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.
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
Supervisory Committee
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)
Abstract
Supervisory Committee
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
(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.
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:
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
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
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
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
Figure 3-8: Changes of accumulating relative angles with different numbers of cluster centres and ith cluster for a convex half sphere surface in 3D bar chart. ... 43
Figure 3-9: Changes of maximum accumulating relative angles and their first and second
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
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 6-9: 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
FigureA1- 3: The linear function of feed rates and an offset contributed by the edge forces Fyc ... 143
FigureA1- 4: The linear function of feed rates and an offset contributed by the edge 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-ball-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-flat-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
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
Figure A4- 19: Slot machining: Chip volume simulations changing with rotation angle θ 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
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
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.
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
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.
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-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
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
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
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 1-1is 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].
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
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
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 layersalong 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
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
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
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.
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.
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
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
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
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
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.
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.
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.
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.
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
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 velocityfor 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
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
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.
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
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.
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.
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.
