c 2008 abhijit bhattacharyyaufdcimages.uflib.ufl.edu/uf/e0/02/26/63/00001/bhattacharyya_a.pdf ·...

PREDICTIVE FORCE MODELING OF PERIPHERAL MILLING

By

ABHIJIT BHATTACHARYYA

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

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c© 2008 Abhijit Bhattacharyya

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ACKNOWLEDGMENTS

Dr. John K. Schueller, my revered advisor, made it possible for me to go back to

school. His method of training appears to be unique. He would ask only the simplest of

questions which sent me hunting for deeper answers. He let me set my own pace, provided

a kind of intellectual freedom that was the envy of my fellow students in the laboratory,

and encouraged me to keep my work uncomplicated. It is impossible for me to repay my

debt to him.

Dr. Brian P. Mann has been an inspiration as well as a coach. His work ethic and

self-discipline puts his graduate students to shame.

Dr. Fred J. Taylor has been most generous to me for reasons that are quite

unfathomable. He succeeded in teaching me a semblance of digital signal processing, a

subject whose non-trivial nature is not lost upon me. In addition to the depth of his

teachings, he combines a breadth of vision with a sense of humor that endears him to the

student community.

Dr. John C. Ziegert introduced me to the fascinating subject of uncertainty. The

imprint of his teaching is evident all over this document. While teaching manufacturing to

senior undergraduate students, it was my effort to adopt his methods of instruction as best

one could hope to imitate.

Dr. Tony L. Schmitz taught me the fundamentals of the structural dynamics of

production machinery. He suggested the idea of letting me teach the senior undergraduate

class. He has more physical stamina than most people, allowing him to work long hours to

produce a large volume of valuable contributions to the literature.

Dr. Norman G. Fitz-Coy has a way of influencing the minds of his students that

is hard to explain. He gave me a perspective into the subject of dynamics that is not

possible to obtain from the textbooks. His lectures titillate the senses. It was my good

fortune to have been his student, watching his blackboard from the backbench, and giving

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him a hard time with an unending string of questions. Never did he betray any irritation,

adding his warm smile to his answers, instead.

Mr. Michael Braddock taught me more about machine shop practice than any other

person. His answers to my amateurish questions were always illuminating. His generosity

in permitting me to work in his laboratory is most gratefully acknowledged.

Many other teachers, staff and friends at the university contributed to my education.

They are too many to identify individually but the names of Dr. Nicolae D. Cristescu, Dr.

Raphael T. Haftka, Dr. Chen-chi Hsu, Dr. Andre Khuri, Dr. Wallace Gregory Sawyer,

Dr. David W. Hahn, Dr. Hitomi Yamaguchi Greenslet, Dr. Nagaraj K. Arakere, Dr. S.

Balachandar, Dr. Gene W. Hemp, Dr. Louis N. Cattafesta III, Dr. Z. Hugh Fan, Dr.

Gerald Bourne, Dr. Roberto Albertani, Dr. Fereshteh Ebrahimi, Dr. Tim Dalrymple,

Ms. Maud Fraser, Ms. Leslie A. Owen, Ms. Debra Anderson, Ms. Susan J. Studstill, Ms.

Genevieve Blake, Mr. Mark Riedy, Ms. Rebecca Hoover, Ms. Pam Simon, Ms. Sonya

Clements, Ms. Salena Robinson, Ms. Shirley Robinson, Ms. Jennifer Brown, Ms. Teresa

Mathia, Ms. Jan Rockey, Mr. Vann Chesney, Mr. John West, Mr. Jeff Studstill, and Mr.

Ron Brown, deserve to be recorded.

Ms. Renee Barfield and her pleasant crew, in general, and Ms. Beverly Wallace and

Ms. Dorothy Schuller in particular, worked diligently to keep our surroundings hygienic.

They ensured a clean house to which we could come to work every morning, and added

generous doses of laughter to provide the human touch that is so necessary to maintain

sanity in the workplace. The team is too large for individual names to be mentioned, but I

owe my gratitude to them all for their friendship.

Most of my students in the senior undergraduate manufacturing classes had difficulty

realizing that I actually cared for them, mistaking my efforts to provide them with a set

of practically useful skills for a lack of understanding of their needs. The conflict was

natural, but it is my hope that most of them took away some tangible benefits from their

forced association with me. Indeed, many of them have acknowledged my efforts in their

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anonymous feedback forms, as well as expressed their happiness in person, though some

have been less than happy. They taught me more important lessons than they learnt from

me. Here, I wish to acknowledge them all for playing a role in my education.

The students at the Machine Tool Research Center provided an atmosphere in which

it was a pleasure to work. So too did the students of the Tribology Laboratory and the

Center for Intelligent Machines and Robotics. Many friendships were struck. It matters

little that most of them are not named here. Watching them at close proximity offered

me insights into how young minds function. The constant attention and assistance that

I have received from my very good friends, Scott W. T. Payne, Vadim J. Tymianski, Dr.

Mohammad H. Kurdi, Duke Hughes, Michelle Zahner, Christopher Zahner, and Jean

Francois Kamath, have made it possible for me to complete this work. Their selflessness

knows no bounds.

Last, but not least, the generosity of the people of the United States of America must

be acknowledged. They have treated me as their own, provided me with the opportunity

to get an education in this wonderful country, and offered me the respect due to a visiting

guest. They have indulged me in every which way, chauffeuring me around, serving me at

the dining table, accompanying me to the theaters and the stadiums, letting me drive their

cars, and even inviting me into their homes. It has been my effort to conduct myself with

dignity. I hope I did not disappoint my hosts.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

CHAPTER

1 OVERVIEW AND MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1 Motivation for Predictive Force Modeling of Peripheral Milling . . . . . . . 251.2 Review of the Open Literature . . . . . . . . . . . . . . . . . . . . . . . . . 261.3 Justification and Scope of the Work . . . . . . . . . . . . . . . . . . . . . . 34

2 DESCRIPTION OF THE PERIPHERAL MILLING PROCESS . . . . . . . . . 37

2.1 The Machine and the Cutting Tool . . . . . . . . . . . . . . . . . . . . . . 372.2 Chip Area at the Tool Workpiece Interface . . . . . . . . . . . . . . . . . . 382.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 CHIP GEOMETRY IN HELICAL PERIPHERAL MILLING . . . . . . . . . . . 43

3.1 Tool Chip Contact Configurations . . . . . . . . . . . . . . . . . . . . . . . 443.2 Analytical Expressions for θL, θT , and Chip Width, b . . . . . . . . . . . . 46

3.2.1 Heaviside Unit Step Function Formulation . . . . . . . . . . . . . . 463.2.2 Fourier Trigonometric Series Formulation . . . . . . . . . . . . . . . 47

3.3 Analytical Expressions for Chip Thickness . . . . . . . . . . . . . . . . . . 493.4 Effect of Process Parameters on Chip Thickness and Chip Width . . . . . 503.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 MECHANISTIC FORCE MODEL FOR STRAIGHT FLUTED ENDMILLS . . 57

4.1 Force Model for a Single Tooth . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Average Force Based Estimates of Kt and Kn . . . . . . . . . . . . . . . . 594.3 Multiple Tooth Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4 Effects of Tooth Runout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Cutting Coefficient Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.6 Force Prediction Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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5 FORCE MODEL CALIBRATION FOR STRAIGHT FLUTED ENDMILLS . . 65

5.1 The Partial Radial Immersion Experiment . . . . . . . . . . . . . . . . . . 655.2 Experimental Extraction of Cutting Coefficients . . . . . . . . . . . . . . . 665.3 Variances of Model Input Parameters . . . . . . . . . . . . . . . . . . . . . 685.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 PROPAGATION OF UNCERTAINTIES THROUGH THE FORCE MODELFOR STRAIGHT FLUTED ENDMILLS . . . . . . . . . . . . . . . . . . . . . . 75

6.1 Propagation of Uncertainties Through a Mathematical Model . . . . . . . 756.2 Variances of Cutting Coefficients and Effective Feed Rates . . . . . . . . . 77

6.2.1 Propagation of Average Force Uncertainty to Cutting Coefficients . 776.2.2 Propagation of Radial Runout Uncertainty to Effective Feeds . . . . 78

6.3 Propagation of Uncertainties Through the Cutting Force Model . . . . . . 786.3.1 Sensitivity Coefficients of Component Uncertainties . . . . . . . . . 786.3.2 Propagation of Type A Uncertainties . . . . . . . . . . . . . . . . . 796.3.3 Propagation of Type B1 Uncertainties . . . . . . . . . . . . . . . . . 806.3.4 Propagation of Type B2 Uncertainties . . . . . . . . . . . . . . . . . 80

6.4 Expanded Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.4.1 Expanded Uncertainty Coverage Factor Type A . . . . . . . . . . . 816.4.2 Expanded Uncertainty Coverage Factor Type B . . . . . . . . . . . 836.4.3 Overall Expanded Uncertainty . . . . . . . . . . . . . . . . . . . . 83

6.5 Force Prediction with 95% Confidence Interval . . . . . . . . . . . . . . . . 846.6 Chapter Summary and Outlook for the Forthcoming Chapters . . . . . . . 84

7 INSTANTANEOUS RIGID FORCE MODEL FOR HELICAL PERIPHERALMILLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.1 Force Model for a Single Toothed Cutter . . . . . . . . . . . . . . . . . . . 927.2 Modeling for Multiple Teeth . . . . . . . . . . . . . . . . . . . . . . . . . . 947.3 Formulation for Cutting Coefficient Identification . . . . . . . . . . . . . . 957.4 Verification of the Analytical Solution . . . . . . . . . . . . . . . . . . . . . 97

7.4.1 The Degenerate Case of Straight Fluted Cutters . . . . . . . . . . . 977.4.2 Comparison with a Numerical Solution for Helical Endmills . . . . . 98

7.5 Experimental Determination of Model Input Parameters . . . . . . . . . . 997.6 Variances of Model Input Parameters . . . . . . . . . . . . . . . . . . . . . 1017.7 Propagation of Input Parameter Uncertainties Through the Force Model . 103

7.7.1 Propagation of Type A Uncertainties . . . . . . . . . . . . . . . . . 1037.7.2 Propagation of Type B1 Uncertainties . . . . . . . . . . . . . . . . . 1047.7.3 Expanded Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.8 Force Prediction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.9 Properties of the Analytical Force Model . . . . . . . . . . . . . . . . . . . 106

7.9.1 Gibbs-Wilbraham Distortion . . . . . . . . . . . . . . . . . . . . . . 1077.9.2 Computer Implementation Issues . . . . . . . . . . . . . . . . . . . 1077.9.3 Relative Merits of the Two Variants . . . . . . . . . . . . . . . . . . 108

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7.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8 EFFECTS OF DIFFERENTIAL TOOTH PITCH ON THE HELICAL FORCEMODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.1 Formulation for Differential Pitch Effects on Force Components . . . . . . 1288.2 Experimental Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . 1288.3 Force Prediction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9 EFFECTS OF RADIAL RUNOUT ON THE HELICAL FORCE MODEL . . . 135

9.1 Formulation for the Effects of Runout on Force Components . . . . . . . . 1369.2 Verification With a Numerical Solution . . . . . . . . . . . . . . . . . . . . 1369.3 Experimental Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . 1369.4 Force Prediction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

10 FORCE MODELING WITH INSTANTANEOUS CUTTING COEFFICIENTS 149

10.1 Instantaneous Cutting Coefficients . . . . . . . . . . . . . . . . . . . . . . 14910.2 Variances of Model Input Parameters . . . . . . . . . . . . . . . . . . . . . 15110.3 Propagation of Input Parameter Uncertainties Through the Force Model . 15210.4 Force Prediction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15210.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

11 DYNAMIC INFLUENCES IN FORCE MEASUREMENTS . . . . . . . . . . . 162

11.1 Description of the Force Measurement Chain . . . . . . . . . . . . . . . . . 16211.2 Frequency Response of the Force Measurement Chain . . . . . . . . . . . . 16211.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

12 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

12.1 Stability and Surface Location Problem Formulation . . . . . . . . . . . . 16812.2 Augmented Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

13 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

APPENDIX

A ZELLNER’S REGRESSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

B DERIVATION OF FOURIER COEFFICIENTS . . . . . . . . . . . . . . . . . . 182

B.1 Fourier Coefficients for θL . . . . . . . . . . . . . . . . . . . . . . . . . . . 182B.2 Fourier Coefficients for θT . . . . . . . . . . . . . . . . . . . . . . . . . . . 183B.3 Fourier Coefficients for b in Type I Cutting . . . . . . . . . . . . . . . . . . 184B.4 Fourier Coefficients for b in Type II Cutting . . . . . . . . . . . . . . . . . 185

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REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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LIST OF TABLES

Table page

2-1 Computation of entry and exit angles (Ref. Fig. 2-5) . . . . . . . . . . . . . . 40

3-1 Construction of Fourier coefficients Lk,Mk, Tk, Rk, Bk and Ck . . . . . . . . . . 51

5-1 Experimental conditions: straight fluted endmill . . . . . . . . . . . . . . . . . 72

5-2 Mean cutting constants with the straight fluted endmill . . . . . . . . . . . . . 72

5-3 Symmetric variance-covariance matrix of cutting constants for experimentalconditions of Table 5-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6-1 Summary of experimental conditions used for verification of force predictions,holding the cutting conditions of Table 5-1 fixed . . . . . . . . . . . . . . . . . 85

7-1 Parameters for comparison with the numerical solution . . . . . . . . . . . . . 111

7-2 Experimental conditions: 3-fluted, 45 helix endmill . . . . . . . . . . . . . . . 111

7-3 Mean cutting constants with the 3-fluted, 45 helical endmill . . . . . . . . . . 111


7-5 Conditions for experimental verification of force predictions with the 3-fluted,45 helix endmill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8-1 Experimental conditions: differential pitch cutter . . . . . . . . . . . . . . . . 131

8-2 Mean cutting constants with differential pitch cutter . . . . . . . . . . . . . . 131


8-4 Conditions for experimental verification of force predictions with differentialpitch cutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9-1 Experimental conditions: cutter with radial runout . . . . . . . . . . . . . . . 140

9-2 Mean cutting constants with cutter having radial runout . . . . . . . . . . . . 140


9-4 Conditions for experimental verification of force predictions with radial runout 140

10-1 Summary of experimental conditions used to verify force predictions based oninstantaneous Ktc,nc,ac, with the conditions of Table 9-1 held fixed . . . . . . . 154

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LIST OF FIGURES

Figure page

1-1 Face milling and peripheral milling (endmilling) . . . . . . . . . . . . . . . . . 36

1-2 Straight fluted and helical fluted endmills . . . . . . . . . . . . . . . . . . . . . 36

2-1 Peripheral milling on a vertical milling machine . . . . . . . . . . . . . . . . . 41

2-2 Endmill terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2-3 Instantaneous chip area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2-4 Circular path approximation for chip thickness . . . . . . . . . . . . . . . . . . 42

2-5 Up-milling and down-milling configurations . . . . . . . . . . . . . . . . . . . . 42

3-1 Development of a helical cutting edge and the corresponding uncut chip . . . . 52

3-2 Progress of tool chip contact zone as the helical endmill rotates . . . . . . . . 52

3-3 Evolution of θL, and θT as functions of θp . . . . . . . . . . . . . . . . . . . . . 53

3-4 Concept of averaged mean chip thickness in helical milling . . . . . . . . . . . 54

3-5 Evolution of mean chip thickness for different helix angles . . . . . . . . . . . . 54

3-6 Variation of the averaged mean chip thickness, hm, with radial immersion andaxial depth of cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3-7 Variation of the averaged mean chip thickness, hm, with helix angle . . . . . . 55

3-8 Chip width evolution for varying helix angles at 50% RI . . . . . . . . . . . . 56

3-9 Chip width evolution for varying helix angles at 25% RI . . . . . . . . . . . . 56

4-1 Uncut chip area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4-2 Transformation of forces from a rotating frame (t, n) to a fixed frame (x, y). . . 63

4-3 Idealization of radial runout and its effect on the feed per tooth . . . . . . . . 64

4-4 Force prediction example for a straight fluted endmill . . . . . . . . . . . . . . 64

5-1 Experimental set-up: straight fluted endmill . . . . . . . . . . . . . . . . . . . 73

5-2 Experimental estimation of entry angle using a phasor signal . . . . . . . . . . 74

5-3 Linear regression fitting of cutting coefficients: dry milling of low carbon steelusing a straight fluted endmill . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6-1 Variation of sensitivity coefficients with θp . . . . . . . . . . . . . . . . . . . . 86

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6-2 Variation of the effective degrees of freedom of ucA(fx,y) . . . . . . . . . . . . . 87

6-3 Variation of component combined uncertainties and overall expanded uncertainty 87

6-4 Example of 95% confidence interval placement on predicted forces . . . . . . . 88

6-5 Force prediction verification: fT = 0.150 mm/tooth, 50% RI, up-milling,runout 15 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88



6-8 Force prediction verification: fT = 0.200 mm/tooth, 75% RI, cut starts withh = 0, runout 15 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6-9 Force prediction verification: fT = 0.150 mm/tooth, 75% RI, cut starts withh = 0, runout 15 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7-1 Differential projected chip areas in helical peripheral milling . . . . . . . . . . 112

7-2 Projected differential frontal chip area . . . . . . . . . . . . . . . . . . . . . . 113

7-3 Projected differential axial chip area . . . . . . . . . . . . . . . . . . . . . . . 113

7-4 Variation of the integrals I2, I2, and I3 . . . . . . . . . . . . . . . . . . . . . . 114

7-5 Verification of the analytical model: the degenerate case of zero helix . . . . . 115

7-6 Comparison of analytical and numerical solutions for 30 helix, 75% RI . . . . 116

7-7 Residuals showing the difference between the analytical and numerical solutionsfor 30 helix, 75% RI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7-8 Analytical solutions for various conditions in helical peripheral milling . . . . . 118

7-9 Experimental set-up: diameter 12.7 mm, 3-fluted endmill with 45 helix . . . . 119

7-10 Linear regression fitting of cutting coefficients: dry milling of 6061-T6 aluminumalloy, using a 3-fluted, 45 helix, endmill . . . . . . . . . . . . . . . . . . . . . 119

7-11 Predicted vs. experimental forces: Average cutting coefficient model, 45 helix,fT = 0.050 mm/tooth, zero runout, 25% RI, down-milling . . . . . . . . . . . 120



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7-15 Predicted vs. experimental forces: Average cutting coefficient model, 45 helix,fT = 0.050 mm/tooth, zero runout, 5% RI, down-milling . . . . . . . . . . . . 122

7-16 Predicted vs. experimental forces: Average cutting coefficient model, 45 helix,fT = 0.200 mm/tooth, zero runout, 5% RI, down-milling . . . . . . . . . . . . 122

7-17 Predicted vs. experimental forces: Average cutting coefficient model, 45 helix,fT = 0.100 mm/tooth, zero runout, 75% RI, cut begins with h = 0 . . . . . . 123

7-18 Predicted vs. experimental forces: Average cutting coefficient model, 45 helix,fT = 0.200 mm/tooth, zero runout, 75% RI, cut begins with h = 0 . . . . . . 123

7-19 Predicted vs. experimental forces: Average cutting coefficient model, 45 helix,fT = 0.100 mm/tooth, zero runout, 25% RI, up-milling . . . . . . . . . . . . . 124

7-20 Predicted vs. experimental forces: Average cutting coefficient model, 45 helix,fT = 0.100 mm/tooth, zero runout, 10% RI, up-milling . . . . . . . . . . . . . 124

7-21 Gibbs-Wilbraham distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7-22 Convergence and computational burden of the Analytical Fourier solution . . . 126

8-1 Differential tooth pitch example . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8-2 Linear regression fitting of cutting coefficients: dry milling of 6061-T6 aluminumalloy, using a differential pitch, 30 helix, endmill . . . . . . . . . . . . . . . . 132

8-3 Differential effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 30 helix, fT = 0.200 mm/tooth, 5% RI, down-milling . . . . . . . . . 133

8-4 Differential effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 30 helix, fT = 0.100 mm/tooth, 20% RI, down-milling . . . . . . . . . 133

8-5 Differential effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 30 helix, fT = 0.100 mm/tooth, 25% RI, up-milling . . . . . . . . . . 134

8-6 Differential effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 30 helix, fT = 0.050 mm/tooth, 50% RI, up-milling . . . . . . . . . . 134

9-1 Comparison with numerical solution with runout included . . . . . . . . . . . 141

9-2 Linear regression fitting of cutting coefficients: dry milling of 6061-T6 aluminumalloy, using an equispaced tooth, 45 helix, endmill, 10 µm runout . . . . . . . 142

9-3 Runout effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 45 helix, fT = 0.100 mm/tooth, runout 10 µm, 5% RI, down-milling . 142

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9-4 Runout effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 45 helix, fT = 0.100 mm/tooth, runout 10 µm, 5% RI, up-milling . . 143

9-5 Runout effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 45 helix, fT = 0.100 mm/tooth, runout 10 µm, 10% RI, down-milling 143





9-10 Runout effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 45 helix, fT = 0.025 mm/tooth, runout 10 µm, 50% RI, up-milling . . 146

9-11 Runout effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 45 helix, fT = 0.050 mm/tooth, runout 10 µm, 75% RI, cut endswith h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9-12 Runout effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 45 helix, fT = 0.050 mm/tooth, runout 10 µm, 75% RI, cut startswith h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9-13 Runout effect. Predicted vs. experimental forces: Average cutting coefficientmodel, 45 helix, fT = 0.050 mm/tooth, runout 10 µm, 100% RI slotting . . . 147



10-1 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.100 mm/tooth, runout 10 µm, 5% RI, down-milling . . . . . 155

10-2 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.100 mm/tooth, runout 10 µm, 5% RI, up-milling . . . . . . . 155

10-3 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.100 mm/tooth, runout 10 µm, 10% RI, down-milling . . . . 156


14

10-5 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.050 mm/tooth, runout 10 µm, 20% RI, up-milling . . . . . . 157



10-8 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.025 mm/tooth, runout 10 µm, 50% RI, up-milling . . . . . . 158

10-9 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.050 mm/tooth, runout 10 µm, 75% RI, cut ends with h = 0 . 159

10-10 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.050 mm/tooth, runout 10 µm, 75% RI, cut starts with h = 0 159

10-11 Predicted vs. experimental forces: Instantaneous cutting coefficient model,45 helix, fT = 0.050 mm/tooth, runout 10 µm, 100% RI, slotting . . . . . . . 160



11-1 FRFs of the force measuring chain: Real and Imaginary parts . . . . . . . . . 165

11-2 FRFs of the force measuring chain: Magnitude and Phase . . . . . . . . . . . 166

11-3 Effect of force measuring chain FRFs on measured force signals . . . . . . . . 167

12-1 Dynamic chip thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

12-2 Scheme for generation of stability contour using TFEA . . . . . . . . . . . . . 176

15

LIST OF SYMBOLS

Aa projected axial chip area

Ac chip area

Adj. R2 adjusted coefficient of determination

Af projected frontal chip area

B block diagonal matrix of single equation least squares estimates of cutting

constants

B0 average Fourier coefficient in the trigonometric series expression for chip

width

Bk coefficients for cosine terms in the Fourier series expression for chip width

C shorthand for cos θ

Ck coefficients for sine terms in the Fourier series expression for chip width

D diameter of the endmill

FD dimensionless factors in the dynamic component of the cutting force

Fn normal (radial) component of the cutting force

FS dimensionless factors in the static component of the cutting force

Ft tangential component of the cutting force

Ftotal total force in the plane of cutter rotation

Fx,y,z x,y,z components of the instantaneous cutting force due to one tooth

FX,Y,Z x,y,z components of the instantaneous cutting force due to multiple teeth

F x,y,z x,y,z components of the average cutting force due to one tooth

F(x,y)D x,y,z components of the dynamic cutting force due to one tooth

F(x,y)S x,y,z components of the static cutting force due to one tooth

H(...) Heaviside unit step function

I1,2,3 integrals used in the evaluation of cutting coefficients in helical milling

Kac linearized axial cutting coefficient

Kae linearized axial edge coefficient

16

Ki ν × 1 vector of logarithms of cutting coefficient responses

Kn linearized normal (radial) cutting coefficient for straight fluted endmills

Knc linearized normal (radial) cutting coefficient for helical endmills

Kne linearized normal (radial) edge coefficient for helical endmills

Kt linearized tangential cutting coefficient for straight fluted endmills

Ktc linearized tangential cutting coefficient for helical endmills

Kte linearized tangential edge coefficient for helical endmills

L0 average Fourier coefficient in the trigonometric series expression for θL

Lk coefficients for cosine terms in the Fourier series expression for θL

Mk coefficients for sine terms in the Fourier series expression for θL

N number of teeth in a cutter, or the number of inputs used to evaluate a

measurand indirectly

R2 coefficient of determination

RI radial immersion

Rk coefficients for sine terms in the Fourier series expression for θT

S shorthand for sin θ

T0 average Fourier coefficient in the trigonometric series expression for θT

Tk coefficients for cosine terms in the Fourier series expression for θT

U( ) overall expanded uncertainty

V (β) symmetric moment matrix of two-stage Aitken estimators

Xi ith input parameter used to evaluate a measurand indirectly

Y measurand whose value is evaluated indirectly as a function of input

parameters

a commanded (nominal) axial depth of cut

ar radial depth of cut

b instantaneous chip width

dfeff |ucA( ) effective degrees of freedom of the combined uncertainty of Type A

17

df |u( ) effective degrees of freedom of the uncertainty of sample measures of

cutting constants

fr feed per revolution

fT commanded (nominal) feed per tooth

fTieffective feed per tooth for the ith tooth

fx,y,z sample measures of x,y,z components of the instantaneous cutting force

due to one tooth

fx,y,z sample measures of x,y,z components of the average cutting force due to

one tooth

g symbol denoting a function

h instantaneous chip thickness (a function of θ)

h average chip thickness

hm mean instantaneous chip thickness in helical milling (a function of θp)

hm averaged mean instantaneous chip thickness in helical milling

hs static chip thickness

k subscript denoting the kth elements such as the kth elements in the

Fourier summations

p subscript denoting the pth tooth

s(t,n,a)(x,y,z) sensitivities of cutting coefficients with respect to average force

components

s(x,y,z)(Kt,n,a) sensitivities of cutting forces with respect to cutting coefficients

s(x,y,z)f sensitivities of cutting coefficients with respect to effective feeds

sij sensitivities of cutting force components with respect to cutting constants

t time

t,n subscripts denoting tangential and normal (radial) components

u( ) standard uncertainty

u( , ) covariance

18

uc( ) combined uncertainty

x,y,z subscripts denoting x, y, and z components

xi sample measures of input parameter, Xi

y sample measures of measurand, Y

βi 2× 1 vector of unknown constant parameters (the cutting constants)

βi single equation least squares estimates of βi

εi random error vector of the ith response

χi pitch angle of the ith tooth

∆h dynamic chip thickness

∆x x-component of dynamic chip thickness

∆y y-component of dynamic chip thickness

Γt,n,a cutting constants

γt,n,a sample measures of cutting constants

κA,B1,B2 coverage factors for component expanded uncertainties

λ helix angle

ν number of experimental data sets

ν1,2 shorthand for trigonometric expressions involving θst and θex

θ angular orientation of an arbitrary point on the cutting edge in the

tool-chip contact zone

θex exit angle of the leading point of the tooth

θL angular orientation of the leading point of tool-chip contact zone

θp cutter rotation angle, i.e., the angular orientation of the leading point of

the reference helical tooth (the pth tooth)

θspan total angular span of tool-chip contact

θst entry angle of the leading point of the tooth

θT angular orientation of the trailing point of the tool-chip contact zone

ρ radial runout of successive teeth

19

τ tooth passing period

ς some fixed value of the argument of a function

Ω angular spindle speed

ξh shorthand for terms involving θL and θT in the expression of hm

ξ1,2,3 shorthand for terms involving θL and θT in the expressions of force

components in helical milling

Ψt,n,a cutting constants

ψt,n,a sample measures of cutting constants

ζ intermediate variable directed along the z-axis in the tool-chip contact

zone

20

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

PREDICTIVE FORCE MODELING OF PERIPHERAL MILLING

By

Abhijit Bhattacharyya

August 2008

Chair: John K. SchuellerCochair: Brian P. MannMajor: Mechanical Engineering

Milling is one of the most important subtractive manufacturing processes. Cutting

force predictions in milling are useful for the structural design of machine tools, selection

of optimum cutting parameters, design of workholding fixtures, tool stress analysis, spindle

bearing design, and the real time monitoring of tool wear and breakage. Force models

are also used in predicting the stability of the milling process, surface location error

predictions, as well as surface finish predictions.

In this dissertation, closed form analytical mechanistic cutting force models, using

linearized lumped parameter cutting coefficients, are considered. Existing analytical

mechanistic force models of helical peripheral milling, which use linearized cutting

coefficients, either utilize four different sets of analytical expressions to describe the forces

for one complete cutter rotation, or do not have closed form solutions. Numerical models

are computationally-intensive, fail to provide the insights that analytical models do, and

do not permit symbolic manipulation. In this work, two equivalent versions of closed form

analytical expressions for chip thickness and chip width are developed using Heaviside

unit step function and Fourier series approaches. The distinguishing feature is that single

expressions describe the chip thickness and chip width during the entire cutter rotation.

These expressions are then applied to develop single, analytical closed form expressions for

each of the three orthogonal components of the cutting force, in a fixed coordinate frame,

21

for helical peripheral milling. The model can be calibrated using partial radial immersion

experiments.

A procedure has been developed to calculate the variances in measured model input

parameters. The propagation of uncertainties through the model is determined by Type

A and Type B evaluations to develop an overall expanded uncertainty for placement of

confidence intervals on cutting force predictions. The availability of single expressions for

force components permits the derivation of compact expressions for sensitivity coefficients

for use in the uncertainty analysis.

Extensive experimental tests, as well as comparisons with established numerical

models, verify the fidelity of the predictions. The model is extended to be able to predict

cutting forces in cases of runout or differential pitch cutters. A refined formulation using

instantaneous cutting coefficients, instead of the usual method of average coefficients, has

also been developed, further increasing the accuracy of force predictions. Experiments

show that the refined model is able to predict the force patterns and magnitudes very

accurately for the entire range of radial immersions.

22

CHAPTER 1OVERVIEW AND MOTIVATION

Peripheral milling is a subtractive process for the manufacture of discrete prismatic

parts. A large variety of components which are used in aircraft, automobiles, ships,

railroads, medical equipment, space vehicles, power generation and process equipment,

as well as tools for the production of electronic components and plastic parts, could not

be economically manufactured without employing some variant of the peripheral milling

process. Material is removed from a raw workpiece by interrupted machining, using a tool

having well defined cutting edges, in the form of chips (swarf), to produce a finished part

having a desired shape and size. This research falls within the subject area of machining of

metallic alloys, but the machining of non-metallic materials such as composites, wood, and

plastics is also important.

The Statistical Abstract (2008a) of the U.S. Census Bureau shows that metalworking

machinery shipments in the United States in the year 2005 totaled $2.80 billion, out of

which $2.08 billion were metal cutting type of machines, while the rest were of metal

forming type. The Statistical Abstract (2008b) of the U.S. Census Bureau shows that

gross value of new orders and exports for U.S. machine tools in the year 2006 totaled $4.38

billion (27,288 units), out of which $3.70 billion (23,670 units) were metal cutting type

of machines, while the rest were of metal forming and other manufacturing technology

type. Though these numbers appear to be modest, the impact of these equipment on

the economy is all encompassing, affecting virtually every product manufactured in the

modern world.

Approximately 33% of the above mentioned $2.08 billion worth of metalworking

machines, shipped in 2005 in the U.S., were of the type which use milling as the main

machining process (milling machines and machining centers). Machines listed in the above

report under the headings ‘boring and drilling machines’, ‘other metal cutting machine

tools’, ‘remanufactured metal cutting machine tools’, and ‘station type machines’ also

23

employ milling in addition to other machining processes. Furthermore, milling is one of

the main processes of machining used in machines listed under the heading ‘gear cutting

machines’ in the above report. These facts emphasize the fact that milling is one of the

most important machining processes.

In milling, the workpiece is fed across a rotating milling cutter to produce prismatic

parts. Two common categories of milling are facemilling and peripheral milling (or

endmilling). Figure 1-1 shows the difference between these two types. Facemilling

generates a surface that is perpendicular to the axis of cutter rotation. Peripheral milling

produces side walls generated by the periphery of the cutter parallel to the axis. Usually, a

secondary surface is also generated which is akin to facemilling, as shown in the figure. In

pure peripheral milling, only side cutting occurs. Arbor mounted slab mills often operate

in this mode. Variants of these processes include ball endmilling, thread milling, and

plunge face milling. Milling cutters may be straight fluted or helical (Fig. 1-2).

The demand for complex prismatic parts is ever increasing because such parts offer

the designer a lot of flexibility. Hence, milling has assumed a central role among machining

processes. Modern milling processes combine capabilities of high material removal rates

with accurate surface placement, fine surface finish and close tolerances on surface flatness.

In the commercial aerospace industry, monolithic machining of very large aluminum

alloy parts has gained popularity. Monolithic part design confers economic advantages

by reducing the number of parts, eliminating expensive assembly and improving part

accuracy. The trade off involves high speed machining of thin ribs using slender end

mills. The machinist faces a problem having a combination of a flexible tool and a flexible

workpiece. The attendant loss of stability limits material removal rates and stability

boundaries have to be correctly determined to find zones of stable machining where high

material removal rates may be possible (Tlusty, 2000).

The boom in the plastic consumer durables market has fostered a mushrooming mold

and die industry. High-speed free-form milling of ferrous alloys, using slender endmills,

24

is the process of choice. Often, the end-milling process is required to finish hard dies to

avoid expensive manual labor involved in finish polishing, and to eliminate rejects that

may otherwise result from post machining heat treatment. The high strength of such

materials coupled with quickly generated wear lands (Nelson et al., 1998) result in high

specific cutting pressures so that the cutters experience very high cutting forces. Other

important applications of milling include the machining of titanium alloys for defense

and aerospace applications (Leigh et al., 2000), and high velocity machining of aluminum

alloys for automobile components. Many of the experiments reported in this document

were conducted on aluminum alloys. Miniature machining is increasing in importance.

Small diameter endmills are limited by lateral force, making it important to estimate force

components while selecting machining parameters (Schueller et al., 2007).

1.1 Motivation for Predictive Force Modeling of Peripheral Milling

The selection of optimized machining parameters is a recurring problem in machine

shops. Any specific milling operation has an associated set of constraints within which

the parameters must be selected to optimize the chosen objectives such as material

removal rate, or the accuracy of surface placement. An optimum selection is a must for

competitive manufacturing. Traditionally, rules of the thumb and experience were used to

select machining parameters, but these are suboptimal procedures because the method is

not scientific. Exploration of parameter space by physical trial and error is prohibitively

expensive. Simulation of machining process is an economical alternative. This requires

predictive modeling of various aspects of the process such as cutting forces.

An analysis of the milling process involves the prediction of cutting forces, stability

limits, surface placement accuracy, and surface finish. Kurdi (2005) has presented a

method of optimization of the surface location and material removal rate in milling

while considering the uncertainty in the milling model. Sampling methods were used

(Latin Hypercube and Monte Carlo) to place confidence intervals on the stability limits.

Confidence intervals were not placed directly on the predicted cutting forces. There are

25

applications where knowledge of the cutting force is directly applied in making decisions.

Cutting force values are required for the estimation of power consumed, the design of

workholding fixtures, tool stress analysis, and spindle bearing design. An understanding

of the cutting forces is crucial for the prediction of stability, surface placement, and

surface finish in machining operations. Force based sensing of real time tool wear, and

tool breakage, is a common application. For these reasons, modeling of milling forces

is of practical importance. In addition, the ability to provide a defensible uncertainty

statement to accompany cutting force predictions is beneficial in practice. It enables the

process planner to decide the usefulness of model based force predictions in any specific

application.

1.2 Review of the Open Literature

The subject of force modeling for peripheral milling has a long history and a

large body of literature has accumulated over the years. This underlines the fact that

there are many complexities associated with the modeling process and the problem is

multifaceted. This research is concerned with mechanistic force modeling. In such models,

the geometry of the chip is described using the chip thickness and the chip width, which

will be described in detail later. These two factors enable the calculation of the chip

area which is multiplied by suitable lumped parameter cutting coefficients to yield the

relevant components of the cutting force. The cutting coefficients capture the effects of

factors other than chip geometry such as the workpiece material properties, tool chip

interface friction, use of cutting fluids, the machining parameters, etc., and are also

known by various other names such as the specific cutting pressure, or the specific force.

Formulations for cutting coefficient identification are an important aspect of mechanistic

force models. The review of the literature that follows concentrates on mechanistic static

force models, though some references are made to other types of models.

Sawin (1926) presented one of the first models that appeared in the literature

addressing both, straight fluted cutters, as well as helical cutters. This was a mechanistic

26

model in which the cutting force was considered to be a function of the geometry. A

cutting coefficient was used which was related to the ultimate tensile strength of the

workpiece material. This coefficient was considered to be inversely proportional to the

fourth root of the maximum chip thickness. The circular path approximation of the true

trochoidal path was used to determine the instantaneous chip thickness.

Salomon (1926) developed a model in which the work done by a straight fluted

cutter was considered to be a function of the geometry. A power law was used to relate

the specific cutting pressure to the instantaneous chip thickness. The circular path

approximation was used to determine the instantaneous chip thickness.

Klien (1937) developed tangential force equations for straight fluted and helical

cutters using the Fourier series. The cutting coefficient was considered to be a constant for

a given workpiece material.

Martellotti (1941; 1945) studied the kinematics of milling, stated the circular tool

path approximation, and developed expressions for the true chip thickness considering the

true trochoidal tool path. The chip thickness, and power consumed in down milling was

shown to be greater than in up milling. However, down milling was shown to have other

advantages such as improved tool life.

Kienzle (1952) defined a parameter which is the force required to machine a chip of

size 1 mm2. He defined the cutting coefficient as a product of this parameter and a power

of the chip thickness which is similar to Solomon’s model.

Sabberwal (1961) argued that for a given set of machining parameters the work

done in material removal should be independent of the helix angle. On this basis he

expected the instantaneous specific cutting pressure to be independent of the helix angle

of the cutter. He verified this observation for machining of high tensile steel (EN28)

with different cutters having helix angles ranging from 0 to 30. He also found the

observation to be true for cutting mild steel (0 and 50 helix) and aluminum (0 and 45

helix). Based on this observation, it is common practice to extract cutting coefficients

27

experimentally for helical cutters by using average forces based on equations representing

straight fluted cutters. Recently, Ku (2006) has provide some evidence that cutting

coefficients could depend on helix angle. Such comparisons are contingent upon the fact

that all other factors remain the same, especially the other geometrical parameters of the

cutters.

Sabberwal (1961) modeled the tangential cutting forces in helical milling by

multiplying the frontal chip area with the specific cutting pressure. However, the normal

cutting force was not modeled, nor was the axial force. Therefore, this model was suitable

for power calculations only. The feed force and the lateral force could not be determined.

An important contribution made by Sabberwal (1961) was to highlight the existence

of different tool engagement conditions for helical milling. He determined two different

types of cutting depending on the whether or not the trailing edge of the tooth entered

the cut before the leading edge left the cut. Each of these two types of cutting was

identified as having three phases as the cut progressed. During the first phase, the chip

width increased, stayed constant in the second phase, and decreased in the third phase.

In a later work, Tlusty and MacNeil (1975) named the two types of cut as Type I and

Type II. The three phases were termed A, B, and C. This is the terminology used in this

document in the following chapters. Tlusty and MacNeil (1975) also presented the first

analytical solution for the two components of cutting force in the plane of cutter rotation

for helical peripheral milling. The tangential cutting force was considered proportional to

the chip area with the specific cutting force, K, being a function of workpiece material,

tool geometry, and the average chip thickness. The radial force (which is generally called

the normal force in this document) was assumed to be 30% of the tangential force. This

was a two dimensional model.

The tangential and normal (radial) force components are defined in a rotating frame

of reference attached to the rotating cutter. In the Tlusty and MacNeil (1975) model,

these were transformed into components in a non-rotating frame through a rotation

28

matrix. It is convenient to measure forces in the non-rotating frame. The effects of

multiple teeth can also be easily added. This was a rigid model, i.e., the dynamic effects

on chip thickness were not considered. The weakness of this model was that there were

three different equations required to defined each force component over the entire cutter

rotation. Moreover, the expression for the Phase B was different for Type I and Type II

cutting, i.e., the model had a total of four expressions for each force component to cover

all possible conditions. One of the strengths of the model was that it was able to analyze

the changes in force during the transients.

The aforementioned models were one or two dimensional, describing components of

the cutting force in the plane. This is an adequate description of forces in pure peripheral

milling with straight fluted tools, which is essentially orthogonal machining. However,

for helical milling, the axial component of the force must also be modeled for a complete

description. This is an instance of oblique machining. Usui et al. (1978a,b) presented

a method in which this oblique cutting tool was modeled as a single point tool being a

collection of orthogonal cutting conditions along the cutting edge. They used an energy

method to predict force components. The input parameters for the model are the shear

angle, the friction angle, and the shear stress. The outputs are the shear energy and

the friction energy based on which the cutting forces are obtained. Tsai (2007) has

developed a version of this model suitable for rotating cutters which can be applied to a

helical milling situation. The difficulty with this approach is that the input parameters,

namely the shear angle and the friction angle, are difficult to identify in a given machining

situation. For this reason, the mechanistic type of model, where the cutting coefficients

can be experimentally determined, is generally preferred. However, this approach could

be useful in predicting forces during cutter design, because at that stage, the tool has not

been manufactured, and experimental determination of cutting coefficients is impossible.

Piispanen (1937, 1948) and Merchant (1945) had developed the concepts of shear angle

and friction angle for orthogonal machining. Altintas and Lee (1996) presented a general

29

three dimensional model for the mechanics of helical endmilling in which the cutting

coefficients were obtained from orthogonal machining data, i.e., the shear and friction

angles were used. Again, this has the advantage of predicting the coefficients for the

benefit of the tool designer, but suffers from the lack of reliable orthogonal machining

data. In this model differential cutting forces were integrated numerically.

Kline (1982) developed a comprehensive mechanistic force model moving ahead of the

rigid force model by considering cutter deflections, cutter runout, and modeling the true

trochoidal tool path. This resulted in a series of papers. In DeVor and Kline (1980) the

cutting coefficients were experimentally estimated using average forces. The coefficients

were called empirical constants, but it was shown that they depended on machining

parameters. The constant associated with tangential force component was called the

unit horsepower or the specific cutting energy. In fitting the constants in the model to

experimental average force data, while machining 4340 steel at 320 BHN, the multiple

correlation coefficients (R2 values) were 0.984 for KT (the tangential coefficient) and 0.936

for KR (the fraction of KT which yields the normal coefficient). In addition to the forces,

the force center, force distribution, and cutter deflection were predicted using numerical

methods. A numerical method was also used to track the true trochoidal tool path. By

doing so it was possible to make a distinction between up milling and down milling forces,

which was not possible with the circular path approximation. Force predictions were found

to agree with experiment within 5 to 10%. The correspondence was better at small radial

depths of cut, but deteriorated as the radial depth of cut increased. An important finding

concerned cutter deflections. It was shown that for a wide variety of cutting conditions

which produced the same resultant force, the magnitude and direction of deflections varied

widely. Kline et al. (1982) developed a further version of this model capable of predicting

force characteristics in cornering cuts during which the radial engagement keeps changing.

This was also a numerical model. They mentioned that such cornering cuts are commonly

encountered in aerospace machining operations. Kline and DeVor (1983) showed that

30

runout increases the average chip thickness for those teeth which have higher runout,

increases the ratio of the maximum to average force, and shifts the frequency content of

the force signal away from the tooth passing frequency to the spindle rotational frequency.

The ratio of the runout to the feed rate was identified as an important parameter which

determines the effect of runout on the cutting force system.

Sutherland and DeVor (1986) extended Kline’s work and suggested a flexible

model including runout. This is a numerical model in which an effective feed rate was

computed which balances the cutting forces and the resultant system deflections. They

discovered that system deflections temper the effects of runout to reduce the peak force

and maximum surface error.

Yellowley (1988) noted that milling torque and forces could be represented in terms

of Fourier series. Yellowly developed the Fourier representation of forces in the plane

for straight fluted cutters. He also developed a symbolic representation for the resultant

torque by considering the tooth to be made up of elemental straight cutting edges having

an angular separation. However, he did not compute the Fourier coefficients, nor did he

compute cutting forces for the helical mill.

Montgomery and Altintas (1991) presented a numerical dynamic model for helical

milling using which forces could be predicted under static or dynamic conditions. The

trochoidal tool path was used to determine the uncut chip thickness. The geometry of

tool and workpiece motion was captured using structural modeling. This was also a two

dimensional model. Another two dimensional model for helical milling was proposed by

Altintas and Spence (1991) for a computationally efficient scheme of force prediction. This

was a closed form analytical solution in which three different expressions had to be used

as the cut progressed through the three phases of cutting. To derive average forces for

calculating the cutting coefficients, the expressions were simplified under the assumption of

a zero helix based on Sabberwal’s (1961) observation that the helix angle had no influence

on cutting coefficients. A generalized mechanistic model for helical milling forces was

31

presented by Engin and Altintas (2001) which took into consideration the trochoidal tool

path. The final expressions for the forces are in the form of integrals which have to be

evaluated by the user for a specific set of cutting conditions.

Abrari and Elbestawi (1997) published a three dimensional closed form solution where

the forces were expressed as a linear combination of a set of basis functions. However,

the simplicity of using linearized cutting coefficients was lost as these were replaced

with a matrix which incorporated the helix angle. Ehman et al. (1997) have provided

a comprehensive review of various force models that appeared in the literature before

1997. Their review also encompasses many aspects of dynamic force modeling. In this

dissertation, the emphasis is on static force modeling. A two dimensional, Fourier series

based solution presented by Schmitz (2005) and Schmitz and Mann (2006) includes the

effect of the helix angle by dividing the cutter into thin axial slices and summing the

effects to obtain the total force. Recently, a three dimensional model was developed by

Mann et al. (2008) using an equivalent complex Fourier series representation of forces to

facilitate symbolic manipulation. This model was applied to stability and surface location

analysis. In the paper, the integrations have not been symbolically solved. If that is done,

this could possibly provide a closed form solution for the force model itself.

Several authors have reported on the use of differential pitch cutters, especially for

mitigating the effects of regenerative chatter as well as reducing dimensional surface errors

(e.g. Slavicek, 1965; Vanherck, 1967; Tlusty et al., 1983; Shirase and Altintas, 1996).

However, a force model for a differential pitch helical endmill, which has a single closed

form analytical expression covering the entire domain of one cutter rotation, has not been

reported in the literature.

Kline and DeVor (1983) showed that runout increases the average chip thickness for

those teeth which have higher runout, increases the ratio of the maximum to average force,

and shifts the frequency content of the force signal away from the tooth passing frequency

to the spindle rotational frequency. The ratio of the runout to the feed rate was identified

32

as an important parameter which determines the effect of runout on the cutting force

system.

Armarego and Deshpande (1991) presented a numerical model considering cutter

eccentricity. Their model could predict forces well except in those cases where cutter

deflections were large, for which case they had a deflection model. Schmitz et al. (2007)

investigated the effects of runout on surface finish, surface location, and stability using a

force model which is a time-domain simulation description. Li and Li (2004) proposed

a theoretical force model with cutter runout and trochoidal tool trajectory. They

discretized the cutter into slices and sum the effects of force due to each slice to obtain the

complete force. Junz Wang and Zheng (2003), and Ko et al. (2002) identified cutting force

coefficients in the presence of runout.

For realistic dynamic modeling of milling processes, the consideration of runout

is an important factor. Recognizing this fact, Altintas and Chan (1992) presented a

digital simulation model for dynamic milling in which they included the effects of runout.

Weinert et al. (2007) have shown that runout affects surface quality even in stable

machining. Wang et al. (1993) presented an in-process control methodology to compensate

for surface finish imperfections due to cutter runout in milling processes. The runout

geometry was identified from cutting force measurements and the manipulation of radial

depth of cut to follow the inverse trajectory of runout. Thus the effects of runout were

counteracted for surface finish improvement.

The subject of cutting coefficients has also received much attention in the literature.

Gradisek et al. (2004) has presented a very comprehensive study of cutting coefficients

in which the dependence of these coefficients on various process parameters has been

experimentally studied in detail. One of the important conclusions was that the

coefficients do not differ in up milling and down milling. Whereas most of the studies

listed here use an averaged value of cutting coefficients, this is only an approximate way

to handle the problem. Since the coefficients change with process parameters, they evolve

33

as the cut progresses, since the chip thickness is constantly evolving in milling, unlike

in turning operations where it is a constant. So the concept of instantaneous cutting

coefficients is sometimes invoked (Wan et al., 2007).

The dimensions of the cutting coefficients are those of pressure [N/mm2]. For this

reason, they are variously referred to as specific cutting pressures (Jayaram et al., 2001;

Perez et al., 2007), specific force coefficients (Gradisek et al., 2004), specific cutting

force (Ko and Cho, 2005), or specific cutting energy (Kline, 1982) in the literature.

Occasionally, they are also called cutting constants (Altintas and Spence, 1991). However,

in this document the term ‘cutting constants’ is used for a different purpose in the

sense that cutting constants are true invariants, whereas the other terms such as cutting

coefficients, specific force, etc., are process parameter dependent. In any case, the meaning

of terminology is usually clear from the context.

1.3 Justification and Scope of the Work

Whereas all these different formulations of the model have been very successful

in predicting the variations of the static (periodic) component of the force in helical

endmilling, most of the analytical models are somewhat difficult to use in applications

such as stability studies or surface placement analysis. The main reason is that a single,

fully analytic expression for the variation of the cutting forces, has not been presented in

closed form. For this reason, either the numerical models have been applied to applications

or the analytical model for straight fluted endmills has been widely used (e.g. Altintas

and Budak, 1995; Mann et al., 2005; Insperger, 2003). However, experimental verification

is usually done using helical endmills (Mann et al., 2003). Helical endmills are preferred

because they distribute the force over a longer cutting edge length, reducing the local

pressure. For this reason, this research is directed towards developing a fully analytical

closed form force model which would have a single expression for each force component

that is valid for the entire cutter rotation.

34

A review of the literature also indicated an absence of uncertainty analysis in forces

predicted by available force models which use measured values of input parameters to

make force predictions. So, the prediction of cutting forces in peripheral milling, to be

accompanied with a statement of confidence, was taken up as one of the subjects of this

research. An analytical method of propagation of uncertainty in model input parameters

was chosen. Such an analysis is readily implemented and yields results very quickly due

to very little computational burden. This analysis is straightforward when applied to a

force model for straight fluted endmills but becomes algebraically unwieldy when applied

to available force models for helical peripheral milling. In practice, peripheral milling

with helical endmills is the norm. This was another motivation for developing a new force

model which facilitates such an uncertainty analysis for helical peripheral milling.

35

Ω

workpiece

feed

Endmill

feed

workpiece

Facemill

Ω

Primary surface generated Secondary surface generated

Primary

surface

generated

Figure 1-1. Face milling and peripheral milling (endmilling) processes

(a) Straight fluted endmills (b) Helical fluted endmills

Figure 1-2. Straight fluted and helical fluted endmills. Two fluted and four fluted cuttersare shown for each of the two varieties.

36

CHAPTER 2DESCRIPTION OF THE PERIPHERAL MILLING PROCESS

This chapter provides a brief qualitative description of the peripheral milling process

accompanied with an explanation of the terminology. The raw workpiece and the cutting

tool are held in a machine which provides the relative motion and power to remove

material in the form of chips. The geometry of the chip holds the key to an understanding

of the process. The chapter concludes by mathematically characterizing the chip for the

simplest possible tool geometry.

2.1 The Machine and the Cutting Tool

Figure 2-1 shows a schematic sketch of a peripheral milling set-up on a 3-axis vertical

milling machine. The endmill, also called the tool or cutter, is held in a toolholder,

called the chuck, which is mounted on a powered machine spindle. The workpiece is fed

across the tool such that there is a linear relative motion between the endmill and the

part, in addition to the rotation of the tool. The cutter rotates at the spindle speed, Ω.

Movements of the X-Y-Z axes provide the relative motion between workpiece and tool.

Machines with additional rotary axes are very common. Material is removed in the form

of chips (swarf) via successive passes of cutting edges (teeth) across the workpiece. In

peripheral milling, the aim is to produce sidewalls generated by the sides of the milling

cutter, though face milling action also takes place, i.e., a surface parallel to the face of the

endmill is also generated. Occasionally, pure peripheral milling may be performed with

only side cutting.

The endmills are rotary, multipoint cutting tools having either helical or straight

flutes (zero helix). The geometry of helical peripheral milling is illustrated in Fig. 2-2

which shows the helix angle, λ, and the diameter, D. The commanded (nominal) axial

depth of cut, a, the radial depth of cut, ar, the feed per tooth, fT , and the spindle speed,

Ω, are fixed parameters set by the machinist. The cutting edges are called teeth with the

chip spaces being called gullets or flutes. The term flute is also loosely used to refer to

37

the teeth, with the context making it clear whether it refers to the tooth or the spaces

between teeth. The radial immersion, RI, is a derived parameter. It is expressed as a

fraction or a percentage and is defined as the ratio

Radial immersion, RI , ar

D(2–1)

2.2 Chip Area at the Tool Workpiece Interface

The instantaneous area of tool chip contact is called the uncut chip area or simply the

chip area, Ac. The instantaneous axial depth of cut is called the chip width. Figure 2-3

illustrates the definitions of the instantaneous chip area, instantaneous chip thickness, and

instantaneous chip width for a straight fluted endmill and a helical endmill.

The chip area is not constant and changes as the cutter rotates. To calculate the

instantaneous chip area, the chip thickness, h, and chip width, b, have to be calculated.

For a straight fluted endmill, the chip width is a fixed constant and equals the commanded

(nominal) axial depth of cut, a. For a helical endmill, the chip width varies as the cutter

rotates. In milling, the chip thickness, h, varies as the cut progresses. The chip thickness

is defined by projecting the chip on the plane of rotation. For straight fluted cutters, the

projection is a straight line allowing the specification of an instantaneous chip thickness.

For helical cutters, the projection is an area of non uniform thickness. It is convenient to

specify a mean instantaneous chip thickness, hm.

The relative motion of the cutting edge with respect to the workpiece is a combination

of the cutter rotation and the translating feed. The true path of the milling tooth in the

plane, with respect to the workpiece, is trochoidal. Martellotti’s (1941) simplified circular

tool path approximation (Fig. 2-4) is used in this document because it is common practice

and yields good results. The circular path approximation had been used by earlier

researchers, such as Salomon (1926) and Sawin (1926). The cutter rotation may be tracked

using a rotation angle, θ, defined with respect to an arbitrary reference line. At any

38

angular orientation, θ, the chip thickness for milling with a straight fluted cutter is

h =

fT sin θ, when the tooth is engaged in the cut

0, otherwise(2–2)

Cutter-workpiece engagement configurations are shown in Fig. 2-5. In up-milling, the

cut begins with zero chip thickness, whereas in down milling the cut ends with a zero chip

thickness. RI values exceeding 50% result in mixed-mode configurations of two possible

types. The cut may either begin with zero chip thickness, or end with zero chip thickness.

A radial immersion of 100% results in machining of double walled channels and is called

slotting. The radial immersion can also be related to the angle the tool tip starts and

exits the cut, designated θst and θex, respectively, through a geometrical calculation. The

domain of θst and θex is [0, π]. Expressions for calculations of entry and exit angles are

given in Table 2-1.

Tooth engagement extent, [θst, θex], for a straight fluted cutter having been defined,

the chip thickness Eq. 2–2 is rewritten formally as

h =

fT sin θ, θ ∈ [θst, θex]

0, otherwise(2–3)

For a straight fluted cutter, the instantaneous uncut chip area is

Ac =

afT sin θ, θ ∈ [θst, θex]

0, otherwise(2–4)

Martellotti (1941) proposed that the average undeformed chip thickness could be

related to the components of the cutting force. The concept of average chip thickness

is invoked to facilitate the experimental determination of cutting coefficients, as well as

average cutting forces. Using Eq. 2–2 the average chip thickness for cutting with straight

39

fluted endmills is

h =1

θex − θst

∫ θex

θst

fT sin θdθ =fT (cos θst − cos θex)

θex − θst

(2–5)

2.3 Chapter Summary

The peripheral milling process was described. The terminology was outlined. Straight

fluted endmills are a special case of helical endmills with zero helix. Definitions of chip

thickness, chip width, and chip area were given and expressions for these parameters were

developed for straight fluted endmills. It was shown that the instantaneous chip thickness

is non-uniform in helical milling, which requires the calculation of a mean instantaneous

chip thickness. It was also shown that the instantaneous chip width is a variable in helical

milling. Expressions for chip thickness and chip width for helical milling are derived in the

next chapter. These calculations are necessary for force modeling.

Table 2-1. Computation of entry and exit angles (Ref. Fig. 2-5)

Radial Up/Down Entry angle Exit angleimmersion milling θst θex

RI ≤ 0.5 Up-milling π2

+ arcsin(1− 2RI) π

RI ≤ 0.5 Down-milling 0 arccos(1− 2RI)

RI > 0.5 h = 0 at entry 0 π2

+ arcsin(1− 2RI)

RI > 0.5 h = 0 at exit arccos(1− 2RI) π

RI = 1.0 Full slotting 0 π

40

+Y

+X

+Z

Machine bed

Endmill

Spindle

Workpiece

ΩΩ

Chuck

Figure 2-1. Schematic sketch of a peripheral milling set-up on a vertical milling machine.

λ

Cutting

edge

Workpiece

οD Ω

a

ar

Cutting

edge

Endmill

fT

shape of

uncut chip

Flute

space or

gullet

Workpiece

feed

direction

Figure 2-2. Endmill terminology: Helix angle, λ, and diameter, D, define the toolgeometry. The commanded (nominal) axial depth of cut, a, the radial depth ofcut, ar, the feed per tooth, fT , and the spindle speed, Ω, are fixed machiningparameters. A two fluted, right handed helical endmill is shown.

41

hΩ

feed

Ω

a

bbh

m

(a) (b)

Figure 2-3. Instantaneous chip area (bold outlines), instantaneous chip thickness, h, andinstantaneous chip width, b, in peripheral milling using (a) straight flutedendmills and (b) helical fluted endmills. The helical cutter yields a variablechip width as it rotates, and a non uniform instantaneous chip thickness. So, amean instantaneous chip thickness, hm, is defined for the helical cutter.

h(θ) = f sinθ

θ

WorkpieceWorkpikpi

Ω

f

feed

T

uncut

chip

h

fT

θ

position of the cutter

when the previous

tooth was in contactposition of the cutter

when the current

tooth is in contact

Curate trochoids approximated as

circular arcs

Tchip shape

Figure 2-4. Circular path approximation for chip thickness calculation.

exθ = π

θst

θex

θ(t)

θ = 0st θ(t)

Up

milling

Down

milling

Figure 2-5. Examples of cutter-workpiece engagement showing up-milling (conventionalmilling) and down-milling (climb milling) configurations for RI < 50%. Theangular orientation of any point on the cutting edge, θ, is a function of timedue to cutter rotation. The leading point of the tooth enters the cut at θst andexits the cut at angle θex which are fixed angular positions in space in theplane of cutter rotation.

42

CHAPTER 3CHIP GEOMETRY IN HELICAL PERIPHERAL MILLING

The aim of this chapter is to analytically capture the shape and size of the chip as

the cutter rotates. This knowledge will be applied in later chapters in the determination

of closed form analytical expressions for the cutting force components, which is one of

the main goals of this research. The expressions for chip thickness will also be used in

characterizing lumped parameter cutting coefficients in the cutting force model. For

straight fluted cutters, the expressions are well known (Tlusty, 2000) and are stated in the

previous chapter. This chapter demonstrates the complexities in analytically describing

the chip in helical peripheral milling. The distinguishing feature of the development is that

single closed form analytical expressions describe the chip over the entire cutter rotation.

Two different, but completely equivalent, versions of the analytical expressions are

presented. One is based on the Heaviside unit step function approach, and the other is

a Fourier trigonometric series model. When the expressions are used to describe cutting

forces, they result in two different, but equivalent versions of cutting force models. The

physics of the process is described in exactly the same manner, but these two versions

have a different mathematical structure which have certain implications for the user.

In the Heaviside unit step function based model an exact representation of the

force is obtained, and this model is useful for most applications. The Fourier series

model requires an infinite sum for exact results, but truncated series sums converge very

quickly. The convergence properties are discussed in more detail in chapter 7. When

symbolic manipulations of the analytical expressions are required for any application, the

mathematical structure of the two different models may dictate the choice for the user.

The derivatives of the Heaviside step function based expressions, with respect to the cutter

rotation angle, are discontinuous. If continuous derivatives are required, the Fourier series

based formulation must be used.

43

3.1 Tool Chip Contact Configurations

The angular position of the leading point of the pth tooth, taken from an arbitrary

reference, is designated θp(t), which is a function of time, t. It may be expressed as:

θp(t) =

(2πΩ

60

)t +

2π

Np (3–1)

where

Ω = the spindle speed in revolutions/minute (rpm), and

N = the number of teeth in the cutter

p = an integer representing the pth tooth

In steady state, under conditions of stable machining, the time evolution of dependent

variables such as chip thickness, chip width, and cutting forces, is cyclic in time. It suffices

to study the evolution of all dependent variables of the problem over a period of one cutter

revolution. In using the various models developed in this document, one would merely

substitute

θp(t) = Ωt, t ∈ [0,∞) (3–2)

to obtain time evolution of the dependent variables. The problem is solved for the

principal values of

θp ∈ [0, 2π)

and any further reference to time is suppressed. The argument of all subsequent

trigonometric functions is θp, shifted by appropriate constants, depending on the situation.

The term “instantaneous” will be taken to refer to the current angular position of the pth

tooth, θp.

Figure 3-1 shows the development of a helical cutting edge and the corresponding

uncut chip. This sketch is also useful for ease of visualization of some subsequent figures.

The tool chip contact zone is shown as a gray line in the developed sketch. λ is the helix

angle of the cutter. The angular orientation, θ, of any point on the tool chip contact zone

or any point on the cutting edge, is taken from an arbitrary reference.

44

Figure 3-1 shows the angular orientations of the leading and trailing points of the tool

chip contact zone which are at θL and θT , respectively. These two angular orientations are

functions of θp. In this document, these functional dependencies are analytically expressed

in single equations in closed form and valid for the entire cutter rotation. Practically

useful expressions for chip thickness, chip width, cutting forces, and cutting coefficients

are derived in the form of single analytical expressions in closed form as functions of θL

and θT , and hence as functions of θp. Closed form analytical expressions are useful in

providing insight into the physical process, can be easily programmed into calculators for

quick calculations, and do not suffer from numerical divergence. In addition, symbolic

manipulations are possible.

A representative tool chip contact configuration is shown in Fig. 3-1 in which the

cylindrical surface of the endmill and the uncut chip have been developed on to a plane.

Sabberwal (1961) and Tlusty and MacNeil (1975) identified different configurations of tool

chip engagement in helical milling according to which two possible types of cutting that

may occur, each having three distinct phases.

Figure 3-2 shows these possible tool chip contact configurations. In Type I cutting,

the chip width attains a value equal to the commanded axial dept of cut at some point

during the cut, i.e., b = a, for some portion of the cut. In Type II cutting, b < a, always.

Each of these types of cutting is divided into three phases. In Phase A, the chip width

increases from zero to some maximum value as the cut progresses. In Phase B the chip

width remains constant at this maximum value for some time. In phase C the chip width

reduces gradually to zero when the entire cutting edge leaves the cut. The following

conditions determine whether the cutting is of Type I or Type II

Type I cutting: a tan λ ≤ D

2(θex − θst) (3–3)

Type II cutting: a tan λ >D

2(θex − θst) (3–4)

45

Each type of cutting is divided into three phases, called A,B and C, after Tlusty and

MacNeil (1975) who developed separate analytical expressions for cutting forces for the

different phases. Their result has four separate sets of equations, one each for the phases

A and C, and two separate equations for phase B corresponding to Type I and Type II

cutting.

The solution presented here treats the three phases as a single one. Algebra takes

care of the two types of cutting governed by Eqs. 3–3 and 3–4. A single expression each

for θL and θT is developed which covers all the different possible too chip contact zone

configurations.

By inspection of the geometry shown in Figure 3-2, the evolution of the intermediate

variables θL, and θT can be studied as functions of θp in Type I and Type II cutting.

The relationships are sketched in Fig. 3-3 in which the relation between b and θp is also

sketched since it is obtained directly by inspection. Now, it is just a matter of writing

single analytical expressions to characterize these sequences of straight line segments

which are periodic over one cutter rotation. Upon inspecting Fig. 3-3, it can be seen that

the various line segments in θL and θT start and end at the same values of the argument

θp. So, it turns out that the analytical expressions are the same for Type I and Type II

cutting, and the user does not have to worry about this distinction.

3.2 Analytical Expressions for θL, θT , and Chip Width, b

In this section compact expressions for θL, θT , and chip width, b will be developed as

a function of θp. Two alternative formulations are presented here, the Heaviside unit step

function formulation and the Fourier series formulation.

3.2.1 Heaviside Unit Step Function Formulation

Inspection of Fig. 3-3 reveals that the functions θL, θT , and b consist of finite line

segments in the domain θp ∈ [0, 2π). The equations of these line segments can be written

by multiplying the equation of the corresponding straight line (which has infinite extents)

with a function which has unit value in the domain of the line segment, and vanishes

46

outside that domain. Such a unit function is merely the difference of two appropriate

Heaviside unit step functions. The following definition of the Heaviside unit step function

is adopted

H (θp − ς) =

1, θp > ς

0, θp < ςθp ∈ [0, 2π) (3–5)

where ς is some fixed value of θP . The function as single valued everywhere in the domain.

The unit step function based expressions for θL and θT are written by inspection of

Fig. 3-3. The equations for the individual straight line segments for θL and θT are given in

Appendix B. The equations of the straight lines are multiplied by a suitable difference of

the step functions. The resulting functional relationships are

θL = θp [H (θp − θst)−H (θp − θex)] + θex[ H (θp − θex)−H (θp − θex − 2a tan λ/D)] (3–6)

θT =

(θp − 2a tan λ/D)[ H (θp − θst − 2a tan λ/D)−H (θp − θex − 2a tan λ/D)]

+θst H (θp − θst)−H (θp − θst − 2a tan λ/D)

(3–7)

The chip width, b, is obtained by inspection of geometry in Fig. 3-2, and may be

expressed as a function of θp through the intermediate variables θL and θT

b = [(D/2) cot λ] θL − θT (3–8)

3.2.2 Fourier Trigonometric Series Formulation

The patterns for θL, θT , and b repeat once every revolution (Fig. 3-3), yielding a

fundamental period of 2π, and making it possible to write these functions in trigonometric

series with appropriate Fourier coefficients (Kreyszig, 2006) as follows

θL = L0 +∞∑

k=1

Lk cos(kθp) + Mk sin(kθp) (3–9)

θT = T0 +∞∑

k=1

Tk cos(kθp) + Rk sin(kθp) (3–10)

where L0, Lk, Mk and T0, Tk, Rk are the relevant Fourier coefficients.

47

Equation 3–8 from the Heaviside formulation still holds. Therefore, for the purpose

of computation, a separate expression for b need not be derived. However, in some

applications, especially if the intent is to study the variation of b alone1, it is more

economical to compute just a single set of Fourier coefficients, instead of two sets for

θL and θT . Thus, b may be expressed in a trigonometric series:

b = B0 +∞∑

k=1

Bk cos(kθp) + Ck sin(kθp) (3–11)

where B0, Bk, Ck are the relevant Fourier coefficients.

Based on the variations given in Fig. 3-3, the Fourier coefficients are derived by

piecewise integrations over the period θp ∈ [0, 2π) using standard procedures (details in

Appendix A). Average Fourier coefficients in Eqs. 3–9 - 3–11 are:

L0 =1

2π

[(θ2

ex − θ2st)

2+

2a tan λ

Dθex

](3–12)

T0 =1

2π

[(θ2

ex − θ2st)

2+

2a tan λ

Dθst

](3–13)

B0 =a(θex − θst)

2π(3–14)

Results of the calculations for Lk,Mk, Tk, Rk, Bk and Ck are listed in Table 3-1. For

instance, the coefficient, Lk, may be formed by inspection of column 2 of Table 3-1 which

shows that Lk has four terms in the summation

Lk = −θst

πksin[kθst]− 1

πk2cos[kθst] +

1

πk2cos[kθex] +

θex

πksink[θex + (2a tan λ)/D] (3–15)

1 Yang et al. (2005) and Xu et al. (1998) have offered interesting studies of depth-of-cutvariations in endmilling and ball endmilling, respectively.

48

The same approach is used for Mk, Tk, Rk, Bk and Ck. The trigonometric series

formulation warrants the use of an infinite sum. In practice, a partial Fourier summation

may be used, which necessitates the truncation of the series. The user chooses the number

of terms in the summation to obtain a suitable convergence depending upon the accuracy

requirements of any specific application.

3.3 Analytical Expressions for Chip Thickness

In this section, analytical expressions for the mean instantaneous chip thickness,

hm, and the averaged mean chip thickness, hm, are derived for helical peripheral

milling. Consider Fig. 2-3 which shows a sketch of a chip in helical milling. The mean

instantaneous chip thickness, hm, may be computed by taking the mean over the entire

angular extent spanned by the chip

hm =1

θT − θL

θT∫

θL

fT sin θdθ (3–16)

Upon simplification,

hm = fT sin

(θL + θT

2

)sinc

(θT − θL

2

), fT ξh (3–17)

where the symbol ξh is shorthand notation and the sampling function (sine cardinal) is

defined as

sinc(ς) , sin(ς)

ς(3–18)

The mean instantaneous chip thickness, hm, varies as the tool moves across the chip.

Figure 3-4 shows the evolution of hm over the complete tooth passage across the uncut

chip. The averaged mean chip thickness, hm, is found by averaging hm over the span of

angular tool-chip contact (θspan)

hm =1

θspan

θex+ 2a tan λD∫

θst

hm(θp)dθp =fT

θspan

θex+ 2a tan λD∫

θst

ξhdθp (3–19)

49

where

θspan , θex +2a tan λ

D− θst (3–20)

In this manner, the mean instantaneous chip thickness and the averaged mean

instantaneous chip thickness have been expressed analytically.

3.4 Effect of Process Parameters on Chip Thickness and Chip Width

Figure 3-5 shows an example of the evolution of the mean chip thickness, hm, as

the cutter rotates. As the helix angle increases, the span of angular tool-chip contact

increases. This results in a lowering of hm. The sensitivity of hm to helix angle is fairly

low.

Figure 3-6 illustrates the variation of hm as a function of radial immersion and axial

depth of cut for a given helix angle. The sensitivity to axial depth of cut is very small.

Figure 3-7 illustrates the variation of hm as a function of radial immersion for different

helix angles. The effect of the helix angle on the averaged mean chip thickness is amplified

at higher levels of radial immersion.

The evolution of chip width is studied using two examples. The results of calculation

for these examples are given in Figs. 3-8 and 3-9. As the helix angle increases, the angular

span of tooth chip contact increases, and at some helix angle, the cut transitions from

Type I to Type II.

3.5 Chapter Summary

In this chapter, the two intermediate variables θL and θT have been identified in

terms of which the chip width and chip thickness in helical peripheral milling have been

analytically expressed. These intermediate variables are themselves expressed in terms

of the independent variable θp so that analytical expressions for chip width and chip

thickness are available as a function of θp. Two different formulations of the analytical

expressions have been derived which are exactly equivalent.

The identification of the variables θL and θT makes it possible to write single

expressions for the chip thickness and chip width in helical peripheral milling which

50

are valid for the entire cutter rotation. The Type I and Type II cutting distinction, as well

as the different phases A, B, and C in tool chip engagement, are automatically taken care

of by the algebra. This method of writing single analytical expressions valid for the entire

cutter rotation will be used in later chapters for calculations of cutting forces and cutting

coefficients. This is one of the main contributions of this research.

The reason for calculating the chip thickness and chip width in helical milling is to be

able to use these expressions in deriving a mechanistic model for cutting forces. The next

chapter is devoted to introducing the reader to the mathematical structure of such models.

Table 3-1. Construction of Fourier coefficients Lk,Mk, Tk, Rk, Bk and Ck

Terms comprisingthe Fourier sum

Coefficients of the terms in col.1 of this table

Lk Mk Tk Rk Bk Ck

sin[k(θst)] − θst

πk− 1

πk2 − θst

πk0 0 −D cot λ

2πk2

cos[k(θst)] − 1πk2

θst

πk0 θst

πk−D cot λ

2πk2 0

sin[k(θex)] 0 1πk2 0 0 0 D cot λ

2πk2

cos[k(θex)]1

πk2 0 0 0 D cot λ2πk2 0

sin[k(θst + 2a tan λD

)] 0 0 0 − 1πk2 0 D cot λ

2πk2

cos[k(θst + 2a tan λD

)] 0 − θex

πk− 1

πk2 0 D cot λ2πk2 0

sin[k(θex + 2a tan λD

)] θex

πk0 θex

πk1

πk2 0 −D cot λ2πk2

cos[k(θex + 2a tan λD

)] 0 0 1πk2 − θex

πk−D cot λ

2πk2 0

51

ΩΩ

development of

the uncut chip

development of the

helical cutting edge

exθstθ

Tθ

Lθp,θ

θ

θ

tool chip

contact

zone

uncut chip

helical

cutting

edge

λ

pθ

workpiece

endmill

a

a

Figure 3-1. Development of a helical cutting edge and the corresponding uncut chip. Theangular orientation, θ, of any point on the tool chip contact zone, or on thecutting edge, is reckoned from an arbitrary reference. Leading and trailingpoints of the contact zone are at θL and θT , respectively. The tool tip is at θp.The entry and exit angles (θst and θex) are fixed orientations, a is thecommanded axial depth of cut, and λ is the helix angle. A representative toolchip contact configuration is shown.

ba

λ

θ

exθstθ

b

T pθL,

b

pθLT

Type

I c

uttin

g

θ θθ θ

T pθL,θ θ

chip

θ

bb

T pθL

b

pθLT

Phase A (increasing b)

Typ

e II

cuttin

g

Phase B (constant b) Phase C (decreasing b)

θ θ θθ

a

exθstθλ

T pθL,θ θ

development

of uncut chip

development of the

helical cutting edge

tool chip

contact

zone

Figure 3-2. Progress of tool chip contact zone as the helical endmill rotates. In Type Icutting, b = a, in Phase B of the cut. In Type II cutting, b < a, always.

52

stθ

pθ

stθ

exθ

exθ

Lθ

D

2 a tan λ exθ

+

st pθ

st

ex

ex

Lθ

D

2 a tan λ

exθ

+

Type I cutting Type II cutting

θ

θ

θ θ

Tθ

Tθ

stθpθexθ

stθ

exθ

D

2 a tan λ stθ

+ D

2 a tan λ exθ

+

stθpθexθ

st

ex

D

2 a tan λ stθ

+D

2 a tan λ exθ

+

θ

θ

pθ

a

b

pθ

a

b

stθ exθ

D

2 a tan λ stθ

+D

2 a tan λ exθ

+

stθ exθ

D

2 a tan λ stθ

+ D

2 a tan λ exθ

+

Figure 3-3. Evolution of the intermediate variables θL, and θT as functions of θp in Type Iand Type II cutting. Evolution of the chip width, b, is also shown. Thefunctional relationships are obtained by inspection of Fig. 3-2

53

θp

hm

θp

hm

θp

hm

Figure 3-4. Concept of the averaged mean chip thickness, hm, for helical peripheralmilling. The mean chip thickness, hm, is averaged over the entire angular spanof tool-chip contact to compute hm. A case of down-milling with a two-flutedendmill is illustrated.

0 50 100 1500

0.02

0.04

0.06

0.08

0.1

θp (deg)

hm

(m

m)

50% RI

0 20 40 60

0

0.02

0.04

0.06

0.08

0.1

helix angle (deg)

50% RI

0 50 100 1500

0.02

0.04

0.06

0.08

0.1

hm

(m

m)

20% RI

0 20 40 600

0.02

0.04

0.06

0.08

0.1

helix angle (deg)

20% RI

hm

(m

m)

hm

(m

m)

0° helix

30° helix

45° helix

60° helix

θp (deg)

Figure 3-5. Evolution of the mean chip thickness, hm, as the cutter rotates, for differenthelix angles at two different radial immersions, and the corresponding averagedmean chip thickness, hm. In this example, fT = 0.10 mm/tooth and a = 4 mm.The action of a single tooth is depicted.

54

00.2

0.40.6

0.81

0

2

4

6

80

0.2

0.4

0.6

0.8

1

Radial immersion

Axial depth of cut (mm)

h−

m

f−

T

Figure 3-6. Variation of the averaged mean chip thickness, hm, with radial immersion andaxial depth of cut for a given helix angle (45). In this plot, hm is normalizedwith the feed per tooth, fT . The variation is the same for up-milling anddown-milling, and independent of endmill diameter.

0 25 50 75 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

h− m

fT

Radial immersion (%)

0°

30°

45°

60°

Helix angle

−

Figure 3-7. Variation of the averaged mean chip thickness, hm, with helix angle for the fullrange of radial immersion, with 4 mm axial depth of cut. hm is normalizedwith the feed per tooth, fT . The variation is the same for up-milling anddown-milling, and independent of endmill diameter.

55

θ (deg)p

0

2

4

b (

mm

) λ = 0.01°

zero helix

λ = 30°

0

2

4

b (

mm

) λ = 45° λ = 60°

0 100 200 300

0

2

4

b (

mm

) λ = 67°

0 100 200 300

λ = 70°

θ (deg)p

Figure 3-8. Chip width evolution for varying helix angles over one cutter rotation for asingle fluted endmill, based on the analytical solution (RI = 50%, a = 4 mm,D = 12 mm, up-milling).

0

2

4

b (

mm

) λ = 0.01° λ = 30°

0

2

4

b (

mm

) λ = 45° λ = 57°

0 100 200 300

0

2

4

b (

mm

) λ = 60°

0 100 200 300

λ = 70°

zero helix

θ (deg)p θ (deg)p

Figure 3-9. Chip width evolution for varying helix angles over one cutter rotation for asingle fluted endmill, based on the analytical solution (RI = 25%, a = 4 mm,D = 12 mm, up-milling).

56

CHAPTER 4MECHANISTIC FORCE MODEL FOR STRAIGHT FLUTED ENDMILLS

The purpose of this chapter is to provide the reader with a background on the method

of mechanistic force modeling and the associated terminology. The mechanistic model

chosen for this demonstration is a two dimensional force model for peripheral milling with

straight fluted endmills which is widely used (Tlusty, 2000). It captures the components

of the cutting force in the plane of cutter rotation. For milling with straight flutes, with

only peripheral cutting, the axial force component vanishes. So a two dimensional model

suffices.

In a mechanistic force model, the geometry of the chip area is determined. The

chip area is multiplied by appropriate cutting coefficients to obtain the respective force

components. These cutting coefficients are lumped parameters. They capture the effects

of the material properties of the workpiece materials being machined, tribological aspects

of tool work interface friction including the application of cutting fluids, the impact of

the specific tool geometry such as the rake and clearance angles, and the dependence

on cutting conditions such as the machining parameters used. Such models need to be

experimentally calibrated for any specific application on a given machining set-up, which is

the subject of discussion in the next chapter.

The mechanism of machining of ductile metallic alloys involves cutting (shearing at

interatomic planes) and plowing actions. Shearing action is the dominant mode of power

consumption in general applications. For micromachining applications, especially when

the order of magnitude of the feed is the same as the edge rounding radius of the sharp

cutting edges, the plowing action can consume comparable amounts of power. The lumped

parameter cutting coefficients which are related to the chip area capture the cutting

(shearing) action well and may be called (shear) cutting coefficients. Plowing is better

modeled as being proportional to the length of the cutting edge in contact with the chip,

57

and the related lumped parameter cutting coefficients are called edge coefficients1. In this

document, the plowing mechanism is ignored. Hence, the models outlined here should not

be used directly in micromachining applications, nor for feed rates which are so low as to

be of order of magnitude similar to the edge radii of the sharp cutting edges.

A mechanistic model presented by Sabberwal (1961) related the tangential force

to the chip area with the cutting coefficient being the constant of proportionality. The

coefficient was called the specific pressure, and was experimentally demonstrated to be

independent of the helix angle for a few work materials. This idea is used as the basis of

the model presented in this chapter.

4.1 Force Model for a Single Tooth

For straight fluted cutters, the instantaneous tangential cutting force, Ft, and the

instantaneous normal cutting force, Fn, on any given tooth are empirically computed

by writing these as functions of the uncut chip area (Tlusty, 2000). The axial chip area

is zero. Hence, the axial cutting force component vanishes. The problem is rendered

2-dimensional for straight fluted cutters. The components of the force in the plane need

only be considered. Figure 4-1 shows the instantaneous orientations of these two force

components. The orientations of these forces change with the rotation of the cutter, i.e.,

these components are expressed in a rotating coordinate frame attached to the endmill.

Eq. 2–4 yields the uncut chip area as Ac = bh(= afT sin θ). Force components on the pth

tooth, as a function of its angular orientation, θp, may be written as

1 Edge coefficients are denoted with a subscript e but are ignored in this document.The cutting coefficients are denoted by a subscript c in later chapters when helical millingis considered. In this chapter, the subscript c is dropped for simplicity but it should beunderstood that they are cutting coefficients, as opposed to edge coefficients. This alsoserves to distinguish the coefficients in this document, those without the subscript c areassociated with straight fluted cutters, and those having subscript c are associated withhelical cutters.

58

Ft

Fn

=

bh 0

0 bh

Kt

Kn

= afT

sin θp 0

0 sin θp

Kt

Kn

(4–1)

where Kt and Kn are lumped parameter cutting coefficients which depend on the

tool-workpiece-cutting fluid combination, as well as on machining parameters such as

cutting speed and chip thickness. They may be obtained by experiment during the process

of model calibration for any specific situation. Kt and Kn are defined in a rotating frame

of reference, and are called the tangential and normal cutting coefficients, respectively.

In a fixed frame of reference, the components of the cutting force experienced by a

single tooth are obtained using the transformations shown in Fig. 4-2, resulting in the

following familiar (Tlusty, 2000) relationships for the force components when the tooth is

in the cut

Fx

Fy

=afT

2

sin 2θp 1− cos 2θp

1− cos 2θp − sin 2θp

Kt

Kn

, θp ∈ [θst, θex] (4–2)

and Fx,y ≡ 0 when θp /∈ [θst, θex] (tooth out of the cut).

4.2 Average Force Based Estimates of Kt and Kn

Sabberwal (1961) showed that the mean cutting coefficient could be related to the

average values of the cutting force. For a single tooth, the averaged components of the

cutting force, F x, and F y, may be expressed as,

F x

F y

=

afT

2(θex − θst)

θex∫

θst



dθp

Kt

Kn

(4–3)

Upon simplification

F x

F y

=afT

2 (θex − θst)

ν1 ν2

ν2 −ν1

Kt

Kn

(4–4)

59

where, for convenience, ν1 and ν2 are shorthand for the following expressions

ν1 = sin(θst + θex) sin(θex − θst) (4–5)

ν2 = (θex − θst)− cos(θst + θex) sin(θex − θst) (4–6)

Inverting

Kt

Kn

=2 (θex − θst)

afT (ν21 + ν2

2)

ν1 ν2

ν2 −ν1

F x

F y

(4–7)

where the average force components are obtained from experiments, and all other terms on

the right hand side of the above equation are known.

4.3 Multiple Tooth Formulation

For a multiple toothed cutter, individual force signatures due to each tooth are

qualitatively identical, except that they are shifted in phase by the pitch angle with

respect to the reference (pth) tooth, and differ in magnitude since the associated feed per

tooth differs due to radial runout or differential pitch effects. The components of the total

force, FX,Y , are a summation of the force components of the individual teeth

FX

FY

=

N∑i=1

Fx

(θp − χi, fTi

)

N∑i=1

Fy

(θp − χi, fTi

)

(4–8)

where

FX and FY are the components of the total force,

χi = pitch angle of the ith tooth w.r.t. the pth tooth,

fTi= the feed/tooth associated with the ith tooth,

N = the number of teeth in the cutter,

and p is arbitrary (1 ≤ p ≤ N), being just a reference.

4.4 Effects of Tooth Runout

Kline and DeVor (1983) identified the ratio of the runout to the feed rate as an

important parameter which determines the effect of runout on the cutting force system.

60

The radial runout influences the effective feed rate experienced by an individual tooth.

Upon clamping the endmill in the toolholder and clamping the toolholder assembly on

the spindle, a static measurement of runout can be taken using a dial indicator. The total

indicated reading (T.I.R.) can be used as a measure of the radial runout. The relative

runout between successive teeth, ρ, governs the effective feed experienced by each tooth.

The tooth having the highest point would experience the maximum effective feed per

tooth. These ideas are illustrated in Fig. 4-3 in which a two fluted cutter is considered for

the sake of simplicity.

Let fT1 and fT2 be the effective feeds per tooth experienced by the two teeth, and fT

be the nominal (commanded) feed per tooth. The following relations hold

fT1 + fT2 = 2fT (4–9)

fT1 − fT2 = |2ρ| (4–10)

with the notation convention such that fT1 > fT2. Solving Eqs. 4–9 and 4–10 simultaneously

fT1 = fT + |ρ| (4–11)

and

fT2 = fT − |ρ| (4–12)

For a cutter with more than two teeth, the expressions for effective feed can be derived

using similar arguments.

4.5 Cutting Coefficient Model

For fixed levels of cutting speed and axial depth of cut, the cutting coefficients depend

on the the chip thickness. A logarithmic relationship of the general form K = eΓ(h)Ψ,

61

proposed by Sabberwal (1961) is adopted, leading to

ln Kt

ln Kn

=

1 ln h 0 0

0 0 1 ln h

Γt

Ψt

Γn

Ψn

(4–13)

where Γt,n and Ψt,n are cutting constants whose values depend on the combination of tool

material and work material, the specific cutting geometry, as well as cutting conditions,

such as the type of cutting fluid being used. These cutting constants are true constants,

as opposed to the cutting coefficients which have just been modeled as being functions of

chip thickness. In general, the cutting coefficients are functions of machining parameters,

i.e., the speed, feed, and the axial as well as radial depths of cut (Gradisek et al., 2004).

In this document, the coefficients are evaluated experimentally for fixed values of cutting

speed and axial depth of cut. Hence, they depend only on the feed and radial depth of cut,

which, together, determine the chip thickness.

4.6 Force Prediction Example

Based on the above force model, the two components of the cutting force may

be predicted after experimentally determining the cutting constants/coefficients. An

illustration of the predicted forces is given in Fig. 4-4. The method of calibrating the

model through experimental input parameter determination is discussed in the next

chapter.

4.7 Chapter Summary

The two dimensional mechanistic force model for peripheral milling with straight

fluted endmills was presented in order to familiarize the reader with the mathematical

structure of a mechanistic force model. The effects of a single tooth were first obtained.

For multiple toothed endmills, a summation is used with appropriate phase shifts for

successive teeth. This rigid force model does not include forced vibration or tool detection

effects, but includes the effects of relative radial runout of the teeth. The concept of

62

lumped parameter cutting coefficients was invoked. It was stated that they are a

function of machining parameters and are related to cutting constants, which are true

invariants. These cutting coefficients and cutting constants have to be experimentally

determined. Experimental errors are to be expected and random effects of variations of

material, friction, etc., are involved, which introduce uncertainties in measured values. The

determination of the cutting coefficients and cutting constants, along with the associated

variances, is the subject of the next chapter.

Ω

feed a

bh

FtFn

workpiece

tn

Fn

Ω

endmillF

θθp

t

Figure 4-1. The uncut chip area, Ac = bh, is related to the instantaneous tangential forcecomponent, Ft, and the instantaneous normal force component, Fn. A righthanded, straight fluted endmill having two teeth is illustrated in down millingconfiguration.

F

θ

y

x

workpiece

t

n

θworkpiece

p

feed

t

Fn

Ft

Fn

FnxFny F

tx

Fty

Ω

endmill

Ω

Figure 4-2. Transformation of forces from a rotating frame (t, n) to a fixed frame (x, y).

63

Ω

Effective feed per tooth

Work

f fT1 T2

f f T T

(a): No runout: f = f = f T2 T1 T

Ω

f f T T

(b): With runout: f = f = f T2 T1 T

fT2T1

Ω

ΩNo runoutendmill

fEffective feed per tooth

T.I.R

/ /

Figure 4-3. Idealization of radial runout and its effect on the feed per tooth. The nominal(commanded) feed per tooth is fT . The total indicated reading (T.I.R.) oversuccessive teeth is a measure of the relative runout between the two teeth. Therunout is exaggerated for illustrative purposes. A case of up-milling with a twofluted cutter, having equispaced teeth, is shown.

−100

0

100

200

300

Fx (

N)

e

0 90 180 270 360−100

0

100

200

300

Fy (

N)

θp

(deg)

Feed force

Lateral force

Figure 4-4. Force prediction example: 2-fluted endmill having diameter 12.7mm, feed 0.150mm/tooth, 50% radial immersion, axial depth-of-cut 0.5 mm, cutting speed 72m/min, up-milling, nominal runout 0.015 µm. Work material: low carbon steel(HV170). Tool material: solid carbide. Cutting constants:Γt = 7.179, Ψt = −0.4145, Γn = 7.006, Ψn = −0.5203.

64

CHAPTER 5FORCE MODEL CALIBRATION FOR STRAIGHT FLUTED ENDMILLS

The aim of this chapter is to familiarize the reader with the method of calibration

of a mechanistic force model. The two dimensional cutting force model described in the

previous chapter is experimentally calibrated. The model input parameters which are

required to make force predictions are the values of the cutting coefficients (or the cutting

constants) and the values of relative radial runout between successive teeth.

Experimental determination introduces uncertainties in the model parameters. These

uncertainties in the input parameters propagate through the force model. The analysis

of uncertainty propagation is the subject of the next chapter. However, the variances

of the model input parameters are determined in this chapter. These variances are a

result of random and systematic effects in experimental measurements. Those effects are

identified, and the uncertainties attached to the model input parameters are quantified in

this chapter.

5.1 The Partial Radial Immersion Experiment

Figure 5-1 shows the experimental set-ups for cutting force measurement and radial

runout measurement. Cutting coefficients were extracted for dry machining low carbon

steel (hardness HV170) using a 25% radial immersion experiment. A straight fluted,

uncoated, solid carbide endmill of SGS1 make, catalog no. 30423, 12.7 mm (1/2”)

diameter, having 2 teeth, was used. The machine was a Mikron UCP Vario 5-axis

machining center. A standard collet chuck was used. Spindle adaption was hollow

shank, taper and face contact HSK-63A. Upon clamping the tool in the spindle, the

endmill was indicated using a dial gage, of Teclock make, having a least count of 2.5 µm

(0.0001”). The nominal total indicated reading (T.I.R) was 15 µm. A three component

1 In this document, all commercial products are identified for the sake of completeness,and to enable other investigators to replicate the experiments. This does not constituteendorsement of any of these products.

65

dynamometer, model 9257B of Kistler make, was used, along with charge amplifiers,

to record two components of the force, i.e., the feed force Fx and the lateral force Fy.

Data was acquired using a National Instruments data acquisition card at a sampling

frequency of 60 kHz. The raw data was denoised using a commercial, wavelet based filter,

available in the MATLABr software of The MathWorks, Inc. This software was also

used to process the data and present the results graphically. Bias correction for DC offset

was manually performed for each set of data obtained. A Tetrahertz Technologies laser

tachometer was used to record a synchronizing phasor signal to help estimate the point

of entry into the cut for one of the flutes in the endmill. Figure 5-2 shows a sketch of

the orientation of the phasor signal with respect to the feed direction. This provides a

reference angular position in the rotation so that the theoretical and experimental force

signals can be aligned.

Cutting coefficients were extracted based on a set of experiments with 25% radial

immersion. The fixed conditions of the experiments are given in Table 5-1. The average

chip thickness can be varied by changing the feed based on Eq. 2–5. Values of cutting

coefficients were recorded for each level of average chip thickness, and linear regression

was used to establish relationships between them. Using these cutting coefficients,

force predictions were made for up milling (50% RI), and mixed mode (75% RI), and

experimentally verified for different values of feed.

5.2 Experimental Extraction of Cutting Coefficients

Equations 4–2 and 4–7 indicate that the cutting coefficients are possibly correlated.

So, the covariances associated with Kt and Kn need to be considered. Experiments were

conducted at seven different feeds, in the range 0.050 − 0.250 mm/tooth, the time traces

of force components were recorded, and average force values, (fx, fy), were computed. The

experimental estimates of the average cutting force components are represented using the

corresponding lower case letters to distinguish sample measures from population measures.

At each feed rate, the experiment was repeated five times. Forces were averaged over one

66

rotation, on a per tooth basis. To reduce the variability, the average was taken for two

successive rotations. Values of Kt and Kn, corresponding to each feed rate, were derived

using Eq. 4–7. Seven sets of data, with five replications in each set, were collected for

fitting a linear model according to Eq. 4–13. Two parameters, the slope and the intercept,

were fitted to 35 points in each regression yielding 35 − 2 = 33 degrees of freedom (Taylor

and Kuyatt, 1994, pp.9) for the standard uncertainties associated with each of the four

cutting constants Γt,n and Ψt,n.

The cutting constants Γt,n an Ψt,n of Eq. 4–13 may be obtained using multiresponse

linear regression. Kurdi (2005) has solved a multiresponse regression problem having a

similar mathematical structure, using the theory presented by Zellner (1962). The method

fits a linear regression model and enables the evaluation of the variance-covariance matrix

between the responses, and the variance-covariance matrix of the random error in the

regression model, which permits the extraction of the variance-covariance matrix of the

cutting constants (details in Appendix A). This information is required for the evaluation

of uncertainty from random effects in the measuring process for the cutting constants,

when making predictions using the force model.

Zellner’s (1962) method of estimating the multiple response parameters, and the

variance covariance matrix of these estimators, was used in calculations. The fitted

regression lines are displayed in Fig. 5-3. Based on the regression, the estimated cutting

constants, Γt,n and Ψt,n, are calculated and given in Table 5-2.

The nominal values of cutting coefficients, obtained from the experiment, may be

computed by using the nominal values of the cutting constants in the Eq. 4–13

Kt = exp(Γt)(h)Ψt = 1312(h)−0.4145 [N/mm2] (5–1)

Kn = exp(Γn)(h)Ψn = 1103(h)−0.5203 [N/mm2] (5–2)

with the average chip thickness, h, expressed in [mm].

67

The reader should note that the above expressions are derived based on experiments

in which the domain of h is

0.0239 [mm] ≤ h ≤ 0.1194 [mm] (5–3)

Hence, the expressions should strictly be used only for this range of h. However, a certain

amount of extrapolation may be acceptable in practical situations.

5.3 Variances of Model Input Parameters

The sources of uncertainty are first identified. They are classified as being of

random or systematic origin. Appropriate methods are applied for the evaluation of

the uncertainties arising out of each source. The analytical method of uncertainty analysis

presented in this research is based on the voluminous 1995 Guide to the Expression of

Uncertainty in Measurement (GUM) (ISO, 1995). The NIST Technical Note 1297, 1994

Edition (Taylor and Kuyatt, 1994), is a more compact guideline authored by two of the

primary authors (Kacker et al., 2007) of the GUM. This document is open to the general

public. So, most references are made to the NIST document. These documents classify

uncertainties into two different components. Uncertainty components which are evaluated

by statistical methods are classified as Type A, whereas those which are evaluated by

other means are classified as Type B.

The measured values of the lumped parameter cutting coefficients, and cutting

constants, are subject to random effects owing to random variations in material properties,

tool chip interface friction, etc. This random effect can be quantified using statistical

evaluation. The variances of the cutting constants are estimated using a Type A

evaluation which is based on a statistical analysis of the measurement data.

The variance-covariance matrix of the cutting constants, Γt,n and Ψt,n, as obtained

using Zellner’s (1962) method, is given in Table 5-3. The diagonal elements are the

68

variances, and the off-diagonal elements are the respective covariances2 . These variances

quantify the random effects in the estimation of the the cutting constants. The variances

of the cutting constants are designated as having been evaluated using a Type A analysis.

Measuring instruments can cause systematic errors. In this case there are two

measurement devices. The dynamometer, amplifier, and data acquisition card system

forms a chain for measurement of the cutting force components. A dial indicator, having

a finite resolution, has been used to measure the radial runout of successive teeth. The

effects of these two measuring devices on the variances of model input parameters are

quantified using the Type B evaluation. To distinguish between the two measurements,

the variances of measured forces are designated as being evaluated using a Type B1

analysis, and the variances of measured radial runouts of successive teeth are designated as

being evaluated using a Type B2 analysis.

These variances in the model input parameters (cutting constants) due to random

effects are obtained directly by using the Type A evaluation. However, the variances

in the model input parameters (cutting coefficients and effective feed per tooth) due

to the systematic effects are not obtained directly using the Type B evaluation. The

measurement values which are affected by systematic errors are the average force

components, and the radial runouts. The Type B1 and B2 evaluations yield the variances

of these measurements as will be shown in what follows. These variances need to be

propagated to the cutting estimated values of cutting coefficients and effective feeds per

tooth. The task of propagating the variances is accomplished in the next chapter. In this

chapter, only the variances of the average forces and radial runout are determined using

the Type B1 and B2 evaluations, respectively.

2 The large number of significant figures are carried so as not to be affected by round offerrors (Dieck, 1997). The Schwarz inequality, |covar(g1, g2)|2 ≤ var(g1)var(g2), is used as across check for the numerical calculations (Taylor, 1997).

69

The force measuring dynamometer, amplifier, and the digital data acquisition card

form a chain for measurement of cutting forces, based on which the cutting coefficients

are evaluated. So, force measurement can impose systematic errors. Systematic effects

in the force measurement process are assessed using a Type B1 analysis based on an

assumed distribution of the combined uncertainties in the average force measurements. An

underlying normal distribution is assumed. Instrument manufacturer’s recommendations

are used to quantify variances of the measured forces.

The variances in measured values of average force components, u2(fx,y,z), may be

calculated based on estimates of standard uncertainties in force measurements provided

by the instrument manufacturer (Cadille, 2008) who is certified to ISO 9001 and ISO

17025 (for calibration). According to the manufacturer’s certificate, the total uncertainty

of force measurement using multicomponent dynamometers having piezoelectric charge

devices, is calculated as√

u2(force range) + u2(charge). For forces of 0...60,000 lbf, the

uncertainty is 0.5%, and for charge of 0...50,000 pC, uncertainty is 0.5%, which yields√

(0.5)2 + (0.5)2 ' 0.707%. For the charge amplifier in the measuring chain, the typical

uncertainty in these ranges is < 0.5% and should be added to the 0.707%. The worst case

uncertainty is then stated at 0.707 + 0.5 = 1.207%. Based on this certificate, values of

u(fx,y,z) are set at 1.207% of the nominal values of the average force components, i.e.,

u(fx,y,z) ' 0.01207fx,y,z (5–4)

The uncertainty associated with runout measurement is captured using a Type B2

evaluation as discussed in Taylor and Kuyatt (1994). The dial indicator used in runout

measurement resolves to 0.0025 mm. The cosine error of the lever type indicator was

neglected in this analysis. Based on a rectangular (uniform) distribution of the half

interval, the variance of the radial runout measurement (Taylor and Kuyatt, 1994) is

u2(ρ) =

(0.0025√

3

)2

(5–5)

70

5.4 Chapter Summary

The experimental determination of model input parameters for the two dimensional

cutting force model for straight fluted endmills was presented. Uncertainties associated

with the measurement processes were quantified. The two coefficients Kt and Kn

are possibly correlated. So, in addition to their variances, their covariance needs to

be determined. The covariance of cutting coefficients due to random effects was not

calculated directly because of the structure of the cutting coefficient model. The cutting

coefficients are nonlinear functions of h (or h). A nonlinear curve fitting procedure is

required to find the covariances, which is difficult. The linear relation ship between the

logarithms of Kt,n and h (or h) is exploited. This yields the variances and covariances of

cutting constants (instead of cutting coefficients) by linear regression using Zellner’s (1962)

method. Thus the random effects are captured.

The variances of cutting coefficients and effective feeds per tooth, due to systematic

effects, are captured indirectly. The variances of the measured average cutting forces were

estimated based on the instrument manufacturer’s estimates which assume an underlying

normal distribution for the variances. The variance of runout measurement was estimated

assuming an underlying rectangular distribution. These variances need to be propagated

to the variances in estimated values of the model input parameters (Kt, Kn, and fTi).

The idea of quantifying uncertainties in cutting coefficients, for their propagation

through models for stability and surface location analysis, has appeared in the literature

(Kurdi, 2005; Duncan et al., 2006). However, the idea of quantifying variances in model

input parameters for placing confidence intervals on predicted cutting forces, and the

determination of the variance-covariance matrix of cutting constants to capture the

random effects for the logarithmic cutting coefficient model chosen here, are contributions

of this research. The propagation of the model input parameter uncertainties through the

cutting force model is the subject of the next chapter.

71

Table 5-1. Experimental cutting conditions: dry machining of low carbon steel (HV170)with a straight fluted, uncoated, solid carbide endmill

Endmill diameter Cutting speed No. of teeth Axial depth of cut

12.7 mm 72 m/min 2 (equispaced) 0.5 mm

Table 5-2. Mean values of estimated cutting constants for cutting conditions of Table 5-1

Γt Ψt Γn Ψn

Estimated parameters: 7.179 − 0.4145 7.006 − 0.5203Goodness of fit: Adj. R2 = 0.943 Adj. R2 = 0.903

Table 5-3. Symmetric variance-covariance matrix of cutting constants for experimentalconditions of Table 5-1

Γt Ψt Γn Ψn

Γt 0.002561 0.000879 0.002046 0.000702Ψt 0.000312 0.000702 0.000249Γn 0.007161 0.002456Ψn 0.000873

72

Figure 5-1. Experimental set-up showing the straight fluted endmill held in a collet chuckand mounted on the vertical spindle, the laser tachometer, and the workpiecemounted on the dynamometer which is held on the machine table. The set-upfor radial runout measurement using the dial indicator is also shown.

73

One tooth aligned with the black-white

transition on the reflecting surface

marked on the toolholder

feedworkpiece

Fixed orientation of the phasor signal

with respect to the feed direction

Figure 5-2. Experimental estimation of the angular orientation of a tooth at the entry of acut using a synchronizing phasor signal. A case of up-milling with 50% radialimmersion is illustrated.

8

8.5

9

ln [

Kt (

N/m

m2)]

Adj. R2 = 0.943

−3.5 −3 −2.5 −2

8

8.5

9

ln [

Kn (

N/m

m2)]

ln [ h (mm)]

Adj. R2 = 0.903

_

Figure 5-3. Linear regression fitting of cutting coefficients, as a function of the averagechip thickness, for dry milling of low carbon steel (HV170), using a straightfluted solid carbide endmill, having 2 equispaced teeth, and a nominal runoutof 15 µm. Experimental points are denoted by small circles.

74

CHAPTER 6PROPAGATION OF UNCERTAINTIES THROUGH THE FORCE MODEL

FOR STRAIGHT FLUTED ENDMILLS

Experimental calibration of the force models introduces uncertainties in the model

input parameters. The analysis of the propagation of these uncertainties through the force

model is one of the important contributions of this research. Uncertainty propagation can

be studied by analytical methods or by sampling methods such as the Monte Carlo. In

this document, the analytical method of propagation has been adopted which is facilitated

by the closed form solutions of the force models. In this chapter, model parameter

uncertainties are propagated through the two dimensional force model for straight fluted

endmills. This serves to demonstrate the technique. The same procedure will be used in

subsequent chapters where closed form force models are developed for helical peripheral

milling.

Though there do not appear to be any studies in the literature that have quantified

the uncertainties associated with predicted instantaneous cutting forces for peripheral

milling, examples of applications in which such an uncertainty analysis is important

include the real time monitoring of tool wear and the automated sensing of tool breakage,

both of which are based on force sensing principles. In the metal cutting community, there

is interest in quantifying uncertainties associated with model based predictions of different

aspects of the machining process. For instance, in his investigation of the optimization of

the milling processes under uncertainty, Kurdi (2005) has studied the variances of cutting

coefficients, and placed confidence intervals on stability boundaries.

6.1 Propagation of Uncertainties Through a Mathematical Model

The sources of uncertainty were identified in the last chapter. Variances of cutting

constants, average forces, and radial runouts were determined by Type A, B1, and B2

evaluations, respectively.

In this chapter, individual standard uncertainties are propagated through the model

to obtain an estimate of the combined uncertainties. The combined uncertainty of

75

predicted force components due to the random effects on cutting constants is designated

ucA(fx,y,z). The combined uncertainty of force components due to the effect of the

force measuring instrumentation is designated ucB1(fx,y,z) , and that due to the runout

measuring instrument is designated ucB2(fx,y,z). According to Taylor and Kuyatt (1994),

when a measurand Y is not measured directly, but is found from N other quantities

X1, X2, . . . XN using a functional relation

Y = g(X1, X2, . . . XN) (6–1)

then an estimate of the measurand, or output quantity Y , denoted by the lowercase y, is

found using input estimates x1, x2, . . . , xN for the N input quantities X1, X2, . . . XN using

the functional relationship

y = g(x1, x2, . . . xN) (6–2)

The combined standard uncertainty of the measurement result y, denoted by uc(y) is

taken to represent the estimated standard deviation of y, and is the positive square root of

the estimated variance u2c(y) given by

u2c(y) =

N∑i=1

(∂g

∂xi

)2

u2(xi) + 2N−1∑i=1

N∑j=i+1

∂g

∂xi

∂g

∂xj

u(xi, xj) (6–3)

where u2( ) are variances and u( , ) are covariances of the input estimates.

Taylor and Kuyatt (1994) refer to Eq. 6–3 as the law of propagation of uncertainty.

The combined standard uncertainties of the predicted forces are derived using this law for

which the functional relation ‘g’ is given by Eq. 4–2, which defines the function governing

the force components. The sensitivity coefficients are the partial derivatives of the force

components with respect to the model input parameters which have uncertainties attached

to them, namely, the cutting constants, the cutting coefficients, and the effective feed

rates.

76

6.2 Variances of Cutting Coefficients and Effective Feed Rates

In section 5.3 of chapter 5, it was observed that the variances of model input

parameters due to random effects were directly available from the Type A evaluation

in the form of a variance-covariance matrix for the cutting constants Γt,n and Ψt,n.

However, the variances of model input parameters due to systematic effects were not

available directly from the Type B1 and B2 evaluations. The Type B1 evaluation provided

an estimate of the variances of the measured average forces Eq. 5–4, and the Type

B2 evaluation gave an estimate of the variances of measured values of radial runout of

successive teeth Eq. 5–5. These variances need to be propagated to obtain estimates of

the variances of cutting coefficients and effective feeds per tooth, which are model input

parameters required for force predictions using Eqs. 4–2 and 4–8.

6.2.1 Propagation of Average Force Uncertainty to Cutting Coefficients

Equation 4–7 yields the following sensitivity coefficients for use in propagation of the

uncertainty in average force measurements to the uncertainties in cutting coefficients

stx

sty

snx

sny

=

∂Kt/∂F x

∂Kt/∂F y

∂Kn/∂F x

∂Kn/∂F y

=2(θex − θst)

afT (ν21 + ν2

2)

ν1

ν2

ν2

−ν1

(6–4)

Using the above sensitivities in Eq. 6–3 the variances of cutting coefficients, due to

the systematic effects, can be obtained

u2(kt)

u2(kn)

=

s2tx s2

ty 0

s2nx s2

ny 0

u2(fx)

u2(fy)

(6–5)

The values of u(fx,y) in the Eq. 6–5 above are set at 1.207% of the nominal values

of the average force components based on the instrument manufacturer’s estimates as

explained in detail in chapter 5. For simplicity, any possible correlation between Kt and

Kn from measurement channel cross talk is neglected, i.e., u(kt, kn) is set to zero.

77

6.2.2 Propagation of Radial Runout Uncertainty to Effective Feeds

Equations 4–11 and 4–12 yield the following sensitivity coefficients for use in

propagation of the uncertainty in relative runout to the uncertainties in effective feeds

∂fT1

∂ρ= 1 and

∂fT2

∂ρ= −1 (6–6)

Equations 5–5 and 6–6 may be used in Eq. 6–3 to estimate the variances of the

effective feed for each individual tooth

u2(fT1) =

(∂fT1

∂ρ

)2

u2(ρ) =

(0.0025√

3

)2

(6–7)

u2(fT2) =

(∂fT2

∂ρ

)2

u2(ρ) =

(0.0025√

3

)2

(6–8)

6.3 Propagation of Uncertainties Through the Cutting Force Model

The three components of uncertainty in input parameters have to be propagated

through the force model Eq. 4–2. Hence, their sensitivities must be computed.

6.3.1 Sensitivity Coefficients of Component Uncertainties

Sensitivity coefficients used to determine the Type A component uncertainties are

s11

s12

s13

s14

s21

s22

s23

s24

=

∂Fx/∂Γt

∂Fx/∂Ψt

∂Fx/∂Γn

∂Fx/∂Ψn

∂Fy/∂Γt

∂Fy/∂Ψt

∂Fy/∂Γn

∂Fy/∂Ψn

=bfT

2

Kt sin 2θp

[Kt ln h] sin 2θp

Kn(1− cos 2θp)

[Kn ln h](1− cos 2θp)

Kt(1− cos 2θp)

[Kt ln h](1− cos 2θp)

−Kn sin 2θp

−[Kn ln h] sin 2θp

(6–9)

where Kt and Kn are parameterized in Γt,n and Ψt,n, according to Eq. 4–13.

78

Sensitivities coefficients with respect to the cutting coefficients are used in the

determination of the Type B1 component uncertainties and may be found using Eq. 4–2

sxKt

sxKn

syKt

syKn

=

∂Fx/∂Kt

∂Fx/∂Kn

∂Fy/∂Kt

∂Fy/∂Kn

=afT

2

sin 2θp

1− cos 2θp

1− cos 2θp

− sin 2θp

(6–10)

Sensitivity coefficients with respect to the effective feeds are used in the determination

of Type B2 component uncertainties. These are found using Eqs. 4–2, 2–5, and 4–13

sxf

syf

=

∂Fx/∂fT

∂Fy/∂fT

=a

2



(1 + Ψt)Kt

(1 + Ψn)Kn

(6–11)

The sensitivities in Eqs. 6–9 - 6–11 are functions of θp. For illustrative purposes, they

are plotted in Fig. 6-1 for a particular combination of machining parameters.

6.3.2 Propagation of Type A Uncertainties

Let lowercase γt,n and ψt,n be the input estimates for the values of input quantities

Γt,n and Ψt,n, respectively. The lowercase representation is used here to indicate sample

measures which are used as estimates of the population measures Γt,n and Ψt,n. These

estimates are available from Table 5-2.

The estimated variances of the cutting constants denoted by u2( ) and the

estimated covariances of the cutting constants denoted by u( , ) are available from

Table 5-3. These uncertainties have to be propagated to the predicted forces using the

sensitivities, sij, given by Eq. 6–9.

These sensitivities are themselves a function of the cutting constants. hence, they

are evaluated at the estimated parameters γt,n and ψt,n. The combined component

uncertainties of predicted forces for the pth tooth, ucA(fx,y), due to the uncertainties in the

cutting constants, are obtained using the sensitivities, sij, in the propagation law Eq. 6–3

79

which assumes the following form

u2cA

(fx)

u2cA

(fy)

=

s211

s212

s213

s214

s221

s222

s223

s224

u2 (γt)

u2 (ψt)

u2 (γn)

u2 (ψn)

+2

s11s12 s11s13 s11s14 s12s13 s12s14 s13s14

s21s22 s21s23 s21s24 s22s23 s22s24 s23s24

u(γt, ψt)

u(γt, γn)

u(γt, ψn)

u(ψt, γn)

u(ψt, ψn)

u(γn, ψn)

(6–12)

6.3.3 Propagation of Type B1 Uncertainties

The variances of cutting coefficients due to average force measurement uncertainties

are available from Eq. 6–11 based on the Type B2 evaluation. These uncertainties may

be propagated to the predicted forces using Eq. 4–2 and the sensitivities expressed in Eq.

6–10 to yield the combined component uncertainties of predicted forces for the pth tooth,

ucB1(fx,y), by applying Eq. 6–3

u2cB1

(fx)

u2cB1

(fy)

=

s2xKt s2

xKn

s2yKt s2

yKn

u2(kt)

u2(kn)

(6–13)


The uncertainties in effective feed per tooth due to radial runout measurement

uncertainties are available from Eqs. 6–7 and 6–8 based on the Type B2 evaluation. These

uncertainties may be propagated to the predicted forces using Eq. 4–2 and the sensitivities

expressed in Eq. 6–5 to yield the combined component uncertainties of predicted forces for

80

the pth tooth, ucB2(fx,y), by applying Eq. 6–3

u2cB2

(fx)

u2cB2

(fy)

= u2(fT )

[sxf ]2

[syf ]2

(6–14)

where the subscript on fTp is dropped for convenience.

6.4 Expanded Uncertainty

The objective of the exercise is to specify the 95% confidence intervals on predicted

forces. Coverage factors for component expanded uncertainties are first computed based

on the associated probability distributions of the component combined uncertainties. To

determine the appropriate coverage factor for expanded uncertainty corresponding to a

95% confidence interval, an appropriate convolution of the various probability distributions

needs to be considered (Turzeniecka, 2000; Fotowicz, 2006).

6.4.1 Expanded Uncertainty Coverage Factor Type A

The Type A standard uncertainty components, owing to the variances and covariances

in the cutting constants, are combined using the law of propagation of uncertainty to yield

the combined standard uncertainties, ucA(fx), and ucA(fy), which are assumed to have

an approximately normal distribution. This happens when the conditions of the Central

Limit Theorem are met. When this is the case, ucA(fx,y) themselves have negligible

uncertainty (Taylor and Kuyatt, 1994, pp.4), and a ±2ucA(fx,y) width about the nominal

value defines an interval in which the measurement result is believed to lie with a level of

confidence of approximately 95%, i.e., the coverage factor for the expanded uncertainty is

κA = 2.

If ucA(fx,y) themselves have non-negligible uncertainty, a conventional procedure may

be used to produce a coverage factor that produces an interval having the approximate

level of confidence desired. The effective degrees of freedom for the combined uncertainties

are estimated based on the Welch-Satterthwaite (W-S) expression (Taylor and Kuyatt,

81

1994, pp. 8-9, Eqs. B-1 and B-2) as follows

dfeff |ucA(fx) = [u?cA(fx)]

4 ×[s411u

4(γt)

df |u(γt)

+s412u

4(ψt)

df |u(ψt)

+s413u

4(γn)

df |u(γn)

+s414u

4(ψn)

df |u(ψn)

]−1

(6–15)

dfeff |ucA(fy) = [u?cA(fy)]

4 ×[

s421u

4(γt)

df |u(γt)

+s422u

4(ψt)

df |u(ψt)

+s423u

4(γn)

df |u(γn)

+s424u

4(ψn)

df |u(ψn)

]−1

(6–16)

and

dfeff |ucA(fx,y) 6 df |u(γt)+ df |u(ψt)

+ df |u(γn) + df |u(ψn) (6–17)

where dfeff |ucA(fx) and dfeff |ucA(fy) are the effective degrees of freedom for the combined

Type A component uncertainties of fx and fy respectively. The superscript ?, used in the

terms u?cA(fx,y) in Eqs. 6–15 and 6–16, indicates that no correlation is considered among

the inputs in calculating these combined uncertainties, i.e. the covariance terms in Eq.

6–12 are not considered, as suggested by Willink (2007) in his generalization of the W-S

expression for use with correlated uncertainty components.

To determine the cutting constants in chapter 5, experiments were conducted at

seven different feeds. At each feed, five replications were taken. Two parameters (slope

and intercept) were fitted to 7 × 5 = 35 points in each regression, yielding 35 − 2 = 33

degrees of freedom for the standard uncertainties associated with each of the four cutting

constants Γt,n, and Ψt,n (Taylor and Kuyatt, 1994, pp.9).

The coverage factor for the Type A component expanded uncertainty is determined

based on the Student’s t-distribution for any desired confidence interval, and may be

read off the Table B.1 in Taylor and Kuyatt (1994) knowing the value of dfeff |ucA(fx,y).

The effective degrees of freedom are functions of the angular position of the pth tooth,

θp. Therefore, the coverage factor is also a function of θp. The functional dependence is

illustrated for an example case in Fig. (6-2). For this particular case, it may be noted

that the effective degrees of freedom are quite large (≥ 31), so that the coverage factor

approaches 2. By inspection of the figure, κA ' 2.04 may be taken as a good conservative

82

value of the coverage factor for the Type A component expanded uncertainty for a 95%

confidence interval for both, the x and y components of the force.

6.4.2 Expanded Uncertainty Coverage Factor Type B

For the Type B1 uncertainties due to systematic effects in the force measuring chain,

underlying approximately normal distributions are assumed for the combined component

uncertainties. Based on this the coverage factor for the Type B1 component expanded

uncertainty is κB1 ' 2 for a 95% confidence interval.

For the Type B2 uncertainties due to systematic effects of the runout measuring

device, the rectangular distribution is assumed with infinite degrees of freedom. Based on

this the coverage factor for the Type B2 component expanded uncertainty is κB2 ' 1.65

for a 95% confidence interval.

6.4.3 Overall Expanded Uncertainty

This problem has both Type A and Type B uncertainties with different types of

underlying probability distributions (approximately normal and rectangular). To calculate

the overall expanded uncertainty, a convolution of these probability distributions has

to be considered, which is a difficult problem. Turzeniecka (2000) has suggested various

approximate methods of calculating the expanded uncertainty in such situations. The best

choice depends on the relative magnitudes of the Type A and Type B uncertainties. In the

problem at hand, this ratio is not fixed. This fact is clarified with an example shown in

Fig. 6-3. Moreover, the ratio is a function of θp, and process parameters such as the feed

and radial immersion. For such a situation, Turzeniecka (2000) has suggested the root sum

of squares (RSS) method as a good solution. The expanded uncertainty is the RSS1 of the

component expanded uncertainties

U(fx,y) =√

κ2Au2

cA(fx,y) + κ2B1

u2B1

(fx,y) + κ2B2

u2B2

(fx,y) (6–18)

1 Turzeniecka (2000) has named it the vector sum method

83

The predicted forces are

Predicted forces ' Fx,y ± U(fx,y)

where Fx,y are calculated based on Eq. 4–2.

The above analysis considers the forces and uncertainties associated with the

engagement of a single tooth. For multiple tooth engagement, the forces due to each

tooth are merely summed. However, the uncertainties must be obtained by the root

sum of squares method by considering the sum of the squares of each of the component

combined uncertainties for every tooth individually, and taking the square root of this

overall sum.

All the uncertainties, combined as well as expanded, are functions of the angular

position of the pth tooth, an example of which is given in Fig. 6-3. For this same example

confidence intervals are placed on predicted forces and shown in Fig. 6-4.

6.5 Force Prediction with 95% Confidence Interval

Various combinations of radial immersion and feed rates were used to make the force

predictions as given in Table 6-1. Confidence intervals were placed on predicted forces

based on the uncertainty analysis. Experimental force signals are superimposed on the

predicted force signals to display the results graphically in Figs. 6-5 - 6-9.

6.6 Chapter Summary and Outlook for the Forthcoming Chapters

The purpose of this chapter was not to study force models, but to demonstrate

the propagation of uncertainties in model input parameters through a force model.

An analytical uncertainty analysis procedure was applied to the practical problem of

determining confidence intervals on predicted cutting forces in peripheral milling. This

covers a gap in the literature in the field of machining. One application, where cutting

force sensing is used for feedback, is the real time monitoring of tool wear and tool

breakages. In comparing the actual cutting forces with predicted (expected) forces, the

confidence bounds on the predicted forces need to be considered. One has to be careful

84

in attributing deviations from nominal predicted forces to wear or breakage, when these

deviations lie within the confidence regions.

Though the uncertainty analysis was performed for a cutting force model applicable

for straight fluted endmills, the procedure is quite general. For helical endmills, the basic

force model will differ, but the uncertainty analysis method will remain exactly the same.

This technique of uncertainty propagation through a milling force model is one of the

contributions of this research. The procedure for determining the uncertainties, and

the corresponding confidence intervals, is readily implemented. The ability to provide a

defensible uncertainty statement to accompany cutting force predictions has a practical

benefit. It enables the process planner to decide the usefulness of model based force

predictions in any specific application. The success of the analytic uncertainty analysis

procedure hinged on the fact that closed form analytical expressions were available for the

cutting forces as well as cutting coefficients. Even so, for the two dimensional force model

for straight fluted cutters, the uncertainty analysis was quite algebraically involved.

In the forthcoming chapters closed form analytical solutions will be developed for

helical peripheral milling. The models will be mechanistic having the same structure as

the force model described in these last three chapters. The closed form solutions will also

permit the use of the same uncertainty propagation procedure as was adopted in this

chapter.

Table 6-1. Summary of experimental conditions used for verification of force predictions,holding the cutting conditions of Table 5-1 fixed

Resultsdisplayed in

Feed(mm/tooth)

Radialimmersion (%)

Up/down milling or mixedmode (> 50% RI)

Figure 6-5 0.150 50 Up millingFigure 6-6 0.100 50 Up millingFigure 6-7 0.050 50 Up millingFigure 6-8 0.200 75 mixed mode, h = 0 at the entryFigure 6-9 0.150 75 mixed mode, h = 0 at the entry

85

−600

−400

−200

0

200

400

(a) Type A

s

11

s12

s13

s14

s21

s22

s23

s24

−0.05

0

0.05

0.1

Se

nsi

tiv

ity

co

eff

icie

nts

(a

pp

rop

ria

te u

nit

s)

(b) Type B1

sxKt

sxKn

syKt

syKn

0 90 180 270 360

0

500

1000

θp

(deg)

(c) Type B2

sxf

syf

Figure 6-1. Variation of sensitivity coefficients with angular position of the pth tooth: feed0.150 mm/tooth, 50% radial immersion, up-milling, nominal runout 15 µm.Other conditions as in Table 5-1.

86

30

40

50

60

df e

ff

df

eff|ucA(f

x)

dfeff|ucA

(fy

)

0 90 180 270 3601.98

2

2.02

2.04

2.06

θp (deg)

κA

κAx

κAy

Figure 6-2. Variation of the effective degrees of freedom of combined uncertainties of TypeA, and the corresponding coverage factor for a 95% confidence interval, withangular position of the pth tooth: feed 0.150 mm/tooth, 50% radial immersion,up-milling, nominal runout 15 µm. Other conditions as in Table 5-1.

0

5

10

15

Co

mb

ine

d u

nce

rta

inty

(N

)

ucA

(fx)

ucB1

(fx)

ucB2

(fx)

ucA

(fy)

ucB1

(fy)

ucB2

(fy)

0 90 180 270 3600

5

10

15

θp

(deg)

Ex

pa

nd

ed

un

cert

ain

ty (

N)

U(Fx)95% CI

0 90 180 270 360

θp

(deg)

U(Fy)95% CI

Figure 6-3. Variation of component combined uncertainties, and overall expandeduncertainty for a 95% confidence interval, with angular position of the pth

tooth: feed 0.150 mm/tooth, 50% radial immersion, up-milling, nominalrunout 15 µm. Other conditions as in Table 5-1.

87

−100

0

100

200

300

Fx (

N)

Nominal predicted force

95% CI bounds

0 90 180 270 360−100

0

100

200

300

Fy (

N)

θp

(deg)

Figure 6-4. Example illustrating the placement of 95% confidence interval bounds onpredicted cutting forces: feed 0.150 mm/tooth, 50% radial immersion,up-milling, nominal runout 15 µm. Other conditions as in Table 5-1.

−200

0

200

400

Fx (

N)

0 90 180 270 360−200

0

200

400

θp (deg)

Fy (

N)

experiment

theory: upper 95% CI

theory: lower 95% CI

Figure 6-5. Predicted vs. experimental force signals: feed 0.150 mm/tooth, 50% radialimmersion, up-milling. Nominal runout 15 µm. Other conditions as in Table5-1.

88

−200

0

200

400

Fx (

N)

0 90 180 270 360−200

0

200

400

θp (deg)

Fy (

N)

experiment




−200

0

200

400

Fx (

N)

0 90 180 270 360−200

0

200

400

θp (deg)

Fy (

N)

experiment




89

−200

0

200

400

Fx (

N)

0 90 180 270 360−200

0

200

400

θp (deg)

Fy (

N)

experiment



Figure 6-8. Predicted vs. experimental force signals: feed 0.200 mm/tooth, 75% radialimmersion, cut starts with h = 0. Nominal runout 15 µm. Other conditions asin Table 5-1.

−200

0

200

400

Fx (

N)

0 90 180 270 360−200

0

200

400

θp (deg)

Fy (

N)

experiment



Figure 6-9. Predicted vs. experimental force signals: feed 0.150 mm/tooth, 75% radialimmersion, cut starts as up-milling. Nominal runout 15 µm. Other conditionsas in Table 5-1.

90

CHAPTER 7INSTANTANEOUS RIGID FORCE MODEL FOR HELICAL PERIPHERAL MILLING

In this chapter a new model for cutting forces in helical peripheral milling is

developed. The model has distinguishing features that imbue it with certain advantages

when compared with other models available in the literature. Since this is a mechanistic

model, the chip area has to be related to the cutting coefficients. The model is three

dimensional because the axial force does not vanish in helical milling. It is a rigid model

since the effects of tool or part deflections on the chip area are ignored. For simplicity, the

effect of radial runout of successive teeth is ignored. That effect will be incorporated in the

next chapter.

Figure 7-1 shows the scheme which is chosen to relate the three differential cutting

force components to the respective areas. The differential projected frontal chip area, dAf ,

is related to the differential tangential force component, dFt, and the differential normal

force component, dFn. The differential projected axial chip area, dAa, is related to the

differential axial force component, dFa. In contrast to the typical approach (e.g. Engin

and Altintas, 2001) the rake face chip area (the gray region in Fig. 7-1) is not used in the

model. A particular feature

of the model is that the axial projected chip area is related to the axial force

component, unlike the general practice of relating the rake face chip area. This ensures

that the axial force component automatically vanishes as the helix angle goes to zero in a

straight fluted cutter. It is not necessary to force the axial cutting coefficient to go to zero

for straight fluted cutters.

The three dimensional force model for helical peripheral milling presented in this

chapter differs from prior work in several ways. Each force component has a single, closed

form expression which is valid for the entire cutter rotation, and the structure of linearized

cutting coefficients is retained. Any arbitrary value of radial immersion, and hence even

small immersions, can be used to experimentally determine the cutting coefficients.

91

7.1 Force Model for a Single Toothed Cutter

Empirical relationships yield the tangential, normal, and axial components of the

differential elements of force for the mechanistic model

dFt

dFn

dFa

=

dAf 0 0

0 dAf 0

0 0 dAa

Ktc

Knc

Kac

(7–1)

where Ktc, Knc and Kac are linearized cutting coefficients which may be obtained

experimentally during the process of model calibration for any specific situation.

The rotating (tangential and normal) components are related to components in a

fixed coordinate frame via a rotation matrix (Fig. 4-2), while the axial component remains

decoupled

dFx

dFy

dFz

=

dFtx + dFnx

dFty + dFny

dFa

=

cos θ sin θ 0

sin θ − cos θ 0

0 0 1

dFt

dFn

dFa

(7–2)

The reader will note that the value of the determinant of the 3×3 matrix in the above

equation equals −1. The (x, y, z) coordinates form a left handed system. This is a result

of the way the (x, y) coordinates are laid out in Fig. 4-2. The sense of the cutter rotation

is clockwise. The positive z−axis is aligned with the sense of rotation. This choice is

arbitrary, and has no physical significance.

Substituting from Eq. 7–1 and integrating yields the total forces

Fx

Fy

Fz

=

∫cos θdAf

∫sin θdAf 0

∫sin θdAf − ∫

cos θdAf 0

0 0∫

dAa

Ktc

Knc

Kac

(7–3)

where the integrations are carried out over the appropriate limits. These limits are

explicitly shown in a subsequent step, after applying certain transformations.

92

To compute the integrals∫

sin θdAf and∫

cos θdAf in Eq. 7–3, a local coordinate, ζ,

has been invoked in Fig. 7-2. By inspection of the geometry

dζ =

(b

θT − θL

)dθ (7–4)

Again, from Fig. 7-2, the differential element of frontal chip area, dAf , is

dAf = hdζ = fT sin θdζ (7–5)

where the local chip thickness at θ is obtained using the circular path approximation,

h = fT sin θ Eq. 2–2.Eliminating dζ using Eq. 7–4 yields dAf as a function of θ

dAf =

(bfT

θT − θL

)sin θdθ (7–6)

Based on Fig. 7-3, the differential element of the projected axial chip area, dAa, is

dAa = hD

2dθ =

DfT

2sin θdθ (7–7)

The three integrals∫

sin θdAf ,∫

cos θdAf and∫

dAa may now be computed:

∫sin θdAf =

(bfT

θT − θL

) θT∫

θL

sin2 θdθ (7–8)

∫cos θdAf =

(bfT

θT − θL

) θT∫

θL

sin θ cos θdθ (7–9)

∫dAa =

DfT

2

θT∫

θL

sin θdθ (7–10)

These three results may be substituted into Eq. 7–3 to obtain, upon simplification

Fx

Fy

Fz

= fT

bξ1 bξ2 0

bξ2 −bξ1 0

0 0 Dξ3

Ktc

Knc

Kac

(7–11)

93

where

ξ1 = sin

(θT + θL

2

)cos

(θT + θL

2

)cos

(θT − θL

2

)sinc

(θT − θL

2

)(7–12)

ξ2 =1

2[1− cos (θT + θL) sinc (θT − θL)] (7–13)

and

ξ3 = sin

(θT + θL

2

)sin

(θT − θL

2

)(7–14)

where the sampling function (sinc) has been defined earlier in Eq. 3–18.

In the above equations, the variables θL, θT and b are functions of θp. The functional

expressions for θL, θT and b have been derived in chapter 3. The Eq. 7–11 represents the

force components on a single tooth, as a function of θp, in closed form.

7.2 Modeling for Multiple Teeth

By analogy to section 4.3 the components of the total force are a summation of the

force components of the individual teeth

FX

FY

FZ

=

N∑i=1

Fx

(θp − χi, fTi

)

N∑i=1

Fy

(θp − χi, fTi

)

N∑i=1

Fz

(θp − χi, fTi

)

(7–15)

where

FX , FY and FZ are the components of the total force,

χi = pitch angle of the ith tooth w.r.t. the pth tooth,

fTi= the feed/tooth associated with the ith tooth,

N = the number of teeth in the cutter,

and p is arbitrary (1 ≤ p ≤ N), being just a reference.

For uniformly spaced teeth, χi has a constant value. For differential tooth spacing the

tooth pitch angles, χi, are directly obtained from the specifications of the endmill. The

94

effective feeds per tooth, fTi, are found by dividing the feed per revolution in proportion

with the angular tooth spacings of successive teeth.

7.3 Formulation for Cutting Coefficient Identification

Cutting coefficients are empirical and are determined experimentally. Martellotti

(1941) proposed that the average undeformed chip thickness could be related to the

components of the cutting force. Sawin (1926), Salomon (1926), and Sabberwal (1961)

showed that the cutting coefficient varies with the chip thickness.

The coefficients also depend on other process parameters such as cutting speed

(Shin and Waters, 1997) and tool geometry (Jayaram et al., 2001). For simplicity, the

cutting speed, axial depth of cut, and tool geometry are kept fixed in the experiments

reported in this paper. Hence, the results reported here hold only for the specific type

of tool geometry used, the stated cutting speed, and the axial depth of cut used in the

experiments.

Based on Sabberwal (1961), the coefficients may be expressed as exponential functions

of the averaged mean chip thickness, hm, having the general form K = eΓ(hm)Ψ. Thus, the

following set of relations may be written

ln Ktc

ln Knc

ln Kac

=

1 ln hm 0 0 0 0

0 0 1 ln hm 0 0

0 0 0 0 1 ln hm

Γtc

Ψtc

Γnc

Ψnc

Γac

Ψac

(7–16)

where Γtc,nc,ac and Ψtc,nc,ac are cutting constants. The values of these constants depend on

the combination of tool material and work material, the specific cutting geometry, as well

as cutting conditions, such as the type of cutting fluid being used.

The cutting coefficients described above can be experimentally extracted based on

a small set of cutting tests. The coefficients Ktc,nc,ac corresponding to a given feed per

95

tooth, fT , may be calculated based on average cutting force components for a chosen set

of cutting conditions. For these same conditions, the averaged mean chip thickness, hm,

may be computed using Eq. 3–19. The exercise is repeated for a set of different values of

fT . Thus, a mapping is established, generating a functional dependence of Ktc,nc,ac on hm.

This data may be used to obtain appropriate fits to find the cutting constants in Eq. 7–16.

For a single helical tooth, the components of the cutting force may be averaged over

one cutter revolution. The averaged components of the cutting force, F x, F y, and F z, may

be expressed as,

F x

F y

F z

= fT

1

2π

2π∫

0

bξ1 bξ2 0

bξ2 −bξ1 0

0 0 Dξ3

dθp

Ktc

Knc

Kac

(7–17)

Using shorthand notation

F x

F y

F z

=fT

2π

I1 I2 0

I2 −I1 0

0 0 DI3

Ktc

Knc

Kac

(7–18)

where

I1 ,2π∫

0

bξ1dθp, I2

2π∫

0

bξ2dθp, I3 ,2π∫

0

ξ3dθp (7–19)

Solving for the coefficients yields

Ktc

Knc

Kac

=2π

fT

I1I21+I2

2

I2I21+I2

20

I2I21+I2

2

−I1I21+I2

20

0 0 1DI3

F x

F y

F z

(7–20)

where F x,y,z are to be obtained based on experiments, and the integrals need to be

evaluated for the corresponding experimental conditions. The integrands in Eq. 7–19 have

closed form expressions using Eqs. 7–12 - 7–14 and Eqs. 3–6 - 3–8 or 3–9 - 3–11. For a

96

particular set of parameters, the integrals I1,2,3 have been evaluated and are plotted in Fig.

7-4 for the purpose of illustration.

In this section, a procedure has been developed for the identification of cutting

coefficients. The formulation is indifferent to whether the cutting coefficients are a

function of helix angle. The expressions are readily computed for any value of radial

immersion. Hence, low immersion tests can be used to determine the cutting coefficients.

For any given combination of workpiece material, tool material, tool geometry, and cutting

conditions such as use of cutting fluid, the cutting coefficients may be experimentally

derived. The coefficients are variable as they are a function of chip thickness. So, cutting

constants have been invoked which are true invariants. In the forthcoming sections the

constants are experimentally determined and the force model is verified.

7.4 Verification of the Analytical Solution

In this section, the analytical solution proposed here is verified. First, the solution

is checked for the case of zero helix angle to see if the equations reduce to the case of

the straight flute solution of chapter 4. Next, the results of the analytical solution are

compared with a numerical solution available in the literature from Tlusty (1985, 2000).

7.4.1 The Degenerate Case of Straight Fluted Cutters

The degenerate case of straight fluted mills is verified by setting the helix angle to

zero in all the relevant force expressions. First, the following relations must be noted by

inspection of the geometry

θT − θL = 0 (7–21)

θL + θT =

2θp, θp ∈ (θst, θex)

0, elsewhere(7–22)

b =

a (a constant), θp ∈ (θst, θex)

0, elsewhere(7–23)

97

Substituting these values in Eq. 7–11 results in the familiar relationships Eq. 4–2for the force components when the tooth is in the cut [θp ∈ (θst, θex)]

Fx

Fy

Fz

=afT

2

sin 2θp 1− cos 2θp 0

1− cos 2θp − sin 2θp 0

0 0 0

Ktc

Knc

Kac

(7–24)

This verifies that the expressions in Eq. 7–11 are correct for the degenerate case,

λ = 0. In computer implementation, these equations are not well behaved, due to the cot λ

term which grows without bound as λ → 0. The formulation is rendered nominally useless

for λ = 0. However, the equations work accurately for extremely small values of helix

angle so that the helix angle is zero for all practical purposes. Figure 7-5 demonstrates the

results of computation for a straight fluted endmill using λ = 0.0000001. By inspection

of the force signals it can be concluded that the results of this analytical formulation are

accurate for straight fluted cutters.

7.4.2 Comparison with a Numerical Solution for Helical Endmills

Tlusty (1985, 2000) has presented a numerical model, based on the method proposed

by DeVor and Kline (1980) and Kline et al. (1982), in which the cutter is broken up into

infinitesimal slabs axially. The effects of each slab are numerically integrated to obtain the

complete force. Tlusty’s (2000) algorithm was used to generate the numerical results for

helical mills displayed in this section.

Figure 7-6 shows a comparison of the numerical solution and the proposed analytical

solution for a 30 helix cutter for 75% radial immersion, with the cut starting with h = 0.

The results agree very closely.

The residuals are plotted in Fig. 7-7 which shows that the difference between the

numerical and analytical solutions is negligible. This is a representative result.

Tlusty (1985, 2000) has published such plots for different combinations of radial

immersion and up/down-milling. Those numerical results are reproduced in Fig. 7-8 using

98

the analytical solutions of this chapter. These plots verify the correctness of the analytical

solution for a 30 helix angle.

When slotting with four teeth, the total tooth engagement angle is 180 and half the

cutter is always engaged. This implies that two flutes are always immersed in the cut. If

they are equispaced, the lag of corresponding points on successive teeth is 90. Therefore,

the total contribution of the two teeth to the chip thickness is constant. The total axial

engagement of the teeth that are in contact with the chip also remains constant. It

is expected that the total force in the plane[√

F 2x + F 2

y

]has a constant magnitude,

independent of the helix angle. Therefore, full immersion slotting, with a 4-fluted cutter,

forms a good test to examine the accuracy of the current formulation. The last plots

in Fig. 7-8 show the results of slot milling with a cutter having 30 helix angle and 4

equispaced teeth. The analytical solutions show the expected constant force magnitude.

7.5 Experimental Determination of Model Input Parameters

The model input parameters are experimentally determined using a partial immersion

experiment. The experimental method is the same as described in chapter 5 except for the

details.

Figure 7-9 shows the experimental set-up. Cutting coefficients were extracted for

dry machining the aluminum alloy 6061-T6 using a 50% radial immersion experiment. A

Kennametal solid carbide endmill, style HPF37A, with titanium diboride coating, having

45 helix, 12.7 mm diameter, and 3 flutes, was used. A Schunk TRIBOSr polygonal

clamping toolholder was employed to minimize runout. Spindle adaption was hollow

shank, taper and face contact HSK-63A. Upon clamping the tool in the spindle, no runout

was measurable when the endmill was indicated using a dial gage having a least count of

2.5 µm (0.0001”), which means that the theoretical formulation that assumes zero runout

can be experimentally verified. Other experimental conditions were the same as those used

for the experiment described in chapter 5. Figure 7-9 also shows a snapshot of a set of

recorded force signals.

99

For fixed levels of cutting speed and axial depth of cut, the averaged mean instantaneous

chip thickness, hm, is the only independent variable, for any given value of radial

immersion. The fixed conditions of the experiments are given in Table 7-2. A reasonably

large spectrum of hm is swept in the 50% radial immersion experiment. This value

was chosen merely for convenience. Any partial immersion experiment will do, so long

as the feed rate is varied enough to capture the desired range of hm over which the

coefficients need to be estimated. One of the features of this model is that a full immersion

experiment is not required to estimate the cutting coefficients. Gradisek et al. (2004) have

shown that the cutting coefficients are insensitive to the milling configuration, being the

same for up milling, down milling, and mixed mode milling. Using coefficients extracted

based on down milling experiments, force predictions have been made for down milling,

up milling, and mixed mode (> 50% radial immersion), and experimentally verified in a

forthcoming section.

Equations 7–11 and 7–20 indicate that the cutting coefficients in the plane (Ktc

and Knc) are possibly correlated. The axial force and the axial cutting coefficient are

independent. In considering the uncertainties associated with the predictions of Fx and

Fy, the variances, as well as the covariances associated with Ktc and Knc need to be

considered, i.e., it is a multi response, multivariate problem. In predicting the axial force

component, the variance associated with Kac suffices, i.e., it is a univariate problem.

The 50% radial immersion experiments were conducted at eight different feed rates,

equispaced within the range 0.025 - 0.200 mm/tooth. Time traces of force components

were recorded, and average force values, (fx, fy, fz), were computed. At each feed rate,

the experiment was repeated five times. Forces were averaged over one rotation, on a

per tooth basis. To control the variability, the average was taken over 100 successive

rotations. Values of cutting coefficients, corresponding to each feed rate, were derived

using Eq. 7–20. Thus, eight sets of five data points each were collected to fit a linear

model Eq. 7–16. Two parameters (slope and intercept) were fitted to 40 points, yielding

100

38 degrees of freedom for the regression (Taylor and Kuyatt, 1994, pp.9) for the standard

uncertainties associated with each of the four cutting constants Γtc,nc and Ψtc,nc.

The cutting constants Γtc,nc and Ψtc,nc of Eq. 7–16 may be obtained using multi

response linear regression, whereas the cutting constants Γac and Ψac can be obtained

using simple linear regression. For the multi response linear regression, Zellner’s (1962)

method of estimating the multiple response parameters (the two-stage Aitken estimators),

and the variance covariance matrix (the moment matrix) of these estimators was used,

following the same method as outlined in chapter 5. The linear regression lines are plotted

in Fig. 7-10. The estimated cutting constants Γtc,nc and Ψtc,nc are given in Table 7-3. A

linear fit was not found suitable for the axial cutting constants (R2 = 0.144 in linear

regression). The mean value of Kac = 395.7 N/mm2 was fitted, resulting in the values

Γac = ln(395.7) = 5.981 and Ψac = 0.

Thus, for any particular value of the averaged mean chip thickness, hm (expressed in

[mm]), the nominal values of cutting coefficients obtained experimentally are

Ktc = eΓtc(hm)Ψtc = 366.8(hm)−0.4906 [N/mm2] (7–25)

Knc = eΓnc(hm)Ψnc = 137.6(hm)−0.6339 [N/mm2] (7–26)

Kac = 395.7 [N/mm2] (7–27)


The method of assessing the variances of model input parameters, which has been

discussed in chapter 5, applies here. In this case there are two sources of uncertainty.

The first is the random effects of the variation of workpiece material properties, tool chip

interface friction coefficients, etc., on the cutting constants. This effect is evaluated using

a Type A analysis. The second source is the systematic effect of the force measuring

instrumentation chain which is evaluated using a Type B1 analysis. The runout is

not considered in this version of the model, though it was measured (and found to be

nominally zero on the experimental set-up).

101

Based on a Type A evaluation, the variances and covariances of the cutting constants,

Γtc,nc and Ψtc,nc, due to random effects, are obtained from the regression analysis described

in the previous section, and are given in Table 7-4. The diagonal elements are the

variances, and the off-diagonal elements are the respective covariances. Additionally,

the variance of the independent cutting constant, Γac, due to random effects, was found to

be 0.00247562.

Systematic effects in the measurement process are assessed using a Type B1 analysis

in the same manner as in chapter 5, except for differences in the details. Equation 7–20

yields the following sensitivity coefficients for use in propagation of the uncertainty in

average force measurements to the uncertainties in cutting coefficients

stx

sty

snx

sny

saz

=

∂Ktc/∂F x

∂Ktc/∂F y

∂Knc/∂F x

∂Knc/∂F y

∂Kac/∂F z

=2π

fT

I1/(I21 + I2

2 )

I2/(I21 + I2

2 )

I2/(I21 + I2

2 )

−I1/(I21 + I2

2 )

1/(DI3)

(7–28)

Using the above sensitivities the variances of cutting coefficients, due to the

systematic effects, can be obtained

u2(ktc)

u2(knc)

u2(kac)

=

s2tx s2

ty 0

s2nx s2

ny 0

0 0 s2az

u2(fx)

u2(fy)

u2(fz)

(7–29)

The values of u(fx,y,z) in the Eq. 7–29 above are set at 1.207% of the nominal values

of the average force components based on the instrument manufacturer’s estimates as

explained in detail in chapter 5. For simplicity, any possible correlation between Ktc and

Knc from measurement channel cross talk is neglected, i.e., u(ktc, knc) is set to zero.

102

7.7 Propagation of Input Parameter Uncertainties Through the Force Model

Sensitivities of force components with respect to the cutting constants are required

to propagate the Type A uncertainties in cutting constants through the force model.

Sensitivities of cutting force components with respect to the cutting coefficients are

required for propagation of the Type B1 uncertainties through the force model.

7.7.1 Propagation of Type A Uncertainties

The sensitivities with respect to the cutting constants are found by using Eq. 7–11

together with Eq. 7–16

s11

s12

s13

s14

s21

s22

s23

s24

s35

s36

=

∂Fx/∂Γc

∂Fx/∂Ψtc

∂Fx/∂Γnc

∂Fx/∂Ψnc

∂Fy/∂Γtc

∂Fy/∂Ψtc

∂Fy/∂Γnc

∂Fy/∂Ψnc

∂Fz/∂Γac

∂Fz/∂Ψac

= fT

bξ1Ktc

bξ1Ktc ln hm

bξ2Knc

bξ2Knc ln hm

bξ2Ktc

bξ2Ktc ln hm

−bξ1Knc

−bξ1Knc ln hm

Dξ3Kac

Dξ3Kac ln hm

(7–30)

where the cutting coefficients are parameterized in the cutting constants according to Eq.

7–16.

These sensitivities are expressed in terms of the cutting coefficients in the above

equation because the expressions remain compact when written in this manner. They are

functions of the angular position of the pth tooth, θp.

The variances of cutting constants due to random effects, available from the Type A

evaluation, may be propagated to the predicted forces using Eq. 7–11 and the sensitivities

expressed in Eq. 7–30 to yield the combined uncertainties ucA(fx,y,z) of predicted forces for

103

a single tooth by applying Eq. 6–3

u2cA

(fx)

u2cA

(fy)

u2cA

(fz)

=

s211 s2

12 s213 s2

14 0 0

s221 s2

22 s223 s2

24 0 0

0 0 0 0 s235 s2

36

u2 (γtc)

u2 (ψtc)

u2 (γnc)

u2 (ψnc)

u2 (γac)

u2 (ψac)

︸︷︷︸estimated variances

+2

s11s12 s11s13 s11s14 s12s13 s12s14 s13s14 0

s21s22 s21s23 s21s24 s22s23 s22s24 s23s24 0

0 0 0 0 0 0 s35s36

×

u (γtc, ψtc)

u (γtc, γnc)

u (γtc, ψnc)

u (ψtc, γnc)

u (ψtc, ψnc)

u (γnc, ψtc)

u (γac, ψac)

︸︷︷︸estimated covariances

(7–31)

where lowercase γtc,nc,ac and ψtc,nc,ac are input estimates for the values of input quantities

Γtc,nc,ac and Ψtc,nc,ac, respectively, and sij are the sensitivity coefficients evaluated at the

estimated parameters γtc,nc,ac and ψtc,nc,ac.


The sensitivities with respect to the cutting coefficients are found using Eq. 7–11

sxKt

sxKn

syKt

syKn

szKa

=

∂Fx/∂Ktc

∂Fx/∂Knc

∂Fy/∂Ktc

∂Fy/∂Knc

∂Fz/∂Kac

= fT

bξ1

bξ2

bξ2

−bξ1

Dξ3

(7–32)

104

The variances of cutting coefficients due to measurement uncertainties are available

from Eq. 7–29 based on the Type B1 evaluation. These uncertainties may be propagated

to the predicted forces using Eq. 7–11 and the sensitivities expressed in Eq. 7–32 to yield

the combined uncertainties ucB1(fx,y,z) of predicted forces for a single tooth by applying

Eq. 6–3

u2cB1

(fx)

u2cB1

(fy)

u2cB1

(fz)

=

s2xKt s2

xKn 0

s2yKt s2

yKn 0

0 0 s2zKa

u2(ktc)

u2(knc)

u2(kac)

(7–33)

7.7.3 Expanded Uncertainty

In the Type A evaluation of the variances and covariances of the cutting constants

was carried out under the assumption of a normal distribution for the measurements and

their combined uncertainties. A sufficiently large number of experiments were conducted

to yield 43 degrees of freedom for each regression. This means that the degrees of freedom

for the standard uncertainties of the cutting constants are large enough to justify the

choice of the usual coverage factor of 2 for the expanded uncertainty for a 95% confidence

interval based on the Student’s t-distribution (Taylor and Kuyatt, 1994), i.e., κA = 2.

The Type B1 evaluation of the standard uncertainties of dynamometer force

measurements was carried out based on the manufacturer’s certificate under the

assumption of a normal distribution for the measurements and their combined uncertainties.

It is assumed that the manufacturer’s estimate is based on a sufficiently large number of

observations to justify the choice of the usual coverage factor of 2 for the expanded

uncertainty for a 95% confidence interval based on the Student’s t-distribution, i.e.,

κB1 = 2. For both the sources of uncertainty, the underlying distributions are normal.

The expanded uncertainty may be computed as the root sum of squares of the component

expanded uncertainties. The total expanded uncertainty is

U(f) =√

κ2Au2

cA(f) + κ2

B1u2

cB1(f) (7–34)

105

The predicted forces are

Predicted forces = Fx,y,z ± U(fx,y,z)

where Fx,y,z are calculated based on Eq. 7–11.

7.8 Force Prediction Results

Force predictions were made for the various combinations of radial immersion and

feed rates given in Table 7-5. Three different cutting configurations were tested namely,

down-milling, up-milling, and mixed mode milling (> 50% radial immersion). The cutting

speed and axial depth of cut were kept fixed Table 7-2.Figures (7-11 - 7-20) show the results of force predictions in which upper and lower

bounds of 95% confidence interval of predicted forces are displayed graphically, along with

the experimentally obtained signals, which verify the predictions. Reliable predictions

were obtained for various combinations of feed, radial immersion, and up or down milling

configurations. The high fidelity of the axial force component prediction indicates that

considering the axial force to be proportional to the projected axial area is justified. The

cutting coefficients, derived on the basis of down milling, sufficed for up milling and mixed

mode predictions, as anticipated in Gradisek et al. (2004).

In general, the predictions were found to be quite reliable. However, certain

deficiencies are noticeable. The peak forces are not predicted very well in quite a few

of the cases. The model overpredicts the peak forces. For high immersion case (such as

75%RI), the experimental signals are somewhat skewed with respect to the predicted force

patterns. These deficiencies of the model are addressed in chapter 10 where a modification

is proposed which corrects these shortcomings.

7.9 Properties of the Analytical Force Model

In this section some of the mathematical properties of the analytical solution are

discussed. The two variants of the model, one based on the Heaviside step function model,

and the other based on the Fourier series solution, are compared and contrasted.

106

7.9.1 Gibbs-Wilbraham Distortion

The functions θL and θT have jump discontinuities Fig. 3-3. Therefore, the

Gibbs-Wilbraham phenomenon affects the Fourier reconstruction (Gibbs, 1898, 1899)

of these functions. The partial Fourier sums do not converge to the function values.

At the point of discontinuity, the convergence is to the midpoint of the jump, but the

reconstructed function oscillates wildly in the neighborhood of the jump discontinuity.

Taking a large number of terms to form the partial Fourier sum drives the oscillations very

close to the jump discontinuities. The oscillations do not vanish, no matter how many

terms are summed. In the neighborhood of the jump discontinuity, they overshoot the

target by about 9% (Brandolini and Colzani, 1999), independent of the number of terms in

the partial Fourier sum. Thus, function values of reconstructed θL and θT are uncertain to

within 9% in the neighborhood of the jump discontinuities. The problem does not affect

the reconstruction of b as it has no jump discontinuities. For this reason, the computed

values of Fx and Fy are inaccurate in a small neighborhood of the jump discontinuities of

θL and θT .

Figure 7-21 shows the effects of increasing the number of terms in the partial Fourier

sums. Even though θL and θT oscillate, the impact on Fx, Fy, and Ftotal is seen to be

very small because the chip width goes to zero at the points where θL and θT oscillate.

In all the calculations of force that have been made by the authors, the smoothness of

Fx, Fy, and Ftotal was never observed to be significantly affected by the Gibbs-Wilbraham

phenomenon affecting θL and θT . Since the chip width always goes to zero at the point

of these jump discontinuities of θL and θT , this factor does not play a significant role in

practical implementation of the code.

7.9.2 Computer Implementation Issues

A discussion on the convergence of the analytical solution and the computational

burden is in order. This is especially so because the Fourier formulation consists of an

infinite series. The Heaviside formulation does not have any convergence issues while the

107

Fourier formulation does. In practice, the infinite trigonometric series is implemented in

the computer using a partial Fourier summation. The question of convergence is easily

settled by the user who specifies the tolerance within which the magnitudes of the force

needs to be computed. One way to do this would be to compute the residuals at every

point in the discretized computational domain. The Heaviside solution or the numerical

solution may be used to calculate the residuals. The number of terms in the partial sums

may be gradually increased in suitable steps until the average residual is within the user

specified tolerance. The operation is then terminated.

Figure 7-22 displays the convergence characteristics of the analytical Fourier solution,

including the processing time. Convergence is achieved with less than 400 terms in the

partial Fourier sum. The computational burden for the Heaviside method is comparatively

less due to the absence of series summation.

7.9.3 Relative Merits of the Two Variants

The Heaviside method is preferable when the objective is to minimize computational

time. In general, this may be the deciding factor. The Heaviside method also yields

the exact result as it has no truncation error. If only a crude estimate of the forces is

required, either method may be used. In any application, if the force expressions need

to be symbolically differentiated, the user must bear in mind that the derivative of the

Heaviside step function is the Dirac delta. At the discontinuity, the derivative is not

defined. However, the force has a derivative at that point. Therefore, in such an instance,

the Fourier formulation may be preferred. Integration does not pose difficulties in either of

these formulations.

The Heaviside step function is implemented in a variety of ways in software.

Some packages do not allot a function value at the discontinuity. Others provide an

average value of half at this point. The programmer must be aware how this affects the

computation.

108

For the zero helix case, the Fourier method is more robust. One may assign as small

a value to the helix angle as one pleases, provided it is non-zero, and the code functions.

The step function method will misbehave in some software packages for angles less than

a certain threshold very near zero, depending on how the code is written. The code

functions correctly for helix angles as small as 1 × 10−5 degree as seen in the example in

Fig. 7-5.

For applications requiring symbolic manipulations, the Heaviside method is

convenient and compact. If accuracy requirements are high, the Fourier method becomes

impractical because the number of terms to be manipulated is very large.

7.10 Chapter Summary

A single, analytical closed form expression has been derived for each of the three

components of the cutting force, in a fixed reference frame, for helical peripheral

milling. A key advantage of the method is that the algebra takes care of all the different

geometrical possibilities (Type I, Type II cutting, and Phases A, B, C) automatically.

Separate expressions are not required, and there is no need to keep track of the cutting

type or phase in computations. These single expressions will allow analysts to conveniently

use them in analytical or semi-analytical applications where symbolic manipulations

are required to be performed. An example of this is the derivation of the sensitivity

coefficients in the uncertainty analysis.

The results have been shown to be valid for the entire parameter space covering

helical as well as straight fluted peripheral milling, partial or full immersion cutting,

multiple teeth in the cut, as well as for up-milling, down-milling and slotting by

comparison with established numerical methods.

Experimental validation was performed by calibrating the model for the aluminum

alloy 6061-T6 using a 45 helix endmill. The fidelity of predictions was shown to be high.

In deriving the axial cutting force component, the projected axial chip area has been

considered, which leads to good predictions of the axial cutting force.

109

One of the features of the model is that the zero helix assumption is not required to

be invoked while extracting the cutting coefficients. Another feature is that a full slotting

cut is not necessary to extract the cutting coefficients. Any value of partial immersion

suffices. This can lead to savings in test material when cutting expensive workpiece

material, or using machines with limited spindle power.

In computer calculations, for the degenerate case of straight flute cutters, a very

small helix angle must be specified, as the expressions are not well behaved for zero helix

angle. However, analytically, it has been shown that the expressions reduce to the familiar

relationships for zero helix, showing that the formulation remains valid. Therefore, the

expressions may be used in applications involving analytic manipulations, such as finding

derivatives or integrals.

This is a rigid model which does not account for radial runout or the effects of tool

or workpiece deflections on the cutting force. The experimental verification was done

with an endmill having teeth with equal pitch (angular spacing), i.e., the three teeth

were spaced at 120 to each other. It was noted that the model overpredicts peak forces,

and does not predict the skewness of the force signals which are experimentally observed

especially in high immersion experiments. These deficiencies are addressed in chapter 10

where the model is modified. However, before making the modification, procedures for

incorporating the effects of differential tooth spacing and radial runout of successive teeth

are considered. Extensions of this model to include these two effects are included in the

next two chapters, respectively.

110

Table 7-1. Parameters in the numerical solution of Tlusty (1985, 2000)

Endmilldiameter

Helixangle

Feed pertooth

No. ofteeth

Cutting coefficientsKtc Knc

30 mm 30 0.10 mm 4 2000 N/mm2 0.3Ktc

Table 7-2. Experimental cutting conditions: dry machining of the aluminum alloy 6061-T6with a TiB2 coated solid carbide endmill

Endmill diameter Helix angle Cutting speed No. of teeth Axial depth of cut

12.7 mm 45 240 m/min 3 4 mm


Γtc Ψtc Γnc Ψnc

Estimated parameters: 5.905 − 0.4905 4.925 − 0.6336Goodness of fit: Adj. R2 = 0.981 Adj. R2 = 0.992


Γtc Ψtc Γnc Ψnc

Γtc 0.00100886 0.00033512 -0.00000994 -0.00000330Ψtc 0.00011722 -0.00000330 -0.00000115Γnc 0.00068626 0.00022796Ψnc 0.00007974

111

Table 7-5. Summary of experimental conditions used for verification of force predictions,with the cutting conditions of Table 7-2 held constant

Resultsdisplayed in

Feed(mm/tooth)

Radialimmersion (%)

Up/down milling or mixed mode(> 50% RI)

Figure 7-11 0.050 25 Down millingFigure 7-12 0.200 25 Down millingFigure 7-13 0.100 10 Down millingFigure 7-14 0.200 10 Down millingFigure 7-15 0.050 5 Down millingFigure 7-16 0.200 5 Down millingFigure 7-17 0.100 75 mixed mode, cut starts with h = 0Figure 7-18 0.200 75 mixed mode, cut starts with h = 0Figure 7-19 0.100 25 Up millingFigure 7-20 0.100 10 Up milling

dA

dA

b hm

Ω

feed

θd

f

a

dFa

dFt

ndF

Figure 7-1. Differential projected chip areas in helical peripheral milling: The differentialprojected frontal chip area, dAf , is related to the differential tangential forcecomponent, dFt, and the differential normal force component, dFn. Thedifferential projected axial chip area, dAa, is related to the differential axialforce component, dFa. The rake face chip area (the gray region which islabeled with b and hm) is not used in the model. A two fluted, right handedhelical endmill is illustrated.

112

a

λ

θ

b

pθLθTθθ

ζ= 0

ζ= b

ζ

development of

the uncut chip

development

of the helical

cutting edgeTθf sin

T

Lθf sinT

dζ

dAf

θf sin T

ζ

Figure 7-2. Differential element of the projected frontal chip area, dAf .

ΩD

pθ

Lθ

Tθ θ

feed

o/

θ

T

θ

dθ

h = f sin θ

dAa

D/2

chip

Figure 7-3. Differential element of the projected axial chip area, dAa.

113

−4

−2

0

Th

e in

teg

ral,

I

(m

m)

1

0

5

10

15

Th

e in

teg

ral,

I

(mm

)2

00.5

1

02

46

8

−1

−0.5

0

Radial immersionAxial depth of cut (mm)

Th

e in

teg

ral,

I3

Figure 7-4. Variation of the integrals described in Eq. 7–19 for down milling using a 45

helix angle. The sign of I1 changes in up milling, with I2 and I3 remaining thesame. The integrals are independent of endmill diameter.

114

0

1000

2000

F x (

N)

0

1000

2000

F y (

N)

0 100 200 300 0 100 200 300

(deg)pθ (deg)p θ

Figure 7-5. Verification of the analytical model for the degenerate case of straight flutedcutter using λ = 0.00000001 to simulate the zero helix case for asingle-toothed endmill. The conventional solution plots (dashed lines) wereobtained by using Eq. 7–24 whereas Eq. 7–11 was used for the currentanalytical solution plots (solid lines). Parameters used: D = 30 mm, fT = 0.10mm/tooth, a = 10 mm, RI = 25%, Kt = 2000 N/mm2, Kn = 0.30Kt.

115

0

2000

4000

6000

8000

10000T

ota

l Fo

rce

(N

)

Numerical solution

0

2000

4000

6000

8000

10000

To

tal F

orc

e(N

)

Analytical solution (Fourier)

0 50 100 150 200 250 300 350

0

2000

4000

6000

8000

10000

Angular position of the pth

tooth, θp (deg.)

To

tal F

orc

e (

N)

Analytical solution (Heaviside)

a = 10 mm

a = 25 mm

a = 50 mm

Figure 7-6. Comparison of the closed form analytical solution with a numerical solutionpublished by Tlusty (1985, 2000) for a case of 0.75% RI, with h = 0 at theentry of the cut. Parameters used are given in Table 7-1. The total force inthe plane is displayed Ftotal =

√F 2

x + F 2y .

116

0 50 100 150 200 250 300 350

−0.5

0

0.5

Percent difference between analytical (Fourier) and numerical solution

Difference (%)

0 50 100 150 200 250 300 350

−0.5

0

0.5

Percent difference between analytical (Heaviside) and numerical solution

Difference (%)

0 50 100 150 200 250 300 350

−0.5

0

0.5

Percent difference between the analytical solutions (Fourier and Heaviside)

Angular position of the pth tooth, θ

p (deg.)

Difference (%)

Figure 7-7. Residuals showing the percentage difference between the analytical andnumerical solutions displayed in Fig. 7-6

117

0

2000

4000

Fourier solution

To

tal F

orc

e(N

) RI 0.25, Up milling

Heaviside solution

RI 0.25, Up milling

0

5000

10000RI 0.50, Up milling

To

tal F

orc

e(N

) RI 0.50, Up milling

0

5000

10000

To

tal F

orc

e(N

) RI 0.50, Down milling RI 0.50, Down milling

0

5000

10000

To

tal F

orc

e(N

)

RI 0.75

a = 10 mm

a = 25 mm

a = 50 mmRI 0.75

a = 10 mm

a = 25 mm

a = 50 mm

0 100 200 3000

5000

10000

To

tal F

orc

e(N

)

p (deg.)

Full immersion slotting

0 100 200 300

θp (deg.)

Full immersion slotting

θ

Figure 7-8. Analytical solutions for various conditions in helical peripheral milling.Parameters are given in Table 7-1. The results match closely with thenumerical solutions published by Tlusty (1985, 2000), with differences beingwithin 0.5%. The total force in the plane is shown Ftotal =

√F 2

x + F 2y .

118

Figure 7-9. Experimental set-up showing the diameter 12.7 mm, 3-fluted endmill with 45

helix held in a polygonal chuck with the black and white reflecting bandwrapped around it, the laser tachometer, and the workpiece mounted on thedynamometer. A screenshot of the LabViewr virtual instrument used torecord force data is also shown.

6

7

8

log

[K

tc (

N/m

m2

)]

Adj. R2

= 0.981

6

7

8

log

[K

nc (

N/m

m2

)]

Adj. R2 = 0.992

−4.5 −4 −3.5 −3 −2.5 −2

6

7

8

log [ h (mm)]−

m

log

[K

ac

(N

/mm

2)]

Linear regression fit unsuitable (R = 0.144);2

Constant mean value of fitted Kac

Figure 7-10. Linear regression fitting of cutting coefficients, as a function of the averagedmean chip thickness, for the experimental conditions given in Table 7-2 using50% radial immersion. Experimental points are displayed as ♦.

119

−150

0

400

Fx (

N) Feed force

−150

0

400

Fy (

N)

Lateral force

0 90 180 270 360−150

0

400

θp (deg)

Fz (

N)

Axial force

Predicted upper 95% C.I.Predicted lower 95% C.I.Experimental verification

Figure 7-11. Predicted vs. experimental force signals: feed 0.050 mm/tooth, 25% radialimmersion, down-milling. Other conditions as in Table 7-2.

−300

0

750

Fx (

N) Feed force

−300

0

750

Fy (

N)

Lateral force

0 90 180 270 360−300

0

750

θp (deg)

Fz (

N)

Axial force



120

−200

0

300

Fx (

N)

Feed force

−200

0

300

Fy (

N)

Lateral force

0 90 180 270 360−200

0

300

θp (deg)

Fz (

N)

Axial force



−300

0

400

Fx (

N)

Feed force

−300

0

400

Fy (

N)

Lateral force

0 90 180 270 360−300

0

400

θp (deg)

Fz (

N)

Axial force



121

−125

0

125

Fx (

N)

Feed force

−125

0

125

Fy (

N)

Lateral force

0 90 180 270 360−125

0

125

θp (deg)

Fz (

N)

Axial force



−250

0

225

Fx (

N)

Feed force

−250

0

225

Fy (

N)

Lateral force

0 90 180 270 360−250

0

225

θp (deg)

Fz (

N)

Axial force



122

−250

0

650

Fx (

N)

Feed force

−250

0

650

Fy (

N)

Lateral force

0 90 180 270 360−250

0

650

θp (deg)

Fz (

N)

Axial force


Figure 7-17. Predicted vs. experimental force signals: feed 0.100 mm/tooth, 75% radialimmersion, cut begins with h = 0. Other conditions as in Table 7-2.

−400

0

900

Fx (

N)

Feed force

−400

0

900

Fy (

N)

Lateral force

0 90 180 270 360−400

0

900

θp (deg)

Fz (

N)

Axial force


Figure 7-18. Predicted vs. experimental force signals: feed 0.200 mm/tooth, 75% radialimmersion, cut begins with h = 0. Other conditions as in Table 7-2.

123

−225

0

550

Fx (

N)

Feed force

−225

0

550

Fy (

N) Lateral force

0 90 180 270 360−225

0

550

θp (deg)

Fz (

N)

Axial force


Figure 7-19. Predicted vs. experimental force signals: feed 0.100 mm/tooth, 25% radialimmersion, up-milling. Other conditions as in Table 7-2.

−150

0

350

Fx (

N)

Feed force

−150

0

350

Fy (

N) Lateral force

0 90 180 270 360−150

0

350

θp (deg)

Fz (

N)

Axial force Predicted upper 95% C.I.Predicted lower 95% C.I.Experimental verification

Figure 7-20. Predicted vs. experimental force signals: feed 0.100 mm/tooth, 10% radialimmersion, up-milling. Other conditions as in Table 7-2.

124

0

20

40

b (

mm

)

0

50

100

L

0

50

100

θ

(deg)

T

5000

6000

7000

To

tal F

orc

e (

N)

p

0

20

40

0

50

100

020406080

0 100 200 3005000

6000

7000

p

k =100 k = 500

0 100 200 300

0 100 200 3000 100 200 300

0 100 200 3000 100 200 300

θ (deg)θ (deg)

θ

(deg)

Figure 7-21. Gibbs-Wilbraham distortion effects. k is the number of terms in the partialFourier sum for b, θL, and θT . Distortion effects on θL and θT , close to thejump discontinuities, are observed to be fairly significant for k = 100.Propagation of the effect to the cutting force is negligible for all practicalpurposes because the chip width, b, vanishes at the points of the jumpdiscontinuities of θL and θT . The total force in the plane is shownFtotal =

√F 2

x + F 2y . There is no distortion effect on b as it does not have any

jump discontinuities.

125

1000

2000

3000

Fto

tal(N

)

k =5 average Residual =13.564 N

CPU time =0.85185 s

1000

2000

3000

Fto

tal(N

) k =10 average Residual = 4.4737 N

CPU time =1.1219 s

1000

2000

3000

Fto

tal(N

) k =50 average Residual =1.2373 N

CPU time =1.3235 s

1000

2000

3000

Fto

tal(N


CPU time =1.5984 s

1000

2000

3000

Fto

tal(N


CPU time =1.9425 s

0 50 100 150 200 250 300 350

1000

2000

3000

Fto

tal(N

)

θp

(deg.)

k =500 average Residual = 0.7086 N

CPU time =2.3484 s

Figure 7-22. Convergence and computational burden of the Analytical Fourier solution(dashed lines). The index ‘k’ refers to the number of terms in the partialFourier summation. Computation time includes computation of the numericalsolution (solid lines). A PC with a 1.6 MHz processor was used. Thecomputations refer to the case of a = 10, RI = 0.75, h = 0 at the entry of thecut, with other parameters listed in Table 7-1. The total force in the plane isshown Ftotal =

√F 2

x + F 2y .

126

CHAPTER 8EFFECTS OF DIFFERENTIAL TOOTH PITCH ON THE HELICAL FORCE MODEL

In this chapter, the instantaneous rigid force model for helical endmilling is extended

to include the effects of differential tooth pitch. It was noted in chapter 1 that several

authors have reported on the use of differential pitch cutters for mitigating chatter and

reducing dimensional surface errors (e.g. Slavicek, 1965; Vanherck, 1967; Tlusty et al.,

1983; Shirase and Altintas, 1996). The extension of the helical force model of chapter 7 to

cover differential pitch effects is very straightforward, merely requiring the computation of

the effective feed for each individual tooth, and the phase angle at which each tooth enters

the cut. This information is applied in Eq. 7–15 to obtain the results. The calibration of

the model follows exactly the same procedure as in chapter 7. In the experimental set-up

used for model calibration, the dial indicator having a resolution of 2.5 µm (0.0001”) could

not resolve the runout. Hence, the effects of runout were not required to be considered.

Runout effects, if any, could be treated using the method to be described in the next

chapter, which is similar to the procedure outlined in chapter 4.

A note on the commercial availability of differential pitch cutters is in order. Face

mills with indexable inserts are commonly manufactured with unequal pitch (e.g., the

series 2J6B, 2J4B, and 2J2B of Ingersoll make, series 8000 VA19 of Stellram make,

and series 245, and 490 of Sandvik Coromant make, etc.). On the other hand, solid

endmills with differential pitch are less common, though their availability is improving

(e.g. the HPHV and HPHVT series of Kennametal make, the Plura series of Sandvik

Coromant make; a four fluted endmill of Futura Carbide SRL, Italy make, etc.). For the

experimental results reported in this chapter, a differential pitch endmill of Kennametal

make was used (details in Table 8-1). These were the same tools using which Powell

(2008) reported the results of a time domain simulation of stability boundaries using

variable pitch endmills.

127

A note on the terminology is warranted. ‘Differential pitch’, ‘unequal tooth spacing’,

and ‘variable pitch’ are three terms that are used synonymously in this document to refer

to the unequal angular separation of successive teeth. Figure 8-1 shows an example of

differential tooth pitch.

8.1 Formulation for Differential Pitch Effects on Force Components

To use the multiple tooth formulation Eq. 7–15, the effective feed per tooth, fTi,

and the phase shift, for each tooth entry, χi, are required. Let fr denote the linear feed

per revolution of the cutter. For the 4-fluted cutter used in the experiments, the tooth

pitch spacing was 97-83-97-83, resulting in the effective feeds for the four teeth being

fT1 = (97/360)fr, fT2 = (83/360)fr, fT3 = (97/360)fr, and fT4 = (83/360)fr, respectively.

The respective pitch angles follow the angular separation pattern. This information is used

in Eq. 7–15, along with the details of the cutter geometry and cutting parameters listed in

Table 8-1, to yield the predicted cutting forces after experimental model calibration.

8.2 Experimental Model Calibration

The experimental set-up was the same as that in the experiment of chapter 7, except

that the endmill and toolholder were different (Table 8-1). A 10% partial radial immersion

test was used to extract cutting coefficients.

Experiments were conducted at nine different feed rates, within the range 0.025 -

0.250 mm/tooth, the time traces of force components were recorded, and average force

values, (fx, f y, f z), were computed. At each feed rate, the experiment was repeated

5 times. Forces were averaged over one rotation on a per tooth basis. To reduce the

variability, the average was taken for 50 successive rotations. Values of Ktc, Knc, and

Kac, corresponding to each feed rate, were derived using Eq. 7–20. Thus, nine sets of five

data points each were collected for fitting a linear model according to Eq. 7–16. Two

parameters (slope and intercept) were fitted to 45 points, yielding 43 degrees of freedom

for each regression. The cutting constants Γtc,nc and Ψtc,nc of Eq. 7–16 were obtained using

Zellner’s (1962) multi response linear regression, whereas the cutting constants Γac and

128

Ψac were obtained using simple linear regression. The fitted regression lines are displayed

in Fig. 8-2.

Based on the regression, the estimated cutting constants Γtc,nc and Ψtc,nc are

calculated and given in Table 8-2. A linear fit was not found suitable for the axial

cutting constants (small values of R2 in linear regression). The mean value of Kac was

fitted. Thus, for any particular value of the averaged mean chip thickness, hm (expressed

in [mm]), the nominal values of cutting coefficients obtained experimentally are

Ktc = 518.0(hm)−0.3375 [N/mm2] (8–1)

Knc = 289.7(hm)−0.3905 [N/mm2] (8–2)

Kac = 379.5 [N/mm2] (8–3)

The variance-covariance matrix for the cutting constants, Γtc,nc and Ψtc,nc, obtained

using Zellner’s (1962) method are given in Table 8-3. The variance of the independent

cutting constant Γac was found to be 0.03940389. Thus, the variances of model input

parameters, due to random effects, are available for the Type A uncertainty evaluation.

For the Type B1 evaluation of uncertainty, for the variances of cutting coefficients

due to systematic effects, the procedure outlined in chapter 7 holds good. Since there is no

runout, the Type B2 evaluation for variances of effective feed is not required.

The propagation of uncertainties in model input parameters follows on the same

lines as in chapter 7. A sufficiently large number of experiments allows the choice

of κA1 = 2 as the expanded uncertainty coverage factor for the combined Type A

component uncertainties for a 95% confidence interval. The underlying approximate

normal assumption for the force measurement uncertainties allows the choice of coverage

factor κB1 = 2 as the expanded uncertainty coverage factor for the combined Type A

component uncertainties for the 95% confidence interval. The root sum of squares method

is used to calculate the overall expanded uncertainty.

129


Figures (8-3) - (8-6) show the results of force predictions for the differential pitch

endmill. Upper and lower bounds of 95% confidence interval of predicted forces are

displayed graphically, along with the experimentally obtained signals, which verify the

predictions.

Reliable predictions were obtained for various combinations of feed, radial immersion,

and up or down milling configurations given in Table 8-4. The deficiencies of the model

which were highlighted in the previous chapter, still hold, i.e., in some cases, the peak

forces are over predicted. The cutting coefficients, derived on the basis of down milling,

sufficed for up milling and mixed mode predictions. The differential pitch gives rise to

a phasing of the forces according to the angular position of the tooth. The resulting

variations in feed per tooth gives rise to variations in the amplitude of the forces from

tooth to tooth.

8.4 Chapter Summary

The helical force model was calibrated for a differential pitch cutter following the

same method as in chapter 7. The effects of differential tooth spacing were incorporated

in a straightforward manner using the appropriate values of the effective feed, and the

phase angle of tooth entry, for each individual tooth. The procedure for propagation of

uncertainties in model input parameters through the force model is the same as in chapter

7. The force predictions were experimentally verified. The helical force model is further

extended to include the effects of radial runout of successive teeth in the next chapter.

130

Table 8-1. Experimental conditions: dry cutting of aluminum alloy 6061-T6; Kennametalmake uncoated solid carbide endmill having differential pitch (97-83-97-83);Schunk make hydrogrip chuck


19.05 mm 30 250 m/min 4 4 mm


Γtc Ψtc Γnc Ψnc Γac Ψac

6.250 −0.3375 5.669 −0.3905 5.939 0.000


Γtc Ψtc Γnc Ψnc

Γtc 0.00039685 0.00011114 0.00091285 0.00025564Ψtc 0.00003239 0.00025564 0.00007451Γnc 0.00340083 0.00095239Ψnc 0.00027760

Table 8-4. Summary of experimental conditions used for verification of force predictionswith radial runout, holding the cutting conditions of Table 8-1 fixed

Resultsdisplayed in

Feed(mm/tooth)

Radialimmersion (%)

Up/down milling or mixedmode (> 50% RI)

Figure 8-3 0.200 5 Down millingFigure 8-4 0.100 20 Down millingFigure 8-5 0.100 25 Up millingFigure 8-6 0.050 50 Up milling

131

97

97 83

83oo

oo

Figure 8-1. Example of differential tooth pitch, also called unequal tooth spacing, orvariable pitch. The specific tooth spacing shown (97-83-97-83) belongs tothe cutter used in the experiments whose results are reported in this chapter.

5

6

7

8

9

ln [

Ktc

(N

/mm

2)]

Adj. R2 = 0.987

5

6

7

8

9

ln [

Kn

c (

N/m

m2

)]

Adj. R2

= 0.927

5 −4.5 −4 −3.5 −3 −2.55

6

7

8

9

ln [h−

m]

ln [

Ka

c (

N/m

m2

)]

Linear regression fit unsuitable (R =0.047);2

Constant mean value of K fittedac

Figure 8-2. Linear regression fitting of cutting coefficients, as a function of the averagedmean chip thickness, for dry milling of 6061-T6 aluminum alloy, using adifferential pitch, 4-fluted, 30 helix, solid carbide endmill having zero nominalrunout. Experimental points are denoted by “M”.

132

−350

0

450

Fx (

N) Feed force

−350

0

450F

y (

N)

Lateral force

0 180 360 540 720−350

0

450

θp (deg)

Fz (

N)

Axial force

Predicted upper 95% C.I.

Predicted lower 95% C.I.

Experimental verification

Figure 8-3. Differential pitch effects. Predicted vs. experimental force signals: 4-fluted, 30

helix cutter with 97-83-97-83 tooth spacing resulting in amplitude andphase variations of forces from tooth to tooth; feed 0.200 mm/tooth, 5% radialimmersion, down-milling. Other conditions as in Table 8-1.

−250

0

600

Fx (

N) Feed force

−250

0

600

Fy (

N)

Lateral force

0 180 360 540 720−250

0

600

θp (deg)

Fz (

N)

Axial force




Figure 8-4. Differential pitch effects. Predicted vs. experimental force signals: 4-fluted, 30

helix cutter with 97-83-97-83 tooth spacing resulting in amplitude andphase variations of forces from tooth to tooth; feed 0.100 mm/tooth, 20%radial immersion, down-milling. Other conditions as in Table 8-1.

133

−200

0

650

Fx (

N)

Feed force

−200

0

650F

y (

N)

Lateral force

0 180 360 540 720−200

0

650

θp (deg)

Fz (

N)

Axial force




Figure 8-5. Differential pitch effects: Predicted vs. experimental force signals: 4-fluted, 30

helix cutter with 97-83-97-83 tooth spacing resulting in amplitude andphase variations of forces from tooth to tooth; feed 0.100 mm/tooth, 25%radial immersion, up-milling. Other conditions as in Table 8-1.

−200

0

400

Fx (

N)

Feed force

−200

0

400

Fy (

N)

Lateral force

0 180 360 540 720−200

0

400

θp (deg)

Fz (

N)

Axial force




Figure 8-6. Differential pitch effects: Predicted vs. experimental force signals: 4-fluted, 30

helix cutter with 97-83-97-83 tooth spacing resulting in amplitude andphase variations of forces from tooth to tooth; feed 0.050 mm/tooth, 50%radial immersion, up-milling. Other conditions as in Table 8-1.

134

CHAPTER 9EFFECTS OF RADIAL RUNOUT ON THE HELICAL FORCE MODEL

In this chapter, the instantaneous rigid force model is extended to include the effects

of radial runout of successive teeth. The accuracy of toolholding and spindle mounting

in modern machine tools has improved tremendously in recent years, especially with

the advent of advanced toolholding methods such as polygonal chucks and hydrogrip

chucks, coupled with precise spindle adaption, such as the HSK interface which provides

simultaneous taper and face butting for the toolholder when clamped in the spindle. The

precision ground solid endmills are manufactured to extremely close tolerances. As a

result, it is possible to hold the tool point runout to a very low value upon assembly of

the endmill in the holder and mounting on the machine spindle. On good setups, it is

easy to keep the radial runout down to just a few microns. Under practical conditions

on the shop floor, it is difficult to reduce the runout to zero. Such an effort would not be

economically justifiable for most situations. Therefore, it is necessary to include the effects

of tool runout for the force model to have practical value.

In chapter 1 it was mentioned that there are many force models available in the

literature which have successfully predicted forces with runout taken into consideration.

Here, the purpose is to provide a closed form analytical force model, which has a single

equation covering the entire cutter rotation, which will predict forces under conditions

of non-zero runout. A straightforward extension of the model proposed in chapter 7

achieves this objective by suitably calculating the effective feed for each individual tooth

to be applied to Eq. 7–15. An additional model input parameter is introduced, i.e., the

effective feed of each tooth, fTi, which is assessed by an actual measurement of the runout

upon tool assembly and mounting on the spindle. This input is affected by uncertainty of

the measured value of runout which is achieved using a measuring device such as a dial

indicator. This introduces the Type B2 evaluation of uncertainty in this input parameter.

The treatment is analogous to the case of the straight fluted model of chapter 4.

135

9.1 Formulation for the Effects of Runout on Force Components

The formulation of section 4.4 of chapter 4 applies. For simplicity, a two fluted

cutter is considered. All the four Eqs. 4–9 - 4–12 apply, with the terms having the same

meanings. The Eqs. 4–11 and 4–12 yield the effective feeds of the two teeth given the

measured value of radial runout, ρ, between these two teeth. For a cutter with more than

two teeth, the expressions for effective feed can be derived using similar arguments. These

effective feeds are used in Eq. 7–15 to yield the force components for a cutter with radial

runout.

9.2 Verification With a Numerical Solution

The results of the analytical solution are compared with the numerical solution of

Tlusty (1985, 2000) for a 2-fluted endmill with 10 µm runout. For a particular set of

parameters, the comparison is shown in Fig. 9-1. The difference between the two solutions

is less than 0.5%.

9.3 Experimental Model Calibration

The experimental set-up was the same as that in the experiment of chapter 7, except

that the endmill and toolholder were different (Table 9-1). Upon clamping the tool in the

spindle, the static radial runout was measured by indicating the endmills with a dial gage

having a least count of 2.5 µm (0.0001”). The nominal value of the measured runout was

10 µm. A 25% partial radial immersion test was used to extract cutting coefficients.

Experiments were conducted at nine different feed rates, within the range 0.025 -

0.250 mm/tooth, the time traces of force components were recorded, and average force

values, (fx, f y, f z), were computed. At each feed rate, the experiment was repeated

5 times. Forces were averaged over one rotation on a per tooth basis. To reduce the

variability, the average was taken for 50 successive rotations. Values of Ktc, Knc, and

Kac, corresponding to each feed rate, were derived using Eq. 7–20. Thus, nine sets of five

data points each were collected for fitting a linear model according to Eq. 7–16. Two

parameters (slope and intercept) were fitted to 45 points, yielding 43 degrees of freedom

136

for each regression. The cutting constants Γtc,nc and Ψtc,nc of Eq. 7–16 were obtained using

Zellner’s (1962) multi response linear regression, whereas the cutting constants Γac and

Ψac were obtained using simple linear regression. The fitted regression lines are displayed

in Fig. 9-2.

Based on the regression, the estimated cutting constants Γtc,nc and Ψtc,nc are

calculated and given in Table 9-2. A linear fit was not found suitable for the axial

cutting constants (small values of R2 in linear regression). The mean value of Kac was

fitted. Thus, for any particular value of the averaged mean chip thickness, hm (expressed

in [mm]), the nominal values of cutting coefficients obtained experimentally are

Ktc = 375.4(hm)−0.4634 [N/mm2] (9–1)

Knc = 121.3(hm)−0.6666 [N/mm2] (9–2)

Kac = 428.5 [N/mm2] (9–3)

The variance-covariance matrix for the cutting constants, Γtc,nc and Ψtc,nc, obtained

using Zellner’s (1962) method are given in Table 9-3. The variance of the independent

cutting constant Γac was found to be 0.00678924. Thus, the variances of model input

parameters, due to random effects, are available for the Type A uncertainty evaluation.

For the Type B1 evaluation of uncertainty, for the variances of cutting coefficients

due to systematic effects, the procedure outlined in chapter 7 holds good. In addition,

the Type B2 evaluation has to be performed to extract the variances of the effective feed

due to uncertainty in the measured value of the radial runout. This analysis has been

performed in chapters 5 and 6 based on which the variances in the effective feeds of the

two teeth are given by Eqs. 6–7 and 6–8, and are noted below for ready reference

u2(fT1) = u2(fT2) =

(∂fT1,2

∂ρ

)2

u2(ρ) =

(0.0025√

3

)2

The propagation of Type A and Type B1 uncertainties through the force model

follow the same procedure as in chapter 7. The additional task here is the propagation

137

of the Type B2 uncertainties. The procedure is the same as that developed in chapter 6.

The sensitivities of the cutting forces to the effective feed rates are determined. These

sensitivities are used to propagate the uncertainties in the effective feed rates to yield the

combined component uncertainties (Type B2) of the predicted cutting force components.

The sensitivities of cutting force components with respect to the effective feeds are

found using Eq. 7–11

sxf

syf

szf

=

∂Fx/∂fT

∂Fy/∂fT

∂Fz/∂fT

=

bξ1 bξ2 0

bξ2 −bξ1 0

0 0 Dξ3

(1 + Ψtc)Ktc

(1 + Ψnc)Knc

Kac

(9–4)

The uncertainties in the effective feed rates of the ith tooth, as expressed in Eq. 6–7

and 6–8, may be propagated to the predicted forces using Eq. 7–11 and the sensitivities

expressed in Eq. 9–4 to yield the combined uncertainties ucB2(fx,y,z)

u2cB2

(fx)

u2cB2

(fy)

u2cB2

(fz)

= u2(fT )

[sxf ]2

[syf ]2

[szf ]2

(9–5)

where the subscript on fTiis dropped for convenience.


Figures (9-3) - (9-15) show the results of force predictions for the 2-fluted, 45 helix,

cutter having a nominal radial runout of 10 µm. The forces are predicted reasonably well

for various combinations of feed, radial immersion, and up or down milling configurations

given in Table 9-4. However, the deficiency in the model shows up in several cases where

the peak forces are over predicted, and the experimental force patterns are skewed with

respect to the predicted patterns. The expected effect of the runout to feed ratio is clearly

observed. The larger this ratio, the larger is the variation in forces from tooth to tooth.

138

9.5 Chapter Summary

The effects of radial runout were incorporated into the closed form analytical force

model by calculating the effective feed for each tooth using the same procedure as was

adopted for the straight fluted force model. The general method of computing the

variances of model input parameters, and the propagation of uncertainties through the

model allow the placing of confidence intervals on force predictions in the same way as

described in earlier chapters. Experimental verification was provided for predicted forces.

For certain cases, especially with high radial immersions, the forces were over predicted by

the model, and the skewness in the experimental force signals was not well predicted. This

shortcoming of the force model was also observed in the results of the last two chapters.

Even so, the model developed in this chapter may be adequate for many applications

where an approximate estimate of the forces and the force patterns may suffice. In the

next chapter, the primary reason for this weakness of the model is discussed, and the

model is modified to rectify it of this defect.

139

Table 9-1. Experimental cutting conditions: dry cutting of aluminum alloy 6061-T6; TiB2

coated solid carbide endmill (Kennametal catalog No. HPF45A750S2150)having two equispaced teeth; Polygonal chuck (Schunk catalog No. 203794)


19.05 mm 45 250 m/min 2 4 mm


Γtc Ψtc Γnc Ψnc Γac Ψac

5.928 −0.4634 4.799 −0.6666 6.060 0.000


Γtc Ψtc Γnc Ψnc

Γtc 0.00037172 0.00011714 0.00017171 0.00005411Ψtc 0.00003885 0.00005411 0.00001795Γnc 0.00184500 0.00058139Ψnc 0.00019285

Table 9-4. Summary of experimental conditions used for verification of force predictionswith radial runout, holding the cutting conditions of Table 9-1 fixed

Resultsdisplayed in

Feed(mm/tooth)

Radialimmersion (%)


Figure 9-3 0.100 5 Down millingFigure 9-4 0.100 5 Up millingFigure 9-5 0.100 10 Down millingFigure 9-6 0.200 10 Down millingFigure 9-7 0.050 20 Up millingFigure 9-8 0.050 50 Down millingFigure 9-9 0.100 50 Down millingFigure 9-10 0.025 50 Up millingFigure 9-11 0.050 75 mixed mode, cut ends with h = 0Figure 9-12 0.050 75 mixed mode, cut starts with h = 0Figure 9-13 0.050 100 SlottingFigure 9-14 0.100 100 SlottingFigure 9-15 0.200 100 Slotting

140

−500

0

500

1000

Fx

,y [

N]

Fx numericalFy numerical

0 90 180 270 360−500

0

500

1000

θp [deg]

Fx

,y [

N]

Fx analyticalFy analytical

Figure 9-1. Comparison of the closed form analytical solution with a numerical solutionpublished by Tlusty (1985, 2000) for a 2-fluted endmill with 10 µm runout,0.50% RI, down-milling, a = 4 mm, fT = 0.100 mm/tooth, D = 19.05 mm,λ = 45, Ktc = 2000 N/m2, Knc = 0.3Ktc. The total force in the plane isdisplayed Ftotal =

√F 2

x + F 2y .

141

5

6

7

8

9

ln [

Ktc

Adj. R2

= 0.992

5

6

7

8

9

ln [

Kn

c

Adj. R2

= 0.981

−4.5 −4 −3.5 −3 −2.5 −25

6

7

8

9

ln [ h ]−

m

ln [

Ka

c (N

/mm

)]

2

Linear regression fit unsuitable (R = 0.223);2

Constant mean value of K fittedac

(N

/mm

)]

(N

/mm

)]

22

Figure 9-2. Linear regression fitting of cutting coefficients, as a function of the averagedmean chip thickness, for dry milling of 6061-T6 aluminum alloy, using anequispaced tooth, 2-fluted, 45 helix, TiB2 coated, solid carbide endmill havinga nominal radial runout of 10 µm. Experimental points are denoted by “#”.

−200

0

300

Fx (

N)

Feed force

−200

0

300

Fy (

N)

Lateral force

0 180 360 540 720−200

0

300

θp (deg)

Fz (

N)

Axial force




Figure 9-3. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.100 mm/tooth, 5% radial immersion, downmilling. Nominal runout 10 µm. Other conditions as in Table 9-1. Moderaterunout to feed ratio (0.10) results in moderate variation in forces from tooth totooth.

142

−150

0

300

Fx (

N)

Feed force

−150

0

300

Fy (

N)

Lateral force

0 180 360 540 720−150

0

300

θp (deg)

Fz (

N)

Axial force




Figure 9-4. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.100 mm/tooth, 5% radial immersion, upmilling. Nominal runout 10 µm. Other conditions as in Table 9-1. Moderaterunout to feed ratio (0.10) results in moderate variation in forces from tooth totooth.

−250

0

400

Fx (

N)

Feed force

−250

0

400

Fy (

N)

Lateral force

0 180 360 540 720−250

0

400

θp (deg)

Fz (

N)

Axial force



143

−350

0

500

Fx (

N)

Feed force

−350

0

500F

y (

N)

Lateral force

0 180 360 540 720−350

0

500

θp (deg)

Fz (

N)

Axial force




Figure 9-6. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.200 mm/tooth, 10% radial immersion, downmilling. Nominal runout 10 µm. Other conditions as in Table 9-1. Smallrunout to feed ratio (0.05) results in small variation in forces from tooth totooth.

−1500

600

Fx (

N)

Feed force

−1500

600

Fy (

N)

Lateral force

0 180 360 540 720−150

0

600

θp (deg)

Fz (

N)

Axial force


Figure 9-7. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helix cutter;feed 0.050 mm/tooth, 20% radial immersion, up milling. Nominal runout10 µm. Other conditions as in Table 9-1. Large runout to feed ratio (0.20)results in large variation in forces from tooth to tooth.

144

−200

0

600

Fx (

N)

Feed force

−200

0

600F

y (N

)

Lateral force

0 180 360 540 720−200

0

600

θp (deg)

Fz (

N)

Axial force


Figure 9-8. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.050 mm/tooth, 50% radial immersion, downmilling. Nominal runout 10 µm. Other conditions as in Table 9-1. Largerunout to feed ratio (0.20) results in large variation in forces from tooth totooth.

−250

0

700

Fx (

N)

Feed force

−250

0

700

Fy (

N)

Lateral force

0 180 360 540 720−250

0

700

θp (deg)

Fz (

N)

Axial force





145

−1500

600

Fx (

N)

Feed force

−1500

600F

y (N

)Lateral force

0 180 360 540 720−150

0

600

θp (deg)

Fz (

N)

Axial force


Figure 9-10. Runout effects: Predicted vs. experimental force signals: equispaced, 2-fluted,45 helix cutter; feed 0.025 mm/tooth, 50% radial immersion, up milling.Nominal runout 10 µm. Other conditions as in Table 9-1. Large runout tofeed ratio (0.40) results in large variation in forces from tooth to tooth.

−1500

600

Fx (

N)

Feed force

−1500

600

Fy (

N)

Lateral force

0 180 360 540 720−150

0

600

θp (deg)

Fz (

N)

Axial force


Figure 9-11. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.050 mm/tooth, 75% radial immersion, cutends with h = 0. Nominal runout 10 µm. Other conditions as in Table 9-1.Large runout to feed ratio (0.20) results in large variation in forces fromtooth to tooth.

146

−150

0

500

Fx (

N)

Feed force

−150

0

500

Fy (

N)

Lateral force

0 180 360 540 720−150

0

500

θp (deg)

Fz (

N)

Axial force




Figure 9-12. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.050 mm/tooth, 75% radial immersion, cutstarts with h = 0. Nominal runout 10 µm. Other conditions as in Table 9-1.Large runout to feed ratio (0.20) results in large variation in forces fromtooth to tooth.

−150

0

500

Fx (

N)

Feed force

−150

0

500

Fy (

N)

Lateral force

0 180 360 540 720−150

0

500

θp (deg)

Fz (

N)

Axial force




Figure 9-13. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.050 mm/tooth, 100% radial immersionslotting. Nominal runout 10 µm. Other conditions as in Table 9-1. Largerunout to feed ratio (0.20) results in large variation in forces from tooth totooth.

147

−250

0

700

Fx (

N)

Feed force

−250

0

700

Fy (

N)

Lateral force

0 180 360 540 720−250

0

700

θp (deg)

Fz (

N)

Axial force




Figure 9-14. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.100 mm/tooth, 100% radial immersionslotting. Nominal runout 10 µm. Other conditions as in Table 9-1. Moderaterunout to feed ratio (0.10) results in moderate variation in forces from toothto tooth.

−500

0

1000

Fx (

N)

Feed force

−500

0

1000

Fy (

N)

Lateral force

0 180 360 540 720−500

0

1000

θp (deg)

Fz (

N)

Axial force




Figure 9-15. Radial runout effect. Predicted vs. experimental force signals: equispaced,2-fluted, 45 helix cutter; feed 0.200 mm/tooth, 100% radial immersionslotting. Nominal runout 10 µm. Other conditions as in Table 9-1. Smallrunout to feed ratio (0.05) results in small variation in forces from tooth totooth.

148

CHAPTER 10FORCE MODELING WITH INSTANTANEOUS CUTTING COEFFICIENTS

So far, all force predictions have been made using fixed values of the cutting

coefficients, Ktc,nc,ac, which were calculated based on an averaged mean chip thickness,

hm. The force predictions, thus obtained, have certain deficiencies, especially in the cases

of high radial immersion, where peak forces are overpredicted, and the skewed patterns

of the experimental forces are not well predicted by the theory. These defects in the force

model of chapter 7, which was based on average cutting coefficients, are addressed in

this chapter. Instantaneous cutting coefficients are used to refine the force model to yield

accurate force predictions.

10.1 Instantaneous Cutting Coefficients

It has been noted earlier that the cutting coefficients are not true constants but

depend on process parameters. Only the cutting constants are truly invariant. As the

tooth progresses through the cut, the instantaneous value of the mean chip thickness,

hm, evolves. So, it is expected that the cutting coefficients also change because they are

dependent on the instantaneous hm. The functional relation governing this dependence has

to be deduced.

Since the cutting constants are invariants, it is plausible that the relationship between

the instantaneous cutting coefficients and the instantaneous hm has the form K = eΓhΨm,

for fixed values of cutting speed, axial depth of cut, and other fixed cutting conditions

such as the type of cutting fluid being used. The values of the cutting constants,

Γtc,nc,ac and Ψtc,nc,ac, may be found using the average forces as outlined in chapter 7.

The underlying assumption is that the values of the cutting constants identified in this

manner can be used to calculate the instantaneous cutting coefficients. The validity of

this assumption will be tested by comparison of experimental force signals with predicted

forces based on the instantaneous cutting coefficients. The effect of considering the

dependence of Ktc,nc,ac on hm is studied in this chapter. The effects of radial runout are

149

also considered so that uncertainties of Type A, B1, and B2 are involved. The helical force

model is suitably modified.

The instantaneous cutting coefficients are functions of θp, and may be expressed

through the intermediate variables θL and θT using Eq. 3–17

Ktc = exp(Γtc)(fT ξh)Ψtc = exp(Γtc)

[fT sin

(θL + θT

2

)sinc

(θT − θL

2

)]Ψtc

(10–1)

Knc = exp(Γnc)(fT ξh)Ψnc = exp(Γnc)

[fT sin

(θL + θT

2

)sinc

(θT − θL

2

)]Ψnc

(10–2)

Kac = exp(Γac)(fT ξh)Ψac = exp(Γac)

[fT sin

(θL + θT

2

)sinc

(θT − θL

2

)]Ψac

(10–3)

where the sampling function (sinc) has been defined in Eq. 3–18.

The relation between instantaneous cutting coefficients and instantaneous cutting

force components may be obtained by inverting the cutting force Eq. 7–11 to yield

Ktc

Knc

Kac

=1

fT

ξ1/[b(ξ21 + ξ2

2)] ξ2/[b(ξ21 + ξ2

2)] 0

ξ2/[b(ξ21 + ξ2

2)] −ξ1/[b(ξ21 + ξ2

2)] 0

0 0 1/(Dξ3)

Fx

Fy

Fz

(10–4)

The above equation raises an interesting possibility. If the instantaneous forces

could be measured using a single experimental cut, the instantaneous values of the

cutting coefficients could be obtained. These values can be directly related to the

instantaneous hm to obtain the relation between the cutting coefficients and hm.

There are certain drawbacks of this scheme. First, the instantaneous forces cannot be

measured as accurately as average forces because the experimental force signals have

high frequency oscillations (wiggles) of small magnitude superimposed on them due to

measuring instrument dynamics. This effect is discussed in chapter 11. The wiggles do not

affect average force values appreciably. Second, the inverted matrix in Eq. 10–4 becomes

numerically ill conditioned for small values of hm.

150


The variances of cutting constants, (obtained by the Type A evaluation) and the

effective feeds (obtained by the Type B2 evaluation) remain the same as in chapters

7 and 9 respectively. However, the variances of cutting coefficients (obtained using a

Type B1 evaluation) are no longer constant numbers, but are functions of θp because

the instantaneous forces, and the sensitivities of cutting coefficients to the instantaneous

forces, are functions of θp.

Equation 10–4 yields the following sensitivity coefficients for use in propagation of

the uncertainty in instantaneous force measurements to the uncertainties in instantaneous

cutting coefficients

stx

sty

snx

sny

saz

=

∂Ktc/∂Fx

∂Ktc/∂Fy

∂Knc/∂Fx

∂Knc/∂Fy

∂Kac/∂Fz

=1

fT

ξ1/[b(ξ21 + ξ2

2)]

ξ2/[b(ξ21 + ξ2

2)]

ξ2/[b(ξ21 + ξ2

2)]

−ξ1/[b(ξ21 + ξ2

2)]

1/(Dξ3)

(10–5)

Using the above sensitivities the variances of cutting coefficients, due to the

systematic effects, can be obtained

u2(ktc)

u2(knc)

u2(kac)

=

s2tx s2

ty 0

s2nx s2

ny 0

0 0 s2az

u2(fx)

u2(fy)

u2(fz)

(10–6)

which is analogous to Eq. 7–29 where the variances of average force components are

replaced by variances of instantaneous force components, u2(fx,y,z). The values of u(fx,y,z)

in the Eq. 10–6 above are set at 1.207% of the nominal values of the instantaneous force

components based on the instrument manufacturer’s estimates, as explained in detail

in chapter 5. For the sake of simplicity, any possible correlation between Ktc and Knc

151

from measurement channel cross talk is neglected, i.e., u(ktc, knc) is set to zero. Thus the

variances in cutting coefficients have been obtained using a Type B1 evaluation.

10.3 Propagation of Input Parameter Uncertainties Through the Force Model

The expressions for the sensitivity coefficients to be used in the propagation of input

parameter uncertainties turn out to be the same as those for the case of the average

cutting coefficients formulation Eqs.

The sensitivity coefficients having been obtained, the uncertainties in model input

parameters are propagated using Eqs. 7–31, 7–33, and 9–5 to obtain the combined

component uncertainties ucA(fx,y,z), ucB1

(fx,y,z), and ucB2(fx,y,z), respectively, where

the values of u2(ktc,nc,ac) obtained from Eq. 10–6 are to be used in Eq. 7–33, and

instantaneous cutting coefficients are to be used in all relevant expressions instead of

fixed values of Ktc,nc,ac.


Force predictions were made for several combinations of parameters as given in Table

10-1. The results are displayed in Figs. (10-1) - (10-13). Compared with the predictions

made under the same conditions in chapter 9, where the cutting coefficients had fixed

values based on the averaged mean chip thickness, the match with experimental results is

observed to be much closer using the instantaneous cutting coefficient model.

The deficiencies noted in the earlier model of chapters 7, 8, and 9 are rectified by

the use of instantaneous cutting coefficients. Noteworthy is the fact that magnitudes

of the forces, as well as the patterns are predicted accurately. The skewness of the

experimental forces in high immersion experiments is also successfully captured. A glaring

demonstration of this is provided by a comparison of the figures in this chapter with the

corresponding figures in the earlier chapter 9. Furthermore, in general, the experimental

force patterns lie within the 95% confidence interval bounds when the instantaneous

cutting coefficients are used. All these pieces of evidence strengthen the claim of the

correctness of the closed form analytical cutting force model.

152


The closed form analytical solution for cutting forces in helical peripheral milling

was augmented with the introduction of the instantaneous cutting coefficients. This

follows naturally from the realization that the mean chip thickness, hm, evolves as the

cutter rotates. Since the cutting coefficients depend on hm, it is important to capture

this variation to provide a closer representation of the physics of the problem than is

possible by using an average value of cutting coefficients based on the averaged mean chip

thickness, hm. At those points in the angular orientation of the cutter where hm > hm,

the actual values of Ktc,nc,ac drop down. Hence, the peak forces in the cycle are lower than

those predicted using the averaged values of Ktc,nc,ac, and vice versa. The variation of hm

also explains the skewness in the cutting force patterns.

The incorporation of the instantaneous Ktc,nc,ac in the force model was seen to be

straightforward due to the structure of the model, and so was the uncertainty analysis.

The choice of the model depends on the user. If very high accuracy of magnitude and

form are required of the force predictions, the instantaneous coefficient formulation may be

used. It was also seen that for high radial immersions, the instantaneous coefficient model

may be preferable. For most purposes, the averaged coefficient model of chapter 7 would

suffice.

153

Table 10-1. Summary of experimental conditions used to verify force predictions based oninstantaneous Ktc,nc,ac, with the conditions of Table 9-1 held fixed

Resultsdisplayed in

Feed(mm/tooth)

Radialimmersion (%)


Figure 10-1 0.100 5 Down millingFigure 10-2 0.100 5 Up millingFigure 10-3 0.100 10 Down millingFigure 10-4 0.200 10 Down millingFigure 10-5 0.050 20 Up millingFigure 10-6 0.050 50 Down millingFigure 10-7 0.100 50 Down millingFigure 10-8 0.025 50 Up millingFigure 10-9 0.050 75 mixed mode, cut ends with h = 0Figure 10-10 0.050 75 mixed mode, cut starts with h = 0Figure 10-11 0.050 100 SlottingFigure 10-12 0.100 100 SlottingFigure 10-13 0.200 100 Slotting

154

−200

0

300

Fx (

N)

Feed force

−200

0

300

Fy (

N)

Lateral force

0 180 360 540 720−200

0

300

θp (deg)

Fz (

N)

Axial force




Figure 10-1. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.100 mm/tooth, 5% radial immersion, down milling. Nominalrunout 10 µm. Other conditions as in Table 9-1. Moderate runout to feedratio (0.10) results in moderate variation in forces from tooth to tooth.

−150

0

300

Fx (

N)

Feed force

−150

0

300

Fy (

N)

Lateral force

0 180 360 540 720−150

0

300

θp (deg)

Fz (

N)

Axial force


Figure 10-2. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.100 mm/tooth, 5% radial immersion, up milling. Nominalrunout 10 µm. Other conditions as in Table 9-1. Moderate runout to feedratio (0.10) results in moderate variation in forces from tooth to tooth.

155

−250

0

400

Fx (

N)

Feed force

−250

0

400F

y (

N)

Lateral force

0 180 360 540 720−250

0

400

θp (deg)

Fz (

N)

Axial force





−350

0

500

Fx (

N)

Feed force

−350

0

500

Fy (

N)

Lateral force

0 180 360 540 720−350

0

500

θp (deg)

Fz (

N)

Axial force




Figure 10-4. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.200 mm/tooth, 10% radial immersion, down milling. Nominalrunout 10 µm. Other conditions as in Table 9-1. Small runout to feed ratio(0.05) results in small variation in forces from tooth to tooth.

156

−150

0

450

Fx (

N)

Feed force

−150

0

450F

y (

N)

Lateral force

0 180 360 540 720−150

0

450

θp (deg)

Fz (

N)

Axial force




Figure 10-5. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.050 mm/tooth, 20% radial immersion, up milling. Nominalrunout 10 µm. Other conditions as in Table 9-1. Large runout to feed ratio(0.20) results in large variation in forces from tooth to tooth.

−200

0

600

Fx (

N)

Feed force

−200

0

600

Fy (

N)

Lateral force

0 180 360 540 720−200

0

600

θp (deg)

Fz (

N)

Axial force




Figure 10-6. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.050 mm/tooth, 50% radial immersion, down milling. Nominalrunout 10 µm. Other conditions as in Table 9-1. Large runout to feed ratio(0.20) results in large variation in forces from tooth to tooth.

157

−250

0

700

Fx (

N)

Feed force

−250

0

700F

y (

N)

Lateral force

0 180 360 540 720−250

0

700

θp (deg)

Fz (

N)

Axial force





−150

0

400

Fx (

N)

Feed force

−150

0

400

Fy (

N)

Lateral force

0 180 360 540 720−150

0

400

θp (deg)

Fz (

N)

Axial force




Figure 10-8. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.025 mm/tooth, 50% radial immersion, up milling. Nominalrunout 10 µm. Other conditions as in Table 9-1. Large runout to feed ratio(0.40) results in large variation in forces from tooth to tooth.

158

−150

0

600

Fx (

N)

Feed force

−150

0

600F

y (

N)

Lateral force

0 180 360 540 720−150

0

600

θp (deg)

Fz (

N)

Axial force Predicted upper 95% C.I.



Figure 10-9. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.050 mm/tooth, 75% radial immersion, cut ends with h = 0.Nominal runout 10 µm. Other conditions as in Table 9-1. Large runout tofeed ratio (0.20) results in large variation in forces from tooth to tooth.

−150

0

500

Fx (

N)

Feed force

−150

0

500

Fy (

N)

Lateral force

0 180 360 540 720−150

0

500

θp (deg)

Fz (

N)

Axial force




Figure 10-10. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.050 mm/tooth, 75% radial immersion, cut starts with h = 0.Nominal runout 10 µm. Other conditions as in Table 9-1. Large runout tofeed ratio (0.20) results in large variation in forces from tooth to tooth.

159

−150

0

600

Fx (

N)

Feed force

−150

0

600F

y (

N)

Lateral force

0 180 360 540 720−150

0

600

θp (deg)

Fz (

N)

Axial force




Figure 10-11. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.050 mm/tooth, 100% radial immersion slotting. Nominalrunout 10 µm. Other conditions as in Table 9-1. Large runout to feed ratio(0.20) results in large variation in forces from tooth to tooth.

−250

0

700

Fx (

N)

Feed force

−250

0

700

Fy (

N)

Lateral force

0 180 360 540 720−250

0

700

θp (deg)

Fz (

N)

Axial force




Figure 10-12. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.100 mm/tooth, 100% radial immersion slotting. Nominalrunout 10 µm. Other conditions as in Table 9-1. Moderate runout to feedratio (0.10) results in moderate variation in forces from tooth to tooth.

160

−450

0

1000

Fx (

N)

Feed force

−450

0

1000

Fy (

N)

Lateral force

0 180 360 540 720−450

0

1000

θp (deg)

Fz (

N)

Axial force




Figure 10-13. Predicted vs. experimental force signals: equispaced, 2-fluted, 45 helixcutter; feed 0.200 mm/tooth, 100% radial immersion slotting. Nominalrunout 10 µm. Other conditions as in Table 9-1. Small runout to feed ratio(0.05) results in small variation in forces from tooth to tooth.

161

CHAPTER 11DYNAMIC INFLUENCES IN FORCE MEASUREMENTS

Measurement of cutting forces using the measurement chain formed by the dynamometer,

cable, amplifier, and data acquisition card, was one of the most important aspects of this

research. The purpose of this chapter is to add certain qualitative comments on force

measurements using this measuring chain. This allows the reader to hold the experimental

force signals in proper perspective.

The ideal force measuring instrument (completely in-phase, unit magnitude response,

with infinite bandwidth) is not available. The dynamics of the force measuring chain

makes it hard to obtain clean force signals. The complete characterization of this

process demands a long and careful procedure (e.g. Castro et al. (2006); Tounsi and Otho

(2000)). In this chapter, the frequency response function (FRF) of the force measurement

chain is considered to examine the influencing dynamics. Experimental FRFs of the

force measurement chain are examined to make qualitative remarks on the fidelity of

experimental force measurements.

11.1 Description of the Force Measurement Chain

The force measuring chain consisted of a multicomponent (3-component) dynamometer

model 9257B, a cable model 1687B5, a charge amplifier model 5019, all of Kistler make,

and a 12-bit data acquisition card model 6062E of National Instruments make. This

dynamometer is based on quartz piezoelectric force sensors and is capable of measuring

forces in the range 5 − 5000 N with a work volume of 170 × 100 × 25 mm. The amplifier

has a built in low pass filter (1/5 to 1/3 of measured resonant frequency).

11.2 Frequency Response of the Force Measurement Chain

The impact testing method was used to obtain the FRFs shown in Fig. 11-1. An

instrumented piezoelectric impact hammer model 086C05, of PCB Piezotronics make, was

used to excite the dynamometer with an impact signal in each of the x, y, and z directions,

and the output force, obtained through the force measurement chain, was recorded.

162

Direct frequency response functions (real and imaginary parts) were obtained using

commercial software, brand name TXF, from Manufacturing Research Laboratories Inc.,

using standard procedures. These FRFs are shown in Fig. 11-1 in the form of real and

imaginary components. This data was converted to magnitude and phase components and

is shown in Fig. 11-2. When the part to be machined is mounted on the dynamometer,

the mass loading effect changes the frequency response function. Two artifacts were used

which were representative of the parts machined while calibrating and verifying the force

models. A lightweight aluminum artifact was clamped on the dynamometer to mimic the

parts used to calibrate the helical force model, and a heavy steel part was mounted which

had been used to calibrate the straight fluted force model. A visual inspection shows that

the FRFs are affected (Figs. 11-1 and 11-2).

To study the influence of the force measurement chain on the input forces, it is

convenient to consider the magnitude and phase of the FRFs in Fig. 11-2. It can be

seen that the magnitude the of FRFs is close to unity for frequencies below 1000 Hz,

but the FRF is not exactly flat in this range. Similarly, the phase is close to zero in this

range of frequencies, but the FRF is not flat. The frequency content of the input forces

in this range will be amplified according to the magnitude response, and will suffer a lag

according to the phase response.

To get a qualitative sense of these influences, the frequency content of a hypothetical

force signals with tooth passing frequency of 60 Hz is examined in Fig. 11-3. At the

integral multiples of the tooth passing frequency, the input signal has frequency content.

The effect of the non unit FRF magnitude at these frequencies will amplify the low power

content and distort the output signal. This also explains the appearance of high frequency

wiggles in the experimental force signals.

The effect of the non zero phase at the tooth passing frequencies and their harmonics

is to introduce a lag in the response of these frequencies. This will create an additional

distortion in the signal.

163

Information available in the experimental FRFs could be used to correct the output

signals. For instance, the specific magnitudes of the FRF at the tooth passing frequencies

are known, so that their amplification factors are known. The effect of phase lag has to

be correctly accounted for. This correction, though mathematically feasible, is rendered

non-trivial by the fact that the FRF is itself a measured quantity.

Measurements introduce uncertainties. The uncertainty in the experimental FRF

creates a further difficulty because it is a bivariate problem. The FRF has a real part

and an imaginary part. So, the method of quantifying its uncertainty is specialized.

This quantification of uncertainties in experimental FRFs has been studied by Kim

and Schmitz (2007). These three factors, the magnitude response, phase response, and

uncertainties in measured values of the experimental FRFs are further complicated by the

realization that the FRF itself is not a constant. As the milling cutter removes material,

the part loses mass, and this changes the FRF constantly.


The influence of the dynamics of the force measurement chain was qualitatively

studied using experimental FRFs of the measurement chain. The non ideal frequency

response of the chain introduces spurious high frequency content in the measured signal.

The tooth passing frequencies and their harmonics are the main driving frequencies.

For state of the art high speed milling spindles, the rotational speeds range between

20,000 - 40,000 rpm. At a speed of 20,000 rpm, a four fluted endmill yields a fundamental

tooth passing frequency of 1332 Hz. This pushes at the bandwidth of the dynamometer

used in this research. So, at such speeds, the fidelity of force measurements using this

dynamometer needs to be closely examined.

For the experiments reported in this document, the maximum fundamental tooth

passing frequency was 300 Hz. At this frequency, the response of the chain is reasonably

flat, both, in magnitude and phase. However, the harmonics are distorted and give rise to

wiggles on the force signals. Qualitatively, the calibration of the force model is not affected

164

to any serious extent because the average force is used to compute the cutting coefficients.

The high frequency wiggles oscillate about the mean force signal. So, the averaging is

not affected to any significant extent. Model calibration is reliable and yields good force

predictions.

−30

−20

−10

0

10

20

Re

(FR

F),

[N

/N]

Bare dynamometer

0 2500 5000

−40

−30

−20

−10

0

10

frequency, [Hz]

Im(F

RF

), [

N/N

]

With aluminum artifact

0 2500 5000

frequency, [Hz]

With steel artifact

0 2500 5000

frequency, [Hz]

Figure 11-1. Direct Frequency Response Functions (FRFs) of the force measuring chainwith the dynamometer mounted on flanges, fixed on a mounting plate, andclamped to the machine table: In addition to the FRF of the baredynamometer, two other sets of FRF were recorded to show the influence ofthe mass loading due to the part being clamped on to the dynamometer whenit is being machined. The lightweight (0.4 kg) aluminum artifact isrepresentative of the machined parts used to verify the helical force model,whereas the heavy (1.8 kg) steel artifact was the part used to verify thestraight fluted force model.

165

0

10

20

30

40

Ma

gn

itu

de

, [N

/N]

Bare dynamometer

0 2500 5000−15

−10

−5

0

5

frequency, [Hz]

Ph

ase

, [ra

d]

0

10

20

30

With aluminum artifact

0 2500 5000−6

−4

−2

0

2

frequency, [Hz]

0

5

10

15

20

25

With steel artifact

0 2500 5000−8

−6

−4

−2

0

2

frequency, [Hz]

Figure 11-2. Direct FRFs of the force measuring chain showing magnitude and phaseobtained from the experimentally determined real and imaginary parts of theFRFs shown in Fig. 11-1.

166

0

5

10

15

20

FR

F m

ag

nit

ud

e, [

N/N

]

Magnitude of FRF with Steel artifact

0 1000 2000 3000 4000 50000

0.2

0.4

0.6

Po

we

r (a

rbit

rary

un

its)

Power spectrum: hypothetical input force, 60 Hz tooth passing freq.

0

5

frequency, [Hz]

Figure 11-3. Demonstration of the effect of non unit magnitude of the FRF at the pointswhere the input force signal has frequency content.

167

CHAPTER 12FUTURE WORK

In this dissertation, the analytical model is developed by considering the tool,

workpiece, and machine structure to be rigid. A flexible model needs to be developed

to include the effects of cutter and workpiece deflections. The use of this force model in

other applications such as stability analysis, surface location error, and surface profile error

needs to be explored. The model also needs to be applied to surface finish predictions.

12.1 Stability and Surface Location Problem Formulation

To motivate an application of this force model in stability and surface location

analysis, an example problem is now formulated. The Temporal Finite Element Method

(TFEA) is capable of solving for stability and surface location error simultaneously.

Previously, the problem was solved using straight fluted geometry (Bayly et al., 2003;

Mann, 2003; Mann et al., 2005). Recently, the effects of helical geometry have been

included in the TFEA model. The problem was solved for a compliant workpiece rigid

tool model in Patel et al. (2008), and a rigid workpiece compliant tool model in Mann

et al. (2008). The TFEA solution of the problem for a single degree of freedom (SDOF)

compliant structure can be obtained using standard procedure (Mann, 2003; Mann et al.,

2005). Using the helical force model of chapter 7, the problem is formulated completely

below for a future researcher to solve it by developing some efficient computational

technique.

Figure 12-1 illustrates the concept of dynamic chip thickness in peripheral milling. In

the ideal case, the (static) chip thickness is hs, in the absence of regeneration of waviness

of the machined surface. Based on the circular path approximation one may write

hs = fT sin θ (12–1)

Regeneration causes a dynamic component ∆h to be imposed on hs. Thus, the actual

chip thickness is

168

h = hs︸︷︷︸static

+ ∆h︸︷︷︸dynamic

(12–2)

The breakup of the dynamic chip thickness, ∆h, into its components ∆x and ∆y is

illustrated in Fig. 12-1.

If τ is the tooth passing period, then

∆x = x(t)− x(t− τ) (12–3)

∆y = y(t)− y(t− τ) (12–4)

where x and y are functions of time, t.

An expression for ∆h may be written based on the geometry displayed in Fig. 12-1

∆h = ∆x sin θ + ∆y cos θ (12–5)

Once the dynamic chip thickness is characterized, the force model is developed to

include its effects. The infinitesimal projected (frontal) area of the uncut chip, dAf , at an

arbitrary angular orientation, θ, is

dAf = (fT sin θ + ∆x sin θ + ∆y cos θ) dζ

The following transformation may be used to eliminate the intermediate variable ζ in the

above equation

dζ =b

(θT − θL)dθ

leading to the relation

dAf =b

(θT − θL)(fT sin θ + ∆x sin θ + ∆y cos θ) dθ (12–6)

where θL and θT are the angular orientations of the leading and trailing edges of the chip

in the tool-chip contact zone, and b is the instantaneous projected (frontal) axial depth of

cut.

169

Using the empirical proportional cutting force model, the tangential and normal

components of the differential force may be related to the tangential and normal linearized

cutting coefficients Ktc and Knc

dFt

dFn

= dAf

Ktc

Knc

(12–7)

The tangential and normal components of the cutting force, in the plane perpendicular

to the cutter axis, are converted to components dFx and dFy, where the non-rotating (x, y)

coordinate system has its origin on the cutter axis. The components are related via a

rotation matrix

dFx

dFy

=

dFtx + dFnx

dFty + dFny

=

cos θ sin θ

sin θ − cos θ

dFt

dFn

(12–8)

Integration yields the total forces on a single tooth

Fx =

∫dFx =

∫(cos θdFt + sin θdFn) (12–9)

Fy =

∫dFy =

∫(sin θdFt − cos θdFn) (12–10)

where the integrations are carried out over the appropriate limits. These limits are

explicitly shown in a subsequent step, after applying certain transformations.

Substituting from Eq. 12–7 into the above two equations recasts the force relationships

in terms of the elemental differential frontal chip area, dA

Fx

Fy

=

∫cos θdAf

∫sin θdAf

∫sin θdAf − ∫

cos θdAf

Ktc

Knc

(12–11)

A shorthand notation is introduced for convenience

S , sin θ and C , cos θ (12–12)

170

Using the expression for dAf from Eq. 12–6, the two integrals∫

sin θdA =∫ SdAf

and∫

cos θdA =∫ CdAf may now be rewritten

∫SdAf =

b

(θT − θL)

[fT

∫S2dθ + ∆x

∫S2dθ + ∆y

∫SCdθ

](12–13)

∫CdAf =

b

(θT − θL)

[fT

∫SCdθ + ∆x

∫SCdθ + ∆y

∫S2dθ

](12–14)

Substituting from Eqs. 12–13 and (12–14) into Eq. 12–11 yields

Fx

Fy

=

FxS

FyS

︸︷︷︸static

+

FxD

FyD

︸︷︷︸dynamic

(12–15)

The static component of the force is

FxS

FyS

︸︷︷︸static

=bfT

(θT − θL)

FS11 FS12

FS21 FS22

Ktc

Knc

(12–16)

where

FS11 =

θT∫

θL

SCdθ (12–17)

FS12 =

θT∫

θL

S2θdθ (12–18)

FS21 =

θT∫

θL

S2θdθ (12–19)

FS22 =

θT∫

θL

−SCdθ (12–20)

171

The dynamic component of the force is

FxD

FyD

︸︷︷︸dynamic

=b

(θT − θL)

FD11 FD12

FD21 FD22

∆x

∆y

(12–21)

where

FD11 = Ktc

θT∫

θL

SCdθ + Knc

θT∫

θL

S2dθ (12–22)

FD12 = Ktc

θT∫

θL

C2dθ + Knc

θT∫

θL

SCdθ (12–23)

FD21 = Ktc

θT∫

θL

S2dθ −Knc

θT∫

θL

SCdθ (12–24)

FD22 = Ktc

θT∫

θL

SCdθ −Knc

θT∫

θL

S2dθ (12–25)

Consider the problem of modeling a peripheral milling process as a single degree

of freedom, linear-spring, mass, and viscous-damper system. Either the structure or the

tool may be considered compliant (Mann, 2003). Let the system be compliant in a single

direction and the x axis be aligned in this direction. For a cutter with a single tooth, Eq.

12–16 yields the static force component in the x direction

FxS1 =b

(θT − θL)[KtcFS11 + KncFS12] (12–26)

In computing the dynamic chip thickness, ∆y ≡ 0. The dynamic component of the

force Eq. 12–21 reduces to

FxD1 =b

(θT − θL)FD11∆x (12–27)

For such a system, a summation of forces yields the equation of motion

mxx(t) + cxx(t) + kxx(t) = FxS1 + FxD1 (12–28)

172

where the coefficients on the left hand side mx, cx, and kx have their usual meanings,

being the modal mass, viscous damping coefficient, and stiffness, respectively. Henceforth,

the suffix “x” will be dropped for simplicity, and these will be referred to as m, c, and k.

Based on (Mann, 2003, pp.17), the following observations may be made on the forcing

function in Eq. 12–28

• If the parametric excitation and the time-delayed relative displacement term, FxD1,were absent, the term FxS1 would provide periodic forcing resulting in a periodicsolution.

• The factor b(θT−θL)

FD11, in the dynamic forcing term FxD1, is a periodic coefficient ofthe dependent variable x.

• Stability is determined by the growth or decay of perturbations about the periodicmotion.

Equation 12–28 represents a time delayed differential system. It does not have a

closed form solution (Mann, 2003). Hence, temporal finite element analysis (TFEA) is

applied to obtain stability and surface placement information. The general procedure for

solution is given in Mann (2003) and Mann et al. (2005), and the reader is referred to

these publications for clarification of the notations that follow. The solution procedure is

also shown in the in the form of a flowchart in Fig. 12-2. Using this procedure the TFEA

solution is formulated as follows

N jpi

=

tj∫

0

[mφi(σj) + cφi(σj) +

k +

(b(σj)FD11(σj)

θT (σj)− θL(σj)

)φi(σj)

]ψp(σj)dσj (12–29)

P jpi

=

tj∫

0

[b(σj)FD11(σj)

[θT (σj)− θL(σj)]

]φi(σj)ψp(σj)dσj (12–30)

Cjp =

tj∫

0

−fT

[b(σj)FS11(σj)

θT (σj)− θL(σj)

]ψp(σj)dσj (12–31)

The dynamic forcing term appearing within the parenthesis in Eq. 12–29, and within

the brackets in Eq. 12–30, is evaluated using Eq. 12–22. The periodic (static) forcing term

173

appearing within the brackets in Eq. 12–31, is evaluated using Eq. 12–17. This completes

the formulation of the problem.

12.2 Augmented Force Model

The cutting coefficients Ktc, Knc and Kac characterize the forces due to the shearing

mechanism. In certain applications, such as micromilling, the ploughing forces can no

longer be neglected and their effects must be accounted for. The ploughing forces depend

on the length of the tool edge in contact with the chip. The associated edge coefficients

may be termed Kte, Kne and Kae. In this augmented model, the Eq. 7–1 becomes

dFt

dFn

dFa

=

KtcdAf

KncdAf

KacdAa

+

KtedS cos λ

KnedS cos λ

KaedS sin λ

(12–32)

where dS is the elemental edge length in the tool-chip contact zone. The expression for dS

is obtained by reference to Fig. 7-2

dS = dζ sec λ (12–33)

Following the same derivation procedure as in chapter 7, a result analogous to Eq.

7–11 can be obtained. Now, six coefficients have to be extracted experimentally using

three equations. Experiments can be conducted at two different values of radial immersion

to obtain a set of six linearly independent equations. The set of equations can be solved

for six coefficients. In this manner, the augmented force model can be developed.

174

fT

Ω

feed

workpiece

endmill

θ

nominal

surface

hs

∆h

∆x

∆yθ

h = hs + ∆h

actual

(wavy) surface

h θ

Figure 12-1. Dynamic chip thickness ∆h. The components of the dynamic chip thickness,∆x and ∆y, and their relation to ∆h, are shown in the geometricalconstruction. hs is the ideal (static) chip thickness in the absence ofregeneration of waviness of the machined surface. A case of up (conventional)milling is illustrated.

175

0 0 jt t t= +

( ) ( )

( ) ( ) ( )2 4

2 1 3 : 2 1 4.. ..

2 1 1 : 2 1 4 i

j

p

j jw ith N

j j×

× − + × − +

× − + × − +

( ) ( )

( ) ( ) ( )2 4

2 1 3 : 2 1 4.. ..

2 1 1 : 2 1 4 i

j

p

j jw ith P

j j×

× − + × − +

× − + × − +

:A

:B

Populate

Populate

1j j= +

j E=is ?

i

j

pN i

j

pP

Symbolic

i

j

pNi

j

pP

( )eval j ( )eval j

, , ,t nK K E RI

&Compute

store, ,c jt tτ

jt

Compute 1Q A B−= as a

function of &RPM ADOC

( )( )( )maxCM abs eig Q=Compute as a

function of &RPM ADOC

No

Yes

( )

( )

2 2

2 2

1: 2, 1: 2

1: 2, 2 1: 2 2

Populate A with I in rows cols

Populate B with in rows cols E E

×

×

Φ × + × +

( )& :2 2Initialize matrices A B size E

where E is the number of elements

× +

( )3E say=

E

,W b from loop

, ,m k c

( )&eval storeΦ

Φ

0

0

0

jSet

t

=

=

0t

Start

End1Plot contour CM =

on RPM vs ADOC axes

, 1

&

SDOF n

Loop over

RPM ADOC

=

Figure 12-2. Scheme for generation of stability contour using TFEA for a structure withSDOF compliance. The reader may refer to Mann (2003) or Mann et al.(2005) for clarification of the terminology.

176

CHAPTER 13CONCLUSION

A single, analytical closed form expression has been derived for each of the three

components of the cutting force, in a fixed coordinate frame, for helical peripheral

milling. A key advantage of the method is that the algebra takes care of all the different

geometrical possibilities (Type I, Type II cutting, and Phases A, B, C) automatically.

Separate expressions are not required, and there is no need to keep track of the cutting

type or phase in computations. These single expressions will allow analysts to conveniently

use them in analytical or semi-analytical applications where symbolic manipulations

are required to be performed. An example of this is the derivation of the sensitivity

coefficients in the uncertainty analysis.

The results have been shown to be valid for the entire parameter space covering

helical as well as straight fluted peripheral milling, partial or full immersion cutting,

multiple teeth in the cut, as well as for up-milling, down-milling and slotting by

comparison with established numerical methods. Experimental validation was performed

by calibrating the model for the aluminum alloy 6061-T6 using a 45 helix endmill. The

fidelity of predictions was shown to be high. In deriving the axial cutting force component,

the projected axial chip area has been considered. It was shown that this method leads to

good predictions of the axial cutting force.

One of the features of the model is that the zero helix assumption is not required to

be invoked while extracting the cutting coefficients. Another feature is that a full slotting

cut is not necessary to extract the cutting coefficients. Any value of partial immersion

suffices. This can lead to savings in test material when cutting expensive workpiece

material, or using machines with limited spindle power.

In computer calculations, for the degenerate case of straight fluted cutters, a very

small helix angle must be specified, as the expressions are not well behaved for zero

helix angle. However, analytically, it has been shown that the expressions reduce to

177

the familiar relationships for zero helix, showing that the formulation remains valid.

Therefore, the expressions may be used in applications involving analytic manipulations,

such as finding derivatives or integrals. A detailed analysis has been provided for the

propagation of uncertainties in the model input parameters through the force model. The

availability of single expressions for the force components permitted the derivation of

compact expressions for the sensitivities. As a result, the algebraic manipulations were

not unwieldy. The analytical method of uncertainty analysis has the advantage of easy

implementation and places low computational burden. Such a defensible uncertainty

statement accompanying the cutting force predictions is beneficial in practice. It enables

the process planner to decide the usefulness of model based force predictions in any

specific application. This demonstrates one of the advantages of the new closed form

analytical force model in applications.

Though some of the predictions made by the helical force model of chapter 7, using

average cutting coefficients, were only approximate, with the model over predicting peak

forces for certain situations such as high immersion milling, or failing to accurately predict

the skewness of the force components, the instantaneous cutting coefficient model of

chapter 10 was able to solve all these issues successfully. In most applications, the model

using average cutting coefficients would be adequate, but when very accurate prediction is

required, the instantaneous cutting coefficient model is available.

Remarks have been made regarding the future directions of work. Two problems

have been formulated for future research. One is a stability and surface location problem

using TFEA. The other is an augmented force model using edge coefficients in addition to

cutting coefficients.

178

APPENDIX AZELLNER’S REGRESSION

Following the method employed by Zellner (1962), the multi response model for

the simultaneous estimation of the cutting constants Γt,n and Ψt,n, of Eq. 4–13, may be

written as

Ki = Hiβi + εi i = t, n (A–1)

where ν is the number of experimental data sets, Ki is a ν × 1 vector of the (logarithms of

the) cutting coefficient responses, βi is a 2× 1 vector of unknown constant parameters (the

cutting constants), εi is a random error vector associated with the ith response, and

Ht = Hn =

[1

ν×1lnhν×1

](A–2)

wherelnh

is the vector of (the logarithms of) the average chip thickness at which the

responses are observed.

In matrix notation, the Eq. A–1 may be rewritten with H appearing in block

diagonal form:

lnKtν×1

lnKnν×1

︸︷︷︸K

=

Htν×2

0

0 Hnν×2

︸︷︷︸H

Γt

Ψt

Γn

Ψn

︸︷︷︸β

+

εtν×1

εnν×1

︸︷︷︸ε

(A–3)

Zellner (1962) has worked out an example in which he has shown how to estimate the

multiple response parameters the two-stage Aitken estimators and obtain the variance

covariance matrix the moment matrix of these estimators. That example is symbolically

reproduced here with suitable changes in notation. Rewriting Eq. A–3 in a simplified

manner, the system to be estimated may be represented as

Kt

Kn

=

Ht 0

0 Hn

βt

βn

+

εt

εn

(A–4)

179

First, the single-equation least squares estimates are obtained in the usual way. These

estimates are

βt =

βt0

βt1

=

Γt

Ψt

and βn =

βn0

βn1

=

Γn

Ψn

To obtain the disturbance covariance matrix, it is convenient to write

[Kt Kn

]=

[Ht Hn

]

βt 0

0 βn

+

[ut un

]

or K = HB + U

Then

UT U = (K −HB)T (K −HB) = KT K −BT HT HB

=

KTt Kt KT

t Kn

KTn Kt KT

n Kn

−

βTt HT

t Htβt βTt HT

t Hnβn

βTn HT

n Htβt βTn HT

n Hnβn

= (ν − 2) sµµ′where sµµ′ is an estimate of the (unknown) expectation value σµµ′ = E (uµuµ′), and

µ, µ′ = t, n. This last matrix may be inverted to obtain (ν − 2)−1sµµ′. Knowing sµµ′

enables one to obtain the following symmetric moment matrix of the two-stage Aitken

estimators

V (β) =

sttHTt Ht stnHT

t Hn

sntHTn Ht snnHT

n Hn

−1

(A–5)

whose diagonal elements are the estimated coefficient estimator variances, and off-diagonal

elements are estimated covariances.

180

The two-stage Aitken coefficient estimates are

β = V (β)

sttHTt Kt + stnHT

t Kn

sntHTn Kt + snnHT

n Kn

(A–6)

For the experiments analyzed in this paper, Ht = Hn, making the two-stage Aitken

estimators the same as the single equation least squares estimators, but the variance-covariance

matrix, V (β), is not diagonal, i.e., the covariances are non-zero.

181

APPENDIX BDERIVATION OF FOURIER COEFFICIENTS

Fourier coefficients L0, Lk,Mk, T0, Tk, Rk, B0, Bk, and Ck, are derived by inspection of

Fig. 3-3, writing the functional relationships valid for the fundamental period, θp ∈ [0, 2π),

and integrating appropriately using standard Fourier series procedures (Kreyszig, 2006).

B.1 Fourier Coefficients for θL

The variation of θL in Type I and Type II cutting is the same as is readily observed

by inspection of the Fig. 3-3. The functional relationship between θL and θ1 , within the

fundamental period, [0, 2π), is

θL =

0, θ ∈ [0, θst)

θ, θ ∈ [θst, θex]

θex θ ∈ [θex, θex + (2a tan λ)/D]

0, θ ∈ [θex + (2a tan λ)/D, 2π)

(B–1)

Fourier coefficient L0

L0 =Area under the (θL − θ) curve

2π(B–2)

A straightforward calculation yields Eq. 3–12.

Fourier coefficient Lk

Lk =1

π

2π∫

0

θL(θ) cos θdθ =1

π

θex∫

θst

θ cos θdθ +

θex+ 2a tan λD∫

θex

θex cos θdθ

Upon simplification

Lk =

−θst

πksin[kθst]− 1

πk2cos[kθst]

+1

πk2cos[kθex] +

θex

πksink[θex + (2a tan λ)/D]

(B–3)

1 in the derivations, the subscript, p, is dropped from θp for the sake of convenience.

182

Fourier coefficient Mk

Mk =1

π

2π∫

0

θL(θ) sin θdθ =1

π

θex∫

θst

θ sin θdθ +

θex+ 2a tan λD∫

θex

θex sin θdθ

Upon simplification

Mk =

− 1

πk2sin[kθst] +

θst

πkcos[kθst]

+1

πk2sin[kθex]− θex

πkcosk[θst + (2a tan λ)/D]

(B–4)

B.2 Fourier Coefficients for θT

Figure 3-3 shows that the variation of θT is the same in Type I and Type II cutting.

The functional relationship between θT and θ, within the fundamental period, [0, 2π), is

θT =

0, θ ∈ [0, θst)

θst, θ ∈ [θst, θst + (2a tan λ)/D]

θ − (2a tan λ)/D, θ ∈ [θst + (2a tan λ)/D, θex + (2a tan λ)/D]

0, θ ∈ (θex + (2a tan λ)/D, 2π)

(B–5)

Fourier coefficient T0

T0 =Area under the (θT − θ) curve

2π(B–6)


Fourier coefficient Tk

Tk =1

π

2π∫

0

θT (θ) cos θdθ =1

π

θst+2a tan λ

D∫

θst

θst cos θdθ+

θex+ 2a tan λD∫

θst+2a tan λ

D

(θ − 2a tan λ

D) cos θdθ

Upon simplification

Tk =

−θst

πksin[kθst]− 1

πk2cosk[θst + (2a tan λ)/D]

+θex

πksink[θex + (2a tan λ)/D]+

1

πk2cosk[θex + (2a tan λ)/D]

(B–7)

183

Fourier coefficient Rk

Rk =1

π

2π∫

0

θT (θ) sin θdθ =1

π

θst+2a tan λ

D∫

θst

θst sin θdθ+

θex+ 2a tan λD∫

θst+2a tan λ

D

(θ − 2a tan λ

D) sin θdθ

Upon simplification

Rk =

θst

πkcos[kθst]− 1

πk2sink[θst + (2a tan λ)/D]

+1

πk2sink[θex + (2a tan λ)/D] − θex

πkcosk[θex + (2a tan λ)/D]

(B–8)

B.3 Fourier Coefficients for b in Type I Cutting

The variation of b in Type I cutting is observed by inspection of the Fig. 3-3. The

functional relation between b and θ, within the fundamental period, [0, 2π), is

b|Type I =

0, θ ∈ [0, θst)

D cot λ2

[θ − θst], θ ∈ [θst, θst + (2a tan λ)/D]

a, θ ∈ [θst + (2a tan λ)/D, θex]

D cot λ2

[−θ + θex + (2a tan λ)/D], θ ∈ [θex, θex + (2a tan λ)/D]

0, θ ∈ (θex + (2a tan λ)/D, 2π)

(B–9)

Fourier coefficient B0|Type I

B0|Type I =Area under the Type I (b− θ) curve

2π(B–10)


Fourier coefficient Bk|Type I

Bk|Type I =1

π

2π∫

0

b(θ)|Type I cos θdθ =1

π

θst+2a tan λ

D∫

θst

D cot λ

2[θ − θst] cos θdθ +

θex∫

θst+2a tan λ

D

a cos θdθ

+

θex+ 2a tan λD∫

θex

D cot λ

2[−θ + θex + (2a tan λ)/D] cos θdθ

184

Upon simplification

Bk|Type I =D cot λ

2πk2

− cos[kθst] + cos[kθex]

+ cosk[θst + (2a tan λ)/D] − cosk[θex + (2a tan λ)/D]

(B–11)

Fourier coefficient Ck|Type I

Ck|Type I =1

π

2π∫

0

b(θ)|Type I sin θdθ =1

π

θst+2a tan λ

D∫

θst

D cot λ

2[θ − θst] sin θdθ +

θex∫

θst+2a tan λ

D

a sin θdθ

+

θex+ 2a tan λD∫

θex

D cot λ

2[−θ + θex + (2a tan λ)/D] sin θdθ

Upon simplification

Ck|Type I =D cot λ

2πk2

− sin[kθst] + sin[kθex]

+ sink[θst + (2a tan λ)/D] − sink[θex + (2a tan λ)/D]

(B–12)

B.4 Fourier Coefficients for b in Type II Cutting

The variation of b in Type II cutting is observed by inspection of the Fig. 3-3. The

functional relation between b and θ, within the fundamental period, [0, 2π), is

b|Type II =

0, θ ∈ [0, θst)

D cot λ2

[θ − θst], θ ∈ [θst, θex)

D cot λ2

[θex − θst], θ ∈ [θex, θst + (2a tan λ)/D]

D cot λ2

[−θ + θex + (2a tan λ)/D], θ ∈ [θst + (2a tan λ)/D, θex + (2a tan λ)/D]

0, θ ∈ (θex + (2a tan λ)/D, 2π)

(B–13)

Fourier coefficient B0|TypeII

B0|TypeII =Area under the Type II (b− θ) curve

2π(B–14)


185

Fourier coefficient Bk|Type II

Bk|Type II =1

π

2π∫

0

b(θ)|Type II cos θdθ =1

π

θex∫

θst

D cot λ

2[θ − θst] cos θdθ

+

θst+2a tan λ

D∫

θex

D cot λ

2[θex − θst] cos θdθ

+

θex+ 2a tan λD∫

θst+2a tan λ

D

D cot λ

2[−θ + θex + (2a tan λ)/D] cos θdθ

Upon simplification

Bk|Type II =D cot λ

2πk2

− cos[kθst] + cos[kθex]

+ cosk[θst + (2a tan λ)/D] − cosk[θex + (2a tan λ)/D]

(B–15)

Fourier coefficient Ck|Type II

Ck|Type II =1

π

2π∫

0

b(θ)|Type II sin θdθ =1

π

θex∫

θst

D cot λ

2[θ − θst] sin θdθ

+

θst+2a tan λ

D∫

θex

D cot λ

2[θex − θst] sin θdθ

+

θex+ 2a tan λD∫

θst+2a tan λ

D

D cot λ

2[−θ + θex + (2a tan λ)/D] sin θdθ

Upon simplification

Ck|Type II =D cot λ

2πk2

− sin[kθst] + sin[kθex]

+ sink[θst + (2a tan λ)/D] − sink[θex + (2a tan λ)/D]

(B–16)

The observation is that the coefficients have the same algebraic form for both, Type I

and Type II cutting. They need not be listed separately in Table 3-1.

186

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BIOGRAPHICAL SKETCH

The author was born in Bhilwara, India, on March 8, 1962. He was raised in India

where he obtained a Bachelor of Technology degree in mechanical engineering from the

Indian Institute of Technology, Kanpur, in 1984. He then worked in sales and applications

of cutting tools, forming tools, and wear-resistant parts. In 2003, he decided to go back

to school to augment his education. He is currently a doctoral candidate the University of

Florida.

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