Toolpath and Cutter Orientation Optimization in 5-Axis CNC ...

Toolpath and Cutter Orientation Optimization in 5-Axis CNC Machining of Free-form

Surfaces Using Flat-end Mills

by

Shan Luo

BSc, Jianghan University, 2008

MSc, Wuhan University of Technology, 2011

A Dissertation Submitted in Partial Fulfillment

of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in the Department of Mechanical Engineering

Shan Luo, 2015

University of Victoria

All rights reserved. This dissertation may not be reproduced in whole or in part, by

photocopy or other means, without the permission of the author.

ii

Supervisory Committee

Toolpath and Cutter Orientation Optimization in 5-Axis CNC Machining of Free-form

Surfaces Using Flat-end Mills

by

Shan Luo

BSc, Jianghan University, 2008

MSc, Wuhan University of Technology, 2011


Dr. Zuomin Dong, (Department of Mechanical Engineering) Supervisor

Dr. Martin Byung-Guk Jun, (Department of Mechanical Engineering) Co-supervisor

Dr. Keivan Ahmadi, (Department of Mechanical Engineering) Departmental Member

Dr. Sue Whitesides, (Department of Computer Science) Outside Member

iii

Abstract


Dr. Zuomin Dong, (Department of Mechanical Engineering) Supervisor

Dr. Martin Byung-Guk Jun, (Department of Mechanical Engineering) Co-supervisor

Dr. Keivan Ahmadi, (Department of Mechanical Engineering) Departmental Member

Dr. Sue Whitesides, (Department of Computer Science)

Outside Member

Planning of optimal toolpath, cutter orientation, and feed rate for 5-axis Computer

Numerical Control (CNC) machining of curved surfaces using a flat-end mill is a

challenging task, although the approach has a great potential for much improved

machining efficiency and surface quality of the finished part. This research combines and

introduces several key enabling techniques for curved surface machining using 5-axis

milling and a flat end cutter to achieve maximum machining efficiency and best surface

quality, and to overcome some of the key drawbacks of 5-axis milling machine and flat

end cutter use. First, this work proposes an optimal toolpath generation method by

machining the curved surface patch-by-patch, considering surface normal variations

using a fuzzy clustering technique. This method allows faster CNC machining with

reduced slow angular motion of tool rotational axes and reduces sharp cutter orientation

changes. The optimal number of surface patches or surface point clusters is determined

by minimizing the two rotation motions and simplifying the toolpaths. Secondly, an

optimal tool orientation generation method based on the combination of the surface

normal method for convex curved surfaces and Euler-Meusnier Sphere (EMS) method

for concave curved surfaces without surface gouge in machining has been introduced to

achieve the maximum machining efficiency and surface quality. The surface normal

based cutter orientation planning method is used to obtain the closest curvature match

and longest cutting edge; and the EMS method is applied to obtain the closest curvature

match and to avoid local gouging by matching the largest cutter Euler-Meusnier sphere

with the smallest Euler-Meusnier sphere of the machined surface at each cutter contact

iv

(CC) point. For surfaces with saddle shapes, selection of one of these two tool orientation

determination methods is based on the direction of the CNC toolpath relative to the

change of surface curvature. A Non-uniform rational basis spline (NURBS) surface with

concave, convex, and saddle features is used to demonstrate these newly introduced

methods. Thirdly, the tool based and the Tri-dexel workpiece based methods of chip

volume and cutting force predictions for flat-end mills in 5-axis CNC machining have

been explored for feed rate optimization to achieve the maximum material removal rate.

A new approach called local parallel slice method which extends the Alpha Shape

method - only for chip geometry and removal volume prediction has been introduced to

predict instant cutting forces for dynamic feed rate optimization. The Tri-dexel workpiece

model is created to get undeformed chip geometry, chip volume, and cutting forces by

determining the intersections of the tool envelope and continuously updating the

workpiece during machining. The comparison of these two approaches is made and

several machining experiments are conducted to verify the simulation results. At last, the

chip ploughing effects that become a more serious problem in micro-machining due to

chip thickness not always being larger than the tool edge radius are also considered. It is

a challenging task to avoid ploughing effects in micro-milling. A new model of 3D chip

geometry is thus developed to calculate chip thickness and ploughing volume in micro 5-

axis flat-end milling by considering the minimum chip thickness effects. The research

forms the foundation of optimal toolpath, cutter orientation, cutting forces/volume

calculations, and ploughing effects in 5-axis CNC machining of curved surfaces using a

flat-end mill for further research and direct manufacturing applications.

v

Table of Contents

Supervisory Committee ...................................................................................................... ii

Abstract .............................................................................................................................. iii

Table of Contents ................................................................................................................ v

List of Tables ................................................................................................................... viii

List of Figures .................................................................................................................... ix

Acknowledgments............................................................................................................. xv

Introduction ................................................................................................. 1 Chapter 1:

1.1 Background and Motivation ............................................................................... 1

1.1.1 Toolpath and Orientations............................................................................... 1

1.1.2 Machine Dynamics ......................................................................................... 4

1.2 Research Contributions ....................................................................................... 6

1.3 Dissertation Outline .......................................................................................... 11

Literature Review...................................................................................... 15 Chapter 2:

2.1 Toolpath Planning ............................................................................................. 15

2.1.1 Surface Division Machining Toolpath .......................................................... 17

2.1.2 Steepest-directed and Iso-cusped (SDIC) Method ........................................ 18

2.1.3 Accessibility-map (A-map) Method ............................................................. 19

2.2 Tool Orientation ................................................................................................ 20

2.2.1 Principal Axis Method (PAM) ...................................................................... 21

2.2.2 Euler-Meusnier Sphere (EMS) Curvature Match ......................................... 22

2.2.3 C-space Based Tool Orientation Methods .................................................... 23

2.3 Machine Dynamics ........................................................................................... 24

2.3.1 Toolpath and Tool Orientation Optimization by Dynamic Constraints........ 24

2.3.2 Chip Volume in 5-axis CNC Machine .......................................................... 25

2.3.3 Cutting Force in 5-axis CNC Machine ......................................................... 28

Optimization of 5-Axis CNC Toolpath and Cutter Orientation for Chapter 3:

Machining Free-form Surfaces ......................................................................................... 31

3.1 Machining Surfaces Patch by Patch Using the Fuzzy Cluster Method ............ 32

3.1.1 Fuzzy C-means Clustering Method .............................................................. 33

3.1.2 Generation of Surface Patches by Surface Normal Vector Distances .......... 35

3.2 Optimization of the Number of Surface Patches .............................................. 37

3.3 Optimal Toolpath Generation ........................................................................... 45

3.3.1 Surface Patch Boundary Definition by Alpha Shape .................................... 45

3.3.2 Toolpath Generation ..................................................................................... 46

3.4 Conclusions ....................................................................................................... 47

Optimal Tool Orientation Generation ....................................................... 48 Chapter 4:

vi

4.1 The Euler-Meusnier Sphere (EMS) Method for Tool Orientation in a Concave

Surface .......................................................................................................................... 48

4.1.1 Principal Curvature Calculation for a NURBS Surface ................................ 50

4.1.2 Two rotation Angles Identification ............................................................... 52

4.2 Optimal Tool Orientation .................................................................................. 55

4.3 Conclusions ....................................................................................................... 58

Chip Volume and Cutting Force Calculations in 5-axis CNC Machining of Chapter 5:

Free-form Surfaces Using Flat-end Mills ......................................................................... 59

5.1 Formulation of Swivel Head 5-axis CNC Tool Motion ................................... 61

5.2 Chip Volume Calculation ................................................................................. 62

5.2.1 The Alpha Shape Method ............................................................................. 63

5.2.1.1 Intersections of two ellipses .................................................................. 63

5.2.1.2 Volume calculation by the Alpha Shape method .................................. 66

5.2.1.3 The algorithm of chip volume calculation ............................................ 67

5.2.2 Local Parallel Sliced Method ........................................................................ 75

5.2.2.1 Chip load model .................................................................................... 75

5.2.2.2 Chip volume by local parallel sliced method ........................................ 77

5.2.2.3 Cutter-workpiece engagement maps ..................................................... 79

5.3 Cutting Force Model ......................................................................................... 82

5.4 Case Studies and Results .................................................................................. 84

5.4.1 Examples of Chip Volume Simulation by the Alpha Shape Method ........... 84

5.4.2 Simulation Results of Chip Volume and Cutting Forces by Local Parallel

Sliced Method ........................................................................................................... 90

5.5 Experiment Verification.................................................................................... 94

5.6 Conclusions ....................................................................................................... 96

The Tri-dexel Method of Chip Volume and Cutting Forces Calculation Chapter 6:

and Simulation for Free-form Surfaces in 5-axis CNC Machining with Flat-end Mills .. 97

6.1 Tri-dexel Method for Chip Volume and Cutting Force Calculation ................. 98

6.1.1 Tri-dexel Workpiece ..................................................................................... 98

6.1.2 Chip Volume Model ................................................................................... 100

6.1.2.1 Tool Projections on the Tri-dexel Workpiece ..................................... 100

6.1.2.2 Boolean operation and chip thickness generation ............................... 101

6.1.3 Chip Volume Calculation ........................................................................... 104

6.2 Cutting Forces Prediction ............................................................................... 111

6.3 Case Studies and Results ................................................................................ 113

6.4 Experimental Verification ............................................................................... 117

6.5 Conclusions ..................................................................................................... 120

Conclusions and Future Work ................................................................ 122 Chapter 7:

7.1 Conclusions ..................................................................................................... 122

7.2 Future Work .................................................................................................... 126

vii

Bibliography ................................................................................................................... 130

Appendix1 ....................................................................................................................... 140

Appendix2 ....................................................................................................................... 145

Appendix3 ....................................................................................................................... 148

Appendix4 ....................................................................................................................... 153

A4.1 Introduction ........................................................................................................ 154

A4.2 Ploughing effects in 5-axis Micro Flat-end Milling........................................... 156

A4.2.1 Chip Geometry of a 5-axis Micro Flat-end Mill ......................................... 156

A4.2.2 Chip ploughing area/volume by local parallel sliced method ..................... 158

A4.2.3 Case Studies and Results ............................................................................ 160

A4.3 Ploughing Effects in 3-axis Micro Ball-end Milling ......................................... 162

A4.3.1 Chip Geometry in Micro Ball-end Milling ................................................. 162

A4.3.2 Ploughing Volume Calculation for Ball-end Milling ................................. 164

A4.3.3 Chip Thickness Calculation Considering Runout Effects........................... 168

A4.3.4 Ploughing Volume Calculation Algorithm Ignoring Runout Effects ......... 170

A4.3.5 Ploughing Volume Simulation .................................................................... 173

A4.3.6 Experimental Setup ..................................................................................... 179

A4.3.7 Experimental Result .................................................................................... 180

A4.4 Conclusion ......................................................................................................... 185

viii

List of Tables

Table 1: The relation of optimal cluster numbers and termination criterion ε for a NURBS

surface ................................................................................................................ 42

Table 2: The relation of optimal cluster numbers and termination criterion ε for the

convex half sphere surface ................................................................................. 44

Table 3: Relationship of surface features, curvatures, gouging and the tool orientation

methods .............................................................................................................. 56

Table 4: Cutting parameters for slot machining in the 3-axis micro-milling ................. 140

Table 5: The parameters for four groups’ experiments .................................................. 181

ix

List of Figures

Figure 1-1: The research roadmap ...................................................................................... 6

Figure 2-1: Iso-parametric toolpath for NURBS surface .................................................. 16

Figure 2-2: Iso-planar toolpath for curved surface ........................................................... 16

Figure 2-3: Surface patches by cluster centers [11] ......................................................... 18

Figure 2-4: The A-map for tool orientation [32] .............................................................. 20

Figure 2-5: Coordinate systems and lead-tilt angles [13] ................................................. 21

Figure 2-6: Triad formed by principal curvature directions and the surface normal [34] 22

Figure 2-7: Euler- Meusnier sphere [39] .......................................................................... 22

Figure 2-8: Gouge-free condition [39] .............................................................................. 23

Figure 2-9: C-space for orientation parameters. (a) Discretized 2-D orientation space

(white area shows safe orientation space). (b) 3D C-space for one toolpath

[43] ................................................................................................................ 24

Figure 2-10: Accessibility cones on the CC point mesh [52] ........................................... 25

Figure 2-11: Tool motions along a pre-defined trajectory in five-axis machining and

corresponding swept profiles: (a) Cutter geometric definition; (b) Cutter

motion track and swept profiles (red lines); (c) Generated swept volume [57]

....................................................................................................................... 27

Figure 2-12: Tool engagement regions and decomposed motion [67] ............................. 29

Figure 2-13: Distribution of chip thickness (a) Horizontal feed; (b) Vertical feed [68]... 29

Figure 3-1: Surface cluster centres and relative angles of surface normal vectors ........... 34

Figure 3-2: (a) The 2D distribution of cluster centres for a NURBS surface in the Fuzzy

Clustering Toolbox; (b) The demonstration of cluster centres and their

surface normal in 3D in MATLAB ............................................................... 35

Figure 3-3: Surface divisions with tool orientations for a NURBS surface from 1 cluster

to 10 clusters .................................................................................................. 37

Figure 3-4: Relative angle φ and accumulating relative angle α ...................................... 38

Figure 3-5: Changes of accumulating relative angles with different numbers of cluster

centres and ith

cluster for a NURBS surface in 3D bar chart. ........................ 40

Figure 3-6: Changes of maximum accumulating relative angles and their first and second

order derivatives for a NURBS surface ......................................................... 41

Figure 3-7: Surface divisions with tool orientations for a convex half sphere surface from

1 cluster to 10 clusters. .................................................................................. 42


centres and ith

cluster for a convex half sphere surface in 3D bar chart. ....... 43

x


order derivatives for a convex half sphere surface. ....................................... 44

Figure 3-10: Surface patch boundaries generated by the alpha shape method with

different probe radius and boundaries shown in 2D and 3D for a convex half

sphere. ............................................................................................................ 45

Figure 3-11: (a) 5 cluster centres of a convex half sphere generated by the clustering

toolbox; (b) Toolpath generation for one surface patch ................................ 47

Figure 4-1: Machined surfaces and cutter Meusnier sphere ............................................. 50

Figure 4-2: Inclination angle α confirmation .................................................................... 53

Figure 4-3: Tool orientation in the Meusnier sphere method ........................................... 54

Figure 4-4: The relation of tool axis with the surface normal and the smallest principal

curvature direction. ........................................................................................ 54

Figure 4-5: A 3D NURBS solid model with concave, convex, and saddle shapes. ......... 55

Figure 4-6: (a) Divisions on grid points of the NURBS surface in 3D; (b) Surface features

in 2D .............................................................................................................. 56

Figure 4-7: (a) Optimal tool orientations for the NURBS surface; (b) Display of the new

tool orientations, surface normal, and minimal surface curvature directions 57

Figure 5-1: The tool motion in the local coordinate system and illustration of rotation

angles. ............................................................................................................ 62

Figure 5-2: Intersections of two ellipses for a tool at two continuous NC positions ........ 64

Figure 5-3: Tetrahedron in a parallelepiped...................................................................... 66

Figure 5-4: Three cases for machining a free-form surface .............................................. 68

Figure 5-5: (a) Tool simulation in Case 1 of the first toolpath machining; (b) The chip

area for the first toolpath on the plane z=0 .................................................... 69

Figure 5-6: Case 2: The chip area for a single toolpath on the plane z=0 in 2D .............. 70

Figure 5-7: Case 2: The chip area for a single toolpath in 3D .......................................... 70

Figure 5-8: Case 2: Valid chip outline by layers in a single toolpath ............................... 71

Figure 5-9: Case 3: (a) Tool motion in the second toolpath; (b) Removed chip in two

adjacent NC points ........................................................................................ 71

Figure 5-10: The chip area for one toolpath considering its neighboring toolpath on the

plane z=0 in case 3 ........................................................................................ 72

Figure 5-11: (a) The valid chip outline generation in two continuous toolpaths (b) Valid

chip outline points; (c) Solid chip shape by the Alpha Shape method .......... 73

Figure 5-12: The tool moves along two NC points from Γ{Ci, j+1, Ψi, j+1} =(0.2, 0.5, 0.2,

4.5°, 4.5°) to Γ{Ci+1, j+1, Ψi+1, j+1}=(0.1, 0.5, 0.5, 6.5°, 6.5°) in the jth

+1

toolpath: (a) Side boundaries in the tool motion at Γ{Ci+1, j+1, Ψi+1, j+1}; (b)

Bottom and top boundaries in the tool motion at Γ{Ci+1, j+1, Ψi+1, j+1}; (c) Side

boundaries in the tool motion at Γ{Ci, j+1, Ψi, j+1} (d) Bottom and top

boundaries in the tool motion at Γ{Ci, j+1, Ψi, j+1}. ......................................... 74

xi

Figure 5-13: Determination of instantaneous chip thickness: (a) Tool motions at two

adjacent NC points; (b) Tool projections on A-A section ............................. 76

Figure 5-14: (a) Chip shape outline points; (b) Sliced chip area for layers; (c) Chip

volume consists of sliced parallelepipeds ..................................................... 78

Figure 5-15: Chip thickness on different layers ................................................................ 79

Figure 5-16: Cutter-workpiece engagement domain in 2D .............................................. 80

Figure 5-17: Cutter-workpiece engagement domain from a removed chip volume: (a) 9

slices with 60 interval points; (b) 15 slices with 100 interval points ............ 81

Figure 5-18: (a)-(c) Displays how the sliced volume is gradually removed in the free-

form surface machining ................................................................................. 82

Figure 5-19: Cutting geometry of a flat-end mill.............................................................. 84

Figure 5-20: (a) Simulation of machining a 3D curve on a free form surface, workpiece

size: 50×50×20 mm3, tool diameter: 10 mm; (b) The simulation of tool

motions in MATLAB. ................................................................................... 85

Figure 5-21: Chip volume simulation for the first toolpath .............................................. 86

Figure 5-22: Chip volume simulation for the second toolpath ......................................... 87

Figure 5-23: Chip volume simulation for a single curve .................................................. 87

Figure 5-24: Chip volume comparison of the first toolpath with and without considering

the edge of the workpiece .............................................................................. 89

Figure 5-25: Volume comparison of the second toolpath with and without considering the

first toolpath. ................................................................................................. 90

Figure 5-26: Simulated cutting forces in X, Y and Z directions for the whole toolpath .. 91

Figure 5-27: Predicted X, Y and Z forces for five revolutions in 5-axis CNC machining

with a flat-end mill ........................................................................................ 91

Figure 5-28: (a) Resultant forces changing with machining time; (b) Chip volume

changing with machining time ...................................................................... 92

Figure 5-29: Comparison of chip volume by the Alpha Shape method and the tool profile

based method ................................................................................................. 93

Figure 5-30: Comparison of NC points got by MasterCAM and the uniform interpolation

method ........................................................................................................... 94

Figure 5-31: Measured and predicted cutting forces changing with rotation angles in three

revolutions. .................................................................................................... 95

Figure 6-1: The Tri-dexel workpiece model in 3D. ........................................................ 100

Figure 6-2: Boolean subtraction and chip thickness generation ..................................... 102

Figure 6-3: Chip thickness for non-uniform distributed chip geometry ......................... 104

Figure 6-4: Chip thickness on the Tri-dexel workpiece.................................................. 105

Figure 6-5: The non-uniform distributed chip shape ...................................................... 106

Figure 6-6: The uniform distributed chip shape and redefined chip thickness ............... 108

Figure 6-7: Non-uniform and uniform distributed valid chip profile points .................. 109

Figure 6-8: Chip thickness for the uniform distributed valid chip geometry ................. 109

xii

Figure 6-9: Cutting simulation of tool removing in the Tri-dexel workpiece ................ 110

Figure 6-10: Varied depth of cut in the workpiece method ............................................ 111

Figure 6-11: Cutting force model of a flat-end mill ....................................................... 113

Figure 6-12: Comparison of simulated cutting forces by the workpiece and the tool based

methods ....................................................................................................... 114

Figure 6-13: (a)-(c) Simulated cutting forces by the Tri-dexel workpiece method; (b) (e)-

(g) Simulated cutting forces by the tool based method ............................... 115

Figure 6-14: Resultant cutting forces by the workpiece method .................................... 116

Figure 6-15: Comparison of chip volume by the tool based method and the workpiece

method ......................................................................................................... 116

Figure 6-16: The pocket toolpath .................................................................................... 118

Figure 6-17: (a) Measured resultant cutting forces changing with machining time; (b)

Predicted chip volume changing with machining time ............................... 119

Figure 6-18: Comparison of simulation and experimental resultant forces in 3-axis

milling ......................................................................................................... 120

Figure 7-1: A CFRP 3D chip model ............................................................................... 128

Figure 7-2: (a)-(b) Removed fiber on the parallel direction; (c)-(d) Removed fiber on the

vertical direction .......................................................................................... 129

FigureA1- 1: Average cutting forces .............................................................................. 141

FigureA1- 2: The linear function of feed rates and an offset contributed by the edge

forces Fxc ..................................................................................................... 142


forces Fyc ..................................................................................................... 143


forces Fzc ...................................................................................................... 144

FigureA2- 1: Distributions of relative angles with different numbers of cluster centres.

......................................................................................................................................... 146

FigureA2- 2: (a) Relation of cluster centre numbers and the maximum and average

relative angles; (b) the change rates of cluster centre numbers and maximum and average

relative angles. ................................................................................................................ 147

FigureA3- 1: Ball-end milling (a) tilt angle=1°; (b) tilt angle=5.78°;(c) tilt angle=10°;

flat-end milling (d) tilt angle=1°; (e) tilt angle=5.78°;(f) tilt angle=10° ..... 148

FigureA3- 2: (a) comparison of machining time with different tilt angles between ball-

end milling and flat-end milling .................................................................. 149

FigureA3- 3: (a) ball-end milling in several toolpaths, D=5mm, tilt angle=5.78°; (b) flat-

end milling in several toolpaths, D=5mm, tilt angle=5.78°;(c) ball-end

milling in one toolpath, D=50mm, tilt angle=0°; (d) flat-end milling in one

toolpath, D=50mm, tilt angle=90°............................................................... 150

FigureA3- 4: (a) comparison of machining time with different tool diameters between

ball-end milling and flat-end milling ........................................................... 151

xiii

Figure A4- 1: Determination of the instantaneous chip thickness in the 5-axis micro flat-

end milling: (a) Tool motions at two adjacent NC points; (b) Ploughing and

shearing areas in tool projections on the A-A section ................................. 157

Figure A4- 2: (a) Ploughing and shearing volume; (b) Ploughing and shear areas on

layers ........................................................................................................... 160

Figure A4- 3: A free-form surface in micro-milling with a flat-end mill ....................... 161

Figure A4- 4: The interpolated toolpath ......................................................................... 161

Figure A4- 5: Comparison of the total, ploughing and shearing volume ....................... 162

Figure A4- 6: A 3D chip geometry of a micro ball-end mill feed in the horizontal

direction ....................................................................................................... 163

Figure A4- 7: The projection in the slice plane when the angle ϕ is zero ...................... 164

Figure A4- 8: Coordinate rotation for upward direction machining ............................... 167

Figure A4- 9: Small segments of a curve in cubes ......................................................... 168

Figure A4- 10: A 3D curve machining ........................................................................... 168

Figure A4- 11: Process faults with parallel offset runout ............................................... 169

Figure A4- 12: The ploughing and shearing volume calculation flowchart ................... 172

Figure A4- 13: Two different toolpaths: a) Straight lines and down-ramping, b) A straight

line ............................................................................................................... 173

Figure A4- 14: The changes of shearing and ploughing volumes with the height of kth

slice z(k)for slot machining ......................................................................... 174

Figure A4- 15: The Voxel and Boolean method: Chip volume simulation for (a) Slot

machining; (b) Straight lines and down-ramping machining ...................... 175

Figure A4- 16: Slot machining: Chip volume simulations changing with the number of

samples. Spindle speed=40,000 rpm, depth of cut=0.1mm, ft =0.75 µm/tooth

..................................................................................................................... 176

Figure A4- 17: Straight line and down-ramping machining: chip volume simulations

changing with rotation angle θ. Spindle speed=20,000 rpm, depth of cut=0.2-

0.7mm, ft =1.5 µm/tooth .............................................................................. 177

Figure A4- 18: Slot machining: Chip volume simulations changing with rotation angle θ

ignoring runout. Spindle speed=40,000 rpm, depth of cut=0.1mm, ft =0.75

µm/tooth ...................................................................................................... 178


considering runout, ε=0.01µm, spindle speed=40,000 rpm, depth of

cut=0.1mm, ft =0.75 µm/tooth ..................................................................... 178

Figure A4- 20: Experimental setup of micro-milling operations [7] .............................. 179

Figure A4- 21: Measured resultant cutting forces with machining times ....................... 181

Figure A4- 22: Measured resultant cutting forces for the slot machining. Spindle

speed=40,000 rpm, depth of cut=0.1mm, ft =0.75 µm/tooth ....................... 183

xiv

Figure A4- 23: The surfaces generated by the ball end milling processes: (a) Depth of cut

dc=100 µm, ft =0.75 µm/tooth; (b) dc=200 µm, ft =0.75 µm/tooth; (c) dc=150-

600 µm, ft =1.5 µm/tooth; (c) dc=200-700 µm, ft =1.5 µm/tooth ................. 184

Figure A4- 24: Topography of the machined surfaces in a 3D surface measurement

machine ....................................................................................................... 185

xv

Acknowledgments

I am grateful to my supervisors, Professor Zuomin Dong and Professor Martin B.G. Jun,

for their generous support, encouragement, kindness, understanding, and awesome

supervision. I thank them for revealing to me the fascinating world of tool-part geometry

and dynamics of 5-axis CNC machining.

I would also like to acknowledge my friends and colleagues in the Advanced

Manufacturing Research Laboratory at the University of Victoria: Yanqiao Zhang,

Abdolreza Bayesteh, Salah Erfurjani, Farid Ahmed, Max Rukosuyev, and Junghyuk Ko

from whom I learned a lot over the past four years.

Finally, I would like to thank my family, Yanchang Luo, Guilin Cheng, Kai Luo, and

Jinrong Cheng for their love and support throughout the lengthy process of my PhD work,

and for their patience and guidance at the difficult moments during my life.

1

Introduction Chapter 1:

1.1 Background and Motivation

1.1.1 Toolpath and Orientations

Compared with traditional 3-axis CNC machining, 5-axis CNC milling provides better

tool accessibility, thus increasing material removal rate, reducing machine setup time,

and producing better surface quality for sculptured surfaces. The CNC

toolpath/orientation planning involving the identification of optimal tool orientation for

5-axis CNC machining is much more complicated than the traditional CNC toolpath

planning for 3-axis machining. 5-axis CNC machining matters more than ever before to

many industries from automobile industries, aerospace, energy to mould industries [1].

Dramatic tool orientation changes as machining a surface with large curvature have

become a significant issue, due to slow rotational axis movements and the less rigid

machine-cutter-part system. A 5-axis CNC machine is less rigid than the corresponding

three-axis counterpart, due to the two additional rotational axes. 5-axis CNC machining

uses five synchronized motions to reach different portions of the machined surface.

However, these 5 axes of motion are not created as equal. The first three axes are

normally accomplished by the conventional linear motions of a conventional 3-axis CNC

machine with higher stiffness and response time due to the rigid machine tool structure

and the larger electric drives. The last two axes of rotation are commonly accomplished

by two smaller drives on the mill head. These less rigid axes of rotation motions also

present slower rate of change and response time. Furthermore, drastic changes in cutter

2

orientation lead to undesirable surface problems such as overcutting, overlap, and

changing cutting forces.

Good rigidity and high precision can satisfy high accuracy demand during machining.

The shorter tool length in 5-axis CNC machine inherently reduces the rigidity and

feedrate compared with 3-axis CNC machines. The increased rigidity of CNC machine

provides better cutting capability and performance and retains accuracy and repeatability

at the highest levels. Yet, the rigidity is based on the machine body and rotational heads.

It is difficult to change the rigidity after CNC machines are designed. Therefore, it is

better to consider other approaches to improve the cutting performance.

For each tool location in a toolpath, there are numerous choices for the selection of a

cutter inclination. Most existing methods for tool orientations are relative to the surface

normal vector at every cutter contact (CC) point [7]. However, there is a drawback for the

surface normal accessibility. For a machined surface area with large curvature, the tool

orientation tends to suffer drastic changes that lead to larger velocity, acceleration, and

jerk on the rotational axes of the machine. Drastic changes in cutter orientation lead to

undesirable surface problems such as overcutting and overlap and unsmooth cutting force

[8, 9]. Therefore, smooth tool motions are necessary. Tool orientation variation and the

change from one CC point to the next should be minimized. To avoid dramatic tool

orientation changes, it is beneficial to generate a fast execution toolpath and small

changes of tool postures by machining surfaces patch-by-patch with similar surface

orientation, identified by the fuzzy clustering method and similar surface normal

variations. Chapter 3 will give more details about toolpaths generated by the fuzzy

clustering technique and the surface normal method.

3

Today, to avoid cutter-part surface interference/gouge at large curvature areas and to

simplify toolpath/orientation planning, a small diameter ball-end mill is commonly used

during machining [2]. This leads to low machining efficiency and large cusps for areas of

the surface with small curvature. Large diameter end cutters present a more rigid and

capable tool with a varying cutter curvature from the radius of the cutter to infinity (in

principle) to support better cutter-part curvature match, leading to much improved

machining efficiency and surface quality [3]. Therefore, it is more sensible to select flat-

end mills for sculptured surface machining. However, flat-end mills cannot easily avoid

curvature gouging problems. It is still challenging to tool orientations using a flat-end

cutter for sculptured surfaces without gouging generation in 5-axis CNC machining. The

control and planning of the tilt angles of the rotational cutter are much more challenging

due to the complex cutter and part surface interaction in 5-axis machining, particularly

when a flat-end mill is used. To improve machining efficiency and the surface quality of

the finished part, the flat-end mill will be focused on in this research, and new methods

will be introduced for gouge avoidance in concave surface milling and for complicated

chip volume and cutting forces calculations.

Currently, commercial Computer Aided Manufacturing (CAM) software can generate

toolpaths automatically. However, the software still has some problems generating

optimal and flexible tool orientations for sculptured surfaces. To avoid gouges, CAM

software tends to select a small diameter ball-end mill for machining which causes low

machining efficiency. Furthermore, CAM system requires the user to select a tool

orientation following a trial and error approach [4-6]. Firstly, cutter orientations are

created by a user-defined strategy like “surface-normal machining” and “tilted through

4

curve”; the toolpath has to be simulated and modified if gouges occur. The traditional

trial and error approach to avoid gouges is inconvenient; therefore, a new approach

should be generated to avoid gouging automatically. In this work, an optimal tool

orientation based on the combination of the Euler-Meusnier Sphere (EMS) method and

the surface normal method to avoid gouges and improve machining efficiency will be

discussed in Chapter 4 to avoid gouges and improve machining efficiency.

1.1.2 Machine Dynamics

Researches of machining dynamic play a significant role when high efficiency is required

[10-12]. The kinematics of tool motions is the most investigated aspect when smooth tool

orientation changes are needed. The tilt and lead angles affect mechanics and dynamics

of the machining process in terms of cutting forces, cutting forces coefficients, torque,

chip thickness, stability, and tool breakage [13]. In this research, instant cutting forces

and cutting volume predictions are mainly considered to optimize the last remaining

planning variable feed rate to achieve high machining efficiency and surface quality in 5-

axis CNC machining using flat-end mills.

Most of the previous research on 5-axis machining has focused on the geometric

aspects such as toolpath/tool orientation generation and machine dynamics aspects

separately [14]. But not too much previous research considers the combination of

geometry and dynamics. Cutter-part surface geometry is linked to dynamics through chip

volume for sculptured surfaces in 5-axis CNC machining. However, there are many new

problems generated when geometry and dynamics are considered together. For dynamics,

it is better to select a toolpath that has maximum cutting force/volume while the height of

cusps remains within the specified tolerance zone. For 5-axis CNC machining, cutting

5

forces are predicted with respect to inclination and lead angles. Cutting forces are

changed with varied tool orientations even if the cutting parameters such as feed rate,

depth of cut, and spindle speed are the same. There is an existing conflict: two rotation

angles for the maximum cutting forces may not be same with rotation angles obtained by

the geometry method for gouging avoidance. Therefore, optimal toolpath and tool

orientation should be selected to obtain the maximum cutting force and removed material

volume while the height of cusps is under the given machining tolerance.

In Vericut, cutting conditions are shown in the status display and available when

stepping through the program using NC Program Review. The feature shows detailed

information about the cutter’s engagement with material, including: axial depth, radial

width, volume removal rate, chip thickness, maximum surface speed, and contact area.

Lots of studies about ball-end mill cutting forces have been done in recently; however,

few studies have been done for flat-end mills in 5-axis CNC machining.

5-axis CNC machining is widely used to produce various components with complex

geometry while potentially providing better tool accessibility to complex surfaces,

producing more accurate curved surfaces, increasing material removal rate, and reducing

machine setup time [15]. For 3-axis CNC machining using a flat-end mill, chip thickness

is constant along the axial direction, and chip volume calculation is relatively simple by

discretizing the tool along the axial direction. However, in 5-axis CNC machining using a

flat-end mill, the contact area between the cutter and machined surface changes all the

time due to the inclination and rotation angles. The varying contact area causes

challenges with calculating chip volume and engagement zone. Knowing values of

removed chip volume can help choose optimal cutting parameters. Therefore, chip

6

thickness and chip volume calculation using flat-end mills in 5-axis CNC machining

should be studied to offer another approach to select optimal cutting parameters.

Predicting cutting forces is significant in the planning process. Cutting force estimates

are useful when choosing optimal cutting parameters such as feedrate, depth of cut to

improve the machining efficiency, and surface quality. The cutting force calculations can

also be used for cutter deflection, tool breakage detection, and process planning. In this

work, two numerical methods will be developed to calculate chip volume and cutting

forces in Chapters 5 and 6.

1.2 Research Contributions

This research aims at introducing new enabling techniques for the combined optimal

toolpath, cutter orientation, and chip volume/cutting force calculations for optimal feed

rates to maximize machining efficiency and obtain better surface quality in 5-axis CNC

machining of curved surfaces using flat-end mills.

Figure 1-1: The research roadmap

7

Figure 1-1is the research roadmap that summarizes how research contributions fit into

the overall effort to obtain high machining efficiency and good surface quality. The

research firstly considers slow responses and weakness of machine tool rotation axes to

generate optimal tool path by machining surfaces into patches, ensuring machining

efficiency and machine-cutter-part system stiffness. It also explores the largest cutting

edge and best curvature match for optimal tool orientation to obtain maximum removal

material with no gouge generation. Lastly, it covers to chip volume and cutting force

modeling and calculations using machining dynamics models to optimize the last

remaining planning variable, feed rate, to accomplish high machining efficiency and

surface quality for 5-axis machining of curved surfaces using a flat-end mill. The

following is a list of contributions toward methods of optimal toolpath/orientation

generation and chip volume/cutting force prediction in 5-axis CNC milling using a flat-

end mill that have been made over this work:

Optimal toolpath generation: To avoid sharp cutter orientation changes by

machining surfaces patch-by-patch with similar surface orientation, an optimal

toolpath identified by fuzzy clustering technique and surface normal variations

control method was proposed to generate fast CNC machining (see Section 3.1 of

Chapter 3). The optimal number of surface patches or surface point clusters is

identified by minimizing accumulating changes of relative angles, discussed in

Section 3.2. To generate closed and smooth boundaries, the computational geometry

method of Alpha Shape is used to find and connect the mesh points on the border of

each surface patch. This work was presented at the 2014 Virtual Machining Process

Technology Conference [16].

8

Optimal tool orientation generation: An optimal and flexible tool orientation

method based on the combination of Euler-Meusnier spheres (EMS) method and

surface normal variations control method is developed in Section 4.1 of Chapter 4.

Better cutter-surface curvature matches and gouge avoidance at the cutter contact

point (CCP) are obtained by applying the EMS principle to determine the optimal

cutter orientation at each cutter contact point on the toolpath for concave surfaces;

the surface normal variation control method is used for convex surfaces due to its

higher efficiency and no gouging issue; selection of one of these methods in tool

orientation determination for saddle shapes is based on the direction of the CNC

toolpath relative to the surface curvature change.

In 5-axis CNC machining, maximum feed rates can achieve the highest machining

efficiency. However, feed rates are always changed, as the synchronized lineal and

rotational movements of rotation axes, and the complicated cutter-part contact geometry.

It becomes complicated to select optimal feed rates for free-form surface machining in 5-

axis CNC machines using flat-end mills. In this work, chip volume and cutting force

predictions will be proposed for the feedrate optimization. Compared to ball-end mill

machining, the chip volume and cutting force prediction in 5-axis CNC machining with

flat-end mills are much more challenging due to the complexity of cutter-part surface

geometry interaction. A ball-end mill has constant curvature so the cutter location is

easier to be determined, while the curvature for a flat-end mill at each CC point varies

with different tilt and lead angle in 5-axis CNC machines; therefore, the discrete method

for chip volume and cutting force calculation with flat-end mills is much more

complicated than ball-end mills. To overcome these challenges, this author developed

9

two numerical approaches to generate chip model and calculate chip volume and cutting

forces. These developments induce several more contributions to the field:

Chip volume calculation by Alpha Shape method: A computational geometry-

based Alpha Shape method is applied to model the volume and shape of the removed

chip during 5-axis milling (see section 5.2.1 of Chapter 5). The 3D chip modeling

requires identifying the chip boundary that defined by a valid tool geometric outline

at two continuous NC points. Since it is difficult to calculate the intersections of two

arbitrary cylinders using closed-form analytical model through translations and

rotations, a numerical method has been used in this work to obtain the intersections

of two arbitrary cylinders by dividing them into many thin layers along the z axis

direction. Three cases of toolpaths are considered to obtain the chip volume and

simulate the real tool motions. The Alpha Shape method provides an efficient and

robust way to calculate chip volume for arbitrary tool orientations because a series of

complicated trigonometric equations, to get intersections of tool motions at two

arbitrary positions, are replaced by a numerical method in ALGORITHM (see

Section 5.2.1.3 of Chapter 5). This work was presented at the 2015 Virtual

Machining Process Technology Conference [17].

Chip volume/cutting force calculations by the tool based method: Alpha Shape

method can display solid chip shape and calculate chip volume with a fast computing

time; however, chip thickness and cutting forces cannot be calculated by this method.

A new approach— the local parallel sliced method (see Section 5.2.2) is then

presented to obtain cutter-workpiece engagement domains, where the depth of cut

and cutting flutes entering/exiting the workpiece are required to predict instant

10

cutting forces. Local parallel sliced method divides the cutter into many slices

perpendicular to the tool axis along the local coordinate system. On each layer, the

removal chip area is a polygon shape generated by connecting two neighbouring

edge points on the current and previous tool edges. The total chip volume is obtained

by adding all polygon areas along axial direction.

The tool profile based method can save computing time to calculate chip volume and

cutting forces. However, it cannot be used in the pocket toolpath. That is why another

approach—the workpiece based method is proposed.

Chip volume/cutting force calculations by Tri-dexel workpiece method: The Tri-

dexel workpiece method (presented in Section 6.1 of Chapter 6) is robust for use with

any kinds of toolpaths to predict chip volume and cutting forces; it gets chip volume

and cutting forces through the intersection of the tool envelope and continuously

updated the workpiece rather than from the tool intersections at four continuous

positions in two neighboring toolpaths. The removed volume is obtained by subtracting

the cutter-workpiece engagement zone. To reduce the complexity of 3D Boolean

subtraction, the Tri-dexel workpiece is sliced into many 2D laminated layers along z-

axis direction in Section 6.1.2. Chip volume can be obtained from the non-uniform

distributed chip model by the intersections of the tool envelope and the workpiece, but

cutting force calculations cannot be applied by this model. Extending the non-uniform

distributed chip model that can only predict chip volume, a uniform distributed chip

model has been added to calculate cutting forces by finding the same column index of a

flat-end mill. The workpiece method is robust for use with to any toolpath to predict

cutting forces; however, the computing time is much longer than the tool profiled

11

method since the workpiece is updated with many line segment operations such as

intersection and subtraction at every tool motion in the whole toolpath.

Chip ploughing volume prediction: It is a challenging task to avoid ploughing

problems in micro-machining. When the cutter crosses the minimum chip thickness

boundary, it enters into the ploughing zone with no material removed. Therefore, it is

important to know the ploughing effects in micro-milling. Chip ploughing volume

prediction for 5-axis micro flat-end milling is presented in Section A4.2. The tool

based model proposed in Section 5.2.2 is used to calculate chip thickness and

ploughing volume. Ploughing zone is the area where the chip thickness is less than

the minimum chip thickness; while in the shearing zone, the chip thickness is larger

than the minimum chip thickness. Chip geometry and chip ploughing volume for a

micro ball-end mill are discussed in Section A4.3. Different cutting conditions, such

as feed rate, spindle, and depth of cut, are tested in a 3-axis micro CNC machine with

a ball-end mill to better understand the ploughing effects in micro machining and to

increase cutting efficiency. Two different CNC toolpaths are used to simulate the

machining process and to obtain the relation between chip ploughing volume and

rotation angle.

1.3 Dissertation Outline

This dissertation presents work in improving current toolpath and orientation methods

and exploring the combination of cutter-workpiece geometry and machine dynamics in 5-

axis CNC machining using flat-end mills. This research emphasizes optimal

toolpath/orientation generation and the development and implementation of numerical

12

approaches to calculate chip volume and cutting forces for feed rate optimization by a

flat-end mill in 5-axis CNC machining.

Chapter 2 is the literature reviews for three aspects: toolpath planning, tool orientation

methods, and machine dynamics.

Chapter 3 starts with presenting an optimal toolpath by machining a surface patch-by-

patch using fuzzy clustering techniques and similar surface normal variations control.

This reduces the range of the rotational axes’ motions and helps to avoid sharp cutter

orientation changes. This chapter also gives a discussion on optimal number of surface

patches establishment by minimizing accumulating relative angle. At the end of this

chapter, the computational geometry method of Alpha Shape is discussed to generate

closed and smooth boundaries of surface patches.

Chapter 4 presents an optimal tool orientation based on the combination of the EMS

method and the surface normal variable control method. The EMS method considers the

best curvature match to achieve maximum removal material with no gouge generation.

The surface normal variable control method can also obtain the highest machining

efficiency by the largest cutting edge. A NURBS with three surface features such as

concave, convex, and saddle is selected to give a detailed explanation of the optimal

toolpath approach. Typically, the EMS method is applied to concave parts to avoid local

gouges. The highest efficient tool orientation for a convex surface is along surface normal

directions. For a saddle surface, the EMS method or surface normal method is selected by

machining directions.

Optimal feed rate can be determined by chip volume and cutting forces. However, it is

complicated to calculate chip volume and cutting forces as machining free-form surfaces

13

using flat-end mills in 5-axis CNC machining due to the two rotational angles and

flexible changes of tool curvature. It is also challenging to apply the analytical method to

get intersections of two flat-end mills at arbitrary directions. To overcome these

problems, Chapter 5 presents a completely new numerical tool based approach to predict

chip volume and cutting forces. Extending the Alpha Shape method, which can only

predict cutting chip geometry, a parallel slice local volume modeling approach has been

added to predict cutting forces. An experiment for the research of cutting volume and

cutting forces in 3-axis micro CNC machine was conducted. The simulation results for 5-

axis machining were verified by machining experiments through specifying the two

rotation angles to zeros. The simulated and measured forces are shown in reasonably

good agreement in both the trend and magnitudes if the runout effects are ignored.

Chapter 6 improves the chip volume and cutting force predictions in any kinds of

toolpaths by demonstrating a Tri-dexel workpiece method. The tool based method

presented in Chapter 5 can provide fast computing time, but it has limitations in the

application of pocket toolpath. The comparisons of these two numerical approaches are

made by a same case study. Extending the non-uniform distributed chip model that can

only predict chip volume, a uniform distributed chip model has been presented to

calculate cutting forces by finding the same column index of a flat-end mill.

Appendix4 is a relative study of chip ploughing volume in Micro-milling. It is a

challenging task to avoid ploughing problems in micro-machining. When the cutter

crosses the minimum chip thickness boundary, the tool would enter into the ploughing

zone with no material removed. The model proposed in Section 5.2.2 for macro 5-axis

flat end milling works in micro-milling to calculate chip thickness and ploughing volume.

14

The ploughing effects for 3-axis micro ball-end milling are also introduced in this

chapter. Two algorithms in this work are demonstrated to get the ploughing volume. To

better understand the ploughing volume problem in micro machining and to increase

cutting efficiency, an experiment testing different axial depths of cut and feed rates was

conducted.

Finally, conclusions and recommendations for future work are discussed in Chapter 7.

15

Literature Review Chapter 2:

Toolpath/orientation generation and machine dynamics in 5-axis CNC machining is an

established field and many researchers have already made significant contributions to this

area. The literature summarized in Section 2.1 that several traditional toolpath generation

methods and some new toolpath generation techniques have been developed to improve

machining efficiency and surface quality. Section 2.2 discusses tool orientation

methodologies along with optimization methods that would overcome some limitations.

There are many researches have studied ball-end mill cutting forces in recent years;

Section 2.3 discusses these contributions. One challenge of adopting ball-end mill

machining is time consumption and poor surface quality. Flat-end mills with flexible

curvature changes can help engineers overcome these limitations, but very limited studies

on 5-axis CNC machining using flat-end mills have been carried out due to the

complexity of cutter-part surface geometry interaction.

2.1 Toolpath Planning

Studies on toolpath generation for CNC machine have been conducted for many years.

Traditionally, there are several toolpath generation approaches, such as the iso-planar

[18], iso-parametric [19], and iso-scallop [20].

The iso-parametric approach is widely applied in freeform surfaces [19, 21-23]. There

are two variables to define freeform surfaces along toolpath and toolpath interval

directions. During toolpath planning, one parameter is changed while the other is fixed.

This method has short computing time but long machining time [24]. From Figure 2-1, it

can be seen that iso-parametric toolpaths are commonly much denser in areas of the

16

surface with small curvatures due to the non-uniform transformations between the

parametric and Euclidean space [25]. The iso-planer method is commonly used in CAM

programs due to its robustness and simplicity [26-28]; however, it cannot control the

cusps height, since the toolpath is generated by intersections of parallel planes and the

machined surface, which can be seen Figure 2-2. The iso-cusped method is an improved

version of the iso-parametric and iso-planar methods by increasing productivity and

avoiding toolpath redundancy [29]. For iso-cusps method, it must have a first toolpath as

the reference; other toolpaths are computed on the offset surface to make sure the height

of cusps is same as the reference toolpath. Although the overall toolpath length is reduced

through constant cusps, the iso-cusps method surfers complicated computation and errors

accumulation.

Figure 2-1: Iso-parametric toolpath for NURBS surface

Figure 2-2: Iso-planar toolpath for curved surface

17

Some new toolpath generation techniques have been developed to improve machining

efficiency and surface quality.

2.1.1 Surface Division Machining Toolpath

Machining a surface patch-by-patch is based on dividing the surface into regions by

specified features, and machining each region separately [1]. There are some studies

about toolpath generation based on regions. Ding [26] used the isophote method to

partition a surface into different areas by the angle between the surface normal and that of

the intersecting planes to reduce redundant tool paths. This method makes the toolpath

side steps to be adaptive to the surface geometry features, reducing the total toolpath

length and increasing machining efficiency. However, it was a challenge to connect the

toolpaths of two neighbouring regions to obtain a much smoother surface. Lee [5]

classified a freeform surface according to principal surface curvatures to find optimal tool

orientations. The surface points were sorted into four different types such as convex,

concave, hyperbolic, and parabolic. A flat-end mill was used to machine convex and flat

regions; a ball-end mill was selected to machine small curvature regions. However, tool

changes should be minimized, due to the non-profit added operations. Chevy Chen

proposed a toolpath method based on fuzzy cluster points and the Voronoi diagram [30].

This toolpath was applied to divide the sculptured surface into surface patches. All the

points in each patch have similar surface features such as surface shape and

machinability. The sculptured surface was first classified into convex, concave, and

saddle shapes according to Gaussian/mean curvatures of the surface. After the rough

subdivision, two fuzzy pattern clustering methods were used for the fine surface

subdivision. Cluster centers in a particular surface shape region were first identified by

18

subtractive fuzzy clustering method; the fuzzy C-mean method was then used to optimize

the locations of cluster centers. Voronoi diagram that generates the boundaries using the

formed cluster centers was finally used to define the surface patches. For the tool

orientation, the rotational axis was fixed by the surface normal direction at the cluster

centre in each surface patch, which can be seen in Figure 2-3. However, Chen’s method

was only applied in 3 ½ ½ -axis CNC machines. Cutter orientations cannot be changed

smoothly and automatically like 5-axis CNC machines, thus this method requires longer

machining set up time.

Figure 2-3: Surface patches by cluster centers [11]

2.1.2 Steepest-directed and Iso-cusped (SDIC) Method

Chevy Chen [31] integrates the steepest-directed and iso-cusped (SDIC) toolpath

generation methods to machine a sculptured surface to a specified surface tolerance with

a minimum of machining time. It is a global method to generate toolpath for 3-axis CNC

machine. However, Chen used these methods for convex surface without considering

19

gouging problems. For convex surfaces, cutter locations are along surface normal

directions without gouging generation. That is due to the Meusnier spheres of the cutter

and the machined surface being located on opposite sides of the tangent plane, no

curvature gouge problems existing. The SDIC method is efficient to generate toolpaths

for a convex surface in 3-axis CNC machining. Yet in 5-axis CNC machining, this

method is not useful anymore because the two rotational axes allow more accessible

machining areas. 5-axis machining is able to reduce the machining time by adjusting

inclination and rotation angles. However, gouge problems should be considered as well

for toolpath planning.

2.1.3 Accessibility-map (A-map) Method

Li [32] proposed an accessibility map (shown in Figure 2-4) of the tool at a cutter

contact point to define the range for the cutter without any cutter-part surface

interference, and thus generating small cutter orientation change and reducing the total

toolpath length. However, when the surface curvature changes dramatically from one

area to the other, the propagated toolpaths are far away from the initial toolpath and the

tool orientation along the feed direction may not be globally smooth due to the correction

process for achieving error control. Therefore, this method needs to generate several

initial toolpaths spreading over the machined surface and then generate adjacent

toolpaths. However, it may increase the complexity of toolpath planning, since different

initial toolpaths are selected as the references to propagated toolpaths.

20

Figure 2-4: The A-map for tool orientation [32]

2.2 Tool Orientation

In 5-axis CNC machining, three coordinate systems are used to display the geometry

of cutter and part surface. Tool positions and orientations are defined in the tool

coordinate system (TCS). The tool position means the tool center point. It is also called

cutter location point or CL point, while the tool orientation is referred to the tool axis

vector. Local coordinate system (LCS) is placed at cutter contact (CC) points with feed

direction (F), normal vector (N) and the cross of feed and normal direction (C). CNC

machines can only read NC data which is specified in the machine coordinate system

(MCS) [33]. In 5-axis CNC milling, the tool posture consists of tool positions and

orientations. Tool orientations are defined by lead and tilt angles which are measured by

surface normal vectors. The lead angle is the rotation of the tool axis about the cross-feed

direction and the tilt angle is the rotation of the feed direction, which can be seen in

Figure 2-5.

21

Figure 2-5: Coordinate systems and lead-tilt angles [13]

2.2.1 Principal Axis Method (PAM)

Principal Axis Method (PAM) is based on a surface-cutter curvature match at cutter

contact points [34-36]. When the tool is tilted along the feed direction, the minimum tool

curvature is matched to the maximum surface curvature at the CC point. An osculating

plane (shown in Figure 2-6) is a plane that contains the CC point and its surface normal

vector. The curvature is changed from maximum principal curvature to minimum as the

osculating plane is rotated around the normal axis. In Figure 2-6, the two principal

directions and surface normal vector are orthogonal with each other. However, PAM only

considers the cutter contact point; the cutting edge of the tool may penetrate the design

surface and then cause gouges. To remove rear gouging, the tool is tilted until gouging is

eliminated or reduced to a specified tolerance zone, and thus it is suitable for open face

freeform surface [37]. It results in curvatures that are no longer matched and the

effectiveness of the PAM is reduced at the CC point.

22

Figure 2-6: Triad formed by principal curvature directions and the surface normal [34]

2.2.2 Euler-Meusnier Sphere (EMS) Curvature Match

Wang [38] presented a 3D model which is based on the new Euler-Meusnier Sphere

(EMS) concept (shown in Figure 2-7) from a generic mathematical and geometric model

of the cutter and surface geometry to avoid gouging for concave surfaces. Given a point

on a surface, there are many normal curvatures at this point in various directions.

Meusnier spheres at this point are determined by these curvatures. The largest and

smallest Meusnier spheres are obtained by the minimum and maximum principle

curvatures.

Figure 2-7: Euler- Meusnier sphere [39]

23

The total elimination of curvature gouges can only be accomplished by ensuring that

there is no overlap between the volumes defined by the largest and smallest Meusnier

spheres of the cutter and the machined surface.

Figure 2-8: Gouge-free condition [39]

The EMS curvature match method is a good way for tool orientations to avoid

gouging; however, it may not be optimal for a non-uniform curvature surface and

sometimes this may not be able to iso-cusps machining. The feed direction in iso-cusps

machining cannot always follow the same direction of minimum principal curvature of

the surface. For concave surfaces, curvature match becomes much more difficult from the

bottom to top.

2.2.3 C-space Based Tool Orientation Methods

The machining configuration space (C-space) is used to find optimal tool orientations

by different machining constraints [9, 40, 41]. This method considers local, rear, and

global gouges in machining[42]. The C-space (shown in Figure 2-9) is the tool tilting and

inclination parameter areas without gouging generation [43]. After construction of the C-

space, there is an optimization process to select smaller tilt angles and the minimum

24

changes of tool orientations. There are many researches about C-space. Lee [7] presented

the orientation domain to avoid local and rear gouges, where the optimal solutions are

close to the boundaries of the C-space. The optimization goal is to maximize scallop

height and minimize tilt and inclination angles. Lu [44] developed a 3D gouge-free C-

space method which is based on minimizing the time travel distance to smooth the tool

orientation changes. Wang [43] proposed a new C-space algorithm to generate a toolpath

with gouge free and maximum angular velocity for 5-axis sculptured surfaces machining.

However, it does not consider the minimum cusp height which is needed for the toolpath

optimization. Although C-space is able to monitor all the possible tool orientations, it

requires lots of computing time to reach the optimal solutions.

Figure 2-9: C-space for orientation parameters. (a) Discretized 2-D orientation space

(white area shows safe orientation space). (b) 3D C-space for one toolpath [43]

2.3 Machine Dynamics

2.3.1 Toolpath and Tool Orientation Optimization by Dynamic Constraints

There are many attempts at optimizing toolpath/orientation with different dynamic

constraints [45-49] such as the velocity stability, cutting forces, feed rates, and torque of

25

a 5-axis machine tool. Farouki [50] proposed an approach to calculate toolpath feed rate

by considering the maximal torque and power of the tool. López de Lacalle [51] used the

prediction of deflection forces as a criterion for the best choice of toolpaths. It provides

the possibility of selecting tool orientations with low deflection forces for geometrical

requirements. Bi introduced an accessibility cone to optimize cutter orientation along

both feed and cross-feed directions [52]. The accessibility cone (shown in Figure 2-10) is

a set of tool orientations from which the cutter contact point is accessed by the cutter

without gouging. This optimization method considers stability of feed velocities and the

smoothness of cutting force at mesh points and only the accessibility cones are needed to

compute, and thus increasing computation efficiency.

Figure 2-10: Accessibility cones on the CC point mesh [52]

2.3.2 Chip Volume in 5-axis CNC Machine

Computing an actual shape of removed material is still challenging [53-56]. There are

some new methods applied to resolve this problem. Sweep volume based on solid method

was introduced by Leuven [57]. Undeformed chip shape can be constructed from the

boundaries of instantaneous engagement domain between a flat-end mill and the

26

workpiece [58]. There are two main approaches to calculate the removed chip volume

[57]: (a) computation of swept volume by the tool profile along NC trajectory and (b)

implementation of the Boolean intersection and subtraction of the tool envelope with the

workpiece. The workpiece based methods to calculate removed material in 5-axis

machining is still challenging due to the non-robust 3D Boolean subtraction operation

and complicated process of updating the workpiece [59]. Sweep volume is a tool

representation method introduced by Weinert [53] using solid modelling technique. A

moving frame in 5-axis tool motions was introduced for the solid sweep volume. Swept

profiles were first generated along the NC trajectory. After moving the profiles, a closed-

form envelope surface was created. It can be seen in Figure 2-11. This solid-based

method can obtain a much more precise cutting volume than the discrete method.

However, the swept profiles are complicated to obtain as the cutter has both translational

and rotational motions.

27

Figure 2-11: Tool motions along a pre-defined trajectory in five-axis machining and

corresponding swept profiles: (a) Cutter geometric definition; (b) Cutter motion track and

swept profiles (red lines); (c) Generated swept volume [57]

Traditionally, the swept volume is obtained approximately by the sum of pure

translational volume and pure rotational volume in 5-axis CNC machining. The results,

even under the specified tolerance, are not exactly equal to the removed cutting volume.

The depth of cut has not yet been included in the method, since swept volume only

considers the top, bottom, and side of a milling cutter. Lee proposed a method to generate

swept volume of a tool by calculating envelope profiles with Gauss map [60]. Yet, the

approach is only applicable to convex set with piecewise C1–continuous motion. The

trajectory of tool motions in the swept volume method is piecewise C1–continuous or

smoothness. It requires that the first order derivative of the trajectory exists and is

continuous. If the tool motion is smooth, the velocity of the tool can be used to get the

Ball-end

Flat-end mill

APT-like cutter

Bull-nose mill

28

swept profile. Otherwise, the swept profile cannot be found. The swept volume method

displays the shape of removed material and can be used for NC verification. However, it

cannot produce the value of chip volume and chip thickness at each NC point to calculate

cutting forces and select optimal cutting parameters. On the other hand, Ferry [61]

generated a swept volume by collecting solid models of the tool together at various NC

points along the tool trajectory. The swept volume was subtracted from the workpiece to

get the finished part. The Parallel Slicing Method (PSM) was used by Ferry to create

cutter-workpiece engagement maps for 5-axis flank machining, with the information of

engagement angles and depth of cut, which is the requirement for predicting cutting

forces. The PSM can obtain the removed volume; however, it is a computationally

inefficient approach to do Boolean operations for achieving the solid model of cutter-

workpiece engagement.

2.3.3 Cutting Force in 5-axis CNC Machine

Predicting the cutting force is significant in the process planning process [11, 33, 49,

62-64]. Cutting volume is the total chip volume of each CC point. Chip thickness is an

important factor to get chip volume and cutting forces. The most popular analytical

method to calculate chip thickness in ball-end milling is the sine product assumption [65]

in which the chip thickness t is simplified as t=f×sinϕ×sink, f is the feed per tooth, ϕ is

the immersion angle, k is the rotation angle along the tool axis. However, this method

causes model errors in axial and tangential directions, especially for small depth of cut

within 10% of cutter radius [66].

Tao Huang [67] proposed that the chip thickness can be calculated by the sum of two

individual cutting conditions with only tilt or lead angle, seen in Figure 2-12.

29

Figure 2-12: Tool engagement regions and decomposed motion [67]

Ferry and Altintas [68] presented a method to compute the chip thickness in 5-axis

flank milling by distributing the chip thickness into horizontal and vertical feed

components, which can be seen in Figure 2-13. B. Ozturk [69] mentioned the boundaries

of engagement regions of the ball-end mill and the workpiece to predict cutting forces

more accurately. However, those methods do not calculate the chip thickness considering

both tilt and lead angles.

Figure 2-13: Distribution of chip thickness (a) Horizontal feed; (b) Vertical feed [68]

30

For 5-axis CNC machine, cutting forces and cutting volume depend on two rotational

angles, feedrate, depth of cut, toolpath, chip thickness, cutting coefficients, and entry and

exit angles. Chip thickness and cutting forces prediction for 5-axis CNC ball-end milling

has been studied by many researchers [70]. Harshad [71] proposed an analytical method

to predict an uncut chip geometry including chip thickness, length, and width by

instantaneous shear angle in a ball-end milling process. Cutting forces were predicted

considering strain and temperature and shear strength by the Johnson-Cook material

model [71]. Bouzakis [72] developed an algorithm to consider the machining surface

topography, the chip formation, cutting forces, and the corresponding cutting tool

deflection with ball-end mills. Various cutting parameters as surface roughness, feedrate,

radial depth of cut, and tool axis inclination angle were investigated to get chip geometry

and cutting forces. Zhang [73] used the Dexel approach to get cutter-workpiece

engagement for chip thickness and cutting force calculation by finding start and exit

angles of discs through the spherical part of the tool. However, very limited studies on 5-

axis CNC machining using flat-end mills have been carried out [74], due to the

complexity of cutter-part surface geometry interaction. A ball-end mill has constant

curvature, therefore chip geometry is easier to obtain, while the curvature for a flat-end

mill at each cutter contact (CC) point varies with different tilt and lead angles during 5-

axis machining. Thus, the discrete method for chip volume calculation with a flat-end

mill is much more complicated.

31

Optimization of 5-Axis CNC Toolpath and Cutter Chapter 3:

Orientation for Machining Free-form Surfaces

An early version of the work proposed in this chapter was presented at the 2014 Virtual

Machining Process Technology conference:

Shan Luo, Zuomin Dong, and Martin B. Jun. Optimization of 5-axis CNC toolpath and

cutter orientation for machining curved surfaces. In Virtual Machining Process

Technology Conference, 2014 [16].

The work presented in this chapter has improved the method further by adding the

process to optimize the number of surface patches.

The original intent of this research was to generate a new 5-axis CNC toolpath and

cutter orientation planning method using fuzzy clustering technique to reduce the range

of the rotational axes’ motions and to avoid sharp cutter orientation changes. A generic

curved surface with half spherical shape and a free-form surface were divided into

patches using the combination of fuzzy cluster method and surface normal variation

control method, discussed in Section 3.1. Fuzzy cluster method is used to get the number

of cluster centres or surface patches; it is discussed in Section 3.1.1. Surface patch

generation is discussed in Section 3.1.2 by selecting surface grid points with similar

surface normal vector distances and creating an accessibility cone area with a set of

surface normal vectors. The optimal number of surface patches or surface point clusters is

identified and discussed in great detail in Section 3.2. Surface patch boundaries using

computational geometry method is introduced in Section 3.3.

32

In this research, a method to reduce the range of the two axes of rotation motions is

adopted. 5-axis CNC toolpaths are generated for different portions of the machine

surface, in which the efficient tool orientation has limited variation. This is generated by

dividing the surface into multiple patches in which all machined points share similar

surface normal directions. The method is accomplished through the application of the

fuzzy clustering method. In earlier research, Chen has used the fuzzy clustering method

to generate 3 ½ ½ axis CNC toolpath [75], where the surface is divided into some easy-

to-machine surface patches by cluster points. For each surface patch, the rotational axes

are set up and fixed by the surface normal direction at cluster point. So it is easy if each

surface patch has the same tool orientation. Once the rotational axes are fixed, the static

and dynamic stiffness are increased as a rigid body without the bearing connection.

Fuzzy clustering method used in 3 ½ ½ -axis CNC machine can be improved in 5- axis

CNC machines. Although the machine stiffness is stronger and the cost is cheaper, it is

time-consuming in 3 ½ ½ -axis machining, due to setups of the rotation axis for different

surface patches machining. To increase machine efficiency, 5-axis CNC machines can be

used in the idea of surface patches division. In 5-axis CNC machining, surface patches

are not only generated according cluster points but also similar surface normal vectors at

all machined points in one surface patch.

3.1 Machining Surfaces Patch by Patch Using the Fuzzy Cluster Method

In this work, a mesh of discrete grid points is used to represent a curved surface

through a half sphere surface example. The grid points are clustered into an adequate

number of regions by the fuzzy clustering method. The fuzzy “similarity” of two surface

points is defined by the weighted combination of a difference of their surface normal

33

direction and their physical distance or Euclidean distance. The new method divides the

machined surface into patches with limited surface normal variation through three steps:

a) finding cluster centres using Fuzzy C-means clustering (FCM) method; b) identifying

the most similar cluster centre for all surface grid points to form the surface point cluster;

and c) mixing boundary points between two neighbouring surface patches.

3.1.1 Fuzzy C-means Clustering Method

Fuzzy C-means clustering (FCM) is an iterative optimization algorithm to minimize

the objective cost function of J [76]:

2

2

1 1 1 1

( , ) ( ) ( , ) ( )n c n c

m m

ij i j ij i j

i j i j

J u V u d X V u X V

(3.1)

where n and c are the number of data points and cluster centres, respectively; uij

represents the membership of ith

data to jth

cluster centre; m is the fuzzy partition matrix

exponent index; Vj is the jth

cluster centre as shown in Figure 3-1; d (Xi, Vj) is the

Euclidean distance between the ith

data point and jth

cluster centre. X =(x, y, z) represents

the location of a data point in Cartesian coordinates, and V = (u, v, w) represents the

location of a cluster centre. The distance between a data point at ith

data point and

the jth

cluster centre in 3D is defined by:

2 2 2

( , )i j i j i j i jd X V x u y v z w

(3.2)

34

Figure 3-1: Surface cluster centres and relative angles of surface normal vectors

This work utilized the Fuzzy Clustering Toolbox in MATLAB, with predefined

parameters such as the number of cluster centres, the fuzzy partition matrix exponent m,

maximum number of iterations, and minimum amount of improvement. Cluster centres

for a NURBS surface are generated in the Figure 3-2 (a). The output of fuzzy C-mean

algorithm is a matrix of cluster centre coordinates. After that, the cluster centres are put

into the 3D surface data set and their surface normal are then obtained to generate surface

patches. The 3D cluster centres of the NURBS surface and their surface normal vectors

are displayed in the Figure 3-2(b).

35

Figure 3-2: (a) The 2D distribution of cluster centres for a NURBS surface in the Fuzzy

Clustering Toolbox; (b) The demonstration of cluster centres and their surface normal in

3D in MATLAB

3.1.2 Generation of Surface Patches by Surface Normal Vector Distances

After cluster centres are identified, all surface mesh points are classified by the

weighted combination of difference of their surface normal direction d(Yi, Wj) and

physical distance d(Xi, Vj). The formula of the combined distance is defined by:

1 1 1 1

[ ( , ) (1 ) ( , )] [ (1 ) ]n c n c

n c i j i j i j i j

i j i j

D d Y W d X V Y W X V

(3.3)

Let Y= (x′, y′, z′) be surface unit normal vector of a mesh point in 3D space, it can be seen

in the Figure 3-1. W = (u′, v′, w′) is surface normal vector of a cluster centre. The distance

between surface normal vectors at the ith

data point and the jth

cluster centre is given by:

2 2 2

( , )i j i j i j i jd Y W x u y v z w

(3.4)

The weighting factor α provides an effect on the clustering. The value of α is varied with

different surface shapes. For instance, α is equal one in a half sphere surface. But for a

36

curved surface with parallel surface normal, the physical distance is considered as well,

as surface normal distance does not work very well for the clustering.

In the combined distance matrix Dn×c, entries in the ith

row are distances of surface

normal vectors between a mesh point and the jth

cluster centre. The minimal distance in

the ith

row is found first by Eq. (3.5) and its index represents the cluster centre which is

closest to the ith

data point. According to this method, all data points can get their own

closest cluster centre. In other words, each cluster centre has its data points obtained by

the shortest combined distance.

1

min( )n

ic

i

c D

(3.5)

where, Dic is the ith

row of the distance matrix Dn×c; c represents the distance between one

data point and its closest cluster centre. In Figure 3-1, the ith

mesh point Xi is in the jth

surface patch where the cluster centre is Vj. All mesh points will find their closest cluster

centre by minimizing the surface normal vector distance. Different surface patches are

created by adding points that have similar surface normal vector distances. The number

of surface patches or the number of cluster centres need to be given. Figure 3-3 shows the

surface divisions with tool orientations for NURBS surface from 1 cluster to 10 clusters.

Different colors represent different surface patches.

37

Figure 3-3: Surface divisions with tool orientations for a NURBS surface from 1 cluster

to 10 clusters

3.2 Optimization of the Number of Surface Patches

Patch-by-patch machining is used to minimize the motions of the two rotational axes,

avoiding dramatic cutter orientation changes during the 5-axis CNC machining. The more

the surface patches we have, the fewer cutter orientation changes there should be.

However, when there are too many surface patches, the effect of surface patches to cutter

orientation changes is not obvious. Toolpath generation would also become much more

complicated when the number of surface patches is increased. It is thus beneficial to find

an optimal number of surface patches to minimize rotation motions and simplify toolpath

generation.

38

Figure 3-4: Relative angle φ and accumulating relative angle α

Accumulating relative angle can be minimized to find an optimal cluster number. In

Figure 3-4, it shows relative angle φij is the angle of surface normal between two

continuous cutter contact points. It is defined by the cross and dot products of two surface

normal vectors Yi, j and Yi+1, j:

, 1,1

, 1,

tan ( )i j i j

ij

i j i j

Y Y

Y Y

(3.6)

Accumulating relative angle αk is the sum of relative angles in kth

patch, k=1, 2…c, c is

the number of cluster centres. It can be expressed by Eq. (3.7).

1 1

s t

k ij

i j

(3.7)

where, s and t are the number of rows and columns in the kth

patch.

The method to get an optimal cluster number for a curved surface is an iterative

optimization algorithm by minimizing the objective function of the change rate of the

second derivative of accumulating relative angle τk. The second derivative of

accumulating angle indicates its stability of changes. Eq. (3.8) shows that the change rate

τk is defined by the second derivative of four accumulating angles in the first, last, kth

, and

kth

+1 patches. The iteration of optimization will stop when the change rate τk is less than

39

a termination criterion ε which is between 0 and 1. The value of ε is given by the user.

The optimal number of clusters is obtained when the change of accumulating relative

angles is largest and stable. The criterion of stable change is that there are three

consecutive τ under the termination criterion ε.

1

1

k kk

c

(3.8)

Changes of accumulating relative angles with different number of cluster centres and

ith

cluster for NURBS surface are shown in the 3D bar chart in Figure 3-5, using three

variables: accumulating relative angles, the number of cluster centres, and the ith

cluster.

The first row along the axis of number of clusters is the maximum accumulating relative

angles from 1 cluster to 10 clusters. The ith

cluster axis shows the distributions of

accumulating relative angle in each surface patch. When machining the entire surface in

one patch, the maximum accumulating relative angle is around 2300°, whereas the

smallest maximum accumulating relative angle about 400° appears when 10 patches are

applied. The larger the maximum accumulating relative angle we have, the more tool

orientation changes there should be. Compared with the changes of maximum

accumulating relative angle from 6 to 10 cluster centres, it can be seen that there are not

too many changes for the changes of maximum accumulating relative angle as the

number of cluster centres is increased. However, when the number of cluster centres is

less than 6, the maximum accumulating relative angle has a fairly large decrease from

2400° to 400°, while the number of cluster centre increases from 1 to 5.

Figure 3-5 shows the relation of cutter orientation changes and cluster centres from the

changes of accumulating relative angles. However, the optimal number of clusters is not

obvious in the bar chart. The line chart in Figure 3-6 shows 40 maximum accumulating

40

relative angles and their first order and second order derivative change with different

number of clusters. In the blue line with rhombus, the accumulating relative angles

decrease with the increase of number of clusters. But it reduces very slowly after 5

clusters. It has similar results with the 3D bar diagrams: maximum accumulating relative

angles have a sharp reduction as the number of cluster centres increases from 1 to 5.

After 5, the relative angles are changed very slowly.


centres and ith

cluster for a NURBS surface in 3D bar chart.

The first order derivative can show the change of accumulating relative angles. In

Figure 3-6, the red line with rectangles is the first order derivative of accumulating

relative angles and it shows the largest first order derivative happens as the number of

cluster centres changes from one to two. That means cutter orientation changes can be

decreased by the surface patch method. However, the first order derivative cannot

41

indicate how fast the change could be. Therefore, the second order derivative is used to

define the speed and stability of changes to accumulating relative angles.


order derivatives for a NURBS surface

The green line with triangles in Figure 3-6 is the second derivative of accumulating

relative angles. The change rate τk is then obtained by the second derivative of

accumulating relative angles through Eq. (3.8) to find the optimal number of cluster

centres. The optimal number of cluster centres is depended on the termination criterion ε

specified by user. The Table 1 shows the optimal cluster numbers of the NURBS surface

are changed with different termination criterion ε. It can be seen that optimal cluster

number rises as ε decreases. However, after ε reduces to 0.08, the optimal cluster number

remains at 12, which means that it is unnecessary to reduce the termination criterion ε to

get more surface patches, reducing tool orientation changes, when the number is larger

than 12. Therefore, the largest optimal number of cluster centres is 12.

42

Table 1: The relation of optimal cluster numbers and termination criterion ε for a NURBS

surface

ε 1 0.25 0.2 0.15 0.14 0.08 0.05 0.03

Optimal cluster

numbers 1 2 5 5 7 12 12 12

The convex half sphere is another example to verify the surface patch method. Figure

3-7 shows the surface is divided into different patches from 1 to 10. The surface divisions

are almost same for 2 and 3 clusters. It is due to the surface normal at the tip of the half

sphere is zero, causing the number of surface patches is one less than the number of

clusters as it is larger than 2.

Figure 3-7: Surface divisions with tool orientations for a convex half sphere surface from

1 cluster to 10 clusters.

A 3D bar chart in Figure 3-8 is made to show the changes of accumulating relative

angles with different number of cluster centres and ith

cluster for the convex half sphere.

43

In this example, the half sphere surface is divided into different numbers of patches. The

largest number of surface patches is 30.


centres and ith

cluster for a convex half sphere surface in 3D bar chart.

The line chart in the Figure 3-9 shows the maximum accumulating relative angles and

their first and second derivative changes for the half sphere surface. From the line of

maximum accumulating relative angles, it can be seen that there are steady changes of

maximum accumulating relative angles after 10 clusters.

44


order derivatives for a convex half sphere surface.

The same method is used to get the optimal number of cluster centres for the half

sphere by the second order derivative of maximum accumulating relative angles. The

Table 2 shows different optimal cluster numbers are varied with different termination

criterion ε. For instance, the optimal number is 10 as the value of ε is 0.2, which can be

seen from the line chart of changes for maximum accumulating relative angles in the

Figure 3-9. The optimal cluster number is no longer increased as the termination criterion

ε is decreased to 0.04. It means the largest optimal cluster number is 22.

Table 2: The relation of optimal cluster numbers and termination criterion ε for the

convex half sphere surface

ε 0.3 0.2 0.15 0.08 0.04

Optimal cluster numbers 4 10 17 22 22

45

3.3 Optimal Toolpath Generation

3.3.1 Surface Patch Boundary Definition by Alpha Shape

After clustering the mesh data, boundary points between two neighbouring surface

patches are mixed to make regions jointed seamlessly. However, boundaries of patches

are not defined. To generate closed and smooth boundaries, the method of Alpha Shape is

used to find and connect the mesh points on the border of each surface patch.

Figure 3-10: Surface patch boundaries generated by the alpha shape method with

different probe radius and boundaries shown in 2D and 3D for a convex half sphere.

Alpha Shape method can identify the mesh points on the border of one surface patch

according to a probe radius [77], denoted by R. Delaunay triangulation is carried out first.

And then the radius of circumcircles of simplices is specified in triangulation.

Circumradius of simplices, which is less than the probe radius R, is selected to get valid

46

vertices of free boundary facets that are inside the probe radius. After that, free boundary is

used to get the coordinates of boundary points which are the vertices of the free boundary

facets. Finally, an arc over the surface of a sphere between two points is adopted to

connect boundary points and create boundary lines.

The value of the probe radius R is found by trial-and-error method. When R is infinite,

the basic Alpha Shape is a convex hull, which can be seen in Figure 3-10. The smaller the

probe radius the more precise the boundary should be. But the value of R cannot be too

small, or it cannot include all grid points for the surface patch.

3.3.2 Toolpath Generation

The surface normal vectors at cluster centres are calculated and shown with arrows in

the Figure 3-11 (b). There are five cluster centres to create four surface patches due to the

surface normal at the top of the sphere being zero; therefore, no surface patch is created

for the cluster centre at the top of the sphere. After surface divisions, the iso-parametric

toolpath is generated for each surface patch. Tool orientations are decided by surface

normal at each mesh point. In this convex surface example, the surface curvature and the

tool curvature are opposite, no gouging would be generated, and the most efficient cutter

orientation is along the surface normal direction. Figure 3-11 (b) shows the iso-

parametric toolpath for one surface patch. Toolpaths for other patches can be generated

similarly.

47

Figure 3-11: (a) 5 cluster centres of a convex half sphere generated by the clustering

toolbox; (b) Toolpath generation for one surface patch

3.4 Conclusions

An optimal toolpath by machining a surface patch-by-patch with points of similar

surface normal orientations is presented to reduce the range of rotation motion and avoid

sharp cutter orientation changes. This method is based on the fuzzy clustering technique

and the similar surface normal variation control method. The optimal number of surface

patches is identified considering both changes of accumulating relative angles to

minimize the two rotation motions and simplify toolpath generation. Two types of

surfaces are demonstrated for surface divisions and the identification of optimal number

of surface patches. Furthermore, Alpha Shape method based on the probe radius is used

to define patch boundaries. The Iso-parametric CNC toolpath is generated due to its

simplicity and surface normal vectors are the most efficient cutter orientations for convex

surfaces.

48

Optimal Tool Orientation Generation Chapter 4:

This chapter presents an optimal and flexible tool orientation methodology based on the

combination of Euler-Meusnier spheres (EMS) method and surface normal in 5-axis

CNC free-form surface machining using a flat-end mill in Sections 4.1 and 4.2. The

largest cutting edge and best curvature match are considered to achieve maximum

removal material and better surface quality. Tool orientations generated by the EMS

method depend on surface principal curvatures. The motivation of principal curvature

calculations, discussed in Section 4.1.1, is to get the two cutter rotational angles

illustrated in Section 4.1.2. A NURBS surface with concave, convex, and saddle features

is applied in Section 4.2 to show how proper tool orientation methods are selected to

avoid gouges and improve machining efficiency. Typically, EMS is a method for concave

surfaces to avoid local gouging by matching the largest cutter Euler-Meusnier sphere

with the smallest Euler-Meusnier sphere of the surface at each cutter contact (CC) point.

Surface normal is the most efficient tool orientation approach for convex surfaces, due to

the largest Euler-Meusnier sphere is generated at the surface normal direction without

generating gouges. Selection of one of these methods in tool orientation determination for

saddle shapes is based on the direction of the CNC toolpath in relative to the surface

curvature change.

4.1 The Euler-Meusnier Sphere (EMS) Method for Tool Orientation in a

Concave Surface

The EMS method provides a generic local solution for gouge detection and

elimination in sculptured surface machining. In Figure 4-1, it can be seen that if the

49

Meusnier sphere of cutter is larger than that of the workpiece, gouge would be generated.

The criterion of gouge free is that match the largest Euler-Meusnier Sphere of the cutter

into the smallest Euler-Meusnier Sphere. Wang [82] used this method to avoid local

gouging only for concave surfaces. In this research, a new toolpath would be generated

based on the combination of the EMS method and surface normal for both concave and

convex surface shapes. To avoid gouging, the EMS method would be used for tool

orientations. In a surface patch, the iso-parametric toolpath is then generated due to

simple approach.

For convex shapes, the surface curvature and the tool curvature are opposite (shown in

Figure 4-1), therefore no gouging is generated, and the most efficient cutter orientation is

along the surface normal direction. This is due to the cutter Meusnier sphere being the

biggest along the surface normal direction, producing more removal material. For

concave surfaces, if the Meusnier sphere of the cutter is larger than that of the workpiece,

there is gouge generated.

The EMS method is applied to get ideal cutter orientations for concave surfaces. The

surface normal variable control method is used to cutter orientations due to its high

efficiency for convex surfaces.

50

Figure 4-1: Machined surfaces and cutter Meusnier sphere

4.1.1 Principal Curvature Calculation for a NURBS Surface

Curvature is used to describe how a surface changes its shape. Given a point on a

surface, there are many normal curvatures at this point in various directions. The

principal curvatures are the extremal curvature values, which are denoted by kmin and

kmax. The maximum and minimum principal curvatures (kmin and kmax) are perpendicular.

Both of them depend on the first and second partial derivatives of the surface. In

mathematical terms, the directions and values of principal curvatures are the eigenvectors

and the corresponding eigenvalues of the symmetric linear map LP, which is based on the

first and second fundamental forms.

In differential geometry, the first fundamental form is the inner product on the tangent

space in 3-D Euclidean space [83]. For a surface S (u, v), the first fundamental form is

denoted by I.

51

2 2I 2Edx Fdxdy Gdy (4.1)

2

2

,

, ,

,

u u u

u v v u

v v v

E S S S

F S S S S

G S S S

(4.2)

Su and Sv are two tangent vectors on tangent space.

,u v

S SS S

u v

(4.3)

The surface unit normal vector n is:

u v

u v

S Sn

S S

(4.4)

The coefficients of second fundament form at a given point are obtained by projections of

second partial derivatives of S onto the normal line. They can be expressed by:

L = Suu ∙ n,M = Suv ∙ n, N = Svv ∙ n

(4.5)

The matrixes of first and second fundamental form in the basis (Su, Sv) of the tangent

plane are I and II, respectively.

I= , II=E F L M

F G M N

(4.6)

A new matrix LP called shape operator is formed to get the principal curvatures.

1I IIPL

(4.7)

where, I-1

is the inverse matrix of I. The directions and values of principle curvatures are

the eigenvectors and eigenvalues of the shape operator LP.

Gaussian curvature and mean curvature are denoted by K and H, given by the following

equations:

2

2 2

2,

2( )

LN M EN GL FMK H

EG F EG F

(4.8)

52

The maximum and minimal principal curvature Kmax and Kmin are obtained from Gaussian

curvature and mean curvature:

2

max

2

min

K H H K

K H H K

(4.9)

The principal directions for maximum and minimal principal curvature are Kdmax and

Kdmin, which expressed by:

max min

max min

d d

d d

EN GLK K

FN GM

EM FLK K

FN GM

(4.10)

4.1.2 Two rotation Angles Identification

The criterion of the EMS method to avoid gouges is to match the largest cutter

Meusnier sphere with the smallest Meusnier sphere of the surface at each CC point. In

Figure 4-2, P is the CC point, O is the centre of smallest Meusnier sphere of the

machined surface, A is the bottom centre of the flat-end mill; n is the surface normal

vector; 𝐭 is the axis direction of the tool; OP is the radius of the smallest Meusnier Sphere

of the workpiece, which is the reciprocal of the largest surface curvature at the CC point

P, denoted by R; AP is the radius of the cutter, which is given as r. From the geometry of

cutter and Meusnier sphere, the inclinational angle α can be obtained from the following

equation:

1sin ( )

r

R

(4.11)

The radius of the smallest Meusnier sphere R is determined by the maximum surface

curvature. Therefore, the surface maximum principle curvatures are required to get the

53

inclination angle α. Radius of curvature is the reciprocal of the surface curvature at each

cutter contact point. The principal curvatures are eigenvalues of the shape operator LP

(given in the Eq. (4.7)). From Eq. (4.9), it can be seen that the maximal and minimal

principal curvatures can be also calculated by Gaussian and mean curvatures.

Figure 4-2: Inclination angle α confirmation

The lead angle α has been determined by the largest surface curvature. On the other hand,

for better curvature match and machining efficiency, the tool axis should be in the plane

A which is defined by the smallest principal curvature direction of the surface and the

surface normal at the cutter contact point O, shown in Figure 4-3.

54

Figure 4-3: Tool orientation in the Meusnier sphere method

From the relation of the tool axis and surface normal and smallest principal curvature

direction shown in the Figure 4-4, it can be seen that the tool axis direction can be

obtained once the minimal principal curvature direction and surface normal at cutter

contact point O are confirmed. In the Figure 4-4, OC is the surface normal, denoted by n,

OD is the minimal principal curvature direction at the point O, denoted by Kdmin, OE is

the tool axis expressed by t. t is obtained by the Eq. (4.12).

min( ) ( )dOE OC CE OC OD OC n K n

(4.12)

Figure 4-4: The relation of tool axis with the surface normal and the smallest principal

curvature direction.

55

4.2 Optimal Tool Orientation

A NURBS surface illustrated in the Figure 4-5 is used in the paper to show optimal

tool orientations in different surface features. To find geometric parameters such as

surface points, principal curvatures, and surface normal, the mathematical model of a

NURBS surface is required. The surface equation is represented as [84-86]:

1 1

, , , ,

1 1min max min max1 1

, , ,

1 1

( ) ( )

( , ) ( , )

( ) ( )

l m

i j i j i k j l

i i

l m

i j i k j l

i i

h P N u N v

S u v u u u v v v

h N u N v

(4.13)

where, u and v are two independent parameters, Pi, j are the x, y, z coordinates and hi, j are

a set of (l+1) by (m+1) control points in the homogeneous coordinates; Ni, k and Nj, l are

the blending function in u and v directions.

Figure 4-5: A 3D NURBS solid model with concave, convex, and saddle shapes.

Surface shape is identified using Gaussian and mean curvatures which are given in Eq.

(4.8). The machined surface is roughly divided into concave, convex, and saddle shapes

by Gaussian curvature and mean curvature. The relationship between surface features and

curvatures can be seen in the Table 3.

56

For a concave shape, Gaussian curvature is positive and mean curvature is negative;

for a convex shape, both Gaussian and mean curvatures are positive. A saddle shape is

special with curves up in one direction, and curves down in a different direction. It means

saddle points can become concave points and convex points by different machining

directions. For instance, in Figure 4-6, a NURBS surface consists of three surface

features: concave (cyan squares), convex (pink stars), and saddle (blue circles). It can be

seen that saddle points in Figure 4-6 (a) are concave points if machining the surface along

u direction and they are convex ones if machining along v direction.

Table 3: Relationship of surface features, curvatures, gouging and the tool orientation

methods

Surface

features

Gaussian

curvature

CGaussian

Mean curvature

Cmean

Gouging

possibility

Tool orientation

methods

Concave CGaussian>0 Cmean<0 Certain EMS

Convex CGaussian>0 Cmean>0 Impossible Surface normal

Saddle CGaussian<0 Cmean<0/Cmean>0 Uncertain EMS/Surface

normal

Figure 4-6: (a) Divisions on grid points of the NURBS surface in 3D; (b) Surface features

in 2D

57

The tool orientation methods are then selected once the surface features, curvatures,

and machining direction are confirmed. For convex shapes, surface normal at each

convex point is the best choice for the tool orientation with the highest machining

efficiency and without gouges generation. For concave shapes, the Euler-Meusnier

Sphere (EMS) method is used to avoid gouging problems for flat-end mills. Tool

orientations for saddle shapes can be applied to the surface normal variable control

method and the EMS method depending on the selected machining direction.

Figure 4-7 (a) shows optimal tool orientations by the combination of the EMS and

surface normal methods. Saddle points are considered as concave points as the toolpath is

along u direction. In Figure 4-7 (b), black arrows denote new tool orientations. It may be

surface normal or the tool axis obtained from Eq. (4.12) in the EMS method. Red arrows

represent surface normal vectors and blue arrows are the minimal principal curvature

directions.

Figure 4-7: (a) Optimal tool orientations for the NURBS surface; (b) Display of the new

tool orientations, surface normal, and minimal surface curvature directions

58

4.3 Conclusions

This chapter presents an optimal tool orientation method in a 5-axis CNC machine

using a flat-end mill. The optimal tool orientation is obtained by the combination of the

Euler-Meusnier Sphere (EMS) method and the surface normal variable control method to

avoid gouges and improve machining efficiency. The EMS method is applied to concave

parts to avoid local gouges by matching the largest Euler-Meusnier Sphere of the cutter to

the smallest Euler-Meusnier Sphere of the machined surface. The highest efficient tool

orientations for a convex surface are along surface normal directions. For saddle surfaces,

the EMS method or the surface normal variable control method is selected by the selected

machining direction.

59

Chip Volume and Cutting Force Calculations in 5-axis Chapter 5:

CNC Machining of Free-form Surfaces Using Flat-end Mills

An early version of the work proposed in this chapter was presented at the 2015 Virtual

Machining Process Technology Conference:

Shan Luo, Zuomin Dong, and Martin B. Jun. Chip volume calculation and simulation in

5-axis CNC machining with flat-end mills. In Virtual Machining Process

Technology Conference, 2015 [17].

The work presented in this chapter has improved the technology further by exploring a

new approach to calculate cutting forces.

Optimal feed rate enables to maximize removed material. Chip volume and cutting

force predictions will be introduced in this chapter to optimize the last remaining

planning variable feed rate. This chapter starts by describing formulation of a swivel head

5-axis CNC tool motion in section 5.1. Section 5.2 presents two convergence studies to

show how chip volume is calculated. A computational geometry-based Alpha Shape

method is applied to resolve the 3D cutter-workpiece intersection problems, solid chip

shape, and calculate chip volume, which were discussed in Section 5.2.1. Extending the

Alpha shape method that can only predict cutting chip geometry, a parallel slice local

volume modeling approach has been added to predict cutting forces as well. It is

discussed in Section 5.2.2. The method is introduced to obtain the chip thickness and

cutting forces within the cutter-workpiece engagement zone, discussed in Section 5.2.2.2.

The modeled cutting chip introduced in Section 5.2.2.1 is sliced into a number of parallel

planes which are perpendicular to the tool axis. The intersections of the cutter and the

60

workpiece are obtained by accumulating intersections of two ellipses on each of these

slices, which are discussed in Section 5.2.2.3. In Section 5.3, the cutting flutes entering

and exiting to the workpiece and the depth of cut are obtained to predict the cutting

forces. The method can be applied to various machine configurations, thus providing an

efficient and robust method for calculating chip volume and cutting forces for arbitrary

tool orientations. The new approach also considers depth of cut and scallop height

between two adjacent toolpaths.

To demonstrate the validity and capability of these new methods, simulation of the

cutting and chip forming process of 5-axis CNC machining on a free-form surface has

been carried out, shown in Section 5.4. Cutting force predictions are made for Al 6061

with a two-flute carbide flat-end cutter. In Section 5.5, physical validation experiment in

controlled conditions has been carried out on a 3-axis micro CNC machine with the two

cutter rotation angles set to be zero. The predicted and measured cutting forces are in

reasonably good agreement both in trend and magnitude. Uniform interpolation was

applied at two continuous NC points by a distance of feed per tooth to reduce modeling

error and get more precise cutting forces. The presented chip volume and cutting force

method can be used to perform cutting force estimations for generating optimal toolpath

and orientation during 5-axis milling. These approaches are applicable in arbitrary tool

orientations and consider depth of cut. Compared to the analytical swept volume method,

they are not restricted to be applied only in piecewise C1–continuous motions. The local

parallel sliced method requires longer computational time than traditional analytical

methods, but it supports the ultimate goal of chip modeling and chip volume calculation,

which results in accurate dynamics cutting force prediction.

61

5.1 Formulation of Swivel Head 5-axis CNC Tool Motion

For 5-axis CNC machines, there are different kinematic configurations, such as swivel

head and rotary table [78]. The swivel head consists of two types of structures C-A and

C-B (A, B, and C are the rotational axes about the x, y, and z axes, respectively). In this

work, a swivel head 5-axis CNC machine is considered to study tool motions and chip

volume generations. As the kinematic transformation usually varies with different

machine configurations, a 5-axis CNC machine with a swivel head configuration, where

the C-axis is primary and the A-axis is secondary, is considered. As shown in Figure 5-1,

the local coordinate system (Ol, Xl, Yl, Zl) is at the cutter contact CC point Ol. The

normalized projection of cutter feed direction is denoted by Xl, Zl is the unit normal

vector of the surface, Yl is cross-feed, and Yl= Xl × Zl. The trajectory of the tool moving

along two NC points PCAM and PCAL in the machine coordinate system (MCS) and local

coordinate system (LCS) can be transformed using homogeneous translation and rotation

matrix multiplication as in Eq. (5.1):

( , , ) ( , ) ( , )CAM CAL

P Trans x y z Rot z Rot x P

(5.1)

The homogeneous transformation for the translation by Δx, Δy, Δz in the x, y, z direction

is denoted by Trans(Δx, Δy, Δz) and the rotation about z and x axes are denoted by Rot(z,

β), Rot(x, α), respectively. Replacing PCAM, PCAL, Trans(Δx, Δy, Δz), Rot(z, β) and Rot(x,

α) with their elements allows Eq. (5.2) to be expressed as:

1 0 0 cos sin 0 0 1 0 0 0

0 1 0 sin cos 0 0 0 cos sin 0

0 0 1 0 0 1 0 0 sin cos 0

1 0 0 0 1 0 0 0 1 0 0 0 1 1

CAnew

CAnew

CAnew

X x X

Y y Y

Z z Z

(5.2)

For a flat-end mill, the cutter geometry is a cylinder, which can be represented as follows:

62

cos

sin

X r

T Y r

Z z

(5.3)

where, r is the tool radius, θ is the tool rotation angle about z axis, θ ∈ [0, 2π], z ∈ [h1,

h2], and the length of the tool is defined by h1 and h2.

Figure 5-1: The tool motion in the local coordinate system and illustration of rotation

angles.

5.2 Chip Volume Calculation

Maximum chip volume is a goal in machining planning processes, since the larger

chip volume is removed, the higher efficiency would be generated. Total chip volume is

calculated by local and global methods. Locally, the chip model should be generated by

the tool geometry, depth of cut, and feed rates. Chip thickness is an important parameter

to get chip volume. Globally, for each CC point, the entry and exit angles are varied in a

63

sculptured surface. For a flat-end mill, the chip thickness is constant along axis direction

in three-axis CNC machining. It becomes much more difficult in 5-axis CNC machining.

Due to the inclination and ration angles, the contact area between the cutter and part

surface changes all the time, causing challenges to get the chip thickness and engagement

area in cutting process. In this chapter, two approaches, the Alpha Shape method and the

local parallel sliced method, are presented to calculate chip volume.

5.2.1 The Alpha Shape Method

An Alpha Shape defined in computational geometry is a family of piecewise linear

simple curves in the Euclidean plane associated with the shape of a finite set of points

[79]. The Alpha Shape associated with a set of points is a generalization of the concept of

the convex hull. The Alpha Shape method is a well-established technique in

computational geometry for triangulation, boundary and area/volume of an Alpha Shape

[77]. In this work, the 3D Alpha Shape method is used to model a chip shape and

calculate the chip volume during the 5-axis milling. The model requires the chip

boundary defined by a valid tool geometric outline at two continuous NC points.

5.2.1.1 Intersections of two ellipses

In defining the Alpha Shape chip profile, the geometric outline of the cutting tool is

formed by the intersections of the tool at two adjacent cutter-contact (CC) positions along

the feed direction. Assuming the flat-end mill can be modeled as a cylinder, the chip

geometry in 5-axis CNC machining can thus be modeled through the intersection of two

cylinders at two continuous cutter-contact points, and the volume of the two intersected

cylinders can be obtained. Since it is difficult to calculate the intersections of two

64

arbitrary cylinders using close-form analytical model through translations and rotations, a

numerical method has been used in this work to obtain the intersections of two arbitrary

cylinders by dividing them into many thin layers along the z axis direction. A vertical

cylinder oriented at an arbitrary angle produces a projection as an ellipse onto a plane, as

shown in Figure 5-2. Therefore, the projections of two intersecting cylinders are two

intersecting ellipses on each layer (shown in Figure 5-2). Intersections of two ellipses are

then accumulated by layers, consisting intersections of the two cylinders.

Figure 5-2: Intersections of two ellipses for a tool at two continuous NC positions

From Eq. (5.2), the following can be obtained:

cos sin cos sin sinCAnewX X Y Z x

(5.4)

sin cos cos cos sinCAnewY X Y Z y

(5.5)

sin cosCAnewZ Y Z z

(5.6)

65

Assuming the plane z=0 is the top of the workpiece, the plane z= h1 is the bottom of the

tool at a NC point. After transformation, the projection of a cylinder is ellipse on the

plane Z=h (h∈ [h1, 0]). For the ellipse, all z values are given as h. Substitute ZCAnew=h

into Eq. (5.6), it can be obtained that:

sin

cos

h Y zZ

(5.7)

From Eqs. (5.4), (5.5) and (5.7), the x and y values of points on the ellipse are derived as:

sin

cos sin cos sin sincos

ellipse

h Y zX X Y x

(5.8)

sinsin cos cos cos sin

cosellipse

h Y zY X Y y

(5.9)

For the ith

and ith

+1 NC points, they are denoted by (xi, yi, zi, αi, βi) and (xi+1, yi+1, zi+1,

αi+1, βi+1), respectively. Therefore translation steps Δxi, Δyi, Δzi can be formulated as:

Δxi= xi+1 - xi, Δyi= yi+1 - yi, Δzi= zi+1 - zi

(5.10)

Substituting Eq. (5.3) into Eqs. (5.8) and (5.9), it can get the ellipse function at the ith

NC

point,

sin sin

cos cos sin sin cos sin sincos

i i

iellipse i i i i i i

i

ih r zX r r x

(5.11)

sin sincos sin sin cos cos cos sin

cos

i i


i

ih r zY r r y

(5.12)

Using Eqs. (5.11) and (5.12), an ellipse on the plane z=hi during any tool motion can be

formed. Between two continuous tool motion points, the removed chip area is the

intersect area of two ellipses along moving direction. As shown in Figure 5-2, the ellipse

on the left is the intersections of the tool surface and plane z=0 at the ith

NC point, and

the ellipse on the right is the ith

+1 ellipse.

66

5.2.1.2 Volume calculation by the Alpha Shape method

The Alpha Shape method defines the volume of a basic Alpha Shape for a set of 3D

points by Delaunay triangulation, according to a probe radius, denoted by α. More

specifically, Delaunay triangulation is carried out first to a 3D point set, and then the

radius of circumcircles of simplices in triangulation is identified. If the radius of

circumcircles is less than the probe radius, valid vertices of free boundary facets are

found. Finally, the volume of Alpha Shape can be calculated from valid vertices. Solid

Alpha shapes created by Delaunay triangulation are composed by many tetrahedrons. The

volume of an Alpha shape is obtained by accumulating volumes of all tetrahedrons which

are part of the parallelepipeds. The volume of a tetrahedron can be obtained by the

volume of a parallelepiped.

Figure 5-3: Tetrahedron in a parallelepiped

In the Figure 5-3, the volume of a 3D parallelepiped has been given by the scalar triple

product of three vectors defined by four vertices A, B, C, D as follows:

( )parallelepipedV AD AB AC

(5.13)

The volume of a tetrahedron (consisted by orange lines in the Figure 5-3) is:

67

6

parallelepiped

tetrahedron

VV

(5.14)

5.2.1.3 The algorithm of chip volume calculation

ALGORITHM (Generation of chip volume between four NC points in two continuous

toolpaths)

Input:

Γ{Ci, j, Ψi, j}, Γ{Ci+1, j, Ψi+1, j}, Γ{Ci, j+1, Ψi, j+1}, Γ{Ci+1, j+1, Ψi+1, j+1}: four NC points in

two continuous toolpaths from NC-data. In the jth

toolpath, Ci, j and Ci+1, j are the ith

and ith

+1 cutter contact (CC) points; Ψi, j and Ψi+1, j are their corresponding rotational

angles. Ci, j+1 and Ci+1, j+1 are the ith

and ith

+1 CC points in the jth

+1 toolpath; Ψi, j+1

and Ψi+1, j+1 are their corresponding rotational angles.

R: the tool radius

h1: the tool height

α: the probe radius

Output:

Chip volume and all profile points for the Alpha Shape method to create the shape of a

removed chip.

Step 1: Define coordinate frames

Generate a cylinder with an arbitrary axis from Eq. (5.3).

Step 2: Divide four intersecting cylinders by layers to get their intersections

Three cases are considered for calculating chip volume from machining a complex

surface, as shown in Figure 5-4.

68

Case 1 is about the tool motion in the first toolpath. The bottom of a cutter is not

totally used to remove material due to the tool compensation. A chip area can be

generated by moving the tool in a distance of feed per tooth along a feed direction. As

mentioned above, in 5-axis CNC machining, the projection of a flat-end mill in a plane

which is parallel with the plane XOY is an ellipse. Therefore, the geometry of a chip

area in the first toolpath is constituted by the tool surface at two continuous NC points

and one edge of the workpiece, shown in Figure 5-5 (a). As seen in Figure 5-5 (b), the

chip area P1P2P3 is a half crescent area. P1 is the intersection of tool projections at two

continuous NC points Γ{Ci, Ψi}, Γ{Ci+1, Ψi+1} on the plane Z=0, P2 and P3 are

intersections of the tool projections and the workpiece edge denoted by OX.

Figure 5-4: Three cases for machining a free-form surface

69

Figure 5-5: (a) Tool simulation in Case 1 of the first toolpath machining; (b) The chip

area for the first toolpath on the plane z=0

Case 2 is a curve machining or a single toolpath machining (shown in Figure 5-4).

Removed chip is only created by the tool surface at two continuous NC points. To

calculate chip volume, two intersecting tool surfaces are firstly divided by layers to get

their intersections (shown in Figure 5-6); two intersecting ellipses which can be

obtained from Eqs. (5.11) and (5.12) are then used to generate a crescent shape chip

area shown in the Figure 5-6 on each layer. Figure 5-7 illustrates the chip area in a 3D

chip shape. Finally, intersections on all layers are collected to get the valid chip

outline shown in Figure 5-8 (a).

70

Figure 5-6: Case 2: The chip area for a single toolpath on the plane z=0 in 2D

Figure 5-7: Case 2: The chip area for a single toolpath in 3D

71

Figure 5-8: Case 2: Valid chip outline by layers in a single toolpath

Figure 5-9: Case 3: (a) Tool motion in the second toolpath; (b) Removed chip in two

adjacent NC points

72

Figure 5-10: The chip area for one toolpath considering its neighboring toolpath on the

plane z=0 in case 3

Case 3 is about the tool motion in two continuous toolpaths shown in Figure 5-9. In

real free-form surface machining, there is a scallop height between two continuous

toolpaths due to the machining tolerance and tool compensation. This can be seen in

Figure 5-10, where removed chips in the jth

+1 toolpath are less than that in a single

toolpath in Case 2 since part of material in the jth

+1 toolpath is already removed by

the tool in the jth

toolpath. The Figure 5-10 shows how chip geometry is generated.

More specifically, the intersection of tool outlines at two continuous NC points Γ{Ci, j,

Ψi, j} and Γ{Ci+1, j, Ψi+1, j} represents removed material in the jth

toolpath, denoted by a

crescent area P5P3P4. P4 and P5 are intersections of tool projections at NC points Γ{Ci,

j, Ψi, j} and Γ{Ci+1, j, Ψi+1, j}. In the jth

+1 toolpath, as the tool moves from point Γ{Ci,

j+1, Ψi, j+1} to Γ{Ci+1, j+1, Ψi+1, j+1}, the removed chip area is P1P2P3. P1 is the

intersection of tool projections on the plane Z=0 at two NC points Γ{Ci, j+1, Ψi, j+1} to

Γ{Ci+1, j+1, Ψi+1, j+1}; P2 and P3 are intersections of tool projections at Γ{Ci, j+1, Ψi, j+1}

73

and Γ{Ci+1, j+1, Ψi+1, j+1} in the jth

+1 toolpath, and the tool projection at Γ{Ci+1, j, Ψi+1,

j} in the jth

toolpath. Intersections on all layers are collected to get the valid chip

outline shown in Figure 5-11 (a). A 3D point set of valid chip in Figure 5-11 (b) is

obtained to get the solid chip shape and volume by the Alpha Shape method illustrated

in Figure 5-11 (c).

Figure 5-11: (a) The valid chip outline generation in two continuous toolpaths (b) Valid

chip outline points; (c) Solid chip shape by the Alpha Shape method

Step 3: Determine the valid boundaries for a valid chip profile

From step 2, it is known that side boundaries for tool at Γ{Ci+1, j+1, Ψi+1, j+1}, and Γ{Ci,

j+1, Ψi, j+1} in the jth

+1 toolpath are the accumulating intersection of tool at Γ{Ci, j+1, Ψi,

j+1}, Γ{Ci+1, j+1, Ψi+1, j+1}, and tool at Γ{Ci+1, j, Ψi+1, j}. Side boundaries are shown in

Figure 5-12 (a) and (c). In Figure 5-12 (b), the bottom boundary for the tool at Γ{Ci+1,

74

j+1, Ψi+1, j+1} is composed by points at the bottom of the tool surface and between the

two side boundaries. The top boundary is consisted by ellipse points between the two

side boundaries on the plane z=0. It can be seen that the side, top, and bottom

boundaries create an outline of a removed chip. The valid boundaries for cylinder at

Γ{Ci, j+1, Ψi, j+1} in the jth

+1 toolpath is similar with that at Γ{Ci+1, j+1, Ψi+1, j+1}, which

can be seen in Figure 5-12 (c) and (d). A 3D data point set of valid chip outline can be

obtained after all boundaries are identified, which is shown in Figure 5-11 (b).

Figure 5-12: The tool moves along two NC points from Γ{Ci, j+1, Ψi, j+1} =(0.2, 0.5, 0.2,

4.5°, 4.5°) to Γ{Ci+1, j+1, Ψi+1, j+1}=(0.1, 0.5, 0.5, 6.5°, 6.5°) in the jth

+1 toolpath: (a) Side

boundaries in the tool motion at Γ{Ci+1, j+1, Ψi+1, j+1}; (b) Bottom and top boundaries in

the tool motion at Γ{Ci+1, j+1, Ψi+1, j+1}; (c) Side boundaries in the tool motion at Γ{Ci, j+1,

Ψi, j+1} (d) Bottom and top boundaries in the tool motion at Γ{Ci, j+1, Ψi, j+1}.

75

Step 4: Use the Alpha Shape method to get chip volume

The Alpha Shape method is an open source in MATLAB to give the area or volume of

a basic Alpha Shape for a 2D or 3D point set. In the codes, input is a probe radius and a

coordinate matrix of size N×3, which are the 3D points consisting of the outline of a

removed chip. Output is the volume and the plot of triangulation of the Alpha Shape. The

shape of removed chip is generated by triangulation of the 3D point set from step 3. In

Figure 5-11 (c), the removed chip is composed by many tetrahedrons and the chip

volume is obtained by the sum of volumes of all tetrahedrons, which are calculated using

Eqs. (5.13) and (5.14). Inputting a 3D point set shown in Figure 5-11 (b) and a probe

radius, a solid crescent chip shape (shown in Figure 5-11 (c)) is then obtained.

5.2.2 Local Parallel Sliced Method

5.2.2.1 Chip load model

The Alpha Shape method can be used to acquire the geometric profile of the removed

chip. However, it cannot provide information on the thickness of the chip and

instantaneous cutting forces. Chip profile points in the Alpha Shape method are mesh

points between intersections of tool profiles at two continuous NC points. Another tool

profile based method is the local parallel sliced method, which divides the cutter into

many slices perpendicular to the tool axis along the local coordinate system. It can

calculate the chip thickness and cutting force by finding the intersections of a line such as

Ci+3 Pi+3 which passing the tool center (shown in Figure 5-13 (a)) and the previous tool

profile.

76

Figure 5-13: Determination of instantaneous chip thickness: (a) Tool motions at two

adjacent NC points; (b) Tool projections on A-A section

Figure 5-13 (a) illustrates the calculation of chip thickness in a 5-axis CNC machining

using a flat-end mill. The tool is divided into many slices to get the chip volume by

accumulating small parallelepipeds along the axial depth of cut and engagement angle.

Let Ol’-Xl’-Yl’-Zl’ be the previous tool position and orientation, Ol-Xl-Yl-Zl represent the

current tool position and orientation after a distance of feed per tooth. Figure 5-13 (b)

shows the instantaneous chip thickness distribution on the ith

layer. Pi1 and Pi2 are

intersections of current and previous tool edges. Ci and Ci’ are the current and previous

tool centres on the ith

layer. Pi, k is the point on the current tool’s cutting edge determined

by equations (5.11) and (5.12); CiPi, k is a vector line runs from the current tool centre to

the tool edge. Pi, k’ is the intersection of line vector CiPi, k and the previous tool edge on

77

the ith

layer. Chip thickness for the kth

interval point can be defined as the distance

between Pi, k and Pi, k’, denoted by:

( , , ) ( , , ) '( , , )ct z k P z k P z k

(5.15)

where, z is the height along the tool axis at the point P, ϕ is the engagement angle or

immersion angle.

5.2.2.2 Chip volume by local parallel sliced method

On each slice, a polygon area is created by connecting the current and the previous

tool edge points between the two intersections Pi1 and Pi2 along the feed direction (shown

in Figure 5-13 (b)). In Figure 5-14 (a), red points are cutter-workpiece intersections at

current position; green circles are corresponding intersections at previous position.

Figure 5-14 (b) shows the removal volume is divided into 9 layers along the direction

which is perpendicular to the current tool axis. On each layer, the removal chip area is a

polygon shape generated by connecting two neighbouring edge points on the current and

previous tool edges. In Figure 5-14 (c), it can be seen that a chip shape is composed by

many parallelepipeds. The total chip volume is obtained by adding the volume of all

parallelepipeds along axial direction. The equation of total chip volume is defined by:

1 1

, , 1 , ,

1 1

M N

total i k i k i k i k

k i

V P P P P z

(5.16)

where, M is the number of interval points on each layer, N is the number of slices; Δz is

the integrating height along the current tool axis.

78

Figure 5-14: (a) Chip shape outline points; (b) Sliced chip area for layers; (c) Chip

volume consists of sliced parallelepipeds

Figure 5-15 shows how the chip thickness changes with the number of interval points on

different layers. From the Eq. (5.15), it can be seen chip thickness is relative to the axial

depth of cut and engagement angle.

79

Figure 5-15: Chip thickness on different layers

5.2.2.3 Cutter-workpiece engagement maps

A cutting force model requires getting engagement areas by discretizing the cutter into

slices in the tool coordinate system. The immersion angle is measured clockwise from the

y-axis in local coordinate system. In Figure 5-13 (b), the engagement or immersion angle

ϕi, k at the point Pi, k is the angle between two vectors Ci Pi, k and Ci Pi1. It can be

determined by the following equation:

, 11

,

, 1

cosi i k i i

i k

i i k i i

C P C P

C P C P

(5.17)

The cutter-workpiece engagement maps [61] are generated by transforming a removal

chip shape at a particular tool motion from global coordinate system to local coordinate

system. The 3D chip shape, as illustrated in Figure 5-14, is projected to the plane YlZl,

80

which goes through the current tool axis. The boundaries of the engagement domain are

the intersections of the tool envelopes at two continuous tool motions. The engagement

domain consists of many pieces of rectangles (shown in Figure 5-16), which are

convenient to predict cutting forces. Each rectangle represents an immersion angle at a

specified height along the tool axis. It is given by four parameters: ϕst, ϕex, dz, Z. ϕst and

ϕex are entry and exit angles; dz is the integrating height or the height of a rectangle and Z

is the distance from the tool tip to the bottom of rectangles. Entry and exit angles are

required for calculating cutting forces. They can be obtained from Eq. (5.17).

Figure 5-16: Cutter-workpiece engagement domain in 2D

Figure 5-17 shows the engagement domain at the 300th

NC point in a single toolpath

machining. The number of layers and rectangles depend on the resolution defined by user.

This figure illustrates the difference in the engagement domain resolution achieved by

various numbers of slice planes and interval points on each slice. Figure 5-17 (a) shows

the removal volume is generated by 9 slice planes with a 0.33 mm separation between

planes, whereas Figure 5-17 (b) shows the engagement domain is divided into 15 slices.

In this case, the distance between planes is 0.2 mm. The higher number of slices, the

81

more accurate the chip shape boundary is. The edge of engagement domain with high

resolution is more smoothly and less “blocky” than with lower number of slices.

However, a high resolution case requires more significant computing time. It took

approximately 4 minutes to calculate chip volume/cutting forces and determine the

engagement domain for around 1500 NC points using a removal volume of 9 slice planes.

By increasing the number of slices to 15, the computing time was expanded to 8 minutes.

The algorithm was written in MATLAB. MATLAB is inefficient in doing loop

calculations compared to code written in C++ or C#. It is one reason why this method is

time-consuming.

Figure 5-17: Cutter-workpiece engagement domain from a removed chip volume: (a) 9

slices with 60 interval points; (b) 15 slices with 100 interval points

Figure 5-18 shows how the sliced volume is gradually removed in the free-form

surface machining. There are three NC points are demonstrated to the removed volume.

The number of slices is different due to the varying depth of cut.

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Figure 5-18: (a)-(c) Displays how the sliced volume is gradually removed in the free-

form surface machining

5.3 Cutting Force Model

Accurate modeling of cutting force is the focus of machine dynamics research. It is

also the foundation for predicting cutting force to determine the optimal cutting

parameters, such as feed rate and depth of cut, to improve the machining efficiency while

satisfying surface quality requirements. The cutting force is normally modeled as three

components, radial (Fr), axial (Fa), and tangential (Fr), (as shown in Figure 5-19) for the

thin layer of cutting under consideration, dz, expressed as [47-49]:

( , , )

( , , )

( , , )

r rc c re

a ac c ae

t tc c te

dF K t z k dz K dz

dF K t z k dz K dz

dF K t z k dz K dz

(5.18)

83

where, Krc, Kac, and Ktc are the radial, axial and tangential cutting force coefficients,

respectively; and Kre, Kae, and Kte are the edge force coefficients, dz is the integrating

height, tc (z, θ, k) is the instantaneous undeformed chip thickness at a NC point (x, y, z,

α,β), represented by Eq. (5.15).

In local coordinate system (LCS), the deferential forces in x, y and z direction are

obtained by the following:

, ,

, ,

cos sin

sin cos

xl t i k r i k

yl t i k r i k

zl a

dF dF dF

dF dF dF

dF dF

(5.19)

where, φ is immersion angle defined by Eq. (5.17). Forces in the local coordinate system

are then transformed into world coordinate system (WCS) to compare forces measured by

dynamometer.

1 1( , ) ( , )

xw xl

yw yl

zw zl

dF dF

dF Rot x Rot z dF

dF dF

(5.20)

Rot(z, β) and Rot(x, α) are rotation matrixes got from Eq. (5.2). Differential cutting forces

for discretized engagement cutting edge elements are then summed by integrating the

differential forces along the immersion angle and axial depth of cut to obtain the total

forces for each given toolpath segment.

84

Figure 5-19: Cutting geometry of a flat-end mill

5.4 Case Studies and Results

5.4.1 Examples of Chip Volume Simulation by the Alpha Shape Method

In the previous section, the Alpha Shape method to generate chip shape and calculate

chip volume is proposed. In this section, a 5-axis CNC machine with swivel head

configuration (AC type) is used to simulate tool motions. Tool motions along a pre-

defined trajectory can be calculated from Eq.(5.2). To display removed chips and get

precise chip volume simulation, a free form surface shown in Figure 5-20 (a) is machined

by a two flutes flat-end mill, with a tool diameter of 10 millimeters. The workpiece size is

with a length and width of 50 millimeters respectively, and a height of 20 millimeters.

The depth cut varying from 0.1 mm to 3 mm. In the milling process, the cutting

parameters are selected by the workpiece material and tool size. The spindle speed is

85

selected as 1000 rpm. For each revolution, the feed per tooth can be calculated by the

following expression [80]:

t

ff

S N

(5.21)

where, f is the feed rate, S is the spindle speed; N is the number of flutes.

The test NC program used in this paper is generated by commercial CAM software.

The toolpath generation method is followed by iso-cusps method. Tool orientations are

surface normal directions.

Figure 5-20: (a) Simulation of machining a 3D curve on a free form surface, workpiece

size: 50×50×20 mm3, tool diameter: 10 mm; (b) The simulation of tool motions in

MATLAB.

The number of NC points is respected with the feed per tooth and the length of a

toolpath. NC points with cutter location and orientation are input in ALGORITHM; the

probe radius is 4 mm. The output of the algorithm is chip volume and chip shape. Figure

86

5-20 (b) shows the tool motions with various orientation angles in a pre-defined

trajectory in MATLAB. The tool profiles are divided into two parts by a plane z=0.

Assuming the workpiece is under the plane z=0. Depth of cut is the height from the

centre of the tool’s bottom to the top of the workpiece.

Removed chips are the part of the tool’s swept volume below the plane z=0. It can be

seen in the Figure 5-11 (c), the shape of a removed chip is a 3D solid crescent, consisted

by many tetrahedrons. The simulation of all chips in a toolpath is then generated by

accumulating a crescent shape at each NC point.

Overall, there are two different situations for a free-form surface machining. For the

first and last toolpaths, chip geometry is generated by tool projections at two continuous

NC points and the edge of the workpiece. The chip volume simulation for the first

toolpath is indicated in the Figure 5-21. In different NC points, chip shapes are various

due to changed tool orientations and depth of cut.

Figure 5-21: Chip volume simulation for the first toolpath

87

Figure 5-22: Chip volume simulation for the second toolpath

Figure 5-23: Chip volume simulation for a single curve

Machining continuous toolpaths is the second situation. Chip geometry is dependent

upon four tool motions at four NC positions in two continuous toolpaths. In Figure 5-22,

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chip volume simulation in the second toolpath of a free-form surface is shown. Chip

geometry at a NC point is a part of a crescent.

A single toolpath machining or curve machining is different with surface machining.

For a single toolpath machining, a chip shape is only respective with the intersections of

two tool motions at two continuous NC points. Therefore, the chip shape at any NC

points is a crescent. In surface machining, the case of a single toolpath machining is an

ideal situation. In Figure 5-23, it shows that chip volumes from the 163rd

NC point to the

200th

obtained from the single toolpath machining method are larger than that got by two

continuous toolpaths machining approach in Figure 5-22. However, it cannot be achieved

due to the machining tolerance and the tool compensation in the real CNC machining.

In Figure 5-24, a comparison is made between chip volume of the first toolpath with

and without considering the edge of the workpiece. It is a single toolpath machining

situation as the first toolpath without considering the edge of the workpiece. The values

of chip volume obtained from the single toolpath machining method are bigger than that

achieved by the first toolpath machining method. At the beginning of the machining

process, the volume got from the two different methods is not changed too much due to

the small depth of cut. Large depth of cut causes big gap of chip volume obtained by the

two approaches during the middle and the end of the machining process.

Chip volume computed from the single toolpath machining method for the second

toolpath of the surface is still more than that obtained by the method considering two

continuous toolpaths. The comparison result can be seen in the Figure 5-25. From the

graph, it can be seen that chip volume changes smoothly in this case due to the smooth

surface curvature changes. Tool orientations depend on surface normal; sometimes chip

89

volumes have drastic changes in the areas with large curvature. Dramatic changing chip

volumes indicates that cutting forces change a lot as well, since cutting forces are

proportional with chip volume [81]. The new challenge is how to get an optimal tool

orientation to make chip volumes change smoothly in future work.

Figure 5-24: Chip volume comparison of the first toolpath with and without considering

the edge of the workpiece

90

Figure 5-25: Volume comparison of the second toolpath with and without considering the

first toolpath.

5.4.2 Simulation Results of Chip Volume and Cutting Forces by Local

Parallel Sliced Method

A simulated test is performed for the same free-form surface shown in Figure 5-20.

The toolpaths for the whole surface were generated in MasterCAM software with 15,430

NC points. For each toolpath, there are approximately 1478 NC points employing one-

way toolpath with lifts. The workpiece size and cutting parameters to get chip volume

remain the same with the case study for the Alpha Shape method. The spindle speed is

selected as 1000 rpm. The feed rate can be calculated from Eq. (5.21). Its value is 0.034

mm per tooth. For each revolution, there are around 80 sampling points selected to

calculate cutting forces. Figure 5-26 illustrates simulated cutting forces in X, Y and Z

91

directions for one toolpath. Figure 5-27 shows the simulation results of the instantaneous

cutting forces changed with the rotation angles in five revolutions.

Figure 5-26: Simulated cutting forces in X, Y and Z directions for the whole toolpath

Figure 5-27: Predicted X, Y and Z forces for five revolutions in 5-axis CNC machining

with a flat-end mill

92

The resultant cutting force acting on the tool is calculated by:

2 2 2

x y zR F F F

(5.22)

Figure 5-28 (a) shows the predicted resultant cutting forces changed with machining

times. Figure 5-28 (b) illustrates the chip volume changes at different machining times.

From the comparisons of Figure 5-28 (a) and (b), it can be seen that chip volume has

similar changes to resultant cutting forces. Therefore, chip volume is another significant

index in the machining process planning to select optimal feed rate, spindle speed and

depth of cut.

Figure 5-28: (a) Resultant forces changing with machining time; (b) Chip volume

changing with machining time

Figure 5-29 shows the comparison of chip volume by the Alpha Shape method and the

tool profile based method. It can be seen that the Alpha Shape method results in smoother

and more accurate volume than the tool profile method. Tetrahedron and triangulation

used in the Alpha Shape method make the chip volume calculation more precise. In the

tool profile method, many rectangle blocks are accumulated to get chip volume, which

93

makes the volume and cutting forces obtained from the tool profile method look very

“blocky”.

Figure 5-29: Comparison of chip volume by the Alpha Shape method and the tool profile

based method

To calculate cutting forces, the distance between two NC points should be the value of

feed per tooth or feed per revolution. However, G-code generated by commercial

software is not uniform due to the machine tolerance and tool compensation and it is not

distributed by feed rate. In Figure 5-30, it shows that the diamond points are NC points

generated by MasterCAM. Some of them are closely spaced at large curvature areas,

others are not so close. An interpolation method called “interparc” is used to interpolate a

set of 2D or 3D points at fixed distance. Interparc is an open source in MATLAB using

94

an ODE solver. The new interpolated NC points—tool positions are distributed uniformly

along the toolpath. Tool orientations can be interpolated using the same approach.

Figure 5-30: Comparison of NC points got by MasterCAM and the uniform interpolation

method

5.5 Experiment Verification

For the validation of the proposed chip volume and cutting force modeling

approaches, an experiment has been conducted on a free-form surface by using a flat-end

mill. Due to the lack of a 5-axis CNC machine, a 3-axis CNC ALIO vertical micro-

milling machine was used instead to verify the predicted cutting forces by enabling two

rotational angles to be zeros. A Kistler table dynamometer (MiniDyn 9256C1) was used

for measuring instantaneous cutting forces. The Al 6061 workpiece was cut without

lubricant by a 4-flute carbide flat-end mill with a diameter of 1/8″. The experiment was

carried out with varying axial depth of cut (0.1-1.2 mm) at 10,000 rev/min spindle speed

and 0.02 mm feed per tooth. The sampling rate was 100 kHz, which is the maximum

capacity of the DAQ board.

95

Figure 5-31 shows the comparison of measured and simulated cutting forces in three

revolutions. It can be seen that the magnitudes of the predicted cutting forces in x, y and z

directions are in reasonably good agreement to the experimental ones if the runout effects

are not considered. Cutting force calculations are based on the instantaneous chip

thickness. The accuracy of predicted chip thickness can be guaranteed after the validation

of simulated cutting forces. The discretized chip volume is obtained by the scalar triple

product of chip thickness, integrating axial depth of cut, and radial edge contact length

between two continuous interval points on each layer. Therefore, chip volume validation

can be evaluated by measured cutting forces.

Figure 5-31: Measured and predicted cutting forces changing with rotation angles in three

revolutions.

96

5.6 Conclusions

A new numerical tool-based method to calculate chip volume and predict cutting

forces in a 5-axis CNC machining with a flat-end mill has been presented in this work.

The Alpha Shape method is only used to calculate chip volume; therefore, the local

parallel sliced method is added to predict cutting forces by identifying cutter-workpiece

engagement domain where the cutting flutes enter and exit the workpiece and depth of

cut are required to cutting force calculation. These approaches have widely applications

and are not restrict to continuous C1 toolpath.

An experiment was conducted to verify the simulation results of cutting volume and

cutting forces in 3-axis micro CNC machine through specifying the two rotation angles to

be zeros. Measured forces are shown in reasonably good agreement with simulated ones

if the runout effects are ignored.

The local parallel sliced method offers a new way to obtain cutter-workpiece

engagement domain and cutting forces for a given NC file in 5-axis machining with a

flat-end mill. There are also some improvements for further research:

Optimal tool orientation should be found to get smooth changing volumes.

Analytical method to get intersections of the cutter and the workpiece could

improve computational speed.

97

The Tri-dexel Method of Chip Volume and Cutting Chapter 6:

Forces Calculation and Simulation for Free-form Surfaces in 5-

axis CNC Machining with Flat-end Mills

The tool based method presented in Chapter 5 limits in some tool paths, such as the

pocket tool path to calculate chip volume and cutting forces for feed rate optimization. To

generate a method to predict chip volume and cutting forces for any tool path, a Tri-dexel

workpiece method is created by establishing intersections of the tool envelope and the

workpiece at every tool motion in this chapter. To decrease the complexity of 3D

Boolean subtraction, 2D Boolean subtraction is used alternatively by dividing the Tri-

dexel workpiece into many layers along z-axis direction in Section 6.1.2. The workpiece

is always updated for the next tool operation. In Section 6.1.3, the non-uniform and

uniform distributions of a chip shape were discussed. The chip shape generated by the

Tri-dexel workpiece method is non-uniform distributed. It does not affect the chip

volume calculation; however, cutting force prediction requires a chip shape to be

distributed uniformly. Therefore, chip profile points are redefined by finding same

column index of the cylinder tool to get a uniform distributed chip shape.

In Section 6.2, cutting force model for the workpiece based method was discussed.

Section 6.3 is one example same with Section 5.4 in Chapter 5 but using different method

to predict chip volume and cutting forces for a free-form surface in 5-axis CNC milling

with a flat-end mill. This simulation was conducted on the AL 6061 workpiece material

to demonstrate the validity of the proposed method. A benchmark experiment test in a 3-

axis micro-milling machine is used to verify the predicted chip volume and cutting forces

98

on a flat surface using a constant depth of cut and the pocket toolpath in Section 6.4.

Force predictions are in good agreement with the measured data both in magnitude and

trend if the runout effects are ignored.

6.1 Tri-dexel Method for Chip Volume and Cutting Force Calculation

6.1.1 Tri-dexel Workpiece

There are several models to represent volumetric models in the NC simulation process,

such as the voxel model and dexel model [87]. The dexel model represents an object with

a grid of long columns compacted together extending along z-axis direction, while the

voxel model consists of many small cubes in a regular lattice [88]. The difference

between dexel and voxel model is the object of z-axis. In voxel model, the height of

model is divided into many small pieces. For dexel model shown in Figure 6-1, the

volume along z-axis is continuous without separating into pieces. voxel representation

has advantages in the Boolean operation over the dexel model because Boolean

operations are conducted at the level of primitive volumetric element; but it is time

consuming, since it requires data on every solid cubes. The dexel model enables higher

efficiency computation than the voxel model, as it does not require data on every section

of model in z-axis direction which the voxel model has to consider [59].

There are many studies about the applications of dexel and voxel representations.

Benouamer [89] presented multi-dexel model to do NC milling simulation. Every dexel

includes values of entry and exit angles and the material property. But the multiple usage

of the single-dexel model caused topological inconsistencies and ignored small objects if

the size of dexel is too large. Hook [90] proposed a dexel data structure to simulate free-

99

form. Each dexel is defined by ray intersection. However, in Hook’s data, structure was

limited to the viewing direction. The view cannot be changed once the dexel data

structure has been built. Huang [91] improved Hook’s approach by developing a Tri-

dexel model to support dynamic viewing transformations and an assessment of dimension

errors.

The voxel model is robust and can apply to many CAD and NC simulation software

[92]. Karunakaran [93] used octree solid representation which is an adaptive version of

the voxel model to do the volumetric NC simulation. The voxel model was divided into

eight parts recursively to simulate cutting process and optimize the cutting parameters to

satisfy the cutting force constraints. Wastra [94] developed a 3D voxel structure to obtain

removed volume from the raw stock in the prototyping system. The voxel representation

used simple data structure to generate fast updating the workpiece. However, a huge

memory space was required to storing the model data, if the accuracy of the model was

improved by large size of voxels.

In this work, an improved Tri-dexel model (shown in Figure 6-1) is applied as a

workpiece model defined by many rectangles extending along the z-axis. Tri-dexel

locations are confirmed by a 2D grid in the xy-plane and physically extend the z-axis of

the Tri-dexel coordinate system. Grid points are uniformly distributed along x, y and z

axes by distances dx, dy, and dz respectively. The size of each Tri-dexel cube dx, dy, and

dz are determined by a user specified tolerance. The higher the tolerance, the more

accurate calculation of chip volume and cutting forces there will be. However, high

tolerance causes long computing time. To resolve this problem, the regular Tri-dexel

mode is improved by slicing the Tri-dexel workpiece into many 2D laminated planes. All

100

Boolean intersections and subtractions are performed on the laminated planes and the

plane heights are given by user. In the Tri-dexel model, each slice shares the same height

information. It is unnecessary to store the data of height information for every Tri-dexel

cell which could save storage memory and generate fast updating workpiece. Vertices in

the Tri-dexel workpiece model are the blue points shown in Figure 6-1. Line segments

between two neighboring vertices can be obtained through the given locations of vertices.

The Boolean subtraction of cutter volumes from the workpiece is equivalent to removed

line segments which are located inside the cutter envelope.

Figure 6-1: The Tri-dexel workpiece model in 3D.

6.1.2 Chip Volume Model

6.1.2.1 Tool Projections on the Tri-dexel Workpiece

To reduce the complexity of 3D Boolean subtraction, 2D laminated planes Boolean

subtraction is used by combining all planes along z-axis direction. The Tri-dexel model

101

of the workpiece is divided into many layers. The number of layers depends on the depth

of cut and the resolution defined by user. On each layer, it consists of m by n grid points.

The number of layers is depended on the tolerance defined by user. Chip volume and

cutting forces are relative to chip thickness, obtained by moving the tool along a distance

of feed per tooth. To simulate the machining process and find chip thickness, a layer of

the workpiece is used to display the generation of chip thickness and the Boolean

subtraction. The projection of a flat-end mill on a plane is an ellipse. The ellipse is

relative to two neighbouring NC points, which are denoted by (xi-1, yi-1, zi-1, αi-1, βi-1) and

(xi, yi, zi, αi, βi). x, y, z are coordinates of the tool, α and β are two rotational angles. The

equations of the ellipse at the ith

NC point can be obtained from [17]:

sin sin

cos cos sin sin cos sin sincos

i i


i

ih r zX r r x

(6.1)

sin sin

cos sin sin cos cos cos sincos

i i


i

ih r zY r r y

(6.2)

min( 0)iellipse i iZ h Z h

(6.3)

where, r is the tool radius, θ is the immersion angle, αi is lead angle, βi is tilt angle, hi is

height of the plane, Δxi and Δyi are translation steps along x and y axes at the ith

NC point.

6.1.2.2 Boolean operation and chip thickness generation

The simulation of cutting process is equivalent to the Boolean subtraction of tool

volumes from the machined workpiece. Figure 6-2 shows chip thickness generation and

the 2D Boolean subtraction. As the tool moves from the previous position P0 to current

position P1, new intersections of the Tri-dexel workpiece and current tool’s boundary are

found and stored in the current list. They are denoted by C1, C2 ... Cj, j is the number of

102

intersections. Line segments which run from the current tool centre to the points from

current list are connected to get the intersections with the previous tool edge. These

intersections are stored in the previous list, denoted by P1, P2…Pj. From here, a polyline

arc-shape along the tool edge is generated by connecting intersections in the previous and

current lists. The polyline arc-shape is regarded as the chip area on each slice. It can be

calculated by adding all areas of small polygons, such as the polygon C1 C2 P2 P1 show in

Figure 6-2. As the polygons are very small, they can be considered as rectangles to

calculate the area.

On the kth

removal chip slice, the chip area Ak is obtained by the accumulating of many

small polygons Cj Cj+1 Pj+1 Pj, j is the number of intersections in the current list. The chip

area Ak can be got from the following equation:

1 1 1

1

( )N

k j j j j

j

A C C P C

(6.4)

where, CjCj+1 is the integrated tool edge length, Pj+1Cj+1 is the chip thickness tj+1.

Figure 6-2: Boolean subtraction and chip thickness generation

103

The material in the polyline arc-shape is removed. From the geometry of the tool and

the workpiece, it can be seen that workpiece line segments which are inside the current

tool projection are trimmed. Therefore, the problem becomes finding entities that are

inside an ellipse and then deleting line segments composed by these entities. Points inside

an ellipse can be obtained by the following inequality:

2 2

2 21

x y

a b

(6.5)

where, a and b are the semi-major axis and semi-minor axis of an ellipse of the current

tool projection.

Lines connected by points from current and previous lists are chip thickness, denoted

by CjPj in Figure 6-2. Chip thickness can be obtained once the intersections of the

workpiece and current and previous tool edges are confirmed. Let (xcj, ycj, zcj) be the

coordinates at the current tool projection point Ci; (xpj, ypj, zpj) represents the coordinates

of the previous tool projection point Pj. The chip thickness tj in the 3D Euclidean space

is:

2 2 2( ) ( ) ( )j cj pj cj pj cj pjt x x y y z z

(6.6)

The same method can be used to get chip thickness on different layers of a chip shape.

Figure 6-3 illustrates how chip thickness changes with the points in the current list on

various layers.

104

Figure 6-3: Chip thickness for non-uniform distributed chip geometry

6.1.3 Chip Volume Calculation

It is essential to continually subtract the intersections of the tool at two adjacent

motions from the raw stock in order to get a final chip shape and predict cutting forces as

realistically as possible. Chip thickness is updated and removed by every tool milling

along the feed direction. The removal volume can be thought of as the Boolean

intersections of the tool envelope with the workpiece. To get the chip volume, the

workpiece is firstly divided into many parallel slices. Figure 6-4 shows the slice volume

on the Tri-dexel workpiece; the chip area and chip thickness are then calculated on each

slice from Eqs (6.4) and (6.6). Finally, total chip volume at the ith

NC point is obtained by

accumulating all chip areas from layers by Eq. (6.7).

1 1 1

1 1

M N

i j j j j

k j

V C C P C h

(6.7)

where, Δh is the integrating height, i is the number of NC points, j is the number of

cutter-workpiece intersections on each slice, k is the number of slices.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55

Ch

ip t

hic

knes

s (m

m3 )

Number of interval points

layer1

layer2

layer3

layer4

layer5

layer6

layer7

layer8

layer9

layer10

105

In 5-axis CNC machining using a flat-end mill, the machined workpiece on each slice

is different, due to the two rotational angles. The data on each layer, such as unremoved

and removed workpiece line segments, cutter-workpiece intersections, and chip thickness

are stored separately. The Tri-dexel workpiece is updated for every milling operation. NC

points are generated according feed per tooth. For surface machining, there are several

cases for the last layer of the workpiece slices between two continuous NC points

depending on the depth of cut. If the depth of cut between two tool motions changes too

much, causing the numbers of slices at position (xi-1, yi-1, zi-1, αi-1, βi-1) and (xi, yi, zi, αi, βi)

to be different, the workpiece should be divided into two types. The fewer number of the

workpiece slices between the two tool positions is found as a reference to divide the

workpiece. The first type is to keep all workpiece data from the previous tool position if

workpiece slices are less than or equal with the fewer number of workpiece slices. For

others, their workpiece data are from the beginning where no line segments are removed.

Figure 6-4: Chip thickness on the Tri-dexel workpiece

Figure 6-5 (a) shows cutter-workpiece engagement by layers; Figure 6-5 (b) illustrates

that the chip shape consists of many non-uniform distributed polygons. This is due to line

106

segments from P1 to the current tool edge (shown in Figure 6-2) not being uniformly

distributed, causing non-uniform distributed immersion angles. It is good enough to

obtain chip volume from these non-uniform distributed polygons; however, it cannot

obtain accurate cutting forces since cutting force are calculated by accumulating

differential radial, axial, and tangential forces along the immersion angle and axial depth

of cut. If immersion angles are not distributed uniformly, the integrating forces from

layers are hard to calculate.

Figure 6-5: The non-uniform distributed chip shape

To resolve this problem, new cutter-workpiece intersections are obtained by finding

boundaries of a chip shape. Firstly, uniformly distributed current tool profile points which

are under the first slice plane or the plane Z=0 are founded. These points are not

107

composed of the real removed chip shape, since some of them have already been

removed by the tool at previous position. Therefore, it is better to get the boundaries of

the valid chip shape by finding intersections of the tool at the current and previous

positions. The method to find the boundaries of the tool at two continuous NC points has

been proposed in the Section 5.2.1. The points under the plane Z=0 are stored in the

under points list by column index. Points in the same column of the cylinder tool share

the same column index. Column indexes of two intersections lines composed by the

current and previous tool intersections are obtained as the boundary conditions to get

valid chip profile points. The index of valid under points shown in the Figure 6-6 (a)

should be between the two column indexes of the two intersection lines. Finally, the valid

current tool profile points (red squares in Figure 6-6 (b)) can be obtained by indexes of

the valid under points. The same approach can be used to get the valid previous tool

profile points (blue squares in Figure 6-6 (b)). From Figure 6-6 (b), it can be seen that a

chip shape consists of valid previous and current tool profile points that are uniformly

distributed. On each slice, immersion angles, which are between the line segments

intruding from each ellipse centre to valid tool profile points, are distributed uniformly as

well. The chip thickness is redefined by connecting valid current and previous tool profile

points.

108

Figure 6-6: The uniform distributed chip shape and redefined chip thickness

Figure 6-7 shows the non-uniform and uniform distributed valid chip profile points.

Green star points and black dot points are the current and previous valid chip profile

points, respectively. Red squares are the uniform distributed new current valid chip

profile points. Different kinds of methods to get distributed chip profile do not affect the

total chip volume, but cutting forces would be affected due to the distribution of

immersion angles.

109

Figure 6-7: Non-uniform and uniform distributed valid chip profile points

In Figure 6-8, chip thickness changed with interval points on different layers is

illustrated. The values of chip thickness on each slice are coincident with the chip shape

shown in Figure 6-6.

Figure 6-8: Chip thickness for the uniform distributed valid chip geometry

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

1 2 3 4 5 6 7 8 9 10111213141516171819202122

Ch

ip t

hic

knes

s (m

m3)

Number of interval points

layer1

layer2

layer3

layer4

layer5

layer6

layer7

layer8

layer9

layer10

110

Figure 6-9 shows the simulation of the tool is removing material from the workpiece.

It can be seen that the density of the removed workpiece is different which is due to the

depth of cut changing for the whole toolpath.

Figure 6-9: Cutting simulation of tool removing in the Tri-dexel workpiece

An example of cutter profiles at some tool motions illustrates the changed depth of

cut, shown in the Figure 6-10. There are three cases for depth of cut affecting number of

layers of the workpiece. Firstly, if the depth of cut increases, a new layer of the

workpiece without any subtraction operation is added to the workpiece at the forward

tool motion. Subtraction would be operated in the new layer of the workpiece, and

information of line segments are saved for the next tool operation. If the depth of cut does

not change too much, the numbers of workpiece layers are the same as the last tool

motion. Finally, if the depth of cut decreases, only part of the workpiece would

participate in Boolean intersecting and subtracting operations; line segments in the

participated workpiece are updated after every Boolean operation. The other layers of the

workpiece at the previous tool motion, whose heights are larger than the maximum height

111

of the tool at current position, would not perform the Boolean operation until the depth of

cut is bigger than its maximum height.

Figure 6-10: Varied depth of cut in the workpiece method

6.2 Cutting Forces Prediction

Predicting cutting force accurately is significant to the machine dynamics research and

it is the foundation to determine optimal cutting parameters. The cutting force prediction

mainly consists of the instantaneous undeformed chip thickness calculation and cutter-

workpiece engagement feature extraction such as entry/exit angles. A numerical

technique is used to slice the cutter into many discs and sum the differential cutting

forces along the immersion angle and axial depth of cut for each tool motion along a

toolpath.

For 5-axis CNC machine, cutting forces are relative to chip thickness, cutting

coefficient, feed rate, and two rotation angles. Chip thickness is also a significant

parameter for chip volume. The following steps demonstrate the chip volume calculation:

1) Model the removed chip geometry and calculate chip thickness

112

2) Obtain the engagement area to calculate immersion angle or start and exit

angles

3) Calculate cutting forces in tangential, radial, and axial directions with cutting

coefficients in local coordinate system (LCS)

4) Transform cutting forces from local coordinate system (LCS) to world

coordinate system (WCS)

For a given NC point on the flat-end milling, the three differential cutting forces radial

(Fr), axial (Fa) and tangential (Fr) are given by the following equation [47-49]:

( )

( )

( )

r rc j re

a ac j ae

t tc j te

dF K t K dz

dF K t K dz

dF K t K dz

(6.8)

where, Krc, Kac, and Ktc are the radial, axial, and tangential cutting force coefficients, and

Kre, Kae, and Kte are the edge force coefficients, determined by experimental tests and the

workpiece material properties. tj is the instantaneous undeformed chip thickness given in

Eq. (6.6); dz is the integrating height along z-axis.

In the feed coordinate system, cutting forces are obtained by transforming the

differential radial, axial, and tangential forces using the immersion angle ϕ:

, ,

, ,

cos sin

sin cos

x t i k r i k

y t i k r i k

z a

dF dF dF

dF dF dF

dF dF

(6.9)

Finally, differential forces in the feed coordinate system are summed for all layers in a

toolpath segment.

113

Figure 6-11: Cutting force model of a flat-end mill

6.3 Case Studies and Results

To compare cutting forces by the Tri-dexel workpiece method and the tool profile

based method in the previous paper [17], cutting parameters and the workpiece size are

identical to the case study applied in the tool profile based method. A two flute flat-end

mill with a diameter of 10 mm is used to machine the free-form surface shown in Figure

5-20 (a) at 1000 rev/min spindle speed and 0.034 mm feed per tooth. The length, width,

and height of the workpiece size are 50 × 50 × 20 mm. The depth of cut is changing from

0.1 mm to 3 mm. There are around 1487 NC points generated in MasterCAM for one

toolpath. Figure 6-12 shows the comparison of predicted cutting forces by the workpiece

based and the tool based methods changing with rotation angles in five revolutions. It can

be seen that the simulated cutting forces obtained by the workpiece based method show a

reasonable agreement with that the ones obtained by the tool based method both in trend

and magnitude.

114

Figure 6-12: Comparison of simulated cutting forces by the workpiece and the tool based

methods

The predicted cutting forces by two approaches along x, y, and z directions in the

whole toolpath are also compared in Figure 6-13. From the comparison of cutting forces

obtained by the workpiece method and the tool profile method, it can be seen that the

results are very similar. For the workpiece method, the tool is sliced into layers by planes

which are perpendicular with z-axis, while in the tool profile based method, the planes

selected to slice the tool are perpendicular with the tool axis at different cutter locations.

Therefore, the ways to divide the tool into many layers do not affect the cutting forces

calculation.

115

Figure 6-13: (a)-(c) Simulated cutting forces by the Tri-dexel workpiece method; (b) (e)-

(g) Simulated cutting forces by the tool based method

The resultant cutting force acting on the tool can be obtained from (5.20). Figure 6-14

shows the resultant cutting forces changed with machining times for the whole toolpath

by Tri-dexel workpiece method. In Figure 6-15, it compares the chip volume obtained by

two different methods. It seems that chip volume obtained by the workpiece method has

more smooth changes than the tool based method since the intersections of the tool and

the workpiece obtained by the workpiece method are much more precise if the resolution

of workpiece grids is high, additionally with longer computing time. From Figure 6-14

and Figure 6-15, it can be seen that simulated chip volume has similar changes to

resultant cutting forces. Therefore, chip volume can be regarded as another important

parameter in the machining process planning to select optimal feed rate, spindle speed,

and depth of cut.

116

Figure 6-14: Resultant cutting forces by the workpiece method

Figure 6-15: Comparison of chip volume by the tool based method and the workpiece

method

0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

Machining time (s)

Re

su

lta

nt cu

ttin

g fo

rce

(N

)

0 8 16 24 32 40 480

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Machining time (a)

Ch

ip v

olu

me

(m

m3)

Volume by tool based method

Volume by workpiece method

117

The computing time used in the workpiece method is much longer than in the tool

profiled method since the workpiece updates with many line segments operations, such as

intersection and subtraction. This must be considered at each tool motion in the

workpiece method. However, the tool profile based method cannot be used in the pocket

toolpath because it requires the same number of NC points in two neighboring toolpaths

to get intersections of tool motions at four positions. Pocket milling cannot guarantee

generation of the same number of NC points in two continuous toolpaths, since the tool

cuts the material inside of the workpiece along an arbitrarily closed boundary. The range

of the closed boundary is gradually increased, causing the number of NC points in the

outside toolpath to be larger than that in the inside toolpath. Zig-zag and one way

toolpaths do not have these kinds of problems. They are linear toolpaths and can generate

same number of NC points in two neighboring toolpaths. The workpiece method is robust

for use with any kind of toolpaths. It gets chip volume and cutting forces through the

intersection of the tool envelope and the workpiece rather than from the tool intersections

at four continuous positions in two neighboring toolpaths.

6.4 Experimental Verification

A benchmark experiment [95] has been used to verify the simulation cutting force and

chip volume modeling methods on a flat surface with the pocket toolpath (shown in

Figure 6-16). The depth of cut is 0.2 mm. The experiment was conducted in a 3-axis

ALIO micro-milling machine with a spindle speed of 30,000 rpm and a feed rate of 1

μm/tooth. A four-flute flat-end mill with a diameter of 2 mm was used to cut an AL 6061

workpiece in the air without lubricant. The size of the workpiece is 10 × 15 × 5 mm as

the width, length, and height, respectively. The benchmark data is collected by a 3-axis

118

Kistler table dynamometer (MiniDyn 9256C1).The 5-axis cutting forces modeling

method based on the Tri-dexel workpiece can also work in 3-axis milling with the two

rotational angles set to be zero.

Figure 6-16: The pocket toolpath

Resultant cutting force acting on the tool is obtained by:

2 2 2

x y zR F F F

(6.10)

Figure 6-17 (a) illustrates the measured resultant force changed with machining time,

while Figure 6-17 (b) shows the chip volume changed with machining time. From the

comparison, it can be seen that the trend of chip volume was similar to the experimental

resultant cutting force. The calculation of chip volume is faster and easier than cutting

force. Therefore, chip volume is another good index to choose optimal cutting parameters

such as feed rate, depth of cut, and spindle speed in the machining process planning.

119

Figure 6-17: (a) Measured resultant cutting forces changing with machining time; (b)

Predicted chip volume changing with machining time

Figure 6-18 shows the comparison of predicted and measured cutting forces for

machining the whole flat surface. The broken green line is the simulation resultant cutting

force, and the black line is the experimental resultant force. The predicted resultant

cutting force was accurately predicted by the cutting coefficients generated from the

experiment tests. The trend and magnitudes of estimated resultant cutting force were in

reasonably good agreement to the measured force if the runout problems are ignored.

120

Figure 6-18: Comparison of simulation and experimental resultant forces in 3-axis

milling

6.5 Conclusions

This chapter presents an improved Tri-dexel workpiece method of chip volume and

cutting force predictions to overcome the limit application of the pocket toolpath by the

tool based approach proposed in Chapter 5. A Tri-dexel workpiece model is generated to

for the predict removal material volume and cutting forces by updating the machined

workpiece and subtracting the cutter-workpiece engagement zone. The 3D Tri-dexel

workpiece is sliced into many 2D laminated layers to reduce the complexity of 3D

Boolean operations. On each slice, the instantaneous chip thickness is determined by the

intersections of the tool cutting edge and the workpiece line segments. A uniform

distributed chip model has been proposed to calculate cutting forces by finding same

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

Machining time (S)

Resultant

forc

e (

N)

Cutting forces change with machining time

Simulation resultant force

Experimental resultant force

121

column index of the flat-end mill. The comparison of the tool profile based method and

the workpiece based method to calculate chip volume and cutting forces are discussed by

a same case study presented in Section 5.4.2 of Chapter 5. Simulations of cutting forces

and chip volume for 5-axis have been carried out by the Tri-dexel workpiece method. A

benchmark validation experiment in controlled cutting conditions has been used on a 3-

axis micro CNC machine. The simulation results for 5-axis CNC machining were verified

by 3-axis machining experiments through setting the two cutter rotation angles at zero.

The simulated results were in reasonably good agreement with the experiment cutting

forces.

122

Conclusions and Future Work Chapter 7:

7.1 Conclusions

This research focuses on the new methods for determining optimal toolpath, cutter

orientation, and feed rate planning based on calculated chip volume and cutting forces in

5-axis CNC machining using a flat-end mill. The research considers cutter-part surface

geometry, chip volume and cutting force predictions. Chapter 3 presents an optimal

toolpath generation by dividing a machining surface patch-by-patch using the fuzzy

clustering technique and similar surface normal variables control to avoid dramatic tool

orientation changes. Chapter 4 proposes an optimal tool orientation method based on the

EMS and the surface normal method to avoid gouges and obtain highest machining by

the largest cutting edge and best curvature match. Chapters 5 and 6 develop two

approaches to calculate chip volume and cutting forces for feed rate optimization.

Machining experiments are conducted to verify the simulation results.

In Chapter 3, machining surface patch-by-patch with points of similar surface normal

orientation is adopted to reduce the range of rotation motion and avoid sharp cutter

orientation changes. This method is based on the fuzzy clustering technique. The

optimized distance is the weighted combination of 3D Euclidean distance of surface

normal vectors and physical distance between mesh points and cluster centres. The value

of weighting factor α is varied with different shapes of the surface. An optimal number of

surface patches are identified considering both changes of accumulating relative angles to

minimize the two rotation motions and simplify toolpath generation. Furthermore, the

Alpha Shape method based on the probe radius is used to define patch boundaries. The

123

Iso-parametric CNC toolpath is generated due to its simplicity and surface normal vectors

are the most efficient cutter orientations for convex surfaces. The proposed method in this

chapter provides an alternative to generate 5-axis CNC toolpaths and cutter orientation

for convex curved surfaces.

Chapter 4 develops an optimal tool orientation in a 5-axis CNC machining. The

flexible combination of the EMS method and the surface normal variable control method

has been used to obtain optimal tool orientations to avoid gouges and improve machining

efficiency. The mathematical model of a NURBS surface is provided to get surface

features including concave, convex, and saddle points, principal curvatures, and surface

normal direction. The relationship of surface features, curvatures, gouges, and tool

orientation methods is presented. The EMS method is applied to concave parts to avoid

local gouges. The tilt rotational angle is decided by matching the largest cutter Euler-

Meusnier sphere with the smallest Euler-Meusnier sphere of the surface at each cutter

contact (CC) point, and the lead angle is relied on the surface smallest principal curvature

direction. The highest efficient tool orientation for a convex surface is along the surface

normal direction. The tool orientation for saddle surfaces can be applied to the EMS

method or the surface normal method depending on the selected machining direction.

A general discussion for predicting chip volume and cutting forces to optimize feed

rate in a 5-axis CNC free-form surface machining with a flat-end mill has been presented

in Chapter 5. The Alpha Shape method and the local parallel sliced method have been

used to obtain removed chip volume. The Alpha Shape method provides an efficient and

robust calculation of chip volume for arbitrary tool orientations because a series of

complicated trigonometric equations, to get intersections of tool motions at two arbitrary

124

positions, are replaced by a numerical method in ALGORITHM. Although the Alpha

Shape method is able to calculate chip volume and display solid chip shapes with a fast

computing time, it cannot be used to get chip thickness and predict cutting forces. The

local parallel sliced method can obtain the cutter-workpiece engagement domain where

the cutting flutes enter and exit the workpiece and depth of cut are required to calculate

cutting forces. These approaches have wide applications and do not restrict to continuous

C1 toolpath. To verify the proposed Alpha Shape method and get precise chip volume

simulation, NC programs for machining a free-form surface is developed to demonstrate

cutting volumes. Three cases to calculate chip volume are considered. The first two cases

are about machining the surface for the first toolpath and one of two continuous

toolpaths. The third one is a single toolpath machining or curve machining. Comparisons

between these three cases are conducted.

The presented cutting force model can predict the 5-axis flat-end milling process

accurately based on the chip thickness obtained by the local parallel sliced method. An

experiment for the research of cutting volume and cutting forces in 3-axis micro CNC

machine was conducted. The simulation results for 5-axis machining were verified by

machining experiments through specifying the two rotation angles to be zeros. Measured

forces are shown in reasonably good agreement with simulated ones if the runout effects

are ignored. Uniform interpolation was applied at two continuous NC points by a distance

of feed per tooth to reduce the modeling error and get more precise cutting forces. Chip

volume is a significant parameter in the machining process, due to the similar changes

that occur with resultant cutting forces.

125

The local parallel sliced method offers a new way to obtain cutter-workpiece

engagement domain and cutting forces for a given NC file in 5-axis machining with a

flat-end mill. This approach is robust because it can be applied to/used for various types

of cutters and sculptured surfaces without additional analysis. It also gives user a degree

of flexibility to choose between computational speed, accuracy, and a combination of

both by different resolutions.

In Chapter 5, the tool based method to calculate chip volume and cutting forces is

limited in the application of the pocket toolpath. Therefore, an improved method for

modeling chip geometry and numerically calculating chip volume and cutting forces with

a flat-end mill is proposed in Chapter 6. This work demonstrates the feasibility of

generating a Tri-dexel workpiece model for the purpose of predicting removal material

volume and cutting forces by updating the machined workpiece and subtracting the

cutter-workpiece engagement zone. The 3D voxel workpiece is sliced into many 2D

laminated layers to reduce the complexity of 3D Boolean operations. On each slice, the

instantaneous chip thickness is determined by the intersections of the tool cutting edge

and the workpiece line segments. Extending the non-uniform distributed chip model that

can only predict chip volume, a uniformly distributed chip model has been added to

calculate cutting forces by finding same column index of the flat-end mill. The

comparison of the tool profile based method and the workpiece based method to calculate

chip volume and cutting forces are also introduced by a same case study in the paper

[17]. 5-axis simulation of cutting forces and chip volume has been carried out for milling

cases that tool orientations and depth of cut change continuously by the voxel workpiece

method. A physical benchmark validation experiment in controlled cutting conditions has

126

been used on a 3-axis micro CNC machine. The simulation results for 5-axis CNC

machining were verified by 3-axis machining experiments through setting the two cutter

rotation angles to zero. The simulation and experiment results illustrate that the voxel

workpiece model is capable of prediction since the predicted results were in reasonably

good agreements with the experiment cutting forces.

7.2 Future Work

There is lots of work that could be further conducted in this research area, as follows:

In this dissertation, two approaches are introduced to calculate chip

volume and cutting forces in Chapter 4 and Chapter 5 without considering runout

effects. In the future, further considerations are required to predicting more

precise chip volume and cutting forces by considering the effects of machine

vibration and tool wear-out problems. Calculating cutting force is the first step for

the research of machining dynamics. Another important consideration is the

analysis and modeling of relative static and dynamic deformations for the cutting

tool. The tool chatter vibrations affect the accuracy of chip thickness generation

during machining operations.

There are many researches about the feed rate optimization for the ball-

end mill, but very limit studies in the 5-axis CNC machining using flat-end mills

due to the complicated cutter-workpiece contact geometry. In this work, chip

thickness, chip volume, and cutting force calculations for flat-end mills in 5-axis

CNC machining are presented by numerical methods. It would be easy in the

future to develop a feed rate optimization algorithm based on chip volume and

127

cutting forces predictions by considering chatter stability and tool wear out

effects.

In Chapters 4 and 5, numerical methods are used to obtain intersections of

the cutting tool and the workpiece. Future consideration is needed to generate an

analytical method to get intersections of the cutter and the workpiece, which

would help improve computational speed.

The algorithm of cutting process simulation in this work was written in

MATLAB. It is inefficient and time-consuming to do loop calculations in

MATLAB compared to code written in C++ or C#. It cannot show the continuous

tool moving of cutting process simulation by animation in MATLAB. In future,

the codes can be optimized and simplified by using C++ or C# to improve the

computational efficiency and show the animation of NC milling simulation. An

interface could be also generated for user friendly to change cutting parameters,

such as tool diameter, feed rate, spindle speed, number of flutes, etc. to get

different simulation results.

Carbon fiber–reinforced polymer (CFRP) material is widely applied in

aerospace and automotive industries due to its high strength and light weight. The

workpiece method to calculate chip volume and cutting force in Chapter 5 can be

used for CFRP trimming. Carbon fiber is made by many layers with different

directions on each layer. It can be seen in Figure 7-1. The workpiece method

enables to adjust the direction of line segments along the fiber directions on each

layer; therefore, it is possible to calculate chip thickness, chip volume, and cutting

forces for carbon fiber by the similar method proposed in this work. Figure 7-1

128

shows a 3D chip model of CFRP in different layers. Figure 7-2 illustrates

removed chips on the parallel and vertical directions. Chip thickness is obtained

by removing the tool a distance of feed per tooth. The angle between fiber

direction and the tool moved direction is called relative angle. Exploring the

relation between relative angle and machining efficiency and surface quality

would be an interesting area to explore. It is useful to select an optimal direction

to cut carbon fiber for better surface quality and high machining efficiency. The

workpiece approach can also calculate the length of fiber which is removed by

layers, shown in Figure 7-2.

Figure 7-1: A CFRP 3D chip model

129

Figure 7-2: (a)-(b) Removed fiber on the parallel direction; (c)-(d) Removed fiber on the

vertical direction

130

Bibliography

1. Lasemi, A., D.Y. Xue, and P.H. Gu, Recent development in CNC machining of

freeform surfaces: A state-of-the-art review. Computer-Aided Design, 2010. 42(7):

pp. 641-654.

2. Wang, Y.J., Curvature Gouge Detection and Prevention in 5-axis CNC Machining

2007.

3. Fard, M.J.B. and H.Y. Feng, Effect of tool tilt angle on machining strip width in five-

axis flat-end milling of free-form surfaces. International Journal of Advanced

Manufacturing Technology, 2009. 44(3-4): pp. 211-222.

4. Rao, A. and R. Sarma, On local gouging in five-axis sculptured surface machining

using flat-end tools. Computer aided design, 2000. 32(2000): pp. 409–420.

5. Lee, Y.S. and H. Ji, Surface interrogation and machining strip evaluation for 5-axis

CNC die and mold machining. International Journal of Production Research, 1997.

35(1): pp. 225-252.

6. Kim, C.B., S. Park, and M.Y. Yang, Verification of NC toolpath and manual and

automatic editing of NC code. International Journal of Production Research, 1995.

33(3): pp. 659-673.

7. Lee, Y.-S., Admissible tool orientation control of gouging avoidance for 5-axis

complex surface machining. Computer aided design, 1997. 29(7): pp. 507-521.

8. Choi, B.K. and R.B. Jerard, Sculptured surface machining: theory and applications.

1998, Dordrecht; London Kluwer Academic.

9. Li, S.X. and R.B. Jerard, 5-axis machining of sculptured surfaces with a flat-end

cutter. Computer-Aided Design, 1994. 26(3): pp. 165-178.

10. Erdim, H., I. Lazoglu, and B. Ozturk, Feedrate scheduling strategies for free-form

surfaces. International Journal of Machine Tools & Manufacture, 2006. 46(7-8): pp.

747-757.

11. Ozturk, E. and E. Budak, Modeling of 5-axis milling processes. Taylor & Francis

Group, 2007. 11: pp. 287–311.

12. Erdim, H., I. Lazoglu, and M. Kaymakci, free-form surface machining and

comparing feedrate scheduling strategies. Machining Science and Technology, 2007.

11(1): pp. 117-133.

131

13. Ozturk, E., L.T. Tunc, and E. Budak, Investigation of lead and tilt angle effects in 5-

axis ball-end milling processes. International Journal of Machine Tools &

Manufacture, 2009. 49(14): pp. 1053-1062.

14. Boz, Y., H. Erdim, and I. Lazoglu, Modeling Cutting Forces for Five Axis Milling of

Sculptured Surfaces. Advanced materials research: pp. 701-712.

15. Lin, T., J.W. Lee, and E.L.J. Bohez, A new accurate curvature matching and optimal

tool based five-axis machining algorithm. Journal of Mechanical Science and

Technology, 2009. 23(10): pp. 2624-2634.

16. Shan Luo, Zuomin Dong, and M.B.G. Jun. Optimization of 5-axis CNC toolpath and

cutter orientation for machining curved surfaces. in Virtual Machining Process

Technology conference. 2014.

17. Shan Luo, Zuomin Dong, and M.B.G. Jun. Chip volume calculation and simulation

in 5-axis CNC machining with flat-end mill. in Virtual Machining Process

Technology conference. 2015.

18. Han, Z. and D.C.H. Yang, Iso-phote Based Tool-path Generation for Machining

Free-form Surfaces. Journal of manufacturing science and engineering, 1999.

121(4): pp. 656-664.

19. Loney, G.C. and T.M. Ozsoy, NC machining of free form surfaces. Computer aided

design, 1987. 19(2): pp. 85-90.

20. Suresh, K. and D.C.H. Yang, Constant Scallop-height Machining of Free-form

Surfaces. Journal of engineering for industry, 1994. 116(2): pp. 253-259.

21. Feng, H.-Y. and H. Li, Constant scallop-height toolpath generation for three-axis

sculptured surface machining. Computer-Aided Design, 2002. 34(9): pp. 647-654.

22. Yi, X.Z., et al., Toolpath generation of 5-axis CNC whirlwind milling model for

freeform surfaces. 2005 IEEE International Conference on Mechatronics and

Automations, Vols 1-4, Conference Proceedings, ed. J. Gu and P.X. Liu. 2005, 4.5.

537-540.

23. Yuwen, S., et al., Iso-parametric toolpath generation from triangular meshes for free-

form surface machining. Adv Manuf Technol, 2006. 28(2006): pp. 721-726.

24. Sarma, R. and D. Dutta, The Geometry and Generation of NC Toolpaths. Journal of

mechanical design (1990), 1997. 119(2): pp. 253-258.

25. Jee, S. and T. Koo. Tool-path generation for nurbs surface machining in Proceedings of

the American Control Conference. 2003.

132

26. Ding, S., et al., Adaptive iso-planar toolpath generation for machining of free-form

surfaces. Computer-Aided Design, 2003. 35(2): pp. 141-153.

27. Tam, H.y., H. Xu, and Z. Zhou, Iso-planar interpolation for the machining of implicit

surfaces. computer aided design, 2002. 34(2002): pp. 125-136.

28. Ren, F., Y.W. Sun, and D.M. Guo, Combined reparameterization-based spiral

toolpath generation for five-axis sculptured surface machining. International Journal

of Advanced Manufacturing Technology, 2009. 40(7-8): pp. 760-768.

29. Lo, C.C., A new approach to CNC toolpath generation. Computer aided design,

1998. 30(8): pp. 649-655.

30. Chen, Z.C., Z.M. Dong, and G.W. Vickers, Automated surface subdivision and

toolpath generation for -axis CNC machining of sculptured parts. Computers in

Industry, 2003. 50(3): pp. 319-331.

31. Chen, Z.C., Optimal and Intelligent Multi-Axis CNC Toolpath Generation for

Sculptured Part Machining. 2004, university of victoria. p. 4.5.

32. Li, L.L., et al., Generating tool-path with smooth posture change for five-axis

sculptured surface machining based on cutter accessibility map. International

Journal of Advanced Manufacturing Technology, 2011. 53(5-8): pp. 699-709.

33. Plakhotnik, D. and B. Lauwers, Computing of the actual shape of removed material

for five-axis flat-end milling. Computer aided design. 44(11): pp. 1103-1114.

34. RAO, N., F. ISMAIL, and S. BEDI, Tool path planning for five-axis machining

using the principal axis method. Int. J. Mach. Tools Manufact, 1997. 37(7): pp.

1025-1040.

35. P.Gray, S.Bedi, and F.Ismail, rolling ball method for 5-axis surface machining.

Computer aided design, 2003. 35(2003): pp. 347-357.

36. Rao, N., F. Ismail, and S. Bedi, Toolpath planning for five-axis machining using the

principal axis method. International Journal of Machine Tools & Manufacture,

1997. 37(7): pp. 1025-1040.

37. Hosseinkhani, Y., J. Akbari, and A. Vafaeesefat, Penetration-limination method for

five-axis CNC machining of sculptured surfaces. International Journal of Machine

Tools & Manufacture, 2007. 47(10): pp. 1625-1635.

38. Wang, Y.J., Z. Dong, and G.W. Vickers, Euler-Meusnier Sphere Based Milling

Cutter Model for Curvature Gouge Avoidance in Curved Surface Machining. Society

of Manufacturing Engineers, 2008: pp. 1-9.

133

39. Wang, Y.J., Z.M. Dong, and G.W. Vickers, A 3D curvature gouge detection and

elimination method for 5-axis CNC milling of curved surfaces. International Journal


40. Olling, G., et al., Five-axis Control Sculptured Surface Machining Using Conicoid

End Mill, in Machining Impossible Shapes. 1999, Springer US. pp. 366-375.

41. Chiou, C.J. and Y.S. Lee, A shape-generating approach for multi-axis machining G-

buffer models. Computer-Aided Design, 1999. 31(12): pp. 761-776.

42. Jun, C.S., K. Cha, and Y.S. Lee, Optimizing tool orientations for 5-axis machining

by configuration-space search method. Computer-Aided Design, 2003. 35(6): pp.

549-566.

43. Wang, N. and K. Tang, Automatic generation of gouge-free and angular-velocity-

compliant five-axis toolpath. Computer-Aided Design, 2007. 39(10): pp. 841-852.

44. Lu, J., et al., A three-dimensional configuration-space method for 5-axis tessellated

surface machining. International journal of computer integrated manufacturing,

2008. 21(5): pp. 550-568.

45. Yoon, J.-H., H. Pottmann, and Y.-S. Lee, Locally optimal cutting positions for 5-axis

sculptured surface machining. Computer-Aided Design, 2003. 35(1): pp. 69-81.

46. Ho, M.-C., Y.-R. Hwang, and C.-H. Hu, Five-axis tool orientation smoothing using

quaternion interpolation algorithm. International Journal of Machine Tools and

Manufacture, 2003. 43(12): pp. 1259-1267.

47. Makhanov, S.S. and W. Anotaipaiboon, Advanced Numerical Methods to Optimize

Cutting Operations of Five-Axis Milling Machines, 5, Editor. 2007, Springer Series

in Advanced Manufacturing.

48. Ozturk, E. and E. Budak, Modelling of 5-axis milling process. Machining Science

and Technology, 2007. 11(3): pp. 287-311.

49. Budak, E., E. Ozturk, and L.T. Tunc, Modeling and simulation of 5-axis milling

processes. CIRP annals, 2009. 58(1): pp. 347-350.

50. Farouki, R.T., Y.F. Tsai, and C.S. Wilson, Physical constraints on feedrates and feed

accelerations along curved toolpaths. Computer Aided Geometric Design, 2000.

17(4): pp. 337-359.

51. Lacalle, L.N.L.p.d., et al., Toolpath selection based on the minimum deflection

cutting forces in the programming of complex surfaces milling. Computer aided

design, 2007. 47(2007): pp. 388–400.

134

52. Bi, Q.Z., et al., Wholly smoothing cutter orientations for five-axis NC machining

based on cutter contact point mesh. Science China-Technological Sciences, 2010.

53(5): pp. 1294-1303.

53. Weinert, K., et al., Swept volume generation for the simulation of machining

processes. International Journal of Machine Tools and Manufacture, 2004. 44(6):

pp. 617-628.

54. Blackmore, D., M.C. Leu, and L.P. Wang, The sweep-envelope differential equation

algorithm and its application to NC machining verification. Computer-Aided Design,

1997. 29(9): pp. 629-637.

55. Martin, R.R. and P.C. Stephenson, Sweeping of three-dimensional objects.

Computer-Aided Design, 1990. 22(4): pp. 223-234.

56. Abdel-Malek, K. and H.-J. Yeh, Geometric representation of the swept volume using

Jacobian rank-deficiency conditions. Computer-Aided Design, 1997. 29(6): pp. 457-

468.

57. Du, S., et al., Formulating swept profiles for five-axis tool motions. International

Journal of Machine Tools & Manufacture, 2005. 45(7-8): pp. 849-861.

58. Ozturk, B. and I. Lazoglu, Machining of free-form surfaces. Part I: Analytical chip

load. International Journal of Machine Tools and Manufacture, 2006. 46(78): pp.

728-735.

59. Lee, S. and A. Nestler, Virtual workpiece: workpiece representation for material

removal process. The International Journal of Advanced Manufacturing Technology,

2012. 58(5-8): pp. 443-463.

60. Lee, S.W. and A. Nestler, Complete swept volume generation, Part I: Swept volume

of a piecewise C1-continuous cutter at five-axis milling via Gauss map. Computer-

Aided Design. 43(4): pp. 427-441.

61. Ferry, W. and D. Yip-Hoi, Cutter-Workpiece Engagement Calculations by Parallel

Slicing for Five-Axis Flank Milling of Jet Engine Impellers. Journal of

manufacturing science and engineering, 2008. 130(5): pp. 51011.

62. Boz, Y., H. Erdim, and I. Lazoglu. Modeling Cutting Forces for Five Axis Milling of

Sculptured Surfaces. 2011: TRANS TECH PUBLICATIONS LTD.

63. Fussell, B.K., R.B. Jerard, and J.G. Hemmett, Modeling of cutting geometry and

forces for 5-axis sculptured surface machining. Computer aided design, 2003. 35(4):

pp. 333-346.

64. Altintas, Y., Manufacturing automation: metal cutting mechanics, machine tool

vibrations, and CNC design 2012, Cambridge ; New York

135

65. Engin, S. and Y. Altintas, Mechanics and dynamics of general milling cutters.: Part I:

helical end mills. International Journal of Machine Tools and Manufacture, 2001.

41(15): pp. 2195-2212.

66. Liang, X.-G. and Z.-Q. Yao, An accuracy algorithm for chip thickness modeling in

5-axis ball-end finish milling. Computer-Aided Design, 2011. 43(8): pp. 971-978.

67. Huang, T., X. Zhang, and H. Ding, Decoupled chip thickness calculation model for

cutting force prediction in five-axis ball-end milling. Int J Adv Manuf Technol, 2013.

2013.

68. Ferry, W.B. and Y. Altintas, Virtual five-axis flank milling of jet engine impellers -

Part I: Mechanics of five-axis flank milling. Journal of manufacturing science and

engineering, 2008. 130(1).

69. Ozturk, B. and I. Lazoglu, Machining of free-form surfaces. Part I: Analytical chip

load. International Journal of Machine Tools & Manufacture, 2006. 46(7-8): pp.

728-735.

70. Lamikiz, A., et al., Cutting force estimation in sculptured surface milling.

International Journal of Machine Tools & Manufacture, 2004. 44(14): pp. 1511-

1526.

71. Sonawane, H.A. and S.S. Joshi, Analytical modeling of chip geometry and cutting

forces in helical ball end milling of superalloy Inconel 718. CIRP Journal of

Manufacturing Science and Technology, 2010. 3(3): pp. 204-217.

72. Bouzakis, K.D., P. Aichouh, and K. Efstathiou, Determination of the chip geometry,

cutting force and roughness in free form surfaces finishing milling, with ball end

tools. International Journal of Machine Tools and Manufacture, 2003. 43(5): pp.

499-514.

73. Zhang, L., Process modeling and toolpath optimization for five-axis ball-end milling

based on tool motion analysis. The International Journal of Advanced

Manufacturing Technology, 2011. 57(9-12): pp. 905-916.

74. Wang, Y., Z. Dong, and G. Vickers, A 3D curvature gouge detection and elimination

method for 5-axis CNC milling of curved surfaces. The International Journal of

Advanced Manufacturing Technology, 2007. 33(3-4): pp. 368-378.

75. Chen, Z.Z.C., Z.M. Dong, and G.W. Vickers, Automated surface subdivision and

toolpath generation for 3 1/2 1/2-axis CNC machining of sculptured parts.

Computers in Industry, 2003. 50(3): pp. 319-331.

76. Collazo-Cuevas, J.I., et al. Comparison between Fuzzy C-means clustering and

Fuzzy Clustering Subtractive in urban air pollution. in Electronics, Communications

and Computer (CONIELECOMP). 2010.

136

77. Edelsbrunner, H. and E.P. M, Three-dimensional alpha shapes. ACM Trans. Graph.,

1994. 13(1): pp. 43-72.

78. Chiou, C.J. and Y.S. Lee, Swept surface determination for five-axis numerical

control machining. International Journal of Machine Tools and Manufacture, 2002.

42(14): pp. 1497-1507.

79. Edelsbrunner, H., D. Kirkpatrick, and R. Seidel, On the shape of a set of points in the

plane. Information Theory, IEEE Transactions on, 1983. 29(4): pp. 551-559.

80. Mayor, J.R. and A.A. Sodemann, Intelligent Tool-Path Segmentation for Improved

Stability and Reduced Machining Time in Micromilling. Journal of manufacturing

science and engineering, 2008. 130(3): pp. 31121.

81. Shan Luo, et al., Chip Ploughing Volume Calculation and Simulation in Micro Ball-

End Mill Machining, in 8th International Conference on Micromanufacturing. 2013:

Victoria, Canada.

82. Wang, Y.J., Z. Dong, and G.W. Vickers, A 3D curvature gouge detection and

elimination method for 5-axis CNC milling of curved surfaces. International Journal


83. Li, X., Curvature analysis and geometric description of landforms using MATLAB,

in Environmental Science and Information Application Technology (ESIAT). 2010:

Wuhan. p. 712 - 715.

84. Chiou, J.C.J. and Y.-S. Lee, Five-Axis High Speed Machining of Sculptured

Surfaces by Surface-Based NURBS Path Interpolation Computer-Aided Design &

Applications, 2007. 4(5): pp. 639-648

85. Liang, H.B. and X. Li, A 5-axis Milling System Based on a New G code for NURBS

Surface. 2009 Ieee International Conference on Intelligent Computing and Intelligent

Systems, Proceedings, Vol 2, ed. W. Chen, S.Z. Li, and Y.L. Wang. 2009, 5. 600-

606.

86. Yangtao Li Arc-Length Parameterized NURBS Toolpath Generation and Velocity

Profile Planning for Accurate 3-Axis Curve Milling in Mechanical Engineering.

2012, Concordia University Montreal. p. 86.

87. Ren, Y., S.K. Lai-Yuen, and Y.-S. Lee, Virtual prototyping and manufacturing

planning by using tri-dexel models and haptic force feedback. Virtual and Physical

Prototyping, 2006. 1(1): pp. 3-18.

88. Zhu, W. and Y.-S. Lee, Dexel-based force-torque rendering and volume updating for

5-DOF haptic product prototyping and virtual sculpting. Computers in Industry,

2004. 55(2): pp. 125-145.

137

89. Benouamer, M.O. and D. Michelucci, Bridging the gap between CSG and Brep via a

triple ray representation, in Proceedings of the fourth ACM symposium on Solid

modeling and applications. 1997, ACM: Atlanta, Georgia, USA.

90. Hook, T.V., Real-time shaded NC milling display. SIGGRAPH Comput. Graph.,

1986. 20(4): pp. 15-20.

91. Huang, Y. and J.H. Oliver, NC milling error assessment and toolpath correction, in

Proceedings of the 21st annual conference on Computer graphics and interactive

techniques. 1994, ACM.

92. Muller, H., et al., Online sculpting and visualization of multi-dexel volumes, in

Proceedings of the eighth ACM symposium on Solid modeling and applications.

2003, ACM: Seattle, Washington, USA.

93. Karunakaran, K.P. and R. Shringi, A solid model-based off-line adaptive controller

for feed rate scheduling for milling process. Journal of Materials Processing

Technology, 2008. 204(1): pp. 384-396.

94. Walstra, W.H., W.F. Bronsvoort, and J.S.M. Vergeest, Interactive simulation of

robot milling for rapid shape prototyping. Computers & Graphics, 1994. 18(6): pp.

861-871.

95. Bayesteh, A. and M.B.-G. Jun, Feed rate optimization issues in micro-milling, in

Proceedings of NAMRI/SME. 2013. p. 1-8.

96. Cheng, X., et al., Design and development of a micro polycrystalline diamond ball

end mill for micro/nano freeform machining of hard and brittle materials. Journal of

micromechanics and microengineering, 2009. 19(11): pp. 115022.

97. Chen, M.J., et al., Research on the Influence Factors for the Deflection of Micro-

ball-end Cutter in Micro-end-milling Process, in Advances in Materials

Manufacturing Science and Technology Xiv, T. Huang, et al., Editors. 2012, Trans

Tech Publications Ltd: Stafa-Zurich. pp. 84-87.

98. Wang, J.-J.J. and C.M. Zheng, Identification of shearing and ploughing cutting

constants from average forces in ball-end milling. International Journal of Machine

Tools & Manufacture, 2002. 42(6): pp. 695-705.

99. Liu, X., et al., A Geometrical Simulation System of Ball End Finish Milling Process

and Its Application for the Prediction of Surface Micro Features. Journal of

manufacturing science and engineering, 2006. 128(1): pp. 74-85.

100. Bono, M. and J. Ni. Experimental analysis of chip formation in micro-milling. 2002:

SOC MANUFACTURING ENGINEERS.

138

101. Rahman, M., W.Y. San, and L. Kui, Micro-Ball Endmilling of Tungsten Carbide for

Micro-Molding and Prototyping Applications. Key engineering materials, 2012: pp.

591-594.

102. Ikawa, N., S. Shimada, and H. Tanaka, Minimum thickness of cut in

micromachining. Nanotechnology, 1992. 3(1): pp. 6.

103. Weule, H., V. Huntrup, and H. Tritschler, Micro-Cutting of Steel to Meet New

Requirements in Miniaturization. CIRP Annals - Manufacturing Technology, 2001.

50(1): pp. 61-64.

104. Vogler, M.P., R.E. DeVor, and S.G. Kapoor, On the Modeling and Analysis of

Machining Performance in Micro-Endmilling, Part I: Surface Generation. Journal of

manufacturing science and engineering, 2005. 126(4): pp. 685-694.

105. Ramos, A.C., et al., Characterization of the transition from ploughing to cutting in

micro machining and evaluation of the minimum thickness of cut. Journal of

Materials Processing Technology, 2012. 212(3): pp. 594-600.

106. Jun, M.B.G., et al., Investigation of the Dynamics of Microend Milling---Part I:

Model Development. Journal of manufacturing science and engineering, 2006.

128(4): pp. 893-900.

107. Bayesteh, A., D. Gym, and M.B.G. Jun. 2-Dimensional Ploughing Simulation Model

Development in Micro Flat End Milling. in International Conference on

Micromanufacturing. 2013.

108. Liu, X., R.E. DeVor, and S.G. Kapoor, An Analytical Model for the Prediction of

Minimum Chip Thickness in Micromachining. Journal of manufacturing science and

engineering, 2005. 128(2): pp. 474-481.

109. Lim, E.M., et al., the prediction of dimensional error for sculptured surface

productions using the ball-end milling process .1. Chip geometry analysis and cutting

force prediction. International Journal of Machine Tools & Manufacture, 1995.

35(8): pp. 1149-1169.

110. Tsai, C.-L. and Y.-S. Liao, Prediction of cutting forces in ball-end milling by means

of geometric analysis. Journal of Materials Processing Technology, 2008. 205(1-3):

pp. 24-33.

111. Kline, W.A. and R.E. Devor, The effect of runout on cutting geometry and forces in

end milling. Int.J.Mach. Tool Des. Res, 1983. 23(2/3): pp. 123-140.

112. Brissaud, D., et al., Influence of the ploughing effect on the dynamic behaviour of

the self-vibratory drilling head. CIRP annals, 2008. 57(1): pp. 385-388.

139

113. Altintas, Y., Manufacturing Automation: Metal Cutting Mechanics, Machine Tool

Vibrations, and CNC Design. 2nd ed. 2012, Cambridge ; New York Cambridge

University Press. xii, 366 p.

140

Appendix1

Cutting Constants and Edge Constants Calculations

This following section introduces the method to calculate cutting constants (Krc, Kac, Ktc)

and edge constants (Kre, Kae, Kte) using experiment data with different feed rate. This is

required in Sections 6.2, 6.3 and 6.4. The experiment was conducted in a 3-axis ALIO

micro-milling machine with a spindle speed of 12,000 rpm and different feed rates from

700 mm/rev to 1200 mm/rev. A four-flute flat-end mill with a diameter of 2 mm was

used to cut an AL 6061 workpiece with slot machining. The depth of cut is 0.8 mm. The

cutting forces were measured and the average forces were given in Table 4. Assuming the

force model is given in Eq.(A1.1) [113]:

t tc te

r rc re

a ac ae

F K ah K a

F K ah K a

F K ah K a

(A1.1)

Table 4: Cutting parameters for slot machining in the 3-axis micro-milling

feedrate

(mm/rev)

spindle

speed

(rpm)

feed per

tooth

(mm/tooth)

depth

of cut

(mm)

immersio

n angle

(°)

average

Fx (N)

average

Fy (N)

average

Fz (N)

700 12000 0.0146 0.8 180 -24.6393 2.8470 2.7050

800 12000 0.0167 0.8 180 -25.4856 3.5036 3.3974

900 12000 0.0188 0.8 180 -25.6002 3.9009 3.5197

1000 12000 0.0208 0.8 180 -25.7261 4.0828 3.6230

1100 12000 0.0229 0.8 180 -25.9135 4.4313 3.4066

1200 12000 0.0250 0.8 180 -26.2096 4.5467 3.5082

141

Full immersion milling experiments are most convenient. Here the entry and exit angles

are equal 0 and π respectively. Full immersion conditions are applied into Eq. (A1.2)

[113], the average forces per tooth period are simplified as

4

4

x rc re

y tc te

z ac ae

Na NaF K h K

Na NaF K h K

Na NaF K h K

(A1.2)

where, N is number of cutting flutes, c is the depth of cut, Krc, Kac and Ktc are radial, axial

and tangential cutting constants. The average cutting forces in x, y and z direction are

obtained from Table 4. They are plot in the FigureA1- 1.

FigureA1- 1: Average cutting forces

Compare Eq. (A1.1) and Eq. (A1.2), it can get the radial, axial and tangential cutting

constants:

44

; ;yx z

rc tc rc

FF FK K K

Na Na Na

(A1.3)

-30

-25

-20

-15

-10

-5

0

5

10

1 2 3 4 5 6

Average Fx

Average Fy

Average Fz

142

The linear interpolation function of average force in x-axis is obtained from the six

groups’ data of average forces and feed rates, shown in FigureA1- 2. The interpolation

function is given in Eq. (A1.4).


forces Fxc

y 127.7x 23.1xc xeF x F

(A1.4)

Substitute the values of 𝐹𝑥𝑐 and 𝐹𝑥𝑒

from Eq. (A1.4) to Eq. (A1.3) and Eq. (A1.2), it can

get the values of Krc and Kre:

4 4 127.7159.6

4 0.8

4 23.122.7

4 0.8

xcrc

xere

FK

Na

FK

Na

(A1.5)

0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028-26.4

-26.2

-26

-25.8

-25.6

-25.4

-25.2

-25

-24.8

-24.6

-24.4linear function of feed rate and an offset contributed by the edge forces Fxc

feed per tooth (mm/tooth)

Fx (

N)

143

The linear interpolation function of average force in y-axis is shown in FigureA3- 3. The

interpolation function for average force 𝐹�� is given in Eq. (A1.6).


forces Fyc

y =157.98x+0.76yc yeF x F

(A1.6)

Substitute the values of 𝐹𝑦𝑐 and 𝐹𝑦𝑒

from Eq. (A1.6) to Eq. (A1.3) and Eq. (A1.2), it can

get the values of Ktc and Kte:

4 4 157.98197.5

4 0.8

4 0.760.75

4 0.8

yc

tc

ye

te

FK

Na

FK

Na

(A1.7)

0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.0282.5

3

3.5

4

4.5

5linear function of feed rate and an offset contributed by the edge forces Fyc


Fy (

N)

144

The linear interpolation function of average force in z-axis is shown in FigureA1- 4. The

interpolation function for average force 𝐹�� is given in Eq. (A1.8).


forces Fzc

y 57.2x+2.2zc zeF x F

(A1.8)

Substitute the values of 𝐹𝑧𝑐 and 𝐹𝑧𝑒 from Eq. (A1.8) to Eq. (A1.3) and Eq. (A1.2), it can

get the values of Kac and Kae:

4 57.256.2

4 0.8

2 2 2.21.4

4 0.8

zcac

zeae

FK

Na

FK

Na

(A1.9)

0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.0282.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7linear function of feed rate and an offset contributed by the edge forces Fzc


Fz (

N)

145

Appendix2

Another Method to Identify Optimal Number of Surface Patches

Section 3.2 presents an optimization method to find optimal number of surface patches by

minimizing the objective function of the change rate of second derivative of

accumulating relative angle. An algorithm was developed to give the mathematic method

to identify optimal number of surface patches. Before this approach was proposed,

another abstract method using the distributions of relative angles defined by the angle

surface normal at cluster centre and grid points in every surface patch.

Relative angle d(Yi, Wj) (shown in Figure 3-1) is used to find optimal number of surface

patches, which is defined by the cross and dot products of surface normal vectors Yi at the

ith

grid point and Wj at the jth

cluster centre:

1( , ) tan ( )

i j

i j

i j

Y Wd Y W

Y W

(A2.1)

In FigureA2- 1, distributions of the relative angles with different number of cluster

centres are shown using two variables, relative angles and number of mesh points for a

convex half sphere with 30 surface patches. When machining the entire surface in one

patch, the largest relative angle is around 150°, whereas the smallest relative angle

appears when 30 patches are used. The larger the relative angle the more tool orientation

changes it has. Compared graphs with 20 and 30 cluster centres, it can be seen that there

are not too many changes for the distributions of relative angles as the number of cluster

centre is increased dramatically. However, when the cluster centres is less than 10, the

relative angle has a fairly large decrease from 150° to 30°, while the number of cluster

centre increases from 1 to 10.

146

FigureA2- 1: Distributions of relative angles with different numbers of cluster centres.

Gradient of deviations for relative angles and number of cluster centres is used to define

the rate of cutter orientation changes. In FigureA2- 2 (b), it shows the largest gradient

happens when the number of cluster centres changes from one to two. It means cutter

orientation changes can be decreased by the surface patch method. The gradients for the

lines of maximum and average relative angle become steady, as the number of cluster

centres is 10, which means there is unnecessary to increase the number of cluster centres

to reduce cutter orientations when the number is larger than 10. Therefore, the optimal

number of cluster centres is 10.

0 20 40 60 80100120140160180

10

30

50

100

150

180

Relative angles (°)

Nu

mb

er

of m

esh

po

ints

Without surface division

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

2 cluster centers

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

3 cluster centers

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

4 cluster centers

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

5 cluster centers

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

6 cluster centers

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

10 cluster centers

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

20 cluster centers

0 20 40 60 80100120140160180

10

30

50

100

150

180


Nu

mb

er

of m

esh

po

ints

30 cluster centers

147

FigureA2- 2: (a) Relation of cluster centre numbers and the maximum and average

relative angles; (b) the change rates of cluster centre numbers and maximum and average

relative angles.

The optimal number of surface patches obtained by this method is same with the one got

by the approach proposed in Section 3.2 as the termination criterion ε is 0.2. Extending

the method introduced in this section that only can give one optimal number of surface

patches by identifying the largest gradient of deviations for relative angles and number of

cluster centres, the approach proposed in Section 3.2 gives user a degree of flexibility to

choose the optimal cluster numbers of NURBS surface by different termination criterion

ε. Anyway, the method introduced in this section is a good try and pre-study to get better

way demonstrated in Section 3.2 to find optimal number of surface patches.

148

Appendix3

Comparisons of Ball-end Milling and Flat-end Milling

This section presents the comparisons of machining time and cusps or gouges between a

ball-end mill and a flat-end mill. In Section 0, it gives a brief explanation why a flat-end

mill is better than a ball-end mill in surface quality and machining efficiency. In this

section, some simulations in CAM software with different tilt angles and tool diameters

in 5-axis CNC machining are conducted to get the data and prove why a flat-end mill is

selected in the research. In FigureA3- 1, it shows ball-end and flat-end milling with

different tilt angles. The tool diameters of the ball-end mill and the flat-end mill are both

5 mm.

FigureA3- 1: Ball-end milling (a) tilt angle=1°; (b) tilt angle=5.78°;(c) tilt angle=10°;

flat-end milling (d) tilt angle=1°; (e) tilt angle=5.78°;(f) tilt angle=10°

149

An optimal tilt angle got by the EMS method from Section 4.1.2 is used to avoid gouges,

which is 5.78°. From FigureA3- 2 (a), it can be seen that a flat-end mill has shorter

machining time than a ball-end mill when the tool diameter and tilt angle are the same. It

also demonstrates that machining time is rising as tilt angle increased. FigureA3- 2 (b)

shows the comparison of gouges or cusps generated by ball-end and flat-end mills with

different tilt angles. Traditionally, to avoid gouges, a small diameter ball-end mill which

is smaller than the minimal curvature of the machined surface is selected. However, it

causes low machining efficiency and large cusps. Cusps generated by a ball-end mill is

independent with tilt angles, keeping constant no matter how tilt angle changes. In

FigureA3- 2 (b), it shows there are no gouges generated when the tilt angle is the optimal

angle of 5.78° obtained by the EMS method. If a tilt angle is larger than the optimal

angle, gouge would be generated; while the tilt angle is smaller than the optimal angle, it

leads to cusps with a flat-end mill.

FigureA3- 2: (a) comparison of machining time with different tilt angles between ball-

end milling and flat-end milling

150

Overall, a ball-end mill always causes constant cusps, due to the constant curvature; a

flat-end mill can eliminate cusps and gouges when the tilt angle is optimal based on

curvature match.

Tool diameter is another significant parameter to control cusps and gouges. FigureA3- 3

shows the 5-axis machining and toolpaths with various tool diameters and tilt angles.

Two different sizes of machining tools are used to compare the machining time. The

diameter of the concave half cylinder is 50 mm. A large diameter tool of 50 mm which

matches the curvature of the concave half cylinder is used as the extreme condition to

achieve the most fast machining time with proper tool rotational angles.

FigureA3- 3: (a) ball-end milling in several toolpaths, D=5mm, tilt angle=5.78°; (b) flat-

end milling in several toolpaths, D=5mm, tilt angle=5.78°;(c) ball-end milling in one

toolpath, D=50mm, tilt angle=0°; (d) flat-end milling in one toolpath, D=50mm, tilt

angle=90°

151

FigureA3- 4: (a) comparison of machining time with different tool diameters between

ball-end milling and flat-end milling

In FigureA3- 3 (a) and (b), the tool diameter is 5 mm and the tilt angle is the optimal

angle for the flat-end mill to avoid gouges. (c) and (d) show large diameter of ball-end

and flat-end milling with one toolpath to get fast CNC machining, no gouges or cusps

generated. To get the highest machining efficiency, the tilt angle for the ball-end mill is

0° and for the flat-end mill is 90° as the tool diameter is matching with the concave half

cylinder surface. The concave half cylinder surface is a special surface for ball-end

milling without gouges or cusps generation as the tool diamater matches surface

curvature. For other surfaces, cusps problems always exist in the ball-end mill machining.

From FigureA3- 2 and FigureA3- 4, it can be seen that a flat-end mill can eliminate

gouges by changing tilt angles or tool diameters, but for a ball-end mill, cusps cannot be

avoided expect the tool diameter is same with a concave half cylinder surface. To avoid

surface gouges at large curvature areas, and to simplify the toolpath/orientation planning,

a small diameter ball-end mill is commonly used during machining, this leads to low

machining efficiency, less rigid machine-cutter-part system, and large cusp for areas of

152

the surface with small curvature. Therefore, a flat-end mill is selected in 5-axis CNC

machining due to its flexibility and high efficiency.

153

Appendix4

Numerical Simulation of Chip Ploughing Volume in Micro Milling

The majority of this chapter is devoted to model 3D chip geometry used to accurately

calculate the chip ploughing volume to support needed toolpath adjustments for ensured

accuracy of the finished part.

This chapter starts by giving a short introduction of ploughing effects in micro milling

in Section A4.1. Chip ploughing becomes a more serious problem in micro-machining

due to its relatively large size with respect to the finished part, negatively impacting on

the accuracy of the finished surface. In Section A4.2, the chip model proposed in Chapter

5 is used to predict chip ploughing and shearing volume in 5-axis CNC micro milling

with flat-end mills. Section A4.3 discusses the ploughing effects in 3-axis micro ball-end

milling. A 3D model is presented in Section A4.3.1. In Section A4.3.2, a new method is

developed to compute the chip ploughing volume by dividing the modeled chip into

many discrete pieces over a ploughing dominated region and a shearing dominated

region. Chip thickness calculation considering runout effects was discussed in Section

A4.3.3. Section A4.3.4 presented an algorithm for ploughing volume calculation. The

study in Section A4.3.5 discussed two simulation methods, analytical method and Dexel

method, to calculate chip ploughing volume in the two given toolpaths. Different axial

depths of cut, spindle speed, and feed rate are tested to study the ploughing effects, which

were discussed in Section A4.3.6. Experiments of cutting force measurement are also

compared with the simulation results in Section A4.3.7.

154

A4.1 Introduction

With the advantages of higher accuracy and reduced costs, micro-milling is able to

produce various miniature components with complex geometry. Mechanical micro-

machining plays more roles to many industries than ever before from biomedical,

electronics, automotive industry, and aerospace applications [96-101].

Certain macro cutting mechanisms are no longer applicable to micro-milling anymore

due to the much smaller feed per tooth than the tool’s cutting edge radius. The minimum

chip thickness has not been considered in the macro machining process. Furthermore, the

small micro tools leads to low cutter stiffness, elevated tool wear, and breakage in

machining a hardened workpiece with improper machining parameters [100].

It is a challenging task to avoid ploughing problems in the micro-milling process due

to the shape of micro-mills, such as flat-end mills and ball-end mills. When the cutter

crosses the minimum chip thickness boundary, the tool enters into the ploughing zone

with no material removal. The uncut chip volume varies with the cutting edge and the

depth of cut in the axial direction. The variation results in an increase of the ploughing

area, which can cause increased thrust force. It is vital to better understand the relations

between ploughing and minimum chip thickness in micro-milling to improve the

machining efficiency and obtain a better finish surface.

A couple of studies have been conducted to investigate the effects of minimum chip

thickness and ploughing effects with micro end mills in 3-axis milling. It is difficult to

avoid ploughing effects in micro-milling with low feed rate and small uncut chip

thickness. Unlike traditional macro-milling, chip thickness is not always larger than the

cutting edge radius in micro-machining. Ikawa [102] defined the minimum chip thickness

155

as the minimum undeformed chip thickness at a cutting edge under perfect performance

of the machine tool without system deflection and tool wear out. Ploughing domain or

zone is the area in which the chip thickness is less than the minimum chip thickness. In

micro-milling, the minimum chip thickness is relative with cutting edge radius. The

influence of minimum chip thickness is significant if the cutting tool enters into a

ploughing zone, since no material would be removed in ploughing zone [103]. The

effects of minimum chip thickness have been studied by many researchers. Vogler [104]

discussed the effects of minimum chip thickness by cutting experiments. It was

discovered that chip formation occurs only as the chip thickness is larger than the

minimum chip thickness and no chip is formed if the feed rate is low and the minimum

chip thickness is not exceeded. Ramos [105] investigated that ploughing effects influence

the chip formation process, burr formation, surface roughness, and residual stress. The

minimum chip thickness is decreased while the cutting velocities are enlarged and it is

increased with a higher cutting tool edge radius. Wang analyzed the dual effects of

shearing and ploughing effects in terms of axial depth of cut and tool radius through a set

of slot machining tests, while the shearing and ploughing areas remained constant due to

the straight toolpath [98]. Jun reported that chip formation mechanism has changed in

micro-end milling, due to the change of relative size of the cutting edge radius to the chip

thickness. The effects of minimum chip thickness and ploughing to cutting dynamics by a

chip thickness model were investigated and the elastic recovery and elastic-plastic in the

ploughing process was considered [106]. However, the volume of uncut chip has not

been studied. Bayesteh [107] developed a dual-dexel model to calculate ploughing area

156

and volume by considering the minimum chip thickness in 3-axis CNC micro milling

with flat-end mills.

There are many researches about ploughing effects in 3-axis micro-milling but very

limited studies in 5-axis flat-end micro machining due to the complexity of cutter-

workpiece contact geometry. The object of this work is to develop a new 3D chip model

with micro flat-end mills to accurately calculate the chip ploughing volume and cutting

forces. Chip volume prediction for a micro ball-end mill is also considered in this work.

To better understand the chip ploughing behavior under different cutting conditions and

to obtain a generic solution to improve machining accuracy and efficiency, different axial

depths of cut and feed rates are tested to control the ploughing area. Toolpaths are

generated to simulate the machining process and determine the relationship between chip

ploughing volume and machining time.

A4.2 Ploughing effects in 5-axis Micro Flat-end Milling

A4.2.1 Chip Geometry of a 5-axis Micro Flat-end Mill

Chip thickness and volume calculation for macro 5-axis flat-end milling by local parallel

sliced method is presented in the Section 5.2.2 of Chapter 5. This method can also be

used in micro 5-axis flat-end milling.

157

Figure A4- 1: Determination of the instantaneous chip thickness in the 5-axis micro flat-

end milling: (a) Tool motions at two adjacent NC points; (b) Ploughing and shearing

areas in tool projections on the A-A section

Figure A4- 1 (a) shows the process of modeling chip geometry in a 5-axis CNC

machining using a micro flat-end mill. Chip thickness is obtained by identifying

intersections of tool edges at the previous tool position (denoted by Ol’-Xl’-Yl’-Zl’) and

the current tool position (represented by Ol-Xl-Yl-Zl). Numerical method is used to get the

intersections of tool edges and chip thickness by slicing the tool into many slices along

the direction which is vertical to the tool axis. Figure A4- 1 (b) shows the instantaneous

chip thickness distribution on one layer. Ci and Ci’ are the current and the previous tool

centres on the ith

layer. Pi, k is the kth

interval point on the current tool’s cutting edge

determined by equations (5.11) and (5.12). Pi, k’ is the intersection of line segment CiPi, k

158

and tool edge at previous position. Chip thickness tc for the kth

interval point can be

obtained as the distance between Pi, k and Pi, k’ by Eq. (5.15).

Ploughing zone happens as chip thickness is less than the minimum chip thickness. In

Figure A4- 1 (b), the ploughing area is shown in the blue shade domain. In the shearing

area, chip thickness is larger than the minimum chip thickness. The minimum chip

thickness denoted by tcmin is related to the tool edge radius re [108]:

min 0.3c e

ft r

S N

(A4.1)

where, f is the feed rate, S is the spindle speed, N is the number of flutes.

A4.2.2 Chip ploughing area/volume by local parallel sliced method

Figure A4- 2 (b) illustrates a chip shape that is divided many layers along the direction

which is perpendicular to the current tool axis. From Figure A4- 1 (b), it can be seen that

on each layer, the removal chip area is a polygon shape generated by connecting two

neighbouring edge points, Pi, k and Pi, k’, on the current and previous tool edges. The

ploughing area is obtained by connecting edge points which chip thickness is less than

the minimum chip thickness tcmin. It is expressed in the following equation:

1

, , 1 , , , , 1 , ,

1

s M

ploughing i k i k i k i k i k i k i k i k

k k t

A P P P P P P P P

(A4. 2)

where, M is the number of interval points on each layer; t and s are the index of edge

points which chip thickness starts and exits to be less than the minimum chip thickness.

The shearing area shown in Figure A4- 1 (b) is the area where chip thickness is larger

than the minimum chip thickness.

1

, , 1 , ,

1

t

shearing i k i k i k i k

k s

A P P P P

(A4. 3)

159

From the Section 5.2.2 of Chapter 5, it is already known a chip shape is composed by

many parallelepipeds. Total chip volume is obtained by adding the volume of all

parallelepipeds along an/the axial direction. Total ploughing volume is integrated by

adding all parallelepipeds which the length along radial direction is smaller than the

minimal chip thickness. The equation of total chip ploughing volume is defined by:

1 1 1

, , 1 , , , , 1 , ,

1 1 1

s N M N

ploughing i k i k i k i k i k i k i k i k

k i k t i

V P P P P z P P P P z

(A4. 4)

where, M is the number of interval points on each layer, N is the number of slices; Δz is

the integrating height along the current tool axis.

The shearing volume can be obtained by adding all parallelepipeds which the length

along radial direction is larger than the minimal chip thickness.

1 1

, , 1 , ,

1 1

t N

shearing i k i k i k i k

k s i

V P P P P z

(A4. 5)

The total volume is the sum of ploughing volume and shearing volume:

total ploughing shearingV V V

(A4. 6)

160

Figure A4- 2: (a) Ploughing and shearing volume; (b) Ploughing and shear areas on

layers

A4.2.3 Case Studies and Results

In this section, a 5-axis micro CNC machine is used to simulate chip ploughing volume

and ploughing cutting forces. A free form surface shown in Figure A4- 3 is machined by

a two-flute flat-end mill, with a tool diameter of 1/32″. The length, width and height of

the workpiece size are 5 × 5 × 3 mm. The depth cut varying from 0.1 mm to 2 mm. The

spindle speed is selected as 30,000 rpm, and the feed rate is 0.004 mm/tooth, the

minimum chip thickness is 0.0012 mm.

161

Figure A4- 3: A free-form surface in micro-milling with a flat-end mill

The one-way toolpaths for the free-form surface are generated in CAM software with

surface normal as the tool orientation method. NC points got by CAM software are

required to be interpolated with uniform distance of feed per tooth to calculate cutting

forces and chip volume. A toolpath with interpolated NC points is shown in Figure A4- 4.

There are around 3000 NC points generated in this toolpath.

Figure A4- 4: The interpolated toolpath

0

5

10

15

20

1

2

3

4-1.4

-1.2

-1

-0.8

-0.6

162

Ploughing and shearing volume are obtained from Eqs. (A4. 4) and (A4. 5). Total volume

is the sum of ploughing and shearing volume. Figure A4- 5 shows the total ploughing and

shearing volume changes with machining time for the whole toolpath.

Figure A4- 5: Comparison of the total, ploughing and shearing volume

A4.3 Ploughing Effects in 3-axis Micro Ball-end Milling

A4.3.1 Chip Geometry in Micro Ball-end Milling

The ball-end milling process is widely used in machining dies and molds for

automotive, medical, and aerospace components with sculptured surfaces. Due to the

constant radius between the cutter contact point and the center of the cutter, it is relatively

easy to generate the toolpath and to calculate the cutting forces and material remove rate

(MRR) for a ball–end mill. Lim [109] presented a 2D chip model to analyze the chip

0 0.5 1 1.5 2 2.5 30

0.0005

0.001

0.0015

0.002

0.0025

0.003

Machining time (s)

Volu

me (

mm

3)

Volume changed with machining time

Ploughing volume

Shearing volume

Total volume

163

engagement surface and calculate the undeformed radial chip thickness with ball-end

mill. This model was further extended to a new 3D geometry model related to the rank

angle, shear plane area, chip thickness presented by Tsai [110]. An extended 3D chip

geometry model with ploughing and shearing areas has been developed as shown in

Figure A4- 6. The slice plane is along the radial plane rather than horizontal direction.

The study on chip geometry involves the undeformed radial chip thickness in the cutting

plane, the rotation angle and inclination angle, the surface generated by previous tooth

path and the present machined surface. In order to calculate the volume of the chip in

micro ball–end milling, a single horizontal cut with an axial depth of cut equal to the

cutter radius has been proposed.

Figure A4- 6: A 3D chip geometry of a micro ball-end mill feed in the horizontal

direction

164

A4.3.2 Ploughing Volume Calculation for Ball-end Milling

In Figure A4- 6, the chip is divided into many slices along radial direction. When the

inclination angle ϕ is 0º, we can get the projection of the chip as shown in Figure A4- 7.

Each layer of the chip has a shearing dominated area and a ploughing dominated area. As

the chip thickness is less than minimum chip thickness, the tool is in the ploughing area.

In : The projection in the slice plane when the angle ϕ is zero, O1A is the radius of the

cutter, denoted by R. OO1 is the feed per tooth ft, which can be obtained from:

t

ff

RPM N

(A4. 7)

where, f is the feed rate in mm/sec. N is the number of flutes. OE is denoted by r (θ (i, j,

k)). The immersion angle at rotational angle θi for flute j and the kth

slice element θ (i, j,

k) is measured by clockwise from y axis. We can get the unknown side from two given

length of sides and the angle between the two known sides:

2 2 2( ( , , ))+2r( ( sin, , )) ( ( , , ))t tR f r i j k i j k f i j k

(A4. 8)

Figure A4- 7: The projection in the slice plane when the angle ϕ is zero

165

In Figure A4- 7, EF or tc is the undeformed radial chip thickness. From the geometry, it

can be known that:

( ( , , )) ( ( , , ))ct i j k R r i j k

(A4. 9)

In the slice plane, the chip thickness is related to the angle ϕ (ϕ is the angle down the ball

that locates the ith

disc), as

( , ) co( s, , ) ( ( , , ))c ct i j k t i j k

(A4. 10)

Substitute Eq. (A4. 8) to (A4. 10), we can get:

2 2 2 2( , , ) ( sin ( , , )( , ) sin ( , , s))coc t t tt i j k R R f i j k f f i j k

(A4. 11)

Because of the helix flutes of the micro ball-end mill, a point on the axis of the cutting

edge will be lagging behind the end point of the tool. Thus, it requires considering the lag

angle. If the helix angle on the tool is β, the new immersion angle should be:

2 ( )

( , , ) ( 1) tani

z ki j k j

N R

(A4. 12)

where, N is the number of flutes, R is radius of the kth

slice, z(k) is height of the kth

slice

from bottom of the cutter, which can be determined as follows:

( ) (1 cos( 0.5) )z k R k

(A4. 13)

where, Δϕ is the incremental inclination angle.

The chip model divides the chip into many small pieces of cuboid shape. When chip

thickness tc(θ, ϕ) is less than minimum chip thickness tmin, the ploughing volume Vp for a

tooth is calculated using the volume sum of the cuboids.

2

2

0

( , , )( , ) θex

st

P cV A L t i j k R

(A4. 14)

166

where, the entry and exit angles are θst =0 and θex =π respectively, Δθ is the incremental

rotation angle.

When the feed direction is not horizontal, but upwards as shown in Figure A4- 8, the

chip geometry is different. The local coordinate system xyz is rotated by the x axis, which

makes y′ axis parallel with the feed direction; z′ axis is perpendicular with the feed

direction. In Figure A4- 8, from the geometry analysis, it can be known that the rotation

angle α from z to z′ is the inclination angle of the machined surface. P is a point on the

cutting edge; its coordinate is expressed as following:

cos cos

sin cos

sin

x

y

z

P R

P R

P R

(A4. 15)

P in the local coordinate is transformed to P′ in the new coordinate as:

1 0 0 cos cos

0 cos sin sin cos

0 sin cos sin

x

y

z

P R

P R

RP

(A4. 16)

In the x′ y′ z′ coordinate system, assuming P′ is:

P ′= (Rcosθ′cosϕ′, Rsinθ′cosϕ′, -Rsinϕ′)

(A4. 17)

From Eq. (A4. 16), we can calculate that

cos cos Rcos cos

sin cos cos sin sin Rsin cos

sin cos sin sin cos sin

x

y

z

P R

P R R

R R RP

(A4. 18)

Therefore, θ′ and ϕ′ can be obtained from (12):

1

1

sin (sin cos sin sin cos )

cos coscos ( )

cos

(A4. 19)

Then, the chip thickness tc in Eq. (A4. 11) is modified as:

167

2 2 2 2( , , ) ( sin ( , , ) sin ( , , ))( , ) cosc t t tt i j k R R f i j k f f i j k

(A4. 20)

Figure A4- 8: Coordinate rotation for upward direction machining

When the cutter moves upwards or downwards with a tilt angle, the chip thickness

calculation is similar with the slot machining. Yet, the new immersion angle θ′ and

inclination ϕ′ should be obtained by the coordinate rotation. As a 3D curve machining,

the curve is divided by many small pieces of segments. Each segment can be regarded as

a linear line if the curved segment is small enough. The two vertices of each segment are

distributed into the furthest two vertices of a cube. For instance, as shown in Figure A4-

9, curve AC is divided into two segments AB and BC. In coordinate XYZ system,

assume the coordinate for A is (xi, yi, zi); the coordinate for B is (xi+1, yi+1, zi+1). For

triangular geometry calculation, the tilt angle α can be obtained:

11

2 2 2

1 1 1

sin( ) ( ) ( )

i i

i i i i i i

z z

x x y y z z

(A4. 21)

168

Figure A4- 9: Small segments of a curve in cubes

Substitute α into Eqs. (A4. 19) and (A4. 20), the chip thickness for 3D curve machining

can be obtained, and ploughing and shearing volume can be obtained from Eq. (A4. 14).

Figure A4- 10 shows the 3D curve machining. When inclination angle α is less than

π/2, the tool moves upwards direction; and vice versa, as α is more than π/2, the feed

direction is downwards.

Figure A4- 10: A 3D curve machining

A4.3.3 Chip Thickness Calculation Considering Runout Effects

When process faults are considered, the radius of a particular tooth such as jth

tooth at the

ith

axial disk is R (i, j) due to process faults. R(i, j) is given by [111],

tan 2

( , ) ( ) cos[ ( ) ( 1) ]R i j r k z k jR N

(A4. 22)

169

where, ε is the parallel offset runout, shown in Figure A4- 11; r(k) is radius of the kth

slice, which can be determined as follows:

( ) (1 sin( 0.5) )r k R k

(A4. 23)

z(k)tanβ/R is the angle measured back from tooth 1 to the jth

tooth, λ is the angle between

the direction of the offset and the nearest tooth. λ is not easy to obtain, therefore in the

paper, we assume λ is zero. (j-1)*2π/N is the angle measured back as the tooth

engagement wraps up the helix.

Figure A4- 11: Process faults with parallel offset runout

The chip thickness with runout is:

(( , ) ( , ), ) 1, ( , )cnew ct i j k R i j jt R i

(A4. 24)

If j=1, then

( , 1) ( , )R i j R i N

(A4. 25)

When the runout is larger than the feed per tooth, only the high side of the tool would be

cut [111].

170

A4.3.4 Ploughing Volume Calculation Algorithm Ignoring Runout Effects

There are two methods for calculating the chip volume. One is based on the combination

of Tri-dexel and solid Boolean operation. The other is by getting the entry and exit angles

from the Tri-dexel method, then having the volume integration along the sliced

differential elements from the bottom of the tool toward the final axial depth of cut at

each incremental rotation and inclination. Although both methods can calculate the chip

volume, the former cannot be used to obtain the ploughing and shearing volume due to its

long calculation time and memory leak problem.

In this work, a two-step method is used for the calculation of the ploughing volume: a)

calculate the reference immersion angle based on entry angle and exit angle for each

cutting contact (CC) point; and b) from the local view, calculate the ploughing volume

and shearing volume according to the axial depth of cut, feed per tooth, start angle and

end angle.

A general procedure of the ploughing volume simulation program is given in the

flowchart as shown in Figure A4- 12. For step a), a Tri-dexel model of the workpiece was

created, and the input variables set by the user are workpiece width, height and thickness,

chip thickness calculation scale factor, G-code program, axial depth of cut, the number of

teeth, feed rate, spindle speed, cutter diameter.

The toolpath was divided by many segments after the input of G-code. All of the

segments were read and saved in the memory. Then, the tool was moved to the position

of cutter contact points and Boolean subtraction and intersection were used to calculate

the chip total volume [95]. Finally, in the axial depth of cut plane, define a start point,

171

connect the central point and the start point as the exit angle and then use the

computational geometry method to find the largest angle to get the exit angle.

For step b), the depth of cut, feed per tooth, exit and entry angle, obtained in step a),

and the helix angle are used to calculate the chip thickness. Present immersion angle is

defined by exit and entry angle. When the present immersion angle is between zero and

the reference immersion angle, the tool works and chips can be generated; otherwise, the

cutting flute does not contact the workpiece. If the chip thickness is less than minimum

chip thickness, calculate the ploughing volume, in this case, the shearing volume is zero.

Vice versa, when the chip thickness is larger than minimum chip thickness, it just

requires calculating the shearing volume.

172

Figure A4- 12: The ploughing and shearing volume calculation flowchart

173

A4.3.5 Ploughing Volume Simulation

Two CNC toolpaths were generated using MasterCAM to simulate the chip ploughing

volume. It can be seen in Figure A4- 13. In machining slots, the relation between the

depth of cut and shearing/ploughing volume are shown in Figure A4- 14. The workpiece

material is aluminum 6061 alloy. The workpiece is machined by a two-flute micro ball-

end mill with a tool diameter of 1/16″. The edge radius re of the tool is about 2 μm. The

minimum chip thickness tmin is related to the tool edge radius re, which can be got from

Eq. (A4.1).

The minimum chip thickness in the case study is 0.6 μm; the feed per tooth is 0.75μm;

and the depth of cut is 200 μm. The simulation reveals that the shearing volume decreases

with the increased depth of cut r(k); reversely, the ploughing volume grows as r(k)

increases. Thus, the maximum ploughing area appears at the bottom of the cutter.

Figure A4- 13: Two different toolpaths: a) Straight lines and down-ramping, b) A straight

line

174

Figure A4- 14: The changes of shearing and ploughing volumes with the height of kth

slice z(k)for slot machining

The total chip volume was simulated by two different methods. Figure A4- 15 (a) and

(b) show the simulation results to calculate the chip volume by Tri-dexel and Boolean

operation method for slot machining and straight lines and down-ramping machining.

The left area in the Figure A4- 15 (a) and (b) displays the changes of G-code. The middle

area shows the 3D simulation of machining. This software can get the 3-axis online chip

volume, which is shown at the bottom.

0 0.05 0.1 0.15 0.20

2

4

6

8x 10

-7

Depth of cut z(k) [mm]

Plo

ughin

g v

olu

me [m

m3]

Plot of ploughing volumes with depth of cut in one revolution

0 0.05 0.1 0.15 0.20

0.5

1

1.5x 10

-5

Depth of cut z(k) [mm]

Shearing v

olu

me [m

m3]

Plot of shearing volumes with depth of cut in one revolution

175

Figure A4- 15: The Voxel and Boolean method: Chip volume simulation for (a) Slot

machining; (b) Straight lines and down-ramping machining

When/while machining slots, the total chip volume is constant due to the constant

depth of cut. As shown in Figure A4- 15 (a) and Figure A4- 16, both simulation methods,

the Voxel combined with Boolean and the Voxel combined with integration, have

generated similar results. For the first method, the chip volume is multiplied by a scale

factor of 50 to reduce the number of subsegments of the toolpath and to save the

calculation time. Therefore, the total chip volume obtained by the Voxel and Boolean

operation is approximately 6×10-4

mm3. The results presented here correspond to the

results generated by Voxel using the integration method, which is about 6×10-4

mm3

as

well.

176

Figure A4- 16: Slot machining: Chip volume simulations changing with the number of

samples. Spindle speed=40,000 rpm, depth of cut=0.1mm, ft =0.75 µm/tooth

For the toolpath of straight line and down-ramping, constant volume is generated

during the straight line section. For the down-cutting section, the total chip volume is

supposed to increase lineally. However, the depth of cut is varied. It is calculated by the

positions of current and previous cutter contact points in the Voxel and Boolean method.

Some deviations exist in CAD model and could cause errors in the Voxel and Boolean

method. There are many deviations as obtaining the entry and exit angles from the Voxel

and Boolean method. Therefore, some deviations exist in the Voxel and integration

method and that is the reason why the chip volume for down-cut was not linearly

increased in the second method.

The phenomenon of ploughing leads to material deformation and side edges

generation. It is important to investigate the ploughing volume and then predict cutting

0 20 40 60 80 100 120 140 1600

0.5

1x 10

-4

Number of Samples

Plo

ughin

g V

olu

me [

mm

3]

0 20 40 60 80 100 120 140 1600

2

4

6x 10

-4

Number of Samples

shearing V

olu

me [

mm

3]

0 20 40 60 80 100 120 140 1600

2

4

6x 10

-4

Number of Samples

Tota

l C

hip

Volu

me [

mm

3]

177

process to improve machining reliability and accuracy. Figure A4- 16 and Figure A4- 17

show that ploughing volume is proportional to the shearing volume with similar changes.

The relation between chip volume and the rotation angle θ in ten revolutions is illustrated

in Figure A4- 18 and Figure A4- 19.

Eq. (A4. 22) shows that the radius of a particular tooth is relative to the Z axis of the

tool rather than the rotation angle of the cutter. Therefore, as different teeth of the cutter

are engaged in the machining, the rotation angle does not affect the run out on chip load

[111]. When parallel offset runout is discussed, the ploughing and shearing volume is

different at each flute cutting for a two-flute tool. In Figure A4- 19, it can be seen the

shearing volume cut by a tooth is larger than that by another tooth.

Figure A4- 17: Straight line and down-ramping machining: chip volume simulations

changing with rotation angle θ. Spindle speed=20,000 rpm, depth of cut=0.2-0.7mm, ft

=1.5 µm/tooth

0 100 200 300 400 5000

2

4x 10

-5

Number of Samples

Plo

ughin

g V

olu

me [

mm

3]

0 100 200 300 400 5000

0.5

1

1.5x 10

-3

Number of Samples

shearing V

olu

me [

mm

3]

0 100 200 300 400 5000

0.5

1

1.5x 10

-3

Number of Samples

Tota

l C

hip

Volu

me [

mm

3]

178


ignoring runout. Spindle speed=40,000 rpm, depth of cut=0.1mm, ft =0.75 µm/tooth


considering runout, ε=0.01µm, spindle speed=40,000 rpm, depth of cut=0.1mm, ft =0.75

µm/tooth

0 10 20 30 40 50 60 700

1

2x 10

-7

Rotation angle (rad)

Vo

lum

e (

mm

3)

Plot of total ploughing volumes with rotation angle

0 10 20 30 40 50 60 700

0.5

1x 10

-5


Vo

lum

e (

mm

3)

Plot of total shearing volumes with rotation angle

0 10 20 30 40 50 60 700

1

2x 10

-5


Vo

lum

e (

mm

3)

Plot of total shearing and ploughing volumes with rotation angle

0 10 20 30 40 50 60 700

1

2x 10

-7


Vo

lum

e (

mm

3)

Plot of total ploughing volumes with rotation angle

0 10 20 30 40 50 60 700

1

2x 10

-6


Vo

lum

e (

mm

3)

Plot of total shearing volumes with rotation angle

0 10 20 30 40 50 60 700

2

4x 10

-6


Vo

lum

e (

mm

3)

Plot of total shearing and ploughing volumes with rotation angle

179

A4.3.6 Experimental Setup

The verification experiments have been performed without lubricant (dry conditions)

on a three-axis CNC ALIO vertical micro-milling machine. The spindle speed range of

the machine is from 10,000 to 80,000 rpm. The experiment setup is shown in Figure A4-

20. A Kistler table dynamometer (MiniDyn 9256C1), connected with the feed table, was

used to measure instant cutting forces. All of the instantaneous cutting forces are

magnified by an amplifier and then displayed in a data acquisition (DAQ) board. The

current position and cutting forces in three axes could be shown in the DAQ board.

Figure A4- 20: Experimental setup of micro-milling operations [7]

In this experiment, a 1/16″ two-flute ball end mill was used. The workpiece material is

Aluminum 6061, and the parallelepiped workpiece is 38 mm long, 20 mm wide, and 9

mm in height. Sampling rate or sampling frequency (fs) defines the number of samples

per second. It can be got from Eq. (A4. 26). In the experiment, the sampling rate is 100

KHz, which is the upper band limit of the signal.

Sampling rate

Samples per intervalRPM

(A4. 26)

The resultant force is the sum of Fx, Fy, and Fz:

222

zyx FFFR

(A4. 27)

180

Two different toolpaths, shown in Figure A4- 13, were machined at different cutting

parameters such as depth of cut, spindle speed, and feed rate. The first toolpath was

consisted of two segments of straight lines and a down-ramping. The second was just a

slot cut. The experiments were divided into four groups with two different spindle

speeds: 20,000 and 40,000 rpm, and two different feed rates: 0.75 and 1.5µm/tooth.

The depth of cut for group a) and b) were constant: 0.1 and 0.2 mm respectively. The

depth of cut was changed from 0.15 to 0.6 mm for group d) and from 0.2 to 0.7 mm for

group d).

A4.3.7 Experimental Result

Figure A4- 21 shows the measured resultant cutting forces. The parameter settings

were illustrated in Table 5. Comparing the simulated total chip volume (shown in Figure

A4- 18) and the resultant cutting force, their changes seem to be similar. When

kinematics and certain properties of the milling process were considered, for the slot

tests, the experiment data of the resultant cutting force were not constant due to vibration,

but with some fluctuation within reasonable range. The signal from the dynamometer was

multiplied by a scale factor of ten. In Figure A4- 21 (a), when the depth of cut was 0.1

mm, with spindle speed of 40,000 rpm and feed rate of 0.75 feed per tooth, the average

resultant cutting force was approximately 2.5 N. Compared with group b), the other

cutting parameters were the same, except the axial depth of cut increased to 0.2 mm, the

average resultant cutting force was about 3N. Therefore, the cutting force is linearly

changed with changes of depth of cut, when slots are machined. However, if the toolpath

is a curve, this assumption is not followed any more. For instance, in Figure A4- 21 (c)

and (d), the resultant cutting forces were approximately 4N as the depth of cut was

181

0.15mm and 0.2 mm respectively. The forces were not constant as machining the second

straight line sections, due to the vibrations and the side burrs formed by the ploughing

effect. The generation of chip thickness during machining process causes tool chatter

vibrations. The excessive vibrations accelerate tool wear and increase cutting forces and

lead to poor finish surface.

Table 5: The parameters for four groups’ experiments

Group

number Toolpath

Depth of

cut (mm)

Spindle

speed (rpm)

Feed per

tooth (μm)

Feed rate

(rev/sec)

a Straight line 0.1 40,000 0.75 1

b Straight line 0.2 40,000 0.75 1

c Straight line 0.2 40,000 2 2.6

d Straight-down line 0.15-0.6 20,000 1.5 1

e Straight-down line 0.2-0.7 20,000 1.5 1

Figure A4- 21: Measured resultant cutting forces with machining times

182

An accurate prediction of total chip volume for a micro-ball end mill is necessary to

predict the cutting forces in process planning, and to study the interaction between the

tool and the milling process and errors left on the finish surface.

The ploughing and shearing volumes depend on the rotation angle θ and the depth of

cut. The differential ploughing and shearing volume are integrated along the in-cut

portion of the tool from rotation and inclination directions to get the total chip volume

generated by flutes.

Figure A4- 18 and Figure A4- 22 illustrate simulated chip volumes and the measured

resultant cutting force for ten revolutions of the slot machining. It can be observed that

the total chip volume shown in Figure A4- 18 is the absolute value of sinusoidal variation

with the rotation angle. The simulation did not consider the tool wear and vibrations. The

shape of the curves of measured resultant force and simulated chip volume is similar. The

mechanistic model also assumes that the ploughing forces are proportional to the

ploughed volume of the material. However, Figure A4- 22 shows that the magnitudes of

resultant cutting force between two flutes of the tool are different. This is due to one

tooth having more in-cut portion with the workpiece than the other and therefore more

cutting force is generated. Comparing Figure A4- 19 and Figure A4- 23, as the process

faults are considered, the shapes of shearing and total volume are similar with resultant

force.

183

Figure A4- 22: Measured resultant cutting forces for the slot machining. Spindle

speed=40,000 rpm, depth of cut=0.1mm, ft =0.75 µm/tooth

The surfaces generated in the micro ball-end milling are inspected by a microscope.

Ploughing is related to moderate wear, cutting process damping, and tool stability. The

ploughing effect happens when the flank face of the tool contacts with the machined

surface[112]. It leads to rough surface and burr formation. A significant number of side

burr generation is associated with the heavy ploughing resulting from the large edges.

Figure A4- 23 (a) shows that the surface produced by a milling operation, with a 100 µm

depth of cut and feed rate of 0.75 µm per flute, has less burrs and ploughing effects than

the surface with the same feed rate but larger depth of cut (shown in Figure A4- 23 (b)).

It demonstrates that the ploughing effect is subject to the depth of cut: a larger depth of

cut is associated with a significant ploughing effect. Furthermore, at a high feed rate, the

ploughing effects are less observable. Comparing the surface results in Figure A4- 23 (a)

and (b) with that in Figure A4- 23 (c) and (d), it can be seen that smaller depth of cut and

higher spindle speed obtains a better finish surface and less burrs occur. The parameters

in Figure A4- 23 (a) and (b) are from groups a) and b) in Table 5. The parameters in

Figure A4- 23 (c) and (d) are from groups d) and e) in Table 5.

0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

Resultant

Forc

e [

N]

[rad]

Experiment resultant force in ten revolutions

184

Figure A4- 23: The surfaces generated by the ball end milling processes: (a) Depth of cut

dc=100 µm, ft =0.75 µm/tooth; (b) dc=200 µm, ft =0.75 µm/tooth; (c) dc=150-600 µm, ft

=1.5 µm/tooth; (c) dc=200-700 µm, ft =1.5 µm/tooth

Overall, the ploughing phenomenon is expected to decrease as the feed rate is

increased. In micro-ball end milling, the ploughing effects are easier seen due to the tool

geometry. The burrs are more significantly seen in one side of the machined surface. That

is because the ploughing effects are more obvious at cutting entry than at exit.

The surface roughness is obtained by a 3D surface measurement machine. Sq is the

root mean square height. It is calculated by the standard deviation for the amplitudes of

the surface.

The surface quality of side wall in Figure A4- 24 (c) is better than that in Figure A4-

24 (b) due to the high feed rate. Comparing the side walls with good surface quality, the

ploughing effect of group c is less obvious than group b. When comparing Figure A4- 24

185

(a) and (b), it can be known that the depth of cut also affects the side wall roughness.

Larger depth of cut leads to higher surface roughness and poor surface quality.

Figure A4- 24: Topography of the machined surfaces in a 3D surface measurement

machine

A4.4 Conclusion

The tool based method of chip thickness and volume calculations proposed in Section

5.2.2 is used to calculate chip ploughing volume in micro 5-axis flat-end milling in this

work. A 3D geometry model and a discrete chip volume calculation method for the

ploughing volume using a micro ball-end mill are also introduced in this chapter. To

better understand the ploughing volume problem in micro-machining and to increase

cutting efficiency, different axial depths of cut and feed rates in a 3-axis micro CNC

machine were tested to reduce the ploughing area. There are two different CNC toolpaths

used to simulate the machining process and to obtain the relation between chip ploughing

volume and rotation angle.

186

The simulation results were verified by machining experiments. The cutting tests on

Aluminum 6061 parts led to the following conclusions:

Resultant cutting force was proportional to the removed chip volume.

High feed rate can diminish ploughing.

A larger tool edge radius is associated with a significant ploughing effect.

This approach supports the generation of more efficient and accurate toolp

aths for micro-milling.

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