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Doktorarbeit / Dissertation, 2016
208 Seiten, Note: Distinction
Abstract
List of Figures
List of Tables
1 Introduction
1.1 Background and Motivation
1.2 Organization of Thesis
2 Literature Review
2.1 Introduction
2.2 Basic Terminologies Associated with Pocket Machining
2.3 Tool Path Requirements for High Speed Pocket Machining
2.4 Organization of the Literature Review
2.5 Noteworthy Literature Reviews
2.6 Various Types of Pockets and Pocket Machining
2.7 Conventional Tool Path Strategies
2.7.1 Directional Parallel
2.7.2 Contour Parallel (Boundary Parallel or Offset) Tool Path
2.7.3 Space Filling Curves
2.8 Corner Machining Tactics
2.9 Advance Tool Path Strategies for HSM
2.9.1 Mapping Based Approaches for Tool Path Generation
2.9.2 Medial Axis Transform Based Method for Tool Path Generation
2.9.3 Clothoidal Spiral Tool Paths
2.9.4 Spiral Tool Paths Based on the Solution of PDE
2.9.5 Trochoidal Tool Paths
2.9.6 Interpolating Tool Paths Based on Bezier, B-spline and NURBS
2.9.7 Miscellaneous Tool Path Planning Strategies
2.10 Current Status of Development in Tool Path Strategies
2.11 Summary Table
2.12 Observations
2.13 Objective of Present Research
3 Spiral Tool Path Based on PDE and NURBS
3.1 Introduction
3.2 Methodology
3.2.1 The Algorithm for Generating Spiral Tool Path for Star-Shaped Polygon Using PDE
3.3 Extending the Method for Non-star-shaped Polygon and Free-form Curves
3.4 Results and Discussion
3.4.1 Effect of Mesh Size on Tool Path.
3.4.2 Effect of Permissible Error and Number of Degree Steps
3.5 Conclusions
4 Study of Elliptical-pocket Machining
4.1 Introduction
4.2 Experimental Details
4.2.1 Tool Path Strategy and Pocket Geometry
4.2.2 Tooling Details and Machining Conditions
4.3 Experimental Plan Procedure
4.4 Results and Discussion
4.4.1 Tool Path Length
4.4.2 Cutting Time
4.4.3 Percentage Utilization of a Tool (PUT)
4.4.4 Average Surface Roughness (Ra)
4.5 Conclusions
5 Quantitative Comparison of Pocket Geometries and Pocket Decomposition
5.1 Introduction
5.2 Dimensionless Number (DN) for Comparing Pocket Geometries
5.2.1 Analogy of Reynolds Number
5.2.2 Percentage Utilization of a Tool (PUT) as a Measure of Effectiveness of Spiral Tool Path
5.2.3 The Concept of Dimensionless Number (DN)
5.2.4 Various Ratios and Their Effects
5.2.5 Dimensionless Number, DN
5.2.6 Modified DN for Spiral Tool Path (DNspiral)
5.3 Results and Discussion
5.4 Pocket Decomposition
5.4.1 Decomposition of a Polygon Geometry
5.5 Free-form Pocket Decomposition
5.6 Decomposition of a pocket with an island
5.7 Conclusions
6 Study of speed, feed and step-over in pocket milling
6.1 Introduction
6.2 Experimental Investigation
6.2.1 Selection of Process Variables, Responses, Workpiece/Tool Material and Tool Path Strategy
6.3 Experimental Setup
6.3.1 Fixture Design
6.3.2 Designing the Experiments
6.3.3 Selection of Sampling Frequency
6.4 Results and Discussion
6.4.1 Cutting Time
6.4.2 Surface Roughness
6.4.3 A Method of Analysing Cutting Forces
6.5 A Crucial Test Before linear Cutting Forces Experiments
6.6 Conclusions
7 Overall Results and Discussion
7.1 Overall Results and Discussion
8 Conclusions
8.1 Conclusions
8.2 Scopes of Future Research
9 References
Appendix 1: Underestimation of Machining Time ( )
Appendix 2: Chemical Analysis of AISI P20
Appendix 3: Certificate of Experimental Readings
Appendix 4: Specifications of Vertical Machining Centre PX10.
Appendix 5: Specifications of KISTLER 9272
Appendix 6: List of Research Publication based on Ph.D. work
This book is largely based on a Ph.D. thesis entitled “Tool Path Development for CNC Pocket Machining Using Partial Differential Equation and NURBS Curve” submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy by Patel Divyangkumar Dharmendra and supervised by Dr. D. I. Lalwani (Asociate Professor) at Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Surat - 395 007, Gujarat, India.
COPYRIGHTS @ 2018
This document is copyrighted material. Under copyright law, no part of the document may be produced without the expressed written permission of the author and the supervisor.
I take this opportunity to express my sincere gratitude to the all-pervading Supreme Personality of Godhead, Lord Sri Krishna who is seated in everyone’s heart as the super- soul, from whom comes all knowledge, remembrance and forgetfulness, who is the reservoir of all pleasure, without whose sanctions not even a blade of grass can move, for providing me sufficient strength, knowledge and inspiration to carry out this work.
I would also like to express my deep gratitude to Dr. D. I. Lalwani, Associate Professor, in Mechanical Engineering Department, SVNIT, Surat for his valuable guidance, motivation, co-operation with an encouraging attitude. He revised my manuscripts with me, word by word. Even though at many times I felt that it was enough and would like to submit, he patiently went through my lengthy manuscripts, uncovering problems here and there, refining the manuscripts to perfect art pieces. He is the person whom I am infinitely indebted
I take this opportunity to express my sincere gratitude to the authorities of SVNIT and the Department of Mechanical Engineering for giving me this opportunity to study and avail the facilities of this Institution.
I am also very much thankful to Prof. Vijay Chaudhary (HOD, Mechanical Department, CHARUSAT), for allowing me to use CNC machine and Kistler dynamometer for force measurement. I am also very much grateful to Dr. Harshit K. Dave for kindly allowing me to use laboratory facility. Also, I would like to thank Mr. Hiren Patel, Mr. Mrugesh Patel, Mr. Umesh Thakkar, Mr. Mukesh Patel and Akash Panday for their help.
I would also like to express my sincere gratitude to Prof. A. K. Shukla, Prof. H. K. Raval, Prof. K. P. Desai and Prof. A. A. Shaikh for giving their valuable time and suggestions at every stage of my research. During our M.tech, A. A. Shaikh Sir asked the students “Is it possible to develop and the equation of a polygon?” This statement gave me a concept that something like this is also possible. Later, I developed an equation of polygon in polar form and used it for generating spiral tool path.
I am also very much thankful to Dr. J. V. Menghani for asking me a question in my credit seminar that “Is there any method or technique or number for comparing different types of pockets?” At that time I gave some convincing reply but later that question triggered the concept of Dimensionless Number (DN) for quantitatively comparing different pockets.
I am also very much in debt to H. G. Caitanya Avatar Das and H. G. Ishana Gaura Das and all my friends for kindly allowing me to stay in their room whenever I came to Surat for my Ph.D. work. I would also like to thank Vishwambhara Das for providing nice prasadam during my stay in Surat. I am also very much thankful to Principal of SVIT, Dr. J. V. Deshkar and my colleagues, namely, Dr. P. V. Ramana, Dr. P. B. Tailor, S. P. Patel, D. S. Shah, Sekar S., Hardik Dodiya and the entire staff of Mechanical Engineering Department, SVIT for their overall support and encouragement.
I am also very much thankful to my wife Nidhi Gaurangi Devi Dasi and my sons Madhusudana and Baladeva for tolerating, and co-operating me during my Ph.D. work. I am also very much in debt to my parents and family members, (namely Nimesh Patel and family, Prakash Patel and family, Mukesh Patel and family, Sankar H. Patel and family) for their support in various ways. The list is endless but I am thankful to everyone who helped me directly or indirectly. I pray that all of them may achieve the ultimate benefit, supreme destination and the worshipable object of their life, i.e., shelter at the lotus feet of Sri Sri Radha and Krishna.
Dr. D. D. Patel
A 2.5D pocket machining is widely used in aerospace, shipyard, automobile, dies and mold industries. In machining of 2.5D pockets, conventional tool path strategies, such as directional parallel tool path and contour parallel tool path, are widely used. However, these tool paths significantly limit the machining performance in terms of machining time, surface finish and tool wear because of repeated machining direction alteration, stop-and-go motion, sharp velocity discontinuity, and frequent repositioning, retraction, acceleration and deceleration of a tool.
To overcome the above-mentioned problems of conventional tool path strategies, an attempt has been made to generate a spiral tool path for 2.5D pocket machining. The spiral tool path is developed using second order elliptic Partial Differential Equation (PDE) and NURBS and it is free from sharp corners inside the pocket region. The spiral tool path begins as a smooth spiral in the pocket interior then progresses outward and finally morphs to the pocket boundary. The algorithm for the spiral tool path is coded in MATLAB® for pre-processing, solving and post-processing of elliptic PDE. The successful generation of spiral tool path depends on various algorithm parameters such as mesh size, permissible error and number of degree-steps. The effect of these parameters on spiral tool path generation is studied and the best values are reported. Further, the developed algorithm is implemented and experimentally validated on pockets that are bounded by a non-star-shaped polygon with bottle-necks, free-form curve and pocket with an island.
The shape of pocket geometry, tool path strategy and various machining parameters (speed, feed rate and depth of cut) affect machining performance. A large amount of the literature related to pocket machining deals with tool path generation and the effect of various machining parameters. However, the effect of the shape of a pocket geometry and tool path strategy on the performance of pocket machining is scarcely reported. Hence, an attempt has been made to investigate the effect of aspect ratio (i.e., changing the shape of a pocket), feed rate and tool path strategies (zig-zag, spiral and contour parallel) on tool path length, cutting time, percentage utilization of a tool and average surface roughness in machining of AISI 304 stainless steel using design of experiments (DOE). A novel concept of Percentage Utilization of a Tool (PUT) is developed to study the effect of tool path strategy and aspect ratio (shape of pocket geometry). It was found from the experimental investigation that the spiral tool path suffers a problem of over machining, i.e., increased tool path length because of squeezing of tool path in a narrow or a bottle-necked region.
From the findings of above experimental investigation, it was anticipated that there is a need to develop a method (or technique) for comparing different pocket geometry quantitatively and predict the effect of pocket geometry on pocket machining. A novel approach is reported for quantitative comparison of different pocket geometries using a dimensionless number, ‘DN’. The concept and formula of DN are developed and DN is calculated for various pocket geometries. The guidelines for comparing pocket geometries based on DN and PUT are reported. The results show that DN can be used to predict the quality of tool path prior to tool path generation.
Further, an algorithm to decompose pocket geometry (parent geometry) into sub-geometries is developed that improves the efficiency of spiral tool path for bottle-neck pockets (or multiple-connected pocket). The algorithm uses another dimensionless number ‘HARIN’ to compare parent pocket geometry with sub-geometries; HARI is the abbreviation of ‘Helps in Appropriate Rive-lines Identification’ and suffix ‘N’ stands for a number, hence it is spelled as ‘HARI number’. The developed algorithm is particularly useful where the region of the bottle-neck cannot be visualized easily. The results indicate that decomposing pocket geometry with the new algorithm improves HARIN and removes the effect of bottle-necks. Furthermore, the algorithm for decomposition is extended for pockets that are bounded by free-form curves.
In addition, experimental investigation of speed, feed and step-over on cutting time, surface roughness and cutting forces in the pocket machining of AISI P20 is carried out using spiral tool path. Also, a new method to select the sampling frequency of dynamometer and analysis the cutting force in machining of pocket using spiral tool path are reported.
Fig. 1.1 Connectivity among different chapters
Fig. 2.1 Representation of pocket machining operation
Fig. 2.2 The engagement angle α of a tool path, (a) a straight segment of the tool path and (b) a curved segment of the tool path [12]
Fig. 2.3 Changes of engagement angle α at different types of tool motions [17]
Fig. 2.4 Machining temperature in machining at high cutting speeds [21]
Fig. 2.5 Classification of pockets: (a) plain, (b) free-form walls, (c) virtual or corner pocket, (d) multi-entrance faces, (e) multi-faced bottom, and (f) free-form-bottom pocket [37]
Fig. 2.6 Different types of polygons (a) a simple polygon, (b) a polygon with a hole, (c) a convex polygon, (d) a monotone polygon, (e) an orthogonal polygon, (f) a spiral polygon and (g) star-shaped polygons [43]
Fig. 2.7 Representation of 2.5D pocket in terms of (a) elemental machining surfaces, (b) adjacency graph and (c) floor [13]
Fig. 2.8 (a) Zigzag tool path (b) Zig tool path (c) Smooth zigzag tool path [19]
Fig. 2.9 Repositioning and retraction of cutter to cover the entire pocket [49]
Fig. 2.10 Contour parallel tool path (a) without island [9] (b) with island [65]
Fig. 2.11 Conventional pair-wise offset approach [67]
Fig. 2.12 Offset polygon using winding number (a) offset each edge (b) extend the offset edges (c) calculate the winding number (d) offset tool path of polygon [67]
Fig. 2.13 Uncut region in pocket machining [63]
Fig. 2.14 Tool path using Voronoi diagram for machining of free-form pocket, (a) segmenting a contour (or boundary of pocket), (b) construction of Voronoi diagram and (c) generation of tool paths using Voronoi diagram [76]
Fig. 2.15 Pixel-based engagement simulation [26]
Fig. 2.16 Machining simulation (a) contour parallel and (b) wavelet-based method [81]
Fig. 2.17 Space filling curve
Fig. 2.18 Space-filling curves (a) Peano (b) Hilbert's (c) Moore's [93, 94, 99]
Fig. 2.19 Corner machining tactics, (a) single strategy and (b) double loop strategy [46]
Fig. 2.20 Tool path generation by Laplace based spiral contouring method (a) Pocket with an arbitrary boundary (b) Mapping of the pocket to a parametric unit square of (u,v) (c) Tool path for pocket machining [47]
Fig. 2.21 Steps involved in conformal map based spiral tool path generation [125]
Fig. 2.22 Mapping based spiral tool path, (a) the pocket region to be machined. (b) triangulated pocket, (c) guided spiral on the disk and (d) generated spiral tool path of pocketing [9]
Fig. 2.23 Medial axis skeleton using (a) grass fire model [137] and (b) maximal disk model [138]
Fig. 2.24 Tool path generated by MAT, (a) Digital photo and boundary points, (b) MA and medial axis circles and (c) MA and offset curves [134]
Fig. 2.25 G1 continuous spiral tool path based on medial axis [45]
Fig. 2.26 Clothoidal spiral tool path [19]
Fig. 2.27 Spiral tool path based on solution of PDE [4]
Fig. 2.28 Trochoidal tool path [160]
Fig. 3.1 (a) Star-shaped geometry and (b) Non-star-shaped geometry [188]
Fig. 3.2 Solution of PDE for principal eigenvalue over star-shaped polygon
Fig. 3.3 Shift origin to (xpole, ypole) and polar equation of star-shaped polygon
Fig. 3.4 Normalized solution of PDE over star-shaped polygon
Fig. 3.5 Iso-contours for a specified step distance along (Lmax, θmax)
Fig. 3.6 Spiraling between iso-contours
Fig. 3.7 Shift origin of coordinate system back to its original place
Fig. 3.8 Experimental validation (a) regular polygon, (b) irregular polygon, (c) decomposition of complex pocket bounded by straight lines, and (d) decomposition of free-form pocket.
Fig. 3.9 Spiral tool path for a non-star-shaped polygon
Fig. 3.10 Spiral tool path for a non-star-shaped free-form pocket
Fig. 3.11 (a) Mapping based spiral tool path [9] (b) Proposed spiral tool path based on PDE
Fig. 3.12 Spiral tool path for a pocket with island
Fig. 3.13 Effect of mesh size (Hmax) on spiral tool path
Fig. 3.14 (a) Number of triangles (sub-domains) vs. Hmax and (b) Relative error (%) vs. Hmax for N = 3, 4, 5, and 6, and L = 50.
Fig. 3.15 (a) Effect of permissible error (1/ PU) and (b) Effect of number of degree steps (NDS) on spiral tool path
Fig. 4.1 Zig-zag (top), spiral (middle) and contour parallel (bottom) tool path for ellipse with aspect ratio of 0.25
Fig. 4.2 Zig-zag (top), spiral (middle) and contour parallel (bottom) tool path for ellipse with aspect ratio of 0.5
Fig. 4.3 Zig-zag (top), spiral (middle) and contour parallel (bottom) tool path for ellipse with aspect ratio of 0.75
Fig. 4.4 Carbide end mill cutter with four flutes and diameter of 8 mm
Fig. 4.5 Workpiece during (left) and after (right) machining
Fig. 4.6 Main effect plot for tool path length
Fig. 4.7 Interaction effect plot for tool path length
Fig. 4.8 Actual vs. predicted value of tool path length
Fig. 4.9 Main effect plot for cutting time
Fig. 4.10 Interaction effect plot for cutting time
Fig. 4.11 Actual vs. predicted value of cutting time
Fig. 4.12 Main effect plot for percentage utilization of a tool
Fig. 4.13 Interaction effect plot for percentage utilization of a tool
Fig. 4.14 Actual vs. predicted values for percentage utilization of a tool
Fig. 4.15 Main effect plot for surface roughness (Ra)
Fig. 4.16 Interaction effect plot for surface roughness (Ra)
Fig. 4.17 Actual vs. predicted values for surface roughness (Ra)
Fig. 4.18 Surface roughness contour in aspect ratio (A) and feed rate (B) plane for zig-zag tool path
Fig. 4.19 Surface roughness contour in aspect ratio (A) and feed rate (B) plane for spiral tool path
Fig. 4.20 Surface roughness contour in aspect ratio (A) and feed rate (B) plane for contour tool path
Fig. 5.1 Spiral tool path on a pocket geometry, (a) before pocket decomposition and (b) after pocket decomposition [12]
Fig. 5.2 Pocket geometries along with EAC, EPC, MEC and LEC, (a) triangular shaped, (b) plus shaped and, (c) and (d) octagonal shaped with spike protruding outside and inside respectively.
Fig. 5.3 (a) to (g) trend of Do, D1, D2, D3, D4, D5 and Ce along with PUT respectively for various pocket geometries
Fig. 5.4 (a) to (e) Spiral tool path, LEC, EAC, EPC and MEC for different regular polygonal geometries and (f) plot showing comparison of DN, DNspiral and PUT
Fig. 5.5 (a) to (e) Spiral tool path, LEC, EAC, EPC and MEC for different plus shaped geometries and (f) plot showing comparison of DN, DNspiral and PUT
Fig. 5.6 (a) to (e) Spiral tool path, LEC, EAC, EPC and MEC for different triangular shaped geometries and (f) plot showing comparison of DN, DNspiral and PUT
Fig. 5.7 (a) to (e) Spiral tool path, LEC, EAC, EPC and MEC for different rectangular shaped geometries and (f) plot showing comparison of DN, DNspiral and PUT
Fig. 5.8 (a) to (e) Spiral tool path, LEC, EAC, EPC and MEC for different irregular shaped geometries and (f) plot showing comparison of DN, DNspiral and PUT
Fig. 5.9 (a) to (e) Spiral tool path, LEC, EAC, EPC and MEC for different elliptical shaped geometries and (f) plot showing comparison of DN, DNspiral and PUT
Fig. 5.10 Plot of DN and PUT in decreasing order of DN
Fig. 5.11 Polygon before decomposition with possible spit-lines for typical bottle-neck corner
Fig. 5.12 Spiral tool path on decomposed polygon
Fig. 5.13 Examples of pocket decomposition on ‘H’, ‘S’ and ‘M’ shaped pockets.
Fig. 5.14 Split line (rive-line) determination using and spiral tool path for two different free form pockets (i.e., (a) and (b)) after pocket decomposition
Fig. 5.15 A Pocket with an island.
Fig. 6.1 Standard end mill cutter layout
Fig. 6.2 Carbide end mill cutters with 4 flutes and diameter 8 mm
Fig. 6.3 PX 10 CNC SIEMENS with SINUMERIC 802 D SL controller
Fig. 6.4 A piezo-based dynamometer (Make: KISTLER®; Model: 9272)
Fig. 6.5 Dimensional details of KISTLER® 9272
Fig. 6.6 Fixture for mounting workpiece on a dynamometer.
Fig. 6.7 Base plate
Fig. 6.8 Strap clamp
Fig. 6.9 V shape locator
Fig. 6.10 Mounting the workpiece on the fixture.
Fig. 6.11 Surftest SJ-210 P, Mitutoyo ®
Fig. 6.12 Main effect plot for cutting time
Fig. 6.13 Main effect plot for surface roughness
Fig. 6.14 Square and circular solts
Fig. 6.15 Cutting forces in X (blue), Y (green) and Z (black) direction
Fig. 6.16 Cutting forces in X (blue), Y (green) and Z (black) direction- zoomed
Fig. 6.17 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 1(i.e., Speed =60, Feed = 0.088 and Step over = 10)
Fig. 6.18 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 2 (i.e., Speed = 80, Feed = 0.088 and Step over = 10)
Fig. 6.19 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 3 (i.e., Speed = 60, Feed = 0.152 and Step over = 10)
Fig. 6.20 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 4 (i.e., Speed = 80, Feed = 0.152 and Step over = 10 )
Fig. 6.21 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 5 (i.e., Speed = 60, Feed = 0.088 and Step over = 30)
Fig. 6.22 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 6 (i.e., Speed =80, Feed = 0.088and Step over = 30)
Fig. 6.23 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 7 (i.e., Speed = 60, Feed = 0.152 and Step over = 30)
Fig. 6.24 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 8 (i.e., Speed = 80, Feed = 0.152 and Step over = 30)
Fig. 6.25 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 9 (i.e., Speed =70, Feed = 0.12 and Step over = 20)
Fig. 6.26 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no. for std order 10 (i.e., Speed =70, Feed = 0.12 and Step over = 20)
Fig. 6.27 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no for machining of a square slot
Fig. 6.28 Fx, Fy and Fz vs. Sample no. and Avg Fx, Fy and Fz (per rotation) vs. rotation no for machining of a circular slot
Fig. 6.29 Radial and tangential forces in end milling process [243]
Fig. 6.30 Dial gauge to verify inclination of tool
Table 2-1 Merits and demerits of different tool path strategies
Table 3-1 Equation of lines with start points and end points for line segment with respect to the range of θ
Table 4-1 Recommended cutting conditions [214, 215]
Table 4-2 Factors with their respective levels for a 33 design in brief
Table 4-3 Design matrix with responses
Table 4-4 Range of responses
Table 4-5 ANOVA (partial sum of square) for tool path length after removing insignificant terms
Table 4-6 ANOVA (partial sum of square) for cutting time after removing insignificant terms
Table 4-7 ANOVA (partial sum of square) for percentage utilization after removing insignificant terms
Table 4-8 ANOVA (partial sum of square) for surface roughness (Ra) after removing insignificant terms
Table 5-1 DN, DNspiral, I and PUT for different pocket geometries
Table 5-2 Range of DN and its interpretation
Table 6-1 Chemical composition of AISI P20
Table 6-2 Tool geometry of end mill cutter
Table 6-3 Tool manufacturer’s recommended cutting speed and feed
Table 6-4 Factors with their respective levels (coded and actual) for a 23 design with center level
Table 6-5 List of various sampling frequency
Table 6-6 Sampling frequency for various speed
Table 6-7 Design matrix
Table 6-8 ANOVA (partial sum of square) for cutting time after removing insignificant terms
Table 6-9 ANOVA (partial sum of square) for surface roughness after removing insignificant terms
In CNC milling operations, a 2.5D pocket machining is used for manufacturing of many mechanical parts and it is one of the main operations compared to other milling operations. A CNC machining center can be classified as 2.5, 3, 4, 5 or 6-axis machine based on the number of axes a machine can interpolate simultaneously. 2.5D machining can be carried out by interpolating two axes (e.g., X and Y) simultaneously and the third axis (Z) is used for positioning. The particular task of 2.5D machining is also known as pocket machining [1]. In 2.5D pocket machining, the material is usually removed in more than one layer, using a tool path strategy, from inside of a pre-defined contour (boundary) of a pocket between two horizontal planes until the entire pocket is formed [2]. Pocket machining has remarkable applications in the aerospace, shipyard, automobile, dies and mold industries [3-5]. One of the most occurring industrial milling tasks is pocket machining [6]. More than 80% of all mechanical parts can be machined by applying the concepts of pocket machining [7, 8].
Conventional tool path strategies for 2.5D pocket machining, such as directional parallel and contour parallel, are widely used and available in many commercial CAD/CAM softwares, namely, Unigraphics, Delcam, Mastercam, Cimatron, etc., because these tool path strategies are computationally tractable and geometrically appealing [9]. However, these conventional tool path strategies significantly limit the machining efficiency in terms of machining time, surface finish and tool wear because of repeated alteration of machining direction, stop-and-go motion, sharp velocity discontinuity, and frequent repositioning, retraction, acceleration and deceleration of a tool.
The above problems of conventional tool path strategies can be overcome by spiral tool path strategies reported by various researchers. The spiral tool path, proposed by Bieterman and Sandstrom [4] and obtained by solving Partial Differential Equation (PDE), is one of the spiral tool path strategies. They obtained spiral tool path by post processing the solution of elliptic PDE. The elliptic PDE equation, whose order of equation is even, has nice smoothing properties. The constant-value contours (i.e., u), obtained from the solution of PDE have progressively less local maximum curvatures as the value of u varies from the peak to the boundary. This property of PDE enables a tool path to begin as a spiral in a pocket interior and gradually morphs to the shape of the pocket.
However, there are two main issues associated with the spiral tool path that is obtained by solving elliptic PDE, namely, (a) the solution function of the elliptic PDE does not have uniform gradient and (b) it is difficult to guarantee that bounded distance between two consecutive tool path contours is less than the desired step-over [10]. In the present work, a post-processing algorithm is developed to obtain spiral tool path using PDE and NURBS to deal with above two issues. NURBS provide a unified mathematical basis for representing analytic shapes, such as conic sections and quadric surfaces, in addition to free-form entities, such as car bodies and ship hulls [11]. Many modern CNC controllers, such as, not limited to, GE Fanuc’s Series 15-B, 16-C, 16i-MA, Sinumerik 840D and Tosnuc 999 supports NURBS interpolator. The developed spiral tool path can be converted into appropriate control points and knot vectors, and fed to the controller to carry out the machining.
The various spiral tool path strategies, proposed by many researchers, suffer from a common drawback, i.e., over machining (increased tool path length) in narrow or bottlenecked region of a pocket. Held and Spielberger [12] reported that the shape of the pocket has a great influence on the suitability of a spiral tool path. However, methods or techniques to compare different types of pocket geometries quantitatively and to decompose a pocket geometry with an aim to improve machining performance are scarcely available. The background related to various issues that are discussed above provides the motivation for our research work.
Hence, the thesis address four separate issues, namely, (i) development of an algorithm for spiral tool path strategy for pocket machining using PDE and NURBS curve that overcomes the problem of assuring bounded distance between consecutive iso-contours, (ii) investigating the effect of shapes of pockets and tool path strategies on performance in pocket machining, (iii) developing a technique (method) for quantitatively comparing different pocket geometries and subsequent decomposition into sub-pockets (if required) and (iv) investigating the effect of speed, feed and step-over on cutting time, surface roughness and cutting forces.
The thesis is presented in eight chapters. Chapters three to six discuss the above mentioned four issues related to pocket machining respectively. In the beginning of these chapters, a brief review of literature is presented, followed by work carried out to address a particular issue. Next, the results and discussion are presented in the same chapter and conclusions are drawn. A brief outline of each chapter is discussed below and connectivity among different chapters is shown in Fig. 1.1.
- Chapter one provides a background and motivation of the subject area of CNC pocket machining.
- Chapter two presents an extensive literature review related to tool path development for 2.5D high-speed pocket machining and the objectives of the present research.
- Chapter three discusses the development of spiral tool path using second order elliptic Partial Differential Equation (PDE) and NURBS. Implementation of the developed spiral tool path is demonstrated on regular polygonal pockets, irregular polygonal pocket, pockets bounded by free-form curve and pockets with an island.
- Chapter four describes the experimental investigation of the effect of aspect ratio (i.e., ratio of minor axis to major axis of an ellipse), feed rate and tool path strategies on tool path length, cutting time, percentage utilization of a tool and average surface roughness (Ra) in machining of elliptical pocket using Design of Experiments on AISI 304 stainless steel.
Abbildung in dieser Leseprobe nicht enthalten
Fig. 1.1 Connectivity among different chapters
- Chapter five discusses the need for a method or technique to quantitatively compare different pocket geometries using a Dimensionless Number, ‘DN’. Further, the algorithm to decompose a pocket into sub-pockets is developed that increases the efficiency of spiral tool path (reduced tool path length and improved surface finish) when bottle-neck is present in the geometry.
- Chapter six presents the findings of an experimental investigation of speed, feed and step-over on cutting time, average surface roughness and cutting forces in the pocket milling of AISI P20 using spiral tool path obtained through PDE and NURBS.
- Chapter seven summarizes overall results and discussion of the research work.
- Chapter eight presents conclusions drawn from the present research work and scopes of future research.
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Literature related to 2.5D pocket machining has recently shown an increased interest from researchers and engineers and has motivated them to look for new methods that will increase productivity and reliability [13, 14]. Selection of appropriate tool path strategy is an important decision in pocket machining because it affects machining process in terms of machining time, surface finish, material removal rate, tool life, tool wear, etc. However, it is difficult to find information on particular types of tool path strategy because the available literature varies both in content and focus.
The three difficulties in making use of the knowledge that is reported by various researchers are: (i) a large number of papers published related to 2.5D pocket machining, (ii) the pocket machining is a broad area, hence, the research papers related to pocket machining may focus on a specific topic from one field or discuss a broad range of topic spanning several areas and (iii) the pocket machining is studied by specialists from different fields, sometimes using subject-specific (and potentially unfamiliar) terminology. Therefore, the above difficulties make the subject matter more complicated and challenging.
In this chapter, first, a brief description of the basic theory of pocket machining, associated terminologies and tool path requirement for High Speed Machining (HSM) are presented. Second, an extensive literature review is carried out related to tool path strategies for 2.5D high-speed pocket machining and tool paths are categorized according to the subject matter. Further, along with a review of each tool path, main researchers who are working in the field, major advances and discoveries along with the merits and demerits are reported. In addition, summary of the merits and demerits of different tool paths is provided in tabular format at the end of the literature review. Futher, the current status of development in tool path strategies is reported and the objectives of the present research are presented.
A workpiece (a component or a part) is a piece of raw material (semi-finished or un-finished) that is machined using a material removal process (e.g., milling) to obtain desired shape, size and surface finish. A Pocket machining (Fig. 2.1) refers to creating an inner empty volume starting from a workpiece face. The pocket machining is similar to conventional milling but differs mainly in two ways, i.e., (i) to begin machining, the tool has to move and cut in the direction normal to a pocket (also called plunging), hence, the tool should have that capability and (ii) the pocket walls constrain the tool path at the corner of pocket that results in finite radius (radius of cutter) at the corners [15]. The created empty volume is called as a pocket. A Tool (a cutting tool or a cutter) refers to a milling cutter (usually an end-mill) that is used to create a pocket in the workpiece.
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Fig. 2.1 Representation of pocket machining operation
Material Removal Rate (MMR) is the rate at which the material is removed (i.e., ratio of the volume of material removed divided by the machining time) from the workpiece. MMR depends on various parameters such as speed, feed, depth of cut, tool path strategy, step-over, engagement angle, etc. In a milling operation, the term speed can refer to spindle speed or cutting speed. Spindle speed is the rate at which the spindle (or tool) rotates about its own axis and is measured as number of revolutions per minute or rotation per minute (rpm) whereas cutting speed (or surface speed) refers to the relative velocity between the workpiece and the cutting tool, and is measured in meters per minute (m/min), millimeters per minute (mm/min) or surface feet per minute (sfm). The cutting speed is proportional to the spindle speed and the diameter of a cutter. Feed or feed rate is the relative velocity at which the cutter advances into the workpiece and in given direction. The feed rate is measured in millimeters per minute (mm/min) or inches per minute (in/min). Sometimes, the feed is also defined as the distance traveled by a workpiece/tool per revolution of a cutter or per tooth of a cutter. The depth of cut is the normal distance between the machined and un-machined surface and is measured in millimeters (mm) or inches (in). In other words, it indicates the penetration of the tool below the original un-machined work surface. Tool path is a trajectory of a center of a tool (also called cutter location path) along which the tool moves to create a pocket. Topal [16] defined step-over as “a milling parameter that defines the distance between two neighboring passes (of tool path) over the workpiece. It is usually given as a percentage of the tool diameter and usually called step-over ratio.” Held and Spielberger [12] defined engagement angle (α) as “ the angle which spans the part of the tool surface that performs the cutting” as shown in Fig. 2.2 and Fig. 2.3. Engagement angle depends on tool path, step-over and unmachined profile of a pocket. The machining parameters, namely, speed, feed, depth of cut, tool path strategy, step-over, engagement angle, etc., affect machining time, cutting forces, surface roughness, tool wear (tool life), accuracy and efficiency of the tool path. Hence, these parameters should be carefully selected and controlled.
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Fig. 2.2 The engagement angle α of a tool path, (a) a straight segment of the tool path and (b) a curved segment of the tool path [12]
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Fig. 2.3 Changes of engagement angle α at different types of tool motions [17]
In general terms, efficiency is the ability to do things well and successfully without wasting materials, energy, efforts, money, time, etc., and it is often measurable. It is generally defined as the ratio of a useful output to an input. Whereas, efficiency in the context of tool path in pocket machining indicates the ability of a tool path strategy to remove the material from a workpiece in the most efficient manner when compared to other tool path strategies. Again, the term ‘most efficient manner’ is subjective and is loosely used in literature. For a given tool path strategy, it may refer to the ability of a tool path strategy to machine a pocket with shorter tool path length, lesser machining time, better surface finish and reduced tool wear (i.e., greater tool life), ability to clear the material at corner, ability of a tool path to reduce variation in step-over, ability of a tool path to deal with bottle necks in a pocket, etc. Interestingly, many researchers try to show that their tool path strategy is better and more efficient compared to other tool path strategies without considering the fact that the effectiveness of tool path is greatly influenced by the shape of the pocket. For example, it can be shown that zig-zag tool path strategy is more efficient (in terms of machining time) as compared to spiral tool path strategy for High Speed Machining of a rectangular pocket with a very high aspect ratio (length >> width) and the opposite is true for a circular pocket.
Quite a few of the many definitions in the literature describe HSM as end milling with small diameter tools (<=10 mm) at high rotational speeds (>=10,000 rpm); power is greater than 10 HP; and material removal rate is considerably higher than that is achieved by conventional techniques [18, 19]. Different definitions of HSM are based on a combination of various parameters such as high spindle speed, high feed rate, specific tools, specific tool motion, high cutting speed, high material removal rate, machine dynamics, etc. Popma [20] pointed out that these definitions for HSM are inadequate because defining spindle speed range as the high-speed range is very ambiguous. Further, the author reported that with the evolution of high speed machines, higher machining speeds have become possible and the trend of rising cutting speeds and feed rates has not yet ended. Hence, these are not the good criterion for a definition. Moreover, the range of cutting speed and feed rates also depends on workpiece and tool material combination. Finally, Popma [20] agreed with the definition, given by Smith from the University of Florida, “One speaks of high-speed machining when the tooth passing frequency of the tool approaches the natural frequency of the machine-tool system.”
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Fig. 2.4 Machining temperature in machining at high cutting speeds [21]
Fig. 2.4 shows that as the cutting speed increases, the tool-workpiece interface temperature increases up to a certain point and then reduces. This is due to a fact that in intermittent cutting operation, as the cutting speed increases, very little time is available for heat dissipation at the tool-work interface and 95 to 98% of cutting heat is taken away by the chip [22]. The history of HSM is outlined in references [21, 23, 24]. Many references related to the introduction, application, capabilities, advantages and disadvantages, technical challenges, process-parameters optimization and the role of the engagement angle for HSM are found in references [20, 21, 23-27]. In recent years, high speed machining has become popular to minimize machining time, improve surface quality and maintain high material removal rates [13, 28, 29]. The applications of HSM have increased due to the need for high precision and high accuracy machining and machining of difficult-to-cut materials [30].
In high speed machining, the tool path (slide motions) have to be smooth to achieve the required surface quality and tool life. Before discussing tool path for HSM, one should know and understand the difference between smooth tool path and smooth motion. Mathematically, a smooth tool path considers only the geometry which means the second derivative with respect to a geometrical parameter (the displacement for example) whereas a smooth motion deals with the temporal movement (i.e., the second derivative with respect to the time) [31]. In HSM, a tool path strategy greatly affects machining time, surface finish and tool wear which means that for the removal of the same amount of material, the type of tool path utilized will produce significantly different results in terms of machining time, surface finish and tool wear [4, 32].
The high cutting speed and high feed rate in HSM require tool path that has a constant chip load and usually a very small step-over. Moreover, a tool path should avoid sharp turns [19, 27], otherwise, unexpected tool breakage, which results from exceeding a tool’s permissible loading conditions, not only costs money but also disrupts the machining process [33]. Therefore, in HSM, smaller step-over with a lighter cuts increment are preferred.
A tool path with strong curvature (or variation of curvature) at high feed rates incurs severe acceleration/deceleration of the machine axes [4, 34]. Depending on the axis inertia and torque capacity of the axis drive motors, the tool path with strong curvature may be difficult or impossible to execute at the nominal feed rate, and significant deviations (contour errors) from the desired path may result [34]. Another approach is to keep feed rate high in the region that does not involve changing direction and lower feed rate for changing direction, but this involves changing the feed rate more often. Feed rate optimization may allow the program to maintain a higher average feed rate where the profile of the tool path changes frequently [4, 35].
In this chapter, to begin with, some of the noteworthy review papers that are published by various researchers related to tool path strategies for pocket machining are reported in section 2.5. Various types of pocket and pocket machining are discussed in section 2.6. Section 2.7 deals with literature review related to conventional tool path strategies such as directional parallel, boundary parallel and space-filling curves. Section 2.8 discusses various corner machining tactics. A literature review related to various advanced tool path strategies for HSM is reported in section 2.9. Lastly, the current status of development in tool path strategies is reported and a summary table is provided for ready reference.
Various tool path strategies are broadly classified based on the types of curve used or methods used for generating tool path. However, there is some overlapping between different classifications. In addition, an attempt has been made to discuss the merits and demerits of the work carried out by other researchers. We have avoided complicated mathematics associated with tool path generation in the literature review. Also, rather than discussing history, application, advantages, mathematics, etc., in details, references are provided to preserve the flow of content. Multiple entries (of references) are made for those readers who are interested in more detail of a particular topic and wish to acquire material on a particular subject matter.
Literature related to tool path strategies of pocket machining from 1989 to 1994 was classified by Dragomatz and Mann [36]. Hatna et al., [37] reported many references related to geometric and technological issues related to 2.5D pocket machining; they considered the relationship among the pocket shape, cutter diameter, machine tool and cutting conditions to optimize cost and lead time of pocket machining. A literature review related to “offset curves” is reported by Maekawan Takashi [38] and Pham [39], and development of CNC machining of free-form surfaces is reported by Lasemi et al. [40]. Schulz and Moriwaki [24] reviewed literature related to development in HSM before 1992. Held [41] carried out literature survey and provided full algorithms related to the development of directional parallel and contour parallel before 1991. Ding et al. [42] reviewed HSM strategies adopted by commercial CAD/CAM systems such as Catia, Cimatron, EdgeCAM, Esprit, FeatureCAM, GibbsCAM, HyperMill, MasterCAM, PowerMILL, Pro/Engineer, QuickMILL, SurfCAM, Tebis, Topsolid, Unigraphics, WorkNC.
The shapes of pocket vary greatly from one component to another depending upon its application within a whole component. Moreover, mathematical description of a pocket can vary from simple to a complex representation [37]. Some well-known descriptions of pockets and pocket machining are discussed below.
Hatna et al. [37] defined and categorized pockets into six main types as shown in Fig. 2.5. They are: (a) plain, (b) free-form-walls, (c) virtual or corner pocket, (d) multi-entrance faces, (e) multi-faced bottom, and (f) free-form-bottom pocket. The details of each pocket type are available in the reference [37].
If the boundary of the pocket is a free-form curve, it is called free-form pocket. When the boundary of the pocket is bounded by straight line segments it is called polygon. However, when the boundary of the pocket is a combination of straight lines segments, arc and/or free-form curve, we here refer as a mixed pocket. A polygon is called as “a simple polygon” or “simply connected” if the polygon is non-self-intersecting and it has no holes (or island).
Polygons can also be classified as non-convex (Fig. 2.6 (a)) and convex (Fig. 2.6 (c)) polygons. If at least, one pair of two points (for example point ‘p’ and ‘q’ in Fig. 2.6 (a)) exists such that they are not visible from each other, the polygon is called non-convex polygon. In other words, if all the internal angles of a polygon are less than 180º, the polygon is said to be convex else non-convex. All regular polygons such as equilateral triangle, square, etc., are convex.
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Fig. 2.5 Classification of pockets: (a) plain, (b) free-form walls, (c) virtual or corner pocket, (d) multi-entrance faces, (e) multi-faced bottom, and (f) free-form-bottom pocket [37]
Polygons are also classified as orthogonal, star-shaped, spiral and monotone polygons and shown in Fig. 2.6. A polygon is considered as a monotone (Fig. 2.6 (d)) with respect to a line if the projections of the vertices of the polygon occur in the same order as the order of polygon on that particular line. If the polygon obtained using only horizontal or vertical line segments, it is considered as an orthogonal polygon (Fig. 2.6 (e)). A polygon is considered as a spiral (Fig. 2.6 (f)) if there exist exactly one concave and one convex sub-chain. If a point exists in the interior of a polygon such that all the edges of the polygon are visible from that point, it is considered as a star-shaped polygon (Fig. 2.6 (g)).
Banerjee et al. [13] also defined pocket as “A typical pocket geometry with length (l), width (w), and height (h) can be represented in terms of elemental machining surface: Floor, Corner, and Wall, as shown in Fig. 2.7 (a) with their connectivity shown with the adjacency graph in Fig. 2.7 (b).” Fig. 2.7 (c) shows that floor machining operation involves the maximum material removal and machining time compared to the other elemental machining surfaces, i.e., wall and corners.
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Fig. 2.6 Different types of polygons (a) a simple polygon, (b) a polygon with a hole, (c) a convex polygon, (d) a monotone polygon, (e) an orthogonal polygon, (f) a spiral polygon and (g) star-shaped polygons [43]
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Fig. 2.7 Representation of 2.5D pocket in terms of (a) elemental machining surfaces, (b) adjacency graph and (c) floor [13]
The mathematician at Boeing Company, Bieterman M. B. and Sandstrom D. R. [4] defined pocket machining as “Removal of material from stock, layer by layer until pockets are formed and a manufactured part emerges. The tool path for a layer of a pocket is the centreline path along which a tool—an endmill—is fed as its rotating teeth cut the material” and shown in Fig. 2.1. Similar definitions are also found in other references [2, 44].
The pockets may or may not have an island. The island is an obstacle that is to be avoided by a cutting tool. The presence of islands greatly affects the process of tool path generation [37]. References for description of pocket and pocket machining are available in references [2, 4, 13, 15, 37, 43, 44].
A common problem in pocket machining is to find a suitable tool path [36, 45]. The conventional tool path strategies can be classified as directional parallel, contour parallel, and space-filling curve. Each strategy has its own merits and demerits.
A direction parallel tool path is a linear tool path that uses line segments that are parallel to an initially selected reference line. The optimum selection of the reference line depends on the shape of the pocket. It is the simplest type of tool path and coordinates of tool path are easily obtained. A directional parallel tool path can be further classified as (a) Two-way tool path (zigzag), (b) One-way tool path (zig) and (c) Smooth zigzag.
The zigzag tool path strategy (Fig. 2.8 (a)) removes the material from the pocket by moving the tool in a zigzag fashion. Two potential problems in the zigzag tool path are its intensive machining direction alteration (up-milling and down-milling), and stop-and-go motion. Intensive machining direction alterations may produce non-uniform machined surface quality and affect tool life. Further, stop-and-go motions consume more time and dynamically undesirable, and may leave tool marks on the boundaries of pockets and islands. In addition, the tool always collides with the walls of the pocket and islands; therefore, a final contouring pass around the pocket boundary is usually needed to achieve a satisfactory surface quality [19, 44, 46].
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Fig. 2.8 (a) Zigzag tool path (b) Zig tool path (c) Smooth zigzag tool path [19]
The zig machining is derived from zigzag machining. It performs cutting only in one direction as shown in Fig. 2.8 (b). The main advantage of zig machining is that consistent up-milling or down-milling is maintained. Usually, zig tool path gives a better surface quality than zigzag tool path. One of the major concerns in zig is excessive retraction motion and plunge feed of the tool. The tool retraction and plunge feed not only increase the length of tool path and machining time but also reduce cutter life [19, 44, 46, 47]. Smooth zigzag tool path as shown in Fig. 2.8 (c) is similar to zigzag tool path except that the corners are smoothened so that stop-go motion of the tool can be avoided. It also greatly reduced the acceleration and deceleration time when the tool has to change its direction but consistent up-milling or down-milling is not maintained [19, 44, 46, 47].
Yao and Joneja [48] classified movement of cutter with respect to workpiece (Fig. 2.9) as (a) effective move (i.e., cutter moves along the path to cover the region that has not been fully covered), (b) repositioning move (i.e., cutter moves along the path that has been fully covered before) and (c) retraction move (i.e., cutter is lifted up to a clearance height and then moved to another position using a rapid movement and then lowered down (plunging) to perform another cutting motion. Sometimes to cover the entire pocket with directional parallel strategy, retraction and/or repositioning of the tool become inevitable [49]. However, for HSM retraction or repositioning of the tool should be avoided as far as possible. Due to above-mentioned problems directional parallel tool path strategy is not suitable for HSM.
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Fig. 2.9 Repositioning and retraction of cutter to cover the entire pocket [49]
Park and Choi [50], Tang et al. [51], Tang [52] and Arkin et al. [1] proposed algorithms to minimize the number of tool path elements and tool retraction. They cited many references related to developed of an algorithm for optimization of pocket machining using zigzag tool path. The references related to directional parallel tool path strategy are given in references [1, 19, 32, 44, 46-59].
Contour parallel tool path is obtained by offsetting the boundary of a pocket internally and the boundary of an island (if present) outwardly until the entire region of the pocket is filled as shown in Fig. 2.10. The contour parallel tool path is generally obtained by algorithms based on (a) pair-wise intersection [44, 60, 61], (b) Voronoi diagram [44, 62] and (c) Pixel-based method [44, 63, 64].
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Fig. 2.10 Contour parallel tool path (a) without island [9] (b) with island [65]
In a pair-wise intersection method as shown in Fig. 2.11 and Fig. 2.12, a boundary of a pocket is offsetted inward and island outward. At concave corners (i.e., interior angle greater than 180) the offset segments are extended and connected to produce the final contour profiles. At convex corners (i.e., interior angle less than 180) the offset segments are trimmed and connected to produce the final contour profiles. An appropriate algorithm is used to remove invalid loops [66]. The offsetting, trimming and extending processes are repeatedly executed on each layer of offset segments and unnecessary loops are removed until required tool path is obtained. Chen and McMains [67] modified pair-wise intersection method and carried out polygon offsetting by computing winding numbers (Fig. 2.12). Their method is simple, robust and produces correct and logically consistent results. The linking of these offset segments and removal of invalid loops to get the desired tool path is a complicated issue and time-consuming [44, 46, 66, 68]. Some of the potential problems (uncut region) in this approach are undesired grooving cut, multiple tool entries, thin wall and poor surface finish as shown in Fig. 2.13 [63].
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Fig. 2.11 Conventional pair-wise offset approach [67]
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Fig. 2.12 Offset polygon using winding number (a) offset each edge (b) extend the offset edges (c) calculate the winding number (d) offset tool path of polygon [67]
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Fig. 2.13 Uncut region in pocket machining [63]
Voronoi diagram is one of the most useful data structures in computational geometry and it is applied to diverse fields of science and engineering. In the Voronoi diagram, a plane is partitioned mutually disjoint sub-areas, and every sub-area is associated with one contour line segment [69]. Then in each offset segments is trimmed its intersection using Voronoi diagram of the original pocket boundary. This approach is accepted to be more efficient and robust as the steps in offsetting the tool path segments can be subdivided in an organized manner [44, 62, 66, 68].
The first undisputed presentation of the Voronoi diagram concepts appeared in the works of Dirichlet (1850) and Voronoi (1908) [70, 71]. A brief historical perspective and development of Voronoi diagram could be found in references [71, 72]. A detailed survey of the Voronoi diagram is found in the references [73, 74]. The steps to generate tool path using Voronoi diagram for machining of free-form pocket are: (i) segment each curve element in a pocket (and island boundaries, if present) into a set of curve segments as shown in Fig. 2.14 (a), (ii) construct Voronoi diagram for the set of curve segments as shown in Fig. 2.14 (b) and (iii) generate tool paths using Voronoi diagram as shown in Fig. 2.14 (c). The limitation of Voronoi diagram is that it may cause instability when applied to the boundary of a point sequence curve (PS-curve) containing near circular portions [44, 68, 75].
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Fig. 2.14 Tool path using Voronoi diagram for machining of free-form pocket, (a) segmenting a contour (or boundary of pocket), (b) construction of Voronoi diagram and (c) generation of tool paths using Voronoi diagram [76]
The pixel-based methods are robust but need a large amount of memory as well as computation time to achieve a desired level of precision [63, 68, 75]. A pixel-based simulation procedure for cutter engagement is shown in Fig. 2.15. The gray region indicates material to be machined and the cutter engagement (θ) is to be calculated. The workpiece is represented as an array of bits. Any unmachined area is represented bitwise by ‘0s’, and the machined area is represented by a bitmap containing ‘1s’. The mask for the tool geometry is represented by a bitmap containing ‘1s’ within the circular region of the tool and ‘0s’ elsewhere. The cutting process is then simulated by masking the tool bitmap onto the workpiece bitmap. This step requires translating the tool to the proper location and replacing the corresponding elements of the workpiece with the result of a logical ‘‘OR’’ operation between the tool and workpiece [26].
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Fig. 2.15 Pixel-based engagement simulation [26]
To overcome the above various difficulties associated with contour parallel tool path such as detection and removal of invalid loops, to obtain contour parallel tool path for free-form pocket, ability to deal with an island inside the pocket, the problem of self-intersection and dis- continuities, etc., some algorithms are proposed and discussed below. Choi and Park [68] developed a fast and robust pair-wise interference detection (PWID) test for removal of invalid loops from PS-curve. Kalmanaovich and Nisnevich [77] proposed a swift and stable polygon growth and broken line offset method for removing invalid loops that do not fail because of missing segments encountered during offsetting. Rohmfeld [78] proposed an IGB (Invariant Gauss-Bonnet values) offset method for removal of plane curves loops by scanning of interval sequences. Veeramani and Gau [79] developed tool path using multiple cutting tool sizes for 2.5D pocket using Voronoi mountain. Arya et al. [80] present a polynomial-time approximation algorithm for the multiple-tool milling problem to mill the desired region with minimum cost. Jeong and Kim [65] found the method for generating tool paths using discrete distance maps which effectively deals with a free form shaped pockets with multiple islands. Narayanaswami and Pang [5, 81] introduced a wavelet based multi-resolution technique in which high-resolution wavelet offset curves are used close to the contour while low-resolution wavelet offset curves are used farther away from the contour. Fig. 2.16 shows machining simulation comparison between (a) contour parallel and (b) wavelet-based method. They reported that wavelet-based tool path strategy improves machining efficiency.
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Fig. 2.16 Machining simulation (a) contour parallel and (b) wavelet-based method [81]
Molina-Carmona et al. [82] suggested a morphological offset method for pocket machining that deals with problems associated with self-intersection and dis-continuities. Dhanik and Xirouchakis [83] introduced a fast marching method for generation of contour parallel tool path for an arbitrarily shaped pocket. Their method does not require detecting self-intersection and can handle pockets, both with and without an island. Kim [84] introduced the tool path generation method for contour parallel machining based on an incomplete two-manifold mesh model, namely, an inexact polyhedron. His method consists of three major steps: (1) machining area detection, (2) 2D offset, and (3) path linking. Machining areas obtained from a mesh model are closed 2D lines. Thus, offsetting closed lines is essential to generate mesh-based contour parallel tool path. The main point of his proposed algorithm was that every point is set to be offset using bisectors, and then invalid offset lines, which are not to participate in offsets, are detected in advance and handled with an invalid offset edge handling algorithm to generate raw offset lines without local invalid loops. References for contour parallel tool path: [5-7, 9, 15, 26, 27, 31, 44, 46, 55, 60-79, 81-92]
The relative merits of direction parallel and contour tool paths are studied by various researchers and discussed below. Kim and Choi [87] reported machining efficiency comparison between one-way zig, pure zig-zag, smooth zig-zag and contour parallel tool path for molds and die manufacturing. They reported that smooth zig-zag tool path is most efficient regardless of feed-rates and path intervals. However, according to a study by EI-Midany et al. [86], the best tool path depends upon the geometry of the part, physical characteristic of CNC machine tool (accelerator and decelerator, continuous path, look ahead, and etc.) and cutting conditions (tool diameter, feed rate, and etc.).
The contour parallel and direction parallel tool paths gained nearly universal acceptance for 2.5D pocket machining. However, machining efficiency in terms of machining time, surface finish and tool wear are greatly limited because of stop-and-go motion, sharp velocity discontinuity, repeated machining direction alteration and frequent repositioning, retraction, acceleration and deceleration of the tool. Xu et al. [9] reported contour parallel tool path inherits the corners of the pocket boundary into the tool path. When a tool encounters sharp corners, the tool has to decelerate to almost zero velocity, change its direction and accelerates as it comes out of the corners. Thus, to account for drive constraints such as velocity, acceleration and jerks, the feed rate has to be decreased resulting in a significant loss in productivity [27, 31]. Bieterman and Sandstrom [4] pointed out that significant reduction in tool life may occur because of overheating caused due to a sudden increase in tool’s engagement with unmachined stock at corners (Fig. 2.3). Due to above-mentioned reasons the directional parallel tool path and contour parallel tool path has limited application in HSM [85].
The development and history of space filling curve is discussed in references [93, 94]. Sarma [95] defined space filling curve as a continuous mapping of a unit line segment onto the unit square. Several types of space-filling curves exist based on the types of initiators (Fig. 2.17 (a)) and generators (Fig. 2.17 (b)) used to create the space-filling curve with any resolution (Fig. 2.17 (c)). Space filling tool path strategy avoids the directionality by frequent changes in orientation of the spanning elements by means of recursive algorithms [96, 97]. The curve does not need additional linkage among cutter path segments [49, 95]. Widely known space filling curves for tool path planning is the recursive Peano’s curve, Hilbert’s curve and Moore’s curve as shown in Fig. 2.18.
The main disadvantages of space filling curve are poor surface finish, a large number of sharp corners, frequent stop-and-go motion and, increase in tool path length and machining time. Because of the above-mentioned disadvantages space filling curve did not become popular and is not used in HSM [98]. References for space-filling curve: [49, 93-104].
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Fig. 2.17 Space filling curve
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Fig. 2.18 Space-filling curves (a) Peano (b) Hilbert's (c) Moore's [93, 94, 99]
Directional parallel, contour parallel and space filling tool paths with sharp corners will lead to abrupt changes in the cutting force and reduced feed rates (due to stop-and-go motion) which are not desirable for HSM [105-107]. The corners of tool paths for pocket machining can be optimized in order to allow the machine tools to use a higher average feed rate [108].
In HSM, to reduce acceleration and deceleration of tool and to maintain smooth cutting processes, the tool path should avoid sharp corners and sharp curvatures. The tool-workpiece contact varies as the tool encounters a corner. As, the tool approaches a sharp concave corner (i.e., interior angle less than 180), the engagement angle of tool and workpiece increases, and results in the sudden rise of cutting resistance in the corner region. This sudden rise in cutting resistance leads to undesirable effects such as high cutting forces, an increase in wear of cutting tool and surface roughness, machine chatter, reduction in tool life and sometimes even tool breakage [46].
Mostly, a tool path is described by linear segments leading to tangent discontinuities at the corners. [31]. To overcome this problem corner machining techniques are proposed. The two main methods available to handle the corner (i.e., smoothing the tool path) are global smoothing and the local smoothing [109]. If the tool path is composed of a large number of short segments, then it is possible to approximate all segments by a curve, also called global smoothing. If the tool path is composed of long segments, it is necessary to follow specifically these segments and thus, each transition has to be smoothened locally, hence called local smoothing. Both global and local smoothing are widely studied [109].
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Fig. 2.19 Corner machining tactics, (a) single strategy and (b) double loop strategy [46]
Choy and Chan [46] developed single loop and double loop strategies as shown in Fig. 2.19 for removing material at pocket corners and to reduce the cutting resistance encountered at the corner. Pateloup et al. [108] suggested a corner optimization techniques that is based on the computation of corner radii, the kinematic behavior of the machine tool and radial depth of the cut variation. Zhao et al. [110] proposed a tool path optimization for sharp corners in pocket machining by inserting two types of biarc transition segments in contour parallel tool paths. The purpose of adding biarc transition segment is to remove the residual material and thereby enabling step-over to be as high as possible to reduce machining time. A reduction in cutting speeds at the corners or providing a corner loop results in increased machining time hence reduces the efficiency [19, 46]. Banerjee et al. [111] proposed a process planning strategy for corner machining based on looping tool path strategy. Studies related to cutting forces when the tool encounters corners are discussed in references [105, 112-114]. Ernesto and Farouki [115] proposed a method to modify a CNC part program that includes sharp tool path corners, such that it enable faster execution of corner and maintains a prescribed geometrical contour error and bounds on the machine axis acceleration. Dotcheva and Millward [116] investigated the geometric relationship among the cutting tool, corner-machining operation and machined surface, and developed a mathematical model that describes the different cutting phases during the corner cutting. They also developed a new ‘intolerance’ mechanism for corner machining to obtain the required part accuracy.
References for corner machining tactics: [2, 19, 31, 35, 44, 46, 105-123]
Mapping based technique for tool path generation involves mapping the shape of the pocket or surface to a known shape such as square, rectangle, circle, etc. The tool path is generated on the known shape and it is inversely mapped to the shape of the pocket or surface.
Oulee et al. [124] presented boundary conformed tool path generation using Laplace based parametric redistribution method. Their approach combines the numerical Laplace solution for initial parameterization and a parametric redistribution algorithm for tool path generation. Chung and Yang, [47] developed a Laplace based spiral contouring method for machining of pockets with or without an island as shown in Fig. 2.20. The generated tool paths do not have thin walls or left over tool marks.
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Fig. 2.20 Tool path generation by Laplace based spiral contouring method (a) Pocket with an arbitrary boundary (b) Mapping of the pocket to a parametric unit square of (u,v) (c) Tool path for pocket machining [47]
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Fig. 2.21 Steps involved in conformal map based spiral tool path generation [125]
Sun et al. [126] developed contour parallel offset tool path for machining of trimmed surfaces based on conformal mapping with free boundary. Sun et al. [125] also used conformal map based approach to generate a spiral tool path for machining of sculptured surfaces. Their method uses a conformal map to establish a relationship between 3D physical surface and planar circular region. The spiral tool path is obtained on the circular region. Then through an inverse mapping, the planar spiral is defined by a mathematical function in 3D physical space as shown in Fig. 2.21.
Xu et al. [9] developed a mapping-based spiral cutting tool path for pocket machining. In this method, they mapped the machined region onto a circular domain by means of mesh mapping that reduces the task of tool path generation from the geometrically complex pocket region to a topologically simple disk. A guided spiral is constructed on this disk, the guided spiral is inversely mapped into the interior of the pocket and then a smooth low curvature spiral is derived as shown in Fig. 2.22.
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Fig. 2.22 Mapping based spiral tool path, (a) the pocket region to be machined. (b) triangulated pocket, (c) guided spiral on the disk and (d) generated spiral tool path of pocketing [9]
. The tool-path obtained is free from sharp corners and allows cutting of the pocket without tool retractions. However, it cannot handle island and pocket that are too concave. The limitation of this kind of tool path is that it is not suitable for a pocket that is too concave (i.e., interior angle greater than 180) in nature. Because the concave pocket leads to longer tool path as constant step-over is not maintained and hence, there is variation in chip load. In addition, bottlenecking may occur in some narrow regions when using this method for certain shapes of the pocket. References for mapping based tool path: [9, 47, 55, 124-133]
The medial axis transform (MAT), also called skeleton, was first introduced as a description of shape by Blum in 1967 [134-136]. The classical definitions include grass-fire model (Fig. 2.23 (a)) and maximal disk model (Fig. 2.23 (b)). The grass-fire model means that an object’s boundary is taken as an initial fire front that propagates within the object’s interior region. Points where the fire front folds or interacts with itself are retained as the skeleton points. The maximal disk model means that the medial axis of a planar domain is the locus of the center of a maximal disc, which touches the boundary in at least two points.[134].
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Fig. 2.23 Medial axis skeleton using (a) grass fire model [137] and (b) maximal disk model [138]
Smogavec and Zalik [139] proposed a fast algorithm for constructing approximate medial axis of the polygon using Steiner points. Aichholzer et al. [140] presented a simple, efficient and stable method for computing the medial axis for free-form shapes. Dorado [135] proposed the construction of medial axis of a planar region by offset self-intersections. Han et al. [141] developed medial axis for a planar domain with infinite curvature boundary points. Cao and Liu [134] developed offset tool path for curved boundaries in the planar domain using medial axis as shown in Fig. 2.24.
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Fig. 2.24 Tool path generated by MAT, (a) Digital photo and boundary points, (b) MA and medial axis circles and (c) MA and offset curves [134]
Held and Spielberger [45] introduced a new algorithm for generating a spiral tool path (Fig. 2.25) for high-speed machining of pockets without islands. The boundary is restricted to straight-line segments and circular arcs. The tool path is generated by interpolating growing disks placed on the medial axis of the pocket. It starts inside the pocket and spirals out to the pocket boundary. The tool path generated is free from self-intersections and the machining is done without tool retractions. The user has to select the start point and cutting width (step-over) of the tool path. The tool path length and machining time may increase if the bottleneck is present in the pocket geometry and/or a sub-optimal start point is selected. To deal with this problem, Held and Spielberger [142] introduced a geometric heuristic for decomposing a complex pocket geometry into simpler sub-pockets and selection of start point of the tool path. They reported that their algorithms reduces total tool path length, reduces the variation of the curvature and of the engagement angle, and decreases ratio between the maximum and the minimum step-over distance.
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Fig. 2.25 G1 continuous spiral tool path based on medial axis [45]
Chen et al. [143] introduced an aggressive machining method for the pocket machining. In their method, the major portion of the pocket is removed with the largest possible cutter available, for machining. The tool path for a specified tool (largest possible diameter) is quickly generated using the pocket’s medial axis transform, which creates the paths for the tool to machine the largest accessible space for pocketing and guarantees the tool to be free of gouging and interference. After this, an optimization model of selecting multiple cutters and generating their NC path (tool path), is built in order to achieve the highest efficiency of the aggressive rough machining. Elber et al. [144] utilized medial axis transform for high speed machining of pockets. Their tool path provides C1 continuity and the tool path is especially suited for machining of elongated narrow pockets.
The tool path generated by Medial Axis Transform (MAT) has very important characteristics such as uniqueness, invertibility, symmetry, topological equivalence and one to one correspondence. Hence, for every boundary curve, there is a unique MAT [134]. However, while machining using this tool path during HSM, certain problem may arise such as retraction and/or repositioning of the tool, stop-and-go motion; frequent deceleration and acceleration of tool when tool encounters the corners. References for MAT based tool paths: [45, 92, 134-141, 143-152]
The Spiral of Cornu is named after the French scientist Marie Alfred Cornu (1841 - 1902). He studied this curve, also known as a Clothoid or Euler's Spiral, in connection with diffraction. Highway and railway designers frequently use clothoid splines as center lines in route location [19, 153]. The characteristic property of Clothoid spiral is that their curvature is a linear function of the arc length, or, in other words, the curvature of the curve is proportional to the length of the curve measured from the origin of the spiral [19]. The parametric representation of cornu spiral is found in references [154]. Another important property of clothoid spiral is that the second derivative varies linearly with the length of the curve. Hence, the curve can be used to generate paths with constant cutter engagement values as well as maintain smooth movements while performing cornering and linking movements [48].
Pamali [19] used Clothoidal spirals (Fig. 2.26) to generate smooth tool paths for HSM. Yao and Joneja [48] combined Archimedean spiral and Clothoidal spiral to generate tool path with constant cutter engagement values and covers a 2D region in an efficient manner. Dripke et al. [155] proposed an approach to interpolation tool path trajectories with piecewise defined clothoid. McCrae and Singh [156] developed a stable and efficient algorithm that fits a sketched piecewise linear curve using a number of Clothoid segments with G2 continuity based on specified error tolerance. References for Clothoidal spiral tool path: [19, 48, 153-158]
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Fig. 2.26 Clothoidal spiral tool path [19]
Biterman and Sandstrom [4] introduced a spiral tool path for pocket machining by solving an elliptic Partial Differential Equation (PDE) boundary value problem. The tool path starts as a spiral from the center of pocket geometry and takes the shape of the pocket boundary as shown in Fig. 2.27. They reported that their tool path increases tool life by 50% on a titanium-cutting experiment and reduces machining time up to 30%. Their method is not suitable for pockets that are too concave pocket or not star-shaped [45]. However, they suggested that using higher Eigen value of PDE solution can be used to generate tool path for the pockets that are too concave [4]. Banerjee et al. [13] modified the approach of Bieterman and Sandstrom by using biarc and arc spline for generating a morphed spiral tool path for floor machining of 2.5D pockets. Their method is capable of handling island inside the pocket. However, Banerjee’s method of dealing an island inside the pocket may result in over machining (increased tool path length) if the island is not centrally placed. They have reported 32% and 40% improvement in productivity with two different feed rate strategies when compared with commercial CAM software [13].
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Fig. 2.27 Spiral tool path based on solution of PDE [4]
Spiral tool path by solving PDE is a promising technique for HSM and substantial improvement in tool life and cutting time is observed. However, very scarce literature is available for development of spiral tool path by solving PDE. One of the reasons is that the gradient of the solution function (‘ u ’) of eigenvalue problem is not uniform in the solution domain and it is the main difficulty for assuring a bounded distance between consecutive iso-contours (because uni-distance ‘ u ’ values will not lead to uni-distance iso-contours). References for tool path based on PDE: [4, 13]
A trochoidal tool path is defined as the combination of a uniform circular motion with a uniform linear motion as shown in Fig. 2.28. This tool path employs a sort of constant looping motion to avoid fully loading the cutting tool and thereby allowing higher feed rates [159]. The trajectory radius is continuous and the tool always moves along a curve of constant radius making it suitable for HSM [159, 160]. Literature related to the analysis of trochoidal tool paths, its merits and demerits are found in references [159, 161]. The brief description of work that is carried out by other researchers related to trochoidal tool path is discussed below.
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Fig. 2.28 Trochoidal tool path [160]
Rauch et al. [160] proposed improved trochoidal tool path generation using process constraints modeling. Ibaraki et al. [92] presented a systematic trochoidal tool path generation strategy by using process constraints modeling for high speed machining of 2.5D pocket. Otkur and Lazoglu [162] proposed an approach to model trochoidal machining. They employed analytical model for defining the engagement and a force model for the prediction of the cutting forces depending on the engagement. Further, they used a numerical model to examine double trochoidal machining. Ferreira and Ochoa [147] proposed a method for generating trochoidal tool paths for 2.5D pocket machining using a medial axis transform. They first represented the pocket and islands as polygons, and then used medial axis transform to calculate a series of points. They sorted the points and grouped them by an algorithm that generates the trochoidal path whenever the desired radial depth of cut is attained. They reported that machining time is reduced through a pixel-based simulation and adjusting the tool path to the remaining material. Wu et al. [163] suggested that trochoidal machining is highly suitable for machining of metals (such as nickel-based super alloys) that have low thermal conductivity because the load on the tool and cutting temperature are greatly reduced as the tool gets enough time to get cooler. A trochoidal tool path is excessively long with a portion of circular motion that does not perform cutting, which needs to be offsetted by faster-cutting feed and speed [159]. But this intrinsic limitation of trochoidal tool path proves to be an advantage while machining metals that have low thermal conductivity. References for trochoidal tool path: [92, 147, 159-169]
Linear and circular interpolation algorithms are used in conventional CNC machines but they are not adequate enough for achieving the desired precision machining of free-form geometries. Therefore, parametric representation of the tool path with NURBS curves was developed [162, 170, 171]. Modern CNC (Computer Numerical Control) machines such as, not limited to, GE Fanuc’s Series 15-B, 16-C, 16i-MA, Sinumerik 840D and Tosnuc 999 supports NURBS interpolator. These controllers not only provide linear/circular interpolations but also offer parametric interpolations for curves such as Bezier, B-spline, and NURBS curves [172, 173]. Some researchers [174, 175] shown that parametric interpolations can reduce feed rate fluctuations and chord errors, and can reduce machining time in comparison with linear/circular interpolations [119].
As discussed earlier, to meet the kinematic constraints of HSM, it is essential that the tool path is smooth with at least G1-continuous. It would be even better if it is C2-continuous since C2-continuity enables tool movement without jolting. The achieved degree of continuity depends on the types of tool path segments that the NC controller supports: Clearly, a tool path that has straight line segments and circular arcs can be made G1-continuous. To achieve C2-continuity, additional tool path segments would be necessary, such as NURBS, B-splines or clothoid [45].
Bouard et al. [2] proposed a new method of tool path computation using uniform cubic B-spline curves based on optimization with constraints (pocket boundary and cutting forces) to obtain a C2 continuous tool path. Deli and Laishui, [176] developed NURBS interpolator based on the adaptive feed rate control. They used look-ahead method and adjusted feed-rate in advance as the curvature varies and the variation ratio of curvature, which makes machining motion quite smooth. Pateloup et al. [35] proposed a new method of 2D curve interpolation using non-uniform cubic B-splines. They adapted this curve for the interpolation of sequences of straight lines and circular arcs. The main purpose of their method is to calculate C2 continuous curves that adapt to high feed rate pocket machining. As the tool path is generally defined by the line segments and circular arcs, they proposed the method for approximating a sequence of line segments and circle arcs using B-spline curves. They concluded that their method is useful for the smoothening of the tool path and reducing machining time. Lartigue et al. [177] presented a method for generating CNC tool path for a smooth free-form surface in terms of planar cubic B-spline curves. Shih and Chuang [178] developed tool path with one-side offset approximation of freeform curves for interference-free NURBS machining. References for interpolating tool paths based on Bezier, B-spline and NURBS: [2, 35, 45, 119, 123, 145, 162, 170-181]
Chatelain et al. [182] proposed an adaptive spiral tool path strategy. They reported 16% reduction in machining time for high speed roughing of light alloy aerospace components. Lee [106] proposed a novel contour offset approach to generate spiral tool path with constant scallop height for finished machining of sculptured surfaces with minimum tool retraction. Thus, it minimizes the fluctuation of cutting forces and possibility of chipping on the cutting edge of the tool. Their approach maintains scallop height to ensure a higher level of efficiency and quality in HSM. Xiong et al. [85] proposed a curvilinear tool path generation for pocket machining using the level set method.
With the advancement of technology and advent of high speed machines, conventional tool-paths generation methods show limitations [106, 107, 183]. Many new approaches to tool path generation for high speed machining are widely explored and there is a growing literature on the investigation of new tool paths including the spiral tool path and the trochoidal tool path [85, 184]. Spiral tool paths and trochoidal tool paths are two main tool path strategies used for HSM [184]. The spiral and trochoidal tool paths can reduce the machining time due to the continuity of tool locus [184].. The advantages of spiral tool path over conventional tool path are reported in many references (not limited to) [4, 9, 47, 185, 186].
Description of pocket (shape, size, presence of an island) and tool path strategy greatly influences the overall performance of pocket machining using HSM [4, 9, 30, 32, 37, 47, 187]. Held and Spielberger [12] pointed out that the shape of the pocket greatly affects the suitability of spiral tool path for HSM. That is if the pocket is very long and narrow (e.g., a rectangle with high aspect ratio) or contains bottlenecks then one spiral may not be efficient to cover the entire pocket. In such situation, it would be better to machine a pocket by decomposing the pocket geometry into sub-geometries. Many references for decomposing a geometry are found in reference [43] but literature related to decomposition of the pocket geometry with the aim to improve overall machining efficiency are scarcely available. Held and Spielberger [142] introduced a geometric heuristic method for decomposing a complex pocket geometry into simpler sub-pockets based on Voronoi diagram for high-speed machining of multiply-connected pockets. They reported that decomposing complex pocket geometry into simpler sub-pockets can reduce the total length of tool path, reduce the variation of the curvature and decreases ratio between the maximum and minimum step-over distance especially of those pocket with some narrow and bottlenecked (neck) region.
It has been observed during the literature survey that different tool path strategies are suitable for different pocket geometry. For example, a rectangular pocket with very high aspect ratio will be machined efficiently with a zigzag tool path (or smooth zigzag), a pentagonal pocket with a morph spiral tool path and a circular pocket with a true Archimedean spiral. Hence, the efficiency of tool path cannot be established independently of pocket geometry. Thus, there is need of a technique (method) to compare different pocket shape quantitatively. Interestingly, very little is known on quantitative comparison of pocket geometry.
The merits and demerits of different tool path strategies are summarized in Table 2-1. The references related to different tool path strategies are given again for ready reference.
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