Mechanics and Dynamics of Multi-Point Threading ofThin-Walled Oil PipesbyMohammad Rezayi KhoshdarregiBSc. Mechanical Engineering, Sharif University of Technology, 2010MASc. Mechanical Engineering, The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)July 2017c©Mohammad Rezayi Khoshdarregi, 2017AbstractThe pipelines used in the offshore extraction of oil and gas are connected by threadedjoints. Any geometrical error or vibration marks left on the thread surface duringthe machining process can lead to stress concentration and fatigue failure of thejoint. Such instances in the past have led to massive oil leakage and environmentaldisasters.Threading is a form cutting operation resulting in wide chips with complex ge-ometries. Multi-point inserts used in mass production can have different customprofiles on each tooth. The chip thickness as well as the effective oblique cuttingangles, cutting force coefficients, and direction of local forces vary along the cut-ting edge. Since the tool moves one thread pitch over each spindle revolution, thevibration marks left by a tooth affect the chip thickness on the following tooth.Threading of oil pipes imposes additional complexities due to the flexural vibra-tions of thin-walled pipes, which lead to severe chatter instability.This thesis develops a novel and generalized model to formulate, simulate, andoptimize general multi-point threading processes. A systematic semi-analyticalmethodology is first proposed to determine the chip geometry for custom multi-point inserts with arbitrary infeed strategies. A search algorithm is developed tosystematically discretize the chip area along the cutting edge considering the chipflow direction and chip compression at the corners. The cutting force coefficientsare evaluated locally for each element, and the resultant forces are summed up overthe engaged teeth.Multi-mode vibrations of the tool and pipe are projected in the direction oflocal chip thickness, and the dynamic cutting and process damping forces are cal-culated locally along the cutting edge. A novel chip regeneration model for multi-point threading is developed, and stability is investigated in frequency domain usingNyquist criterion. The process is simulated by a time-marching numerical methodiibased on semi-discretization. An optimization algorithm is developed to maximizeproductivity while respecting machine’s limits. The proposed models have beenverified experimentally through real scale experiments.The algorithms are integrated into a research software which enables the indus-try to optimize the process ahead of costly trials.iiiLay SummaryPipelines used in offshore oil extraction are subject to severe loads and abrasiveenvironment, making them susceptible to failure and leakage. Investigations haveshown that the thread connection between the pipes is the weakest point in thepipeline. The accuracy and surface quality of the threads have direct impact on thereliability of the connection.The threads are generated by incrementally removing material from the pipeto get the final thread shape. This thesis studies the threading process of oil pipesand develops mathematical and physical models to explain the behaviour of thisoperation. The developed models can simulate the process ahead of costly trialsand recommend conditions to achieve highest quality, productivity, and safety ofthe process.ivPrefaceThis research has been defined and carried out at Manufacturing Automation Lab-oratory under supervision of Professor Yusuf Altintas to address the challenges ob-served in threading thin-walled oil pipes. The practical aspect of the problem wasput forward by the Research and Development Centre of TenarisTAMSA (Veracruz,Mexico), one of the major manufacturers of steel tubes for the global oil and gasindustry. Research methodologies were proposed by the author and approved bythe supervisor.• The cutting force validation experiments presented in Chapter 3 were mostlycarried out during author’s five-month industrial internship at Metal CuttingResearch Department at Sandvik Coromant, Sandviken, Sweden. The exper-iments were entirely planned and analyzed by the author, and the companyprovided the cutting tools and machines.• A concise version of Chapter 3 has been published in [1], “Rezayi Khoshdar-regi, M., and Altintas, Y., 2015, Generalized modeling of chip geometry andcutting forces in multi-point thread turning, International Journal of MachineTools and Manufacture, 98, pp. 21-32”. The manuscript was written by thefirst author and edited by the supervisor.• The experimental modal analyses and chatter tests presented in Chapter 5were carried out at the R&D center of TenarisTAMSA, Veracruz, Mexico.Experiment planning, measurements, and data analysis were performed bythe author, and the company provided the cutting tools and pipes.• Concise versions of Chapters 4 and 5 have been submitted (at the time ofdefence) as two related papers to ASME Journal of Manufacturing Scienceand Engineering, 1) “Khoshdarregi, M., and Altintas, Y., 2017, Dynamicsvof multi-point threading: Part I: general formulation”, 2) “Khoshdarregi, M.,and Altintas, Y., 2017, Dynamics of multi-point threading: Part II: applicationto thin-walled oil pipes”.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Mechanical Behaviour of Threaded Connections . . . . . . . . . . . 62.3 Mechanics of Form Cutting Operations . . . . . . . . . . . . . . . . 72.4 Process Dynamics and Chatter Stability . . . . . . . . . . . . . . . 92.5 Turning Thin-walled Workpieces . . . . . . . . . . . . . . . . . . . 112.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12vii3 Chip Geometry and Cutting Forces . . . . . . . . . . . . . . . . . . . 133.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Discrete Representation of the Cutting Edge . . . . . . . . . . . . . 133.3 Chip Geometry for General Engagement . . . . . . . . . . . . . . . 153.4 Chip Geometry for Partial Root Engagement . . . . . . . . . . . . . 203.4.1 Re-evaluation of The Left Engagement Point (psc) . . . . . . 203.4.2 Re-evaluation of The Right Engagement Point (pec) . . . . . 223.4.3 Updating the Upper and Lower Bands . . . . . . . . . . . . 223.4.4 Thread Profile After Each Cut . . . . . . . . . . . . . . . . 223.4.5 Lower Band in Partial Engagement . . . . . . . . . . . . . . 243.5 Systematic Chip Discretization . . . . . . . . . . . . . . . . . . . . 243.6 Cutting Force Calculation . . . . . . . . . . . . . . . . . . . . . . . 283.6.1 Local Oblique Cutting Angles . . . . . . . . . . . . . . . . 293.6.2 Local Cutting Force Coefficients . . . . . . . . . . . . . . . 323.6.3 Total Cutting Forces . . . . . . . . . . . . . . . . . . . . . 353.7 Experimental Validation of Mechanics Model . . . . . . . . . . . . 363.7.1 Semi-Orthogonal Identification Tests . . . . . . . . . . . . . 373.7.2 Validation of Threading Force Prediction . . . . . . . . . . 393.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Dynamics of Multi-Point Threading . . . . . . . . . . . . . . . . . . . 474.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Chip Regeneration Mechanism . . . . . . . . . . . . . . . . . . . . 474.3 Calculation of Dynamic Cutting Forces . . . . . . . . . . . . . . . . 484.3.1 Dynamic Chip on The Upper Band . . . . . . . . . . . . . . 484.3.2 Dynamic Chip on The Lower Band . . . . . . . . . . . . . . 514.3.3 Dynamic Chip on The First Tooth . . . . . . . . . . . . . . 524.3.4 Total Dynamic Cutting Forces . . . . . . . . . . . . . . . . 524.4 Calculation of Process Damping Forces . . . . . . . . . . . . . . . 534.5 Total Forces on The Insert . . . . . . . . . . . . . . . . . . . . . . . 564.6 Stability Analysis in Frequency Domain . . . . . . . . . . . . . . . 574.7 Dynamic Equation of Motion in Time Domain . . . . . . . . . . . . 594.8 Time-Marching Numerical Simulation . . . . . . . . . . . . . . . . 614.8.1 Remarks: Cutter Disengagement Due to Large Vibrations . . 65viii4.9 Sample Simulation Results and Discussions . . . . . . . . . . . . . 654.10 Numerical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 684.10.1 Efficiency of Time-marching Numerical Methods . . . . . . 684.10.2 Frequency Resolution in Nyquist Stability Analysis . . . . . 714.11 Process Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 734.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Threading Thin-Walled Oil Pipes . . . . . . . . . . . . . . . . . . . . . 775.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Structural Dynamics of Cylindrical Shells . . . . . . . . . . . . . . 775.3 Response of Cylindrical Shells to Threading Loads . . . . . . . . . 785.3.1 Dynamic Equation of Motion . . . . . . . . . . . . . . . . . 795.3.2 Sample Time Simulation Results . . . . . . . . . . . . . . . 845.4 Application: Threading Oil Pipes . . . . . . . . . . . . . . . . . . . 855.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 865.4.2 Dynamic Behaviour Across Clamping Chucks . . . . . . . . 885.4.3 Effect of Jaw Configuration . . . . . . . . . . . . . . . . . . 895.5 Mode Shape Analysis of Oil Pipes . . . . . . . . . . . . . . . . . . 905.5.1 Finite Element Mode Shape Extraction . . . . . . . . . . . . 905.5.2 Experimental Mode Shape Extraction . . . . . . . . . . . . 925.6 Chatter Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6.1 Threading Over Several Passes . . . . . . . . . . . . . . . . 995.6.2 Threading at Different Infeed Values . . . . . . . . . . . . . 1025.6.3 Threading With V-profile Insert . . . . . . . . . . . . . . . 1045.6.4 Remarks: Change in Pipe Dynamics During Threading . . . 1045.7 Chatter Suppression Strategies . . . . . . . . . . . . . . . . . . . . 1065.7.1 Effect of Additional Damping . . . . . . . . . . . . . . . . 1075.7.2 Using Different Spindle Speeds For Subsequent Passes . . . 1095.8 Threading Toolbox Simulation Software . . . . . . . . . . . . . . . 1105.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . 1136.1 Summary and Contributions . . . . . . . . . . . . . . . . . . . . . 1136.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . 115ixBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A Mode Shape Extraction From ANSYS . . . . . . . . . . . . . . . . . . 125xList of TablesTable 3.1 Parameters in the semi-orthogonal identification tests . . . . . . . 38Table 3.2 Threading with V-profile insert at radial infeed of 0.15 mm/pass . 42Table 3.3 Threading with V-profile insert at radial infeed of 0.3 mm/pass . 42Table 3.4 Threading with V-profile insert at flank infeed of 0.15 mm/pass . 43Table 3.5 Threading V-profile at alternate flank infeed of 0.15 mm/pass . . 43Table 3.6 Threading with single buttress insert at 0.075 mm/pass . . . . . . 45Table 3.7 Measured forces in threading with twin buttress insert . . . . . . 45Table 4.1 Dynamic parameters of the workpiece used in simulations. . . . . 66Table 4.2 Minimum discretization intervals for time-marching simulations . 69Table 4.3 Optimization constraints for the case study with V-profile insert . 74Table 4.4 Optimization results and the limiting factors . . . . . . . . . . . 74Table 5.1 Measured dynamic parameters of the dominant mode in pipes D7 93Table 5.2 Comparison of experimental and predicted stability conditions . . 101Table 5.3 Predicted stability charts at different infeed values . . . . . . . . 104Table 5.4 Predicted stability charts for three-point V-profile insert . . . . . 105Table 5.5 Dynamic parameters of pipe D13 before and after threading . . . 106xiList of FiguresFigure 1.1 Threaded connection between oil pipes . . . . . . . . . . . . . 1Figure 1.2 Different types of infeed plans in thread turning operations . . . 3Figure 1.3 Local forces at different locations along a threading chip. . . . . 4Figure 1.4 Chatter vibrations and the resultant surface finish in threading. . 5Figure 3.1 Tool and insert coordinate systems. . . . . . . . . . . . . . . . . 14Figure 3.2 Definitions of radial infeed and depth of cut . . . . . . . . . . . 16Figure 3.3 Calculation of lower band for different cases . . . . . . . . . . . 17Figure 3.4 Chip boundaries and thread profile in partial root engagement . . 21Figure 3.5 Overlapping of chip flow lines due to chip interference . . . . . 25Figure 3.6 Search algorithm and forming the chip discretization lines. . . . 26Figure 3.7 Sample chip discretization results . . . . . . . . . . . . . . . . 28Figure 3.8 Sensitivity of chip modelling to the discretization length . . . . 28Figure 3.9 Local oblique cutting parameters and cutting forces . . . . . . . 29Figure 3.10 Sample effective oblique angles for V-profile and buttress inserts 31Figure 3.11 Local and total forces exerted on the tool . . . . . . . . . . . . . 32Figure 3.12 Edge vectors defining the chip elements. . . . . . . . . . . . . . 33Figure 3.13 Ploughing and the effect of edge radius. . . . . . . . . . . . . . 35Figure 3.14 Experimental setup for the semi-orthogonal tests . . . . . . . . 37Figure 3.15 Experimentally identified cutting force coefficients . . . . . . . 39Figure 3.16 Experimental setup for the threading experiments . . . . . . . . 39Figure 3.17 Measured cutting forces for single V-profile insert . . . . . . . . 41Figure 3.18 Measured cutting forces for single and twin buttress inserts . . . 44Figure 4.1 Chip regeneration mechanism in multi-point thread turning. . . . 49Figure 4.2 Dynamic chip area due to the current and previous vibrations. . 50xiiFigure 4.3 Chip regeneration on the first tooth. . . . . . . . . . . . . . . . 52Figure 4.4 Local process damping forces in threading. . . . . . . . . . . . 54Figure 4.5 Semi-discretization of delay differential equations . . . . . . . . 62Figure 4.6 Insert geometry and workpiece dynamics used in the simulation 65Figure 4.7 Simulated stability charts for three-point V-profile insert . . . . 67Figure 4.8 Comparison of Semi-discretization and Euler’s methods . . . . 69Figure 4.9 Sensitivity of the stability lobes to numerical discretization . . . 70Figure 4.10 Nyquist plots with different frequency resolutions . . . . . . . . 72Figure 4.11 Simulated results for the optimized plan . . . . . . . . . . . . . 75Figure 5.1 Vibration modes of a clamped thin-walled workpiece. . . . . . . 78Figure 5.2 Response of cylindrical shells to machining loads. . . . . . . . . 79Figure 5.3 Simulated shell response under threading loads . . . . . . . . . 85Figure 5.4 Experimental setup for the chatter threading tests . . . . . . . . 86Figure 5.5 Pipes and the cutting inserts used in the experimental tests. . . . 87Figure 5.6 Measurement devices used in the experimental threading tests . 87Figure 5.7 Effectiveness of chuck clamping in vibration isolation . . . . . . 88Figure 5.8 Effect of jaw configuration on pipe dynamic . . . . . . . . . . . 89Figure 5.9 Finite element modal analysis of pipe D13 . . . . . . . . . . . . 91Figure 5.10 Grids for experimental mode shape analysis. . . . . . . . . . . . 92Figure 5.11 Circumferential mode shape analysis of pipe D13 . . . . . . . . 94Figure 5.12 Axial mode shape analysis of pipe D13 . . . . . . . . . . . . . 95Figure 5.13 Circumferential mode shape analysis of pipe D7 . . . . . . . . . 96Figure 5.14 Axial mode shape analysis of pipe D7 . . . . . . . . . . . . . . 97Figure 5.15 Parameters affecting chatter stability in threading oil pipes. . . . 98Figure 5.16 Measured FRFs before each threading set . . . . . . . . . . . . 99Figure 5.17 Experimental results for threading test Set 1 . . . . . . . . . . . 100Figure 5.18 Experimental results for threading test Set 2 . . . . . . . . . . . 101Figure 5.19 Experimental results for threading test Set 3 . . . . . . . . . . . 102Figure 5.20 Experimental results at different infeed values . . . . . . . . . . 103Figure 5.21 Experimental results with three-point V-profile insert . . . . . . 105Figure 5.22 Change in the FRF of pipe D13 after a threading pass . . . . . . 106Figure 5.23 Effect of the damping ring on pipe dynamics . . . . . . . . . . . 107Figure 5.24 Effect of the damping ring on stability . . . . . . . . . . . . . . 108xiiiFigure 5.25 Effect of adding rubber between the pipe and the clamping jaws 109Figure 5.26 Effect of using different speeds for chatter suppression . . . . . 110Figure 5.27 User interface of the developed Threading Toolbox software . . 112xivList of Symbols and AbbreviationsSymbolsA0 time-invariant state matrixAc area of chip elementAcm cross section of compressed material under flank faceAdl , Adu dynamic chip area on lower and upper bandsa thread depthBc current state matrixBd delayed state matrixbl ,bu local width vectors on lower and upper bandsCm,1 modal structural damping matrixCm,2 modal process damping matrixCp equivalent process damping coefficient matrixdw workpiece diameterdu discretization lengther unit radial direction vectorFc total cutting force vectorFd dynamic cutting force vectorFdl , Fdu local dynamic cutting force vectors on lower and upper bandsFf c,Frc,Ftc feed, radial, and tangential cutting forcesFGr generalized radial force vectorFp process damping force vectorFp f process damping force in feed directionFpt process damping force in tangential directionFp,L local process damping force vectorxvFs static cutting force vectorFs,L local static cutting force vectorFsm modal static cutting force vectorFu orthogonal friction forceFv orthogonal normal forcefa axial feedratefcr radial force at cutting pointG matrix of relative FRFs between tool and workpieceGr generalized radial dynamics of cylindrical shellGp three-dimensional FRF matrix at point pGpk,qn cross FRF between DOFs pk and qnGt ,Gw FRF matrices of tool and workpieceg j boolean function for tooth entryh local chip thickness vectorh¯ local average chip thicknessh˜ local candidate chip thickness vectorhdl ,hdu dynamic chip thickness on lower and upper bandshp thread pitchi index of lower band pointic number of waves in circumferential patternik index of lower band point paired with point k of upper bandim index of centre point of moving windowip index of previous selected point of lower bandj tooth numberjl number of half-waves in axial patternKdc,Kdd equivalent current and delayed dynamic force coefficient matricesKe equivalent cutting force coefficient vector for chip elementK f c,Krc,Ktc feed, radial, and tangential cutting force coefficientsKmc,1 modal squared natural frequency matrixKmc,2 modal current dynamic cutting force coefficient matrixKmd modal delayed dynamic cutting force coefficient matrixKsp process damping indentation force coefficientKu,Kv friction and normal cutting force coefficientsxviKuv vector of orthogonal cutting force coefficientsk index of cutting edge pointkc1 Kienzle coefficientks,i modal stiffness of mode iL length of workpiece after chuckL¯l ,L¯u local unit vector tangent to to lower and upper bandsLl ,Lu local vectors tangent to lower and upper bandsLw width of wear land on flank facem number of discretizations around circumference of workpiecemc Kienzle coefficientmi modal mass of mode imt , mw number of modes of tool and workpieceNc shape functionjNe number of elements on tooth jNp total number of discrete edge pointsNt total number of teeth on insertNl,Nu number of discrete points of lower and upper bandsn spindle speedng total number of grid pointsnp pass numberP discrete representation of cutting edge in global coordinatesPIC discrete representation of cutting edge in insert coordinatesjnpPl,jnpPu discrete representation of lower and upper bands on tooth j and pass npPlL,PlM,PlR discrete representation of left, middle, and right segments of lower bandjnpP f discrete representation of thread profile on tooth j and pass npP f s shifted previous profilepec , pep current and previous intersection points on right edgepsc , psp current and previous intersection points on left edgepe,u1 , pe,u2 first and second candidate points on right engagement (upper band)ps,u1 , ps,u2 first and second candidate points on left engagement (upper band)pe,l temporary candidate for right engagement (lower band)ps,l temporary candidate for left engagement (lower band)Qr generalized radial vibration vectorxviiq relative vibration vector between tool and workpieceqcr radial vibration at cutting pointqi radial vibration at point iqt vibration vector of toolqw vibration vector of workpieceRkl,i residue of mode i in FRF between points k and lrε edge radius of cutting edgeS static forcing function vectorSl,Su projection of candidate chip thickness on lower and upper bandsT spindle periodTIG insert to global transformationTLG local (TFR) to global transformationTob orthogonal to oblique transformationTsc transformation from shell coordinates to tool CSt timet0 initial timetw wall thickness of workpieceU f c,Urc,Utc unit direction vectors of local cutting forcesU f l,U f u chip thickness direction on lower and upper bandsUˆt ,Uˆw mass-normalized mode shape matrix of tool and workpieceUp,i mass-normalized mode shape vector of point p in mode iUˆr mass-normalized radial mode shape matrixu subscript for friction orthogonal directionupk,i mass-normalized eigen value of kth DOF of node p in mode iVcm volume of compressed material under flank faceV f axial velocity vector due to tool feedVt total cutting velocity vectorVw circumferential velocity vector at workpiece surfacev subscript for normal orthogonal directionXYZ global CSX ′Y ′Z′ insert CSyw coordinate of workpiece surface line in insert CSYt coordinate of tip of final tooth in insert CSxviiiα local effective rake angleα0 nominal rake angle of insertα¯ local unit vector normal to rake faceβl,βu corner angles on lower and upper bandsβs summation of corner anglesΓ combined modal displacement vector of tool and workpieceγ local effective clearance angleγ0 nominal clearance angle of cutting edgeγ¯ local unit vector normal to clearance face∆a radial infeed∆t time step∆ω frequency resolutionε f x axial infeed offsetζ damping ratioη chip flow angleθ local approach angleθFL flank angleθM deviation angle in modified flank infeedΛ characteristic functionλ inclination angleµ Coulomb friction coefficient between tool and workpieceρ helix angle of thread, shim angleϕ circumferential angleΨ modal displacement vectorΨr modal radial displacement vectorΩ state vectorω frequencyωc chatter frequencyωn natural frequencyωsp spindle rotation frequencyxixAbbreviationsCAD Computer-Aided DesignCAM Computer-Aided ManufacturingCS Coordinate SystemDDE Delay Differential EquationDOF Degree of FreedomFEM Finite Element MethodFFT Fast Fourier TransformFRF Frequency Response FunctionMAL Manufacturing Automation LaboratoryMPa Mega PascalODE Ordinary Differential EquationSD Semi-discretizationTFR Tangential, Feed, RadialTPI Thread Per InchxxAcknowledgmentsFirst and foremost, I would like to sincerely thank my supervisor, professor YusufAltintas, who has been the greatest source of support, encouragement, and inspira-tion for me over the past seven years of my Master’s and PhD studies at the Man-ufacturing Automation Laboratory. He has created an environment in MAL whereeveryone feels safe, valued, and motivated.I am also thankful to the members of my PhD supervisory committee, Dr. Hsi-Yung Feng, Dr. A. Srikantha Phani, and Dr. Reza Vaziri for their valuable input ondifferent technical aspects of this thesis.This research has been partly sponsored by TenarisTAMSA, Veracruz, Mexico.I owe special thanks to Dr. Ramo´n Aguilar, manager of Process Development andMechanical Solutions Group, for providing industrial input and coordinating myon-site experiments during my visit. I am also grateful to the engineers and ma-chining technicians especially Omar, Antonio, Naum, and Octavio for their kindhelp with the experiments.Most of the cutting tools used for the experiments in this research have beenkindly provided by Sandvik Coromant, Sweden. I am deeply grateful to the en-gineers and technicians at Coromant for sharing their experience and knowledgeduring my five-month industrial internship in the company. Special thanks to Mr.Mikael Lundblad and Dr. Anders Liljerehn for scheduling my experiments and of-fering consultation. I thank Sandvik Coromant for providing housing, and NSERC-CANRIMT for covering my travel expenses during the internship.I thank DMG Mori for providing a Mori Seiki NT3150 Mill Turn Centre toMAL. The machine has been extensively used for preliminary verification of themodels in this thesis.The past seven years at MAL have been full of great memories. I feel absolutelylucky to have got the chance to work with many passionate students and scholars.xxiI am particularly grateful to Dr. Keivan Ahmadi, former post-doctoral fellow, forpreparing a preliminary code which I used as a starting point for my research. Ialso thank our software engineer, Mr. Byron Reynolds, for his collaboration indeveloping a user interface for my threading simulation engine.Last but not least, I am indebted to my family for their unconditional supportthroughout all these years, for believing in me, and for allowing me to go after mydreams no matter where they take me.xxiiTo my family for their unconditional support...xxiiiChapter 1IntroductionThreaded connections are widely used in industry in making impermanent joints.Internal and external threads are generated either by forming (plastic deforma-tion) [2] or thread cutting (tapping [3], machining [4]). While higher surface hard-ness can be achieved in thread forming, cutting operations are more suitable forprecision applications. In contrast to conventional tapping, thread machining suchas turning and milling operations provides easier chip removal and more controlover the cutting parameters, thus results in improved accuracy, surface quality, andproductivity.Drill pipes, tubes, and casings used in the exploration and extraction of oil andgas from deep offshore reservoirs are connected through threaded joints as illus-a)b)ButtressProfile310147'1.58mm2.98mmR=0.5 mm60V60Profilec)FAFR FTFigure 1.1: a) Threaded connection between oil pipes, b) schematic threadturning operation, c) sample API buttress and API V60 profiles.1Chapter 1. Introductiontrated in Figure 1.1.a. The pipelines must typically reach few kilometres deep inthe ocean where they are subject to high pressure, torque, cyclic stresses, varyinginternal pressure, and severe abrasive wear. Based on finite element and exper-imental investigations, the threaded joints are the weakest point in the pipelines.The threads are cut using turning operations (Figure 1.1.b) on large scale indus-trial lathes. Any form error or chatter marks on the thread surface left during themachining operation can initiate fatigue failure and leakage of the joint.Thread specifications and tolerances are regulated mainly based on Spec 5B ofAmerican Petroleum Institute (API) standard [5]. Figure 1.1.c shows two types ofthreads commonly used in oil and gas industry: API V60 profile for shoulderedconnections, and API Buttress profile for casings and tubes. These two profilesare extensively used in this thesis for demonstrations, but the developed models aregeneralized and can be used for any custom profile.As shown in Figure 1.1.c, the depth of the thread is typically around 1.5 mm to3 mm. Due to the limited power of machines, structural flexibilities of the setup,and chip removal problems, the entire depth of the thread cannot be cut in oneaxial travel (pass) of the cutting tool along the pipe axis. Normally, the thread iscut over 6-12 passes, the first few of which are for roughing, and the final passesare for finishing. As illustrated in Figure 1.2.a for a V-profile thread, there arefour main strategies which determine how much and in which direction the cuttingtooth penetrates into the thread over each pass. The arrows in the figure representthe infeed direction. Each of these strategies, also called Infeed Plans, has certainadvantages and disadvantages. Radial infeed is the simplest strategy and can beperformed by all conventional lathes. It also results in cancellation of axial forcesdue to equal chip thickness on side edges. Figure 1.2.b shows the cross section ofthe actual chip for the first, intermediate, and final passes when threading with aV-profile insert at radial infeed of 0.15 mm/pass. It can be seen that towards finalpasses, a long section of the tooth is engaged in the cut, which leads to poor chipformation.Flank infeed, on the other hand, cuts with only one side of the tooth. Eventhough the thickness of the chip is twice as the one in radial infeed (assuming samechip area), the cutting section has a smaller width and a more straight profile. Theproblem, however, is that not only the insert wears out unevenly but also the twosides of the finished thread will have different surface characteristics. Alternate2Chapter 1. IntroductionRadial Infeed Flank Infeed Modified Flank3°- 5°a)b) c)Alternate FlankFigure 1.2: a) Different types of infeed plans, b) cross section of actual chip(microscope image), c) sample multi-point threading insert (NingboSanhan).flank infeed is designed to mitigate this problem by alternating between the sidesof the tooth and cut surface.The main issue in both flank and alternate is that over each pass one side of thecutting edge (free edge) is constantly rubbing against the thread surface, resultingin elevated temperatures, faster tool wear, and degraded surface quality. Modifiedflank solves this problem by combining the advantages of the other three strategies;due to cutting with both edges, the wear and surface quality is relatively even onboth sides. Since the chip on one side is very thin and easy to deform, the chipevacuation is also superior to radial infeed. Most modern machine tools can auto-matically perform all these infeed strategies. Similar infeed plans can be definedfor other types of profiles to change the chip load distribution. In this thesis, ageneralized model has been developed which can systemically determine the chipgeometry for any custom profile and arbitrary infeed strategy.In order to improve the productivity of threading operations, especially in massproduction such as oil and gas industry, it is common to use multi-point inserts(Figure 1.2.c). In one axial travel of the tool along the workpiece, the first fewteeth perform rough cutting and the final teeth gradually finish the surface. In somecases, more than one pass might be still needed. Unlike single-point threading,multi-point inserts are designed for only a specific pitch. Custom infeed strategiescan be integrated in the design of multi-point inserts by shifting each tooth in theaxial (feed) direction relative to its previous tooth. This axial shift does not change3Chapter 1. IntroductionXYFfcFrcFtcFfcFtcFrcFigure 1.3: Local forces at different locations along a threading chip.the pitch of the thread since the insert still travels one pitch per spindle revolution.Figure 1.3 illustrates a sample threading chip and the local cutting forces. Thechip thickness as well as the cutting force coefficients, oblique cutting angles, andthe direction of local forces can vary significantly along the cutting edge. In thisthesis, a systematic chip discretization method is proposed which allows local eval-uation of cutting forces. The developed technique can form the elements basedon the local chip flow direction while considering the chip compression at sharplycurved segments.Due to the structural flexibilities of the tool and workpiece, any change in thecutting forces can cause the setup to vibrate. As illustrated in Figure 1.4, the currentvibrations and the vibration marks left from the previous cut lead to variation inthe instantaneous chip thickness. The closed loop interaction between the cuttingforces and the structure can lead to unstable chip regeneration (regenerative chatter).Dynamics and stability of multi-point threading is different than regular turning inthat each tooth is affected by the previous vibration marks left by a different tooth.Due to the variation of local approach angle in threading inserts, the effect of currentand previous vibrations must be analyzed locally at each point along the cuttingedge.Relative vibrations between the tool and workpiece also result in additionaldamping in the system. The effect of process damping is especially more signif-icant when the rotation frequency of the spindle is considerably smaller than thestructural vibration frequency. Thread turning operations are often run at low spin-dle speeds (below 1000 rpm) since the tool has to move one thread pitch over eachspindle revolution. Process damping has therefore a significant effect on chatterstability limits in threading operations. Threading of oil pipes imposes additional4Chapter 1. IntroductionXYhVtStable Light Chatter Heavy ChatterhFigure 1.4: Chatter vibrations and the resultant surface finish in threading.complexities due to the shell dynamics of thin-walled pipes. This thesis developsa generalized dynamic model and predicts the chatter stability diagrams for threadturning with custom multi-point inserts subject to three-dimensional flexibilities ofthe tool and workpiece. The model is extended to threading thin-walled cylindricalshells as well.The rest of this thesis is structured as follows. Chapter 2 reviews some of theprevious literature related to threading operations. Chapter 3 presents the mechan-ics of multi-point threading process. A generalized methodology is first proposedto determine the chip boundaries for custom multi-point inserts. The chip is dis-cretized along the cutting edge, and the cutting forces are evaluated locally for eachchip element. The mechanics model is validated experimentally.Generalized dynamics of multi-point threading is developed in Chapter 4. Thedynamic equation of motion is derived in time and frequency domains, and thestability of the process is investigated. An optimization algorithm is presented tofind the optimum infeed settings subject to user-defined constraints.The model is extended in Chapter 5 to threading thin-walled workpieces withdominant shell vibrations. The proposed dynamic model is validated experimen-tally through extensive tests on real scale oil pipes. The thesis is concluded inChapter 6 by summarizing the research contributions and possible future directions.5Chapter 2Literature Review2.1 OverviewThis chapter reviews some of the past research related to threading operations.Section 2.2 discusses the effect of machining parameters on mechanical charac-teristics of threaded connections. Section 2.3 presents the previous methodologiesproposed for calculation of chip geometry and cutting forces in general form cut-ting operations. Dynamics and chatter stability of turning and threading operationsare discussed in Section 2.4, followed by more specific application to thin-walledcylindrical workpieces in Section 2.5. The chapter is concluded by the summary inSection 2.6.2.2 Mechanical Behaviour of Threaded ConnectionsResearchers have extensively studied mechanical characteristics of threaded jointsin the pipelines. Using finite element analysis, Shahani and Sharifi [6] showed thatthe threads near the shoulder and free end of drill pipes bear the maximum load. Luand Wu [7] carried out fractographic analysis on the joints between drill pipes, andobserved that fatigue cracks nucleated mostly at the root of the first thread. Yuenet al. [8] studied stress distribution in the thread connection during the make andbreak process of oil pipes and showed that the reliability of the connection can beimproved by decreasing the local stress concentration and improving the rigidity ofthread surface. Abrahmi et al. [9] analyzed the effect of threading process on themechanical and tribological behaviour of triangular threads. They concluded that6Chapter 2. Literature Reviewcompared to machining operations, rolling processes lead to improved mechanicalresistance of the thread joints. However, the accuracy and efficiency of rollingprocess is not satisfactory for oil and gas applications.Researchers have also studied the effect of machining parameters on thread per-formance. Fetullazade et al. [10] carried out several sets of thread turning experi-ments on SAE 4340 steel at different values of cutting speed, depth of cut, and toolwear. They measured the residual stresses at the root of the machined thread us-ing electro-chemical layer removal technique, and observed residual stress rangingfrom 600MPa to 1450MPa as compared to the material tensile strength of 850MPa.They also found strain hardening in the range of 320HV-430HV at the thread rootas compared to the base hardness of 260HV. Akyildiz and Livatyali [11] inves-tigated the effect of machining parameters on fatigue strength of test specimensthreaded on a turning machine. They analyzed the endurance limit of the threadedparts using a rotary fatigue test machine, and concluded that higher cutting veloc-ity and larger tool wear improves the fatigue strength of the threads due to strainhardening.2.3 Mechanics of Form Cutting OperationsThe toothed profile of the cutter in form cutting operations such as threading, hob-bing and gear shaping can generally result in multi-flank chips. Klocke et al. [12]developed a finite element-based machining simulation to model the chip forma-tion, thermal and stress distribution, and tool wear along the cutting edge in multi-flank form cutting. Cutter-workpiece engagement can be obtained using solid ordiscrete geometric modelling kernels [13]. Bouzakis [14] used a solid kernel inSolidWorks R© to model the chip geometry in hobbing. Although solid modellingprovides highly accurate representation due to its analytical approach, the compu-tational load is intensive. Discrete representation of the engagement using meshedgeometries, on the other hand, can provide fast yet reliable approximate solutions.Brecher et al. [15] modelled the chip geometry in bevel gear cutting using the dis-crete volume representation of the tool and workpiece. Erkorkmaz et al. [13] usedmulti-dexel volume representation to extract the cutter-workpiece engagement ateach time step during gear shaping operation. Unlike the methods used in the re-search cited above, this thesis uses the kinematics of the process and geometry ofthe tool to semi-analytically determine the boundaries of the chip on each tooth in7Chapter 2. Literature Reviewthreading with multi-point inserts.Thread milling is used mainly in generating internal threads especially for appli-cations involving asymmetrical workpieces or hard-to-cut materials. Araujo et al. [4]simplified thread milling as a common end milling operation by ignoring the feed-ing motion of the tool along the hole axis (z-direction). Fromentin and Poulachon[16, 17] derived a mathematical model to describe the tool envelope profile, cuttingangles, and uncut chip thickness in thread milling. Jun and Araujo [18] developeda force model for “thrilling” operation, which performs drilling and threading withthe same tool. Wan and Altintas [19] studied the mechanics and dynamics of threadmilling processes, and modelled the varying cutter-workpiece engagement basedon the kinematics of the process. They also predicted process stability along thehelical threading path using semi-discretization method [20]. Arajuo et al. [21]presented geometrical and cutting force analysis for thread milling of API threads,and analyzed the surface roughness at different vertical positions for different fee-drates. Unlike for thread milling and other form cutting operations, the mechanicsof general thread turning processes have not been studied systematically before.Researchers have used different approaches for modelling the chip geometry intypical turning operations. Eynian and Altintas [22] divided the chip area based onthe linear and curved edges of the insert. Reddy et al. [23], Lazoglu et al. [24], andOzlu and Budak [25] obtained a more accurate approximation through discretiza-tion of the chip along the cutting edge. The chip geometry in thread turning is morecomplicated and cannot be modelled using the approaches cited above. Akyildizand Livatyali [26] presented a force calculation method for thread turning based onchip discretization, but their method is limited to V-profile with radial infeed. In ad-dition, they used a linear cutting force coefficient model, and observed considerablediscrepancy between the simulated and measured forces especially towards the finalpasses. Akyildiz [27] reported the change in the shear angle [28] over subsequentthreading passes as the main source of this discrepancy. Kafkas [29] carried out anexperimental study on cutting forces in thread turning, and observed an increase inthe cutting force coefficients over deeper passes; chip interference was claimed tobe the reason for this behaviour.This thesis develops a generalized systematic model which can determine thechip geometry and predict the cutting forces for thread turning with custom multi-point inserts and arbitrary infeed plans.8Chapter 2. Literature Review2.4 Process Dynamics and Chatter StabilityInstability of machining processes due to self-excited vibrations, known as chatter,was first recognized and modelled by Taylor [30], Tobias [31], Tlusty [32], andMeritt [33]. They described the dynamics of the process as delay differential equa-tions. Fundamentals of machining dynamics have been summarized by Schmitzand Smith in [34]. Researchers have developed numerous methods to solve theprocess dynamic equations and predict chatter stability for different operations, asreviewed by Altintas and Weck [35]. Minis and Yanushevsky [36] proposed a fre-quency domain solution based on Fourier analysis and Floquet’s theory. Altintasand Budak [37] approximated the direction factors by their average, and proposedan analytical zero-order solution. Insperger and Stepan [20, 38] developed a timedomain semi-discretization method which approximates the delay differential equa-tion by a series of ODEs. In this method, the delay state and the time-varyingcoefficient matrices are approximated numerically. Ding et al. [39, 40] proposedfull-discretization and numerical integration methods to analyze the stability of theprocess. Asl and Ulsoy [41] solved the linear delay differentials using Lambertfunctions while Butcher et al. [42] employed Chebyshev polynomials and collo-cation methods. Mann et al. [43] used Temporal Finite Element Analysis (TFEA)for simultaneous prediction of stability and surface location error in milling opera-tions. Eksioglu et al. [44] implemented an extended version of semi-discretizationmethod to predict the stability and surface location errors in machining thin-walledworkpieces. Honeycutt and Schmitz [45] proposed a metric for automated stabilityidentification in time domain simulations based on periodic sampling of signals.Frequency domain solutions provide a faster stability prediction model while timedomain solutions can provide both stability and time simulation of the process.This thesis uses frequency domain and semi-discretization methods to analyze andsimulate the dynamics of multi-point threading process.Dynamics of typical turning operations have been extensively studied by re-searchers. Ozlu and Budak [25] developed an analytical chatter stability model forturning and boring operations based on a simple chip discretization method. Intheir proposed technique, all chip elements are formed parallel to the feed direc-tion even at the curved nose of the insert. Dynamics of multi-point threading hassome similarities to parallel turning in that more than one cutting edge is engagesimultaneously. Lazoglu et al. [46] derived the dynamic model for parallel turning9Chapter 2. Literature Reviewwhen the two tools cut different surfaces of the workpiece. Budak and Ozturk [47]developed a frequency domain stability model for parallel turning on the same sur-face. In both [46] and [47], the two cutting tools were positioned on the oppositesides of the workpiece (180◦ apart), and it was shown that if the natural frequenciesof the two tools are close, chip regeneration can be altered and productivity can beimproved compared to single turning. Brecher et al. [48] modelled the dynamics ofparallel turning as a function of the circumferential angle between the two turrets(cutting tools). They showed that in the cases where the two tools are dynamicallycoupled, the circumferential angle can be used to optimize the process stability.One of the main challenges in stability prediction for turning operations suchas thread turning is the accurate modelling of process damping effect. Researchershave used indentation models along with experimental calibrations to quantify theprocess damping contribution. Albrecht [49] considered the roundness of the cut-ting edge and modelled the ploughing under the clearance face of the tool. Sissonand Kegg [50] analyzed the effect of edge radius on additional damping at lowspeed cutting. Shaw and DeSalvo [51] studied the plastic flow under the flankface of worn tools, and suggested that process damping force in the feed direc-tion is proportional to the volume of the indented material. Chiou and Liang [52]modelled the volume of the compressed material as a function of the ratio of thevibration velocity to cutting velocity, and calculated the process damping forcesusing an experimentally identified damping coefficient. Clancy and Shin [53] fur-ther extended this model and presented a three-dimensional mechanistic frequencydomain chatter stability for face turning processes including the flank wear effect.Altintas et al. [54] used a fast piezo actuator to generate controlled vibrations andidentified the process damping coefficients. Budak and Tunc [55] presented an iden-tification method based on the results of stable and chatter tests, and Ahmadi andAltintas [56] proposed a technique using output-only modal analysis. Ahmadi [57]further expanded the process damping terms by considering the nonlinearities ofthe damping effect. Tyler et al. [58] proposed an analytical multi-degree of free-dom process damping model for turning operations which considers the effect ofboth depth of cut and cutting velocity. They were able to predict the stability lim-its using only one empirical coefficient. They also concluded that identification ofprocess damping coefficients in multi degree of freedom systems must be carriedout based on the most flexible vibration mode.10Chapter 2. Literature ReviewWhile typical turning operations have been extensively studied in the past, thedeveloped models cannot be used for multi-point thread turning processes. Notonly is the chip geometry more complicated, but also the regeneration mechanismis different in that the effect of previous vibrations left by each tooth is seen byanother tooth over the next spindle revolution. This thesis develops a novel modelto describe the dynamics of multi-point threading with custom profiles.2.5 Turning Thin-walled WorkpiecesTurning of thin-walled workpieces imposes additional complexities due to the non-linear low-damped shell modes of the tube. Flexural vibrations in shells have beenextensively studied in the past as summarized by Meirovitch [59] and Leissa [60].Several nonlinear models have been proposed based on Love’s equations for elas-ticity [61] and Donnell’s shallow shell theory [62]. The case of cylindrical shell,in particular, involves greater complexity due to the cross coupling of different ax-ial and circumferential modes, as studied by Evensen [63, 64] and Dowell [65].However, analytical models become impractically complex when considering theboundary conditions and varying geometry of the workpiece in turning operations.Rahman and Ito [66] and Lai and Chang [67] showed that the frequency and di-rection of mode shapes can change significantly as a function of the contact lengthbetween the three-jaw clamping chuck and the workpiece. Finite element modelsalong with experimental measurements have been the main practice in analyzingthe dynamics of thin-walled workpieces.Lai and Chang [67] studied chatter stability in turning thin-walled cylindricalshells and showed that tubes allow smaller depth of cut compared to solid bars evenif they have the same moment of inertia. Dospel and Keskinen [68] derived theequations of motion for a rotating cylindrical shell subject to machining operations.Chanda et al. [69] analyzed the stability in turning cylindrical shells using semi-discretization method [38]. Chen et al. [70] employed optimization techniquesto find the optimum cutting conditions in turning thin-walled workpieces. Fischerand Eberhard [71] designed an adaptronic chisel retrofitted with vibration sensorsto damp out the shell vibrations in real time and achieved higher stability limits.Mehdi et al. [72, 73] presented a numerical model to simulate the turning process ofthin-walled workpieces. Lorong et al. [74] considered the varying dynamics of thethin-walled workpiece during turning operation using finite element and numerical11Chapter 2. Literature Reviewsimulation. They showed that the pattern and angle of vibration marks change alongthe tube axis due to varying dynamics.There has not been any research in the past focused on the dynamics of threadingof thin-walled workpieces. This thesis employs finite element and experimentalmodal analyses combined with the proposed dynamic models to investigate chatterstability in threading real scale oil pipes.2.6 SummaryChip geometry and regeneration mechanism in multi-point threading are very dif-ferent than those in typical turning operations. There is currently no research avail-able on the generalized mechanics and dynamics of thread turning. This thesisaims at filling this gap and proposes novel generalized models for calculation ofchip boundaries, cutting forces, and chatter stability in multi-point threading. Ap-plication to thin-walled oil pipes is studied.12Chapter 3Chip Geometry and Cutting Forces3.1 OverviewThis chapter proposes a semi-analytical generalized model which can systemat-ically determine the chip geometry and cutting forces for any given multi-pointthreading insert and infeed settings. The chip is first discretized along the cuttingedge using a systematic technique. The local chip thickness and cutting force co-efficients are determined for each element, followed by the calculation of the localand total cutting forces.Unlike most CAD/CAM software packages which calculate the tool-workpieceengagement (chip geometry) based on the intersection of the full CAD models,the proposed method integrates the kinematics of the process and the mathematicalrepresentation of the geometries. As a result, not only is the algorithm stand-aloneand more accurate, but also the processing time is significantly shorter than fullintersection methods.3.2 Discrete Representation of the Cutting EdgeIn the proposed methodology, all curves including the cutting edge and chip bound-aries are represented by a series of discrete points laid evenly along the curve. Asillustrated in Figure 3.1, two coordinate systems (CS) are used to define the setupgeometry. These two CSs share the same origin, which is chosen arbitrarily at anylocation on the insert. The XYZ axes of the tool CS (global) are aligned with theaxial, radial, and tangential directions of the workpiece, respectively. In order to13Chapter 3. Chip Geometry and Cutting ForcesXYZXZTool CSYXYZX Y Z Insert CSρρ '''' ' 'helix angleTool FeedX'Y'Z'ρXYhpFigure 3.1: Tool and insert coordinate systems.match the helix angle of the thread path, shims are used to rotate the insert aroundthe radial (Y ) axis equal to the helix angleρ =−tan−1(hp/pidw) (3.1)where hp is the thread pitch and dw is the workpiece diameter. The minus sign in Eq.(3.1) is to account for the negative rotation based on the defined CS (Figure 3.1).The transformation from the insert CS to the global CS is thus obtained as:TIG = cosρ 0 sinρ0 1 0−sinρ 0 cosρ (3.2)Consider the custom multi-point insert shown in Figure 3.2.a. Assume that thecutting edge of the insert has been designed using custom linear, circular, and splinecurve segments. As the first step, the entire cutting edge is continuously interpolatedby discrete points at constant intervals of du, which has been chosen as 10µm inthis thesis based on the cutting edge length (1 mm-5 mm) in common threadingoperations.The geometry of the cutting edge represented in the insert CS is stored as anarray of Np number of points with ascending values of x′:PIC =Px′Py′Pz′IC={x′1 x′2 × × x′Np}{y′1 y′2 × × y′Np}{z′1 z′2 × × z′Np}IC,(3×Np)(3.3)14Chapter 3. Chip Geometry and Cutting ForcesThe edge points PIC are transformed from the insert CS to the global CS using theTIG transformation in Eq. (3.2)PGC =PxPyPzGC= TIGPIC (3.4)Hereafter, all geometric parameters are represented in the global CS unless other-wise stated.3.3 Chip Geometry for General EngagementThis section presents the general chip calculation methodology based on the as-sumption that each tooth cuts a fresh profile and does not intersect with the pre-vious thread surface. In other words, both sides of the cutting tooth intersect withthe outer cylinder of the workpiece surface (Figure 3.2.b). The case of partial rootengagement is presented in Section 3.4 as an extension to this general procedure.Figure 3.2.b illustrates the tool-workpiece engagement for a sample threadingtooth. The chip geometry is determined as the area confined between the upperand lower bands. The upper band is formed during the current cut thus followsthe shape of the current tooth. The lower band consists of several segments; thetwo linear edges on the left and right correspond to the workpiece surface, andthe middle segment is the thread surface generated by the previous tooth. In caseof single-point inserts, the middle segment is the thread profile from the previouspass. Assume the multi-point threading insert shown in Figure 3.2.a has Nt numberof teeth with custom profiles. The tip of the final tooth, which performs the deepestcut, has radial coordinate of Yt = max(Py). In order to carry out the threadingoperation over pass np, the radial penetration of the final tooth into the workpieceis incremented by infeed value of ∆a relative to the previous pass. The tool travelsparallel to the workpiece axis (X) at the axial feed rate of fa. The achieved threaddepth after pass np is calculated as:a(np) = a(np−1)+∆a(np) (3.5)where a(np−1) is the thread depth at the previous pass, and ∆a(np) is the currentradial infeed. The outer surface of the workpiece can be represented in the current15Chapter 3. Chip Geometry and Cutting ForcesPulowerbandupperbandXYChipCuttingEdgepsp pepPlpscP(k)P(k+1) P(k)P(k+1)P(k)peca)final profileXY2 3Insert1a( ) Δa( )a( -1)np npnpTool FeedYyw tWorkpiece b)Figure 3.2: a) Definitions of radial infeed and depth of cut, b) discrete repre-sentation of the cutting edge and chip boundaries.CS as a horizontal line (Figure 3.2.a):yw = Yt−a(np) (3.6)In order to determine the engagement points between the cutting edge and the work-piece surface, all the points on the cutting edge, P(k), are swept from smallest tolargest x values, and every time two consecutive points lie on the two sides of theworkpiece surface, it marks an intersection point. The engagement point on the left( jpsc) and right ( jpec) of tooth j are located using the sign change in Py− yw:i fPy(k)− yw < 0Py(k+1)− yw > 0}→ jpsc lies between P(k) and P(k+1)(left engagement point)i fPy(k)− yw > 0Py(k+1)− yw < 0}→ jpec lies between P(k) and P(k+1)(right engagement point)(3.7)The exact location of the intersection points are obtained by interpolating betweenP(k) and P(k+1) and knowing that psc,y = pec,y = yw. In Figure 3.2.b, psp and pepmark the intersection points over the previous cut, and are assumed to be knownfrom similar analysis.The discrete point representation of the upper band on tooth j over pass np,i.e. jnpPu =[jnpPux,jnpPuy,jnpPuz]T, is obtained by extracting part of the cutting edgewhich lies between the current intersection points jpsc and jpec (Figure 3.2.b). Cal-16Chapter 3. Chip Geometry and Cutting Forcesa) case 1first cutXY2 3Insert1PlPuTool FeedXY1 2 3Workpiece (N -1) ht pΔa( )n p PuPlInsertTool Feedρhp(N -1)thpXZTool Feedb) case 2d) case 4pecpscXY1 2 3InsertTool FeedhpPlMPuhpTool FeedρhpXZρhpPlRpep pecPlLpscpspc) case 3Tool FeedPun =1pn -1pn pε ( )n pfx ε ( )n -1 pfxPlXYΔa( )n p(current)(previous)(first)Figure 3.3: Calculation of lower band for different cases, a) first pass, firsttooth, b) next teeth, c) next passes for single insert, d) next passes formulti-point insert.culation of the lower band jnpPl , on the other hand, depends on the tooth number,pass number, and whether the insert is single-point or multi-point. All the possiblecombinations can be categorized into four cases shown in Figure 3.3. The lowerband in each case is calculated as follows.Case 1 – First Tooth ( j = 1), First Pass (np = 1), Single- or Multi-Point InsertAs illustrated in Figure 3.3.a, the lower band of the chip in the first cut is a straightline corresponding to the workpiece surface. The discrete representation of thelower band is obtained by linear interpolation between the left (psc) and right (pec)intersection points using Nl = round( |psc−pec|/du) number of points, where du is17Chapter 3. Chip Geometry and Cutting Forcesthe discretization length and chosen the same as for the upper band, i.e. du= 10µm.Case 2 – Next Teeth ( j > 1), Any Pass (np ≥ 1), Multi-Point InsertAs shown in Figure 3.3.b, the lower band of the chip has three segments: left( jnpPlL), middle (jnpPlM), and right (jnpPlR). The middle lower band is obtained byshifting the upper band of the previous tooth ( j−1np Pu) in the X direction equal to onethread pitch hp:jnpPlM =j−1np Pux+hpj−1np Puyj−1np Puz+hp tanρj,npj > 1np ≥ 1(3.8)Note that due to the inclination angle ρ around Y axis (Figure 3.1), the z componentof the shifted upper band has been adjusted using the term hp tanρ .The left segment, which corresponds to the workpiece surface, is obtained bylinear interpolation between psc and psp using du = 10µm. The right segment issimilarly interpolated between pep and pec, and the complete lower band on toothj is obtained by combining all three segments, i.e jnpPl =[jnpPlL,jnpPlM,jnpPlR].Case 3 – Subsequent Passes (np > 1), Single-Point InsertAfter finishing the first pass, the tool retracts and moves back to the starting position(tip of the workpiece). Before cutting the new pass, the radial depth and axialshift of the tool must be set based on the infeed plan (Figure 3.3.c). Since thecoordinate system is attached to the insert, the middle lower band 1npPlM is obtainedby transforming the upper band of the previous pass (1np−1Pu) from the previous CSto the current CS:1npPlM =1np−1Pux+[ε f x(np)− ε f x(np−1)]1np−1Puy−∆a(np)1np−1Puz+[ε f x(np)− ε f x(np−1)]tanρj,npNt = 1np > 1(3.9)where ∆a(np) = a(np)−a(np−1) is the radial infeed relative to the previous depth,and ε f x(np) is the axial shift of the tool in pass np measured relative to the axialstart position of the tool in the first pass (Figure 3.3.c). ε f x(np) can be calculated18Chapter 3. Chip Geometry and Cutting Forcesfor different infeed plans as (see Figure 1.2.a):Radial Infeed: ε f x(np) = 0 (np > 1)Flank Infeed: ε f x(np) = a(np). tan(θFl)Alternate Flank Infeed: ε f x(np) = (−1)np.a(np). tan(θFl)Modified Flank Infeed: ε f x(np) = a(np). tan(θFl−θM)(3.10)where θFl is the flank angle and equals to half of the nose angle in V-profile. θM ∼3◦-5◦ is the deviation angle in Modified Flank (Figure 1.2.a). In Eq. (3.9), the term[ε f x(np)− ε f x(np−1)]calculates the axial shift (x component) of the tool in thenew pass relative to the previous pass, and the term[ε f x(np)− ε f x(np−1)]tanρadjusts the z component due to the inclination (shim) angle of the insert.Similar to Case 2, the left and right linear segments are interpolated between thecurrent and previous intersection points, and the complete lower band is obtainedby combining the three segments.Case 4 – First Tooth ( j = 1), Subsequent Passes (np > 1), Multi-Point InsertIn this case, the lower band on the first tooth is formed by the the final tooth( j= Nt) during the previous pass. As illustrated in Figure 3.3.d, the last upper band( j=Ntnp−1Pu =[j=Ntnp−1Pux,j=Ntnp−1Puy,j=Ntnp−1Puz]T) is shifted onto the first tooth and trans-formed from the previous CS to the new CS:j=1np PlM =j=Ntnp−1Pux− (Nt−1)hp+[ε f x(np)− ε f x(np−1)]j=Ntnp−1Puy−∆a(np)j=Ntnp−1Puz− (Nt−1)hp tanρ+[ε f x(np)− ε f x(np−1)]tanρj,npj = 1Nt > 1np > 1(3.11)where Nt is the total number of teeth engaged in the cut during the previous pass.The term (Nt−1)hp shifts the previous upper band in the axial direction onto thefirst tooth, and (Nt−1)hp tanρ adjusts the z component due to the shim angle. Theterms ∆a(np) and[ε f x(np)− ε f x(np−1)]adjust the radial and axial shift of thecurrent CS relative to the previous CS, and[ε f x(np)− ε f x(np−1)]tanρ correctsfor the insert’s inclination angle.Similar to Cases 2 and 3, the linear segments on the left and right are interpo-lated between the current and previous intersection points, and the complete lower19Chapter 3. Chip Geometry and Cutting Forcesband is obtained by combining the three segments.3.4 Chip Geometry for Partial Root EngagementAs illustrated in Figure 3.4, multi-point inserts are commonly designed such thatfew teeth perform rough cutting of the root, and the following teeth finish the threadprofile. Partial engagement is especially common for buttress threads as the sideedges are nearly parallel to the infeed (radial) direction; cutting the sides with allteeth may lead to impractically small chip load thus severe ploughing and poorsurface finish.In the case of partial root engagement, at least one side of the tooth cuts insidethe previous thread profile (Figure 3.4.a). Partial engagement on each tooth can bedetected using the following conditions:1) psc,x > psp,x (left)and/or2) pec,x < pep,x (right)→ Partial Engagement (3.12)where psc, pec, psp, and psp are the intersection points calculated based on theassumptions of general engagement (Eq. (3.7)). If the conditions in Eq. (3.12)hold, psc and/or pec must be re-evaluated to find the correct engagement pointsbetween the tooth and workpiece.3.4.1 Re-evaluation of The Left Engagement Point (psc)The following steps must be taken to find the correct left engagement point (con-vexity of the profile is assumed):Step 1. Calculate the upper and lower bands of the chip assuming complete en-gagement as explained in Section 3.3.Step 2. Find the first and second far left points (smallest x coordinates) on theupper band, marked as ps,u1 and ps,u2 in Figure 3.4.c.Step 3. Construct the equation of the line connecting ps,u1 and ps,u2.Step 4. Find the point ps,l on the lower band which has smallest x coordinate andsatisfies ps,l(x)> ps,u1(x).20Chapter 3. Chip Geometry and Cutting Forces1Workpieceps,u1End of Search End of SearchXY2InsertTool Feed1WorkpiecePu PuPl PlXY2InsertTool Feed1WorkpiecePf Pfpscpecpsp peppeppeppsppspPuPuPlPl psc pecps,u1ps,u2pe,u1pe,u1pe,u2pe,lps,lps,lpe,u1pe,u1p ec=pe,u2pe,lps,u1ps,u2ps,u1 psc=ps,la)b) c) d) e)f)Figure 3.4: Determining the chip boundaries and thread profile in the case ofpartial root engagement.Step 5. Project ps,l in theY direction onto the line (ps,u1,ps,u2) from Step 3. Checkif ps,l is located under this line segment (or its extension) and also betweenps,u1 and ps,u2 in the Y direction, i.e.ps,l(y)<ps,u2(y)−ps,u1(y)ps,u2(x)−ps,u1(x) (x−ps,u1(x))+ps,u1(y)andps,u1(y)< ps,l(y)< ps,u2(y)(3.13)If both conditions in Eq. (3.13) hold, psc = ps,u1 is chosen as the correct leftintersection point, and the search is completed. Otherwise, ps,u1 is deletedfrom the upper band, and ps,u2 becomes the new ps,u1. The search restartsfrom Step 2 with the new ps,u1 and continues until the correct intersection isfound.21Chapter 3. Chip Geometry and Cutting Forces3.4.2 Re-evaluation of The Right Engagement Point (pec)The procedure follows similar steps presented above but the conditions must beadjusted for the right side edge. As marked in Figure 3.4.d, assume pe,u1 and pe,u2are the first and second far right points (largest x coordinates) on the upper band.pe,l is the point on the lower band with the largest x coordinate which satisfiespe,l(x)< pe,u1(x). pe,l is projected in the Y direction onto the line connecting pe,u1and pe,u2. If pe,l(y)<pe,u2(y)−pe,u1(y)pe,u2(x)−pe,u1(x) (x−pe,u1(x))+pe,u1(y)andpe,u1(y)< pe,l(y)< pe,u2(y)(3.14)the search is completed and pec = pe,u1 is chosen as the correct right intersectionpoint (Figure 3.4.e); otherwise, pe,u1 is deleted and the search restarts with the newpoints.3.4.3 Updating the Upper and Lower BandsOnce the correct intersection points are found using the above procedure, part ofthe previously calculated upper band (based on Section 3.3) which lies betweenthe updated psc and pec is extracted as the tooth-workpiece engagement (upperband). Calculation of the lower band requires knowing the previous thread profile;as shown in Figure 3.4.f, in case of partial engagement, the resulting profile consistsof several segments corresponding to the current and one or more of the precedingcuts. The following section presents a general methodology to obtain the threadprofile.3.4.4 Thread Profile After Each CutFor each of the four cases defined in Section 3.3 (Figure 3.3), the previous threadprofile is shifted from the previous tooth (or pass) onto the current tooth. The newprofile, jnpP f , follows the shape of the current tooth in the engaged sections (upperband) and keeps the previous profile for the uncut parts (Figure 3.4.f). Calculationof the thread profile for each case is presented below.Case 1. – First cut ( j = 1,np = 1)22Chapter 3. Chip Geometry and Cutting ForcesThe thread profile in this case is the same as the upper band of the first tooth,i.e. (j=1np=1P f)=(j=1np=1Pu)(3.15)Case 2. – Next teeth ( j > 1)The thread profile is shifted from the previous tooth to the current tooth (seeFigure 3.3.b):jnpP f =j−1np P f + hp0hp tanρ C1←−∀k : psp(x)<(j−1np P f ,x(k))<psc(x)or pec(x)<(j−1np P f ,x(k))<pep(x)jnpP f =jnpPuC2←− ∀k : psc(x)<(j−1np P f ,x(k))<pec(x)(3.16)where hp is the thread pitch and ρ is the shim angle. C1 is the conditionfor the uncut segments, and C2 corresponds to the engaged segments (Figure3.4.f).Case 3. – Next passes (np > 1), single-point insertThe thread profile from the previous pass is transformed from the previousCS to the current CS based on the radial and axial infeed settings (see Fig-ure 3.3.c):1npP f =1np−1P f + ε f x(np)− ε f x(np−1)−∆a(np)[ε f x(np)− ε f x(np−1)]tanρ ←C11npP f =1npPu ←C2(3.17)where C1 and C2 are the conditions defined in Eq. (3.16), and ∆a and ε f x arethe radial and axial infeed settings defined in Section 3.3.Case 4. – Next passes (np > 1), first tooth ( j = 1), multi-point insertIn this case, the thread profile from the previous pass is shifted from the finaltooth onto the first tooth and transformed from the previous CS to the current23Chapter 3. Chip Geometry and Cutting ForcesCS (see Figure 3.3.d):1npP f =Ntnp−1P f + −(Nt−1)hp+[ε f x(np)− ε f x(np−1)]−∆a(np)(−(Nt−1)hp+ [ε f x(np)− ε f x(np−1)]) tanρ ←C11npP f =1npPu ←C2(3.18)where Nt is the total number of teeth engaged in the cut during the previouspass, and C1 and C2 are the conditions defined in Eq. (3.16).3.4.5 Lower Band in Partial EngagementIn the case of partial engagement, the lower band follows part of the previous threadprofile engaged in the current cut (see Figure 3.4.a,f), i.e.jnpPl = P f s←∀k : psc(x)<P f s,x(k)<pec(x) (3.19)where P f s is the shifted previous profile calculated in the first equation (withoutcondition) in Eqs. (3.16), (3.17), and (3.18) corresponding to each case.It should be noted that the chip evaluation procedure presented in this sectioncan be readily used for the cases where one side of the insert is intersecting with theworkpiece surface and the other side is cutting inside the thread profile.3.5 Systematic Chip DiscretizationFigure 3.5.a illustrates the chip flow lines in cutting a buttress profile. It can beseen that around the corners where two edges meet, the flow lines overlap. This isdue to chip interference and happens when the material removed by different edgescompress into each other. Figure 3.5.b shows a sample case of stress distribution(using AdvantEdge) in cutting a V-profile thread. It can be seen that there is higherstress concentration at the nose due to chip interference.Remark. The chip flow angle at the corners is also affected by the chip interfer-ence. However, since the cutting forces are generated very close to the cutting zone,the change in the chip flow angle due to interference causes less than 1% error inthe force modelling, and thus has been ignored in this thesis.The following sections present a systematic discretization method which forms24Chapter 3. Chip Geometry and Cutting Forcesa) b)Figure 3.5: a) Overlapping of chip flow lines due to chip interference, b) stressdistribution in cutting V60◦ thread.the chip elements along the cutting edge taking into account the chip flow direc-tion and chip interference. The proposed method draws the discretization linesby running a search algorithm which pairs proper points on the upper and lowerbands. Since the procedure is carried out for each tooth individually, the super-script j (tooth number) and subscript np (pass number) are dropped in the followingderivations for simplicity.As illustrated in Figure 3.6, at every point Pu(k) on the upper band, where k =1,2, ...,Nu, the local vector Lu(k) tangent to the upper band is calculated as:Lu(k) = Pu(k+1)−Pu(k) (3.20)Similarly, at every point Pl(i) on the lower band, where i = 1,2, ...,Nl , the localtangent vector is Ll(i) = Pl(i+ 1)−Pl(i). For a fixed point Pu(k) on the upperband and for any point Pl(i) on the lower band, vector h˜(k, i) which connects Pu(k)to Pl(i) is calculated as (Figure 3.6):h˜(k, i) = Pl(i)−Pu(k) (3.21)Scalar projection of h˜(k, i) on the upper band is obtained using inner product asSu(k, i) = L¯u(k) · h˜(k, i) (3.22)where L¯u(k) is the unit average tangent vector at Pu(k):L¯u(k) =Lu(k−1)+Lu(k)|Lu(k−1)+Lu(k)| =Pu(k+1)−Pu(k−1)|Pu(k+1)−Pu(k−1)| (3.23)25Chapter 3. Chip Geometry and Cutting ForcesPulowerbandupperbandXY(k)(i)= (i )(k-1)(k+1)h(k,i)~ h(k,i+1)~h(k,i-1)~θθ=0θMovingWindowL (k)uL (i)ls (k,i-1)uβ (k,i+1)u β (k,i+1)lPl(i+1)Pl(i-1)PlPl kPlPuPuPuFigure 3.6: Search algorithm and forming the chip discretization lines.The unit average vector L¯l(i) tangent to the lower band at Pl(i) is calculated similarto Eq. (3.23). Projection of h˜(k, i) on the lower band is obtained by inner productas Sl(k, i) = L¯l(i) · h˜(k, i). The angles βu (on the upper band) and βl (on the lowerband) between the discretization line h˜(k, i) and the positive direction of L¯u(k) andL¯l(i), respectively, are calculated as:βw(k, i) = cos−1(Sw(k, i)∣∣h˜(k, i)∣∣), w= u, l →{−180◦ < βu < 00 < βl < 180◦(3.24)where βu lies in the third or forth quadrants of the trigonometric circle, and βl liesin the first or second quadrants. In order to have the discretization lines normalto both the cutting edge and lower band, βu and βl must be ideally −90◦ and 90◦,respectively. In this case,βs(k, i) = βu(k, i)+βl(k, i) = 0 (3.25)However, due to the varying curvature of the edge profile as well as chip interfer-ence in the corners, satisfying the condition in Eq. (3.25) may not be possible. Thevalue of |βs(k, i)| quantifies the deviation of the discretization line from orthogonal-ity, and thus can be used as a pairing criterion; every point Pu(k) on the upper bandis paired with a point Pl(i) on the lower band which results in the smallest absolutevalue of βs(k, i).In order to improve the reliability of the discretization scheme, additional con-straints are imposed on the angles βu and βl; the discretization lines are allowed tohave maximum deviation of 30◦ from orthogonality to the upper band and 60◦ from26Chapter 3. Chip Geometry and Cutting Forcesorthogonality to the lower band. The pairing criteria can therefore be summarizedas:Pu(k)→ Pl(i) :min{|βu(k, i)+βl(k, i)|}|βu(k, i)+90|< 30◦|βl(k, i)−90|< 60◦(3.26)Since for every point Pu(k) on the upper band, the proper point Pl(i) on the lowerband is expected to be found at a close vicinity of Pu(k), the search algorithm is runonly over a moving window as illustrated in Figure 3.6. This window extends 5%1 of the length of the lower band around Pl(im), where im is the index of the pointwhich divides the lower band at the same ratio that Pu(k) divides the upper band,i.e.:imNl=kNu→ im = NlNu k (3.27)where Nu and Nl are the total number of points on the upper and lower bands,respectively. In order to avoid intersection of the discretization lines, the searchalgorithm skips the points of the lower band which lie before the last paired pointPl(ip). The moving window can therefore be described as:Pu(k)→ Pl(i) : max(ip, im−0.05Nl)< i< im+0.05Nl (3.28)where ip is the index of the last paired point on the lower band, and im is defined inEq. (3.27).The procedure presented above is run for each point Pu(k) on the upper band,and connects it to a proper point on the lower band. As a result, the number ofdiscretization lines (and thus chip elements) is equal to the number of points on theupper band. Depending on the geometry of the chip and curvature of the threadprofile, several discretization lines might be connected to the same point on thelower band, and on the other hand, some points of the lower band might not beconnected to any discretization line.Figure 3.7 shows sample results for the multi-point V-profile and buttress insertswith the discretization length of du = 10µm. It can be seen that the chip has beenuniformly discretized even at the highly curved segments where chip interference issignificant. For verification, the total area calculated as the summation of individual1Due to the convex shape of thread profiles, the proper discretization line always lie within the5% moving window. Hence, further extension of the window does not improve the accuracy.27Chapter 3. Chip Geometry and Cutting Forces1Pass#1220.10.20.30.40.50.60.70.80.91Y[mm]Figure 3.7: Sample chip discretization results (du= 10µm). 0123450.0010.010.11Error in Chip Area (%) Discretization Length (du) [mm] Figure 3.8: Sensitivity of chip modelling to the discretization length (du) (In-sert: three-point V-profile, chip thickness per tooth: 0.3 mm).chip elements (Eq. (3.42)) has been compared against the CAD models in NX (byintersecting the cross sections of the insert and the workpiece). In all cases, the erroris below 0.5%. Figure 3.8 shows the sensitivity of the calculated chip area to thediscretization length (du) for the three-point V-profile insert shown in Figure 3.7. Itcan be seen that reducing the discretization length below 10µm does not affect theaccuracy significantly, thus is not practical.3.6 Cutting Force CalculationThis section presents a systemic methodology to calculate the local and total cuttingforces.28Chapter 3. Chip Geometry and Cutting ForcesRTFVUFVFUVCαλη rakefaceha)VVwf XYαγαγ VtLVtclearancefacecuttingedgerakefacelocal normalplanesb)Figure 3.9: a) General definition of oblique cutting parameters, b) localoblique vectors.3.6.1 Local Oblique Cutting AnglesMechanics of oblique cutting is illustrated in Figure 3.9.a. The rake angle α isdefined as the angle between the rake face and the vector normal to the cut surface.Inclination λ is the relative angle between the cutting edge and the vector normalto the cutting velocity in the plane of cut. Clearance γ is the angle between theclearance face and the cut plane. η is the chip flow angle, which is assumed to bethe same as inclination, i.e. η = λ , as suggested by Stabler [75].As illustrated in Figure 3.9.b, the cutting velocity in thread turning consistsof two components corresponding to the rotation of the workpiece (Vw) and axialfeedrate of the cutting tool (V f ):Vt = Vw+V f ←{Vw = [0,0,−(pidwn)/60]T (circumferential)V f = [hp. n/60,0,0]T (axial)(3.29)where dw is the diameter of the workpiece, n [rev/min] is the spindle speed, and hpis the thread pitch. In most regular turning operations, the axial feed (Vf ) is rela-tively small and can be ignored compared to the circumferential velocity. In threadturning, however, the tool has to travel axially one pitch over each spindle revo-lution; depending on the workpiece diameter, pitch of the thread, and the spindlespeed, the axial feedrate can be considerable in thread turning. Both componentsare considered in this thesis for generality.29Chapter 3. Chip Geometry and Cutting ForcesAs illustrated in Figure 3.9.b, the angle between the velocity vector Vt and thecutting planes varies along the threading tooth. In order to calculate the effectiveoblique angles systematically, three orthogonal unit vectors have been defined lo-cally at each point along the cutting edge: vector L¯ tangent to the cutting edge,vector α¯ normal to rake face, and vector γ¯ normal to the clearance face. At eachpoint Pu(k) on the cutting edge (Figure 3.6), the unit tangent vector L¯(k) is obtainedas (repeated from Eq. (3.23)):L¯(k) =Pu(k+1)−Pu(k−1)|Pu(k+1)−Pu(k−1)| (3.30)The rake and clearance unit vectors are calculated from their projections on the XYplane and Z axis:Γ¯(k) = TIG.Γ¯IC(k) = TIG.Γ¯xy(k)+ Γ¯z(k)∣∣Γ¯xy(k)+ Γ¯z(k)∣∣ , Γ¯= α¯, γ¯ (3.31)where α¯xy(k) = (sinα0)[cosθ(k),−sinθ(k),0]Tα¯z(k) =[0,0,(cosα0)]T (3.32) γ¯xy(k) = (cosγ0)[−cosθ(k),sinθ(k),0]Tγ¯z(k) =[0 0 (−sinγ0)]T (3.33)where α0 and γ0 are the nominal rake and clearance angles of the insert, and TIG isthe insert-to-global coordinate transformation matrix defined in Eq. (3.2). θ(k) isthe local approach angle defined in Figure 3.6.b and calculated as:θ(k) = tan−1(Pu,y(k+1)−Pu,y(k)Pu,x(k+1)−Pu,x(k))(3.34)considering the four quadrants of the trigonometric circle. Based on the geometricdefinitions in Figure 3.9.a, the effective local oblique angles at each point Pu(k)along the threading tooth are calculated as:rake angle: α(k) =−pi2+ cos−1(α¯(k) ·(L¯(k)× Vt|Vt |))(3.35)30Chapter 3. Chip Geometry and Cutting Forces0 1 2 3−202468EffectiveLAnglesL[deg]EdgeLLengthL[mm]0 1 2 3−4−202468EffectiveLAnglesL[deg]EdgeLLengthL[mm]RakeL00V-Profile Insert Buttress Insert0 2 4−4−202468EffectiveLAnglesL[deg]EdgeLLengthL[mm]a) c)b) d)Nose0 2 4−202468EffectiveLAnglesL[deg]EdgeLLengthL[mm]RakeL00clearancerakeinclinationRootRakeL30RakeL30clearancerakeinclinationclearancerakeinclinationclearancerakeinclinationFigure 3.10: Sample effective oblique angles for V-profile and buttress insertswith rake angles of α0 = 0◦ and α0 = 3◦ (workpiece diameter: 40 mm,thread pitch: 5 mm, clearance angle: 5◦)inclination angle: λ (k) =−pi2+ cos−1(L¯(k) · Vt|Vt |)(3.36)clearance angle: γ(k) =pi2− cos−1(γ¯(k) · Vt|Vt |)(3.37)where (·) and (×) denote inner and cross products, respectively.Figure 3.10 shows the calculated oblique angles along sample V-profile andbuttress inserts (dimensions given in Figure 1.1.c). The diameter of the workpieceand the thread pitch are assumed to be 40 mm and 5 mm, respectively, and theinserts have nominal clearance angle of 5◦. The oblique angles have been calculatedfor two values of rake angle: α0 = 0◦ and α0 = 3◦. It can be seen that the effectiveangles vary considerably along the cutting edge.31Chapter 3. Chip Geometry and Cutting ForcesXYFXY VtL-VtL-a)b)FAFRFTfcFrcFtcFfcFtcFrcUfcUrcUtcUtcUfcUrcFigure 3.11: a) Local and total forces exerted on the tool, b) unit directionvectors of the local cutting forces.3.6.2 Local Cutting Force CoefficientsFigure 3.11.a illustrates the local tangential, feed, and radial cutting forces exertedon the tool at two locations along the cutting edge. For each chip element k, the unitvectors defining the direction of the local forces can be calculated as (Figure 3.11.b):Utc(k) =Vt|Vt | (3.38)U f c(k) =Vt× L¯(k)∣∣Vt× L¯(k)∣∣ (3.39)Urc(k) =(Utc(k)×U f c(k)) · sgn(U f c,x(k)) (3.40)where Vt and L¯(k) are the velocity and local edge vectors defined in Eqs. (3.29)and (3.30), respectively. The magnitudes of the local force components are:32Chapter 3. Chip Geometry and Cutting ForcesupperbandlowerbandXYPuPu(k+1)(k)h(k+1)h(k)(ik)(i +1)kPlPlb u(k)b l(k)Figure 3.12: Edge vectors defining the chip elements.Ftc(k)Ff c(k)Frc(k)= Ktc(k)K f c(k)Krc(k) ·Ac(k) (3.41)where Ac(k) is the area of the chip element k, and Ktc, K f c, and Krc are the localcutting force coefficients. As illustrated in Figure 3.12, the area of the chip elementk can be calculated using the cross product of the element’s edge vectors:Ac(k) =12{|h(k)×bl(k)|+ |h(k+1)×bu(k)|} (3.42)where h(k) and h(k+1) are the thickness vectors, and bl(k) and bu(k) are the widthvectors on the upper and lower bands, respectively. These vectors are calculated as(Figure 3.12):h(k) = Pl(ik)−Pu(k) (3.43)h(k+1) = Pl(ik+1)−Pu(k+1) (3.44)bu(k) = Pu(k+1)−Pu(k) (3.45)bl(k) = Pl(ik+1)−Pl(ik) (3.46)where ik is the index of the point (on the lower band) paired with the point Pu(k)of the upper band (see Section 3.5). As marked in Figure 3.12, the cross productsin Eq. (3.42) give the area of the lower and upper triangles, respectively. Usingthis approach, the area is calculated accurately even in the case of distorted or threeedge elements.For each chip element k, the local cutting force coefficients in Eq. (3.41) are33Chapter 3. Chip Geometry and Cutting Forcesobtained as: Ktc(k)K f c(k)Krc(k)= Tob(k)[Ku(k)Kv(k)](3.47)where Tob(k) is the local orthogonal-to-oblique transformation, and Ku and Kv arethe friction and normal cutting force coefficients, respectively ([76]). Tob(k) mapsthe parameters from the UV to TFR coordinates shown in Figure 3.9. Due to thevariation of oblique angles along the cutting edge, this transformation must be cal-culated locally for each chip element k as:Tob(k) = cosλ (k) sinλ (k) 0−sinλ (k) cosλ (k) 00 0 1 cosα(k) 0 sinα(k)0 1 0−sinα(k) 0 cosα(k) 0 1sinη(k) 0cosη(k) 0= sinλ (k)sinη(k)+ cosλ (k)sinα(k)cosη(k) cosλ (k)cosα(k)cosα(k)cosη(k) −sinα(k)−cosλ (k)sinη(k)+ sinλ (k)sinα(k)cosη(k) sinλ (k)cosα(k)(3.48)where α(k) and λ (k) are the local effective rake and inclination angles calculatedin Eqs. (3.35) and (3.36), respectively; η(k) is the local chip flow angle, which isassumed equal to the inclination angle, i.e. η(k) = λ (k) [75].Figure 3.13 illustrates the cutting edge and the chip removal mechanism. Typ-ical threading inserts have edge radius of around 50µm 2. During the cutting pro-cess, some of the material is compressed under the round section of the cuttingedge, resulting in higher friction and normal forces. The effect of ploughing be-comes more significant when the chip thickness is equal to or smaller than the edgeradius. In threading buttress profile, the chip thickness on the sides is typicallyfew microns while the chip load at the root can be as large as 0.5 mm. Therefore,linearization of cutting force coefficients is not a realistic assumption. This thesisuses nonlinear Kienzle force model [77]; the friction (Ku) and normal (Kv ) cuttingcoefficients are modelled as (see Figure 3.15):Ki = kc1,i (h¯)−mc,i , i= u,v (3.49)2Measured by the author for several inserts during his industrial internship at Sandvik Coromant,Sweden, 2014.34Chapter 3. Chip Geometry and Cutting Forceschiptool tiptrajectoryrecoveredmaterialhVtploughedmaterialinsertFigure 3.13: Ploughing and the effect of edge radius.where kc1 and mc are the Kienzle coefficients, and h¯ is the average chip thicknesscalculated for each element k as:h¯(k) =12(|h(k)|+ |h(k+1)|) (3.50)where h(k) and h(k+ 1) are defined in Eqs. (3.43) and (3.44), respectively. Thecutting coefficients Ku and Kv in Eq. (3.49) depend on the workpiece material, insertcoating, edge radius, chip thickness, cutting speed, and other tool-workpiece char-acteristics. Finite element modelling can be used to predict these coefficients semi-analytically using slip line field [78]. In this thesis, however, in order to minimizethe errors originating from the cutting coefficients, Ku and Kv have been identifiedexperimentally as presented in Section 3.7.1.3.6.3 Total Cutting ForcesThe local static cutting forces for each chip element k are obtained from Eqs. (3.41)and (3.47):Fs,L(k) =Ftc(k)Ff c(k)Frc(k)= Tob(k)[Ku(k)Kv(k)].Ac(k) (3.51)35Chapter 3. Chip Geometry and Cutting Forceswhere Ac(k) and [Tob(k)]3×2 are defined in Eqs. (3.42) and (3.48), respectively.The local forces are projected in the global coordinates as:Fs(k) =Fs,x(k)Fs,y(k)Fs,z(k)= TLG(k)Ftc(k)Ff c(k)Frc(k) (3.52)where [TLG(k)]3×3 is the local-to-global transformation at the location of elementk:TLG(k) =[{Utc(k)},{U f c(k)},{Urc(k)}] (3.53)where Uic, i = t, f ,r, are the local unit direction vectors (Figure 3.11.b) defined inEqs. (3.38)-(3.40). The resultant cutting force vector on each tooth j is calculatedby summing all the elemental forces on the tooth, i.e:jFs=jFs,xjFs,yjFs,z=jNe∑k=1jFs(k)=jNe∑k=1{Ftc(k)Utc(k)+Ff c(k)U f c(k)+Frc(k)Urc(k)}j(3.54)where jNe = jNu is the number of chip elements (points on the upper band) ontooth j. Finally, the total cutting forces exerted on the insert are obtained as:Fs =Nt∑j=1( jFs) (3.55)where Nt is the total number of teeth engaged in the cut.3.7 Experimental Validation of Mechanics ModelIn order to validate the proposed force prediction model, numerous sets of threadingexperiments have been conducted using different inserts and infeed plans. Sampleresults are presented in this section. 33All experiments in this section were carried out at Sandvik Coromant in Sweden during theauthor’s industrial internship in 2014.36Chapter 3. Chip Geometry and Cutting ForcesXYF vFutwh bfaXYa) b)c)θFigure 3.14: Semi-orthogonal tests, a) experimental setup (insert: SandvikCoromant), b) chip geometry, c) friction and normal forces.3.7.1 Semi-Orthogonal Identification TestsIn order to identify the cutting force coefficients, a set of semi-orthogonal cut-ting tests have been performed using a V-profile threading insert. As shown inFigure 3.14.a, a workpiece with concentric tubes of 1.95 mm wall thickness hasbeen prepared by opening circular slots on the face of a solid cylinder. The work-piece material is AISI 1045, and the insert is Sandvik Coromant 22V401A0503EV-profile (60◦). The cutting tool has been positioned such that only one straightedge of the insert cuts the workpiece, and the round nose is out of cut. Each tubehas been cut at different values of axial feedrate fa in the range of 0.056-0.615mm/rev. Cutting speed in all experiments is 150 m/min.As illustrated in Figure 3.14.b, the width (b) and thickness (h) of the chip ineach test is calculated as:h= fa sinθ , b= tw/sinθ (3.56)where tw = 1.95 mm is the wall thickness and θ = 60◦ is the approach angle 4.The forces have been measured using a 3-axis Kistler 9121 turning dynamometer,and data acquisition has been implemented in CUTPRO R©MALDAQ software [79].4Due to the fixed orientation of the tool and dynamometer, positioning the cutting edge normalto the feed direction was not possible37Chapter 3. Chip Geometry and Cutting ForcesTable 3.1: Parameters and measured forces in the semi-orthogonal identifica-tion tests (material: AISI 1045, cutting speed: 150 m/min, insert: Sand-vik Coromant V-profile.fa [mm/rev] h [mm] b [mm] Fx [N] Fy [N] Fz [N] Fu [N] Fv [N]0.056 0.048 2.252 274 161 383 318 3830.063 0.055 2.252 297 173 417 343 4170.080 0.069 2.252 356 208 509 412 5090.098 0.085 2.252 396 233 582 460 5820.126 0.109 2.252 453 268 708 527 7080.154 0.133 2.252 502 302 826 585 8260.196 0.170 2.252 528 322 970 618 9700.252 0.218 2.252 586 367 1184 691 11840.308 0.267 2.252 754 458 1465 882 14650.392 0.339 2.252 782 495 1776 925 17760.615 0.533 2.252 954 617 2562 1135 2562The axes of the dynamometer are aligned with the global coordinates, i.e. XYZ inthe axial, radial, and tangential directions, respectively. The normal force (Fv) andfriction force (Fu) (Figure 3.14.c) are determined as:Fu = Fx sinθ +Fy cosθ , Fv = Fz (3.57)Table 3.1 provides the settings and the measured forces.For each test, cutting force coefficients Ku and Kv are obtained by dividing thecorresponding forces by the chip area, i.e. Ki = Fi/(bh), i= u,v. Figure 3.15 showsthe cutting force coefficients plotted as a function of chip thickness. It can be seenthat the trend is highly nonlinear due to the effect of ploughing (Figure 3.13) espe-cially below 50µm. Least square method has been used to fit a nonlinear Kienzlemodel (Eq. 3.49) to this data, resulting in:kc1,u = 1204.3MPa , mc,u = 0.384kc1,v = 691.6MPa , mc,v = 0.534(3.58)These coefficients are used in all the threading experiments presented in the nextsection.38Chapter 3. Chip Geometry and Cutting ForcesKv/=/1204.3/h-0.384Ku/=/691.6/h-0.534025005000750010000125000.000 0.050 0.100 0.150 0.200 0.250Cutting//Force//Coeff./[N/mm2 ]Chip/Thickness/)ha/[mm]Normal/)vaFriction/)uaFigure 3.15: Experimentally identified cutting force coefficients (material:AISI 1045, cutting speed: 150 m/min).Figure 3.16: Setup for the threading experiments. Workpiece material: AISI1045, diameter: 176 mm, cutting speed: 150 m/min.3.7.2 Validation of Threading Force PredictionFigure 3.16 shows the setup used for the threading tests. The workpiece is AISI1045 solid cylinder with diameter of 176 mm; it has been restricted by a tailstockat the end to avoid deflections and chatter vibrations. The cutting speed and threadpitch in all experiments are 150 m/min and 5 mm5, respectively. The forces havebeen measured using a 3-axis Kistler turning dynamometer with the load capacityof 25 KN, and data acquisition has been implemented in CUTPRO R© MALDAQsoftware [79].5More accurately, the pitch is 5 TPI (thread per inch), i.e. 25.4/5=5.08 mm (API Standard [5]).39Chapter 3. Chip Geometry and Cutting ForcesV-ProfileFor this set of experiments, Sandvik Coromant 266RG-22V401A0503E V-profileinsert with nose angle of 60◦ and nose radius of 0.5 mm has been used. Fig-ure 3.17.a compares the predicted and measured cutting forces in the tangential,radial, and axial directions for 14 passes of 0.15 mm/pass radial infeed. The chipthickness at the nose is equal to the infeed, i.e. 0.15 mm, and the chip load on thesides is 0.15 sin(30◦) = 0.075 mm. The axial forces are relatively small due to thesymmetry of the insert profile. It can be seen that all the predicted forces agreewith the measurements within 95% accuracy. Figure 3.17.b shows the results forthe same operation but using 6 passes of 0.3 mm/pass. The accuracy of predictionsin this case lies within 80% for all passes. The details are provided in Tables 3.2and 3.3.It can be observed that in both Figures 3.17.a and b, the discrepancies betweenthe predicted and measured forces grow as the insert penetrates deeper into thethread. This is caused by two main reasons:1. At deeper passes, a longer part of the tooth is engaged in the cut resulting inwider chip and more severe chip interference.2. Due to severe ploughing of material under the cutting edge in the precedingpasses, surface hardness increases incrementally as a result of strain hard-ening, which has not been modelled in this thesis. Chip thickness (infeed)is particularly smaller in deeper passes (to limit the forces), leading to sev-erer ploughing in deeper passes. The strain hardening effect can be includedby updating the cutting force coefficients over each pass based on the newcharacteristics of the cut surface. This can be done by predicting the surfacehardness using finite element simulations. Alternatively, the cutting forcecoefficients subject to strain hardening can be identified experimentally byrepeated orthogonal cutting tests on the same surface.The accuracy of the proposed mechanics model can be improved by modelling theeffect of stain hardening.Figure 3.17.c compares the simulated and measured forces for the same insertwith flank infeed of 0.15 mm/pass. The resultant chip load is 0.15 mm on one sideof the insert and zero on the other side. The detailed infeed parameters and forces40Chapter 3. Chip Geometry and Cutting Forces0300600900120015000 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25ForceR[N]DepthR[mm]a)ASingleAV-profile,ARadialAInfeedA0.15Amm/passsimexp TangentialRadialAxial050010001500200025000 0.25 0.5 0.75 1 1.25 1.5 1.75 2ForceR[N]Depth [mm]b)ASingleAV-profile,ARadialAInfeedA0.3Amm/passsimexpTangentialRadialAxial-500-300-10010030050070090011000 0.25 0.5 0.75 1 1.25 1.5 1.75ForceR[N]DepthR[mm]c)ASingleAV-profile,AFlankAInfeedA0.15Amm/passsimexpTangentialRadialAxial -300.0-100.0100.0300.0500.0700.0900.00 0.25 0.5 0.75 1 1.25ForceR[N]DepthR[mm]d)ASingleAV-profile,AAlternateAFlankA0.15Amm/passsimexp TangentialRadialAxialFigure 3.17: Simulated and measured cutting forces for threading with singleV-profile insert (Sandvik Coromant 22V401A0503E, workpiece mate-rial: AISI 1045, diameter: 176 mm, thread pitch: 5 mm, cutting speed:150 m/min)are provided in Table 3.4. Even though the simulation and experiment still agreewithin 90%, the discrepancy is slightly larger compared to the same operation withradial infeed (Figure 3.17.a). This is due to the fact that in flank infeed plan, oneside of the insert is constantly rubbing on the thread surface, resulting in higherforces.Figure 3.17.d shows the simulated and measured forces for the case of alternateflank infeed with 0.15 mm/pass. The details are provided in Table 3.5. The axialoffset has been set in the same direction as the feed for even pass numbers, andopposite to feed for odd passes (see Figures 1.2.a and 3.3.c). It can be seen thatthere is a close agreement over even passes; for odd pass numbers, however, thesimulation predicts relatively large axial forces (due to single-edge cutting) whilethe measured axial forces are almost zero. This means that the free side of the insert,which is supposed to be sliding freely on the thread surface, is in fact experiencing41Chapter 3. Chip Geometry and Cutting ForcesTable 3.2: Simulated and measured forces in threading with V-profile insert;radial infeed plan at 0.15 mm/pass (Figure 3.17.a).Infeed Depth Simulation ExperimentPass ∆a [mm] a [mm] Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N]1 0.15 0.15 5 153 208 -2 153 1992 0.15 0.30 8 213 335 0 208 3333 0.15 0.45 10 249 421 3 245 4234 0.15 0.60 12 284 506 7 291 5205 0.15 0.75 14 320 591 9 321 5976 0.15 0.90 16 355 675 9 364 6957 0.15 1.05 18 391 760 10 405 7748 0.15 1.20 20 426 845 11 450 8739 0.15 1.35 21 461 929 13 492 96310 0.15 1.50 23 497 1014 14 535 105211 0.15 1.65 25 533 1099 16 572 113012 0.15 1.80 27 568 1183 17 607 120613 0.15 1.95 29 603 1268 17 651 128914 0.15 2.10 31 640 1353 26 675 1355Table 3.3: Simulated and measured forces in threading with V-profile insert;radial infeed plan at 0.3 mm/pass (Figure 3.17.b).Infeed Depth Simulation ExperimentPass ∆a [mm] a [mm] Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N]1 0.3 0.3 5 227 478 5 255 4162 0.3 0.6 9 330 816 17 362 7503 0.3 0.9 12 422 1101 29 483 10594 0.3 1.2 15 514 1386 40 612 13705 0.3 1.5 18 606 1671 51 738 16766 0.3 1.8 22 699 1955 63 861 197742Chapter 3. Chip Geometry and Cutting ForcesTable 3.4: Simulated and measured forces in threading with V-profile insert;flank infeed plan at 0.15 mm/pass (Figure 3.17.c).Infeed Depth Simulation ExperimentPass ∆a [mm] a [mm] Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N]1 0.15 0.15 5 153 208 -1 160 1962 0.15 0.30 -9 213 330 -17 208 3223 0.15 0.45 -46 238 400 -49 225 4004 0.15 0.60 -86 263 466 -92 261 4775 0.15 0.75 -128 287 532 -130 300 5996 0.15 0.90 -169 312 598 -168 330 6317 0.15 1.05 -209 336 664 -208 365 7168 0.15 1.20 -251 361 730 -237 396 7799 0.15 1.35 -292 385 796 -269 422 85310 0.15 1.50 -333 410 862 -309 454 93211 0.15 1.65 -374 435 928 -344 481 1008Table 3.5: Simulated and measured forces in threading with V-profile insert;alternate flank infeed at 0.15 mm/pass (Figure 3.17.d).Infeed Depth Simulation ExperimentPass ∆a [mm] a [mm] Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N]1 0.15 0.15 1 140 219 -2 160 2012 0.15 0.30 -14 193 349 -21 210 3643 0.15 0.45 53 218 423 -6 258 4354 0.15 0.60 -82 243 496 -86 269 4835 0.15 0.75 128 267 567 -2 335 6206 0.15 0.90 -155 292 643 -147 364 6657 0.15 1.05 204 316 712 -13 440 8098 0.15 1.20 -226 340 785 -185 450 845severe rubbing. The rubbing force counteract the cutting forces of the engagedside, resulting in smaller total axial force. The friction and normal rubbing forcescause an increase in the tangential and radial forces as well. Flank rubbing mustbe reduced through insert design and has not been modelled in this thesis. Theagreement between the simulation and experiment over even pass numbers stillprove that the model can predict the cutting forces accurately if rubbing is avoided.43Chapter 3. Chip Geometry and Cutting ForcesSingle Buttress InsertIn the second set of threading experiments, Sandvik Coromant single buttress in-sert (266RG-22BU01A050E) has been used to cut 17 passes at radial infeed of0.075 mm/pass. The resultant chip thickness on the left and right sides of the toothare 13µm and 4µm, respectively (see Figure 1.1.c). Figure 3.18.a compares thesimulated and measured forces, and Table 3.6 provides the detailed data. Whilethere is a close agreement over the first half of passes, the deviation grows towardsthe end. The insert was examined after the experiment, and large shiny patcheswere found on the side flank faces. These shiny areas are caused due to the severegrinding between the flank face and the thread surface, which result in additionalforces. Rubbing must be reduced through insert design and is not modelled here.Twin Buttress InsertIn the final set, a Sandvik Coromant Tmax Twin-Lock two-point API buttress insert(R166.39G-24BU12-050) has been used to cut the same buttress profile but usingtwo teeth. The radial depth of the second tooth with respect to the first tooth is0.26 mm, and the infeed values over the first, second, and third passes are 0.46 mm,0.44 mm, and 0.40 mm, respectively. Based on the insert design, the first toothmainly performs rough cutting of the root, and the second tooth performs finishingof the thread profile (see Figure 3.4.a). Figure 3.18.b compares the simulated andmeasured forces, and Table 3.7 provides the detailed data. It can be seen that the020040060080010000 0.25 0.5 0.75 1 1.25ForceR[N]DepthR[mm]a) Single Buttress, Radial Infeed 0.075 mm/passsimexp TangentialRadialAxial01000200030000.25 0.5 0.75 1 1.25RForce[N]DepthR[mm]simexpAxialButtress Twin, Radial InfeedTangentialRadialb) Figure 3.18: Simulated and measured cutting forces for threading with singleand two-point API buttress inserts (Sandvik Coromant API ButtressFull Form and Twin, material AISI 1045, cutting speed: 150 m/min,workpiece diameter: 176 mm, thread pitch: 5 mm)44Chapter 3. Chip Geometry and Cutting ForcesTable 3.6: Simulated and measured forces in threading with single buttressinsert; radial infeed at 0.075 mm/pass (Figure 3.18.a).Infeed Depth Simulation ExperimentPass ∆a [mm] a [mm] Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N]1 0.075 0.075 8 326 374 2 313 2952 0.075 0.150 13 436 569 8 490 5213 0.075 0.225 18 441 588 17 524 5974 0.075 0.300 21 442 596 27 518 6105 0.075 0.375 23 443 602 31 513 6276 0.075 0.450 26 444 608 33 511 6467 0.075 0.525 29 446 614 33 503 6628 0.075 0.600 32 447 620 28 504 6719 0.075 0.675 35 448 626 42 517 69510 0.075 0.750 37 449 632 47 527 72011 0.075 0.825 40 450 638 38 537 75012 0.075 0.900 43 451 645 32 540 76813 0.075 0.975 46 452 651 23 547 79514 0.075 1.050 48 453 657 20 551 81715 0.075 1.125 51 454 663 12 569 84216 0.075 1.200 54 456 669 17 566 86017 0.075 1.275 57 457 675 21 578 887Table 3.7: Simulated and measured forces in threading with twin buttress in-sert (Figure 3.18.b).Infeed Depth Simulation ExperimentPass ∆a [mm] a [mm] Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N]1 0.46 0.46 7 1463 2289 -45 1226 22042 0.44 0.90 81 1517 2546 27 1350 26603 0.40 1.30 164 1480 2569 57 1405 2881simulation and experiment agree within 85% accuracy for all passes.3.8 SummaryThis chapter presents a generalized and semi-analytical approach to model the chipgeometry and predict the cutting forces in thread turning with custom multi-pointinserts. The boundaries of the chip are determined based on the insert geometry,infeed settings, and kinematics of the process. A systematic search algorithm has45Chapter 3. Chip Geometry and Cutting Forcesbeen proposed to discretize the chip area along the cutting edge considering the lo-cal chip flow direction and the effect of chip interference. Cutting force coefficientsare evaluated locally for each chip element, and the total cutting forces are deter-mined by summation of the element forces. The proposed mechanics model hasbeen verified experimentally for V-profile, single buttress, and twin buttress insertswith different infeed plans; the methodology and results have been published in [1].46Chapter 4Dynamics of Multi-Point Threading4.1 OverviewThis chapter investigates the general dynamics of multi-point thread turning op-erations. Section 4.2 studies the chip regeneration mechanism, followed by mod-elling the dynamic forces in Section 4.3 to 4.5. Equations of motion in frequencyand time domain (modal space) are derived in Sections 4.6 and 4.7, respectively.A time-marching numerical simulation method based on semi-discretization andSimpson’s integration rule is presented in Section 4.8, followed by sample resultsin Section 4.9. The proposed mechanics and dynamics models are used to developa process optimization algorithm in Section 4.11, and the chapter is concluded inSection 4.12.4.2 Chip Regeneration MechanismThe static cutting force vector Fs in Chapter 3 has been derived based on the nom-inal (planned) infeed settings. Relative vibrations between the tool and workpieceresult in two additional components: 1) dynamic cutting force Fd due to chip thick-ness variation, 2) process damping force Fp due to dynamic indentation of the flankface into the cut surface. The total force vector exerted on the tool at time t is:{Fc(t)}3×1 = Fs(t)+Fd(t)+Fp(t) (4.1)where the static forceFs(t) has been written as a time-dependent variable to accountfor the transient condition at the start of the operation (see Section 4.5 for details).47Chapter 4. Dynamics of Multi-Point ThreadingAssume that the deflection of the tool and workpiece at the cutting point dueto the structural flexibilities are represented in the three-dimensional coordinates as{qt(t)}3×1 and {qw(t)}3×1, respectively. Relative vibration between the tool andworkpiece is obtained as:q(t) = qt(t)−qw(t) =qt,x(t)−qw,x(t)qt,y(t)−qw,y(t)qt,z(t)−qw,z(t)=qx(t)qy(t)qz(t) (4.2)It is assumed that the width of the insert is considerably smaller than the lengthof the workpiece, hence q(t) is the same for all teeth. The vibration vector atone spindle revolution before is represented as q(t−T ), or in Laplace domain asq(s)e−T s, where T is the spindle period.Figure 4.1.a illustrates the effect of current and previous vibrations on the chipthickness. Regeneration mechanism in multi-point threading is different than reg-ular turning in that the vibration marks left by each tooth affect the chip thicknesson a different tooth. It involves additional complexities due to the fact that theteeth may have different profiles (see Chapter 3). The closed loop dynamics of chipregeneration in each pass is illustrated in Figure 4.1.b. ∆a and ε f x are the infeedsettings for the pass, q(s) is the relative vibration vector in Laplace domain, andG(s) represents the structural dynamics of the tool-workpiece setup.Dynamic chip thickness and the resultant dynamic forces are derived in thefollowing section.4.3 Calculation of Dynamic Cutting ForcesFigure 4.2 illustrates a sample threading chip and the effect of current and previousvibrations. At each point along the cutting edge, only the vibration component inthe local feed direction affects the chip thickness. A systematic methodology isproposed in this section to calculate the dynamic forces along the threading insert.4.3.1 Dynamic Chip on The Upper BandAs illustrated in Figure 4.2, the upper band of the chip corresponds to the currenttooth-workpiece engagement. For each chip element k along the cutting edge, the48Chapter 4. Dynamics of Multi-Point Threading+Δa,εfx ToothF1 Ft1F2FNtFq(s)e-TsthreadFprofilefromFpreviousFpassG(s)newFthreadFprofileq(s)q(s)q(s)InsertInfeedToothF2InfeedToothFNtnewFthreadFprofilefinalFthreadFprofile(overFtheFcurrentFpass)delaychipFgeometry ForceFModelToothF2ToothFNtForceFModelForceFModelchipFgeometrychipFgeometrye-TsZYXY2 3Insert1Tool FeedWorkpiecevibration marksfrom previous passpreviousvibration markscurrentvibration marksq(t)a)b)Figure 4.1: Chip regeneration mechanism in multi-point thread turning.unit vector in the local feed (chip thickness) direction can be obtained as:U f u(k) =bu(k)×Vt|bu(k)×Vt | (4.3)where× denotes cross product, bu(k) is the element’s edge vector on the upper band(Eq. (3.45)), and Vt is the total velocity vector calculated in Eq. (3.29). The localdynamic chip thickness on the upper band, i.e. hdu(k, t), is obtained by projectingthe relative vibration vector in the direction of local chip thickness:hdu(k, t) = q(t) ·U f u(k) =U f u,x(k)qx(t)+U f u,y(k)qy(t)+U f u,z(k)qz(t) (4.4)49Chapter 4. Dynamics of Multi-Point ThreadingXYPuPlVtXYcurrentvibrations vibrationspreviousvibrationshhlocal effectivechip thicknessVtb l(k)Uf l(k)hdl(k,t)Adl(k,t)b u(k)hdu(k,t)Adu(k,t)q(t)q(t)Ufu(k)Figure 4.2: Dynamic chip area due to the current and previous vibrations.or in matrix form:hdu(k, t) = {U f u(k)}Tq(t) (4.5)Note that hdu(k, t) can take negative values when the tool and workpiece deflectaway from each other. The dynamic chip area on the upper band (Figure 4.2) canbe calculated as:Adu(k, t) = |bu(k)|hdu(k, t) = |bu(k)|{U f u(k)}Tq(t) (4.6)where bu(k) is the width vector on the upper band (Eq. (3.45)). Negative dynamicchip thickness results in negative dynamic chip area until the cutter disengages fromthe workpiece due to large vibrations (see Section 4.8.1). Using Eq. (3.51), thedynamic cutting force vector on chip element k due to the current vibrations isobtained as:Fdu(k, t) = [Tob(k)]3×2[Ku(k)Kv(k)].Adu(k, t) (4.7)where Tob(k) is the local orthogonal-to-oblique transformation (Eq. (3.48)), andKu(k) andKv(k) are the local friction and normal cutting force coefficients (Eq. (3.49)),respectively.Remark: Based on Kienzle model (Eq. (3.49)), Ku and Kv are functions ofinstantaneous chip thickness. This means that vibrations affect not only the chiparea but also the cutting force coefficients. As a result, Ku and Kv in Eq. (4.7)50Chapter 4. Dynamics of Multi-Point Threadingare time dependent. In order to reduce the numerical computation, it is assumedin this thesis that vibrations are sufficiently smaller than the static (nominal) chipthickness. Hence, the dependency of cutting force coefficients on dynamic chipthickness is neglected.4.3.2 Dynamic Chip on The Lower BandAs illustrated in Figures 4.1 and 4.2, the effect of previous vibrations appear asdynamic chip on the lower band. Since these vibration marks are generated by theprofile of the previous tooth, q(t−T ) must be projected in the local feed directionat each point along the previous tooth. Since the lower band on the current toothmatches the cutting edge of the previous tooth, the dynamic chip thickness due toprevious vibrations can be calculated as (Figure 4.2):hdl(k, t) =−{U f l(k)}Tq(t−T )=−[U f l,x(k)qx(t−T )+U f l,y(k)qy(t−T )+U f l,z(k)qz(t−T )] (4.8)where the minus sign is to account for the fact that positive vibrations over theprevious cut (overcutting) result in reduction in the chip thickness on the currenttooth. U f l(k) in Eq. (4.8) is the unit vector in the local feed direction on the lowerband:U f l(k) =bl(k)×Vt|bl(k)×Vt | (4.9)where bl is the width vector defined in Eq. (3.46). For each chip element k along thecutting edge, the dynamic chip area due to the previous vibration marks (Figure 4.2)can be calculated as:Adl(k, t) = |bl(k)|hdl(k, t) =−|bl(k)|{U f l(k)}Tq(t−T ) (4.10)and the corresponding dynamic cutting force vector on the lower band is:Fdl(k, t) = [Tob(k)]3×2[Ku(k)Kv(k)].Adl(k, t) (4.11)51Chapter 4. Dynamics of Multi-Point ThreadingXY21Tool Feedvibration marksfrom previous passindependentof currentvibrationslargevibrationslargevibrationslargevibrationsFigure 4.3: Chip regeneration on the first tooth.4.3.3 Dynamic Chip on The First ToothThe first tooth has a different chip regeneration mechanism. Assume that the pre-vious pass was stable with relatively small vibrations. As illustrated in Figure 4.3,if the current pass causes large vibrations, the resultant marks appear on both thelower and upper bands of all teeth except the first tooth. Vibration marks from theprevious pass act as disturbance on the first tooth (Figure 4.1.b), but they do notcontribute to the closed loop chip regeneration in the current pass.In this thesis, the dynamics and stability is analyzed based on the assumptionthat the previous pass has been stable with small vibrations. As a result, the dynamicchip on the first tooth is generated by only the current vibrations (upper band), aspresented in Section 4.3.1.Remark: Note that since all teeth are rigidly connected to each other, the chip onthe first tooth is still indirectly affected by the chip regeneration on the other teeth.Large dynamic forces on even one tooth can lead to large vibrations on all teeth.4.3.4 Total Dynamic Cutting ForcesThe local dynamic cutting force vector is calculated for each element k by addingthe components of the upper and lower bands derived in Eqs (4.7) and (4.11), re-spectively:Fd(k, t) = [Tob(k)]3×2[Ku(k)Kv(k)]. (Adu(k, t)+Adl(k, t)) (4.12)52Chapter 4. Dynamics of Multi-Point ThreadingThe local force vectors are projected in the global coordinates and summed up overeach tooth j:jFd(t) =jNe∑k=1{TLG(k)Fd(k, t)}=jNe∑k=1{TLG(k)Tob(k)Kuv(k) [|bu(k)|hdu(k, t)+ |bl(k)|hdl(k, t)]}(4.13)where Kuv = [Ku,Kv]T , and transformation matrix TLG is defined in Eq. (3.53).Using Eqs. (4.5) and (4.8), the dynamic force vector can be written in terms of thecurrent and previous vibrations as:jFd(t) =[ jKdc(t)]3×3{q(t)}3×1− [ jKdd(t)]3×3{q(t−T )}3×1 (4.14)where jKdc(t) and jKdd(t) are the equivalent current and delayed dynamic forcecoefficients:[ jKdc(t)]3×3 = jNe∑k=1[TLG(k)Tob(k)Kuv(k) |bu(k)|U f u(k)][ jKdd(t)]3×3 =[0]3×3 if j = 1jNe∑k=1[TLG(k)Tob(k)Kuv(k) |bl(k)|U f l(k)]if j > 1(4.15)and jNe is the number of chip elements on tooth j. Note that the first tooth has beenexcluded from the delayed term.4.4 Calculation of Process Damping ForcesAs illustrated in Figure 4.4.a, process damping forces are generated due to the in-dentation of the flank face of the tool in the undulated cut surface when the toolis vibrating in the direction of chip thickness. As modelled by Shaw and DeSalvo[51], the component in the feed direction, Fp f , is proportional to the volume of thecompressed material (Vcm) under the flank face of the tool:Fp f = KspVcm (4.16)53Chapter 4. Dynamics of Multi-Point ThreadingindentedmaterialLwchiptool tiptrajectoryflank facehnormalvibrationsFpfVtXYLwFpf vibrationsq(t) flankwearFpta) b)FptFTRFigure 4.4: Local process damping forces in threading.where Ksp is the material-dependent indentation force coefficient obtained experi-mentally. The normal force, Fp f , induces a dynamic friction force in the tangentialdirection:Fpt = µFp f (4.17)where µ is the Coulomb friction between the tool and work material, which isassumed as 0.3 for typical metal cutting operations.Figure 4.4.b illustrates the local process damping forces on a threading tooth.Similar to the static and dynamic cutting forces, the direction and magnitude of pro-cess damping components vary along the cutting edge. As proposed by Chiou andLiang [52], the volume of the compressed material is proportional to the vibrationvelocity in the direction of chip thickness (feed). The relative vibration velocitybetween the tool and workpiece is obtained from Eq. (4.2) as:q˙(t) =dq(t)dt= q˙t(t)− q˙w(t) (4.18)For each element k along the cutting edge, q˙(t) is projected in the direction of localchip thickness as:{U f u(k)}T {q˙(t)}=U f u,xq˙x+U f u,yq˙y+U f u,zq˙z (4.19)where U f u(k) is calculated in Eq. (4.3). The cross section of the indented materialat the location of element k is obtained using the projected vibration velocity ([52]):Acm(k) =Lw2(k)2|Vt | U f u(k)T q˙(t) (4.20)54Chapter 4. Dynamics of Multi-Point Threadingwhere Vt is the total cutting velocity vector, and Lw(k) is the local width of thewear land on the flank face (Figure 4.4.b). The process damping model is sensitiveto Lw, therefore Lw is normally measured with a microscope before and after theexperiments. Lw is around 100µm in typical inserts. The local volume of theindented material is calculated as:Vcm = |bu(k)|Acm(k) (4.21)where bu(k) is the width of the element on the cutting edge (Eq. (3.45)). Hence,the local process damping forces in the feed and tangential directions are:Fp f (k) = Ksp|bu(k)|Lw2(k)2|Vt | U f u(k)T q˙(t)Fpt(k) = µFp f (k)(4.22)where Ksp and µ are the indentation and Coulomb friction coefficients, respec-tively. Considering the directions of these two components (Figure 4.4.b), the pro-cess damping force vector represented in the local TFR coordinate system can bewritten as:Fp,L(k) =Fpt(k)Fp f (k)0TFR=µ10Fp f (k) (4.23)Fp,L(k) is projected to the global coordinates using TLG(k) transformation matrix(Eq. (3.53)). The total process damping force vector on tooth j is then obtained bysummation of the element forces:jFp(t) =jFp,xjFp,yjFp,z=jNe∑k=1([TLG(k)]3×3Fp,L(k))=[ jCp]3×3{q˙(t)}3×1 (4.24)where jCp is the equivalent process damping coefficient matrix for tooth j:[ jCp]3×3 = jNe∑k=1Ksp|bu(k)|Lw2(k)2|Vt | TLGµ10U f u(k)T (4.25)55Chapter 4. Dynamics of Multi-Point Threading4.5 Total Forces on The InsertIn order to model the transient condition at the start of the operation when the teetharrive in the cut one by one, a boolean (unit step) function g j is defined for eachtooth j as:g j(t) = 1(t− ( j−1)T ) ={1 if t ≥ ( j−1)T0 otherwise(4.26)where T is the spindle period. Reference time t = 0 is when the first tooth engagesin the workpiece, after which a new tooth comes into the cut upon each spindlerevolution. Since most machines perform a rapid retract at the end of the threadingcycle, the transient condition at the exit has not been considered in this thesis.Assuming that the width of the threading insert is considerably smaller thanthe length of the workpiece, the total forces on the tool can be calculated by lumpsummation of the static, dynamic, and process damping forces on all teeth j =1,2, . . . ,Nt :{Fc(t)}[3×1] =Nt∑j=1{g j(t)[ jFs+ jFd(t)+ jFp(t)]} (4.27)Using the expressions obtained for these components in Eqs. (3.54), (4.14), and(4.24), the total force vector can be summarized and re-written in terms of vibrationvector q(t) as:Fc(t) = Fs(t)+ [Kdc(t)]q(t)− [Kdd(t)]q(t−T )+ [Cp(t)] q˙(t) (4.28)whereFs(t) =Nt∑j=1g j(t).jNe∑k=1[TLG(k)]3×3[Tob(k)]3×2{kc1uh¯(k)−mcukc1vh¯(k)−mcv}︸ ︷︷ ︸Ke(k)As(k)(4.29)[Kdc(t)]3×3 =Nt∑j=1(g j(t)jNe∑k=1(Ke(k) |bu(k)|U f u(k)T))(4.30)56Chapter 4. Dynamics of Multi-Point Threading[Kdd(t)]3×3 =Nt∑j=2(g j(t)jNe∑k=1(Ke(k) |bl(k)|U f l(k)T))(4.31)[Cp(t)]3×3 =Nt∑j=1g j(t). jNe∑k=1Ksp|bu(k)|Lw2(k)2|Vt | TLGµ10U f u(k)T(4.32)where Kdc, Kdd , and Cp are the equivalent dynamic coefficient matrices corre-sponding to the current, delayed, and process damping forces on the whole insert.4.6 Stability Analysis in Frequency DomainAssume the three-dimensional frequency response functions (FRF) of the tool (Gt)and workpiece (Gw) at the cutting location are expressed in Laplace domain as:Gi(s) = Gxx,i(s) Gxy,i(s) Gxz,i(s)Gyx,i(s) Gyy,i(s) Gyz,i(s)Gzx,i(s) Gzy,i(s) Gzz,i(s)3×3, i= t,w (4.33)where the terms on the main diagonal correspond to the direct FRFs, and the off-diagonal terms account for the cross couplings. All the FRFs are obtained either byFE modal analysis or experimental hammer tests (see Section 5.5). Critical stabilityof the process is analyzed based on the most severe loading, which happens when allteeth are engaged in the cut. In this case, g j(t) = 1 for all teeth, and the coefficientmatrices in Eqs. (4.30)-(4.32) become constants. Hence, the total force vectorcalculated in Eq. (4.28) can be represented in Laplace domain as:Fc(s) = Fs/s+[Kdc]q(s)− [Kdd]e−T sq(s)+ [Cp]sq(s) (4.34)where T is the spindle period. Force Fc exerted on the tool and the reaction force−Fc exerted on the workpiece result in the three-dimensional vibrations of the tooland workpiece:qt(s) = [Gt(s)]3×3{Fc(s)}3×1 (4.35)qw(s) = [Gw(s)]3×3{−Fc(s)}3×1 (4.36)57Chapter 4. Dynamics of Multi-Point ThreadingRelative vibration vector is calculated as (Eq. 4.2):q(s) = qt(s)−qw(s) = [G(s)]3×3Fc(s) (4.37)where G is the relative structural dynamics between the tool and workpiece:G(s) = Gt(s)+Gw(s) (4.38)The closed loop equation of chip regeneration is obtained by substituting Fc(s)from Eq. (4.34) into Eq. (4.37):q(s) = G(s)([Kdc]− [Kdd]e−T s+[Cp]s)q(s) (4.39)where the static force Fs has been dropped as it does not contribute to chip regener-ation (perturbation theory). The characteristic equation of the process is therefore:∣∣I3×3−G(s)([Kdc]− [Kdd]e−T s+[Cp]s)∣∣s= jω = 0 (4.40)where I3×3 is the identity matrix and j is the unit imaginary number. Stability ofthe process is analyzed in frequency domain using Nyquist criterion; for a giventool-workpiece setup, pass number, infeed settings, and spindle speed, the coeffi-cient matrices are calculated from Eqs (4.30)-(4.32). The value of the characteristicfunction Λ( jω),Λ( jω) = I3×3−G( jω)([Kdc]− [Kdd]e−T jω +[Cp] jω)(4.41)is then calculated for different frequencies (ω) and plotted on the Real-Imaginaryplane. Stability is determined based on the encirclement of the origin (Nyquistcriterion). Sample results and numerical discussions are presented in Section 4.9.Remarks1. If the structural dynamics of the workpiece (G(s)) varies along the axial di-rection, critical stability must be analyzed based on the most dynamicallyflexible point.2. Stability analysis is performed for each pass individually; if the structural58Chapter 4. Dynamics of Multi-Point Threadingdynamics of the workpiece changes due to material removal, the updatedFRFs can be used for the next passes.4.7 Dynamic Equation of Motion in Time DomainIn order to simulate the response of the system during the entire operation, thissection sets up the dynamic model in time domain. The equations are derived inmodal space since the resultant matrices are decoupled and sparse. In order todecrease the computation load, the structural dynamics of the tool and workpieceare approximated using mt and mw number of dominant modes, respectively. Thereduced transfer functions (FRFs) of the tool and workpiece can be constructed inLaplace domain as [44]Gi(s) = Uˆi(Is2+2ζiωn,is+ωn,i2)−1 UˆTi , i= t,w (4.42)where I[mi×mi] (i = t,w) is the identity matrix, and ζi[mi×mi] and ωn,i[mi×mi] are thediagonal damping ratio and natural frequency matrices, respectively. Uˆi[3×mi] (i =t,w) is the mass-normalized mode shape matrix of the tool and workpiece at thecutting location:Uˆi =ux,1,iuy,1,iuz,1,iux,2,iuy,2,iuz,2,i · · ·ux,mi,iuy,mi,iuz,mi,i3×mi, i= t,w (4.43)where each column is a mode shape vector showing the relative displacements ofthe cutting point in the three directions (global XYZ) when the system vibrates inthat mode. Vibrations of the tool and workpiece can be transformed to the modalspace as:{qi(t)}[3×1] ={Uˆi}[3×mi] {Ψi(t)}[mi×1] , i= t,w (4.44)where qi and Ψi are the displacement vectors in the physical and modal spaces, re-spectively. The relative vibration vector between the tool and workpiece (Eq. (4.2))is:q(t) = qt(t)−qw(t) = UˆtΨt(t)− UˆwΨw(t) (4.45)59Chapter 4. Dynamics of Multi-Point ThreadingUsing Eq. (4.28), the total force vector is written in terms of modal displacements:Fc(t) = Fs(t)+ [Kdc(t)](UˆtΨt(t)− UˆwΨw(t))− [Kdd(t)](UˆtΨt(t−T )− UˆwΨw(t−T ))+[Cp(t)](UˆtΨ˙t(t)− UˆwΨ˙w(t))(4.46)Dynamic equations of motion for the tool and workpiece (in Laplace domain) areobtained by combining Eqs. (4.35), (4.36), (4.42), and (4.44):{ (Is2+2ζtωn,ts+ωn,t2)Ψt(s) = UˆTt Fc(s)(Is2+2ζwωn,ws+ωn,w2)Ψw(s) =−UˆTwFc(s)(4.47)The modal displacement vectors of the tool and workpiece are put together in asingle vectorΓ (t) ={Ψt(t)Ψw(t)}mt+mw, (4.48)and the force expression from Eq. (4.46) is substituted into Eq. (4.47). The com-bined modal dynamic equation of the tool and workpiece in the time domain and inmatrix form is:Γ¨ (t)+(Cm,1+Cm,2(t))Γ˙ (t)+(Kmc,1+Kmc,2(t))Γ (t)+Kmd(t)Γ (t−T ) =Fsm(t)(4.49)whereCm,1 =[2ζtωn,t 0[mt×mw]0[mw×mt ] 2ζwωn,w], Cm,2(t) =−[UˆTt−UˆTw][Cp(t)][Uˆt −Uˆw]Kmc,1 =[ω2n,t 0[mt×mw]0[mw×mt ] ω2n,w], Kmc,2(t) =−[UˆTt−UˆTw][Kdc(t)][Uˆt −Uˆw]Kmd(t) =[UˆTt−UˆTw][Kdd(t)][Uˆt −Uˆw]Fsm(t) =[UˆTt−UˆTw]Fs(t)(4.50)60Chapter 4. Dynamics of Multi-Point ThreadingCm,i, Kmc,i (i=1,2), and Kmd are square matrices of size (mt +mw)× (mt +mw),and Cp, Kdc, and Kdd have 3× 3 dimensions; mode shape matrices Uˆi,(i = t,w)are of size 3×mi, and Fsm(t) and Fs(t) are column vectors of size (mt +mw)× 1and 3×1, respectively.It is numerically more efficient to solve the equation of motion in state space. Ifthe state vector is defined as:Ω(t) ={Γ (t)Γ˙ (t)}2(mt+mw)×1, (4.51)Eq. (4.49) can be transformed to state space as:Ω˙(t) = A0Ω(t)+Bc(t)Ω(t)+Bd(t)Ω(t−T )+S(t) (4.52)where A0, Bc(t), and Bd(t) are the constant, current, and delayed state matrices(each of size 2(mt+mw)×2(mt+mw)), respectively, and column vector S(t) is thepiecewise1 static forcing function (length 2(mt+mw)):A0 =[0 I−Kmc,1 −Cm,1], Bc(t) =[0 0−Kmc,2(t) −Cm,2(t)]Bd(t) =[0 0−Kmd(t) 0], St(t) =[{0}[(mt+mw)×1]{Fsm(t)}[(mt+mw)×1]] (4.53)where all the 0 and I entries are of size (mt+mw)×(mt+mw), and the other entriesare defined in Eq. (4.50). Since the static force (S(t)) has been included, thesolution of Eq. (4.52) consists of the vibrations as well as the static deflections ofthe tool and workpiece. Hence, the time domain solution allows us to predict notonly the stability of the process but also the accuracy and surface location errors inthe machined threads.4.8 Time-Marching Numerical SimulationIn order to simulate the full time history of the process, this section presents a time-marching numerical method to solve Eq. (4.52) consecutively at each time step.Semi-discretization technique proposed by Insperger and Stepan [20] is combined1Due to the transient condition at the beginning of the process.61Chapter 4. Dynamics of Multi-Point Threadingt i-mt i-m+1t i+1t it i-m+2Ti-mi-m+1ΩΩΩi-m+2ΩiΩi+1ΩtFigure 4.5: Semi-discretization of delay differential equations [20].with the Simpson’s three-point integration rule to numerically evolve the solutionover time. As demonstrated in Figure 4.5, each spindle revolution (period T ) isdivided into m intervals of ∆t = T/m. In each time interval t0 < t < t0 +∆t, thelast three terms in Eq. (4.52) are approximated by their values at t = t0. The delaydifferential equation (DDE) then turns into an ordinary differential equation (ODE)over the interval, the solution of which can be obtained analytically. Since A0 is atime-invariant matrix, the solution to the ODE in the current interval can be writtenas [40]:Ω(t) = eA0(t−t0)Ω(t0)+∫ tt0eA0(t−τ)H(τ)dτH(τ) = Bc(τ)Ω(τ)+Bd(τ)Ω(τ−T )+S(τ)(4.54)whereΩ(t0) is the state value at initial time t0. The integral in Eq. (4.54) is approx-imated using Simpson’s integration rule; at each time step ti = t0 +(i− 1)∆t,(i =2,3, ...), the state value Ω(ti) is obtained from the available function values H atthree integration points (ti−2, ti−1, ti):Ω(ti) = eA0(ti−ti−2)Ω(ti−2)+∫ titi−2eA0(ti−τ)H(τ)dτ= e2A0∆tΩ(ti−2)+(2∆t)6{e2A0∆tH(ti−2)+4eA0∆tH(ti−1)+H(ti)}(4.55)Equation (4.55) is an implicit relation sinceH(ti) requires the valueΩ(ti) (Eq. 4.54).If all the H functions in Eq. (4.55) are expanded using Eq. (4.54), Ω(ti) can be62Chapter 4. Dynamics of Multi-Point Threadingobtained explicitly as:Ω(ti) =(I− ∆t3Bc(ti))−1 [e2A0∆tΩ(ti−2)+∆t3e2A0∆t {Bc(ti−2)Ω(ti−2)+Bd(ti−2)Ω(ti−2−m)+S(ti−2)}+4∆t3eA0∆t {Bc(ti−1)Ω(ti−1)+Bd(ti−1)Ω(ti−1−m)+S(ti−1)}+∆t3{Bd(ti)Ω(ti−m)+S(ti)}](4.56)where m= T/∆t is the number of time steps in one spindle revolution, and I is theidentity matrix of size 2(mt+mw)×2(mt+mw). For simulation of the first spindlerevolution, all the delay states (initial conditions) are assumed to be zero, i.e.∀i,0≤ i≤ m−1 : Ω(ti−m) = {0}[2(mt+mw)×1] (4.57)Equation (4.56) cannot be used for calculation of Ω(t1) since coefficient matricesare not defined at t < 0. The time-marching simulation is initialized by calculatinga dummy stateΩ(t0+∆t/2) using two-point trapezoidal integration:Ω(t0+∆t2) = eA0∆t/2Ω(t0)+(∆t/2)2{eA0∆t/2H(t0)+H(t0+∆t2)}, (4.58)which along with Eq. (4.57) leads to the following explicit expression:Ω(t0+∆t2) =(I− ∆t4Bc(t0+∆t2))−1 [eA0∆t/2Ω(t0)+∆t4eA0∆t/2 {Bc(t0)Ω(t0)+S(t0)}+∆t4{S(t0+∆t2)}] (4.59)Ω(t1) is then calculated by three-point Simpsons rule using the function values attimes t0, t0+∆t/2, and t1:Ω(t1) = eA0∆tΩ(t0)+∆t6{eA0∆tH(t0)+4eA0∆t/2H(t0+ ∆t2 )+H(t1)} (4.60)63Chapter 4. Dynamics of Multi-Point Threadingor in explicit form:Ω(t1) =(I− ∆t6Bc(t1))−1 [eA0∆tΩ(t0) +∆t6eA0∆t {Bc(t0)Ω(t0)+S(t0)}+4∆t6eA0∆t/2{Bc(t0+∆t2)Ω(t0+∆t2)+S(t0+∆t2)}+∆t6{S(t1)}](4.61)Once the state vector Ω(ti) is calculated, the vibration vector of the tool and work-piece can be extracted from Eqs. (4.44), (4.48), and (4.51). The resultant forces ateach time step are simulated using Eq. (4.28).Remark. The time domain solution can be used to determine the stability of theprocess by simulating the vibrations. It is important to ensure that instability of thepredicted vibrations is due to the instability of the process and not the numericalmethod. In this thesis, the minimum required time discretization for numericalstability has been found by trial and error, but the stability region can be foundanalytically using advanced theories of numerical methods.Comparison With Euler SolutionIn order to examine the efficiency and stability of the presented numerical method,the equation of motion (Eq. (4.49)) has been solved using Euler numerical methodas well. If the derivative terms are approximated as:Γ¨ (ti) =Γ (ti+2)−2Γ (ti+1)+Γ (ti)∆t2(4.62)Γ˙ (ti) =Γ (ti+1)−Γ (ti)∆t, (4.63)and substituted in Eq. (4.49), the explicit time-marching expression for the statevector is obtained as:Γ (ti+2) = E3Γ (ti+1)+E2Γ (ti)+E1Γ (ti−m)+E0 (4.64)64Chapter 4. Dynamics of Multi-Point ThreadingXY1 2 3InsertTool Feed600R=0.50.3mm 0.3mm5mm3mminfeed0 500 100000.511.52xq10−7Magnitudeq[m/N]Frequencyq[Hz]a) b)Figure 4.6: Simulation setup: a) sample multi-point V-profile insert, b) FRFof the workpiece in the radial direction.where E3 = 2I−∆t(Cm,1+Cm,2(ti))E2 =−I+∆t(Cm,1+Cm,2(ti))−∆t2(Kmc,1+Kmc,2(ti))E1 =−∆t2Kmd(ti)E0 = ∆t2Fsm(ti)(4.65)All the initial conditions are assumed to be zero.4.8.1 Remarks: Cutter Disengagement Due to Large VibrationsIn the case of large vibrations, one or more teeth can disengage from the workpiecefor part of the vibration cycle. At each time ti, if the vibrations separating the tooland workpiece are larger than the chip thickness on tooth j, the boolean functiong j(ti) (Eq. (4.26)) is set to zero until the tooth comes back in the cut. In the casewhen all teeth jump out of cut, all the forcing coefficients in Eq. (4.50) become zeroexcept the structural matrices Cm,1 and Kmc,1. In this case, both the tool and work-piece undergo free vibration state until they engage again. A sample simulationinvolving out-of-cut jumps is presented in the next section (Figure 4.7.e).4.9 Sample Simulation Results and DiscussionsThis section highlights important technical and numerical remarks in modelling andsimulation of multi-point threading operations. A sample V-profile insert shown inFigure 4.6.a has been used for the simulations. The pitch of the thread is 5mm,and all passes are performed at 1850 rpm spindle speed. The workpiece material65Chapter 4. Dynamics of Multi-Point ThreadingTable 4.1: Dynamic parameters of the workpiece used in simulations.ωn [Hz] ζ ks [N/m]Mode 1 300 2% 1.5×108Mode 2 700 1% 3.0×108is AISI1045 steel with cutting force coefficients identified in Section 3.7.1. Theindentation coefficient used in the process damping model is Ksp = 4×1013 N/m3.As shown in Figure 4.6.b, it is assumed that the structural dynamics of the work-piece is dominated by two modes in the radial direction. Table 4.1 provides thedynamic parameters for these two modes.Nyquist criterion in frequency domain has been used to analyze the stability ofthe process for each pair of infeed value (∆a) and spindle speed (n). Figure 4.7.ashows the predicted stability chart for the first pass, where the area below the lobesrepresent the stable region. In order to capture the shape of the lobes with moredetails, a grid of 100 infeed values and 500 spindle speeds has been used. On a PCwith i5 core and 3.10 GHz CPU, it takes about 10 minutes to generate this chartwith such a dense grid (50,000 points). In practice, however, the absolute stabilitylimits can be detected reliably in less 30 seconds using a coarser grid.In order to verify the frequency domain stability chart in Figure 4.7.a, the time-marching numerical method presented in Section 4.8 has been used to simulate therelative displacement between the tool and workpiece over 50 spindle revolutions.Figures 4.7.b-d show the results for three infeed values close to the stability limitat 1850 rpm spindle speed. It can be seen that the point below the lobes (1.6 mm,1850 rpm) results in decaying vibrations while the point above (2.0 mm, 1850 rpm)leads to instability. The process is marginally stable at (1.8 mm, 1850 rpm), whichfully agrees with the frequency domain lobes. Figure 4.7.e shows the system re-sponse to an aggressive infeed of 3 mm. Even though the process is unstable, thevibrations cannot grow indefinitely due to cutter disengagement (Section 4.8.1).It can be observed in Figure 4.7.b that in addition to the high frequency vibra-tions, there is a 0.02 mm average offset in the simulated displacements. This isdue to the static deflection of the workpiece under the cutting forces, and resultsin undercutting the thread profile over the current pass. The uncut material addsto the chip load and cutting forces in the following pass; the static deflections over66Chapter 4. Dynamics of Multi-Point Threading0 1000 2000 3000 4000 50000.511.520 1000 2000 3000 4000 50000.511.50 1000 2000 3000 4000 50000.20.40.60.811.20 1000 2000 3000 4000 50000.20.30.40.50.60.7Pass]2]cprevious]depth:]0.7]mmD Pass]3]cprevious]depth:]1.4]mmDPass]4]cprevious]depth:]1.9]mmD Pass]5]cprevious]depth:]2.4]mmDSpindle]Speed][rpm] Spindle]Speed][rpm]Spindle]Speed][rpm]Spindle]Speed][rpm]Infeed][mm]Infeed][mm]Infeed][mm]Infeed][mm]Pass]1Spindle]Speed][rpm]Infeed][mm]0 1000 2000 3000 4000 50000.511.522.530 20 40−0.04−0.020Displacement][mm]Spindle]Revolution0 20 40−0.06−0.04−0.020Displacement][mm]Spindle]Revolution0 20 40−0.200.2Displacement][mm]Spindle]Revolution1.6]mm,]1850]rpm1.8]mm,]1850]rpm2.0]mm,]1850]rpmaDfD gDhD iDSimulationSimulationSimulationbDcDdD0 20 40−4−2024Displacement][mm]Spindle]Revolution3.0]mm,]1850]rpmSim.eDFigure 4.7: Stability charts and time simulations for threading with three-point V-profile insert (material: AISI 1045).67Chapter 4. Dynamics of Multi-Point Threadingthe final pass directly translates to defects and inaccuracy in the thread profile. Inpractice, the infeed value over the final pass is chosen relatively small to reducethese errors.Stability over each pass depends on the previous thread depth. As marked bysmall circles in Figures 4.7.a and f, assume that the infeeds for both the first andsecond passes are chosen as 0.7mm (to avoid large cutting forces). The stabilitycharts for the third and fourth passes are shown in Figures 4.7.g and h, respectively.The absolute stability limit at 1850 rpm is only 0.6mm, which means cutting withonly two teeth (#2 and #3); engaging tooth #1 leads to instability for any infeedvalue. Considering a safety margin, the infeeds for both third and fourth passesare chosen as 0.5mm. Assume that the desired final depth for this application is2.6mm. Figure 4.7.i shows the stability chart for the final pass with the previousdepth of 2.4mm. The final infeed is thus 0.2mm, which leads to cutting with onlytooth #3.4.10 Numerical Remarks4.10.1 Efficiency of Time-marching Numerical MethodsSemi-discretization and Euler’s methods (Section 4.8) have been used to simulatethe stable point (1.6 mm, 1850 rpm) in Figure 4.7.b over 50 spindle revolutions.The two methods have been compared based on two criteria:1. Stability: Minimum number of discretization intervals m (per spindle revolu-tion) required to correctly predict the stability of the threading process.2. Convergence: Minimum number m such that further increase in m does notchange the amplitudes and shape of the simulated vibrations considerably.Table 4.2 summarizes the comparison results, and the simulated vibrations areshown in Figure 4.8. It can be seen that SD method is stable with m = 70 whileEuler’s method requires at least m= 57,000. For convergence, SD method requiresm= 115 and can simulate the process over 50 spindle revolutions in 0.06 seconds.Euler’s method, on the other hand, requires at least m = 230,000 and 35 secondsto simulate the same process. This is due to the fact that SD technique solves68Chapter 4. Dynamics of Multi-Point Threading0 50−0.1−0.0500.050.1DisplacementD[mm]SpindleDRevolution0 50−0.04−0.03−0.02−0.010DisplacementD[mm]SpindleDRevolution0 50−0.04−0.03−0.02−0.010DisplacementD[mm]SpindleDRevolution0 50−0.04−0.03−0.02−0.010DisplacementD[mm]SpindleDRevolution0 50−0.04−0.03−0.02−0.010DisplacementD[mm]SpindleDRevolution0 50−2−1012xD1010DisplacementD[mm]SpindleDRevolutiona)DEulerb)DSemi-discretizationm=115m=70m=45m=34,000 m=57,000 m=230,000stabilitystabilityconvergenceconvergenceNumericallyUnstableNumericallyUnstableFigure 4.8: Comparison of Semi-discretization and Euler’s methods for time-marching numerical simulation (structural frequencies: 300 Hz and700 Hz, spindle speed: 1850 rpm, infeed: 1.6 mm).Table 4.2: Minimum discretization intervals for stability and convergence oftime-marching simulations.Stability Convergence Simulation Timem m (50 rev, convergence)Semi-discretization 70 115 0.06 sEuler’s method 57,000 230,000 35 sthe approximated ODEs analytically in each time step while Euler’s method solvesalgebraic equations by approximating all the derivative terms numerically.In conclusion, not only SD method is faster than Euler discretization (by a factorof 500 in this example) but also it requires significantly less memory due to smallernumber of time steps.Remark 1. Since the presented numerical method in Section 4.8 is used to de-termine the instability of the machining process, it is important to ensure that the69Chapter 4. Dynamics of Multi-Point Threadingnumerical method itself is not unstable. Hartung et al. [80] proved mathematicallythat there exist a minimum discretization number (m) which guarantees the stabil-ity and convergence of the semi-discretization method. The minimum m dependson the process parameters and cannot be determined analytically. The suggestedapproach is to first simulate the vibrations for low infeed values where the pro-cess is stable, find the minimum discretization number for stability of the numericalmethod, and then generate the stability charts.Remark 2. Sensitivity of the predicted stability lobes (Figure 4.7.a, three-pointV-profile insert) to the discretization number m is analyzed in Figure 4.9. The ’ex-3200 3250 3300 3350 34001.61.822.22.42.62.8m=100m=200 m=20002000 3000 4000 50001.61.822.22.4m=100m=200m=2000Spindle Speed [rpm]Infeed [mm]Spindle Speed [rpm]Infeed [mm]a)b)(exact)Figure 4.9: Sensitivity of the stability lobes to numerical semi-discretization,a) stability lobes for the three-point V-profile insert, material:AISI 1045, b) comparison for different number of discretization m.70Chapter 4. Dynamics of Multi-Point Threadingact’ stability chart has been generated using m = 2000 points. The stability chartsfor m = 100 and m = 200 are also shown in Figure 4.9. It can be seen that whilethere is a slight difference between the exact solution and the lobes for the casewith m = 100, using m = 200 gives nearly identical stability lobes as m = 2000.Therefore, using m> 200 has no practical effect on the predicted stability charts inthis operation.4.10.2 Frequency Resolution in Nyquist Stability AnalysisAs presented in Section 4.6, stability of the process is analyzed by plotting Λ( jω),(repeated from Eq. 4.41)Λ( jω) = I3×3−G( jω)([Kdc]− [Kdd]e−T jω +[Cp] jω)(4.66)for ω ∈ (ω1 : ∆ω : ω2), where ∆ω is the frequency resolution (increment). If ∆ωis not sufficiently small, Nyquist criterion may lead to false stability prediction.To demonstrate this, consider a sample point (3 mm, 1500 rpm) in Figure 4.7.a.Figure 4.10.a shows Λ( jω) for ω in the range of 0-1000 Hz with 0.1 Hz increments.It can be seen that Λ( jω) forms a spiral shape with varying diameter; the circlingis mostly due to the phase contribution of the complex exponential terme−T jω = cosTω− j sinTω, (4.67)which has a periodic frequency equal to the spindle frequencyωsp =2piT(4.68)where T is the spindle period. This means that as ω is swept over the range offrequencies, Λ( jω) completes a full circle over each spindle frequency. The diam-eters of the circles are mainly determined by the FRF of the structure (G( jω)) ateach frequency. The bigger circles, which determine the stability of the process, areformed as ω approaches the resonance of the structure. It is therefore crucial thatthe swept frequencies include all the dominant structural modes.Figure 4.10.b and c show the plotted Λ( jω) for the same range of frequenciesbut with resolutions of 2 Hz and 5 Hz, respectively. It can be seen that despitethe undulations, the plot with ∆ω = 2 Hz still encircles the origin, thus correctly71Chapter 4. Dynamics of Multi-Point Threading−0.5 0 0.5 1 1.5 2 2.5−2−1.5−1−0.500.5ImagRealFrequencyIResolution:I0.1IHz−0.5 0 0.5 1 1.5 2 2.5−2−1.5−1−0.500.5ImagRealFrequencyIResolution:I2IHz−0.5 0 0.5 1 1.5 2 2.5−2−1.5−1−0.500.5ImagRealFrequencyIResolution:I5IHza)b) c)ModeI1(300IHz)ModeI2(700IHz)Figure 4.10: Nyquist plots with different frequency resolutions (structural fre-quencies: 300 Hz and 700 Hz, spindle speed: 1500 rpm, infeed: 3mm).implies instability. However, if 5 Hz resolution is used (Figure 4.10.c), Λ( jω) nolonger crosses the negative side of the imaginary axis, and the process is falselypredicted as stable.In order to capture the shape of the circles reliably, at least 10 to 20 data pointsare required along each circle. Hence, the frequency resolution must be smallerthan∆ωmax =ωsp10=2pi/T10(4.69)where T is the spindle period. If the measured FRF of the structure has beenrecorded with a coarser resolution, it must be numerically interpolated before us-ing Nyquist analysis. In the presented example with spindle speed of 1500 rpm,ωsp = 157 rad/s = 25 Hz, thus ∆ωmax = 2.5 Hz. In analyzing low speed threadingof oil pipes (120 rpm), the frequency resolution has to be as low as 0.2 Hz.72Chapter 4. Dynamics of Multi-Point Threading4.11 Process OptimizationOne of the main capabilities of the integrated systematic model developed in thisthesis is that not only can it simulate the process for a given condition, but alsoit can work backwards and help with process planning. An iterative optimizationengine has been developed to automatically determine the number of passes andinfeed values for maximum productivity while respecting user-defined constraints.For a given tool-workpiece setup, the operator selects the desired final thread depthand the cutting speed. The following practical constraints can be imposed:• Minimum chip thickness: to avoid severe ploughing• Maximum chip thickness: to avoid chip evacuation issues• Tangential force on tooth: to avoid tooth breakage• Radial force: to limit workpiece deflections• Axial force: to avoid pulling the workpiece out of the chuck• Spindle torque/power: to avoid machine stall• Stability margin: e.g. 20% below the marginal stability limitThe optimization module uses a binary search algorithm to find the maximum in-feed which satisfies the defined constraints. Optimization starts with the first pass,and for the first iteration, the lower and upper limits are set equal to the minimumand maximum allowed infeed values, respectively. The resultant chip thickness,forces, torque, and power are then calculated for the upper limit. If any of theseoutputs violate the constraints, the solution is rejected, and the upper limit is up-dated to the average of the previous lower and upper limits. The search continuesuntil the size of the search window is smaller than few microns. Considering thetypical infeed values in threading (smaller than 1 mm), the solution can always befound in less than 10 iterations (210 > 1000). The converged value for the infeed isthen compared to the predicted stability limit (at the desired spindle speed) in thecurrent pass. If it is unstable, the stability limit (minus the user-defined margin)must be chosen as the optimized infeed. After the first pass, new passes are addedand optimized one by one until the desired thread depth is reached.73Chapter 4. Dynamics of Multi-Point ThreadingTable 4.3: Optimization constraints for the case study with three-point V-profile insert (material: AISI 1045).Chip Thickness Forces Spindle StabilityMin Max Axial Radial Tangential Torque Power Margin0.04 mm 0.3 mm 1000 N 2000 N 5000 N 80 Nm 20 KW 20%Table 4.4: Optimization results and the limiting factors (material: AISI 1045,insert: three-point V-profile).Pass 1 Pass 2 Pass 3 Pass 4Selected Infeed 0.9 mm 0.8 mm 0.48 mm 0.42 mmLimiting Factor Max Chip Thickness Max Torque Stability Final DepthThe case study presented in Section 4.9 (Figure 4.6) has been optimized toachieve final thread depth of 2.6 mm subject to the constraints defined in Table 4.3.The selected infeed values and their corresponding limiting factors are providedin Table 4.4. Figure 4.11 shows the calculated values and stability charts for theoptimized process. As marked by LF1 in Figure 4.11.a, the first pass has beenlimited by the maximum allowed chip thickness (0.3 mm), resulting in total infeedof 0.9 mm. The second pass has been capped at infeed of 0.8 mm based on themaximum spindle torque (Figure 4.11.e). For the third pass, stability is the limitingfactor; based on Figure 4.11.i, infeed values up to 0.6 mm are stable. Consideringthe required 20% stability margin, the infeed for the third pass has been determinedas 0.48 mm. Finally, the desired thread depth can be achieved over the fourth passusing infeed of 0.42 mm, which satisfies all the constraints. On a PC with i5 coreand 3.10 GHz CPU, optimization of each pass takes less than 10 seconds.4.12 SummaryThis chapter studies the dynamics and stability of multi-point thread turning oper-ations. Chip regeneration model for custom multi-point inserts has been developedby studying the effect of current and previous vibrations. Dynamic cutting forcesand process damping forces have been calculated by projecting the vibrations in thedirection of local thickness at each point along the cutting edge. Dynamic equationof motion has been derived in the time and frequency domains, and stability of the74Chapter 4. Dynamics of Multi-Point Threadingg R A x779g79R79A79xChipDThicknessD[mm]PassDNumberg R A x79x796798gInfeedD[mm]PassDNumberg R A x7gRADepthD[mm]PassDNumberg R A x7jg7gjR7RjPowerD[KW]PassDNumberg R A xx76787g77TorqueD[Nm]PassDNumberg R A x−6777−x777−R7777ForceD[N]PassDNumberg777 R777 A777 x777 j77779jgg9jRR9jAPassDgSpindleDSpeedD[rpm]InfeedD[mm]g777 R777 A777 x777 j77779jgg9jRPassDRDFpreviousDdepth:D799DmmSSpindleDSpeedD[rpm]InfeedD[mm]g777 R777 A777 x777 j77779R79x796798gg9RPassDADFpreviousDdepth:Dg97DmmSSpindleDSpeedD[rpm]InfeedD[mm]g777 R777 A777 x777 j77779R79x796798PassDxDFpreviousDdepth:DR9g8DmmSSpindleDSpeedD[rpm]InfeedD[mm]ToothDg ToothDRToothDAThresholdThresholdThresholdThresholdThresholdThresholdDFTangentialSTangentialRadialAxialThresholdDFRadialSThresholdDFAxialSThresholdThresholdThresholdThresholdaS bScSdSeS fSgS hSiS jSLF4LF2LF3LF1Figure 4.11: Simulated results for the optimized plan (material: AISI 1045,insert: three-point V-profile).75Chapter 4. Dynamics of Multi-Point Threadingprocess is predicted using Nyquist criterion. A time-marching numerical methodhas been presented to simulate the process during the entire operation. Finally, thedeveloped models have been implemented in an optimization engine to maximizeproductivity while respecting user-defined constraints.76Chapter 5Threading Thin-Walled Oil Pipes5.1 OverviewThis chapter extends the dynamic model developed in Chapter 4 to threading thin-walled workpieces. Structural dynamics of clamped cylindrical shells are brieflydiscussed in Section 5.2, and the dynamic equation of motion for thin-wall thread-ing is derived in Section 5.3. The remainder of the chapter is dedicated to experi-mental validation of the proposed dynamic model. Extensive threading tests havebeen conducted on real scale oil pipes in TenarisTAMSA, Veracruz, Mexico1. Theexperimental setup is first introduced in Section 5.4, followed by finite element andexperimental modal analysis of the pipes in Section 5.5. Sample chatter experi-ments are presented and compared against simulations in Section 5.6. Section 5.7presents approaches for chatter suppression, and the developed threading simula-tion engine is presented in Section 5.8. The chapter is summarized in Section 5.9.5.2 Structural Dynamics of Cylindrical ShellsFigure 5.1 illustrates different vibration modes associated with a clamped cylindri-cal workpiece. The dominant vibration pattern depends on the ratio of the diame-1Process planning, measurements and analyses have been carried out by the author during hisindustrial visit in August 2016.77Chapter 5. Threading Thin-Walled Oil PipesBeam4Modes(Cantilever)Axial4Patterns(Shell4Modes)Circumferential4Patterns(Shell4Modes)XφLi44=2c i44=3c i44=4cj4=1l j4=2lj4=3lj4=1l j4=2l j4=3lFigure 5.1: Vibration modes of a clamped thin-walled workpiece.ter (dw) to wall thickness (tw) as [81]:dwtw< 5 → beam mode> 60 → shell mode≈ 30 → beam + shell(5.1)For a cylindrical shell with the length of L, the instantaneous radial vibration ateach point on the workpiece surface can be written in general form as [65]:qr(x,ϕ, t) =∞∑jl=1∞∑ic=2A jl ,ic(t)cos(icϕ)sin( jlxLpi+θ jl)+∞∑jl=1∞∑ic=2B jl ,ic(t)sin(icϕ)sin( jlxLpi+θ jl)(5.2)where t denotes time, and 0≤ x≤ L and 0≤ ϕ ≤ 2pi are the axial and circumferen-tial coordinates of the point, respectively (Figure 5.1); ic = 2,3, ... is the number ofwaves (lobes) in the circumferential pattern, and jl = 1,2, ... is the number of half-waves in the axial pattern. A jl ,ic , B jl ,ic , and θ jl are determined from the boundaryconditions.Dynamic response of thin-walled workpieces during machining operation is dis-cussed in the following section.5.3 Response of Cylindrical Shells to Threading LoadsThe dynamic threading model developed in Chapter 4 is based on the assumptionthat the workpiece behaves as a cantilever beam. The model is extended in thissection to threading thin-walled workpieces by addressing the following two maindifferences:78Chapter 5. Threading Thin-Walled Oil Pipes• In machining cylindrical shells, radial flexibilities of the workpiece are themost dominant source of regenerative vibrations. All other flexibilities canbe neglected.• Due to the low-damped shell mode patterns, cutting forces result in differentvibration amplitudes around the circumference of the workpiece. These resid-ual vibrations affect the chip thickness when the corresponding point arrivesin the cutting region. In order to model the process, instantaneous vibrationsat all points must be determined.The dynamic model presented below evaluates the shell vibrations in each crosssection individually based on the local structural dynamics.5.3.1 Dynamic Equation of MotionAs illustrated in Figure 5.2, assume that the circumference of the workpiece is dis-cretized by m number of points. The instantaneous radial vibration at each point kis denoted by scaler qk(t) (k= 1,2, ...,m), where outward vibrations are consideredpositive. The generalized radial vibration vectorQr(t) is formed by stacking all thepoint vibrations:Qr(t) =q1(t)q2(t):qm(t)[m×1](5.3)At each time t, the radial vibration at the cutting point can be obtained as:qcr(t) =Nc(t)Qr(t) (5.4)ZYq crq2q3q4q1=q mq m-1......t=0Figure 5.2: Response of cylindrical shells to machining loads.79Chapter 5. Threading Thin-Walled Oil Pipeswhere the row vector {Nc(t)}[1×m] is the instantaneous shape function at the cuttingpoint. All entries of Nc(t) are zero except for the entry corresponding to the pointin the cutting zone, which is set to one. If necessary, higher order continuity canbe achieved by interpolating (weighted average) over few points around the cuttingregion. Based on the defined tool coordinate system shown in Figure 5.2 (same inprevious chapters), the three-dimensional vibration vector of the workpiece at thecutting location can be represented as:qw(t) =0−qcr(t)0 (5.5)Since the tool is considered rigid, qt(t) = {0,0,0}T . The three-dimensional relativevibration vector between the tool and workpiece is (Eq. 4.2):q(t) = qt(t)−qw(t) =0qcr(t)0= er qcr(t) (5.6)where er = {0,1,0}T is the unit vector in the radial direction at the cutting point.Combining Eqs. (5.4) and (5.6) yields:q(t) = erNc(t)Qr(t) (5.7)The resultant three-dimensional cutting forces on the tool is obtained by substitutingq(t) in Eq. (4.28):Fc(t)=Fs(t)+Kdc(t)erNc(t)Qr(t)−Kdd(t)erNc(t)Qr(t−T )+Cp(t)erNc(t)Q˙r(t)(5.8)where the static force vector Fs and dynamic matrices Kdc, Kdd , and Cp are de-fined in Eqs. (4.29)-(4.32), respectively. The reaction force −Fc is exerted on theworkpiece but only the radial force component, denoted as fcr(t), affects the chipregeneration process:fcr(t) =−{0,1,0}{Fc(t)}=−eTr Fc(t) (5.9)80Chapter 5. Threading Thin-Walled Oil PipesThe generalized force vector {FGr(t)}[m×1], which consists of the instantaneousradial forces at all m discrete points around the circumference, can be obtainedusing the shape functionNc:{FGr(t)}[m×1] =NTc (t) fcr(t) (5.10)which combined with Eq. (5.9) yields:FGr(t) =−NTc (t)eTr Fc(t) (5.11)All entries of {FGr(t)} are zeros except for the entry corresponding to the point(or points, depending on the chosen shape function) at the cutting region. Thegeneralized vibration and force vectors are related to each other in Laplace domainas:Qr(s) = Gr(s)FGr(s) (5.12)where [Gr(s)][m×m] is the generalized structural dynamics of the workpiece in theradial direction. Each entry i j (i, j = 1,2, ...,m) in Gr(s) matrix contains the FRFfrom the radial force at point j to the radial vibrations at point i. Assume thatthe structural behaviour of the workpiece can be approximated by mw number ofcircumferential shell modes. Similar to Eq. (4.42), the reduced generalized matrixGr(s) can be constructed in Laplace domain as [44]Gr(s) = Uˆr(Is2+2ζωns+ωn2)−1 UˆTr (5.13)where I[mw×mw] is the identity matrix, and ζ[mw×mw] and ωn[mw×mw] are the diagonaldamping ratio and natural frequency matrices, respectively. Uˆr[m×mw] is the mass-normalized radial mode shape matrix:Uˆr =u1,1u2,1:um,1u1,2u2,2:um,2 · · ·u1,mwu2,mw:um,mwm×mw(5.14)where each column corresponds to a circumferential shell mode. As illustratedin Figure 5.1, each circumferential pattern with ic (ic = 2,3, ...) number of waves81Chapter 5. Threading Thin-Walled Oil Pipes(lobes) has an arc wavelength of 2pi/ic with the corresponding mode shape vector:Uˆ(ic) = |Aic|cos(icϕ+ϕ0,ic), 0≤ ϕ ≤ 2pi= |Aic|cos(ic(k−1)2pim +ϕ0,ic), 1≤ k ≤ m(5.15)where |Aic| is the mass-normalized amplitude, and ϕ0,ic is a constant determinedbased on the boundary conditions. Extraction of mode shapes using FE and experi-mental methods is discussed in Section 5.5.The generalized radial vibration vectorQr(t) is transformed to the modal spaceusing the mode shape matrix Uˆr as:Qr(t) = UˆrΨr(t) (5.16)where {Ψr(t)}[mw×1] is the modal radial displacement vector. Combining Eqs (5.12),(5.13), and (5.16), and following similar derivations presented in Eqs. (4.47)-(??),the dynamic equation of motion in terms of the modal displacement vector can bewritten in Laplace domain as:(Is2+2ζωns+ωn2)Ψr(s) = UˆTr FGr(s) (5.17)The generalized force vector FGr is written in terms of modal displacement vector(Ψr) by combining Eqs. (5.8), (5.11), and (5.16):FGr(t) =−NTc (t)eTr Fc(t)=−Tsc(t)TFs(t)−Tsc(t)TKdc(t)Tsc(t)UˆrΨr(t)+Tsc(t)TKdd(t)Tsc(t)UˆrΨr(t−T )−Tsc(t)TCp(t)Tsc(t)UˆrΨ˙r(t)(5.18)where[Tsc(t)][3×m] = {er}[3×1]{Nc(t)}[1×m] (5.19)is defined as the equivalent transformation from the generalized shell coordinates(in the radial direction) to the three-dimensional tool CS at the cutting point. Fi-nally, the dynamic equation of motion for the instantaneous response of the cylin-drical shell under the threading loads can be written in time domain modal space by82Chapter 5. Threading Thin-Walled Oil Pipescombining Eqs. (5.17) and (5.18):Ψ¨r(t)+(Cm,1+Cm,2(t))Ψ˙r(t)+(Kmc,1+Kmc,2(t))Ψr(t)+Kmd(t)Ψr(t−T )=Fsm(t)(5.20)whereCm,1 = [2ζωn][mw×mw] , Cm,2(t) = Tsc(t)TCp(t)Tsc(t)Kmc,1 = [ω2n ][mw×mw] , Kmc,2(t) = Tsc(t)TKdc(t)Tsc(t)Kmd(t) =−Tsc(t)TKdd(t)Tsc(t)Fsm(t) =−Tsc(t)TFs(t)(5.21)where the static force vector Fs and dynamic matrices Kdc, Kdd , and Cp are definedin Eqs. (4.29)-(4.32), respectively.Since Eq. (5.20) has the exact form as Eq. (4.49), all the state space deriva-tions and time-marching numerical simulation techniques presented in Section 4.8can be readily used to solve for Ψr(t). Once the modal displacement vector Ψr(t)is obtained, the radial vibrations at each point k (k = 1,2, ...,m) can be calculatedusing Eq. (5.16). The resultant axial, radial, and tangential forces, which includeboth the static and dynamic components, are evaluated from Eq. (5.8). The instan-taneous radial vibration at the cutting point, which generates the finish surface, canbe obtained using the shape function in Eq. (5.4).Remark 1. Due to the time dependency of the shape function Nc(t) (and thusTsc(t)), the stability of the process cannot be analyzed in frequency domain usingNyquist criterion. Instead, time domain techniques such as semi-discretization [20]and full-discretization [39] can be implemented by stacking all the state vectorsover one spindle revolution. The stability of the process is then analyzed based onthe eigen values of the augmented matrix [44], which has a size of [2(m×mw)×2(m×mw)]. However, if the vibration frequency is significantly greater than thespindle rotation frequency, a large discretization number (m) is required to capturethe wave length of the vibrations on the workpiece surface. In this case, the re-sultant augmented matrix becomes impractically large, and the eigen values cannotbe evaluated using normal computers. If this is the case, full time history of theprocess must be simulated using the numerical time-marching techniques (Section4.8), and stability is investigated based on the decay or growth of the vibrations.83Chapter 5. Threading Thin-Walled Oil PipesRemark 2. If the vibration frequencies are at least two order of magnitudesgreater than the spindle rotation frequency, the effect of residual shell vibrationson stability may be neglected. In this case, the vibration marks are densely packedaround the circumference, and instability can occur locally. For these systems,frequency domain stability analysis developed in Chapter 4 can be used along withthe FRF at the most dynamically flexible point around the circumference.5.3.2 Sample Time Simulation ResultsIn order to demonstrate the response of cylindrical shells under threading loads, thecase study presented in Section 4.9 is revisited here. Assume that the three-pointV-profile insert shown is Figure 4.6.a is used to thread a thin-walled workpiecewith diameter of dw = 70 mm. The structural dynamics of the workpiece in theradial direction at the most dynamically flexible point around the circumference isprovided in Figure 4.6.b and Table 4.1. Assume that modes 1 and 2 have two-lobe(ic = 2) and three-lobe (ic = 3) circumferential patterns, respectively (see Figure5.1). The maximum amplitude of each mass-normalized mode shape vector i canbe calculated based on the dynamic parameters in Table 4.1 as:|Uˆi|= 1√mi =ωn,i√ks,i(ωn in rad/s) (5.22)which gives 0.154 and 0.254 for modes 1 and 2, respectively. Using Eq. (5.15), themode shape vectors can be obtained as:Uˆ1(ic = 2) = 0.154 cos((k−1)4pim), 1≤ k ≤ mUˆ2(ic = 3) = 0.254 cos((k−1)6pim), 1≤ k ≤ m(5.23)where m = 10,000 discretization points have been used for the simulations in thissection.The process has been simulated using the same cutting conditions as in Fig-ure 4.7.b, i.e. radial infeed of 1.6 mm and spindle speed of 1850 rpm. Figure 5.3.ashows the simulated vibrations at the cutting point (Eq. (5.4)) over the first 50spindle revolutions. It can be seen that while the operation is stable, the vibrationscontinue to exist even after the process reaches “steady” condition. The generated84Chapter 5. Threading Thin-Walled Oil Pipes0 10 20 30 40 50SpindlerRevolution-0.03-0.02-0.0100.01Displacementr[mm]0 0.01 0.02 0.03Timer[s]-0.03-0.02-0.0100.01VibrationrMarksr[mm]GeneratedrSurfaceVibrationsratrCuttingrPointφ=72φ=120φ=360a7b7c7Figure 5.3: Simulated shell response under threading loads, a) vibrations atthe cutting point, b) generated surface over the first revolution, c) sam-ple instantaneous shell deformation (scaled, spindle speed: 1850 rpm,ωn,1 = 300 Hz, ωn,2 = 700 Hz).surface over the first spindle revolution along with the corresponding shell defor-mation (Eq. (5.3)) at few instances are shown in Figure 5.3.b and c, respectively.It can be seen that the deformations continuously vary through the mixed two-lobeand three-lobe patterns.5.4 Application: Threading Oil PipesIn order to validate the dynamic models developed in Chapters 4 and 5, extensivethreading experiments have been conducted on real scale oil pipes. The remainder85Chapter 5. Threading Thin-Walled Oil Pipes160" (4.064 m)86.25" (2.205 m)~12" (~304.8 mm)a) Rear chuck b) Front endc)Figure 5.4: Experimental setup: a,b) front and rear clamping chucks in MazakSLANT-TURN 60 lathe, c) distance between clamping locations.of the chapter is dedicated to presenting the results and discussions.5.4.1 Experimental SetupFigures 5.4 shows a clamped pipe on Mazak SLANT-TURN 60 machining centre.The pipe is 4 m long, and it is held by two three-jaw chucks, one close to the frontend and the other one between the centre and the rear end. Two types of pipes withdiameters of 7′′ (177.8 mm) and 1338′′(339.7 mm) have been used for the experi-ments2. For simplicity, these pipes are referred to as D7 and D13, respectively. Thewall thickness is originally 0.5′′ (12.7 mm), which is tapered down to 0.43′′ (10.9mm) with a small taper angle of 34′′per ft (1◦47′) before threading. The pipe mate-rial is custom steel with hardness in the range of 240-250 BHN (23-25 HRC). Asshown in Figures 5.5.a and b, the clamping jaws for these two pipes have differentcontact length, resulting in different structural behaviour and mode shapes of thepipes. Two types of inserts have been used in the experiments (Figures 5.5.c andd):• 5-point buttress (Ceratizit, 4.371-CE-LP025) with pitch of 5 TPI (5.08 mm)and depth of 0.062′′ (1.57 mm).2Imperial units are more common in oil and gas industry. The dimensions in this chapter arepresented in inches, and the equivalents in millimetres (rounded) are provided in the parentheses.86Chapter 5. Threading Thin-Walled Oil Pipesa)cClampingcofcpipecD13 b)cClampingcofcpipecD73-pointcV-profile(SandvikcCoromant)5-pointcbuttress(Ceratizit)c)d)Figure 5.5: Pipes and the cutting inserts used in the experimental tests.• Three-point V-profile (Sandvik Coromant R166.39G-24RD13-080) with pitchof 8 TPI (3.175 mm) and depth of 0.071′′ (1.8 mm).The measurement setup is shown in Figure 5.6. The eccentricity of the pipeis measured using a dial gage and has been kept below 0.030′′ (0.76 mm) duringthe initial set up. An instrumented hammer and accelerometers have been usedfor modal analysis, and sound data during chatter tests have been collected with amicrophone. A laser displacement sensor has been mounted normal to the pipe axisto measure the dynamic eccentricity and vibrations.3.a)c)b)d)Figure 5.6: Measurement devices: a) laser displacement sensor, b) dial gauge,c) microphone, d) instrumented hammer and accelerometer.3The laser sensor was not used during the machining operation due to chip entangling concerns.87Chapter 5. Threading Thin-Walled Oil Pipes5.4.2 Dynamic Behaviour Across Clamping ChucksIn order to investigate the effect of chucks on vibration isolation, pipe D7 has beenhit at the tip with an instrumented hammer. The vibrations have been measured atthree points along the pipe axis (Figure 5.7.a): P1 and P2 right in front and behindof the front chuck, and P3 behind the rear chuck. Figure 5.7.b and c compare themeasured FRFs at these three points. It can be seen that the vibrations behind thefront chuck (P2) are even larger than those at P1. This is due to the axial patternof the dominant mode shape, which is analyzed in more details in Section 5.5.2.Nevertheless, this observation confirms that the three-jaw chucks cannot effectivelyrestrict the flexibilities of the pipe at the clamping location. Similar conclusionswere deduced from the hammer tests on pipe D13.820 840 860 880 900 92001234xz10−7Magnitudez[m/N]Frequencyz[Hz]P1P2P31000 2000 3000 4000 500000b511b522b533b5xz10−7Magnitudez[m/N]Frequencyz[Hz]P1P2P3FRFz)zPipezD7P1AccFrontChuckRearChuckP2P3ao bocoFigure 5.7: Effectiveness of chuck clamping in vibration isolation (pipe D7):a) measurement points, b) FRFs between the tip and the measurementpoints, c) comparison of FRFs at the dominant mode.88Chapter 5. Threading Thin-Walled Oil Pipes5.4.3 Effect of Jaw ConfigurationDuring the threading operation and as the spindle rotates, the structural dynamics ofthe pipe at the cutting location can vary as a function of jaw positioning (boundaryconditions). In order to investigate the significance of jaw configuration, hammertests have been performed at two extreme cases shown in Figures 5.8.a and b. Themeasured FRFs at the tip of pipe D7 are compared in Figures 5.8.c and d. It canbe seen that case 1, in which one jaw aligns with the cutting point, results in thelowest stiffness. This is due to the fact that in case 1 there is minimum support atthe opposite side of the cutting point to limit the deflections. The measured FRFsfor pipe D13 led to the same conclusion. Hence, all the stability charts presented inthis chapter have been generated based on case 1 configuration, which is the mostflexible setup.In order to determine the source of flexibilities leading to chatter vibrations,structural behaviour of the pipe is studied in the next section.Acc AccCaseq1 Caseq2850 900 950 100000.511.522.53xq10−6Magnitudeq[m/N]Frequencyq[Hz]Caseq1Caseq2aP bPcP1000 2000 3000 4000 500000.511.522.53xq10−6Magnitudeq[m/N]Frequencyq[Hz]dPFRFq(qPipeqD7q-untaperedPCaseq1Caseq2Figure 5.8: Effect of jaw configuration on pipe dynamics: a) Case 1: jawaligned with the measurement point, b) Case 2: measurement point be-tween two jaws, c) direct FRFs at the tip, d) comparison of FRFs at thedominant mode.89Chapter 5. Threading Thin-Walled Oil Pipes5.5 Mode Shape Analysis of Oil PipesCasings and tubes used in pipelines have diameter-to-thickness ratios typically inthe range of 15-40, which makes them exhibit dominant shell modes. Finite elementanalysis of sample pipes is presented in the following section.5.5.1 Finite Element Mode Shape ExtractionPipe D13 with thickness of 0.5′′ (12.7 mm) and clamping stickout of 12′′ (304.8mm) has been modelled in ANSYS using 8-node SHELL93 elements with six de-grees of freedom at each node. The nodes corresponding to the contact area betweenthe jaws and the pipe have been fixed, and modal analysis based on Block Lancozmethod has been performed. Figures 5.9.a-c show the first three mode shapes of theclamped pipe.Assume that each node p has six degrees of freedom denoted as pk, (k= x,y,z,θx,θy,θz).For each mode i, the mass-normalized mode shape vector corresponding to DOFsof node p, i.e.Uˆp,i =upx,iupy,iupz,iupθx ,iupθy ,iupθz ,i, (5.24)can be extracted from ANSYS using the script provided in Appendix A. Considertwo arbitrary nodes p and q, each having six degrees of freedom represented re-spectively as pk and qn (k,n ∈ x,y,z,θx,θy,θz). If the first mw number of modes areextracted from the FE analysis, the cross FRF between the arbitrary DOFs pk andqn can be approximated by summing the contribution of all mw modes asGpk,qn( jω) =mw∑i=1[upk,i ·uqn,i(ωn,i2−ω2)+2 jζiωn,iω](5.25)where ζi and ωn,i are the damping ratio and natural frequency of mode i. ζi mustbe estimated (or identified experimentally) and cannot be modelled from FE modalanalysis. In Eq. (5.25), if pk = qn, the resultant frequency function (Gpk,pk) is calledthe direct FRF at point p in the direction of DOF k.90Chapter 5. Threading Thin-Walled Oil PipesE 1EEE 2EEE 3EEEEE/511/52xu1E−6Frequencyu[Hz]Magnitudeu[mNN] ExpFEModeu1:u685uHzquuuuuuuuuuuuuuuyModeu2:u787uHzquuuuuuuuuuuuuuuyModeu3:u1427uHzquuuuuuuuuuuuuuuyay bycy dyFRFumuPipeuD13uquntaperedyi =2gc j =1l i =3gc j =1li =4gc j =1lFigure 5.9: Finite element modal analysis of pipe D13 (stickout: 12′′(304.8 mm), thickness: 0.5′′ (12.7 mm)): a-c) first three mode shapes,d) comparison of the FE and measured radial FRFs at the tip.Figure 5.9.d compares the FE and measured direct FRFs in the radial directionat the tip of pipe D13 with stickout of 12′′ (304.8 mm). The finite element FRF hasbeen constructed using the first 10 modes and with assigned damping ratio of 0.5%(Eq. (5.25)). It can be seen that there is a considerable discrepancy between the FEand measured FRFs. This is mainly due to the fact that the jaws in the actual setupcannot fully restrict the pipe at the clamping points, as discussed in Section 5.4.2.Accurate FE modelling of the pipe dynamics requires advanced contact analysis,which is not the focus of this thesis. In order to validate the proposed threadingmodels more accurately, the stability charts in this chapter are generated based onthe measured FRFs. Experimental modal analysis of the pipe is presented in thefollowing sections.Remark 1. If the depth of the thread is considerable compared to the thickness ofthe pipe, the dynamic parameters can change over each pass as a result of materialremoval. The new FRF after each pass can be obtained using advanced analyticalFRF updating methods [82]. Alternatively, if the pipe is modelled in FE usingsolid elements, the geometry after each threading pass can be re-meshed, and thenew FRFs can be extracted. The stability model proposed in Chapters 4 and 5 can91Chapter 5. Threading Thin-Walled Oil Pipesaccept different FRFs for different passes. However, it is assumed here that the pipedynamics do not change considerably over the threading process. This assumptionis discussed in more details in Section 5.6.4.Remark 2. Stability limit for each pass is determined based on the most dynam-ically flexible point along the pipe axis, which can happen at a location other thanthe tip due to the shell behaviour of pipes.5.5.2 Experimental Mode Shape ExtractionAs illustrated in Figure 5.10, the circumference of the pipes D7 and D13 have beenmarked at every 45 degrees at several locations along the pipe axis including fewpoints behind the chuck. The pipes have been hit with an instrumented hammer ateach grid point, and the resultant vibrations at the tip have been measured with anaccelerometer (in the radial direction).Assume that the grid point at the location of accelerometer is called point 1.The direct FRF at this point can be approximated by contribution of mw number ofdominant modes as:G11( jω) =mw∑i=1[1/mi(ωn,i2−ω2)+2 jζiωn,iω](5.26)where mi (i: mode number) is the modal mass, and ωn,i and ζi are the naturalfrequency and damping ratio. These modal parameters are identified using leastsquare method [83] by fitting Eq. (5.26) to the measured direct FRF. Table 5.1provides the identified parameters for the most dominant modes of pipes D7 andD13.X YZZAcc5432 1Figure 5.10: Grids for experimental mode shape analysis.92Chapter 5. Threading Thin-Walled Oil PipesTable 5.1: Measured dynamic parameters of the dominant mode in pipes D7(stickout: 11′′ (279.4 mm)) and D13 (stickout 12′′ (304.8 mm)), bothtapered.natural frequency damping ratio modal stiffness modal massωn [Hz] ζ ks [N/m] m [Kg]Pipe D7 910 0.02% 1.2×109 36.7circumferential mode shape: -0.435, -0.018, 0.399, 0.076, -0.487axial mode shape (top): 0.435, 0.269, 0.081, -0.038, -0.045axial mode shape (bottom): -0.433, -0.222, -0.037, 0.081, 0.088Pipe D13 840 0.3% 2.9×108 10.5circumferential mode shape: 0.356, -0.277, -0.023, 0.194, -0.332axial mode shape (top): 0.356, 0.252, 0.172, 0.107, 0.083, 0.053, 0.034axial mode shape (bottom): 0.332, 0.238, 0.165, 0.103, 0.074, 0.046, 0.022The cross FRF between the hitting point k and the accelerometer (point 1) con-sists of the same modes as G11 but with different amplitudes (contributions):Gk1( jω) =mw∑i=1[Rk1,i(ωn,i2−ω2)+2 jζiωn,iω](5.27)where Rk1,i is called the residue of mode i in the FRF between points k and 1. Rk1,ivalues are identified using least square method by fitting Eq. (5.27) to the measuredcross FRFs. Once the modal parameters and the residue values are known, themass-normalized mode shape vector (in the radial direction) for each mode i can beobtained as:Uˆr,i =u1,iu2,i:ung,i (5.28)where ng is the total number of grid points (FRFs), and the entries are calculated as[83]:u1,i =1√miuk,i =√mi Rk1,i← 2≤ k ≤ ng(5.29)93Chapter 5. Threading Thin-Walled Oil Pipeswhere mi and Rk1,i are identified from the measured FRFs (Eqs. (5.26) and (5.27)).The experimental mode shapes must be interpolated by m number of discretizationpoints before being used in time-marching simulations (Sections 4.8 and 5.3).Figure 5.11 demonstrates the circumferential mode shape analysis at the tip ofpipe D13. Only half of the circumference is analyzed due to symmetry (the jawconfiguration is also symmetric). The magnitude and imaginary parts of the FRFsat the dominant mode (840 Hz) are compared in Figures 5.11.c and d, respectively.Relative displacements of the grid points in the radial direction can be compared09018045135830 835 840 845 850−505xH10−7ImaginaryH[m/N]FrequencyH[Hz]0Hdeg135Hdeg90Hdeg180Hdeg45Hdeg830 835 840 845 85002468xH10−7MagnitudeH[m/N]FrequencyH[Hz]0Hdeg180Hdeg90Hdeg45Hdeg135Hdeg11000 2000 3000 4000 500002468xH10−7MagnitudeH[m/N]FrequencyH[Hz]a= b=c= d=e= f=PipeHD13H(tapered=PipeHD13H(tapered=iHH=3cFigure 5.11: Circumferential mode shape analysis of pipe D13 (at the tip):a) measurement points, b) radial direct FRFs at all points, c-d) mag-nitude and imaginary components of the FRFs at the dominant mode,e-f) circumferential pattern of the dominant mode.94Chapter 5. Threading Thin-Walled Oil Pipesfrom the magnitude plot, and the relative phase can be deduced from the phase plot.For example, the points at the top (0 deg) and bottom (180 deg) have nearly samemagnitude but opposite phase. The identified mode shape components are providedin Table 5.1 and plotted in a polar plane in Figure 5.11.e. It can be seen that pipeD13 exhibits its highest dynamic flexibility in three-lobe (ic = 3) circumferentialpattern.Axial mode shape analysis of pipe D13 is presented in Figure 5.12. The radialFRFs have been measured along the top and bottom of the pipe at the locationsshown in Figure 5.12.a. The measured FRFs around the dominant mode (840 Hz)are compared in Figures 5.12.b and c. It can be seen that all the grid points vibratein-phase, and the amplitudes of vibrations continuously decrease from the tip to-wards the chuck. The FRFs along the bottom have nearly the same magnitude andimaginary components, thus have not been plotted for conciseness. The identifiedmode shapes along the top and bottom of the pipe are illustrated in Figure 5.12.d.83z 835 84z 845 85z 855z2468xnIz−7Magnituden[mHN]Frequencyn[Hz]83z 835 84z 845 85z 855−6−4−2zxnIz−7Imaginaryn[mHN]Frequencyn[Hz]zT4T8TI2TI4TI7T2zTzT4T8T I2TI4TI7T2zT−2z −I5 −Iz −5 zTTTTz4T8TI2TI4TI7T2zTambm cmdm emPipenDI3[taperedmjn=IlFRFsnAlongnThenTop FRFsnAlongnThenTopFigure 5.12: Axial mode shape analysis of pipe D13: a) measurement points,b-c) magnitude and imaginary components of the radial direct FRFs atthe dominant mode, d-e) axial pattern of the dominant mode.95Chapter 5. Threading Thin-Walled Oil PipesIt can be seen that the dominant mode of pipe D13 has 1st-bending ( jl = 1) axialpattern.Similarly, circumferential and axial mode shape analyses for pipe D7 are pre-sented in Figures 5.13 and 5.14, respectively. Based on the identified parameters,pipe D7 exhibits highest dynamic flexibility in 2-lobe (ic = 2) circumferential pat-tern (Figure 5.13.e) and 2nd-bending ( jl = 2) axial pattern (Figure 5.14.f). Themode shape components for the most dominant modes of pipe D7 and D13 areprovided in Table 5.1.Remark. Since both pipes D7 and D13 exhibit highest dynamic flexibility at the090 180451350Ndeg135Ndega. b.c. d.e. f.PipeND7N8tapered.1000 2000 3000 4000 50000123xN10−6MagnitudeN[m/N]FrequencyN[Hz]PipeND7N8tapered.970 975 980 98500.511.52xN10−6MagnitudeN[m/N]FrequencyN[Hz]45Ndeg180Ndeg90Ndeg970 975 980 985−2−1012xN10−6ImaginaryN[m/N]FrequencyN[Hz]0Ndeg180Ndeg90Ndeg135Ndeg45Ndeg2.5iNN=2cFigure 5.13: Circumferential mode shape analysis of pipe D7 (at the tip):a) measurement points, b) radial direct FRFs at all points, c-d) mag-nitude and imaginary components of the FRFs at the dominant mode,e-f) circumferential pattern of the dominant mode.96Chapter 5. Threading Thin-Walled Oil Pipesh4/8/T4/T6/azbz czdz ezfz gzPipe[D7Htaperedz97h 975 98h 985hhA5TTA5Bx[Th−6Magnitude[[moN]Frequency[[Hz]h/4/8/T4/T6/97h 975 98h 985hImaginary[[moN]Frequency[[Hz]Bx[Th−6hA5TTA54/h/T4/T6/97h 975 98h 985hhA5TTA5Bx[Th−6Magnitude[[moN]Frequency[[Hz]h/4/8/T4/T6/−T5 −Th −5 h///97h 975 98h 985−B−TA5−T−hA5hImaginary[[moN]Frequency[[Hz]x[Th−6hA54/8/h/T4/T6/8/j[=BlAlong[the[Top Along[the[TopAlong[The[Bottom Along[The[BottomFigure 5.14: Axial mode shape analysis of pipe D7: a) measurement points,b-c) magnitude and imaginary components of the radial direct FRFsat the dominant mode along the top, d-e) FRFs along the bottom, f-g)axial pattern of the dominant mode.97Chapter 5. Threading Thin-Walled Oil Pipestip, the stability charts presented in the next section have been generated based onthe FRFs at the tip.5.6 Chatter ExperimentsFigure 5.15 categorizes different parameters affecting process stability in threadingoil pipes. Stability limits subject to change in these parameters have been simulatedand validated experimentally. The cutting force coefficients identified in Eq. (3.58)have been used in the simulations. The indentation coefficient and wear land inprocess damping model (Eq. 4.25) have been estimated as Ksp = 4× 1013 N/m3and Lw = 0.12mm, respectively [22]. The predicted and experimental results forthe threading tests are presented in the following sections.Remark 1. Pipe D7 and D13 are machined at 120 rpm and 250 rpm, respectively.This means that the spindle rotation frequency in all experiments is less than 5 Hzwhile the vibration frequencies are around 1000 Hz. Hence, the frequency domainstability analysis developed in Section 4.6 can be used to predict the stability limits.Remark 2. Setting up a pipe for threading experiments required a time-consumingprocedure including mounting (with eccentricity control), surface cleaning, and ta-pering. After each set of experiments, the threaded section (or layer) was machinedaway, and if possible, the next tests were carried out on the same piece withoutunclamping the pipe. As a result, the FRFs and the stability charts presented in thefollowing sections are different for different tests. In each case, the experimentalresults have been compared against the corresponding stability lobes.ParametersGaffectingGchatterFlexibilityPipeGThickness Stickout ClampingInfeedGzDepth)SpindleGSpeedInsertGGeometry5P-BUG 3P-VGzSandvik)zCeratizit)Figure 5.15: Parameters affecting chatter stability in threading oil pipes.98Chapter 5. Threading Thin-Walled Oil Pipes5.6.1 Threading Over Several PassesIn the first set of experiments, pipe D13 with thickness of 0.43′′ (10.9 mm) hasbeen threaded at spindle speed of 120 rpm (cutting speed of 130 m/min). Threepasses with radial infeed of 0.020′′ (0.508 mm)/pass have been cut using Ceratizit5-point buttress insert. The sound data was collected using a microphone inside themachine. After finishing all three passes, the threaded layer was removed by axialturning, resulting in smaller wall thickness. Similar to the first set, another two setsof three passes at 0.020′′ (0.508 mm)/pass were cut on the same piece.Figure 5.16 shows the radial FRF at the tip of the pipe before conducting eachthreading set. The predicted stability chart for the first set is shown in Figure 5.17.a,where the experiment point (120 rpm, 0.508 mm) has been marked by a star. Sta-bility lobe for each pass has been capped at the infeed value where the maximumthread depth for the insert is reached. Figures 5.17.b and c show the measuredsound signal and its frequency contents, and the resultant surface finish is shownin Figure 5.17.d. Same plots for the second and third sets are presented in Figures5.18 and 5.19, respectively. The predicted and experimental stability conditions forall passes are compared in Table 5.2. The dominant natural frequency (before eachset) and the observed chatter frequency (if any) during each pass are also provided.Based on the predicted stability lobes, all passes of set 1 are stable, and allpasses of set 3 are unstable (heavy chatter). These predictions agree with the exper-imental results. The threading passes over set 2 exhibited marginal stability or lightchatter; it can be seen in Figure 5.18 that the model predicts similar behaviour as01234xr10Setr1D1395rHz)Setr3D1004rHz)Setr2D1180rHz)200 400 600 800 1000 1200 1400−6Magnituder[m/N]Frequencyr[Hz]FRFr-rPiperD13Figure 5.16: Measured FRFs at the tip of pipe D13 before each threading set.99Chapter 5. Threading Thin-Walled Oil Pipes0 50 100−0.04−0.0200.02Timeq[s]SoundqSignalq[V] dkqStable0 1000 20000123xq10−4FFTqofqSoundq[V]Frequencyq[Hz]bk ck5-pointqbuttressak0 100 200 300 40000.511.5SpindleqSpeedq[rpm]Infeedq[mm]Set 1UnstableRegionStableRegion1qtooth2qteeth3qteeth4qteethploughingpocketsPass 1Pass 2Pass 3Figure 5.17: Experimental results for threading test Set 1 (pipe D13, 120 rpm,infeed: 0.020′′ (0.508 mm)/pass, insert: Ceratizit 5-point buttress):a) stability charts and the experiment point (star), b-c) recorded soundsignal and its frequency contents, d) surface finish.the experiment point is very close to the stability lobes, especially over the first andsecond passes of set 2.Remarks: Ploughing Pockets in Stability ChartThe number of engaged teeth at each infeed value has been provided in the stabilitychart in Figure 5.17.a. It can be seen that at spindle speeds over 300 rpm, engagingmore than one tooth leads to instability. Due to the large engagement length, cuttingwith multiple teeth is only feasible at lower spindle speeds where process dampingsignificantly increases the stability limits. However, spindle speed must be above100 rpm to avoid chip shearing problems.As illustrated in Figure 5.17.a, at infeed values where an additional tooth en-gages in the cut, the stability lobes show a deep unstable pocket, which are marked100Chapter 5. Threading Thin-Walled Oil Pipes0 100 200 300 40000.511.5SpindleoSpeedo[rpm]Infeedo[mm]UnstableRegionStableRegionSet 2dHoLightoChatteraH0 50 100−0.04−0.0200.02Timeo[s]Passo1Passo2Passo3SoundoSignalo[V]bH0 1000 20000123xo10−4FFToofoSoundo[V]Frequencyo[Hz]1140oHzcH5-pointobuttressPass 1Pass 2Pass 3Figure 5.18: Experimental results for threading test Set 2 (pipe D13, 120 rpm,infeed: 0.020′′ (0.508 mm)/pass, insert: Ceratizit 5-point buttress):a) stability charts and the experiment point (star), b-c) recorded soundsignal and its frequency contents, d) surface finish.Table 5.2: Comparison of experimental and predicted stability conditions fordifferent passes (pipe D13, 120 rpm, infeed: 0.020′′ (0.508 mm)/pass, in-sert: Ceratizit 5-point buttress) (S: stable, MS: marginally stable, C: chat-ter, LC: light chatter)Pass 1 Pass 2 Pass 3ωn [Hz] sim exp ωc [Hz] sim exp ωc [Hz] sim exp ωc [Hz]Set 1 1395 S S - S S - S S -Set 2 1180 S LC 1173 MS MS - C C 1140Set 3 1004 C C 1001 C (heavy chatter) C (heavy chatter)101Chapter 5. Threading Thin-Walled Oil Pipes0 100 200 300 40000.511.5SpindlezSpeedz[rpm]Infeedz[mm] UnstableRegionStableRegionSet 30 5 10−0.2−0.100.1Timez[s]SoundzSignalz[V]Passz10 1000 200000.010.02FFTzofzSoundz[V]Frequencyz[Hz]1001zHzd3zHeavyzChattera3b3 c35-pointzbuttressPass 1Pass 2Pass 3Figure 5.19: Experimental results for threading test Set 3 (pipe D13, 120 rpm,infeed: 0.020′′ (0.508 mm)/pass, insert: Ceratizit 5-point buttress):a) stability charts and the experiment point (star), b-c) recorded soundsignal and its frequency contents, d) surface finish.as “ploughing pockets”. This is mainly due to the large ploughing forces as a resultof small chip thickness on the added tooth. Since cutting force coefficients in thisthesis have been calculated using nonlinear Kienzle model (Eq. (3.49)), the effectof ploughing has been reflected in the stability charts.5.6.2 Threading at Different Infeed ValuesThe effect of infeed value on chatter stability has been examined on pipe D13 us-ing Ceratizit 5-point buttress insert. Three experiments at infeed values of 0.020′′(0.508 mm), 0.025′′ (0.635 mm), and 0.035′′ (0.889 mm) have been conducted at120 rpm (cutting speed of 130 m/min). Only one pass has been cut at each infeedvalue, and the threaded layer has been removed after each pass. Based on the insertgeometry, the first two infeed values engage two teeth while the final experimentcut with three teeth.Figure 5.20.a shows the measured radial FRFs at the tip before conducting each102Chapter 5. Threading Thin-Walled Oil PipesbDDD CDDD hDDD 7DDD TDDDDD-Tbb-TCxSbD−6MagnitudeS[mPN]FrequencyS[Hz]SetShf8D8SHz:SetSCf879SHz:D bDD CDD hDD 7DDD-CD-7D-6D-8bb-Cb-7SpindleSSpeedS[rpm]InfeedS[mm]DUnstableRegionStableRegionSetSbSetSCSetSh:SexperimentSpointsD C 7 6 8−D-b−D-DTDD-DTD-bTimeS[s]D bDDD CDDDDC768xSbD−hFFTSofSSoundS[V]FrequencyS[Hz]SoundSSignalS[V]FRFSRSPipeSDbhSetShSetShSetSCStabilitySCharta:b: c:d:TRpointSbuttressfh8hSHz:f77hSHz:SetSbfbh9TSHz:Figure 5.20: Experimental results at different infeed values (pipe D13,120 rpm, insert: Ceratizit 5-point buttress): a) measured FRFs be-fore each set, b) stability chart and the corresponding experiment point(star) for each set, c-d) sound signals and their frequency contents.experiment. The predicted stability charts are shown in Figure 5.20.b. The arrowsconnect the experiment points (marked by stars) to their corresponding stabilitylobes. It can be seen that the model predicts the process to be stable at infeed valuesof 0.020′′ (0.508 mm) and 0.025′′ (0.635 mm), and unstable at 0.035′′ (0.889 mm).The recorded sound signals and their frequency contents are shown in Figures 5.20.cand d. As compared in Table 5.3, the predicted and experimental stability conditionsagree at all three infeed values.Remark. Due to the excessively large cutting forces in the third test (infeed of0.035′′ (0.889 mm)), the structural mode of the clamping chuck has been excitedas well. The frequency contents in Figures 5.20.c show chatter frequencies due to103Chapter 5. Threading Thin-Walled Oil PipesTable 5.3: Experimental and predicted stability conditions at different infeedvalues (pipe D13, 120 rpm, insert: Ceratizit 5-point buttress) (S: stable,C: chatter)Pass 1Infeed ωn [Hz] sim exp ωc [Hz]Set 1 0.020′′ (0.508 mm) 1395 S S -Set 2 0.025′′ (0.635 mm) 849 S S -Set 3 0.035′′ (0.889 mm) 808 C C 383, 773both chuck mode (383 Hz) and pipe mode (773 Hz).5.6.3 Threading With V-profile InsertIn order to verify the chatter stability model for V-profile threads, several tests havebeen conducted on pipe D7 with thickness of 0.3′′ (7.6 mm) using Sandvik Coro-mant three-point V-profile insert. One pass with infeed value of 0.025′′ (0.635 mm)has been cut at 250 rpm (cutting speed of 140 m/min). The original stickout of thepipe was 11′′ (279.4 mm). After the first test, the threaded section was cut away,leaving 9′′ (228.6 mm) stickout. A threading pass with the same infeed was cuton the new section. The procedure was repeated once more at 7′′ (177.8 mm) pipestickout.Figure 5.21.a shows the radial FRF at the tip of the pipe before conductingeach test. The predicted stability charts are shown in Figure 5.21.b, where theexperiment point (250 rpm, 0.635 mm) is marked by a star. The sound signals andtheir frequency contents are shown in Figures 5.21.c and d.The simulation and experimental results are compared in Table 5.4. It can beseen that the model correctly predicts the stability conditions in all three tests withV-profile insert.5.6.4 Remarks: Change in Pipe Dynamics During ThreadingIn order to investigate the effect of material removal on pipe dynamics, hammertests have been conducted on pipe D13 before and after a threading operation. Theoriginal pipe thickness is 0.43′′ (10.9 mm), and the final thread depth (buttress) is0.035′′ (0.89 mm). Figure 5.22 compares the FRF at the dominant mode for the104Chapter 5. Threading Thin-Walled Oil Pipeshbbb "bbb ,bbb 7bbb Rbbbbh",xOhb−-MagnitudeO[msN]FrequencyO[Hz]SO:OStickoutSetOhSetO"SetO,b " 7 -−bIh−bIbRbTimeO[s]SetOhSetO"SetO,b hbbb "bbbbbIRhhIR""IRxOhb−,FFTOofOSoundO[V]FrequencyO[Hz]SetO"9C-,OHz)SetO,SetOhO9D-bOHz)b hbb "bb ,bb 7bb RbbbIRhhIRSpindleOSpeedO[rpm]InfeedO[mm]SetO,9exp:Ostable)UnstableRegionSetO"9exp:Ounstable)SetOh9exp:Ounstable)hOtooth"Oteeth,OteethStableRegion9SOCkpOC-DOHz)9SOPkpOhbb,OHz)9SOhhkpODC-OHz)FRFOlOPipeODPSoundOSignalO[V]StabilityOCharta)b)c)d),lpointOVOprofileFigure 5.21: Experimental results with three-point V-profile insert at differ-ent pipe stickout (pipe D7, 250 rpm, infeed: 0.025′′ (0.635 mm), in-sert: Sandvik Coromant 24RD13-080): a) measured radial FRFs be-fore each set, b) stability charts and the experiment point (star), c-d)recorded sound signals and their frequency contents.Table 5.4: Experimental and predicted stability conditions for three-point V-profile insert (pipe D7, infeed: 0.025′′ (0.635 mm)) (s: stable, C: chatter).Pass 1Pipe Stickout ωn [Hz] sim exp ωc [Hz]Set 1 11′′ (279.4 mm) 896 C C 860Set 2 9′′ (228.6 mm) 968 C C 963Set 3 7′′ (177.8 mm) 1003 S S -105Chapter 5. Threading Thin-Walled Oil Pipes00.511.5xF10−6MagnitudeF[m/N]AfterBefore820 830 840 850 860 870FrequencyF[Hz]FRFF-FPipeFD13Figure 5.22: Change in the FRF of pipe D13 after a threading pass (threaddepth: 0.035′′ (0.889 mm), buttress profile)Table 5.5: Dynamic parameters of pipe D13 before and after threading (wallthickness: 0.43′′ (10.9 mm), thread depth: 0.035′′ (0.889 mm), buttress).natural frequency damping ratio modal stiffness modal massωn [Hz] ζ Ks [N/m] m [Kg]Before 849 0.16% 3.0×108 10.5After 839 0.16% 2.3×108 8.0original and threaded pipe, and the modal parameters are compared in Table 5.5.It can be seen that even though the depth of the thread is less than 10% of thewall thickness, the dynamic stiffness at the dominant mode has decreased by nearly40% after threading. This is mainly due to the sensitivity of shell modes to wallthickness. The natural frequency has shifted only about %1.Based on the experimental investigations, the pipe dynamics can change at least10%-20% over each pass. Advanced FRF updating methods [82] can be employedto further improve the stability predictions especially over deeper passes.5.7 Chatter Suppression StrategiesThread turning is a restrictive operation in that the width of cut is dictated by thethread profile. Assuming that the insert and workpiece are already chosen, the onlyprocess parameters which can be selected by the operator are the spindle speedand infeed values. In multi-point inserts, however, the infeed settings affect the106Chapter 5. Threading Thin-Walled Oil Pipeschip geometry on the first tooth only. It is more effective to integrate the infeedstrategies into the insert design; for example, the total width of cut can be reducedby implementing flank infeed or partial root engagement on some teeth (Figure 3.4).Strategies which enhance the dynamic stiffness of the system or disrupt thechip regeneration process can effectively increase the stability limits in threadingoperations. Two sample approaches are presented in the following sections.5.7.1 Effect of Additional DampingAs shown in Figure 5.23.a, a rubber ring has been mounted tightly inside pipe D13.The wall thickness of the pipe has been intentionally reduced to 0.25′′ (6.35 mm) toamplify the flexibilities. The radial FRF at the tip with and without the rubber ringare compared in Figure 5.23.b. Based on modal analysis of the FRFs, the modaldamping at the dominant mode (640 Hz) has increased from 0.06% to 0.95%, i.e.more than 15 times. It can be seen in Figure 5.23.c that both FRFs have the samevalues at lower frequencies, which means that the damping ring has not changedthe static stiffness.A threading pass with infeed of 0.020′′ (0.508 mm) has been cut at 120 rpm200 220 240 260012xR10−7MagnitudeR[m/N]FrequencyR[Hz]DampingRing500 1000 150002468xR10−6MagnitudeR[m/N]FrequencyR[Hz]WithRing-622RHzPWithoutRRing-640RHzPFRFR-RPipeRD13bPcPaPFigure 5.23: Effect of the damping ring on pipe dynamics (pipe D13, wallthickness: 0.25′′ (6.35 mm)): a) damping ring mounted inside, b) mea-sured radial FRFs at the tip with and without the ring, c) comparisonof the FRFs at low frequency region.107Chapter 5. Threading Thin-Walled Oil PipesSpindle8Speed8[rpm]0 100 200 300 4000F511F5 UnstableRegionStableRegionWithoutRingWithRingInfeed8[mm]Stability8Charta.0 2 4 6−0F2−0F100F1Time8[s]Sound8Signal8[V]WqORingWithRingb.0 1000 20000123x810−3FFT8of8Sound8[V]Frequency8[Hz]WqORing45808Hz.WithRingc .Figure 5.24: Effect of the damping ring on stability (pipe D13, Ceratizit 5-point buttress insert, 120 rpm): a) stability charts and the experimentpoint (star), b-c) recorded sound signals and their frequency contents.using Ceratizit 5-point buttress insert. Figure 5.24.a shows the predicted stabil-ity lobes with and without the damping ring. The experiment point (120 rpm,0.508 mm) is marked by a star. It can be seen that the damping ring significantlyincreases the stability limits. The recorded sound signals and their frequency con-tents are shown in Figures 5.24.b and c. The experimental results confirm that theprocess becomes fully stable when the damping ring is included.It should be noted, however, that additional damping affects the stability limitsonly if it stiffens the dominant mode. This is the case when using the dampingring since the dominant modes have shell behaviour. As shown in Figure 5.25.a,instead of the damping ring, a thick layer of rubber has been placed between thepipe and the clamping jaws. Figure 5.25.b compares the FRF at the tip with andwithout the rubber. It can be seen that while some of the modes have been dampedout significantly (possibly beam modes), the added rubber has had very little effecton the dynamic stiffness at the dominant mode. This is again due to the fact that108Chapter 5. Threading Thin-Walled Oil PipesRubberRubber 00.511.52xq10−6WithoutRubber2.5800 850 900 950 1000 1050 1100 1150Magnitudeq[m/N]Frequencyq[Hz]FRFq-qPipeqD7b7a7WithRubberFigure 5.25: Effect of adding rubber between the pipe and the clamping jaws,a) pipe D7 with the added rubber, b) measured radial FRFs at the tipwith and without the rubber.the dominant mode has a shell behaviour, and adding rubber at the chuck does notabsorb the shell vibrations at the tip.5.7.2 Using Different Spindle Speeds For Subsequent PassesIn typical turning operations, chip regeneration can be disrupted by continuous spin-dle speed variation. In threading processes, however, the axial feed is locked to thespindle speed based on the thread pitch. Due to the limited bandwidth of servodrives, change in the spindle speed can cause lead errors in the thread path. Whilesome modern machine tools claim to have achieved the required precision, spindlespeed variation during threading is still not recommended.However, using a similar idea, the effect of vibration marks from the previouspasses can be attenuated by selecting slightly different spindle speeds for subse-quent passes. The feasibility of this approach has been tested experimentally inthreading pipe D7 with Ceratizit 5-point buttress insert. In the first set, two passeswith infeed value of 0.020′′ (0.508 mm)/pass were cut both at spindle speed of250 rpm. The first pass was stable and the second pass was unstable. The processwas repeated on a similar pipe with the same infeed values, but this time the secondpass was performed at 225rpm, which eliminated chatter. Figure 5.26 shows therecorded sound signals and their frequency contents for the second passes in eachset. It can be seen that the process has not exhibited chatter vibrations in the caseof using different spindle speeds (Figure 5.26.c).109Chapter 5. Threading Thin-Walled Oil Pipes12 13 14 15−0.04−0.0200.02TimeF[s]PassF2Different9250/225Frpm6Same9250/250Frpm6SoundFSignalF[V]a61000 2000024xF10−4FFTFofFSoundF[V]FrequencyF[Hz]b6FFSameF9250/250Frpm61000 2000024xF10−4FFTFofFSoundF[V]FrequencyF[Hz]c6DifferentF9250/225Frpm6F9968FHz6Figure 5.26: Effect of using different speeds for chatter suppression (pipe D7,infeed: 0.020′′ (0.508 mm), Ceratizit 5-point buttress insert, Set 1: twopasses both at 250 rpm, Set 2: first pass at 250 rpm and second pass at225 rpm): a) recorded sound data during the second pass in each set,b-c) frequency contents of the sound signal for each case.5.8 Threading Toolbox Simulation SoftwareThe threading models developed in this thesis have been implemented in a simula-tion engine along with a user interface in MATLAB. The research software, calledThreading Toolbox, has been released as one of the modules of CUTPRO R© Vir-tual Machining software. It is currently being used by the collaborating company(TenarisTAMSA) in analyzing the threading process of oil pipes.Figure 5.27 shows the interface of the developed threading toolbox. The geom-etry of the custom multi-point insert is imported as a DXF file directly from a CADmodel. The workpiece material can be selected from the available library, or thecutting force coefficients can be directly entered based on linear or Kienzle forcemodels. The software operates in two modes: simulation and optimization. In thesimulation mode, the user selects the infeed settings, i.e. infeed strategy, number ofpasses, and infeed per pass. The threading engine then simulates the following out-puts for each tooth and over each pass: chip geometry (including engagement lengthand chip area), axial/radial/tangential cutting forces, spindle torque and power. The110Chapter 5. Threading Thin-Walled Oil Pipesnumerical results can be exported as a CSV file to MS Excel for further analysis.Alternatively, the software can be used in optimization mode for process plan-ning. Instead of choosing the number of passes and infeed values, the user se-lects the final thread depth and imposes optional constraints on minimum/maxi-mum chip thickness, cutting forces, spindle torque/power, and stability margin (seeSection 4.11 for constraints). The threading engine then finds the required numberof passes and infeed values to achieve highest productivity while respecting all thedefined constraints.The developed research software can be used not only for process planning inmanufacturing units but also as a design tool for optimization of threading inserts.5.9 SummaryThis chapter investigates the threading process of thin-walled workpieces with spe-cific application to oil pipes. Dominant mode shapes of sample pipes have been de-termined using finite element and experimental modal analysis. Extensive threadingexperiments have been conducted on real scale oil pipes, and the results have beencompared against the predictions.It has been shown that the proposed model can reliably predict the stability ofthe process for different setup dynamics, cutting conditions, and insert geometries.Sample approaches for chatter suppression have been suggested, and their effective-ness has been demonstrated experimentally. All the developed models have beenimplemented in a simulation engine which can be used for process planning andinsert design.111Figure 5.27: User interface of the developed Threading Toolbox software package.112Chapter 6Conclusions and Future Directions6.1 Summary and ContributionsAccuracy and surface quality of threads is crucial in applications such as offshorepipelines where strong sealing is required. Geometrical errors and vibration marksleft during the machining process can lead to stress concentration, fatigue failure,and leakage of the joint. A model which can predict the behaviour of the processahead of time provides a systematic tool to optimize the cutting tools and processparameters without costly trials.There are numerous challenges in modelling the mechanics and dynamics ofmulti-point thread turning. Since threading is a form cutting operation, the resul-tant chip can have complex geometries. Not only the chip thickness but also theeffective oblique cutting angles and direction of local forces can vary significantlyalong the cutting edge. Due to varying local approach angle, relative vibrationsbetween the tool and workpiece can also have different effect on the local chipthickness. Chip regeneration mechanism in multi-point threading is different thanregular turning; since the tool moves one thread pitch over each spindle revolution,the previous vibration marks left by each tooth affects the chip thickness on a dif-ferent tooth. Threading thin-walled oil pipes involves additional complexities dueto the structural beam and shell mode vibrations of the pipe.This thesis develops a generalized model to determine the chip geometry, cut-ting forces, and stability during threading of oil pipes. The main contributions canbe summarized as:113Chapter 6. Conclusions and Future Directions• A novel semi-analytical methodology has been developed to model the chipgeometry for multi-point inserts with custom tooth profiles and arbitrary in-feed plans. The proposed method categorizes possible cases of tool-workpieceengagements, and uses the kinematics of the process and geometrical param-eters of the insert to determine the boundaries of the chip. Special cases suchas partial root engagement can be modelled as well.• A systematic discretization method has been developed to divide the chiparea along the cutting edge based on the chip flow direction. The algorithmis efficiently run only on a small moving window, and can consider the effectof chip compression (interference) at highly curved segments of the profile.• A novel cutting force calculation model for threading inserts has been pro-posed. The local effective oblique cutting angles are systematically deter-mined by defining local coordinate systems and oblique vectors. Cuttingforce coefficients are evaluated locally for each chip element using nonlin-ear Kienzle force model and local orthogonal-to-oblique transformations.• A generalized chip regeneration model for multi-point threading has beendeveloped. The three-dimensional vibrations of the tool and workpiece havebeen projected in the local chip thickness direction at each point along thecutting edge. The effect of current and previous vibrations on chip thick-ness have been modelled for the general case where the teeth have differentprofiles.• The delay differential equation representing the three-dimensional dynamicsof multi-point threading has been developed. Process damping forces are cal-culated locally by projecting the vibration velocity in the direction of chipthickness at each point along the cutting edge. Dynamic cutting forces areevaluated from chip thickness variation due to the current and previous vi-brations. State space representation of dynamic equation of motion in modalspace has been derived. A time-marching numerical method based on semi-discretization and Simpson’s integration rule has been presented to simulatethe vibrations and dynamic forces during the threading operation.• A chatter stability model for multi-point threading has been developed. Giventhe insert geometry and structural dynamics of the tool and workpiece, the114Chapter 6. Conclusions and Future Directionsmodel can generate stability diagrams for each pass showing the maximumallowed infeed at each spindle speed.• A novel dynamic model for threading thin-walled workpieces have been pro-posed by modelling the effect of residual shell vibrations on chip thickness.The circumference of the workpiece is discretized, and the dynamic equationof motion is solved in modal space to find the instantaneous vibrations at eachpoint along the circumference.• An optimization algorithm has been developed to plan the number of passesand infeed settings for each pass. The algorithm can consider user-definedconstraints on: spindle torque/power, axial/radial/tangential forces on eachtooth, and minimum/maximum chip thickness on each tooth. The process isoptimized based on stability, productivity, uniform load distribution over dif-ferent passes, tooth breakage avoidance, chip evacuation, and surface quality.• As a practical application, threading real scale oil pipes has been investigatedexperimentally. It has been shown that the developed models can predict thestability of the process with reasonable accuracy.• A threading simulation software (Threading Toolbox) has been developed forindustrial use. It can be used not only for process planning but also as a designtool for optimization of the cutting inserts and tooling systems.6.2 Future Research DirectionsThe models presented in this thesis are the first iteration in generalized modellingof thread turning operations. There are still several aspects of this research whichcan be further refined or extended:• Cutting force coefficients in this thesis are identified mechanistically throughcutting experiments. Semi-analytical methods such as slip line field [78] andfinite element models can be used to predict the coefficients based on mate-rial’s characteristics and cutting edge parameters.• Chip evacuation is a serious issue in threading. While chip geometry anddiscretization has been presented in this thesis, the three-dimensional chip115Chapter 6. Conclusions and Future Directionsflow [84] was not modelled. Finite element methods can be used to simulatethe chip formation process.• The accuracy of cutting force prediction can be improved by including theeffect of strain hardening [10] over subsequent passes. Finite element modelsmust be developed to study the effect of ploughing on surface hardness andcutting force coefficients.• Thermal models [85] can be developed to investigate temperature distributionand optimize the insert coating accordingly.• Structural dynamics of thin-walled pipes were mainly identified experimen-tally in this thesis. 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Cambridge UniversityPress.→ cited on p. 92, 93[84] Jawahir, I., and van Luttervelt, C., 1993. “Recent developments in chip controlresearch and applications”. CIRP Annals - Manufacturing Technology, 42(2),pp. 659 – 693. → cited on p. 116[85] Islam, C., Lazoglu, I., and Altintas, Y., 2015. “A three-dimensional transientthermal model for machining”. Journal of Manufacturing Science and Engi-neering, 138(2), Sept., pp. 021003–021003–17. → cited on p. 116124Appendix AMode Shape Extraction FromANSYSAfter building the geometry and meshing, copy the following code in the SessionEditor of ANSYS! SOLVE THE EVPANTYPE, 2 ! Modal ana l ys i sMODOPT,LANB,20 ! Block Lancoz , number o f modesEQSLV,FRONT ! F ron ta l SolverMXPAND,20 ! Expand number o f modesSOLVEFINISH/POST1SET, LIST! ! Output the eigen values/ output , , f r qset , l i s t/ output , termThe following lines write the eigen values and the mass-normalized mode shapevectors to the text filensel , s , node , , 2 ! TCPnsel , a , node , , 4 ! Repeat f o r a l l nodes of i n t e r e s t/ output , , e ig∗do , i ,0 ,10 ! no of modesset , , i/ page , , ,200prd isp∗enddo/ output , term125
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Mechanics and dynamics of multi-point threading of thin-walled oil pipes Rezayi Khoshdarregi, Mohammad 2017
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Title | Mechanics and dynamics of multi-point threading of thin-walled oil pipes |
Creator |
Rezayi Khoshdarregi, Mohammad |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | The pipelines used in the offshore extraction of oil and gas are connected by threaded joints. Any geometrical error or vibration marks left on the thread surface during the machining process can lead to stress concentration and fatigue failure of the joint. Such instances in the past have led to massive oil leakage and environmental disasters. Threading is a form cutting operation resulting in wide chips with complex geometries. Multi-point inserts used in mass production can have different custom profiles on each tooth. The chip thickness as well as the effective oblique cutting angles, cutting force coefficients, and direction of local forces vary along the cutting edge. Since the tool moves one thread pitch over each spindle revolution, the vibration marks left by a tooth affect the chip thickness on the following tooth. Threading of oil pipes imposes additional complexities due to the flexural vibrations of thin-walled pipes, which lead to severe chatter instability. This thesis develops a novel and generalized model to formulate, simulate, and optimize general multi-point threading processes. A systematic semi-analytical methodology is first proposed to determine the chip geometry for custom multi-point inserts with arbitrary infeed strategies. A search algorithm is developed to systematically discretize the chip area along the cutting edge considering the chip flow direction and chip compression at the corners. The cutting force coefficients are evaluated locally for each element, and the resultant forces are summed up over the engaged teeth. Multi-mode vibrations of the tool and pipe are projected in the direction of local chip thickness, and the dynamic cutting and process damping forces are calculated locally along the cutting edge. A novel chip regeneration model for multi-point threading is developed, and stability is investigated in frequency domain using Nyquist criterion. The process is simulated by a time-marching numerical method based on semi-discretization. An optimization algorithm is developed to maximize productivity while respecting machine's limits. The proposed models have been verified experimentally through real scale experiments. The algorithms are integrated into a research software which enables the industry to optimize the process ahead of costly trials. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-07-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0349093 |
URI | http://hdl.handle.net/2429/62441 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2017-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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