Generalized Modelling of Flexible Machining System With

GENERALIZED MODELLING OF FLEXIBLE MACHINING SYSTEM WITH

ARBITRARY TOOL GEOMETRY

Zekai Murat Kilic

B.S., Middle East Technical University, 2006

M.S., Middle East Technical University, 2009

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES

(Mechanical Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

April 2015

The final shape of mechanical parts is mainly determined through turning, boring, drilling and milling operations. The prediction of the cutting forces, torque, and power of the machining process, and surface errors and vibration marks left on the parts is required to plan the machining operations and achieve shorter production cycle times while avoiding damage on the part, tool and machine. Past research has focused on developing dedicated mathematical models for each machining operation and tool type. However, the tool geometry and configuration of the machining set-up varies widely depending on the part geometry and application. This thesis presents a generalized mathematical model of machining operations carried out using geometrically defined cutting edges.

The mechanics of cutting between the tool edge and the work material are modelled to predict the friction and normal forces on the rake face of a single cutting edge. The combined static and dynamic chip thickness is modelled as a function of tool geometry, the kinematics of machining operation and the relative regenerative vibrations between the tool and workpiece.

The cutting forces are transformed to process coordinates by considering the orientation of cutting edge and the kinematics of the machining operation, and are applied on the structural dynamics of the machine tool and workpiece by distribution along the cutting tool–workpiece contact zone.

The cutting forces, vibrations, chatter stability and surface errors are simultaneously predicted in a semi-discrete time domain.

The geometry and force transformation models are unified in a parametric, mathematical model which covers all cutting operations. The application of the proposed model is demonstrated on turning, drilling and milling operations; multifunctional tools that combine ii

drilling-boring and chamfering in one operation; and two parallel face-milling cutters machining

a plate from both sides. The proposed mathematical models are experimentally validated by

comparing the measured forces, surface errors, vibrations and chatter stability charts against

simulations.

The thesis shows the first unified, generalized mathematical modelling of metal cutting

operations in the literature. The proposed model is expected to widen the application of science- based machining process simulation, planning and optimization methods in the virtual production of parts.

iii

Preface

This Ph.D. dissertation proposes a generalized mathematical model which unifies turning,

boring, drilling and milling operations, and is based on the papers that are already published, and

on manuscripts currently prepared for publication in peer-reviewed journals. The presented work

in these publications is carried out by the Ph.D. candidate in his Ph.D. research under the

supervision of Dr. Yusuf Altintas. Relative contributions of the authors for each of the research

articles are explained in this section.

The research in Chapter 3 is partially related to the study that is presented in Ref. [1]. The candidate is the co-author of this article focusing on a method for generalizing the machining operation by assuming two-dimensional insert geometry. Chapter 3 further describes the development and projection of the generalized method to the solid tools, and to the indexable cutters with three-dimensional insert geometry.

Chapter 4 focuses on the basic mathematical model presented in Ref. [2]. The generalized method is extended to include turning, boring, drilling and milling applications. The results are presented in Chapter 6.

Chapter 5 advances the model published in Refs. [2,3]:

• Ref. [2] sets up the general equation of motion for the end milling operation. The Ph.D.

candidate contributed to the solution and experimental proof of the distributed dynamics

of structure-milling process interaction which is described by the periodic, delay-

differential equations. Caner Eksioglu, former M.A.Sc. student in Altintas lab, initiated

the basic mathematical model required to represent the flexible end milling operation

with distributed dynamics along the axial depth of cut. This contribution received ASME

Blackall Machine Tool and Gage Award as the best journal paper in manufacturing

engineering in 2013.

• Ref. [3] forms the concept of general equation of motion for all turning, boring, drilling

and milling operations. The candidate is the main contributor of this study, and the

supervisor presented the concept in the academy (CIRP) in a brief paper. The candidate

advanced the general mechanics and dynamic model to include all tool and cutting

operations in this thesis, which will be submitted for journal review.

• There will be a three part journal paper that shows: 1) The combined formulation of

mechanics and dynamics of general cutting process, 2) adaptation of general model to

predict the forces, torque, power and surface location errors in turning, drilling, boring

and milling operations, 3) generalized stability analysis of all operations in semi-discrete

time and frequency domains with experimentally supported case studies. The candidate is

the primary contributor to all of the research here under the supervision of Dr. Altintas.

• Several applications of the proposed general modeling are given in Chapter 6. The basic

modeling concept to predict the cutting forces for double-sided, parallel milling operation

covered in Chapter 6 was presented in a conference article [4]. The complete model

including the dynamics and stability will be submitted as a journal article. The candidate

contributed to the modeling, analysis and solution and received experimental support

from Dr. Derek Luo of ITRI, Faby Feng of Apply Zeta, and Kevin Kao of KOVA cutter

in Taiwan. Ref. [5] presents the force and stability prediction of multifunctional cutters.

Dr. Min Wan analyzed the geometry of the multifunctional cutters and setup the equation

of motion in the modal domain. The candidate made contributions to the experiments and

the validation of results by applying the formulation as reported in Chapters 3-5. v

Table of Contents

Abstract ...... ii

Preface ...... iv

Table of Contents ...... vi

List of Tables ...... ix

List of Figures ...... x

List of Symbols and Abbreviations ...... xvii

Acknowledgements ...... xxxiv

Dedication ...... xxxvi

Chapter 1: Introduction ...... 1

Chapter 2: Literature Survey ...... 4

2.1 Overview ...... 4

2.2 Mechanics of Metal Cutting ...... 4

2.3 Basic Kinematics of Regenerative Machining Operation ...... 5

2.4 Cutting Forces ...... 8

2.4.1 Mechanistic Identification ...... 9

2.4.2 Generalized Cutting Force Prediction ...... 10

2.5 Modelling of Generalized Tool Geometry ...... 11

2.6 Simultaneous Machining Modelling ...... 13

2.7 Dynamics of Machining Operations ...... 14

2.8 Surface Form Error ...... 16

2.9 Summary...... 17 vi

Chapter 3: Generalized Geometric Model of Cutting Tools ...... 19

3.1 Generalized Geometric Parameters of Arbitrary Tools ...... 19

3.2 Generalized Modeling of Cutting Edge Coordinates on the Tool Body ...... 32

3.2.1 Indexable Cutter ...... 32

3.2.2 Solid Tool ...... 62

3.3 Computation of the Effective Tool Geometry ...... 67

3.3.1 Velocity of Cutting Edge of the Tool ...... 67

3.3.2 Tool Cutting Planes ...... 69

3.3.3 Effective Tool Angles ...... 74

Chapter 4: Generalized Modelling of Cutting Mechanics ...... 81

4.1 Chip Geometry ...... 81

4.2 Modeling of Cutting Forces...... 92

4.3 Force Transformations for Arbitrary Tools ...... 95

4.3.1 Rake face UV (friction-normal) to Cutting Edge RTA (radial-tangential-axial) ...... 96

4.3.2 RTA to Edge Reference Frame (xRyRzR), and to Process Frame (xyzθ) ...... 98

4.4 Transformation of the Vibration Vector to the Dynamic Chip Thickness ...... 105

4.5 Representation of General Force Vector ...... 107

4.5.3 Multiple-Level Tool with Lumped Force at Each Station/Level ...... 113

4.6 Summary...... 117

Chapter 5: Generalized Modelling of Cutting Dynamics ...... 118

5.1 General Equation of Motion ...... 118

5.2 Solution of the Equation of Motion in Modal Domain ...... 124

5.3 Simplified Solution Methods for Special Cases ...... 149 vii

5.3.1 Low Radial Immersion with Free Vibration Condition (e.g. Double-Sided Milling) 150

5.4 Summary...... 154

Chapter 6: Applications ...... 155

6.1 Overview ...... 155

6.2 Application-1: Slender, Cylindrical Helical End Mill with Regular Pitch Angles [2] ...... 155

6.3 Application-2: Regular Pitch Serrated Cylindrical Helical End Mill [21] ...... 180

6.4 Application-3: Two-Level Multifunctional Drilling/Boring Cutter [5] ...... 191

6.5 Application-4: Drilling with a Multifunctional, Indexable Cutter ...... 203

6.6 Application-5: Double-Sided Parallel Milling ...... 210

6.7 Application-6: Prediction of Cutting Forces for a Two-Insert Indexable End Mill ...... 221

6.8 Application-7: Three Holders with the Same General Purpose Insert ...... 221

6.9 Application-8: Parallel Turning with Two Tools ...... 225

Chapter 7: Conclusions and Future Research Directions ...... 230

7.1 Conclusions ...... 230

7.2 Future Research Directions ...... 232

References ...... 234

Appendix A Transforming the basis from Frame-1 to Frame-2 ...... 245

viii

List of Tables

Table 4.1 Inputs at the k=19-th axial segment of the 3-tooth serrated cylindrical end mill...... 85

Table 6.1 Information of the cylindrical end mill ...... 156

Table 6.2 Radial runout measurement at five axial locations using a dial gage with 5-micrometer

tolerance...... 156

Table 6.3 Mechanistic identification of the milling operation. Tool information in Table 6.1. . 167

Table 6.4 Identified modal parameters in x0 and y0 directions...... 171

Table 6.5 Information of the serrated end mill ...... 181

Table 6.6 Identified modal parameters in x and y directions (taken from Table 2 of Ref. [21]). 189

Table 6.7 Identified modal parameters for Sandvik and Kennametal multifunctional tools...... 200

Table 6.8 Measured angular location, pitch angle and radial runout of the left- and right-side

cutters...... 212

Table 6.9 Average forces derived from experiment and theoretical calculations...... 215

List of Figures

Figure 2.1 Schematics of chip removal at differential cutting edge element ...... 5

Figure 2.2 Dynamic interaction between tool and workpiece...... 6

Figure 3.1 Left hand turning side-view schematics ...... 21

Figure 3.2 Indexable drill schematics ...... 22

Figure 3.3 Geometry of serrated end mill that is used for rough milling operations ...... 23

Figure 3.4 General end mill geometry schematics...... 24

Figure 3.5 Parameters for the drilling operation at initial time instant (t=0) ...... 27

Figure 3.6 Rake face vectors at element Sjk ...... 31

Figure 3.7 Locating the local rake face frame in the insert reference frame ...... 34

Figure 3.8 Locating the local cutting edge frame ...... 35

Figure 3.9 Step-1 of positioning the insert on the holder ...... 37

Figure 3.10 Insert placement on the holder ...... 37

Figure 3.11 KOVA XNEX080608 insert...... 38

Figure 3.12 Assigning the insert reference frame ...... 38

Figure 3.13 Modelling of the rake face of the insert...... 40

Figure 3.14 Highlighted two cutting edge sections (ces-5 and ces-7) with flat rake faces ...... 41

Figure 3.15 Rake face approximation of curved rake face ...... 41

Figure 3.16 Local coordinates of the selected point e along the cutting edge section-1 ...... 42

Figure 3.17 Cutting edge-section-2 (Figure 3.13b) with curved rake face ...... 44

Figure 3.18 Rake face geometry along the modelled cutting edge (bold curve): (a) Rake face

vectors; (b) rake face normal vectors; (c) rake face and rake face normal vectors ...... 44

Figure 3.19 Indexable milling cutters ...... 45 x

Figure 3.20 Transformation from Frame-i to Frame-0 of the cylindrical face mill cutter ...... 46

Figure 3.21 Transformation from Frame-i-Right to Frame-0 of the double milling tool ...... 46

Figure 3.22 Transformation from Frame-i-Left to Frame-0 of the double milling tool ...... 47

Figure 3.23 Indexable U502 drilling cutter. (Courtesy of KOVA.)...... 47

Figure 3.24 Transformations from Frame-i to Frame-0 for the indexable drilling tool ...... 47

Figure 3.25 Sample two-insert end mill from Sandvik-Coromant ...... 49

Figure 3.26 Relative locations of the origins of cutter, insert and first rake face section of the first

cutting edge section...... 50

Figure 3.27 Cutting edge and rake face sections of the insert ...... 55

Figure 3.28 Insert edge geometry of the two-inserted end mill ...... 56

Figure 3.29 Chip length along the first 0.030-mm section of the cutting edge ...... 57

Figure 3.30 Multifunctional cutters ...... 58

Figure 3.31 Measured cutting edge geometry of Kennametal drill ...... 59

Figure 3.32 Modelled Kennametal cutter: (a) Axial location; (b) angular location...... 60

Figure 3.33 Modelled cutting edge geometry of the Sandvik TM880 cutter...... 61

Figure 3.34 Locating the Frame-D on the solid tool starting from Frame-0 ...... 63

Figure 3.35 Surface profilometer measurement data of the serrated end mill ...... 64

Figure 3.36 Cutting edge angle of the serrated end mill ...... 65

Figure 3.37 Resultant cutting directions for two consecutive edges at segment k of milling cutter

moving in x0 direction...... 68

Figure 3.38 Tapered ball end mill to show Pf, Pfe and Pr planes ...... 70

Figure 3.39 Tapered ball end mill to show A , Pn, Ps and Pr planes ...... 72

Figure 3.40 Cutting edge angle calculation: (a) Right-hand milling; (b) Left-hand milling...... 76 xi

Figure 3.41 Cutting edge angle calculation of the drilling operation...... 77

Figure 3.42 Calculations of (a) Cutting edge inclination angle and (b) normal rake angle...... 78

Figure 3.43 Effective tool angles of the two-insert end mill...... 80

Figure 3.44 Effective tool angles of U140 face mill ...... 80

Figure 3.45 Effective tool angles along the cutting edge (in one serration period) of the serrated

end mill [21]...... 80

Figure 4.1 Algorithm for predicting the actual static chip thickness and tooth passing delay

period Tjki ...... 84

Figure 4.2 Serrated cylindrical end mill with the cross section at k-th segment...... 85

Figure 4.3 Schematics for the 2-insert drill with axial runout...... 89

Figure 4.4 Parallel turning schematics: (a) Radial discretization; (b) effect of axial offset and

axial runout between cutters...... 90

Figure 4.5 Transformation of rake face forces (Fu and Fv) into the forces along the R,T and A

axes of the cutting edge plane Ps. (Adapted from [1,127].) ...... 96

Figure 4.6 Transformations from RTA to xRyRzR, and from xRyRzR to x0y0z0 for basic machining

operations. (Images adapted from Sandvik and Ref.[3].) ...... 102

Figure 4.7 Right hand turning and parallel turning schematics ...... 103

Figure 4.8 Double-sided milling schematics...... 105

Figure 4.9 Double-sided milling tool geometry [4]: (a) Left-side cutter; (b) right-side cutter. .. 105

Figure 4.10 Modelling of cutting forces ...... 111

Figure 4.11 Radial force distribution on the flexible tool-workpiece zone ...... 113

Figure 4.12 Sandvik TM880 multifunctional tool with D=2 dynamic levels...... 115

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Figure 4.13 The double-sided milling cutter with forces applied at the opposite sides of the

workpiece...... 116

Figure 4.14 Cutting forces of parallel turning operation...... 117

Figure 5.1 Interpolation of the delay state using five neighboring states...... 130

Figure 5.2 The neighboring states and the Lagrange- 4-th order interpolation ...... 133

Figure 5.3 Surface location error for the milling operation ...... 147

Figure 6.1 Geometry along cutting edge of cylindrical end mill: (a) Angular location; (b) relative

angular location, (c) radial runout; (d) radius...... 158

Figure 6.2 Modelled 3D locations in the Frame-0 of cylindrical end mill...... 159

Figure 6.3 Tool entry and exit parameters for milling operation ...... 161

Figure 6.4 (a) Schematics of 5% down milling. (b) In-cut and out-of-cut regions...... 161

Figure 6.5 Effects of radial tunout for 5% down milling of cylindrical helical end mill, on (a) tool

entry angle; (b) Maximum effective chip load along the cutting edge...... 162

Figure 6.6 Zoomed tool-workpiece contact zone (from Figure 6.4b) at 1.2-mm axial depth, for

tooth/edge numbers: (a) #1; (b) #2; (c) #3; (d) #4...... 164

Figure 6.7 Effect of runout on: (a) effective chip load; (b) actual instantaneous pitch angle. ... 165

Figure 6.8 Dynamometer direct force transfer function: ...... 168

Figure 6.9 Experiment vs. predicted forces with runout effect...... 169

Figure 6.10 FRF measurements along the tool length: (a) Test locations; (b) Feed direction

magnitude plot; (c) Normal direction magnitude plot...... 171

Figure 6.11 Sketch of Figure 6.10a...... 171

Figure 6.12 Results of discrete-time simulation ...... 178

Figure 6.13 Discrete time force simulation for the experiments (A) and (B) of Figure 6.12a ... 179 xiii

Figure 6.14 Serrated end mill flute geometry ...... 182

Figure 6.15 Serrated end mill: (a) angular location; (b) relative angular location...... 182

Figure 6.16 Serrated end mill: (a) Modelled 3D geometry; (b) close-up 3D view...... 183

Figure 6.17 Computed tool geometry of serrated end mill ...... 183

Figure 6.18 (a) Radial runout for the first 8-mm depth; (b-f) Maximum effective chip load along

the cutting edge for five cases...... 184

Figure 6.19 Force simulations compared to Cutpro [118] ...... 186

Figure 6.20 Force simulation compared to the experiment plots of [21] ...... 187

Figure 6.21 Comparison of the stability prediction to the results presented in [21]...... 190

Figure 6.22 Sandvik multifunctional tool: (a) 3D geometry; (b) Radial depth of the midpoints of

the segments along the cutter body...... 192

Figure 6.23 Angular location (a) and relative angular location (b) of element Sjk...... 193

Figure 6.24 Axial location: (a) Level-1; (b) Level-2...... 193

Figure 6.25 (a) Axial runout: (a) Level-1; (b) Level-2...... 193

Figure 6.26 Pitch angle: (a) Feedrate neglected (fr 0); (b) fr=0.1-mm...... 195

Figure 6.27 Effective tool geometry calculated from≈ the rake face orientation ...... 195

Figure 6.28 Cutting force coefficients (a,b,c) and edge coefficients (d,e,f) in RTA directions. 196

Figure 6.29 Element-available-to-cut (a) and missed-cut (b,c,d,e) conditions for Sandvik

multifunctional tool...... 198

Figure 6.30 Predicted cutting forces for Sandvik multifunctional tool ...... 199

Figure 6.31 Sandvik multifunctional tool stability predictions compared to experiments ...... 202

Figure 6.32 3D locations of the segments on the Kennametal cutter body ...... 204

Figure 6.33 Angular location (a) and relative angular location (b) of element Sjk...... 205 xiv

Figure 6.34 (a) Axial location (height), and (b) corresponding axial runout...... 205

Figure 6.35 Pitch angle: (a) Norunout/zero feedrate; (b) fr=0.1-mm/rev ...... 205

Figure 6.36 Effective tool geometry ...... 206

Figure 6.37 Effect of the rubbing of the pilot drill on the cutting forces of Kennametal drill. .. 207

Figure 6.38 Predicted and measured cutting forces at period-B of Figure 6.37...... 207

Figure 6.39 Cutting forces without rubbing effect. Pilot hole diameter is larger than 10-mm. .. 208

Figure 6.40 Predicted and measured cutting forces (close-up of Figure 6.39)...... 208

Figure 6.41 Stability test with Kennametal drill: (a) Stability chart compared to experiments; (b)

Forces with the resulting surface finish photo of the selected unstable cutting test [5]...... 209

Figure 6.42 Measurement of radial runouts on the double-sided milling tool...... 210

Figure 6.43 Double-sided milling tool: (a) In operation; (b) with brazed inserts; (c) with

detachable inserts...... 211

Figure 6.44 Double-sided milling tool: (a) Components‒see Figure 4.8; (b) operation is ~20%

down milling...... 212

Figure 6.45 Supporting edge line (tool holder reference line) shown on: (a) Left-side and (b)

right-side cutters with labeled insert numbers...... 213

Figure 6.46 Double-sided milling experiment setup: (a) Part cross section; (b) 3D view...... 213

Figure 6.47 Predicted (solid line) and experimental (broken line) cutting forces ...... 214

Figure 6.48 Experiment setup with flexible workpiece ...... 216

Figure 6.49 Pre-set spacer width of the double-sided milling tool...... 216

Figure 6.50 Engagement condition of double-sided cutting tests...... 216

Figure 6.51 Tool entry (a) and exit (b) angles for the indexable double-sided milling tool...... 217

Figure 6.52 First two vibration mode shapes from finite element simulation ...... 219 xv

Figure 6.53 Measurement at selected point (at A and B distances) on the steel part...... 219

Figure 6.54 Real and imaginary parts of measured and fitted FRF...... 219

Figure 6.55 Predicted limit depth of cut per side for stable double sided milling...... 220

Figure 6.56 Identified cutting edge of the two-insert end mill ...... 220

Figure 6.57 Predicted forces compared to the experimental results reported by Eynian [75]. ... 221

Figure 6.58 KOVA U140 face mill geometry. (Courtesy of KOVA.) ...... 222

Figure 6.59 Face milling operation cutting force prediction...... 223

Figure 6.60 KOVA U502 drill geometry. (Courtesy of KOVA.) ...... 224

Figure 6.61 Drilling operation cutting force predictions...... 224

Figure 6.62 Parallel turning setup ...... 228

Figure 6.63 Comparison of the predicted stability chart (modelled) to the literature...... 229

xvi

List of Symbols and Abbreviations

Nq, ∑ (input) : Function of double summation from k=1..q for each j=1..N jk,

( ⋅ )lumped : Lumped force vectors at one point along tool-workpiece contact zone

⋅⋅⋅ ( )xyz,,( ) ( ) : Magnitude components of a vector

⋅ ( )jk : Matrix/vector related to element Sjk

PPrn ( ⋅⋅) , ( ) : Projection of vector on Pr-plane and on Pn-plane

⋅⋅ ⋅ ( )external,,( ) bored-hole( ) drilled-hole : Refer to external radius of workpiece, inner radius of bored-

hole and inner radius of drilled-hole

⋅ ( )[rows x] cols : Way to show the dimensions of a matrix/vector. rows and cols refer to number of

rows and columns

⋅⋅ ( )exact, ( ) approx : Exact and approximate matrices

11 11 ( xzee,0, ) : Location of point-e in rake face section-1 in Figure 3.27b

[..]specific : Matrix values computed for specific geometry geometry

⋅ : Absolute value of the elements of the input vector

⋅ : Euclidean norm (magnitude) of vector

0 : Zeros matrix/vector

2D, 3D : Two- and three-dimensional

A, B: Dimensions showing the modal test location on flexible workpiece

A0 : Constant state-transition matrix xvii

Ajk : Local axial location of element Sjk ap: Depth of cut ap1 , ap2 : Depth of cut at tool#1 and tool#2 arccos ( ⋅ ) , arcsin ( ⋅ ) : Trigonometric inverse-cosine and inverse-sine functions

ast,jk , aex,jk : Entry and exit distance at element Sjk atan2 (inputX, inputY ) : Trigonometric atan2 function

Aγ , Aα : Rake face and clearance face (flank) b : Largest multiple of modified delay term ( n*)

B1(t) , B2,jk(t) : Time dependent current and delayed state matrices

a pa ccki,,ki γ ki : Arrays of actual feed per edge, chip load and decrease in chip load at segment-k at

specific time instant

pa cjki : Instantaneous chip load of element Sjk

a cjki : Instantaneous feed per edge of element Sjk cjk : Nominal feed per edge of element Sjk

C, c : Number of cutting edge sections and index of selected cutting edge section

C1,i , C2,i , C3,jki : Matrices that represent dynamic forces

C4,i : Vector that represents static forces

CAD : Computer-Aided Design ceil ( ⋅ ) : Operator to round the input number to its closest larger integer value ces: Cutting edge section cj(t) ,c0,j (), tc d,j () t Instantaneous regenerative, static and dynamic chip loads of tooth-j xviii

cr,jki : Numerator coefficient of weight term for interpolating instantaneous delay of element Sjk

CW, CCW: Clockwise, counterclockwise

D : Total number of dynamic stations/levels

D1,i-3 , D2,jk,i-3 : Merged matrices of current and delayed dynamic forces within one spindle period de : Last radial segment index of the e-th dynamic level

DF , Df : State transition matrices for forced and free vibration periods dr : Denominator coefficient of weight term for interpolating instantaneous delay of element Sjk ds: Differential thickness along flute direction of serrated end mill dz: Constant thickness of each differential cutting edge segment-k (1..N) e : Index for dynamic station/level e : Index of selected point on the rake face section-c,r

( ⋅ ) e , exp(⋅ ) : Matrix exponential

Ei-3 : Merged static force vectors within one spindle period eig ()⋅ : Eigenvalues of the input matrix

d e jki : General transformation vector from tool reference coordinates to the negative-R ek: Surface location error at axial segment-k

c c c Fx,jki , Fy,jki , Fz,jki : Shear force components in Frame-0

d Fjki : Dynamic force multiplier of the dynamic chip load

ed Fjki : Dynamic ploughing (process damping) force vector f t,j : Feedrate per tooth-j

FFii, ()t : Force at i-th time instant (ti)

xix

d fjki : Local delayed dynamic force term of element Sjk element at i-th time instant (ti)

p Fjki : Process damping force multiplier of the velocity

ccc FFFr,jki,, t,jki a,jki : Shear force components in radial, tangential and axial directions

cc FFrta,jki, uv,jki : Shear force vectors in RTA and UV frames

cs cd FFjki, jki : Static and dynamic shear force vectors

es es es FFFr,jki,, t,jki a,jki : Static ploughing force components in radial, tangential and axial directions

es Frta,jki : Static ploughing force vector in RTA frame

cs cd es ed ffffiiii,,, : Static shear, Dynamic shear, dynamic, static ploughing and dynamic ploughing

force terms of equation of motion at i-th time instant (ti)

tw FFjki, jki : Total cutting force on tool and workpiece in process coordinates

t Frta,jk : Total cutting force on tool in RTA

LR FFii, : Total force vector of left- and right-side cutters

st ss FFjki, jki : Total static and steady state cutting forces

F(t), Ftc(), F es () t: (Chapter 2) Instantaneous total static, shear and static ploughing forces f(xr) : Neighbouring state values for interpolation

Fjki : Instantaneous cutting force

c es FFjki, jk : Shear and static ploughing components of cutting force floor ()⋅ : Function to round the input down to closest integer fr, ft : Translational feed per revolution, and feed per tooth

Frame-2, Frame-3 : Coordinate frames defined for mechanics of oblique cutting (Figure 4.5)

Frame-c,r : Local rake face frame

Frame-D , Frame-c,r,e : Local cutting edge design frame

Frame-i : Insert reference frame

Frame-R : Tool reference frame

Frame-s : Insert pocket (seat) frame

FRF : Frequency response function or Transfer function

Fu, Fv: Rake face forces in friction and normal directions

cc FFu,jki, v,jki : Shear force in friction and normal directions g1,jk : Element-available-to-cut condition of element Sjk

LL L L gg1,jk,,, 2,jki g 3,jki g 4,jki,r : Binary in-cut condition parameters related to left-side cutter

RR R R gg1,jk,,, 2,jki g 3,jki g 4,jki,r : Binary in-cut condition parameters related to right-side cutter g1,k , g3,ki : Element-available-to-cut and missed-cut arrays of segment-k at specific time instant g2,jki : Instantaneous edge-in-cut condition for element Sjk g3,jki : Instantaneous missed-cut condition parameter of the element Sjk based on the runout g4,jki,r : Binary parameter to decide if r-th radial zone is in-cut

st d hhjki, jki : Instantaneuos static and dynamic components of chip thickness of element Sjk h(τ) : Intermediate function for shortening integral form of Eq.

st (hjki )zero : Static chip thickness when runout is zero at segment-k runout h, lc: Uncut chip thickness, projected chip length hjki , Ac,jki : Instantaneous uncut chip thickness and uncut chip area of element Sjk xxi

I : Identity matrix i : Index for time instants i0 ,,jk00: Local unit axes vectors of Frame-0 i2 , j2 , k2 : Axis vectors of Frame-2

111 i22,, jk 2: Axis vector representation of Frame-2 relative to Frame-1 iii ic,r,, jk c,r c,r : Local unit axis vectors of Frame-c,r defined in Frame-i iii iii iDD,, jk D ( ic,r,e,, jk c,r,e c,r,e ): Local unit axis vectors of Frame-D (Frame-c,r,e) defined in

Frame-i

000 iDD,, jk D: Local unit axes vectors of Frame-D defined in Frame-0 id1, id2, id3: Index points used for separating free and force vibration zone during one spindle

rotation ii ,,jkii: Local unit axes vectors of Frame-i

000 iii,, jk i: Local unit axes vectors of Frame-i defined in Frame-0 int ( ⋅ ) : Operator to round the input number to its closest integer value

000 iRR,, jk R: Local unit axes vectors of Frame-R (tool reference frame) defined in Frame-0 is ,,jkss: Local unit axes vectors of Frame-s

J : Total number of paired free-forced vibration periods j: Index of current edge/tooth/flute ja : Index of edge in chip thickness calculation algorithm jout: Selected output edge for the chip thickness calculation algorithm

xxii

ccc KKKr,jki,, t,jki a,jki : Cutting force coefficients in radial, tangential and axial directions

eee KKKr,jki,, t,jki a,jki : Edge coefficients in radial, tangential and axial directions k: Index of axial or radial segment-k

K: Total number of axial or radial segments of tool

Kc(p1,..pL): Cutting force coefficient as a function of L parameters (p1,..pL)

ce KKjki, jk : Cutting force and edge coefficients

Ksp : Material-dependent specific contact force

cc KKu,jki, v,jki : Cutting force coefficients in friction and normal directions kk, x,ntt ,t y,n ,t : Modal stiffnesses in x and y directions for nt-th tool vibration mode

L1: Length of cutting edge section-1 in Figure 3.16 lc,jk : Projected uncut chip length along T direction of element Sjk

Lw : Flank wear land of tool m, mt, mw: Total number of vibration modes, number of tool modes and number of workpiece

modes max ( ⋅ ) : Maximum function among edges (j) and time instants (i) ji, max ( ⋅ ) : Maximum scalar value of the inputted matrix

Mf , Kf , Cf : Mass, stiffness, damping matrices of full model min( ...) , max( ...) : Minimum and maximum value functions for the input parameter of all j=1..N j j

teeth at the segment k mjki : Delay state index of element Sjk at time ti

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mod(input1,input2): Common residue function mt1 , mt2: Number of vibration modes of tool#1 and tool#2 in Section 6.9 n : The maximum delay state index value n*: Modified delay term

N: Total number of teeth/edges/flutes of tool

Na , Na,new : Actual and updated edges-in-cut array at segment-k at specific time instant

Na,k: Actual number of cutting edges that remove material from the workpiece at segment-k njki : Instantaneous number of edges skipping the cut after the element Sjk np11: Number of points along cutting edge section-1 in Figure 3.16 ns : Number of discrete time steps in one spindle period ns: (Appendix A) Number of rotation steps ns: Number of rotation steps to locate the cutting edge nt : Index for tool vibration mode nw : Index for workpiece vibration mode

P : Number of simulated spindle rotations

P, μ : Process damping multiplier vector and Coulomb friction coefficient

Pf: Assumed working plane at the selected location on the cutting edge

Pfe,d, Pfe,m: Working planes of drilling and milling tools

Pfe: Working plane at the selected location on the cutting edge

Pn: Cutting edge normal plane at the selected location on the cutting edge

Pr: Tool reference plane at the selected location on the cutting edge

Ps: Cutting edge plane at the selected location on the cutting edge

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Q(s), F(s) : Displacement and force vectors in Laplace domain

Q(t), F(t) : Displacement and force vectors in time domain q: Number of cutting edge segments along the tool-workpiece contact zone

Qf(t), Ff(t) : Displacement and force vectors for full model in time domain qki ,(qktT i− jki ): Relative physical displacement vector and delayed vibration vector q ki : Relative physical velocity vector

1 R2 : Rotation matrix from Frame-2 to Frame-1

121212 rrrxyz,,: Cartesian components of position vector of origin of Frame-2 relative to Frame-1

12 rtest : Frame-1 representations of test vector when procedures A1, A2, B1, B2, C1 and C2 ( )( ⋅ )

used

2 rtest : Frame-2 test vector to check accuracy of rotations in Appendix A

s Rjk : Local radius measured along flute direction of serrated end mill

0ir : Local translation vector from origin of Frame-0 to origin of Frame-i iDr : Local translation vector from origin of Frame-i to origin of Frame-D iir : Local translation vector from origin of Frame-i to origin of Frame-i

0D 0D 0D rrrxyz,,: Magnitude components of position vector in Frame-D

0i 0i 0i rrrxyz,,: Magnitude components of position vector in Frame-i

00 RRxy, : Magnitude components of radius vector rj,,jκ jj: Parameters for locating insert on multifunctional cutter (Figure 3.31)

12r : Vector of origin of Frame-2 represented in Frame-1 xxv

R, r : Number of rake face sections in the selected cutting edge section-c and index of selected

rake face section r, s: Refers to selected point and total number of points along periphery of cutter contacting

flexible workpiece

R: Calculated number to decide the quality of curve fitting (maximum value 1)

R0: Nominal radius of the tool

0 D 0 0 r , RD , TD Translation vector, rotation matrix and homogeneous transformation matrix from

Frame-D to Frame-0

0 i 0 0 r , Ri , Ti : Translation vector, rotation matrix and homogeneous transformation matrix from

Frame-i to Frame-0

0 R 0 0 r , RR , TR Translation vector, rotation matrix and homogeneous transformation matrix from

Frame-R to Frame-0

2 r : Vector of origin of Frame-2 represented in Frame-2 c,r c,r,e c,r c,r r , Rc,r,e , Tc,r,e : Translation vector, rotation matrix and homogeneous transformation matrix

from Frame-c,r to Frame-i

Ri, Rf: Initial and machined workpiece radii i c,r i i r , Rc,r , Tc,r : Translation vector, rotation matrix and homogeneous transformation matrix from

Frame-c,r to Frame-i

0 R jk : Radius vector of Sjk element in Frame-0

Rjk: Local radius (scalar) of element Sjk in Frame-0

Rmax,k : Maximum radius at segment-k

xxvi

Rotx ()⋅ , Roty ()⋅ , Rotz ()⋅ : Coordinate rotation functions defined in Appendix A

Rt,jk : Moment arm at element Sjk

RTA: Cutting edge coordinate frame

RTA: Radial-tangential-axial

S(t) : Static force vector with process damping s: Laplace-domain operator sameNotSame: Parameter to check if actual edges-in-cut array updated during chip thickness

calculation algorithm

Sjk: Cutting edge element of edge-j at segment-k

SLE: Surface location error

1 T2 : Homoegeneous transformation matrices procedures A1, A2, B1, B2, C1 and C2 used ( )( ⋅ )

1 T2 : Homoegeneous transformation matrix from Frame-2 to Frame-1

i i Tc,r,e , TD : Homogeneous transformation matrix from Frame-c,r,e (Frame-D) to Frame-i

c Tjki : Cutting torque

TTRI,jk,L, 0R,jki,, L ,, TT RI,jk,R 0R,jki R : Transformation matrices for left- and right- side cutters

TRI,jk : Transformation matrix from RTA to Frame-R

TIU, jk : Transformation matrix from UV to RTA t, ti : Continuous time parameter, and discrete time at instant i

T0M : Transformation matrix from process to tool coordinate system

T0R,jk : Transformation matrix from RTA to Frame-0 t1, t2, r1, r2, r3: Labels for representing new frames in Figure 3.10 and Figure 3.34 xxvii

Tj: Delay period that is the time passed since the cut of previous tooth-(j+1) until the cut of

current tooth-j (s)

Tjk(t) : Instantaneous ime delay of element Sjk tjk, n1, jk , n1c, jk , n2,jk : Local unit vectors of element Sjk in tangent, rake face, corrected rake face

and rake face normal directions

Tjki : Instantaneous delay period of element Sjk

Tmax : Maximum delay period along the tool’s edges during one spindle rotation

Trans ()⋅ : Coordinate translation function defined in Appendix A

Ts : Period (in [s]) of one spindle rotation

U, Uf : Mode shape matrices of reduced-order model and of full model

Ue , Fe(t) , Qe(t): Extended mode shape matrix, force vector and displacement vector covering

both cutting and non-cutting zone u : Mode shape vector contribution at segment-k on n -th (tool) mode k,nt ,t t u : Mode shape vector contribution at segment-k on n -th (workpiece) mode k,nw ,w w

Unc , Fnc(t) : Mode shape matrix and force vector covering only non-cutting zone

Ut, Uw : Mode shape matrices of tool and of workpiece

Ut1 , Ut2: Mode shape matrices of tool#1 and tool#2 in Section 6.9

UV: Rake face friction-normal

UV: Rake face friction-normal coordinate frame uuuu,,,: Components of u x,k,ntttt ,t y,k,n ,t z,k,n ,t θ,k,n ,t k,nt ,t uuuu,,,: Components of u x,k,nwwww ,w y,k,n ,w z,k,n ,w θ,k,n ,w k,nw ,w

xxviii

vf , vf : Translational feed vector and its magnitude vf,d , vf,m : Translational feed vectors for drilling and milling

0 vc,jk : Cutting velocity of element Sjk on tool relative to workpiece

0 vc,jk : Local cutting speed of tool relative to workpiece

0 vr,jk : Resultant cutting vector of element Sjk vc,jk : Cutting speed at element Sjk wh,jki , whr,jki: Half-size weight matrices (r=1..5) for delay approximation in the time domain

cutting force simulation wjki , wr,jki , wr,jki : Sub-components (r=1..5) of weight matrix of the Lagrange interpolation

polynomial

Wjki : Weight function matrix of the Lagrange interpolation polynomial xyz222,, : Coordinates of a point in Frame-2

121xyz,, 212 : Coordinates of a point in Frame-2 relative to origin of Frame-1 x0, y0, z0: Cartesian coordinates in tool reference frame (Frame-0) xF : Parameter for nonlinear cutting force xjki : Relative time of delay state

I xjki : x-component of relative physical vibration vector in radial direction xki, yki, zki, θki: Components of relative physical vibration vector xr , xs : Relative times of neighbouring state

xxix

xR,j(t), xR,j ()tT− j Instantaneous relative tool/workpiece current displacement at current tooth-j,

and delay displacement at previous tooth-(j+1) xyz : Coordinates of process coordinate frame

ya,k : Actual surface distance at axial segment-k yd,k : Desired surface distance at axial segment-k yL4(xjki) : Approximate value of the delayed state

0 Zk, zk : Local position of the midpoint of the k-th segment

0 zt1, zt2, zw : z -displacement of tool#1, tool#2 and workpiece

β serr : Helix angle of serrated end mill

βj: Helix angle of tooth-j

γγκfpr,,: Euler angles to locate cutting edge on tool

γ jki : Instantaneous decrease in chip load of element Sjk

Γ(s) : Modal displacement vector in Laplace domain

Γ(t) , ΓΓ(),tt  (): Modal displacement, velocity and acceleration vectors in time domain

Γc(t-Tjki) : Delay combined modal displacement

    Γc,i ,ΓΓc,i , c,i (or ΓΓΓ(ttt i ), ( i ), ( i )) : Combined modal displacement, velocity and acceleration

vectors at i-th time instant (ti)

γn ,λs ,, ηαn : (Chapter 2) Normal rake, cutting edge inclination, chip flow and clearance angles

  Γt(t) , ΓΓtt(),tt (): Tool’s modal displacement, velocity and acceleration vectors in time domain

  Γw(t) , ΓΓww(),tt (): Workpiece’s modal displacement, velocity and acceleration vectors in time

domain xxx

Δ : Delta sign to account for chip regeration on the component

Δqjki : Instantaneous dynamic chip load of element Sjk

Δt : Sampling time of time domain simulation

4 ε j : Axial runout of tooth-j relative to fourth insert

meas er,jk : Measured radial runout

meas er,k : Measured radial runout vector at segment-k

εr,k : Normalized radial runout vector at segment-k

εr,j: Radial runout at tooth-j

εr,jk , εa,jk : Radial and axial runouts of element Sjk

ζ, ωn : Modal damping ratio and natural frequency matrices

ζc, ωn,c : Combined modal damping ratio and natural frequency matrices

ζ , ω : Modal matrices of n -th tool mode ntt ,t n,n ,t t

ζ , ω : Modal matrices of n -th workpiece mode nww ,w n,n ,w w

ζt, ωn,t : Tool’s modal damping ratio and natural frequency matrices

ζw, ωn,w : Workpiece’s modal damping ratio and natural frequency matrices

ηjk , βn,jki : Chip flow angle and projected friction angle

Θ(t) : Modal state vector of the tool-workpiece system

Θ(t - T (t )) , Θ : Delayed modal state vector i jk i i,Tjki

Θfh : Full time history vector of modal states

Θi : Modal state vector of the tool-workpiece system at i-th time instant

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serr,s serr kr,jk ,kr,jk : Cutting edge angles of serrated end mill, measured along flute and mapped along

tool axis

* kr,jk,,λγ s,jk n,jk : Effective tool angles (true cutting edge angle, cutting edge inclination and

normal rake angle) of element Sjk

κrj , κr,jk : Approach angles for tooth-j and for element Sjk

μ: (Chapter 2) Overlap factor

σ p(tT i− jki ) : Elements of approximation of delayed modal state vector (p=1..2m)

σ p (t) : Elements of modal state vector (p=1..2m)

τ : Integral time operator

τs,jki ,,φβn,jki a,jki : Shear stress, shear angle and friction angle of cutting mechanics model

ϒi : Merged modal state vectors within one spindle period

Φ(s) : Transfer function matrix in Laplace domain

ϕj(t): Instantaneous angular position of tooth-j measured from normal axis (rad)

ϕjki : Instantaneous angular position of element Sjk in [rad]

ϕjki,L ,,,ψ jk,Lφjki,R ψ jk,R : Angles of left- and right- side cutters

Φp : Period-transition matrix of the disrete system

(Φp )multiple : Period-transition matrix when there are more than one free and forced vibration regions

region

ϕp,j: Pitch angle of tooth-j (rad)

ϕp,jk , Tjki : Time invariant pitch angle and delay period for zero-runout case

xxxii

ϕp,jki : Instantaneous pitch angle of element Sjk

ϕp,ki : Array of instantaneous pitch angle at segment-k

ϕpd,j: Designed pitch angle of tooth-j

ϕst,jk , ϕex,jk : Local entry and exit angles of element Sjk

ϕst,r , ϕex,r : Entry and exit angles of r-th radial zone

 Ψ(ti) , Ψ()ti : Relative displacement and velocity vectors

Ψi , Ψd,jki : Overall displacement and delay vectors at segment-k

rel ψjk ,ψ jk : Angular and relative angular locations of Sjk element

Ψki , Ψk(ti-Tjki) : Current and delayed relative displacement vectors at segment-k

rel ψr,k ,ψk : Arrays of angular location and relative angular location at segment-k

ω0 : Rotation vector of tool relative to workpiece

ω(t) : Time varying angular spindle speed

Ω: Angular velocity of tool relative to workpiece in [rev/min]

ω: Constant angular spindle speed (rad/s)

Ωw: Angular velocity of workpiece

11 θe : Slope at point-e in rake face section-1 in Figure 3.27b

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Acknowledgements

I am sincerely grateful to Dr.Yusuf Altintas, my research supervisor, for his patient guidance and motivational support of this research work. In Manufacturing Automation Laboratory

(MAL), he has created a comfortable, family-like environment which is very important while studying far away from homes and families. I am happy that he encouraged me to do internship, which helped me to see the practical aspects of my research work.

I am thankful to Dr.Derek Luo, director of ITRI-IMTTC in Taiwan, for hosting me during my internship on the double-sided milling project and for his encouragement. I am also grateful to Faby Feng (Apply Zeta Technology, Taiwan) for his continuous technical support on the experiments that are conducted in Taiwan and for his contributions on this research. I would also like to thank Eason Lai and Yiming Chen, engineers of ITRI-IMTTC, for helping in design of double-sided milling experiment setup and for their tireless help with the experiments. I am also thankful to Dr.Darjen Peng, Kenzi Liao, and to all brilliant engineers of ITRI-IMTTC for their support and discussions.

I am very thankful to Dr.Min Wan, Dr.Keivan Ahmadi, Caner Eksioglu and to all previous and current members of MAL for all great discussions and shared knowledge. I will miss the times that I spent with the friendly, world-class researchers of MAL.

I would like to express my thanks to the industrial supporters of my laboratory; MTTRF

Foundation for donating the Mori Seiki NMV5000DCG machine tool; Sandvik and Kennametal for generously donating the cutting tools. I am also grateful to the president, Kevin Kao, and staff of KOVA company for supplying tools for my experiments in Taiwan and for helping me understand the geometry of different tools.

I am thankful to the NSERC-CANRIMT for the support during my internship. xxxiv

Thank you, Dr.Michelle Tan, for thoroughly checking the grammar of my thesis. I would also like to thank Ozge Goktepe and Dr.Orkun Ozsahin for their inputs on my writing.

I am indebted to my mother, Dr.Nedret Kilic, my father, Dr.S.Engin Kilic, and my sister,

Dr.Pelin Kilic, for all their guidance and for always supporting me.

I would like to thank Yuchien Hsu for her love and invaluable support during the last two years.

I am truly thankful to all my Vancouver friends for all the nice memories.

I am very happy that it was possible to be actively involved in the board of directors of

ASME-UBC Student Section (2011-2014), in the board of directors of Turkish-Canadian Society

(2014) and in Vancouver's Tango community. These volunteering experiences helped me develop professional skills and let me meet great people.

xxxv

Dedication

This thesis is dedicated to my family.

xxxvi

Chapter 1: Introduction

Machining operations are widely used in industry in producing the final shapes on the parts.

A typical metallic part may need metal cutting operations with geometrically defined cutting

edges (i.e., turning, boring, drilling, milling and tapping), electrochemical metal removal, or abrasive operations (e.g., grinding). The objective of this thesis is to develop generalized mathematical models which can be used to simulate and optimize metal cutting operations ahead of currently practiced, high-cost physical trials.

Turning, boring, drilling, and milling are most commonly used operations in the metal

cutting industry. Depending on the shape of the workpiece, each operation may need tool

geometries which vary widely. For example: a solid end mill may have tapered body with ball

end, and serrated, helical flutes spaced at varying pitch angles. An indexed cutter may have a

unique body shape with distributed inserts both in radial and axial directions. A boring bar may

have one insert at its tip, or may have a boring head with distributed inserts in heavy duty finishing of cylinder bores. A multifunctional tool may drill, bore and chamfer large holes in a

single operation in an automotive application. The current industry mostly relies on the past

experience in designing tools and their use in manufacturing. This is an unproductive and costly

trial and error approach. Researchers have been developing mathematical models of each

operation individually, and allowing the prediction of metal cutting forces, vibrations, torque and

power ahead of costly physical trials. However, the past research mainly focused on fixed

operation type (i.e. turning or milling) with fixed tool geometries. Since the part may need

hundreds of different tools and variety of operations, there is a strong need to develop a unified,

generalized mathematical model that can be adapted to cover all tool geometries and machining

operations. 1

This thesis starts with introducing the modelling of metal cutting mechanics between the cutting edge and chip removed from the workpiece, which is common for all metal cutting operations. The metal cutting process forces at the cutting edge is later transformed to tool- workpiece and machine tool coordinate systems by considering the kinematics of each operation type (i.e. turning, boring, drilling, milling) and tool geometry. The relative vibrations between the tool and workpiece are also considered in the generalized model, which can be used to simulate any machining operation with tools having arbitrary geometry. The proposed model can be used to predict the performance of various tool geometries, machine tool configurations and tool path strategies ahead of costly physical trials.

Henceforth the thesis is organized as follows.

The previous literature in modelling of metal cutting mechanics and machining vibrations is reviewed in Chapter 2. The mechanics of orthogonal cutting theory and oblique transformation to predict cutting forces in three dimensional applications are briefly surveyed, while most relevant literature used in the model development is cited in dedicated chapters in extensive detail.

The generalized geometric model of cutting edges, tools with solid body or cutters with inserts is presented in Chapter 3. A differential cutting edge is oriented to form oblique geometry, i.e., rake, inclination and approach angles. The differential cutting edge elements are distributed along the cutter body by considering its local geometry such as radius, radial and axial runouts, and axial location along the tool axis. Insert orientations on the cutter body is modelled in accordance with ISO standards. Several turning, boring, drilling and milling geometries are presented to illustrate the applicability of proposed general model on a variety of machining applications.

The generalized mechanics of cutting is modelled in Chapter 4. The chip thickness distribution along the cutting edge is modeled by considering both the tool geometry and kinematics of specific machining operation. The regenerative vibrations between the tool and workpiece and the time delay terms determine the chip thickness distribution, and are determined by the tool geometry and process kinematics.

The generalized mechanics model is extended to consider the interaction between the machining operation and the structural dynamics of machine tool/workpiece in Chapter 5. Linear equation of motion at a differential cutting zone is modelled. The external load is modelled as a cutting force, which can have periodic amplitude and affected by the present and past vibration amplitudes. Past vibration amplitudes may bring multiple delays to the equation of motion due to varying pitch angle or runouts. The dynamics of the process is modeled in semi-discrete time domain. The process states, such as force, vibration, surface form error, torque, and power are predicted by solving the semi–discrete time domain model of the machining process which may be defined by a set of time varying, periodic differential equations with a single or multiple time delays.

The proposed, generalized model is experimentally proven in Chapter 6. Turning and boring with single cutting edge; boring with multiple inserts; drilling with inserted or solid body; milling with solid tools and indexable cutters; and complex operations such as multifunctional tools, parallel turning and milling applications are presented. The experimental verification includes the prediction of cutting forces, chatter stability and surface location errors.

The thesis is concluded with a summary of contributions and suggestions for future research directions in Chapter 7 .

Chapter 2: Literature Survey

2.1 Overview

The literature and the state-of-art in modeling machining operations are reviewed in this chapter. Mechanics of cutting, prediction of forces and surface error, and the solution of cutting dynamics and chatter stability are covered. Further relevant literature is presented in main chapters where specific contributions in generalized mathematical modeling of machining operations are presented.

2.2 Mechanics of Metal Cutting

The metal chip is plastically sheared from the workpiece material in shear zone, and the deformed chip slides on the rake face of the tool experiencing both sticking and sliding friction as shown in Figure 2.1 [6]. The shear angle, average shear stress and average friction between the moving chip and stationary tool surface are used to predict the cutting forces, torque and power in orthogonal cutting where the cutting edge is perpendicular to the cutting velocity, (i.e.,

λs = 0 in Figure 2.1a). When the cutting edge is inclined relative to the cutting velocity, the process becomes oblique with cutting forces in radial, tangential and axial (RTA) directions (i.e.,

λs ≠ 0 in Figure 2.1a). While orthogonal cutting process is used to identify the fundamental

process parameters such as shear stress, shear angle and friction angle (shown by βa in Figure

2.1a), most cutting tools used in practice have oblique tool geometry.

Figure 2.1 Schematics of chip removal at differential cutting edge element: (a) Oblique cutting with rake face

forces shown; (b) orthogonal plane view.

2.3 Basic Kinematics of Regenerative Machining Operation

When there are relative vibrations between the workpiece and tool structures, the chip thickness oscillates at a frequency close to one of the natural modes of the structure (Figure 2.2) .

The phase or time delay between the waves on the upper and lower surface of the chip leads to machining instability, referred as “chatter”.

Figure 2.2 Dynamic interaction between tool and workpiece.

Since Taylor [7] explained the chatter of the tool as one of the most difficult problems facing the machinist; there have been numerous research papers focusing on understanding and the solution of vibrations of machine tool and chatter. Ehmann et al. [8] gave an extensive review of

the research done until 1997.

Altintas and Weck [9] discuss the significant research (until 2004) in modelling the stability

of metal cutting. Tlusty and Polacek [10] and Tobias et al. [11,12] modelled the chip

regeneration which then results in chatter. The basic mechanism of chatter vibration is explained

as the present tooth leaving a material on the workpiece, and next tooth removing the remaining

material. Tobias [13] proposes the overlap factor which takes the possibility when material left

by present tooth cannot be completely removed by the next tooth. This overlap factor is

especially important for cutters with high corner radius used at high feed operations [14]. Merritt

[15] proposed a chatter feedback loop model for defining the regenerative chip thickness

behavior.

The regenerative chip load of tooth-j ( ctj()) is as follows:

ctcj()= 0,j () t + xt R () −⋅µ xtT R ( − j ), (2.1)

where the static chip load of tooth-j is ct0,j(). Dynamic chip load ( ctd,j()) depends on the relative

tool/workpiece displacement at current tooth-j ( xtR,j()), and the delay displacement of previous

tooth-(j+1) ( xR,j() tT− j ):

ctxtd,j()= R,j () −⋅µ xtT R,j ( − j ) (2.2)

where µ is the overlap factor [13]. It is assumed in this thesis that present tooth removes all

material left by previous tooth, therefore µ =1. Tj is defined as the delay period that is the time passed since the cut of previous tooth-(j+1) until the cut of current tooth-j.

Although the static chip load is constant for turning, boring and drilling operations, it is time

varying in milling operations. Martellotti [16] approximated the actual trochoid milling tool path

by a series of circular arcs (cycloid); then, static chip thickness is related to feedrate by a simple

equation:

ct0,j()= f t,j ⋅ sinf j () t, (2.3)

where ft,j is the feedrate per tooth-j of the tool and φj()t is the angular position of tooth-j

measured from normal axis. Long et al. [17] and Faassen et al. [18] computed the chip thickness

for trochoid path analytically; and Montgomery and Altintas [19] digitized the part surface to

calculate the exact dynamic chip thickness numerically.

Regular tools have equal tooth spacing whereas the general irregular tools have unequal

tooth spacing. Pitch angle of tooth-j is the angle between the current tooth-j and the previous tooth-(j+1):

φφp,j= j+1()tt − φ j (), (2.4)

and it is directly related to the delay period for the constant angular spindle speed (ω ):

φ = p,j Tj ω . (2.5)

The radial or axial runout of the tool depends on insert/flute size variation, spindle out-of-

roundness, improper tool balancing, dimensional tolerance of insert seats (pockets) [20]; on

designed location of inserts and pockets [5]; on designed variation of the edge geometry [21]. In

milling, for instance, radial runout of previous tooth-(j+1) and radial runout of present tooth-j

affect the static chip load of Equation (2.3) as follows [20]:

ct0,j()=⋅ f t,j sinf j () t +−εε r,j r,j+1 . (2.6)

For small radial runout εr,j and εr,j+1 for present and previous teeth, respectively. For high

runout values the tooth-j would not be in cut if the radial runout of previous tooth is enough to remove the chip load of the present tooth:

εr,j+1>⋅ft t,jsinf j ( ) +ε r,j . (2.7)

A special algorithm for calculating the chip load is given in Refs. [20,22] to compute the

chip load for high runout tools.

2.4 Cutting Forces

The chip load that is found in Equation (2.1) is projected onto the chip thickness direction

based on the kinematics of the operation. Fu et al. [20] presented the geometrical transformations

for the milling kinematics. Similarly, Armarego [23] has formed the kinematics of the oblique

geometry of twist drill.

Cutting force has been an area of extensive research (see Figure 4 of Ref.[8]). Merchant [24]

introduced the widely used thin-zone shear model that considers the tool with sharp edge, i.e.,

the cutting force is assumed only by the shearing action. As noted by Hitomi [25] and Shaw [26],

Masuko (1953) derived an additional force due to indentation and rubbing at the edge of the tool.

Albrecht [27] also reported the same mechanism and called it ploughing force. Ploughing force is practically negligible for the sharp tools [26]. Then, cutting force is composed of two components: Shearing and ploughing,

Ft()= Fc () t + F es () t (2.8)

Koenigsberger and Sabberwal [28] validated the dependence of the cutting force to area of the chip section (chip area) by repeating the cutting tests with 5 different cutting edge angles (or corner angles). Instantaneous shear force is related to the instantaneous chip area ( Atc ()) with the following function [28]:

Ftc()= K c12 ( pp , ,.. p L )⋅ At c () (2.9) where the cutting coefficient or specific cutting pressure (Kc) is a function of different L number of cutting parameters ( p1 to pL ). It is also reported by Koenigsberger and Sabberwal [28] that the cutting coefficient increases with decreasing chip thickness (also called feedrate).

Static ploughing forces are modeled using the chip length and the so called edge force coefficients [29]. In general, ploughing forces account both for the static indentation of the tool into the uncut chip [25] and the rubbing of the flank of the tool on the wavy machined surface.

First component is called the static ploughing forces or edge forces, and second component is called the dynamic ploughing forces or the process damping forces [30]. More literature related to ploughing is presented in Chapter 4.

2.4.1 Mechanistic Identification

Mechanistic identification is a curve fitting of cutting force coefficients to the experimentally measured cutting force data collected with a specific tool. Kline et al. [31] fitted tangential and radial miling force coefficients to a second-order polynomial function of radial

depth of cut, axial depth of cut and feed. Altintas [6] modelled the cutting force coefficients as a

linear function of feed; this model is more convenient in solving chatter stability than the

nonlinear models. Chandrasekharan et al. [32] presented a mechanistic method to identify the

cutting coefficients for drilling by collecting the cutting force data during the entry to cutting. A

detailed survey of mechanistic cutting force coefficient methods are given in [33].

2.4.2 Generalized Cutting Force Prediction

Armarego (Equation (11) in [34]) correlated the cutting force to tool geometry and to cutting

conditions defined by the depth of cut, the feedrate and the speed. Based on the thin-zone shear

theory of Merchant [24], the orthogonal and oblique cutting forces can be represented as

functions of shear stress, shear angle, friction angle (shown by βa in Figure 2.1a) and tool geometry [35]. Armarego [36] explains how the cutting force is related to the shear plane parameters using the oblique thin shear zone analysis.

Armarego and Whitfield [29] introduced the concept of single edge oblique machining operation, and stated that all practical machining operations can be modeled using this model.

They also added the edge force into the force model and correlated it to the chip length.

Armarego and Deshpande [37] set the goal to unify the cutting force model for the practical

machining operations such as turning, drilling and milling; their study gave the idea of

computing the oblique cutting forces based on the rake face along the cutting edge of arbitrary

tool. They also introduced the idea to build up a common database for the shear parameters that

are the shear stress, shear angle and friction angle. Based on Refs. [29,36], Budak et al. [35]

generalized the milling force using the mechanics of oblique cutting and predicted the cutting

forces on the orthogonal cutting plane (RTA plane) along the cutting edge of the milling tool.

Modelling the shear cutting forces this way sets the basic idea of general machining model. This 10

model takes the oblique geometry and cutting conditions as input and predicts the cutting force

coefficients along the cutting edge using an available shear parameter database for the tool- workpiece material combination. Yucesan and Altintas [38] modelled the rake face and flank geometry of ball end mill. They applied the kinematics of ball end mill to represent the forces, then they identified the cutting coefficients from the experiments. Furthermore, Engin and

Altintas [39] modelled the cutting edge geometry of general end mill, and mechanistically identified the cutting coefficients. They also modelled the indexable end mill by rotating the 2D- geometry inserts to locate on tool holders [40]. Altintas and Lee [41] used the theory of Ref. [35]

to estimate the cutting forces and stability of ball end milling operation, without any need of

mechanistic identification tests. Subsequently, they developed a similar model for helical end

mills [42].

2.5 Modelling of Generalized Tool Geometry

Armarego [34] discussed about the generalized or unified model in his extensive review of

different operations. Figure 15 of Ref. [34] shows the goal to unify all the turning, boring,

drilling, milling and tapping operations into one model. There are two important studies in

modelling the general geometry: Chandrasekharan et al. [43] proposed a general model for

computing the drilling force with arbitrary drill geometry. Their model determines the effective

tool geometry (normal rake, cutting edge inclination and cutting edge angles) at each discrete

oblique segment along cutting edge. Yucesan and Altintas [38] modeled the rake face and

clearance face (flank) of ball end mill tool using five vectors at each point along the cutting edge:

Cutting edge tangent, rake face normal, rake face, clearance face normal and clearance face

vectors. Although these two models give a great insight into modeling the cutting operation for

drilling and ball end milling, still there is no mathematical model that combines multiple 11

machining operations. This thesis models the geometry of arbitrary cutter, and determines the cutting forces at each segment using the mechanics of oblique cutting for all metal cutting operations with varying tool geometries.

Lee and Lin [44] also used a similar model to Ref. [38] but only with the rake face geometry. They only used this geometric model to improve the tool path design for the ball end milling operations. Fontaine et al. [45] modeled the ball end mill geometry and predicted cutting force for an arbitrary feed direction; they used thermo-mechanical process model.

Tsai and Liao [46] also proposed a cutting force model for ball end mill geometry considering vertical and horizontal feed directions. Unlike dividing the cutting edge into discrete disk segments, their method needs to compute chip areas along the cutting edge (see also [47]).

This type of cutting force prediction is harder to generalize for other cutting operations.

Recently, Tuysuz et al. [48] proposed a model for ball end milling based on the five-axis milling model of Ozturk and Budak [49], and additionally included the indentation effect at the tool tip. They used disk elements to discretize the cutting edge. Thus, their method of predicting indentation forces is more applicable for general cutting operation.

The first study combining the geometric and force modeling is from Kaymakci, Kilic and

Altintas [1]. They included the standard indexable cutter geometry set by ISO standards [50,51].

Their model is only limited to two-dimensional rake face geometry, and only applied to indexable cutters. Campocasso et al. [52] modelled the geometry of a general turning insert.

They used homogeneous transformation matrices to locate the insert on tool holder. In [53], they further extended their method to generalize mechanics for turning and drilling operations.

Altintas and Kilic [3] developed a generalized dynamics model to include the regenerative

vibrations for all metal cutting operations. Otto et al. [54] introduced a fast unified stability prediction method for the machining operations.

Wan et al. [5,55] modelled the multifunctional and thread milling operations using the similar geometric modeling and force prediction concept. Tunc et al. [56] modelled the five-axis milling operation using an arbitrary geometry tool and predicting the force by thermo- mechanical process model. Recently, Ahmadi and Savilov [57] modeled the arbitrary drilling cutter based on the concept developed by [1,3,43]. This thesis takes their model one step forward to include arbitrary tool geometry that is used for general machining operation.

2.6 Simultaneous Machining Modelling

Multiple tools are used in simultaneous machining of features on a single part. Modelling of dynamic chip thickness for serial machining operation (only one tool in cut at a time) is done by using single pair of relative tool-workpiece vibration. However, simultaneous machining operations involve more than one tool, thus the dynamic chip thickness depends on multiple pairs of relative tool-workpiece vibrations. Lazoglu et al. [58] modelled the dynamics of simultaneous (parallel) turning operation in modal domain, such that they could consider the flexibility at different parts on workpiece; they presented a general model for two turning tools, and included the flexibility of both tools and workpiece; they solved the equation of motion in time-domain. Budak and Ozturk [59] also solved the same system in frequency domain based on the method of Altintas and Budak [60]. They observed an increase in stability for specific depth of cut values; this is later supported by Ding et al. [61]. Budak and Ozturk applied the same methodology to parallel milling operations as well [62]; they solved the equation using conventional time-domain solution. Budak et al. [63] solved the same system in frequency

domain. They used a similar modal model to [58] and [64], to consider the flexible dynamics

along different sections of tool-workpiece contact zones.

A specific application of parallel milling double-sided milling operation, where two face milling tools simultaneously cut from each side of a flexible part. Shamoto et al. [65] designed a system of two spindles and set a difference between rotation speeds of dual spindles and eliminated the forces and vibrations along flexible direction of the workpiece. They used the same idea as in Ref. [66], where the authors designed tool pitch angles to eliminate the unsable vibrations. Shamoto et al. extended the stability model to general multitool systems in [67].

Although the flexibility of the part changes along the tool-workpiece contact, they averaged the dynamics along the contact. Olgac and Sipahi [68] developed an algorithm to solve the

equivalent algebraic form of the characteristic equation of simulatenous machining operation.

Brecher et al. [69] simulated the dual-spindle milling system in tim-domain; they investigated the

effects of relative angular offset and the speed-difference, and validated by experiments.

2.7 Dynamics of Machining Operations

Machining operation involves the interaction of tool-workpiece structural dynamics with the chip generation process (Figure 2.2). Vibrations of the structure creates a regeneration cycle of chip thickness. Chatter vibration amplitudes are against the presumed small vibration assumption.

Solution by Tlusty and Ismail [70] considers the case when the tool instantaneously jumps out-

of-cut which happens when the vibration amplitude exceeds the instantaneous chip load. Non- linear solution methods can be replaced by linear solutions if only the stable or marginally stable solutions are sought. Minis and Yanushevsky [71] proposed a frequency domain method using

Fourier analysis and Floquet’s theory; they solved the equation numerically. Altintas and Budak

[60] used a similar method, and introduced an analytical frequency domain solution to predict 14

the stability lobes of machining for the cylindrical milling cutters. They represented the cutting

force with its average value and assumed single chatter vibration frequency. In addition to being

very fast, this method is accurate for milling operations with regular pitch angles and high radial

depth of cut to diameter ratio. For low radial depth of cut to diameter ratio, an improved method

is presented by Budak and Altintas [72] and by Merdol and Altintas [73]. Altintas et al. [66]

presented a method for stability analysis with unequal pitch tools. Li et al. [74] proposed a fast,

analytical solution to obtain exact solution for stability of boring bars. Similar concept is used by

Eynian and Altintas [75] to model milling tools with rotating dynamics.

Zero order analytical frequency domain solution of Budak and Altintas gives a fast analysis of stability. Due to the time-varying nature of dynamic chip thickness, higher order methods increase the accuracy of predictions [76]. Bachrathy and Stepan [77] introduced an efficient method for higher order frequency domain solution. Insperger and Stepan [78,79] developed a semi discrete-time domain solution (semi-discretization method) which can approximate the delay-differential equations of milling by a series of ordinary-differential equations [80]; only delayed states and time-periodic coefficients of the milling equation are approximated. The resulting simulation accuracy is higher than the conventional time-domain simulation of delay- differential equations. Ding et al. [81] isolated the major component of the system dynamics matrix, and approximated the rest of the terms in the integral part of the approximate difference equation; they obtained a faster solution. Sims et al. [82] discuss the accuracy of semi-

discretization method for variable pitch milling tools. Asl and Ulsoy [83] solved linear delay-

differential equations using the matrix Lambert function. Yi and Ulsoy [84] further extended the

method to general system of delay-differential equations, and they extended the method to

simulate stability of machining operation [85] by considering multiple vibration modes. Butcher 15

et al. solved the milling dynamics equation using the Chebyshev polynomials and collocation

method [86,87]. Ding et al. [88] also used Chebyshev points while interpolating the delay term

using barycentric Lagrange interpolation polynomials.

2.8 Surface Form Error

Stability and surface form error (called surface location error, or surface error in the

literature) are usually considered during the design of machining operations. Surface error can be

defined as the difference between commanded and actual locations of the machined surface; it

results from the relative vibrations of the flexible tool-flexible workpiece structure under the influence of dynamic cutting forces. For end milling operation, Kline et al. predicted the surface errors using static deflection model for the tool and workpiece [89]; they also used the cutting force model of Kline and DeVor [22] that considers radial runout of the tool. Ko et al. [90]

included the runout effect in the cutting force model, and used the experimental cutting force

data to predict the runout parameters. Wan et al. [91] also showed a model on how to obtain the runout parameters form the cutting tests. In another study, Wan et al. compared different runout models [92], and found out that radial runout model of Kline and DeVor [22] is accurate despite its simplicity. Sutherland and DeVor presents an improved method for predicting the surface error considering the tool runout [93]. Smith and Tlusty [94] discussed the evolution of the

cutting force models and their effect on surface error prediction. Li and Shin [95] used a modal

model with complex mode shapes obtained at 5 measurement points along the flexible end mill-

flexible workpiece, and solved the equation of motion in time domain using Duhamel

integration. They successfully predicted the surface errors under the cutting forces. Schmitz and

Mann [96] predicted surface location error in frequency domain. They applied Fourier analysis to

the cutting force and obtained the surface error while also including helix angle effect of the end 16

mill. Bachrathy et al. [97] also used Fourier transformation to analyze the surface error for

helical end mills. Pirtini and Lazoglu [98] constructed a static drilling force model to simulate

machined hole profile by considering forced vibrations. Bayly et al. [99] developed a time-

domain model to predict the lobed holes; they solved the equations in discrete-time domain.

Roukema and Altintas [100] introduced an extensive time-domain simulation model for drilling; their model is also capable of predicting machined hole shape. In their paper, they give a detailed literature review on modelling of drilling operations.

Time-domain analyses are used for simulating the dynamic forces, relative vibrations of tool-workpiece system, and surface errors [19,101,102]. Bachrathy et al. [103] introduced a nonlinear solution method for milling operations with high amplitude; they also argued about two-delay contact at the tool rake face. Insperger et al. [104] predicted surface error with

conventional time-domain simulation, while Mann et al. [105] developed a fast algorithm to

predict the surface error in time-domain.They used the steady state component of the solution

method proposed by Bayly et al. [106]. Ding et al. [107] used the steady state part of the state

transitions. Eksioglu, Kilic and Altintas [2] used semi-discretization, and simulated general end

milling operation by simultaneously predicting the forces, vibrations and surface errors.

2.9 Summary

Almost all of the presented relevant literature in machining modelling is on milling operations. The method presented in this thesis utilizes the extensive literature in milling, and

combine it with turning, boring and drilling studies. The generalized model presented in the

remaining chapters of this thesis is capable of not only covering all individual modelling results given in the literature, but without needing to reconstruct the model for each specific machining

process. The model developed by the author is general, and can be parametrically configured to solve all machining operations with varying tool geometry and configurations.

Chapter 3: Generalized Geometric Model of Cutting Tools

This chapter generalizes the geometric model of indexable and solid tools that are used in

machining operations. First, the geometric parameters are identified. 15 parameters are chosen to

define geometry of an arbitrary tool. Then, cutting edge tangent, rake face tangent and rake face

normal vectors are defined along the cutting edge of the cutter. Finally, effective tool geometry is

derived for general operation. The method is demonstrated on example indexable and solid tools.

It is noted that there are various terminologies used for tools in the literature and industry. In general, the term “tool” is used to refer to a body that holds cutting edge(s). Correspondingly, single point tools have only one cutting edge or tooth, and multi-point tools have multiple cutting edges. While tools with multiple cutting edges ground on one body are called solid tools (e.g., twist drills, solid end mills with fluted cutting edges), tools with added “inserts” are called indexable cutters (e.g., indexable milling cutters and boring heads).

3.1 Generalized Geometric Parameters of Arbitrary Tools

The proposed mechanics and dynamics model developed in the thesis is based on the generalized, parametric definition of cutter geometries. While single point turning and boring operations have one tooth, there may be multiple layers of teeth or varying flute geometry on multi-point tools such as drills, indexable cutters, fluted end mills and boring heads.

Discretization of cutting edges: As shown in Figure 3.1, Figure 3.2, Figure 3.3 and Figure

3.4, the cutting edge of the cutter is discretized into K number of segments with a constant differential thickness dz for turning, drilling, roughing end mill and general milling tools, respectively. Geometry of a milling tool is discretized in its axial direction; turning tool geometry is discretized in radial direction of workpiece; geometry of boring and drilling tools are 19

discretized in their radial direction. Thus, for milling tools, these segments look like discs with

axial thickness, whereas turning, boring and drilling cutters are discretized by segment-like rings

with radial thickness. Each discrete cutting edge element is coded as Sjk for edge-j at k-th

axial/radial segment of a tool with j=1..N number of teeth and k=1..K number of segments.

The axial (for milling) or radial (for turning/boring/drilling) location of each differential

cutting edge segment-k is measured from the zero depth of cut point, i.e., measured from the

bottom side of the turning (Figure 3.1a) and milling (Figure 3.3a and Figure 3.4a) tools, and

measured from the outer diameter of the bored or drilled hole (Figure 3.5c). Each differential

cutting edge segment has a constant thickness of dz with constant geometry.

The total number of cutting edges on the tool is N, while the actual number of edges that

remove material from the workpiece at axial or radial segment-k is Na,k. As seen in Figure 3.2a,c,

element S21 of the insert-2 at radial segment-1 of the drilling cutter do not exist. Although the

total number of inserts is 2, the actual number of cutting edges at radial segment-1 is 1. Edge

index , j (j=1..N) is used for numbering the edges (also called teeth, flutes or inserts) of the

cutter. Edge-j increases from the bottom to upwards in axial direction for milling tools; and from

outside towards the cutter center for drilling tools. At the same axial or radial segment-k, for right hand cutter with clockwise (CW) rotation when looked from drive side, the edge number-j is increased in CW direction (see, e.g., Figure 3.3b or Figure 3.4c). For left hand cutter, j increases in counterclockwise (CCW) direction. Thus, if the current edge in cut is j, then the previous edge in cut is j+1 and the next edge in cut is j-1.

Figure 3.1 Left hand turning side-view schematics: (a) Edge is discretized in radial direction from workpiece

axis; (b) general side view during cutting; (c) Close-up of element S1k .

The geometric parameters of turning tools are stored as a function of radial location starting from the machined workpiece radius (Rf in Figure 3.1c). The boring and drilling tool parameters are derived as a function of radial location starting from the outside circle of the hole, and the cutting edge is discretized at dz intervals along the radius of the tool (Figure 3.2c). The discretization is carried out along the tool axis (or rotation axis), i.e., along the axial direction of the tool in milling (Figure 3.3a, Figure 3.4a). Each cutting edge element Sjk at each discrete segment-k is defined by 15 geometric parameters:

• Element available-to-cut condition,

• Location in Cartesian coordinates (x0, y0 and z0),

• Axial depth and axial runout,

• Radius, radial depth and radial runout,

• Location angle, relative location angle and designed pitch angle,

• True cutting edge angle, cutting edge inclination angle and normal rake angle,

These 15 parameters are computed for each tool-machining operation (i.e., turning, drilling, boring and milling) for the generalized mathematical model developed in the thesis.

Figure 3.2 Indexable drill schematics: (a) Side view; (b) bottom view with insert numbers; (c) bottom with

the radial discretization.

Figure 3.3 Geometry of serrated end mill that is used for rough milling operations: (a) Axial segment

numbering; (b) radial locations (including runout) at k-th axial segment.

Contact conditions between the work material-cutting edges: Depending on the engagement geometry of the part (workpiece) with the tool, there are three parameters defined for the contact condition: 1- Element-available-to-cut is related to the physical availability of the element Sjk; 2- edge-in-cut condition depends on the engagement boundaries; 3- missed-cut

condition depends on the tool runout, i.e., on the chip load variation between the edges at same

segment-k.

Figure 3.4 General end mill geometry schematics. (a) Side view with the axial segments; (b) side view with cutting forces and chip geometry; (c) top view with edge numbers and angular parameters; (d) top view with radial runouts. (Adapted from Eksioglu et al. [2].)

The availability to cut the work material: It is assumed that all edges (j=1..N) of the tool

would engage the workpiece.When the cutter is discretized into K number of segments, there

may not be any geometry parameter assigned for some element Sjk (see Figure 3.2c). Segment-k 24

of tooth j is flagged by time invariant binary parameter g1,jk, i.e., if the contact exists g1,jk=1; and if it does not g1,jk=0. Actual number of cutting edges (Na,k) is found by summing up the element-

available-to-cut array at each segment-k:

N Nga,k= ∑ 1,jk . (3.1) j

Coordinates of cutting edge on the tool: Cartesian coordinate locations (x0y0z0) are

0 assigned for each element Sjk in tool reference frame. z is set as tool axis for milling (Figure

3.3a) and drilling/boring (Figure 3.2a) operations, and it is set as workpiece rotation axis for

turning (Figure 3.1a) operation. x0 and y0 are located radially using the radius and angular location of the element Sjk.

The axial/radial position (Zk) of the midpoint of the edge segment k is:

(k − 1) Zz= d . (3.2) k 2

The number of differential cutting edge segments that are in contact with the work material

is q; and it is smaller than the total number of segments: qK≤ . The number of the contacting

cutting edge segments (q) depends on the type of operation. For milling, segments are aligned in

z0 (tool axis) direction. For turning and boring, the tool is radially immersed into the workpiece,

thus the cutting edge segments are aligned with radial direction of the workpiece. For turning

(Figure 3.1c), the depth of cut is the difference of the initial and final external radii of the

= − workpiece, ap( RR if)external . For boring, the depth of cut is the difference of the initial and

= - final radii of the bored hole in the workpiece, ap( RR fi)bored-hole . Drilling operation (Figure

3.5a) is similar to the boring process, and the cutting edge segments are aligned with radial

direction of the tool. The depth of cut is the difference of the initial and final radii of the drilled

= - hole, ap( RR fi)drillled-hole . The number of cutting edge segments in the tool-workpiece contact zone is:

ap q = (3.3) dz

The cutting edge segment k on the cutting edge j is referred as element Sjk with a unique geometry, and there are Nq× amount of cutting edge elements on the tool.

Angular location of cutting edge: Angular location (ψ jk ) of the element Sjk is independent

of the time and the rotation of the cutter or workpiece. As shown in drilling cutter of Figure 3.5b,

0 0 it is defined as the angle measured from +y axis to the radius vector ( R jk ), which is defined on

0 0 0 x y plane. Positive direction of the ψ jk angle is CW around the rotation axis (z ) of tool or

rel workpiece axis, depending on which one is rotating. Relative angular location ψ jk is measured

0 from radius vector ( R11 ) of the S11 element or the first segment (k=1) on the first edge (j=1),

rel which leads to ψ11 = 0 . The relative angular location of element Sjk, defined relative to element

rel S11, becomes ψjk= ψψ jk − 11 .

Local radius of cutting edge: The local radius Rjk of the cutting edge element Sjk is defined

as the distance from the edge-segment to the tool center in milling; from the edge-segment to the

workpiece center in single point turning (Figure 3.1c) and boring tools; as the distance from the

edge-segment to the hole center for drill (Figure 3.5a) and indexable boring heads. The local

radius of the tool does not change during the operation. 26

Figure 3.5 Parameters for the drilling operation at initial time instant (t=0). (a) Side view with the radius

0 vector ( R1k ) of segment-k on insert-1; (b) top view with the angular and relative angular locations.

Local entry and exit angles of cutting edge: Local entry angle (φst,jk ) is the angle of the tool at which the element Sjk engages with the workpiece and starts to cut. Similarly, the local

exit angle ( φex,jk ) is the angle of the tool at which the element Sjk leaves the cut. For

conventional turning, boring and drilling operations of the tool is always engaged with the workpiece, thus φst,jk = 0 and φex,jk = 2π . For general milling operation (Figure 3.4c), these

angles depend on actual radius (including the tool’s radial runout) and the translational feed per

revolution of the cutter [22]. Although not included in this thesis, the tool may experience

multiple local entry and exit angles during one rotation if the part geometry changes.

Feed per revolution of the tool: Translational feed per revolution (fr) of the tool is used in

determining the instantaneous chip load (i.e. thickness) at time t. The tool runout and feedrate

determine the chip load and the delay period between the subsequent tooth at each time instant.

Also, the entry and exit angles of each element Sjk are found using the tool runout and the feedrate. As a special case, if there is no runout (e.g., if all εr,jk = 0 in Figure 3.4d) and if all

elements are available-to-cut (i.e., all g1,jk=1), feed per tooth (ft) is calculated as ft= fr/N.

Instantaneous edge-in-cut condition parameter: The cutting edge elements become in-

and out-of-cut especially during interrupted cutting operation like milling. In cut conditions are

determined by the time-dependent binary parameter g2,jki of the cutting edge element Sjk, as a function of local entry and exit angles as :

1,φst,jk≤≤ φφ jki ex,jk g2,jki =  (3.4) 0, otherwise

For the uninterrupted turning, boring and drilling operations, the g2,jki is always 1.

Instantaneous angular position of the cutting edge segment: The instantaneous angular

position ( φjki ) of the element Sjk is evaluated as a function of tool-workpiece contact and

spindle’s angular position (see Figure 3.4c), as:

φψωjki= jk +⋅t i (milling, boring and drilling) , (3.5)

where ω (rad/s) and Ω (rev/min) is the spindle speed; ti (s) is the time at discrete time interval counter i. Instantaneous angular position is generally different than angular location which is

independent of tool rotation. However, if the cutter does not rotate (as in turning and boring), the

instantaneous position and angular location become equal for all time instants,

φψjki= jk (turning, boring) . (3.6)

Tool runout: The radial runout (see Figure 3.3b) εr,jk of the element Sjk is modeled as part

of the local radius of tool, and defined as εr,jk=RR jk − min ( jk ) . For milling, other than 3- and j

more axis cutting, the axial runout is only useful for deciding which segments of the edge are in-

cut for the current depth of cut. For turning with multiple tools (Figure 4.4b), boring and drilling

(Figure 4.3), the feed direction is aligned with the tool axis, hence the the axial runout εa,jk of

the element Sjk becomes important. The local axial location Ajk of the element Sjk is used for

deciding the axial runout at each segment, εa,jk=max ( AA jk) − jk . j

Instantaneous missed-cut condition parameter: The radial runout of the element Sjk

compared to the previous edge Sj+1,k may result in zero chip generation (missed-cut condition

[108]), since the previous edge might have already removed a higher amount of chip (negative chip thickness means no cutting). The zero chip occurrence at segment-k may be caused by several successive cutting edges n after the edge S , i.e., no cutting occurred for the past jki j+njki ,k

“njki-1” edge-passes [22]. The limits of njki is 1≤≤nNjki . A binary time dependent parameter

g3,jki is introduced to account for the runout effect on the chip removal. If chip generation occurs

at the element Sjk at the i-th time instant, g3,jki is set as 1, otherwise 0.

Instantaneous pitch angle: The pitch angles are important to model the cutting edge

geometry for the cutters with multiple cutting edges. Helical or indexable cutters may have axial

cutting edge segments that may be positioned at unequal radius around the body of the tool, due

to the design or presence of radial runout. Due to the edge-in-cut condition, the pitch angle may be quite different at each cutting edge segment along the axis of the cutter. The general term for the pitch angle is defined as φφ= − φ [22]. The maximum possible value of the pitch p,jki j+njki ,ki jki

angle is φp,jki = 2π and the minimum possible value is the no runout condition,

φφp,jki= j+1,ki − φ jki (assuming that the previous edge j+1 is in-cut, i.e. g1,j+1,k=1). Parameter njki is

iteratively computed by considering the runout and the feed per revolution of the cutter. Details

are given in Chapter 4.

Instantaneous delay period: The delay period Tjki of the cutting edge element Sjk at each time instant is derived from the rotational speed of the tool or workpiece and the instantaneous

pitch angle as:

φ = p,jki Tjki ω (3.7)

which may vary periodically in addition to varying along the cutting edge.

0 Cutting speed: Relative cutting speed ( vc,jk ) at element Sjk of the tool relative to workpiece

is defined using the rotational speed of the tool or workpiece and the local radius:

0 vRc,jk [m/min] = 2π ⋅jk [m] ⋅Ω [rev/min]. (3.8)

Geometric definition of the cutting edge: The cutting edges are usually not straight for the roughing tools, hence they can be represented by local, unit tangent vector t jk at the midpoint of the element Sjk. The cutting edge tangent vector t jk lies on the rake face ( Aγ in Figure 3.6b) which can be defined by a vector (not parallel to the cutting edge) n1, jk on the rake face, or by the rake face normal vector n2,jk . In case of vector n1, jk , its direction should be corrected to n1c, jk and made perpendicular to the cutting edge as:

(tnjk×× 1,jk) t jk n1c, jk = (3.9) (tnjk×× 1,jk) t jk

Crossing the cutting edge and corrected rake face plane vectors gives the rake face normal vector which is directed into the rake face (similar to the models in Refs. [38,44]):

nnt2,jk= 1c,jk × jk . (3.10)

As shown in Figure 3.6 rake face vectors are the axes of the cutting edge frame (Frame-D), which is described in Figure 3.8.

Figure 3.6 Rake face vectors at element Sjkn arbitrary tool; (b) close-up view of local cutting edge frame. 31

A note on the modelled milling operations: In pocket milling operation (2+1-axis cutting)

[109], the feed direction is always perpendicular to the tool axis. Thus, the tool moves only in x

or y directions of the machine tool. In this thesis, additional movement in z or rotational axes

(A,B,C) are not taken into account. However, the same developed methods can be applied to 3-,

4- and 5-axis tools moving in arbitrary feed directions [48,49]. In this thesis, milling cutter discretization is conventionally done in tool axis (z0) direction. The geometry of each of the discrete cutting edge is also referred as local cutting-edge geometry or simply, local geometry.

The local geometry of the milling cutter can be stored as a function of the axial height. For the standard milling cutters, for example, Engin and Altintas derived analytical functions [39]. For general/arbitrary cutter it is assumed that the local geometry is available only numerically

[21,56,110]. At each axial height for the k-th (k=1..K) segment of the j-th edge/teeth/insert/flute

(j=1..N) of the milling cutter the geometrical parameters are stored.

3.2 Generalized Modeling of Cutting Edge Coordinates on the Tool Body

The generalized geometric model of tools is built first by defining rake face of individual cutting edge or insert geometry in local insert frame. Then, origin of local insert frame is translated to the position on the tool holder. Finally, new frame is rotated to obtain the designed orientation of the cutting edge. The method is applicable to the kinematics of turning, boring, drilling and milling operations.

3.2.1 Indexable Cutter

There are two steps for locating the cutting edge of the indexable cutter: 1. Transformation from the insert reference coordinate frame (Frame-i) to the cutting edge coordinate frame; 2.

Insert is translated and rotated to match the desired orientation on the cutter. The first

transformation locates the cutting edge relative to the local insert frame, whereas the latter

transformation locates the insert on the holder.

Locating the cutting edge on the insert: The conventional inserts have relatively simple geometry and can be approximated as 2D flat face with constant rake face plane geometry [1].

The modern insert designs have non-straight cutting edges, to reduce the vibrations and forces,

and to improve the surface quality. Thus, the edge geometry of modern inserts are not on the same plane on the entire edge [111]. For complicated or unknown geometry, if the 3D model is

not available, a contact-type instrument (e.g., a Coordinate Measurement Machine/CMM) or tool

microscope may be used to measure the edge geometry of the insert.

With the local cutting edge tangent vector and the local rake face ( Aγ ) known at the

selected point along the cutting edge, the transformations are in two steps as follows:

Step-1: The rake face may or may not be constant along the cutting edge, even when the

cutting edge is on the same plane (see example insert geometry in Figure 3.15). As shown in

Figure 3.7a, the 3D cutting edge of the insert is divided into C number of cutting edge section planes, and the selected section-c (c=1..C) is divided into R number of flat-rake face sections.

Thus, the number of the rake face sections is equal or larger than the number of cutting edge sections on the insert. Each selected point in the rake face section-c,r (r=1..R) lies on the same local flat rake face Aγ (Figure 3.7b). The orientation of the local rake face frame (Frame-c,r) of

the selected point in the insert reference frame (Frame-i) is determined by the following

homogeneous transformation:

Figure 3.7 Locating the local rake face frame in the insert reference frame. (a) Dividing the cutting edge into

flat rake face sections. (b) Close-up view of the selected rake face section on the cutting edge

i i c,r i Rrc,r Tc,r =  (3.11) 0 1 1x3 4x4

This is done by translation vector (irc,r shown in Figure 3.7b) and rotation transformation (

i Rc,r ). Rotation transformation is found directly by the unit axis vectors (shown in Figure 3.7b)

i i of the local rake face frame. Tangent vector ( ic,r ), rake face normal vector ( jc,r ) and rake face

i vector ( kc,r ) are expressed relative to Frame-i. Then, the rotation matrix of Equation (3.11) is:

Rii=  i i jk i c,r c,r c,r c,r . (3.12)

Thus, Equation (3.11) is rewritten by inserting Equation (3.12):

i i i i c,r i ic,r jkr c,r c,r Tc,r =  (3.13) 00 0 1

Figure 3.8 Locating the local cutting edge frame: (a) Highligted rake face section of Figure 3.7a with the

selected point circled; (b) close-up view of the rake face section with selected point-e.

Step-2: Step-1 transformation from the insert reference frame to the local rake face frame is

shown in Figure 3.7b. Step-2 is to transform to local cutting edge design frame (Frame-D). The

transformation from Frame-c,r to the selected point-e in section-c,r is shown in Figure 3.8.

General form of the transformation matrix is:

c,r c,r c,r,e c,r Rrc,r,e Tc,r,e = . (3.14) 0 1 1x3 4x4

The transformation is only translational if the cutting edge is a straight line in the rake face

section (e.g., see Figure 3.11); it is both translational and rotational if the cutting edge has at least

two lines or at least one arc (e.g., see Figure 3.27b). Rotation transformation within Frame-c,r is

i i on local rake face ( Aγ ), which is defined by local tangent ( ic,r ) and rake face ( kc,r ) vectors

(Figure 3.7b). Therefore, rotation of the Step-2 transformation is only around the rake face

i normal vector ( jc,r ). 35

i Combining the two transformations of Equations (3.11) and (3.14) gives Tc,r,e , which orients the local cutting edge frame (Frame-c,r,e or generally, Frame-D) at the selected point-e to the insert frame (Frame-i):

i i c,r Tc,r,e= TT c,r ⋅ c,r,e (3.15)

Then, the unit axis vectors at the rake face of the selected point are found by expressing

Equation (3.15) similar to the form of Equation (3.13):

i i i iD i iDD jkr D TD =  (3.16) 00 0 1

i i i i where the local tangent ( iD or ic,r,e ), rake face normal vector ( jD or jc,r,e ) and rake face i i ( kD or kc,r,e ) vectors are expressed in insert reference coordinates (Frame-i).

Locating the inserts on the tool holder: The cutting edge frame orientation is obtained by transforming from the insert frame (Frame-i) to tool coordinate frame (Frame-0) by using homogeneous transformation matrices as follows:

• The origin of the fixed tool reference frame (Frame-0) is selected. Then, the desired

location for insert j is set. As shown in Figure 3.9a, for simplicity, the desired orientation

of the insert is taken same as the pocket (seat) frame (Frame-s). Initially, the insert is

aligned with the fixed frame. After the transformations, the insert frame (Frame-i) is

represented with respect to Frame-0 (Figure 3.9b).

• The origin of the insert frame is translated to the seat location of insert j on the cutter

frame. This is done by the translation vector ( 0ir ) as shown in Figure 3.10a.

• The insert frame is rotated around its axes for the angular orientation of the insert in the

insert seat of the tool body [1]. This is shown in Figure 3.10b.

The details of the transformation matrices are given in Appendix A. Transformations in this section are independent of the insert type, and only depend on the tool holder type and its seat orientation.

Figure 3.9 Step-1 of positioning the insert on the holder: (a) Desired insert orientation is selected; (b) Final orientation of the insert frame (Frame-i) matches the pocket (seat) frame (Frame-s).

Figure 3.10 Insert placement on the holder: (a) by translation in step-2; (b) by rotation in step-3. t1 is to label new frame after first translation. r1 and r2 are labels for the new frames after the first and second rotations.

If the unit axes vectors (Figure 3.9b) of the final orientation are identified from CAD drawing, the transformation matrix (from Frame-i to Frame-0) is directly obtained:

0 0i 0 Rri Ti =  (3.17) 0 1 1x3 4x4

0 0 0 where the rotation matrix is a combination of the ii , ji and ki vectors (Figure 3.9b)

represented in Frame-0:

R00=  i 0 jk 0 i ii i. (3.18)

Figure 3.11 KOVA XNEX080608 insert: (a) General view; (b) cutting edge highlighted; (c) rake face

highlighted; (d) flank highlighted. (Courtesy of KOVA.)

Figure 3.12 Assigning the insert reference frame: (a) Front plane (b) insert reference plane; (c) insert

coordinate frame; (d) front view. (Courtesy of KOVA.)

Case studies for indexable cutter model:

Case-1: Insert used with different tool holders: KOVA product insert (Figure 3.11a)

with XNEX080608 code is used for various cutting operations. Assuming the top side of the

insert is blocked by the tool holder, the bottom-side of the cutting edge of the insert is

highlighted (Figure 3.11b). Depending on the type of the operation, any section of the

highlighted section may be in-cut. The purpose of the geometry model is to locate the rake face

(Figure 3.11c): First on the insert, and then on the cutter body. Thus, the insert reference frame is

decided first. Flank is also shown in Figure 3.11d, and its geometry can be modelled similarly to

rake face [38].

Insert reference coordinate frame: As shown in Figure 3.12c, the geometric center of the

insert is selected as the center of the insert’s reference frame, with axis directions (Figure 3.12d)

consistent with the insert shape: Front plane (Figure 3.12a) is selected parallel to the reference plane of the insert; reference plane (Figure 3.12b) of the insert coincides its mid-plane.

Locating the rake face on the insert: Seven cutting edge sections (C=7 in Figure 3.7a) are

defined on this insert (Figure 3.13a-b). Top-left side edge is labelled 1, and the rest is labelled in

CCW direction (Figure 3.13b). Totally 21 approximate rake face sections are defined on the

seven cutting edge sections, and the rake face geometry is interpolated along the cutting edge

(Figure 3.13c).

Cutting edge sections 1, 3, 5 and 7 (ces-1, ces-3, ces-5 and ces-7) do not have curved rake face; thus, each of them is defined by single rake face section (see Figure 3.14). 39

Cutting edge sections 2, 4 and 6 (labeled ces-2, ces-4 and ces-6 in Figure 3.13b) have curved

rake face geometry, thus local flat rake face sections are used for the approximation. Rake face

approximation of cutting edge section-6 (Figure 3.15a) is shown in Figure 3.15b. Each cutting

edge section with curved rake face geometry is approximated using six rake face section planes.

Rake face approximation of cutting edge section-2 (labeled ces-2 in Figure 3.13b) is shown in

Figure 3.19 with the six corresponding rake face coordinate frames.

Cutting edge section-1 (c=1 in Figure 3.6a) is defined at the left-side of the insert (as in

Figure 3.16). Since the rake face includes the cutting edge at all points, only one rake face section is defined in this section. The coordinate frame of the rake face section-1 of the cutting edge section-1 is named Frame-1,1. The relative position vector ir 1,1 of the origin of “Frame-

1,1” is defined in Frame-i of the insert, and is found from the CAD model as:

T ir 1,1 =−−[ 5.797 2.978 3.089] (mm) . (3.19)

(a) (b) (c) Figure 3.13 Modelling of the rake face of the insert: (a) General front view (b) labelled cutting edge sections

(ces-1 to ces-7); (c) local rake face vectors along the cutting edge.

Figure 3.14 Highlighted two cutting edge sections (ces-5 and ces-7) with flat rake faces

(a) (b)

Figure 3.15 Rake face approximation of curved rake face: (a) ces-6 plane; (b) local flat rake faces.

The relative axis unit vectors are defined in Frame-i:

i T i1,1 =−−−[ 0.257 0.122 0.959] ; i T k1,1 = [0.862 0.419− 0.284] ; (3.20) i i ii T j1,1=×=− k 1,1 ij 1,1; 1,1 [ 0.436 0.900 0.00252] .

ii i Here, the ij1,1, 1,1 and k1,1 correspond to the cutting edge tangent vector, the rake face

normal vector and the rake face vector of the rake face section-1 of the cutting edge section-1,

respectively. The transformation of the Frame-1,1 to Frame-i is carried out by using the located origin of the Frame-1,1 given from Equation (3.13):

i i i i 1,1 i i1,1 jkr 1,1 1,1 T1,1 =  (3.21) 00 0 1

To define point-e on the cutting edge, additional translation Trans(0,0,x1,1,e) is needed for the length of the straight edge:

ii i jkr i i 1,1 ii i j i k i r 1,1,e  1,1 1,1 1,1 ⋅=Trans( 0,0, x1,1,e ) 1,1,e 1,1,e 1,1,e , 00 0 1 0 0 0 1 

i i i i 1,1,e i1,1,e jkr 1,1,e 1,1,e i 1,1,e =T1,1 ⋅Trans( 0,0, x ) . (3.22) 00 0 1

where the index for selected point along the cutting edge is e. If the number of points along

cutting edge section-1 is np11, and the total length of the section-1 is L1, then the point along the

edge is (Figure 3.16):

L xe1,1,e = ⋅ 1 (3.23) np11 −1

The transformation from the “Frame-1,1,e” to the Frame-i becomes,

i i 1,1,e TT1,1,e= 1,1 ⋅Trans( 0,0, x ) (3.24)

Figure 3.16 Local coordinates of the selected point e along the cutting edge section-1

The measured position and rake face vectors along all the 21 points are measured and stored in

ii:: i jkr i : i c,r c,r c,r c,r (3.25) for each of the r-th rake face section of the c-th cutting edge section ( c∈{1,2...7}, r ∈{1,2...6} ).

Then, the transformation matrices are found throughout the entire edge. The transformation matrix for the r-th rake face section of the c-th cutting edge section is:

i i i i c,r i ic,r jkr c,r c,r Tc,r =,cr∈∈ {1,2...7}, {1,2...6} . (3.26) 00 0 1

Similar translational transformations are performed at each rake face section. Then, each point along the cutting edge is identified by applying Equation (3.24) at each section. Identified rake face vectors (Figure 3.18a) and rake face normal vectors (Figure 3.18b) of the cutting edge section (highlighted in Figure 3.11b). When the insert rake face geometry is fully identified

(Figure 3.18c), it can be located on different kinds of tool holders depending on the operation type.

Figure 3.17 Cutting edge-section-2 (Figure 3.13b) with curved rake face

(a) (b) (c)

Figure 3.18 Rake face geometry along the modelled cutting edge (bold curve): (a) Rake face vectors; (b) rake

face normal vectors; (c) rake face and rake face normal vectors.

Indexable face milling tool: A cylindrical right-handed face milling cutter (Figure 3.19a), a double right-hand milling cutter (Figure 3.19b) and a double left-hand milling cutter (Figure

3.19c) are used to demonstrate the placement of inserts on the tool body. In Figure 3.20a,

Coordinate transformation from insert reference frame (Frame-i) to tool reference frame (Frame-

0) is shown for an example insert on the tool body. As shown in Figure 3.20b-c, the transformation of coordinate frame involves both translation and rotation depending on the orientation of the pocket geometry.

Transformation for right-hand side cutter of the double milling tool, from Frame-i-Right

(insert frame of right-side holder) to Frame-0, is shown in Figure 3.21a-b. Transformation for left-hand side cutter of the double milling tool, from Frame-i-Left (insert frame of left-side holder) to Frame-0, is shown in Figure 3.22a. Transformation bor both cutters of double-milling tool is shown in Figure 3.22b.

(a)

(b)

(c)

Figure 3.19 Indexable milling cutters: (a) U140 cylindrical face mill; (b) double milling right-hand side cutter;

Figure 3.20 Transformation from Frame-i to Frame-0 of the cylindrical face mill cutter: (a) Top view; (b)

front view; (c) isometric view. (Courtesy of KOVA.)

Figure 3.21 Transformation from Frame-i-Right to Frame-0 of the double milling tool: (a) Top view; (b) 3D

view. (Courtesy of KOVA.)

Indexable drilling tool: A right-hand drilling tool (Figure 3.23) with the same type of inserts

is used as a case study. As shown in Figure 3.24a-b, both translation distance and rotation angles are different for peripheral (insert#1) and central (insert#2) inserts of the drill cutter.

Figure 3.22 Transformation from Frame-i-Left to Frame-0 of the double milling tool: (a) Top view; (b) 3D

view. (Courtesy of KOVA.)

Figure 3.23 Indexable U502 drilling cutter. (Courtesy of KOVA.)

Figure 3.24 Transformations from Frame-i to Frame-0 for the indexable drilling tool: (a) Top view at the

insert cross section; (b) 3D view. (Courtesy of KOVA.)

Case-2: Indexable two-insert end mill: A sample indexable R390-020A20L-11L end mill

(with R390-11 T3 04M inserts) from Sandvik-Coromant1 is used as a case study, see Figure

3.25. The cutter frame (Frame-0) is the fixed coordinate frame of the tool holder, and named as

0 0 0 x y z frame. The translational feed motion of the cutter is aligned with i0 unit axis vector

0 0 (positive x -direction), and the tool rotation axis is the k0 unit axis vector (positive z -direction).

0 0 The velocity vector is always on x y plane. The plane including the velocity and feed vectors,

defined at the selected point on the cutting edge, is the assumed working plane Pf. This plane is

0 0 also parallel to the x y plane at the zero depth of cut position of the cutter, as is shown in Figure

3.25.

The first step is to determine the desired location and orientation of the insert on the holder

(as shown in Figure 3.9a and Figure 3.25a-b). Insert pocket (seat) frame (Frame-s) on the holder

is assumed to match the desired orientation of the insert frame (Frame-i), as is shown in Figure

3.9b and Figure 3.25b-c. This assumption is to avoid an extra transformation from insert pocket

frame to desired location of the insert frame. The unit axis vectors of the insert frame, Frame-i,

0 0 0 0 0 0 are ii, ji, and ki (or is, js, and ks of pocket/seat frame, Frame-s) with the origin positioned on

the tool holder, are shown in Figure 3.26. Initial orientation of the insert frame (Frame-i) is the

same as the tool reference frame (Frame-0), as is shown in Figure 3.9a.

1 For example: http://www.sandvik.coromant.com/en-gb/products/pages/productdetails.aspx?c=R390- 020A20L-11L#/?active=detail

Figure 3.25 Sample two-insert end mill from Sandvik-Coromant (a) Position of the insert on the tool holder;

(b) Frame-s on the tool; (c) Frame-i from the back; (d) Frame-i from the front. The second step is to place

the insert origin at the insert pocket. In Figure 3.10a and Figure 3.26, 0ir is the position vector of the origin of the Frame-i relative to the origin of the Frame-0 expressed in Frame-0:

0i 0i 0i 0i ri=rrrx0 ⋅+ y0 ⋅+ jk z ⋅ 0, (3.27)

000 where i00,, jk 0 are the axis unit vectors represented in Frame-0 in xyz,, directions.

0i 0i 0i 0i rrxy, and r z are the magnitude components of the position vector r . In homogeneous vector

form, the position vector 0ir of Equation (3.27) can be written as:

T 0ir =  0irrr 0i 0i . (3.28) xyz

Third step is the rotation of the insert frame (Figure 3.10b). The Euler angle rotations for the

insert frame matching the pocket frame orientation are carried out in the following order [1]:

t1 • Negative- γ f rotation around the z -axis of the new frame after the first translation; as

shown in Figure 3.10a, since there is no rotation at the first translation step, the new

frame (Frame-t1) has the same unit axis vectors as Frame-0;

Figure 3.26 Relative locations of the origins of cutter, insert and first rake face section of the first cutting edge

section.

r1 • negative- γ p rotation around the x -axis of the new frame after the first rotation;

r2 • negative-κr rotation around the y -axis of the new frame after the second rotation.

The order of rotations is generally important, but for small angles (less than 30 [deg]), this effect is negligible [53]. The transformation from Frame-i to the Frame-0 is:

0 0 0 0i  ii ii j iκ i r 0i 0i 0i iii jκ ir  =Trans(,,)rrrxyzRotz (−−−γγκ f ) Rotx ( p ) Roty ( r ) (3.29) 0 0 0 1  00 0 1

The position vector of the origin of the Frame-i in Frame-i is zero:

T iir = [000] . (3.30)

The axis vectors are represented in Frame-i as:

TT T iii= [100] ; jk= [ 010] ; i= [ 001.] (3.31) 50

Although there is research to find the analytical expression with large rotation angles [112], approximation is possible with multiple ns step rotations as:

0 0 0 0i iii jκ ir 0i 0i 0i ns =Trans(rrrx , y , z )( Rotz(−−−γγκ fs n )Rotx( ps n )Roty( rs n )) . (3.32) 00 0 1

0 Generally the transformation is performed by Ti matrix:

0 0 0 0iii  ii j i k i r0 iii jk i r  = Ti , (3.33) 0 0 0 1 00 0 1  where

0 0i 0i 0i Ti=Trans(rrr xyz , , )Rotz(−−−γγκ f )Rotx( p )Roty( r ) . (3.34)

The negative rotations are due to the inverse rotations which start from the original tool frame orientation (see Figure 3.10b). The form of the homogeneous transformation matrix in

Equation (3.34) is:

0 0i 0 Rri Ti =  (3.35) 01x3 1

0 where Ri is the rotation matrix from the insert frame to the tool frame, and is given as:

cosγγ− sin 0 1 0 0  cosκ 0 sin κ ffr r 0   Ri= sinγγ ff cos 0 0 cos γ pp− sin γ  0 1 0 ,   0 0 1 0 sinγγ cos −sinκκrr 0 cos pp (3.36) cosκγ cos+ sin κγγ sin sin cos γγ sin−+ sin κγ cos cos κγγ sin sin r f rpf pf r f rpf 0 −+κγ κγγ γγ κγ+ κγγ Ri = cosrf sin sin r sinpf cos cos pf cos sin rf sin cos rpf sin cos . κγ − γ κγ sinrp cos sin p cosrp cos

The reverse rotation matrix (from Frame-0 to Frame-i) is given by:

T 00 RRii= ( ) ;  cosκr cos γ f+ sin κγ rpf sin sin γ −+ cos κγ rf sin sin κγ rp sin cos γ f sin κ r cos γ p (3.37)  i γγ γ γ − γ R0 = cospf sin cosp cos f sin p. −+κ γ κγ γ κγ+ κγ γ κ γ sinr cos f cos rp sin sin( f sin rf sin cos rp sin cos f cos r cos p

The inverse of the rotation matrix is normally equal to its transpose. However, this may not hold true for an arbitrary rotation matrix. In that case, an approximation is needed based on the identified Euler angles. The axis vectors of the insert coordinate frame (Frame-i) are used to for the approximation. From the 3D drawing of the cutter (Figure 3.26), the exact vector axis representation in Frame-0 are:

1  0  0  0.9874  0    00     iii=0 ; jk =  1 ; i =  0 ; i i =  0.1580  ; k i=  0.1742  ;        0  0  1  0.00344   0.9847  (3.38) −0.1550 0 0 00  ji=×= k i ij ii; 0.9723 .  −0.1720

The position vector 0ir is already given by manufacturer or obtained by measurements.The exact transformation matrix is:

0 0 0 0i ii  ii j i k i r 0 iii jk i r  =(Ti ) ⋅ . 0 0 0 1 exact 00 0 1  0.9874− 0.1550 0 0ir x 1000 0i 0 0i  0.1580 0.9726 0.1742 ry (Rri ) 0100 = exact ⋅ ; (3.39) 0i 0010 0.003440− 0.1720 0.9847 rz 01x3 1  0001 0 0 01 0.9878− 0.1549 0 0  Ri = 0.1580 0.9726 0.1741 . ( )exact  0.003442− 0.1719 0.9847

The 3-unknown Euler angles are fitted by matching 5-equations of the parametric matrix

(Equation (3.36)) and the numerical matrix (Equation (3.39)):

−=−=sin(ggpp ) 0.1720; 9.9035[deg];

sin(κrp )cos(g )= 0.00344  κr = atan2( 0.00344,0.9847) = 0.2002 [deg]; (3.40) cos(κrp )cos(g )= 0.9847 

cos(ggpf )sin( )= − 0.1550  g f =−=atan2( 0.1550,0.9723) − 9.0544 [deg]. cos(ggpf )cos( )= 0.9723 

Although a non-linear iterative fitting method with 9-equations of Equation (3.36) and 3- unknown Euler angles may also be used, it is observed that using the method in Equation (3.40)

gives satisfyingly accurate result. The approximate rotation matrix compares well to the exact

one:

0 Ri=Rotz( −⋅γγκ fpr ) Rotx( −⋅ ) Roty( − ) ( )approx 0.9874−− 0.1550 0.0305 (3.41) 0  Ri = 0.1580 0.9728 0.1693 . ( )approx  0.00344− 0.1720 0.9851

The homogeneous transformation matrix is thus reassigned using the approximate rotation

matrix:

0 0i (Rri ) T0 = approx (3.42) i  01x3 1

Locating the edge on the insert: Although exact dimensions of the insert’s edge and tool

would be preferred for accurate analysis, a 2D model could still give a good estimation for

process prediction [1]. Since the transformation from the Frame-0 to Frame-i is done, the transformation from the Frame-i to the local cutting edge needs to be carried out. Since the insert

of Figure 3.25 does not have a flat-face, either an approximation is needed or every selected point along the cutting edge is treated individually. For the design of this insert, the assumption of three (C=3) cutting edge sections with flat rake faces (R=1) is valid (Figure 3.27a), i.e., each cutting edge section has single rake face section. Then, each rake face section is formulated within itself. Before transforming to the local cutting edge, transformation to the reference of the local rake face is needed. An example close-up view of the first local rake face section is shown in Figure 3.27b. Cutting edge is approximated by line-arc-line function series in this section. The reference of the first section of the rake face (r=1) on the first cutting edge section (c=1) of the insert is named as Frame-1,1 (Figure 3.27b). The transformation from the Frame-1,1 to the

Frame-i is:

i i i i 1,1 1,1 1,1 i1,1 j 1,1 k 1,1 ri i1,1 jk 1,1 1,1 r  = T1,1 . (3.43) 0 0 0 1 00 0 1

i Similar to Equation (3.35), the transformation T1,1 has the following form:

i i 1,1 i Rr1,1 T1,1 = . (3.44) 01x3 1

Here, the relative position vector ir 1,1 is defined in Frame-i. It is the vector from the origin

i of the Frame-i to the first rake face frame (Frame-1,1). R1,1 is the rotation matrix of the first rake face frame after the translation. Similar to locating the insert frame on the cutter (Equation (3.42)

), an Euler angle identification is needed here for each rake face section along the cutting edge.

i i 1,1 (Rr1,1) Ti = approx . (3.45) 1,1  01x3 1

Figure 3.27 Cutting edge and rake face sections of the insert: (a) Flat rake face sections along the cutting

edge; (b) example close-up view of the first local rake face section.

i Obtaining the transformation matrix T1,1 usually requires information about the three dimensional geometry of the tool. Since the local rake face plane is flat, there is no more transformation in the local y-axis (j1,1) of the first rake face frame of the first cutting edge

section. The rotation from Frame-1,1 to Frame-D (and for other rake faces) is inherently an Euler

angle rotation. The transformation to the cutting edge frame, Frame-D, of the local cutting edge

is a 2D transformation. The cutting edge section on a constant rake face plane is assumed to

consist of a combination of the 2D lines and arcs. For example, rake face section-1 is defined as the combined line-arc-line segments, as is shown in Figure 3.27b. Thus, for point-e along the cutting edge in rake face section-1, the translation and rotation are combined using the local

11 11 11 coordinates ( xzee,0, ) and slope θe :

1,1 =11 11 ⋅−θ11 TD Trans( xzee ,0,) Roty( e) . (3.46)

(a) (b) (c) (d)

wiper edge Figure 3.28 Insert edge geometry of the two-inserted end mill: (a) Modelled 3D cutting edge; (b) modelled rake face geometry; (c) radius along the cutting edge; (d) relative location angle along the cutting edge.

Then, the transformation from the Frame-D to the Frame-0 is the series multiplication of all

transformation matrices. The transformation matrix from Frame-D to Frame-0 is:

0 0 i 1,1 TD= TT i 1,1 TD . (3.47)

0 Again similar to the Equation (3.35), the transformation TD has the following form:

0 0D 0 RrD TD = . (3.48) 01x3 1

Here, the relative position vector 0Dr (Figure 3.27a) of the origin of the cutting edge frame at the selected point-e is defined in Frame-0, and is defined as the vector from the origin of the

0 Frame-0 to the local cutting edge frame (Frame-D). RD is the rotation matrix to represent the

design frame axes in the cutter frame axes. Modelled 3D geometry of the cutting edge is given in

Figure 3.28a. Rake face and rake face normal vectors are shown in Figure 3.28b at each sampled

point along the cutting edge of the insert. Radius (Figure 3.28c) and angular location (Figure

3.28d) are defined from the 3D geometry. Wiper edge geometry (Figure 3.28b) is also included

in the model.

(a) (b)

Figure 3.29 Chip length along the first 0.030-mm section of the cutting edge: (a) 0.001-mm segment thickness;

(b) 0.010-mm segment thickness.

Cutting edge length is derived from the 3D geometry (Figure 3.28a). For small depths of cut,

the effect of the cutting edge length is given in Figure 3.29. For a 0.030-mm axial depth of cut, if

the depth increment (segment thickness) is taken at dz=0.001-mm intervals, the total cutting edge

length is 1.064-mm. Figure 3.29a shows the chip length distribution along the first 0.030-mm

section of the cutting edge for dz=0.001-mm thickness. But, if the depth increment is taken to be

dz=0.010-mm, the total cutting edge length becomes 0.2480-mm. Figure 3.29b shows the chip

length distribution for dz=0.010-mm thickness. Prediction of the cutting edge length is reduced

to less than half the value of the more accurate value (1.064-mm) when 10 times finer segment

thickness is used. This significant reduction shows the signinificant effect of the wiper edge

section (see Figure 3.28b) underneath the insert (the initial line section in Figure 3.27b).

Although the wiper section makes a small contribution to the chip thickness, it has a large

contribution to the chip length. The wiper edge has a major effect for the first 0.010-mm depth

(see Figure 3.29a) which dominates the chip length distribution.

Case-3: Multifunctional single level indexable cutter: Multifunctional cutters combine at least two conventional operations in one operation. Figure 3.30a shows a single-level multi-step

drilling cutter from Kennametal with HTSR043R025M tool holder, DFR030204GD-KC7020 inserts, and with a twist drill (pilot drill) at center; Figure 3.30b shows a two-level combined

drilling, step-boring, countersinking and chamfering cutter from Sandvik with TM880 tool

holder and 5513-020 series inserts.

Since there is no available CAD geometry of the inserts and cutters, the measured data is used

for identification of the rake face geometry. Optical microscope is used outside the machine, and

the tool holder is measured with the inserts installed on it. The cutting edge along the indexable

section (without the pilot drill) is measured. The sample 3D geometry of the inserts located on

the tool holder body is given in Figure 3.31a-b. Since the rake angle of the inserts is zero, the

same figures also represent the rake face geometry.

(a) (b)

Z Z Top level 1 3 1 2 Bottom level Bottom level 3 2 X 4 X 4 Bottom view Bottom view Figure 3.30 MultifunctionalBottom view cutters: (a) Multi-step drilling tool Bottomfrom Kennametal view ; (b) Two-level cutter from

Sandvik. (Adapted from Wan et al. [5].)

Figure 3.31 Measured cutting edge geometry of Kennametal drill: (a) Top and front views ofeach insert; (b)

transformation of insert frames to their locations on tool frame.

The insert geometry is formed from the optical microscope measurement. As shown in

Figure 3.31a, the origin of the insert frame coordinates is set at the insert tip point. Radial location of the origin is measured by a microscope, and the axial location at the tip of each insert is measured by a dial gage instrument (as in measurement of axial runout of inserts).

Transformation of the insert #1 is shown in Figure 3.31b, and the other three inserts are located on the cutter in a similar way. Transformation steps are as follows:

4 0 • Translation Trans(0,0,ε j ) in tool z -axis for the axial runout relative to the fourth insert (

4 ε4 = 0 );

0 • Rotation Rotz(jj ) by the location angle jj around tool z -axis. jj is measured from the

+x0-axis in CCW direction;

• Translation Trans(rj,0,0) by the radial offset rj from the tool center in the x-axis of the

new frame;

• Rotation Roty( −κ j ) by the κ j rotation angle which is measured around the y-axis of the

new frame in CCW direction. This κ j angle corresponds to α angle of Figure 2a of [5].

The resulting Frame-i (insert frame) is found by multiplying the relative transformations in the order:

0 0 0 0iii  ii j i k i r0 iii jk i r  = Ti  (3.49) 0 0 0 1 00 0 1 

The transformation matrix is found by the kinematic chain:

04 Ti=Trans( 0,0,εj j) ⋅ Rotz( jj) ⋅ Trans(r ,0,0) ⋅− Roty( κ j) . (3.50)

i The transformation TD at each element Sjk of the cutting edge is known from the ( )jk measurements. The representation of the cutting edge tangent, rake face normal, rake face and position vectors in Frame-0 is:

0 0 0 0D  D  iDDD j k r (iDD) ( jkr) ( D) ( )jk( ) jk( ) jk( ) jk 0i jk jk jk ( )jk  = TTiD . (3.51)  ( )jk  0 0 0 100 0 1

Figure 3.32 Modelled Kennametal cutter: (a) Axial location; (b) angular location.

(a) (b)

Figure 3.33 Modelled cutting edge geometry of the Sandvik TM880 cutter: (a) Drawing; (b) schematics.

D Cutting edge tangent (iD ) , rake face normal ( jD ) , rake face (kD ) and position r jk jk jk ( )jk vectors of the element Sjk are given as follows:

TT TD T (iDD) = [100;] ( jkr) = [ 010;] ( D) = [ 001;] = [ 000] (3.52) jk jk jk ( )jk

0 The cutting edge is represented in the Frame-0 by the cutting edge tangent iD , rake ( )jk

0 0 0D face normal jD , rake face kD and position r vectors of the element Sjk. ( )jk ( )jk ( )jk

Axial location (Figure 3.32a) and angular location (Figure 3.32b) are identified from the 3D geometry (Figure 3.31b).

Case-4: Multi-functional two-level cutter: Sandvik TM880 multifunctional tool with drilling, boring, countersinking and chamfering functions is shown in Figure 3.30a. It has two levels (see Figure 3.33a) to give the depth to the machined countersunk hole. The mathematically modelled geometry (Figure 3.33b) is obtained by measuring the cutting edge under an optical 61

microscope. The axial runout of the insert #3 is measured by a dial gage when the cutter installed

on the spindle.

3.2.2 Solid Tool

Unlike the indexable cutters which require the definition of the inserts and their placements

on the tool body, the cutting edges are an integral part of the solid tool geometry. The cutting edge is defined relative to the tool reference coordinate frame. The transformation matrices are given in Appendix A. The method of locating the cutting edge is as follows:

• The location of the fixed tool reference coordinate frame, e.g., center of the tool’s bottom

geometry (Figure 3.34a), is selected for the cutter;

• The cutting edge element Sjk ( kK∈{1, 2, .. } , jN∈{1, 2, .. } ) is placed at the axial segment

0 0 (Zk=kdz) as shown in Figure 3.34b; radial location (x ,y ) is defined by using rotational

(Figure 3.34c) and translational (Figure 3.34d) transformations;

• The cutting edge element Sjk is rotated using local helix angle (Figure 3.34e), rake angle

(Figure 3.34f) and cutting edge angle (Figure 3.34g) to obtain the local rake face

geometry (Figure 3.34h).

It may be difficult to represent some of the tools’ geometry analytically (as in Ref. [39]),

but they can be digitized and represented numerically [110].

Figure 3.34 Locating the Frame-D on the solid tool starting from Frame-0: Steps (a)-(d) serve to locate the

x0y0z0 coordinates; steps (e)-(h) are for rotational alignment of the local cutting edge frame. 63

Case-5: Serrated cylindrical, helical end mill: Serration waves, which are constructed from spline points, are ground on the helical flutes of roughing end mills [21]. The sample serrated end mill is Sandvik R216.33-20040-AC38U H10F. The serration profile of its cutting

edge is measured by a surface profilometer. The stylus is set to zero at the bottom of the flute and

the radial variation is traced upwards along the flute at ds intervals. Surface profilometer

s measurement data for one serration period is given in Figure 3.35. Radius ( Rjk ) is measured at

each point along the flute (Figure 3.35a).

The corresponding sampling distance dz along the cutter axis is given by:

dz ds = (3.53) cos β serr

serr s where β is the helix angle of the serrated end mill. The radial variation ( Rjk ) of flutes are

mapped in the tool coordinate frame; radius ( Rjk ) at each segment along tool axis is given in

Figure 3.35b.

Figure 3.35 Surface profilometer measurement data of the serrated end mill: (a) Radius along the flute; (b)

axial and radial distance from the tool tip along the tool axis.

Figure 3.36 Cutting edge angle of the serrated end mill: (a) Along the flute; (b) mapped along the tool axis.

serr,s Cutting edge angle (kr,jk of Figure 3.36a) along the flute is interpolated from the measured

serr profile. It is mapped along tool axis using Equation (3.53) to obtain kr,jk of Figure 3.36b.

Each cutting edge segment, with differential height (dz) is placed on the tool body using

Denavit-Hartenberg (also called “moving axes approach” [113]) transformations, as follows:

• Translation Trans(0,0,kdz) in the tool axis (z0 of the Frame-0) at the axial height kdz.

• Rotation Rotz( π −ψ ) around z0-axis. If the helix and pitch (φ ) angles are constant, 2 jk p

and R0 is the nominal radius of the tool, the angular location angle ψ jk of flute j and edge

segment k is given as:

tan β serr ψφjk= p ⋅( j −−1d) ⋅kz (3.54) R0

• Translation Trans(Rjk,0,0) along the radius vector: Tranforms the coordinate frame to the

position along the cutting edge (see Figure 3.34e).

• Rotation Rotx( −β serr ) around x-axis of the new frame: Aligns the cutting edge with the

helical flute of the tool.

• Rotation Rotz( −γ serr ) around z-axis of the new frame: Aligns the coordinate frame with

the rake face of the cutting edge.

serr • Rotation Roty( −kr,jk ) around y-axis of the new frame: This final transformation aligns

the coordinate frame with the cutting edge design frame (Frame-D) of the tool.

Transformation from Frame-D to Frame-0 (tool reference frame) is thus (see Figure 3.34h):

0 0 0 0D  D  iDDD j k r( iDD) ( jkr) ( D) ( )jk( ) jk( ) jk( ) jk D jk jk jk ( )jk  = T0 . (3.55)  ( )jk  0 0 0 1 00 0 1

The complete transformation matrix to place the edge element Sjk on the tool body is found by the following kinematic chain:

TD =Trans 0,0,kz d ⋅ Rotz π −⋅y TransR ,0,0 ⋅ ( 0 ) ( ) ( 2 jk ) ( jk ) jk (3.56) serr serr serr Rotx(−βγ) ⋅ Roty( −⋅−) Rotz( kjk )

D Cutting edge tangent (iD ) , rake face normal ( jD ) , rake face (kD ) and position r jk jk jk ( )jk vectors of the element Sjk are:

TT TD T (iDD) = [100] ;( jkr) = [ 010] ;( D) = [ 001;] = [ 000] (3.57) jk jk jk ( )jk

Thus, the cutting edge of the element Sjk is represented in the Frame-0 (tool reference

0 0 0 coordinates) by the cutting edge tangent iD , rake face normal jD , rake face kD ( )jk ( )jk ( )jk and position 0Dr vectors. ( )jk

3.3 Computation of the Effective Tool Geometry

Based on the geometry definition supplied by ISO standards [51], the local tool angles (or effective geometry) are found by using the feed and velocity vectors of the cutting edge segment

[1]. They are defined in the order of relevance as follows:

3.3.1 Velocity of Cutting Edge of the Tool

Velocity of the element Sjk is defined by cutting velocity and feed vectors. Their superposition gives the resultant cutting vector.

0 Cutting velocity ( vc ): It is defined as the tool velocity relative to the workpiece. The position vector 0Dr of the origin of the local cutting edge coordinates relative to the cutter frame is similar to Equation (3.28):

T 0Dr =  0Drrr 0D 0D . (3.58) xyz

0Dr is used for describing the cutting velocity direction. The rotation axis is the tool axis for milling, drilling and boring operations. The workpiece axis is the rotation axis for turning operation. For right-hand milling (Figure 3.3a), drilling (Figure 3.5a) or boring operation, the relative rotation is CW when looked from the tool side. For left-hand turning (Figure 3.1a) operation, the relative rotation is also CW when looked from the tool side. Thus, the rotation vector ω0 defined in the tool frame (Frame-0) as:

-ω ⋅k0 (right-hand cutting) ω0 =  . (3.59)  ω ⋅k0 (left-hand cutting) where (rad/s) (or Ω [rev/min]) is the relative angular speed of the tool with respect to the workpieceω . The magnitude of the cutting velocity vector is given in Equation (3.8). The local

0 radius vector R jk of the element Sjk is defined in Frame-0, and evaluated as: 67

0 0D Rjk=×× k 0 ( rk0 ). (3.60)

The cutting velocity of the tool is realized from the radius vector and the rotation vector as:

000 vRc=ω × jk . (3.61)

0 0 0 where the radius vector R jk is in the x y plane:

T R0= RR 000 (3.62) jk x y

Feed vector: The direction of the feed vector vf is defined in Frame-0 (tool coordinate frame) with magnitude vf (m/min) which is product of fr (mm/rev) and Ω (rev/min). The feed

is assumed to be in positive-x0 direction in milling, in negative-z0 direction in turning, boring and

drilling operations, as follows:

 f Ω⋅10-3 i if milling  ( r0) vf =  (3.63) - f Ω⋅10-3 k if turning,boring or drilling  ( r0)

Figure 3.37 Resultant cutting directions for two consecutive edges at segment k of milling cutter moving in x0

direction.

Resultant cutting vector: The definition of the resultant relative velocity of the tool

includes the feed direction. As shown in Figure 3.37, if the feed motion is comparable to the

cutting velocity, then the resultant cutting direction is not same for the two consecutive cutters.

Generally, with the exception of thread cutting operation, the magnitude of the feed motion is small compared to the cutting speed. Thus, the resultant cutting direction is assumed to be the

same as the cutting velocity. This implies that the planes defining the tool-in-hand and tool-in-

use systems are the same, except for the working plane (Pfe) [51]. Since the working plane (Pfe) includes the directions of cutting velocity and feed vectors, its orientation depends on the feed direction. For thread cutting tools, the feed amplitude has the same order as the cutting speed.

Hence, the assumption of equal tool-in-hand and tool-in-use systems is no longer valid for thread cutting application.

3.3.2 Tool Cutting Planes

Tool cutting planes are demonstrated by using tapered ball end mill (Figure 3.38a) which is

capable of 5-axis machining. As shown by green circle (Figure 3.38a-b), the selected point along

cutting edge is at its spherical (bottom) section.

Assumed working plane (Pf) and working plane (Pfe): Although Pf and Pfe planes both

include the cutting velocity and feed vectors, they are in general different planes [51]. Assumed

working plane is shown in Figure 3.38b. It is parallel to x0y0 plane with an offset in tool axis (z0).

Working plane is parallel to tool axis for drilling operation, and it is perpendicular to the tool axis for 2-axis milling operation (Figure 3.38b). For 3-axis milling, the working plane orientation becomes position-dependent due to the z0 component of the feed velocity.

Figure 3.38 Tapered ball end mill to show Pf, Pfe and Pr planes: (a) General view; (b) working planes for

milling and drilling operations; (c) Tool reference plane (Pr) with Frame-R.

0 0 While the Pf assumes that the feed direction is only in the x y plane of Frame-0, Pfe takes the

actual feed direction into account. For 2-axis milling operation, the translation feed vector ( vf,m

0 0 in Figure 3.38a-b) is in x y plane. Thus, as shown in Figure 3.38b, Pf and Pfe,m are the same

planes for 2-axis milling. For turning, boring and drilling, feed vector ( vf,d in Figure 3.38a-b) is

0 towards the negative-z direction. Thus, as seen in Figure 3.38b, assumed working plane (Pf)

does not represent the working plane (Pfe,d) of drilling. Therefore, the more general Pfe plane is used for calculating the effective tool geometry.

0 First defining vector of the working plane (Pfe) is the tool axis z direction as it is

0 0 0 perpendicular to the cutting velocity. For 2-axis (x y plane) milling, the radius vector R jk of the

element Sjk is the first defining vector. For turning, boring and drilling, the radius vector is

relative to the workpiece rotation axis or the center of the drilled/bored hole. The second defining

vector is the feed vector. Pfe plane is perpendicular to the tool reference plane. 70

For 2-axis milling, the working plane includes the cutting velocity, radius vector and xR-axis

of the tool reference plane frame (Frame-R in Figure 3.38c). For turning, boring and drilling

operations, the working plane includes the cutting velocity, radius vector and negative zR-axis of the tool reference plane (Pr in Figure 3.38c) frame. 5-axis milling can also be analyzed similarly.

Its working plane includes the cutting velocity, radius vector and the feed axis direction on the Pr

frame (Frame-R).

Tool reference plane (Pr): Pr plane (see Figure 3.38c and Figure 3.39e) is perpendicular to

0 the cutting velocity vector which is defined by the radius vector R jk and the unit tool axis k0

0 0 0 0 vector in the z direction. The corresponding axis unit vectors ( iR , jR and kR ) of the Pr

coordinate frame (Frame-R) represented in Frame-0 are shown in Figure 3.38 by xR,yR and zR

directions.

0 0 R jk v 000i=; j =−=c ; kk . (3.64) R00 R R0 Rvjk c

0 Similar to Equation (3.39) the transformation matrix TR from Frame-R to Frame-0 is

obtained using the axis unit vectors of Equation (3.64):

0 0 0 0R 0 iRR jkr R TR = . (3.65) 00 0 1

The position vector 0Rr of the origin of the Frame-R is the same as the position vector 0Dr

of the origin of the Frame-D, because their origins coincide (compare Figure 3.38a and Figure

3.39a):

T 0Rr= 0D rr;. 0R =  0Drrr 0D 0D (3.66) xyz 71

Figure 3.39 Tapered ball end mill to show Aγ, Pn, Ps and Pr planes: (a) General view same as in Figure 3.38a;

(b) Cutting edge frame (Frame-D); (c) Cutting edge normal plane (Pn) and Frame-3; (d) related rake face

normal view; (e) cutting edge plane (Ps).

0 Then, the rotation matrix can be extracted from TR by the analysis as in Equation (3.35):

Rr0 0R T0 = R (3.67) R T 0 1

Rake face (Aγ): The local cutting edge frame, Frame-D, coincides with the rake face (Aγ) plane, as is shown in Figure 3.39b, where clarance face (flank, Aα) is also shown. iD is the unit cutting edge tangent vector; jD is the unit rake face normal vector (directed into the rake face);

kD is the unit rake face vector, which is tangent to the rake face but perpendicular to the cutting

edge:

TT T iDD= [100] ; jk= [ 010] ; D= [ 001.] (3.68)

The corresponding coordinate system is named as Frame-3 (Figure 4.5), with the following

mapping of the axis directions:

x3≡ yy D3; ≡− xz D3 ;. ≡ z D (3.69)

The cutting edge tangent iD vector is the same as t jk of Equation (3.9). The rake face

normal jD direction is also linked to Equation (3.9). The rake face direction kD is the same as

Equation (3.10).

Cutting edge normal plane (Pn): As shown in Figure 3.39d, Pn plane is defined as

perpendicular to the cutting edge tangent iD vector (see Figure 3.39c), thus it includes the rake

2 2 face normal jD and rake face kD vectors. Pn is equivalent to the x z plane of the Frame-2

3 3 (Figure 4.5). y and x directions of Frame-3 (Figure 4.5) are perpendicular to the Pn plane. The

yDzD components of the cutting edge design coordinates can be used for projecting vectors on the

Pn plane.

Cutting edge plane (Ps): Ps includes the cutting edge tangent vector and it is perpendicular

0 to tool reference plane (Pr). It includes the cutting velocity vector vc . The representation of the

0 cutting edge tangent vector in the cutter reference Frame-0 iD is found using the Equation

(3.55):

00 iD= Ti DD ⋅ . (3.70)

The cutting edge tangent unit vector iD represented is from Equation (3.68).

3.3.3 Effective Tool Angles

* * Local true cutting edge angle (kr,jk ): Tool cutting edge angle (kr,jk ) is defined as the angle between the tool cutting edge plane Ps and the assumed working plane Pfe measured in the

tool reference plane Pr [51]. Calculation of the cutting edge angle depends on the Pfe plane

orientation, and thus dependent on the feed vector direction. Since for all operations, the cutting

velocity is perpendicular to the Pr plane, the Pfe plane intersects the Pr plane at the feed vector direction. Thus, the corresponding method for milling is to project the cutting edge tangent xD- axis vector on the Frame-R. Subsequently, find the angle that the projected vector makes with the feed direction.

0 The representation of the cutting edge tangent vector on Frame-0, iD is found in Equation

D R (3.70). This vector is in x direction in Figure 3.39e. For its Frame-R representation ( iD ), the

transformation from Frame-D to Frame-R is done:

R R0 iTiD0= ⋅ D (3.71)

where, the inverse transformation from Frame-0 to Frame-R is found from Equation (3.65),

−1 R0 TT0R= ( ) . (3.72)

Or, from the homogeneous matrix algebra [114],

TT 0 0 0R R (RRR) −( Rr) T0 =  (3.73) T 0 1

P R R R r Pr plane is formed by the x z components of the Frame-R. So the projected vector ( iD )

R is obtained by setting the y-component of the transformed vector iD to zero. If the transformed

vector is

RR R R iiDD= ( ) ( i D) ( i D) 0, (3.74) xyz

then, the projected vector is

P RRr  R iiDD= 0 i D 0. (3.75) ( ) ( )xz( )

Two-axis Milling: It is shown in Figure 3.38b that if the working plane (Pfe) is parallel to

x0y0 plane for 2-axis milling, the feed vector direction can be assumed as the positive xR

direction. Figure 3.40a is a 2D representation of the Pr plane in Figure 3.39e. Cutting edge

D tangent vector in x direction (Figure 3.39e) is projected onto Pr plane in Equation (3.75). As

shown in Figure 3.40a, the cutting edge angle is defined as the angle between the projected

cutting edge tangent vector and the xR-axis:

P *Rr κr=arccosii RD ⋅( ) , (3.76) 

T where, iR = [100] . Thus,

 R iD * ( )x κr = arccos . (3.77) 22 RR iiDD+ . ( )xz( )

* The range of arccos function is [0,π], so Equation (3.76) covers all the possible κr values.

R R But, if iD has negative z -axis component, the cutting edge physically makes contact with the 75

back side, thus it is not in-cut. It is thus more convenient to use atan22 function for a general

cutter case:

* RR κr= atan2ii DD , . (3.78) ( )zx( )

Calculation of the cutting edge angle in Equation (3.78) only applies for right-hand milling

(Figure 3.40a), where the tool axis rotation is in CW direction. In the case of left-hand milling

(Figure 3.40b), the tool rotates in the CCW direction. As a result, the calculation of the cutting edge angle is based on the negative cutting edge tangent vector.

* RR κr=−−atan2ii DD , . (3.79) ( )zx( )

Figure 3.40 Cutting edge angle calculation: (a) Right-hand milling; (b) Left-hand milling.

2 More information at: http://www.mathworks.com/help/matlab/ref/atan2.html

Figure 3.41 Cutting edge angle calculation of the drilling operation.

Turning, Boring and Drilling: For turning, boring and drilling operations, the feed

direction is the negative zR direction. The definition of the cutting edge angle is the angle

R R between the negative cutting edge tangent (− iD ) vector and the negative z direction. Hence,

as shown in Figure 3.41, the cutting edge angle formula is a rotated version of Equation (3.78):

*3π  RR κr=−−−atan2iiDD , (3.80) 2 ( )xz( )

Local tool cutting edge inclination angle ( λs,jk ): Tool cutting edge inclination angle ( λs ) is

measured on the cutting edge plane Ps. It is the angle between the cutting edge and the tool

reference plane Pr [51]. The cutting edge tangent and the cutting velocity vectors are both included in the Ps plane, and the cutting velocity is perpendicular to the Pr plane. As shown in

Figure 3.42a, the cutting edge inclination angle can be defined as the complementary angle

0 (adding up to 90-deg in total) of the angle between negative cutting edge tangent ( − iD ) and the

cutting velocity directions. 77

Figure 3.42 Calculations of (a) Cutting edge inclination angle and (b) normal rake angle.

0 The dot product between the negative cutting edge tangent ( − iD ) and the cutting velocity directions gives the complementary angle of the cutting edge inclination angle:

−⋅00 π ( ivDc) λs = − arccos. (3.81) 2 v0 c

Alternatively, same angle can be found by taking the dot product of the cutting edge tangent vector and the yR-axis vector of the Frame-R:

π λ =−⋅arccos 00ij. (3.82) s2 ( DR)

Since the range of the arccos function is [0,π], the range for Equation (3.81) is [-π/2,π/2], which encompasses all the possible values.

Local tool normal rake angle (γ n,jk ): Tool normal rake angle ( γ n ) is defined as the angle between the rake face Aγ and the tool reference plane Pr as measured in the cutting edge normal

plane Pn [51]. Alternatively, the same angle is obtained if the cutting velocity is projected onto Pn plane (Frame-D); the angle that the projected velocity makes with the rake face normal is found as shown in Figure 3.42b. The cutting velocity can be first transformed to the Frame-D as inverse operation of Equation (3.70):

D0 D 0 vc= Tv 0c ⋅ , (3.83) where, the inverse transformation from Frame-0 to Frame-D is found from Equation (3.47):

−1 D0 TT0D= ( ) . (3.84)

P D D D0 n Pn plane is formed by the y z components of the Frame-D. So the projected vector ( vc )

D0 is obtained by setting the x-component of the transformed vector vc to zero. If the transformed cutting velocity is:

D0 D0 D0 D0 vvcc= ( ) ( v c) ( v c) , (3.85) xyz then, the projected vector is:

P D0n  D0 D0 ( vc) = 0.( vv cc) ( ) (3.86) yz

The positive normal rake angle is defined as the angle between the negative of the projected cutting velocity and the rake face normal jD vector. It is found by taking the inverse-sine of the negative z-component:

 D0 − vc ( )z γ n = arcsin . (3.87) 22 D0 D0 ( vvcc) + ( ) yz

The range of arcsin function is [-π/2,π/2], so Equation (3.87) covers all the possible γ n values.

Sample effective geometry data are given in Figure 3.43, Figure 3.44 and Figure 3.45 for indexable cutter (two-insert end mill and U140 face mill) and solid tool (serrated end mill), respectively.

Figure 3.43 Effective tool angles of the two-insert end mill.

Figure 3.44 Effective tool angles of U140 face mill

Figure 3.45 Effective tool angles along the cutting edge (in one serration period) of the serrated end mill [21].

Chapter 4: Generalized Modelling of Cutting Mechanics

The generalized tool geometry presented in the previous chapter is used to model the chip geometry and corresponding cutting forces in this chapter. The kinematics of turning, boring, drilling and milling operations are developed to transform the forces and vibrations at the desired tool and part locations.

4.1 Chip Geometry

The chip thickness and length are the basis for predicting the cutting forces. The cutting forces are then used to determine resulting vibrations and the machining errors. Chip thickness (

hjki ) is defined as the material removed in the radial R direction of the cutting edge element Sjk;

st d it consists of static ( hjki ) and dynamic ( hjki ) components:

st d hhhjki= jki + jki . (4.1)

The static chip thickness is formed by the relative rigid body motion of cutting edge and

a workpiece elements, and evaluated from the feed per edge ( cjki ) of the axial or radial segment

* (k) of tooth (j) at time instant (i), with true cutting edge angle (kr,jk ) as:

 a* cjki ⋅−sinfjki gkjki⋅sin r,jk if milling st ( ) hjki =  , (4.2)  ca*−⋅gksin if turning,boring or drilling  ( jk jk) r,jk

a where the actual feed per edge ( cjki ) and decrease in chip load (γ jki ) depend on the runout of the cutting edges. In milling, the chip load depends on the angular location of the edge due to the feed movement in radial direction; whereas turning, boring and drilling operations have constant

a chip load, due to the feed motion in axial direction. Hence, cjk and γ jk are time invariant as is

noted in Equation (4.2). 81

Static chip thickness and tooth passing delay with zero runout: If there is no runout, the

actual feed per edge for the Sjk element becomes time invariant and takes the value of the

a nominal feed per edge, ccjki= jk . Since there is no deviation from the chip engagement due to

runout, variation in the chip load becomes γ jki = 0 , and the static chip thickness is rewritten using

Equation (4.2):

 * cjk ⋅sinfjki ⋅ sinkr,jk if milling hst =  . (4.3) ( jki )zero * runout  cjk⋅sink r,jk if turning,boring or drilling

The tooth passing delay period Tjki for multi-point tools also becomes time invariant and

only depends on the designed pitch angle φp,jk of the edge element Sjk:

φp,jk T = . (4.4) jk 2π Ω 60

Static chip thickness and tooth passing delay with runout effect: A block diagram is

prepared to show the influence of the axial runout (for turning, boring and drilling) or radial

runout (for milling) on the chip thickness and the time delay term. Similar to tooth period, the

delay period of element S is defined as the time passed after the last edge S cut, which is jk j+njki ,k

at the same axial or radial segment. Here, njki-1 is the number of edges skipping the cut before tooth-j at segment-k and time instant-i. If njki=Na,k (Equation (3.1)), then only one edge is cutting at that segment instantaneously. Hence, the chip is removed like in a “no-runout” case. The prediction of actual static chip thickness and tooth passing delay period (Tjki ) from the runout

geometry is based on the previous studies [20,22,108,115]. Due to different kinematic

configuration, milling operation and turning, boring and drilling operations are presented

separately. The resultant feed direction of milling is on the x0y0-plane. Thus, only the radial runout is affecting the chip thickness. If the tool is also moving in the z0-axis, the axial runout is

additionally required to evaluate a resultant runout in the feed direction. The feed direction of

turning, boring and drilling operations are in the z0-axis. Thus, the axial runout affects the actual

chip thickness.

Chip thickness derivation is based on the translational feed motion of the cutter per one

spindle revolution. Cutting edge elements at each discrete segment-k of the tool share the removed chip section according to the runout at the edges.

Example case with a radial runout: The algorithm presented in Figure 4.1 is demonstrated

on a cylindrical serrated end mill, as shown in Figure 4.2 (taken from Ref.[21]). The tool geometry is sampled in axial direction (z0) with dz thick segments. Each depth of cut geometry is

approximated at its middle axial distance (0.5kdz-0.5dz). For this case, 1.9-mm depth of cut is

selected (Figure 4.2). Segment thickness dz is 0.1-mm, thus k=19. The k-th axial segment

information is given as in Table 4.1.

Initial observation at k-th segment (Figure 4.2) is as follows: Since the 1st edge has the most

rd runout (εr,1k = 0.1766-mm ), it will remove most of the material. The 3 edge has the least runout

(εr,3k = 0-mm ), it removes the least material among the 3 edges.

Figure 4.1 Algorithm for predicting the actual static chip thickness and tooth passing delay period Tjki .

Figure 4.2 Serrated cylindrical end mill with the cross section at k-th segment.

Table 4.1 Inputs at the k=19-th axial segment of the 3-tooth serrated cylindrical end mill.

Element-available-to-cut condition array g1,k = [111]

Radial runout vector εr,k = [0.1766 0.1688 0.0] mm Relative angular locations vector (measured from 1st edge rel ψki = [0 120 240] deg in CW direction) Actual number of edges in cut at k-th segment N =3 Nga,k=∑ 1, j k =++=111 3 j=1 Spindle speed Ω=1000 − rev/min Feedrate per one spindle revolution fr =0.120 − mm Edge for which the output is requested jout = 2 Instantaneous angular position of the output edge φ2ki =114.6 − deg (measured from +y0 axis in CW direction)

Initial case: Since all the cutting edge elements (S1k, S2k and S3k) are available-to-cut (all

g1, j k =1 for j=1,2,3), initial instantaneous missed cut condition is set to 1 for all teeth:

g3,ki = [111]. Edges in cut array Na is assumed to include all edges: Na = [123] . No edge

is assigned into the new edges in cut array Na,new yet: Na,new = []. It is assumed in the beginning

that the assigned arrays are correct and will not be updated during the loop, and there is a binary

flag parameter is initially set to zero: “sameNotSame=0”. If this “sameNotSame“ changes to 1

during the first loop, then the computation will continue to the second loop (see Figure 4.1).

First loop: Starting with the first edge ( jNaa=1.. ) in the edges in-cut list, the pitch angles for

each edge are calculated from the angular locations of the edges in cut:

f=−=ψ ψforjN 1.. , (4.5) p,NNaa ( j ),kiaa ( j +1),k Naa ( j ),k a a

φ = which yields p,ki [120 120 120]1xN (deg), as expected for regular pitch cutter. Actual feed

a per edge array cki is found using the pitch angle array:

fp,ki ca = f , (4.6) ki2π r

a = − pa which leads to cki [0.040 0.040 0.040] mm . Actual chip load ( cki ) at the inputted

instantaneous angular position (φ2ki ) is:

pa a ccki =ki ⋅sinφ 2ki , (4.7)

pa with cki = [36.4e-3 36.4e-3 36.4e-3] -mm. Reduction in chip load array ( γki ) is found by

considering the effect of the runout of the previous in-cut tooth (ε ) to the current one ( r,Naa ( j +1),k

ε ): r,Naa ( j ),k 86

γε= − ε, (4.8) NNaa(j ),ki r,aa (j +1),k r, Naa (j ),k

which gives γki =−−[ 0.0147 0.1618 0.1766] -mm. Negative value of γ jki means that element Sjk

removes more chip than the actual value. If the reduction in chip load ( γ jki ) is more than the

actual chip load for element Sjk, then its missed cut condition parameter ( g3, jki ) is set to zero:

pa checkg jki ≥==cjki ; if yes :set g3,jki 0; if no: keep g3,jki 1. (4.9)

This updates the missed cut array: g3,ki = [110] . Thus, the third edge is not able to cut at

this time step. Edges-in-cut array is updated according to the missed cuts, Na,new = [1 2] , i.e.,

only edges j=1 and j=2 are in cut. Since the updated and original g3,ki are different, the flag

parameter is updated: “sameNotSame=1”. The “sameNotSame“ parameter changed to 1 during the first loop, then the computation continues to the second loop (see Figure 4.1).

Second loop: The calculation of the second loop starts with setting the actual edges in cut

array to the previously updated edges-in-cut array: Na = [1 2] . Number of edges in cut is

Na = 2 . Since edge number 3 (j=3) is not in the Na list, parameters related to j=3 have no

meaning, but the vector sizes are still kept at 3 for consistency. Pitch angles for each edge in-cut

( ja =1 and ja = 2 ) are found similar to the first loop:

fp,N (1),ki=−==−ψψ NN (2),k (1),kforj a 1,f p,1ki 120 deg, a aa (4.10) f =−+ψψ 2π forj = 2,f =− 240 deg, p,Na (2),ki NN aa (1),k (2),k a p,2ki

φ = which leads to the updated pitch angle array: p,ki [120 240 0]1xN deg. The 0 value in the

rd array is not used in the calculations since the 3 edge is not cut ( g3,3ki = 0 ). Using Equation (4.6)

a , the actual feed per edge array is: cki = [0.040 0.080 0] − mm . Actual chip load is found from

pa Equation (4.7) at the instantaneous location angle: cki = [36.4e-3 72.7e-3 0.0] -mm. The

decrease in chip load is calculated similar to Equation (4.8) using the actual edges in cut:

γN(1),ki=−== εε NN (2),k (1),kforj a 1, γ 1ki − 0.0147 mm, a aa (4.11) γ=−== εεforj 2, γ 0.0147 mm, Na(2),ki NN aa (2),k (1),k a 2ki

which leads to the updated array: γki =[ −0.0147 0.0147 0] -mm. Carrying out a similar evaluation as in Equation (4.9) updates the missed cut array as:

checkg ≥==cpa ; if yes :set g 0; if no:set g 1; jki jki 3,jki 3,jki (4.12) g3,ki = [1 1 0.]

The updated missed cut array is the same as the previous one (result of the first loop in

Equation (4.9)), and the flag parameter for the convergence of iteration is set to

sameNotSame=0.

At the given angle (φ2ki =114.6 − deg ), the algorithm gives the following output at segment-k

for the selected edge (jout=2):

a Actual feed per edge:c2ki = 0.080-mm; Decrease in chip load:g 2ki = 0.0147-mm; pa a Actual chip load:cc2ki =⋅=2ki sinf 2ki 0.0727-mm; pa Effective chip load: c2ki -g 2ki =0.0580-mm; (4.13) Missed cut condition:g3,2ki = 1;

Actual pitch angle:fp,2ki = 240-deg; π fp,2ki 240 Actual delay period:T = = 180 = 0.04-s. 2ki 2πΩ⋅ 2π 1000 60 60

Figure 4.3 Schematics for the 2-insert drill with axial runout.

Figure 4.3 shows the schematics, radial geometry discretization and the axial runout

definition for the drill. Axial runout definition is similar to the radial runout definition of the

milling cutter (Figure 4.2). Radial geometry discretization for turning, boring and drilling cutter is equivalent to the axial geometry discretization for the milling cutter. The algorithm for

determining the actual chip load is exactly the same for the milling tool, except the calculation in

Equation (4.7). The chip load is constant for all rotation angles due to the feed motion in the z0

axis:

pa a ccki = ki (4.14)

Figure 4.4 Parallel turning schematics: (a) Radial discretization; (b) effect of axial offset and axial runout

between cutters.

Parallel turning operation shown in Figure 4.4 has a similar structure as the drilling operation, except that the radial geometry discretization (Figure 4.4a) starts from the diameter of the machined part of the workpiece (smaller diameter) and increases towards to the nonmachined diameter (larger diameter). The chip load with runout effect (Figure 4.4b) is the same as the drilling operation.

Chip length: When the axial segment height is fixed to dz, then for the true cutting edge

* angle kr,jk at the element Sjk, the chip length is defined as the projected length of the cutting edge along the tangential (T) direction (Figure 2.1a):

dz l = c,jk * . (4.15) sinkr,jk

st d Chip thickness ( hjki ): The static ( hjki ) and dynamic ( hjki ) chip thickness terms are time- dependent due to the runout. The static component depends on the feed amount and the geometry of the tool, whereas the dynamic component depends on the flexibility of the system and the geometry, as presented in later chapters. The instantenous chip thickness is evaluated as:

st d hhhjki= jki + jki . (4.16)

Chip Area: The time dependent chip area at the element Sjk is found from the product of the chip thickness and the chip length:

Ac,jki= hl jki ⋅ c,jk . (4.17)

4.2 Modeling of Cutting Forces

Cutting force is separated into chip shearing and ploughing components [25,27,116]. If all

the tool angles are identified, then the shearing forces can be predicted using the shear

parameters that are identified from orthogonal cutting experiments [35,36]. The angles derived in

Chapter 3: are used to predict the rake face cutting force coefficients. While the shearing forces

are defined in UV coordinate frame, the ploughing forces are calibrated in RTA frame. The

mechanism of ploughing is related to the tool indentation into the workpiece in cutting edge

plane (Ps), which includes the radial (R) and tangential (T) directions. Thus, it is assumed that there is no ploughing force component in axial (A) direction. Huang et al. [30] separated the ploughing forces into static (edge force [34]) and dynamic (process damping [117]) parts. Edge

forces (static ploughing forces) in radial and tangential directions are assumed to be linearly

proportional to the chip length [29] with edge force coefficients as multipliers. One way to

predict the edge coefficients is to create a generic database [34,118] for the tool-workpiece material combination and tool coating. A second way is to calibrate for each individual cutting operation. For example, Atabey et al. [47] calibrated the edge forces of boring operation for fixed depth of cut, fixed spindle speed and varying feedrate. Dynamic component of the ploughing force is linear to the extruded workpiece volume under the tool flank [119]. The dynamic part

(process damping) is analytically modeled by Chiou and Liang [120] for small amplitude vibrations of tool relative to workpiece. Recent methods are available for large edge radius tool under large amplitude vibrations [121,122].

Cutting forces without process damping: Shearing force is constituted by a combination of the friction (Fu) and the normal (Fv) forces on the rake face. Without the process damping 92

c (dynamic ploughing) forces, the rake face forces are separated into shearing ( Fjki ) and static

es ploughing ( Fjki ) components, as:

c es c e FFFKhvjki=+= jki jki( jki,, c,jkγγ n,jk) ⋅ A c,jki + Kv( c,jk, n,jk) ⋅ l c,jk , (4.18)

c where the shear component ( Fjki ) is expressed as a linear function of the chip area:

c c es e c es Fjki= KhlFKlFF jki ⋅⋅ jki c,jk;; jk = jk ⋅ c,jk jki = jki + F jk , (4.19)

where chip length ( lc,jk ) is given in Equation (4.15). Due to the variation of cutting velocity, the

normal rake angle and chip thickness along the geometry, the cutting coefficient

c e Kh( jki,, v c,jkγ n,jk ) , and the edge coefficient Kv( c,jk,γ n,jk ) are geometry-dependent. Moreover,

c Kh( jki,, v c,jkγ n,jk ) is also time-variant due to the time dependence of the chip thickness

(especially for the milling operation). Although the chip size effect is not significant when using

the linear force models, the effect would be more pronounced when nonlinear force models are

used, such as the Kienzle model (see [123]):

cexF Fjki= Kv( c,jk,,γγ n,jk) ⋅( h jki) ⋅+ l c,jk Kv( c,jk n,jk) ⋅ l c,jk , (4.20)

Range of parameter xF is typically: 01<≤xF . The linear force model in Equation (4.18) is obtained, if xF =1 in Equation (4.20). Equations (4.18) and (4.20) are nonlinear due to their dependence on the total chip thickness, but they can be linearized if the dynamic component (

d st hjki ) of the chip thickness is dropped for a rigid case, leaving only the static term ( hhjki≈ jki ).

The cutting forces in Equations (4.18) and (4.20) are thus linearized around static part of

instantaneous chip thickness:

c st e Fjki= Kh( jki,, v c,jkγγ n,jk) ⋅⋅+ h jki l c,jk Kv( c,jk,; n,jk) ⋅ l c,jk − (4.21) c st xF 1 e Fjki= Kv( c,jk,γγ n,jk) ⋅⋅ x F( h jki) ⋅ h jki ⋅ l c,jk + Kv( c,jk,. n,jk) ⋅ l c,jk 

Cutting coefficients can be identified mechanistically by fitting the coefficients specifically for each cutting operation empirically from cutting tests, which is costly [6,32,124]. A more effective method is to design an orthogonal cutting database with shear angle, shear stress and friction coefficients specific to each tool-workpiece material combination [34]. The orthogonal parameters can be transformed into the oblique geometry of any machining operation and the tool geometry. For example, for AISI 1045 steel, the fitted equations for the local (of element Sjk at time ti) shear stress τs,jki (MPa), shear angle φn,jki (rad) and average friction angle βa,jki (rad),

st as functions of static chip thickness hjki (mm) and cutting velocity vc,jk (m/min) are [118]:

st ts,jki =450.3 +⋅ 0.4vc,jk + 227.5 ⋅hjki ; st φn,jki =arctan( 0.4 + 0.0005 ⋅vc,jk +⋅ 0.6hjki ) ; (4.22) β =26.8- 0.0313⋅+v 11.77 ⋅⋅hst π ; a,jki ( c,jk jki ) 180 st limits :hjki ∈∈[ 0.08, 0.32] -mm; vc,jk [ 50, 150] -m/min; γ n,jk ∈[ 0, 0] -rad.

Refs. [35,36] derive the total force acting on the rake face based on the orthogonal cutting principal of Merchant [24]. The total shearing force is projected onto the friction u and normal v directions using the friction βa,jki angle:

cc cc FFu,jki= jki ⋅⋅sin ββa,jki ; FFv,jki= jki cos a,jki . (4.23)

The projected cutting forces are expressed in terms of the chip area Ac,jki of the element Sjk at time ti and the cutting coefficient:

cc cc FKu,jki= u,jki ⋅⋅AAc,jki ; FKv,jki= v,jki c,jki . (4.24)

Equation (4.24) represents the shear component of the force expressed in Equation (4.21).

The chip area ( Ac,jki ) is given in Equation (4.17). Using the orthogonal to oblique cutting transformation [6,35,36], the cutting coefficients of Equation (4.24) are obtained for the cutting edge element Sjk at time ti, as:

t1− tan22ηβ sin c s,jki jkn, jki Ku,jki = sin βa,jki; 2 22 cosλφs,jk sin n,jki cos ( φβγn,jki+−+ n, jki n, jk ) tan ηjk sin βn, jki (4.25) t1− tan22ηβ sin (d)kz c s,jki jkn, jki Kv,jki = cos βa,jki. 222 cosλφs,jk sin n,jki cos ( φβγn,jki+−+ n, jki n, jk ) tan ηjk sin βn, jki

The projection of the friction angle on the cutting edge normal plane (Pn) is:

βn,jki = arctan( tan βηa,jki ⋅ cos jk ) . (4.26)

Stabler’s rule [125] is assumed for the chip flow angle: η jk ≅ λs,jk . This is a fair assumption for large depth of cut where the chip flow angle approaches the cutting edge inclination angle

[124,126].

4.3 Force Transformations for Arbitrary Tools

Forces predicted on the rake face are transformed to the machine coordinates with the following order:

• From the rake face UV frame to the cutting edge RTA frame;

R R R • From the cutting edge RTA frame to the edge reference x y z frame;

R R R 0 0 0 • From the edge reference x y z frame to the tool reference x y z frame;

0 0 0 • From the tool reference x y z frame to the machine reference xyz frame.

Figure 4.5 Transformation of rake face forces (Fu and Fv) into the forces along the R,T and A axes of the

cutting edge plane Ps. (Adapted from [1,127].)

4.3.1 Rake face UV (friction-normal) to Cutting Edge RTA (radial-tangential-axial)

Starting from the rake face force vector, the transformations are sequenced in accordance with Figure 4.5:

• Negative chip flow angle ( −η jk ) rotation around V-axis;

III • Negative normal rake angle rotation (- γ n,jk ) around the new y-axis (Y );

II • Negative cutting inclination angle (- λs,jk ) around the new z axis (Z );

• RTA frame (xIyIzI frame) is obtained.

Resulting forces in RTA frame is represented by the rake face forces in UV frame, as:

cc  Frta,jki=  TF IU,jk⋅ uv,jki , (4.27) 3x1 3x2 2x1

T T c=  ccc c=  cc with Frta,jkiFFF r,jki t,jki a,jki and Fuv,jkiFF u,jki v,jki . Similarly, for the cutting coefficients, Equation (4.27) can be rewritten using Equation (4.24), as:

cc cc FKrta,jki=⋅=⋅ rta,jkiAA c,jk;; FK uv,jki uv,jki c,jk (4.28) cc  Krta,jki=  TK IU,jk⋅ uv,jki . 3x1 3x2 2x1

T T c=  ccc c=  cc with Krta,jkiKKK r,jki t,jki a,jki and Kuv,jkiKK u,jki v,jki . The transformation for the element Sjk is,

cosγη cos −sinγ n,jk jk n,jk TIU,jk= sinλη s,jk sin jk+ cos λγ s,jk sin n,jk cos η jk cos λ s,jk cos γ n,jk . (4.29)  −+cosληs,jk sin jk sin λγ s,jk sin n,jk cos η jk sin λ s,jk cos γ n,jk

If the cutting coefficients in RTA frame are identified mechanistically from cutting tests, they would be projected onto the UV frame (on rake face), as is similar to Equation (4.27):

cc−1 KTuv=( IU,jk) ⋅ K rta . (4.30) specific geometry

The edge forces or ploughing forces are defined in the RTA frame. The occurrence of ploughing forces is due to the indentation of the cutting edge into the workpiece, thus the creation of a normal force in R direction. A corresponding friction force acts in T direction, which is the same as the cutting velocity direction. Theoretically, the direction A has zero ploughing force as no indentation movement occurs towards the sides of the cutting edge. The edge forces are written from the Equations (4.15) and (4.19) as:

es e  dz FKrta,jk = rta,jk ⋅ , (4.31)  3x1 3x1 * sinkr,jk 97

T T es=  es es es e=  eee where, Frta,jkFFF r,jk t,jk a,jk . The edge coefficients are Krta,jkiKKK r,jk t,jk a,jk

e and Ka,jki ≅ 0 . The edge coefficients can also be identified from the generic database, which is

obtained by conducting orthogonal cutting tests [34]. Similar to fitting shear parameters, edge coefficient equations (in N/mm) for AISI 1045 steel, for example, are as follows [118]:

e 32 Kt,jk =9.0688e-6 ⋅( vc,jk ) - 1.0382e-2⋅( vvc,jk ) + 1.9856 ⋅c,jk - 55.431; 32 Ke =2.9929e-4 ⋅ v- 9.8351e-2⋅ vv + 10.299 ⋅- 294.96; r,jk ( c,jk ) ( c,jk ) c,jk (4.32) e Ka,jk = 0 (as expected); st limits :hjki ∈∈[ 0.08, 0.32] -mm; vc,jk [ 50, 150] -m/min; γ n,j k ∈[0, 0] -deγ.

t With no vibrations in the system, the total cutting force Frta,jk acting on the element Sjk of

the tool at time ti is found in RTA frame from Equation (4.28) and Equation (4.31):

t c es FFFrta,jki= rta,jki + rta,jk ; tc e FKrta,jki= rta,jki ⋅hljki⋅ c,jk +⋅ Krta,jk lc,jk ; (4.33)

hjki 1 FKtc= ⋅dzz +⋅ Ke d. rta,jki rta,jki **rta,jk sinkkr,jk sin r,jk

4.3.2 RTA to Edge Reference Frame (xRyRzR), and to Process Frame (xyzθ)

The tool coordinate frame (x0y0z0) is same as the process coordinate frame (xyzθ) for

drilling, turning, boring and three axis milling operations. For 5-axis milling, there would be

additional lead and tilt angle rotation transformations [49]. Torsional (θ) degree of freedom of

the tool (for milling, drilling, boring) or workpiece (for turning, boring, drilling) is added as the

4th degree of freedom. The corresponding torque is calculated using the force in T direction,

which is in the velocity direction and is perpendicular to the radius vector (Figure 4.5).

Forces in Radial-Tangential-Axial (RTA or xIyIzI) coordinate system is transformed into

process reference coordinates in the xyzθ frame. The intermediate step is the cutting edge reference coordinates in the xRyRzR frame. RTA (xIyIzI) is a right-hand coordinate system, and the

cutting edge forces of all the machining operations are represented within this frame. Figure 4.6

shows the schematics of the turning, boring, drilling and milling operations. Since the cutting

edge rake face (Figure 4.5) is standard for all cutters, the orientation of RTA (xIyIzI) along the

cutting edge is the same for all the operations. Force transformation has the same form for all the

operations:

TT⋅ cc0R,jki RI,jk FFjki = ⋅ rta,jki , (4.34) 00Rt, jk

c c c where the forces in tool reference frame include the forces in 3 axes ( Fx,jki , Fy,jki , Fz,jki ) and the

c torque Tjki on the element Sjk at time ti:

T c=  c c cc FjkiF x,jk i F y,jk i FT z,jk ii jk . (4.35)

Rt, jk in Equation (4.34) is the moment arm (radius) of the cutting edge element Sjk.

c c c Components of force vector are the feed ( Fx,jki ), normal ( Fy,jki ) and axial ( Fz,jki ) forces and

c torque (Tjki ) on element Sjk at time ti. TRI,jk is the transformation matrix from RTA to cutting

R R R 0 0 0 edge reference x y z , and T0R,jki projects the forces onto the tool reference frame x y z . The

transformation for each operation can be derived from the given orientations shown in Figure

4.6.

Transformations for the rotating right hand boring and right hand drilling tools:

Following from Figure 4.6d and Figure 4.6e, the transformations for the rotating boring and drilling operations are:

−−coskk** 0 sin r,jk r,jk TRI,jk = 010. (4.36)  ** sinkkr,jk 0− cos r,jk

T0R,jki transforms force and deformation from cutting edge reference to tool coordinates:

sinφφjki− cos jki 0  T0R,jki = cosφφjki sin jki 0 , (4.37)  0 01 where φjki is the instantaneous angular position of the element Sjk at time ti:

ti φjki =∫ ω(tt )d +ψ jk . (4.38) 0

ω()t is the angular speed of the cutter (in rad/s). ψ jk is the angular location of element Sjk and is relative to the +y0 axis (measured in CW direction).

Transformations for the nonrotating right-hand boring cutter: Traditional right-hand boring (Figure 4.6b) is a special case of the rotating boring operation. The equations are kept the same as in Equations (4.36) and (4.37), but the rotation angle term is removed:

φjki =ψ jk . (4.39)

The nominal value for the instantaneous angular position is 0-deg.

100

Transformations for the left-hand turning cutter: Left hand turning (Figure 4.6a) is the external scheme of the right hand boring operation. The model is the same as in Equations (4.36) and (4.37) for a rotating boring-cutter, but the rotation angle is removed from the instantaneous location angle term of Equation (4.38):

φjki =ψ jk . (4.40)

The nominal value for the angular position is -90 degrees.

Transformations for the right-hand milling cutter: Right hand milling (Figure 4.6c) has a similar transformation matrix TRI,jk as the right hand turning operation:

−−sinkk** 0 cos r,jk r,jk TRI,jk = 010. (4.41)  ** coskkr,jk 0− sin r,jk

The T0R,jki transformation is as the same as the drilling operation:

sinφφjki− cos jki 0  T0R,jki = cosφφjki sin jki 0 , (4.42)  0 01 with the same instantaneous location angle definition:

ti φjki =∫ ω(tt )d +ψ jk . (4.43) 0

The rotation angle term in Equation (4.43) is written in general form to include spindle

2π speed variation. For the constant rotational speed, ω()t = Ω : 60

2π φjki = Ω⋅ti+ψ jk . (4.44)  60 

101

Figure 4.6 Transformations from RTA to xRyRzR, and from xRyRzR to x0y0z0 for basic machining operations.

(Images adapted from Sandvik and Ref.[3].) 102

Transformations for the right-hand turning cutter: Right hand turning cutter is shown in

Figure 4.7a. The TRI,jk transformation is written as:

coskk** 0− sin r,jk r,jk TRI,jk = 01 0. (4.45)  ** sinkkr,jk 0 cos r,jk

The T0R,jki transformation is as same as the traditional right-hand boring operation.

sinφφjki− cos jki 0  T0R,jki = cosφφjki sin jki 0 (4.46)  0 01

With the constant angular location:

φjki =ψ jk (4.47)

The nominal value for the angular position is 0-deg (Figure 4.7a), as similar to the traditional right-hand boring operation.

Figure 4.7 Right hand turning and parallel turning schematics. (Images adapted from Sandvik.)

103

A note for the parallel turning operation: The nominal angular positions of tool #1 and

tool #2 of the parallel turning operation (Figure 4.7b) are 90 and 270 degrees, respectively.

Transformations for the left-hand milling cutter: Left-hand milling cutter is used for the application of proposed general mathematical model on a double milling application. As shown in Figure 4.8, left- and right-side indexable cutters are labeled according to left-hand and right-

hand milling action, respectively.

The transformations for the left-hand cutter (Figure 4.9a) are given as [4]:

** −−sinkkr,jk 0 cos r,jk sinφφjki,L cos jki,L 0  TTRI,jk,L = 0 1 0 ; 0R,jki, L = cosφφjki,L sin jki,L 0 , (4.48)  ** − coskkr,jk 0 sin r,j k 0 01

and for the right-hand cutter (Figure 4.9b):

** −−sinkkr,jk 0 cos r,jk sinφφjki,R− cos jki,R 0  TTRI,jk,R = 0 1 0 ; 0R,jki, R = cosφjki,R sinφ jki,R 0 . (4.49)  **  coskr,jk 0− sinkr,jk 0 01

The instantaneous angular positions for left- and right-side cutters are defined as:

2π φjki,x = Ω⋅ti+ψ jk,x (4.50)  60 

where, for the left-side cutter: x=L, and for the right-side cutter: x=R.

104

Figure 4.8 Double-sided milling schematics.

Figure 4.9 Double-sided milling tool geometry [4]: (a) Left-side cutter; (b) right-side cutter.

4.4 Transformation of the Vibration Vector to the Dynamic Chip Thickness

The relative vibrations between the tool and workpiece are needed to calculate the dynamic

cutting forces and chatter stability. A vibration vector defined in machine xyzθ frame at k-th

segment at time ti is:

T θki= [xyz ki ki kiθ ki ] . (4.51)

The dynamic chip load ∆q jki of j-th tooth at segment-k is dependent on the difference of the surface left at the previous tooth pass:

∆=−qjki qq ki k()tT i − jki , (4.52) 105

where the delay period Tjki for the cutting edge element Sjk is both geometry- (at tooth-j and

segment-k) and time (ti) -dependent.

The vibrations in the tool reference frame are projected in the chip thickness direction. The

positive chip thickness direction is in the negative-radial (R) direction (-xI) of Frame-I (xIyIzI

frame) as shown in Figure 4.5. Thus. the dynamic chip thickness is:

dI hxjki= −∆ jki . (4.53)

The effect of the regenerative vibrations of the tool-workpiece system is projected onto the

negative-R direction. The feed (i.e. rigid body motion) may contribute to the regenerative

vibration in the torsional direction ( ∆θki ) in some applications, such as drilling and plunge milling [128], but this effect is negligible compared to the structural vibrations of the system.

However, the torsional-axial dynamic coupling for most of the drilling tools (e.g. twist drills) is strong, with the torque consequently causing an axial deflection [129]. At time ti, the following

I transformation orients the regenerative vibration vector ∆q jki at k-th segment to the −∆xjki

vector of the element Sjk:

Id −∆xjki =e jki ⋅∆q jki , (4.54)

where the general transformation vector from tool reference coordinates to the negative-R is [3]:

d T ejki =[−100] ⋅⋅(TT0R,jki RI,jk ) T0M (4.55)

Since the translational axes of process (xyzθ) and tool (x0y0z0) coordinate systems are taken

to be the same (in this thesis), the transformation is,

1000  T0M = 0100. (4.56) 0010

106

The last column is zero since the effect of torsional-axial coupling due to the feed (rigid body motion) is neglected. The dynamic chip thickness is represented by the relative vibrations between the tool and workpiece structure, which can be estimated from Equations (4.53), (4.54)

and (4.55) as:

d T hjki =[ −100]⋅⋅(TT0R,jki RI,jk ) Tq0M⋅∆ jki . (4.57)

The total chip thickness is represented in the general form from Equation (4.16):

st d hhjki= jki +e jki ⋅∆q jki . (4.58)

4.5 Representation of General Force Vector

The force vector is composed of shearing, ploughing and process damping components [2].

The shearing force is needed to plastically deform the material and form the chip, hence it is the major force with static and dynamic parts, as is expressed in Equations (4.18) and (4.19). The shearing force is written in general form by substituting Equation (4.58) into Equation (4.33), and into Equation (4.34) successively:

st+ d ⋅∆ TT0R,jki⋅ RI,jk (hjkie jkiq jki ) FKcc= ⋅⋅ dz . (4.59) jki rta,jki * 00Rt, jk sinkr,jk

The shearing force can be represented by the static cs and dynamic cd shear components as:

c cs cd FFFjki= jki + jki; st TT0R,jki⋅ RI,jk hjki FKcs = ⋅⋅c d;z (4.60) jki rta,jki * 00Rt, jk sinkr,jk d TT0R,jki⋅ RI,jk e jki⋅∆q jki FKcd = ⋅⋅c d.z jki rta,jki * 00Rt, jk sinkr,jk

The dynamic component is written in a separated force coefficient and displacement terms: 107

cd d FFjki= jki ⋅∆ q jki , (4.61)

where the dynamic force component is:

d TT0R,jki⋅ RI,jk e jki FKdc= ⋅⋅ dz . (4.62) jki rta,jki * 0 Rt, jk 0 sinkr,jk

es The ploughing force Fjki is the constant term of Equation (4.33) and does not depend on chip thickness, It is expressed from Equations (4.33) and (4.34) as:

TT0R,jki⋅ RI,jk 1 FKes = ⋅⋅e dz . (4.63) jki rta,jki * 0 Rt, jk 0 sinkr,jk

Process damping terms are caused by the plastic deformation of the edge of the tool pressing

into the workpiece material [119]. The average force is modeled as the friction and normal terms

in T and R directions, respectively [120]. As shown in Refs. [2,3], the process damping is

cd generalized similar to the dynamic shearing force term ( Fjki of Equation (4.60)) as:

2d TT0R,jki⋅ RI,jk KLsp we jki⋅q ki FPed =⋅ ⋅⋅ dz (4.64) jki 0* 00Rt, jk 4⋅vc,jkisink r,jk

T where P={10µ } and µ is the Coulomb friction coefficient. Ksp is the material-dependent

specific contact force; Lw is flank wear land length of the tool. Although it is not within the

scope of this thesis, Equation (4.64) could be extended to more advanced models, such as the

models proposed in Refs. [121,122]. Similar to the dynamic shear force, the process damping

component is written in a separate force coefficient and velocity terms (see Equation (4.61)) as:

ed p Fjki= Fqjki ⋅  ki , (4.65)

where the average force component is:

108

2d TT0R,jki⋅ RI,jk KspL we jki FPp =⋅ ⋅⋅ dz . (4.66) jki ⋅ * 0 Rt, jk 0 4 vc,jki sinkr,jk

Then, the total force in machine coordinate frame (xyzθ) acting at the tool-workpiece contact

is represented in the general form:

FF=++c Fes Fed . (4.67) ji ji ji ji shearing static ploughing process damping

Equation (4.67) represents the force acting on the tool:

t FFjki= jki . (4.68)

Hence, the force acting on the workpiece is in the opposite direction:

w FFjki= − jki . (4.69)

For the general case, the total cutting force is given in Equation (4.67). There are two special

cases: Rigid system and stable cutting. If the tool-workpiece system was rigid, the vibration

related displacement and velocity terms are set as zero, as shown in Equation (4.67). Thus, the static forces are:

st cs es FFFjki= jki + jki . (4.70)

If the tool-workpiece is flexible and the process is stable, then the delay terms are set to zero

as shown in Equation (4.67), giving the steady state forces:

ss cs es ed FFFFjki=++ jki jki jki . (4.71)

The differential force Fjki is applied at the cutting edge element Sjk of the arbitrary cutter.

Although the differential component is general, forming the general cutting force vector Fi

depends on the geometry and dynamics of the cutting operation.

109

4.5.1 Flexible system with lumped force and dynamics at one tool-workpiece contact

point

This is the most common occurrence when the dynamics is assumed constant along the tool- workpiece contact. This assumption is valid when the following conditions are true:

• The depth of cut is small;

• The workpiece rigidity is high compared to tool rigidity;

• Small variation of dynamics along the tool-workpiece contact.

The cutting force vector at time ti is simply the summation of all the cutting edge elements at

the point representing the average dynamics:

Nq, =  FFi∑ gg 1,jk 2,jki g 3,jki jki , (4.72) jk, 4x1

where q is the number of levels for the depth of cut ap:

a q = p . (4.73) dz

Figure 4.10a shows the lumping of cutting forces for the cylindrical end mill. The forces are

predicted at each axial segment-k and all of the forces are lumped at the single point with the

averaged dynamics. Since the variation of dynamics along the tool-workpiece is assumed to be

low, the forces can be lumped at any point, preferably at the point of the dynamic identification

(e.g., hammer test). The same principle is applied to the turning, boring and drilling operations.

The forces generated at all the radial segments are lumped at the point where the dynamics are

identified.

110

Figure 4.10 Modelling of cutting forces: (a) Lumped force when low flexilibility variation; (b) axially distributed force when high flexibility variation (adapted from Ref.[2]); (c) distributed force when high

flexibility variation along tool profile.

111

4.5.2 Flexible system with distributed force along tool-workpiece contact

If the flexibility of the system is varying over the tool-workpiece contact (as in Figure 4.10b-

c), lumping the forces at a single point may not fully represent the system dynamics, especially

when finding machining surface errors.

Axially distributed force: In the case of a long end mill with flexible tool-workpiece

dynamics along the contact zone, the forces are axially distributed along the contact length

(Figure 4.10b). This is common for finishing operations with large depth of cut, ap. The force

vector is formed by keeping the forces on each differential segment-k:

gg1,j1 2,j1i g 3,j1iF j1i  gg1,j2 2,j2i g 3,j2iF j2i N   Fi = ∑ . (4.74) j gg g F 1,jk 2,jki 3,jki jki   gg g F 1,jq 2,jqi 3,jqi jqi (4q) x1

Radially distributed force: In the case of face milling operation, the axial depth of cut is

small and the cutter is assumed to have constant dynamics along the tool-workpiece contact. The

variation of dynamics comes from the flexible workpiece, and each differential dynamic section

is radially positioned as shown in Figure 4.10c. The tool-workpiece is assumed to have constant

flexibility along axial direction. As shown in Figure 4.11a, at each instance of cutting the force is

lumped along axial direction (shown by Fji). At the current position of the cutter on the workpiece, if the flexible tool-workpiece contact is divided into s number of radial sections, the

force vector is defined as:

112

Figure 4.11 Radial force distribution on the flexible tool-workpiece zone: (a) Instantaneous axially lumped

forces; (b) limit angles for angular distribution.

gg1,jk 2,jki g 3,jki g 4,jki,1F jki  gg1,jk 2,jki g 3,jki g 4,jki,2F jki  Nq,  [F ] = ∑ , (4.75) i (4s) x1 gg g g F jk, 1,jk 2,jki 3,jki 4,jki,r jki   gg g g F 1,jk 2,jki 3,jki 4,jki,s jki (4s) x1

where the new function at point-r in the workpiece contact zone is defined as:

1 if ElementSrjk falls in the -th radial zone at time ti g4,jki,r =  . (4.76) 0 if ElementSrjk does not fall in the -th radial zone at time ti

The r-th zone is defined by the angles bounding the radial sections in the tool-workpiece

contact (Figure 4.11b). The entry and exit angles (φst,r and φex,r ) for each zone-r can be assigned

using unequal or equal step angles depending on the measured dynamics of the structure.

4.5.3 Multiple-Level Tool with Lumped Force at Each Station/Level

If the tool has more than one dynamic levels (or stations), then the forces are lumped locally at each dynamic level [5].

113

Multifunctional tool: Multifunctional drilling/boring cutters have D levels (minimum D is

2) in contact with the workpiece. In this case, the dynamics is assumed to be constant within

each dynamic level. The force vector is formed by lumping the force terms at each level

accordingly. Forces at each level are lumped according to the limit radial level value (de with

e=1..D), as:

Nd, 1 ∑ gg1,jk 2,jki g 3,jkiF jki j,k= 1 Nd, 2 ∑ gg1,jk 2,jki g 3,jkiF jki = + jd,k1 1 • Nd, Fi = e . (4.77) ∑ gg1,jk 2,jki g 3,jkiF jki = + jd,ke-1 1 •  •

Nd, D ∑ gg1,jk 2,jki g 3,jkiF jki = + jd,kD 1 [4⋅D] x1

The dynamic level limit vector ( d ) is used for storing the last radial segment index of each

level:

d=[dd12 d e d D] , (4.78)

with de being the last radial segment index of the e-th dynamic level, e.g., radial segments from

“d1+1” to “d2” is lumped onto dynamic level-2 (e=2). If the number of dynamic levels is one

(i.e., D=1), then the total number of radial segments in contact (q) is assigned to d1=q and the

Equation (4.77) reduces to the form of Equation (4.72).

114

Figure 4.12 Sandvik TM880 multifunctional tool with D=2 dynamic levels.

For example Sandvik multifunctional tool has 2 dynamic levels, therefore D=2. As shown in

Figure 4.12, for Level-1 the forces from radial segment-1 to radial segment-d1 are lumped on

Level-1 dynamics; forces from d1+1 to d2=K are lumped onto Level-2 dynamics.

Double-sided milling cutter: There are two sides in the double-sided milling cutter (Figure

4.13). Each side of the cutter has s number of sections (Figure 4.11) to approximate the dynamics along the tool-workpiece contact zone. The force vector is formed by writing the similar force vectors for both left- and right-side cutters, and joining them, as follows:

115

Figure 4.13 The double-sided milling cutter with forces applied at the opposite sides of the workpiece.

FL F = i ; i R Fi (2⋅ 4s) x4 ggxx g x g xF x 1,jk 2,jki 3,jki 4,jki,1 jki xx x x x gg1,jk 2,jki g 3,jki g 4,jki,2F jki Nq,  x  Fi = ∑ ; (4.79) jk, ggxx g x g xF x 1,jk 2,jki 3,jki 4,jki,r jki   ggxx g x g xF x 1,jk 2,jki 3,jki 4,jki,s jki (4s) x1 L if left-side cutter x.=  R if right-side cutter

Multiple-tool turning system (Parallel turning): Parallel turning application (Figure 4.14) is very similar to multifunctional cutter, except that it has two tools with individual dynamics for the tool #1 (j=1) and tool #2 (j=2):

116

Figure 4.14 Cutting forces of parallel turning operation.

q ∑ gg1,1k 2,1ki g 3,1kiF 1ki F = k . (4.80) i q ∑ gg g F 1,2k 2,2ki 3,2ki 2ki k [4⋅ 2] x1

Each tool experiences a lumped force, therefore acts like a separate dynamic level, as is shown in Figure 4.12. Following the principles of Figure 4.10a, forces on the radial segments of each tool are summed up at their own station.

4.6 Summary

Chip geometry and tool angles at each edge element Sjk is used for computing the

differential machining force. Shearing forces are transformed from rake face to process

coordinate system. Ploughing forces are transformed from the RTA frame to process coordinate

system. The forces are integrated along the cutting tool edges which are engaged with the

workpiece. The flexibilities of the tool-workpiece are included in the model.

117

Chapter 5: Generalized Modelling of Cutting Dynamics

The previous chapters have focused on the modelling of tool and chip geometry, kinematics of chip generation and corresponding cutting forces with and without vibrations; these are the foundations of cutting dynamics model. Cutting forces excite the structural dynamics of the tool and part. This chapter models the interaction between the generalized metal cutting process and the vibrating structure. The resulting delayed differential equations are solved in semi-discrete time domain to simulate the dynamic cutting forces, surface form errors and vibration amplitudes.

5.1 General Equation of Motion

The classical form for relating the force ( F()s ) and deformation ( Q()s ) in the Laplace (s)- domain can be expressed by:

Q()s[4q× 1]= ΦF () ss [4 qq ×× 4 ] () [4 q 1]. (5.1)

The transfer function matrix Φ()s of the structure is represented by the identified modal parameters of the system [64]:

2 2− 1T Φ()s[4qq× 4 ] =++ U ( I ss 2 ζωn ω n ) U , (5.2) where I is the identity matrix, ζ is the diagonal damping ratio matrix, ω is the []mm× []mm× n[mm× ] diagonal natural frequency matrix, and U[4qm × ] is the mass normalized mode shape matrix for the m number of identified tool and workpiece vibration modes. The transformation from physical to modal domain is carried out by:

QU()ss= ⋅Γ (), (5.3)

118

where Γ[m× 1] is the modal displacement vector expressed explicitly using Equations (5.1) and

(5.2):

2 2− 1T Γ()ss=++ (I 2ζωnn s ω ) U F () s. (5.4)

The reduced order modal model equation (Equation (5.4)) is converted into the time domain as:

2T Γ()t+ 2ζωnn ΓΓ  () tt += ω () U F () t. (5.5)

The physical displacement vector Q()t is obtained using Equation (5.3):

QU()tt= ⋅Γ (). (5.6)

If the machine tool’s modeled mass, stiffness and damping matrices are available, the equation of motion of the full model is:

  MQff()t++ CQ ff () t KQ ff () tt = F f (), (5.7) where subscript-f represents the parameters related to full order model of the machine tool. By applying the modal transformation:

QUff()tt= ⋅Γ (). (5.8)

The modal displacement vector Γ()t of the full order model is assumed to be equal to the one of the reduced order model presented in Equation (5.3). The Equation (5.7) is then reduced to the same equation as Equation (5.5):

2T Γ()t+ 2ζωn ΓΓ  () tt += ω n () U ff F () t. (5.9)

As long as the represented vibration modes are the same in reduced and full order models, their final modal equation would be the same (see Section 4.3 of Ref. [130]).

119

5.1.1 Reduced Dynamic Equation at the Non-cutting Zone

The analysis of interaction between the tool and workpiece structure maybe required both at

the cutting and non-cutting zones. The extended mode shape matrix Ue is defined to cover the

non-cutting and cutting zone points:

U Ue = . (5.10) Unc

Similarly, the full order force vector Ff ()t is renamed as extended force vector Fe ()t and

defined as follows:

F()t Fe ()t = . (5.11) Fnc ()t

If there are no external cutting forces acting in the non-cutting zone, the related force vector

is set to zero, F0nc ()t = , which yields to the extended form of Equation (5.9) as:

2TF()t Γ()t+ 2ζωnne ΓΓ  () tt += ω () U . (5.12) 0

The mode shape matrix Ue is extended by including the contribution of the non-cutting zone

to each existing mode. Moreover, more modes can be added to include the modes that do not

contribute to the dynamic cutting force, which can then be used to predict the vibrations of the

machine tool structure at the critical locations. The physical displacement vector then covers all

cutting and noncutting zone points as:

QUee()tt= ⋅Γ (). (5.13)

5.1.2 Dynamic Interaction Between the Tool and Workpiece Structures

The cutting forces act on both tool and workpiece structures at the contact zone (Equation

(5.5)):

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2T Γ t()t+ 2ζω t n,t ΓΓ t () t += ω n,t t () tt U t F (); (5.14) 2T Γ w()t+ 2ζω w n,w ΓΓ w () t +=− ω n,w w () tt U w F (), where the cutting force ( F()t ) acts both on the tool and workpiece with the same magnitude, but in opposite directions.. If q contacting points at the tool workpiece interface, with 4 flexibility directions at each point, are considered, the mass normalized mode shape matrices for the tool (

Ut ) and workiece ( Uw ) are:

uu1,1,t 1,n ,t u 1,m ,t tt      U = uu u ; t k,1,t k,ntt ,t k,m ,t      uu u q,1,t q,ntt ,t q,m ,t 4q×m t (5.15) uu1,1,w 1,n ,w u 1,m ,w ww      U = uu u . w k,1,w k,nww ,w k,m ,w      uu u q,1,w q,nww ,w q,m ,w 4q×mw

The mode shape vector contributions at axial/radial segment-k on nt -th (tool) and nw -th

(workpiece) modes are:

T u=uuuu ; k,nt ,t{ x,k,n tttt ,t y,k,n ,t z,k,n ,t θ,k,n ,t} (5.16) T u = uuuu . k,nw ,w{ x,k,n wwww ,w y,k,n ,w z,k,n ,w θ,k,n ,w} where x,y and z are the translation and θ is the torsional directions in the process coordinate system.

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Natural frequency and modal damping ratio matrices are defined for each tool and

workpiece for the corresponding number of vibration modes mt and mw (total vibration modes

m=mt+mw):

ωζn,1,t 001,t       ωζ=  ωζn,n ,t ;;= n ,t  n,t  ttt        00ωζn,m ,t m ,t  ttm ×m m ×m tt tt (5.17) ωζn,1,w 001,w       ωζ= ωζn,n ,w ; = n ,w . n,w  www        00ωζ  n,mww ,w  m ,w  mww ×m mww ×m

Reference coordinate directions of tool and workpiece are set to be the same. The equation

of motion (Equation (5.14)) is combined in one matrix form as:

2 T ΓΓ tt()tt2ζωt n,t 0  () ωn,t 0 Γtt()t U + += F()t . (5.18) ΓΓ ()tt0 2ζω  () 2 Γ ()t T www n,w 0 ωn,w w −Uw

The tool and workpiece coordinates are combined in short but general form as:

2T Γ c,i+2ζω c n,c ΓΓ c,i += ω n,c c,i U F i , (5.19)

where FFii= ()t is the force vector at time ti , and the combined tool and workpiece modal coordinate vectors are:

Γti()t ΓΓ ti()tt  ti () ΓΓΓc,i = ;; c,i =  c,i = , (5.20) Γwi()t ΓΓ wi()tt  wi ()

with the corresponding combined modal parameter matrices:

ζ0t ω0n,t ζω= ; = . (5.21) c 0 ζ n,c 0 ω w mmx n,w mmx 122

The combined mass normalized mode shape matrix is defined as:

UU=[ tw − U]. (5.22)

5.1.3 Cutting Force Representation

cs cd The structural response can be analyzed by considering the static fi , dynamic fi , delayed

d es ed fjki , ploughing fi and process damping fi force vector components at time ti individually, as

is shown in Equation (5.19):

0[4(km− 1) x ] Nq,  cs cd d es ed Fffi= i+ iΓ c,i −∑  fjki[4xm ]⋅ ΓΓ c(tT i − jki) ++ff i i c,i (5.23) jk,  0 [4(qk− )x m ]

cs The static component Fjki of the shearing force at k-th axial/radial segment can be localized from Equation (4.60) as:

0[4(k− 1)x 4] Nq,  cs cs fi [4q x1] = ∑ gg1,ki 2,jki g 3,jkiIF [4x 4] jki (5.24) jk,  0[4(qk− )x 4]

For the most general case, the tooth passing delay Tjki is a function of both geometry and

time. From Equations (4.61) and (4.62), the local delayed force term at the element Sjk and time ti

can be expressed as:

0[4(km− 1) x ] Nq,  cd d ffi = ∑ jki[4 xm ] ; (5.25) jk,  0[4(qk− )x m ]

0 [4(km− 1) x ] 0 −  [4(k 1)x 4] dd fjki[4xm ]= gg 1,ki 2,jki g 3,jkiI [4x 4] F0 jki [4x 4(k−− 1)] I [4x 4] 0 [4x 4(qk ))][ U t− U w ].(5.26)   0 0[4(qk− )x m ] [4(qk− )x 4]

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The ploughing term is derived from Equation (4.63):

0[4(k− 1)x 4] Nq,  es es fi [4q x1] = ∑ gg1,ki 2,jki g 3,jkiIF [4x 4] jki . (5.27) jk,  0[4(qk− )x 4]

The process damping component is obtained from Equation (4.66):

0[4(k− 1)x 4] Nq,  ed = p − fi ∑  I[4x 4] F0jki [4x 4(k−− 1)] I [4x 4] 0 [4x 4(qk ))][ U t U w ] . (5.28) jk,  0[4(qk− )x 4]

The force components are for a general structure with axially and radially distributed delay and dynamics. The forces are later projected to the desired locations in special machining applications with multifunctional cutter, double milling cutter and parallel turning tools.

The generalized linear dynamic equation of Equation (5.19) is solved in time-domain using a variation of the discrete time method developed by Stepan et al.[81,131]. The stability of cutting operations, cutting forces, torque and power, structural vibrations, and surface deflection errors are predicted in discrete time intervals. Only the delayed states are approximated, while the other time terms are kept continuous [81].

5.2 Solution of the Equation of Motion in Modal Domain

The general equation of motion represented in Equation (5.19) is converted into state space form to apply the time-marching solution method. An unconditionally A-stable 4th order method is used for solving the time history [132]. Delayed state vector is approximated using 4th order

Lagrange polynomials. The convergence rate of the solution method is checked by testing the increase in eigenvalue accuracy versus the decrease in sampling time size.

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5.2.1 State space representation

The second-order dynamic equation of motion Equation (5.19) is reduced to first-order form

 by using two states. Here, displacement Γc ()t and velocity Γc ()t states are used. The modal state vector of the tool-workpiece system at a time t is:

Γc ()t Θ()t =  . (5.29) Γ c ()t 2(mmtw+× ) 1

Equation (5.19) is then written in state-space form in continuous time domain as:

Nq, Θ()t= AΘ0 ( t i− 3 ) + B 1 () tt Θ () −∑ { B2,jk () ttTt Θ ( −+ jk ())} S () t, (5.30) jk, where the A0 is constant matrix, B1()t and B2,jk ()t are the time dependent current and delayed state matrices, and S()t is static force vector with process damping. The state matrices are derived from Equation (5.19) as,

0  0I T A =  ; S()t = U ; 0 2 t (ffcs ()t+ es ())d tz −−ωnn2 ζω 2(mmtw+× )  + T 2(mmtw ) −Uw 2(mmtw+× ) 1 00  TT  B1()t = UUttcd ed ; ff(tz )d ( tz )d TT  −−UUww  2(mmtw+× ) 2(mmtw+ ) 00  0[4(km− 1) x ] B ()t = UT   . 2,jk t ⋅ f0d  T jki[4 xm ] −U   w 0 [4(qk− )x m ] 2(mmtw+× ) 2(mmtw+ ) (5.31)

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5.2.2 General Solution Method in Semi Discrete-Time Domain

Following the method presented by Aguilar [132], the modal state vector Θ()t is solved in

∆t discrete time intervals in three intermediate steps within time frame of tii−3 ≤≤ tt for

in=4,7,...,s + 1. The solution is carried out until the spindle period index ns reached for the

general cases, but shorter (regular pitch milling tools) or longer (multi-functional drills) periods

are used in special cases. The start time of each interval is ti−31= ti +( − 4) ⋅∆ t with initial time t1 ,

and the final time of the simulation for one spindle period is t= tn +() ⋅∆ t.The solution in n1s + 1 s

each time interval is [81]:

t Nq, A03()tt− i− A0 (t−t ) ΘΘ()tt= e (i−3 ) +∫ eBΘ1 ()ττ ()−∑ { B2,jk () ττ Θ ( −+Tjk ()) τ} S ()d ττ (5.32) ti−3 jk,

Sampling time ∆t is selected with respect to the vibration wave with the highest natural

frequency of the combined tool-workpiece structure. The sampling time is set to:

π ∆=t . (5.33) 3⋅ max (ωn,c )

The integral part of Equation (5.32) is approximated by using two intermediate states, i.e. at

A ⋅−()tt i −1 and i − 2 [132]. Since e 03i− part can be computed analytically, Equation (5.32) is put into following form:

t A03⋅−()tti− A0 (t−t ) ΘΘ(tt )−= e (i−3 )∫ e h (tt )d ; t i−3 (5.34) Nq, hB()τ=1 () ττΘ () −∑ { BΘ2,jk () ττ ( −+Tjk ()) τ} S (). τ jk,

The following three equations are obtained by applying unconditionally A-stable, fourth- order integration method based on Simpson’s rule [132]:

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ti−1 A0⋅∆2 t A0(t i1− −t ) ΘΘ(tti1−− )−= e (i3 )∫ e h (tt )d ti−3 ∆ t A0(tt i1−−− i3) AA0(tt i1−−−− i2) 0(tt i1−− i1) ≅ eh (t− ) ++ 4e hh ( tt −−) e ( ); 3 i3 i2 i1

ti A0⋅∆2 t A0i(t −t ) ΘΘ(tti )−= e (i2− )∫ e h (tt )d ti−2 ∆ (5.35) t A0i(tt− i2−−) AA0i(tt −− i1) 0i(tt i) ≅ eh (t−− ) ++ 4e hh ( tt ) e ( ) ; 3 i2 i1 i

ti A0⋅∆3 t A0i(t −t ) ΘΘ(tti )−= e (i3− )∫ e h (tt )d ti−3

A0(tt i− i3− ) A0(tt i− i2− ) 3∆t ehh (tt−− )++ 3e ( ) ... ≅ i3 i2 . 8 AA0i(tt−− i1− ) 0i(tt i) ...++ 3ehh (tti1− ) e (i )

By substituting hh(tti1−− ), ( i2 ) and h()ti3− from Equation (5.34) at discrete-time intervals, the states become:

⋅ ∆ ⋅∆ eAA002 ttB Θ+4e BΘ + BΘ- ⋅∆ ∆t 1,i-3 i-3 1,i-2 i-2 1,i-1 i-1 ΘΘ-eA0 2 t = + ... i-1 i-3 AA⋅2 ∆tt⋅∆ 3 e 00B Θ + 4e B Θ +B Θ ( 2,i-3 i-3,Tjk,i-3 2,i-2 i-2,Tjk,i-2 2,i-1 i-1,Tjk,i-1 ) A ⋅∆2 t ∆t e 0 S + ... + i-3 ; 3 A0⋅∆t +⋅4e SSi-2 + i-1 ⋅ ∆ ⋅∆ eAA002 ttB Θ+4e B Θ + BΘ- 1,i--- 2 i 2 1,i−− 1 i 1 1, ii A ⋅∆2 t ∆t 0 = Nq, + Θi -e Θi-2 ⋅ ∆ ⋅∆ ... 3 ∑ eAA002 ttB Θ ++4e B Θ BΘ { 2,i-2 i-2,Tjk,i-2 2,i-1 i-1,Tjk,i-1 2,i i,Tjk,i } jk, A ⋅∆2 t ∆t e 0 S + ... + i-2 ; 3 A0⋅∆t 4e⋅+SSi-1 i

A0⋅∆3 t ΘΘi-e i-3 =

AAA00⋅32 ∆t⋅ ∆ tt 0⋅∆ e B1,i-3Θ i-3 ++3e B1,i-2 Θ i-2 3e B1,i-1 Θ i-1+ BΘ 1,i i - 3∆t  Nq, + ∆ ∆∆ 8 ∑ eAAA0032tB Θ ++3e ttBΘ 3e 0B Θ +B Θ { 2,i-3 i-3,Tjk,i-3 2,i-2 i-2,Tjk,i-2 2,i-1 i-1,Tjk,i-1 2,i i,Tjk,i } jk,

AA00⋅∆32tt⋅∆ 3∆t eSSi-3 +⋅ 3e i-2 + ...+ , (5.36) 8 A0⋅∆t 3e⋅+SSi-1 i

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where the notation is simplified for current terms, e.g., BB1,i1−−= 1()t i1 , BB2,jk,i-3= 2,jk()t i- 3 , and are represented as follows for the delayed terms:

Nq, B Θ= ∑ BΘ(t ) ( tTt- ( )); 2,i i,Tjki 2,jk i i jk i jk, Nq, B Θ= ∑ BΘ()(t t- Tt ()); 2,i-1 i-1,Tjk,i-1 2,jk i-1 i-1 jk i-1 jk, (5.37) Nq, B Θ= ∑ BΘ()(t t- Tt ()); 2,i-2 i-2,Tjk,i-2 2,jk i-2 i-2 jk i-2 jk, Nq, B Θ= ∑ BΘ()(t t- Tt ()). 2,i-3 i-3,Tjk,i-3 2,jk i-3 i-3 jk i-3 j,k

The number of discrete states (discretizations) is an integer multiple of 3. If motion in one spindle period (one rotation of cutter or workpiece) is simulated, the number of discretizations

(ns) is:

Ts ns =3 ⋅ int, (5.38) 3∆t where Ts is the period of one spindle rotation. Since the division inside int operator may not be exact, sampling time is updated as: ∆→t Tnss/ . For a common (called general tool system here) application, the maximum delay occurring in the system is expected to be periodic at every spindle rotation (i.e., turning), which would be equal to the maximum delay period as:

Tmax= max( T jk ( t ); j= 1 Nk ; = 1  qt ; = 0 Ts ) , (5.39) which gives the spindle period delay index as:

Ts ns = ceil. (5.40) ∆t

Depending on the application, the maximum delay period may vary from the tooth passing period (i.e., regular pitch milling) to an infinite number. However, assuming that all parameters

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of the operation are periodic at each spindle rotation, the maximum delay angle is set to 2π .

Each of the delayed state vectors, Θ , Θ , Θ and Θ are approximated i-3,Tjk,i-3 i-2,Tjk,i-2 i-1,Tjk,i-1 i,Tjki

using five neighbouring points, as is shown in Figure 5.1. The delay state index mjki of element

Sjk at time ti corresponding to each delay period is set as:

Tjki mjki = ceil, (5.41) ∆t

where the input of ceil function is rounded off to the closest larger integer value. The maximum

value (n) of the delay state index is:

n= max( mjki ; j= 1 Nk ; = 1  qi ; = 1 ns +1) . (5.42)

The states of current period are linked to states of previous tooth period through a state

transition equation. The intermediate states ( in=4,7,...,s + 1) is expressed from Equation (5.36) in

general form:

Θ(ti-3- Tt jk ( i-3 ))  Θi  Nq, Θ(ti-2- Tt jk ( i-2 )) C ⋅Θ =⋅+ CΘ∑ C ⋅+C, (5.43) 1,i i-1 2,i i-3 3,jki Θ(t- Tt ( )) 4,i Θ jk, i-1 jk i-1 i-2  Θ(tTti- jk ( i ))

where time- and geometry-dependent matrices are:

4∆∆tt -exp(AB0∆t ) 1,i-2 I- B1,i-1 0[2mm x2 ] 33 ∆∆tt4 ∆ t CAB1,i =-exp(20∆+tt ) I 1,i-2 -exp(AB0∆ ) 1,i-1 I - B1,i ; (5.44) 33 3  9∆t 93 ∆∆ tt -exp(2AB0∆tt ) 1,i-2 -exp(AB0∆ ) 1,i-1 I - B1,i 8 88[32⋅⋅mm x32 ]

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Figure 5.1 Interpolation of the delay state using five neighboring states.

∆t exp(2AB0∆+t ) I 1,i-3 3 C2,i = 0; (5.45)  3∆t exp(3AB∆+t ) I 0 1,i-3 8 [32⋅ mm x2 ]

∆∆tt4 ∆ t 2AA00∆∆tt  -e B2,jk,i-3 - e B2,jk,i-2 - B02,jk,i-1 [2mm x2 ]   33 3  ∆t4 ∆∆ tt  2AA00∆∆tt C3,jki = 0[2mm x2 ] -e B2,jk,i-2 - e BB2,jk,i-1 - 2,jki  3 33 ; (5.46)   3993∆∆∆∆tttt322AAA∆∆∆ttt -e000B -e B -e BB -   2,jk,i-3 2,jk,i-2 2,jk,i-1 2,jki  8888[32⋅⋅mm x42 ]

∆t [exp(2AS0∆tt ) i-3 +⋅ 4 exp( AS0 ∆ ) i-2 + S i-1] 3 ∆t C=[exp(2AS∆tt ) +⋅ 4 exp( ASS ∆ ) + ] . (5.47) 4,i 3 0 i-2 0 i-1 i  3∆t [exp(3AS∆t ) +⋅ 3 exp(2 AS ∆ tt ) +⋅ 3 exp( ASS ∆ ) + ] 0 i-3 0 i-2 0 i-1 i 8 [32⋅ m x1]

All the state transitions within one spindle period Ts is merged in one matrix equation as:

Nq, ϒϒ= + ⋅+ iDD 1,i-3∑ 2,jk,i-3 i-3 E i-3 , (5.48) jk,

130

where the state vector ϒi , previous state vector ϒi-3 and static force vector Ei-3 at previous state are defined with an inclusion of all the states within the maximum delay period (the spindle period for the general case):

Θi-n-4  Θi-n-3   Θ Θi-n-1 i-mjki -5     Θi-n Θ -1  i-mjki -4 (C1,i C4,i )   [2m⋅ 3] x 1    ϒϒ= = = i Θi-3 ;;i-3 Θ Ei-3 . (5.49)  i-mjki +2 Θi-2   Θ  i-mjki +3 0 Θi-1  [2mn (- 1)] x 1 Θ Θi-m +4  i [2mn (+ 2)] x 1 jki [2mn (+ 2)] x 1   Θi-4 Θ i-3 [2mn (+ 2)] x 1

Maximum delay index (n) may be different than the spindle period index ns in some cases.

However, for the general case, the maximum delay is set to spindle period: nn= s . In Equation

(5.48) the constant matrix is:

0 CC-1 [2m⋅ 3]x ( 1,i 2,i )[2m⋅ 3]x [2mn⋅+ ( 1)] [2m⋅ 1]   D = . (5.50) 1,i-3  0I[2mn (− 1)x [2mn ( −− 1)]x[2 mn ( 1)] ⋅ 2m 3]   [2mn (++ 2)]x[2 mn ( 2)]

131

Before writing the matrix D2,jk,i-3 related to the delay states, the delay states vector has to be expressed in terms of the previous state vector ( ϒi-3 ), and by means of using an approximated weight function matrix Wjki :

Θ(ti-3- Tt jk ( i-3 ))  Θ(ti-2- Tt jk ( i-2 )) = W ⋅ ϒ . (5.51) jki[2m⋅+ 4x 2 mn ( 2)]i-3 [2mn ( + 2)] x 1 Θ(ti-1- Tt jk ( i-1 )) Θ(tTt- ( )) i jk i [2m⋅ 4] x 1

The matrix related to delay states is then expressed as follows:

CCW-1 ( 1,i 3,jki jki )[2m⋅ 3]x [2mn⋅+ ( 2)]   D2,jk,i-3 = . (5.52) 0 [2(1)x2(2)]mn- mn+   [2mn (++ 2)]x[2 mn ( 2)]

Fourth-order Lagrange interpolation for the delay state: The interpolation of delay states in Equation (5.51) is carried out by a 5-point (4-th order) Lagrange interpolation scheme

(Figure 5.2a). All of the delay states ( Θ(tTti− jk ( i )), Θ(ti-1- Tt jk ( i-1 )), Θ(ti-2- Tt jk ( i-2 )) and

Θ(ti-3- Tt jk ( i-3 )) ) are approximated similarly. A special case is shown in Figure 5.2b when the delay corresponds to the maximum delay of the operation. There are five points needed around the delay state. Thus, to account for this special case, (i-n-4)-th state is added to the calculations.

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Figure 5.2 The neighboring states and the Lagrange- 4-th order interpolation: (a) xr values from reference

state (i-mjki-1); (b) a special case for maximum delay term.

As illustrated in Figure 5.1 and Figure 5.2a, the delay state Θ(tTti− jk ( i )) is approximated using the five neighbouring discrete states from Θ to Θ using the weighting im−−jki 1 im−+jki 3

function matrix w jki at time ti:

Θim−− 1 jki Θim− jki −= ⋅Θ −+ Θw(tTti jk ( i ))  jki imjki 1 2mm x(2⋅ 5)  . (5.53) Θ im−+jki 2  Θ −+ imjki 3 wwwwww=  jki 1,jki2mm x2 2,jki2mm x2 3,jki2mm x2 4,jki2mm x2 5,jki2mm x2

Each delay state vector is assumed to have the same weight in modal space, as:

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wIr,jki=w r,jki ⋅ [2mm x2 ] , (5.54)

where r=1,2…5, and I[2mm x2 ] is identity matrix with size 2m. For each modal state of

Θ(tTti− jk ( i )) , the weight terms wr,jki at time ti for the element Sjk are set as:

5 yL4( x jki )≅ ∑ w r,jki fx () r , (5.55) r=1

where the function fx()r takes on the values at the neighboring delay states (from im−−jki 1 to

im−+jki 3). The approximate value yxL4() jki of the delay state corresponds to each modal state

of Θ(tTti− jk ( i )) . In Figure 5.2a, when the approximation is shown by taking the (i-mjki-1)-th

state as a reference, the Equation (5.55) is expressed using the Lagrange 4-th order interpolation

as:

5 5 xxjki− s yL4( x jki )≅ ∑ fx ()r ∏ ; r=1 s=1 xxrs− sr≠ (5.56) 5 cr,jki yL4() x jki ≅ ∑ fx(r ). r=1 dr

The weights are wr,jki= cd r,jki r , where:

55 cr,jki=−=−∏∏ x jki xd s; r xx r s . (5.57) ss=11= sr≠≠sr

Tjki is the instantaneous delay period for the k-th segment of j-th cutting edge. Because of the the effects of runout and spindle speed modulation, the delay period may be time-dependent.

Also, because of the variable geometry of the cutter, the delay may also depend on the cutting edge and segment. For the reference State-1: tt= , the corresponding relative times ( x ref i−− mjki 1 r

134

values in Figure 5.2a) of Equation (5.57) are found by subtracting the reference state from the actual values:

xt= −= t; x 0; 1 i−− mjki 1 ref 1

x21= x +∆ tx;; 2 =∆ t x31= x +∆2; tx 3 =∆ 2; t (5.58) x41= x +∆3; tx 4 =∆ 3; t x51= x +∆4; tx 5 =∆ 4. t

Since the time interval is fixed, the time delay values are also fixed. From Equation (5.57):

5 4 d1=−=−−−−=∆∏ xx 1s( xxxxxxxx 12131415)( )( )( ) 24 t ; s=1  (  (( (  (  ( ( ( s≠1 −∆tt−∆24tt−3 ∆ −∆ 5 4 d2=∏ xx 2s − =( xxxxxxxx 21232425 −)( −)( −)( −) =−∆6; t s=1 ( (( ( ( ( ( ( s≠2 ∆tt−∆tt−∆2 −3 ∆ 5 4 d3=∏ xx 3s −=( xxxxx 31323 −)( −)( −xxx435)( −=∆) 4; t (5.59) s=1 ( ((( ( (( ( s≠3 2∆∆tt−∆tt −2 ∆ 5 4 d4=∏ xx 4s − =( xxxxxxxx 41424345 −)( −)( −)( −) =−∆6; t s=1 ( ((( ( (( ( s≠4 32∆∆tt∆tt −∆ 5 4 d5=∏ xx 5s −=( xxxxxxxx 51525354 −)( −)( −)( −) =∆24 t . s=1 ((( (((( ( s≠5 432∆t ∆ t ∆∆ tt

Neighboring states are chosen such that delay state falls between 2-nd and 3-rd states

(Figure 5.2a). Relative time of delay state is:

xjki=( m jki + 1) ∆− tTjki . (5.60)

From Equation (5.57) the coefficients of weight function for cutting edge element Sjk at time ti are:

5 cr,jki=∏ xx jki − s , (5.61) s=1 sr≠

135

where differences with time reference are:

 xjki− x 1 =( m jki + 1) ∆− tTjki − 0; x jki − x 1 = ( m jki + 1) ∆− tTjki ;  xjki− x 2 =( m jki + 1) ∆− tTjki −∆ tx; jki − x 2 = m jki ∆− tT jki;  xjki− x 3 =( m jki + 1) ∆− tTjki − 2 ∆ tx ; jki − x 3 = ( m jki − 1) ∆− tT jki ; (5.62) xjki− x 4 =( m jki − 2) ∆− tTjki ;

xjki−= xm 5(3 jki −).∆−tTjki

Putting Equations (5.59), (5.61) and (5.62) into Equation (5.56) gives the approximate value for the modal states σ p ()t (for p=1,2…2m) of the delay state vector Θ()t in Equation (5.53):

5 σσp(tTwx i−≅ jki) ∑ r,jki ⋅ p( r ); r=1 w= c/; dxt= ;; xt= xt = ; xt= ; xt= ; (5.63) r,jki r,jkir 1 im−−jki 1 2 im −jki 3 im −+jki 1 4 im −+jki 2 5 im −+jki 3

σ1()t  ≅ Θ()t = σ p ()t ,  ≅  σ 2m ()t which results in Equation (5.53):

5 Θ(tTi−≅ jki) ∑ wΘ r,jki ⋅( x r ) . (5.64) r=1

Similarly, the weighting functions at other time instants are found and other delay state vectors are formed as in Equation (5.53):

5 Θ(tTi- 1- jki) ≅ ∑ wΘ r,jk,i-1( xt r -∆ ); r=1 5 Θ(tTi- 2- jki) ≅∑ wΘ r,jk,i-2( x r -2;∆ t) (5.65) r=1 5 Θ(tTi- 3- jki) ≅∑ wΘ r,jk,i-3( x r -3.∆ t) r=1

The following expressions are written similar to Equation (5.53):

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Θim-- 2 jk,i-1 Θim-- 1 jk,i-1 -= ⋅Θ - Θw(ti-1 Tt jk ( i-1 ))  jk,i-1 imjk,i-1 ; 2mm x(2⋅ 5)  Θ im-jk,i-1+ 1  Θ -+ imjk,i-1 2

Θim-- 3 jk,i-2 Θim-- 2 jk,i-2 -= ⋅Θ -- Θw(ti-2 Tt jk ( i-2 ))  jk,i-2 imjk,i-2 1 ; 2mm x(2⋅ 5)  Θ imjk,i-2  Θ -+ imjk,i-2 1

Θim-- 4 jk,i-3 Θim-- 3 jk,i-3 -= ⋅Θ -- Θw(ti-3 Tt jk ( i-3 ))  jk,i-3 imjk,i-3 2 ; 2mm x(2⋅ 5)  Θ im--jk,i-3 1  Θ - imjk,i-3  (5.66) wwwwwwjk,i-1=  1,jk,i-1 2,jk,i-1 3,jk,i-1 4,jk,i-1 5,jk,i-1 ;  wwwwwwjk,i-2=  1,jk,i-2 2,jk,i-2 3,jk,i-2 4,jk,i-2 5,jk,i-2 ;  wwwwwwjk,i-3=  1,jk,i-3 2,jk,i-3 3,jk,i-3 4,jk,i-3 5,jk,i-3 .

The delay state approximation is put into the general form as in Equation (5.51): By deriving

the index of each delay state vector in the previous state vector ( ϒi-3 ). The first index of previous state vector is “i-n-4” and the final state is “i-3”. Thus, for example, for the vector in Equation

(5.53), the start index is “i-mjki-1” and the final is “i-mjki+3”. Then, the first level of weight

matrix Wjki has

{(im−jki − 1) −−− (i n 4)} ⋅ 2m ; (5.67) (nm−jki +⋅32) m

preceding columns of zeros, and

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{(i− 3) −− (imjki + 3)} ⋅ 2 m ; (5.68) (mmjki −⋅62) succeeding columns of zeros after the delay approximation matrix w jki . Similarly, the second level of weight matrix Wjki has dimensions (nm-jk,i-1 +⋅22) m and (mmjk,i-1 -52) ⋅ . The third level has dimensions (nm-jk,i-2 +⋅12) m, and the fourth level has dimenstions (nm-jk,i-3 ) ⋅2 m and (mmjk,i-3 -32) ⋅ with preceding and succeeding columns of zeros before their corresponding delay approximation matrices, w jk,i-1 , w jk,i-2 and w jk,i-3 . The weight matrix Wjki of the general form in Equation (5.51) is thus written as:

 0w- ⋅ jki 0-⋅  2xmnm( jk,i-3 ) 2 m 2xmm( jk,i-3 32) m  0- +⋅  w0jk,i-1 - ⋅ 2xmnm( jk,i-2 12) m  2xmm( jk,i-2 42) m Wjki = . (5.69) 0 w0jk,i-2  2xmnm( -jk,i-1+⋅ 22) m 2xmm( jk,i-1- 52)⋅ m     0- +⋅ w0jk,i-3 -⋅  2xmnm( jki 32) m  2xmm( jki 62) m

Time-domain stability simulation: The solution to the equation of motion of the tool- workpiece system with the defined cutting forces is:

ϒϒi=⋅+DE i-3 i-3 i-3 , (5.70) with

Nq, DDi-3= 1,i-3 + ∑ D 2,jk,i-3 . (5.71) jk,

The system is assumed to vibrate around its equilibrium state if the quasi-static Ei−3 terms in

Equation (5.70) is dropped. Based on Floquet’s theory on periodically changing systems,

138

Insperger and Stepan [131] proposed a fast stability prediction for the equation of motion of machining operation. Depending on the periodicity of the system (tooth passing period for regular pitch cutters, spindle period for general tools), stability is predicted by checking eigenvalues of the state transition matrix of the system within its periodicity. To be consistent with general cutters, the periodicity is set as the spindle rotation period: Tss= nt ∆ . Thus, ns time

points are simulated in total:

ϒϒi=D i-3 ⋅ i-3 ϒϒi3+ =⋅=⋅D i i DD i i-3 ⋅ ϒ i-3 ϒϒ++=D ⋅ = DD + ⋅⋅ ϒ i6 i3 i+3 i3 i i (5.72) • • ϒϒ=D ⋅ =D D⋅⋅⋅⋅ DD ⋅ ⋅ϒ. in+s in + s- 3 in+ s- 3 in+ ss- 3 in+- 6 i3+ i i

Since the geometry of the cutter is assumed general, and the spindle speed variation is

assumed to have periodicity of spindle, the motion involves periodically changing parameters.

Thus, the state transition matrix repeats itself after each spindle period: DD= . Floquet i in+ s

theory can be applied by evaluating the eigenvalues of period-transition matrix ( Φp ) of the

discrete system as:

Φ =D D⋅⋅⋅⋅ DD ⋅ . (5.73) p in+−ss 3 in +− 6 i3 + i

If the dominant eigenvalue (with largest absolute value) is outside the unit circle, the

operation is unstable. The operation is stable otherwise, i.e., if the eigenvalues are less than

unity:

>1 (unstable)  max( eig(Φ=

139

The stability lobes are constructed by repeating the eigenvalue evaluation at the interested

range of spindle speeds and dephts of cut.

Force/vibration simulation: The same scheme in Equation (5.72) is used for full time domain simulation of the system, which leads to the prediction of vibrations, forces, power, torque and surface errors of the machining operation. State transition matrices are minimized to the lowest possible size and is decided by the maximum delay index in the system, n of

Equation (5.42). For the general case, if periodicity of the system is the spindle rotation period, all the state transition matrices in one spindle/cutter/workpiece rotation (

D, D⋅⋅⋅⋅ DD, and E, E⋅⋅⋅⋅ EE, ) must be computed. Since these i+nss -3 i+n -6 i+3 i i+nss -3 i+n -6 i+3 i transition matrices repeat themselves after each spindle period, computing them for one rotation is sufficient. For the general case, the size of the transition matrix is set by the maximum delay

state (n), and it is smaller than the spindle period index (ns). Similar to Equation (5.72), if the quasi-static component Ei3− is included, the transition until the delay amount is as below:

ϒϒi=⋅+DE i3−− i3 i3 −; ϒϒi3+ =⋅+DE i i i; ϒϒi6+ =D i+3 ⋅ i+3 + EDD i+3 = i+3 ⋅( i ⋅+ ϒ i EEDD i) + i+3 = i+3 ⋅⋅+ iϒ i( DEE i+3 ⋅+ i i+3 );

ϒϒ+ =D ⋅ + E = D ⋅( D ⋅⋅+ Dϒ( D ⋅+ EE)) + E i9 i+6 i+6 i+6 i+6 i+3 i i i+3 i i+3 i+6 (5.75) =DDDi+6 ⋅ i+3 ⋅⋅+ iϒ i( DDEDE i+6 ⋅ i+3 ⋅+ i i+6 ⋅ i+3 + E i+6 );  ϒi+n* = Di+n*−3⋅ϒϒ i+n*−3 = D i+n*−3 D i+n*−6⋅⋅⋅⋅ DD i+3 ⋅ i i +...... +(DDi+n*−3 i+n*−6⋅⋅⋅ DEDD i+3 i + i+n*−3 i+n*−6⋅⋅⋅ DE i+6i+3 +⋅⋅+ DE i+n*−3 i+n*+ E i+n*−3 ).

Here, the spindle index ns term may not be an integer multiple of the delay index n term.

Thus, as in Equation (5.75), modified delay “n*” term assures the integer number of transitions,

by setting n* closest 3-multiplier of n while less than or equal to it: 140

nn*=−≤ mod( n ,3); nn * . (5.76)

The first part of the state equation Equation (5.75) is the state transition matrix, which is similar to Equations (5.72) and (5.73). The second part of the equation is the quasi-static component, and it is important for determining the resulting vibrations of the tool-workpiece system. Equation (5.75) represents the state transition with the maximum delay amount.

Continuing the state transitions for the next cycles until the spindle period gives:

ϒϒi+n*=D i+n*-3 ⋅ i+n*-3 = D i+n*-3 D i+n-6⋅⋅⋅⋅ DD i+3 ⋅ iϒ i +...

... +(DDi+n*-3 i+n*-6⋅⋅⋅ DEDD i+3 i + i+n*-3 i+n*-6⋅⋅⋅ DE i+6 i+3 +⋅⋅+ DE i+n*-3 i+n*+ E i+n*-3 );

ϒϒi+2n*=DD i+2n*-3 i+2n*-6⋅⋅⋅⋅ D i+n*+3 ⋅ D i+n* i+n* +...

... +⋅(DDi+2n*-3 i+2n*-6 ⋅⋅Di+n*+3 ED i + i+2n*-3 D i+2n*-6⋅⋅⋅ D i+n*+6 E i+n*+3+⋅⋅+ D i+2n*-3 Ei+2n*+ E i+2n*-3 );

ϒi+3n* = DDi+3n*-3 i+3n*-6⋅⋅⋅⋅ D i+2n*+3 ⋅ D i+2n*ϒ i+2n* +...

... +(DDi+3n*-3 i+3n*-6⋅⋅⋅ D i+2n*+3 EDD i + i+3n*-3 i+3n*-6⋅⋅⋅ D i+2n*+6 E i+2n*+3+⋅⋅+ D i+3n*-3EE i+3n*+ i+3n*-3 ); (5.77) 

ϒϒi+bn*=DD i+bn*-3 i+bn*-6⋅⋅⋅⋅ D i+(b-1)n*+3 ⋅ D i+(b-1)n* i+(b-1)n* +...

 DDi+bn*-3 i+bn*-6⋅⋅⋅ D i+(b-1)n*+3 EDD i+ i+bn*-3 i+bn*-6 ⋅⋅⋅ Di+(b-1)n*+6 E i+(b-1)n*+3 +⋅⋅+.. ... +  ;  DEi+bn*-3 i+bn*+ E i+bn*-3  ϒ = DD⋅⋅⋅⋅ D ⋅ Dϒ +... i+ns i+nss -3 i+n -6 i+n s -bn*+3 i+bn* i+bn*

... +(DDi+n -3 i+n -6⋅⋅⋅ D i+n -n*+3 EDDi + i+n -3 i+n -6⋅⋅⋅ D i+n -bn*+6 E i+n -bn*+3+⋅⋅+ DE i+n -3 i+n+ E i+n -3 ), ss s ss s s ss s where term b is for the last portion of the spindle rotation period. It is the largest multiple of modified delay term n* while less than spindle period term ns :

b= floor( nns *) . (5.78)

The floor function rounds the input down to the closest integer.

Algorithm for force/vibration simulation: The current state vector ϒi of Equation (5.49) starts with the initial vector form for i=4. Initial condition vector for ϒi-3 at i=4 is set to zero:

141

Θ−n 0   Θ1n− 0 ϒϒ11=  ; =  . (5.79)   Θ0 0   Θ1 [2mn (+ 2)] x 1 0 [2mn (+ 2)] x 1

Additionally, ϒ4 is taken as zero initial condition, and all the states in ϒ4 are set to zero:

Θ3n− 0   Θ4n− 0 ϒϒ44=  ; =  . (5.80)   Θ3 0   Θ4 [2mn (+ 2)] x 1 0 [2mn (+ 2)] x 1

Assuming that the n* is not equal to ns, the first set of n * 3 transitions from ϒ4 to ϒ4+n*

(setting i=4 in Equation (5.77)) are set as:

1st

ΘΘ3−− n transition 3 n+n*  amount   ΘΘ−−n* 4 n 3 4 n+n*  →  . (5.81)    ΘΘ3 3+n*  ΘΘ   4 4+n* ϒϒ4 4+n*

If the delay term n and modified delay term n* are the same, only the last two state vectors

ϒ4 ( Θ3 and Θ4 ) are carried forward to be the first two state vectors of the ϒ4+n* . In general,

4− (3 −+nn *) + 1; (5.82) nn−+*2

number of state vectors (at least 2, for nn* = ) are carried forward. This is important information when the state vectors are stitched to each other after each transformation. The rest of the transformations continue until the last one, as:

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st nd 1 2 th b ΘΘ3−− n transition 3 n+n* transition Θ3 − n+2n* Θ3− n+bn* amount amount  ΘΘ−−nn** Θ − Θ − 4 n 334 n+n* 4 n+2n* 4 n+bn* → →   →→  . (5.83)   ΘΘ3 3+n* Θ3+2n* Θ3+bn* ΘΘ Θ Θ 4 4+n*  4+2n*   4+bn* ϒϒ4 4+n* ϒ4+2n* ϒ4+bn*

In total, b ⋅ n * 3 transformations are conducted from ϒ4 to ϒ4+bn* . The last transformation

may be needed to complete the spindle period:

(b +1)th

transition Θ3− n+n Θ3− n+bn* amount s  (ns − bn*) Θ Θ4− n+bn* 4− n+ns 3   →  . (5.84)  Θ Θ 3+bn* 3+ns  Θ4+bn* Θ (( 4+ns ϒ  ( ( 4+bn* ϒ 4+ns

For the final (b+1)-th transformation, last

(4+bn *) −( 3 −+ n n ) + 1; s (5.85) bn*2+− n ns +

state vectors of ϒ are carried forward to the first state vectors of ϒ . There is a special 4+bn* 4+ns condition, for example, when maximum delay period is the same as the spindle period, or when the tool has single tooth: b=1 and n=n*=ns. In this case, the transformation step in Equation

(5.84) is not needed, and only the 1st step in Equation (5.83) is conducted.

Finally, after stitching the consecutive state vectors ( ϒ , ϒ ,.. ϒ , ϒ ) to each 4 4+n* 4+bn* 4+ns

other, overall modal displacement and velocity history in one spindle period is obtained:

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Θ3  Θ4 Θ =  . (5.86) sp  Θ 3n+ s  Θ 4n+ s

The rest of the simulation (after one spindle period) is carried out in a similar fashion by

assigning Θ and Θ as the initial condition of the next period. If P number of spindle 3n+ s 4n+ s

rotations are simulated, the full history of the modal states are:

Θ3  Θ4 Θ =  . (5.87) fh  Θ 3+ Pns  Θ 4+ Pns

From Equation (5.29), at each simulated time point ti from t to t , the relative 3 4+ Pns

 deformations Ψ()ti and velocities Ψ()ti of the tool-workpiece system are found by back modal transformation of Equation (5.6), as:

Γci()t Θ()ti =  ; Γ ci()t 2(mmtw+× ) 1

ΨΓ()tti =[UUt −⋅ w] ci( ); (5.88)

ΨΓ ()tti =[UUt −⋅ w]  ci( ),

where the vector is composed of the vibration of each axial/radial segment:

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x1i  y1i  z 1i  q1i    xki  yki  Ψ()ti = z . (5.89) ki  qki    xqi  yqi  zqi  qqi [4q x 1]

The individual tool and workpiece deformation and velocity of the vibration are:

Q(t ) =U⋅ΓΓ ( tt ); Q ( ) =−⋅ U ( t ); tii t ci w w ci (5.90)  Qt(tii ) =U t⋅ΓΓ ci ( tt ); Q w ( ) =−⋅ U w ci ( t ).

Cutting forces (including the torque) along the contact zone is found at each time step ti from Equation (5.23), as:

0[4(km− 1) x ] Nq,  cs cd d es ed F()ti= ff i+ iΓ c,i −∑  fjki[4xm ]⋅ ΓΓ c(tT i − jki) ++ff i i c,i . (5.91) jk,  0 [4(qk− )x m ]

At time ti, the approximation of delay term Γc(tT i− jki ) in Equation (5.91) is determined similarly as in Equation (5.53):

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Γc,i−− m 1 jki Γc,i− m jki Γ −= ⋅Γ −+ c(tT i jki) w h,jki c,i mjki 1 ; mmx(⋅ 5)  Γ c,i−+ mjki 2  Γ −+ c,i mjki 3 wwwwww= ; (5.92) h,jki h1,jkimmxxxxx h2,jkimm h3,jkimm h4,jkimm h5,jkimm

wIhr,jki=w r,jki ⋅ [mmx ],

where the weight matrices ( wh,jki and whr,jki ) are half the size of the ones given in Equations

(5.53) and (5.54). r=1,2…5 and I[mmx ] is the identity matrix with size m. The weight terms wr,jki

at time ti for the element Sjk are derived in Equation (5.63).

Surface location error simulation: The surface location error (SLE) is the difference

between the desired surface left by the rigid motion of the tool and the actual surface left by the

flexible motion of the tool [133]. It is meaningful to define it only for stable operations, since

unstable cutting will result in exponentially increased surface roughness and the process would

be unacceptable in production.

Figure 5.3 shows the SLE for the undercutting and overcutting cases of milling operation:

Part surface is undercut (Figure 5.3a) if the intended amount of material left is larger than the

actual amount of machined material; it is overcut (Figure 5.3b) if the intended removed material

is less than the actual material removed. If the feed travel motion of the cutter is ignored, then the

0 SLE depends on the radius and the normal (y ) component of the relative vibration Ψ()ti of tool- workpiece system (Equation (5.88)).

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Figure 5.3 Surface location error for the milling operation. (a) Undercut surface; (b) Overcut surface. (c)

significant transient vibrations (data from [96]).

SLE is calculated in the steady state region. SLE is defined only at the tool workpiece contact at the start and exit of the operation. Then, the maximum y0 location of the cutting edge

element Sjk is the distance from the center to the machined surface. If only the rigid body motion is considered, the desired surface distance yd,k (at axial segment-k) from the center would be the maximum of all the teeth cutting in one rotation period in the steady state region:

 up milling: max(Rjk cos(φ jki ))  ji, yd,k =  , (5.93) down milling: max(−Rjk cos(φ jki ))  ji,

0 where the instantaneous angular position φjki is measured from the +y axis in CW direction; the

0 radius (Rjk) of the element Sjk includes the radial runout ( εr,jk ). If the vibrations in y -direction 147

are added, then the actual surface distance ya,k from the center would be the maximum of all

teeth during one rotation:

 up milling: max(Ryjk cos(φ jki ) + ki )  ji, ya,k =  . (5.94)  down milling: max(−+(Ryjk cos(φ jki ) ki ))  ji,

Then the surface location error ek at axial segment-k is defined as the difference between the

desired and actual surface [104]:

eyk= d,k − y a,k . (5.95)

A positive ek implies that the workpiece is undercut (Figure 5.3a), whereas a negative value

shows that the workpiece is overcut (Figure 5.3b). As shown in example case Figure 5.3c, the

transient vibration period when the cutter is entering the cut may introduce more error on the

workpiece surface. This effect is more pronounced for high feed and high speed cutting

operations, because the cutter may travel a long distance in the transient region. To be consistent,

SLE is defined at the steady state vibration period of the cutting operation. .

Steady state vibrations: The instantaneous location angle information for each time step ti

for the milling operation is stored during each revolution. For stable cutting, when sufficient

amount of rotations are covered in the simulations, the vibrations reach steady state and will

continue repeating the same motion profile.

Ding et al. [107] employed an analytical method to predict the steady state vibrations using the state transition equations. Assuming that the maximum delay is the same as the spindle period (n=n*=ns), Equation (5.75) is rewritten to cover all spindle period:

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ϒϒi=⋅+DE i3−− i3 i3 −; ϒϒi3+ =⋅+DE i i i; ϒϒi6+ =D i+3 ⋅ i+3 + EDD i+3 = i+3 ⋅( i ⋅+ ϒ i EEDD i) + i+3 = i+3 ⋅⋅+ iϒ i( DEE i+3 ⋅+ i i+3 );

ϒϒi9+ =D i+6 ⋅ i+6 + E i+6 = D i+6 ⋅( D i+3 ⋅⋅+ D iϒ i( D i+3 ⋅+ EE i i+3)) + E i+6 (5.96) =⋅⋅⋅+⋅⋅+⋅+DDDi+6 i+3 iϒ i( DDEDE i+6 i+3 i i+6 i+3 E i+6 );  ϒ = D⋅ϒϒ = D D⋅⋅⋅⋅ DD ⋅ +... i+ns i+ns−3 i+n s−3 i+n ss−3 i+n −6 i+3 i i ... + DD⋅⋅⋅ DEDD + ⋅⋅⋅ DE +⋅⋅+ DE+ E . ( i+nss−3 i+n −6 i+3 i i+nss−3 i+n −6 i+6 i+3 i+nss−3 i+n i+n s−3 )

Similar to Equation (5.73), the state transition over a spindle period is expressed by including the quasi-static terms:

ϒϒ=Φ ⋅ +Φ ; i+ns p i s Φ =D D⋅⋅⋅⋅ DD ⋅ ; (5.97) p in+ss- 3 in+- 6 i3+ i Φ =DDDEDD⋅⋅⋅ + ⋅⋅⋅ DE +⋅⋅+ DE+ E. s i+nss -3 i+n -6 i+3 i i+nss -3 i+n -6 i+6 i+3 i+nss -3 i+n i+n s -3

Setting the state vectors equal at the steady state response,

ϒ= ϒϒ = , (5.98) i+ns i s simplifies Equation (5.97) and the steady state vector ϒs can be written as:

−1 ϒs=(I −Φ ps) ⋅Φ (5.99)

The limitation of using Equation (5.75) to predict the surface error is the assumption that the number of delay terms and spindle period terms are equal. If the delay terms are less than the spindle period, Equation (5.77) could be used for obtaining a similar expression to Equation

(5.97).

5.3 Simplified Solution Methods for Special Cases

Although the presented general solution covers all possible machining cases, the solution can be simplified for special cases such as milling with low radial immersion case, where the

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cutter out-of-cut time is considerably large. The solution can also be simplified for milling with

regular (uniform) pitch cutters.

5.3.1 Low Radial Immersion with Free Vibration Condition (e.g. Double-Sided Milling)

The cutter is not in-cut during the full period of the spindle rotation The formulation in this

section is based on the methods presented by Bayly et al. [106] and Ding et al. [134].

At a time instant ti, if both g1,jk (element-available-to-cut) and g2,jki (edge-in-cut) are zero among the cutting edges Sjk, then the cutter is practically out-of-cut and experiences free

vibrations. All the force terms in Equation (5.30) are dropped, leaving only the system dynamics related component:

Θ()tt= AΘ0 ( i3− ), (5.100)

which is the free vibration equation, and its solution is analytically obtained from Equation

(5.32) as:

A03()tt− i− ΘΘ()tt= e (i−3 ). (5.101)

In order to have efficient simulation, the cutter location angle is adjusted such that during

each delay period of the cutter, the cutter enters cut once and exits cut once.

Adjusting the cutter location angle: The cutter location angle is adjusted such that the first

state (state-1) becomes the start point of the in-cut region. This first state is called id1. Then, the

second state id2 is selected as the last state in the in-cut region. The third state id3 is the last out-

of-cut state.

If the cutting during each tooth period is analyzed, the cutter is adjusted to have an in-cut

region from id1 to id2-4, and an out-of-cut region from id2-3 to id3. If there is no id2 assigned, the

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cutter is always in cut. There may be more than one id2 assigned, thus implying that the cutter

goes into in-cut and out-of-cut more than once during one spindle rotation.

Step-1: State transition in the in-cut region: The in-cut region is defined from the state id1

to state id2-3. If a nonzero force is always acting on the tool-workpiece system, then the solution presented in Equation (5.70) is valid in this region (for i=id1,id1+3,…id2-3). From Equation

(5.72) the state transition for the dynamic component is written as:

ϒϒ=D ⋅ ; id11 id -3 i-3 ϒϒ=⋅=⋅D DD ⋅ ϒ; id1+ 3 id 11 id id 11 id -3 id 1 -3 • (5.102) • ϒϒ=D ⋅ = DD⋅⋅⋅⋅ D ⋅ D ⋅ϒ. id2− 3 id 2 − 6 id 2 − 6 id 2 −− 6 id 2 9 id 1+ 3 id 11 id

The first step presented in Equation (5.102) is the same as the original method and no computational advantage exists. Rewriting the state transition gives a shorter form:

ϒϒid− 3=D F ⋅ id ; 21 (5.103) DD= D⋅⋅⋅⋅ D ⋅ D, F id22−− 6 id 9 id 1 + 3 id 1

where DF is the state transition matrix for the forced vibration period. In the second step, the

state id2-3 is linked to the state id3 in one single equation, thus reducing the computation time.

Step-2: State transition in the free vibration region: The first step involves the state

transition (from ϒ to ϒ ) in the forced vibration region and is the same as the original id1 id2 − 3

procedure. Second step employs Equation (5.101) to link the state vectors of the free vibration region (from ϒ to ϒ ): id2 − 3 id3

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ϒϒ= D , (5.104) id32 f id− 3 where Df is the transition matrix in the noncutting duration of time. The state vectors are from

Equation (5.49):

ΘΘid−− n 1 id−− n 4  32   Θ Θ id3 − n id2 −− n 3           Θ − Θid−− n 2 id2 4 3  Θid− 3 Θid−− n 2  ϒ = 2 ;ϒ = 3  id32id− 3 . (5.105) ΘΘid− 2 id−− n 1  23   Θ Θ id2 − 1 id3 − n        Θid− 1 Θid− 4  3 2  Θ Θ  id3 id2 − 3 

In Equation (5.105), the states above the dashed line of ϒ are same as the states below the id3 dashed line of ϒ . These states are related by using identity matrix: id2 − 3

ΘΘid−− n 1 I[2mx 2 m ] id−− n 1 33 ΘΘI id33−− n [2mx 2 m ] id n   =   , (5.106)   ΘΘid−− 4 I[2mx 2 m ] id 4 22 ΘΘI  id22−− 3 [2mx 2 m ] id 3 where each I[2mx 2 m ] is a 2m-by-2m identity matrix. Free vibration states starts from id2-3 until the state id3. Initial state vector Θ is transferred to other states in the free vibration region by id2 − 3 the matrix:

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A0∆t Θ − e id2 2   ∆ Θ A0 2 t id2 − 1 e  =  ⋅ Θ . (5.107)  ( id2 − 3 ) − +∆ Θ − A03(id id 2 2) t id3 1 e Θ A (id− id +∆ 3) t id3 e 03 2

Thus, combining the matrices of Equations (5.106) and (5.107) gives the transition matrix of

Equation (5.104) in the free vibration region:

Θ id3 −− n 1   I  Θid−− n 4    2 Θ I id3 − n       0           Θid− 4     2       I  Θ Θid− 3 id3 −− n 2 2 = A0∆t ⋅ . (5.108) e Θ −− Θid− 2 id3 n 1 2  A0 2∆t  e Θ − Θid− 1 id3 n 2 0  A0 3∆t e     Θ Θ −  id2 − 4 id3 1  − +∆   A03(id id 2 3) t Θ Θ e id2 − 3 id3 

Combining the forced and free vibration regions: After finding the forced vibration and free vibration state transitions, the total expression in one in-cut and out-of-cut period is found by putting Equation (5.103) into Equation (5.104):

ϒϒ=⋅⋅DD , (5.109) id31 f F id

which gives the same result with Equation (5.73) for single in-cut and out-of-cut region pair. For single tooth cutting with low radial immersion, the period-state transition matrix would be:

Fp=DD fF ⋅ . (5.110)

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The stability of the operation is described similar to Equation (5.74). If multiple in-cut and out-of-cut regions are involved during one rotation period, then the solution needs to be modified in order to continue the state transition from one region to another. The final state transition would look like:

(Fp )multiple =⋅⋅(DDf,J F,J) ( D f,J-1 ⋅ D F,J-1)( DD f,1 ⋅ F,1) , (5.111) regions

where J is the total number of paired free-forced vibration periods. Equation (5.111) gives same result with Equation (5.110) for single region, J=1.

5.4 Summary

This chapter presents the derivation of the equation of motion and its solution for general cutting operations. The presented model covers turning, boring, drilling and milling operations with arbitrary tool geometry and multiple tools. The chatter stability of general cutting operation, and the cutting states such as forces, vibrations and surface location errors can be predicted. The application of the generalized model on various practical machining operations is presented in the next chapter.

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Chapter 6: Applications

6.1 Overview

The proposed new, generalized mathematical model of machining processes can be used to

model and predict the performance of various turning, boring, drilling and milling operations. In

this chapter, the model is applied to several industrial machining operations. The validity of the

proposed model is experimentally verified in milling with solid end mills, single and multiple

indexable milling operations, drilling, boring, and in parallel machining operations. The

validations are carried out by comparing the simulated and experimentally observed cutting

forces, surface form errors, and chatter stability diagrams.

6.2 Application-1: Slender, Cylindrical Helical End Mill with Regular Pitch Angles [2]

The parameters of the end mill is given in Table 6.1. The end mill has a long fluted section

with varying flexibility along the tool-workpiece contact [64]. The runouts and the FRFs are

measured along the tool axis. Radial runout is measured when the tool is on the spindle, by using

a dial gage. In Table 6.2, minimum value of the measured runout values is normalized to zero.

Runout of each element Sjk along the cutting edge is interpolated from the measured runout, and

meas is assigned to er,jk .

The generalized geometric, mechanics and dynamics modeling of the end mill is illustrated.

The geometry is modeled by 15 parameters that are described in Section 3.1. The mode shapes along the tool axis are used to identify the distributed vibration amplitudes at cutting edge segments. The corresponding cutting forces, surface location errors and chatter stability conditions are predicted by the model with experimental validations.

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Table 6.1 Information of the cylindrical end mill

Type of end mill Regular pitch cylindrical end mill Manufacturer and model SGS Series 1EL-3/4 end mill Diameter (nominal) 19.05-mm (3/4-in) Length of fluted section 76.2-mm (3-in) Stick out from the shrink fit tool holder 110-mm Number of flutes N=4 Rake angle Not given by manufacturer Helix angle 30-deg (given by manufacturer) Cutting edge angle 90-deg Only manufacturing error (measured data Radial runout in Table 6.2, interpolated data in Figure 6.1c) Axial runout Negligible

Table 6.2 Radial runout measurement at five axial locations using a dial gage with 5-micrometer tolerance.

Axial distance from 0 (tool tip) 9 19 29 39 the tool tip (mm) Measured j=1 5 0 10 20 17 radial runout j=2 19 10 7 0 4 (micrometer) for all teeth j=3 33 28 21 15 3 (j=1,..4) j=4 22 25 29 22 13

Geometry: The end mill is axially discretized with dz=0.2-mm segments. Only 38 mm depth of the fluted section would be in cut. Therefore, the total number of axial segments yields

to K=190:

38 38 K = = =190 . (6.1) dz 0.2

The end mill has four flutes (N=4). The 15 geometric parameters are defined at K locations

of each flute. Each geometric parameter is represented by a column, and each level is represented

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by a row. Thus, the whole geometry is stored in K-by-15N matrix (i.e., 190 x 15N ). Since all the cutting edges are available to cut, then g1 parameter is unity for all cutting edge elements:

g1, jk =1for all jNkK = 1.. and = 1.. . (6.2)

0 0 The axial location of each edge segment zk ( zZkk= in Equation (3.2) is measured from the tooltip, and defined at the midpoint of the axial segment-k of the tool:

Zkk =( − 0.5) d z. (6.3)

The angular location ψ jk (Figure 6.1a) of each cutting edge element Sjk is defined by the helix angle β j , the axial location Zk and the designed pitch angle φpd,j :

Z ψφ=−−( j 1) k tan β, (6.4) jk pd,j π j where the helix and pitch angles are uniform in this application:

2ππ β= 30deg ; f= = for alljN= 1.. . (6.5) j pd,j N 2

rel The relative location angle ψ jk (Figure 6.1b) is measured from the first S11 element in CW direction, as:

rel ψjk= ψψ jk − 11. (6.6)

The nominal radius (R0) of all cutting edges is the same for the cylindrical end mill:

R0 = 9.525 mm for alljNkK= 1.. and = 1.. . (6.7)

Radial runout along the cutting edge is found by linear approximation between the

meas measurement locations and stored in er,jk parameter. The highest measured radial runout value is 33- μm at the 3rd tooth at the tool tip (Table 6.2). Assuming that the maximum radius is the

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nominal radius (R0), the radius (Rjk) of each cutting edge element is found by normalizing the

meas measured runout er,jk to their maximum value. The radius Rjk (mm) at element Sjk is obtained

as shown in Figure 6.1d:

meas− 3 RRjk=−− 0(33e r,jk ) ⋅ 10 . (6.8)

meas The measured ( er,k ) and normalized ( εr,k ) radial runout vectors at axial segment-k are defined as:

emeas= eee meas meas meas r,k r,1k r,jk r,Nk ; (6.9)  er,k= ε r,1k εε r,jk r,Nk .

Figure 6.1 Geometry along cutting edge of cylindrical end mill: (a) Angular location; (b) relative angular

location, (c) radial runout; (d) radius. 158

The normalized radial runout vector at each axial segment-k is set to have minimum value of zero, as:

meas meas εεr,k= r,k − min( εr,k ) , (6.10) j

where the min(⋅ ) function takes the minimum of all teeth (j=1..N) values at axial segment-k j

(k=1..K). Figure 6.1c shows the radial runout ( εr,jk ) values after normalizing minimum value of

each axial segment-k to zero.

0 0 The locations ( xjk , yjk ) of the cutting edge element Sjk is derived from the angular location

ψ jk and the radius Rjk:

00 xRjk=⋅=⋅ jksinyy jk ; yR jk jkcos jk . (6.11)

0 0 0 Each location for element Sjk is represented by point ( xjk , yjk , zjk ) in the tool reference

coordinate frame (Frame-0), as is shown in Figure 6.2.

Figure 6.2 Modelled 3D locations in the Frame-0 of cylindrical end mill.

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* The helical cylindrical end mill has pre-set true cutting edge angle ( kr,jk ), cutting edge

inclination angle ( λs,jk ) and normal rake angle ( γ n,jk ) for each element Sjk:

ls,jk=β j; l s,jk = 30deg;

g n,jk = 0deg; (6.12) * kr,jk = 90deg; for each element Sjk .

The geometry matrix is formed according to the listed variables in Section 3.1. The cutting

force algorithm directly uses this geometry matrix, along with two more instantaneous condition

parameters, which are the edge-in-cut (g2,jki) condition from Equation (3.4) and the missed-cut

(g3,jki) condition as explained in Section 3.1. If the cutter rotates at constant speed Ω (rev/min),

then the instantaneous angular position measured from +y0 axis is:

2π φψjki= jk +Ωti . (6.13) 60

Since the chip thickness is changing between a minimum and maximum value during the

tool rotation, the effect of runout at each time instant (ti) is different. As a result, both the chip thickness and delay period become time-dependent. The cutting force is evaluated after calculating the instantaneous chip thickness using the feed per revolution and radial engagement of the tool.

Chip engagement calculation: For the 5% down milling case, the cutting edge element Sjk

is supposed to cut if its instantaneous angular position (φjki ) falls between the range of the entry

and exit angles (Equation (3.4)). Engagement condition of element Sjk at axial segment-k is shown in Figure 6.3 for milling operation. Figure 6.3a shows the radial entry (ast,k) and exit

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0 (aex,jk) distances from the cutter center in y direction. The engagement location depends on the maximum radius (Rmax,k), which is the maximum of the teeth radii (j=1..N) at segment-k:

RRmax,k = max ( jk ) . (6.14) j

Effect of vibration in y0 is assumed negligible. The regular no runout case is shown in Figure

6.3a. The formulation is done for more general case with runout (Figure 6.3b). A 5% down milling geometry is shown in Figure 6.4. The radial runout also affects the cutter entry and exit angles due to the missed-cut (g3,jki) condition parameter (shown in Figure 6.4b).

Figure 6.3 Tool entry and exit parameters for milling operation: (a) No runout case; (b) Effect of runout.

Figure 6.4 (a) Schematics of 5% down milling. (b) In-cut and out-of-cut regions.

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The entry and exit angles for element Sjk are evaluated at segment-k from Figure 6.3b and

Figure 6.4a as:

a φ = arccosst,k ; st,jk  Rjk (6.15) aex,jk φ =arccos − . ex,jk  Rjk

where, radial entry and exit distances (ast,k and aex,k) are the relative positions of the workpiece

edges measured from the cutter center in y0 axis. For 5% down milling, the radial entry distance

is 90% of the Rmax,k stated by tool envelope:

 aRst,k=−−⋅ max,k(2 R max,k ) ( 5%) ; ast,k =− 0.9 R max,k . (6.16)

The entry distance ( ast,k ) is negative due to the definition in Figure 6.3a. The maximum radius at k-th segment is larger than or equal to the actual radius (Rjk) of element Sjk. Then, the

radial exit distance for each element Sjk is equal to the actual radius:

aRex,jk= jk . (6.17)

Figure 6.5 Effects of radial tunout for 5% down milling of cylindrical helical end mill, on (a) tool entry angle;

(b) Maximum effective chip load along the cutting edge.

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The entry and exit angles are evaluated from Equation (6.15), as:

R φ =arccos − 0.9max,k ; st,jk  Rjk (6.18) Rjk φφ=−=arccos ; 180deg. ex,jk ex,jk Rjk

The predicted cutter entry angle variation for 5% down milling is shown in Figure 6.5a.

Cutter exit angle is 180-deg at all points along the cutting edge. The effect of the runout on the

chip load can be seen from Figure 6.5b during one spindle rotation. Maximum effective chip

load along the cutting edge (k=1..190) of each tooth (j=1..4) with dz=0.2-mm. In Figure 6.5b, for

feed fr=1.0-mm, total effective chip load at each segment-k almost keeps constant (~0.42-mm).

The algorithm presented in Section 4.1 is applied for the actual static chip thickness simulation.

The edges j at each segment-k have different runout values. Therefore, each axial segment-k requires a separate analysis.

Analyses of edge-in-cut and missed-cut conditions are also combined in the simulation. For example, at 1.2-mm depth (k=6; kdz=1.2-mm), Figure 6.1a shows that the tooth j=3 has the highest runout, and tooth j=1 has the least runout. Hence, tooth j=3 cuts the most amount of chip whereas tooth j=1 cuts the smallest amount of chip. The flutes remove chips with the sequence of

4-3-2-1. In Figure 6.6, effect of runout on in-cut condition is simulated result for all teeth at the

1.2-mm depth (k=6). As tooth j=3 (Figure 6.6c) cuts during all instants in the engagement zone, j=4 (Figure 6.6d) has the second most cut amount, and teeth j=2 (Figure 6.6b) and j=1 (Figure

6.6a) cuts the least.

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Figure 6.6 Zoomed tool-workpiece contact zone (from Figure 6.4b) at 1.2-mm axial depth, for tooth/edge

numbers: (a) #1; (b) #2; (c) #3; (d) #4.

pa φ The effective chip load ( cjki −γ jki ) and the actual pitch angle p,jki for each tooth at 1.2-mm depth and fr=1.0-mm is presented for one spindle period in Figure 6.7a and Figure 6.7b, resepectively. As seen in Figure 6.7b, Tooth-4 has 270-deg pitch angle when exiting the cut.

That is due to the material left from the previous teeth (j=1 and j=2). Tooth-3 cuts the most material. Tooth-3 also cuts the left portion by tooth-4 (shown in Figure 6.6d) and it gives a 360- deg pitch angle instantaneously.

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Figure 6.7 Effect of runout on: (a) effective chip load; (b) actual instantaneous pitch angle.

Cutting force prediction with runout: Cutting coefficients in RTA-frame are identified mechanistically for this application [6] and listed in Table 6.3. All the true cutting edge angles

* kr, jk along each cutting edge element Sjk are 90-deg (see Equation (6.12)). Then, transformation

from the RTA (cutting edge coordinate frame) to Frame-R (insert coordinate frame) is constant, as from Equation (4.41):

−10 0  TRI = 010. (6.19) 00− 1

The transformation from Frame-R to tool reference coordinate frame (Frame-0) is written

from Equation (4.42), as:

sinφφjki− cos jki 0  T0R,jki = cosφφjki sin jki 0 , (6.20)  0 01

where the instantaneous angular position is expressed (Equation (6.13)) for a constant rotation.

Due to vibration free nature of static condition, the static cutting force (Equation (4.70)) is

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assumed not to be affected by the vibrations of the tool-workpiece structure. The static differential force at each element Sjk along the depth of cut is:

st cs es FFFjki= jki + jki , (6.21)

where the static component of shearing force can be written from Equation (4.60) as:

⋅ st TT0R,jki RI,jk hjki FKcs = ⋅⋅c dz jki rta,jki * . (6.22) 00Rt, jk sinkr,jk

The transformation TRI,jk from RTA to Frame-R is constant at all points. The moment arm

* Rt, jk is the radius (Rjk of Figure 6.1d). The true cutting edge angle is kr,jk = 90deg for all points.

c The cutting coefficient vector ( Krta,jki ) is constant at all points and time instants (given in Table

st 6.3). The axial segment thickness dz=0.2-mm and the actual chip thickness ( hjki ) is found from

Equation (4.2) for milling. Equation (6.22) is rewritten as:

TT⋅ cs 0R,jki RI c st FKjki = ⋅rta ⋅⋅h jki dz . (6.23) 00Rjk

The static ploughing component of Equation (6.21) in simplified form is evaluated from

Equation (4.63):

TT⋅ es 0R,jki RI e FKjki = ⋅⋅rta dz , (6.24) 00Rjk

e where the edge force coefficient vector Krta is constant at all points and time instants (given in

Table 6.3). Integration of the total cutting force acting at the tool-workpiece contact gives the

total lumped cutting force Fi at each discrete time instant (ti) from Equations (4.72), (6.21),

(6.23) and (6.24), as:

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Table 6.3 Mechanistic identification of the milling operation. Tool information in Table 6.1.

Operation scheme 5% down milling Workpiece and cutter material Aluminum 7050 and uncoated carbide Spindle speed 12,700-rev/min Axial depth of cut 25-mm Number of sample constant feed tests 4 Feed per revolutions of each sample test 0.840, 0.920, 1.00 and 1.02-mm/rev Identified cutting coefficients Krc=106, Ktc=696, Kac=344-MPa Identified edge force coefficients Kre=13.4, Kte=12.4, Kae=0-N/mm Curve fitting accuracy R2=99%

 Nq, TT0R,jki⋅⋅ RI  TT0R,jki RI F=∑ gg g ⋅KKc ⋅+h st  ⋅e ⋅dz , (6.25) i 1,jk 2,jki 3,jki 00RRrta jki 00rta jk, jk jk

c with the total x ( Fx,i ),y ( Fy,i ),z ( Fz,i ) and torque ( Ti ) components:

T = c FiF x,i F y,i FT z,i i . (6.26)

The total cutting force is simulated for 3 spindle rotation periods for the given conditions in

Figure 6.9 at depth of cut 25-mm (ap=25mm, dz=0.2mm). Then, q=125 from Equation (4.73), spindle speed Ω=12700-rev/min, and feed per revolution fr=1.0-mm/rev. The sampling time is

∆=t 4.1442e-5[s] . The spindle rotation period is:

60 60 T = = =4.7244 × 10−3 [s] . (6.27) s Ω 12700

Each spindle rotation has the following number of simulation points (ns), as:

T nn=s ; = 114 . (6.28) ss∆t

The time instant (ti) is defined for 3 spindle periods as:

ti= i ⋅∆ ti, = 1,2...,3 nss ,3 n+ 1. (6.29)

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Figure 6.8 Dynamometer direct force transfer function:Magnitude plot; (b) Phase plot; (c) Magnitude

response in the vicinity of spindle rotation frequency; (d) in the vicinity of tooth passing frequency.

The direct magnitude (Figure 6.8a) and phase (Figure 6.8b) response of Kistler 9257 table

type force dynamometer in each x, y and z directions are measured from 0 to 1500-Hz (modal

hammer sensitivity 2.38 mV/N).

The force measurement is acceptable for the cutting force input under 600-Hz, since the

magnitude response plot is close to unity, i.e., the distortion is negligible in this range (see Figure

6.8c). The cutting forces in machine coordinates are compared with the experimentally measured

forces in Figure 6.9, for fr=1.0-mm; depth of cut=25-mm; 12700-rev/min spindle speed; 5% down milling; tool information in Table 6.1; Radial runout in Table 6.2; Cutting coefficients in

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Table 6.3. Kalman filter of our laboratory’s software is used for removing dynamometer distortion from x and y forces [135].

Figure 6.9 Experiment vs. predicted forces with runout effect.

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Spindle rotation frequency is 12,700/60=211-Hz and tooth passing frequency is 211*4=844-

Hz, which is greater than the dynamometer’s 600 Hz bandwidth. As seen in x force plot of

Figure 6.9, the collected data has distortion due to the 844-Hz tooth passing frequency. Thus, the

fundamental harmonics related to the spindle rotation frequency (211-Hz) is captured accurately.

However, as seen in Figure 6.8d, there is a heavy distortion mode around the tooth passing

frequency (844-Hz) due to the response peak in x direction. In Figure 6.9, Kalman filter is used

to remove the distortion modes from the force in x and y directions [135]. Although the filter in x

direction gives a clear output, the y direction filter does not change the results as much. The

reason is that the distortion in y-direction is around 1055-Hz (see Figure 6.8a).

Dynamics of the tool workpiece system: The relative vibrations between the slender end

mill and rigid workpiece occur due to the periodic cutting forces distributed at the contact zone.

Figure 6.10 shows the Frequency Response Functions (FRF) of the tool. Machine tool is Mori

Seiki NMV5000DCG; modal hammer sensitivity is 2.38-mV/N; Polytec laser Doppler velocity

sensor sensitivity is 10-V/mm. Test locations on the tool are shown in Figure 6.10a: The impact

location is point-3; output is measured at points 1 to 4. Figure 6.10b and Figure 6.10c give the

magnitude plots of FRF measurements in feed (x0) and normal (y0) directions, respectively.

Modal parameters are identified using Cutpro [118]. Only tool related vibration modes (with subscript t) are considered. Modal damping ratio ( ζt ) and natural frequency ( ωn,t ) matrices of

Equation (5.17) are formed using the identified modal parameters in Table 6.4. Mode shape Ut

matrix is formed by linear interpolation between the measurement locations. As shown in Figure

6.11, mode shape values at the measurement locations (Table 6.4) are interpolated at the mid-

locations of each axial segment-k (1..K). 170

Figure 6.10 FRF measurements along the tool length: (a) Test locations; (b) Feed direction magnitude plot;

Figure 6.11 Sketch of Figure 6.10a.

Table 6.4 Identified modal parameters in x0 and y0 directions.

Feed direction Normal direction Number of vibration 8 8 modes Natural frequencies 76,139,261,405,570,780,825,1143 190,265,424,566,794,830,1144,1508 [Hz] Damping ratios [%] 3.50,4.89,4.55,6.45,3.17,3.18,1.10,3.34 4.90,3.12,1.88,3.13,2.53,0.92,2.98,5.36 Mass normalized Mode #1: 0.0574,0.0540,0.0566,0.0532 Mode #9: 0.0781,0.0846,0.0947,0.104 mode shape vectors Mode #2: 0.0613,0.0577,0.0629,0.0684 Mode #10: 0.110,0.088,0.0904,0.0882 for each vibration Mode #3: 0.144,0.120,0.123,0.107 Mode #11: 0.154,0.127,0.138,0.103 mode [ 1/ kg ] in x Mode #4: 0.272,0.170,0.213,0.163 Mode #12: 0.470,0.362,0.333,0.226 and y directions. Mode #5: 0.501,0.369,0.342,0.242 Mode #13: 1.71,1.18,0.910,0.589 Mode #6: 1.61,1.05,0.939,0.577 Mode #14: 1.06,1.01,0.733,0.448 Mode #7: 1.26,1.14,0.784,0.556 Mode #15: 2.04,1.41,0.753,0.257 Mode #8: 2.01,1.56,0.800,0.238 Mode #16: 1.22,0.812,0.226,0.0153

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Axial (z) and torsional (θ) directions are rigid, hence their mode shape values are set to zero

in Equation (5.16), as:

uu=0; = 0 z,k,ntt ,t θ,k,n ,t . (6.30)

Mode shape values in x and y directions ( u and u ) at each axial segment-k are x,k,nt ,t y,k,nt ,t

interpolated from Table 6.4; the hammer test locations are shown in Figure 6.10a. The mode

shape vector u is formed at each level-k (k=1..K) and for each mode nt (1..mt) with the total k,nt ,t

number of tool modes mt=16 (8 modes each direction). Hence, the tool mode shape matrix Ut (4q

x mt) for the operational depth of cut (ap=qdz) is obtained. For example, if ap=25-mm depth of cut and dz=0.2-mm segment thickness, then the number of contact elements is q=125, and the mode shape matrix would be 500 x 16. Since the workpiece is rigid, all the modal parameters of the tool represents the combined system: m=mt; U=Ut; ωωn,c= n,t ; ζζct= .

The cutting force is axially distributed along the tool workpiece contact following from

Equation (5.23). Process damping is neglected; the force at each cutting edge element Sjk is formed and the force vector at each axial level is summed up to obtain the distributed force vector (Equation (4.74)), as:

gg1,j1 2,j1i g 3,j1iF j1i  gg1,j2 2,j2i g 3,j2iF j2i N   Fi = ∑ . (6.31) gg1,jk 2,jki g 3,jkiF jki j   gg g F 1,jq 2,jqi 3,jqi jqi (4q) x1

Rewriting the force in general form (Equation (5.23)) without the process damping leads to:

172

0[4(km− 1) x ] Nq,  cs cd d es Ffi= i+ f iΓΓ c,i −∑ fjki[4 xm ]⋅− c(tT i jki) +f i . (6.32) jk,  0 [4(qk− ) x m ]

cs The static component of the shearing force vector ( fi ) at time ti is formed from the general

cs representation of Equation (5.24), where the force Fjki at each segment-k is derived from

Equation (4.60):

gg g Fcs 1,j1 2,j1i 3,j1i j1i gg g Fcs 1,j2 2,j2i 3,j2i j2i N  f cs = ∑  i cs . (6.33) j gg1,jk 2,jki g 3,jkiF jki    gg g Fcs 1,jq 2,jqi 3,jqi jqi (4q) x1

es Similarly, the static ploughing force vector ( fi ) at time ti is obtained from the general

es formula Equation (5.27), where the force Fjki at each segment-k is obtained from Equation

(4.63):

gg g Fes 1,j1 2,j1i 3,j1i j1i gg g Fes 1,j2 2,j2i 3,j2i j2i N  f es = ∑  i es . (6.34) j gg1,jk 2,jki g 3,jkiF jki    gg g Fes 1,jq 2,jqi 3,jqi jqi (4q) x1

cd Dynamic component ( Fjki ) of the shearing force of element Sjk at time ti has two components (as seen in Equation (4.61)): One related to the current vibrations, and the other one

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related to the delay vibrations. Rewriting the Equation (4.61) using the relative vibration vector of Equation (5.89) leads to:

cd d FFjki= jki(ΨΨ ki −− k()tT i jki ) , (6.35)

where the relative displacement Ψki vector and the delay vector Ψk()tT i− jki at segment k are:

x xtk() i− T jki ki   − yki ytk() i T jki ΨΨki = ;(ktT i−= jki )  . (6.36) zki ztk() i− T jki     θki θ ()tT− [4 x 1] k i jki [4 x 1]

The overall displacement Ψi vector becomes:

Ψ1    . (6.37) Ψi = Ψki   Ψ qi (4q x1)

The overall delay vector Ψd,jki for segment-k is defined by only keeping the element at segment-k and setting the rest to zero, as:

0[4(k− 1) x1]  ΨΨd,jki= k()tT i − jki . (6.38)  0 − [4(qk ) x1] (4q x1)

Equations (6.37) and (6.38) are used to rewrite Equation (6.35), as:

cd d  Fjki= F0 jki [4 x 4(k−− 1)] I [4 x 4] 0 [4 x 4(qk ))](ΨΨ i− d,jki ) , (6.39) which is reorganized with the modal transformation (Equation (5.88)), as:

cd d  Fjki= F0 jki [4 x 4(k−− 1)] I [4 x 4] 0 [4 x 4(qk ))] U t−⋅ U w(ΓΓ c()t i − c ( tT i − jki )) . (6.40)

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Since the only non-zero elements in Ψd,jki belong to the segment-k, the delay modal vector

Γc()tT i− jki is written using delay period Tjki of tooth-j in Equation (6.40) at the segment-k.

cd The first component fi of the dynamic shear force vector is the multiplier of the current modal displacement vector Γc,i in Equation (6.40):

gg g FId  0 U 1,j1 2,j1i 3,j1i j1i [4x 4] [4x 4(q− 1))] [4 qm x ] d  gg1,j2 2,j2i g 3,j2iF0 j2i [4x 4] I [4x 4] 0 [4x 4(q− 2))] U [4 qm x ] N  f cd = ∑ , (6.41) i [4xqm] d j gg g F0 I 0 U 1,jk 2,jki 3,jki jki [4x 4(k−− 1)] [4x 4] [4x 4(qk ))] [4 qx] m   d gg1,jq 2,jqi g 3,jqiF jqi0 [4x4(q− 1)] IU [4x4] [4qm x ] [4xqm] or simply:

gg g FId  0 1,j1 2,j1i 3,j1i j1i [4x 4] [4x 4(q− 1))] d  gg1,j2 2,j2i g 3,j2iF0 j2i [4x 4] I [4x 4] 0 [4x 4(q− 2))] N  f cd = ∑ ⋅ U . (6.42) i [4xqm] d [4qm x ] j gg g F0 I 0 1,jk 2,jki 3,jki jki [4x 4(k−− 1)] [4x 4] [4x 4(qk ))]   d gg1,jq 2,jqi g 3,jqiF0 jqi [4x 4(q− 1)] I [4x 4]  [4qq x4 ]

The second component of the dynamic shear force vector includes the delay modal

displacement vector Γc()tT i− jki :

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0[4(km− 1) x ] Nq,  d ∑ fjki[4xm ]⋅−Γ c(tT i jki ) =... jk,  0 [4(qk− )x m ] (6.43) 0[4(km− 1) x ] Nq,  d ... ∑ gg g F0−− I 0⋅⋅ UΓ tT − . 1,jk 2,jki 3,jki jki [4x 4(k 1)] [4x 4] [4x 4(qk ))] c( i jki )[m x1] jk,  0[4(qk− )x m ] 

Thus, all the components of Equation (6.32) are obtained. The rest of the formulation is the

same as the solution of Equation (5.19).

Prediction of stability chart: The chatter stability of the cylindrical end mill with the

parameters given in Table 6.3 is verified. Only the lateral (x,y) directions of the cutter are flexible, and the size of the modal matrix U is halved by removing the terms related to the rigid z and torsional directions.

The sampling time is selected by considering the highest natural frequency and the desired number of points in the engagement zone.

The sampling time is initially chosen to be 6 times faster than the maximum natural frequency of the tool workpiece system, i.e., max(ωn,c ) = 2π ⋅ 1508 rad/s (see Table 6.4), hence

∆= ( t)1 1.1052e-4 s. It is also desired to have at least 8 sample points in the tool workpiece

engagement zone. For 5% down milling, the nominal entry and exit angles are 154.1-deg and

180-deg respectively. Then, the ratio of cutting time in one rotation period is

360 (180−= 154.1) 13.9 . If 8 sample points are taken in the cutting time, then in one rotation

period, there are ns =13.9 ⋅≅ 8 112 (integer value) sample points. For example, for 8000 rev/min

spindle speed, one cutter rotation period is Ts =60 8000 = 7.5e-3 s. The sampling time is

176

∆= = ( t)2 Tnss0.696e-4 [s], which is almost half the value of the sampling time from the first

∆= criterion: ( t)1 1.1052e-4 . The number of discretization is chosen as: ns=112. From Equation

(5.38), ns has to be an integer multiple of 3, therefore ns is updated to the nearest higher integer

multiple of 3: ns =114 . Updating the number of discrete states in one period also updates the

sampling time: ∆=t Tnss =0.678e-4 [s]. The procedure is applied to all cases.

The thickness of axial segments is set to dz=1-mm, which leads to K=38 segments for a 38-

mm maximum simulated depth of cut.

The stability is predicted between the speed range of 8000-rev/min to 16000-rev/min at 100- rev/min increments. Scanned axial depth of cut are from 10-mm to 38-mm with 1-mm

increments. The critically stable depth of cut is detected at one speed when an unstable

eigenvalue (larger than 1) is reached. The cutting forces and surface finish are measured to assess

the stability during experiments, as is shown in Figure 6.12. Stability chart is predicted separately when the effect of runout is included in one case, and neglected in the other case

(Figure 6.12a). The simulation without runout is compared to the frequency domain solution

[136]. Radial runout increases the stability limit due to the change in delay periods along the cutting edge. The resulting disturbance on the chip regeneration increases the stable depth of cut limit. Wan et al. [115] reported similar results on the effect of runout on stability.

While the chatter frequency is predicted to occur at 1152-Hz, it occurred at 1245-Hz during the milling experiments. The difference may be due to slight measurement error during impact testing, and also due to variations in the contact stiffness of spindle bearings and tool-holder- spindle interface joints during machining. For example, the presence of split modes at 790 and

1144 Hz leads to identification errors during impact testing.

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Figure 6.12 Results of discrete-time simulation: (a) Stability predictions; (b) SLE measurement; (c) Predicted

and measured SLE; (d) Example displacement simulation.

The surface location error (SLE) is predicted from a window of 50 spindle rotation periods

during stable machining after the transient vibrations diminish in seven spindle periods. (P=50 in

Equation (5.87)), and is shown in Figure 6.13 for unstable (A) and stable (B) cases. The chatter frequency is predicted as 1152-Hz in Figure 6.13 whereas the measurement (Figure 6.12a) gives

1245-Hz.

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Figure 6.13 Discrete time force simulation for the experiments (A) and (B) of Figure 6.12aExperimental

validation of SLE has been carried out by machining a reference surface just before the actual machined area. The reference surface with 30-mm axial depth and 5 mm radial width has been pre-machined in multiple passes with very small forces to create smooth surface which is not touched by the tool during test cuts Also, another control surface with 5-mm depth is formed just below the machined area. The milled surface for SLE verification is measured by a dial-gage

instrument at 3 different locations (Figure 6.12b) in the helix angle direction. Three measurement

sets are presented in Figure 6.12c; zero SLE of this graph corresponds to the reference surface

after compensating for the radial offset.

The predicted static flexibilities using the modal model at the 3rd and 4th points (Figure

6.10a) on the cutter are 1.60e-7 and 2.86e-7 m/N, respectively. However, the measured static

179

flexibilities at the same points are found to be 1.90e-7 and 3.40e-7 m/N, respectively. The static

stiffness errors in the direction normal to the surface are compensated by creating a highly

damped (50%) fictitious mode at a low frequency (10-Hz), which shifted the SLE upwards.

SLE is predicted by the steady-state section of discrete-time displacement simulation, an example of which is shown in Figure 6.12d, at 13-th axial segment along the tool axis. Predicted

SLE is compared to the measurement in Figure 6.12c. Minima of the predicted and measured

SLE are 28 and 56-micrometer respectively. The maxima of predicted and measured SLE are 75-

and 95-micrometer respectively. It seems that the flexibility of the actual system during the

machining at this spindle speed has less damping and stiffness than the measured system.

6.3 Application-2: Regular Pitch Serrated Cylindrical Helical End Mill [21]

Serrated end mill, or the end mill with undulated cutting edge [137], is designed to increase

the chatter stability of roughing operations. A three fluted serrated end mill is modelled. The

model is validated with the experiments presented by Merdol and Altintas [21] and with the

exact numerical simulation given by Cutpro [118].

Geometry: As presented in Section 3.2.2, the geometry of the serrated flutes is measured as

a radial distance from the tool axis, as is shown in Figure 3.35a. Measurement along the flute is

then projected to the tool axis (Figure 3.35b). The nominal geometry of the end mill is given in

Table 6.5.

serr,s serr Cutting edge angle (kkr,jk = r,j (d)ks) at midpoint of each axial segment-k is approximated numerically between the two neighboring measured points (from Figure 3.35a):

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2ds k serr,s ≅ arctan , (6.44) r,jk ss RRj,k+1- j,k-1

where dsz= d cos β serr from Equation (3.53). The approximate cutting edge angle along the flute

is given in Figure 3.36a. Cutting edge angle information is projected onto tool axis in Figure

3.36b and Figure 6.14b. Radial runout εr,jk (Figure 6.14a) is obtained by normalizing each axial segment-k to its minimum (similar to Equation (6.10)) using the local radius (Figure 3.35b).

Angular location and radius of each element Sjk is defined as the location of each element Sjk

relative to the origin of the tool reference frame. In Equation (3.55), the position vector 0Dr of

the coordinate system of each cutting edge element Sjk is obtained from the transformations of

Figure 3.34a-d. The axial segment thickness is set to dz=0.1-mm; the geometry is interpolated at

the midpoint of each segment. The maximum depth of cut is set to 39.2-mm, which makes

K=39.2/0.1=392 axial levels in total. The angular location (ψ jk ) (Figure 6.15a) is found using the

0 0 x y coordinates (in tool reference frame) of the element Sjk:

Table 6.5 Information of the serrated end mill

Type of end mill Regular pitch cylindrical serrated end mill Manufacturer and model Sandvik R216.33-20040-AC38U H10F Diameter (nominal) 20-mm Length of fluted section 38-mm Number of flutes N=3 Designed pitch angle 120-deg Rake angle / Helix angle γ serr =10-deg / β serr =40-deg serr serr Cutting edge angle, kkr,jk= r,j (d)kz Varying due to serrated profile (Figure 6.14b) Radial runout Figure 6.14a. Axial runout Negligible

181

y =π − atan20D0Drr , jk 2 ( y x ) . (6.45)

Relative angular location is defined relative to the element S11 in CW direction. The modelled 3D geometry of cutting edge in Frame-0 (tool reference frame) is shown in Figure

6.16a. Close-up 3D view is given in Figure 6.16b with tooth numbers (j=1,2,3) shown.

The remaining rotational transformations (Figure 3.34e-h) are done to obtain the cutting

i j k edge tangent ( D )jk , rake face normal ( D )jk , and the rake face ( D )jk vectors (Equation (3.56)).

Tool angles are computed from the formulation given in Section 3.3. The true cutting edge angle, cutting edge inclination angle and normal rake angle are presented in Figure 6.17.

Figure 6.14 Serrated end mill flute geometry: (a) Normalized radial runout; (b) cutting edge angle.

Figure 6.15 Serrated end mill: (a) angular location; (b) relative angular location.

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Figure 6.16 Serrated end mill: (a) Modelled 3D geometry; (b) close-up 3D view.

Figure 6.17 Computed tool geometry of serrated end mill: (a) True cutting edge angle ; (b) Cutting edge

inclination angle; (c) Normal rake angle .

Computation of the tool angles along the cutting edge leads to the prediction of forces and stability for the given conditions.

Force prediction: The actual chip load distribution is found by considering the serrations as radial runouts. An example of radial runout for the first 8-mm depth along the cutting edge is shown in Figure 6.18a. Maximum effective chip load simulations in Figure 6.18b-f show that the chip load reaches the feedrate at most of the cutting section due to the serrated profile. Only one tooth is in-cut (while the other two miss the cut) at the sections where chip load reaches the

183

feedrate (fr) value. Cutting engagement, depth of cut and feedrate of five simulations are given in

Figure 6.18.

Figure 6.18 (a) Radial runout for the first 8-mm depth; (b-f) Maximum effective chip load along the cutting

edge for five cases: (b) 50% down m., ap=8-mm, fr=0.24-mm/rev; (c) 50% down m., ap=8-mm, fr=0.30-

mm/rev; (d) 50% up m., ap=4-mm, fr=0.24-mm/rev; (e) 50% up m., ap=6-mm, fr=0.24-mm/rev; (f) Slot milling,

ap=14-mm, fr=0.12-mm/rev.

184

Cutting forces are distributed along the cutting edge and lumped at the tool tip (Figure 4.2).

Equation (6.25) is used for predicting the static cutting forces with the application of

c e transformation matrix ( TRI,jk ) and the cutting ( Krta,jki ) and edge ( Krta,jk ) coefficient vectors The matrix ( TRI,jk ) is used to transform the forces from the geometry dependent cutting edge frame

(RTA) to the edge reference coordinate frame (R) (Equation (4.41)) :

−−sinkk** 0 cos r, jk r, jk TRI, jk = 010. (6.46)  ** coskkr, jk 0− sin r, jk

* where the variation of true cutting edge kr, jk angle along the tool axis is shown in Figure 6.17a.

c c Cutting coefficients Ku,jki and Kv,jki on the rake face (UV-Frame) are found from Equation

(4.25) as:

t− 22ηβ c s,jki 1 tanjk sin n, jki Ku,jki = sin βa,jki; 2 22 cosλφs,jk sin n,jki cos ( φβγn,jki+−+ n, jki n, jk ) tan ηjk sin βn, jki (6.47) t− 22ηβ c s,jki 1 tanjk sin n, jki (d)kz Kv,jki = cos βa,jki , 222 cosλφs,jk sin n,jki cos ( φβγn,jki+−+ n, jki n, jk ) tan ηjk sin βn, jki

where the shear parameters are approximated using the orthogonal cutting tests. The workpiece

material is Aluminum 7050-T7451. Low speed Al7050-T7451 orthogonal cutting database of

Cutpro [118] is used with the following shear variables:

t=266 + 0.0437 ⋅v +⋅ 174 hst +⋅0.896 γ⋅180 ; s,jki c,jk jki n,jk π st π φγn,jki =(19.4 +⋅ 42.0 hjki +⋅0.02 ⋅vc,jk ) ⋅ +0.384n,jk ; 180 (6.48) βγ=25.8- 1.28⋅hst - 0.0075 ⋅v ⋅π +0.181⋅ ; a,jki ( jki c,jk ) 180 n,jk st limits :hjki ∈∈[ 0.04, 0.20] -mm; vc,jk [ 14.3, 200] -m/min; γ n,jk ∈[ 0, 0.5235] -rad .

185

Shear stress τs,jki (MPa), shear angle φn,jki (rad) and average friction angle βa,jki (rad) of

st Equation (6.48) are stored as functions of static chip thickness hjki (mm), cutting velocity vc,jk

(m/min) and rake angle γ n,jk (rad). The cutting coefficients are transferred to RTA frame using

Equation (4.28), where the TIU, jk matrix transforming rake face (UV-frame) to RTA frame is from Equation (4.29). The edge force coefficients are directly found in RTA frame using the orthogonal cutting force database. The edge force coefficients defined in Al7050-T7451 database of Cutpro [118] are given as:

Figure 6.19 Force simulations compared to Cutpro [118]: (a) 50% down m., ap=8-mm, fr=0.24-mm/rev; (b)

50% down m., ap=8-mm, fr=0.30-mm/rev; (c) 50% up m., ap=4-mm, fr=0.24-mm/rev; (d) 50% up m., ap=6-

mm, fr=0.24-mm/rev.

186

Kve =20.9- 0.0152⋅- 0.358⋅⋅g 180 ; t,jk c,jk n,jk π Kve =33.6- 0.00598⋅- 0.151⋅⋅g 180 ; r,jk c,jk n,jk π (6.49) e Ka,jk = 0 (as expected for orthogonal cutting); st limits :hjki ∈∈[ 0.04, 0.20] -mm; vc,jk [ 14.3, 200] -m/min; g n,jk ∈[ 0, 30] -deg .

Cutting forces are predicted at dz=0.1-mm increments with Δt=1.25e-4 [s] sampling time; spindle speed: Ω=1600-rev/min; workpiece material: Al7050-T7451. Cutting engagement, depth of cut and feedrate of four simulations are given in Figure 6.19.

Figure 6.20 Force simulation compared to the experiment plots of [21]

187

Predicted forces are consistent with the numerical, time domain simulation results obtained

from Cutpro [118] in Figure 6.19, and with the experimental cutting forces [21] in Figure 6.20.

The phase difference between predicted forces and measured forces (Figure 6.20) can be neglected due to no information on dynamometer and measurement characteristics. Cutting engagement, depth of cut and feedrate of four simulations of Figure 6.20 are same as in Figure

6.19.

Stability chart: Tool is assumed to have lumped dynamics at its tip where the total forces are applied. The total cutting forces are expressed from Equation (6.32), as:

Nq, lumped cs,lumped+ cd,lumpedΓΓ −d ⋅−+es,lumped Ffi = ii fc,i∑ { f jki[4 xm ] c(tT i jki )} fi, (6.50) jk,

lumped where the total force vector is F . The static component of the shearing force ( f cs ) is ( i )[4× 1] i

modified to combine all the axial segments (from k=1..q) along the tool workpiece contact:

Nq, cs,lumped = cs (fFi ) ∑ gg1,jk 2,jki g 3,jki( jki ) . (6.51) [4x1] jk, 4x1

The ploughing force vector is also combined using the Equation (6.34):

Nq, es,lumped = es (fFi ) ∑ gg1,jk 2,jki g 3,jki( jki ) . (6.52) [4x1] jk, 4x1

The component related to current state of the dynamic shear force vector is written from

Equation (6.41). Since the modal identification is done at the tool tip, the size of the mode shape

is U[4 xm ] . The Equation (6.41) is modified by multiplying each segment by the lumped

dynamics mode shape matrix, as:

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gg g FUd 1,j1 2,j1i 3,j1i j1i [4xm ] gg g FUd 1,j2 2,j2i 3,j2i j2i [4xm ] N  f cd = ∑  i [4xqm] d . (6.53) j gg1,jk 2,jki g 3,jkiFU jki [4xm ]    gg g FUd 1,jq 2,jqi 3,jqi jqi [4xm ] [4xqm]

The lumped force vector is obtained by summing the contributions of all segments:

Nq, fcd,lumped = gg g FUd . (6.54) ( i ) ∑ 1,jk 2,jki 3,jki( jki) [4xm ] [4xm] jk, 4x4

Following the Equation (6.54), the component related to the delay state of the dynamic shear

force vector is obtained by combining the contribution of all segments, as shown in Equation

(6.43):

Nq,,Nq ∑∑fdd⋅−ΓΓtT = gg g FU⋅ ⋅−tT . (6.55) { jki[4xm ] c( i jki )} { 1,jk 2,jki 3,jki jki [4qm x ] c( i jki )[m x1]} jk,,jk

The rest of the formulation is the same as in Section 5.2. The hammer test is conducted at the tool tip, and tool information is found in Table 6.5. Modal parameters of the tool dynamics are taken from [21], and listed in Table 6.6. Since the two x and y directions do not have coupled dynamics, their mass normalized mode shape values are found as (Equation (5.16)):

Table 6.6 Identified modal parameters in x and y directions (taken from Table 2 of Ref. [21]).

Feed direction Normal direction Number of vibration modes 7 7 Natural frequencies [Hz] 480,530,893,1216,1433,2085,2691 470,533,884,1155,1270,1946,2670 Damping ratios [%] 1.52,2.80,3.03,3.33,4.11,2.94,2.65 2.81,2.67,2.41,1.61,2.80,2.44,1.89 Normalized mode shape values [ Mode #1: 0.2740; Mode #2: 0.1920; Mode #8: 0.1020; Mode #9: 0.2820; 1/ kg ] Mode #3: 0.5080; Mode #4: 1.5100; Mode #10: 0.5600; Mode #11: 0.6320; Mode #5: 0.9250; Mode #6: 1.7480; Mode #12: 1.0400; Mode #13: 1.9600; Mode #7: 1.7160. Mode #14: 1.6070.

189

ωω n,ntt ,t n,n ,t uux,k,n ,t = ,nntt= 1,2...7; y,k,n ,t = ,= 8,2...14, (6.56) ttkk x,ntt ,t y,n ,t

where the 1st mode to 7th mode of Table 6.6 are listed for x-direction, and the 8th mode to 14th

mode are listed for y-direction. and are the modal stiffnesses of x and y directions kx,nt ,t ky,nt ,t from Table 2. The vibration modes with the highest natural frequency have little effect on the overall flexibility, therefore reduced dynamics is formed by neglecting modes 6, 7, 13 and 14, as

is found in Table 6.6.

The stability of slot milling of Aluminum 7050 is simulated for the feedrate fr=0.12-mm/rev,

and is compared with the experimental and numerical results given by Merdol and Altintas [21].

Spindle speed is scanned from 4000 to 5500-rev/min with 100-rev/min increments. The depth of

cut is scanned from 7 to 11-mm with 1-mm increments at each speed. The sampling frequency is

set to 3-times the maximum natural frequency, and the axial segment thickness is set to dz=0.1-

mm.

Figure 6.21 Comparison of the stability prediction to the results presented in [21]. 190

Simulation results are shown in Figure 6.21. Full dynamics (Table 6.6) and reduced

dynamics (Table 6.6 when 6th,7th,13th and 14th modes removed) models give similar results. As seen in Figure 6.21, the predicted results are slightly lower than the results presented in the

original study reported in “Fig.13 of Ref. [21]”, the latter employing numerical, time domain

simulation that includes nonlinear effects such as vibrations affecting the chip thickness. In

addition, the numerical simulation assumes instability if the vibrations exceed more than 20% of

the forced vibrations [21]. However, the stability decision method presented in this thesis is

analytical and based on the eigenvalues.

6.4 Application-3: Two-Level Multifunctional Drilling/Boring Cutter [5]

A multifunctional (boring and drilling combined) tool case presented by Wan et al. [5] is

used to illustrate the application of the proposed general formulation on a cutter with complex

geometry and multiple functionalities. The cutter is a custom made multifunctional drill with the

Sandvik TM880 type body. It has two drilling inserts at the bottom level (Level-2) and two step-

boring inserts at the top level (Level-1) with different dynamics. Only the lateral flexibility is

critical for this cutter; it is rigid in the torsional-axial directions. The toolholder is Sandvik

Coromant HydroGrip (HSK-63A interface) 392.410CGA-63 20 088B.

Geometry: 3D geometry of the cutting edge on the cutter body is shown in Figure 6.22a.

Level-1 inserts are #1 and #2, and Level-2 inserts are #3 and #4. As shown in in Figure 6.22b,

radial depth vs. radius of the midpoints is a linear plot. 9.9-mm radial depth corresponds to cutter center segment, and 0.1-mm corresponds to the perimeter segment of the cutter.

191

(a)

Figure 6.22 Sandvik multifunctional tool: (a) 3D geometry; (b) Radial depth of the midpoints of the segments

along the cutter body.

The geometry and locations of the two inserts at the bottom side (Level-2) of the cutter are

identified from tool microscope measurements and is given in Figure 6.22a. The radial depth is discretized at dz=0.2-mm segment increments. Thus, the 10-mm radius is divided into

K=10/0.2=50 elements.

Figure 6.23a is the angular location of element Sjk measured from +y0 axis of the cutter

reference frame, and Figure 6.23b is the relative angular location is measured from S11 element in CW direction. Axial location (Ajk) of each element is measured from the tool tip. As shown in

Figure 6.24, the two levels are given separately for easier visualization. It is assumed that there is

no axial difference (Figure 6.24a) between the 1st and 2nd inserts at Level-1. Thus, there is no runout between Level-1 inserts (Figure 6.25a). The axial height difference between the tips of 3rd

and 4th inserts of Level-2 are measured after installing the cutter on the spindle. The chip engagement is shared in the overlapping zone (Figure 6.24b) of the Level-2 inserts. Figure 6.25b shows the axial runout as a result of the axial height difference in the overlapping zone (see

Figure 6.24b) of Level-2 inserts.

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Figure 6.23 Angular location (a) and relative angular location (b) of element Sjk.

Figure 6.24 Axial location: (a) Level-1; (b) Level-2.

Figure 6.25 (a) Axial runout: (a) Level-1; (b) Level-2.

Mechanics of cutting: The peripheral radius section starts at k=1 element, and k=50 element corresponds to the center segment in Figure 6.22b. The number of dynamic levels D in Figure 193

4.12 depends on the depth of cut, qdz. The depth of cut qdz is defined as the difference in the

outer radius of cutter and the final radius of the drilled/bored hole; the number of contact

elements is q. If the cutter is fully in cut with both levels, then D=2. If only the bottom or top

level inserts are in cut, then D=1.

The following three cases are considered:

• When the cutter starts the engagement only at the bottom side, Level-2 inserts are in cut. The

cutter acts as a regular drilling cutter. The cutter radius for the bottom section is 7-mm and

the number of elements is redefined as K=7/dz=35 for dz=0.2-mm segment thickness.

Number of dynamic levels is D=1. Depending on the number of contact elements q for the

depth of cut qdz, the limit for the level is d1=q.

• After dilling 14-mm depth hole, the topside Level-1 inserts engage with the workpiece. The

cutter radius is 10-mm and the number of elements is the same as the original value (K=50).

The number of dynamic levels is D=2. The depth of cut is qdz. The limit for the second level

is d2=q. The limit for the first level is constant since Level-1 is always fully engaged if

bottom inserts are in cut. For 3-mm radius area, the limit for first level is d1=3/dz=15.

• If the predrilled hole is larger than 7-mm, then only Level-1 inserts are in cut. The cutter acts

as a regular boring cutter. The cutter radius for the top section is 10-mm and the number of

elements is redefined as K=3/dz=15. The number of dynamic levels is D=1. Depending on

the number of contact elements q for the depth of cut qdz, the limit for the level is d1=q.

The geometric parameters are evaluated for all the elements (Sjk) of the cutter at both levels.

The axial runout of the Level-1 inserts are zero (Figure 6.25a), and the runout for the Level-2

inserts are given in Figure 6.25b. The designed pitch angle (Figure 6.26a) is derived when the

feedrate effect is neglected, i.e., when assuming that there is no overlap or runout between the 194

inserts. It is found by using the angular location of Figure 6.23. The actual pitch angle due to runout is presented in Figure 6.26b; it is determined when the overlapping zone is considered for fr=0.1-mm commanded feedrate.

The cutting edge tangent, rake face normal and rake face vectors are identified from the

measured geometry. The effective tool geometry is identified from the theory presented in

Section 3.3 and is shown in Figure 6.27. Cutting coefficients in RTA directions are presented in

Figure 6.28a-c, for machining AISI 1045 steel at 1000-rev/min cutting speed and 0.1-mm/rev

feedrate; edge coefficients in RTA arepresented in Figure 6.28d-f.

Figure 6.26 Pitch angle: (a) Feedrate neglected (fr 0); (b) fr=0.1-mm.

≈

Figure 6.27 Effective tool geometry calculated from the rake face orientation: (a) True cutting edge angle; (b)

Cutting edge inclination angle; (c) normal rake angle at each element Sjk.

195

Figure 6.28 Cutting force coefficients (a,b,c) and edge coefficients (d,e,f) in RTA directions.

Total static cutting forces acting at the tool-workpiece contact is found by integrating the

ed forces as derived in Equations (4.70) and (4.72). Process damping ( Fjki ) forces are negligible.

Force vector at each segment consists of static shear Equation (4.60) and static ploughing forces

Equation (4.63). The moment arm is the radius at each element: RRt,jk= jk . The static chip 196

thickness of drilling operation is written from Equation (4.2), and the total static cutting force at

element Sjk is expressed as:

 TT0R,jki⋅⋅ RI,jk TT0R,jki RI,jk 1 FKst = ⋅cc a −γ + ⋅⋅ Ke dz . (6.57) jki 00RRrta,jk( jk jk ) 00rta,jk * jk jk sinkr,jk

Transformation matrices at each element Sjk are given in Equations (4.36) and (4.37). For

constant rotation speed Ω (rev/min), the instantaneous angular position is found from Equation

(4.38):

2π φjki =Ω+tijψ k . (6.58) 60

The edge-in-cut (g2,jki) parameter takes the value 1 due to the continuous cutting action of

drilling:

g2,jki =1for allj = 1.. Nk , = 1.. K and i = 1.. ns +1 (one rotation period) . (6.59)

Since the chip engagement is time invariant for drilling operation (Equation (4.2)), the

missed cut condition (g3,jki) parameter becomes time invariant as well: g3,jki=g3,jk. Figure 6.29

shows that for Level-1 inserts (#1 and #2), g1 and g3 are the same due to zero axial runout (see

Figure 6.25a). However, g1 and g3 parameters for Level-2 inserts (#3 and #4) are not the same because of the nonzero axial runout (see Figure 6.25b-e). Missed-cut conditions for the Level-2

inserts depend on feedrate (Figure 6.29d-e shows for fr=0.1-mm/rev). Substituting the force terms from Equation (6.57) into Equation (4.72) gives:

Nq, FFst = gg st i∑ 1,jk 3,jk jki . (6.60) jk, 4x1

197

Figure 6.29 Element-available-to-cut (a) and missed-cut (b,c,d,e) conditions for Sandvik multifunctional tool.

The prediction of cutting forces is validated experimentally. Number of data points per spindle period is kept constant for each simulation, ns=360. The simulation is carried out for one

spindle period: ti= i ⋅∆ ti, = 1,2..., nnss , + 1 . The work materials were AISI 1045 steel (Equation

(4.22)) and Aluminum 7050-T7451 Equation (6.48), and the cutting coefficients were obtained

198

from the Cutpro database [118]. The predicted forces (in tool reference frame) for drilling AISI

1045 steel are compared against experiments, as shown in Figure 6.30. For the test of Figure

6.30a1-a3: Only Level-2 of Sandvik TM880 cutter is in cut; it is full hole drilling; 3000-rev/min spindle speed; no pre-drilled hole with 7-mm radial depth (dz=0.2-mm, q=35), fr=0.1-mm/rev.

For the test of Figure 6.30b1-b3: Both levels (all inserts) of Sandvik TM880 cutter are in cut; it is

combined full hole drilling / stepboring operation; 1000-rev/min spindle speed; 10-mm radial

depth (dz=0.2-mm, q=50); fr=0.1-mm/rev.

Figure 6.30 Predicted cutting forces for Sandvik multifunctional tool: (a) Only Level-2 in cut with full hole

drilling; (b) both levels in cut with combined drilling / stepboring.

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Dynamics: The formulation is shown for the second case, where both levels of the cutter are

in cut. Level-1 inserts cover the outer 3-mm radial section, thus from k=1 to k=15, and d1=15, as shown in Figure 4.12. Level-2 inserts cover the inner 7-mm radial section; starting from

k=d1+1=16 to q. The parameter d2 in Figure 4.12 is dD=q. The force vector along the tool

workpiece contact is expressed from Equation (4.77):

15 ∑ ggF N 1,jk 3,jk jki k1= F = ∑ . (6.61) i q j ∑ ggF 1,jk 3,jk jki k= 16 [4⋅ 2] x1

The rest of the formulation is similar to the formulation given in Section 5.2. The flank wear

land is assumed to be Lw=20-micrometer in the process damping force model. Specific

3 3 indentation force for Aluminum and Steel is taken as Ksp=1.5e5 N/mm [120] and 4e5 N/mm

[119] respectively. Coulomb friction coefficient is assumed as µ =0.3.

Table 6.7 Identified modal parameters for Sandvik and Kennametal multifunctional tools.

Natural Damping Vibration Cutter type frequency ratio Mass normalized mode shape matrix mode (Hz) (%) Sandvik 1 936.66 2.51 uux,1,1,t x,1,2,t 0.801 0   TM880 ( x uu   y,1,1,t y,1,2,t  0 0.814 direction)  uuz,1,1,t z,1,2,t  00 2 928.57 3.93   uu 00 ( y θ,1,1,t θ,1,2,t    UU=t = = 1 kg direction) uu 0.838 0   x,2,1,t x,2,2,t  uuy,2,1,t y,2,2,t 0 0.881  00 uuz,2,1,t z,2,2,t  00 uuθ,2,1,t θ,2,2,t   Kennametal 1 3677 0.51 0 HTS-R (torsional- ux,1,1,t  0 axial, zθ) uy,1,1,t UU= = = −0.1761 1 kg t u  z,1,1,t  2 uθ,1,1,t 188.4 1 kg⋅ m 

200

Instead of lumping the forces at each segment as in Equation (6.31), there are two dynamic levels. The corresponding segments are summed within each dynamic level. The related mode shape matrix includes only two dynamic levels (1 and 2) instead of every radial segment

(1,2...q). Only the principal lateral directions are measured since the torsional axial flexibilities are comparatively more rigid. The principal directions are assumed to be fixed and nonrotating.

The identified modal parameters are listed in Table 6.7. There are two mode shape vectors ue,nt ,t for each dynamic level (e=1,2) and for each mode (nt=1,2) of the cutter. Only the lateral related

ux,e,nt ,t and uy,e,nt ,t are nonzero, and the torsional axial components are zero: uu=0; = 0 z,e,ntt ,t θ,e,n ,t .

Prediction of stability chart is validated experimentally for each case. When only Level-2

(bottom side) is in cut (first case), the process is stable at all speeds for full hole drilling‒see

Figure 6.31a5. When only the top inserts are in cut, the predictions are highly sensitive to the input geometry. There are two lateral stability predictions: The first prediction is when the measured geometry is used (Figure 6.31a1 and Figure 6.31a2), and the second is when the top side inserts are assumed regular-pitch 2-tooth boring cutter (Figure 6.31a3 and Figure 6.31a4).

When only Level-1 inserts (#1 and #2) are in cut, Figure 6.31a3 gives a more accurate estimation than Figure 6.31a1. This shows that the measured geometry is inaccurate and the actual dimensions are closer to that of a regular 2-tooth boring cutter. For the case when all inserts in cut both simulations (Figure 6.31a4 and Figure 6.31a2) give inaccurate results. Cutting tests are all stable in the measured range most probably due to the stiffening effect from the hinge or pinned contact at the cutting zone of the drill cutter. When only the bottom-side (Level-2) inserts are in cut, the simulation and prediction are matching as seen in Figure 6.31a5.

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Figure 6.31 Sandvik multifunctional tool stability predictions compared to experiments: (a1,a2) With the

measured geometry of Level-1 inserts; (a3,a4) with the Level-1 inserts assumed regular 2-tooth boring

geometry; (a5) when only Level-2 inserts are in cut; (b) results of a sample unstable cutting test [5].

During measurements, the cutting force and sound data are collected and the resulting surface is checked to decide if chatter occured. Measured force, sound and surface image of a 202

sample unstable cutting test (from Ref. [5]) is presented in Figure 6.31b. The sample data is for

the case in Figure 6.31a1 and Figure 6.31a3, at 1,200-rev/min unstable cutting test.

When all the inserts are in cut, the stability is predicted with better accuracy if the measured

geometry is used. Therefore, following are concluded:

• The stability prediction depends directly on the geometric identification. Accuracy of

geometry directly affects the delay period estimation which influences the stability

predictions.

• One other reason may be the stiffening effect of the drilling cutter during the operation,

as is argued by Tekinalp and Ulsoy [138].

6.5 Application-4: Drilling with a Multifunctional, Indexable Cutter

A drill with four inserts [5] is used to validate the generalized model for the prediction of drilling operations‒see Figure 6.32. The geometry of each insert is presented in Figure 3.31.

Locations of inserts are measured with an optical tool microscope and the geometry is constructed as shown in Figure 6.32. The diameter of the cutter body is 46-mm with a 10-mm

diameter twist drill with its center having a 118-deg point angle and a 30-deg helix angle

(Kennametal catalog #B513S10000; order #1940741-HSS coated; grade AS3).

The cutting force and chatter stability models are verified within 18 mm radial depth of cut

by excluding the pilot drilling operation. Torsional-axial flexibility of the drill given in Table 6.7

is considered as the dominant source of regeneration [5]. The model is validated in drilling steel

and aluminum alloys.

203

(a)

3 1 Bottom level

2 4 X Bottom view Figure 6.32 3D locations of the segments on the Kennametal cutter body. Insert numbers are labeled. Bottom view Geometry of the tool is defined by the angular location (Figure 6.33a, measured from +y0

axis of the tool reference frame) and axial location (Figure 6.34a) of each element Sjk. The relative angular location (Figure 6.33b) is measured from the reference element S11 in CW

direction. The axial runout (Figure 6.34b) is derived by shifting the minimum axial location

(Figure 6.34a) value to zero at each radial segment-k. The designed pitch angle (Figure 6.35a) depends on the angular location (Figure 6.33b) of the elements, and it is only meaningful for the static condition (when feedrate is zero). It is calculated by assuming that there is no overlap or runout between the inserts. The actual pitch angle additionally depends on the axial runout and feedrate, and is plotted in Figure 6.35b (for 0.1-mm/rev feedrate). The resulting effective tool geometry is given in Figure 6.36.

204

Figure 6.33 Angular location (a) and relative angular location (b) of element Sjk.

Figure 6.34 (a) Axial location (height), and (b) corresponding axial runout.

Figure 6.35 Pitch angle: (a) Norunout/zero feedrate; (b) fr=0.1-mm/rev

A pilot hole with a 10-mm diameter is predrilled before expanding the hole to a 46-mm diameter with indexed region of the cutter. However, since the 10-mm diameter twist drill is not removed prior to cutting experiment, it adds rubbing force to the measurement. 205

Figure 6.36 Effective tool geometry: (a) True cutting edge angle; (b) Cutting edge inclination angle; (c)

normal rake angle.

As seen in Figure 6.37, z0-force does not show this effect, but it is clearly seen in the x0 and

y0-force data. Test conditions (Figure 6.37 and Figure 6.38) for drilling AISI 1045 steel are:

Ω=900-rev/min; 10-mm pilot hole diameter (18-mm depth of cut); fr=0.1-mm/rev; total depth of the drilled-hole is 2-mm (~1.3-s). Simulation conditions (Figure 6.37 and Figure 6.38): dz=0.2- mm (q=90), ns=360.

From the measured x0 and y0 forces, the rubbing effect is clearly seen in the noncutting

period-A, where the rubbing force amplitude is 100-N. In the cutting period-B, the combined

rubbing and shearing action gives 200-N amplitude force. The estimated x0 and y0 cutting forces

in Figure 6.38 have 73-N amplitude. z0-Force has no rubbing effect with zero force in period-A,

therefore the estimated z0-Force in Figure 6.38 is more accurate. When the pilot hole diameter is set larger than the pilot drill diameter (10-mm), the rubbing effect is avoided (see Figure 6.39).

Therefore, the force predictions become more accurate (see Figure 6.40) with some error left,

which may be attributed to the ploughing of the peripheral inserts.

Test conditions (Figure 6.39 and Figure 6.40) for drilling Aluminum 7050-T7451 are:

Ω=300-rev/min; 17-mm pilot hole diameter (14.5-mm depth of cut); fr=0.1-mm/rev; total depth

206

of the drilled-hole is 2-mm (~1.3-s). Simulation conditions (Figure 6.39 and Figure 6.40): dz=0.05-mm (q=290), ns=360.

Figure 6.37 Effect of the rubbing of the pilot drill on the cutting forces of Kennametal drill.

Figure 6.38 Predicted and measured cutting forces at period-B of Figure 6.37.

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Figure 6.39 Cutting forces without rubbing effect. Pilot hole diameter is larger than 10-mm.

Figure 6.40 Predicted and measured cutting forces (close-up of Figure 6.39).

For the stability simulation, only the z0 and torsion directions are solved. In this case, the related state transition matrix in Equation (5.70) becomes time invariant. Therefore, finding the eigenvalues for the constant state transition matrix is sufficient and thus saves significant computation time. Process damping parameters are taken to be the same as the Sandvik TM880 cutter of Section 6.4: Lw=20-micrometer; specific indentation force for aluminum, Ksp=1.5e5

N/mm3; Coulomb friction coefficient, µ =0.3. Simulated torsional-axial stability chart for drilling Aluminum 7050-T7451 is compared to experiments in Figure 6.41a. The scanned depths of cut are from 8 to 18-mm with 0.4-mm increments, and the scanned spindle speeds are from

500 to 5300-rev/min with 15-rev/min increments; a total of 321x26 grid of 8346 points is solved.

208

Figure 6.41 Stability test with Kennametal drill: (a) Stability chart compared to experiments; (b) Forces with

the resulting surface finish photo of the selected unstable cutting test [5].

Cutting forces and sound are measured (total 4 channels) during the tests. Sample (taken from Ref. [5]) force and sound measurement data, and resulting surface finish for an unstable operation, are shown in Figure 6.41b. Forces in y0 and z0 directions are shown for Ω=1000- rev/min and ap=17-mm. The resulting sutface finish has sunray pattern which is a sign for torsional-axial chatter.

209

6.6 Application-5: Double-Sided Parallel Milling

Double-sided milling application is used in the industry for machining flexible, cantilevered

plates (e.g., elevator guide) from both sides. The proposed model is verified in modeling the

geometry of the double cutter and predicting the simultaneous milling of a cantilevered plate

from both sides

Geometry: The inserts are either brazed permanently or clamped to the insert pockets.

Although detachable inserts are more practical, they have less location accuracy in the pocket

(with up 40 µm). The runout of inserts are measured with an optical microscope (KeeJaan KJ-

340) with a 10 µm measurement accuracy, as is shown in Figure 6.42a. The optical camera with a high resolution lens is aimed at the cutting edge of the cutter which is mounted on the spindle of the instrument. The cutter is free to rotate and can translate in the axis towards the column.

The lens can move along the cutter axis. The radial and axial runout references are set by moving the axis and rotating the cutter to match the red broken line on the screen. Figure 6.42b shows the sample output screen view for measuring the cutting edge for insert#1 of right-side cutter. The

cutting tests have been conducted on a highly rigid DIXI DJC-350 CNC machining center at

ITRI-IMTTC (Taiwan) [4].

(a) (b)

Figure 6.42 Measurement of radial runouts on the double-sided milling tool: (a) setup; (b) screen view.

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(a) (b) (c) Figure 6.43 Double-sided milling tool: (a) In operation; (b) with brazed inserts; (c) with detachable inserts.

Milling experiments with an eight-tooth brazed cutter: KOVA company (Taiwan) custom produced the double-sided tool with brazed inserts shown in Figure 6.43b. The cutters are used in parallel milling of a plate shown in Figure 6.43a and Figure 6.44a. Engagement of the down milling operation is approximately 20% (Figure 6.44b). The left-side (KOVA D125-20L-

D109.7-8T) and right-side (KOVA D125-20R-D109.7-8T) cutters have non-detachable, brazed

inserts. The brazed inserts are ground on the cutter body to minimize the radial and axial runouts.

The cutters are installed on keyways of the shank of the custom tool holder, with BT-50

taper at spindle interface. The keyway is the supporting edge when the cutters are mounted on

the body. The supporting edges are aligned at the tool holder reference line (Figure 4.9) for

angular location measurement. Following measured geometry of the left and right-side cutters

are listed in Table 6.8: Measured angular location (with respect to tool holder reference line in

Figure 6.46), corresponding designed pitch angle (Figure 4.9) and radial runout (using setup of

Figure 6.42).

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Table 6.8 Measured angular location, pitch angle and radial runout of the left- and right-side cutters.

Angular location from Pitch angle (deg) Radial runout toolholder reference (micrometer) Insert (deg) Right number Right Left cutter Left cutter Right Left cutter cutter cutter L ψ j,L cutter ψ j,R φp,j,L εr,j R φp,j,R εr,j 1 -8 -9 45 44.5 0 18 2 36.5 36 45 45.5 6 18 3 82 81 45 45.5 6 9 4 127.5 126 44.5 45 22 0 5 172.5 170.5 45.5 44.5 21 4 6 217 216 45.5 45 17 4 7 262 261.5 45 45 17 14 8 307 306.5 44.5 45 0 16

Figure 6.44 Double-sided milling tool: (a) Components‒see Figure 4.8; (b) operation is ~20% down milling.

212

(a) (b)

Figure 6.45 Supporting edge line (tool holder reference line) shown on: (a) Left-side and (b) right-side cutters

with labeled insert numbers.

* The inserts have true cutting edge angle of κr = 15-deg , 0-deg normal rake angle, and 0-deg

inclination angle. The cutting forces are measured by Kistler 9257BA dynamometer during

parallel milling of the AISI 1045 steel rigid workpiece (170-mm length). Cross section of the workpiece/part is given in Figure 6.45a; it is mounted on the dynamometer using an adapter plate with 6 clamps, as is shown in Figure 6.45b. The clamps are fixed with M8 screws which are tightened at controlled 400-kgf.cm torque using a torque wrench.

(a) (b)

Figure 6.46 Double-sided milling experiment setup: (a) Part cross section; (b) 3D view.

213

Figure 6.47 Predicted (solid line) and experimental (broken line) cutting forces: (a) Only left-side milling; (b)

only right-side milling; (c) double-sided milling.

214

Table 6.9 Average forces derived from experiment and theoretical calculations.

Average Average % Prediction error theoretical experiment ((Theory-Exp)/Exp.. force (N) force (N) ..*100) Left-side x0-force 105 118 -11 % Figure 6.47a Left-side y0-force 138 147 -6.1 % Left-side z0-force -133 -132 -0.75 % Right-side x0-force 105 116 -9.4 % Figure 6.47b Right-side y0-force 138 153 -9.8 % Right-side z0-force 133 141 -5.6 % Double-sided x0-force 211 240 -12 % Figure 6.47c Double-sided y0-force 276 305 -9.5 % Double-sided z0-force 0.11 6.1 -98 %

Three cutting tests have been conducted: Only left-side milling, only right-side milling and

double-sided milling. The predictions and experiment results for all tests are given in Figure

6.47a-c and the average forces are compared in Table 6.9. Test conditions of milling AISI 1045

steel part: N=8-tooth; Ω=250-rev/min; fr=0.16-mm/rev; dry cutting (with no cutting fluid);

orthogonal cutting edge geometry (0-deg normal rake and 0-deg cutting edge inclination angle)

* with κr =15-deg cutting edge angle; ap=0.5-mm depth of cut from each side. The geometry is

discretized by dz=0.1-mm axial segment height (q=5-segments in tool-workpiece contact zone).

The time step of the simulation is ∆t =0.6-ms. The orthogonal cutting database for steel is from

Equation (4.22). The angular location, pitch angle and radial runout is found in Table 6.8.

The average forces (Table 6.9) of the single sided cutting are predicted accurately, within

11% error. Prediction accuracy of average forces in z0-direction of double-sided milling tool is

low due to the geometrical differences of left- and right-side cutters. As seen in Figure 6.47, the

peak amplitudes do not match precisely. The orthogonal cutting edge of the cutter creates an

impact force during the entry of the cutter into the plate, thus exciting the dynamometer to

produce peaks. A non-zero cutting edge inclination angle could reduce the effect of the impacts. 215

Figure 6.48 Experiment setup with flexible workpiece: (left) CAD schematics; (right) photo.

Figure 6.49 Pre-set spacer width of the double-sided milling tool.

Figure 6.50 Engagement condition of double-sided cutting tests.

216

Milling experiments with parallel cutters having single tooth: A single detachable insert

(KOVA XNEX080608), which avoids the presence of runout, is used at each side of the double- sided cutter, as is shown in Figure 6.43c. A flexible workpiece set-up is used to verify the prediction of the stability. Constant 400-kgf.cm torque is applied on M8 screws to clamp the workpiece (Figure 6.48); this ensures repeatable dynamics of the workpiece when it is demounted and mounted. A spacer is put in between the cutters to fix the machined width of the workpiece at 23mm, see Figure 6.49. The initial width of the workpiece is 24-mm (Figure 6.48a),

thus the spacer between the cutters pre-sets the axial depth of cut to 0.5-mm from each side.The

positioning of the cutter reference, with respect to workpiece, is set by removing the left-side cutter and measuring the position of the insert tip at the right-side cutter. The radial depth of cut of the operation is set by positioning the insert tip 2-mm below the machined zone of the workpiece. As shown in Figure 6.50, the periphery circle of the tip of the insert (zero depth of cut location) is set 2-mm below the machined zone on the workpiece. The entrance into the cut happens 25-mm above that point for the tip of the insert. According to the engagement condition in Figure 6.50, tool entry and exit angles change along the cutting edge, as shown in Figure

6.51a-b.

Figure 6.51 Tool entry (a) and exit (b) angles for the indexable double-sided milling tool.

217

The cutter center to insert tip distance is 109.7-mm, then the cutter entry and exit angles

along the cutting edge are found as in Equation (6.16). For 25-mm machined zone thickness, the

radial entry and exit distances are identified as:

109.7 a= −=2 52.8 mm; aa =−( −25) = 27.8 mm, (6.62) ex 2 st ex

which leads to the entry (Figure 6.51a) and exit (Figure 6.51b) angles at each axial segment-k,

from Equation (6.15), as:

aa   φφ= arccosst ;= arccos  − ex  st,jk ex,jk  . (6.63) RRjk jk 

The finite element simulation of the cantilevered plate gives dominant bending (762 Hz) and

torsional (2102 Hz) vibration modes, as shown in Figure 6.52. Standard impact hammer test is

done using accelerometer, at the location shown in Figure 6.53 (A=48-mm; B=4-mm). Identified

modal parameters (from Cutpro [118]) of first mode: wn,1,w =739-Hz, ζ1,w =1.2%,

uz,k,1,w = 0.99-[1 kg] ; of second mode: wn,2,w =2089-Hz, ζ 2,w =0.37[%], uz,k,2,w = 2.13-[1 kg]. The difference in identified and predicted natural frequencies can be attributed to the assumption that the clamps are fixed rigidly. Fitted FRF is constructed by identified modal parameters. Figure

6.54a-b show the measured FRF and fitted FRF around the frequencies of first two vibration modes. Identified modal parameters are used during the dynamics analysis.At each time instance, the workpiece is assumed to have 6 dynamic levels, i.e., s=3 in Equation (4.79) for each cutter.

Dynamic level definition is similar to the multifunctional tool case; the difference is that the forces are lumped onto the dynamic levels on the workpiece at each radial location r=1,2,3 (in

L R general r=1..s) on the workpiece. Force functions Fjki and Fjki for left and right side cutters are

218

found from Equation (4.67) using the transformation matrices given in Equation (4.48) and

Equation (4.49).

(a) (b)

Figure 6.52 First two vibration mode shapes from finite element simulation: (a) Bending mode shape at 762-

Hz; (b) Torsional mode shape at 2102-Hz.

Figure 6.53 Measurement at selected point (at A and B distances) on the steel part.

(a) Around first vibration mode (b) Around second vibration mode

Figure 6.54 Real and imaginary parts of measured and fitted FRF.

219

Figure 6.55 Predicted limit depth of cut per side for stable double sided milling.

The predicted stability is given in Figure 6.55 for the depth of cut per side. AISI 1045 steel

part is machined with the radial depth condition given in Figure 6.50. Average FRF (Figure 6.54) is assumed along the tool-workpiece contact zone. AISI 1045 steel database given in Equation

(4.22). KOVA XNEX080608 inserts (Figure 3.11) used with double milling cutter (left-hand

side Figure 3.19c; right-hand side Figure 3.19b).

(a) (b)

Figure 6.56 Identified cutting edge of the two-insert end mill: (a) View in Frame-i; (b) close up view.

220

Figure 6.57 Predicted forces compared to the experimental results reported by Eynian [75].

6.7 Application-6: Prediction of Cutting Forces for a Two-Insert Indexable End Mill

The generalized model is applied on an indexable end mill produced (Sandvik R390-

020A20L-11L) with two inserts having 0.2 mm nose radius (R390-11 T302E-PM-4240). The experimental results were reported by Eynian [75] for this tool, but only the CAD model of a

similar inserts (R390-11T3 04M-PM 1025) with 0.4 mm nose radius is available. The placement

of inserts on the tool body is given in Figure 3.26, and the modelled cutting edge is shown in

insert reference frame (Frame-i) in Figure 6.56a. The cutting edge geometry is modified to have

a nose radius of 0.2 mm as shown in Figure 6.56b. The runout values were not reported in Ref.

[75]. The predicted cutting forces reasonably agree with the experimental measurements reported

by Eynian [75], see Figure 6.57. Simulated spindle speed is Ω=4125-rev/min; number of inserts

is N=2-tooth; feedrate is fr=0.2-mm/rev; number of tool workpiece contact segments is q=ap/dz=0.25/0.001=250.

6.8 Application-7: Three Holders with the Same General Purpose Insert

KOVA insert XNEX080608 (Figure 3.11) has been used on three different tool holders to

illustrate the adaptability of the proposed model on face milling (Figure 3.19b), drilling (Figure

3.23) and double-sided milling (Figure 3.19a). The experimental validation of double-sided

221

milling is presented in Section 6.6 and the applications for face milling and drilling are given below.

Face Milling Cutter: The geometry of the face mill is shown in Figure 6.58. The radial runout of inserts are measured when the cutter was mounted on the spindle. Forces are simulated with ns=360 (number of points in one spindle period), and they are given in Figure 6.59.

Slot milling operation is simulated: Spindle speed is 1000-rev/min; axial depth of cut is 2- mm; feedrate is 1.0-mm/rev for machining AISI 1045 steel (Figure 6.59a-b), and it is 0.5-mm/rev for milling Aluminum 7050-T7451 (Figure 6.59c-d).

Figure 6.58 KOVA U140 face mill geometry. (Courtesy of KOVA.)

222

Figure 6.59 Face milling operation cutting force prediction.

Drilling Cutter: The drilling tool geometry with two inserts is modeled as shown in Figure

6.60. Axial runout is measured after the cutter is installed on the spindle. Force is simulated with

ns=360 (number of points in one spindle period) and is presented in Figure 6.61.

Simulated spindle speed is 1000-rev/min, and feedrate is 0.1-mm/rev. There exists a 0.5-mm diameter pre-drilled pilot hole. Workpiece materials of simulations in Figure 6.61a-b is AISI

1045 steel, and in Figure 6.61c-d is Aluminum Al7050-T7451.

223

Figure 6.60 KOVA U502 drill geometry. (Courtesy of KOVA.)

Figure 6.61 Drilling operation cutting force predictions.

224

6.9 Application-8: Parallel Turning with Two Tools

A cylindrical part is turned from two opposite sides simultaneously in parallel turning application, as reported in Ref. [139]. The regenerative delay between the two tools is usually equal to half of the spindle period, but a more general form is obtained here with the application of Equation (4.52). Assuming constant dynamics along the tool-workpiece contact for each tool, the regenerative, dynamic chip thickness of the two tools can be expressed as:

∆=−q qq(tT − ); 1ki 1i 2 i 2ki (6.64) ∆=−−q2ki qq 2i 1(tT i 1ki ),

where the dynamic chip thickness ∆q1i for tool #1 is related to the delay displacement

q2()tT i− 2ki of tool #2, and the dynamic chip thickness ∆q2i for tool #2 is related to the delay

displacement q1()tT i− 1ki of tool #1. The displacement vector for each tool is expressed from

Equation (4.51), but together with the constant dynamics along tool workpiece contact (k=1..q) for each tool:

T θ1i= [xyz 1i 1i 1iθ 1i ] ; T (6.65) θ2i= [xyz 2i 2i 2iθ 2i ] .

The dynamic shear force vector is rewritten by substituting Equation (6.64) in Equation

(4.61):

Fcd= Fq d ( −− q(tT );) 1ki 1ki 1i 2 i 2ki (6.66) cd d F2ki= Fq 2ki( 2i −− q 1(tT i 1ki ),)

d d where the force component F1ki and F2ki vectors for tool #1 and tool #2 are given in Equation

(4.62). If the flexible workpiece is added to the formula, Equation (6.64) has to be modified since

the delay vibrations of the workpiece affects both tools [63]. For a rigid workpiece, the mode

shape matrix is composed of flexible tool #1 and tool #2, as: 225

U =U[ U] , (6.67) [8x(mmt1+ t2 )] t1 t2

where tool #1 and tool #2 mode shape matrices are Ut1 and Ut2 respectively; mt1 and mt2 are the

number of vibration modes of each tool. Assuming a continuous turning operation, the in cut

parameters for both tools are set as gg2,1ki= 2,2ki =1, as in Equation (4.80):

q ∑ gg1,1k 3,1kiF 1ki k (F ) = . (6.68) i [8x1] q  ∑ gg1,2k 3,2kiF 2ki k

The first 4 rows of the force ( Fi ) vector are related to tool #1, and the last 4 rows are related

to tool #2. The forces generated on each tool are coupled due to the delay effect in Equation

(6.66), which affect the forces expressed in Equation (5.23) as follows:

Static shear force vector from Equation (5.24):

q gg Fcs cs = 1,ki 3,1ki 1ki fi [8x1] ∑ cs . (6.69) k gg1,ki 3,2kiF 2ki

Ploughing force vector from Equation (5.27):

q gg Fes es = 1,ki 3,1ki 1ki fi [8x1] ∑ es . (6.70) k gg1,ki 3,2kiF 2ki

Dynamic shear force component from Equation (5.25):

q gg F0d fUcd = 1,ki 3,1ki 1ki ⋅ i [8x1] ∑ d . (6.71) k 0Fgg1,ki 3,2ki 2ki

Process damping force vector from Equation (5.28):

q gg F0p fUed = 1,ki 3,1ki 1ki ⋅ i [8x1] ∑ p . (6.72) k 0Fgg1,ki 3,2ki 2ki 226

Delay related force vector has an updated definition from Equation (6.66) as:

q d 00 0Fgg1,ki 3,2ki 2ki −∑⋅⋅UUΓΓc(tT i − 2ki ) +⋅⋅c(tT i − 1ki ) , (6.73) gg F0d k 00 1,ki 3,1ki 1ki

where the combined modal displacement vector is composed only of tool vibration modes:

Γt1(t i ) Γci(t ) = . (6.74) Γt2(t i )

The force vector is constructed from Equation (5.23):

q d 00 cs cd 0Fgg1,ki 3,2ki 2ki Ffi= i+ f iΓΓ c,i −∑⋅⋅Uc(tT i − 2ki ) +⋅⋅U Γc(tT i − 1ki ) gg F0d k 00 1,ki 3,1ki 1ki (6.75) es ed  ...++ffi iΓ c,i .

The remaining part of the formulation is the same as the general model given in Section 5.2.

Geometry: Figure 4.14 shows the two dimensional geometry when both tools have identical

nose radii. The transformation matrices for the right hand turning operation of both tools are

found using Equations (4.45) and (4.46). The angular locations for the tools are taken as φ1ki = 90

deg for tool #1 (j=1), and φ2ki = 270deg for tool #2 (j=2). The angular location of each cutting edge element Sjk along each tool is assumed to be constant.

Stability calculation and comparison to the literature: In Figure 4.7, the cutting edge elements S1k on tool #1 and S2k on tool #2 are at the same radial segment-k along the tool workpiece contact. There are three possible delay conditions at the radial segment-k:

227

• If both tools are in cut (g1,1k=g1,2k=1 and g3,1ki=g3,2ki=1), then the delay period of the tools

TT is equal to half of the spindle rotation period at the radial segment-k: TT=ss; = ; 1ki22 2ki

• If only tool #1 is in cut (g1,1k=g3,1ki=1 and g1,2k=g3,2ki=0), then the delay period for

element S1k is equal to the full spindle period: TT1ki= s .

• If only element S2k of tool #2 is in cut (g1,2k=g3,2ki=1 and g1,1k=g3,1ki=0), then the full

spindle rotation period is the delay period for element S2k: TT2ki= s .

Budak and Ozturk [59] reported parallel turning experiments (Figure 6.62), which were used to validate various stability prediction methods reported in the literature [61]. They fixed the milling spindle to install the tool #1 and used the turret chuck to fix the tool #2. The tools are moved in z0 by the two individual feed drives. If the two drives are not synchronized to be located at same axial coordinates, there would be an axial offset between the tools. The effect of this offset on the chip thickness is the same as axial runout, as is shown in Figure 4.4. Delay period of the tools would be affected as well.

Figure 6.62 Parallel turning setup. (Taken from Figure 1 and Figure 2 of Ref. [59].)

228

The predicted stability chart proposed by the generalized model is compared against the

experimental and simulation results reported in Refs. [59,61], see Figure 6.63. Modal parameters

are given below each chart (Figure 6.63a-b). The following assumptions are adopted from the reported tests: Sharp cutter is assumed for the inserts, i.e., the nose radii are zero and the true

* cutting edge angles are κr = 90 deg for each element Sjk. Zero axial offset is assumed between tools; both tools are only flexible in the z0 direction, only the radial cutting force acting in z0

direction affects the dynamics. As a result, only the radial cutting coefficient Krc affects the

stability.

Figure 6.63 Comparison of the predicted stability chart (modelled) to the literature.

229

Chapter 7: Conclusions and Future Research Directions

7.1 Conclusions

Metal cutting tools have edges which may have various shapes such as slanted edge with a nose radius in turning, helical edge in milling or curved lip in drilling operations. Traditional metal cutting studies discretize the curved cutting edge of the tool into small, differential linear segments. The cutting force generated by each differential segment is predicted as a function of chip thickness and cutting force coefficients which depend on the tool-workpiece material pair.

The chip thickness is modeled as a function of rigid body kinematics of individual machining operation, relative vibrations between the tool and workpiece and geometry of each cutting tool.

The total force acting on the tool body is predicted by integrating the distributed differential forces along the entire cutting edge which is engaged with the workpiece. The traditional approach requires dedicated mathematical models for each tool geometry and machining operation.

This thesis introduces a novel approach in unifying all cutting operations with arbitrary tool geometries in one generalized mathematical model. One parametric, mathematical model simulates all machining operations, as opposed to developing dedicated models for each tool geometry and operation; it simulates cutting forces, torque, and power of the machining process; surface form errors on the workpiece; relative tool-workpiece vibrations and chatter stability lobes for all turning, boring, drilling and milling operations.

The generalized mathematical model is achieved with the following contributions to the literature:

The tool’s cutting edges are either discretized as axial disk segments for milling, or as radial ring segments for turning, boring and drilling operations. Each differential cutting edge element 230

is defined by 15 geometric parameters which define the edge geometry, rake face of the tool,

location of the edge in Cartesian coordinate system, axial and radial runouts and the engagement

conditions of the edge with the workpiece. The geometric parameters of the cutting edge is

transformed to tool body by orienting the edge in accordance with the kinematic model of the

operation and tool geometry. The general transformation matrices are configured parametrically to define any tool for any metal cutting configuration. It is shown that solid tools as well as

indexable cutter bodies can be modeled by the proposed approach, which is the foundation in

generalizing the metal cutting operations.

The model is demonstrated for milling and drilling operations, multifunctional drilling/boring tools, and for solid tool with arbitrary (i.e., serrated/undulated) edge of end mills and twist drills.

The chip geometry that is removed by each differential cutting edge element is modeled by considering both rigid body motion and structural vibrations of tool-workpiece system. The runout of the edge is considered. The time delay contributed by the regenerative vibrations is modeled as a function of machining operation and tool geometry. The differential cutting force for the edge segment is modeled by considering both chip shearing and ploughing forces. The differential forces are integrated in tool body coordinates to evaluate the total force acting on the tool and workpiece. It is shown that the proposed model can predict cutting forces for turning, boring, drilling, milling and variety of multifunctional tool geometries.

The relative vibrations between the tool and workpiece can either be modeled at a lumped point (e.g., tip of the end mill) or distributed along the tool-workpiece contact zone (e.g., flexible end mill). The cutting forces can be distributed by considering the mode shapes of the structure.

The equation of motion between the cutting process and machine tool/part is derived by 231

considering single or multiple delays depending on the tool geometry. The set of delay-

differential equations are reduced to a set of first order differential equations, which are solved in

semi-discrete time domain. Both stability as well as cutting forces, surface location errors,

vibration amplitudes and stability diagrams are predicted for arbitrary tool and machining

operations by setting the process parameters.

The proposed method is experimentally proven in turning, boring, drilling and milling. In addition to predicting the classical tools and operations, the thesis includes the applications on a

multifunctional drill with multiple cutting edges and parallel face milling cutters in machining thin workpieces with position dependent dynamics.

The presented genaralized mathematical modeling of tools, operations, cutting mechanics, cutting dynamics and chatter stability is the first research in the literature. The generalized model allows the virtual simulation and optimization of various machining operations in an integrated manner in Compuer Aided Manufacturing (CAM) platforms.

7.2 Future Research Directions

Both the concept and developed mathematical model in generalizing the process mechanics and dynamics are general, and can be extended to other machining operations as follows:

• Current model is for 2-axis machining operations in xy direction of the machine-tool. The

mechanics and dynamics model can be extended to include the additional feed direction

(i.e., 3-axis operations), and to include varying feed along the tool axis (i.e., 5-axis

operations). The fundamental mathematical models presented in this thesis required only

coordinate transformations from the feed-cutting velocity plane to machine tool frame,

which requires kinematic model of the machine.

232

• The generalized model can be adopted to simulate broaching operations by modeling the

broach edge geometry and spacing of the broaching teeth only.

• As illustrated in the thesis, multifunctional tools can easily be modeled by the proposed

general model. Complex operations such as turn-milling can also be simulated by the

proposed generalized model by incorporating the forward and inverse kinematics of these

machines.

• The cutting mechanics model can be improved by considering fklank wear, back cutting

and tool indentation mechanics as suggested by Tuysuz et al. [48] for balll end milling

operations.

• The solution can be extended to include nonlinear chip thickness and velocity effects,

based on the works of Montgomery and Altintas [19], Campomanes and Altintas [101]

and Ratchev et al. [140]. Estimating the vibration amplitude is important for different tool

and edge geometry, and it can help to predict tool life, force and vibration during the

design.

• Variable delay along the cutting edge (the most general case) is presented in this thesis.

The delay can be non-periodic such as in thread-milling or chatter suppression through

spindle speed variation [141].

• Extension of the model to the following can be considered: i) Tools with rotating

vibration mode directions [75]; ii) thread cutting operation where feed is comparable to

cutting velocity [55], i.e., the resulting cutting vector is a function of feed direction; iii)

machining involving multiple rotating bodies (e.g., turn-milling application) where

deriving the delay term is itself a research item.

233

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Appendix A Transforming the basis from Frame-1 to Frame-2

The Frame-2 origin and vectors can be represented in Frame-1 coordinates using translational and rotational transformations.

A.1 Pure translation

If the relative Frame-1 position of the origin of the Frame-2 is only translational, then the following transformation is used:

T T 111x, y , z ,1= T 1 xyz , , ,1 . (A.1) 2 2 2 2[ 222]

111 ( xyz222,,) is the Frame-1 representation of the ( xyz222,,) vector that is represented in

1 Frame-2. The homogeneous transformation matrix T2 only translates the pre-multiplied frame

(Frame-2) to the new frame (Frame-1)

1 121212 T2= Trans( rrr xyz , , ) , (A.2)

121212 where the rrrxyz,, are the relative x,y,z components in the Frame-1 of the origin of the Frame-

2. The translation matrix equation of Equation (A.2) is:

100a  010b Trans(abc , , ) = . (A.3) 001c  0001

The a,b,c components are the x,y,z axis translations of the pre-multiplied frame (Frame-2) to

transform it to the new frame (Frame-1).

For example, if the origin r2 of the Frame-2 represented in Frame-2 is:

T r2 = [0001] , (A.4)

then, its relative location in Frame-1 is found by Equation (A.1): 245

10012r x 0  01012r 0 12r=y ⋅=rr2; 2 . (A.5) 12 0 001 rz  1 000 1

The resulting vector is as expected:

T 12r =  12rrr 12 12 1 . (A.6) xxz

A.2 Pure rotations

The transformation of the Equation (A.1) can be used for rotations around one of the x,y,z axes of the Frame-1. The functions for the rotation around x,y, and z axes are named Rotx,Roty and Rotz. If Frame-2 is obtained by rotating the Frame-1 around x-axis by α angle, the Rotx transformation is applied:

10 0 0  0 cosαα− sin 0 Rotx(α ) = . (A.7) 0 sinαα cos 0  00 0 1

T The last column of Rotx matrix is [0001] because of the pure rotation and no translation. If Frame-2 is obtained by rotating the Frame-1 around y-axis by φ angle, the Roty transformation is applied:

cosφφ 0 sin 0  0 100 Roty(φ ) = . (A.8) −sinφφ 0 cos 0  0 001

If Frame-2 is obtained by rotating the Frame-1 around z-axis by ψ angle, the Rotz transformation is applied:

246

cosψψ− sin 0 0  sinψψ cos 0 0 Rotz(ψ ) = . (A.9) 0 0 10  0 0 01

αφψ,, angles are also called Euler angles [114]. The order of rotations is important because each transformation is done with respect to the previous frame. For large angle rotations, the difference is more pronounced than the small angle rotations. When the rotations are done in infinitesimal way, the accuracy is increased. The three procedures are compared in the following sections.

A.3 Procedure A: Large angle rotations

The following two rotations are considered:

1. Rotating first rotate 30 [deg] around x-axis of Frame-1, and then rotate 60 [deg] around

the y-axis of the intermediate frame, Frame-i:

1 T2 =Rotx(30[deg])⋅ Roty(60[deg]); ( )A1 0.500 0 0.866 0  (A.10) 1 0.433 0.866 -0.250 0 T2 = ; ( )A1 -0.750 0.500 0.433 0   0 0 0 1

2. Rotating first rotate 60 [deg] around x-axis of Frame-1, and then rotate 30 [deg] around

the y-axis of the intermediate frame, Frame-i*:

1 T2 =Roty(60[deg])⋅ Rotx(30[deg]); ( )A2 0.5000 0.4330 0.7500 0  (A.11) 1  0 0.8660 -0.5000 0 T2 = . ( )A2 -0.8660 0.2500 0.4330 0   0 0 0 1

2 A test vector rtest is chosen in Frame-2 to check the accuracy of the transformation matrices: 247

2T rtest = [2340]. (A.12)

From two different order transformations of Equations (A.10) and (A.11) the test vector can be represented in Frame-1,

12 1 2 12 1 2 rtest= Tr 2 test;; r test= Tr 2 test ( )A1( ) A1 ( )A2( ) A2 4.464 5.299     (A.13) 12 2.464 12 0.5981  rrtest = ;.test = ( )A1 1.732 ( )A2 0.7500      0  0 

The two orders give quite different vectors. The off-diagonal terms of the two

transformations of Equations (A.10) and (A.11) are not equal, and they have a large difference.

The difference is expected to decrease when the angles get smaller.

A.4 Procedure B: Small angle rotations

For small angles, the following two ways would give almost the same transformations:

1. Rotating first rotate 10 [deg] around x-axis of Frame-1, and then rotate 20 [deg] around

the y-axis of the intermediate frame, Frame-i;

2. Rotating first rotate 20 [deg] around x-axis of Frame-1, and then rotate 10 [deg] around

the y-axis of the intermediate frame, Frame-i*.

Resulting transformations are:

0.9397 0 0.3420 0 0.9397 0.0594 0.3368 0  110.0594 0.9848 -0.1632 0  0 0.9848 -0.1736 0 TT22= ; = . (A.14) ( )B1 -0.3368 0.1736 0.9254 0 ( )B2 -0.3420 0.1632 0.9254 0   0 0 0 1  0 0 0 1

The corresponding transformed test vectors are:

248

12 1 2 12 1 2 rtest= Tr 2 test;; r test= Tr 2 test ( )B1( ) B1 ( )B2( ) B2 3.247 3.404   12 2.420 12 2.259 rrtest = ;.test = ( )B1 3.549 ( )B2 3.507    0  0 (A.15)

The vectors are closer to each other. To make it more accurate, the rotations can be done in

multiple steps.

A.5 Procedure C: Rotating in multiple steps

If the ns is the number of rotations for a rotation matrix, then the original input angles should be divided to ns and the transformation is done ns times. For example, to implement Procedure A

in multiple steps, the following steps should be performed ns times:

1. Rotating first rotate 30/ns [deg] around x-axis of Frame-1, and then rotate 60/ns [deg]

around the y-axis of the intermediate frame, Frame-i;

2. Rotating first rotate 60/ns [deg] around x-axis of Frame-1, and then rotate 30/ns [deg]

around the y-axis of the intermediate frame, Frame-i*.

For 10-steps of rotation, ns =10 :

1 ns T2s=( Rotx(30nn [deg])⋅ Roty(60 s [deg])) ; ( )C1 1 ns T2s=( Roty(60nn [deg])⋅ Rotx(30 s [deg])) ; (A.16) ( )C2 0.5114 0.2224 0.8301 0 0.5114 0.2655 0.8173 0  1 0.2655 0.8778 -0.3988 0 1 0.2224 0.8778 -0.4243 0 T2 = ;.T2 = ( )C1 -0.8173 0.4243 0.3898 0 ( )C2 -0.8301 0.3988 0.3898 0   0 0 0 1  0 0 0 1

Then, the transformation matrices become more comparable and the dependence on the

order is weaker. Corresponding test vector representations in Frame-1 also become very close to

each other: 249

12 1 2 12 1 2 rtest= Tr 2 test;;r test= Tr 2 test ( )C1( ) C1 ( )C2( ) C2 5.010 5.088   (A.17) 12 1.569 12 1.380 rrtest = ;.test = ( )C1 1.197 ( )C2 1.095    0  0

A.6 Finding the transformation matrix using the known vectors

The 4-by-4 homogenous matrix transformation from Frame-2 to Frame-1 can be found if the representation of the four vectors of Frame-2 are known in Frame-1. The three axis vectors of

Frame-2 are the easiest ones to use, as:

TT T i22= [1000] ; jk= [ 0100] ; 2= [ 0010] . (A.18)

00 0 Their vector representations ijii, and k i in Frame-1 are assumed as known. The fourth vector would be position of the origin of Frame-2:

T r2 = [0001.] (A.19)

Its vector representation 12r in Frame-1 is also assumed known. Then, the transformation

1 matrix T2 from Frame-2 to Frame-1 is found by joining the four vectors:

1i: 1 j : 1 k : 12 r = T 1⋅= ijk :::;::: r2 ijk r2 I ; 2 2 2  2 22 2 22 2 4 (A.20) T1=  1 i:: 1 jkr 1 : 12 . 2 22 2

The form of homogeneous transformation matrix is:

1 12 1 Rr2 T2 = . (A.21) 01x3 1

Although the inverse of the rotation matrix is its transpose:

−1T 21 21 RR12= ( ) ; RR 12= ( ) , (A.22)

250

then, the inverse of the homogeneous transformation matrix is:

TT −1 R1− Rr 1 12 R2− Rr 21 2 TT21=;; T 2 = ( 22) ( ) T2= 11. (A.23) 12( ) 11 01x3 1 01x3 1

251