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 |  موضوع: كتاب Advanced Numerical Methods to Optimize Cutting Operations of Five-Axis Milling Machines الجمعة 12 يناير 2024, 1:37 am | |
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أحضرت لكم كتاب
Advanced Numerical Methods to Optimize Cutting Operations of Five-Axis Milling Machines
With 113 Figures and 16 Tables
Stanislav S. Makhanov, Weerachai Anotaipaiboon
و المحتوى كما يلي :
Contents
List of Figures . X
List of Tables XV
1 Introduction . 1
1.1 Motivation and Structure of the Book 1
1.2 CAD/CAM Formats . 3
1.3 Short Literature Survey 4
References 16
2 Introduction to Five-Axis NC Machining 25
2.1 Five-Axis NC Machining Concepts . 25
2.2 NC Part Programming . 28
2.3 Classification of Five-Axis Machines . 34
2.4 Five-Axis Machine Kinematics 37
2.5 Five-Axis Machining Example . 43
References 48
3 Fundamental Issues in Tool Path Planning . 51
3.1 Surface Representation . 51
3.2 Machining Strip Width Estimation 53
3.3 Optimal Tool Orientation and Gouging Avoidance . 60
3.4 Kinematics Error 63
3.5 Tool Path Generation 66
References 68
4 Space-Filling Curve Tool Paths 73
4.1 History of Space-Filling Curves and Their Applications . 73
4.2 Tool Path Optimization 75
4.3 Tool Path Generation using Adaptive Space-filling Curves 77
4.3.1 Grid Construction . 77X Contents
4.3.2 Space-Filling Curve Generation . 78
4.3.3 Tool Path Correction 80
4.4 Examples and Discussion . 83
References 94
5 Tool Paths in Adaptive Curvilinear Coordinates . 97
5.1 Introduction 97
5.2 A Historical Note on Grid Generation 98
5.3 Variational Grid Generation for Tool Path Optimization 101
5.3.1 Preliminary Examples . 101
5.3.2 Variational Method and Functionals . 102
5.3.3 The Harmonic Functional 109
5.3.4 Examples of the Tool Path Optimization . 110
5.4 Application of Harmonic Functional to Tool Path Generation 116
5.5 Space-Filling Curve Generation on Block Structured Grid 124
5.6 Examples and Discussion . 125
Appendix: Derivation of Computational Formulas for
Adaptive-Harmonic Grid Generation . 133
References 144
6 Optimization of Rotations . 151
6.1 Introduction 151
6.2 Kinematics Error and Angle Variation . 155
6.3 Optimization Problem . 160
6.4 Optimization Problems: Examples and Practical Machining . 162
6.5 Uniform Angular Grids . 168
6.6 Uniform Angular Grids: Numerical and Machining
Experiments 176
Appendix: The APT cutter . 182
References 183
7 Theory of Optimal Setup for Five-Axis NC Machining . 185
7.1 Introduction 185
7.2 Tool Trajectory Analysis . 189
7.2.1 Invariant Parameters 189
7.2.2 Workpiece Setup and the Tool Trajectory 190
7.3 Least-Squares Optimization and Dependent Variables 192
7.3.1 Least-Squares Optimization . 192
7.3.2 Dependent Variables . 193
7.4 Examples and Discussion . 194
7.4.1 Numerical Method . 194
7.4.2 Examples 195
References 202
Index 205List of Figures
2.1a MAHO 600E milling machine . 26
2.1b Workpiece, clumping device, and rotary tables of MAHO
600E . 26
2.1c HERMLE UWF902H milling machine 27
2.2 Zigzag and spiral tool paths 28
2.3 Kinematic chain diagram of machine in Fig. 2.1a 35
2.4 Example of 2-0 machine 36
2.5 Example of 1-1 machine 37
2.6 Example of 0-2 machine 38
2.7 Simple shape parametric surface . 44
2.8 Zigzag tool path for surface (2.12) (Fig. 2.7) 44
2.9 Cutting simulation of surface (2.12) (Fig. 2.7) with 10×10
tool path . 46
2.10 Cutting simulation of surface (2.12) (Fig. 2.7) with 20×20
tool path . 46
2.11 Bezier surface . 48
2.12 Tool trajectories with loops of 20×20 tool path for surface in
Fig. 2.11 . 48
2.13 Cutting simulation of surface in Fig. 2.11 with 20×20 tool
path . 49
3.1 Examples of cutting tools 54
3.2 Geometric analysis of the cutting operations 55
3.3 Machining strip width estimation 55
3.4 Machining strip width estimation method of Lee and Ji 59
3.5 Tool gouging 61
3.6 Kinematics error between two cutter contact points 64
3.7 Angle adjustment for the 2-0 machine 65
3.8 Kinematics error reduction by angle adjustment . 67
3.9 Overlapping of machining strips on adjacent tool paths . 68
3.10 Example of isoparametric tool path 69XII List of Figures
4.1 3 iterations of the Peano’s space-filling curve 74
4.2 6 iterations of the Hilbert’s space-filling curve . 74
4.3 Grid construction for SFC tool path . 78
4.4 Small circuits and dual graph construction 79
4.5 Illustration of SFC tool path construction 81
4.6 Machining strips on adjacent tool paths generated by using
SFC . 82
4.7 Modification of tool path trajectory at the turns 82
4.8 Trajectories of the cutter’s effective cutting edge before and
after applying the tool path correction . 83
4.9 Overlaying of two isoparametric tool paths for the surface in
Example 4.1 85
4.10 SFC tool path for the surface in Example 4.1 . 86
4.11 SFC tool paths in Example 4.1 generated with and without
merging dependency . 87
4.12 Practical machining using the SFC tool path without
correction 87
4.13 Practical machining using the SFC tool path with correction 88
4.14 Overlaying of two isoparametric tool paths for the surface in
Example 4.2 89
4.15 SFC tool path for the surface in Example 4.2 . 90
4.16 Overlaying of two isoparametric tool paths for the surface in
Example 4.3 91
4.17 SFC tool path for the surface in Example 4.3 . 92
4.18 Simulation result of five-axis machining with SFC tool path
in Unigraphics 18 . 93
5.1 Surface with a curvilinear zone of large gradients 101
5.2 Curvilinear grid which can be converted into a tool path to
machine the surface in Fig. 5.1 102
5.3 A grid which can be converted into a tool path for a complex
shaped region . 103
5.4 Tool path as a mapping from the computational to the
parametric (physical) domain . 104
5.5 Constructing surface T (ξ, η) from patches 105
5.6 The scallop height evaluation for convex surfaces 108
5.7 Tool path generation with adaptation to a weight function
and the boundary . 111
5.8 Grids obtained by constraint minimization and unconstrained
minimization . 112
5.9 A test Bezier surface . 113
5.10 The machined Bezier surface 113
5.11 Conventional tool path and tool paths from grid generation . 114
5.12 Example of regular grid 117
5.13 Examples of block-structured grids 118List of Figures XIII
5.14a Example of adaptive-harmonic grid 119
5.14b Control function 120
5.15 Correspondence of node numbers for a mapping of the unit
square in the (ξ, η) plane on to the quadrilateral cell 1 of the
grid in the (u, v) plane . 121
5.16 Partitioning of block-structured grid for refinement 124
5.17 Undirected graph construction for SFC tool path generation 126
5.18 Covering of grid in Fig. 5.17 by small circuits . 127
5.19 Unimodal surface with exponential peak along a line . 127
5.20 Curvilinear grid adapted to the unimodal surface with
exponential peak along a line . 128
5.21 Zigzag and SFC tool paths based on isoparametric and grid
generation methods 129
5.22 Convergence rate of the grid generation technique in term of
the number of iterations for Example 5.4 . 131
5.23 Grid refinement . 132
5.24a Curvilinear grid adapted to the surface in Example 5.6 in
(u, v) domain . 133
5.24b Curvilinear grid adapted to the surface in Example 5.6 in
workpiece coordinate system 134
5.25a SFC tool path for the surface in Example 5.6 in (u, v) domain 135
5.25b SFC tool path for the surface in Example 5.6 in workpiece
coordinate system . 136
5.26 Simulation result of five-axis machining with SFC tool path
in Unigraphics 18 for the surface in Example 5.6 137
5.27 Convergence rate of the grid generation technique in term of
the number of iterations for Example 5.6 . 138
5.28 Single blade of an impeller 139
5.29 Curvilinear grid adapted to part of a surface of the blade in
(u, v) domain . 140
5.30 Curvilinear grid adapted to part of a surface of the blade in
workpiece coordinate system 141
5.31 SFC tool path for milling of the broken blade in (u, v) domain 142
5.32 SFC tool path for milling of the broken blade in workpiece
coordinate system . 143
5.33 Virtual cutting of the blade in Unigraphics 18 . 144
6.1 An experimental part surface S1 152
6.2 Conventional tool path simulated by the virtual machine and
surface machined by HERMLE UWF902H 153
6.3 Optimized tool path simulated by the virtual milling machine
and optimized surface machined by HERMLE UWF902H . 154
6.4 Nonlinearity of the tool path due to rotations in the
workpiece coordinates 155XIV List of Figures
6.5 A loop-like trajectory induced by large gradients of the
rotation angles 156
6.6 The “repaired” trajectory 157
6.7 The set of feasible rotations . 161
6.8 A graph corresponding to the set of the feasible rotations . 163
6.9 Trajectories corresponding to (1) bbase and (2) π − bbase and
the optimized trajectory composed from (1) and (2) . 163
6.10 Trajectories corresponding to (1) abase (2) abase − π (3)
abase + π (4) abase − 2π and the optimized trajectory
composed from the trajectories (1) (2) and (4) 164
6.11 Around or across the hill? 165
6.12 The kinematic error as a function of a2 and b2 166
6.13a Conventional tool path for S2 on HERMLE UWF902H
showing large overcuts . 171
6.13b The tool path optimized with regard to the total error . 172
6.13c The tool path optimized with regard to the overcut error . 173
6.14a Conventional tool path for S2 on MAHO 600E 173
6.14b Optimization with regard to the total error . 174
6.14c Optimization with regard to the overcut error, S2 on MAHO
600E . 174
6.15a Without optimization, S2 on MAHO 600E (corresponds to
Fig. 6.14a) . 175
6.15b Optimization with regard to the total error, S2 on MAHO
600E (corresponds to Fig. 6.14b) 175
6.15c Optimization with regard to the overcut error, S2 on MAHO
600E (corresponds to Fig. 6.14c) 176
6.16a Tool path and tool orientations, S2 on MAHO 600E, before
insertion . 177
6.16b Tool path and tool orientations, S2 on MAHO 600E,
conventional point insertion . 177
6.16c Tool path and tool orientations, S2 on MAHO 600E, angular
grid insertion . 178
6.17a Spatial grid for S2 on MAHO 600E (corresponds to Fig.
6.16b) . 180
6.17b Angular grid insertion for S2 on MAHO 600E (corresponds
to Fig. 6.16c) . 180
6.18 Tool path for surface S2 on MAHO 600E after the optimal
sequencing . 181
6.19 Tool path for surface S2 on MAHO 600E after the optimal
sequencing and inserting one point into the largest loop 181
6.20 APT cutter geometry 182
7.1 Changing an initial workpiece setup by rotating around the
z1-axis and the y1-axis . 186
7.2 Introductory example of optimal setup . 188List of Figures XV
7.3 Dependent parameters T12,z + T23,z = c of the 2-0 machine . 194
7.4 Two test surfaces for optimal setup 196
7.5 Comparison of tool path trajectories in various optimal
setups for sweep surface on the 2-0 machines 199
7.6 Comparison of tool path trajectories in various optimal
setups for sweep surface on the 1-1 machines 200
7.7 Comparison of tool path trajectories in various optimal
setups for sweep surface on the 0-2 machines 201
7.8 The sweep surface machined with different setups . 202List of Tables
2.1 List of word address codes 30
2.2 List of G-code functions 31
2.2 List of G-code functions (cont.) . 32
2.3 List of M-code functions 33
2.4 List of CL points for tool path in Fig. 2.8 45
2.5 G-codes for tool path in Fig. 2.8 . 47
4.1 Performance of the SFC tool paths versus the isoparametric
tool paths in terms of tool path length . 84
4.2 Performance of the SFC tool paths versus the isoparametric
tool paths in terms of estimated machining time . 94
5.1 Convergence of the method λp versus θ . 111
5.2 Accuracy of the machined surface . 115
5.3 Convergence of the algorithm h versus 116
5.4 Comparison of tool paths based on variational grid generation
techniques versus the isoparametric tool path scheme 130
6.1 Kinematics error for the optimized and non-optimized tool
path, surface S1 on MAHO 600E 167
6.2 Overcut error optimization, surface S2 on HERMLE
UWF902H 169
6.3 Overcut error optimization, surface S2 on MAHO 600E 170
6.4 Error versus number of inserted points, the basic grid size
15×20 . 177
7.1 Minimal sets of optimization parameters . 194
7.2 Performance of the optimization method, the sweep surface . 197
7.3 Performance of the optimization method, the two-bell surface 197
7.4 Performance of the optimization measured by the required
number of CC points 198
Index
accessibility, 14
address code, see word address code
angle adjustment, 64
angle variation, 158
angular jump, 64
angular speed, 154
APT, see Automatic Programmed
Tools
Automatic Programmed Tools, 29, 178,
182
CNC, see machines, CNC
cover and merge algorithm, 78, 88
curvature interference, see gouging
curvilinear grid, see grid
cutter, 25
ball-end, 53
flat-end, 53
toroidal, 53
cutter contact, 27, 53
cutter location, 25, 54
cutting direction, 54
discrete functional, 120
effective cutter radius, 60
effective cutting shape, 10, 54
finish machining, 1
five-axis machines, 25
classifications, 34–37
kinematics, 37–43
forward step error, see kinematics error
G-code, 29–30
global interference, 12
gouging, 7, 11, 60
grid, 97
block-structured, 117
convex, 100
degenerate, 99
generation, see grid generation
refinement, 100
regular, see grid, structured
structured, 99, 117
unstructured, 100
grid divergence, 111
grid generation, 9, 99
algebraic methods, 117
computational region, 103
differential methods, 117
parametric region, 103
variational methods, 102, 117
harmonic functional, 98, 110, 115
approximation, 110
inclination angle, 10, 54
minimum, 60
kinematics error, 63, 151, 185
minimization, 66, 160
total, 155
M-code, 32
machine coordinate system, 39
reference point, 39
machines206 Index
CNC, 4, 25
five-axis, see five-axis machines
NC, 25
machining strip, 53–58
machining time, 88
NC
block, 29
machines, see machines, NC
program, 25, 28–34
nonlinear trajectory, 38, 47
overcut, 10, 60, 159
parametric surface, 51
curvature
Gaussian, 52
mean, 52
normal, 52
principal, 52
normal vector, 51
point classification, 52
stationary point, 151
part program, see NC, program, 27
point insertion, 66
uniform angular grid, 168
postprocessing, 28
rear gouging, 12
regional milling, 8
rotary axis, 25
rotation angle, 28, 64
angular jump, see angular jump
degree of freedom, 162
rough cutting, 1
setup
machine configuration, 185
workpiece, 185, 190
solid modeling, 13
space-filling curves, 73
adaptive, 76, 77
Hilbert’s curve, 73
non-recursive, 74
Peano’s curve, 73
recursive, 74
tool path, see tool path, space-filling
curve
tilt angle, 10, 54
tool orientation, 27, 39, 66
optimal, 60–63
vector, 160
tool path, 27
correction, 76, 80, 84
generation, 66–68
iso-planar, 7
iso-scallop, 7
isoparametric, 7, 66
length, 88
optimization, 28, 75
space-filling curve, 9, 77
generation, 77–83
spiral, 7, 27
zigzag, 7, 27, 43
tool trajectory, 63, 158, 186
dependent parameters, 186, 193
invariant parameters, 186, 189
trimmed surface, 97
undercut, 10, 159
uniform angular grid, see point
insertion, uniform angular grid
variational methods, 99
visibility, 14
Winslow functional, 98, 105, 110
word address code, 28
coordinate functions, 30
feed functions, 32
format, 29
miscellaneous functions, see M-code
preparatory functions, see G-code
speed functions, 32
tool functions, 32
workpiece, 25
workpiece coordinate system, 39
yaw angle, 10
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