Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Quiring, Shelby Alexandra
Master of Applied Science - MASc
University of British Columbia
Turning operations are commonly limited by chatter vibrations, in which some combinations of spindle speed, feed rate, and depth of cut cause the tool or workpiece to vibrate uncontrollably. Chatter limits material removal rates, can damage the tool and workpiece, and is highly undesirable. This thesis presents a novel actuator that passively damps workpiece vibrations when turning highly flexible workpieces, such as a long slender rod, or a hollow tube, where the flexibility of the workpiece is the dominant source of chatter. Traditionally, CNC machines and tools are designed to be highly rigid, ostensibly to minimize chatter. However, for flexible workpieces, a rigid tool is not optimal for chatter minimization. This thesis proposes that a flexible tool with a high damping ratio can significantly improve the chatter stability of the tool-workpiece system when the tool’s natural frequency and stiffness are tuned optimally. The proposed actuator has two subsystems, comprising a primary compliant damped tool tip stage, mounted on a secondary linear actuation stage. The tool compliance is delivered by stacking a number of plate springs, which leads to tool flexibility only in the flexible radial direction of the workpiece. This is the direction where the regenerative chip thickness causing chatter is produced. The high damping ratio is delivered by closely-spaced plates in an oil bath, where shearing a thin oil film between plates provides linear damping with no static friction. A stepper motor linear actuator compensates for the static deflection errors from the flexible tool deforming under cutting loads. The mathematical model of the system consists of a workpiece and tool each with single degree of freedom dynamics, and a digital controller that compensates for the deflection errors. The actuator and supporting electronics have been designed, manufactured, and instrumented with vibration sensors. The performance of the actuator has been experimentally evaluated by turning a steel test workpiece. It is shown that the chatter stability is improved by at least 10 fold, and the static deflections within the bandwidth of the controller are compensated with a tool-tip position variation within ±3𝜇m at steady state.
Compliant Tool Modelling and Designfor Chatter Suppression in HighlyFlexible TurningbyShelby Alexandra QuiringB.A.Sc., The University of British Columbia, 2023A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2025© Shelby Alexandra Quiring 2025The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Compliant Tool Modelling and Design for Chatter Suppression in HighlyFlexible Turningsubmitted by Shelby Alexandra Quiring in partial fulfillment of the re-quirements forthe degree of Master of Applied Sciencein Mechanical EngineeringExamining Committee:Hongshen Ma, Professor, Mechanical Engineering, UBCSubstitute Examiner, in place of SupervisorXiaoliang Jin, Associate Professor, Mechanical Engineering, UBCSupervisory Committee MemberRyozo Nagamune, Professor, Mechanical Engineering, UBCSupervisory Committee MemberAdditional Supervisory Committee Members:Yusuf Altintas, Professor, Mechanical Engineering, UBCSupervisoriiAbstractTurning operations are commonly limited by chatter vibrations, in whichsome combinations of spindle speed, feed rate, and depth of cut cause thetool or workpiece to vibrate uncontrollably. Chatter limits material removalrates, can damage the tool and workpiece, and is highly undesirable.This thesis presents a novel actuator that passively damps workpiece vibra-tions when turning highly flexible workpieces, such as a long slender rod, ora hollow tube, where the flexibility of the workpiece is the dominant sourceof chatter. Traditionally, CNC machines and tools are designed to be highlyrigid, ostensibly to minimize chatter. However, for flexible workpieces, arigid tool is not optimal for chatter minimization. This thesis proposes thata flexible tool with a high damping ratio can significantly improve the chat-ter stability of the tool-workpiece system when the tool’s natural frequencyand stiffness are tuned optimally.The proposed actuator has two subsystems, comprising a primary compliantdamped tool tip stage, mounted on a secondary linear actuation stage. Thetool compliance is delivered by stacking a number of plate springs, whichleads to tool flexibility only in the flexible radial direction of the workpiece.This is the direction where the regenerative chip thickness causing chatter isproduced. The high damping ratio is delivered by closely-spaced plates in anoil bath, where shearing a thin oil film between plates provides linear damp-ing with no static friction. A stepper motor linear actuator compensates forthe static deflection errors from the flexible tool deforming under cuttingloads. The mathematical model of the system consists of a workpiece andtool each with single degree of freedom dynamics, and a digital controllerthat compensates for the deflection errors. The actuator and supportingelectronics have been designed, manufactured, and instrumented with vi-bration sensors. The performance of the actuator has been experimentallyevaluated by turning a steel test workpiece. It is shown that the chatterstability is improved by at least ten fold, and the static deflections withinthe bandwidth of the controller are compensated with a tool-tip positionvariation within ±3µm at steady state.iiiLay SummaryManufacturing metal parts commonly requires cutting material away fromlarge blank pieces of metal to form the desired part shape. Maximizingthe speed of material removal is desirable for production efficiency. Thesecutting processes can suffer from an adverse vibration phenomenon knownas chatter, which limits material removal rates, and can damage parts andmachines. This work focuses on the design of a vibration-absorbing cuttingtool intended for use on large lathes cutting highly flexible workpieces. Ingeneral, tools and machines are designed to be as rigid as possible. The keyadvancement of this work is that the tool must instead be flexible and heavilydamped to achieve chatter improvements when the flexible workpiece is theprimary source of chatter. A prototype tool has been designed and tested toverify the theoretical advancement. This tool provided an experimentally-demonstrated material removal rate improvement of >10x over a traditionalrigid tool.ivPrefaceThis thesis is a result of the research and development conducted by theauthor, Shelby Alexandra Quiring, under the supervision of Professor YusufAltintas of UBC Mechanical Engineering and Dr. Dan O¨stling and AndersLiljerehn of Sandvik Coromant. This work was accomplished as a collabora-tion between the Manufacturing Automation Laboratory of the Universityof British Columbia and Sandvik Coromant Trondheim.Dr. Altintas provided supervision, guidance, funding, editing, and the lit-erature review. Dan and Anders provided engineering advice and indirectfinancial support via Sandvik Coromant’s support of the Manufacturing Au-tomation Laboratory. I (Shelby Quiring) completed all modelling, design,build, and experimental verification activities in this thesis.No generative AI was used in any of the research or thesis writing.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Research Focus . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Orthogonal Cutting Chatter Models . . . . . . . . . . . . . . 42.1.1 Flexible Workpiece . . . . . . . . . . . . . . . . . . . 42.1.2 Flexible Tool . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Flexible Workpiece And Flexible Tool (2DOF) . . . . 112.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 163 Compliant Compensated Tool Modelling . . . . . . . . . . . 183.1 Tool Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Parameter Optimization . . . . . . . . . . . . . . . . . . . . 223.3 Tool Mass Limiting Effect . . . . . . . . . . . . . . . . . . . 263.4 Static Compensation . . . . . . . . . . . . . . . . . . . . . . 273.5 Workpiece Position Dependence Investigation . . . . . . . . . 303.6 Single-Axis Model Simulation . . . . . . . . . . . . . . . . . 35vi4 Compliant Compensated Tool Design . . . . . . . . . . . . . 414.1 Coordinate System Definition . . . . . . . . . . . . . . . . . 414.2 Target Design Parameters . . . . . . . . . . . . . . . . . . . . 434.3 Tool Architecture Overview . . . . . . . . . . . . . . . . . . . 454.3.1 Mechanical Architecture . . . . . . . . . . . . . . . . 454.3.2 Electrical Architecture . . . . . . . . . . . . . . . . . 504.4 Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . . 524.4.1 Compliant Spring Design . . . . . . . . . . . . . . . . 524.4.2 Damper Design . . . . . . . . . . . . . . . . . . . . . 614.4.3 Tool Tip Design . . . . . . . . . . . . . . . . . . . . . 764.4.4 External Housing Design . . . . . . . . . . . . . . . . 784.4.5 Linear Actuation Stage Design . . . . . . . . . . . . . 794.5 Electrical Design . . . . . . . . . . . . . . . . . . . . . . . . . 864.5.1 Linear Actuator Selection . . . . . . . . . . . . . . . . 864.5.2 Displacement Sensor Selection . . . . . . . . . . . . . 884.5.3 Other Electrical Equipment . . . . . . . . . . . . . . 914.5.4 Controller Design . . . . . . . . . . . . . . . . . . . . 994.6 Tool System Identification . . . . . . . . . . . . . . . . . . . 1074.6.1 Linear Actuator Plant Identification via Open-LoopSine Sweep . . . . . . . . . . . . . . . . . . . . . . . . 1074.6.2 Tool Tip Dynamics Identification via Tap Testing . . 1095 Experimental Validation . . . . . . . . . . . . . . . . . . . . . 1135.1 Test Workpiece Design . . . . . . . . . . . . . . . . . . . . . 1135.2 Test Workpiece System Identification . . . . . . . . . . . . . 1175.3 Experimental Design . . . . . . . . . . . . . . . . . . . . . . 1195.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 1225.5 Predicted vs Experimental Stability Improvements . . . . . . 1336 Conclusion and Recommendations . . . . . . . . . . . . . . . 1396.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.2 Recommendations for Future Work . . . . . . . . . . . . . . 141Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144AppendicesA Active Tool-Tip Control Strategies . . . . . . . . . . . . . . . 147A.1 Tool Tip Active Control . . . . . . . . . . . . . . . . . . . . . 147viiA.1.1 System Dynamics With Control . . . . . . . . . . . . 147A.1.2 Controller Design . . . . . . . . . . . . . . . . . . . . 151A.1.3 Active Control Challenges . . . . . . . . . . . . . . . 155viiiList of Tables2.1 Modal parameters used for example FRF. . . . . . . . . . . . 143.1 Modal parameters used for example FRF. . . . . . . . . . . . 213.2 Modal parameters fixed for parameter optimization analysis. 223.3 Modal parameters fixed for position dependence analysis. . . 323.4 Modal and cutting parameters fixed for simple model Simulinksimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Descriptions of each simulation run. . . . . . . . . . . . . . . 364.1 Target Design Specifications. . . . . . . . . . . . . . . . . . . 444.2 Inner pinwheel flexure spring parameters. . . . . . . . . . . . 604.3 Linear sheet damper parameters. . . . . . . . . . . . . . . . . 724.4 Outer pinwheel flexure spring parameters. . . . . . . . . . . . 814.5 Select specifications for linear actuator. . . . . . . . . . . . . 874.6 Select specifications for displacement sensor system. . . . . . 894.7 Motor Driver Specifications. . . . . . . . . . . . . . . . . . . . 924.8 Electrical System Power Requirements. . . . . . . . . . . . . . 944.9 Block Diagram Constant Definitions. . . . . . . . . . . . . . . 1004.10 Position Control Loop Characteristic Values. . . . . . . . . . 1024.11 Sine Sweep Parameter Results. . . . . . . . . . . . . . . . . . 1094.12 Identified Tool Parameters. . . . . . . . . . . . . . . . . . . . 1115.1 Experimental Test Matrix. . . . . . . . . . . . . . . . . . . . . 1225.2 Identified Effective Workpiece and Tool Parameters, SDOFFits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3 Predicted Stability Limits from Tlusty’s Law. . . . . . . . . . 137A.1 Active chatter controller transfer functions. . . . . . . . . . . 151ixList of Figures1.1 Example of turned surface with chatter (left) and without(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Single-Degree-of-Freedom Orthogonal chatter diagram withflexible workpiece. . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Example Stability Lobe Diagram . . . . . . . . . . . . . . . . 72.3 Example single-degree-of-freedom force-displacement FRF. Thestability-limiting negative real part is highlighted. . . . . . . . 82.4 Block diagram dynamics representation of single-degree-of-freedom orthogonal chatter model, with flexible workpiece. . . 92.5 Single-Degree-of-Freedom Orthogonal chatter diagram withflexible tool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 2-Degree-of-Freedom Orthogonal chatter diagram with flexi-ble workpiece and flexible tool. . . . . . . . . . . . . . . . . . 112.7 Block diagram dynamics representation of 2-degree-of-freedomorthogonal chatter model, with flexible workpiece and flexibletool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Example 2-degree-of-freedom force-displacement FRF. Thestability-limiting minimum negative real part is highlighted. . 153.1 Example 2DOF FRF with flexible, lightly-damped workpieceand highly flexible, heavily-damped tool. . . . . . . . . . . . . 183.2 Example 2DOF FRF with flexible, lightly-damped workpieceand highly flexible, heavily-damped tool. . . . . . . . . . . . . 213.3 Mimimum Negative Real Part Heatmap vs stiffness ratio andnatural frequency ratio, ζn,ω = 0.005 . . . . . . . . . . . . . . 243.4 Mimimum Negative Real Part Heatmap vs stiffness ratio andnatural frequency ratio, ζn,ω = 0.02 . . . . . . . . . . . . . . . 253.5 Mimimum Negative Real Part Heatmap vs stiffness ratio andnatural frequency ratio, ζn,ω = 0.1 . . . . . . . . . . . . . . . 26x3.6 3-degree-of-freedom external turning dynamics model withflexible workpiece, flexible tool, and linear actuation stageusing active control to position the tool. . . . . . . . . . . . . 273.7 Example highly flexible tool FRFs with and without com-pensation. Adding compensation makes the static stiffnesseffectively infinite. . . . . . . . . . . . . . . . . . . . . . . . . 293.8 MDOF workpiece model setup - fixed-fixed beam. . . . . . . . 313.9 Real part of workpiece-only FRFs vs position and frequency. 323.10 Selected workpiece FRFs at varying positions. . . . . . . . . . 333.11 Workpiece, tool, and combined FRFs at centre of workpiece. . 333.12 Real part of workpiece + tool FRFs vs position and frequency. 343.13 Real part of workpiece + tool FRFs vs frequency. This showsall the FRFs from Figure 3.12 superimposed for each position. 343.14 Simulink Diagram . . . . . . . . . . . . . . . . . . . . . . . . 363.15 Simulation 1 result showing tool and workpiece displacement. 373.16 Simulation 2 result showing tool and workpiece displacement. 383.17 Simulation 3 result showing tool and workpiece displacement. 394.1 Coordinate system diagram for CNC Lathe. . . . . . . . . . . 414.2 Tool and Workpiece Displacement Defintions. . . . . . . . . . 424.3 Nested flexure spring architecture. . . . . . . . . . . . . . . . 464.4 Simplified full system diagram, labeled. Green corresponds tocomponents moving with displacement xt2, blue correspondsto xt1, and black is stationary. . . . . . . . . . . . . . . . . . . 474.5 Full tool assembly cross-section, labeled. . . . . . . . . . . . . 484.6 Full tool assembly photos. . . . . . . . . . . . . . . . . . . . . 494.7 Diagram of electrical system. . . . . . . . . . . . . . . . . . . 514.8 Photo of assembled electrical panel. . . . . . . . . . . . . . . . 514.9 Pinwheel spring flexure design, inner flexure (left) and outerflexure (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 524.10 Idealized displacement degrees of freedom diagram for dual-pinwheel flexure. . . . . . . . . . . . . . . . . . . . . . . . . . 534.11 Definition of spring arm dimensions. . . . . . . . . . . . . . . 544.12 Beam spring free-body diagram. . . . . . . . . . . . . . . . . 554.13 Stiffness ratio vs beam cross-section aspect ratio. . . . . . . . 564.14 Stress vs thickness, number of layers for pinwheel spring. . . . 584.15 Inner Pinwheel Spring and Spacer Designs. . . . . . . . . . . 594.16 Damper Endcap Assembly, Exploded View. . . . . . . . . . . 594.17 Fluid piston damper diagram. . . . . . . . . . . . . . . . . . . 61xi4.18 Linear and nonlinear damping characteristic curves showingtangent damping of the nonlinear curve. . . . . . . . . . . . . 624.19 2-plate oil shearing model. . . . . . . . . . . . . . . . . . . . . 634.20 Cross-Section Option A: Cylindrical Piston and Cylinder . . . 644.21 Cross-Section Option B: Interleaved Spirals . . . . . . . . . . 644.22 Cross-Section Option C: Layered Sheets . . . . . . . . . . . . 654.23 Comparison of Damper Cross-Section Options . . . . . . . . . 664.24 Damper Cartridge Assembly - moving system (left) and sta-tionary system (right). . . . . . . . . . . . . . . . . . . . . . . 674.25 Damper sheet comparison - moving sheet (left) and stationarysheet (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.26 Damper Housing. . . . . . . . . . . . . . . . . . . . . . . . . . 694.27 Damper Assembly, Vertical Plane Cross-Section. . . . . . . . 704.28 Damper Assembly, Horizontal Plane Cross-Section. . . . . . . 714.29 Damper build process - layers of moving and stationary sheetsare built up on an assembly fixture plate, separated usinglayers of spacer washers. . . . . . . . . . . . . . . . . . . . . . 724.30 Damper build process - fully-built damper cartridge assemblyconsisting of sheet stack assembly with moving and stationaryframes clamping sheets together. . . . . . . . . . . . . . . . . 734.31 Damper build process - the damper cartridge assembly isplaced in the damper housing. . . . . . . . . . . . . . . . . . . 744.32 Damper oil fill via automotive vacuum pump. . . . . . . . . . 754.33 Full spring-damper tool assembly. Note this photo shows anold single-layer endcap spring design, not the final multi-layerdesign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.34 Reference Toolholder SSDCN 1212F 09-M . . . . . . . . . . . 764.35 Tool head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.36 Tool head isolated in full assembly. . . . . . . . . . . . . . . . 774.37 Stationary outer housing. . . . . . . . . . . . . . . . . . . . . 784.38 Linear actuation assembly, exploded. . . . . . . . . . . . . . . 794.39 Diagram of the linear actuation assembly. Black text indi-cates a stationary component, blue shows components movingwith displacement xt1, and green shows components movingwith displacement xt2. . . . . . . . . . . . . . . . . . . . . . . 804.40 Outer pinwheel caps. . . . . . . . . . . . . . . . . . . . . . . . 814.41 Damper adapter mount connection. . . . . . . . . . . . . . . . 824.42 Motor coupling comparison - flexible option (left) and rigidoption (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 82xii4.43 Pocket recessed around the probe tip to prevent undesirableinductive coupling. . . . . . . . . . . . . . . . . . . . . . . . . 834.44 Sensor target on the damper adapter mount is a circle of16mm, to provide an accurate reading. . . . . . . . . . . . . . 844.45 Setting the offset of the probe tip using precision shim. . . . . 844.46 Sensor S2 installed in the prototype. . . . . . . . . . . . . . . 854.47 LN17D6200-W0601-300MN Linear Actuator (Source: Lin En-gineering) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.48 Manufacturer’s Load-Speed Curve for Linear Actuator (Source:Lin Engineering) . . . . . . . . . . . . . . . . . . . . . . . . . 874.49 ECL150 Sensor Driver (Source: Lion Precision) . . . . . . . . 884.50 Probe Range and Offset Definitions (Source: Lion Precision) 894.51 Definition of displacements and sensors . . . . . . . . . . . . . 904.52 TMS320 Launchpad (Source: Texas Instruments). . . . . . . 914.53 Motor Driver (Source: Lin Engineering). . . . . . . . . . . . . 924.54 ADC Evaluation Board (Source: Analog Devices). . . . . . . 934.55 Computer power supply (Source: FSP Technology). . . . . . 944.56 Protoboard 1, Panel Side. . . . . . . . . . . . . . . . . . . . . 964.57 Protoboard 2, Tool Side. . . . . . . . . . . . . . . . . . . . . . 974.58 Graphic User Interface. . . . . . . . . . . . . . . . . . . . . . 984.59 Control Block Diagram - Summing Forces Perspective. . . . . 994.60 Control Block Diagram - Rearranged for Traditional PositionControl Block Diagram Structure. . . . . . . . . . . . . . . . 1004.61 Bode Plot of controller, open-loop (or loop return) transferfunction, and closed-loop transfer functions in z domain. . . . 1024.62 System FRFs, Kp=1. . . . . . . . . . . . . . . . . . . . . . . . 1034.63 System FRFs, Kp=10. . . . . . . . . . . . . . . . . . . . . . . 1044.64 Phase Margin Sweep. . . . . . . . . . . . . . . . . . . . . . . . 1054.65 Sine Sweep System Identification Results. . . . . . . . . . . . 1084.66 Measured Open-Loop Tool FRF. . . . . . . . . . . . . . . . . 1104.67 Measured Open-Loop Tool FRF, Passive Stage Only. . . . . . 1114.68 Measured Compensated Tool FRF. . . . . . . . . . . . . . . . 1125.1 Experimental Workpiece Design . . . . . . . . . . . . . . . . . 1145.2 Workpiece Mode Shapes . . . . . . . . . . . . . . . . . . . . . 1155.3 Theoretical tool, workpiece, and combined FRFs. . . . . . . . 1165.4 Tap testing locations for the experimental workpiece. . . . . . 1175.5 Experimental workpiece FRFs, with average best fit param-eters mw = 21kg, cw = 406 Ns/m, kw = 5.9e6 N/m. . . . . . 1185.6 Experimental Testing Setup. . . . . . . . . . . . . . . . . . . . 120xiii5.7 Rigid Tool Setup. . . . . . . . . . . . . . . . . . . . . . . . . . 1205.8 Passive Compliant Tool Setup. . . . . . . . . . . . . . . . . . 1215.9 Compliant Compensated Tool Setup. . . . . . . . . . . . . . . 1215.10 Cut test results for feed rate 0.05 mm/rev, spindle speed 175RPM, no coolant. Only depth of cut and cutting tool config-uration are varied. . . . . . . . . . . . . . . . . . . . . . . . . 1235.11 Stability Map for each tool configuration. . . . . . . . . . . . 1245.12 Fourier Transforms of Rigid vs Passive Compliant Accelerom-eter Signals, n = 175 RPM, feed rate = 0.05 mm/rev, radialdepth of cut ≈ 0.2 mm. . . . . . . . . . . . . . . . . . . . . . 1255.13 Chip clumping problem, due to small tool clearance and hotchips melting and embedding in plastic chip cover. . . . . . . 1275.14 Cut test data from displacement sensors in Test No. 19, n =175 RPM, feed = 0.05 mm/rev, depth of cut 0.4 mm. . . . . 1295.15 Cut test data from displacement sensors in Test No. 23, n =175 RPM, feed = 0.05 mm/rev, depth of cut 0.6 mm. . . . . 1315.16 Cut test surfaces, (A) before Test No. 19, (B) after Test No.19 and before Test No. 23, and (C) after Test No. 23. . . . . 1325.17 Chip area model. . . . . . . . . . . . . . . . . . . . . . . . . . 1335.18 Oriented FRF derivation diagram. . . . . . . . . . . . . . . . 1345.19 Effective angle from nose radius. . . . . . . . . . . . . . . . . 1355.20 FRFs for Stability Prediction. . . . . . . . . . . . . . . . . . . 137A.1 3-Degree-of-Freedom Orthogonal chatter diagram with flex-ible workpiece, flexible tool, and linear actuation providingactive control of the tool tip. . . . . . . . . . . . . . . . . . . 147A.2 Block diagram dynamics representation of single-degree-of-freedom orthogonal chatter model, with flexible workpiece,flexible tool, and active chatter control. . . . . . . . . . . . . 150A.3 Example controller FRFs for the follower and 45o lag con-troller. The workpiece natural frequency is indicated. . . . . . 154A.4 Example single-degree-of-freedom effective FRFs, with no con-trol, follower control, and 45o lag control. Both controllersraise the minimum negative real part . . . . . . . . . . . . . . 155xivAcknowledgementsI extend my gratitude to my research supervisor, Dr. Yusuf Altintas, for allof his support, patience, and teachings. I also thank Dan O¨stling and AndersLiljerehn for their oversight and review of my work, and for supporting me incompleting part of my work at the Sandvik Coromant facility in Trondheim.Thank you to the many engineers and technicians of the UBC MECH Ma-chine Shop and the Instrumentation Shop for ongoing advice and access totools and machines with which to build the prototype for this thesis.A significant thank you to my dad Christopher Quiring, for the support inmy pursuit of mechanical engineering.Finally, thank you to the Natural Sciences and Engineering Research Councilof Canada, as well as the British Columbia Graduate Scholarship, for thefinancial support of this research.xvDedicationFor Jordan and Oscar.xviChapter 1Introduction1.1 Research FocusExternal turning is a machining process for axisymmetric metal cutting, andis a foundational process in industrial manufacturing. However, productivityrates (measured in volume of material removed per unit time) are in somecases limited by the stability of the turning process, which is in turn drivenby the flexibility of the machine tool and the workpiece. In general, moreflexible workpieces and/or tools are more prone to an oscillating instability,known as ’chatter’, which results in noise, poor surface finish, and potentialdamage to the tool and/or part.Figure 1.1: Example of turned surface with chatter (left) and without(right).1Chatter is a phenomenon which can affect all machining processes relyingon rotating tools and/or workpieces. In certain conditions, there are well-known established methods of avoiding chatter. In general, making thetool and workpiece as rigid as possible by reducing cantilever overhangs orusing low aspect-ratio tools is a simple and well-known approach for chatterminimization.In some cases, it may be necessary to have an undesirably flexible tool orworkpiece. For example, in the case of internal single-point boring of high-aspect-ratio holes, the tool must be long and thin, making it highly flexible.In these cases, advanced techniques such as installing a tuned-mass-damperinto the tool may be used to significantly improve its dynamic stiffness andtherefore the process stability.This thesis is focused in particular on the inverse case:• External turning,• Of highly flexible workpieces, which are prone to chatter during tradi-tional external turning,• With an arbitrarily rigid (or compliant) tool.In this case, a tuned-mass-damper generally cannot be equivalently installedin the workpiece. The workpiece is also rotating, making it challenging toattach anything that may stiffen or damp it. For specific types of workpieces(eg, simple cylinders), there is existing technology such as follower rests thatmay be mounted to the machine and support the workpiece with rollingcontact. However, these are unsuitable for anything with more complexgeometry, including tapers or stepped diameters.The thesis question is then as follows: Given some highly flexible,chattering workpiece, can the dynamics of the tool be modified via activeor passive means to improve the stability of the combined tool-workpiecesystem?Research in this area has revealed that this is possible, using a hybrid active-passive tool, referred to in this work as a ’Compliant Compensated Tool’.This tool consists of a cutting edge mounted to a highly compliant and2heavily damped mass-spring-damper, which is itself mounted to a linearactuation stage.Chapter 2 outlines background information on the chatter phenomenon,existing methods to combat chatter, and a literature review on complianttools for chatter stabilization. Chapters 3 and 4 outline the modelling anddesign of a compliant compensated tool, respectively. Chapter 5 coversexperimental testing and validation, and Chapter 6 summarizes and providesconclusions.3Chapter 2Background2.1 Orthogonal Cutting Chatter ModelsThe concepts explained in this chapter draw heavily from work done by Dr.Yusuf Altintas, recorded in [4].2.1.1 Flexible WorkpieceFigure 2.1: Single-Degree-of-Freedom Orthogonal chatter diagram withflexible workpiece.4The simplest system to use for investigating the chatter phenomenon is thesingle degree of freedom orthogonal cutting model shown in Figure 2.1, whichconsists of:• An infinitely rigid, fixed tool.• A rotating workpiece with dynamics modelled as a single degree offreedom mass-spring-damper.• The cutting force acting on both bodies, coupling them together.Letting xw represent the lateral displacement of the workpiece in the x-axis,the dynamics of the system can be written as follows:mwx¨w + cwx˙w + kwxw = −F (2.1)Note the force above is negative as it acts in the negative x-axis. Thesimplest representation of the force represents it as a linear function of thechip area:F = Kca0h0 (2.2)WhereKc is an experimentally-determined cutting force coefficient inN/mm2,a0 is the width of cut in mm, and h0 is the feed rate per revolution inmm/rev.In the case of the SDOF dynamics model above, the chip area is changingas the workpiece deflects towards and away from the tool. In this case, thedynamic force can be re-written:F = Kca0(h0 + xw − xw(t− T )) (2.3)Here, T is the period of the spindle rotation. The delayed term xw(t − T )is included because the tool is cutting the same surface it cut exactly onespindle revolution ago, and any deflection on the previous revolution leaves asurface profile that is cut again on the subsequent revolution and contributesto the chip area.5Combining Equations 2.1 and 2.3, the governing equation for the systemcan be written:mwx¨w + cwx˙w + kwxw = −Kca0(h0 + xw − xw(t− T )) (2.4)Taking the Laplace transform of both sides of the equation and re-arrangingyields an expression for the workpiece displacement over nominal feed rate.L (mwx¨w + cwx˙w + kwxw) = L (−Kca0(h0 + xw − xw(t− T )))(mws2 + cws+ kw)Xw(s) = −Kca0(h0(s) +Xw(s)−Xw(s)e−sT)The dynamic compliance of the system is defined to be Φw, for convenience:Φw(s) =1mws2 + cws+ kw1Φw(s)Xw(s) = −Kca0(1− e−sT )Xw(s)−Kca0h0(s)Xw(s)(1 +Kca0(1− e−sT )Φw(s))= −Kca0Φw(s)h0(s)Finally, workpiece displacement over nominal feedrate is given as follows:Xw(s)h0(s)=−Kca0Φw(s)1 +Kca0(1− e−sT )Φw(s) (2.5)Analysis of the characteristic equation can be used to assess system stability.The inclusion of the delay term x(t−T ) in the time domain introduces e−sTin the characteristic equation, adding an infinite number of poles.Significant work has been done characterizing this exact problem. Originallyproposed by Tlusty and Polacek [18] and Tobias and Fishwick [19], it is nowgenerally accepted that there exists a critical limiting width of cut acrit fora given spindle speed T , open-loop dynamics Φ(s), and cutting coefficientKc such that:{a0 < acrit Stable Cutting; No Chattera0 ≥ acrit Unstable Cutting; Chatter6This can be visualized with a Stability Lobe Diagram, an example of whichis given in Figure 2.2.Figure 2.2: Example Stability Lobe DiagramTwo critical features of the Stability Lobe Diagram can be observed:• For certain high spindle speeds, there exist local stability pocketswhere large values of a0 can result in stable cutting operation.• In the low spindle speed range, the stability lobes are tightly packed,and the maximum depth of cut is defined by the minimum limitingdepth of cut on the stability lobes.7Turning is a machining process primarily conducted below 4000 rev/min,placing operation in the ”Low Spindle Speed” range and indicating thatfor these purposes, only the minimum limiting depth for cut alim need beconsidered. Equation 2.6, known as Tlusty’s Law, gives an expression foralim as a function of the cutting coefficient and the structural dynamics:alim = − 12Kc min (Re{Φw(jω)}) (2.6)Therefore, for a given workpiece material defining Kc, the stability is definedby the minimum negative real part of the structural dynamics FrequencyResponse Function (FRF). Note that only the negative real part of theFRF produces positive, physical solutions for the depth of cut. The realand imaginary components of Φw(jω) are plotted in Figure 2.3 with someexample parameters, with the stability-defining minimum negative real parthighlighted.Figure 2.3: Example single-degree-of-freedom force-displacement FRF. Thestability-limiting negative real part is highlighted.8The dynamics of regenerative chatter for a flexible workpiece can also beexpressed using a block diagram, as follows:Figure 2.4: Block diagram dynamics representation of single-degree-of-freedom orthogonal chatter model, with flexible workpiece.2.1.2 Flexible ToolNow, a similar single degree of freedom orthogonal cutting model is consid-ered, shown in Figure 2.5, which consists of:• A tool with dynamics represented by a single degree of freedom mass-spring-damper.• A rigid, rotating workpiece that does not experience lateral deflection.• The cutting force acting on both bodies, coupling them together.Letting xt represent the lateral displacement of the tool in the x-axis, thedynamics of the system can be written as follows:mtx¨t + ctx˙t + ktxt = F (2.7)9Figure 2.5: Single-Degree-of-Freedom Orthogonal chatter diagram withflexible tool.Writing an expression for cutting force similar to Equation 2.3, and repeatingthe previous derivation:F = Kca0(h0 − xt + xt(t− T )) (2.8)mtx¨t + ctx˙t + ktxt = Kca0(h0 − xt + xt(t− T )) (2.9)L (mtx¨t + ctx˙t + ktxt) = L (Kca0(h0 − xt + xt(t− T )))(mts2 + cts+ kt)Xt(s) = Kca0(h0(s)−Xt(s) +Xt(s)e−sT)The dynamic compliance of the system is defined to be Φt, for convenience:Φt(s) =1mts2 + cts+ kt1Φt(s)Xt(s) = −Kca0(1− e−sT )Xt(s) +Kca0h0(s)Xt(s)(1 +Kca0(1− e−sT )Φt(s))= Kca0Φt(s)h0(s)10Finally, tool displacement over nominal feedrate is given as follows:Xt(s)h0(s)=Kca0Φt(s)1 +Kca0(1− e−sT )Φt(s) (2.10)Note that this expression has the same characteristic equation as the singledegree of freedom workpiece chatter model from Equation 2.5, and thereforethe same stability conclusions can be drawn.For turning processes, stability will be governed by alim, which is driven bythe minimum negative real part of the structural FRF. It can be concludedthat structural flexibility from either the workpiece OR tool may contributeto chatter, and that a generalized structural FRF representing the relativeflexibility between the workpiece and tool should be considered.2.1.3 Flexible Workpiece And Flexible Tool (2DOF)Figure 2.6: 2-Degree-of-Freedom Orthogonal chatter diagram with flexibleworkpiece and flexible tool.11A more general two degree of freedom model of orthogonal turning chatterwill be considered, shown in Figure 2.6, which consists of:• A tool with dynamics represented by a single degree of freedom mass-spring-damper.• A rotating workpiece with dynamics represented by a single degree offreedom mass-spring-damper.• The cutting force acting on both bodies, coupling them together.Letting xw and xt represent the displacements of the tool and workpiece inthe x-axis, the dynamics of the system can be written as follows:mwx¨w + cwx˙w + kwxw = −F (2.11)mtx¨t + ctx˙t + ktxt = F (2.12)Let the relative displacement between the tool tip and workpiece surface bedefined:xr = xw − xt (2.13)The dynamic force can now be expressed as a function of the relative dis-placement:F = Kca0 (h0 + xr(t)− xr(t− T )) (2.14)Substituting in Equation 2.13:F = Kca0(h0 + xw(t)− xt(t)− xw(t− T ) + xt(t− T )) (2.15)Now substituting Equation 2.15 into Equations 2.11 and 2.12, and takingthe Laplace transform:mwx¨w + cwx˙w + kwxw = −Kca0(h0 + xw − xt − xw(t− T ) + xt(t− T ))(2.16)mtx¨t + ctx˙t + ktxt = Kca0(h0 + xw − xt − xw(t− T ) + xt(t− T )) (2.17)12(mws2 + cws+ kw)Xw(s) = −Kca0 (Xw(s)−Xt(s)) (1− e−sT )−Kca0h0(s)(2.18)(mts2 + cts+ kt)Xt(s) = Kca0 (Xw(s)−Xt(s)) (1− e−sT ) +Kca0h0(s)(2.19)Using the following substitutions to represent the structural dynamics oftool and workpiece:Φt(s) =1mts2 + cts+ kt, Φw(s) =1mws2 + cws+ kw(2.20)By re-arranging Equations 2.11 and 2.12, the following expression for tooldisplacement as a function of workpiece displacement can be derived:1ΦwXw(s) = − 1ΦtXt(s) (2.21)∴ Xt(s) = − ΦtΦwXw(s) (2.22)Finally, a transfer function for workpiece displacement over nominal feed ratecan be extracted, as in the singe degree of freedom cases, using Equations2.18 and 2.22:Xw(s) = −ΦwKca0(Xw(s)(1 +ΦtΦw))(1− e−sT )− ΦwKca0h0(s)(2.23)Xw(s)(1 + ΦwKca0(1 +ΦtΦw)(1− e−sT )) = −ΦwKca0h0(s) (2.24)Xw(s)h0=−ΦwKca0h01 +Kca0 (Φw +Φt) (1− e−sT ) (2.25)It can be seen from the characteristic equation of Equation 2.25 that thecombined dynamics of workpiece and tool, Φw +Φt, must be considered forevaluating chatter stability in a two degree of freedom system. Substitutingthe combined dynamics into Tlusty’s law of Equation 2.6 yields the followingexpression for limiting depth of cut in a two degree of freedom system. Ablock diagram for this system is given in Figure 2.7.13alim = − 12Kc min (Re{Φw +Φt}) (2.26)Figure 2.7: Block diagram dynamics representation of 2-degree-of-freedomorthogonal chatter model, with flexible workpiece and flexible tool.The structural dynamics of the workpiece, tool, and combined system canbe investigated by plotting the real and imaginary parts of each FRF. Theexample values used for generating this plot are given in Table 2.1. Thesevalues are approximately based on a case with a moderately flexible work-piece and a relatively more stiff tool.Parameter Symbol Value UnitsWorkpiece Modal Mass mw 10 kgWorkpiece Natural Frequency fn,w 159 HzWorkpiece Damping Ratio ζw 0.02 -Tool Modal Mass mt 1 kgTool Natural Frequency fn,t 356 HzTool Damping Ratio ζt 0.02 -Table 2.1: Modal parameters used for example FRF.14Figure 2.8: Example 2-degree-of-freedom force-displacement FRF. Thestability-limiting minimum negative real part is highlighted.Notably, the stability-limiting negative real part of the combined FRF con-tains contributions from both the tool and the workpiece. However, in thiscase where the workpiece modal stiffness is lower than the tool modal stiff-ness, it is the workpiece flexibility that ultimately limits the overall systemstability.152.2 Literature ReviewChatter is a self-excited vibration that occurs during machining, due tothe dynamic interaction between the cutting tool and the workpiece causedby their flexibilities. When the dynamic process becomes unstable, thechip thickness oscillates and grows exponentially, along with the resultingcutting forces and vibrations [18]. Chatter limits productivity, compromisessurface quality, accelerates tool wear, and may damage both the machineand workpiece. The onset of chatter is influenced by spindle speed, depthof cut, and feed rate, and the dynamic stiffness of both the tool and theworkpiece [3].Conventional strategies to mitigate chatter have focused on enhancing thestructural rigidity of the machine tool system and workpiece setup. High-stiffness tool holders, robust spindle systems, and reinforced fixtures areoften employed to increase the dynamic stiffness of the system which leadsto higher depth of cuts. Additionally, stability lobe diagrams (SLDs) areused as predictive tools for selecting stable cutting parameters [2]. The fric-tion between the wavy cut surface and clearance face of the tool increasesthe stability at low speeds which need to be considered in turning and boringoperations [5]. As the dynamics of the system vary along the tool path, thestability lobes are updated to select chatter free speeds [16]. While effectivefor rigid workpieces, these methods are less successful when machining in-herently flexible components such as long rods or hollow tubes, where thedominant source of vibration is the workpiece itself. Flexible workpieces,such as slender shafts, have low bending stiffness and low natural frequen-cies which cause chatter.Chatter can be suppressed by both passive and active damping strategies[12]. Passive damping methods include using tuned mass dampers, con-strained layer damping materials, and compliant structures with inherentdamping characteristics. Tuned mass dampers are designed by adding amass-spring-damping structure with a single degree of freedom dynamicswhose natural frequency is close to the main structure to be damped [15].In turning, the minimum negative part of the receptance of the coupleddynamic system is minimized [1]. Multiple tuned mass dampers can beadded to increase the damping of the structure [22], especially when thenatural mode shifts slightly during machining [14]. These systems are typ-ically maintenance-free and energy-efficient but lack adaptability. Activedamping techniques involve sensors, actuators, and control systems to dy-namically counteract vibrations in real time [12]. Although effective, active16systems are complex and often costly. Active damping requires an actuatorwith high bandwidth that covers the frequency to be damped with a micro-processor that can handle heavy computation load at control intervals lessthan a millisecond [6]. Active control laws try to push the unstable poles ofthe closed loop dynamics of the turning system to stable zones [8]. The con-trol laws used in chatter suppression include sliding mode controllers [10],H infinity [6], and regenerative phase delay control [17].Hybrid solutions that combine passive damping with active compensation,such as semi-active or smart damping systems, are emerging as promisingalternatives. These systems aim to offer high damping with limited com-plexity. Recent work has shown that rethinking tool and support flexibilitycan improve chatter stability in certain contexts. Specifically, introducingcontrolled compliance into the tool system, when properly tuned, can shiftsystem dynamics to avoid chatter [9] [20]. In the case of flexible workpieces,tools with designed flexibility and high damping ratios can absorb energyfrom regenerative vibrations, thereby enhancing stability. Moreover, inte-grating smart actuators that can adapt or compensate for static deflectionscaused by cutting forces adds an extra layer of control. Munoa et al. [13]proposed to use the existing servo drive of the machine as active dampingactuator. They used an accelerometer to measure the vibrations and addedthe acceleration feedback to the closed loop position control of the drive.Alternatively, magnetic bearings of the spindles are also used to activelydamp machine tool vibrations [7] [21]. Mancisidor et al. used an inertialelectro-magnetic actuator placed near the tool to damp the machine toolmodes [11]. Zhang and Sim used piezoelectric actuator to actively dampworkpiece vibrations in milling [23].The literature indicates that while traditional methods of chatter suppres-sion are well-developed for rigid workpieces, they fall short in machiningoperations involving flexible workpieces. Emerging methods that combinepassive damping with smart actuation present a promising direction for im-proving chatter stability. The thesis builds on these concepts by introducinga novel, passively damped actuator with controlled flexibility, high damping,and static deflection compensation, offering a significant advancement in thefield of chatter mitigation for flexible turning applications.17Chapter 3Compliant CompensatedTool ModellingThe proposed ’Compliant, Compensated Tool’ of this thesis consists of a toolwith a low stiffness, low mass, near-critical damping, and a linear actuationstage. A simple diagram of this system is provided in Figure 3.1. Thefollowing sections will justify why this tool architecture is well-suited tosignificantly improving chatter stability in highly flexible workpieces.Figure 3.1: Example 2DOF FRF with flexible, lightly-damped workpieceand highly flexible, heavily-damped tool.183.1 Tool DynamicsAs established in Chapter 1, the problem statement for this thesis is theinvestigation of chatter in highly flexible workpiece turning, so the workpiecematerial and dynamics can be considered ”fixed” requirements, defining Kcand Φw. From Equation 2.26, there are 3 factors defining the stability-limiting minimum depth of cut:• Material Cutting Coefficient, Kc• Workpiece Structural Dynamics, Φw(jω)• Tool Structural Dynamics, Φt(jω)This leaves only manipulation of the tool dynamics Φt as a means for sta-bilizing the flexible workpiece. Considering again a two degree of freedommodel as shown in Figure 2.6, and using the definitions of Equation 2.20:Φw +Φt =1mws2 + cws+ kw+1mts2 + cts+ kt(3.1)Substituting s = jω, ωn,i =√kimi, and ζi =2ci√kimi:=1−mwω2 + jcwω + kw +1−mtω2 + ctω + kt (3.2)=ω2n,wkw1(ω2n,w − ω2)+ 2jζwωn,wω+ω2n,tkt1(ω2n,t − ω2)+ 2jζtωn,tω(3.3)Extracting the real part, per Tlusty’s Law:Re{Φw +Φt} = Re{Φw}+Re{Φt} (3.4)=ω2n,wkw(ω2n,w − ω2)(ω2n,w − ω2)2+ (2ζwωn,wω)2+ω2n,tkt(ω2n,t − ω2)(ω2n,t − ω2)2+ (2ζtωn,tω)2(3.5)19Studying Equation 3.5, several features can be observed:• For each term, only the numerators will determine whether the termis positive or negative, as the denominators contain squared terms.• In this single degree of freedom model, the workpiece real part will benegative above the natural frequency ωn,w, when(ω2n,w − ω2)< 0. Itis in this frequency range that chatter will happen.• In order to make the sum of the real parts as positive as possible,the tool real part should be POSITIVE at the frequencies requiringstabilization. In this SDOF model, that means that ωn,t > ωn,w.• If the tool is lightly damped, the tool term will contribute a largenegative real part when ω is near ωn,t. Therefore, the tool must beHEAVILY damped to prevent decreasing the overall system stability.• For a fixed natural frequency, the real part is inversely proportional tostiffness. Achieving large positive real parts in the tool to offset largenegative real parts in the workpiece will require a LOW tool stiffness.From these observations, it appears that a tool with the following character-istics may improve overall system stability with a highly flexible workpiece:• Tool natural frequency ωn,t larger than the workpiece natural fre-quency of interest ωn,w.• Tool damping ratio ζt close to critically damped.• Tool stiffness kt sufficiently low to generate large positive real partsnear the minimum negative real frequencies of the workpiece.An example of such a tool and workpiece is given in Figure 3.2 below. Thetool and workpiece parameters for this example are chosen somewhat arbi-trarily to illustrate the point, and are given in Table 3.1.20Parameter Symbol Value UnitsWorkpiece Modal Mass mw 10 kgWorkpiece Modal Stiffness kw 1e7 N/mWorkpiece Damping Ratio ζw 0.02 -Tool Modal Mass mt 0.1 kgTool Modal Stiffness kt 6.7e5 N/mTool Damping Ratio ζt 1 -Table 3.1: Modal parameters used for example FRF.Figure 3.2: Example 2DOF FRF with flexible, lightly-damped workpieceand highly flexible, heavily-damped tool.21Several critical features of these FRFs can be observed:• The compliant, low-stiffness tool has large positive real parts at lowfrequencies. Placing the tool natural frequency well above the work-piece natural frequency uses the large positive real part of the tool tooffset the negative real part of the workpiece, significantly improvingoverall stability.• The heavy damping of the tool attenuates the minimum negative realpart of the tool, preventing this point from decreasing stability.The critical result is that a flexible, heavily-damped tool helps to stabilizethe unstable workpiece, and results in a system that is overall more stablethan an infinitely rigid tool. This finding is believed to be a novel contribu-tion in the field.3.2 Parameter OptimizationIt is of interest to determine how varying tool parameters affects the overallsystem stability. For this analysis, single-degree-of-freedom models for themass, stiffness, and damping of both tool and workpiece will be assumed, asdepicted in Figure 2.6. The variables of interest are therefore as follows:• Tool modal parameters mt, ct, kt• Workpiece modal parameters mw, cw, kwIn an effort to reduce the number of independent variables and find moregeneralizable results, the following ratios will be considered:• Tool-workpiece stiffness ratio kt/kw• Tool-workpiece natural frequency ratio ωn,t/ωn,wFor this analysis the following parameters given in Table 3.2 will be fixed.Parameter Symbol Value UnitsTool Damping Ratio ζt 1 -Workpiece Modal Mass mw 10 kgWorkpiece Modal Stiffness kw 3.95e6 N/mTable 3.2: Modal parameters fixed for parameter optimization analysis.22The remaining free parameters over which the analysis will be run are arange of stiffness ratios, natural frequency ratios, and workpiece dampingratios. The analysis steps are as follows:• A range of stiffness ratios, natural frequency ratios, and workpiecedamping ratios are defined.• For a given workpiece damping ratio ζw:– A combination of stiffness ratio and natural frequency ratio isselected.– All modal parameters mw, cw, kw, mt, ct, kt are calculated usingthe defined ratios and the fixed parameters in Table 3.2.– The tool and workpiece force-displacement FRF’s, Φw and Φt,are calculated per Equation 2.20.– The relative tool-workpiece FRF is calculated as Φr = Φw +Φt– The minimum negative real part of the relative FRF is extractedand stored.– Steps (a)-(e) above are repeated for all other combinations ofstiffness ratio and natural frequency ratio.– A colormapped contour plot of minimum negative real part isplotted over the 2D plane of stiffness ratios and natural frequencyratios. Green indicates a more stable system than the baseline,while red indicates a less stable system.Note that in each colormapped plot, the middle yellow color is the baseline,and is normalized to be the minimum negative real part of the workpieceonly. This represents the case of a relative tool-workpiece FRF with a rigidtool.The results of this analysis are shown in Figure 3.3 through Figure 3.5 below.Note each figure uses a different workpiece damping ratio, displayed in thefigure title.23Figure 3.3: Mimimum Negative Real Part Heatmap vs stiffness ratio andnatural frequency ratio, ζn,ω = 0.005Several critical observations can be made from Figure 3.3 through Figure3.5:• Examining first Figure 3.4, with the mid-range damping ratio: Thetop row of each plot represents very high stiffness ratios, where thestability is entirely dominated by the flexible workpiece. These arenormalized to yellow-ish in the colormap, and represent the baselinecase of a rigid tool. It can be observed that for much lower stiffnessratios around kt/kw ≈ 0.1, the relative stiffness of the tool-workpiecesystem improves significantly if the natural frequency ratio ωn,t/ωn,wis greater than 1, supporting earlier conclusions that a highly flexibletool can stabilize an otherwise unstable highly flexible workpiece.24Figure 3.4: Mimimum Negative Real Part Heatmap vs stiffness ratio andnatural frequency ratio, ζn,ω = 0.02• Comparing Figures 3.3 through 3.5: It can be observed that the ef-fect of varying the workpiece damping ratio changes the stiffness ratiothat is optimal for the workpiece. A workpiece with a lower dampingratio requires a lower tool stiffness ratio for peak stabilization, while aworkpiece with a higher damping ratio requires a higher tool stiffness.Regardless of damping ratio, it is consistent that the tool natural fre-quency must be greater than the workpiece natural frequency for thetool compliance to be beneficial.It can also be observed that workpieces with lower damping ratios havehigher potential for stabilization.25Figure 3.5: Mimimum Negative Real Part Heatmap vs stiffness ratio andnatural frequency ratio, ζn,ω = 0.13.3 Tool Mass Limiting EffectGiven the dual requirements of some target stiffness ratio kt/kw and a min-imum natural frequency ratio ωn,t/ωn,w, the allowable range of mass ratiosmt/mw must be limited. By selecting a minimum natural frequency ratiorω ≤ ωn,t/ωn,w and a target stiffness ratio rk = kt/kw and , this range oftool masses can be investigated:rω ≤ ωn,tωn,w=√ktmtmwkw=√rkmwmt(3.6)∴ rm ≡ mwmt≥ r2ωrk(3.7)From parameter optimization, an approximate suggestion for these ratiosmay be rk = 0.1, rω ≥ 2.3. This would provide a mass ratio of rm ≈ 50,26suggesting that this compliant tool stabilization approach could be effectivefor workpieces with an effective modal mass ≈ 50x larger than the effectivemodal mass of the tool. Critically, this informs tool design, suggesting thatminimizing the tool mass will maximize the range of workpieces that can bestabilized.3.4 Static CompensationAs has been shown in preceeding sections, careful selection of tool dynamicscan produce a combined tool-workpiece relative FRF that is more stablethan the workpiece alone. However, the proposed solution of a flexible toolcomes at the steep cost of sacrificed static stiffness. In precision turningapplications, large static tool deflections would lead to unacceptable partform errors. To improve this, a tool modification is proposed in which thetool is mounted to a linear actuation stage that measures low-frequency tooldeflection and advances the tool tip position to compensate for this. Theproposed compliant, compensated tool accomplishes stability improvementwith compliance and damping, and recovers static stiffness via compensa-tion. A simplified diagram of such a tool is given in Figure 3.6.Figure 3.6: 3-degree-of-freedom external turning dynamics model with flex-ible workpiece, flexible tool, and linear actuation stage using active controlto position the tool.27The effect of adding this linear actuation stage is an effective change to thestructural FRF of the tool, Φt. In order to distinguish between the passivetool dynamics and the compensated tool dynamics, Φtc will be defined asthe compensated tool FRF. This new compensated FRF can be derived asfollows, starting with a force balance on the tip of the tool. Here, F refersto the cutting force as defined in Equation 2.15.mtx¨t = −ct(x˙t − x˙a)− kt(xt − xa) + F (3.8)mtx¨t + ctx˙t + ktxt = ctx˙a + ktxa + F (3.9)Taking the Laplace transform:L{mtx¨t + ctx˙t + ktxt} = L{ctx˙a + ktxa + F} (3.10)(mts2 + cts+ kt)Xt(s) =1ΦtXt(s) = (cts+ kt)Xa + F (s) (3.11)Implementing the example integral control law Xa = −Kis Xt:1ΦtXt(s) = −Kis(cts+ kt)Xt(s) + F (s) (3.12)(1Φt+Kis(cts+ kt))Xt(s) = F (s) (3.13)Finally, the new compensated force-displacement FRF of the tool can bewritten:Xt(s)F (s)= Φtc =11Φt+ Kis (cts+ kt)=smts3 + cts2 + (kt +Kict)s+Kikt(3.14)This FRF can be plotted alongside the uncompensated FRF in Figure 3.7to show the improvement in static stiffness with compensation. It can beobserved that the y-intercept, representing the static compliance, is driven tozero with compensation. The frequency range affected by the compensatorcan be tuned by tuning the integral gain Ki for this particular controller.28This improvement in static stiffness can also be verified by evaluating theDC gain of the new compensated tool FRF. It can be found for this choiceof controller that the new compensated DC gain is 0:DC Gain: Φtc|s=0 = (0)mt(0)3 + ct(0)2 + (kt +Kict)(0) +Kikt= 0 (3.15)Figure 3.7: Example highly flexible tool FRFs with and without compen-sation. Adding compensation makes the static stiffness effectively infinite.293.5 Workpiece Position Dependence InvestigationAnalysis up to this point has considered constant SDOF parameters mw,cw, kw for the workpiece. However, in a general turning setup, this is notan accurate representation. A slightly more realistic model is shown inFigure 3.8. Here, a cylindrical workpiece pinned at the chuck and tailstockis shown. This workpiece has lateral displacement dynamics that vary as afunction of position z along the workpiece. This can be represented using afinite-element model of the workpiece.It is important to verify that a compliant tool designed to stabilize the mostflexible part of the workpiece - here, the centre - will not destabilize ornegatively impact the stability of the system at other axial positions. Thisanalysis is conducted in the following steps:• AMDOFmodel of the workpiece is created using several Euler-Bernoullibeam elements.• At each element node, the collocated force-displacement FRF in thelateral (x) direction is calculated.• The real part of the FRF at each frequency and position is plotted inthe 3D plot Figure 3.9, with select contours shown in Figure 3.10.• Compliant tool parameters are selected to improve the first mode ofthe FRF at the least stable part of the workpiece (here, the centre).• The combined FRF of compliant tool + workpiece is re-examined atall positions along the workpiece.30Figure 3.8: MDOF workpiece model setup - fixed-fixed beam.Figure 3.9 shows each FRF vs position and frequency. It can be observedthat the largest (and therefore, stability-dominating) response occurs at thecentre of the beam in the first mode. Figure 3.11 shows the FRFs of theworkpiece, tool, and combined system at the centre of the workpiece, whichthe tool is designed to stabilize. Figures 3.12 and 3.13 show the combinedworkpiece-tool FRFs at all positions. The key results are as follows:• For all positions, the minimum negative real part from the first modeis significantly lifted.• Across all positions, the new stability-limiting minimum negative realpart now comes from the 2nd mode, which is made slightly worse bythe tool.• Overall, the stability of the system has been significantly improved bya factor of 4.4x.A table describing the fixed parameters in this investigation is given in Table3.3.31Parameter Symbol Value UnitsWorkpiece Length L 1 mWorkpiece Diameter D 0.05 mWorkpiece 1st Mode Natural Frequency fn,w 225 HzWorkpiece 1st Mode Damping ζw 0.02 -Tool Mass mt 80 gTool Stiffness kt 1e6 N/mTool Damping Ratio ζt 1 -Min Neg Real Part, Workpiece Only - -0.79 µm/NMin Neg Real Part, Workpiece + Tool - -0.18 µm/NMin Neg Real Part Improvement Ratio - 4.4 -Table 3.3: Modal parameters fixed for position dependence analysis.Figure 3.9: Real part of workpiece-only FRFs vs position and frequency.32Figure 3.10: Selected workpiece FRFs at varying positions.Figure 3.11: Workpiece, tool, and combined FRFs at centre of workpiece.33Figure 3.12: Real part of workpiece + tool FRFs vs position and frequency.Figure 3.13: Real part of workpiece + tool FRFs vs frequency. This showsall the FRFs from Figure 3.12 superimposed for each position.343.6 Single-Axis Model SimulationThese theoretical predictions of stability improvement are tested via a verysimple time-domain simulation using a Simulink model. A diagram of thesystem being modelled is given in Figure 3.1, and a Simulink model repre-senting the system is given in Figure 3.14. Table 3.4 describes the inputparameters for the simulation, while Table 3.5 describes the specific simula-tion cases.The only features of the simulation being varied are the tool stiffness kt,and whether or not the linear actuation compensation is used. All otherparameters are held constant. The addition of the linear actuation controlstage effect is achieved by replacing the passive tool dynamics Φt with thecontrolled tool dynamics Φtc as defined in Equation 3.14.Parameter Symbol Value UnitsWorkpiece Mass mw 10 kgWorkpiece Stiffness kw 2e6 N/mWorkpiece Natural Frequency fn,w 71 HzWorkpiece Damping Ratio ζw 0.02 -Tool Mass mt 100 gTool Damping Ratio ζt 1 -Mass Ratio mtmw 0.01 -Spindle Speed n 1000 RPMFeed Rate h0 0.1 mm/revDepth of Cut a0 0.6 mmCutting Coefficient Kc 500 N/mm2Controller Intergral Gain Ki 300 -Table 3.4: Modal and cutting parameters fixed for simple model Simulinksimulation.35No. Description Variable Value Units1 Rigid Tool kt 2e9 N/m2Compliant ToolNo Compensationkt 1.5e5 N/m3Compliant ToolWith Compensationkt 1.5e5 N/mTable 3.5: Descriptions of each simulation run.Figure 3.14: Simulink Diagram36Figure 3.15: Simulation 1 result showing tool and workpiece displacement.37Figure 3.16: Simulation 2 result showing tool and workpiece displacement.38Figure 3.17: Simulation 3 result showing tool and workpiece displacement.39The outputs of the simulation show the following:Simulation 1: Rigid Tool• Workpiece Displacement :Passively unstable and chattering - exponential growth in oscillations.This is the baseline reference case.• Tool Displacement :Negligible relative to the workpiece displacement.• Relative Displacement :Dominated by the workpiece displacement.Simulation 2: Compliant Tool, No Compensation• Workpiece Displacement :No oscillations, just a gentle movement to a steady-state displacement.This workpiece that was unstable with a rigid tool has been stabilizedby a flexible one.• Tool Displacement :Gentle movement to a very large static displacement.• Relative Displacement :Very large relative displacement between tool and workpiece, mostlydominated by the static displacement of the tool.Simulation 3: Compliant Tool, With Compensation• Workpiece Displacement :Some passively stable oscillations that decay to a steady-state displace-ment. This workpiece is still stabilized with a compliant tool includingcompensation.• Tool Displacement :Some small passively stable oscillations that decay to zero steady-statedisplacement.• Relative Displacement :Dominated by the workpiece displacement.It is concluded that the Simulink model simulation results support the pro-posed chatter stability findings of earlier sections.40Chapter 4Compliant CompensatedTool DesignGiven the promising predictions of the theoretical modelling in Chapter3, it is now of interest to design and build a compliant compensated toolprototype to experimentally verify the improved chatter performance.4.1 Coordinate System DefinitionFor the remainder of this thesis, unless otherwise specified, the coordinatesystem in Figure 4.1 is in effect. The workpiece and tool displacements arefurther defined in Figure 4.2.Figure 4.1: Coordinate system diagram for CNC Lathe.41Figure 4.2: Tool and Workpiece Displacement Defintions.424.2 Target Design ParametersA set of target design parameters for the tool design was determined toguide the design. Note that as this is a one-off proof-of-concept tool, manyof these are not hard requirements, and can be modified. Most critically, thetool dynamics must be tuned to the workpiece, and most other parameterscan be changed so long as this demand is met.The following target design parameters were derived using the general prac-tical constraints of:• The resulting tool components should be practically manufacturable.• The resulting workpiece total length should fit within the centre-to-centre distance of the available lathes for testing, with a maximumworkpiece length of approximately 1m.• The test workpiece should be highly unstable, but at least stableenough that small cuts can be taken with a rigid tool to establishthe stable/unstable boundary of the rigid tool.This results in the parameters set out in Table 4.1. In this table:• ’Nominal Load’ is the load resulting from the largest allowable cut -eg, max feed rate and max depth of cut.• ’Design Peak Load’ is the maximum load at which the tool must bedesigned to operate as intended.• ’Over Load’ is an excessive loading condition that the tool must bedesigned to survive if exposed a small number of times, but the toolneed not operate to typical specs under this load.• ’Nominal Travel’ is the distance the compliant tool spring would stat-ically deflect under the Nominal Load; alternatively, it is the travelthe linear actuation stage would have to move to compensate for theNominal Load from the largest allowable cut. Note that only unidi-rectional travel is strictly necessary, but the system is designed for asymmetric travel range about the neutral position.• ’Operational Travel’ is the distance the linear actuation stage is de-signed to be able to travel within, and is larger than the nominal travelrange.43Parameter Symbol Value UnitsWorkpiece Effective Mass mw 20 kgWorkpiece Effective Stiffness kw 5∗106 N/mWorkpiece Natural Frequency fn,w 80 HzWorkpiece Damping Ratio ζw 0.02 -Tool Effective Mass mt 281 gTool Effective Stiffness kt 4.8∗105 N/mTool Damping ct 738 Ns/mTool Damping Ratio ζt 1 -Maximum Feed Rate h0 0.1 mm/revMaximum Radial Depth of Cut a0 1.5 mmCutting Coefficient, X Kx 1250 N/mm2Cutting Coefficient, Y Ky 2500 N/mm2Cutting Coefficient, Z Kz 1250 N/mm2Design Factor, Peak Operating Load Kpl 1.5 -Design Factor, Overload Kol 2 -Nominal Load, X Fx0 188 NNominal Load, Y Fy0 375 NNominal Load, Z Fz0 188 NDesign Peak Load, X Fxp 281 NDesign Peak Load, Y Fyp 563 NDesign Peak Load, Z Fzp 281 NOver Load, X Fxol 375 NOver Load, Y Fyol 750 NOver Load, Z Fzol 375 NDesign Factor,Nominal to Operating Linear TravelKto 2 -Tool Tip Nominal Travel (Static Deflection) dXn ±500 umLinear Actuation Stage Operational Travel dXo ±1000 umLinear Actuation Minimum Repeatability - 5 umTable 4.1: Target Design Specifications.Beyond the target specifications, the criteria to optimize are maximizingthe stiffness of the linear actuation stage in the X axis, and maximizing thestiffness of the tool tip in all rotational degrees of freedom and in the Y andZ axes.444.3 Tool Architecture Overview4.3.1 Mechanical ArchitectureThe final design of the tool involves 3 separate stages, physically nested inconcentric layers:• Tool Tip Mass-Spring-Damper Stage: (Innermost Layer)– This stage experiences displacement xt2 relative to the outerhousing.– The moving sheets of the layered sheet damper are sumberged ina sealed oil bath and generate damping by moving relative to thestationary damper sheets.– The moving sheets are rigidly connected to flexure springs ateither end of the damper body which provide only 1 degree offreedom with the specified compliant spring constant.– The insert and insert holder bolt onto the end of this damperbody, allowing the tool tip to move in one axis with a calibratedspring stiffness and high amount of damping.• Linear Compensation Stage: (Intermediate Layer)– This stage experiences displacement xt1 relative to the outerhousing.– The damper housing is a key component of this stage. The linearactuator is mounted to the stationary housing and connects tothe damper housing to provide small displacement corrections.– The damper housing uses a flexure spring design to move only inone axis relative to the external stationary housing.• Stationary Outer Housing: (Outermost Layer)– This stage is mounted directly to the toolpost of the machine.– The linear actuator and motor driver electronics are mounted tothe stationary outer housing, which encloses all other layers forprotection.Some images of the tool design from CAD, including a cross-section of thefull assembly, are included in Figures 4.3 and 4.5.45Figure 4.3: Nested flexure spring architecture.46Figure 4.4: Simplified full system diagram, labeled. Green correspondsto components moving with displacement xt2, blue corresponds to xt1, andblack is stationary.47Figure 4.5: Full tool assembly cross-section, labeled.48Figure 4.6: Full tool assembly photos.494.3.2 Electrical ArchitectureThe electrical system consists of the following components and subsystems:• Stepper motor linear actuator• Stepper motor driver• 2x high-resolution inductive linear displacement probes + probe driver• Analog-to-digital conversion board• TMS320 microcontroller• 12V, 5V, and 3.3V power supply• Emergency stop button• 3m cables connecting the electrical panel (external to the machine) tothe tool inside the machine• Additional interface boards for breaking out cable conductorsThe linear actuator, motor driver, and displacement probes are all mountedto the tool prototype inside the machine, while the remaining electricalcomponents are mounted to a flat panel externally. The electrical panel hasconnections to mains power and a USB connection to a test laptop, whichruns a custom Python user interface to control the tool and log data.The electrical system layout is provided in Figure 4.7, and a photo of theassembled electrical panel is provided in Figure 4.8.50Figure 4.7: Diagram of electrical system.Figure 4.8: Photo of assembled electrical panel.514.4 Mechanical Design4.4.1 Compliant Spring DesignThe ideal tool degrees of freedom are:• Compliance in only one linear axis (X) with spring constant k• No compliance in any other degrees of freedom (Y , Z, θx, θy, θz).In this subsection, ’axial’ will refer to the tool axis X, while ’radial’ willrefer to the orthogonal axes Y, Z.To achieve this constraint behaviour, a variation on a spiral flexure bearingis designed, designated the ’pinwheel’ flexure. A single ideal pinwheel flexurebearing permits linear motion in 1 axis while restricting linear motion in theother 2, and allows 2 rotational degrees of freedom. By rigidly connecting2 pinwheel flexure bearings separated by the length of the tool, the 2 angu-lar degrees of freedom are eliminated and the final single-degree-of-freedomassembly is created. There are 2 designs for pinwheel flexure springs in thisprototype, labeled in Figure 4.9.Figure 4.9: Pinwheel spring flexure design, inner flexure (left) and outerflexure (right).52This subsection is primarily concerned with the design of the inner pinwheelflexure, but the outer pinwheel (discussed further in the Linear Actuatordesign section) uses a similar design procedure.A simplified diagram of the the 3 stages connected by 2 springs at eitherend of a body is given in Figure 4.10.Figure 4.10: Idealized displacement degrees of freedom diagram for dual-pinwheel flexure.53The real-world pinwheel flexure cannot be infinitely rigid in the axes itconstrains, so instead the figure of merit for this design is the ratio of radialstiffness to axial stiffness.R =kradialkaxial(4.1)The axial stiffness can be tuned by controlling the thickness and width of thebeams that make up the arms of the pinwheel. Multiple layers of pinwheelsmay also be used.The design considerations for the pinwheel flexure are as follows:• Achieve the desired axial spring constant kaxial• Maximize the radial-to-axial stiffness ratio R• Operate without exceeding the yield stress of the material in the over-load caseFatigue will not be considered here as this is a prototype tool intended forlimited proof-of-concept use, but fatigue would have to be considered in amore general case.The dimensions of each arm of the pinwheel flexure are given in Figure 4.11:Figure 4.11: Definition of spring arm dimensions.54The beam-bending configuration considered for this design is as shown inFigure 4.12.Figure 4.12: Beam spring free-body diagram.The axial stiffness of one arm of the pinwheel can be modelled as a simplefixed-fixed beam bending. The total axial stiffness is the sum of all the arms,where n is the number of layers of the flexure:karm,axial =12EIal3(4.2)Ia =112bt3 (4.3)ktotal,axial = 4nkarm,axial (4.4)55The radial stiffness may be approximated assuming half of the arms arein bending, and half are in tension/compression. These arm stiffnesses arecalculated as follows:karm,tension = karm,compression =EAl(4.5)A = bt (4.6)karm,radial =12EIrl3(4.7)Ir =112tb3 (4.8)ktotal,radial = 2n(karm,tension + karm,radial) (4.9)Figure 4.13 shows the relationship between the stiffness ratio R and thebeam cross-section aspect ratio. There is a large baseline ratio due to thehigh stiffness of the beams in tension, with increasing stiffness ratio as thewidth b increases. It is clear that the beam width b should be maximizedwithin the available volume envelope.Figure 4.13: Stiffness ratio vs beam cross-section aspect ratio.56The largest stress will occur near the fixed ends of the beam across the flatsurface as it bends to allow axial motion. This stress can be calculated asfollows, taking Ftotal to be the total force applied in the axial (X) axis:σ =MyIa(4.10)M = Fal/2, y = t/2 (4.11)Fa =Ftotal4n(4.12)σ =Falt4Ia(4.13)Note that these calculations all assume that the end conditions are bothperfectly fixed, which is not completely true in real life, where the clampingmembers may also deform slightly, and the fastener preload affects the jointstiffness. Some initial prototype testing indicated that this effect was signif-icant, with calculations requiring a correction factor kc of 0.58 to match themeasured results, such that:ka,true = kcka,theory (4.14)Subsequent prototype iterations improved the joint component stiffness, sothe expected correction factor in future iterations could fall within a range.0.58 < kc < 1Space constraints make a convenient length l ≈ 20mm. This drives anapproximate max width b ≈ 6.5mm. Given these parameters, the remainingfree parameters are the thickness t and number of layers n. Figure 4.14below shows peak stress vs thickness for several layer options. Upper andlower expected stiffness limits are plotted to reflect the unknown stiffnesscorrection factor.57Figure 4.14: Stress vs thickness, number of layers for pinwheel spring.It is clear that in order to achieve the desired spring constant and the desiredload capacity without yielding, it is necessary to use multiple layers of theflexure spring. A thickness of 0.35mm is selected as a good tradeoff between:• Trying to minimize the number of layers for ease of assembly• Trying to maintain a minimum safety factor of 1.2 on yield• Splitting the difference between the stiffness limitsTo prevent static friction and rubbing, each spring layer is separated by aspacer layer the leaves the arms free to bend. A figure of the spring andspacer layer designs is given in Figure 4.15:58Figure 4.15: Inner Pinwheel Spring and Spacer Designs.Figure 4.16: Damper Endcap Assembly, Exploded View.59The external perimeter of the pinwheel flexure spring is clamped using twosteel frames connected by 4 M4 fasteners. The final spring design parametersare given in Table 4.2.Parameter Value UnitsBeam Length l 20 mmBeam Width b 6.5 mmSheet Thickness t 0.35 mmNumber of Flexure Layers (per endcap) 9 -Number of Spacer Layers (per endcap) 8 -Predicted Stiffness 5∗105 N/mPredicted Stiffness Ratio R 1800 -Material 65Mn Spring Steel -Material Yield Stress 600 MPaPeak Stress (overload) 400 MPaOverload-to-Yield Safety Factor 1.5 -Table 4.2: Inner pinwheel flexure spring parameters.Note: During experimental testing, the tool was modified by removing 6spring layers from the rear and 3 spring layers from the front. This wasdone to provide some more compliance, at the cost of total load capacity.This decreases the total number of inner flexure springs in the assembly(including both endcaps) from 18 to 9.604.4.2 Damper DesignThe damper is the most critical part of the compliant tool, and representsthe source of most of the technical risk in this project, as well as significantmass and volume in the prototype.The damper has several key requirements:• High damping coefficient with low mass: The tool must be near-critically damped, but the mass must be as small as possible. Highlymass-efficient damping is required.• Linear damping response: To achieve good stabilization, it is im-portant that the damping response be dominantly linear.The modelling of the tool mass-spring-damper up to this point assumes alinear damper. However, many existing damper designs are not optimizedfor linearity. For example: A common damper design is a hydraulic cylinderwith a permeable piston. A photo of such a damper is given in Figure 4.17:Figure 4.17: Fluid piston damper diagram.A damper such as this one, where the damping force comes from a pressuredifference in the fluid across an orifice, has force proportional to velocitysquared (F α x˙2) and is not linear. An illustration of linear and nonlineardamping characteristics is provided in Figure 4.18.61Figure 4.18: Linear and nonlinear damping characteristic curves showingtangent damping of the nonlinear curve.Critically, the effective damping from the nonlinear damper can be approx-imated using the tangent damping from the characteristic curve in a neigh-bourhood around the active velocity range. A non-linear damper with asquared velocity profile has zero tangent damping at low velocities. Sinceheavy effective damping is required to make this system work, a pressure-based fluid damper is unsuitable for this application. Instead, a linear ornearly-linear damper with a robust damping response at low velocities isdesired for good stabilization.The proposed design to achieve a linear response is one in which force isgenerated by shearing oil, instead of from a pressure difference. Other op-tions, such as eddy current damping, were considered but eliminated due tocomplexity and high mass requirements. The damping coefficient c from 2parallel sliding planes with area A with a thin oil film between is derivedbelow. Here, µ is the dynamic viscosity of the fluid in Pa s, τ is the shearpressure in the X direction in N/m2, and u is the velocity of the fluid in theX direction in m/s.62Figure 4.19: 2-plate oil shearing model.τ = µdudy= µx˙h(4.15)F = τA (4.16)F =Aµhx˙ (4.17)∴ c = Aµh(4.18)It is of interest to try to maximize the damping coefficient for a given massand volume of a tool. To achieve this, the following degrees of freedom are:• Maximizing the shear area A• Maximizing the oil viscosity µ• Minimizing the oil film thickness hThe cross-section of moving and stationary components normal to the di-rection of travel can be analyzed to find high-performance options. Severalpotential variations are presented below and compared:63Figure 4.20: Cross-Section Option A: Cylindrical Piston and CylinderFigure 4.21: Cross-Section Option B: Interleaved Spirals64Figure 4.22: Cross-Section Option C: Layered Sheets• Cylindrical Piston with Oil Annulus: This option was the initialone considered and prototyped due to its simplicity, but it is not highlyefficient in its use of the available internal volume, as inspection of thecross-section reveals.• Interleaved Spirals: This option dramatically improves on the useof internal volume for available oil shear area, but physically manu-facturing the spirals and maintaining a consistent gap thickness wouldbe quite challenging, making this option impratical.• Layered Sheets: Finally, this option is both highly efficient in its useof space, and practically manufacturable using simple pieces of sheetmetal.The performance of each cross-section is analysed and shown in Figure 4.23.Assuming the same bounding volume (10mm x 10mm square) for the cross-section, and the same length, the wetted area per unit length and overalldamping coefficient may be calculated. Note that a smaller oil film gapwas assumed for the piston/cylinder design, as it is assumed modern lathes65would be able to produce highly accurate cylindrical forms and thereforeachieve a smaller reasonable gap than a sheet metal assembly.It can be observed that the layered sheets is the best performing option bya significant margin - this design is the one chosen and developed for theprototype.Figure 4.23: Comparison of Damper Cross-Section OptionsThe damper assembly consists of a moving subcomponent and a stationarysubcomponent, with alternating interleaved sheets, all submerged in oil.Figure 4.24 shows the damper cartridge assembly, and Figure 4.26 showsthe damper housing into which the cartridge is installed.66Figure 4.24: Damper Cartridge Assembly - moving system (left) and sta-tionary system (right).An image of the moving and stationary damper sheets is provided in Figure4.25. Small slots are included in the overlapping area of each sheet in orderto aid the distribution of the high-viscosity oil to the middle of the sheets.The first version of the prototype used 0.373mm gaps between the sheetsbecause it allowed the use of precision 1mm spacer washers. However, slightbending and deflection of the damper sheets caused this design to rub. Fol-lowing several subsequent iterations, 1.5mm spacers proved robust and pro-vided enough space that sheet deflection was negligible.67Figure 4.25: Damper sheet comparison - moving sheet (left) and stationarysheet (right).Initial tests of the damping coefficient showed that the effective coefficientwas significantly higher than designed. It is not very clear what causesthis, but it is suspected that a contributing factor may be variable oil gapthicknesses, and screw clamping load compressing the damper cartridge andshrinking oil film gaps. To achieve the correct damping value, the damperwas rebuilt iteratively with fewer and fewer sheets until more reasonabledamping performance was achieved.68Figure 4.26: Damper Housing.69Figure 4.27: Damper Assembly, Vertical Plane Cross-Section.Damping was originally measured using a simple setup on a bench with alaser displacement sensor and a load cell. A sine sweep of load was applied tothe system, and the slope of the resulting load-velocity curve was extracted.This produced Measurement 1. Later, a tap test and curve-fit to FRF datawas used to estimate the damping value again, giving Measurement 2. Thesemeasurements are significantly different, but it is believed that that the taptest data is likely most representative of the true value. Both measurements,along with the damper parameters, are provided in Table 4.3.70Figure 4.28: Damper Assembly, Horizontal Plane Cross-Section.The interleaved-sheet design was very convenient for development testing,as the damping coefficient could fairly easily be changed by rebuilding thedamper with varying numbers of sheets. However, the damping coefficientseems to be highly dependent on the assembly procedure. This design couldbe improved to better control the gaps between the sheets and make it lessassembly-dependent.71Parameter Value UnitsOil Film Thickness 0.625 mmSheet Shear Area 1400 mm2Silicone Oil Viscosity 5000 cStSheet Metal Thickness 0.25 mmSheet Metal Material Carbon Steel -Number of Moving Layers 11 -Number of Stationary Layers 11 -Measurement 1: Measured Damping ct 780 Ns/mMeasurement 2: Measured Damping ct 385 Ns/mTable 4.3: Linear sheet damper parameters.Figures 4.29 - 4.33 show photos of the damper components and the buildprocess.Figure 4.29: Damper build process - layers of moving and stationary sheetsare built up on an assembly fixture plate, separated using layers of spacerwashers.72Figure 4.30: Damper build process - fully-built damper cartridge assem-bly consisting of sheet stack assembly with moving and stationary framesclamping sheets together.73Figure 4.31: Damper build process - the damper cartridge assembly isplaced in the damper housing.74Figure 4.32: Damper oil fill via automotive vacuum pump.Figure 4.33: Full spring-damper tool assembly. Note this photo shows anold single-layer endcap spring design, not the final multi-layer design.754.4.3 Tool Tip DesignThe selected tool tip is modelled after a Sandvik Coromant SSDCN 1212F09-M toolholder, using a SCMT 09T308 insert. An image of the referencetoolholder and the custom design are given below.Figure 4.34: Reference Toolholder SSDCN 1212F 09-MA 45o insert was chosen as it was unclear at the design stage whether thisprototype would be tested on a right-hand or left-hand machine; The sym-metry of this insert makes this irrelevant.The original manufactured design for the tool tip had a removable tool tipbolting to a simple faceplate on the end of the tool, such that the tool tipcould be easily swapped out without major disassembly of the tool. However,this became a weak point in the design that introduced undesirable toolvibration modes in preliminary testing, and the tool and faceplate werelater merged into one part for rigidity. A CAD model of this part and itsposition in the assembly are given in Figures 4.35 and 4.36 below.76Figure 4.35: Tool head.Figure 4.36: Tool head isolated in full assembly.774.4.4 External Housing DesignThe housing consists of 10mm thick anodized aluminum plates that bolttogether to form a rigid frame. Two endcaps clamp the outer pinwheelflexure to either end. To ensure a flat mounting surface for the pinwheelflexure, a locating boss is placed on the top and bottom panels, and amating recess is placed on the side panels for alignment.Figure 4.37: Stationary outer housing.Each side panel also has a pocket and threaded holes. This feature wasoriginally intended to be the mounting interface for the tool on one side,and the mounting interface for the tool electronics panel on the other. Inadvance of testing, it was not known which side of the tool would mount tothe machine, so the pockets are symmetric on each side. The eventual toolmounting scheme did not make use of this feature, but the tool electronicspanel still mounts here.Sets of 3mm dowels are press-fit into each end of the housing to serve asalignment features for the end plates and the motor mount.784.4.5 Linear Actuation Stage DesignThe linear actuation stage consists of:• A stepper motor linear actuator, with a NEMA17 frame and a 1mmpitch lead screw• A coupling between the actuator lead screw and an adapter mount toconnect to the damper housing• The damper adapter mount• 2x pinwheel flexure springs to suspend the damper housing from theexternal housing and permit linear motion in one axis• 2x eddy-current displacement probesA diagram of the linear actuation stage components is provided in Figures4.39 and 4.38.There are two displacement sensors installed in the tool, both mountedrigidly to the stationary machine reference frame. Sensor S1 measures thedisplacement xt1 of the linear actuation stage, while sensor S2 measuresthe displacement of the tool tip stage xt2. Figure 4.39 shows the mountinglocations of each sensor.Figure 4.38: Linear actuation assembly, exploded.79Figure 4.39: Diagram of the linear actuation assembly. Black text indicatesa stationary component, blue shows components moving with displacementxt1, and green shows components moving with displacement xt2.The pinwheel spring design here is similar to the pinwheel springs inside thespring-damper subassembly. However, the linear actuator springs are notdesigned to deliver a specific spring constant - instead, they are designedto maximize off-axis stiffness while keeping the load on the linear actuatorat peak displacement low. The final spring design parameters are given inTable 4.4.80Parameter Value UnitsBeam Length l 38.5 mmBeam Width b 9.25 mmSheet Thickness t 0.5 mmNumber of Flexure Layers (per endcap) 2 -Material 65Mn Spring Steel -Material Yield Stress 600 MPaPeak Stress 200 MPaPeak Displacement 1 mmPeak Load @ Peak Displacement 40 NAxial Stiffness 6.5e4 N/mRadial Stiffness 2.0e8 N/mStiffness Ratio R 3100 -Table 4.4: Outer pinwheel flexure spring parameters.The outer pinwheel flexure spring is clamped to the damper housing by smallcustom caps in the front, and by the damper adapter mount in the back.Figures 4.40 and 4.41 show each of these connections.Figure 4.40: Outer pinwheel caps.81Figure 4.41: Damper adapter mount connection.The damper adapter mount has a large central cavity to make space for thedisplacement sensor S2. The actuator lead screw screws into the actuatorcoupling, which then bolts to the damper adapter mount. The actuatorcoupling and adapter mount are separate components to provide modularity.Two adapter couplings were ordered, one flexible and one solid rigid design,in case manufacturing and assembly tolerance stacks caused binding in thelinear actuation mechanism. However, the linear actuator had significantangular compliance built-in, so the rigid coupling was used in testing.Figure 4.42: Motor coupling comparison - flexible option (left) and rigidoption (right).82The motor is bolted to the motor mount component, which is a hollow alu-minum box with interfaces for the NEMA 17 motor frame and the prototypetool housing. 8x M4 fasteners connect the motor mount to the housing -as many fasteners as could fit were added to maximize the rigidity of thisbolted joint.The displacement sensors each require a cylinder of empty space near theprobe tip that must be free of metals, to avoid undesirable inductive couplingthat would contaminate sensor readings. This clearance is pocketed out ofthe motor mount. Additionally, two small clear polycarbonate windows areplaced on either side of the motor mount, looking at the probe tip. Thiswindow can be removed and shims inserted through it to set the correctprobe offset from the target before tightening the set screw to lock theprobe in place.The sensors also require a minimum 15mm target diameter to accuratelymeasure displacement. The damper adapter mount is the target compo-nent for measuring the linear actuation stage displacement xt1. Two targetareas are designed in, one on each side of the adapter mount, to facilitateplacement of the displacement sensor on either side of the tool.Figure 4.43: Pocket recessed around the probe tip to prevent undesirableinductive coupling.83Figure 4.44: Sensor target on the damper adapter mount is a circle of16mm, to provide an accurate reading.Figure 4.45: Setting the offset of the probe tip using precision shim.84The sensor S2 is mounted to the end frame of the housing along the centralaxis of the tool, and uses a small plastic 3D printed component to route thecable out of the housing while maintaining the minimum 6mm bend radius.Precision shims are used to set the sensor position offset from the targetbefore tightening the sensor clamp.For both sensors, the mounting interface around them is designed such thatovertravel of any components contacts on an internal hardstop surface beforethe probe tip makes contact, to protect the sensors.Figure 4.46: Sensor S2 installed in the prototype.854.5 Electrical Design4.5.1 Linear Actuator SelectionThe linear actuator selected for the project is the LN17D6200-W0601-300MNlinear stepper motor from Lin Engineering. This actuator is a stepper mo-tor with an integral internal lead nut, and a through-axis lead screw. Thisactuator is driven using the R208 Bipolar Stepper Motor Driver, also fromLin Engineering. An image of the actuator is given Figure 4.47. Selectedspecifications for this actuator are given in Table 4.5 below, and the manu-facturer’s load-speed performance curve is given in Figure 4.48.Initially, both fine-pitch lead-screw and piezo linear actuators were consid-ered, given the fine positioning resolution required. However, a piezo actua-tor capable of the range of travel and load capacity required was extremelyexpensive, large, and requiring high voltage.Instead, a lead-screw-based linear actuator was chosen to prevent backdriv-ing of the actuator under cutting loads. This also allows the actuator to holdposition while unpowered. A stepper motor with fine rotational steps waschosen in order to allow very precise positioning adjustments. The integralnut of this actuator provides a very convenient compact solution.Figure 4.47: LN17D6200-W0601-300MN Linear Actuator (Source: LinEngineering)86Figure 4.48: Manufacturer’s Load-Speed Curve for Linear Actuator(Source: Lin Engineering)Parameter Value UnitsFrame Size NEMA 17 -Peak Current 2 ALead Screw Pitch 1 mmFull Steps per Revolution 200 -Travel per Full Step 5 µmTravel per 1/8 Step (finest resolution) 0.625 µmTable 4.5: Select specifications for linear actuator.874.5.2 Displacement Sensor SelectionThe displacement sensors selected for the project are the ECL150 eddy-current displacement from Lion Precision, used with 2 of their U5C probes.Selected specifications for these probes are given in Table 4.6 below. Figuresrepresenting the sensor driver and probes are given in Figures 4.49 and 4.50.The displacement sensor options explored for the purposes of this projectincluded optical, capacitive, and eddy current sensors. Ultimately, the eddycurrent option was the most economical while still achieving the fine reso-lution requirement. Despite being the most economical of the off-the-shelfcommercial options, the cost of this measuring system was approximately$9k CAD. Earlier in the project, some prototyping work was done on acustom inductive-coil-on-PCB displacement sensor solution which showedpromise of being able to work for this application, but this was shelved dueto time constraints.Figure 4.49: ECL150 Sensor Driver (Source: Lion Precision)88Figure 4.50: Probe Range and Offset Definitions (Source: Lion Precision)Parameter Value UnitsProbe Near Gap 0.25 mmProbe Range 1.25 mmRMS Resolution @ 10 kHz 140 nmProbe Diameter 4.9 mmNumber of Channels 2 -Table 4.6: Select specifications for displacement sensor system.A diagram for the displacement sensor placement is given in Figure 4.51,defining the displacements measured by each sensor. Note that only sensorS2 is strictly necessary; the second sensor S1 is included for the purposes ofresearch and system diagnostics only.• Sensor S1 is mounted to the stationary tool post reference frame andmeasures the displacement xt1 of the low-frequency linear compensa-tion stage. This is also the displacement of the linear actuator.• Sensor S2 is mounted to the stationary tool post reference frame andmeasures the displacement xt2 of the tool tip displacement.89Figure 4.51: Definition of displacements and sensors904.5.3 Other Electrical EquipmentMicrocontrollerThe microcontroller used for this project is Texas Instrument’s TMS320F28069,on the LAUNCHXL-F28069M evaluation board platform. This board wasselected because it was already owned and therefore free, and it appearedto fit project requirements. A photo of the board from the manufacturer isprovided in Figure 4.52.Figure 4.52: TMS320 Launchpad (Source: Texas Instruments).The microcontroller performs the following functions, custom-written forthis application in C:• Programs the ADC board via SPI on startup to configure the inputchannels and voltage range.• Queries the ADC board via SPI to get digital displacement measure-ments from both probes.• Calculates motor control effort based on the designed control law andsends a proportional PWM signal to the stepper motor driver.• Reports back to the main computer the ADC measurements overUART.• Runs safety monitoring code to poll the manual e-stop and cut allmotor commands if the e-stop is triggered.91Repeated actions such as polling the ADC and e-stop, calculating controleffort, and reporting to the computer are triggered via an internal CPUtimer interrupt running at 1000 Hz. This was determined to be a samplingrate that was a good balance of data resolution, serial communication errorrates, and control system performance.Motor DriverThe motor driver is the R208 Bipolar Stepper Motor Driver from Lin En-gineering, the same manufacturer as the actuator. An image of the deviceis provided in Figure 4.53, and several specifications are given in Table 4.7.The device is configured to operate in 1/8 stepping mode for this applicationto maximize resolution.Figure 4.53: Motor Driver (Source: Lin Engineering).Parameter Value UnitsMax Continuous RMS Current (Per Phase) 1.4 APeak Current (Per Phase) 2 ASignal Voltage Level 5 VPower Voltage Level 12-24 VMax Step Frequency 500 kHzMicrostepping Increments 1, 1/2, 1/4, 1/8 -Table 4.7: Motor Driver Specifications.92ADC BoardThe output of the Lion Precision sensor system is a ± 5V analog signal,which must be sampled and converted to a digital signal before being used tocalculate control effort. For simplicity, a commercial ADC evaluation boardwas purchased for the task. The evaluation board, part number EVAL-AD7616SDZ from Analog Devices, contains the core ADC chip AD7616,as well as a PCB platform with the required power supply, screw terminalinputs, and broken-out serial programming pins. An image of the evaluationboard from the manufacturer is provided below:Figure 4.54: ADC Evaluation Board (Source: Analog Devices).The ADC board can be configured via serial SPI communications to setwhich of the 16 inputs to sample, the sampling order, and the input voltagerange of each channel. In this application, only 2 inputs are used, andthe input voltage range is set to the maximum ± 10V to allow for someover-range margin on the input signal.93Power SupplyThe power requirements of the system include:System Voltage CurrentMotor Driver Power Inputs 12 - 24 V 1 - 2 AMotor Driver Signal Inputs 5 V <50 mAADC Output Power Supply 3.3 V <50 mATable 4.8: Electrical System Power Requirements.Elements such as the microcontroller receive power over the USB connec-tion and do not require a separate power supply. To support this rangeof necessary system voltages, a standard personal computer power supplywas selected from FSP Technology, part number FSP350-50TAC. This unitconveniently provides separate 12V, 5V, and 3.3V outputs from a commonmains input. An image of the selected supply from the manufacturer isprovided below.Figure 4.55: Computer power supply (Source: FSP Technology).94Custom ProtoboardsTwo small custom protoboards were constructed to provide an interface forwire harnesses connecting the tool and the electrical panel, as well as levelshifting and line driving on the panel side.Protoboard 1 (Panel Side): Protoboard 1 performs the following func-tions:• Provides a pin interface for connecting to the microcontroller• Provides a screw terminal interface for receiving power from the powersupply• Provides an RJ45 jack for connecting to the 3m CATV cable used totransmit motor control signals• Provides a secondary 2-terminal jack for connecting to the 3m 2-conductor power cable used to transmit power to the motor• Level-shifts the 3 motor control signals from the microcontroller (step,direction, enable) from 3.3V to 5V using an isolation chip(ISO7741FQDBQRQ1)• Uses a line-driver chip (SN74LVC3G17DCTR) to drive the motor con-trol signals out along the 3m cables to the toolAn image of Protoboard 1 is provided in Figure 4.56:95Figure 4.56: Protoboard 1, Panel Side.Protoboard 2 (Tool Side): Protoboard 2 performs the following func-tions:• Provides an RJ45 jack for connecting to the 3m CATV cable used totransmit motor control signals• Provides a secondary 2-terminal jack for connecting to the 3m 2-conductor power cable used to transmit power to the motor• Provides an 8-pin connector for transmitting mower and signals tomotor driverAn image of Protoboard 2 is provided in Figure 4.57:96Figure 4.57: Protoboard 2, Tool Side.Graphic User InterfaceIn order to configure and control the system, as well as stream and recorddata, a simple graphic user interface was written in Python, using the tkinterlibrary. A screenshot of the user interface is provided in Figure 4.58. Thisinterface allows the user to:• Start / stop streaming data from the microcontroller to the computer• Display streamed data for each displacement and the control effort vstime on a scrolling plot• Start / stop recording data• Save recorded data locally• Enable / disable the motor• Set the zero offsets of each displacement sensor97• Select between open-loop or closed-loop motor control modes• In open-loop control mode:– Command a selectable number of steps forward or backward– Control the open-loop position command using a manual slider– Control the open-loop traverse velocity using a manual slider• Program the ADC board manually with register addressing• Query the value of each ADC board registerFigure 4.58: Graphic User Interface.984.5.4 Controller DesignThe controller design for the linear actuation stage is a simple proportionalcontroller Kp = 5. This is robust and effective for the purpose of static stiff-ness compensation. However, the controller design process provides somefundamental insight into potential future system improvements, and is de-tailed below.The control loop block diagram is given in Figure 4.59 below. Note that inFigure 4.59, the sum block represents the summing of forces acting on thetool tip, and there exists a negative sign within the feedback path.Figure 4.60 re-arrances the blocks into a more traditional negative-feedbackposition control loop, in which the reference is a static zero position and thecutting force Fcut is a disturbance input.Let the following shorthand definitions be used:Φt(s) =1mts2 + cts+ kt(4.19)P (s) =Ksteps(4.20)Ksensor = Ks (4.21)Figure 4.59: Control Block Diagram - Summing Forces Perspective.99Figure 4.60: Control Block Diagram - Rearranged for Traditional PositionControl Block Diagram Structure.A table of constants to define the block diagram is provided below:Parameter Symbol Value UnitsTool Effective Mass mt 0.228 kgTool Effective Damping ct 385 Ns/mTool Effective Stiffness kt 4.0∗105 N/mDisplacement Probe Gain Kprobe 8000 V/mAnalog-to-Digital Gain KADC 65535 / 20 int/VOverall Sensor Gain Ksensor = KprobeKADC 26214000 int/mLinear Actuator Gain Kstep 1e-6 m/stepTable 4.9: Block Diagram Constant Definitions.Now, the loop return ratio L(s) (also known as loop gain, open-loop transferfunction, or negative loop transmission) can be expressed as the product ofall gain blocks in the loop:L = ΦtKsCP (cts+ kt) (4.22)100The effective tool dynamics are represented as follows:No Control: Φt =xt2Fcut=1ms2 + cs+ k(4.23)With Control: Φtc =xt2Fcut=Φt1 + L(4.24)Observe that the quantity S = 11+L is the sensitivity function from theclassical loop shaping control technique, and it represents the effect of addingfeedback control to this system on the tool dynamics.Finally, let the transfer function Gcl be defined as the closed-loop responseof the position-control block diagram from Figure 4.60, given as the ratioof the tool tip position xt2 to the reference position xr. The bandwidthof this transfer function can be regarded as the frequency range in whichdisturbances from Fcut will be effectively rejected, and should look like alow-pass filter with a cutoff well below the workpiece natural frequency.Gcl =xt2xr=KsCP (cts+ kt)Φt1 + L(4.25)The loop return ratio L(s) can be used to examine stability by extractinggain and phase margins at the phase crossover frequency ωp and the gaincrossover frequency ωc, respectively. The margin definitions are as follows:Gain Margin GM: − |L(jωp)| (4.26)Phase Margin PM: ∠L(jωc) + 180o (4.27)Figure 4.61 plots the frequency response of C, L, and Gcl in the z domain toaccount for the sampling time delay affecting the phase margin. The selectedcontroller of Kp = 5 can be observed to be robustly stable with positivegain and phase margin. This gain was chosen via iterative manual tuningon the hardware, as some small unmodelled nonlinearities in the systemcaused unpredictable effects for Kp = 6 or greater. Table 4.10 outlines thecharacteristics of this control loop.101Figure 4.61: Bode Plot of controller, open-loop (or loop return) transferfunction, and closed-loop transfer functions in z domain.Parameter Value UnitsGain Margin 17.3 dBPhase Margin 83.2 dBBandwidth 22 HzTable 4.10: Position Control Loop Characteristic Values.An experimental tap test is used to identify the FRF of Φtc. This FRF isprovided in a subsequent section in Figure 4.68.102Now, some real-imaginary FRFs can be visualized and evaluated for differentpotential controller values. Figures 4.62 and 4.63 are generated assumingthe simple proportional controllers of C(s) = Kp = 1 and C(s) = Kp = 10.Figure 4.62: System FRFs, Kp=1.Several critically valuable insights can be drawn from these plots:• For Kp = 1, ωc is significantly below the workpiece natural frequency,and the effect of adding control is to drive the system complianceto near-zero within the closed-loop bandwidth - this is the expectedresult.• As Kp increases, the phase margin of L decreases.• For Kp = 10, the controller has two effects:– The expected effect of driving the system compliance to near-zerofor ω << ωc– An unexpected effect of producing a small peak of positive realpart in the system FRF near ωc103Figure 4.63: System FRFs, Kp=10.In the case of controller design for this system, achieving a lower phasemargin is not strictly a downside - it is an advantage that improvesoverall system performance. This is a highly significant finding.In a more typical control system, phase margins of approx. 60o or greateris considered a good rule of thumb for a stable response without too muchresonance in the system response. Using phase margins 0o < PM < 60ostarts to produce significant resonance near ωc, which is usually undesirable.However, in this case, it can be an advantage.To further illustrate this last point, a plot of the workpiece, tool, and closed-loop tool responses are provided for a range of closed-loop phase margins,keeping ωc constant. The controllers here are defined as a proportional gainand pure delay to achieve independently variable gain and phase lag:C(s) = Kpe−Tds (4.28)104Figure 4.64: Phase Margin Sweep.It can be observed that the resonance from lower phase margins tends totranslate to large positive real peaks, which can be used to offset the negativereal peak of the workpiece. This is significant for the following reasons:• Currently, low stiffness is required to produce large positive real partsof the tool FRF. This would allow tools to be built with higher pas-sive stiffnesses, more easily allowing the stabilization of higher chatterfrequencies, and/or providing more mass margin to the tool design.• Using a controller design such as Equation 4.28 allows independentcontrol over the crossover frequency ωc (by controlling Kp) and thephase margin (by controlling Td), which allows independent control ofthe frequency and magnitude of the real peak, which would then beeasily tunable by changing only controller parameters. As the currentdesign requires hardware changes to tune the FRF, having softwaretuneability would be a significant advantage, particularly for adapt-ing varying workpiece dynamics between workpieces, or within oneworkpiece as material is removed.105This low-phase-margin advantage was not discovered until relatively latein the project, after the linear actuator had been specified. The currentlinear actuator uses a lead screw with a small amount of backlash, which issuitable for low-frequency static compensation, but unsuited to the directionreversals of this resonant operational mode. Attempts to implement thisresonant controller on the existing tool were unsuccessful, and ultimatelythe selected controller was Kp = 5, which provided robust performance withthe presence of the small backlash. However, this controller shows significantpromise, and is a key path for future development.1064.6 Tool System Identification4.6.1 Linear Actuator Plant Identification via Open-LoopSine SweepThe plant model for the stepper motor linear actuator was initially modelledas a simple integrator as follows, where the input is a commanded velocity insteps/s, and the output is the linear displacement of the actuator in metres:P (s) =Ksteps(4.29)Following system identification, a small pure delay term was added to themodel, where Td is the delay in seconds. It is believed that this small delayterm arises as the observable effect of some small backlash in the LTI systemmodel.P (s) =Kstepse−Tds (4.30)Critically, the zero-order hold block exists between the signal injection pointand the measurement point, so the identified system must account for thiseffect by fitting a zero-order hold equivalent plant model P (z), defined below.In practice, this is accomplished using the ’c2d()’ command in Python’scontrol package.P (z) = (1− z−1)ZP (s)s= c2d(P, Ts, ’zoh’) (4.31)To identify the system parameters, a constant-amplitude sine sweep signalwas injected into the system after the controller block while operating inopen-loop mode. The sweep was run 3x, and the average of the results wasused for curve fitting. The sweep profile is defined as follows:• Frequency Range: 0 - 120 Hz• Amplitude: 4000 (16.4 steps/s)• Sweep Duration: 30 s• Sweep Rate: LinearLetting this input be defined as u, the identified plant P is therefore:P (jω) =Xt1(jω)U(jω)(4.32)107Figure 4.65 below shows the results of each individual sine sweep, the averageresult, and the best-fit parameter model. Table 4.11 reports the initially-estimated parameters and the final curve-fit parameters. The coherence ofthe input-output signals is also plotted to indicate the frequency range inwhich the results are reliable. The data selected for the curve fit was allfrequencies where the coherence is greater than 0.6. It can be observed thatthe model fits very well for the low-frequency range, with some unmodelledgain at higher frequencies.Figure 4.65: Sine Sweep System Identification Results.108Symbol Initial Value Identified ValueKstep 0.625 µm/step 0.937 µm/stepTd 0 s 0.00043 sTable 4.11: Sine Sweep Parameter Results.4.6.2 Tool Tip Dynamics Identification via Tap TestingTool ModificationFollowing system identification of the tool and workpiece, as well as con-troller tuning, it became clear that the bump of positive real part from theclosed-loop control predicted in Section 4.5.4 was not materializing in theexperimental tool FRFs from tap testing for controller proportional gainsKp < 10. Additionally, some small system nonlinearities caused the sys-tem to become unstable for gains greater than ≈ Kp = 12. Since both thetool and test workpiece had been designed assuming this benefit of closed-loop control, the theoretical performance benefits of the tool design wereweakened.From here, there existed several options for experimental testing:• Proceed to testing with the current tool and workpiece designs, accept-ing that the demonstrated stability improvement would be less thanoptimal.• Re-build the test workpiece to be better suited to the experimentaltool.• Re-build the tool to be better suited to the experimental workpiece.In order to maintain a good technology demonstration, and given that thetool stiffness and damping are modular and easy to modify, it was chosen todecrease the tool stiffness to provide a better stability improvement to thebuilt workpiece.To accomplish this, half of the inner pinwheel flexures were removed from thepassive stage of the tool. This provides greater positive real part passivelyto offset the negative real part of the workpiece, improving stability. Theidentified tool FRFs in the following sections, and all of the experimentaldata, reflect this new built configuration of the tool. Notably, the choiceto remove these springs proportionally lowers the peak load capacity of thetool.109Tap Test ProcedureTap tests on the tool were conducted to verify the open- and closed-loopdynamics. A Kistler instrumented hammer provided the input force, andthe displacement was measured with the integral position sensors within thetool. A figure showing an example tool-tip tap testing setup is provided inFigure 4.66.Figure 4.66: Measured Open-Loop Tool FRF.110Passive Stage Only: Open-Loop MeasurementAs a verification step for the design of the passive spring-damper assembly, atap test is conducted without the linear actuation stage attached, as shownin Figure 4.66. The average FRF is calculated from data using Welch’smethod, and a curve fit is made to a SDOF model with mass, stiffness,and damping. The coherence between the input and output signals is alsoplotted as a function of frequency, where coherence near 1 is desirable. Thecurve fit is calculated using the average FRF data in the frequency rangeof coherence > 0.9. Figure 4.67 shows the data and curve fit, while Table4.12 reports the identified SDOF parameters of the passive stage of the tool.The minimum negative real part of the SDOF fit is -0.6 um/N.Figure 4.67: Measured Open-Loop Tool FRF, Passive Stage Only.Parameter Value UnitsTool Mass mt 228 gTool Stiffness kt 4.0∗105 N/mTool Damping ct 385 Ns/mTable 4.12: Identified Tool Parameters.111Full Tool Assembly: Closed-Loop MeasurementNow, the tap test is conducted in the same axis as shown in Figure 4.66,but with the full tool assembly including the linear actuation stage, which ispowered on and provides low-frequency compensation. Note that here, theinput is still the hammer force, but the output uses the integral displace-ment sensors directly, instead of calculating displacement from accelerometermeasurements. The final compensated tool FRF is shown in Figure 4.68:Figure 4.68: Measured Compensated Tool FRF.Notably, the DC compensation is not very visible in this FRF, despite beingvery clear in person. This may be due to the tap test hammer being anAC-coupled device, not intended to measure such low frequencies well. Theminimum negative real part of this FRF is -1.4 um/N, more than doublethe minimum negative real part of the passive stage only. It is suspectedthat this is due to some compliance in the linear actuation stage, which isnot infinitely rigid.112Chapter 5Experimental ValidationTo characterize the performance improvements of the compliant compen-sated tool over a traditional rigid tool, it is necessary to:• Construct a highly flexible test workpiece.• Conduct cut tests of varying depth of cut using a conventional rigidtool to establish the experimental stability limit.• Conduct cut tests on the same workpiece with the same insert andcutting conditions, using the compliant compensated tool instead, tofind the new stability limit.• Compare the theoretical stability improvements of the compliant com-pensated tool to the experimental results.The following sections detail the design and execution of these tests.5.1 Test Workpiece DesignThe chatter improvements available from a compliant compensated toolcome when the tool dynamic response is well-matched to the workpiecedynamics. Since the compliant compensated tool designed in the precedingchapters is not trivially tuneable, a dedicated test workpiece with matcheddynamics must be designed. In the timeline of the project, the tool require-ments and approximate workpiece design were determined simultaneouslyto ensure that the tool capabilities could be practically demonstrated.To characterize the improvement in turning stability, a test workpiece wasdesigned and built with the following features and design guidelines:• Natural frequency of the workpiece designed to be in a range thatcould be effectively stabilized by the prototype tool113• Critical depth of cut that could cause chatter with a rigid tool <0.5mm radially• Vibrational response in the cut location dominated by the 1st modeof the workpieceAs determined in Chapter 3, the effective mass of the workpiece must beapproximately 50x greater than the effective mass of the tool. Assuming anapproximate mass target of 250g for the tool, this requires at least 12.5kg ofmass in the workpiece. To achieve this high mass while still maintaining aflexible workpiece, a design as shown in Figure 5.1 is proposed, with a large-diameter cylindrical centre section and two smaller-diameter high-aspect-ratio arms on either side.Figure 5.1: Experimental Workpiece Design114Figure 5.2: Workpiece Mode ShapesThe advantages of this design can be observed from the theoretical modeshapes diagram. The part of the workpece that would be used for cut testingwould be only the large-diameter cylindrical part. It can be observed that inthis region, the modal response is dominated by the first mode, preventinginterference from higher modes. Calculating the theoretical FRF at thecentre of the workpiece, the tool, workpiece, and combined responses areplotted in Figure 5.3.The minimum negative real part of the workpiece combined with compen-sated tool (green) is significantly higher than the minimum negative realpart of the workpiece alone (blue), indicating significant chatter stabilityimprovements.Notably, this design has assumed infinite stiffness at the chuck and tailstockinterfaces of the machine, which is not true in practice, and so the realtest workpiece must be tap tested to have its dynamics identified in theexperimental setup.115Figure 5.3: Theoretical tool, workpiece, and combined FRFs.Based on the dimensions of available material at the time of fabrication, theworkpiece was constructed by taking a 1.5m length of 36mm diameter mildsteel, and using Loctite adhesive to glue two 50mm thick discs of 180mmdiameter steel first to each other, and then to the shaft, centred at 400mmfrom the tailstock end of the workpiece. After the incremental removal ofmaterial at the ends of the workpiece to achieve a good dynamic response,the final dimensions are given Figure 5.4.1165.2 Test Workpiece System IdentificationThe tool and workpiece were both tap tested using a Kistler accelerometerand Kistler instrumented hammer. Two input and output locations weremeasured, as indicated in Figure 5.4, with 5-10 taps in each location toget a robust measurement. This was done iteratively, starting with a totalworkpiece length of 1m, and slowly removing material from the ends of theworkpiece to compensate for mounting flexibility until a reasonable responseis achieved. The result of these measurements is the following receptancematrix, H:H(ω) =[H11(ω) H12(ω)H21(ω) H22(ω)][µm/N ] (5.1)Photos of the tap testing setup for the tool are given in Figure 5.4, and theresulting collocated FRFs H11 and H22 are shown in Figure 5.5.Figure 5.4: Tap testing locations for the experimental workpiece.117Figure 5.5: Experimental workpiece FRFs, with average best fit parametersmw = 21kg, cw = 406 Ns/m, kw = 5.9e6 N/m.It can be observed that the magnitudes of each FRF are similar, but theminimum negative real part at point 2 (near the tailstock) is slightly lower,indicating that the cutting stability near point 2 will be slightly worse thannear point 1.1185.3 Experimental DesignTo demonstrate the stability improvements between a highly rigid tool anda compliant compensated tool, cut tests must be conducted on the sametest workpiece with both a rigid tool and the compliant prototype. The toolconfigurations to be tested are as follows:• Rigid Tool• Compliant Passive Tool (no compensation, passive stage only)• Compliant Compensated ToolTo evaluate whether or not chatter is occurring, an accelerometer will befixed to the tool. For cuts with the passive stage only, two accelerometerswill be used, one on the compliant stage and one on the fixed frame. Forcuts with the compensated tool, an accelerometer will be placed on the rigidhousing and the integral displacement sensors will also be used to identifychatter. Photos of each test will be taken to identify any visual chattermarks in the surface.Cuts are taken at constant feed rate and constant depth of cut across theentire cut region of the workpiece. The depth of is started very small, andeach test uses a larger depth of cut until chatter occurs. This is repeatedwith each tool configuration.Note that all depth of cut measurements here are RADIAL, not diametral.For the compliant passive tool without compensation, the nominal depth ofcut is listed, and the estimated effective depth of cut accounting for tooldeflection is given in parentheses, in the format ”nominal (effective)”. Theprocedure for conducting each test is described below.• Zero the tool tip on the outer surface of the cut region of the workpiece.• Move axially away from the chuck to clear the cut section of the work-piece, and then move the tool radially inwards to the desired depth.• Start the spindle and use the automated feed to cut a fixed depth ofcut along the entire cut section of the workpiece.A figure of the test setup is given in Figure 5.6. Figure 5.7 shows the rigidtool test setup, Figure 5.8 shows the passive stage only test setup, and Figure5.9 shows the full compliant compensated tool test setup.119Figure 5.6: Experimental Testing Setup.Figure 5.7: Rigid Tool Setup.120Figure 5.8: Passive Compliant Tool Setup.Figure 5.9: Compliant Compensated Tool Setup.1215.4 Experimental ResultsThe tests, cutting conditions, and test results are documented in Table 5.1.No. ToolFeed Rate[mm/rev]Depth of Cut[mm]Chatter?1 Rigid 0.05 <0.01 No2 Rigid 0.05 0.025 No3 Rigid 0.05 0.04 Yes4 Rigid 0.05 0.05 No5 Rigid 0.05 0.06 No6 Rigid 0.05 0.08 Yes7 Rigid 0.05 0.1 Yes8 Rigid 0.05 0.2 Yes9 Compliant 0.05 0.05 (0.02) No10 Compliant 0.05 0.1 (0.05) No11 Compliant 0.05 0.2 (0.09) No12 Compliant 0.05 0.3 (0.14) No13 Compliant 0.05 0.4 (0.18) No14 Compliant 0.05 0.5 (0.23) No15 Compliant 0.05 0.65 (0.29) No16 Compliant 0.05 0.8 (0.36) No17 Compliant 0.05 0.95 (0.43) No18 Compliant Comp. 0.05 0.2 No19 Compliant Comp. 0.05 0.4 No20 Compliant Comp. 0.05 0.45 No21 Compliant Comp. 0.05 0.5 No22 Compliant Comp. 0.05 0.55 No23 Compliant Comp. 0.05 0.6 No24 Compliant Comp. 0.05 0.7 NoTable 5.1: Experimental Test Matrix.Figure 5.10 shows a summary of the results comparing the rigid tool (left)and compliant compensated tool (right). Significant chatter can be seenon the rigid-cut surfaces, while no chatter is visible on the compliant-cutsurfaces. The stability lobe diagram is calculated using Equation 5.13.122Figure 5.10: Cut test results for feed rate 0.05 mm/rev, spindle speed 175RPM, no coolant. Only depth of cut and cutting tool configuration arevaried.The accelerometer frequency spectra supports this observation, with a largechatter peak and harmonics for the rigid tool, and no significant peaks withthe compliant tool.The passive compliant tool without the compensation stage was also tested.The stability predictions behind the design suggest that the tool should bestable both with and without compensation. Experimental testing confirmed123this, with the passive compliant tool performing just as well in all conditionstested as the compensated compliant tool from a stability perspective. Agraphical representation of the stable depths of cut vs tool type is providedin Figure 5.11. The stability improvement between the standard rigid tooland the compliant compensated tool prototype is at least 10x, and couldbe greater - the compliant compensated tool could not be made to chatterwithin its calculated load capacity.Figure 5.11: Stability Map for each tool configuration.Figure 5.12 shows a comparison of the Fourier transforms of the accelerom-eter signals from the following tests with roughly equivalent effective depthsof cut.• Test No. 8, Rigid Cut Test, Radial Depth of Cut 0.2mm• Test No. 14, Passive Compliant Cut Test, Radial Depth of Cut 0.5mmNominal (estimated 0.23mm effective)It can be observed that the rigid tool has very large peaks at 87 Hz, the fre-quency of the minimum negative real part of the identified workpiece FRF,with some smaller peaks at multiples of this frequency. By contrast, there124is some low-level broadband content from the passive compliant tool ac-celerometer fixed to the moving compliant stage, and hardly any observableresponse from the passive compliant tool rigid frame.Figure 5.12: Fourier Transforms of Rigid vs Passive Compliant Accelerom-eter Signals, n = 175 RPM, feed rate = 0.05 mm/rev, radial depth of cut ≈0.2 mm.Several test notes are as follows:• The workpiece had a small amount of runout, so some cut tests havean observable 3 Hz force fluctuation from the spindle frequency.• No cutting oil was used in any test.• The spindle speed used in each test was 175 RPM.The critical observations from this testing are:• There was hardly any stable depth of cut for the rigid tool, and thecuts that were stable were much less than the nose radius of the insert(0.4 mm).125• The passive compliant tool was stable under all cutting conditionsattempted. Testing was stopped after 0.95mm nominal (0.43mm ef-fective) radial depth of cut just to move on to testing with the linearactuation compensation, not because of a stability limit.• The compliant compensated tool was stable under all cutting condi-tions attempted. The prototype worked well - the stable depth of cutwas larger than predicted, and was >10x larger than the stable depthof cut of the rigid tool. Cut testing was stopped at 0.7mm radial depthof cut due to concerns over load capacity of the flexure springs.• In general, the amplitude of tool tip position variation with the com-pliant compensated tool was < ±8µm, and during steady-state testswith a constant depth of cut (not cutting runout) this was commonlyas low as ±3µm.• The axial length of cuts with the compliant compensated tool waslimited to only a few mm at a time due to chip management challenges(see Figure5.13).One significant challenge in testing specifically the compliant compensatedtool was the combination of:• Low stick-out of the tool head from the rectangular frame• Low melting point of the plastic chip coverA simple 3D printed structure intended to act as a flexible bellows-type coverkeeping chips out of the inside of the tool instead tended to collect and bunchup chips near the cutting head, as hot chips embedded themselves in theplastic and then ensnared more chips in a positive feedback cycle. Somephotos of this phenomenon are shown in Figure 5.13. To manage this, onlya few millimetres of axial cut length could be taken at a time before stoppingthe test to clear out chips.126Figure 5.13: Chip clumping problem, due to small tool clearance and hotchips melting and embedding in plastic chip cover.Some example datasets from the two displacement sensors measuring thetool tip displacement (xt2) and the linear stage displacement (xt1) are ex-amined in more detail. The following two tests are chosen:• Test No. 19, with 0.4mm radial depth of cut, cutting a surface withsignificant runout, shown in Figure 5.14.• Test No. 23, with 0.6mm radial depth of cut, cutting the surface leftby Test No. 19 above, which now has minimal runout, Figure 5.15.127In Test No. 19, the cutting forces are periodic at the spindle rotation fre-quency due to a small amount of runout in the workpiece. This causesperiodic displacement of the tool tip stage. The controller acts to absorbmost of this oscillation via linear actuation compensation (xt1), but someof this oscillation remains in the tool tip position (xt2), which must be keptwithin the allowable part form error tolerance. A large DC offset can be ob-served in xt1, demonstrating compensation of the static cutting force, whilethe tool tip is kept at zero DC offset as intended. Additionally, the followingfeatures can be observed:• (Subfigure A) The proportional controller causes some low-level jitter-ing behaviour, as the tool tip bounces back and forth across the zeroposition with an amplitude of 1-2um. This is slightly audible as a lowbuzz, and can be felt by placing a hand on the actuator. This is dueto the discrete size of the linear actuator steps being 1um, and thedistance required to get to a tool position of zero being <1um. Thiscould be eliminated by implementing a deadband, but was not foundto be a concern during testing.• (Subfigure B) Due to runout in the workpiece surface being cut, theeffective depth of cut varies over the spindle period. This shows up inthe data as a very strong 3Hz signal in both displacement measure-ments. The majority of the runout is compensated for by the linearactuation stage, as can be observed by the 35um amplitude oscillationsin xt1.• (Subfigure C) A small amount of 3Hz signal makes it past the controllerand to the tool tip position, causing an 8um amplitude oscillationhere. This is the main amplitude component of the tool tip positionsignal, and could be mitigated by improved linear actuation stage andcontroller design.• (Subfigure D) To end the cut, the tool is rapidly manually pulled outof the workpiece radially, which drops the depth of cut to zero near-instantaneously. With the current controller design and tuning, thiscauses a small spike of ≈ 30um in the tool tip position. However, notethat this is NOT a groove cut into the workpiece - this is the tooltip moving through air. As is, it is a harmless feature, but a designoptimized to allow better interrupted cut performance would seek tominimize this spike.128Figure 5.14: Cut test data from displacement sensors in Test No. 19, n =175 RPM, feed = 0.05 mm/rev, depth of cut 0.4 mm.129In Test No. 23, the surface being cut is the surface left by Test No. 19, sothe runout is largely gone. The following features can be observed:• During the ’on-ramp’ period where the depth of cut is slowly increasingas the tool feeds into the workpiece, the 3Hz spindle frequency can stillbe observed, as the face of the shoulder being cut into is not perfectlynormal to the axis of rotation.• During the ’Steady-State’ period, the position of xt1 mostly stabilizes,but continues increasing slowly. This is due to chips building up be-tween the tool and the workpiece, applying force to the tool tip thatthe linear actuator must push back against.• Also during the ’Steady-State’ period, the position of the tool tip xt2is maintained within a ±3um amplitude range, a very good and stableresult.Photos of the workpiece surface (A) before Test No. 19, (B) after Test No.19 and before Test No. 23, and (C) after Test No. 23 are given in Figure5.16.130Figure 5.15: Cut test data from displacement sensors in Test No. 23, n =175 RPM, feed = 0.05 mm/rev, depth of cut 0.6 mm.131Figure 5.16: Cut test surfaces, (A) before Test No. 19, (B) after Test No.19 and before Test No. 23, and (C) after Test No. 23.1325.5 Predicted vs Experimental StabilityImprovementsIn orthogonal cutting, the limiting radial depth of cut alim can be predictedsimply using Tlusty’s Law, where Kc is a material-specific force coefficientin N/m2 and Φw and Φt are the workpiece and tool receptance FRFs, re-spectively, in m/N .alim =−12Kcmin(Re{Φw +Φt}) (5.2)In non-orthogonal cutting, a coordinate transform can be used to accountfor the angle of the insert. Figure 5.17 shows the chip area model for thisnon-orthogonal setup.Figure 5.17: Chip area model.133In this non-orthogonal model, c is the feed rate in mm/rev, a is the radialdepth of cut in mm, b is the chip width in mm, and h is the chip thicknessin mm. These quantities can be related by the edge angle of the insert, θ.The Tlusty’s Law calculation may now be used to calculate the limiting chipwidth, and the depth of cut can be back-calculated.b =asin θ, h = c sin θ (5.3)blim =−12Kcmin(Re{Φo}) (5.4)In this new stability limit expression, Φo is the oriented relative transferfunction between tool and workpiece, which is a projection of the relativeflexibilities into an axis normal to the chip width. Given a structural FRFΦ and an angle of force application θ, the oriented transfer function can bederived as follows, based on the diagram in Figure 5.18.Figure 5.18: Oriented FRF derivation diagram.ΦF ′ = x′ (5.5)F ′ = F cos θ, x′ =xcos θ(5.6)ΦF cos θ =xcos θ(5.7)Φo =xF= Φcos2 θ (5.8)134Since the insert used for these tests has a θ = 45o cutting edge angle, it isnecessary to account for this orientation angle. However, for very small cutscomparable to the size of the insert nose radius, the effective angle of theforce applied to the tool-workpiece system is no longer just the insert edgeangle. In these cases, a new effective angle θeff is constructed by drawinga line from the centre of the nose radius to the corner of the chip, as shownin Figure 5.19.Figure 5.19: Effective angle from nose radius.135The effective angle θeff is then calculated as follows:d =√2R−R = (√2− 1)R (5.9)g =d+ atan θ(5.10)tan θeff =ag= tan θad+ a(5.11)θeff = tan−1(tan θad+ a)(5.12)Finally, an expression for the limiting depth of cut alim can be constructed.Note that the depth of cut a appears on both sides of the equation, sinceθeff depends on a, so 5.13 must be solved iteratively for alim.alim = blim sin θeff =− sin θeff2Kcmin(Re{(Φw +Φt) cos2 θeff}) (5.13)Using 5.13, and the SDOF fits to the measured FRFs of the tool and work-piece, the predicted stability-limiting depths of cut can be calculated withand without compensation. The identified modal parameters for the tooland workpiece are provided in Table 5.2, and the combined FRFs are pro-vided in Figure 5.20.Parameter Symbol Value UnitsWorkpiece Effective Mass mw 21 kgWorkpiece Effective Stiffness kw 5.9∗106 N/mWorkpiece Effective Damping cw 406 Ns/mTool Effective Mass mt 228 gTool Effective Stiffness kt 4.0∗105 N/mTool Effective Damping ct 385 Ns/mTable 5.2: Identified Effective Workpiece and Tool Parameters, SDOF Fits.136Figure 5.20: FRFs for Stability Prediction.ConfigurationMin. Neg.Real PartPredictedLimitUnorientedPredictedLimitOrientedExperimentalLimitRigid Tool -2.24 um/N 0.18 mm 0.013 mm 0.06 - 0.08 mmCompensatedCompliant Tool-0.61 um/N 0.65 mm 0.67 mm > 0.7 mmTable 5.3: Predicted Stability Limits from Tlusty’s Law.Table 5.3 lays out the minimum negative real parts of the unoriented FRFs,and the predicted limiting depths of cut. Note the following:137• All of the calculations in this section assume Kc = 1250N/mm2.• The column ’Predicted Limit Unoriented’ uses Equation 5.2.• The column ’Predicted Limit Oriented’ uses Equation 5.13. This rep-resents the actual cutting conditions and is the basis for comparisonwith experimental data.• All depths of cut listed in Table 5.3 are radial, not diametral.There are several key features from Table 5.3:• A simple way to predict the stability improvement ratio would be totake the ratios of the minimum negative real parts, with and with-out compensation. This improvement ratio is 3.7x, which is a goodimprovement, but lower than the > 10x improvement observed in ex-perimental testing. This is due to this method not accounting for thenose radius effects of the insert.• The nose radius of the insert is 0.4mm, which is much larger than theunoriented unstable depth of cut for the rigid tool. This means thatthe nose radius has a significant effect on the stability limit for therigid tool case. This is also supported by the oriented depth of cutlimit being 0.013mm.• There is an observed experimental stable depth of cut for the rigidtool that is greater than the predicted oriented stability limit. Thismay be due to the extremely small depth of cut causing changes inthe material removal mechanism that void the assumptions behindTlusty’s Law (eg, the tool does more rubbing/scraping than cuttingon the workpiece surface).• Conversely, the compliant compensated tool’s largest depth of cut inthe oriented case is larger than the nose radius, which means the noseradius has little effect on this cutting case.• The experimental results are plausibly in line with the predicted ori-ented stability limits, though no unstable depth of cut was actuallyfound for the compliant compensated tool.138Chapter 6Conclusion andRecommendations6.1 ConclusionThis thesis presented a novel actuator to damp the vibrations of highlyflexible parts in turning operations. The proposed compliant compensateddesign uses a stack of flexible plate springs to provide a design stiffnesson one axis, a layered-sheet damper design in an oil bath to provide highdamping, and a stepper motor linear actuation stage to compensate for thestatic deflection of the highly flexible tool, preventing errors on the workpiecesurface.The experimental results showed a notable stability improvement of >10xwhen cutting the test workpiece with the prototype compliant compensatedtool. The test results appear to be in line with predictions from Tlusty’sLaw when nose radius effects are considered (Equation 5.13). The prototypealso demonstrated good DC stiffness with the active compensation of thelinear stage, maintaining a tool tip position of < ±8µm amplitude throughvaried testing conditions, and < ±3µm amplitude for steady-state cuts withconstant cut depth.Additionally, this work proposed a rule of thumb for determining the condi-tions under which a compliant compensated tool is an applicable techniquefor flexible workpiece stabilization. Section 3.3 outlines the ”Tool MassLimiting Effect”, which proposes that if the effective modal mass of theworkpiece mode to be stabilized (ie, the mass from a best-fit SDOF modelof the mode) mw is ≈ 50x larger than the effective modal mass of the toolmt, then there exists values of tool stiffness and damping that would providean effective stabilizing solution.Finally, this work proposed but did not implement a concept for resonantcontrol via low controller phase margins in Section 4.5.4. The compliant139compensated tool prototype has ”fixed” stiffness and damping, in that theseparameters require tool teardown and hardware modification to change.Combined with the fact that the tool parameters must be tuned to theworkpiece, this makes it inconvenient to use this tool for different work-pieces with differing dynamics. The proposed resonant control approachwould allow the frequency and magnitude of the peak of the positive realpart of the tool FRF to be tuned in software, making the tool much moreversatile. The experimental proof of this concept is left to future work.1406.2 Recommendations for Future WorkThe following future work can be carried out for the improvement and op-timization of the proposed design:• Higher Load Capacity Design - Characterize the True Stabil-ity Ceiling:– This work did not find a depth of cut that could cause chatter forthe compliant compensated tool within the expected load limitsof the tool. A tool with a higher load capacity could take largercuts and could find the true stability ceiling of this design.• Better Chip Guard - Enable Longer Cuts:– This tool design used a front bellows made of 3D printed PLA(bio-based thermoplastic polymer), which melted on contact withchips and ensnared chip clusters near the tool head. This limitedthe length of cuts to 3-10mm, depending on the depth of cut.– Heat resistant bellows and tools with chip breaking rake facesmust be used in the future.• Simplified Variation for Roughing - No Compensation:– The majority of the cost and complexity in this project comesfrom the inclusion of the precision linear actuation stage, whichrequires electrical power, a controller, actuator, and sensors.– For applications like roughing where large material removal isdesired but precision is not required, the linear actuation stagecould be eliminated to dramatically simplify the tool.• Advanced Variation - Positive Real Part Generation via Res-onant Control:– While designing low-frequency controllers for static stiffness com-pensation, it was incidentally discovered that a control loop withthe following characteristics produced a very large local positivereal part:∗ A negative-loop-transmission unity-gain crossover frequencynear the workpiece natural frequency.∗ Low but positive phase margin, approximately 30-60 degrees.141– This behaviour was consistently observed in mathematical mod-elling, but could not be replicated on the physical prototype. Itis believed that a small amount of backlash in the chosen linearactuator is the root cause of the lack of replication.– If this behaviour could be physically reproduced, this could im-prove the limits of the tool mass limiting effect by allowing largerpositive real parts without requiring such low tool stiffnesses,making higher natural frequencies more reachable and loweringthe stringent tool mass limits. This would be the one of the mostvaluable next steps for the project.• Advanced Variation - Variable Spring and Damping Rates:– The nature of trying to mitigate chatter produced primarily bythe workpiece dynamics is that the tool dynamics must be tunedto each workpiece.– For this prototype, a specific and dedicated test workpiece wasdesigned to demonstrate proof of concept.– To make the compliant compensated tool concept more versatile,a tool could be designed that had online-tuneable spring rates anddamping rates, and a control algorithm that was able to estimateworkpiece dynamics online and tune the spring and damper ac-cordingly. This would be the one of the most valuable next stepsfor the project.• Dedicated Displacement Sensor Design:– Sensing displacements on the order of microns is quite costly, andthe commercial sensors account for ≈60% of the total cost of theprototype.– The current prototype cost is $16,700 CAD, of which $9500 is thesensors. This cost could be dramatically reduced by developinga custom sensor for the application.– A custom sensor was explored early in the project using Texas In-struments’ LDC1614 inductive-to-digital sensing chip and a PCBcoil sensing element, but was shelved due to project time con-straints.– The dedicated displacement sensor would cost an estimated $100-$200 CAD, potentially bringing the total tool cost to $7900, wheremost of the cost is now the custom-machined components.142• Using Only One Displacement Sensor:– This project used 2 displacement sensors to measure the 2 differ-ent internal displacements of the tool - the tool tip stage, and thelow-frequency compensation stage.– Functionally, only the measurement of the tool tip displacementis required, and the second measurement of the low-frequencycompensation displacement was included for research and diag-nostic value, but could be eliminated in a future revision to savecost and complexity.• Mass Optimization of Mass-Spring-Damper System:– Mathematical modelling predicts that the lower the mass of themass-spring-damper part of the tool, the wider the range of fre-quencies available for chatter stability improvement.– The design of many components in this prototype, such as the0.25mm thick steel interleaved damper sheets, was driven primar-ily by conservative estimates for what thickness of material wouldbe practical to handle for manual assembly, or practical to easilymachine.– There exists significant opportunity for mass-optimization of thesystem to improve performance and significantly shrink the pack-aging volume in future design iterations.• Tool Housing Profile Optimization:– Currently, the tool housing design is very boxy, and the cuttingedge does not extend very far forward of the front face.– This limits the contours that the tool could theoretically be usedto machine.– An improved tool design with a lower profile housing could al-low the tool to better reach into complex geometries for turningprofiles.143Bibliography[1] Y. Altintas, D. Lappin, D. van Zyl, and D. O¨stling. Automaticallytuned boring bar system. CIRP Annals, 70(1):313–316, 2021.[2] Y. Altintas, G. Ste´pa´n, D. Merdol, and Z. Dombo´va´ri. Chatter stabilityof milling in frequency and discrete time domain. CIRP Journal ofManufacturing Science and Technology, 1(1):35–44, 2008.[3] Y. Altintas and M. Weck. Chatter stabiliy of metal cutting and grind-ing. CIRP Annals, 53(2):619–642, 2004.[4] Yusuf Altintas. Manufacturing Automation: Metal Cutting Mechan-ics, Machine Tool Vibrations, And Cnc Design. Cambridge UniversityPress, second edition, 2012.[5] E. Budak and E Ozlu. Analytical modeling of chatter stability in turn-ing and boring operations: A multi-dimensional approach. CIRP An-nals, 56, 2007.[6] F. Chen, M. Hanifzadegan, Y. Altintas, and X. Lu. Active damping ofboring bar vibration with a magnetic actuator. IEEE/ASME Transac-tions on Mechatronics, 20(6):2783–2794, 2015.[7] M. Chen and C.R. Knospe. Control approaches to the suppression ofmachining chatter using active magnetic bearings. IEEE Transactionson control systems technology, 15(2):220–232, 2007.[8] A. Ganguli, A. Deraemaeker, and A. Preumont. Regenerative chatterreduction by active damping control. Journal of Sound and Vibration,300(3-5):847–862, 2007.[9] K Li, F Gu, A Longstaff, and G. Li. An investigation into tool dy-namics adaptation for chatter stability enhancement in the turning offlexible workpieces. The International Journal of Advanced Manufac-turing Technology, 111(11):3259–3271, 2020.144[10] H. Ma, J. Guo, J. Wu, Z. Xiong, and K.M. Lee. An active controlmethod for chatter suppression in thin plate turning. IEEE Transac-tions on Industrial Informatics, 16(3):1742–1753, 2019.[11] I. Mancisidor, X. Beudaert, G. Aguirre, R. Barcena, and J. Munoa.Development of an active damping system for structural chatter sup-pression in machining centers. International Journal of AutomationTechnology, 12(5):642–649, 2018.[12] J. Munoa, X. Beudaert, Z. Dombovari, Y. Altintas, E. Budak,C. Brecher, and G. Stepan. Chatter suppression techniques in metalcutting. CIRP Annals, 65(2):785—-808, 2016.[13] J. Munoa, X. Beudaert, K. Erkorkmaz, A. Iglesias, A. Barrios, andM. Zatarain. Active suppression of structural chatter vibrations usingmachine drives and accelerometers. CIRP Annals, 64(1):385–388, 2015.[14] A. Otto, F.A. Khasawneh, and G. Radons. Position-dependent stabilityanalysis of turning with tool and workpiece compliance. Int. J. Adv.Manuf. Technol., 79(9):1453–1463, 2015.[15] N.D. Sims. Vibration absorbers for chatter suppression: A new ana-lytical tuning methodology. Journal of Sound and Vibration, 301(3-5):592–607, 2007.[16] G. Stepan, A.K. Kiss, B. Ghalamchi, J. Sopanen, and D. Bachrathy.Chatter avoidance in cutting highly flexible workpieces. CIRP Annals,66(1):377–380, 2017.[17] H. Tanaka, F. Obata, T. Matsubara, and H. Mizumoto. Active chattersuppression of slender boring bar using piezoelectric actuators. JSMEinternational journal. Ser. C, Dynamics, control, robotics, design andmanufacturing, 37(3):601–606, 1994.[18] J Tlusty and M Polacek. The stability of the machine tool againstself-excited vibration in machining. ASME J. Manuf. Sci. Eng, 1965.[19] SA Tobias and Wilfred Fishwick. Theory of regenerative machine toolchatter. The engineer, 205(7):199–203, 1958.[20] L. Vela-Mart´ınez, J.C. Ja´uregui-Correa, E. Rubio-Cerda, G. Herrera-Ruiz, and A. Lozano-Guzma´n. Analysis of compliance between thecutting tool and the workpiece on the stability of a turning process.145International Journal of Machine Tools and Manufacture, 48(9):1054–1062, 2008.[21] S. Wan, X. Li, W. Su, J. Yuan, J. Hong, and X. Jin. Active damping ofmilling chatter vibration via a novel spindle system with an integratedelectromagnetic actuator. Precision Engineering, 57:203–210, 2019.[22] Y. Yang, J. Munoa, and Y. Altintas. Optimization of multiple tunedmass dampers to suppress machine tool chatter. International Journalof Machine Tools and Manufacture, 50(9):834–842, 2010.[23] Y. Zhang and N.D. Sims. Milling workpiece chatter avoidance usingpiezoelectric active damping: a feasibility study. Smart materials andstructures, 14(6):65, 2005.146Appendix AActive Tool-Tip ControlStrategiesA.1 Tool Tip Active ControlA.1.1 System Dynamics With ControlThe dynamics of the cut can be changed by actively controlling the tool tipdisplacement. This requires a specialty tool with a built-in actuator capableof moving the tool tip relative to the rest of the machine. This approachuses negative feedback to stabilize the passively unstable chattering sys-tem. In general, this control is possible but impractical, as it requires veryhigh bandwidth and the direct measurement of the workpiece displacement,and so can only be effectively deployed for systems with very low chatterfrequencies. A diagram of such a system is provided in Figure A.1:Figure A.1: 3-Degree-of-Freedom Orthogonal chatter diagram with flexibleworkpiece, flexible tool, and linear actuation providing active control of thetool tip.147This thesis originally explored this direct active control strategy, before piv-oting to the simpler compliant compensated tool.Let the displacement of the actuated tool tip relative to its base be xa.Given this, the relative displacement between the tool tip and work piecesurface can be defined:xr = xw − xt + xa (A.1)The dynamic force can now be expressed as a function of the relative dis-placement:F = Kca0 (h0 + xr(t)− xr(t− T )) (A.2)Substituting in Equation A.1:F = Kca0 (h0 + xw(t)− xt(t) + xa(t)− xw(t− T ) + xt(t− T )− xa(t− T ))(A.3)Taking the Laplace Transform:F (s) = Kca0 (Xw(s)−Xt(s) +Xa(s))(1− e−sT )+Kca0h0(s) (A.4)Now, let the control effort xa be defined as a function of the passive dis-placement of the tool and workpiece. It is assumed for active control thatthis displacement can be either measured or estimated.Xa(s) = C(s) (Xt(s)−Xw(s)) (A.5)Substituting Equation A.5 into the dynamic force:F (s) = Kca0 (Xw(s)−Xt(s)− C(s) (Xw(s)−Xt(s)))(1− e−sT )+Kca0h0(s)(A.6)F (s) = Kca0 (Xw(s)−Xt(s)) (1− C(s))(1− e−sT )+Kca0h0(s) (A.7)Substituting in 2.22:F (s) = Kca0Xw(s)(1 +ΦtΦw)(1− C(s)) (1− e−sT )+Kca0h0 (A.8)148Finally, substituting this dynamic force expression into the 2DOF dynamicsequations of 2.11, and using 2.20:1ΦwXw(s) = −F1ΦtXw(s) = FOnly one of the tool or the workpiece need be considered for stability evalua-tion, as their relative displacements are linearly related by 2.22. Expandingwith the workpiece:1ΦwXw(s) = −Kca0Xw(s)(1 +ΦtΦw)(1− C(s)) (1− e−sT )−Kca0h0(s)Rearranging to find Xw(s)/h0(s):Xw(s)h0(s)=−Kca0Φw1 +Kca0 (Φw +Φt) (1− C(s)) (1− e−sT ) (A.9)The same transfer function can be found for the tool dynamics by once againusing 2.22:Xw(s)h0= − ΦtΦwXw(s)h0=Kca0Φt1 +Kca0 (Φw +Φt) (1− C(s)) (1− e−sT ) (A.10)Several interesting conclusions can be drawn from A.9:• Comparing Equation A.9 with Equation 2.25, it can be observed thatthe effect of adding active control as modelled is to add the term(1− C(s)) in the characteristic equation.• The term containing C(s) in the characteristic equation is multipliedby the delay term, (1−e−sT ), indicating that the control effort xa willbe subject to the same regenerative effect as the workpiece and tooldisplacements.149Let the relative FRF between tool and workpiece WITHOUT active controlbe defined as follows:Φr = Φw +Φt (A.11)From Tlusty’s Law, it is known that the minimum negative real part of theopen-loop force-displacement FRF Φr determines the chatter stability limitof the system. A new ’Effective Open-Loop FRF’ can be defined in thesystem with active control as follows:Φeff = Φr (1− C(s)) = (Φw +Φt) (1− C(s)) (A.12)This insight means that analysis of the minimum negative real part of Φeffmay be used to determine system stability with active control. Given this,Tlusty’s Law can be rewritten as follows:alim =−12Kcmin (Re{(Φw +Φt) (1− C(s))}) (A.13)A block diagram of this dynamic system may be drawn as follows:Figure A.2: Block diagram dynamics representation of single-degree-of-freedom orthogonal chatter model, with flexible workpiece, flexible tool, andactive chatter control.150A.1.2 Controller DesignIt is of interest to evaluate potential controllers C(s) that may stabilize apassively unstable, chattering system. Some controller designs have beenfound to work in simulation, with varying amounts of success. These areoutlined in Table A.1.Name Controller C(s)Follower Controller C(s) = 145o Lag Controller C(s) = e−sTcTable A.1: Active chatter controller transfer functions.Follower Controller The follower controller describes the following con-trol law:C(s) = 1 (Ideal) (A.14)C(s) =ωfs+ ωf(Practical) (A.15)xa = xt − xw (A.16).This would effectively drive the relative displacement between tool tip andworkpiece to zero, preventing regenerative chatter. The practical imple-mentation is a unity-gain low-pass filter, with a gain of 1 below the cutofffrequency ωf . This can be observed by substituting this controller expres-sion back into Equation A.9:Xw(s)h0=−Kca0Φw1 +Kca0 (Φw +Φt) (1− C(s)) (1− e−sT ) (A.17)=−Kca0Φw1 +Kca0 (Φw +Φt) (1− (1)) (1− e−sT ) (A.18)= −Kca0Φw (A.19)The characteristic equation is reduced to 1, and the poles from the delayterm vanish, leaving only the passively stable structural dynamics in thenumerator.151This design performs very well in simulation, however it suffers in practicefrom phase delay in the feedback system. The ideal controller C(s) = 1means there can be no delay between the measurement of xt − ww and theactuation of the tool. This is nonphysical, as there must be come processingdelay, meaning that C(s) is not quite 1, and the regenerative delay poles arenot completely erased, though their effect is still reduced.To achieve near-zero delay in the feedback line, the controller C(s) requiresa very high bandwidth, and therefore a very high sampling rate when im-plemented, restricting this design to only very low-frequency chatter insta-bilities. Using general rules of thumb that digital sampling should occurat >10x bandwidth, and phase rolloff starts at approximately 1/10x band-width, the sampling rate for a follower controller system would need to beat least 100x greater than the chatter frequency.45o Lag Controller Using the definition of the effective FRF from Equa-tion A.12, and the desired goal of minimizing the negative real part of thiseffective FRF, the expression can be expanded into real and imaginary com-ponents, and the real part of the result with control can be inspected:C(s) = CR + jCI (A.20)(Φw +Φt) = Φr = Φr,R + jΦr,I (A.21)Φeff = (Φw +Φt)(1− C(s)) = (Φr,R + jΦr,I)(CR + jCI) (A.22)= Φr,R + jΦr,I − CRΦr,R − jCRΦr,I − jCIΦr,R + CIΦr,I (A.23)= (Φr,R(1− CR) + Φr,ICI) + j (Φr,I(1− CR) + Φr,RCI) (A.24)Extracting only the real part of Φeff :Re{Φeff} = Φr,R(1− CR) + Φr,ICI (A.25)Inspecting Equation A.25 shows the influence of the real and imaginary partsof the controller on the real part of the effective FRF. Some general designprinciples can be extracted:• If CR ≈ 1 for frequencies near the chatter frequency, then the magni-tude of the minimum negative real part will be reduced by bringingΦr,R(1− CR) close to zero.152• If CI ≈ 0 near the chatter frequency, then the second term will alsobe brought near zero.• Alternatively, if the sign of Φr,I near the chatter frequency is known,then selecting a CI near the chatter frequency such that sign(Φr,I) =sign(CI), then additional stability can be added, as this will enforceΦr,ICI > 0.Many controllers that satisfy the list of design principles above may exist.One possible option is the 45o Lag controller design, given below, where ωcrepresents the chatter frequency in rad/s:C(s) = e−sTc , Tc = 45oπ180o1ωc(A.26)The real and imaginary parts of this controller are plotted in Figure A.3below, with the chatter frequency indicated as a vertical line. Observe thatthe first and third design principles listed above are satisfied. Finally, theresulting original FRF Φr and effective FRF with control Φeff are plottedin Figure A.4. It can be observed that the minimum negative real part isdramatically improved with the addition of the 45o lag controller.153Figure A.3: Example controller FRFs for the follower and 45o lag controller.The workpiece natural frequency is indicated.It is also noted that the 45o Lag controller has some advantages over thefollower controller, as the system does not need to react instantaneously, andany delays inherent to the control system can be folded into the controller.The amount of delay in the feedback line, Tf [s], can be measured, andthe controller can be tuned as in Equation A.27 so that the total delayexperienced by the signal matches the 45o target:C(s) = e−s(Tc−Tf ) (A.27)154Figure A.4: Example single-degree-of-freedom effective FRFs, with no con-trol, follower control, and 45o lag control. Both controllers raise the mini-mum negative real partA.1.3 Active Control ChallengesThough active control has significant power to alter the effective dynamicsof the cutting process, it faces a number of hurdles to successful practicalimplementation:• Accurately Measuring or Estimating Relative Displacement:The controller designs discussed in Section A.1 rely on the measure-ment of the relative displacement between tool and workpiece, xt−xw,at the point of contact. In practice, this measurement is very challeng-ing to acquire.– Optical Displacement Measurements: Using an optical mea-surement system is highly susceptible to the optical path beingoccluded by flying chips or coolant. Additionally, any part runout155or cylindricity error is interpreted as relative displacement andsend as a command to the actuator, decreasing part accuracy.Finally, an optical position cannot measure exactly where thetool is currently cutting, and must measure the workpiece somedistance away from the tool, introducing some small potentialerrors in displacement estimate.– Force-Based or Hybrid Observers: A force-based displace-ment observer could be constructed using an in-line load cell inthe tool, however this approach requires knowing the cutting co-efficients and width of cut in order to convert load into displace-ment.• Electrical Noise: Using active control to actively drive the tool tipmotion means that any ambient electrical noise in the room that cou-ples into the sensor signals is then potentially transmitted to the actu-ator command and left as a physical imprint in the part surface finish.This can be somewhat mitigated with strong filtering, at the expenseof some controller bandwidth.• Sampling Rate Requirements: System sampling rates must be inthe range of >10-100x the target chatter frequency, depending on thetype of controller. This limits the maximum possible chatter frequencythat can be stabilized with this approach.• Hardware Nonlinearity / Rate Limits: Structural machine toolvibrations generally reside in a frequency range of 10-1000 Hz, withresonances in the 100’s of Hz being common. Driving an actuatorback and forth at these frequencies means high velocities and very highaccelerations, which may be limited by power amplifier slew rates orother nonlinearities.156
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Altintas, Yusuf, 1954-
Compliant tool modelling and design for chatter suppression in highly flexible turning
Text