The additional factors

are an expansion of the main factors of interest. These expanded factors consist of a

set of positional measures based on what we have termed the Tip Proximate Distance

(TipPD) and Body Proximate Distance (BodyPD). As presented in later sections, we

correctly anticipated that the expanded factor set would reveal useful insight into the

performance drivers in cutter head design. A summary of key definitions is provided

in Table 4.5.

Table 4.5: Key definitions relating to proximate distance

Term Definition

Axially adjacent neighbors The two picks having the least axialdistance from the pick of interest (ide-ally, picks have uniform axial spacing)

Angular proximate distance Angular distance between a pick andeither of its axially adjacent neighbors

Radial proximate distance Height difference (from cutter headaxis of rotation) between a pick andeither of its axially adjacent neighbors

Body-side The direction, along the cutter headaxis, toward which a given pick’s bodyis angled by skewing

Tip-side The direction, along the cutter headaxis, away from which a given pick’sbody is angled by skewing (oppositethe body-side)

45

Figure 4.2: Diagram showing definitions and sample values of tip and body sideproximate distance for a selected pick and its axially adjacent neighbors

Tip-side and body-side, for a given pick, are defined by the skew angle for the

pick as shown in Figure 4.2. As labeled in the figure, the body-side of a pick is the

46

direction along the axis of the drum toward which the body of the pick is angled,

by skewing. The Tip PD is the distance from a given pick to the axially adjacent

neighbor on the tip side, along a specified dimension. Tip Radial PD is the radial

distance between a pick and its axially adjacent neighboring picks. Axial PD is the

axial distance between a pick and its axially adjacent neighboring picks.

Angular PD is the angular distance between a pick and either of its axially

adjacent neighboring picks, measured in the forward direction; i.e. angular PD is

measured in the pick’s direction of travel. This is illustrated in Figure 4.2 for the

body-side PD, where angular distance (dimension broken at ‘aBodyPD’) is measured

the long way around the drum.

Using expanded factors allows us to more effectively test the interaction be-

tween per-pick volume and skew angle. The condensed factor set, described in the

preceding section, does not account for the fact that a given pick may cut more as-

phalt on its body side than it does on its tip side. Similarly, a given cutting volume

may have a different effect on the body side than it does on the tip side.

Table 4.6: Additional experiment variables potentially correlated to bit failures

Variable Name Factor Description

xBodyPD Axial body-side proximate distance

xTipPD Axial tip-side proximate distance

rBodyPD Radial body-side proximate distance

rTipPD Radial tip-side proximate distance

aBodyPD Angular body-side proximate distance

aTipPD Angular tip-side proximate distance

47

4.3 Overall Average Pick Consumption Rates

In an effort to approximate the amount of data that would be needed for

significant results, we collected some preliminary data on overall pick consumption

rate per machine. We collected data from information on pick sales and machine

hours. The data were collected from 3 different customers, using information on pick

sales and machine hours. Due in part to a wide variety of applications, the data we

collected had a very large range. The first customer averaged a consumption rate of

30 picks per hour, the second averaged a consumption rate of 1 pick per hour, and

the third averaged 5 picks per hour. By considering the applications in which these

data were collected, we anticipated a pick consumption rate for our experiment of 20

picks per hour. We will show in a later section that this approximation was much

higher than the actual.

4.4 Independent Sampling and Bias

The assumption that each pick location represents an independent observation

is only approximately true. For example, if a particular pick fails, and the machine

continues to operate, the neighboring picks will experience an increased load until

the failed pick is replaced. This situation will cause pick failures in neighboring pick

positions. By stopping the reclamation machine for inspection and data recording at

frequent intervals, the dependency between wear rate of neighboring picks is reduced.

The solution to this dependency problem is to closely monitor the state of each

pick, and replace it as soon as it fails. However, characteristics of asphalt reclamation

make it very difficult to continuously monitor the state of individual picks. As a

result, we are required to collect data at certain intervals in time (i.e. readout data).

Because the precision of our analysis depends on the time resolution of the data

collected, choosing the length of the time interval requires careful consideration.

The ideal interval would approximately match the life of the shortest lived

pick. However, since we have very little predictive capability, we are unable to di-

rectly determine the actual life of any pick. Our approximation of average overall

48

pick consumption rate is useful in determining an initial target for the readout in-

terval, however, the optimum interval is expected to vary by circumstance. In actual

application, we start with an initial target for readout interval, and then adjust that

interval based on observations in that application.

An alternative approach to reducing inter-pick failure dependency is to replace

picks before they completely fail. This could be accomplished by creating a ‘not-go’

gage to be used in determining a uniform level of wear at which a pick is to be

replaced. While this method has attractive statistical attributes, it is difficult to

apply in practice, considering that multiple failure modes have been identified.

4.4.1 Cut Overlap Bias

It was expected that on some jobs, a machine operator would make multiple

adjacent cuts with the reclamation machine. In this case, the machine’s path may

overlap the trench left by a previous path. The result of this situation is that not all

of the picks on the cutter head will be engaging material, which would be a serious

disruption to our experiment. In order to avoid the resulting bias in our data, we

required the machine operator to make several long passes, without significant overlap.

We then discarded data from picks located within a certain distance of the edge of

the cutter head.

49

Chapter 5

Conducting the Experiment

5.1 Measuring Actual Pick Position and Orientation

The first task in our experimentation was to make accurate location and orien-

tation measurements for each bit on a particular cutter head. We used a Romer/Cim-

core brand articulated arm CMM (model ‘10 foot Infinite Arm’) to take the measure-

ments. These measurements were performed on empty pick mounting blocks, and

then transformed, based on the shape of the style of pick to be used in our study.

This approach is possible because of the relatively low dimensional variation between

picks. Appendix D details the setup, methods, and results of these measurements.

Based on measurement repetition, we estimate that all measurements are accurate

within ±.015 inches.

Table 5.1: Descriptive statistics for bit placement measurements (no edge bits)

Mean Max Min StDevAxial Spacing 0.5” 0.67” 0.32” 0.08”Tip Radius 17.92” 18.01” 17.84” 0.04”Left Skew -11.5◦ -7.3◦ -18.2◦ 2.3◦

Right Skew 6.0◦ 11.6◦ 1.9◦ 2.1◦

Attack Angle 51.9◦ 53.6◦ 50.2◦ 0.7◦

With precise information on pick location and orientation, we were able to

use the previously discussed computer model to calculate the volume per cutter head

51

revolution removed by each pick, for specified operating parameters. The other factors

of interest, listed in Table 4.2 were also readily available from the transformed CMM

measurements. Descriptive statistics for the bits that were used in the analysis are

shown in Table 5.1.

5.2 Data Collection Methods

Data were collected on various job sites, in various applications. Approximate

times and descriptions of the applications in which we conducted tests are shown in

Table 5.2, comprising a total of 50 hours of operation.

Because of the way asphalt milling attachments are operated, different appli-

cations can affect large changes in some of our factors of interest. Most notable is

the effect of machine forward speed on per-bit-volume. In thin, or soft, material the

machine’s forward speed averaged about 25 feet per minute, whereas in hard or thick

material the machine’s forward speed was between 8 and 10 feet per minute. The

actual average forward speed for the three different material types in our testing were

recorded by the operator, and are shown in Table 5.2.

Table 5.2: Descriptions of Testing Applications

TestingTime

ApplicationDescription

Bit Re-placements

ForwardSpeed

12 hours thin, ‘alligatored’ 27 25 ft/min

35 hours deep, hard 55 10 ft/min

3 hours deep, soft 30 20 ft/min

The first application listed in Table 5.2 is described as ‘alligatored.’ This term

refers to a state of asphalt paving where the material is broken into uniform pieces,

but the material is still in place. An example image of alligatored asphalt is included

in Figure 5.1

52

Figure 5.1: Image of asphalt in ‘alligatored’ condition

Data collected during testing were recorded by the machine operator. The

machine operator reported that in softer materials, some of the bits experienced

a significant amount of body wear. In some cases, the operator was required to

replace these bits before they had completely failed, to avoid damaging the block.

These body-wear-failures represent a separate failure mode of which we were not

previously aware. The number of body and tip failures are compared in Table 5.3.

Total descriptive statistics for the bits considered in our analysis, based on cut overlap,

are presented in Table 5.4.

Table 5.3: Number of failures by failure mode and application

Failure ModeApplications Pick Tip Pick BodyMaterial Type 1 7 20Material Type 2 54 1Material Type 3 0 30

53

Table 5.4: Descriptive statistics for bit failures on bits considered in the analysis

tipFails bodyFailsObservations 90 90Total Failures 55 33Zero Counts 43 64Max Failure Count 2 3Mean Failures 0.61 0.37

5.3 Controlling Machine Operating Parameters

During operation, the operator of the host vehicle (a wheeled loader in this

case) attempts to operate the milling machine forward at a constant cutting torque.

Engine rotational speed is the most convenient indicator of cutting torque. The

operator attempts to hold the milling machine’s engine at a constant rotational speed

(measured in RPM). In our experimentation, the engine speed averaged 2300 RPM.

With a 18:1 gear reduction, this results in a cutter head rotational speed of about

130 RPM.

The host vehicle operator is not able to directly control machine advance rate.

If he pushes too hard with the loader, the milling machine’s engine will stall. By fixing

the depth of cut, and by operating with the milling machine’s engine at a speed for

optimal torque, the machine advance rate can only be observed. Forward speeds for

each testing application are shown in Table 5.2.

Although we intended to operate the machine dry (without the use of water

spray), this was not possible because of restrictions from local health departments.

Operating with a light spray of water prevents the formation of hazardous dust. Our

general observations of the machine’s operation indicated that the light water spray

had no significant effect on performance, or on our data.

In our testing, we found that in full depth reclamation, the cutter head en-

counters a significant amount of moisture in the road base. This makes the proportion

of moisture content coming from the spray system small relative to the total amount

54

of moisture experienced by the bits. We were also careful to check the spray system

at regular intervals to make sure that the spray pattern remained uniform. By taking

these precautions, we have limited the potential for bias in the results of the testing.

55

Chapter 6

Analysis of Results

In the following sections, we present results of a detailed study of the effect of

cutter head geometry on bit failure, using methods based on Design Of Experiments

(DOE). Several strong effects have been identified, which offer new insights into pick

failure phenomena on reclamation machines. Some of the results present here verify

commonly held theories, but several new findings run contrary to accepted failure

models.

The results of our testing included a large number of pick body failures. This

was an unexpected result of the experiment which led us to make separate analyses

of the two failure modes. We have therefore separated pick tip failures and pick

body failures into two studies. As described in Section 4.2.1, we have also separated

the two failure modes into condensed-factor and expanded-factor regression models.

Descriptive statistics for bit failures were presented in Table 5.4.

6.1 Note on Regression Model Closeness

The observed response for this experiment, bit failures, is a count variable with

a low mean value. Therefore, in the following work, we model the experimental system

using Poisson regression with factors scaled to range from -1 to 1. Non-contributing

factors were dropped from the models using a backward-stepwise regression method.

A model dispersion parameters φ, is appended to each of the regression results tables

shown below. Appendix C details the motivations and process for developing the

regression models presented here.

57

For the present study (an observational study), the factors being studied have

only natural variation whereas, in a designed experiment, special variation would be

intentionally introduced to the model. For each factor, variation about the mean is

smaller than would be introduced in a designed experiment, resulting in a model with

low predictive power.

Note that certain regions of the models predict very high failure rate. For

example, at high values of aT ipPD, and low values of rT ipPD, the model predicts a

mean rate of 84 pick tip failures (Figure 6.3). This unlikely prediction stems from a

known limitation of observational studies. Specifically, the boundaries of a statistical

model are often not fully explored, when based on an observational study. In Section

4.1, we described the problem as data extrapolation.

We expect that the models developed here account for only a small range of

possible operating conditions, and are therefore unable to make accurate predictions

about actual cutter head performance. However, the statistical significance of individ-

ual factors allows us to make inferences on the strength and direction of relationships

between design parameters and wear rate. We are therefore able to make relative

predictions on performance from controllable parameters.

The p-value is used frequently in the following analysis. P-value is defined

as the probability of finding a stronger effect than the one observed, assuming that

there is no real correlation between the regression factors and the system response.

A p-value of 0.05, for example, indicates that we would have only a 5% chance of

drawing the sample being tested if the null hypothesis (no correlation) was actually

true. In essence, the p-value is a measure of evidence against the null hypothesis.

6.2 Pick Tip Failure Model

6.2.1 Condensed Factors

The condensed factor model for pick tip failures showed no significant corre-

lation. The last factor considered by the backward stepwise routine was the absolute

skew angle, with a P-Value of 0.37. Based on this finding, we proceed to consider the

58

expanded factor model for pick tip failures. See Appendix C for more details on the

analysis of these factors,

6.2.2 Expanded Factors

For the expanded factor model on pick tip failures, several factors were sig-

nificant. Table 6.1 presents results of a backward stepwise regression on pick tip

failures, for both linear effects and first order interactions. The response variable for

this regression is the number of pick tip failures at each pick location.

Predictor variables for this study have been scaled to range between -1 and 1,

and are defined in Section 4.2. Note that variables of the form *TipPD and *BodyPD

are the distance between a pick and its immediately adjacent tip or body side neighbor

(defined by the direction of the skew angle), along a specified dimension. See Figure

4.2 for an illustration of these variables.

Table 6.1: Tip failure regression results, expanded factors and first order interactions

Factor Type Factor Name Coeff P-Value

Main Effects (Intercept) -0.463 0.0654

rTipPD -0.124 0.8651

aBodyPD 0.989 0.1060

aTipPD 1.136 0.0226

Interactions rTipPD x aBodyPD -2.985 0.0230

rTipPD x aTipPD -3.639 0.0510

1 φ = 0.641

In Figure 6.1, we provide histogram plots of predicted failure probability dis-

tribution. Each plot shows the two distributions: one for the low factor value, and

the other for the high factor value. These plots were generated by holding all but

59

one of the linear factors at their mean value, while allowing a single variable to range

between its high and low. The plots show a simple trend of increasing mean failure

probability with increasing factor values.

rTipPD Effects

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Tip Failure Count

Pro

bab

ilit

y

Low rTipPD

High rTipPD

aBodyPD Effects

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Tip Failure Count

Pro

bab

ilit

y

Low aBodyPD

High aBodyPD

aTipPD Effects

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Tip Failure Count

Pro

bab

ilit

y

Low aTipPD

High aTipPD

Figure 6.1: Plot of main effects in pick tip failure regression model

Physical Interpretation – The results of the regression show a significant

connection between both tip-side and body-side angular spacing. This observation

appears to agree with theories relating frontal exposure to tip failure. Specifically,

with greater relative angular spacing, a pick is more likely to be impacted by large

pieces of broken asphalt, as they pass through the reclamation machine. As shown in

Section 2.3.3, having larger relative angular spacing also causes a pick to pass through

a larger cross-section of asphalt as it makes its cutting cycle. We expect that this

condition causes higher loads on the tip of a pick, and contributes to early failure.

An interesting result from this experiment is the discovery of interaction be-

tween relative tip height and angular spacing. Figures 6.2 and 6.3 illustrate the nature

60

of these interaction effects. For these figures, predicted mean failure rates were gen-

erated at the extreme observed values of the two interacting factors. While holding

all other factors at their mean values, we generate four response predictions from

the four possible combinations of the interacting factors. The bubbles are sized to

indicate larger or smaller response.

The particular interaction effects found to be significant, are an indication

that two modes are present in pick tip failures: sudden breakage, and abrasive wear.

Appendix C.4.1 provides additional evidence of multiple failure modes. In deep, hard

applications, pick tips appear to fail predominantly by abrasive mechanisms. An

abrasive, or wear-out failure, is evidenced by a carbide insert that is present, but has

been worn thin and flat.

Previous studies have shown that the rotation of a pick in its holder plays a

major role in prolonging pick life[4]. The USBOM study, referenced in Section 1.3.2,

found that “negative skew” was the primary contributor to pick rotation. For our

experiment, body-side angular spacing is a variant of the negative skew defined in

the USBOM study. Results from the present experiment agree with these previous

findings.

Fundamental principles of mechanical failure suggest that picks with greater

relative tip height would have shorter wear life. However, the rT ipPD × aBodyPD

interaction effect, illustrated in Figure 6.2, shows the opposite relationship. For bits

with high body-side angular spacing, relative tip height has little or no effect on tip

failure (1.28 mean failure rate vs. 0.45 mean failure rate).

Physical Interpretation – We expect that increased frontal exposure, on

the pick’s body side, would correlate well with increased rotational moment. It can

be seen from the plot of rT ipPD× aBodyPD that increased angular spacing on the

body side of the pick, combined with increased relative tip height, reduces tip failure

rate.

61

Figure 6.2: Predicted pick tip failures for extreme combinations of the rT ipPD ×aBodyPD interaction effect

Another consideration in pick rotation, is a phenomenon which we will term

“sticking.” In some curcumstances, tar and dirt tend to cause bits to stick in their

holders, which prevents rotation and induces early failure. To reduce this problem,

most bit manufacturers have added a Belleville type washer between the bit and its

holder. The washer allows the bit a slight amount of movement, up and down in its

holder. Any amount of movement serves to break adhesion between bits and holders,

enabling free rotation. This approach is especially effective where bits experience

interrupted impact loading.

62

Figure 6.3: Predicted pick tip failures for extreme combinations of the rT ipPD ×aT ipPD interaction effect

Physical Interpretation – The plot of rT ipPD× aT ipPD, in Figure 6.3, is

illustrative of bit sticking. We expect that in cases of greater tip-side angular spacing,

bits will experience a larger impact load with each revolution of the cutter head. As

explained above, this impact loading drives an increase in bit rotation. Pick wear

rate is reduced by greater relative tip height, when higher tip-side angular spacing

acts to induces pick rotation (84.42 mean failure rate vs. 0.05 mean failure rate).

6.3 Pick Body Failure Model

6.3.1 Condensed Factors

A stepwise regression for a condensed factor model on pick body failures re-

turned several significant factors. Table 6.2 presents coefficients and significance for

main and interaction effects.

63

Table 6.2: Body failure regression results for condensed factors

Factor Type Factor Name Coeff P-Value

Main Effects (Intercept) -1.148 0.0000

attack 0.816 0.1052

absSkew 1.007 0.0422

volume 1.748 0.0017

Interactions attack x volume -2.441 0.0129

1 φ = 0.898

The coefficients for this experiment indicate that pick body failures are higher

for picks with greater volume removal. Figure 6.4 shows a significant increase in

the predicted probability of failure for the observed range of per-pick volume. Picks

with high per-pick volume typically lead their neighboring picks through the cut,

which means greater exposure to both processed and intact asphalt. We also observe

that, for pick body failures, absolute skew angle (variable absSkew) has a positive

relationship to failure rate. This finding contradicts the findings of other studies[4].

However, there are two major observations that help explain this result.

Physical Interpretation – First, when operating in shallow material, picks

are engaged with the asphalt for a shorter distance on each cutting cycle. This

condition would limit the amount of rotation (in the pick holder) that each pick

experiences. By reducing the amount of rotation on each cutting cycle, the benefits

of skew angle are also reduced. Note that most pick body failures occurred in shallow

applications.

64

attack Effects

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5

Body Failure Count

Pro

bab

ilit

yLow attack

High attack

absSkew Effects

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Body Failure Count

Pro

bab

ilit

y

Low absSkew

High absSkew

volume Effects

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Body Failure Count

Pro

bab

ilit

y

Low volume

high volume

Figure 6.4: Plot of main effects on pick body failure (condensed-factor model)

The second observation relates to pick body exposure. When a particular

bit has a larger skew angle, the body is more exposed to abrasive wear, increasing

the rate of body failures for that position. The main effects plot, Figure 6.4, for

attack angle suggests that increased attack angle is associated with increased body

exposure. However, the interaction between attack angle and per-pick volume shows

a more complex relationship.

Physical Interpretation – Figure 6.5 reveals that for bits with low volume,

attack angle does indeed increase bit failure rate. But, for bits with high per-pick

volume, greater attack angle actually decreases pick failures. This observation can

be explained by examination of Figure 2.5. Larger attack angles move the bit body

farther from the cutting interface. This change in geometry reduces body exposure to

intact asphalt, and we expect this to also reduce body exposure to processed material

flowing between pick bodies.

65

Figure 6.5: Predicted pick body failures for extreme combinations of the attack-angle× per-pick-volume interaction effect

6.3.2 Expanded Factors

Regression on pick body failures, and expanded factors, showed no interaction

effects. Table 6.3 shows that the driving factor in pick body failures is aBodyPD,

the body-side angular spacing.

Table 6.3: Body failure regression results for expanded factors

Factor Name Coeff P-Value(Intercept) -8.914 0.0130xBodyPD 1.449 0.0008rBodyPD 1.200 0.0362rTipPD 1.145 0.0083aBodyPD 8.387 0.0371

1 φ = 0.844

66

The variable, aBodyPD, is essentially a combination of skew angle and per-

pick volume. The body-side component of the variable references the skew angle.

Angular spacing is a primary contributor to per-pick volume. This finding reinforces

the conclusions of the preceding section. Additionally, body-side and tip-side variables

allow us to test the assumption that per-pick volume may have a larger effect on one

side of a pick than on the other (see Section 4.2.1). And, the results presented here

show strong evidence that this assumption is correct.

xBodyPD Effects

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2

Tip Failure Count

Pro

bab

ilit

y

Low xBodypPD

High xBodyPD

rBodyPD Effects

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2

Tip Failure Count

Pro

bab

ilit

y

Low rBodyPD

High rBodyPD

rTipPD Effects

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2

Tip Failure Count

Pro

bab

ilit

y

Low rTipPD

High rTipPD

aBodyPD Effects

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Tip Failure Count

Pro

bab

ilit

y

Low aBodyPD

High aBodyPD

Figure 6.6: Plot of main effects on pick body failure (expanded-factor model)

6.4 Contributors to Tip Radius Variation

With tip height being a strong factor in predicting bit failure, we felt it impor-

tant to analyze the sources of variation in tip height. Although direct measurement

of certain cutter head characteristics would be a more conclusive method of analyzing

tip height, we can make some definite conclusions about leading contributors.

We have identified three main sources of variation in tip height: 1) out of

round cutter head skin, 2) mis-alignment of cutter head drive plate, and 3) random

67

variation in placement of pick holders. The random height variation, resulting from

pick holder placement is apparent in Figure 6.7.

Sources of variation in pick tip height:

1. Out-of-round cutter head skin (welded pipe)

2. Mis-alignment of cutter head drive plate

3. Random variation in placement of pick holders

Figure 6.7: Plot of bit tip height and bit axial location, versus angular position

Out-of-round cutter head skin can easily be detected from direct measurements

made previously on the cutter head. Figure 6.7 shows tip height plotted against the

68

angular position of each pick around the cutter head. This plot can be interpreted

quite simply by considering the manufacturing process for large diameter pipe. This

type of pipe is rolled from sheet steel and welded. From the shape of the plot, we

can safely assume that the two ends of the pipe did not align after rolling. The

sudden change in tip height is exactly aligned with the seam in the pipe, indicating

an eccentricity problem.

Row Slope

Positive

Row Slope

Positive

CUTTER HEAD AXIS

Figure 6.8: Drawing illustrating the definition of the row slope variable

Detecting runout, resulting from a mis-aligned drive plate, proved to be more

difficult. Our analysis uses the fact that all picks are placed by pairs in rows along the

axis of the drum, with uniform angular spacing (approximately six degrees between

each row). This geometric construction is illustrated in Figure 6.8. First, we defined

a row slope variable, which is the angle between the axis of the drum and a line

69

between pick tips in a given row. Next, we compare the row slope of a given row, and

its opposing row (the row 180 degrees around the drum from the given row). If the

drive plate were mis-aligned, we would expect the row slopes to have opposite sign

for each set of opposing rows. A paired T-Test was used to determine whether the

row slopes for each set of opposing rows comes from a different population. For the

cutter head used in the present study, we failed to reject the null hypothesis, meaning

that there was no statistically detectable mis-alignment of the drive plate.

70

Chapter 7

Conclusions and Recommendations

7.1 Contributions

This research project has made significant contributions to the body of knowl-

edge surrounding tool consumption rates for construction equipment. Applications

may even be found in such fields as mining and excavating. The successful appli-

cation of observational analysis techniques to the field of engineering is particularly

noteable. The following list describes some of the major accomplishments.

• The work presented here details an effective approach to adapting statistical

methods, often applied in the fields of medicine and econometrics, to engineer-

ing problems. In particular, we have conducted a modest observational study,

but have been able to extract an amazing amount of information and insight.

We have also identified some of the limitations and potential pit-falls of using

observational experiments in engineering applications.

• As part of our study, we have developed a fast algorithm for calculating per-pick

volume removal in typical cutter head patterns. This may allow volume based

optimization of pick patterns that can be run on individual workstations, in

reasonable amounts of time. Per-pick volume also provides the designer with

a simple proxy for the complex geometrical considerations in analyzing cutter

head performance.

• Per-pick volume was the largest contributor to pick body failure. Although

per-pick volume, as presently formulated, did not appear to be a significant

71

contributor to pick tip wear, we have identified other important contributors.

The most important of these being the angular spacing between neighboring

picks. We have identified some primary sources of manufacturing variation,

and shown that they have statistically significant effect on pick wear. Recom-

mendations have been made for reducing the effect of these factors, including,

manufacturing process, lacing pattern, and pick orientation changes.

• Originally, we planned to focus our study only on pick tip failures. But, early

experimental results revealed that pick body failures accounted for a large por-

tion of overall failures. The quantifiable correlations between design parameters

and this alternate failure mode will allow cutter head designers to better tailor

their designs to specific applications.

7.2 Conclusions

Several important conclusions may be drawn from the results and analysis of

the present study. We have also made some important conclusions about observational

studies. In the following sections, we detail a few of the most significant findings:

• Three main pick failure modes are apparent in the experimental data

• Skew angle may have either a negative or positive effect on failures, depending

on application

• Pick failures follow a Poisson process

• Manufacturing variation proved to be a significant contributor to pick failure

• Observational studies can be performed with minimal resources, but should be

used with caution

7.2.1 Multiple Failure Modes

The data collected in experimentation revealed pick body failures as a signifi-

cant corollary to the pick tip failure mode. Accordingly, we developed separate models

72

for pick tip failures and pick body failures. We also presented, in Sections 6.2.2 and

C.4.1, significant evidence of multiple modes within pick tip failures: abrasive wear,

and tip breakage. These pick tip failure modes were proposed in a previous study[1],

but our analysis revealed that these modes may have different models relating them

to cutter head geometry.

7.2.2 Skew Angle Effect

Previous studies into the effects of skew angle led us to believe that greater

skew would reduce tool consumption rate. In actuality, the present study showed that

for applications in asphalt milling, skew angle can have either a negative or positive

relationship to wear. The direction of the relationship has been shown to depend on

the type of material being processed, and the relative spacing between a pick and its

axially adjacent neighbors.

Findings on skew angle touch on both design and manufacturing considera-

tions. Future cutter head designs could adjust skew angle and skew direction to im-

prove performance in a particular application. The current experiment, and resulting

models, are somewhat limited in their ability to make actual predictions. However,

with an improved model, we could revise manufacturing tolerances to reduce pick

wear resulting from variations in skew angle.

7.2.3 Pick Failures, a Poisson Process

Count data are typically modeled most effectively as a Poisson process. Al-

though the data of this experiment are somewhat underdispersed, Section C.4 presents

strong justification for the use of a Poisson model. Our analysis, using Poisson re-

gression, showed convincing statistical evidence that pick failures are indeed a pure

Poisson process.

7.2.4 Manufacturing Variation

Of the factors resulting from manufacturing variation, we found skew angle

to have very large error. As shown in Section 6.2.2, our model formulations appear

73

to have canceled the effect of this variable. However, skew angle is implicit in the

expanded factor set. The presence of proximate distance factors, in the significant

model, implies that skew angle is a contributing factor to pick tip failure. Section

6.3.1 showed that absolute skew angle was a significant contributor to pick body

failures. Accordingly, better control over the manufacturing process that orients pick

holders is likely to improve wear performance.

The findings presented in Chapter 6 showed relative tip height (rT ipPD) to

be a significant contributor to pick failure. Both failure modes that we have analyzed

show a strong correlation with pick tip radius, relative to neighboring picks. The

analysis of Section 6.4 shows that variation in cutting radius between a pick and its

neighbors results largely from runout at the pick tips. Cutter head runout is the

result of three main conditions:

1. Out of round cutter head skin

2. Mis-alignment of cutter head drive plate

3. Random variation in placement of pick holders

For the cutter head we used in this study, skin cylindricity was the largest

contributor to pick tip runout. Random variation in the placement of pick holder,

during manufacturing, was also clearly present. We were not able to detect any

statistically significant mis-alignment in the cutter head’s drive plate. From these

observations, we conclude that out-of-round cutter head skins contribute to early

pick failure.

7.2.5 Observational Studies

The use of an observational study has provided significant time and cost sav-

ings over the course of this project. We avoided having to build a special test cutter

head by taking advantage of large manufacturing variation. A designed experiment

would have required the test cutter head to be manufactured to an accuracy higher

than can be produced with current processes. In general, the deliberate perturbations

in cutter head geometry would require gage-level tolerances.

74

Observational studies represent real operating conditions. Specifically, many

different factor levels are considered when working with natural variation. This can

improve our ability to detect curvature in the response, within observed factor ranges.

Factors considered in an observational study have a smaller range of values

than would be introduced in a designed experiment. This condition acts to lowered

the significance of effects. Of course, the resulting model is also only valid for the

reduced range of observation.

When working with naturally varying data, important or influential factor

levels may not be observed. In this case, possible but unlikely states of the system are

not explored. For example, particular combinations of factor values may have never

been observed. Predictions from this type of model should be used with caution, as

predictions, even within factor ranges, may be extrapolations from the data.

The size of the sampled data set has a large influence on accuracy and validity

of results. This study had a relatively small sample size, which can lead to factor

confounding. As an example of potential confounding, a few of the picks installed in

our testing machine never failed during the experiment. In this case, we are unable

to determine whether these picks lasted longer because of variations in the picks

themselves, or because of variations in primary experiment factors. However, we can

show that pick manufacturing variation is quite small relative to the primary factors

of our experiment.

Another problem associated with small sample sizes is the increased risk of

predicting in unexplored model space. While observed data covers more of the range

for each factor, there can be unexpected gaps in the data. This may lead to inaccurate

predictions for unlikely states of the observed system.

7.3 Recommendations

The initial motivation for this study was to find the relationship between

pick tip failures and per-pick volume removal. Pick body failures were significantly

effected by volume, but we found no significant correlation with tip failures. However,

the results and analysis of Section 6.3.2 suggest that a slight variation on per-pick

75

volume would produce better results. We recommend that any further studies use

both “tip-side volume” and “body-side volume”, rather than the single variable “per-

pick volume”. This recommendation aligns with the discussion of section 4.2.1.

Based on experimental results, we recommend two separate approaches to

improving bit life: cutter head design changes, and better manufacturing controls.

Initial manufacturing improvement efforts should focus primarily on reducing

cutter head runout. This would best be accomplished by using design and production

methods that reduce the sensitivity of the assembly to variation in cutter head drum

shape. One example of this would be to turn the cutter head skin before welding the

pick mounting blocks in place. This production method would practically eliminate

pick tip runout resulting from skin runout. Random variation in the placement of

blocks could be reduced by using a gentler welding process (i.e. preheat, use short

welds, and anneal).

Design improvements to the cutter head lacing pattern are constrained by com-

peting demands. Reduction of pick tip failure rate requires greater angular spacing

on the body side, as illustrated in Figure 6.2. As described in Section 6.3.2, reduction

of pick body failure rate requires smaller angular spacing on the body side. The main

objective would therefore be to find a minimum balance between pick tip and pick

body failure rates.

In Appendix C.4, we show that pick tip failures occur in two distinct modes.

Any future experiments could easily differentiate between the two failure modes by

observing tip breakage and tip wear-out as separate events. This improvement would

provide important insights into the pick failure phenomenon.

Our final recommendation relates to future studies. One weakness in the

present study was our limited ability to account for material flow between picks. A

more complete model of material flow will be required to fully characterize the effects

of lacing pattern on pick wear. The nature of the application will probably require

some simulation, and greater efforts at direct observation.

76

Chapter 8

Future Work

This study has essentially been a screening experiment. The main purpose

of this work has been to identify factors for future study. In order to make overall

performance predictions, an in-depth designed experiment would be required. A cen-

tral composite design, similar to that performed by the USBOM[4], would require

some modification of analysis methods to be effective for count data. The significant

contributing factors, identified in this study, should be the primary focus of further

studies relating design parameters, manufacturing variation, and specific cutting ap-

plications to general tool wear rates.

8.1 Dataset Size and Randomization

Every experiment could benefit from a larger data set, and this experiment

is no exception. Specifically, we feel that the size of our dataset (number of pick

failures) is small relative to some of the random factors of Table 4.4. This weakness

could lead to a certain amount of confounding among factors. An example of potential

confounding was presented in Section 7.2. This study could help guide future studies

in the amount of testing required to achieve a certain number of pick failures.

8.2 Material Flow Between Picks

The high number of pick body failures, found in this study, suggests some ad-

ditional analysis is needed. Pick body wear failures are generally a result of processed

material flowing between picks. The factors affecting this phenomenon are not well

understood. Future efforts to understand this failure mechanism should probably

77

include both cutter head design characteristics, and asphalt material characteristics.

A list of factors, possibly affecting body failures follows.

• Material flow

– Volume of reworked material

– Clearance for material flow between picks

– Pick tip height above cutter head skin (or cutter drum)

• Pick position and orientation

– Skew angle

– Attack angle

– Pick tip radius from cutter axis of rotation

– Pattern of placement, relative to adjacent picks (or lacing pattern)

• Asphalt properties

– Aggregate size

– Matrix adhesive strength

– Asphalt temperature

– Asphalt and base material moisture content

In our experimentation, we have noticed a trend relating material flow be-

tween picks. When a patch of alligatored material is encountered (see Figure 5.1),

a number of larger pieces of asphalt are pulled through the machine, accompanied

by a significant amount of additional noise. On the read-out interval immediately

following these encounters, there appears to be an additional number of pick failures.

As discussed in Appendix C.4, two modes are apparent in pick tip failures.

We believe exploring this would be a useful extension to the current study. An

hypothetical explanation for this observation relates to lacing pattern. In most lacing

patterns, there are locations on the cutter head where the lacing pattern leaves open

78

regions or gaps. Picks that trail the gap are likely to experience larger loads when

impacted by these large pieces of processed material. We expect this to result in

premature tip failure.

8.3 Predictive Models for Pick Consumption Rate

This research lays the ground work for the development of tools that could

help road maintenance companies better manage milling machines. In particular, it

may be possible to develop an empirical model that would accurately predict ma-

chine performance and operating costs for a particular job. Inaccurate performance

predictions can be very costly, especially for contractors who have legal obligations

to meet the terms of an initial bid.

As discussed in previous sections, picks must be inspected at regular intervals.

A possible application for a general model would be to find the optimum inspection

interval, to minimize downtime.

For the present study, we have allowed the factors in Table 4.4 to vary ran-

domly, and simply observed the factors in Table 4.3. The development of a general

performance model would require explicit controls on these factors, which would add

a significant amount of effort and complexity to data collection methods.

8.4 Design Optimization

The simplified model, developed in the present study, could be adapted for

use in optimizing pick lacing patterns. Constraints on pick location, described in

Section 2.2, make this a combinatorial optimization problem. Picks are required to

be evenly spaced in both axial position and circumferential position. Only two picks

can be aligned in any allowed circumferential position. A cutter head with 94 picks

in the main pattern can be setup as an integer program consisting of 94 numbers in

varying sequence. The objective of the optimization would be to minimize variation

in wear rate between all picks. Some of the constraints on machine performance were

discussed in Section 1.1. The primary restrictions on design changes are 1) clearance

between pick holders, and 2) vibration characteristics.

79

Insufficient clearance between pick holders can make assembly difficult or im-

possible, and can prevent material flow through the pattern. These problems are

avoided by constraining the pattern to have some minimum clearance between picks,

measured in any direction. However, it may be possible to compensate for spacing

requirements by adjusting other controllable parameters. For example, irregularities

in angular spacing could be counteracted by deliberately varying tip height.

Vibration issues arise from two well-understood design problems: load imbal-

ance in the cutter head’s axial direction, and load imbalance tangent to the cutter

head. In Section 2.2, we describe a vibration problem, arising from the grouping of

picks on chevron pattern cutter heads. This grouping causes a tangential loading

imbalance, which induces an undesirable vibration mode where the milling machine

tends to bounce the entire host vehicle.

In a previous experiment, Asphalt Zipper found that designing picks to im-

pact in an alternating fashion, from left to right, induces another undesirable vibration

mode. We describe this problem as an axial loading imbalance, which causes the ma-

chine to “wag” from side-to-side. We have summarized the optimization requirements

in the following list.

Optimization definition...

• objective:

– Minimize inter-pick wear variation

• subject to:

– Lateral loading imbalance factor < Predetermined maximum value

– Tangential loading imbalance Factor < Predetermined maximum value

– Minimum pick spacing distance > Predetermined minimum value

Some combinatorial problems can be effectively solved using a branch-and-

bound method. However, this class of optimization methods require that the ob-

jective function be continuous. As currently defined, the objective function is not

80

continuous. Discontinuities in the present model arise from the the use of several

conditional evaluations on the direction of spacing. Although some statistical opti-

mization methods (i.e. genetic algorithms, or simulated annealing) would be effective,

it may be possible to simplify the model and set limits that would allow the use of

the more efficient gradient-based methods.

8.5 Variations on Volume Calculation

As shown in Section 6.2.2, per-pick volume is a weak predictor of pick tip fail-

ure, while some of the expanded factors were quite strong. This observation indicates

that the volume variable is not well formed. One alternative approach to volume

calculation would be to redefine the per-pick volume by tip-side and body-side, as

defined in Section 4.2.1. It appears, from the results of the present study, that this

approach may improve the fit of per-pick volume and greatly simplify the regression

model.

A simplified model for characterizing wear performance would allow us to im-

plement more practical optimization methods for cutter head design. The model we

have developed is much faster than using a solid modeler. But, a combinatorial opti-

mization problem with 94 parameters could still be very computationally expensive.

A central composite experimental design, on an improved per-pick volume variable,

would probably allow us to use a gradient-based optimization method.

81

Appendix A

Cutter Head Pattern Definitions

This appendix includes tables and plots of cutter head design and manufac-

turing information. Specifically, we have appended tables of pick position and orien-

tation, plots of pick lacing patterns, and plots of volume removal levels for 2 sizes of

cutter head. Angular measures are in radians; linear measurements are in inches.

A.1 Specifications for 30 inch Cutter Head

The following plot shows a flattened pattern for the location of bits listed in

Table A.1.

��

Figure A.1: Flattened plot of designed bit positions, 30 inch cutter head

The following table contains a lacing pattern specification. This specification

would be used in manufacturing 30 inch wide standard cutter heads.

83

Table A.1: Cutter head lacing pattern specification, 30 inch cutter head

AxialPosition

AngularPosition

TipRadius

SkewAngle

AttackAngle

AxialPosition

AngularPosition

TipRadius

SkewAngle

AttackAngle

-0.75 0.898 16.5 -20 47 14.5 5.924 16.5 6 52-0.75 1.795 16.5 -20 47 15 5.027 16.5 6 52-0.75 2.693 16.5 -20 47 15.5 4.129 16.5 6 52-0.75 3.590 16.5 -20 47 16 3.231 16.5 6 52-0.75 4.488 16.5 -20 47 16.5 2.334 16.5 6 52-0.75 5.386 16.5 -20 47 17 1.436 16.5 6 52

0 0.000 16.5 -4 52 17.5 0.539 16.5 6 520.5 0.359 16.5 -4 52 18 5.565 16.5 6 52

1 1.257 16.5 -6 52 18.5 4.668 16.5 6 521.5 2.154 16.5 -6 52 19 3.770 16.5 6 52

2 3.052 16.5 -6 52 19.5 2.872 16.5 6 522.5 3.949 16.5 -6 52 20 1.975 16.5 6 52

3 4.847 16.5 -6 52 20.5 1.077 16.5 6 523.5 5.745 16.5 -6 52 21 0.180 16.5 6 52

4 0.718 16.5 -6 52 21.5 4.308 16.5 6 524.5 1.616 16.5 -6 52 22 5.206 16.5 6 52

5 2.513 16.5 -6 52 22.5 6.104 16.5 6 525.5 3.411 16.5 -6 52 23 3.411 16.5 6 52

6 6.104 16.5 -6 52 23.5 2.513 16.5 6 526.5 5.206 16.5 -6 52 24 1.616 16.5 6 52

7 4.308 16.5 -6 52 24.5 0.718 16.5 6 527.5 0.180 16.5 -6 52 25 5.745 16.5 6 52

8 1.077 16.5 -6 52 25.5 4.847 16.5 6 528.5 1.975 16.5 -6 52 26 3.949 16.5 6 52

9 5.924 16.5 -6 52 26.5 3.052 16.5 6 529.5 5.027 16.5 -6 52 27 2.154 16.5 6 5210 4.129 16.5 -6 52 27.5 1.257 16.5 6 52

10.5 3.231 16.5 -6 52 28 0.359 16.5 4 5211 2.334 16.5 -6 52 28.5 0.000 16.5 4 52

11.5 1.436 16.5 -6 52 29.25 0.898 16.5 20 4712 0.539 16.5 -6 52 29.25 1.795 16.5 20 47

12.5 5.565 16.5 -6 52 29.25 2.693 16.5 20 4713 4.668 16.5 -6 52 29.25 3.590 16.5 20 47

13.5 3.770 16.5 -6 52 29.25 4.488 16.5 20 4714 2.872 16.5 -6 52 29.25 5.386 16.5 20 47

84

A.2 Specifications for 48 inch Cutter Head

The following plot shows a flattened pattern for the number and location of

bits listed in Table A.2.

Figure A.2: Flattened plot of designed bit positions, 48 inch cutter head

The following table contains a lacing pattern specification. This specification

would be used in manufacturing 48 inch wide standard cutter heads.

Table A.2: Cutter head lacing pattern specification, 48 inch

Bit Axial Angular Tip Skew AttackNumber Position Position Radius Angle Angle

31 -0.75 0.0 18 23.5 4718 -0.75 120.0 18 23.5 4744 -0.75 211.8 18 23.5 471 -0.75 303.5 18 23.5 479 0 56.5 18 14.5 52

14 0.5 91.8 18 11.5 5222 1 148.2 18 6 52

85

Table A.2: continued


28 1.5 190.6 18 6 5235 2 240.0 18 6 5241 2.5 282.4 18 6 5248 3 331.8 18 6 524 3.5 21.2 18 6 52

12 4 77.6 18 6 5219 4.5 127.1 18 6 5225 5 169.4 18 6 5232 5.5 218.8 18 6 5238 6 261.2 18 6 5245 6.5 310.6 18 6 5251 7 352.9 18 6 526 7.5 35.3 18 6 52

10 8 63.5 18 6 5216 8.5 105.9 18 6 5223 9 155.3 18 6 5229 9.5 197.6 18 6 5236 10 247.1 18 6 5242 10.5 289.4 18 6 5249 11 338.8 18 6 525 11.5 28.2 18 6 52

13 12 84.7 18 6 5220 12.5 134.1 18 6 5226 13 176.5 18 6 5233 13.5 225.9 18 6 5239 14 268.2 18 6 5246 14.5 317.6 18 6 522 15 7.1 18 6 527 15.5 42.4 18 6 52

11 16 70.6 18 6 5217 16.5 112.9 18 6 5224 17 162.4 18 6 5230 17.5 204.7 18 6 5237 18 254.1 18 6 5243 18.5 296.5 18 6 523 19 14.1 18 6 52

86



47 19.5 324.7 18 6 5240 20 275.3 18 6 5234 20.5 232.9 18 6 5227 21 183.5 18 6 5221 21.5 141.2 18 6 5215 22 98.8 18 6 528 22.5 49.4 18 6 52

50 23 345.9 18 6 5254 23.5 14.1 18 -6 5298 24 324.7 18 -6 5291 24.5 275.3 18 -6 5285 25 232.9 18 -6 5278 25.5 183.5 18 -6 5272 26 141.2 18 -6 5266 26.5 98.8 18 -6 5259 27 49.4 18 -6 52

101 27.5 345.9 18 -6 5294 28 296.5 18 -6 5288 28.5 254.1 18 -6 5281 29 204.7 18 -6 5275 29.5 162.4 18 -6 5268 30 112.9 18 -6 5262 30.5 70.6 18 -6 5258 31 42.4 18 -6 5253 31.5 7.1 18 -6 5297 32 317.6 18 -6 5290 32.5 268.2 18 -6 5284 33 225.9 18 -6 5277 33.5 176.5 18 -6 5271 34 134.1 18 -6 5264 34.5 84.7 18 -6 5256 35 28.2 18 -6 52

100 35.5 338.8 18 -6 5293 36 289.4 18 -6 5287 36.5 247.1 18 -6 5280 37 197.6 18 -6 52

87



74 37.5 155.3 18 -6 5267 38 105.9 18 -6 5261 38.5 63.5 18 -6 5257 39 35.3 18 -6 52

102 39.5 352.9 18 -6 5296 40 310.6 18 -6 5289 40.5 261.2 18 -6 5283 41 218.8 18 -6 5276 41.5 169.4 18 -6 5270 42 127.1 18 -6 5263 42.5 77.6 18 -6 5255 43 21.2 18 -6 5299 43.5 331.8 18 -6 5292 44 282.4 18 -6 5286 44.5 240.0 18 -6 5279 45 190.6 18 -6 5273 45.5 148.2 18 -6 5265 46 91.8 18 -11.5 5260 46.5 56.5 18 -14.5 5269 47.25 0.0 18 -23.5 4782 47.25 120.0 18 -23.5 4795 47.25 211.8 18 -23.5 4752 47.25 303.5 18 -23.5 47

88

The following table contains actual measurements from a 48 inch wide cutter

head. Differences between this table and Table A.2 are the result of manufacturing

variation.

Table A.3: Actual lacing pattern measurements, 48 inch


31 -0.78 204.39 17.90 -47.60 57.6418 -0.72 114.92 17.99 -46.57 57.5944 -0.71 293.91 17.90 -55.75 61.651 -0.65 0.00 18.11 -50.30 61.299 0.00 55.43 18.08 -31.62 55.28

14 0.44 90.34 18.07 -28.52 54.4622 0.83 144.95 17.96 -18.20 53.5628 1.49 185.07 17.86 -9.05 51.4935 1.96 233.71 17.88 -12.15 51.5541 2.48 274.87 17.84 -11.28 52.2048 2.92 323.63 17.92 -15.17 52.254 3.58 21.20 17.99 -11.38 51.64

12 4.03 75.69 17.94 -12.89 51.6819 4.48 124.18 17.96 -12.61 53.3625 4.90 165.22 17.89 -11.93 52.4132 5.44 212.97 17.88 -12.16 52.0238 5.89 254.23 17.85 -10.92 51.2045 6.50 303.03 17.88 -8.71 51.8251 6.99 344.29 17.98 -12.89 52.336 7.61 34.70 17.97 -11.61 51.87

10 8.18 62.35 17.98 -8.15 51.8516 8.60 103.60 17.96 -7.27 52.0523 8.92 151.49 17.95 -11.17 52.6429 9.50 192.18 17.89 -10.90 51.8736 10.07 240.83 17.87 -9.96 52.5742 10.42 282.09 17.88 -14.56 52.2849 11.09 330.68 17.95 -14.22 52.905 11.57 28.12 17.97 -13.27 52.54

13 12.00 82.75 17.95 -12.69 51.9720 12.42 130.83 17.92 -11.27 51.8126 12.93 171.40 17.91 -9.73 51.22

89



33 13.42 220.06 17.90 -11.66 53.0739 14.05 261.06 17.86 -9.75 51.0546 14.52 309.47 17.86 -8.69 51.242 15.04 7.42 18.00 -10.68 52.217 15.44 42.16 17.96 -15.49 53.07

11 15.92 69.61 17.97 -14.51 53.1917 16.49 109.75 17.95 -8.93 51.6524 16.92 158.23 17.93 -9.39 51.9830 17.36 198.89 17.91 -11.03 51.8737 17.92 247.51 17.88 -11.59 52.5543 18.42 288.37 17.90 -14.42 51.413 18.90 14.21 17.99 -10.32 51.63

47 19.47 315.95 17.90 -11.85 51.0240 19.97 268.56 17.89 -10.71 53.3934 20.49 226.77 17.89 -13.49 52.7327 20.99 178.30 17.89 -7.96 50.9521 21.50 137.42 17.89 -7.96 51.1515 22.07 96.40 17.93 -9.47 51.948 22.45 48.76 17.93 -13.48 51.95

50 22.94 337.44 17.95 -13.21 53.5354 23.43 14.31 17.95 4.85 51.0898 23.93 316.08 17.87 6.21 50.8291 24.51 268.12 17.86 9.18 52.2085 24.91 226.78 17.89 4.87 51.3778 25.38 178.10 17.88 4.94 50.7772 25.93 137.03 17.90 5.71 50.5066 26.39 96.76 17.91 3.89 51.5859 27.05 48.16 17.93 8.53 51.57

101 27.39 337.08 17.96 1.92 52.1794 27.91 288.45 17.87 6.16 50.2488 28.40 247.36 17.86 8.22 51.3181 28.83 198.64 17.90 3.73 50.9875 29.33 158.36 17.95 4.42 52.4068 29.93 109.64 17.92 6.51 51.3862 30.38 69.44 17.94 2.96 52.5958 30.87 41.83 17.94 6.20 51.75

90



53 31.30 7.58 18.01 4.75 51.1697 31.83 308.72 17.88 3.17 50.9790 32.46 261.20 17.89 8.95 52.2984 32.83 219.33 17.91 3.74 51.1277 33.41 171.58 17.91 7.07 51.1571 33.86 130.63 17.92 6.98 51.4964 34.37 82.54 17.92 5.11 51.1056 34.87 28.61 17.96 5.82 52.25

100 35.41 330.81 17.96 2.36 53.2093 35.79 281.18 17.86 3.18 50.4687 36.38 240.39 17.89 7.14 52.0680 36.88 192.26 17.92 7.48 51.8074 37.33 150.85 17.95 6.16 51.0567 37.89 103.56 17.93 7.06 52.3761 38.45 63.01 17.95 9.15 53.0757 38.87 35.24 17.95 5.40 52.49

102 39.35 343.71 17.99 3.58 52.3396 39.87 301.94 17.90 4.11 51.3889 40.47 253.64 17.87 11.59 51.1783 40.89 212.66 17.92 5.35 51.8176 41.38 164.90 17.92 6.64 51.7470 41.85 123.62 17.94 5.71 51.6963 42.34 75.97 17.89 5.55 51.5355 42.93 21.56 17.96 8.10 51.8499 43.55 322.33 17.92 7.22 50.7392 43.96 274.51 17.86 9.04 51.9186 44.33 233.34 17.89 5.87 51.2479 44.82 185.16 17.90 6.74 51.7373 45.48 144.79 17.97 8.51 52.9865 46.12 89.72 18.02 30.62 54.6160 46.51 55.03 18.02 29.31 53.9169 47.15 114.78 17.98 52.31 60.8882 47.16 204.06 17.97 44.40 56.5695 47.21 294.29 17.99 46.83 58.9452 47.22 0.34 18.08 55.20 63.46

91

Appendix B

Analysis Code Listings

B.1 MATLAB/Octave Per-Pick Volume Calculation

Following are the MATLAB functions and routines used to calculate the vol-

ume removed by each bit on a cutter head. The indented list, shown in Table B.1,

presents the call structure for functions used in calculating per-pick volume removal.

Code and related information for each of the functions are included in the subsequent

listings.

Table B.1: Function call structure for volume calculation

CutterApp.m

↪→ MainVolume.m

↪→ BitVolume.m

↪→ BitArea.m

↪→ Dominance.m

↪→ SectPt.m

93

Listing B.1: Main volume function - cutter head specific1 % CutterApp .m

%% SYNOPSIS% This s c r i p t c a l c u l a t e s per−pick−volume removal f o r each p ick o f a g iven% cu t t e r head . The s p e c i f i c a t i o n s f o r the cu t t e r head are pu l l e d from a

6 % comma de l im i t ed f i l e . Operat iona l parameters are d i r e c t l y ass i gned to% va r i a b l e s wi th in t h i s f i l e . Resu l t s are wr i t t en to a% constant−column−width t e x t f i l e .%% ARGUMENTS

11 % none%% INPUT% cu t t e r head s p e c i f i c a t i o n f i l e (comma de l im i t ed )% − column 1 = ax i a l p o s i t i on

16 % − column 2 = angular po s i t i on% − column 3 = arc d i s t ance po s i t i on% − column 4 = b i t t i p rad ius% − column 5 = skew ang le% − column 6 = at tack ang le

21 % − column 7 = b i t number ( from eng ineer ing drawings )% − column 8 = b i t type%% OUTPUT% column−a l i gned t e x t f i l e

26 % − column 1 = ax i a l index% − column 2 = per−pick−volume% − column 3 = ax i a l p o s i t i on% − column 4 = angular po s i t i on%

31 % OTHER VARIABLES% anaVar : de s c r i b ed in f o l l ow i n g comments% desVar : [ a x i a l p o s i t i o n , angle around drum , t i p r a d i u s ]% ppVol : [ b i t i n d e x , b i t vo lume , a x i a l p o s i t i o n , drum angle ]%

36

clear a l l ;close a l l ;

41 % ana l y s i s v a r i a b l e sVf = 8 ; % 1 forward v e l o c i t y ( f t /min) ranges from 10 to 40Wr = 150 ; % 2 angular v e l o c i t y o f c u t t i n g drum ( rev /min)Za = 75 ∗ pi /180 ; % 3 ang le o f impact a f f e c t e d zone ( rad )Ro = 0 . 0 8 ; % 4 t i p rad ius o f f s e t ( in )

46 Dc = 8 . 0 ; % 5 depth o f cut ( in )Rn = 18 ; % 6 nominal b i t rad ius ( in )Tr = 0 ; % 7 rad i a l p o s i t i o n a l t o l e rance ( in )Ta = 0 ; % 8 ax i a l p o s i t i o n a l t o l e rance ( in )Tc = 0 ; % 9 c i r cumf e r en t i a l p o s i t i o n a l t o l e rance ( in )

51 Gn = 0 . 5 ; % 10 mean gap between b i t s − pro j e c t ed onto drum ax i ss tep = 10 ; % 11 number o f vo lumetr i c i n t e g r a t i on s t ep snEB = 4 ; % 12 number o f edge b i t s per s i d e

anaVar = [ Vf ;Wr; Za ;Ro ;Dc ;Rn ; Tr ;Ta ; Tc ;Gn; s tep ;nEB ] ;56

% des ign v a r i a b l e s [ a x i a l p o s i t i o n , angle around drum , t i p r a d i u s ]inData = dlmread( ’ 48 Locator 2005 −08−31. csv ’ ) ;inData = sort rows ( inData , 1 ) ;

desVar = [ inData ( : , 1 ) , inData ( : , 2 ) , inData ( : , 4 ) ] ;61 nBits = s ize ( desVar , 1) ;

% ppVol = [ b i t i n d e x , b i t vo lume , a x i a l p o s i t i o n , drum angle ]ppVol = MainVolume ( desVar , anaVar ) ;ppVol = sor t rows ( ppVol , [ 3 , 4 ] ) ;

66

save −a s c i i 48 l o ca to r−ppvol 2005 −08−31. txt ppVol

disp ( ’ b i t i nd ex bit vo lume a x i a l p o s i t i o n drum angle ’ ) ;disp ( ppVol ) ;

94

Listing B.2: Main Volume Calculation Routine% MainVolume .m%% SYNOPSIS% This func t i on c a l c u l a t e s volume of mater ia l removed by each cu t t e r ,

5 % neg l e c t i n g forward movement o f machine wh i l e b i t i s engaged%% ARGUMENTS% desVar : an array o f b i t po s i t i on s , SORTED BY AXIAL POSITION, con s i s t i n g o f% − column 1 = ax i a l p o s i t i on o f b i t , measured from edge o f sk in

10 % − column 2 = angular po s i t i on o f b i t measured from drum ’ s weld seam% − column 3 = rad ius o f b i t t i p from drum ax i s% anaVar : a vec tor o f machine parameters (more d e t a i l s below )%% OUTPUT

15 % volArray : An array o f b i t s , by index , wi th volume , a x i a l pos i t i on , and drum% ang le . Inc ludes a l l b i t s , but has NaN va lue s f o r b i t s t ha t% exper ience edge e f f e c t s ( edge b i t s , and b i t s t ha t dominate edge% b i t s ) .% − column 1 = index based on increa s ing a x i a l p o s i t i on

20 % − column 2 = per−pick−volume fo r each b i t% − column 3 = ax i a l p o s i t i on o f b i t , measured from edge o f sk in% − column 4 = angular po s i t i on o f b i t measured from drum ’ s weld seam%% OTHER VARIABLES

25 % bi tPa t t e rn : [ a x i a l p o s i t i o n , angle around drum , t i p r ad i u s , machine advance ]% ordBi ts : [ a x i a l p o s i t i o n i n d e x , angle around drum ]%% REVISION HISTORY% changed 2005−01−28:

30 % removed automatic d e t e c t i on o f number o f edge b i t s% added a var i ab l e , s e t by the user , f o r the number o f edge b i t s%% changed 2005−12−13:% changed volArray from

35 % [ ax i a l i nde x , b i t vo lume , a x i a l p o s i t i o n , machine advance ]% to% [ ax i a l i nde x , b i t vo lume , a x i a l p o s i t i o n , drum angle ]%% changed 2006−01−26:

40 % changed return va lue to inc lude a l l b i t s , wi th NaN for the volume% of non−c a l c u l a t e d edge b i t s%

function volArray = MainVolume ( desVar , anaVar )45

% name the ana l y s i s v a r i a b l e sVf = anaVar (1 ) ; % 1 forward v e l o c i t y ( f t /min) ranges from 10 to 40Wr = anaVar (2 ) ; % 2 angular v e l o c i t y o f c u t t i n g drum ( rev /min)Za = anaVar (3 ) ; % 3 ang le o f impact a f f e c t e d zone ( rad )

50 Ro = anaVar (4 ) ; % 4 t i p rad ius o f f s e t ( in )Dc = anaVar (5 ) ; % 5 depth o f cut ( in )Rn = anaVar (6 ) ; % 6 nominal b i t rad ius ( in )Tr = anaVar (7 ) ; % 7 rad i a l p o s i t i o n a l t o l e rance ( in )Ta = anaVar (8 ) ; % 8 ax i a l p o s i t i o n a l t o l e rance ( in )

55 Tc = anaVar (9 ) ; % 9 c i r cumf r en t i a l p o s i t i o n a l t o l e rance ( in )Gn = anaVar (10) ; % 10 average gap between b i t s ( p ro j e c t ed onto drum ax i s )s tep = anaVar (11) ; % 11 number o f vo lumetr i c i n t e g r a t i on s t ep snEB = anaVar (12) ; % 12 number o f edge b i t s

60 nBits = s ize ( desVar , 1 ) ;

% ana l y s i s f unc t i on s65

% forward v e l o c i t y ( in /min)anaParam . Vi = Vf ∗ 12 ;

% rad/min70 anaParam .Wa = Wr∗2∗pi ;

95

% depth o f cutanaParam .Dc = Dc ;

75 % advance per r e vo l u t i on ( in )anaParam . Ar = anaParam . Vi/Wr;

% ang le from the nega t i v e o f the advance d i r e c t i on to the cut e x i t ang leanaParam .bMax = acos ( (Rn−Dc) /Rn) + pi /2 ;

80

% maximum domination index d i f f e r e n c eDdMax = (Rn + Tr) − (Rn−Tr) ∗( sin ( pi − . . .

asin ( anaParam . Ar/(Rn−Tr) ∗ sin ( anaParam .bMax) ) − . . .anaParam .bMax) / sin ( anaParam .bMax) ) ;

85 maxDomDist = DdMax/tan ( pi/2 − Za/2) ;anaParam . maxDomIndex = ce i l (maxDomDist/(Gn − 2∗Ta) ) ;

% ang le o f s i d e o f cut p r o f i l e in the f l a t p laneanaParam . domAngle = pi/2 − Za /2 ;

90

% s lope o f s i d e o f cut p r o f i l e in the f l a t p laneanaParam . domSlope = tan ( anaParam . domAngle ) ;

% l i s t o f r o t a t i on ang l e s at which to eva lua t e b i t cross−s e c t i on s95 s ta r tAng l e = pi /2 ;

stepAng = (anaParam .bMax − s ta r tAng l e ) /( s tep − 1) ;eva lAngles = star tAng l e : stepAng : anaParam .bMax ;

anaParam . eva lAng les = evalAngles ’ ; % ( transposed )

100 % arc d i s t ance between b i t cross−s e c t i on sanaParam . stepArc = Rn ∗ stepAng ;

% l i s t o f r a d i i to a spha l t sur face , a long each eva lAng les d i r e c t i onanaParam . surfRad = ( (Rn + Ro) − Dc) . / cos ( anaParam . eva lAng les − pi /2) ;

105

% edges o f cut ( l e f tmo s t and r igh tmos t b i t s )anaParam . edge = [min( desVar ( : , 1 ) ) ,max( desVar ( : , 1 ) ) ] ;

110

% add machine advance d i s t ance at each b i t ’ s impact% [ a x i a l d i s t , drum angle , t i p r ad i u s , machine advance ]

b i tPat te rn = [ desVar , ( anaParam . Vi ∗ desVar ( : , 2 ) / anaParam .Wa) ] ;

115 % add t i p rad ius o f f s e t to b i t r a d i i ( accounts f o r rounded t i p on b i t )b i tPat te rn ( : , 3 ) = b i tPat te rn ( : , 3 ) + Ro ;

% add a x i a l i n d e x and so r t b i t s by order o f impact% ordBi ts = [ ax i a l i nde x , drum angle ]

120 ordBit s = [ ( 1 : nBits ) ’ , b i tPat t e rn ( : , 2 ) ] ;o rdBit s = sor t rows ( ordBits , 2 ) ;

125 % step through b i t s in order o f impactbitVolArray = [ ] ;lastAdvance = 0 ;for j = 1 : nBits

130 % get next impact b i t and make current (move b i t forward to 2nd drum rev )currBi t Index = ordBit s ( j , 1 ) ; % next b i t a x i a l i n d e xcurrBitData = bi tPat t e rn ( currBitIndex , : ) ; % next b i t data rowcurrBitData (4 ) = currBitData (4 ) + anaParam . Ar ; % move next b i t forward

135 % i f not an edge b i t , c a l l volume func t ion and add to b i t volume arrayi f ( currBi t Index > nEB + 1) && ( currBi t Index < nBits−nEB)

bi tVo l = BitVolume ( b i tPattern , currBit Index , currBitData , anaParam) ;bitVolArray = [ bitVolArray ;

currBit Index , bitVol , currBitData (1 ) , currBitData (2 ) ] ;140 else

bitVolArray = [ bitVolArray ;currBit Index , NaN, currBitData (1 ) , currBitData (2 ) ] ;

end

96

145 % update b i t pa t t e rn to new po s i t i on o f current b i tb i tPat te rn ( currBit Index , : ) = currBitData ;

end

150 volArray = bitVolArray ;

97

Listing B.3: Single Pick Volume Routine% BitVolume .m%% SYNOPSIS

4 % Receives data f o r a s i n g l e b i t l o ca t i on , and re turns i t s cut volume .%% ARGUMENTS% b i tPa t t e rn : an array o f b i t po s i t i on s , sor t ed by a x i a l p o s i t i on :% − column 1 = ax i a l p o s i t i on o f b i t , measured from edge o f sk in

9 % − column 2 = angular po s i t i on o f b i t measured from drum ’ s weld seam% − column 3 = rad ius o f b i t t i p from drum ax i s ( ye t to come)% − column 4 = machine advance d i s t ance s ince s t a r t o f t h i s r e vo l u t i on% currBi t Index : index po s i t i on o f current b i t in b i tPa t t e rn ( prev rev )% currBitData : l i s t o f data f o r current b i t in advanced po s i t i on ( curr rev )

14 % anaParam : l i s t o f ana l y s i s paramaters and func t i on s f o r the machine%% OUTPUT% bi tVo l : volume of mater ia l removed by the b i t at currentBi t Index%

19 % OTHER VARIABLES% se c tB i t s : A sub s e t o f the main b i tPa t t e rn data , conta in ing b i t s w i th in% dominance range o f the current b i t . This avo ids having to search% a l l b i t s f o r dominance .%

24

function bi tVo l = BitVolume ( b i tPattern , currBit Index , currBitData , anaParam)

l im i tL = currBi t Index − 2 ∗ anaParam . maxDomIndex ;29 i f l im i tL < 1

l im i tL = 1 ;endl imitR = currBi t Index + 2 ∗ anaParam . maxDomIndex ;i f l imitR > s ize ( b i tPattern , 1 )

34 l imitR = s ize ( b i tPattern , 1 ) ;ends e c tB i t s = b i tPat te rn ( l im i tL : l imitR , : ) ;

% open f i l e f o r debugging39 %f i d = fopen ( ’ area debug1 . t x t ’ , ’ a+t ’ ) ;

%f p r i n t f ( f i d , ’%10.6 f \ t ’ , b i tPa t t e rn ( currBitIndex , 1 ) ) ;

% c a l c u l a t e current b i t ’ s volumebi tVo l = 0 ;

44 currBitArea = 0 ;for i = 1 : ( s ize ( anaParam . evalAngles , 1 ) )

% get next b i t ’ s volumenextBitArea = BitArea ( s e c tB i t s , currBitData , anaParam , . . .

49 anaParam . eva lAng les ( i ) , anaParam . surfRad ( i ) ) ;

% cen t r a l d i f f e r e n c e i n t e g r a t e f o r t h i s segment o f volumebi tVo l = bitVol + anaParam . stepArc ∗ ( nextBitArea + currBitArea ) /2 ;

54 % s h i f t next area to current areacurrBitArea = nextBitArea ;

% pr in t debug in f o%f p r i n t f ( f i d , ’%10.6 f \ t ’ , currBitArea ) ;

59

end

% c l o s e debugging f i l e%f p r i n t f ( f i d , ’\n ’ ) ;

64 %f c l o s e ( f i d ) ;

98

Listing B.4: Calculate Single Pick Cut Cross-Section1 % BitArea .m

%% SYNOPSIS% This func t i on r e c e i v e s a b i t array and current b i t index , and re turns the% cross−s e c t i o n a l area o f the mater ia l removal zone f o r the s p e c i f i e d b i t at

6 % the s p e c i f i e d ang le .%% Green ’ s Theorem%% 1 n / \

11 % area = − SUM ( x [ i ] y [ i +1] − x [ i +1] y [ i ] )% 2 i=1 \ /%% re f e rence page 9 o f l a b notebook fo r more d e t a i l s%

16 % ARGUMENTS% se c tB i t s : an array o f b i t po s i t i on s , so r t ed by a x i a l p o s i t i on :% − column 1 = ax i a l p o s i t i on o f b i t , measured from edge o f sk in% − column 2 = angular po s i t i on o f b i t measured from drum ’ s weld seam% − column 3 = rad ius o f b i t t i p from drum ax i s ( ye t to come)

21 % − column 4 = machine advance d i s t ance s ince s t a r t o f t h i s r e vo l u t i on% currBitData : l i s t o f data f o r current b i t in advanced po s i t i on ( curr rev )% anaParam : l i s t o f ana l y s i s paramaters and func t i ons f o r the machine% currAngle : ang le o f r o t a t i on fo r current b i t% surfRad : rad ius to a spha l t sur face , a long currAngle

26 %% OUTPUT% bitArea : cross−s e c t i o n a l area o f the current b i t ’ s cut , at the current% cu t t e r head ang le o f r o t a t i on%

31 % OTHER VARIABLES% b i t S e c t 3 : sub s e t o f s e c tB i t s , conta in ing b i t s not dominated by any other b i t% in sec tB i t s , or by current b i t% sec tP t s : s e t o f x , y coord ina te s f o r i n t e r s e c t i o n po in t s between cut% p r o f i l e s o f a l l b i t s ( f o r green ’ s theorem )

36 %

function bitArea = BitArea ( s e c tB i t s , currBitData , anaParam , currAngle , surfRad )

41

% i n i t i a l i z e v a r i a b l e snSect1 = s ize ( s e c tB i t s , 1 ) ;s e c tPt s = [ currBitData (1 ) , currBitData (3 ) ] % coords o f 1 s t v e r t e x [ x , y ]bitArea = 0 ;

46

% f ind r e l a t i v e depths (Dr) f o r the sub s e t o f b i t sR = se c tB i t s ( : , 3 ) ; % co l vec to r o f r a d i iA = currBitData (4 ) − s e c tB i t s ( : , 4 ) ; % co l vec to r o f advance d i s t ance sb = currAngle ;

51 Dr = R .∗ ( sin ( pi − asin (A . / R .∗ sin (b) ) − b) / sin (b) ) ;b i tS e c t 1 = [ s e c tB i t s , Dr ] ;b i tS e c t 3 = [ ] ;

56

% Generate a sub s e t o f non dominated b i t sfor i = 1 : nSect1

domChk = 0 ;61 % f ind domination l im i t s

lowLim = i − anaParam . maxDomIndex ;hiLim = i + anaParam . maxDomIndex ;i f i <= anaParam . maxDomIndex

lowLim = 1 ;66 e l s e i f i >= ( nSect1 − anaParam . maxDomIndex)

hiLim = nSect1 ;end

% f ind dominated b i t s71 for j = lowLim : hiLim

99

i f ( i ˜= j )i f Dominance ( b i tS e c t 1 ( i , [ 1 , 5 ] ) , b i tS e c t 1 ( j , [ 1 , 5 ] ) , . . .

anaParam . domAngle ) == 1domChk = 1 ;

76 break ;end

endend

81 % wri t e non−dominated b i t s to an arrayi f domChk == 0

i f b i tS e c t 1 ( i , 5 ) > surfRadb i tS e c t 3 = [ b i tS e c t 3 ; b i tS e c t 1 ( i , : ) ] ;

else86 % inc lude the b i t anyway fo r the qu ick and d i r t y method ( to

% be changed l a t e r )b i tS e c t 3 = [ b i tS e c t 3 ; b i tS e c t 1 ( i , : ) ] ;

endend

91

end

96 nSect2 = s ize ( b i tSect3 , 1 ) ;lastDomChk = 0 ;

for i = 1 : nSect2

101 % righ t−most b i t ( t h i s b i t to the r i g h t o f the current b i t , and not% dominated by the current b i t )

i f b i tS e c t 3 ( i , 1 ) > currBitData (1 )i f Dominance ( b i tS e c t 3 ( i , [ 1 , 5 ] ) , currBitData ( [ 1 , 3 ] ) , . . .

anaParam . domAngle ) == 0106 % in t e r s e c t i o n − t h i s b i t and new b i t

newSectPt = SectPt ( currBitData ( [ 1 , 3 ] ) , b i tS e c t 3 ( i , [ 1 , 5 ] ) , . . .anaParam . domSlope ) ;

%i f newSectPt (2) < surfRad% change the i n t e r s e c t i o n po in t to be with the sur face

111 % of the aspha l t , ra ther than with the ne ighbor ing b i t%endbitArea = bitArea . . .

+ ( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗newSectPt (2 ) . . .− newSectPt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;

116 s e c tPt s = [ s e c tPt s ; newSectPt ] ;break ;

endend

121 % non−dominated b i t si f Dominance ( b i tS e c t 3 ( i , [ 1 , 5 ] ) , currBitData ( [ 1 , 3 ] ) , . . .

anaParam . domAngle ) == 0

% l e f t −most b i t ( t h i s b i t not dominated , next b i t dominated , or126 % a f t e r the current b i t )

i f (Dominance ( b i tS e c t 3 ( i +1 , [ 1 , 5 ] ) , currBitData ( [ 1 , 3 ] ) , . . .anaParam . domAngle ) == 1) | | ( b i tS e c t 3 ( i +1 ,1) > currBitData (1 ) ) ;

% in t e r s e c t i o n − t h i s b i t and new b i t131 newSectPt = SectPt ( b i tS e c t 3 ( i , [ 1 , 5 ] ) , currBitData ( [ 1 , 3 ] ) , . . .

anaParam .domSlope) ;

b itArea = bitArea + . . .( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗newSectPt (2 ) − . . .newSectPt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;

136 s e c tPt s = [ s e c tPt s ; newSectPt ] ;

% in t e r s e c t i o n − t h i s b i t and next b i tnewSectPt = SectPt ( b i tS e c t 3 ( i , [ 1 , 5 ] ) , b i tS e c t 3 ( i +1 , [ 1 , 5 ] ) , . . .

anaParam . domSlope ) ;141 bitArea = bitArea + . . .

( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗newSectPt (2 ) − . . .

100

newSectPt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;s e c tPt s = [ s e c tPt s ; newSectPt ] ;

146 end

% dominated b i t selse

% current b i t has l e f t overhang151 i f i == 1

edgePt = [ anaParam . edge (1 ) , b i tS e c t 3 ( i , 5 ) ] ;b i tArea = bitArea + . . .

( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗ edgePt (2 ) − . . .edgePt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;

156 s e c tPt s = [ s e c tPt s ; edgePt ] ;

% current b i t has r i g h t overhange l s e i f i == nSect2

edgePt = [ anaParam . edge (2 ) , b i tS e c t 3 ( i , 5 ) ] ;161 bitArea = bitArea + . . .

( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗ edgePt (2 ) − . . .edgePt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;

s e c tPt s = [ s e c tPt s ; edgePt ] ;

166 else% ver t e x o f t h i s b i tnewSectPt = b i tSe c t 3 ( i , [ 1 , 5 ] ) ;b itArea = bitArea + . . .

( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗newSectPt (2 ) − . . .171 newSectPt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;

s e c tPt s = [ s e c tPt s ; newSectPt ] ;

% in t e r s e c t i o n − t h i s b i t and next b i tnewSectPt = SectPt ( b i tS e c t 3 ( i , [ 1 , 5 ] ) , b i tS e c t 3 ( i +1 , [ 1 , 5 ] ) , . . .

176 anaParam . domSlope ) ;b itArea = bitArea + . . .

( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗newSectPt (2 ) − . . .newSectPt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;

s e c tPt s = [ s e c tPt s ; newSectPt ] ;181 end

lastDomChk = 1 ;end

end186

% complete s ec tP t s loop191 newSectPt = sec tPt s ( 1 , : ) ;

b i tArea = bitArea + . . .( s e c tPt s ( s ize ( sectPts , 1) , 1) ∗newSectPt (2 ) − . . .newSectPt (1 ) ∗ s e c tPt s ( s ize ( sectPts , 1) , 2) ) ;

b itArea = bitArea /2 ;196 s e c tPt s = [ s e c tPt s ; newSectPt ] ;

% open f i l e to wr i t ef i d = fopen ( ’ s e c tPt s . txt ’ , ’ a ’ ) ;for i = 1 : s ize ( sectPts , 1 )

201 fpr intf ( f i d , ’%f \ t%f \n ’ , s e c tPt s ( i , : ) ) ;endfprintf ( f i d , ’ \n ’ ) ;fc lose ( f i d ) ;

206 % modify l i s t o f s e c tP t s to account f o r a spha l t sur face emergencei f min( s e c tPt s ( : , 2 ) ) < surfRad && s ize ( sectPts , 1) > 2

newSectPts = [ ] ;for i = 1 : ( s ize ( sectPts , 1 ) − 1)

211

p1 = sec tPt s ( i , : ) ;p2 = sec tPt s ( i +1 , : ) ;

i f surfRad > p1 (2) && surfRad > p2 (2)

101

216 %do nothinge l s e i f surfRad < p1 (2) && surfRad < p2 (2)

%add poin t p1 to the new arraynewSectPts = [ newSectPts ; p1 ] ;

e l s e i f p1 (2) > surfRad && surfRad > p2 (2)221 %f ind the i n t e r s e c t i o n between surfRad and vec tor [ p1 , p2 ]

newSectX = p1 (1) + . . .( surfRad − p1 (2) ) ∗( p2 (1 ) − p1 (1) ) /( p2 (2 ) − p1 (2) ) ;

%add p1 and i n t e r s e c t i o n to the new array in ordernewSectPts = [ newSectPts ; p1 ; [ newSectX , surfRad ] ] ;

226 e l s e i f p2 (2) > surfRad && surfRad > p1 (2)%f ind the i n t e r s e c t i o n between surfRad and vec tor [ p1 , p2 ]newSectX = p1 (1) + . . .

( surfRad − p1 (2) ) ∗( p2 (1 ) − p1 (1) ) /( p2 (2 ) − p1 (2) ) ;%add i n t e r s e c t i o n to the new array

231 newSectPts = [ newSectPts ; [ newSectX , surfRad ] ] ;else

% errorend

236 end

% implement green ’ s theorem , i f any non−dominated area e x i s t si f s ize ( newSectPts , 1 ) == 0

bitArea = 0 ;241 else

newSectPts = [ newSectPts ; newSectPts ( 1 , : ) ] ;newBitArea = 0 ;for i = 1 : ( s ize ( newSectPts , 1) − 1)

% port ion o f formula i n s i d e the sum : x [ i ] y [ i +1] − x [ i +1] y [ i ]246 newBitArea = newBitArea + . . .

( newSectPts ( i , 1) ∗newSectPts ( i +1 ,2) − . . .newSectPts ( i +1 ,1)∗newSectPts ( i , 2 ) ) ;

endbitArea = newBitArea / 2 ;

251

end

end

102

Listing B.5: Determine Pick Domination1 % Dominance .m

%% SYNOPSIS% This func t i on take s the po s i t i on o f two b i t t i p s , and determines whether% one i s dominated by the other . The f i r s t b i t i s dominated i f the second b i t

6 % has removed i t ’ s mater ia l on an e a r l i e r pass .%% ARGUMENTS% pos1 : Coordinate s e t f o r b i t 1 t i p l o c a t i on% − element 1 = ax i a l ( x ) t i p po s i t i on o f b i t 1

11 % − element 2 = rad i a l ( y ) t i p po s i t i on o f b i t 1% pos2 : Coordinate s e t f o r b i t 1 t i p l o c a t i on% − element 1 = ax i a l ( x ) t i p po s i t i on o f b i t 2% − element 2 = rad i a l ( y ) t i p po s i t i on o f b i t 2% domAngle : S lope o f the cut p r o f i l e ( r i s e /run in the p lane o f the current

16 % b i t s rad ius vec to r )%% OUTPUT% dominated : i n t e g e r va lue% 0 = b i t 1 i s NOT dominated by b i t 2

21 % 1 = b i t 1 i s dominated by b i t 2%

function dominated = Dominance ( pos1 , pos2 , domAngle )

26 compAngle = atan2 ( ( pos2 (2 ) − pos1 (2 ) ) , ( pos2 (1 ) − pos1 (1 ) ) ) ;

i f ( compAngle > domAngle ) && ( compAngle < ( pi − domAngle ) )dominated = 1 ;

else31 dominated = 0 ;

end

103

Listing B.6: Pick Path Intersection Coordinates% SectPt .m%

3 % SYNOPSIS% This func t i on take s the po s i t i on o f two b i t t i p s , and f i n d s the coord ina te s% the coord ina te s at which the cut p r o f i l e s o f each b i t i n t e r s e c t .% NOTE: Bit arguments ( b i t p o s i t i on vec tor o b j e c t s ) must be supp l i e d in l e f t% to r i g h t order ( i . e . the b i t wi th the sma l l e r x must be passed as

8 % the f i r s t argument )%% ARGUMENTS% ( r i s e /run in the p lane o f the current b i t s rad ius vec tor )% pos1 : Coordinate s e t f o r b i t 1 t i p l o c a t i on

13 % − element 1 = ax i a l ( x ) t i p po s i t i on o f b i t 1% − element 2 = rad i a l ( y ) t i p po s i t i on o f b i t 1% pos2 : Coordinate s e t f o r b i t 1 t i p l o c a t i on% − element 1 = ax i a l ( x ) t i p po s i t i on o f b i t 2% − element 2 = rad i a l ( y ) t i p po s i t i on o f b i t 2

18 % domAngle : S lope o f the cut p r o f i l e ( r i s e /run in the p lane o f the current% b i t s rad ius vec to r )%% OUTPUT% sectArray : x−y coord inate po s i t i on o f the i n t e r s e c t i o n between the two

23 % b i t s ’ cut p r o f i l e%

function sectArray = SectPt ( pos1 , pos2 , cutS lope )28

% swi tch the s i gn o f the cutS lope , s ince the l e f t b i t i s always f i r s tcutS lope = −cutS lope ;

% f ind the x coord inate33 x = ( ( pos2 (2 ) + cutS lope ∗pos2 (1 ) ) − ( pos1 (2 ) − . . .

cutS lope ∗pos1 (1 ) ) ) /(2∗ cutS lope ) ;

% the y coord inate i s found from the x coord inatey = cutS lope ∗x + ( pos1 (2 ) − cutS lope ∗pos1 (1 ) ) ;

38

% return the coord inate arraysectArray = [ x , y ] ;

104

B.2 SolidWorks/VBA Per-Pick Volume Calculation

Listing B.7: Visual Basic, SolidWorks Automation MacroOption Explicit

’ Visua l Basic macro us ing the SolidWorks API to manipulate a s o l i d model’ and f i nd per−p ick volume

5

Sub main ( )

Dim swApp As ObjectDim swModelDoc As ModelDoc2

10 Dim swFeature As FeatureDim swFeatureData As TablePatternFeatureData

Dim dblSwPoints (69 , 2) As DoubleDim varSwPoints As Variant

15 Dim varMassProp As VariantDim dblVolume (69) As DoubleDim i As Integer

Dim dblMachAdv As Double20 Dim dblAdvOffset As Double

Dim db lAx ia lO f f s e t As Double

Dim boo l s t a tu s As BooleanDim l n g s t a tu s As Long

25 Dim l ngEr ro r s As LongDim lngWarnings As Long

varSwPoints = MakePointArray ( dblSwPoints )dblMachAdv = 1 .6 ∗ 0 .0254

30 dblAdvOffset = 6 .5 ∗ 0 .0254db lAx i a lO f f s e t = 0 .5 ∗ 0 .0254

’ g e t a pp l i c a t i on and model35 Set swApp = Appl i ca t ion . SldWorks

Set swModelDoc = swApp . ActiveDoc ( )

’ g e t cut pa t t e rn data o b j e c tSet swFeature = swModelDoc . FeatureByName ( ”CutPattern” )

40 Set swFeatureData = swFeature . GetDe f in i t i on

’ s e t cut pa t t e rn l o c a t i on s f o r f i r s t r e v o l u t i onswFeatureData . pointArray = varSwPointsboo l s t a tu s = swFeature . Mod i fyDe f in i t i on ( swFeatureData , swModelDoc , Nothing )

45

For i = 0 To UBound( dblSwPoints ) − 1

I f i = 9 ThenDebug . Print ”paused at p ick #” & 9

50 End I f

’ move machine advance planeswModelDoc . Parameter ( ”MachAdv@MachAdvPlane” ) . SystemValue =

55 dblSwPoints ( i , 1) + dblMachAdv + dblAdvOffset

’ s i d e s h i f t cut nega t i v e s o l i dswModelDoc . Parameter ( ”AxialPos it ion@CutNegat iveSketch ” ) . SystemValue =

dblSwPoints ( i , 0) + db lAx i a lO f f s e t60

’ r e b u i l d modelboo l s t a tu s = swModelDoc . EditRebui ld3boo l s t a tu s = swModelDoc . EditRebui ld3I f boo l s t a tu s = False Then

65 Debug . Print ” r ebu i l d e r r o r at p ick #” & iDebug . Printboo l s t a tu s = True

105

End I f

70 ’ g e t volumevarMassProp = swModelDoc . Extension . GetMassPropert ies (1 , l n g s t a tu s )dblVolume ( i ) = varMassProp (3 )

’ pause execu t ion fo r s p e c i a l b i t s75 I f i = 22 Or i = 49 Then

Debug . Print ”paused f o r model check ing at p ick #” & iDebug . Print

End I f

80 ’ ad ju s t pa t t e rn array f o r new cut po s i t i onvarSwPoints ( i ∗ 3 + 1) = varSwPoints ( i ∗ 3 + 1) + dblMachAdvswFeatureData . pointArray = varSwPoints

’ r e s e t cut pa t t e rn l o c a t i on s85 boo l s t a tu s = swFeature . Mod i fyDe f in i t i on ( swFeatureData ,

swModelDoc , Nothing )I f boo l s t a tu s = False Then

Debug . Print ” pattern update e r r o r at p ick #” & iDebug . Print

90 boo l s t a tu s = TrueEnd I f

Next i

95 Debug . Print ” ax ia l , machadv , volume”For i = 0 To UBound( dblVolume ) − 1

Debug . Print dblSwPoints ( i , 0) & ” , ” & dblSwPoints ( i , 1) &” , ” & dblVolume ( i )

Next i100

End Sub

105 Function MakePointArray ( dblArray ( ) As Double ) As Variant’ db lArray i s an empty , s i z e d array’ array va lue s i n s e r t e d here as a quick−and−d i r t y method’ f o r g r ea t e r f l e x i b i l i t y , the va lue s shou ld be loaded from a f i l e

110 Dim dblPointArray ( ) As DoubleDim intRows As DoubleDim i n tCo l s As DoubleDim i As IntegerDim j As Integer

115

’ the passed array i s measured to r e s i z e a matching output arrayintRows = UBound( dblArray , 1) + 1in tCo l s = UBound( dblArray , 2) + 1ReDim dblPointArray ( intRows ∗ i n tCo l s − 1)

120

’ machine advance po s i t i o n s f o r each pick ’ s impact event ( meters )dblArray (0 , 0) = 0.01905dblArray (1 , 0) = 0.74295dblArray (2 , 0) = 0.20955

125 dblArray (3 , 0) = 0.55245dblArray (4 , 0) = 0.03175dblArray (5 , 0) = 0.73025dblArray (6 , 0) = 0.32385dblArray (7 , 0) = 0.46355

130 dblArray (8 , 0) = 0.12065dblArray (9 , 0) = 0.64135dblArray (10 , 0) = 0#dblArray (11 , 0) = 0.762dblArray (12 , 0) = 0.22225


140 dblArray (18 , 0) = 0.13335

106

dblArray (19 , 0) = 0.62865dblArray (20 , 0) = 0#dblArray (21 , 0) = 0.762dblArray (22 , 0) = 0.23495










190 dblArray (68 , 0) = 0.17145dblArray (69 , 0) = 0.59055

’ a x i a l p o s i t i on o f each p ick ( meters )dblArray (0 , 1) = 0#

195 dblArray (1 , 1) = 0#dblArray (2 , 1) = 0.0011611dblArray (3 , 1) = 0.0011611dblArray (4 , 1) = 0.0023223dblArray (5 , 1) = 0.0023223



210 dblArray (16 , 1) = 0.0092891dblArray (17 , 1) = 0.0092891dblArray (18 , 1) = 0.0104503dblArray (19 , 1) = 0.0104503

107

dblArray (20 , 1) = 0.0116114215 dblArray (21 , 1) = 0.0116114

dblArray (22 , 1) = 0.0127726dblArray (23 , 1) = 0.0127726dblArray (24 , 1) = 0.0139337dblArray (25 , 1) = 0.0139337









260 dblArray (66 , 1) = 0.0383177dblArray (67 , 1) = 0.0383177dblArray (68 , 1) = 0.0394789dblArray (69 , 1) = 0.0394789

265 ’ z coord inate requ i red f o r SolidWorks APIdblArray (0 , 2) = 0#dblArray (1 , 2) = 0#dblArray (2 , 2) = 0#dblArray (3 , 2) = 0#

270 dblArray (4 , 2) = 0#dblArray (5 , 2) = 0#dblArray (6 , 2) = 0#dblArray (7 , 2) = 0#dblArray (8 , 2) = 0#



285 dblArray (19 , 2) = 0#dblArray (20 , 2) = 0#

108

dblArray (21 , 2) = 0#dblArray (22 , 2) = 0#dblArray (23 , 2) = 0#










335 dblArray (69 , 2) = 0#

’ arrange the po s i t i on array in to a s i n g l e vec tor’ ( r equ i red by SolidWorks API)For i = 0 To intRows − 1

340 For j = 0 To in tCo l s − 1dblPointArray ( i ∗ i n tCo l s + j ) = dblArray ( i , j )

Next jNext i

345 ’ re turn vec tor as var ian t typeMakePointArray = dblPointArray

End Function

109

B.3 Poisson Regression and Plotting Code

Regression analysis of Poisson distributed data required some special statistical

tools. We used Octave to prepare data, and to present results, but the main statistical

work was performed using The R Project for Statistical Computing. We decided it

would be easier to write out each step of the backward stepwise regression, than to

write a custom regression function for R. Following is a short sample from over 4,000

lines of actual code.

Listing B.8: Sample Poisson Regression Analysis############################################################

2 ## combined and expanded f a c t o r models on p ick body #### f a i l u r e s ( o u t l i e r removed ) ##############################################################

# load l i b r a r i e s7 l ibrary ( ’ Z e l i g ’ ) ;

l ibrary ( ’ car ’ ) ;l ibrary ( ’ z i c ount s ’ ) ;

#load data12 var body <− read . table ( ’ var body shor t . txt ’ , header = TRUE) ;

17 ################################################ expanded f a c t o r model , wi th i n t e r a c t i o n s ##

# step 0zp body expd . out <− z e l i g (

22 formula =bodyFai l s ˜attack +xBodyPD +xTipPD +

27 rBodyPD +rTipPD +aBodyPD +aTipPD +attack∗xBodyPD +

32 attack∗xTipPD +attack∗rBodyPD +attack∗rTipPD +attack∗aBodyPD +attack∗aTipPD +

37 xBodyPD∗xTipPD +xBodyPD∗rBodyPD +xBodyPD∗rTipPD +xBodyPD∗aBodyPD +xBodyPD∗aTipPD +

42 xTipPD∗rBodyPD +xTipPD∗rTipPD +xTipPD∗aBodyPD +xTipPD∗aTipPD +rBodyPD∗rTipPD +

47 rBodyPD∗aBodyPD +rBodyPD∗aTipPD +rTipPD∗aBodyPD +rTipPD∗aTipPD +aBodyPD∗aTipPD ,

52 model = ” po i s son ” ,data = var body) ;

110

# step 20 alpha = 0.1zp body expd . out <− z e l i g (

57 formula =bodyFai l s ˜xBodyPD +xTipPD +rBodyPD +

62 rTipPD +aBodyPD +aTipPD +xTipPD∗rBodyPD +xTipPD∗aTipPD ,


# step 24 alpha = 0.05zp body expd . out <− z e l i g (

72 formula =bodyFai l s ˜xBodyPD +rBodyPD +rTipPD +

77 aBodyPD ,model = ” po i s son ” ,data = var body) ;

82 # model d i a gno s t i czp body expd . diag <− glm . diag ( zp body expd . out ) ;glm . diag . p l o t s ( zp body expd . out , zp body expd . diag ) ;

# ou t l i e r t e s t87 o u t l i e r . t e s t ( zp body expd . out ) ;

# wri t e data to f i l e swrite . table (

92 summary( zp body expd . out )$coeff ic ients ,quote=FALSE,sep=” , ” ,f i l e=”beta body expd shor t . txt ” ) ;

write . table (97 summary( zp body expd . out )$deviance . resid ,

quote=FALSE,sep=” , ” ,f i l e=”deviance r e s i d u a l s body expd shor t . txt ” ) ;

write . table (102 zp body expd . out$residuals ,

quote=FALSE,sep=” , ” ,f i l e=” r e s i d u a l s body expd shor t . txt ” ) ;

write . table (107 zp body expd . diag$h ,

quote=FALSE,sep=” , ” ,f i l e=”hat body expd shor t . txt ” ) ;

write . table (112 zp body expd . diag$cook ,

quote=FALSE,sep=” , ” ,f i l e=”cook body expd shor t . txt ” ) ;

117

# p l o t and summarize r e s u l t sx . out <− s e tx ( zp body expd . out ) ;s . out <− sim ( zp body expd . out , x = x . out ) ;plot ( s . out ) ;

122 summary( zp body expd . out ) ;summary( zp body expd . out )$coef f ic ients ;z . out$residuals ;

111

127

#################################################################### expanded−f a c t o r quas ipo i s son d i s p e r s i on parameter es t imat ion ##

132 # step 24 alpha = 0.05zqp body expd . out <− glm(

formula =bodyFai l s ˜xBodyPD +

137 rBodyPD +rTipPD +aBodyPD ,

family = quas ipo i s son ,data = var body) ;

142

############################################################147 ## zero−i n f l a t e d expanded f a c t o r model , wi th i n t e r a c t i o n s ##

z z ip body cond . out <− z i c oun t s (re sp = bodyFai l s ˜ . ,x = ˜ xBodyPD + rBodyPD + rTipPD + aBodyPD ,

152 z = ˜ xBodyPD + rBodyPD + rTipPD + aBodyPD ,data=var body ,d i s t r = ”ZIP” ) ;

157

################################################# condensed f a c t o r model , wi th i n t e r a c t i o n s ##

162 # step 0zp body cond . out <− z e l i g (

formula =bodyFai l s ãttack +

167 absSkew +volume +attack∗absSkew +attack∗volume +absSkew∗volume ,


# step 1zp body cond . out <− z e l i g (

177 formula =bodyFai l s ãttack +absSkew +volume +

182 attack∗absSkew +attack∗volume ,

model = ” po i s son ” ,data = var body) ;

187 # step 2zp body cond . out <− z e l i g (

formula =bodyFai l s ãttack +

192 absSkew +volume +attack∗volume ,

model = ” po i s son ” ,data = var body) ;

197

# model d i a gno s t i c s

112

zp body cond . diag <− glm . diag ( zp body cond . out )glm . diag . p l o t s ( zp body cond . out , zp body cond . diag )

202

# ou t l i e r t e s to u t l i e r . t e s t ( zp body cond . out )

207 # e f f e c t s p lo t , a t t a c k ang lex . low <− s e tx ( zp body cond . out , at tack = −1)x . high <− s e tx ( zp body cond . out , at tack = 1)s . out <− sim ( zp body cond . out , x = x . low , x1 = x . high )summary( s . out )

212 plot ( s . out )

# e f f e c t s p lo t , a t t a c k ang lex . low <− s e tx ( zp body cond . out , absSkew = −1)x . high <− s e tx ( zp body cond . out , absSkew = 1)

217 s . out <− sim ( zp body cond . out , x = x . low , x1 = x . high )summary( s . out )plot ( s . out )

# e f f e c t s p lo t , a t t a c k ang le222 x . low <− s e tx ( zp body cond . out , volume = −1)

x . high <− s e tx ( zp body cond . out , volume = 1)s . out <− sim ( zp body cond . out , x = x . low , x1 = x . high )summary( s . out )plot ( s . out )

227

# ind i v i d u a l l e v e l p l o t ss . low <− sim ( zp body cond . out , x = x . low )s . high <− sim ( zp body cond . out , x = x . high )plot ( s . low )

232 plot ( s . high )

# wri t e r e s u l t s to f i l ewrite . table (

237 summary( zp body cond . out )$coeff ic ients ,quote=FALSE,sep=” , ” ,f i l e=”beta body cond shor t . txt ” ) ;

write . table (242 zp body cond . out$residuals ,

quote=FALSE,sep=” , ” ,f i l e=” r e s i d u a l s body cond shor t . txt ” ) ;

write . table (247 summary( zp body cond . out )$deviance . resid ,

quote=FALSE,sep=” , ” ,f i l e=”deviance r e s i d u a l s body cond shor t . txt ” ) ;

write . table (252 zp body cond . diag$h ,

quote=FALSE,sep=” , ” ,f i l e=”hat body cond shor t . txt ” ) ;

write . table (257 zp body cond . diag$cook ,

quote=FALSE,sep=” , ” ,f i l e=”cook body cond shor t . txt ” ) ;

262

#################################################################### condensed−f a c t o r quas ipo i s son d i s p e r s i on parameter es t imat ion ##

267

# step 3zqp body expd . out <− glm(

formula =bodyFai l s ˜

272 attack +

113

absSkew +volume +attack∗volume ,

family = quas ipo i s son ,277 data = var body) ;

282 ############################################################### zero−i n f l a t e d condensed f a c t o r model , wi th i n t e r a c t i on s ##

# step 3z z ip body cond . out <− z i c oun t s (

287 re sp = bodyFai l s˜ . ,x= ˜ attack + absSkew + volume + attack∗volume ,z= ˜ attack + absSkew + volume + attack∗volume ,data = var body ,d i s t r = ”ZIP” ) ;

114

B.4 Linear Regression and Plotting Code

Our analysis methods for the present study were fairly different from typical

statistical applications. In particular, we required the regression routine to distinguish

between main factors and interaction factors. Because of these differences, we chose

to develop custom regression and plotting routines. The following code is written for

the Octave scripting language (mostly MATLAB compatible).

Table B.2: Function call structure for backward stepwise regression

WearAnalysis.m

↪→ FactGen.m

↪→ MatVol1.m

↪→ MatVol2.m

↪→ MatVol3.m

↪→ EdgeTrim.m

↪→ BackStepRegres.m

↪→ LatexTabMed.m

↪→ PlotData.m

PlotInteract.m

115

Listing B.9: Main analysis function - calls regression% WearAnalysis .m%% SYNOPSIS

4 % This s c r i p t l oads data and c a l l s f unc t i on s to prepare va r i a b l e s , execute a% backward s t epw i s e regres s ion , generate p l o t data , and wr i t e r e s u l t s t a b l e s .% The only input and output f o r t h i s s c r i p t i s through f i l e s .%% INPUT

9 %% cu t t e r head s p e c i f i c a t i o n f i l e (comma de l im i t ed )% − column 1 = ax i a l p o s i t i on% − column 2 = angular po s i t i on% − column 3 = c i r cumf e r en t i a l p o s i t i on ( arc d i s t ance )

14 % − column 4 = b i t t i p rad ius% − column 5 = skew ang le% − column 6 = at tack ang le% − column 7 = b i t number ( from eng ineer ing drawings )% − column 8 = b i t type

19 %% mater ia l 1 wear data (comma de l im i t ed f i l e )%% mater ia l 2 wear data (comma de l im i t ed f i l e )%

24 % mater ia l 3 wear data (comma de l im i t ed f i l e )%% OUTPUT%% column−a l i gned t e x t f i l e

29 % − column 1 = fa c t o r id% − column 2 = fa c t o r name% − column 3 = c o e f f i c i e n t% − column 4 = p−va lue%

34 %% NOTE: p o s i t i v e skew ang le i s in the c l o ckw i s e d i r e c t i on , viewed from above%

clear a l l ;39 close a l l ;

%se t p a t h s ( ’ l i n s e r v1 ’ ) ;

%dbstop i f warning44 %dbstop in mu l t i r e g r e s .m at 120

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% load experiment data %%

49 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% bitPosData = [% 1 a x i a l d i s t ,% 2 drum angle ,

54 % 3 drum arc dis t ,% 4 t i p r ad i u s ,% 5 skew angle ,% 6 ang l e o f a t t a c k ,% 7 bit num

59 % ]bitPosData = dlmread( ’ cut te r measure data . csv ’ ) ;bitPosData = sort rows ( bitPosData , 7 ) ;

% mat0BitFai l = [ bit num , f a i l c o u n t ]64 mat1BitFai l = dlmread( ’ mate r i a l 1 wear data . csv ’ ) ;

mat1BitFai l = sor t rows ( mat1BitFail , 1 ) ;

mat2BitFai l = dlmread( ’ mate r i a l 2 wear data . csv ’ ) ;mat2BitFai l = sor t rows ( mat2BitFail , 1 ) ;

69

mat3BitFai l = dlmread( ’ mate r i a l 3 wear data . csv ’ ) ;mat3BitFai l = sor t rows ( mat3BitFail , 1 ) ;

116

t i pB i tFa i l = dlmread( ’ t i p wear data . csv ’ ) ;74 t i pB i tFa i l = sor t rows ( t i pB i tFa i l , 1 ) ;

bodyBitFai l = dlmread( ’ body wear data . csv ’ ) ;bodyBitFai l = sor t rows ( bodyBitFai l , 1 ) ;

79 % importData = [% 1 bit num ,% 2 a x i a l d i s t ,% 3 drum angle ,% 4 t i p r ad i u s ,

84 % 5 skew angle ,% 6 a t t ack ang l e ,% 7 mat1 fa i l s ,% 8 mat2 fa i l s ,% 9 mat3 fa i l s ,

89 % 10 t i p f a i l s ,% 11 b o d y f a i l s% ]importData = [ bitPosData ( : , [ 7 , 1 , 2 , 4 , 5 , 6 ] ) , . . .

mat1BitFai l ( : , 2 ) , . . .94 mat2BitFai l ( : , 2 ) , . . .

mat3BitFai l ( : , 2 ) , . . .t i pB i tF a i l ( : , 2 ) , . . .bodyBitFai l ( : , 2 ) ] ;

importData = sort rows ( importData , [ 2 , 3 ] ) ;99

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% generate r e g r e s s i on vars %%

104 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% edgeTrimDist = d i s tance i n t e r v a l on edges o f cut in which e f f e c t s are% confounded by cut over lap (measured in inches )edgeTrimDist = 1 . 5 ;

109

% generate f a c t o r and response v a r i a b l e s and names[X, Y, fac t , factNames , respNames ] = FactGen ( importData , edgeTrimDist , ’ body ’ ) ;

114

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% run reg r e s s i on ana l y s i s %%

119 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%n = s ize (X, 1 ) ;k = s ize (X, 2 ) ;

% c a l l r e g r e s s i on rou t ine124 % se t s i g n i f i c a n c e t h r e s ho l d to 1.0 i f you want a standard ml r e g r e s s i on ( as

% opposed to a backward s t epw i s e r e g r e s s i on )[ beta , res , s t a t s , f ac t ,X] = BackStepRegres (X, Y, fac t , factNames , 0 . 1 , 0) ;factNames = factNames ( [ f a c t ( : , 1 ) +1 ] , : ) ;

129

% c a l l p l o t t i n g rou t ine%p lo t ma t l a b (X, Y, beta , res , factNames ) ;%p l o t r o t a t e o c t (X, Y, beta , res , means , factNames ) ;%p l o t s t anda r d o c t (X, Y, beta , res , factNames , respNames ) ;

134 %p l o t v o l c o n t r i b (X, Y, beta , res , factNames , respNames ) ;PlotData (X, Y, beta , res , factNames , respNames ) ;

139 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wri t e t a b l e to f i l e %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% open f i l e to wr i t ef i d = fopen ( ’ m l r r e s u l t s . txt ’ , ’ at ’ ) ;

144

117

% pr in t r e g r e s s i on repor t headerheadFormat = ’%−6s \ t%−26s \ t%8s \ t%7s \n ’ ;bodyFormat = ’%−6s \ t%−26s \ t%8.4 f \ t%6.4 f \n ’ ;

149 p r i n t f ( ’ \n%s \n ’ , respNames {1 ,3} ) ;fpr intf ( f i d , ’ \n%s \n ’ , respNames {1 ,3} ) ;p r i n t f ( headFormat , ’ f a c t o r ’ , ’ d e s c r i p t i o n ’ , ’ c o e f f ’ , ’p−value ’ ) ;fpr intf ( f i d , headFormat , ’ f a c t o r ’ , ’ d e s c r i p t i o n ’ , ’ c o e f f ’ , ’p−value ’ ) ;

154 % pr in t f a c t o r datafor i = 1 : s ize (X, 2 )

p r i n t f ( bodyFormat , factNames{ i , 2} , factNames{ i , 3} , beta ( i , 1 ) , beta ( i , 3 ) ) ;fpr intf ( f i d , bodyFormat , factNames{ i , 2} , factNames{ i , 3} , beta ( i , 1 ) ,beta ( i , 3 ) ) ;

end159

p r i n t f ( ’ \n ’ ) ;fpr intf ( f i d , ’ \n ’ ) ;p r i n t f ( ’%s \n ’ , [ ’R−Square = ’ , num2str( s t a t s (1 ) ) ] ) ;fpr intf ( f i d , ’%s \n ’ , [ ’R−Square = ’ , num2str( s t a t s (1 ) ) ] ) ;

164 p r i n t f ( ’%s \n ’ , [ ’ Adjusted R−Square = ’ , num2str( s t a t s (2 ) ) ] ) ;fpr intf ( f i d , ’%s \n ’ , [ ’ Adjusted R−Square = ’ , num2str( s t a t s (2 ) ) ] ) ;p r i n t f ( ’ \n ’ ) ;

fc lose ( f i d ) ;

118

Listing B.10: Generate regression variables and related information% FactGen .m

2 %% SYNOPSIS% This func t i on accep t s fundamental data from a s p e c i f i c c u t t i n g app l i ca t i on ,% and re turns r e g r e s s i on v a r i a b l e s and r e l a t e d information , f o r the s p e c i f i e d% f a i l u r e mode .

7 %% INPUT%% funData : fundamental data from a s p e c i f i c c u t t i n g app l i c a t i on% column 1 = bit num ,

12 % column 2 = a x i a l d i s t ,% column 3 = drum angle ,% column 4 = t i p r ad i u s ,% column 5 = skew angle ,% column 6 = at tack ang l e ,

17 % column 7 = mat1 fa i l s ,% column 8 = mat2 fa i l s ,% column 9 = mat3 fa i l s ,% column 10 = t i p f a i l s ,% column 11 = b o d y f a i l s

22 %% edgeTrimDist : d i s t ance i n t e r v a l on edges o f cut in which e f f e c t s are% confounded by cut over lap (measured in inches ) . These b i t s% are removed from the ana l y s i s%

27 % mode : f a i l u r e mode s t r ing , c on i s t i n g o f one o f the f o l l ow i n g :% ’ body ’ = body f a i l u r e mode% ’ t ip ’ = t i p f a i l u r e mode%% OUTPUT

32 %% Y: vec tor conta in ing va lue s f o r the dependent or response v a r i a b l e%% X: matrix o f independent v a r i a b l e va lue s ( rows=samples , columns=f a c t o r s )%

37 % fa c t : indexed l i s t o f f a c t o r i n t e r a c t i o n s (number o f rows shou ld equa l% number o f columns in X)% column 1 = fa c t o r number% column 2 = in t e r a c t i on f a c t o r 1 (0 fo r l i n e a r terms )% column 3 = in t e r a c t i on f a c t o r 2 (0 fo r l i n e a r terms )

42 %% factNames : names o f f a c t o r s f o r r epor t ing purposes ( c e l l array )% column 1 = va r i a b l e index% column 2 = va r i a b l e id% column 3 = va r i a b l e name

47 %% example c e l l array : f a c t o r names% −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−% index | v a r i a b l e i d | var iab le name% −−−−−−−+−−−−−−−−−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−

52 % 1 | X01 | drum angle% 2 | X02 | t i p r a d i u s% 3 | X03 | skew ang le% 4 | X01X02 | drum angle X t i p r a d i u s% 5 | X01X03 | drum angle X skew ang le

57 % 6 | X02X03 | t i p r a d i u s X skew ang le% −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−% factNames (1 ,1 :3 ) = [{1} ,{ ’X01 ’} ,{ ’ drum angle ’ } ] ;%% respNames : name of response v a r i a b l e

62 % column 1 = va r i a b l e index% column 2 = va r i a b l e id% column 3 = va r i a b l e name%

67 function [X,Y, fac t , factNames , respNames ] = FactGen ( funData , edgeTrimDist , mode) ;

119

72 %%%%%%%%%%%%%%%%%%% re s p on s e v a r i a b l e %%%%%%%%%%%%%%%%%%%i f mode==’ body ’

respNames ( 1 , 1 : 3 ) = [{1} ,{ ’Y ’ } ,{ ’Body Fa i l u r e s ( octave ) ’ } ] ;Y = funData ( : , 1 1 ) ;

e l s e i f mode==’ t i p ’77 respNames ( 1 , 1 : 3 ) = [{1} ,{ ’Y ’ } ,{ ’ Tip Fa i l u r e s ( octave ) ’ } ] ;

Y = funData ( : , 1 0 ) ;else

return ;end

82

%%%%%%%%%%%%%%%%%%%% mode l i n t e r c e p t %%%%%%%%%%%%%%%%%%%%87 i n t e r c e p t = ones ( s ize ( funData , 1 ) ,1 ) ;

92

%%%%%%%%%%%%%%%%%%%%%% a t t a c k a n g l e %%%%%%%%%%%%%%%%%%%%%attack = funData ( : , 6 ) ;

97

%%%%%%%%%%%%%%%%%% ab s o l u t e s k ew an g l e %%%%%%%%%%%%%%%%%%absSkew = abs ( funData ( : , 5 ) ) ;

102

107 %%%%%%%%%%%%%%%%%%%% mater ia l−n b i t v o l ume %%%%%%%%%%%%%%%%%%%%

% volume func t ion re turns NaN for b i t s c l o s e to the edge , based% on in t e r a c t i on with the edge o f the cut%

112 % we w i l l u s ua l l y be ab l e to re load p r e v i ou s l y c a l c u l a t e d data to same time%% volume rou t ine v a r i a b l e d e f i n i t i o n s% desVar = [ a x i a l d i s t , drum angle , t i p r a d i u s ]% ppVol = [ ax i a l i nde x , b i t vo lume , a x i a l p o s i t i o n , drum angle ]

117 %desVar = funData ( : , [ 2 , 3 , 4 ] ) ;

% mater ia l 1% ca l c u l a t e per−p ick volume

122 %ppVol1 = MatVol1 ( desVar ) ;%ppVol1 = sortrows ( ppVol1 , [ 3 , 4 ] ) ;%save −a s c i i ppVol1 . t x t ppVol1 ;load ppVol1 . txt ;

127 % mater ia l 2%ppVol2 = MatVol2 ( desVar ) ;%ppVol2 = sortrows ( ppVol2 , [ 3 , 4 ] ) ;%save −a s c i i ppVol2 . t x t ppVol2 ;load ppVol2 . txt ;

132

% mater ia l 3%ppVol3 = MatVol3 ( desVar ) ;%ppVol3 = sortrows ( ppVol3 , [ 3 , 4 ] ) ;%save −a s c i i ppVol3 . t x t ppVol3 ;

137 load ppVol3 . txt ;

i f mode==’ body ’% b i t volume fo r body f a i l u r e svolume = mean( [ ppVol1 ( : , 2 ) , ppVol3 ( : , 2 ) ] , 2 ) ;

142 e l s e i f mode==’ t i p ’% b i t volume fo r t i p f a i l u r e svolume = ppVol2 ( : , 2 ) ;

120

elsereturn ;

147 end

152 %%%%%%%%%%%%%%%%%%%% expanded f a c t o r s %%%%%%%%%%%%%%%%%%%%

aBodyPD = [ ] ;aTipPD = [ ] ;rBodyPD = [ ] ;

157 rTipPD = [ ] ;xBodyPD = [ ] ;xTipPD = [ ] ;

% get l e f t edge case162 i f funData (1 , 5 ) < 0 % t i p l e f t

% angulari f funData (2 , 3 ) > funData (1 , 3 )

aBodyPD = [ aBodyPD ; funData (1 , 3 ) − ( funData (2 , 3 ) − 2∗pi ) ] ;167 else

aBodyPD = [ aBodyPD ; funData (1 , 3 ) − funData (2 , 3 ) ] ;endaTipPD = [ aTipPD ; 0 ] ;

172 % rad i a lrBodyPD = [ rBodyPD ; funData (1 , 4 ) − funData (2 , 4 ) ] ;rTipPD = [ rTipPD ; 0 ] ;

% ax i a l177 xBodyPD = [xBodyPD ; abs ( funData (1 , 2 ) − funData (2 , 2 ) ) ] ;

xTipPD = [ xTipPD ; 0 ] ;

else % t i p r i g h t

182 % angulari f funData (2 , 3 ) > funData (1 , 3 )

aTipPD = [ aTipPD ; funData (1 , 3 ) − ( funData (2 , 3 ) − 2∗pi ) ] ;else

aTipPD = [ aTipPD ; funData (1 , 3 ) − funData (2 , 3 ) ] ;187 end

aBodyPD = [ aBodyPD ; 0 ] ;

% rad i a lrTipPD = [ rTipPD ; funData (1 , 4 ) − funData (2 , 4 ) ] ;

192 rBodyPD = [ rBodyPD ; 0 ] ;

% ax i a lxTipPD = [ xTipPD ; abs ( funData (1 , 2 ) − funData (2 , 2 ) ) ] ;xBodyPD = [xBodyPD ; 0 ] ;

197

end

% get f i e l d casefor i = 2 : ( s ize ( funData , 1 ) −1)

202

i f funData ( i , 5 ) < 0 % t i p l e f t

% angulari f funData ( i +1 ,3) > funData ( i , 3 )

207 aBodyPD = [ aBodyPD ; funData ( i , 3 ) − ( funData ( i +1 ,3) − 2∗pi) ] ;

elseaBodyPD = [ aBodyPD ; funData ( i , 3 ) − funData ( i +1 ,3) ] ;

endi f funData ( i −1 ,3) > funData ( i , 3 )

212 aTipPD = [ aTipPD ; funData ( i , 3 ) − ( funData ( i −1 ,3) − 2∗pi )] ;

elseaTipPD = [ aTipPD ; funData ( i , 3 ) − funData ( i −1 ,3) ] ;

end

121

217 % rad i a lrBodyPD = [ rBodyPD ; funData ( i , 4 ) − funData ( i +1 ,4) ] ;rTipPD = [ rTipPD ; funData ( i , 4 ) − funData ( i −1 ,4) ] ;

% ax i a l222 xBodyPD = [xBodyPD ; abs ( funData ( i , 2 ) − funData ( i +1 ,2) ) ] ;

xTipPD = [ xTipPD ; abs ( funData ( i , 2 ) − funData ( i −1 ,2) ) ] ;

else % t i p r i g h t

227 % angulari f funData ( i +1 ,3) > funData ( i , 3 )

aTipPD = [ aTipPD ; funData ( i , 3 ) − ( funData ( i +1 ,3) − 2∗pi )] ;

elseaTipPD = [ aTipPD ; funData ( i , 3 ) − funData ( i +1 ,3) ] ;

232 endi f funData ( i −1 ,3) > funData ( i , 3 )

aBodyPD = [ aBodyPD ; funData ( i , 3 ) − ( funData ( i −1 ,3) − 2∗pi) ] ;

elseaBodyPD = [ aBodyPD ; funData ( i , 3 ) − funData ( i −1 ,3) ] ;

237 end

% rad i a lrTipPD = [ rTipPD ; funData ( i , 4 ) − funData ( i +1 ,4) ] ;rBodyPD = [ rBodyPD ; funData ( i , 4 ) − funData ( i −1 ,4) ] ;

242

% ax i a lxTipPD = [ xTipPD ; abs ( funData ( i , 2 ) − funData ( i +1 ,2) ) ] ;xBodyPD = [xBodyPD ; abs ( funData ( i , 2 ) − funData ( i −1 ,2) ) ] ;

247 end

end

% get r i g h t edge case252 i f funData (1 , 5 ) > 0 % t i p r i g h t

% angulari f funData (end−1 ,3) > funData (end , 3 )

aBodyPD = [ aBodyPD ; funData (end , 3 ) − ( funData (end−1 ,3) − 2∗pi ) ] ;257 else

aBodyPD = [ aBodyPD ; funData (end , 3 ) − funData (end−1 ,3) ] ;endaTipPD = [ aTipPD ; 0 ] ;

262 % rad i a lrBodyPD = [ rBodyPD ; funData (end , 4 ) − funData (end−1 ,4) ] ;rTipPD = [ rTipPD ; 0 ] ;

% ax i a l267 xBodyPD = [xBodyPD ; abs ( funData (end , 2 ) − funData (end−1 ,2) ) ] ;

xTipPD = [ xTipPD ; 0 ] ;

else % t i p l e f t

272 % angulari f funData (end−1 ,3) > funData (end , 3 )

aTipPD = [ aTipPD ; funData (end , 3 ) − ( funData (end−1 ,3) − 2∗pi ) ] ;else

aTipPD = [ aTipPD ; funData (end , 3 ) − funData (end−1 ,3) ] ;277 end

aBodyPD = [ aBodyPD ; 0 ] ;

% rad i a lrTipPD = [ rTipPD ; funData (end , 4 ) − funData (end−1 ,4) ] ;

282 rBodyPD = [ rBodyPD ; 0 ] ;

% ax i a lxTipPD = [ xTipPD ; abs ( funData (end , 2 ) − funData (end−1 ,2) ) ] ;xBodyPD = [xBodyPD ; 0 ] ;

122

287

end

292

%%%%%%%%%%%%%%%%%%%% as s i g n o u t p u t %%%%%%%%%%%%%%%%%%%%X = [ ] ;f a c t = [ ] ;

297

% main f a c t o r sfactNames ( 1 , 1 : 3 ) = [{0} ,{ ’X00 ’ } ,{ ’ i n t e r c e p t ’ } ] ;

X = [X, i n t e r c e p t ] ;f a c t = [ f a c t ; [ 0 , 0 , 0 ] ] ;

302 factNames ( 2 , 1 : 3 ) = [{1} ,{ ’X01 ’ } ,{ ’ a t tack ’ } ] ;X = [X, attack ] ;f a c t = [ f a c t ; [ 1 , 0 , 0 ] ] ;

factNames ( 3 , 1 : 3 ) = [{2} ,{ ’X02 ’ } ,{ ’ absSkew ’ } ] ;X = [X, absSkew ] ;

307 f a c t = [ f a c t ; [ 2 , 0 , 0 ] ] ;factNames ( 4 , 1 : 3 ) = [{3} ,{ ’X03 ’ } ,{ ’ volume ’ } ] ;

X = [X, volume ] ;f a c t = [ f a c t ; [ 3 , 0 , 0 ] ] ;

factNames ( 5 , 1 : 3 ) = [{4} ,{ ’X04 ’ } ,{ ’xBodyPD ’ } ] ;312 X = [X, xBodyPD ] ;

f a c t = [ f a c t ; [ 4 , 0 , 0 ] ] ;factNames ( 6 , 1 : 3 ) = [{5} ,{ ’X05 ’ } ,{ ’xTipPD ’ } ] ;

X = [X, xTipPD ] ;f a c t = [ f a c t ; [ 5 , 0 , 0 ] ] ;

317 factNames ( 7 , 1 : 3 ) = [{6} ,{ ’X06 ’ } ,{ ’ rBodyPD ’ } ] ;X = [X, rBodyPD ] ;f a c t = [ f a c t ; [ 6 , 0 , 0 ] ] ;

factNames ( 8 , 1 : 3 ) = [{7} ,{ ’X07 ’ } ,{ ’ rTipPD ’ } ] ;X = [X, rTipPD ] ;

322 f a c t = [ f a c t ; [ 7 , 0 , 0 ] ] ;factNames ( 9 , 1 : 3 ) = [{8} ,{ ’X08 ’ } ,{ ’aBodyPD ’ } ] ;

X = [X, aBodyPD ] ;f a c t = [ f a c t ; [ 8 , 0 , 0 ] ] ;

factNames ( 1 0 , 1 : 3 ) = [{9} ,{ ’X09 ’ } ,{ ’ aTipPD ’ } ] ;327 X = [X, aTipPD ] ;

f a c t = [ f a c t ; [ 9 , 0 , 0 ] ] ;

% tr im and sca l e da ta[X, Y] = EdgeTrim (X, Y, funData , edgeTrimDist ) ;

332 %for i = 2: s i z e (X,2 )% X( : , i ) = (X( : , i )−((max(X( : , i ) )+min(X( : , i ) ) ) /2) ) /((max(X( : , i ) )−min(X( : , i ) ) ) /2) ;%end

% in t e r a c t i on f a c t o r s337 factNames ( 1 1 , 1 : 3 ) = [{10} ,{ ’X10 ’ } ,{ ’ a t tack x absSkew ’ } ] ;

X = [X, X( : , 2 ) .∗ X( : , 3 ) ] ;f a c t = [ f a c t ; [ 1 0 , 1 , 2 ] ] ;

factNames ( 1 2 , 1 : 3 ) = [{11} ,{ ’X11 ’ } ,{ ’ a t tack x volume ’ } ] ;X = [X, X( : , 2 ) .∗ X( : , 4 ) ] ;

342 f a c t = [ f a c t ; [ 1 1 , 1 , 3 ] ] ;factNames ( 1 3 , 1 : 3 ) = [{12} ,{ ’X12 ’ } ,{ ’ a t tack x xBodyPD ’ } ] ;

X = [X, X( : , 2 ) .∗ X( : , 5 ) ] ;f a c t = [ f a c t ; [ 1 2 , 1 , 4 ] ] ;

factNames ( 1 4 , 1 : 3 ) = [{13} ,{ ’X13 ’ } ,{ ’ a t tack x xTipPD ’ } ] ;347 X = [X, X( : , 2 ) .∗ X( : , 6 ) ] ;

f a c t = [ f a c t ; [ 1 3 , 1 , 5 ] ] ;factNames ( 1 5 , 1 : 3 ) = [{14} ,{ ’X14 ’ } ,{ ’ a t tack x rBodyPD ’ } ] ;

X = [X, X( : , 2 ) .∗ X( : , 7 ) ] ;f a c t = [ f a c t ; [ 1 4 , 1 , 6 ] ] ;

352 factNames ( 1 6 , 1 : 3 ) = [{15} ,{ ’X15 ’ } ,{ ’ a t tack x rTipPD ’ } ] ;X = [X, X( : , 2 ) .∗ X( : , 8 ) ] ;f a c t = [ f a c t ; [ 1 5 , 1 , 7 ] ] ;

factNames ( 1 7 , 1 : 3 ) = [{16} ,{ ’X16 ’ } ,{ ’ a t tack x aBodyPD ’ } ] ;X = [X, X( : , 2 ) .∗ X( : , 9 ) ] ;

357 f a c t = [ f a c t ; [ 1 6 , 1 , 8 ] ] ;factNames ( 1 8 , 1 : 3 ) = [{17} ,{ ’X17 ’ } ,{ ’ a t tack x aTipPD ’ } ] ;

X = [X, X( : , 2 ) .∗ X( : , 1 0 ) ] ;

123

f a c t = [ f a c t ; [ 1 7 , 1 , 9 ] ] ;factNames ( 1 9 , 1 : 3 ) = [{18} ,{ ’X18 ’ } ,{ ’ absSkew x volume ’ } ] ;

362 X = [X, X( : , 3 ) .∗ X( : , 4 ) ] ;f a c t = [ f a c t ; [ 1 8 , 2 , 3 ] ] ;

factNames ( 2 0 , 1 : 3 ) = [{19} ,{ ’X19 ’ } ,{ ’ absSkew x xBodyPD ’ } ] ;X = [X, X( : , 3 ) .∗ X( : , 5 ) ] ;f a c t = [ f a c t ; [ 1 9 , 2 , 4 ] ] ;

367 factNames ( 2 1 , 1 : 3 ) = [{20} ,{ ’X20 ’ } ,{ ’ absSkew x xTipPD ’ } ] ;X = [X, X( : , 3 ) .∗ X( : , 6 ) ] ;f a c t = [ f a c t ; [ 2 0 , 2 , 5 ] ] ;

factNames ( 2 2 , 1 : 3 ) = [{21} ,{ ’X21 ’ } ,{ ’ absSkew x rBodyPD ’ } ] ;X = [X, X( : , 3 ) .∗ X( : , 7 ) ] ;

372 f a c t = [ f a c t ; [ 2 1 , 2 , 6 ] ] ;factNames ( 2 3 , 1 : 3 ) = [{22} ,{ ’X22 ’ } ,{ ’ absSkew x rTipPD ’ } ] ;

X = [X, X( : , 3 ) .∗ X( : , 8 ) ] ;f a c t = [ f a c t ; [ 2 2 , 2 , 7 ] ] ;

factNames ( 2 4 , 1 : 3 ) = [{23} ,{ ’X23 ’ } ,{ ’ absSkew x aBodyPD ’ } ] ;377 X = [X, X( : , 3 ) .∗ X( : , 9 ) ] ;

f a c t = [ f a c t ; [ 2 3 , 2 , 8 ] ] ;factNames ( 2 5 , 1 : 3 ) = [{24} ,{ ’X24 ’ } ,{ ’ absSkew x aTipPD ’ } ] ;

X = [X, X( : , 3 ) .∗ X( : , 1 0 ) ] ;f a c t = [ f a c t ; [ 2 4 , 2 , 9 ] ] ;

382 factNames ( 2 6 , 1 : 3 ) = [{25} ,{ ’X25 ’ } ,{ ’ volume x xBodyPD ’ } ] ;X = [X, X( : , 4 ) .∗ X( : , 5 ) ] ;f a c t = [ f a c t ; [ 2 5 , 3 , 4 ] ] ;

factNames ( 2 7 , 1 : 3 ) = [{26} ,{ ’X26 ’ } ,{ ’ volume x xTipPD ’ } ] ;X = [X, X( : , 4 ) .∗ X( : , 6 ) ] ;

387 f a c t = [ f a c t ; [ 2 6 , 3 , 5 ] ] ;factNames ( 2 8 , 1 : 3 ) = [{27} ,{ ’X27 ’ } ,{ ’ volume x rBodyPD ’ } ] ;

X = [X, X( : , 4 ) .∗ X( : , 7 ) ] ;f a c t = [ f a c t ; [ 2 7 , 3 , 6 ] ] ;

factNames ( 2 9 , 1 : 3 ) = [{28} ,{ ’X28 ’ } ,{ ’ volume x rTipPD ’ } ] ;392 X = [X, X( : , 4 ) .∗ X( : , 8 ) ] ;

f a c t = [ f a c t ; [ 2 8 , 3 , 7 ] ] ;factNames ( 3 0 , 1 : 3 ) = [{29} ,{ ’X29 ’ } ,{ ’ volume x aBodyPD ’ } ] ;

X = [X, X( : , 4 ) .∗ X( : , 9 ) ] ;f a c t = [ f a c t ; [ 2 9 , 3 , 8 ] ] ;

397 factNames ( 3 1 , 1 : 3 ) = [{30} ,{ ’X30 ’ } ,{ ’ volume x aTipPD ’ } ] ;X = [X, X( : , 4 ) .∗ X( : , 1 0 ) ] ;f a c t = [ f a c t ; [ 3 0 , 3 , 9 ] ] ;

factNames ( 3 2 , 1 : 3 ) = [{31} ,{ ’X31 ’ } ,{ ’xBodyPD x xTipPD ’ } ] ;X = [X, X( : , 5 ) .∗ X( : , 6 ) ] ;

402 f a c t = [ f a c t ; [ 3 1 , 4 , 5 ] ] ;factNames ( 3 3 , 1 : 3 ) = [{32} ,{ ’X32 ’ } ,{ ’xBodyPD x rBodyPD ’ } ] ;

X = [X, X( : , 5 ) .∗ X( : , 7 ) ] ;f a c t = [ f a c t ; [ 3 2 , 4 , 6 ] ] ;

factNames ( 3 4 , 1 : 3 ) = [{33} ,{ ’X33 ’ } ,{ ’xBodyPD x rTipPD ’ } ] ;407 X = [X, X( : , 5 ) .∗ X( : , 8 ) ] ;

f a c t = [ f a c t ; [ 3 3 , 4 , 7 ] ] ;factNames ( 3 5 , 1 : 3 ) = [{34} ,{ ’X34 ’ } ,{ ’xBodyPD x aBodyPD ’ } ] ;

X = [X, X( : , 5 ) .∗ X( : , 9 ) ] ;f a c t = [ f a c t ; [ 3 4 , 4 , 8 ] ] ;

412 factNames ( 3 6 , 1 : 3 ) = [{35} ,{ ’X35 ’ } ,{ ’xBodyPD x aTipPD ’ } ] ;X = [X, X( : , 5 ) .∗ X( : , 1 0 ) ] ;f a c t = [ f a c t ; [ 3 5 , 4 , 9 ] ] ;

factNames ( 3 7 , 1 : 3 ) = [{36} ,{ ’X36 ’ } ,{ ’xTipPD x rBodyPD ’ } ] ;X = [X, X( : , 6 ) .∗ X( : , 7 ) ] ;

417 f a c t = [ f a c t ; [ 3 6 , 5 , 6 ] ] ;factNames ( 3 8 , 1 : 3 ) = [{37} ,{ ’X37 ’ } ,{ ’xTipPD x rTipPD ’ } ] ;

X = [X, X( : , 6 ) .∗ X( : , 8 ) ] ;f a c t = [ f a c t ; [ 3 7 , 5 , 7 ] ] ;

factNames ( 3 9 , 1 : 3 ) = [{38} ,{ ’X38 ’ } ,{ ’xTipPD x aBodyPD ’ } ] ;422 X = [X, X( : , 6 ) .∗ X( : , 9 ) ] ;

f a c t = [ f a c t ; [ 3 8 , 5 , 8 ] ] ;factNames ( 4 0 , 1 : 3 ) = [{39} ,{ ’X39 ’ } ,{ ’xTipPD x aTipPD ’ } ] ;

X = [X, X( : , 6 ) .∗ X( : , 1 0 ) ] ;f a c t = [ f a c t ; [ 3 9 , 5 , 9 ] ] ;

427 factNames ( 4 1 , 1 : 3 ) = [{40} ,{ ’X40 ’ } ,{ ’ rBodyPD x rTipPD ’ } ] ;X = [X, X( : , 7 ) .∗ X( : , 8 ) ] ;f a c t = [ f a c t ; [ 4 0 , 6 , 7 ] ] ;

factNames ( 4 2 , 1 : 3 ) = [{41} ,{ ’X41 ’ } ,{ ’ rBodyPD x aBodyPD ’ } ] ;X = [X, X( : , 7 ) .∗ X( : , 9 ) ] ;

432 f a c t = [ f a c t ; [ 4 1 , 6 , 8 ] ] ;

124

factNames ( 4 3 , 1 : 3 ) = [{42} ,{ ’X42 ’ } ,{ ’ rBodyPD x aTipPD ’ } ] ;X = [X, X( : , 7 ) .∗ X( : , 1 0 ) ] ;f a c t = [ f a c t ; [ 4 2 , 6 , 9 ] ] ;

factNames ( 4 4 , 1 : 3 ) = [{43} ,{ ’X43 ’ } ,{ ’ rTipPD x aBodyPD ’ } ] ;437 X = [X, X( : , 8 ) .∗ X( : , 9 ) ] ;

f a c t = [ f a c t ; [ 4 3 , 7 , 8 ] ] ;factNames ( 4 5 , 1 : 3 ) = [{44} ,{ ’X44 ’ } ,{ ’ rTipPD x aTipPD ’ } ] ;

X = [X, X( : , 8 ) .∗ X( : , 1 0 ) ] ;f a c t = [ f a c t ; [ 4 4 , 7 , 9 ] ] ;

442 factNames ( 4 6 , 1 : 3 ) = [{45} ,{ ’X45 ’ } ,{ ’aBodyPD x aTipPD ’ } ] ;X = [X, X( : , 9 ) .∗ X( : , 1 0 ) ] ;f a c t = [ f a c t ; [ 4 5 , 8 , 9 ] ] ;

447

return ;

125

Listing B.11: Trim bit data, removing confounding edge effects% EdgeTrim .m

2 %% SYNOPSIS% This func t i on modi f i e s the X and Y matr ices by removing data r e l a t i n g to% b i t s l o ca t ed wi th in a ce r t a in d i s t ance o f the edge o f the cut . Candidate% b i t s are i d e n t i f i e d us ing fundamental measured data .

7 %% INPUT%% X: matrix o f independent v a r i a b l e va lue s ( rows=samples , columns=f a c t o r s )%

12 % Y: vec tor conta in ing va lue s f o r the dependent or response v a r i a b l e%% funData : fundamental data from a s p e c i f i c c u t t i n g app l i c a t i on% column 1 = bit num ,% column 2 = a x i a l d i s t ,

17 % column 3 = drum angle ,% column 4 = t i p r ad i u s ,% column 5 = skew angle ,% column 6 = at tack ang l e ,% column 7 = mat1 fa i l s ,

22 % column 8 = mat2 fa i l s ,% column 9 = mat3 fa i l s ,% column 10 = t i p f a i l s ,% column 11 = b o d y f a i l s%

27 % edgeTrimDist : d i s t ance i n t e r v a l on edges o f cut in which e f f e c t s are% confounded by cut over lap (measured in inches ) . These b i t s% are removed from the ana l y s i s%% OUTPUT

32 %% X trim : trimmed vers ion o f the X matrix%% Y trim : trimmed vers ion o f the Y matrix%

37

function [ X trim , Y trim ] = EdgeTrim(X, Y, funData , edgeTrimDist )

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% remove edge over lap confounding %%

42 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%X trim = X;Y trim = Y;

l e f tEdge = min( funData ( : , 2 ) ) ;47 r ightEdge = max( funData ( : , 2 ) ) ;

% trim l e f t confounded b i t swhile ( funData (1 , 2 ) < l e f tEdge + edgeTrimDist )

funData ( 1 , : ) = [ ] ;52 X trim ( 1 , : ) = [ ] ;

Y trim ( 1 , : ) = [ ] ;end

% trim r i g h t confounded b i t s57 while ( funData (end , 2 ) > r ightEdge − edgeTrimDist )

funData (end , : ) = [ ] ;X trim (end , : ) = [ ] ;Y trim (end , : ) = [ ] ;

end62

return ;

126

Listing B.12: Perform backward stepwise regression1 % BackStepRegres .m

%% SYNOPSIS% This func t i on performs a backward s t epw i s e r e g r e s s i on . I t i s cu r r en t l y% implemented to wr i t e a f i l e conta in ing a l a t e x t a b l e f o r each s t ep o f the

6 % regre s s i on .%%% INPUT%

11 % Y : vec tor conta in ing va lue s f o r the dependent or response v a r i a b l e%% X : matrix o f independent v a r i a b l e va lue s ( rows=samples , columns=f a c t o r s )%% fa c t : indexed l i s t o f f a c t o r i n t e r a c t i o n s (number o f rows shou ld equa l

16 % : number o f columns in X)% : column 1 −> f a c t o r number% : column 2 −> i n t e r a c t i on f a c t o r 1 (0 f o r l i n e a r terms )% : column 3 −> i n t e r a c t i on f a c t o r 2 (0 f o r l i n e a r terms )%

21 % factNames : names o f f a c t o r s f o r r epor t ing purposes ( c e l l array )% column 1 = va r i a b l e index% column 2 = va r i a b l e id% column 3 = va r i a b l e name%

26 % a : s i g n i f i c a n c e l e v e l ( t y p i c a l l y 0 .05)%% z : ( op t i ona l )% : z=0 −> f o r c e zero i n t e r c e p t ( or non−zero i n t e r c e p t inc luded in X)% : z=1 −> f i t non−zero i n t e r c e p t ( d e f a u l t )

31 %%% OUTPUT%% beta

36 % B hat : l e a s t squares c o e f f i c i e n t% T : t−s tuden t p r o b a b i l i t y f o r each c o e f f i c i e n t% P : p−va lue s f o r each c o e f f i c i e n t%% res

41 % Y hat : p r ed i c t ed response% E : r e s i d u a l s% Z : z−score o f r e s i d u a l s ( f o r normal p r o b a b i l i t y p l o t )% var Y hat : error in p red i c t ed response%

46 % s t a t s% R2 : propor t ion o f exp la ined va r i a t i on ( r−squared )% R2 adj : ad jus td r−squared% t s i g : 0.05 s i g n i f i c a n c e l e v e l f o r c o e f f i c i e n t s%

51 % means ( inc l ud ing i n t e r c e p t )% X mean : f a c t o r mean at each d i s c r e t e response l e v e l% Y mean : s e t o f d i s c r e t e response l e v e l s% Y hat mean : p r ed i c t i on at l e v e l mean% Y hat error : var iance in pred i c t ed response f o r means

56 %% trend% X trend : s e t o f appropr ia te x va lue s f o r each f a c t o r% Y trend : trend o f p r ed i c t ed response at X trend% Y error : standard dev in pred i c t ed response ( Y trend )

61 %%% IMPORTANT NOTE: To f i t an MLR model , the number o f% samples must be g r ea t e r than the number o f independent% va r i a b l e s . As the number o f v a r i a b l e s g e t s c l o s e r to

66 % the number o f samples , the X’X inve r s e matrix may become% uns tab l e and may not e x i s t ( i . e . , s i n gu l a r X’X matrix ) .%

71 function [ beta , res , s t a t s , f ac t ,X] = BackStepRegres (X,Y, fac t , factNames , a , z ) ;

127

% load octave func t i on wrappers f o r c ompa t i b i l i t y with matlab%path ( path , ’ / home/mht/ octave /wrappers ’ ) ;

76

% se t v a r i a b l e sf a c t i n = f a c t ;X in = X;

81 % check order o f i n t e r a c t i on f a c t o r sintOrder = s ize ( fac t , 2 ) −1;

% check fo r zero−i n t e r c ep t−f l a g86 i f nargin < 3

z=1;endi f z == 1

X = [ ones ( [ length (Y) , 1 ] ) , X ] ;91 f a c t = [ [ 0 , 0 , 0 ] ; f a c t ] ;

end

n = length (Y) ; % number o f samplesk = s ize (X, 2 ) ; % number o f f a c t o r s

96

% check to see t ha t n >= k ( samples >= independent vars )i f k>n

error ( ’More v a r i a b l e s than samples , ’ , . . .’ use l e s s v a r i a b l e s and/ or omit i n t e r c e p t ’ )

101 end

% maximum number o f i t e r a t i o n s equa l to number o f v a r i a b l e sstep num = 0 ;

106 while 1

k = s ize (X, 2 ) ; % number o f f a c t o r s

% c o e f f i c i e n t s and p r ed i c t i on s111 X ha l f i nv = inv (X’ ∗ X) ;

X inv = X ha l f i nv ∗ X’ ;B hat = X inv ∗ Y;Y hat = X ∗ B hat ;

116

% model c l o s ene s s (R−Squared , Adjusted R−Squared )SSE = (Y − X ∗ B hat ) ’ ∗ (Y − X ∗ B hat ) ;SST = sum( [ (Y − mean(Y) ) . ˆ 2 ] ) ;R2 = 1−SSE/SST ;

121 i f (0 )R2 adj = 1 − (1 − R2) ∗ ( ( n − 1) /(n − k − 1) ) ;

elseR2 adj = 1 − (1 − R2) ∗ ( ( n − 1) /(n − k ) ) ;

end126

% s t a t i s t i c a l s i g n i f i c a n c e o f c o e f f i c i e n t s (T−Value , P−Value )S2 = SSE/(n−k ) ;var B hat = diag ( X ha l f i nv ) ∗ S2 ;

131 s td B hat = var B hat . ˆ (1/2) ;T = B hat .∗ (1 . / std B hat ) ;t s i g = abs ( t inv ( 0 . 0 5/2 , ( n−k ) ) ) ;P = tcd f (−abs (T) , ( n−k ) ) ∗2 ;

136

% prec i s i on o f p r e d i c t i on svar Y hat = X ∗ X ha l f i nv ∗ X’ ∗ S2 ;

141 % check model assumptions ( r e s i dua l s , normal p r o b a b i l i t y )E = Y − Y hat ; % re s i d u a l sE ord = [E, [ 1 : n ] ’ ] ;E ord = [ [ 1 : n ] ’ , so r t rows ( E ord , 1 ) ] ; % re s i d u a l s ordered by s i z e

128

Z ord = norminv ( ( E ord ( : , 1 ) −0.5) / max( E ord ( : , 1 ) ) , 0 , 1 ) ;146 Z ord = [ E ord ( : , 3 ) , Z ord ] ; % Z−Score f o r E ord

Z = sort rows ( Z ord , 1 ) ;Z = Z ( : , 2 ) ; % Z−Score o f r e s i d u a l s in o r i g i n a l order

% arrange r e g r e s s i on r e s u l t s151 beta = [ B hat , T, P, var B hat ] ;

r e s = [ Y hat , E, Z ] ;s t a t s = [R2 , R2 adj , t s i g ] ;

% wri t e r e g r e s s i on r e s u l t s from t h i s s t ep to l a t e x t a bu l a r f i l e156 LatexTabMed( step num , fac t , factNames , s t a t s (1 ) , s t a t s (2 ) ,beta ( : , 3 ) ,beta ( : , 1 )

) ;

% f ind drop candidatefFact = [ fac t ,P, zeros ( s ize ( fac t , 1 ) , 1 ) ] ;

161 for j = 1 : s ize ( fac t , 1 )[ int , b ] = find ( fFact ( : , 2 : 3 )==fFact ( j , 1 ) ) ;i f length ( i n t )==0

fFact ( j , 5 ) = 1 ;end

166 end[ p , dFact ] = max( [ fFact ( : , 4 ) .∗ fFact ( : , 5 ) ] ) ;

% drop current f a c t o r from fa c t o r l i s t , f a c t o r names , and from X matrixi f fFact ( dFact , 4 ) > a

171 f a c t ( dFact , : ) = [ ] ;X( : , dFact ) = [ ] ;factNames ( dFact , : ) = [ ] ;

elsebreak

176 endstep num = step num + 1 ;

end

181 % wri t e f i n a l r e g r e s s i on r e s u l t s to l a t e x t a bu l a r f i l eLatexTabMed( step num , fac t , factNames , s t a t s (1 ) , s t a t s (2 ) ,beta ( : , 3 ) ,beta ( : , 1 ) ) ;

129

Listing B.13: Produce medium verbosity latex table for each regression step% LatexTabMed .m%

3 % SYNOPSIS% Produce a medium ve r b o s i t y l a t e x t ab l e , conta in ing r e g r e s s i on r e s u l t s f o r% each s t ep in the backward s t epw i s e r e g r e s s i on .%%

8 % INPUT%% stepNum : s t ep number in the s t epw i se r e g r e s s i on%% fa c t : indexed l i s t o f f a c t o r i n t e r a c t i o n s (number o f rows shou ld equa l

13 % number o f columns in X)% column 1 = fa c t o r number% column 2 = in t e r a c t i on f a c t o r 1 (0 fo r l i n e a r terms )% column 3 = in t e r a c t i on f a c t o r 2 (0 fo r l i n e a r terms )%

18 % factNames : names o f f a c t o r s f o r r epor t ing purposes ( c e l l array )% column 1 = va r i a b l e index% column 2 = va r i a b l e id% column 3 = va r i a b l e name%

23 % rSq : r−square va lue f o r current r e g r e s s i on s t ep%% rSq : r−square va lue f o r current r e g r e s s i on s t ep%% pVal : p−va lue f o r each f a c t o r at the current r e g r e s s i on s t ep

28 %% bHat : c o e f f i c i e n t s f o r each f a c t o r at the current r e g r e s s i on s t ep%% OUTPUT%

33 % mlr s t ep x . t e x : t e x t f i l e conta in ing a s p l i t column l a t e x t a b l e o f r e s u l t s%

function funcStat = LatexTabMed( stepNum , fac t , factNames , rSq , aRSq , pVal , bHat ) ;

38

% open f i l ef i d = fopen ( [ ’ m l r s t ep ’ ,num2str( stepNum) , ’ . tex ’ ] , ’wt ’ ) ;

43 % s t a r t t a b l e environmentfpr intf ( f i d , ’ \\ f o o t n o t e s i z e \n ’ ) ;fpr intf ( f i d , ’ \\ t ab l i n e s ep =4.0 pt\n ’ ) ;fpr intf ( f i d , ’ \\ begin { tabu la r }{ l l r c l l r }\n ’ ) ;

48 % wri t e t a b l e headerfpr intf ( f i d , ’ \ t \ t \\multicolumn {7}{ c }{\\ bf Step Number %i }\\\\\n ’ , stepNum) ;fpr intf ( f i d , ’ \ t \\ h l i n e \n ’ ) ;

% wri t e column headers53 headRow = [ ] ;

headRow = [ headRow , ’ {\ bf ID} ’ ] ;headRow = [ headRow , sprintf ( ’ \ t&\t ’ ) ] ;headRow = [ headRow , ’ {\ bf Factor Name} ’ ] ;headRow = [ headRow , sprintf ( ’ \ t&\t ’ ) ] ;

58 headRow = [ headRow , ’ {\ bf P−Value} ’ ] ;fpr intf ( f i d , ’ \ t \ t%s \ t&\t { }\ t&\t%s \\\\\n ’ ,headRow , headRow) ;fpr intf ( f i d , ’ \ t \\ c l i n e {1−3} \\ c l i n e {5−7}\n ’ ) ;

% get t ab l e ’ s row s p l i t po in t63 nRows = s ize ( fac t , 1 ) ;

spl itRow = ce i l (nRows/2) ;

% wri t e t a b l e bodyfor i = 1 : spl itRow

68

% get t ab l e ’ s l e f t columnbodyRowL = [ ] ;bodyRowL = [ bodyRowL , sprintf ( ’{%s } ’ , factNames{ i , 2} ) ] ;

130

bodyRowL = [ bodyRowL , sprintf ( ’ \ t&\t ’ ) ] ;73 bodyRowL = [ bodyRowL , sprintf ( ’{%s } ’ , factNames{ i , 3} ) ] ;

bodyRowL = [ bodyRowL , sprintf ( ’ \ t&\t ’ ) ] ;bodyRowL = [ bodyRowL , sprintf ( ’ {%5.4 f } ’ , pVal ( i ) ) ] ;

% get t ab l e ’ s r i g h t column ( hand l ing odd number o f rows )78 i f ( i+spl itRow ) > nRows

bodyRowR = [ ] ;bodyRowR = [bodyRowR , sprintf ( ’{%s } ’ , ’ ’ ) ] ;bodyRowR = [bodyRowR , sprintf ( ’ \ t&\t ’ ) ] ;bodyRowR = [bodyRowR , sprintf ( ’{%s } ’ , ’ ’ ) ] ;

83 bodyRowR = [bodyRowR , sprintf ( ’ \ t&\t ’ ) ] ;bodyRowR = [bodyRowR , sprintf ( ’{%s } ’ , ’ ’ ) ] ;

elsebodyRowR = [ ] ;bodyRowR = [bodyRowR , sprintf ( ’{%s } ’ , factNames{ i+splitRow , 2} ) ] ;

88 bodyRowR = [bodyRowR , sprintf ( ’ \ t&\t ’ ) ] ;bodyRowR = [bodyRowR , sprintf ( ’{%s } ’ , factNames{ i+splitRow , 3} ) ] ;bodyRowR = [bodyRowR , sprintf ( ’ \ t&\t ’ ) ] ;bodyRowR = [bodyRowR , sprintf ( ’ {%5.4 f } ’ , pVal ( i+spl itRow ) ) ] ;

end93

fpr intf ( f i d , ’ \ t \ t%s \ t&\t { }\ t&\t%s \\\\\n ’ ,bodyRowL , bodyRowR) ;

end

98 % wri t e t a b l e f o o t e rfpr intf ( f i d , ’ \ t \\ h l i n e \n ’ ) ;footRow = [ ] ;footRow = [ footRow , ’ \multicolumn {7}{ l } ’ ] ;footRow = [ footRow , sprintf ( ’ {$Rˆ2$ = %4.3 f , \\ hskip 20pt Adjusted $Rˆ2$ = %4.3 f }

’ , rSq , aRSq) ] ;103 fpr intf ( f i d , ’ \ t \ t%s \\\\\n ’ , footRow ) ;

% end t a b l e environmentfpr intf ( f i d , ’ \\end{ tabu la r }\n ’ ) ;fpr intf ( f i d , ’ \\ normalfont \n ’ ) ;

108

% c l o s e f i l efc lose ( f i d ) ;

131

Listing B.14: Export data for generating plots of regression results% PlotData .m%% SYNOPSIS% Export data f o r genera t ing p l o t s o f r e g r e s s i on r e s u l t s . Requires f unc t i ons

5 % from ’< t h e s i s >/ana l y s i s / octave /m’%% INPUT%% X: matrix o f independent v a r i a b l e va lue s ( rows=samples , columns=f a c t o r s )

10 % Y: vec tor conta in ing va lue s f o r the dependent or response v a r i a b l e%% beta% B hat : l e a s t squares c o e f f i c i e n t% T : t−s tuden t p r o b a b i l i t y f o r each c o e f f i c i e n t

15 % P : p−va lue s f o r each c o e f f i c i e n t% var B hat : var iance o f c o e f f i c i e n t s%% res% Y hat : p r ed i c t ed response

20 % E : r e s i d u a l s% Z : z−score o f r e s i d u a l s ( f o r normal p r o b a b i l i t y p l o t )%% factNames : names o f f a c t o r s f o r r epor t ing purposes ( c e l l array )% column 1 = va r i a b l e index

25 % column 2 = va r i a b l e id% column 3 = va r i a b l e name%% OUTPUT%

30 % mlr data . t x t : column−a l i gned t e x t f i l e% − column 1 . . . nFact = reg r e s s i on f a c t o r s% − column nFact+1 = r e s i d u a l s% − column nFact+2 = z−score o f r e s i d u a l s% − column nFact+3 = response

35 %% respNames : name of response v a r i a b l e% column 1 = va r i a b l e index% column 2 = va r i a b l e id% column 3 = va r i a b l e name

40

function p l o t d a t a f i l e (X, Y, beta , res , factNames , respNames ) ;

45 % open f i l e f o r wr i t i n gf i d = fopen ( ’ mlr data . txt ’ , ’wt ’ ) ;

% generate format and header s t r i n g s50 factHead = factNames {1 ,3} ;

factBody = ’%f ’ ;for i = 2 : ( s ize (X, 2 ) )

factHead = [ factHead , ’ \ t ’ , factNames{ i , 3 } ] ;factBody = [ factBody , ’ \ t%f ’ ] ;

55 end

% add p r ed i c t i on s and r e s i d u a l spresHead = ’ Y hat\tE\ tZ ’ ;presBody = ’%f \ t%f \ t%f ’ ;

60

% add responserespHead = respNames {1 ,3} ;respBody = ’%f ’ ;

65 % wri t e main t a b l emainHead = [ factHead , ’ \ t ’ , presHead , ’ \ t ’ , respHead , ’ \n ’ ] ;mainBody = [ factBody , ’ \ t ’ , presBody , ’ \ t ’ , respBody , ’ \n ’ ] ;fpr intf ( f i d , mainHead ) ;for i = 1 : s ize (X, 1 )

70 fpr intf ( f i d , mainBody , [X( i , : ) , r e s ( i , : ) , Y( i ) ] ) ;end

132

% fac t o r mean at each d i s c r e t e response l e v e l75 fpr intf ( f i d , ’ \n ’ ) ;

fpr intf ( f i d , [ factHead , ’ \ t ’ , respHead , ’ \n ’ ] ) ;factMean = [ ] ;for i = 0 :max(Y)

l o c = find (Y == i ) ;80 i f s ize ( loc , 1 ) ˜= 0

levelMean = mean( [X( loc , : ) ,Y( loc , 1 ) ] , 1 ) ;fpr intf ( f i d , [ factBody , ’ \ t ’ , respBody , ’ \n ’ ] , levelMean ) ;

endend

85

% c l o s e f i l efc lose ( f i d ) ;

133

Listing B.15: Generate interaction plots in SVG format% Plo t In t e r a c t .m

2 %% SYNOPSIS% Export data f o r genera t ing p l o t s o f r e g r e s s i on i n t e r a c t i on r e s u l t s .% Octave only .%

7 % INPUT% beta% column 1 = B hat , l e a s t squares c o e f f i c i e n t% column 2 = T, t−s tuden t p r o b a b i l i t y f o r each c o e f f i c i e n t% column 3 = P, p−va lue s f o r each c o e f f i c i e n t

12 % var B hat : var iance o f c o e f f i c i e n t s% factNames : names o f f a c t o r s f o r r epor t ing purposes ( c e l l array )% column 1 = va r i a b l e index% column 2 = va r i a b l e id% column 3 = va r i a b l e name

17 % fa c t : indexed l i s t o f f a c t o r i n t e r a c t i o n s (number o f rows shou ld equa l% number o f columns in X)% column 1 = fa c t o r number% column 2 = in t e r a c t i on f a c t o r 1 (0 fo r l i n e a r terms )% column 3 = in t e r a c t i on f a c t o r 2 (0 fo r l i n e a r terms )

22 % in t e r : i n d i c a t e whether the model i n c l ude s a non−zero i n t e r c e p t%% OUTPUT% in t e r a c t i o n p l o t . svg : s c a l a b l e vec to r graph ic s f i l e conta in ing i n t e r a c t i on% p l o t s f o r a l l i n t e ra c t i on s , on a s i n g l e page

27 %

function P lo t I n t e r a c t (beta , factNames , f ac t , i n t e r ) ;

close a l l ;32

c = beta ( : , 1 ) ;v = find ( f a c t ( : , 2 ) ˜=0) ;

intCount=s ize (v , 1 ) ;37 pCols = round( sqrt ( intCount ) ) ;

pRows = round( intCount /pCols ) ;i f pCols∗pRows < intCount

pRows = pRows+1;end

42

automat i c r ep l o t =0;gnup l o t r aw ” reset\n ” ;

47

xS ize = pCols ∗240 ;yS ize = pRows∗180 ;strTerm = ’ g n u p l o t s e t te rmina l svg s i z e %i %i fname ” Ar i a l ” f s i z e 14 ’ ;cmd = sprintf ( strTerm , xSize , yS i ze ) ;

52 eval (cmd) ;

g n u p l o t s e t output ’ t i p i n t e r a t i o n p l o t s . svg ’ ;

57 for i =1: length ( v )j = v ( i ) ;

% get vec to r p o s i t i o n s from fa c t o r i nd i c e sixFact1 = find ( f a c t ( : , 1 )==f a c t ( j , 2 ) ) ; % fac t o r 1 array index

62 ixFact2 = find ( f a c t ( : , 1 )==f a c t ( j , 3 ) ) ; % fac t o r 2 array index

% put the l a r g e r f a c t o r f i r s ti f abs ( c ( ixFact2 ) ) > abs ( c ( ixFact1 ) )

ixFact2 = find ( f a c t ( : , 1 )==f a c t ( j , 2 ) ) ; % fac t o r 2 array index67 ixFact1 = find ( f a c t ( : , 1 )==f a c t ( j , 3 ) ) ; % fac t o r 1 array index

end

% generate fac tor−low pr ed i c t i on equat ion c o e f f i c i e n t s

134

72 v l = zeros (2 , length ( c ) ) ;v l ( : , 1 ) = i n t e r ; %in t e r c e p tv l (1 , ixFact1 ) = −1;v l (2 , ixFact1 ) = 1 ;v l ( : , ixFact2 ) = −1;

77 v l ( : , j ) = v l ( : , ixFact1 ) .∗ v l ( : , ixFact2 ) ;

% generate fac tor−high p r ed i c t i on equat ion c o e f f i c i e n t sv h = zeros (2 , length ( c ) ) ;v h ( : , 1 ) = i n t e r ; %in t e r c e p t

82 v h (1 , ixFact1 ) = −1;v h (2 , ixFact1 ) = 1 ;v h ( : , ixFact2 ) = 1 ;v h ( : , j ) = v h ( : , ixFact1 ) .∗ v h ( : , ixFact2 ) ;

87 % generate p r ed i c t i on equat ion s o l u t i o n sp l = v l ∗ c ;p h = v h ∗ c ;

92 % pr in t p l o t va lue s%p r i n t f ( ’ rows=%i , c o l s=%i , index=%i \n ’ , pRows , pCols , i ) ;plotVect = [ v l ( : , ixFact1 ) , p l , p h ] ;p r i n t f ( ’# %s \n ’ , factNames{ j , 3} ) ;p r i n t f ( ’# %s \ t%s \ t%s \n ’ , ’ x ’ , ’ p l ’ , ’ ph ’ ) ;

97 p r i n t f ( ’%f \ t%f \ t%f \n ’ , p lotVect ( 1 , : ) ) ;p r i n t f ( ’%f \ t%f \ t%f \n ’ , p lotVect ( 2 , : ) ) ;p r i n t f ( ’ \n ’ ) ;

% p l o t format and t i t l e102 subplot (pRows , pCols , i ) ;

c l e a r p l o t ( ) ;s t r T i t l e = ’ g n u p l o t s e t t i t l e ”%s ” font ”Aria l , 14” ’ ;cmd = sprintf ( s t rT i t l e , [ factNames{ ixFact1 , 3} , ’ x ’ , factNames{ ixFact2

, 3 } ] ) ;eval (cmd) ;

107 g n u p l o t s e t nokey%ax i s ( ’ square ’ ) ;% g n u p l o t s e t s i z e r a t i o 0.5

% x ax i s format t ing112 g n u p l o t s e t x t i c s border nomirror (”−1” −1, ”0” 0 , ”1” 1) font ”Aria l

, 8”g n u p l o t s e t xrange [ −1 : 1 ] no reve r s e nowriteback

cmd = sprintf ( ’ g n u p l o t s e t x l ab e l ”%s ” 0 , 0 . 7 ’ , factNames{ ixFact1 , 3} ) ;eval (cmd) ;

g n u p l o t s e t xlabel f on t ”Aria l , 10”117

% y ax i s format t ingg n u p l o t s e t ylabel ” Fa i l u r e s ” 0 . 9 , 0g n u p l o t s e t ylabel f on t ”Aria l , 10”

ymin = f loor (min(min ( [ p l , p h ] ) ) ) ;122 ymin = 1.1∗ ymin ;

ymax = ce i l (max(max( [ p l , p h ] ) ) ) ;ymax = 1.1∗ymax ;strYTics = ’ g n u p l o t s e t y t i c s border nomirror ’ ;s t rYTics = s t r c a t ( strYTics , ’ (”%d” %d ,”0” 0,”%d” %d) font ”Aria l , 8” ’ ) ;

127 cmd = sprintf ( strYTics , ymin , ymin , ymax , ymax) ;eval (cmd) ;strYRange = ’ g n u p l o t s e t yrange [ %d : %d ] noreve r s e nowriteback ’ ;cmd = sprintf ( strYRange , ymin , ymax) ;eval (cmd) ;

132 g n u p l o t s e t ylabel f on t ”Aria l , 11”

% p l o t to the f i g u r eplot ( v l ( : , ixFact1 ) , p l , ’ −3;Lo ; ’ , v h ( : , ixFact1 ) , p h , ’ −1;Hi ; ’ ) ;

137 end

automat i c r ep l o t =1;onep lot ( ) ;

135

Appendix C

Regression Procedure

Count data, especially at low mean values, follow the Poisson distribution. Our

dependent variable, pick failures, is a count of failures events, over the experiment as

described in the previous section. Therefore, we have made extensive use of Poisson

regression analysis in the following work. In particular, we have applied a backward

stepwise regression method for building a meaningful model.

In an observational study, the factors being studied have only natural variation

whereas, in a designed experiment, special variation would be intentionally introduced

to the model. For each factor, variation about the mean is smaller than would be

introduced in a designed experiment. This condition has the effect of reducing factor

significance in general, and especially that of higher order interaction terms. In view

of this, we have limited our analysis to linear terms and their first order interactions.

Our initial approach to modeling the experimental data used a linear regres-

sion method. Some of the tools and methods are included, for reference, in another

appendix. The results of the linear regression were similar to those found with Poisson

regression, but the underlying assumptions of linearity were clearly inappropriate.

Regression analysis of non-saturated models requires that the number of sam-

ples must be greater than the number of fitted independent variables. As the number

of variables gets closer to the number of samples, the solution may become unsta-

ble and may not exist (i.e., singular X ′X matrix). Because we have many variables

that are similar to each other, the regression is especially sensitive to the number of

independent variables. By limiting the factor set to first order interactions, we have

avoided problems with matrix singularity.

137

C.1 Regression Model Trimming

P-Value is defined as the probability that the statistical null hypothesis is true

. . . in other words, the probability that there is no correlation between a factor and

the response variable (bit failures). Assuming a 95% confidence interval, any factor

with P-Value greater than 0.05 is considered to be non-significant. In Section 4.2, we

defined two factor sets, of three and nine variables, to be analyzed in relation to pick

failure. Including first order interactions, the two models consist of, respectively, five

and eighteen total variables for study.

In accordance with the law of parsimony, most statisticians hold that a trimmed

regression model is more appropriate than a full model. Trimming a model is accom-

plished through a process of iteratively running the regression analysis with the least

significant factor removed (also termed Backward Stepwise Regression). The process

is stopped when only significant factors remain in the model. Most statisticians rec-

ommend that if factor interactions are included in the model, then their contributing

factors should also be included, whether or not they are significant.

In an effort to gain additional insight into pick failure phenomenon we have

included an expanded-factor variable set. This variable set is comprised of several

variables, contributing to per-pick volume, and relative to skew angle. In order to

reduce multi-collinearity, we have dropped skew angle and volume from the expanded

models. We expected that these additional regression variables would have strong

interactions. In some cases we have encountered interactions between these vari-

ables that are significant, where the underlying linear terms are not. For the results

presented and discussed here we have included these non-significant linear terms.

All models were defined using a p-value threshold of α = 0.05. An example

regression run is provided in Section C.5.

C.2 Factor Scaling and Similarity

Data for this experiment were collected in 3 different applications, as explained

in Section 5.2. Rather than developing a complete predictive model, our purpose here

has been to develop a general overview of the significant contributing factors to pick

138

wear. Because of this limitation on the scope of the analysis, we will work exclusively

with scaled factors. Working with scaled factors greatly simplifies data analysis and

display. Most plots have been generated from predicted values of the scaled regression

model.

We showed in Section 5.2 that the depth and machine advance rate varied

between experiment applications (job sites). Variation in these parameters causes

variation in the factor ‘per-pick volume’. Because per-pick volume has an approxi-

mately linear relationship to the operational parameters of the machine, the scaled

values from the different applications are similar. When comparing results from sim-

ilar material types, the scaled per-pick volumes are nearly identical. From Table 5.3,

we can see that applications one and three experienced mostly pick body failures, and

have similar operational parameters. We have therefore used an average pick volume

between material types one and three for analyzing pick body failures. For analysis of

pick tip failures, we have used the per-pick volume calculated from material type two.

The following tables present descriptive statistics for un-scaled and scaled regression

variables.

Table C.1: Descriptive statistics for un-scaled regression variables

factor min max mean stdev

attack 0.88 0.93 0.91 0.01

absSkew 0.03 0.32 0.15 0.06

volume 2.25 4.88 3.49 0.47

xBodyPD 0.32 0.67 0.50 0.08

xTipPD 0.32 0.67 0.50 0.08

rBodyPD -0.14 0.10 0.00 0.05

rTipPD -0.12 0.14 0.00 0.05

aBodyPD 0.64 5.80 5.01 1.41

aTipPD 0.48 5.57 1.22 1.33

139

Table C.2: Descriptive statistics for scaled regression variables

factor min max mean stdev

attack -1.00 1.00 -0.02 0.45

absSkew -1.00 1.00 -0.16 0.43

volume -1.00 1.00 -0.04 0.34

xBodyPD -1.00 1.00 0.06 0.45

xTipPD -1.00 1.00 0.07 0.46

rBodyPD -1.00 1.00 0.19 0.38

rTipPD -1.00 1.00 -0.10 0.37

aBodyPD -1.00 1.00 0.69 0.55

aTipPD -1.00 1.00 -0.71 0.52

C.3 Regression Results

We conducted two regression analyses, for each of the two main failure modes.

The backward stepwise criteria dropped factors below significance level α = 0.05.

C.3.1 Pick Tip Failures (Condensed Factors)

For the condensed factor model on pick tip failures, the backward stepwise

routine ran to completion without finding any significant correlation. The last factor

considered was absolute skew angle (absSkew), with a P-Value of P = 0.455, and

dispersion parameter φ = 0.686. In order to confirm these results, we re-ran the

regression using a quasi Poisson regression model, which adjusts standard errors down

for underdispersed models. The results from the quasi Poisson regression were similar,

with a P-Value of P = 0.370 for the absolute skew angle.

C.3.2 Pick Tip Failures (Expanded Factors)

The regression results shown in Table C.3 were generated from a backward

stepwise regression with a significance threshold of α = 0.05. Two interaction ef-

fects were found to be significant, and only one of the underlying factors was not

140

individually significant. Figure C.1 shows the change in predicted failure probability

distribution for each main effect. Figure C.2 shows the change in predicted mean

failure rate for each interaction effect.

Table C.3: Regression results for tip failures for expanded factor model

Factor Name Coeff P-Value(Intercept) -0.463 0.0654rTipPD -0.124 0.8651aBodyPD 0.989 0.1060aTipPD 1.136 0.0226rTipPD x aBodyPD -2.985 0.0230rTipPD x aTipPD -3.639 0.05101 φ = 0.641

rTipPD Effects

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Figure C.1: Plot of main effects on pick tip failure for expanded factor model

141

Figure C.2: Plot of interaction effects on pick tip failure for expanded-factor model

C.3.3 Pick Body Failures (Condensed Factors)

Regression on pick body failures, using condensed factors, returned two sig-

nificant predictors, and one significant interaction. Table C.4 lists coefficients and

significance. The dispersion parameter for this model is phi = 0.898.

Figure C.3 shows the change in predicted failure probability distribution for

each main effect. Figure C.4 shows the change in predicted mean failure rate for

the interaction effect. Note that this regression model was formed on the shortened

dataset (minus an outlier) described in a later section.

Table C.4: Regression results for body failures for expanded factor model

Factor Name Coeff P-Value(Intercept) -1.148 0.0000attack 0.816 0.1052absSkew 1.007 0.0422volume 1.748 0.0017attack x volume -2.441 0.01291 φ = 0.898

142

attack Effects

0

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absSkew Effects

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volume Effects

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Figure C.3: Plot of main effects on pick body failure for condensed factor model

Figure C.4: Plot of interaction effects on pick body failure, condensed factor model

143

C.3.4 Pick Body Failures (Expanded Factors)

The regression results shown in Table C.5 were generated from a backward

stepwise regression with a significance threshold of α = 0.05. No interaction effects

were found to be significant. Figure C.5 shows the change in predicted failure prob-

ability distribution for each main effect. Note that this regression model was formed

on the shortened dataset described in a later section.

Table C.5: Regression results for body failures for expanded factor model

Factor Name Coeff P-Value(Intercept) -8.914 0.0130xBodyPD 1.449 0.0008rBodyPD 1.200 0.0362rTipPD 1.145 0.0083aBodyPD 8.387 0.03711 φ = 0.844

xBodyPD Effects

0

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rBodyPD Effects

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rTipPD Effects

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aBodyPD Effects

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Figure C.5: Plot of main effects on pick body failure for expanded factor model

144

C.4 Model Assumptions, Closeness, and Goodness of Fit

In an observational study, the factors being studied have only ordinary vari-

ation whereas, in a designed experiment, special variation would be intentionally

introduced to the model. For each factor, variation about the mean is smaller than

would be introduced in a designed experiment. For linear regressions, the R2 value

describes the proportion of the variance in bit wear attributable to the variance in

the factors of interest. When the factors of interest have smaller variance, they will

be smaller in proportion to any environmental noise. This has the effect of lowering

the R2 values of the regression. We expect this trend to also apply to the Poisson

regression, although there is no R2 value defined for non-linear models.

For Poisson regression models, the goodness of fit is typically described in

terms of overdispersion or underdispersion. Ideally, conditional variance of the re-

gression model is equal to the conditional mean (conditional mean and variance are

parameters of the regression model, not of the raw sample data). However, this

assumption rarely holds for real data. When the conditional variance is less than

(greater than) the conditional mean, the model is underdispersed (overdispersed).

The dispersion parameter for Poisson models, φ, is defined by V = φµ. A common

method for calculating dispersion parameter is to perform a quasi Poisson regression,

which fits the dispersion parameter using a maximum-likelihood method, rather than

fixing it at unity.

Well formed linear regression models exhibit normally distributed residuals,

and can be checked using a normal probability plot. Poisson regression residuals

are not normally distributed. There are however several versions of residuals that

can be used to assess Poisson model adequacy. In the text, “Regression Analysis of

Count Data,” Zorn[8] observes: “For count data there is no one residual that has zero

mean, constant variance, and symmetric distribution. ”This leads to several different

residuals according to which of these properties is felt to be most desirable.” For the

purposes of this study, we have primarily used the Anscombe residual. The Anscombe

residual is expected to be closest to normality for many Poisson regression models,

and can therefore be examined in a normal probability plot.

145

All of the models we have evaluated in this experiment exhibit underdispersion.

Underdispersion causes regression standard errors to be overestimated, therefore, our

models are conservative. However, a synopsis covering common sources of under- or

overdispersion in data provides certain insights into the phenomenon of bit failure.

A number of conditions can cause dispersion shifts in otherwise Poisson data, includ-

ing unobserved heterogeneity, zero-inflated/zero-truncated processes, and contagion

across events.

Overdispersion is often the result of “unobserved heterogeneity.” This cir-

cumstance occurs when observations with identical predictor values have different

observed values. Unobserved heterogeneity will be found where a relevant predictor

variable is missing from the model.

Poisson models with low mean predict a certain number of zero-value obser-

vations however, in many cases, the nature of the data generating process results in

greater or fewer zeros than expected. Zero inflated observations generally result from

special circumstances called dual regime data generating processes. Count data, with

zero-truncation, will result in underdispersed models. Overdispersion is correlated

with zero inflated data.

When positive contagion is present, high values of a particular observation

lead to correspondingly high values in other observations. Negative contagion is the

opposite: high values of one observation affects low values in other observations. As

explained by Zorn[8], “positive contagion leads to count data which are overdispersed;

negative contagion corresponds to underdispersed counts.”

Of the possible sources of underdispersion, negative contagion seems most

relevant. Assuming that contagion were present, we would expect high bit failures

in a particular position to result in lower bit failures in surrounding positions. Such

a phenomenon could be explained by differences in size between new and worn bits.

Bits with higher exposure to wear and breakage will be replaced more often than

their neighbors. When these bits are restored to new, they are taller than, and will

give a certain measure of protection to their already worn neighbors. This assessment

assumes that bits are changed immediately upon failure. If failed bits are not changed

146

promptly, we would expect to observe positive contagion among bits, as a failed bit

increases exposure to its neighbors.

In the following sections, we evaluate each of the regression models defined

above. Our evaluation includes analysis of the dispersion, a check for influential

observations and outliers, and examination of model residuals.

We have checked each regression model for dual regime processes (zero-inflation),

using a zero inflated Poisson regression model. This type of model combines Poisson

regression with a binomial (true/false) regression, and compares results. No evidence

of zero inflation was found in the models presented here.

C.4.1 Tip Failures, Expanded Factors

Regression on tip failures, with expanded factors, returned a dispersion param-

eter of φ = 0.641. In count regression literature, a dispersion parameter of φ = 0.75 is

considered mild underdispersion. This model appears to be strongly underdispersed.

If contagion is playing a role in bit failures, we would expect pick tip wear-out to be

the most strongly affected failure mode, which helps to explain the low dispersion

parameter.

Anscombe Residuals Normal Probability

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Anscombe Residuals

Z-S

co

re

Figure C.6: Normal probability of residuals for expanded-factor regression on picktip failures

147

Another problem with this regression model becomes apparent upon examining

the residuals (Figure C.6). The residuals are obviously not normally distributed, and

in fact, indicate the presence of a bimodal distribution. In a previous section, we

described the pick tip failure mode as a simple wear-out phenomenon. Rather, there

seem to be two different failure modes in this data set. From our observation of

milling machine operation, especially in “alligatored” material, the second failure

mode is most likely sudden tip breakage, resulting from encounters with large rocks.

The problem we are observing here is not unobserved heterogeneity, and can-

not be fixed by adding additional predictor variables. This bimodal distribution is

a variant on the dual regime data generating process, and would ordinarily cause

overdispersion. However, the low dispersion value suggests that the bimodal nature

of the data is weak compared to the size of the response.

Influential observations are displayed in Figure C.7. This plot is based on

Cook’s distance, and can be helpful in identifying potential outliers. To statistically

analyze these influential observations, we performed a Bonferroni Outlier Test. We

found that observation number 47 (bit number 98) was the most extreme, but failed

to meet outlier criteria (i.e. PBonferroni < 0.1).

Influential Observations (Cook's Distance)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 20 40 60 80 100 120

Bit Number

Co

ok S

tati

sti

c

Figure C.7: Influential observations among pick tip failures

148

C.4.2 Body Failures, Expanded Factors

An initial regression run on pick body failures, with both condensed and ex-

panded factors, showed a probable outlier at bit number 32 (observation number 10).

We have therefore dropped this observation from the dataset for all pick body regres-

sion models. Based on Cook’s distance, the adjusted dataset appears to produce a

more reasonably formed model (compare Figures C.9 and C.10). Additionally, the

resulting coefficients appear to be a better match with the theory. The comparison

of Table C.6 indicates that for the shortened dataset, body failures are more closely

correlated with body side geometry.

This regression model resulted in a dispersion parameter of φ = 0.844. While

the Anscombe residuals exhibit a slightly skewed distribution, the results generally

indicate a good fit to the Poisson distribution.

Deviance Residuals Normal Probability

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Deviance Residuals

Z-S

co

re

Figure C.8: Normal probability of residuals for expanded-factor regression on pickbody failures

149


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 20 40 60 80 100 120

Bit Number

Co

ok S

tati

sti

c

Figure C.9: Influential observations among pick body failures (expanded factors,including outlier)


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 20 40 60 80 100 120

Bit Number

Co

ok S

tati

sti

c

Figure C.10: Influential observations among pick body failures (expanded factors)

150

Table C.6: Comparison of coefficients for pick body outlier regression model

Outlier Model Short ModelFactor Name Coeff Factor Name Coeff(Intercept) -16.365 (Intercept) -8.914xBodyPD 1.609 xBodyPD 1.449xTipPD 15.964 rBodyPD 1.200rTipPD -8.588 rTipPD 1.145aBodyPD 17.176 aBodyPD 8.387aTipPD -0.066xTipPD x aBodyPD -17.848rTipPD x aTipPD -11.985

C.4.3 Body Failures, Condensed Factors

A backward stepwise regression on body failures, using condensed factors,

produced a model with dispersion parameter φ = 0.898. This value, being close

to unity, indicates a good fit with the Poisson distribution. Figure C.11 shows a

normal probability plot for the Anscombe residual. Although the Anscombe residual

is not always useful for non-linear regression, the “S” shape of the plot indicates lower

variance in the residuals than expected for a normal distribution.

Figure C.12 presents a plot of Cook’s distance for influential observations.

Comparison with Figure C.10 indicates that the two models on pick body failure are

quite similar. Note that both models use the shortened dataset, as described in the

preceding section.

151

Anscombe Residuals Normal Probability

-5

-4

-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3

Anscombe Residuals

Z-S

co

re

Figure C.11: Normal probability of residuals for condensed-factor regression on pickbody failures


0

0.05

0.1

0.15

0.2

0.25

0 20 40 60 80 100 120

Bit Number

Co

ok S

tati

sti

c

Figure C.12: Influential observations among pick body failures (condensed factors)

152

C.5 Example Backward-Stepwise Regression

The following set of tables contain the steps taken in a backward stepwise

regression. This model regresses condensed factors against pick body failures, with a

significance threshold of α = 0.05. A total of 2 steps were required to arrive at the

final model. Note that the AIC value aids in Poisson model selection, much like R2

aids in linear model selection.

Step Number 0Factor Name Coeff P-Value(Intercept) -1.099 0.0000attack 0.893 0.0754absSkew 1.361 0.0231volume 1.801 0.0012attack x absSkew -1.001 0.3217attack x volume -2.506 0.0453absSkew x volume -0.710 0.61821 AIC = 123.61 φ = 0.871

Step Number 1Factor Name Coeff P-Value(Intercept) -1.092 0.0000attack 0.901 0.0713absSkew 1.277 0.0259volume 1.794 0.0013attack x absSkew -0.794 0.3794attack x volume -2.785 0.01291 AIC = 121.91 φ = 0.850

153

Step Number 2Factor Name Coeff P-Value(Intercept) -1.148 0.0000attack 0.816 0.1052absSkew 1.007 0.0422volume 1.748 0.0017attack x volume -2.441 0.01291 AIC = 120.71 φ = 0.842

154

C.6 Regression Model Variable Values

The following table contains unscaled values for each of the variables used in regression analysis. The values in the table

are both calculated and measured, as described in Chapter 4.

Table C.7: Unscaled regression model variable values

bitNum intercept attack absSkew volume xBodyPD xTipPD rBodyPD rTipPD aBodyPD aTipPD tipFails bodyFails

2 1 0.9112 0.1864 4.2488 0.3941 0.5244 0.0404 0.1411 5.6769 1.0115 0 1

3 1 0.9011 0.1801 3.0404 0.5716 0.4848 0.0878 0.0918 1.0169 1.4983 2 0

4 1 0.9012 0.1986 4.4526 0.4579 0.6584 0.0499 0.0657 5.3321 1.0048 0 1

5 1 0.9170 0.2317 3.4797 0.4289 0.4859 0.0199 0.0230 5.3297 1.0024 0 1

6 1 0.9053 0.2027 4.2778 0.5694 0.6199 -0.0125 -0.0129 5.8005 0.8798 1 1

7 1 0.9263 0.2703 2.8524 0.4800 0.3941 -0.0118 -0.0404 5.8041 0.6063 0 2

8 1 0.9067 0.2353 3.4320 0.4958 0.3781 -0.0249 0.0003 1.2448 5.4517 1 0

10 1 0.9050 0.1423 3.6187 0.4239 0.5694 0.0171 0.0125 5.5633 0.4827 0 0

11 1 0.9284 0.2533 3.5683 0.5696 0.4800 0.0190 0.0118 5.5827 0.4791 0 1

12 1 0.9020 0.2250 3.0091 0.4417 0.4579 -0.0258 -0.0499 5.4370 0.9511 1 0

13 1 0.9070 0.2214 3.1368 0.4154 0.4289 0.0266 -0.0199 5.4441 0.9535 1 0

15 1 0.9065 0.1653 3.1572 0.3781 0.5676 -0.0003 0.0325 0.8315 5.5672 1 0

16 1 0.9084 0.1270 2.9891 0.3158 0.4239 0.0114 -0.0171 5.4474 0.7199 0 0

17 1 0.9015 0.1558 3.6555 0.4332 0.5696 0.0251 -0.0190 5.4369 0.7005 1 0

19 1 0.9313 0.2201 3.5655 0.4237 0.4417 0.0737 0.0258 5.5669 0.8462 1 0

20 1 0.9042 0.1968 3.1062 0.5090 0.4154 0.0161 -0.0266 5.5750 0.8391 1 1

21 1 0.8927 0.1390 3.6901 0.5676 0.5071 -0.0325 0.0004 0.7160 5.5697 1 0

22 1 0.9348 0.3177 3.0646 0.6617 0.3899 0.0970 -0.1184 5.5830 0.9532 0 2

23 1 0.9187 0.1950 2.8531 0.5815 0.3158 0.0569 -0.0114 5.5730 0.8358 0 2

24 1 0.9072 0.1639 3.1569 0.4411 0.4332 0.0170 -0.0251 5.5736 0.8462 2 0

25 1 0.9147 0.2083 2.8883 0.5445 0.4237 0.0151 -0.0737 5.4498 0.7163 0 0

155

Table C.7: continued


26 1 0.8939 0.1699 3.4831 0.4949 0.5090 0.0079 -0.0161 5.4339 0.7081 0 0

27 1 0.8893 0.1390 3.5038 0.5071 0.5064 -0.0004 0.0073 0.7135 5.4373 1 0

28 1 0.8986 0.1580 3.6827 0.4738 0.6617 -0.0170 -0.0970 5.4343 0.7002 0 0

29 1 0.9054 0.1903 3.7880 0.5721 0.5815 0.0202 -0.0569 5.4341 0.7102 1 0

30 1 0.9053 0.1926 3.3064 0.5603 0.4411 0.0365 -0.0170 5.4345 0.7095 0 1

32 1 0.9080 0.2122 3.6785 0.4442 0.5445 0.0241 -0.0151 5.5630 0.8334 0 2

33 1 0.9263 0.2035 3.8634 0.6251 0.4949 0.0412 -0.0079 5.5676 0.8493 0 0

34 1 0.9204 0.2354 3.5652 0.5064 0.5230 -0.0073 -0.0012 0.8459 5.5539 0 0

35 1 0.8997 0.2120 3.5715 0.5173 0.4738 0.0406 0.0170 5.5648 0.8488 0 1

36 1 0.9175 0.1739 3.5746 0.3513 0.5721 -0.0059 -0.0202 5.5631 0.8491 1 0

37 1 0.9172 0.2023 3.5522 0.4965 0.5603 -0.0196 -0.0365 5.5701 0.8487 0 0

38 1 0.8936 0.1905 3.2070 0.6147 0.4442 -0.0298 -0.0241 5.4315 0.7202 1 1

39 1 0.8910 0.1701 3.6941 0.4714 0.6251 -0.0042 -0.0412 5.4384 0.7156 1 0

40 1 0.9318 0.1870 3.4880 0.5230 0.4923 0.0012 -0.0125 0.7293 5.4560 1 0

41 1 0.9111 0.1968 2.9774 0.4398 0.5173 -0.0877 -0.0406 5.4322 0.7184 0 0

42 1 0.9124 0.2542 2.8863 0.6653 0.3513 -0.0657 0.0059 5.4350 0.7201 0 0

43 1 0.8972 0.2516 2.9704 0.4848 0.4965 -0.0918 0.0196 4.7849 0.7131 1 0

45 1 0.9045 0.1521 3.7792 0.4840 0.6147 -0.0980 0.0298 5.5630 0.8516 0 0

46 1 0.8943 0.1517 2.9671 0.5244 0.4714 -0.1411 0.0042 5.2717 0.8448 0 0

47 1 0.8904 0.2068 3.3047 0.4923 0.5716 0.0125 -0.0878 0.8272 5.2663 1 0

48 1 0.9120 0.2647 3.7069 0.6584 0.4398 -0.0657 0.0877 5.2784 0.8510 0 1

49 1 0.9233 0.2482 4.2030 0.4859 0.6653 -0.0230 0.0657 5.2808 0.8481 1 1

50 1 0.9342 0.2305 4.8831 0.4836 0.4958 0.0036 0.0249 5.6396 5.0384 2 0

51 1 0.9133 0.2250 4.0767 0.6199 0.4840 0.0129 0.0980 5.4034 0.7202 1 3

53 1 0.8929 0.0829 4.4607 0.4259 0.5340 0.0703 0.1325 5.6854 1.0272 0 1

54 1 0.8915 0.0846 2.2511 0.4836 0.5004 -0.0036 0.0796 0.6435 1.0162 2 0

55 1 0.9047 0.1414 4.5331 0.5862 0.6168 0.0638 0.0313 5.3335 1.0337 1 1

156



56 1 0.9119 0.1016 3.8695 0.4972 0.5357 0.0422 0.0051 5.3418 1.0088 0 0

57 1 0.9162 0.0942 3.4581 0.4178 0.4829 0.0092 -0.0320 5.7985 0.8994 0 1

58 1 0.9033 0.1083 2.8994 0.4957 0.4259 0.0051 -0.0703 5.8013 0.5978 1 0

59 1 0.9000 0.1489 3.1500 0.6568 0.3408 0.0208 -0.0257 5.4350 1.2406 2 0

61 1 0.9262 0.1598 3.1251 0.5622 0.4178 0.0132 -0.0092 5.5754 0.4847 1 1

62 1 0.9178 0.0517 3.3680 0.4459 0.4957 0.0125 -0.0051 5.5815 0.4819 0 0

63 1 0.8994 0.0969 3.5373 0.4916 0.5862 -0.0460 -0.0638 5.4515 0.9496 0 1

64 1 0.8919 0.0892 3.4103 0.5118 0.4972 0.0024 -0.0422 5.4438 0.9413 0 0

66 1 0.9002 0.0680 4.1498 0.4659 0.6568 0.0144 -0.0208 5.5803 0.8481 2 0

67 1 0.9141 0.1233 3.7761 0.5572 0.5622 -0.0160 -0.0132 5.4578 0.7078 0 1

68 1 0.8967 0.1137 3.3023 0.6024 0.4459 -0.0213 -0.0125 5.4329 0.7017 1 0

70 1 0.9022 0.0996 3.6814 0.4724 0.4916 0.0186 0.0460 5.5626 0.8317 1 1

71 1 0.8987 0.1218 3.5792 0.4526 0.5118 0.0114 -0.0024 5.5684 0.8393 1 0

72 1 0.8813 0.0996 3.3877 0.5488 0.4659 0.0146 -0.0144 5.5664 0.7029 1 0

73 1 0.9246 0.1486 4.6731 0.6589 0.6382 0.0684 -0.0436 5.5787 0.9611 0 2

74 1 0.8909 0.1074 3.8905 0.4562 0.5572 0.0296 0.0160 5.5606 0.8254 1 0

75 1 0.9145 0.0772 4.2167 0.5027 0.6024 0.0443 0.0213 5.5801 0.8503 1 0

76 1 0.9030 0.1159 3.2176 0.4859 0.4724 -0.0036 -0.0186 5.4498 0.7206 0 1

77 1 0.8928 0.1233 3.3104 0.5752 0.4526 -0.0019 -0.0114 5.4499 0.7147 0 0

78 1 0.8861 0.0862 3.4791 0.4625 0.5488 -0.0054 -0.0146 5.4337 0.7168 0 0

79 1 0.9029 0.1176 3.8929 0.4962 0.6589 0.0122 -0.0684 5.4421 0.7045 0 0

80 1 0.9041 0.1306 3.2449 0.4971 0.4562 0.0249 -0.0296 5.4431 0.7226 1 0

81 1 0.8897 0.0650 3.2905 0.4240 0.5027 0.0419 -0.0443 5.4329 0.7031 0 0

83 1 0.9043 0.0934 3.5819 0.4218 0.4859 0.0502 0.0036 5.5678 0.8334 0 1

84 1 0.8923 0.0653 3.8012 0.3720 0.5752 0.0177 0.0019 5.5525 0.8333 0 0

85 1 0.8965 0.0849 3.3825 0.4014 0.4625 0.0341 0.0054 5.5616 0.8495 1 0

86 1 0.8943 0.1025 3.2658 0.3662 0.4962 0.0343 -0.0122 5.5647 0.8410 0 0

157



87 1 0.9087 0.1245 3.6642 0.5897 0.4971 0.0290 -0.0249 5.5713 0.8401 1 0

88 1 0.8955 0.1435 2.9388 0.4948 0.4240 -0.0120 -0.0419 5.5660 0.8502 1 0

89 1 0.8931 0.2022 2.9639 0.6005 0.4218 -0.0310 -0.0502 5.4402 0.7154 1 0

90 1 0.9126 0.1562 2.9883 0.6284 0.3720 0.0099 -0.0177 5.4536 0.7307 0 0

91 1 0.9110 0.1602 2.9193 0.5860 0.4014 -0.0126 -0.0341 5.4461 0.7216 1 0

92 1 0.9061 0.1578 2.3621 0.4156 0.3662 -0.0663 -0.0343 5.4485 0.7184 0 0

93 1 0.8807 0.0556 3.1750 0.3827 0.5897 -0.0921 -0.0290 5.4170 0.7119 0 0

94 1 0.8768 0.1076 3.2082 0.5186 0.4948 -0.0873 0.0120 5.4344 0.7172 1 0

96 1 0.8968 0.0717 3.8501 0.5192 0.6005 -0.0831 0.0310 5.5542 0.8430 1 0

97 1 0.8897 0.0554 3.6090 0.5340 0.6284 -0.1325 -0.0099 5.2560 0.8295 0 0

98 1 0.8870 0.1085 3.5845 0.5004 0.5860 -0.0796 0.0126 5.2669 0.8371 2 0

99 1 0.8854 0.1261 3.4717 0.6168 0.4156 -0.0313 0.0663 5.2495 0.8347 1 0

100 1 0.9285 0.0412 3.3251 0.5357 0.3827 -0.0051 0.0921 5.2744 0.8662 2 0

101 1 0.9105 0.0335 3.6488 0.3408 0.5186 0.0257 0.0873 5.0426 0.8488 1 0

102 1 0.9133 0.0625 3.9062 0.4829 0.5192 0.0320 0.0831 5.3838 0.7290 1 0

158

Appendix D

Cutter Head Measurement

We used a Romer/Cimcore 10 foot Infinite Arm model CMM to measure the

location and orientation of each pick on a selected cutter head. These measurements

were taken on empty holders, and then transformed based on the shape of the pick

to be used in our study. This approach is possible because of the relatively low

dimensional variation between picks. Figure D.1 shows the setup and apparatus used

in collecting measurements.

In order to more accurately represent cutting conditions, we measured the

completely assembled cutter head (with its drive system). We used the drive system

mounting surfaces as measurement reference surfaces. Because the CMM arm could

not reach completely around the cutter head, we were required to reposition the CMM

three times while taking measurements. To accomplish the repositioning, we fastened

four large rigid plates to the cutter head skin at each quadrant. One of the plates

can be seen in Figure D.1.

159

Figure D.1: CMM measurement apparatus and setup

Using Romer/Cimcore software, we created a ‘part program’ consisting of sev-

eral measurement defined features. The program first received several measurements

to create a coordinate system (described in Figure D.2). For each pick measurement,

program data was saved to a spreadsheet, and the program was reset. This resulted

in 110 separate spreadsheets that were combined programmatically for a complete

dataset. A sample spreadsheet, generated by the part program, is attached as Ap-

pendix D.2

160

Support Shaft End Face

Z-Axis

Support Shaft OD

Input Shaft OD

Pick Holder 52

X-Axis

Y-Axis

Figure D.2: Coordinate system created in the CMM part program

The data returned by the part program required post-processing to convert

from the CMM reference frame to measures more commonly used in pick lacing

specifications. Data returned by the part program consisted of clock angle, forward

angle, side angle, block radius, and axial distance. Using several simple trigonometric

operations, we transformed the foregoing variables into Drum Angle, Attack Angle,

Skew Angle, Tip Radius, and Axial Position. Each of these variables are further

defined here, and in Section 2.3.2. A listing of chapter variables, relative to the

geometry presented in Figure D.2, is provided in Section D.1.

Equation D.1 shows the transform for converting clock angle to the angular

position required for a lacing specification. The contributing variables are illustrated

in Figures D.3, D.4, and D.5. This equation depends primarily on the length and

orientation of the pick.

161

lp = la cos (sa)

caa = arctan

[lp sin (fa)

rb + lp cos (fa)

]

φd =

−(ca + caa) + ca1 if ca + caa < 3

−(ca + caa) + ca1 + 2π if ca + caa ≥ 3(D.1)

Where:

la = Actual bit length

lp = Projected bit length

sa = Block side angle

fa = Forward angle

rb = Radial distance to block reference point

ca = Clock angle

caa = Clock angle adjustment

ca1 = Adjusted clock angle for bit 1

φd = Drum angle

Equation D.2 shows the calculation for transforming block radius, rb, into

pick tip radius, R. The geometry for this calculation is quite simple. Contributing

variables are illustrated in Figure D.3.

162

R =[r2b + l2p − 2rblp cos (π − fa)

]1/2(D.2)

Where:

lp = Actual bit length

fa = Forward angle

rb = Radial distance to block reference point

R = Pick tip radius

Figure D.3: Angular construction used to find individual pick tip radius R

Equation D.3 converts a pick mounting block’s forward angle to an attack

angle. Attack angle (φa) is measured from the X-Y plane (pick path plane) to a

projection of the pick axis onto the plane tangent to the pick’s cutting path. Figure

163

D.5 illustrates these plane definitions. Variables involved in this transformation are

shown in figure D.4.

φa = π/2 + caa − fa (D.3)

Where:

fa = Forward angle

caa = Clock angle adjustment

φa = Pick attack angle

Figure D.4: Angular construction used to find each pick’s attack angle φa, and clockangle adjustment caa

164

φs = arctan

[sin (sa)

cos (φa) cos (sa)

](D.4)

Where:

sa = Block side angle

φa = Pick attack angle

φs = Pick skew angle

Block Reference Point

φ

pl

a

s a

AxisBlock Bore

s

l

Block BoreAxis Projection

Figure D.5: Illustration of variables for transforming measured side angle (sa) intoskew angle φs

Some of the features used to calculate each pick’s Skew Angle are shown in

Figure D.5. As described in Equation D.4, skew angle can be determined solely from

a block’s side angle and attack angle. Side angle was directly measured with the

CMM, and attack angle was calculated in Equation D.3.

165

The CMM setup measured mounting block axial position based on the end

surface of the cutter head support shaft. In theory, standard lacing specifications

measure the pick tip axial position from the edge of the cutter head skin (as shown

in engineering drawings). However, in practice, axial position is more appropriately

measured from the zero-bit. The zero-bit is the first pick where the tip is axially

aligned with the left side of the cutter head skin (viewed from the front of the ma-

chine). Equation D.5 converts axial position as measured by the CMM to that of the

lacing specification.

Pick skew angle (φs) is defined in terms of a plane tangent to the path of the

pick tip. This allows us to find axial offset from skew angle and pick length, without

transformation. The pick skew angle was calculated using Equation D.4.

Da = ad0 − [ad − la sin (φs)] (D.5)

Where:

ad0 = Axial distance from CMM origin to zero-bit

ad = Axial distance from CMM origin to block reference point

la = Actual bit length

φs = Pick skew angle

Da = Axial coordinate of pick tip

166

D.1 Variables Summary

Block measurements, based on CMM coordinate system:

ca = Clock angle, measured from X-Z plane (based on bit number 52) to

origin of individual block, around cutter head axis of rotation

caa = Clock angle adjustment, converting from block clock angle to pick tip

clock angle

ca1 = Adjusted clock angle, measured from origin X-Z plane to reference point

of block number 1

fa = Forward angle, measured from radius vector for the block reference

point to the axis of the block’s bore, projected onto the X-Y plane

sa = Block side angle, the smallest angle from the block bore axis to the pick

path plane

rb = Radial distance to block origin, measured from CMM coordinate system

z-axis

ad = Axial distance from cutter head CMM coordinate system origin to block

reference point

ad0 = Axial distance from CMM coordinate system origin to tip of lacing

pattern zero-bit (bit number 9)

Bit parameters, measured in lacing specification coordinate system:

φd = Drum angle (or angular position), measured from cutter head skin

origin-line to individual pick tip

φa = Pick attack angle, measured from pick path plane to pick axis, projected

onto cut tangent plane

φs = Pick skew angle, measured from the X-Y plane to the projection of the

pick axis onto the cut-tangent plane

R = Individual pick tip radius, from rotational axis of cutter head

Da = Axial coordinate of a pick tip location, relative to reference point on

edge of cutter head skin

Pick related parameters:

la = Actual bit length (new), measured from shoulder to tip

lp = Projected bit length (new), actual bit length projected onto pick path

plane (X-Y plane in CMM coordinate system)

167

D.2 Sample Part-Program Report

��

� �� 0.000�� 0.000

� �� 0.000�� 3.137 3.037

��

� �� 0.000�� 0.000

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defining a model for tool consumption rate on asphalt

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