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Physics based turning process simulation 2011
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Title | Physics based turning process simulation |
Creator |
Gencoglu, Ahmet |
Publisher | University of British Columbia |
Date Created | 2011-08-24T22:00:36Z |
Date Issued | 2011-08-24 |
Date | 2011 |
Description | The manufacturing planning of parts is currently based on experience and physical test trials. The parts are modeled, and Numerically Controlled (NC) tool paths are generated in Computer Aided Manufacturing (CAM) environment. The NC programs are physically tested, and if the process faults are found, the NC program is re-generated in the CAM environment. The objective of this thesis is to develop Virtual Turning System that predicts the part machining process ahead of costly physical trials. Tool–workpiece engagement geometry is calculated along the tool path by a proposed polycurve method. The part geometry is imported as a stereolithography (STL) model from the CAM system, and the cross section around the turning axis is reconstructed. The tool and part cross sections are modeled by polycurves, which are constructed by series of arcs and lines. The tool–part geometries are intersected using boolean operations to obtain the engagement conditions. The turning process is modeled by predicting the chip area and equivalent chord angle. The process forces are modeled proportional to the material dependent cutting force coefficients, depth of cut and equivalent chord length that depends on the nose radius and approach angle of the tool. The chatter stability of the process is examined using Nyquist criterion at each tool–workpiece engagement station along the path. The virtual turning simulation simulates the forces and detects the chatter stability, and adjusts the feeds at each tool-part engagement station. The physical turning of parts with arbitrary geometry can be simulated, and cutting conditions that leads to most optimal machining operation is automatically determined without violating the limits of the machine tool and part. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2011-08-24T22:00:36Z |
DOI | 10.14288/1.0072084 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2011-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/36886 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0072084/source |
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PHYSICS BASED TURNING PROCESS SIMULATION by Ahmet Gencoglu A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering) The University of British Columbia (Vancouver) August 2011 c Ahmet Gencoglu, 2011 Abstract The manufacturing planning of parts is currently based on experience and physical test trials. The parts are modeled, and Numerically Controlled (NC) tool paths are generated in Computer Aided Manufacturing (CAM) environment. The NC programs are physically tested, and if the process faults are found, the NC program is re-generated in the CAM environment. The objective of this thesis is to develop Virtual Turning System that predicts the part machining process ahead of costly physical trials. Tool–workpiece engagement geometry is calculated along the tool path by a proposed poly- curve method. The part geometry is imported as a stereolithography (STL) model from the CAM system, and the cross section around the turning axis is reconstructed. The tool and part cross sec- tions are modeled by polycurves, which are constructed by series of arcs and lines. The tool–part geometries are intersected using boolean operations to obtain the engagement conditions. The turning process is modeled by predicting the chip area and equivalent chord angle. The process forces are modeled proportional to the material dependent cutting force coefficients, depth of cut and equivalent chord length that depends on the nose radius and approach angle of the tool. The chatter stability of the process is examined using Nyquist criterion at each tool–workpiece engagement station along the path. The virtual turning simulation simulates the forces and detects the chatter stability, and adjusts the feeds at each tool-part engagement station. The physical turning of parts with arbitrary geom- etry can be simulated, and cutting conditions that leads to most optimal machining operation is automatically determined without violating the limits of the machine tool and part. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Virtual Turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Workpiece Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Tool Representation Model . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Tool–Workpiece Engagements . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Chatter Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.1 Direct and Random Search . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.2 Dichotomous Search and Interval Halving Methods . . . . . . . . . . . . . 21 iii 2.4.3 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Contour Turning Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Angle Definitions for Contour Turning . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Tool–Workpiece Engagements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Cutting Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.1 Cutting Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.2 Orthogonal to Oblique Transformation . . . . . . . . . . . . . . . . . . . . 29 3.3.3 Chip Flow Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.4 Effective Chip Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Tool Motion Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Stability with Process Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5.1 Process Damping Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Feedrate Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6.1 Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6.2 Design Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6.4 Direct Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.6.5 Engagement Based Scheduling . . . . . . . . . . . . . . . . . . . . . . . . 47 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 Turning Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Vericut Application Program Interface . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Polycurve Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2.1 Entity Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Workpiece Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.1 Generator Curve Representation . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.2 Intersection of a Triangle with a Plane . . . . . . . . . . . . . . . . . . . . 55 iv 4.3.3 Generator Curve of a Turning Workpiece . . . . . . . . . . . . . . . . . . 61 4.4 Tool Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.1 Parametric Tool Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.2 User Defined Tool Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.5 Tool–Workpiece Engagement Model . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5.1 Boolean Based Tool–Workpiece Engagements . . . . . . . . . . . . . . . . 67 4.5.2 Processing Engagement Profiles . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Parsing NC-Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6.1 Spindle Speed and Cutting Speed . . . . . . . . . . . . . . . . . . . . . . . 79 4.7 Simulation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.7.1 Constant Engagement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.7.2 Non Axi–symmetric Workpiece . . . . . . . . . . . . . . . . . . . . . . . 81 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Application and Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . 83 5.1 Virtual Turning Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Contour Turning Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.2 Tool–Workpiece Engagements . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2.3 Cutting Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2.4 Feedrate Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 Chatter Stability Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.2 Structural Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 v Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 A.1 Intersection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 A.1.1 Line - Line Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 A.1.2 Circle–Circle Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.1.3 Line - Arc Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 A.1.4 Extension of Intersection Algorithms for Line Segments and Arcs . . . . . 111 A.2 AISI 1045 Cutting Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.2.1 Cutting Edge Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 A.2.2 Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.3 NC-Commands for Hardinge Superslant Lathe . . . . . . . . . . . . . . . . . . . . 122 vi List of Tables Table 2.1 Insert shapes defined in ISO 13399 [1] . . . . . . . . . . . . . . . . . . . . . . 10 Table 4.1 Relational operations with real and floating point numbers . . . . . . . . . . . . 72 Table 5.1 Geometric specifications of the tools used in experiments . . . . . . . . . . . . 86 Table 5.2 Computational time required by the engagement models . . . . . . . . . . . . . 88 Table 5.3 Suggested feedrate values for contour turning experiment . . . . . . . . . . . . 90 Table 5.4 Radial turning chatter experiments . . . . . . . . . . . . . . . . . . . . . . . . 92 Table A.1 Coefficients for fitted surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Table A.2 Cutting coefficients for AISI 1045 steel with GC3205 grade tool [2] . . . . . . 116 Table A.3 Cutting coefficients for AISI 1045 steel with KC8050 grade tool [3] . . . . . . . 118 Table A.4 Coefficients for edge coefficient of AISI 1045 steel with GC3205 grade tool . . 120 Table A.5 Coefficients for edge coefficient of AISI 1045 steel with KC8050 grade tool . . 121 Table A.6 NC-commands implemented in the G-code parser . . . . . . . . . . . . . . . . 122 vii List of Figures Figure 2.1 Predictive models of machining operations for practical applications [4] . . . . 5 Figure 2.2 Virtual turning workflow [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 2.3 Feature primitive representation proposed by Shengfang et al. [6] . . . . . . . . 7 Figure 2.4 Workpiece representation by iso-boundaries [7] . . . . . . . . . . . . . . . . . 8 Figure 2.5 Workpiece representation by a 2D section obtained from a solid modeler [8] . . 8 Figure 2.6 Turning workpiece representation by the generator curve of a rotational part [9] 9 Figure 2.7 Workpiece representation using Dexel elements [10] . . . . . . . . . . . . . . 9 Figure 2.8 Analytical definition of the cutting tool geometry [5] . . . . . . . . . . . . . . 11 Figure 2.9 Outline of a grooving tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Figure 2.10 Modeling of the cutting edge with B-splines [11] . . . . . . . . . . . . . . . . 12 Figure 2.11 Uncut chip area configurations considered by Atabey et al. [12] . . . . . . . . 13 Figure 2.12 Chip geometry produced by a tool having a nose radius and approach angle during cylindrical turning [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Figure 2.13 Chip geometry in contour turning [13] . . . . . . . . . . . . . . . . . . . . . . 15 Figure 2.14 Feature decomposition method proposed by Zhou [5] . . . . . . . . . . . . . . 16 Figure 2.15 Calculation of cutting forces with discrete orthogonal elements [14] . . . . . . 17 Figure 2.16 Approximate chord for a cutter with a straight edge [15] . . . . . . . . . . . . 18 Figure 2.17 Experimental verification of Colwell’s chord approximation [16] . . . . . . . . 19 Figure 2.18 Discretization of chip load by trapezoidal elements [17] . . . . . . . . . . . . . 20 Figure 2.19 Dichotomous searching method [18] . . . . . . . . . . . . . . . . . . . . . . . 21 Figure 2.20 Interval halving method [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 viii Figure 3.1 Contour turning process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Figure 3.2 Tool–motion and surface angles in contour turning . . . . . . . . . . . . . . . 25 Figure 3.3 Tool–workpiece engagements . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Figure 3.4 Comparison of engagement maps in milling and turning . . . . . . . . . . . . 26 Figure 3.5 Geometry of an oblique turning tool [14] . . . . . . . . . . . . . . . . . . . . . 29 Figure 3.6 Cutting forces during oblique cutting . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 3.7 Changing engagement geometry along the tool path in contour turning . . . . . 33 Figure 3.8 Tool motion classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Figure 3.9 Force model modification for left and right handed cuts . . . . . . . . . . . . . 35 Figure 3.10 Chip thickness in contour turning . . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 3.11 Direct search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 3.12 Engagement based feedrate scheduling using search based algorithm . . . . . . 47 Figure 3.13 Modification of tool–workpiece engagement geometry . . . . . . . . . . . . . 48 Figure 4.1 Polycurve entity definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 4.2 Generator curve for a solid part of revolution [9] . . . . . . . . . . . . . . . . . 56 Figure 4.3 Signed distance of a point from a plane in space . . . . . . . . . . . . . . . . . 56 Figure 4.4 Possible cases of triangle–plane intersection . . . . . . . . . . . . . . . . . . . 57 Figure 4.5 Euler angles [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 4.6 Parameters of a rhombic insert . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Figure 4.7 Definition points for rhombic inserts . . . . . . . . . . . . . . . . . . . . . . . 64 Figure 4.8 Geometric definition of a rhombic tool . . . . . . . . . . . . . . . . . . . . . . 66 Figure 4.9 Use of swept polygon for boolean operation [9] . . . . . . . . . . . . . . . . . 67 Figure 4.10 Entity classification for boolean algorithm . . . . . . . . . . . . . . . . . . . . 68 Figure 4.11 In-region test with 250 test points . . . . . . . . . . . . . . . . . . . . . . . . 70 Figure 4.12 Errors in in-region test due to test–ray crossing vertices . . . . . . . . . . . . . 70 Figure 4.13 Boolean operations supporting holes in the workpiece model . . . . . . . . . . 72 Figure 4.14 Positive and negative direction conventions for Green’s Theorem . . . . . . . . 74 ix Figure 4.15 Area of the arc entity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 4.16 Area calculation by Green’s theorem . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 4.17 Entities originating from tool curve are in cut . . . . . . . . . . . . . . . . . . 76 Figure 4.18 Direction convention for the effective chord vector . . . . . . . . . . . . . . . 77 Figure 4.19 Flow–chart for NC-code parser algorithm . . . . . . . . . . . . . . . . . . . . 78 Figure 4.20 Simulation scheme for constant engagement . . . . . . . . . . . . . . . . . . . 81 Figure 4.21 Multiple simulation planes per spindle revolution . . . . . . . . . . . . . . . . 82 Figure 5.1 Structure of the virtual turning application . . . . . . . . . . . . . . . . . . . . 85 Figure 5.2 Initial workpiece geometry for contour turning experiments . . . . . . . . . . . 86 Figure 5.3 Comparison of engagement model with Vericut Solution . . . . . . . . . . . . 89 Figure 5.4 Cutting forces during contour turning . . . . . . . . . . . . . . . . . . . . . . 90 Figure 5.5 Comparison of cutting forces before and after optimization . . . . . . . . . . . 91 Figure 5.6 Direct FRF of the workpiece with different overhang values. . . . . . . . . . . 93 Figure 5.7 Radial turning experiment #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Figure 5.8 Radial turning experiment #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Figure 5.9 Radial turning experiment #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Figure 5.10 Radial turning experiment #4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Figure A.1 Line-Line intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Figure A.2 Circle-Circle intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Figure A.3 Line-Circle intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Figure A.4 Quadratic surface fit for shear stress (AISI 1045 Steel) . . . . . . . . . . . . . 113 Figure A.5 Quadratic surface fit for friction angle (AISI 1045 Steel) . . . . . . . . . . . . 114 Figure A.6 Quadratic surface fit for shear angle (AISI 1045 Steel) . . . . . . . . . . . . . 115 Figure A.7 Comparison of cutting edge coefficients for AISI 1045 steel . . . . . . . . . . 117 Figure A.8 Curve fitting for cutting edge coefficients (GC3205 grade tool and AISI 1045 Steel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 x Figure A.9 Curve fitting for cutting edge coefficients (KC8050 grade tool and AISI 1045 Steel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 xi Glossary Ac Uncut chip area in [mm2] ac Depth of cut in [mm] an Normal rake angle ao Equivalent orthogonal angle ba Average friction angle in orthogonal cutting Dac Difference in the depth of cut parameter D fc Difference in the feedrate parameter fc Tool movement during one complete spindle revolution [mm/rev] h Height of the approximate cutting chord in [mm] hc Chip thickness heff Effective chip thickness in [mm] i Equivalent oblique angle kr Approach angle of the cutting tool Lc Total cutting edge Length l Tool edge length for a Rhombic Tool xii Le f f Length of the approximate cutting chord en Nose angle for a Rhombic Tool fc Shear angle in orthogonal cutting yr Side cutting angle of the cutting tool P Power required during machining in [W] rn Nose radius of the tool n Spindle speed in rpm ys Local surface angle of the in–process workpiece ts Shear yield stress in orthogonal cutting q Approximate chord angle T Torque required during machining in [Nm] Vc Cutting speed or Surface speed in [m/min] w Width of the approximate cutting chord in [mm] xiii Acknowledgements First of all, I would like to express my lifelong gratitude to my supervisor Prof. Dr. Yusuf Altintas for allowing me to be a member of the University of British Columbia and Manufacturing Automation Laboratories. It was my privilege to conduct this re- search under his supervision and I am thankful for his wisdom, guidance and knowl- edge which he shared with me during the course of my studies. I would also like to thank Prof. Dr. Steve Feng, for helping me with my research and for being a source of motivation. The two years I spent in Vancouver has been a wonderful time thanks to my col- leagues and friends. It has been both my honor and pleasure to be working in such a professional and yet fun atmosphere. I would like to thank members of the MAL for their support and friendship. Finally, I would like to thank my brother Aytuğ, my father Aykut, my mother Melek and my fiancee Eda for their everlasting love and support. I would not be able to succeed without them. Therefore I want to dedicate this work to my family. xiv 1 Introduction The present manufacturing industry requires rapid and cost effective design and manufacturing methods to produce new products in small batches. The traditional manufacturing, which requires process planning, physical trial and re–design of the process in few iterations, is no longer suit- able to produce parts cost effectively in short periods. Recently, the design and manufacturing operations have been replaced by their digital simulation, i.e. virtual models, in industry. Virtual manufacturing is classified under two categories: Geometric and physics based simu- lations. Computer Aided Design (CAD) and Computer Aided Manufacturing (CAM) researchers developed graphics based simulation model of the machines and workpiece geometry. The colli- sion of the tool, fixtures and machine tool can be simulated along the tool path graphically, and geometric errors can be corrected ahead of physical machining. The physics based simulation sys- tems need to predict the forces, vibrations, and dynamic behavior of the machine tool and metal cutting process along the tool path. Manufacturing Automation Laboratory at the University of British Columbia has been one of the pioneering research centers which contributed significantly to the virtual simulation of milling processes. This thesis introduces the first Virtual Turning Pro- cess in the same laboratory using a new tool–part intersection model with computationally efficient turning mechanics and dynamics models. Virtual turning has the following modules that have to be computationally efficient with a geometric accuracy within ten micrometers: • Tool–part engagement computation • Process mechanics model 1 • Chatter stability detection • Feedrate scheduling Each module has to be executed at each tool–part engagement station which can reach to over 100,000 on a typical aircraft engine disk. Tool–workpiece engagement is identified by representing the symmetric – cylindrical geomet- ric features by series of arcs and lines. Tool and part features are intersected using Boolean op- erations. The new, two dimensional engagement method is shown to be nine fold faster than commonly used Z–buffer methods based on solid geometry. Geometric accuracy is also improved by analytical calculation of the tool–workpiece engagements. The turning process is commonly modeled by digitizing the curved cutting edge into small, differential elements, i.e. typically 20-30 segments. The chip thickness and the corresponding cutting force, torque and power are predicted for each differential element, and digitally summed to evaluate the total forces at each tool–workpiece engagement station which can be over 100,000. The traditional mechanics model, which requires 20-30 fold increase in computational time due to digital integration, is therefore not suitable to be used in virtual turning. An equivalent cutting edge concept, which was previously introduced by Colwell [15] in 1950s, is used in rapidly evaluating the cutting forces, torque and power in this thesis. The method is proven to have sufficient accu- racy for virtual turning operations. The chatter is detected by applying computationally efficient Nyquist criterion to the dynamics of the turning process at each station. Turning of parts with arbitrary geometry can be simulated, and cutting forces, torque and power can be predicted along the tool path by the proposed Virtual Turning System. Also, the cycle times can be reduced by searching feed and spindle speed values that do not violate the physical limits of the machine such as torque and power. Henceforth, the thesis is organized as follows: The literature related to the cutting mechanics and virtual turning is presented in Chapter 2. The process mechanics, chatter detection and feedrate scheduling models are given in Chapter 3 for contour turning operations. A new tool–part intersection model and its integration to the proposed 2 Virtual Turning system are presented in Chapter 4. The application and experimental verification of Virtual Turning system is presented in Chapter 5. The thesis is concluded in Chapter 6. 3 2 Literature Review 2.1 Virtual Turning An accurate process model has to be developed to simulate the machining performance of a speci- fied machining process. The machining performance can be evaluated either in technical or com- mercial aspects. Technical performance represents the physical outputs of the process such as cutting forces, power, torque and machine tool vibrations. Commercial performance is related to the operational costs, machining costs and productivity. The progress report of the CIRP working group ‘Modeling of Machine Operations’ lists the main motivations for modeling metal cutting operations as follows [4]: • Design of Processes • Optimization of Processes • Control of Processes • Simulation of Processes • Design of Equipment As Figure 2.1 demonstrates, the simulation of a machining process requires the definition of three main components: • Cutting Tool • Workpiece 4 • Machine Tool Ta sk D e! ni to n Re co m m en de d Ap pl ic at io n Machine Tool Speci!cations Jig and Fixtures Static and Dynamic Forces Coolant Application Tool Material Tool Type Turning Milling Drilling Tool Geometry Work Material Machine Tool Size Shape Tolerance Cutting Tool Workpiece Applied Machining Performance (AMP) Optimized Machining Performance Optimum Cutting Conditions Optimum Cutting Tool Selection Figure 2.1: Predictive models of machining operations for practical applications [4] For turning operations, a virtual machining simulation was proposed by Zhou [5]. The ba- sic user input for this virtual machining system is CAD/CAM data, which can be obtained from available software packages. Since most processes are designed using one of these CAD/CAM environments, this is the most convenient choice of input data for the manufacturing industry. The workflow proposed in Figure 2.2 describes the tasks of the virtual turning system as follows: 5 • Geometric modeling of the Tool–Workpiece engagements (TWE Model) • Force prediction model • Optimization Conventional CAD/CAM Software (CATIA, Pro/E, UG) or Developed Virtual Turning System Geometric Solid Modelling Tool-Workpiece Engagement Model (TWE Model) Engagement Information (A, A1, A2, Lc, Lc1, Lc2, q Feedrate, Depth of Cut, Cutting Speed, Stock Diameter Mechanistic Force Prediction Model (MF Model) Optimization based on Maximum Force, Torque, Power, Chip Load, De!ection Cutting forces, Power, Torque De!ection, Chatter Stability Figure 2.2: Virtual turning workflow [5] The tool–workpiece engagements have to be resolved in the initial step of the virtual turning. The second step involves computation of the cutting forces, power and torque requirements for the given engagements. The final step evaluates errors, surface quality, stability of the cut, and optimizes the operation. 2.1.1 Workpiece Representation A workpiece representation should have the following characteristics: • Displayed on the screen efficiently • Simple creation, manipulation and modification 6 XZ z h d2 d1 a) Wireframe Model b) Solid Model Figure 2.3: Feature primitive representation proposed by Shengfang et al. [6] • Easy to implement and low computational load • Memory efficient Some studies have defined the workpiece as a solid model, usingConstructive Solid Geometry [20], Boundary representation, as shown in Figure 2.4 [7], or Solid primitives, as shown in Figure 2.3 [6]. Solid model representations are relatively complex and have large memory requirements. These representations are necessary when dealing with milling operations where the cutting process is de- scribed with the tool motion in three or more axes. However, turning operations are axi–symmetric, 7 Figure 2.4: Workpiece representation by iso-boundaries [7] OP1 OP4 OP2 OP5 OP7 OP3 OP6 OP8 Figure 2.5: Workpiece representation by a 2D section obtained from a solid modeler [8] therefore, it is possible to reduce the turning geometry into a two dimensional problem. Model developed by Zhou et al. [8] proposed to use the cross sectional area of a given solid model to represent the workpiece as seen in Figure 2.5. Zhou used the ACIS solid libraries to manipulate the workpiece model. Because her approach still depends on the capabilities of the solid modeler, it cannot take full advantage of a two-dimensional solution. 8 Figure 2.6: Turning workpiece representation by the generator curve of a rotational part [9] Figure 2.7: Workpiece representation using Dexel elements [10] Li et al. [9] showed that the workpiece in a turning operation can be simply defined as the gen- erator curve of a rotating geometry as demonstrated in Figure 2.6. The curve can be represented by a polygon, an analytical curve including non-linear segments, or by non–linear parametric curves such as splines. A two dimensional workpiece can also be represented by dexels, which are usually used in milling simulations as the z-buffer method as described by Zhang et al. [10], see Figure 2.7. Dexel method can be reduced to represent two dimensional geometry by using a one-dimensional grid in- 9 stead of the two. One dimensional model can reduce the computational time spent by the intersec- tion and modification algorithms. However, dexel based method can only represent the geometry in a discrete manner, and suffers from visualization difficulties. 2.1.2 Tool Representation Model Table 2.1: Insert shapes defined in ISO 13399 [1] !"#$%&' !()*+' ,-.+/0' # !%" 3+ "%" 0+ )* 2+ // 1/ /+ 20 ./ " 2+ & ,-+ $ ,./ $ ,+2 $%" 3+ "" 2/ 2- ( // % ' It is common practice in the industry to use multiple turning tools within a single turning op- eration, therefore, a versatile virtual turning system should be able to simulate the turning process with a wide variety of geometrically defined cutting tools. Kaymakci [1] has investigated the geometry of standard cutters and inserts defined by ISO standards. Typical insert shapes mentioned in ISO 13399 standard are illustrated in Table 2.1. The tool geometry can be represented by the outline of the cutting edge as used by Zhou [5], see Figure 2.8. This study represents the tool profile as a series of lines and arcs which have been analytically defined as functions of typical tool design parameters such as tool nose radius, side and end cutting edge angles. However, this method is not generalized enough to support different 10 rc Tool nose radius sr Side cutting edge angle gr End cutting edge angle L Tool height W Tool width 1. Tool center position O(X0,Y0) 2. Tool nose arc edge e2 upper tangent point Pa Xa = X0 - rc cos(sr) Ya = Y0 + rc sin(sr) 3. Tool nose arc edge e2 lower tangent point Pb Xb = X0 + rc sin(gr) +sin(sr))] tan(sr)] Yd = Y0 + L - rc 5. Tool straight edge e3 upper right point P u = [ W - rc(1+sin(gr)] / cos(sr+gr) Xe=u cos(gr) + Xb Ye=u sin(gr) + Yb Figure 2.8: Analytical definition of the cutting tool geometry [5] r1 r2 Figure 2.9: Outline of a grooving tool tool geometries such as rectangular grooving tools where the cutting edge must be represented by two circular regions, and the interconnecting cutting edge, as shown in Figure 2.9. The cutting tool was defined as a polygon by Li et al. [9], which has the disadvantage of not being able to define non-linear curves, but only straight lines. Polygons increase the memory usage when a large number of small line segments are used to approximate circular regions of the tool. Another possibility for defining the tool geometry is to use parametric non-linear curves. 11 Yussefian et al. [11] proposed to use B–spline curves to represent the turning tool geometry. B– spline curves are defined by their control points, which are arranged to form a Control Points Polygon as shown in Figure 2.10. A clipping algorithm is employed to find the intersections of the tool and the workpiece in order to define the chip boundaries that are used to calculate the resulting cutting forces. Control Points Polygon Control Points B-Spline Figure 2.10: Modeling of the cutting edge with B-splines [11] The cutting tool also can be represented volumetrically. It is the common approach in milling operations to define the cutting tool as a solid model, and generate the volume swept by the tool as it moves along the NC path [21][7]. 2.1.3 Tool–Workpiece Engagements Unlike milling operations, where cutter workpiece engagements are three dimensional problems, turning engagements can be reduced to a two dimensional problem due to the axial symmetry of the process. Various studies aim to calculate the tool-workpiece engagements of the turning operation with a tool having a nose radius (rn) and side cutting angle (yr). 12 Atabey et al. [12] proposed to divide the uncut chip area into geometrical regions. Assuming a feedrate that is smaller than the nose radius of the tool, they reduced the number of possible engagement conditions to five, and used discrete elements to calculate the uncut chip area formed by the nose radius. Four of the five proposed configurations are shown in Figure 2.11. However, due to discretization of the chip load this method is not computationally favorable. Also, the configurations are defined only for straight cylindrical cuts, therefore, the solutions are not valid for contour turning, grooving or other turning operations. Figure 2.11: Uncut chip area configurations considered by Atabey et al. [12] Eynian and Altintas [2] developed a geometrical model of the chip load to predict the process. They considered two configurations, one where the cut is deeper than the nose radius, and another for the the smaller cuts, as shown in Figure 2.12. 13 gr g'r gn (a) (b) x y w w c/2 c/2 a a h h e e r ¡ r ¡ hcusp hcusp c r r Approximate Chord Figure 2.12: Chip geometry produced by a tool having a nose radius and approach angle during cylindrical turning [2] Reddy et al. [13] developed geometrical models of the chip load for contour turning. Similar to Atabey et al. [12], they divided the chip load into zones as shown in Figure 2.13. However, this technique also relies on integration, which could be computationally costly and does not cover all possible configurations (such as the case where the depth of cut is smaller than the tool nose radius arc). Zhou [5] used the ACIS kernel to define tool engagement by defining various engagement features, that are extracted from the workpiece by boolean operations. The analytically defined tool contour is swept along the tool path. Area swept by the tool, namely the tool swept area, is then intersected with the workpiece. Extracted engagements are decomposed by the proposed feature based method as illustrated in Figure 2.14. The feature decomposition is computationally complex and the solid modeler kernel brings additional computational load. Some research has defined the tool workpiece engagement parameters (uncut chip area and cutting edge length) analytically, without any geometrical simulations. Ozdoganlar and Endres [22] developed analytical formulations where the motion in both radial and axial directions are considered. However, this approach does not account for the workpiece geometry, and does not present any possibility to modify and update the in-process workpiece. 14 Ec Ep OpOc Lp Lc RI H G Elemental Lead Angle Zone I Zone II Chip Width c dc Zone III a Op R I Oc f dc dp Ec Ep fx Lp Lc fz d X Z Current Tool Position Previous Tool Position Figure 2.13: Chip geometry in contour turning [13] The method proposed by Li et al. [9] used polygons to define both the workpiece and the area swept by the cutting tool during its motion. The intersecting area is then calculated by using a polygon clipping algorithm. 2.1.3.1 Boolean operations on polygons In the literature, polygon clipping and boolean operations for polygons are widely studied, and efficient algorithms have been developed. Early work concentrated on polygon intersection prob- lems. Shamos and Hoey [23] developed algorithms, which detect whether two given polygons intersect or not. Continuing the work of Shamos and Hoey, Bentley and Ottmann [24] counted the number of intersections between two polygons. Rivero and Feito [25] proposed an algorithm which calculated the boolean results of two polygons by using simplical chains. More complex boolean algorithms that are capable of dealing with general polygons have been recently presented [26, 27]. 15 Figure 2.14: Feature decomposition method proposed by Zhou [5] 2.2 Force Model The most prominent force models assume that the cutting forces are proportional to the uncut chip area and the cutting edge length. The coefficients that relate the chip load to the cutting forces are known as cutting coefficients [14]. There are several methods to evaluate the cutting coefficients, such as orthogonal to oblique transformation or mechanistic approach. In reality the cutting tools are not always ideal and every cutting tool has a nose radius (rn), which thins the chip and changes the force direction. The cutting forces can be calculated by dividing the chip load into small discrete elements, which are assumed to cut the workpiece or- thogonally. Forces are then calculated in the tangential, radial and feed directions as shown in Figure 2.15. These forces are transformed into measurement coordinates by using the local ap- proach angle of each discrete element [14]. Yussefian et al. [11] proposed to use a discretization technique similar to Altintas [14], in order to obtain an effective chip flow angle for the boring process. 16 a) Isometric View b) Top View O O’’ de dFr dFn dFt fc e Figure 2.15: Calculation of cutting forces with discrete orthogonal elements [14] Colwell [15] proposed the use of an equivalent chord, to which the chip flow is assumed to be perpendicular. The approximate chord is shown in Figure 2.16 for a straight edge tool, and in Figure 2.12 for a tool with a nose radius. Eynian and Altintas [2] used a force model, which utilizes the approximate chord proposed by Colwell [15]. The approximate chord is calculated using the tool geometry and the process param- eters such as depth of cut and feedrate as shown in Figure 2.12. The cutting forces are assumed to be equivalent to the cut where the approximate chord acts as the cutting edge. This approximation eliminates the necessity of using discretized elements and is proven to be sufficiently accurate by Eynian and Altintas [2] (see Figure 2.17). 2.3 Chatter Stability Prediction of the regenerative chatter stability is one of the most important challenges in machining operations. Chatter instability in machining causes poor surface finish, tool wear, and damage to 17 Workpiece e a) Cylindrical Turning with a Straight Edge Cutter b) Close-up of the Tool-Workpiece Engagement with Colwell’s Approximate Chord Feed (fc) D epth of Cut (a c ) Fn Fr Ft Current Tool Position Previous Tool Position Figure 2.16: Approximate chord for a cutter with a straight edge [15] the machine tool. Prediction of chatter stability enables the manufacturers to avoid scraped parts, excessive tool wear, and damaged tools. The method for obtaining the analytical stability lobes in orthogonal cutting process was de- veloped by Tobias and Fishwick [28], Tlusty and Polacek [29], and is detailed by Altintas [14]. Ozlu and Budak [17] used this knowledge to develop a stability model for multi-dimension dy- namic systems using a tool having a nose radius, in turning and boring. They discretized the uncut chip by using trapezoidal elements (Figure 2.18) and used a numerical solution for the eigenvalue problem. However, this approach does not take into account the full geometry of the chip (the cusp area) and requires extensive computational time. Eynian and Altintas [2] used the approximate chord model and developed two regenerative chip models. Model I assumes a dynamically changing equivalent chip thickness. Model II defines the cutting forces as a function of both chip area and the chord length. Model II also considers the process damping effect. The stability of the obtained characteristic equations is evaluated by the 18 Fy (Feed direction) 30 1 20 200 400 600 800 1000 Fx (Depth of cut direction) 0 1 2 30 200 400 600 0 1 2 30 500 1000 1500 2000 Fz (Tangential direction) 30 1 20 200 400 600 800 1000 0 1 2 30 200 400 600 0 1 2 30 500 1000 1500 2000 30 1 2 0 200 400 600 800 1000 0 1 2 3 0 200 400 600 0 1 2 3 0 500 1000 1500 2000 30 1 20 200 400 600 800 1000 0 1 2 30 200 400 600 F x [N ] F y [N ] F z [N ] F x [N ] F y [N ] F z [N ] F x [N ] F y [N ] F z [N ] F x [N ] F y [N ] F z [N ] 0 1 2 30 500 1000 1500 2000 depth of cut [mm] depth of cut [mm] depth of cut [mm] (a) (b) (c) (d) Figure 2.17: Experimental verification of Colwell’s chord approximation [16] Nyquist criterion. 2.4 Optimization Algorithms 2.4.1 Direct and Random Search Search based algorithms use an objective function to evaluate the performance of a generated design vectors in the design space. If the optimization problem is constrained, the feasibility of each design vector is also checked. The optimum solution is chosen as the design vector with the highest performance, which does not violate any constraints. 19 b d2 b d1 e1 e2 en+1=c bnosebe be remaining edge of the insert, outside the nose region n+1st element 1st element 2nd element nth element Figure 2.18: Discretization of chip load by trapezoidal elements [17] Direct search method generates and evaluates all possible design vectors within a design space, where the design variables are bounded, and discretized with a certain resolution. Random search method generates the design vectors by choosing a random value for each variable. Basic procedure of search based optimization is described by Rao [18]. 1. Generate the first trial design vector. 2. Verify whether the chosen design vector satisfies the constraints. The equality constraints are checked within a tolerance. If any constraint is violated, repeat from step 1. 3. Evaluate the objective function using the design vector. If the performance is improved in comparison to the current solution, store the values as the current solution, if not discard the design vector, and repeat from step 1. 4. Best design available after a specified number of trial designs is taken as the solution. 20 2.4.2 Dichotomous Search and Interval Halving Methods A multi variable optimization problem can be reduced to a single dimensional optimization prob- lem by fixing the rest of the design variables. The reduced problems can be solved by using simple one dimensional algorithms such as Dichotomous Search or Interval Halving methods [18]. f(x1) x1 x2 f(x2) b b L0/2 L0 xmin xmax Figure 2.19: Dichotomous searching method [18] In the dichotomous search method, two sets of experiments are needed. The design variable is set as close as possible to the mid point of the design space, separated by a chosen value d . The value (d ) should be selected in a way that it causes a significant variation in the objective function, that allows the interval of uncertainty to be halved with each pair of experiments, as shown in Figure 2.19. Another similar one-dimensional optimization algorithm is the Interval Halving method. By dividing the interval of uncertainty into four smaller intervals at each step the interval where the possible optimal point lies can be halved as shown in Figure 2.20. Both of these methods are only applicable to one dimensional optimization problems with a convex objective function. By setting one free design variable at each step, every design variable can be optimized one by one, until an optimum solution is reached. 21 xmin L0 xmax f(x2) f(x0) f(x1) xmin L0 xmax f(x2) f(x0) f(x1) xmin L0 xmax f(x2) f(x0) f(x1) Figure 2.20: Interval halving method [18] 2.4.3 Genetic Algorithms Similar to the search based algorithms, genetic algorithms require the design spaces for each vari- able be expressed in discrete values. The possible intervals for the design variables must be dis- cretized with a chosen resolution which determines the computational efficiency. Genetic algorithms use the natural phenomenons such as, reproduction, cross-over and muta- tion. Depending on the survival of the fittest theory, the optimal solution is obtained in a defined number of generations [18]. 22 3 Contour Turning Process Model Contour turning is a kind of turning process where the cutting tool moves along a curved tool path in the working–plane (X–Z) of the machine tool. It is used to machine free–form surfaces of revolution as illustrated in Figure 3.1. The process parameters, such as feedrate ( fc) and spindle speed (n), are defined in the same manner as they are for a straight cut. However, due to the free–form nature of the operation it is not always possible to define a static depth of cut value. This is true when the final part contour is different than the initial workpiece surface profile. Contour turning is chosen as an example for the generalized turning process. During contour- ing, the cutting parameters and the uncut chip profile may vary along the tool path, therefore, it is necessary to repeat the simulation at small time intervals, leading to a large number of simulation steps. The development of computationally efficient algorithms is essential to reduce simulation time when a large number of tool paths are processed. 3.1 Angle Definitions for Contour Turning Unlike a straight axial or radial turning process, the angle of the tool motion varies during contour turning. The instantaneous angle of the tool motion in the working plane is called the tool–motion angle (ym), as shown in Figure 3.2. The inclination of the workpiece surface also changes along the tool–path depending on the in–process workpiece geometry. The local inclination angle of the surface is called the surface angle (ys), as shown in Figure 3.2. 23 Workpiece Depth of cut (a) Spindle Speed (n) Feedrate (fc) Figure 3.1: Contour turning process 3.2 Tool–Workpiece Engagements The variation of chip load along the tool path must be determined to simulate the physics of cutting operations. The chip load is defined as the region of the tool, which is instantaneously in–cut. This region is also equivalent to the intersection of the cutting tool and the workpiece at any given time during cutting. These intersections are called the tool–workpiece engagements, as shown in Figure 3.3. In milling, the tool–workpiece engagement is the area on the tool surface, which is in contact with the workpiece (Figure 3.4a). These 3D engagements can be mapped into a two–dimensional coordinate system [30]. In turning processes, the engagement profile is the region on the cutting tool’s rake face, which is currently in contact with the uncut workpiece. This contact area is also defined as area swept by the tool edge profile during one complete spindle revolution, as shown in Figure 3.4b. In straight turning processes, the uncut chip area is a function of the feedrate and the depth of cut, and therefore, does not vary along the tool path. Unlike straight turning processes, when the tool path is a contoured profile, the tool–workpiece engagements vary along the tool path. For contour turning simulations, an effective method to 24 Workpiece` ss Previous Tool Position Current Tool Position X Z fc sm Figure 3.2: Tool–motion and surface angles in contour turning a) 3D View b) Top View Tool-Workpiece Engagement Feed Feed Figure 3.3: Tool–workpiece engagements 25 Chapter 3. Process Simulation 45 preferred due to the smaller computational burden, deriving a generalized analytical relation is not always possible. It is important to state that the calculation of the engagement zone, i.e. intersection between cutter and workpiece, is not among one of the objectives of this research, it is rather taken as an input from a CAD system. In addition to available commercial packages that provide cutter/workpiece intersection maps, there is also considerable amount of research conducted in this field. Takata [89] used the Z-buffer method to obtain the cutter/workpiece intersection for machining force pre- R z 0 R Feed Direction R xy [rad] 0 z CEF CEF CEF CEP MAP Flute UNF OLD Figure 3.6 : Cutter - workpiece intersection, CEF mapping. 0 A xi al D ep th o f C ut Immersion Angle Flute CEF Figure 3.7 : Generalized CEF.(a) Engagement maps in milling [30] 77.3 77.477.5 77.6 77.7 77.8 77.9 78 78.1 15.5 15.6 15.7 15.8 15.9 16 16.1 16.2 16.3 Tool-WorkpieceEngagement Axial Direction Z [mm] Ra di al D ire ct io n X [m m ] Ac Current Tool Position Previous Tool Position Tool-Workpiece Engagement e e (b) Engagement maps in turning Figure 3.4: Comparison of engagement maps in milling and turning 26 calculate the time varying tool–workpiece engagements is developed and presented in Chapter 4. 3.3 Cutting Force Model One of the major objectives of turning process simulation is the prediction of cutting forces. The cutting forces are modeled proportional to the uncut chip area, and the total length of the cutting edge that is in–cut. Cutting forces Ft , Fn and Fr act on the tangential, normal and radial directions respectively as shown in Figure 3.6. A widely accepted, semi-mechanistic model is presented by Altintas [14]. Fn = KncAc+KneLc Fr = KrcAc+KreLc (3.1) Ft = KtcAc+KteLc or expressed in matrix form 8>>>><>>>>: Fn Fr Ft 9>>>>=>>>>;= 266664 Knc Kne Krc Kre Ktc Kte 377775 8><>:AcLc 9>=>; (3.2) where Ac is the total uncut chip area or in other words the area enclosed by tool–workpiece engagement profile. Lc is the total length of the tool edge actively in cut. The terms Ktc,Knc,Krc are the cutting force coefficients in the tangential, normal and radial directions respectively. The cutting force coefficients account for the shearing forces during the turning process. Kte,Kne,Kre are called the edge force coefficients which account for the forces which do not contribute to the shearing process. The edge force coefficients are known to be sensitive to changes in tool wear, temperature, tool coating and use of lubrication. 27 3.3.1 Cutting Coefficients 3.3.1.1 Cutting force coefficients Cutting force coefficients for a given material are obtained experimentally by conducting a series of orthogonal cutting tests as explained in Section A.2. Cutting forces are measured to obtain process parameters under different cutting conditions. The material dependent properties are obtained for each workpiece material, and stored in databases as a function of cutting parameters [31]. ts = f (Vc,hc,an) fc = f (Vc,hc,an) (3.3) ba = f (Vc,hc,an) where ts , fc and ba are shear yield stress, shear angle and average friction angle respectively. The cutting parameters are cutting speed (Vc), chip thickness (hc), and normal rake angle (an). 3.3.1.2 Edge force coefficients Edge force coefficients are specific to each workpiece–tool material pair. The edge force coeffi- cients are calibrated using a mechanistic approach, and their average values for a specific work- piece material are stored in databases. The edge force coefficients for AISI 1045 steel paired with two different tool material are presented in Section A.2. 3.3.1.3 Orthogonal cutting In orthogonal cutting, the cutting force coefficients can be expressed as follows: 28 Kt = ts cos(baar) sinfc cos(fc+baar) Kn = ts sin(baar) sinfc cos(fc+baar) (3.4) Kr = 0 However, most turning tools have oblique geometry, therefore, the corresponding cutting force coefficients are calculated by Orthogonal to Oblique Transformation method which is explained in the following section. 3.3.2 Orthogonal to Oblique Transformation !"#$%&$'"$( )*+'$ ,*#%&$'"$( )*+'$ ,*#%-.//"*+ ,#+$%)*+'$ !"#$%&01$ )*+'$ 2031%&01$ )*+'$ !"#$%-.//"*+ ,#+$%)*+'$ 455'%65"*/%57 859$%&0#".9 ! " # ! $ %& ! $ %& # Figure 3.5: Geometry of an oblique turning tool [14] For analyzing the mechanics of oblique cutting, normal rake and oblique angle of the cutting tool must be defined. Since most tools and inserts do not have simple geometries, it is common to define oblique angles by using equivalent angles [32]. For the tool shown in Figure 3.5, equivalent 29 oblique angle (i), equivalent orthogonal angle (ao) and normal rake angle angle (an) are defined by tana0 = tana f cosyr+ tanap sinyr tan i= tanap cosyr+ tana f sinyr (3.5) tanan = tana0 cos i Once all the required cutting parameters are obtained, cutting coefficients for a given material can be calculated as follows: Knc = ts sinfn cos i sin(bnan)q cos2(fn+bnan)+ tan2h sin2bn Krc = ts sinfn cos(bnan) tan i tanh sinbnq cos2(fn+bnan)+ tan2h sin2bn (3.6) Ktc = ts sinfn cos(bnan)+ tan i tanh sinbnq cos2(fn+bnan)+ tan2h sin2bn where the material dependent properties, ts , fc and ba are obtained from Equation 3.3. The chip flow angle is assumed to be equal to the oblique angle due to Stabler’s chip flow rule [33]. i= h (3.7) Also the friction angle and the shear angle are assumed to be the same as orthogonal cutting for oblique cutting. bn = ba (3.8) fn = fc (3.9) 30 In contour turning operations, cutting parameters such as the cutting speed and the chip thick- ness vary with time, therefore, the cutting coefficients are calculated at each simulation step. This allows the cutting coefficients to vary as the chip geometry changes along the tool path. 3.3.3 Chip Flow Angle In an ideal orthogonal turning operation with a straight cutting edge, cutting forces are generated in two principal directions, namely the normal direction, which is perpendicular to the cutting edge, and the tangential direction, which is the direction of the cutting speed. Most turning tools have a nose radius and side cutting edge angle. The direction of the resultant force is affected by the oblique geometry of the tool, therefore, the resultant force has to be ex- pressed in a three dimensional coordinate system. In addition to the normal and tangential forces, radial forces are considered as shown in Figure 3.6. The oblique geometry is accounted for, by using the orthogonal to oblique transformation as explained in Section 3.3.2. It is still necessary, however, to find the direction of the resultant force on the rake face, or in other words the angle of the chip flow. There are numerous models to predict the chip flow angle [17, 12]. However, most of these models discretize the chip load into small elements, and superpose cutting forces generated by these small elements to find the direction of the resultant cutting force. The use of discrete elements increases the computational load, which is a major drawback for simulation purposes. To estimate the chip flow direction in a more efficient manner, the Colwell effective chord approximation is employed in the virtual turning system. 3.3.3.1 Effective chord approximation In order to avoid discretizing the chip geometry into small elements, a single straight cutting edge is assumed as proposed by Colwell [15]. He explains that the effect of the tool nose radius can be modeled by using an approximate cutting edge, that is chosen to be the major diameter of a given tool–workpiece engagement profile, as shown in Figure 2.16 and Figure 3.4b. The length of the approximate chord (Le f f ) and the approximate chord angle (q ) are obtained 31 Fc Fr `a a Fn sr Ft `r Z X Workpiece Cutting Tool Vc Figure 3.6: Cutting forces during oblique cutting from the engagement profile geometry. Normal force is assumed to be acting in the direction perpendicular to the effective chord, while the radial force is assumed to be acting along the chord (Figure 2.16). 3.3.4 Effective Chip Thickness When the tool path is a free–form curve, as in the case of contour turning, the chip thickness varies along the tool path, even if the depth of cut and the feedrate remain constant. The variation of chip thickness is shown in Figure 3.7. The chip thickness (hc) is assumed to be the thickness of the uncut chip profile measured perpendicularly to the approximate chord as suggested by Colwell [15]. When the surface inclination angle (ys) is known, the chip thickness can be calculated as hc = fc sin(q +ys) (3.10) where ys is the instantaneous surface inclination as shown in Figure 3.2. In a general turning process it is assumed that there is no a priori knowledge about the work- 32 Workpiece Figure 3.7: Changing engagement geometry along the tool path in contour turning piece surface profile. A more generalized method for estimating the chip thickness is used. The effective chip thickness is calculated by dividing the uncut chip area, to the total cutting edge length [34]. heff = Ac Lc (3.11) where Lc is the cutting edge length, and Ac is the uncut chip area. 3.4 Tool Motion Classification For a general purpose turning simulation, the force model must be generalized enough to simulate a range of turning operations. Depending on the tool motion, five different cutting operations are defined, similar to the four quadrant division proposed by Reddy [13]. The type of cut is determined by the instantaneous tool motion angle and feedrate (Figure 3.8). The instantaneous tool motion vector is written as 33 Left Handed Cut Grooving - Parting O! Threading No Cut Right Handed Cut fz g¡ ^m gr Z X rn fx fc Figure 3.8: Tool motion classification fc = 8><>: fzfx 9>=>; (3.12) where fz and fx are the axial and radial components of the feed respectively. The angle (ym) and magnitude ( fc) of this vector in relation to the tool geometry defines the type of cut. k fck> rn Feedrate value is larger than the tool nose radius. All three edges of the tool are in contact with the workpiece. This type of feedrates are generally used for threading operations. kr < ym < ke Tool is cutting with the tool nose radius and right edge. This type of cut is classi- fied as a Right Handed Cut. 34 Left Handed Cut Right Handed Cut Fn Fr Fn Fr e e z y x Figure 3.9: Force model modification for left and right handed cuts ke ym pkr Tool is leaving the workpiece, no edges are in contact, therefore, there is no cut. pkr < ym < p+ke Tool is cutting with the tool nose radius and left edge. This type of cut is classified as a Left Handed Cut. p+ke ym kr Tool is cutting with all three edges, this type of cut occurs in grooving or parting–off processes. In these inequalities kr and ke are the edge approach angles for left and right hand side angles respectively, and rn is the tool nose radius. These regions are represented by a circle of radius equivalent to the tool nose radius, rn. The circle is split into four regions by the angles of ke and kr, as shown in Figure 3.8. By placing the feedrate vector fc at the center of this circle, the type of cut is determined depending on the region in which the end point lies. The virtual turning system considers, the left and right handed cuts. The validity and applica- bility of the models with grooving, parting-off, and threading operations must be further studied. When using the approximate cutting edge, the cutting forces are expressed as normal, radial and tangential directions (Fn, Fr, Ft). These forces are transformed into machine–tool coordinate system as: 35 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= [Cnm] T 8>>>><>>>>: Fn Fr Ft 9>>>>=>>>>; (3.13) It can be seen from Figure 3.9 that depending on the type of cut, the transformation matrix [Cnm]T in Equation 3.13 needs to be modified. Following the coordinate system given in Figure 3.9, the transformation matrices for the two cases are written as: [Cnm]T = 8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>: 26666664 cosq sinq 0 0 0 1 sinq cosq 0 37777775 for Left Handed Cut 26666664 cosq sinq 0 0 0 1 sinq cosq 0 37777775 for Right Handed Cut (3.14) 3.5 Stability with Process Damping The chatter stability solution for axial turning process has been developed by Eynian and Altintas [2] who proposed two separate models. The first model considered dynamically changing chip thickness. The second model incorporates the variance in the uncut chip area, and the effective cutting edge during the cutting process. It also considers the process damping forces. In order to develop a computationally efficient model, a modified version of the regenerative chip thickness model is used to determine chatter stability. The model is modified to include process damping forces due to flank penetration into workpiece at slow cutting speeds [2, 35, 36]. Dynamic cutting forces consist of two components; cutting and process damping forces. Cut- ting forces are due to the chip formation process of the turning. These forces are calculated using 36 ess hc Previous Tool Position Current Tool Position X Z fc sm Figure 3.10: Chip thickness in contour turning the mechanistic force model explained in Section 3.3. 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= [F ]Cutting+[F ]Process Damping (3.15) Process damping forces are expressed as a function of the vibration velocity, and the process damping gain matrix [Jv] as follows: [F ]Process Damping = [Jv] 8>>>><>>>>: ẋ(t) ẏ(t) ż(t) 9>>>>=>>>>; (3.16) The calculation of process damping gains [Jv] is explained in Section 3.5.1. Replacing Equa- tion 3.13 and Equation 3.16 in Equation 3.15: 37 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= [Cnm] T 8>>>><>>>>: Fn(t) Fr(t) Ft(t) 9>>>>=>>>>;+[Jv] 8>>>><>>>>: ẋ(t) ẏ(t) ż(t) 9>>>>=>>>>;8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= [Cnm] T 8>>>><>>>>: Kna Kra Kta 9>>>>=>>>>;Lchc(t)+ [Jv] 8>>>><>>>>: ẋ(t) ẏ(t) ż(t) 9>>>>=>>>>; (3.17) The chip thickness changes due to the tool motion perpendicular to the effective chord. The dynamic chip thickness is expressed as a function of tool motion in the X and Z directions and the approximate chord angle. The difference between the previous vibrations and the current vibrations gives the dynamic chip thickness equation as: hc = csinq {[x(t)cosq + z(t)sinq ] [x(t t)cosq + z(t t)sinq ]} (3.18) where t is the spindle period. Dropping the static term, the dynamically varying chip thickness is written in the matrix form hc = ⇢ cosq 0 sinq 8>>>><>>>>: x(t) x(t t) y(t) y(t t) z(t) z(t t) 9>>>>=>>>>; (3.19) replacing this term in Equation 3.17 gives 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= [Cnm] T 8>>>><>>>>: Kna Kra Kta 9>>>>=>>>>;Lc ⇢ cosq 0 sinq 8>>>><>>>>: x(t) x(t t) y(t) y(t t) z(t) z(t t) 9>>>>=>>>>;+[Jv] 8>>>><>>>>: ẋ(t) ẏ(t) ż(t) 9>>>>=>>>>; (3.20) 38 Expressing the dynamic force equation in Laplace domain we obtain 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= (1 e st)[J] 8>>>><>>>>: x(s) y(s) z(s) 9>>>>=>>>>;+ s[Jv] 8>>>><>>>>: x(s) y(s) z(s) 9>>>>=>>>>;8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= ⇥ (1 est)[J]+ sJv ⇤ 8>>>><>>>>: x(s) y(s) z(s) 9>>>>=>>>>; (3.21) Cutting tool vibrations can be expressed as a function of the cutting forces and the frequency response function of the flexible structure. The tool vibrations are expressed as 8>>>><>>>>: x(s) y(s) z(s) 9>>>>=>>>>;= [F(s)] 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>; (3.22) Replacing the cutting tool vibrations into Equation 3.21 yields the equation for dynamic cutting forces as 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>;= ⇥ (1 est)[J]+ sJv ⇤ [F(s)] 8>>>><>>>>: Fx Fy Fz 9>>>>=>>>>; (3.23) The characteristic equation of the system can be obtained by rearranging the above equation det [I]3⇥3 (1 est)[J]+ sJv [F(s)] = 0 (3.24) where 39 J = [Cnm]T 8>>>><>>>>: Kna Kra Kta 9>>>>=>>>>;Lc ⇢ cosq 0 sinq (3.25) The process damping gain matrix Jv is evaluated for each engagement profile, as explained in the following section. For every engagement profile the matrices J and Jv are calculated. When the frequency re- sponse function F(s) is known for the flexible workpiece and tool, the stability of the charac- teristic equation given above can be determined by using the Nyquist criterion. The frequency response function of the structure is measured by experimental modal analysis techniques, such as impact hammer testing. The Nyquist criterion allows the detection of chatter and possible chatter frequencies. 3.5.1 Process Damping Gains The process damping gain matrix is given by Eynian and Altintas [2]. After necessary coordinate transformations, the contribution from a straight edge is given as [Jv]line =L2w KspLline 2Vc 266664 cos2kr 0 sinkr coskr µc coskr (coskr+ sinkr) 0 µc sinkr (coskr+ sinkr) sinkr coskr 0 sin2kr 377775 (3.26) where Lline is the length and kr is the inclination angle of the straight edge. Contribution of an arc is given as 40 [Jv]arc =L2w Ksp 2Vc ˆ q1 q0 266664 cos2kn 0 sinkn coskn µc coskn (coskn+ sinkr) 0 µc sinkn (coskn+ sinkn) sinkn coskn 0 sin2kn 377775rndkn (3.27) [Jv]arc =L2w Ksp 2Vc 266664 kn 2 + sin 2kn 4 0 sin2 kn 2 µc ⇣ kn 2 + sin 2kn 4 + sin2 kn 2 ⌘ 0 µc ⇣ sin2 kn 2 + kn 2 sin 2kn4 ⌘ sin2 kn 2 0 kn 2 sin 2kn4 377775 q1 q0 rn (3.28) where q0 and q1 are the limit angles of an arc and rn is the radius. Ksp is an experimentally identified contact force coefficient, Lw is the flank wear, Vc is the cutting speed and µc is the coefficient of friction which is assumed to be 0.3 for steels [36]. The damping gain coefficients ([Jv]) are calculated by adding the contribution of each edge segment currently in–cut. Cutting edge segments which are in–cut are obtained by using the en- gagement model detailed in Chapter 4. 3.6 Feedrate Scheduling The ultimate goal of virtual machining is to provide the industry with means of increasing produc- tivity, in other words, decreasing the machining cycle times. Conventionally, the cutting parameters are chosen based on on the experience of the machinists and engineers, or values suggested in handbooks and catalogues by the tool manufacturers [30]. The strict tolerances and surface requirements force the industry to use conservative feeds and speeds, causing a loss in productivity. For contour turning, a feedrate scheduling approach is developed where feasible cutting condi- tions, which do not violate the machine tool physics, are identified. 41 3.6.1 Design Variables • Feedrate ( fc): Large feeds increase the cutting forces, surface roughness, and chip thickness. For each workpiece–tool pair suggested maximum feedrate values are provided by the tool manufacturer. The amplitude of the feed mark can be limited by setting a maximum feedrate ( fc)max. When the feed is chosen too small, cutting tool rubs against the workpiece surface, prevent- ing chip formation. Therefore a minimum limit can be chosen by applying a minimum chip thickness criteria. • Spindle Speed (n): The choice of spindle speed influences the tool–life and the required power during machining. Depending on the workpiece and cutting tool material, minimum and maximum limits for the spindle speed are suggested by manufacturer specifications. 3.6.2 Design Space After the boundaries for the design variables are set, the design space is generated by using a specific resolution for each design variable. Smaller resolutions improve the simulation precision, while increasing the computational time. Design space is generated by fc = ⇢ ( fc)min ( fc)min+D fc . . . ( fc)maxD fc ( fc)max n= ⇢ (n)min (n)min+Dn . . . (n)maxDn (n)max (3.29) where ( fc,n)max and ( fc,n)min are the upper and lower boundaries of the variables respectively. D fc, Dn are the resolution for each design variable. 42 3.6.3 Constraints Constraints which limit the machining operations include, chatter stability, machine tool capabil- ities, such as maximum force, power, torque limits, and performance constraints, such as chip breakability and form errors [30],[14]. For contour turning process, the following constraints are considered: • Force Constraints : F Fmax The simulated cutting forces are compared against the maximum allowable forces, and the solution is only accepted if the forces are smaller than the maximum allowable value. • Power Constraint : P Pmax Power required during machining is calculated P= Fy Vc 60 (3.30) P= Fy 2prn 60⇥103 (3.31) whereVc is cutting speed, Fy is the tangential cutting force, n is spindle speed, and P is power in [W]. • Torque Constraint : T Tmax Required spindle torque in turning can be estimated as a function of current tool position and simulated cutting forces, as follows [14]: T = Fy ✓ da 2 ◆ (3.32) where Fy is the tangential cutting force, d is the workpiece diameter, a is the depth of cut in cylindrical turning, and T is torque in [Nm]. For process simulation, Equation 3.32 is rewritten by replacing the workpiece diameter with the current tool position. 43 T = Fy ⇣xtool 2 ⌘ (3.33) where xtool is the radial coordinate of the current tool position that is obtained by parsing the NC-code as explained in Section 4.6. • Chip Thickness: hmin < heff < hmax A minimum chip thickness constraint prevents rubbing of the tool, therefore ensures chip formation. A maximum limit is also chosen as large chip thickness values cause tool chip- ping and tool breakage [30]. Chip thickness is approximated as heff = Ac Lc (3.34) where Ac is uncut chip area and Lc is cutting edge length. • Stability Constraint Chatter stability of the cut is checked using the Nyquist criterion, as explained in Section 3.5. The stability model allows for the detection of any cutting conditions that causes chatter instability. If chatter is detected at an engagement map, a new depth of cut value is recommended, by searching for a value which leads to stable cutting conditions. NC-commands causing chatter instability are marked for revision. 3.6.4 Direct Search Algorithm The cutting forces in contour turning are a function of cutting speed (Vc), feedrate ( fc), depth of cut (ac), and engagement and tool geometries: F = f (Vc, fc,ac,Engagement Geometry,Tool Geometry) (3.35) 44 The term Engagement Geometry is used to indicate that the resulting cutting forces are also a function of the tool, workpiece and tool path geometries. It should be noted that the engagement geometry itself is a function of the design variables. Engagement Geometry= f ( fc,ac) (3.36) Equation 3.35 and Equation 3.36 reveal that the cutting forces are a complex function of the process parameters, and chatter stability is determined only for the simulated cutting conditions. Hence, it is not possible to use conventional and linear optimization techniques. Instead, a feasible solution is searched by using a direct search algorithm as detailed in the following section. A direct search algorithm is used to search for the largest feedrate value, which does not violate the machine tool constraints. The performance of the process is evaluated, and the constraints are checked using the presented process model. Most contour turning geometry consist of smooth and continuous features, therefore the feeds can be modified at big intervals. Each NC command of the part program is assigned a new feedrate value. This approach prevents sudden changes in the feedrate which may violate the dynamics of the machine drives, and cause poor finish. The optimization algorithm (Figure 3.11) is as follows : 1. The entire design space is generated using Equation 3.29. 2. Simulation is run with the maximum allowable feedrate value. Cutting forces, power and torque requirements, and chip thickness are calculated, and stability is checked. 3. If the constraints are satisfied, current feedrate is assigned for the current engagement map. If the constraints are violated, a smaller feedrate value is chosen and the procedure is repeated until a solution, which satisfies all the constraints, is found. After the feedrate scheduling, a secondary step searches the highest feasible spindle speed. The direct search is repeated by using the spindle speed as the design variable. 45 ConstraintsDesign VectorNext Highest Feed Next Highest Speed Process Simulation Process Simulation Process Simulation Feasible Feed Feasible Speed Recommended Depth of Cut Verify Constraint Verify Constraint Chatter Detection Generate Design Space F max > F > F min Pmax > P > P min Tmax > T > T min hmax > hc> hmin Chatter Stability n = { (n) min (n) min + 6n ... (n) m ax - 6n (n) max } or Vc = { (Vc) min (Vc) min + 6Vc ... (Vc) m ax - 6Vc (Vc) max } Speed:( ) Feedrate: fc = { (fc) max (fc) max- 6fc ... (fc) min + 6fc (fc) min } Reduce Depth of Cut Constraints Violated Constraints Violated Constraints Satis!ed Constraints Satis!ed Chatter No Chatter Figure 3.11: Direct search algorithm 46 Initial Cutting Parameters Engagement Calculation Tool -Workpiece Engagements Virtual Turning Modify Engagement Modify Cutting Parameters Force Power Torque Stability Constraints Engagement Based Feedrate Scheduling Solution Modi!ed Parameters Fi na l V er i! ca tio n Engagement Modi!cation No Yes Figure 3.12: Engagement based feedrate scheduling using search based algorithm Finally, the chatter stability is checked for the modified cutting conditions. If the cut is unstable, the depth of cut value is gradually reduced and a new chatter–free condition is identified. The engagement position causing chatter is marked, and the maximum feasible depth of cut value is recommended to the NC planner for revision. 3.6.5 Engagement Based Scheduling At every step of the search algorithm, the cutting forces, power and torque are evaluated, and chatter stability is solved to ensure chatter–free operation, which requires the process simulation 47 to be repeated a large number of times. This is not a feasible approach due to time constraints. The most time consuming task is the calculation of the engagement profiles. Therefore, an analytical solution for engagement modification is developed. Initial engagements are obtained using the engagement model explained in Chapter 4. The change in the engagement parameters is calculated as a function of the change of cutting parame- ters, and the parameters are optimized through the direct search method. The final solution is verified by simulating the process entirely with the chosen cutting param- eters (Figure 3.12). 3.6.5.1 Modified engagement geometry 6wfc 6hfc fc 6fc ^s 6wa2 6wa1 6ha2 6ha1 a) E!ect of feedrate change b) E!ect of depth of cut change ^m ac 6ac Figure 3.13: Modification of tool–workpiece engagement geometry For a given engagement profile, height (h), width (w), and length (Le f f ) values of the approx- imate chord is obtained as explained in the Section 3.3.3.1. Using the initial approximate chord geometry, the modified chord (Figure 3.13) is calculated as follows: 48 Feedrate The difference in the height and width of the approximate chord due to the change in the feedrate (D fc in Figure 3.13a) is approximated as: Dwfc = D fccos(y 0s) (3.37) Dh fc = D fcsin(y 0s) with y 0s = pys where D fc is the difference in the feedrate and (ys) represents the surface angle. The modified approximate edge is now defined as: w0 = Dwfc +w h 0 = Dh fC +h (3.38) L0e f f = p w02+h02 q 0 = tan1 w 0 h0 The uncut chip area is assumed to be linearly proportional to the feedrate ( fc. Therefore the modified chip load can be calculated as A0c = Ac f 0c fc (3.39) Depth of cut The difference in the height and width of the approximate chord due to the change in the depth of cut (Dac in Figure 3.13b) is approximated as: 49 Dwac1 = |Dac tan(y 0skr)|cos(py 0s) Dhac1 = |Dac tan(y 0skr)|sin(py 0s) Dwac2 = Daccos( 3 2 py 0s) Dhac2 = Dacsin( 3 2 py 0s) (3.40) Dwac = Dwac1+Dwac2 Dhac = Dhac1+Dhac2 Writing Equation 3.38 for depth of cut: w0 = Dwac +w h0 = DhaC +h (3.41) L0e f f = p w02+h02 q 0 = tan1 w 0 h0 Similar to Equation 3.39 the chip area is assumed to be linearly proportional to the depth of cut (ac) A0c = Ac a0c ac (3.42) 3.7 Summary Process model for contour turning operations is presented in this chapter. Once the engagement geometry of the cut is known, the cutting forces, power and torque requirements, and the process stability is evaluated using a semi–mechanistic material model. Based on the presented process model a feedrate scheduling method is proposed. Feedrate scheduling aims to improve the over- all productivity by modifying the feedrate and spindle speed commands in the NC-code while ensuring the physical limitations of the machine tool are not violated. 50 4 Turning Simulation Model A process model for contour turning operations is presented in Chapter 3, where the physics of cutting is defined by the tool–workpiece engagements. Generating the engagement maps is the essential task of turning simulations. Twomethods for obtaining the engagement data are presented in this chapter. The first method, which is detailed in the section one of this chapter, obtains the engagement maps by using a commercially available software. Implementation of this method is simple and fast, since all necessary models for tool, workpiece, and numerical control are handled by the external application. However, this approach suffers from compatibility and efficiency problems due to the dependency on a third party software. To eliminate this dependency a new simulation model is developed specifically for contour turning simulations. The model is developed by considering computational implementation. Be- cause the ultimate objective of this system is to form a process planning tool that can be used in a computer environment, computational efficiency, ease of implementation and visualization tasks are important. The general geometric representation adopted by the new simulation model is presented in section two. The workpiece and tool models are described in sections three and four, respectively. Section five details the tool–workpiece intersection model that is based on two–dimensional boolean operations. Process parameters that form the inputs of the force model are obtained by processing the generated tool–workpiece engagement maps. Section six details simulation schemes that allow the user to decide between the simulation resolution and computational time. 51 The last section of the chapter explains briefly how the NC-code is parsed by the virtual turning system. The NC-code is used to generate the tool locations along the tool–path, as well as to determine cutting parameters such as cutting speed, feedrate and spindle speed. 4.1 Vericut Application Program Interface Tool–workpiece engagement profiles are obtained directly from Vericut, which is a machining verification software capable of simulating CNC machine tools. With this software package, an application program interface (API) library is provided under the name of Vericut Development Tools. Version 7.0.3 of the development tools provides an API function which allows the extraction of tool–workpiece engagements during turning operations. A custom API is developed in C++ that runs alongside the NC–simulation. An output file is created, which is processed by the virtual turning system, to estimate the cutting forces, torque and power requirements. The output file created by the Virtual Turning API, contains the resulting tool–workpiece engagement profiles for each simulation step. The main API function which allows the engagement profile of the simulated turning operation to be extracted is the “opapi get turning contact profile” function. Invoking this function at every complete spindle revolution allows the correct tool–workpiece profiles to be obtained. Output data structure The data fields of API output file are detailed as follows Step Number: Number of the current simulation step Loop Number: Number of closed loops defining the engagement profile Spindle Speed: Spindle speed at the current simulation step, in rotations per minute Feedrate: Feedrate at the current simulation step Total Time: Total tool motion time elapsed, in seconds xp : The X coordinate of the point or the centre point of the arc 52 re e P0(x,y)P1(x,y) P0(x,y) a) Line Segment L(P0, P1) b) Arc A(P0, r, e1, e2) Figure 4.1: Polycurve entity definitions zp : The Z coordinate of the point or the centre point of the arc rp : If zero, the point represents a corner of the boundary, if non-zero it is the radius of the arc Dx : The radial tool motion, in diameter coordinates Dz : The axial tool motion 4.2 Polycurve Representation For the new simulation model, a general two dimensional curve representation scheme is proposed to represent the workpiece and tool geometries. Profiles are represented as series of line segments and arcs, which form a closed curve. Inclusion of circular elements in the boundary allows the ge- ometry to be defined accurately, and with less elements, compared to polygons. This representation is referred to as the polycurve representation throughout this text. Geometric models and boolean operations for polygons are extended to accommodate the in- clusion of circular elements. Modifications to the polygon models are explained in the following sections. 4.2.1 Entity Definitions Polycurves are represented by using two geometric entity types; namely line segments and arcs. Each entity is defined by geometric parameters as illustrated in Figure 4.1. Entity data is formatted 53 and stored using the following data structure. 4.2.1.1 Data structure for entities A data structure for the storage of the geometric entities is presented. The data structure combines the geometric properties required to define an arc or line segment, and extra data fields for entity classification. Entity.Type Defines the type of the current entity as ’LINE’ or ’ARC’ Entity.Layer Defines if the entity is part of the workpiece contour (’IPW’), or part of the tool contour (’TOOL’) Entity.x (LINE) {x0,x1} X-coordinates of the end-points Entity.x (ARC) x0 X-coordinates of the center point Entity.y (LINE) {y0,y1}Y-coordinates of the end-points Entity.y (ARC) y0 Y-coordinates of the center point Entity.r (ARC) r Radius of the arc Counter-Clockwise if r > 0 Clockwise if r < 0 Entity.t (ARC) {q1,q2} Start and end angles of the arc Entity.Begin The beginning point of the entity regardless of type Entity.End The ending point of the entity regardless of type 54 4.3 Workpiece Model Workpiece model is an essential component of the virtual turning. Among the variety of workpiece representations, which were presented in Chapter 2, generator curve representation is adopted for the virtual turning system, due to its computational simplicity and efficiency. The main advantage of the generator curve representation is that it represents workpiece geometry in two–dimensions, which facilitates generation, modification and visualization of the in–process workpiece . If the workpiece is not perfectly axi–symmetric, the workpiece model is extended to accom- modate the non-symmetric features of the workpiece as presented in Section 4.7. 4.3.1 Generator Curve Representation Revolved solid geometries can be represented by their axis of revolution, and their cross–sections, as shown in Figure 4.2. This representation is called the Generator Curve Representation [9]. The CAD data of the workpiece is converted to the generator curve representation, before it is used in the virtual turning simulation. STL file format is chosen as the user input because it is a neutral file format that generates water tight solids. STL parts consist of triangular facets, therefore, the edges of an stl solid are linear. The generator curve of the solid part is extracted from the solid model by intersecting the triangular facets with the machining–plane. The axis of revolution for the generator curve repre- sentation coincides with the spindle axis of the machine–tool (Figure 4.2). The elements of an STL solid are triangles in three–dimensional space. Finding the intersection of the solid model with a plane is reduced to the calculation of the intersection between a plane and a triangular facet. The cross–section is reconstructed by collecting the resulting intersection edges. 4.3.2 Intersection of a Triangle with a Plane To classify the intersection problem, the vertices of the triangle must be located with respect to the plane, by calculating the signed distance between the plane and the point (Figure 4.3). 55 Spindle Axis Axis of Revolution Generator Curve Z X Ma chi nin g P lan e Figure 4.2: Generator curve for a solid part of revolution [9] P n̂ Qi P0 D Figure 4.3: Signed distance of a point from a plane in space 56 (a) (b) (c) (d) (e) P P P P P T1 T1 T1 T1 T1 T2 Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Q4 I1 I2 I1 Figure 4.4: Possible cases of triangle–plane intersection Let T1 be the triangle with vertices Q1(x1,y1,z1), Q2(x2,y2,z2), Q3(x3,y3,z3) and P a plane in three dimensional space (Figure 4.4). P is defined as follows: ax+by+ cz+d = 0 (4.1) The distance between point i and the plane is given by the equation D= axi+byi+ czi+dp a2+b2+ c2 (4.2) The distance can also be expressed by using the dot product 57 D= |P0Qi| ⇧ n̂ kn̂k (4.3) where n̂ is the normal vector defining the plane n̂= 8>>>><>>>>: nx ny nz 9>>>>=>>>>;= 8>>>><>>>>: a b c 9>>>>=>>>>; (4.4) Location of a point with respect to the plane is defined by the sign of the distance sign(D) = sign(|P0Qi| ⇧ n̂) (4.5) where P0 is an arbitrary point on the plane. As a convention, this point is chosen as the machine–tool coordinate system origin. • sign(D)> 0 The point is on the positive side of the plane • sign(D)< 0 The point is on the negative side of the plane • sign(D) = 0 The point is on the plane Once all three vertices are located, the intersection problem is solved using the cases below: (a) Two vertices lie on one side of the plane (Q2,Q3), while the third one is on the other side (Q1). The intersection points for the two edges that crosses from one side of the plane to the other are calculated (Figure 4.4a). I1 = P\ |Q1Q2| (4.6) I2 = P\ |Q1Q3| The intersection edge becomes 58 P\T = |I1I2| (4.7) (b) All vertices lie on one side of the plane, or they all lie on the plane itself in which case the intersection is the entire triangular facet (Figure 4.4b). There is no apparent intersection edge. (c) One vertex (Q1) lies on the plane while the other two (Q3,Q2) are on the same side (Fig- ure 4.4c). P\T = Q1 (4.8) Since there is only a single point on the plane, therefore, there are no apparent intersection edges. (d) One vertex (Q1) lies on the plane while the other two (Q2,Q3) lie on different sides . A single intersection point (I1) is calculated. The intersection edge is the line segment between the point on the plane and the calculated intersection point (Figure 4.4d). I1 = P\ |Q2Q3| (4.9) P\T = |Q1I1| (e) Two vertices of the triangular facet lie on the plane (Q1Q3). The intersection edge is the line segment between these two points (Figure 4.4e). P\T = |Q1Q3| (4.10) 59 4.3.2.1 Intersection of a line with a plane For cases (a) and (c), the intersection of a line with the cross–section plane is calculated as follows. Let the normal vector n̂ and origin P0 define a plane in three-dimensional space. Also let Pin and Pout be two points which are on different sides of this plane. Let P0 represent the position vector of the intersection point. Therefore, P0 is a point on both the plane and the line. Writing in vector form, P0 P0 ⇧ n̂= 0 (4.11) The parametric line equation is written P0 = t(PinPout)+Pout (4.12) where t is the line equation parameter with 0< t < 1 Combining Equation 4.12 with Equation 4.11 (t (PinPout)+PoutP0) ⇧ n̂= 0 t (PinPout) ⇧ n̂= (P0Pout) ⇧ n̂ t = (P0Pout) ⇧ n̂ (PinPout) ⇧ n̂ (4.13) Finally, intersection point (P0) is calculated by replacing t in Equation 4.12. P0 = (P0Pout) ⇧ n̂ (PinPout) ⇧ n̂(PinPout)+Pout (4.14) 60 4.3.2.2 Processing intersection data When an edge lies exactly on the cross-section plane (Figure 4.4e), it appears twice in the set of intersection edges, due to the fact that all the edges are shared by exactly two facets. The resul- tant intersection edges are checked, and any redundant edges are removed from the cross–section profile. 4.3.3 Generator Curve of a Turning Workpiece In turning processes, by convention the machining takes place in X–Z plane of the machine tool coordinates, where X-axis is the radial coordinate, and Z-axis is the axial coordinate. This plane is called the machining–plane (Figure 4.2). Therefore, the machining–plane is defined in the machine–tool coordinate system as P0 = (0,0,0) n= {0,1,0}T where Po is the point of origin and n is the normal vector. If the CAD data supplied by the user does not follow this convention, the input data is modified by a model transformation. The workpiece coordinate system is re-oriented using rotations and translations, before the generator curve is extracted. 4.3.3.1 Model transformation Using the Euler Angles convention (Figure 4.5) rotations in the space are described by three angles (a,b ,g). The rotation matrix is written as 61 Nx y z Z X Y _ ` a Figure 4.5: Euler angles [19] R= 266664 cosb cosg cosa sing+ sina sinb cosg sina sing+ cosa sinb cosg cosb sing cosa cosg+ sina sinb sing sina cosg+ cosa sinb sing sinb sina cosb cosa cosb 377775 (4.15) Therefore, the vertices of the solid model is transformed by 8>>>><>>>>: X Y Z 9>>>>=>>>>;= R 8>>>><>>>>: x y z 9>>>>=>>>>;+ 8>>>><>>>>: a b c 9>>>>=>>>>; (4.16) where a,b and c are the translational distances in machine–tool coordinate system (X ,Y,Z) and R is the rotational transformation matrix given in Equation 4.15. 62 4.4 Tool Model 4.4.1 Parametric Tool Modeling Primary geometry of cutting tools is defined by standard types such as diamond, round or rectangle. The exact tool geometry is defined by parameters such as the nose radius (re), nose angle (en), and tool size (l). Parametric tool model follows the same tool selection concept. First, the primary tool type is selected by the user from a list of standard geometries (Table 2.1). Depending on the tool geometry chosen, a set of parameters is used to generate the cutting edge geometry. The oblique angles such as side rake angle (a f ), back rake angle (ap) and approach angle (kr) are imposed by the tool holder geometry. Tool and tool holder specifications are available in manufacturer catalogues. The cutting edge contour is constructed using the input parameters. As an example, the tool generation process for rhombic inserts is given in the following section. Rhombic inserts cover all the diamond shaped cutters with varying nose angles. ( Types C, D, E, F, M and V in Table 2.1) 4.4.1.1 Rhombic inserts Rhombic inserts are diamond shaped tools, which are widely used for radial, axial, and contour turning. They are classified into smaller groups according to their nose angle (en). By defining the nose angle as one of the parameters, a general model for rhombic tools is developed. The parameters of the model are as follows (Figure 4.6): • en: Nose angle • l: Tool edge length • rn: Nose radius • kr: Approach angle 63 S iC ¡ l r ¡ Figure 4.6: Parameters of a rhombic insert iC ¡ l P1 P2 P3 P4 P5 P6 x z gr g ¡ A Close-up view A Figure 4.7: Definition points for rhombic inserts 4.4.1.2 Definition points Points of the cutting edge must be defined, including any point which does not directly contribute to the geometry, but is used as a tool driven point. For example, point P2 in Figure 4.7, which is the apparent intersection of the major and minor cutting edges, is included in the model, although it is not a part of the cutting tool edge geometry. This point is commonly used as a tool–driven point due to its ease of detection during part setup. Let P1 be the origin of the tool coordinate system. The remaining control points are defined as 64 follows (Figure 4.8): P1 = P(0,0) P2 = P cos ✓ 3p 2 e 2 kr ◆ re sin e 2 ,sin✓3p 2 e 2 kr ◆ re sin e 2 ! P3 = P(cos(pkr)re ,sin(pkr)re) P4 = P ⇣ cos ⇣ ke p2 ⌘ re ,sin ⇣ ke p2 ⌘ re ⌘ P5 = P ⇣ P2x+ cos ⇣p 2 kr ⌘ L,P2y+ sin ⇣p 2 kr ⌘ L ⌘ P6 = P ⇣ P2x+ cos ⇣p 2 kr e ⌘ L,P2y+ sin ⇣p 2 kr e ⌘ L ⌘ The limit angles of the arc defining the nose radius are given by q1 = pkr q2 = ke p2 ke = p 2 (kr+ en) where, (Figure 4.7) kr is the approach angle defined by the user ke is the angle between the secondary edge and Z axis en is the nose angle that defines the tool type 4.4.1.3 Cutting edge construction The cutting edge is generated using the entities defined in Section 4.2. For simplicity, only one half of tool edge is modeled, since practically the depth of cut in turning is much smaller than the tool edge length (l). Entities of the tool contour are defined as (Figure 4.8): 65 P1 P3 P5 L1 A1 L2 L3 P6P4 x z r ¡ e e Figure 4.8: Geometric definition of a rhombic tool A1 = A(P1,re ,q1,q2) L1 = L(P3,P5) L2 = L(P4,P6) L3 = L(P5,P6) 4.4.2 User Defined Tool Geometry Any tool geometry that can be converted to the polycurve representation, can be used as the cutting tool. This allows the use of arbitrary tool geometries, provided that the CAD data is available. When a solid model of the tool is provided, the cross–section of the tool on the machining– plane is obtained using the intersection method explained in Section 4.3.2. Alternatively, if the two–dimensional drawing of the tool is provided, CAD data is reformatted into polycurve repre- sentation. 66 O Z X Actual Tool Positions Swept Polygon Generator Polygon Tool-Workpiece Engagement Cusp Geometry Figure 4.9: Use of swept polygon for boolean operation [9] 4.5 Tool–Workpiece Engagement Model 4.5.1 Boolean Based Tool–Workpiece Engagements Li et al. [9] proposed to use boolean operations on polygons to simulate turning process, where both the workpiece and the tool–swept area are defined with polygons. Tool–workpiece engagements are calculated as the intersection of two polygons. The main disadvantage of this model is that, the arcs are discretized by lines. When a tool swept polygon is used as one of the operands the cusp region is ignored (Figure 4.9). As a result, the number of elements in the resulting in–process workpiece model does not increase significantly. However, in virtual turning, the simulation is run at spindle period intervals. Feed marks left on the workpiece are generated, therefore, a large number of elements is required by the polygon representation. The boolean operations are extended for profiles having circular entities to reduce the number of elements needed in tool–workpiece engagement maps, and in–process workpiece. 4.5.1.1 Boolean operations on polycurves The following properties on the operands are assumed. Operand profiles 67 o : Outer Entity i : Inner Entity u : Unshared Entity Tool Workpiece Control Point (Mid Point) Intersection Point Intersection Point Figure 4.10: Entity classification for boolean algorithm • Are closed curves without any self-intersections, which is true for the cross-section of any valid solid object. • Can have holes. • Can be convex or concave. The main steps of the boolean algorithm are as follows: 1. Calculate all occurring intersections between the entities, by running intersection algorithms for Line–Line, Line–Arc and Arc–Arc intersections. A review of these intersection algo- rithms is presented in Section A.1. 2. Split the entities at points of intersection. 3. Classify each entity of one region as being in, out, or on the other region (see Figure 4.10). The classification process is as follows: 68 Entity classification Once all the entities are split at the intersection points, each entity is classi- fied as outer, inner or unshared. A fourth class, shared, is possible with general boolean algorithms but this class does not occur in machining simulations as the tool profile can not be located inside the workpiece. The classification is performed by choosing a test–point on the entity, which is chosen as the mid-point, for convention. Entities of the cutting tool profile are tested against the in–process workpiece profile and vice versa. This test classifies the entities as follows: Outer Entity is outside the other region Inner Entity is inside the other region Shared Entity is on the boundary of the other region and both regions lie on the same side of the entity Unshared Entity is on the boundary of the other region and the regions lie on different sides of the entity In-Region test Point in–region test checks whether a given point lies outside, inside or on a given region. An arbitrary ray originating from within a closed boundary, crosses the closed boundary at an odd number of times. Similarly, an arbitrary ray originating from outside a closed boundary, crosses the boundary at an even number of times. Location of the test point is determined by counting the number or intersections between the test ray and the region boundary. In–region test yields accurate results as long as the generated ray does not intersect with any vertex. When a polygon is represented by its vertices, which is the most common application, a ray passing through a vertex will apparently intersect the polygon twice, resulting in the test to fail (Figure 4.12). To tackle this problem, a test ray which crosses a vertex is assumed to be slightly at the side of the vertex. In this case the ray is shifted by a sufficiently small amount (e). 69 (a) Point in Region Test with Rectangular Grid (b) Point in Region Test with Random Point Outside Inside On Figure 4.11: In-region test with 250 test points Two apparent Intersection (Outside) Three apparent Intersection (Inside) Single Intersection Double Intersection Vertex Figure 4.12: Errors in in-region test due to test–ray crossing vertices 70 The results of the in–region test are verified in Figure 4.11, where 250 points were tested against a sample region. In Figure 4.11a, test points form a regular grid. In Figure 4.11b, the test points are generated randomly. Intersection The engagement profile is the intersection of the current tool profile with the current in–process workpiece cross-section. The intersection of the two profiles is defined as the collection of entities that are classified as either inner or shared from Region A, and the ones that are classified as inner from Region B (Figure 4.13a). A\B= A(inner)[A(shared)[B(inner) (4.17) Difference The resulting in–process workpiece is the last workpiece profile minus the current engagement. Therefore, the in-process workpiece profile is obtained by calculating the difference of in–process workpiece cross-section and the cutting edge profile. The difference of two regions is the collection of entities marked as either outer or unshared from Region A and the entities marked as inner from Region B (Figure 4.13b). AB= (A(outer)[A(unshared))[B(inner) (4.18) 4.5.1.2 Numerical robustness One of the major problems of the boolean algorithms is the floating point errors. Computer algo- rithms are not capable of representing real numbers exactly due to memory limitations and their binary nature [37], therefore, care must be taken during implementation, especially when defining relational operations. Since real numbers cannot be expressed exactly, the relational operations are defined with a tolerance as shown in Table 4.1. 71 Region A Region B a) Region A E Region B b) Region A <Region B Figure 4.13: Boolean operations supporting holes in the workpiece model Table 4.1: Relational operations with real and floating point numbers Relational Operation Real Numbers Floating Point > A> B A> B+ e < A< B A< B e = A= B |AB|< e The geometrical values can be limited to a given precision, such as the maximum number of digits allowed by input files (i.e. stl file). All digitally calculated physical values are rounded off to this chosen precision. e must be chosen as a value sufficiently smaller than the precision of the simulation, and larger than the data–type precision. 72 4.5.2 Processing Engagement Profiles The engagement profiles obtained from the boolean operation are processed to obtain the parame- ters required by the force and stability models. 4.5.2.1 Uncut chip area The area enclosed by an engagement map is calculated using the Green’s Theorem. Green’s theorem: Green’s theorem expresses a double integral as a line integral around its bound- ary. A very common application of this theorem is the calculation of the area enclosed by a closed profile. The general form of the Green’s theorem is expressed as: ˆ ˆ R ✓ ∂ f ∂x + ∂g ∂y ◆ dxdy= ˛ c ( f dygdx) (4.19) For discrete points, such as the ones defining a closed polygon, the Green’s integral (Equa- tion 4.19) takes the form of a summation. Ac = 1 2 n Â i=1 (xiyi+1 xi+1yi) where n+1= 1 (4.20) where n is the number of vertices on the polygon, and (xi,yi) are the coordinates of the ith vertex. This expression gives the area enclosed by a polygon, calculated as a summation of signed triangular areas. Each triangle is formed by the origin of the coordinate system and two consecutive vertices of the polygon (Figure 4.16b). 73 e e Positive Area Neutral Area Negative Area Figure 4.14: Positive and negative direction conventions for Green’s Theorem er e r Apie Acircular Atriangle Figure 4.15: Area of the arc entity P0(0,0) P1 P2 P3 P4 P5 P6 P0(0,0) P1 P2 P3 P4 P5 P6 Positive Area Neutral Area Negative Area a) Polycurve b) Polygon Area c) Polycurve Area P1 P2 P3 P4 P5 P6 Figure 4.16: Area calculation by Green’s theorem 74 Area of polycurves: The area of the engagement profile represented by a polycurve (Figure 4.16a) is calculated in two steps. The first step is to evaluate the polygon area, excluding the arc elements (Figure 4.16b), which is calculated using Equation 4.20. Conventionally, counter–clockwise direc- tion is chosen as positive. In the second step, the contribution of each circular region is calculated and added to the area of the polygon following the same positive direction convention (Figure 4.16c). The area of the circular region is the area of the pie shaped region minus the triangular region, as shown in Figure 4.15. The circular region’s area is calculated as follows: Acircular = ApieAtriangle Acircular = q 2p pr 2 r2 sinq 2 Acircular = r 2 ✓ q sinq 2 ◆ (4.21) The total area can be expressed as the sum of the polygon area and the sum of the circular regions. Ac = Apolygon +Acircular Ac = 1 2 n Â i=1 (xiyi+1 xi+1yi)+ m Â j=1 ✓ r2j ✓ q j sinq j 2 ◆◆ (4.22) where n is the number of vertices in the polygon and m is the number of arc elements in the polycurve. Due to the positive direction convention, the uncut chip area can be negative. Since there are no self–intersections in the engagement profile, the absolute value of the uncut chip area is used. 4.5.2.2 Cutting edge length Edges of the engagement profile are marked as originating from either the tool profile or the in– process workpiece. Edges originating from the tool profile constitute the cutting edge currently in 75 Edges originating from Cutting Tool Edges originating from In-Process Workpiece Figure 4.17: Entities originating from tool curve are in cut cut, therefore, finding the length of the cutting edge is just a matter of calculating the total length of these edges. For a linear edge, the coordinates of the end points are known, therefore the edge length is calculate as follows: Lline = q (x1 x0)2+(y1 y0) (4.23) The length of an circular edge is calculated by using the limiting angles and the radius values. Larc = (q1q0)r (4.24) 4.5.2.3 Approximate chord angle The approximate chord is the major diagonal of the uncut chip area [15]. The approximate chord angle (q) is the angle between the approximate chord and the positive Z axis of the machine tool coordinate system. However, during software implementation, this definition is insufficient because the direction of the effective chord is undetermined. To completely define the approximate chord, a convention for its direction is assumed. The orientation of the chord is given by the line connecting the vertices of the engagement profile that are furthest away from each other, and the direction is chosen, so that it forms an acute angle with the feed vector (Figure 4.18). 76 fcfc fc fc Ce! Ce! a) Cylindrical Turning b) Radial Turning X Z e e Figure 4.18: Direction convention for the effective chord vector { fc} . Ceff > 0 (4.25) where, { fc} is the feed vector and {Ceff} is the approximate chord vector. This convention ensures that the approximate chord vector points away from the nose of the tool, allowing the chord angle (q) to be calculated systematically. 4.6 Parsing NC-Code NC-codes are generated by CAM systems in a machine independent form, namely the APT files. The machine independent codes are converted to machine specific G-codes using post–processors. Turning simulations have the ability to use either APT or G-codes. 77 Load Text File Commands Command & Value Pair Value Motion Interpolation Motion Type Value Current Machine Status Read Line Update Machine Status Scan Key and Values Simulate Motion End of Line End of File Command No No Yes Yes Modal Position End Figure 4.19: Flow–chart for NC-code parser algorithm For virtual turning, G-codes are chosen as the user input. A basic, single purpose parser, which reads NC programs stored as text files, is developed for the GN6 series controller on the Hardinge Superslant lathe. The flow–chart for the parser algorithm is shown in Figure 4.19. Compatible commands, and expected values are given in Section A.3. 78 4.6.1 Spindle Speed and Cutting Speed Spindle speed (n) is the rotational speed of the machine tool spindle, usually expressed in revolu- tions per minute. Cutting speed (Vc), or the surface speed is the tangential velocity of the rotating workpiece at the tool contact point. Modern CNC lathes support both, Constant Spindle Speed and Constant Surface Speed modes. Constant spindle speed mode When the constant spindle speed mode is active, the controller keeps rotational speed of the spindle constant. In contour operations, the tool is free to move in both axial and radial directions. Since the cutting speed is dependent on the tool location in radial axis, it should be calculated for each step of the simulation. Cutting speed is calculated as follows: Vc = npd = npxavg = np (xstart + xend) 2 (4.26) where Vc is cutting speed in [mm/min], n is spindle speed in [rpm]. xstart and xend are the starting and ending locations of the tool in the radial direction of the machine tool coordinate, given in diameter coordinates. Constant surface speed mode The CNC controller adjusts the spindle speed to provide a constant cutting speed. Rearranging Equation 4.26, the spindle speed is calculated for a given tool position as follows: 79 n= pd Vc n= pxavg Vc n= p(xstart + xend) Vc (4.27) 4.7 Simulation Schemes Turning simulations require different levels of complexities for different processes. It is possible for the tool–workpiece engagements to remain constant, or vary in time depending on the tool motion and workpiece geometry. In addition, workpieces can have non axi–symmetric features. Therefore, simulation schemes that allow for a variable level of complexity in virtual turning are proposed. 4.7.1 Constant Engagement During linear cutting motions, it is possible that the tool–workpiece engagement remains constant, therefore, it is redundant to simulate the cutting process at spindle period steps. The entire linear motion command can be simulated by subtracting the tool swept region from the in–process workpiece, which allows the process simulation to be run once for the entire NC- command [9, 5]. The use of tool swept region also generates an in–process workpiece with a smaller number of elements, reducing the memory requirements (Figure 4.9). The algorithm to evaluate tool–workpiece engagement is detailed as follows (Figure 4.20): 1. For each motion command in the NC-Code, the motion is discretized into spindle period simulation steps. However, the cutting forces are calculated only at every nth simulation step, where n is a user-defined value, representing how small or big the time step of the simulation is. 2. After cutting forces are calculated, the area swept by the tool in the next n steps is generated. 80 Step I) Force Calculation Total of swept region for the simulation intervalObtained by removing the swept region from the workpiece. Will not generate high number of cusp regions Step III) In-Process Workpiece Generation Step II) Swept Region Generation Calculated at chosen simulation step intervals Figure 4.20: Simulation scheme for constant engagement 3. The in–process workpiece is generated by using boolean operations between the workpiece and the generated swept area. The process is repeated until the entire tool motion is simu- lated. 4.7.2 Non Axi–symmetric Workpiece Simulation of a workpiece having non axi–symmetric features requires simulation steps smaller than the spindle period to capture the changes within one revolution of the spindle. This is achieved by running the process simulation on multiple simulation planes (Figure 4.21). Each of these simulation planes represents the machining–plane as the spindle rotates. A separate 81 Generator Profile Simulation Planes Figure 4.21: Multiple simulation planes per spindle revolution in–process workpiece profile is stored in memory for each simulation plane, increasing the total computational time linearly . 4.8 Summary In this chapter, a new simulation model based on two dimensional boolean operations is developed. Polygon representation is expanded to include circular edges. This new representation is used to define tool and workpiece cross–section profiles on the machining plane. Tool–workpiece engagement is calculated as the intersection of tool and workpiece profiles, while the in–process workpiece is defined as their difference. The boolean operations on the pro- posed geometric representation are developed. Controller behavior is simulated by parsing the NC program. At each simulation step, tool location is obtained, and engagement profile is calculated by the new tool–workpiece engagement model. Also, simulation schemes which allow the adjustment of simulation resolution is presented. 82 5 Application and Experimental Verification A process model for contour turning is presented in Chapter 3, and simulation models for virtual turning are presented in Chapter 4. A virtual turning simulation, which integrates these models, is developed into a software application. In the first section of the chapter, the details of the computer implementation is presented. In the following sections, the validity of the virtual turning application is verified experimen- tally. A sample part with a free–form surface geometry is machined to verify the tool–workpiece engagements, cutting forces, and feedrate scheduling. Additionally, face turning experiments are conducted to verify the chatter detection capabilities. Process simulations are created in the virtual turning application, and the simulated results are compared against the measurements. 5.1 Virtual Turning Simulation The virtual turning application consists of six modules, which are developed in Matlab environ- ment. Generated data is stored and transferred between the modules using the native data format (.mat files). Simple graphical user interfaces for each module are built using the GUIDE toolbox. Core functions of the system are converted into C language using the CODER toolbox to increase the computational performance. The structure of the virtual turning application is illustrated in Figure 5.1. The following core Matlab modules represent the developed virtual turning system. 83 VTStl generates the generator curve representation from the solid workpiece geometry, as detailed in Section 4.3. Solid model is reoriented, and intersected with the machining plane to obtain the cross–section. VTTool module generates the tool data from the user input, as detailed in Section 4.4. Tool data includes the cutting edge contour and the oblique angles of the tool holder. NCLoader interprets the NC programs written with G-codes, as detailed in Section 4.6. VTTurning generates the tool–workpiece engagement maps, using boolean algorithms as pre- sented in Section 4.5. Engagement map data includes the chip load geometry and the cutting conditions. VTTurning module is responsible for updating the in–process workpiece geom- etry during the cut. Visualization of the in–process workpiece, tool motion and engagement profiles, are also handled. VTResult module calculates the cutting forces, power and torque requirements, and determines the chatter stability using the process model presented in Chapter 3. OptiSearch module improves the feeds and speeds, using the initial engagements generated by the virtual turning system. 5.2 Contour Turning Experiments 5.2.1 Experimental Setup Experiments are conducted on a Hardinge Superslant CNC lathe, which is controlled by a GN 6 Series Fanuc controller. Cutting forces are measured with a three axes Kistler 9121 dynamometer installed on the turret of the lathe. 5.2.1.1 Cutting tools and tool holders Sandvik Coromant VNGA 160408 inserts on a DVJNL 2020K 16 tool holder is used for the con- tour turning experiments. The specifications of the tools and holders are presented in Table 5.1. 84 Tool Parameters Tool Holder Parameters Tool Edge Creation Orthogonal to Oblique Transformation Tool Data Tool Model Input VTTool Output STL data Reorientation Solid-Plane intersection Generator Curve Representation Workpiece Model Input VTStl Output Boolean Operations Motion Interpolation Engagement maps Updated Workpiece Engagement Model Tool Data Tool Locations Cutting Parameters In-process Workpiece Input VTTurning Output Process Model Force Power Torque Chatter Stability Contour Turning Process Model Engagement Maps Structural Flexiblity Material Model Chatter Detection Input VTResult Output Engagement Modi!cation Feeds and Speeds Engagement Feedrate Scheduling Initial Engagement Maps Parameter Boundaries Verify Constraints Input OptiSearch Output Recommended Depth of Cut Text File Parse G-code Tool Locations NC Parser Cutting Parameters Input NCLoader Output Figure 5.1: Structure of the virtual turning application 85 R5 R15 R20 14 45o 45o A B C D EF Figure 5.2: Initial workpiece geometry for contour turning experiments Workpiece material: AISI 1045 Steel Table 5.1: Geometric specifications of the tools used in experiments Nose Edge Approach Side Rake Back Rake Radius Length Angle Angle Angle Cutting Tool Tool Holder rn l kr a f ap CNMA120404 DCLNL 2020K 12 0.4 mm 12 mm 5 6 6 CNMA 120408 DCLNL 2020K 12 0.8mm 12mm 5 6 6 CNMA120416 DCLNL 2020K 12 1.6 mm 12 mm 5 6 6 VNGA 160408 DVJNL 2020K 16 0.8 mm 16 mm 3 4 13 5.2.1.2 Workpiece The initial workpiece is a free–form, AISI 1045 steel part. Initial workpiece geometry is shown in Figure 5.2. Orthogonal cutting data for the AISI 1045 steel is given in Section A.2. The part geometry includes six geometrical features as shown in Figure 5.2: A) Convex circular interpolation 86 B) Linear cut with negative slope C) Concave circular interpolation D) Linear cut with positive slope E) Convex circular interpolation with small radius (Fillet) F) Straight axial cut Workpiece geometry and the tool–paths are generated in UGS NX software. For Vericut simula- tions, geometric models are saved in IGES format, while in virtual turning simulations the STL data format is used. 5.2.1.3 Cutting parameters Cutting parameters for contour turning experiment are chosen as ac = 0.5 mm fc = 0.1 mm rev Vc = 120 m min (5.1) Machining is performed in constant surface speed mode, therefore, the cutting speed (Vc) is controlled by the CNC, while the spindle speed (n) varies during the process. 5.2.2 Tool–Workpiece Engagements Tool–workpiece engagement maps are generated using both the Vericut API, and the boolean– based engagement model presented in Section 4.5. The boolean model is verified by comparing the engagement maps from two models. Figure 5.3 shows that the models generate very similar engagement maps. Small discrepancies emerge due to the difference in geometric representation, and tool–motion interpolation. Both models are found to generate the engagement maps of the simulated process accurately, however, the boolean–based model is significantly faster. Computation times required by the mod- 87 Table 5.2: Computational time required by the engagement models Vericut API Boolean based Number of engagement maps 776 783 Computation time 90 sec 10 sec els on a standard desktop computer are compared in Table 5.2. The proposed polycurve tool–part intersection method is nine fold faster in comparison to existing commercial code. 5.2.3 Cutting Forces Cutting forces are simulated using the virtual turning application. Simulated forces match the experimental data closely, as shown in Figure 5.4. 5.2.4 Feedrate Scheduling The cycle time is reduced by using the proposed engagement–based feedrate scheduling method detailed in Section 3.6. Workpiece, cutting tool, cutting parameters and the experimental setup are given in Section 5.2.1. 5.2.4.1 Constraints Maximum cutting force, torque and power constraints are applied. Fmax = 500N Tmax = 20Nm Pmax = 1000W 5.2.4.2 Results The highest, feasible feedrate for each NC command (Figure 5.2) is searched. The results are shown in Table 5.3. The total cycle time has been reduced from 33 seconds to 8 seconds after the scheduling oper- ation, resulting in a simulated productivity improvement of approximately four times. The simulated cutting forces, power and torque values are shown in Figure 5.5. The simula- 88 ï ï ï ï 13 (QJDJHPHQW0DSV ï ï ï ï ï ï ï ï ï ï 7 (QJDJHPHQW0DSV ïïïï ï ïïï 19 (QJDJHPHQW0DSV Vericut Polycurve Z [mm] X [mm] Figure 5.3: Comparison of engagement model with Vericut Solution See Figure 5.2 for the part geometry. Cutting conditions: ac = 0.5mm fc = 0.1mm/rev Vc = 120m/min 89 20 25 30 35 40 45 50 55 −200 0 200 400 Time [s] Cu tt in g Fo rc e F z [N ] 20 25 30 35 40 45 50 55 −400 −200 0 200 Cu tt in g Fo rc e F y [N ] 20 25 30 35 40 45 50 55 −100 0 100 200 300 Cu tt in g Fo rc e F x [N ] Experiment Simulation EB CA D F Figure 5.4: Cutting forces during contour turning See Figure 5.2 for the part geometry. Workpiece material: AISI 1045 Steel. Cutting conditions: ac = 0.5mm fc = 0.1mm/rev Vc = 120m/min Table 5.3: Suggested feedrate values for contour turning experiment Region Feedrate [mm/rev] A 0.41 B 0.41 C 0.37 D 0.42 E 0.46 F 0.43 90 0 5 10 15 20 25 30 35 40 −400 −200 0 Cu tt in g Fo rc e F y [N ] −600 Flim Feedrate Scheduling Initial 0 5 10 15 20 25 30 35 40 −1200 −1000 −800 −600 −400 −200 0 Po w er [W ] 0 5 10 15 20 25 30 35 40 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 Simulation Time [s] To rq ue [N m ] Figure 5.5: Comparison of cutting forces before and after optimization Initial; see Figure 5.4 for cutting conditions. After feedrate scheduling; see Table 5.3 for the feedrate values. tion shows that tangential cutting force and the power is maximized without violating the chosen constraints. 91 5.3 Chatter Stability Experiments A series of face turning experiments verify the stability solution, which is presented in Section 3.5. The frequency response of the flexible structure is obtained experimentally by impact hammer test- ing. The engagements and stability solution by Nyquist criterion are simulated by virtual turning application. 5.3.1 Experimental Setup Sandvik Coromant CNMA 120408 KR inserts on a DCLNL 2020K 12 tool holder is used. Tool specifications are given in Table 5.1. Workpiece is an AISI 1045 steel bar, with a diameter of 41.275 mm, and an initial length of 150 mm. Three facing experiments are conducted with increasing depth of cut values. The cutting pa- rameters for the experiments are given in Table 5.4. Table 5.4: Radial turning chatter experiments Workpiece: AISI 1045 Steel. See Figure 5.6 for workpiece FRF Experiment n Vc fc ac Chatter [rpm] [m\min] [mm/rev] [mm] [Hz] Radial 1 Variable 75 0.1 1 No Chatter Radial 2 Variable 75 0.1 2 No Chatter Radial 3 Variable 75 0.1 3 800 Radial 4 Variable 75 0.1 4 800 92 5.3.2 Structural Flexibility 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 −5 −4 −3 −2 −1 0 1 x 10 -7 Frequency [Hz] Im ag in ar y Co m po ne nt [m /N ] 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000−10 −8 −6 −4 −2 0 2 x 10 -7 Re al C om po ne nt [m /N ] 150mm 140mm Figure 5.6: Direct FRF of the workpiece with different overhang values. AISI 1045 Steel bar with D= 41.275mm In this set of experiments the tool holder assembly is rigidly clamped. Since the major cutting forces in facing operations act along the rigid tool holder shaft, this component is assumed to be fully rigid. The workpiece is clamped with an overhang to increase the flexibility. Workpiece geometry is given as Dworkpiece = 41.275mm Linitial = 150mm Lfinal = 140mm (5.2) 93 The flexibility of the workpiece is measured at the end of the long workpiece, where the cutting takes place. Since the overhang of the workpiece is reduced from 150mm to 140mm during the experiments, its FRF is measured both before and after the cut, as shown in Figure 5.6. The change in the dynamic response of the workpiece is found to be negligible, therefore, the measurements from the workpiece having 140mm overhang is used to simulate the process. The dynamics of the workpiece are assumed to be identical in X and Y directions. The work- piece is assumed to be rigid along the Z direction (spindle axis). For simplicity, the cross transfer functions were not considered in these experiments. 5.3.3 Results In experiment #1 (Figure 5.7), FFT of the microphone signal does not contain any vibration fre- quencies, and the cutting forces are stable during the cut. No chatter instability is observed. This result verifies the virtual turning simulation, which does not indicate chatter for this process. In experiment #2 (Figure 5.8), simulation predicts small regions of instability, however, these regions are too small for chatter to develop. Between the 7-8 seconds, no chatter is predicted by the simulation, as verified by the amplitude spectrum of the microphone signal. Around the regions where chatter is simulated, the FFT of the microphone signal reveals vibrations around 800 Hz, but no growth of the cutting forces is observed. It is concluded that there are no chatter vibrations, in accordance with the simulation results. In experiment #3 (Figure 5.9), the simulation predicts chatter around 780 Hz, all along the cut. FFT of the microphone signal reveals vibrations around 800Hz, which corresponds to the natural frequency of the first bending mode of the workpiece. The cutting forces slightly grow at several regions of the cut. It is concluded that, there are slight chatter vibrations during this cut. In experiment #4 (Figure 5.10), the microphone signal reveals chatter around 800 Hz, along with growing cutting forces. 800Hz is the first natural frequency of the workpiece as shown in Figure 5.6. Under these cutting conditions the process is unstable, as predicted by the virtual turning simulation. 94 4 5 6 7 8 9 10 11 12 13 14 −600 −400 −200 0 200 400 Cu tt in g Fo rc e F y [ N ] Frequency (Hz) 1000 2000 3000 4000 5000 60000 0.2 0.4 0.6 0.8 Single−Sided Amplitude Spectrum of y(t) Range:8>t>10 |Y (f) | 4 5 6 7 8 9 10 11 12 13 140 10 20 30 M ic ro ph on e Simulated Stable Simulated Force Experimental Force Time [s] Figure 5.7: Radial turning experiment #1 Facing process of AISI 1045 steel bar. Workpiece diameter D= 41.275 Cutting conditions: ac = 1.0mm fc = 0.1mm/rev Vc = 75m/min 95 4 5 6 7 8 9 10 11 12 13 14 −1000 −500 0 500 1000 Cu tt in g Fo rc e F y [N ] 1000 2000 3000 4000 5000 6000 0 0.5 1 1.5 2 2.5 Single−Sided Amplitude Spectrum of y(t) Range:13>t>13.5 Frequency (Hz) 1000 2000 3000 4000 5000 6000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Single−Sided Amplitude Spectrum of y(t) Range:7>t>8 Frequency (Hz) |Y (f) | 4 5 6 7 8 9 10 11 12 13 14 5 10 15 20 25 30 35 M ic ro ph on e Simulated Stable Simulated Chatter Simulated Force Experimental Force Time [s] Figure 5.8: Radial turning experiment #2 Facing process of AISI 1045 steel bar. Workpiece diameter D= 41.275 Cutting conditions: ac = 2.0mm fc = 0.1mm/rev Vc = 75m/min 96 4 5 6 7 8 9 10 11 12 13 ï 0 2000 Cu ttin g Fo rc e F y [N ] 1000 2000 3000 4000 5000 60000 0.5 1 1.5 6LQJOHï6LGHG$PSOLWXGH6SHFWUXPRI\W Range:4>t>5 )UHTXHQF\+] _< I _ 1000 2000 3000 4000 5000 60000 0.5 1 1.5 2 6LQJOHï6LGHG$PSOLWXGH6SHFWUXPRI\W Range:9>t>10 )UHTXHQF\+] _< I _ 4 5 6 7 8 9 10 11 12 13 ï 0 50 100 0 LF UR SK RQ H 6LPXODWHG6WDEOH 6LPXODWHG&KDWWHU 0HDVXUHG)RUFH 6LPXODWHG)RUFH Time [s] Figure 5.9: Radial turning experiment #3 Facing process of AISI 1045 steel bar. Workpiece diameter D= 41.275 Cutting conditions: ac = 3.0mm fc = 0.1mm/rev Vc = 75m/min 97 Frequency (Hz) 4 5 6 7 8 9 10 11 12 13 −3000 Cu tt in g Fo rc e F y [N ] −1000 −2000 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 10 Single−Sided Amplitude Spectrum of y(t) Range:10>t>12 |Y (f) | 4 5 6 7 8 9 10 11 12 13 0 100 200 M ic ro ph on e −100 Simulated Stable Simulated Chatter Measured Force Simulated Force Time [s] Figure 5.10: Radial turning experiment #4 Facing process of AISI 1045 steel bar. Workpiece diameter D= 41.275 Cutting conditions: ac = 4.0mm fc = 0.1mm/rev Vc = 75m/min 98 5.4 Summary A virtual turning application is developed in Matlab environment. Machining simulations are con- structed using the developed modules and the results are compared against experiments. Contour turning experiments are conducted to verify cutting force simulations. The engagement maps are generated accurately by both the third party API and the developed boolean based model, while the latter proved to be nine fold faster. It was shown that the cutting force simulations are sufficiently accurate. Using the engagement based optimization method the machining process was optimized. The machining performance was almost tripled, while satisfying the machine tool and process lim- itations. Also the chatter detection capabilities of the virtual turning application is verified by a series of face turning experiments. 99 6 Conclusion A computationally efficient virtual turning process simulation based on a new tool–workpiece en- gagement model is introduced in this thesis. The system has tool–workpiece engagement, process mechanics and dynamics, and feedrate scheduling modules which constitute the main contributions of the thesis. Tool–workpiece engagement boundaries along the segmented tool path needs to be calculated in order to solve the equations which describe the process physics. While solid model based algo- rithms are most accurate, they take considerable amount of computation time which makes them impractical to use in virtual turning. A typical turning of a gas turbine shaft may require over 100,000 tool-part engagement calculations, which can take hours unless computationally efficient models are introduced. It has been more common to use z–buffer models which are computation- ally faster but the engagement conditions are not identified accurately because the solid models are digitized at discrete mesh points. Since the turning parts are modeled by the revolution of their cross–sections, a simple but efficient polycurve method is introduced in the thesis. Workpiece and tool edge intersection boundaries are analytically calculated by Boolean operations. Using two dimensional analytical geometry allows nine fold increase in computational efficiency while improving the geometric accuracy in comparison to popular z-buffer methods. With the best of author’s knowledge, this thesis introduced the application of polycurve method first time in the virtual machining literature. The process forces must also be rapidly calculated in virtual machining. Past research iden- tified the cutting force distribution along the cutting edge, which required computationally costly digital integration of differential force distribution. Without sacrificing from the accuracy of force 100 prediction, this thesis applied Colwell’s chord angle and line approximation in solving the mechan- ics of cutting. The method has proven to predict the varying forces along a part, while avoiding costly and slow iterative methods. Feeds and speeds of the cutting process are improved while respecting torque, power, force and depth of cut limits. 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Goldberg, “What every computer scientist should know about floating-point arithmetic,” ACM Computing Surveys, 1991. [38] E. W. Weisstein. Line-Line Intersection. [Online]. Available: http://mathworld.wolfram. com/Line-LineIntersection.html [39] ——. Circle-Circle Intersection. [Online]. Available: http://mathworld.wolfram.com/ Circle-CircleIntersection.html [40] ——. Cirlce-Line Intersection. [Online]. Available: http://mathworld.wolfram.com/ Circle-LineIntersection.html 104 Appendices A.1 Intersection Algorithms Using two entity types there are three possible intersection types that may occur in the geometry. • Line – Line Intersection • Line – Arc Intersection • Arc – Arc Intersection Simple geometric algorithms for these intersections are widely available and relatively simple. A.1.1 Line - Line Intersection The intersection of the two lines are calculated [38]. Let two lines be defined by two points. L1 L2P1(x1,y1) P3(x3,y3) P2(x2,y2) P4(x4,y4) Pint(xint,yint) Figure A.1: Line-Line intersection 105 P1 = (x1,y1) P2 = (x2,y2) P3 = (x3,y3) P4 = (x4,y4) (A.1) Writing the linear equations in the general form the lines are represented by L1! A1x+B1y=C1 L2! A2x+B2y=C2 (A.2) where A1 = y2 y1 B1 = x1 x2 C1 = A1x1+B1y1 A2 = y3 y3 B2 = x3 x4 C2 = A2x3+B2y3 (A.3) Two line equation can now be written in the matrix form as 264A1 B1 A2 B2 375 264x y 375= 264C1 C2 375 (A.4) Solving for x and y 264x y 375= 264A1 B1 A2 B2 375 1264C1 C2 375 (A.5) 264x y 375= 1A1B2A2B1 264 B2 B1 A2 A1 375 264C1 C2 375 (A.6) Therefore the point of intersection is 106 r1 P1 P2 P3 P4,1 P4,2 r2 a h b Figure A.2: Circle-Circle intersection Pint = 264B2C1B1C2D A1C2A2C1 D 375 (A.7) with D= A1B2A2B1 (A.8) A.1.2 Circle–Circle Intersection The intersection points of two circles is calculated for two circles with centers P1(x1,y1) and P2(x2,y2) and radii r1 and r2 respectively [39]. The distance between two circles d =||P2P1|| = q (x2 x1)2+(y2 y1)2 (A.9) 107 Depending on this value the intersection conditions are checked d r1+ r2! No Intersection d < |r1 r2|! No Intersection d = 0^ r1 = r2! Cirlces Coincident Using the triangle r21 = h 2+a2 (A.10) r22 = h 2+b2 (A.11) Since a+b= d (A.12) b= da (A.13) it follows that a= (r22 r21d2) 2d (A.14) h= q r21a2 (A.15) ) P3 = (P2P1)ad +P1 (A.16) ) P4 = 264x3± hd (y2 y1) y3⌥ hd (x2 x1) 375 (A.17) 108 e e (x1,y1) y x (xint1,yint1) (xint2,yint2) (x2,y2) Figure A.3: Line-Circle intersection A.1.3 Line - Arc Intersection Intersection of a line and circle is calculated [40]. Let L1 be the line defined by its two points. P1(x1,y1) P2(x2,y2) and the circle centered at the origin with a radius of r. 109 dx = x2 x1 (A.18) dy = y2 y1 (A.19) dr = q d2x +d2y (A.20) D= x1 x2 y1 y2 = x1y2 x2y1 (A.21) The discriminant D⌘ r2d2r D2 (A.22) defines the incidence such as D< 0! No Intersection D= 0! Tangent D> 0! Intersection at two discrete points and the coordinates of the intersection point is given by x= Ddy± sgn⇤(dy)dx p r2d2r D2 d2r (A.23) y= Ddx±|dy| p r2d2r D2 d2r (A.24) where the signum function is described as sgn⇤(x) 8>><>>: 1 for x< 0 1 otherwise 110 A.1.4 Extension of Intersection Algorithms for Line Segments and Arcs Since the intersection algorithms are defined only for complete circles and infinite lines, the cal- culated intersection points must be checked to make sure they lie on the line segments or arcs represented by the entities. This can be accomplished by verifying the intersection point Pint be- longs to both entities. Pint ⇢ Entity 1^Pint ⇢ Entity 2 (A.25) A.1.4.1 Line segments The calculated intersection point Pint(x,y) lies on the line segment if and only if min(x1,x2) xmax(x1,x2) min(y1,y2) xmax(x1,x2) (A.26) For the intersection of two line segments, the intersection point must lie on both line segments to be a valid intersection point. Therefore using the equation above, min(x1,x2) xmax(x1,x2)^min(y1,y2) ymax(y1,y2)^ . . . min(x3,x4) xmax(x3,x4)^min(y3,y4) ymax(y3,y4) (A.27) which can be simplified into max [min(x1,x2),min(x3,x4)]xmin [max(x1,x2),max(x3,x4)] ^ max [min(y1,y2),min(y3,y4)]ymin [max(y1,y2),max(y3,y4)] (A.28) 111 A.1.4.2 Arc The arc entities are defined by a simple circle entity, and the corresponding start and end angles (qi and q f ) respectively. Once the intersection point is located, the corresponding angular position of the intersection point can be calculated by atan2 function. Dx= (x xo) Dy= (y y0) q = tan1 ✓ Dy Dx ◆ (A.29) The point Pi lies on the arc if and only if qi q q f For Line-Arc intersection, this condition must be checked along with the check for the line segment. For Arc-Arc intersection the same condition must be satisfied by both arcs for the inter- section point to be valid. 112 A.2 AISI 1045 Cutting Coefficients 50 100 150 0.1 0.2 0.3 450 500 550 600 Cutting Speed Vc [m/min] Cutting Speed Vc [m/min] Feedrate fc [mm/rev] Feedrate fc [mm/rev] 6 KH DU 6 WU HV V Ѭ s Re lat ive E rro r [ % ] Experimental Fitted Values 50 75 100 150 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.3 -15 -10 -5 0 5 10 15 20 Figure A.4: Quadratic surface fit for shear stress (AISI 1045 Steel) Physics of an oblique turning process can be defined by using parameters such as shear stress (ts), friction angle (ba) and shear angle (fc). These values can be estimated by using data stored in orthogonal cutting databases. For this purpose, the orthogonal cutting data are represented commonly as a function of cutting parameters. For AISI 1045 steel, these values are obtained by varying the chip thickness (hc) and cutting 113 50 100 150 0.1 0.15 0.2 0.25 0.3 25 30 35 40 Fr ict ion A ng le ћ a 50 75 100 150 0.08 0.1 0.120.140.160.18 0.2 0.3 -15 -10 -5 0 5 10 15 20 Experimental Fitted Values Cutting Speed Vc [m/min] Feedrate fc [mm/rev] Cutting Speed Vc [m/min] Feedrate fc [mm/rev] Re lat ive E rro r [ % ] Figure A.5: Quadratic surface fit for friction angle (AISI 1045 Steel) speed (Vc). ts = f (hc,Vc) ba = f (hc,Vc) (A.30) fc = f (hc,Vc) 114 50 100 150 0 0.1 0.2 0.3 25 30 35 Sh ea r A ng le є c 50 75 100 150 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.3 -10 -5 0 5 Experimental Fitted Values Cutting Speed Vc [m/min] Feedrate fc [mm/rev] Re lat ive E rro r [ % ] Cutting Speed Vc [m/min] Feedrate fc [mm/rev] Figure A.6: Quadratic surface fit for shear angle (AISI 1045 Steel) These parameters can be modeled as a quadratic surface as given in Equation A.31 by surface fitting over the experimental data. The coefficients in these equations are calibrated directly from orthogonal cutting experiments [3]. [ts,ba,fc] =C1+C2V 2c +C3Vc+C4h2c +C5hc (A.31) These coefficients are obtained using a least squares method. The obtained results are shown in 115 Figure A.4, Figure A.5, Figure A.6. The coefficients for the fitted surfaces are given in Table A.1. Table A.1: Coefficients for fitted surfaces C1 C2 C3 C4 C5 Error ts 639.03 0.0152 -3.094 30.34 153.04 4.81% fc 15.82 -0.0004 0.13 1.67 25.47 3.53% ba 37.49 0.0006 -0.20 -173.98 72.43 4.82% A.2.1 Cutting Edge Coefficients Cutting edge coefficients, which account for the portion of cutting forces which are not related to the shearing process, are also calibrated by cutting experiments. However, unlike cutting coeffi- cients which are modeled to be only workpiece material dependent, edge coefficients are specific to the workpiece–tool material pair [14]. For AISI 1045 steel two sets of experiments were performed in previous studies. The first set of experiments, have been conducted with Sandvik GC3205 grade inserts. The results are given in Table A.2 [2]. Table A.2: Cutting coefficients for AISI 1045 steel with GC3205 grade tool [2] n Vc Kf e Kfa Kre Kra Kte Kta [rpm] [m/min] [N/mm] [MPa] [N/mm] [MPa] [N/mm] [MPa] 100 12 50 1816 10 -25 43 3512 200 25 28 2458 0 -5 32 3776 400 49 0 4359 -2 188 10 4811 600 74 31 3364 -5 172 25 4152 800 99 57 2509 10 53 37 3623 1000 123 77 1937 23 -52 46 3235 1250 154 79 1773 28 -81 46 3102 1500 185 81 1639 26 -85 39 3294 1750 216 81 1580 27 -115 44 2937 2000 247 83 1544 26 -124 44 2881 2250 278 81 1619 35 -219 41 2929 2500 308 80 1698 44 -293 37 3042 The second set of experiments were conducted by Centmayer who used AISI 1045 Steel work- 116 0 50 100 150 200 250 300 350 −20 0 20 40 60 K r e Cutting Speed (Vc) 0 50 100 150 200 250 300 350 0 50 100 150 K t e 0 50 100 150 200 250 300 350 0 100 200 Comparison of Cutting Edge Coe"cients K f e KC8050 GC3205 2 Figure A.7: Comparison of cutting edge coefficients for AISI 1045 steel piece with Kennametal KC8050 grade cutting tools. Experimental results are presented in Ta- ble A.3 [3]. Since this grade of tool is very similar to the Sandvik GC6050, these coefficients are assumed to be applicable to the GC6040 grade as well. It can be seen that the cutting edge coefficients obtained from each study is very different as shown in Figure A.7. Therefore it is necessary to run cutting tests for each workpiece and tool 117 Table A.3: Cutting coefficients for AISI 1045 steel with KC8050 grade tool [3] Vc Kf e Kte Kre [m/min] [N/mm] [N/mm] [N/mm] 50 35.12 58.033 -1.2361 75 154 118.69 -8.4721 100 147.21 123.93 -9.2675 150 136.72 101.97 -9.3731 material combination. The edge coefficients are also sensitive to the tool coating, tool wear and chipping. In the orthogonal databases, data for commonly used workpiece - tool material pairs can be stored. Such an process will unavoidably, increase the number of orthogonal cutting tests required to construct an orthogonal database. A.2.2 Curve Fitting In order to store these experimental data in a format which can be used in a simulation system, the acquired data is fitted with a curve. However a polynomial curve fitting does not yield satisfactory representation of the cutting coefficients. Therefore a function which can represent the overall shape of the curve was chosen. The fitted curve is in the form of an arctangent function as shown in Equation A.32 Ke =C1 atan ✓ VcC2 C3 ◆ +C4 (A.32) The experimental data (Table A.2 and Table A.3) are fitted by using a least squares method. In order for the fitted curve to preserve the overall shape of the experimental values, intermediate data points are created by interpolation between the given data points. Also, the cutting edge coefficients were assumed to reach a value of zero when the cutting speed goes to zero. 118 0 50 100 150 200 250 300 350 −20 0 20 40 60 K r e Cutting Speed (Vc) [m/min] 0 50 100 150 200 250 300 350 0 20 40 60 K t e 0 50 100 150 200 250 300 350 −50 0 50 100 Curve Fitting for AISI 1045 Steel with GC3205 insert K f e Experimental Data Fitted Curve Figure A.8: Curve fitting for cutting edge coefficients (GC3205 grade tool and AISI 1045 Steel) (Ke)Vc=0 = 0 (A.33) The resulting fitted curves are compared to the experimental values in Figure A.8 and Figure A.9 for GC3205 and KC8050 grades respectively. The coefficients for each tool grade are found as 119 0 20 40 60 80 100 120 140 160 180 200 −15 −10 −5 0 5 Cutting Speed (Vc) [m/min] K r e [ N /m m ] 0 50 100 150 0 20 40 60 80 100 120 140 160 180 200 K t e [ N /m m ] 0 20 40 60 80 100 120 140 160 180 200 0 100 200 Curve Fitting for AISI 1045 Steel with KC8050 insert K f e [ N /m m ] Experimental Data Fitted Curve Figure A.9: Curve fitting for cutting edge coefficients (KC8050 grade tool and AISI 1045 Steel) shown in Table A.4 and Table A.5 respectively. Table A.4: Coefficients for edge coefficient of AISI 1045 steel with GC3205 grade tool C1 C2 C3 C4 Kfe 14.060 69.967 12.979 21.711 Kte 31.833 78.128 20.653 35.946 Kre 12.642 107.350 19.143 13.690 120 Table A.5: Coefficients for edge coefficient of AISI 1045 steel with KC8050 grade tool C1 C2 C3 C4 Kfe 46.486 58.968 4.967 76.219 Kte 39.804 50.355 7.883 57.179 Kre -27.267 -102.09 110.65 21.059 121 A.3 NC-Commands for Hardinge Superslant Lathe The NC-commands and the required data formats for the Hardinge Superslant lathe are obtained from the user manual. The commands listed in Table A.6 are implemented within the NC-parser algorithm. Table A.6: NC-commands implemented in the G-code parser Command Value Type Definition G 00 Modal Rapid Motion G 01 Modal Linear Motion G 02 Modal CW Circular Interpolation G 03 Modal CCW Circular Interpolation G 96 Modal Constant Surface Speed Mode X XXX.xxx Position X–coordinate in diameter mode Z XXX.xxx Position Z–coordinate I XXXX.xxxx Position Circular interpolation arc center (X incremental) K XXXX.xxxx Position Circular interpolation arc center (Z incremental) F XXX.xxx Modal Feedrate S XXXX Modal Spindle speed (or Cutting speed if CSS mode is active T XX Modal Tool selection 122
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