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Design, control and cutting process for a three-degree-of-freedom ultrasonic vibration tool holder Gao, Jian 2020

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Design, Control and Cutting Process for aThree-Degree-of-Freedom Ultrasonic Vibration ToolHolderbyJian GaoB.Sc., Shanghai Jiao Tong University, 2011M.Sc., Shanghai Jiao Tong University, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)March 2020c© Jian Gao, 2020The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Design, Control and Cutting Process for a Three-Degree-of-FreedomUltrasonic Vibration Tool Holdersubmitted by Jian Gao in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Mechanical Engineering.Examining Committee:Dr. Yusuf Altintas, Mechanical Engineering, UBCSupervisorDr. Ryozo Nagamune, Mechanical Engineering, UBCSupervisory Committee MemberDr. Xiaodong Lu, President, Planar Motor Inc.Supervisory Committee MemberDr. John Madden, Electrical and Computer Engineering, UBCUniversity ExaminerDr. Mu Chiao, Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Dr. Juri Jatskevich, Electrical and Computer Engineering, UBCiiAbstractUltrasonic vibration-assisted cutting is a popular unconventional manufacturingprocess with lower cutting forces and less heat generation. Special tools are re-quired to excite high-frequency vibrations at the tool tip during cutting; however,there is no ultrasonic vibration actuated tool holder for general-size milling ordrilling tools reported in the literature. This thesis presents the design of a novelthree-degree-of-freedom (3DOF) ultrasonic vibration tool holder with a sensor-less control system. In addition to proposing a mechatronics design, this thesispresents the cutting dynamics and mechanics exhibited by the developed vibrationtool holder.The 3DOF ultrasonic vibration tool holder is designed for milling and drillingoperations. 3DOF vibrations are generated by the actuator consisting of threegroups of piezoelectric rings actuating in the X-, Y -, and Z-directions at the naturalfrequencies of the structure. The vibrations excited in the XY produce an ellipticallocus to assist milling process. The vibrations along Z-axis are used in drillingoperations.A sensorless method is developed to track and control the frequency and am-plitude of ultrasonic vibrations produced by the 3DOF vibration tool holder duringmachining. A dynamic model of the actuator is first established to obtain a transferfunction between the supply voltage and driving current. An observer with Kalmanfilters in each actuator direction is designed to estimate the vibrations during cut-ting to closed-loop control the amplitude and track the resonanceThe dynamics of the ultrasonic elliptical vibration-assisted milling operationsis analyzed to assess the system stability. The chip thickness is modeled by con-sidering the rigid body motion of the tool, regenerative vibration and ultrasoniciiivibration. The loss of contact between the tool and workpiece at the ultrasonicvibration excitation frequency is considered in evaluating the directional factors.The stability of the system is solved using the semi-discrete time-domain methodand verified experimentally.The effects of ultrasonic vibration assistance in cutting of Ti-6Al-4V are inves-tigated. A plastic chip flow model is developed to predict the stress and temperaturevariations in the primary shear zone. Simulation results show that the temperaturein vibration-assisted cutting is much lower than that for conventional cutting.ivLay SummaryThe aerospace/aviation and biomedical industries demand high-performance cut-ting tools with longer tool life and better surface finish in cutting of difficult-to-cutmaterials such as carbon-fiber-reinforced polymer and titanium alloys. Investiga-tions have shown that ultrasonic vibration-assisted cutting processes have lowercutting forces and temperatures, with longer expected tool life.This thesis expands the applications of ultrasonic vibration-assisted cutting tomilling and drilling by developing a novel three-degree-of-freedom ultrasonic vi-bration tool holder and a corresponding sensorless control system. In addition topresenting a cutting tool design, this thesis investigates the dynamics of ultrasonicvibration-assisted milling. Moreover, the effects of ultrasonic vibration in the cut-ting of Ti-6Al-4V are analytically studied. The research presented in this thesis willbenefit the manufacturing industry in machining advanced hard-to-cut material.vPrefaceThis thesis presents the design, control and cutting process of a novel three-degree-of-freedom (3DOF) ultrasonic vibration tool holder. This research was performedin Manufacturing Automation Laboratory at The University of British Columbiaunder the supervision of Dr. Yusuf Altintas. The contributions of each chapter areas follows:• A version of Chapter 3, which focuses on the mechatronics design of a 3DOFultrasonic vibration tool holder, has been published in [24] [J. Gao and Y. Al-tintas. Development of a three-degree-of-freedom ultrasonic vibration toolholder for milling and drilling. IEEE/ASME Transactions on Mechatron-ics, 24(3):1238-1247, June 2019.]. The author of this thesis was the leadinvestigator and was responsible for the development of the tool holder andits corresponding instrumentation, including a digital controller, conditionercircuits and power amplifiers. Dr. Altintas supervised the project and editedthe manuscript.• A version of Chapter 4, which presents a sensorless control system designfor the 3DOF ultrasonic vibration tool holder, has been published in [26][J. Gao, H. Caliskan, and Y. Altintas. Sensorless control of a three-degree-of-freedom ultrasonic vibration tool holder. Precision Engineering, 58:47-56, 2019.]. The manuscript was written by the author of this thesis and wasedited by Dr. Caliskan and Dr. Altintas. In addition to editing the paper, Dr.Caliskan contributed to the state-space model of periodical signals used toformulate Kalman filters for the phase estimations from the supply voltageto the transforming current and the vibration• A version of Chapter 5, which investigates the dynamics and stability ofthe elliptical vibration-assisted milling process, has been accepted by theCIRP Journal of Manufacturing Science and Technology [J. Gao, Y. Altintas.Chatter stability of synchronized elliptical vibration assisted milling.]. Theauthor of this thesis was the lead author and was responsible for the modelof the elliptical vibration assisted milling dynamics and deriving a stabilitysolution using the semi-discrete method. Dr. Altintas supervised the projectand edited the manuscript.• A version of Chapter 6, which describes the effect of ultrasonic vibration onchip formation in the cutting of Ti-6Al-4V, has been published in [25] [J.Gao and X. Jin. Effects of ultrasonic vibration assistance on chip formationmechanism in cutting of Ti-6Al-4V. Journal of Manufacturing Science andEngineering, 141(12), 10 2019.]. The author of this thesis was the lead in-vestigator and was responsible for analytically modeling the cutting mechan-ics of Ti-6Al-4V under ultrasonic vibration and for conducting experiments.Dr. Jin edited the manuscript. Moreover, the cutting mechanics model of Ti-6Al-4V proposed in Chapter 6 has been applied to milling process to predictcutting forces, and a paper regarding ultrasonic vibration-assisted milling ofTi-6Al-4V has been submitted as [X. Jin, J. Gao, Y. Altintas. Mechanics ofElliptical Vibration Assisted Milling of Titanium Ti-6Al-4V. CIRP Annals-Manufacturing Technology.]. The author of this thesis was responsible forthe model development and conducting experiments.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Ultrasonic Vibration Cutting Tool Design . . . . . . . . . . . . . 72.3 Control for the Ultrasonic Vibration Actuator . . . . . . . . . . . 102.4 Ultrasonic Vibration-Assisted Milling Process and Its Dynamics . 112.5 Mechanics of Vibration-Assisted Cutting Ti-6Al-4V . . . . . . . . 13viii3 3DOF Ultrasonic Vibration Tool Holder Design . . . . . . . . . . . . 173.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Mechanical Structure and Mode Shapes . . . . . . . . . . . . . . 183.3 Lumped Parameter Model of the Actuator . . . . . . . . . . . . . 203.4 Finite Element Model of the Actuator . . . . . . . . . . . . . . . 253.5 Prototype Manufacturing . . . . . . . . . . . . . . . . . . . . . . 263.6 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Sensorless Control of 3DOF Ultrasonic Vibration Tool Holder . . . 344.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Principle of Frequency and Amplitude Control . . . . . . . . . . . 344.3 Dual-loop Control System . . . . . . . . . . . . . . . . . . . . . 374.4 Verification of Control System . . . . . . . . . . . . . . . . . . . 424.5 Two-axis Ultrasonic Vibration Control . . . . . . . . . . . . . . . 474.6 Cutting Experiments . . . . . . . . . . . . . . . . . . . . . . . . 504.6.1 Drilling tests . . . . . . . . . . . . . . . . . . . . . . . . 504.6.2 Milling tests . . . . . . . . . . . . . . . . . . . . . . . . 514.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 Dynamics of Vibration-Assisted Milling . . . . . . . . . . . . . . . . 555.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Milling Process with Elliptical Vibration . . . . . . . . . . . . . . 555.2.1 Chip generation with elliptical vibration assistance . . . . 555.2.2 Time delay in elliptical vibration-assisted milling . . . . . 595.2.3 Tangential tool-workpiece separation . . . . . . . . . . . 615.2.4 Process damping effect . . . . . . . . . . . . . . . . . . . 635.3 Chatter Stability Analysis . . . . . . . . . . . . . . . . . . . . . . 655.3.1 Dynamic milling force . . . . . . . . . . . . . . . . . . . 655.3.2 Stability of milling with ultrasonic vibrations . . . . . . . 665.4 Simulations and Experiments . . . . . . . . . . . . . . . . . . . . 695.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73ix6 Vibration Assistance on Chip Formation of Ti-6Al-4V . . . . . . . . 756.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 Experimental Observations . . . . . . . . . . . . . . . . . . . . . 756.3 Plastic Flow in Primary Shear Zone . . . . . . . . . . . . . . . . 796.4 Simulations and Experiments . . . . . . . . . . . . . . . . . . . . 876.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97xList of TablesTable 2.1 Summary of ultrasonic vibration cutting tool design . . . . . . 9Table 3.1 Piezoelectric material parameters . . . . . . . . . . . . . . . . 23Table 3.2 Lumped parameters . . . . . . . . . . . . . . . . . . . . . . . 23Table 5.1 Modal parameters . . . . . . . . . . . . . . . . . . . . . . . . 70Table 6.1 Conditions of orthogonal cutting experiments . . . . . . . . . 77Table 6.2 Ti-6Al-4V material properties . . . . . . . . . . . . . . . . . . 87Table 6.3 Ti-6Al-4V Johnson-Cook parameters . . . . . . . . . . . . . . 88xiList of FiguresFigure 1.1 Research flowchart . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Research flowchart . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.3 UBC 3DOF ultrasonic vibration tool holder . . . . . . . . . . 4Figure 1.4 UBC 3DOF ultrasonic vibration tool holder on CNC machin-ing center . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 3.1 Conceptual design of 3DOF ultrasonic vibration tool holder. . 18Figure 3.2 Final design of the 3DOF ultrasonic vibration tool holder . . . 19Figure 3.3 Equivalent circuit of piezoelectric actuator for each axis. . . . 21Figure 3.4 Impedance frequency response functions of piezoelectric actu-ator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.5 Mode shapes of the 3DOF vibration actuator in FE simulation 25Figure 3.6 Prototype of 3DOF ultrasonic vibration tool holder . . . . . . 27Figure 3.7 Structural frequency response functions . . . . . . . . . . . . 28Figure 3.8 Slip rings assembly . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.9 Electronic system for X vibration. . . . . . . . . . . . . . . . 30Figure 3.10 Vibrations outputs from the laser vibrometer for X, Y and Z. . 32Figure 4.1 Linear lumped model of the ultrasonic actuator. . . . . . . . . 35Figure 4.2 Dual-loop ultrasonic vibration control system . . . . . . . . . 38Figure 4.3 Closed-loop frequency response functions . . . . . . . . . . . 43Figure 4.4 Turning setup with a laser vibrometer . . . . . . . . . . . . . 44Figure 4.5 Measured transforming gains (ψ) as a function of referencetracking phase (ϕr). . . . . . . . . . . . . . . . . . . . . . . 45xiiFigure 4.6 Single-axis ultrasonic vibration during turning . . . . . . . . 46Figure 4.7 Transforming gains with different feed rates. The turning spin-dle speed is 1500 RPM, and the width of cutting is 1.5 mm. . 47Figure 4.8 Measured FRFs along X and Y when clamping an 8 mm millingtool with 4 flutes . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 4.9 Two-axis ultrasonic vibration crosstalk observer . . . . . . . . 49Figure 4.10 Elliptical loci generated by two-axis ultrasonic vibration con-troller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 4.11 Amplitudes of vibration velocities in drilling tests. The spin-dle speed is 1500 RPM, and the feed rate is 0.1 mm/rev. Theworkpiece material is aluminum alloy 7050. . . . . . . . . . 51Figure 4.12 Thrust forces along axial direction during drilling. The spin-dle speed is 1500 RPM, and the feed rate is 0.1 mm/rev. Theworkpiece material is aluminum alloy 7050. . . . . . . . . . 52Figure 4.13 Milling tests with 3DOF ultrasonic vibration tool holder . . . 54Figure 5.1 Elliptical vibration-assisted milling process. . . . . . . . . . . 57Figure 5.2 Dynamic chip thickness. . . . . . . . . . . . . . . . . . . . . 58Figure 5.3 Time delay for two cases . . . . . . . . . . . . . . . . . . . . 61Figure 5.4 Tangential resultant velocity at tool tip. . . . . . . . . . . . . 62Figure 5.5 Process damping effect . . . . . . . . . . . . . . . . . . . . . 64Figure 5.6 3DOF ultrasonic vibration tool holder on CNC machine . . . 69Figure 5.7 Frequency response functions of the milling tool . . . . . . . 70Figure 5.8 Stability lobes for half immersion up milling of Aluminum . . 71Figure 5.9 Microphone outputs in half-immersion up milling of AL 7050 72Figure 5.10 Stability lobes for half immersion down milling of Steel . . . 73Figure 5.11 Microphone outputs in half-immersion down milling of AISI1045 steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 6.1 Turning experimental setup with an ultrasonic vibration cut-ting tool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 6.2 SEM images of segmented chips . . . . . . . . . . . . . . . . 78xiiiFigure 6.3 SEM images of segmented chips under different cutting con-ditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 6.4 Orthogonal cutting with ultrasonic vibration along tangentialdirection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 6.5 Deformation zone in 2D orthogonal cutting. . . . . . . . . . . 82Figure 6.6 Steady-state shear stress and temperature in the primary shearzone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 6.7 Comparison of pitch lengths between simulations and experi-ments with different cutting speeds and uncut chip thicknesses(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 6.8 Comparison of tangential cutting forces between simulationsand experiments . . . . . . . . . . . . . . . . . . . . . . . . 93xivList of SymbolsA(z) Cross-section area of piezoA0, B0 Vibration amplitudesApdj Effective area for process damping of jth tootha j(t),b j(t) Vibration displacements of jth tooth along tangential and radial directionsb Width of cut in Chapter 6C0, Cd Static capacitance of piezoCm Mechanical equivalent capacitancec, cx, cy Damping coefficientd33 Piezoelectric strain constant along the thickness directionE Young’s modulusFext External forceFf r Friction forceFn Normal thrust forceFr Milling force along radial directionFS Shear forceFt Transforming force in Chapter 4; tangential cutting force in Chapter 5 and 6F pdr , Fpdt Process damping forcesf Excitation frequencyfa Anti-resonance frequencyfe Excitation frequencyfr Resonance frequencyg1, j, g2, j, g3, j,g j Unit step function for tool-workpiece disengagementh0, j jth tooth static uncut chip thicknessxvhr, j Resultant uncut chip thickness of jth toothI Area momentum of inertia in section 3.2; Driving currentIt Transforming currentKM Electromechanical coupling coefficientKt , Kr Cutting force coefficientsKsp Indentation coefficientsk Stiffness in Chapters 3, 4 and 5; thermal conductivity in Chapter 6kx, ky StiffnessL Length of the piezo actuatorLm Mechanical equivalent inductancem MassN Number of flutesNA Vibration magnification factorNT Number of teethq(t) State functionRe Tool edge radiusRm Mechanical equivalent resistancer˙ j Velocity along radial direction for jth toothsr, j Resultant displacement of jth toothT Tooth passing period in Chapter 5; temperature in Chapter 6T0 Room temperatureTm Melting temperatureu(z, t) Displacement of bending modeV , Vx0, Vy0, V0 VoltageVS Nominal cutting speedVr Resultant cutting speedw Vibration displacement in Zx, x˙ Vibration displacement and velocity along Xx0 Vibration amplitude along Xxˆ Model displacement vectorY33 Young’s modulus of piezo along the thickness directionα Rake angleβe Separation anglexviβk Wave coefficient of kth bending modeε Normal strainγ , γp Shear strainγ˙ Shear strain rateµ Friction coefficientψ Force factor of piezo, transforming gainψxx,ψyy,ψxy,ψyx,ψz Transforming gainsρ Mass densityΩ Spindle rotation speedωn Natural frequencyωe Excitation frequencyϕ Phase from voltage to currentφ Shear angleφ(t) Immersion angleφ j Immersion angle of jth toothφp Tooth spacing angleσ Normal stressζ Damping ratioθ , θxy Phase difference between x and y vibrationsτ j Time delay of jth toothτs Shear stress in the primary shear zonexviiList of AbbreviationsAC Alternating currentAD Amplitude detectorADC Analog-to-digital converterCFRP Carbon-fiber reinforced plasticsCNC Computer numerical controlDAC Digital-to-analog controllerDOF Degree of freedomFE Finite elementFFT Fast Fourier transformFRF Frequency response functionMRR Material removal ratePD Phase detectorPI Proportional integralPLL Phase lock loopPWM Pulse-width modulationRPM Revolution per minuteSEM Scanning electron microscopeSP Separation pointVCA Voltage controlled amplifierxviiiAcknowledgmentsFirst and foremost, I want to thank my supervisor, Prof. Yusuf Altintas, for hisunconditional support throughout my Ph.D. years. He offered me a chance to con-tinue my studies as a Ph.D. student at Manufacturing Automation Laboratory whenI was in my low moments. Moreover, his insights and passion for manufacturingresearch always encourage me to pursue further achievements, and this experienceis a lifetime of wealth for me.I want to thank Dr. Xiaodong Lu, who brought me to UBC five years ago tolet me realize the world-class research and gave me solid fundamentals of mecha-tronics. I also want to thank Dr. Xiaoliang Jin and Dr. Hakan Caliskan, who weremy collaborators, to expand my research scope. I remember productive discus-sions with Dr. Caliskan about Kalman filters when I worked on the control systemof the ultrasonic vibration tool holder. Dr. Jin’s abundant knowledge and experi-ence about manufacturing and mechanics brought a chance for me to investigatethe effects of ultrasonic vibration in the cutting of Ti-6Al-4V.I thank all members of the Manufacturing Automation Laboratory and the Pre-cision Mechatronics Laboratory. In the family-like atmosphere, I never hesitatedto ask any questions, and they always offered help and provided guidance.I thank my family for their support and encouragement throughout my life.Most importantly, I want to thank my wife, Lu Pan. I cannot make my life this fartowards the Ph.D. without her. She does provide not only unquestioned support inlife but also the first reader of my technical manuscripts.xixChapter 1IntroductionUltrasonic vibration-assisted cutting is an unconventional manufacturing process.The cutting tool is forced to vibrate at a high frequency (≥ 15 kHz) with a small am-plitude (10-20 µm). Ultrasonic vibration-assisted cutting has been demonstratedto reduce cutting forces and heat generation in comparison to conventional ma-chining operations. As a result, ultrasonic vibration-assisted cutting has becomea popular method in industry, particularly in machining of carbon-fiber reinforcedplastics (CFRP), which is widely used in light-weight, high-strength automotiveand aircraft parts [49, 96]. Ultrasonic vibrations are also widely applied in pre-cision manufacturing of hardened steel molds with mirror surface finishes [79].However, to finish ultrasonic vibration-assisted cutting process, special tools withvibration generations at tool tip are required, which may increase the cost of themanufacturing process.The fundamental feature of the ultrasonic vibration-assisted cutting process isto generate high frequency (≥15 kHz) vibration at the cutting tool tip in orderto make additional intermittent contact between the cutting tool and workpiece.The ultrasonic vibration assistance can be generated along the tangential directionof cutting process and the normal direction, as shown in Figure 1.1 a) and b).Alternatively, the vibration assistance can be generated in both tangential and radialdirections simultaneously to form an elliptical vibration locus to change the frictionbetween chip and tool’s rake face as shown in Figure 1.1c).Piezoelectric actuators are commonly used in ultrasonic vibration-assisted cut-1Vcvibration assistancecutting toolchipVcchipVcchipa) b) c)Figure 1.1: Research flowchartting to deliver vibrations with desired frequencies and amplitudes. These actuatorsgenerally resonate at their natural frequencies to increase the vibration amplitudeat high frequencies. Various ultrasonic vibration-assisted cutting tools have beendeveloped for different machining processes. In the past, single-axis vibration andtwo-dimensional elliptical vibration systems have been investigated. Ultrasonicvibration assistance was first applied to turning operations, which require a sta-tionary cutting tool, due to their lower-complexity vibration tool design [56, 76].Later, two-dimensional elliptical vibration cutting was invented by Shamoto andMoriwaki [78], and then it was applied at ultrasonic vibration [58]. The ellipticalvibration was applied to the turning process in which a mirror surface is appliedto a hardened steel mold. Machining processes using rotary cutting tools, suchas milling and drilling can reach higher productivity for mass production. Thus,machine tool companies have developed tool holders to generate ultrasonic vibra-tions along the rotating spindle axis for rotary cutting operations. For example,DMG MORI developed an ultrasonic vibration-assisted machining center with vi-brations along the spindle axis for grinding metals and drilling CFRP materials[76]. Acoustech Systems presented a single axis ultrasonic vibration tool holderthat is compatible with standard spindle mounting systems (HSK and CAT) fordrilling applications [56]. Thus far, most research on two-dimensional ellipticalvibration cutting has been performed with stationary cutting tools due to the lim-itations of vibration generation devices, and ultrasonic vibration-assisted rotaryprocesses have been primarily studied for single-axis vibration devices.As shown the research flow chart in Figure 1.2, the first aim of this thesis is to2V(t)Cd vt+-It=ψxI(t)Electro-meachnical system control (Chapter 4)ψFt=ψVmkbFext x3DOF Ultrasonic vibration tool holder design(Chapter 3)C1XYRWorkpieceΩVibration locusMilling toolA0B0Dynamics of elliptical vibration assisted milling(Chapter 5)cuttingtoolστφVprimary shear zonecu tintooltool-workpiece contact tool-workpiece separateVibration assistance on chip formation of Ti-6Al-4V(Chapter 6)Figure 1.2: Research flowchartdevelop a three-degree-of-freedom (3DOF) ultrasonic vibration cutting tool holderfor both milling and drilling operations using general-purpose cutting tools. Theproposed holder has a standard mounting interface (HSK or CAT) for CNC (com-puter numerical control) machines; the holder can deliver vibrations in the XYplane to assist milling along the tangential and radial directions and generates vi-brations along the Z-axis for drilling applications. An actuator with three groupsof piezoelectric rings is designed to create vibrations along the three axes, respec-tively. The vibration mode shapes are selected to maximize the amplitudes, and thepiezoelectric components are then parametrically designed using a lumped param-eter model. Finite element analysis (FEA) is used to verify the natural frequenciesand the mode shapes before the holder was manufactured. The prototype of the3DOF ultrasonic vibration tool holder shown in Figure 1.3 was tested with cuttingexperiments. In addition to the actuator design, a corresponding electrical system,including power amplifiers, output signal-conditioning circuits and current sensing3Figure 1.3: UBC 3DOF ultrasonic vibration tool holdercircuits, are developed.The ultrasonic vibrations are produced by resonating the natural modes of thetool holder structure. However, the structural dynamics of the piezo-driven toolholder structure vary under cutting loads, which must be detected to resonate thestructure. The vibration amplitude at ultrasonic frequencies also decreases undercutting load disturbance, which must be maintained at the desired reference level.Conventional closed-loop control methods reported in the literature use vibrationsensor feedback to track the frequency and amplitude of the ultrasonic vibrationsduring cutting with additional piezo stacks or strain gauges mounted on the ac-tuator’s surface. The vibration feedback is used to track the resonance and tocontrol the vibration amplitude; moreover, the feedback can eliminate crosstalkarising between orthogonal piezo actuators in elliptical vibration by compensatingthe excitation voltage [80]. However, these dedicated vibration sensors increasethe cost of the tool and the complexity of signal transmission during rotary cuttingapplications. Therefore, sensorless control methods were implemented to track theresonance frequency and keep desired amplitudes. In contrast to existing methods,a new sensorless closed-loop control method, which can be adaptive to differentcutting tool structural dynamics, is proposed for the developed 3DOF ultrasonic vi-bration tool holder to track the resonance frequencies and amplitudes of ultrasonic43DOF ultrasonic vibration tool holderslip ringsworkpieceFigure 1.4: UBC 3DOF ultrasonic vibration tool holder on CNC machiningcentervibrations during machining. In the control system, the feedbacks are estimatedfrom a dynamic model of the piezo actuator using Kalman filters, and the crosstalkarising in elliptical vibrations is eliminated. This control system was implementedduring cutting using the experimental setup shown in Figure 1.4 for milling anddrilling operations.For the effective operation of the ultrasonic vibration tool, the dynamics of thevibration-assisted process must be modeled and analyzed. Chatter, which is a typeof regenerative vibration that can lead to unstable cutting, should be avoided dur-ing operation. Thus, a dynamics model of elliptical vibration-assisted milling isproposed in the third part of this thesis. The chatter stability is predicted from theproposed dynamics model to select the desired spindle speed and depth of cut. Inmilling, the developed 3DOF ultrasonic vibration tool holder can deliver vibrationsalong the tangential and radial directions at the same excitation frequency to pro-duce an elliptical locus. In the dynamics model, the tangential vibration generateshigh-frequency tool-workpiece separations, which change the dynamic milling co-5efficients and thus affect the milling stability. Additionally, the radial vibrationaffects the indentation between the tool and workpiece periodically, which altersthe process damping and the milling stability.Ultrasonic vibration assistance has been experimentally demonstrated in cut-ting of various materials such as hardened steel, titanium alloy, CFRP, and glass.However, a analytical physical model elucidating the mechanisms of vibration as-sistance has not yet been established in the literature. The mechanism of vibrationassistance may vary for different materials. In this thesis, a titanium alloy, Ti-6Al-4V, is studied as an example to demonstrate the principle of ultrasonic vibrationassistance. The effect of ultrasonic vibration assistance on shear band formationand chip segmentation in orthogonal cutting of Ti-6Al-4V is investigated. A chipflow model is developed to determine the shear flow mechanism in the primaryshear zone and to explain the suppression of adiabatic shear bands when ultrasonicvibration assistance is applied along the tangential cutting direction. The chip ge-ometries and cutting forces predicted by the proposed analytical model are verifiedby orthogonal cutting experiments.The remainder of this thesis is structured as follows. Chapter 2 reviews previ-ous literature pertaining to ultrasonic vibration actuator design, resonance and am-plitude tracking, chatter stability of vibration-assisted cutting processes and modelsof Ti-6Al-4V cutting. The 3DOF ultrasonic vibration actuator design is presentedin Chapter 3, with performance test results obtained for a prototype. A sensorlesscontrol system of the developed 3DOF ultrasonic vibration tool holder is presentedin Chapter 4. Chapter 5 proposes a dynamics model of ultrasonic vibration-assistedmilling to predict the stability of the developed tool holder. Chapter 6 presents theeffects of ultrasonic vibration assistance in cutting of Ti-6Al-4V with a physicalmodel of chip formation. The shear flow in the primary shear zone is analyzedbased on the experimental observations to illustrate the advantages of vibrationassistance. The conclusions obtained from this thesis are presented in Chapter 7along with a discussion of future work.6Chapter 2Literature Review2.1 OverviewThis thesis focuses on the design and control of a novel 3DOF ultrasonic vibrationcutting tool, the dynamics of the elliptical vibration-assisted milling process, andthe mechanics of vibration-assisted cutting of Ti-6Al-4V. This chapter reviews pastresearch related to these areas. Section 2.2 presents the state of the art in ultrasonicvibration cutting tool design, and section 2.3 discusses existing control methodsof ultrasonic vibration during cutting. Milling dynamics models and the studiesrelated to the stability of vibration-assisted cutting are reviewed in section 2.4.Section 2.5 reviews cutting mechanics models of Ti-6Al-4V, with a focus on theformation of adiabatic shear bands, and presents experimental studies on vibration-assisted cutting of Ti-6Al-4V.2.2 Ultrasonic Vibration Cutting Tool DesignSingle-axis and two-dimensional elliptical vibration systems have been investi-gated in both academia and industry. DMG MORI developed an ultrasonic vibration-assisted machining center with vibration along the spindle axis for grinding met-als and drilling CFRP materials [76]. Recently, Acoustech Systems presented asingle-axis ultrasonic vibration tool holder that is compatible with standard spindlemounting systems (HSK and CAT) for drilling applications [56, 84]. The effect of7ultrasonic vibrations on machining performance has been experimentally investi-gated by several researchers using different vibration cutting tools. For example,Li et al. presented the drilling performance of ceramic matrix composites witha single-axis rotary ultrasonic vibration assistance [46], with an observed reduc-tion in cutting forces and a better surface finish. Liu et al. studied single-axisrotary ultrasonic machining for drilling of brittle materials to achieve larger mate-rial removal rates (MRRs) [47]. Lucas et al. used two full ring piezos to excite thecoupled longitudinal-torsional vibration mode of a drill for ultrasonic vibration-assisted drilling [3]. Jin and Murakawa developed a one-dimensional ultrasonicvibration-assisted turning tool based on the Langevin transducer, which was thenused to improve the surface roughness in the turning of hardened steel [34]. Xiaoet al. reported the suppression of chatter during turning by using a single-axis ac-tuator to achieve intermittent contact between the cutting tool and the workpiece[95].Elliptical vibration cutting was proposed by Shamoto and Moriwaki [78] formirror surface machining of hardened steel dies and molds; in this work, theymounted four piezoelectric plates on the surface of the tool to generate an ellipti-cal vibration locus [58]. Later, Shamoto et al. developed a controller for ellipticalultrasonic vibration cutting [80]. The fundamental objective behind imposing el-liptical vibrations on the tool is to reduce the friction force, which in turn reducesthe heat generation and the tool wear [82]. With this approach, carbon-rich dia-mond tools can cut carbon-hungry steel alloys due to the significant reduction inheat. It has been found that the main reason for the tool wear reduction is a pro-tective oxide film formed on the newly-developed surface during the non-contactperiod[22]. Shamoto et al. expanded upon the elliptical ultrasonic vibration assis-tance to develop a 3DOF ultrasonic vibration tool to sculpture free-form surfaceswith a specially designed diamond cutting tool [81][87]. This tool holder applies3DOF ultrasonic vibration for ultra-precision shaving of hardened steel dies, wherethe cutting force amplitudes are much smaller (approximately 2-5 N) than those ob-tained in regular machining applications. Jung et al. used a stationary ultrasonicvibration transducer with ring-type piezo stacks to excite both axial and bendingmodes of the structure for shaping applications [38]. The elliptical vibration cut-ting was first applied to micro texturing of die steel [88]. Later, Guo and Ehmann8Table 2.1: Summary of ultrasonic vibration cutting tool designManufacturing process SDOF assistance 2DOF assistanceTurning [34, 95] [58, 80, 81]Drilling [3, 46, 47, 56, 84] [49]Milling [76]Grinding [76] [27]Shaping [38]Texturing [20, 29, 98]developed an elliptical vibration generator combining two Langevin transducersfor high speed texturing by imposing elliptical vibrations on the tool [29]. Yang etal. analytically designed an elliptical vibration tool based on a portal frame struc-ture using the Euler-Bernoulli beam model for surface texturing processes [98].Dow et al. excited the axial and bending modes of a turning tool at 40 kHz withpiezo plates to generate features in nanocoining processes [20]. Kim and Loh de-signed and implemented an elliptical vibration tool with two parallel piezoelectricactuators for micro grooving and observed a reduction in cutting forces and burrs[41]. Liu et al. developed an elliptical ultrasonic vibration device for CFRP drillingwith diamond abrasive core drills and reported better surface finish with less de-lamination [49]. Geng et al. proposed a 2DOF ultrasonic vibration cutting tool forperipheral grinding of CFRP using half-ring piezos [27].Table 2.1 summarizes the existing ultrasonic vibration cutting tool designs fordifferent cutting processes. Due to the limitations of existing ultrasonic vibrationgeneration devices, most research on two-dimensional elliptical vibration cuttinghas been performed with stationary tools, such as turning, texturing and shapingtools. For rotary cutting processes, studies on ultrasonic vibration assistance havefocused on single-axis vibration devices, except for the works described in [49]and [27]. Moreover, the existing rotary ultrasonic vibration generation devices aredesigned for specific applications. In this thesis, a novel 3DOF ultrasonic vibrationtool holder is presented in Chapter 3, which can be used for milling and drillingwith various cutting tools.92.3 Control for the Ultrasonic Vibration ActuatorDuring vibration-assisted cutting, the ultrasonic vibrations are produced by res-onating the natural modes of tool holder structure. However, the structural dynam-ics of the piezo driven tool holder structure vary under cutting loads, which need tobe detected to resonate the structure. The amplitude of the vibration at ultrasonicfrequency also reduce under cutting load disturbance, which need to be maintainedat the desired, reference level. The conventional close-loop control methods usesensor feedback to track the frequency and amplitude of the ultrasonic vibrationsduring cutting. Shamoto et al. used two more piezo stacks mounted on the actua-tor’s surface as feedback sensors in elliptical vibration-assisted precision diamondturning [80]. The vibration feedback was used to track the resonance and controlthe vibration amplitudes, and they removed the crosstalk between the orthogonalpiezo actuators by compensating the excitation voltage with the sensor feedback.Babitsky et al. also developed an auto-resonant control with multiple sensors totrack the resonance during ultrasonic vibration cutting [6]. Later, Suzuki et al.sculptured the hardened steel surface by adjusting the supply voltage for open loopcontrol of the elliptical vibration amplitudes [88]. They controlled the vibrationamplitude within 4 µm along the thrust direction within 300 Hz bandwidth. Someresearchers focused on the control of rotary ultrasonic vibration tool in the liter-ature. Ketelaer installed a piezoelectric ring with four segments coaxially in anultrasonic vibration tool holder to detect vibrations in multiple directions duringcutting to monitor and control the tool holder [40]. Chen et al. bonded a straingauge on the ultrasonic vibration actuator surface to detect the vibration to trackthe resonance frequency, and they designed a pancake coil to transmit the sensoroutput from the rotary tool [15]. Industrial companies developed sensorless controlmethods for commercial ultrasonic vibration cutting tools. Taga Electrics devel-oped a vibration control system to track the resonance frequency using the supplyvoltage and driving current through electronic circuits for general ultrasonic vibra-tion apparatuses [30]. Later, Taga Electrics used the developed sensorless controlsystem in the ultrasonic vibration cutting process to track the resonance frequencyand monitoring the cutting process [31]. Short developed a closed-loop controlsystem to track the resonance for ultrasonic vibration assisted drilling process by10monitoring the power consumption in power amplifiers [85].The piezoelectric materials exhibit hysteresis and nonlinearities that dependon the driving conditions [28, 52], which affect vibration amplitude generated bythe piezo actuators. Newcomb et al. showed that the linearity of a piezoelec-tric ceramic actuator can be improved if the applied electric charge, rather thanthe applied voltage, is varied to control the actuator’s displacement [60]. Ryba etal. compensated the hysteresis of piezoelectric actuator using a polynomial model[71]. Woronko et al. used a sliding mode controller to handle the nonlinear gain ofthe actuator [94].The use of dedicated sensors in control systems for vibration-assisted cuttingtools increase the cost of the system and the complexity of signal transmissionduring rotary cutting applications. Additional slip rings or coils of the rotary trans-former are necessary for feedback signals transmission to the controller unit. Thus,this thesis proposes a sensorless closed loop control method in Chapter 4 to trackthe resonance frequency and amplitude of ultrasonic vibrations during machining.In addition to the single-axis controller, the crosstalk between two axes is compen-sated using the estimated feedback.2.4 Ultrasonic Vibration-Assisted Milling Process and ItsDynamicsVibration-assisted cutting has been applied to various cutting operations includingturning [79], shaping [29, 38, 88], grinding [27], and drilling [56, 69]. Milling isan intermittent-contact process in which each cutting tooth touches the workpieceperiodically within the entry and exit angles at the tooth-passing frequency. Theexternally applied ultrasonic vibrations generate additional, high-frequency tool-workpiece separations, which change the dynamics of the milling process. The vi-brations are either delivered through the workpiece or the cutting tool. Jin and Xiedeveloped a workpiece carrying piezo vibration stage to achieve vibration assis-tance along the feed or normal direction of milling with varying frequencies (5-11kHz) for micro-cutting of glass and reported an improved surface finish [36]. Chenet al. used a piezo vibration stage to generate synchronized vibrations along thefeed and normal directions at 8 kHz in a micro-milling process to produce micro-11texture patterns on the workpiece surface [16]. Ni et al. applied vibrations at 20kHz along the feed direction from the workpiece side and reported an improvedsurface finish and reduced cutting forces in the milling of Ti-6Al-4V [61]. It ispreferable to deliver vibrations from the tool side in order to machine large work-pieces with ultrasonic vibration assistance. Chen et al. applied ultrasonic vibrationalong the axial direction in helical milling of Ti-6Al-4V and reported reductionsin both the cutting forces and heat generation [14]. Elliptical vibration-assistedcutting was proposed by Shamoto and Moriwaki in which vibrations along the tan-gential and thrust directions are synchronized in diamond turning and milling [78].Then, it was applied to various workpieces at various cutting conditions afterwards.Moriwaki and Shamoto also developed an elliptical vibration milling machine us-ing an eccentric sleeve to deliver vibrations at 167 Hz and reported a better surfacefinish in the milling of hardened steel [59]. Liu et al. applied ultrasonic ellipticalvibrations along the tangential and radial directions in the milling of Ti-6Al-4V[50] and demonstrated that rotary ultrasonic elliptical vibration-assisted millingsignificantly improved the integrity of the machined surfaces. In this thesis, Gaoand Altintas developed an ultrasonic vibration tool holder that can vibrate indepen-dently in three orthogonal directions[24]. A group of half-ring piezos is used toexcite the third bending mode of the holder to generate ultrasonic vibrations alongthe tangential and radial axes in milling operations. The peak-to-peak amplitudeof vibration along the X and Y axes can reach 20 µm at 17 kHz, and the two vi-brations share the same excitation frequency with an adjustable phase difference toform an elliptical vibration locus. The excited elliptical vibration is synchronizedto the spindle rotation, and the tool holder can be installed on a CNC machiningcenter using standard spindle interfaces such as CAT or HSK.The dynamics of the vibration-assisted machining process must be modeled toidentify chatter-free cutting conditions. Although the chatter stability of conven-tional machining operations has been studied extensively in the literature [5, 33],the stability of milling under ultrasonic vibrations has not been sufficiently inves-tigated. In a previous work, Xiao et al. numerically and experimentally studiedthe chatter stability of a 1DOF vibration-assisted turning process and reported alarger depth of cut [95], but they did not predict the stability lobes of the process.Tabatabaei et al. applied a Fourier transform to a pulsating cutting force with a de-12lay of spindle rotation in turning and numerically predicted the stability in the timedomain [89]. Ma et al. excited ultrasonic elliptical vibration on a turning tool andanalyzed the chatter stability in the frequency domain using the Nyquist criteria[51], and they considered a periodic contact time of the actuator by using a Fourierseries, and applied a dynamic orthogonal cutting model to determine the stability.Ko and Tan implemented ultrasonic vibration along the spindle axis during millingand observed a reduction in chatter marks on the surface due to improved chatterstability [42]. Recently, Wan et al. provided the first analytical chatter stabilitymodel of a vibration-assisted milling process using the semi-discrete method [93].In Wan’s study, the vibration was excited along the feed or normal direction fromthe workpiece side, and the high-frequency tool-workpiece separations dynami-cally changed the time delay and hence shifted the stability lobes. Consequently,the authors concluded that the critical depth of cut was not improved.Thus far, no dynamic model has been reported in the literature for spindle-synchronized elliptical vibration-assisted milling processes. In Chapter 5, a modelof elliptical vibration-assisted milling dynamics is proposed to predict the stability,and the elliptical vibration assistance is verified to improve the depth of cut due toadditional high-frequency tool-workpiece separations, achieved by using the 3DOFultrasonic vibration tool holder developed in Chapter 3 and the corresponding con-trol system presented in Chapter 4.2.5 Mechanics of Vibration-Assisted Cutting Ti-6Al-4VThe titanium alloy Ti-6Al-4V is widely used in biomedical and aerospace appli-cations due to its light weight, high temperature strength and corrosion resistance.However, the low thermal conductivity of Ti-6Al-4V limits its machinability be-cause the heat accumulates in the primary shear zone generating adiabatic shearbands and leads to segmented chips in high-speed cutting [18]. Chip segmentationcauses stress and temperature oscillations at the tool edge, leading to faster toolwear and damage to the machined surface [8].Chip segmentation in the cutting of Ti-6Al-4V includes localized shear band-ing and severe fractures [83]. As Recht suggested [70], the adiabatic shear bandingcommonly occurs in high speed cutting of high-strength metal alloys such as hard-13ened steel, titanium, and nickel alloys. When the cutting speed exceeds a criticalvalue, there is not sufficient time for heat to dissipate in the primary shear zone, andconsequently, the thermal softening of the workpiece material overcomes the strainhardening. As a result, adiabatic shear bands form with an instantaneous reductionin shear flow stress. When new material enters the primary shear zone, the shearstress increases, and a new cycle of shear band formation occurs. This oscillation inshear stress causes variations in the material plastic flow in the primary shear zonethus generating segmented chips. Komanduri and Brown investigated machinedchips and reported that the plastic instability in the primary shear zone leads to adi-abatic shear bands, which forms segmented chips in the cutting of Ti-6Al-4V [43].Later, Komanduri and Hou noted that the temperature rise that occurs during thecutting of Ti-6Al-4V causes plastic instability [44], and the relationship betweenthe uncut chip thickness and the temperature in the shear band was determined.Burns and Davies studied the shear band instability by developing governing equa-tions of shear stress and temperature variations in the primary shear zone basedon material plasticity[11, 19], and they concluded that an increased cutting speedleads to a Hopf bifurcation; thus, the shear stress and the temperature in the primaryshear zone transition from constant to periodic steady-state behavior. Molinari etal. modeled the adiabatic shear banding mechanism in the cutting of Ti-6Al-4V,and predicted the width and pitch length of the shear bands [57].Researchers have also investigated the segmentation mechanism by consider-ing the occurrence of fractures in cutting of Ti-6Al-4V. Vyas and Shaw proposedthat chip segmentation may be caused by periodic crack propagation, based onquick-stop micrographs of machined chips [92]. FE methods have been widelyused in the literature to investigate the chip segmentation mechanism related to ma-terial failure under different criteria. Chen et al. implemented the Johnson-Cookconstitutive model to simulate high-speed cutting of Ti-6Al-4V while consideringthe ductile failure of the material [13]. Ozel et al. simulated chip segmentationin cutting of Ti-6Al-4V by using an FE method with a modified Johnson-Cookmodel [66], and the effect of insert coatings on the cutting process was determined.Melkote et al. considered microstructure evolution-induced flow softening in sim-ulations of chip segmentation and grain refinement in shear bands using an FEmethod [54]. Childs et al. performed FE simulations of chip segmentation us-14ing a material flow stress and failure model calibrated from experiments [17], andit was concluded that flow stress models involving large strain softening are notneeded for predicting chip segmentation. Zhang et al. considered stress triaxial-ity in the material failure criteria of Ti-6Al-4V using FE simulations to predict thetemperature and strain [99]. Liu et al. studied chip segmentations in the cutting ofTi-6Al-4V by implementing the Johnson-Cook failure properties combined withJohnson-Cook constitutive model in FE simulations with different tool coatingsand rake angles [48]. Other researchers demonstrated that adiabatic shear bandingand fracture failure are major factors influencing the formation of segmented chipsduring the cutting of Ti-6Al-4V. Lee and Lin suggested that adiabatic shear band-ing is the precursor of a fracture locus due to a localized high strain intensity [45],with the segmentation pitch depending on the frequency of shear band formation.Conventionally, high-strength metal alloys are machined at relatively low cut-ting speeds with a small uncut chip thickness in order to avoid excessive heat ac-cumulation in the primary shear zone [97]. However, the productivity of these ma-chining operations is consequently limited. Various techniques have been reportedto improve the machinability of high-strength metal alloys. For example, Ozel et al.investigated the effect of insert coatings on cutting of Ti-6Al-4V for improving thetool life [66]. Recently, Sagapuram and Viswanathan proposed a viscous slidingmechanism to study the nucleation and evolution of shear banding [72] and investi-gated fluid-like behavior with small viscosity in the bands. Subsequently, Sagapu-ram et al. proposed a passive geometric flow control method to suppress adiabaticshear banding in the cutting of titanium and nickel alloys [73]. In addition, ul-trasonic vibration-assisted cutting is a popular unconventional machining methodwith the advantages of lower cutting forces, better surface finishes and longer toollifetimes [9]. Ultrasonic vibrations with peak-to-peak amplitudes of approximately15 - 20 µm at frequencies above 15 kHz are applied at the tool tip to generate inter-mittent contact between the cutting tool and the workpiece, which enables periodicheat dissipation. Patil et al. modeled the effects of ultrasonic vibration in the cut-ting of Ti-6Al-4V using FE analysis and demonstrated that a reduction in cuttingforces and temperature when ultrasonic vibration is added[67]. Experimental in-vestigations have also been conducted on ultrasonic vibration-assisted cutting oftitanium alloys. For example, Sui et al. reported that the tool life was extended15by 300%, and cutting force was reduced by approximately 50% when ultrasonicvibration assistance was applied in turning [86]. Pujana et al. studied ultrasonicvibration-assisted drilling of Ti-6Al-4V and observed a reduction in cutting forces[69]. Similar observations of reduced cutting forces and heat generation were re-ported by Sanda et al. in the drilling of Ti-6Al-4V stacked with CFRP when ul-trasonic vibration assistance was implemented [74]. Pawar et al. experimentallystudied the axial and torsional vibration-assisted tapping of Ti-6Al-4V and reportedthat both the cutting forces and temperature were reduced by applying vibrations[68].Although it has been demonstrated that ultrasonic vibration assistance can en-hance the cutting performance of Ti-6Al-4V, previous studies in the literature arelimited to experimental observations. Thus far, there has been no report in the liter-ature on how the tool-workpiece separation in ultrasonic vibration-assisted cuttinginfluences the shear banding mechanism. Chapter 6 presents the effect of ultrasonicvibration assistance on shear band formation and chip segmentation in orthogonalcutting of Ti-6Al-4V. A chip flow model is developed to determine the shear flowmechanism in the primary shear zone and to explain the experimental observationsof chip formation when ultrasonic vibration assistance is applied in the tangen-tial direction. The chip geometries and cutting forces predicted by the proposedanalytical model were verified by orthogonal cutting experiments.16Chapter 33DOF Ultrasonic Vibration ToolHolder Design3.1 OverviewA novel three-degree-of-freedom (3DOF) ultrasonic vibration tool holder is de-signed to expand the application of vibration assistance in milling and drilling. Theproposed tool holder can deliver the elliptical vibration in the XY plane to assistmilling operations along the radial and tangential directions through piezoelectricactuators, and it can generate vibration along the spindle axis (Z) to help drillingprocess.The mechanical structure of the actuator is modeled in section 3.2 using thebeam equations to select the vibration modes to resonate the structure. A lumpedparameter model is implemented in section 3.3 to design dimensions of piezoelec-tric components. In section 3.4, a finite element model is used to verify the naturalfrequencies and to refine the mechanical structure. The prototype of the 3DOFultrasonic vibration tool holder is manufactured and presented in section 3.5. Theelectronics to drive and to monitor the piezo actuator, including power amplifiersand conditioning circuits, are presented in section 3.6.17housingpiezo stacksnodemode shapes3DOF vibration actuatorFigure 3.1: Conceptual design of 3DOF ultrasonic vibration tool holder.3.2 Mechanical Structure and Mode ShapesThe ultrasonic vibration tool holder generates vibrations in X , Y and Z directionsusing piezoelectric components. Piezoelectric stacks are excited at the natural fre-quencies of the tool holder structure to amplify the vibrations up to 20-30 µmrange. The actuator is connected to the tool holder housing at the neutral nodes ofvibration mode shapes where the modal displacements are zero as shown in Figure3.1.The final design of the tool holder with 3DOF actuators is shown in Figure 3.2.Three sets of piezo ring pairs, one set for each direction, are integrated into therotating tool holder, which receives power from the stationary source via four sliprings. The piezoelectric stacks are full rings in the Z direction and used to excitethe axial mode of the structure. Two sets of two half rings, which are assembledas a single full ring by the manufacturer, are used per X and Y direction in thedesign as shown in Figure 3.2b). Each half ring has the opposite polarity, and theassembled ring set is covered by a full ring copper electrode on the upper and lowersurface to supply AC voltage. Figure 3.2c) shows the connections of electrodes forX piezo rings as an example. Since the polarity of each half ring is opposite alongthe thickness direction, one half ring set expands and the other set shrinks whenvoltage is applied from the three electrodes, thus creating a moment to excite thebending mode of the holder. The vibration of each direction (X or Y ) is excited18X and Y piezoelectric ringsslip ringsclamp fixtureZYX+--++--Y piezo ringsX piezo ringsa)b)XYZ piezoelectric ringsCNC machine mounting interfaceVxelectrodesX piezo ringsc) half ring setFigure 3.2: Final design of the 3DOF ultrasonic vibration tool holder: a)3DOF ultrasonic vibration tool holder design; b) piezoelectric rings forX and Y vibrations; c) side view of X piezo rings with connections ofelectrodes.independently with the same frequency to create elliptical vibrations. The massesin the front and rear are used to tune the mode shapes of the structure. Differentmilling and drilling tools can be clamped in the front through a standard collet,which is ER-16 in this design.The actuator is designed to resonate the structure, hence the natural frequenciesand mode shapes are parametrically designed based on beam dynamic equations.The axial modes to generate vibrations in Z axis are governed by the followingbeam equation:∂∂x(EA(z)∂w(z, t)∂ z) = ρA(z)∂ 2w(z, t)∂ t2(3.1)where E is Young’s modulus, A is the cross-section area, w is the displacementalong Z, ρ is the mass density, and L is the length of the actuator. The natural fre-quencies in the axial direction are evaluated for a free-free beam with the boundary19conditions ∂w∂x = 0 at z= 0 and z= L as:ωk = kpiL√Eρ,k = 0,1,2... (3.2)The bending mode of the beam is identified by the following Euler’s beammodel:ρA∂ 2u(z, t)∂ t2=− ∂2∂ z2(EI∂ 2u∂ z2) (3.3)where u is the displacement along the radial directions (X or Y ), and I is the areamomentum of inertia. The natural frequencies of the bending modes are evaluatedby using free-free boundary conditions ( ∂2u∂ z2 = 0 and∂ 3u∂ z3 = 0 at z = 0 and z = L)as:ωk = β 2k√EIρA,k = 0,1,2... (3.4)where βk is the wave coefficient for mode k. The wave coefficients for the first fourmodes are given as β1 = 4.73/L, β2 = 7.85/L, β3 = 10.99/L and β4 = 14.14/L[53].Stainless steel 304 is used as the primary material of the 3DOF vibration actu-ator, and the overall length was designed to be less than 200 mm due to the cuttingtool length and stiffness requirements. The diameter of the actuator was targetedto be around 30-45 mm, which is compatible with the spindle interface. The de-sired vibration frequencies have been set to be about 15-20 kHz in three directions;hence the first axial mode was selected to generate vibrations in the Z direction,and the third bending mode was used to produce vibrations in XY plane.3.3 Lumped Parameter Model of the ActuatorA lumped parameter model is used to convert each direction of the actuator motionto an equivalent circuit in order to design the capacitance of the piezo stacks foreach direction (see Figure 3.3) [75]. The capacitance of piezoelectric componentsdue to element’s dielectric properties is represented by C0; Lm is the equivalent in-ductance corresponding to actuator’s mass;Cm is the equivalent capacitance relatedto the stiffness of the actuator; and Rm is the equivalent resistance due to dampingof the system. The relationship between mechanical parameters and electrical pa-20rameters are modeled as:Lm =mψ2; Cm =ψ2k; Rm =cψ2(3.5)where m is the mass of the actuator, k is the stiffness, c is the damping coefficientand ψ is the force factor of the piezoelectric stacks.CmLmRmC0VinFigure 3.3: Equivalent circuit of piezoelectric actuator for each axis.The force factor (ψ) is related to the piezoelectric material properties and ge-ometry, and can be expressed as:ψ =Y33d33At1NA(3.6)where A is the cross-section area of piezoelectric rings, t is the thickness of rings,Y33 is Young’s modulus of the piezoelectric material, and d33 is the piezoelectricstrain constant along the thickness, which is also the polarity direction. NA is thevibration magnification factor between piezo stacks and the cutting tool tip dueto the geometry of the actuator which is experimentally identified as explained insection 4.3.The impedance of the piezo actuator for each direction is modeled by the equiv-alent circuit, and it leads to a resonance frequency ( fr), which corresponds to thenatural frequency of the mechanical structure with the minimum impedance magni-tude, and an anti-resonance frequency ( fa) with a maximum impedance magnitude21(defined in Eq. (3.7) ).fr =12pi√CmLm; fa =12pi√C0+CmLmCmC0(3.7)The electromechanical coupling coefficient (KM), which shows the ability ofthe system to store mechanical energy, is defined as:KM =√1− f2rf 2a(3.8)The resonance frequencies have been selected at around 16.5 kHz in Z axis, and17.5 kHz for the X and Y directions for the actuator by considering the stiffnessrequirements for CFRP and Titanium finish machining and standard tool holderinterface with regular CAT40/HSK63 spindle interface used in industry. Since theaxial vibration along Z direction and elliptical vibration XY plane are used sepa-rately for drilling and milling, the natural frequencies for two modes are selectedwith 1 kHz difference to prevent coupling effects hence avoid exciting the tool inundesired directions. The piezo ring diameter range needs to be under 45 mm tofit the actuator to the targeted spindle interface. The diameter range of the tool isunder 10mm, and the stick out can be tuned to excite the natural frequencies of theactuator. The electromechanical factors (KM) for all three directions have been setto 0.25. The equivalent resistance related to damping is an estimated value for eachdirection. With an estimated modal mass of the actuator (around 1 kg), the equiv-alent inductance Lm and the capacitance of the piezoelectric material are evaluatedfrom Eq.(3.5). Based on Eq.(3.7), the capacitance of the piezoelectric componentsis derived as:C0 =14pi21f 2a − f 2r1Lm(3.9)The capacitance of the piezoelectric components has been decided by the materialproperties and the geometry as:C0 =keε0At(3.10)where ke is the dielectric constant of the material, ε0 is the permittivity of free22Table 3.1: Piezoelectric material parametersParameter Value UnitRelative dielectric constant, ke 1375 N/APiezoelectric strain constant, d33 300 10−12 m/VYoung’s modulus, YE33 63 GPaDensity, ρ 7.6 g/cm3Mechanical quality factor, Qm 1400 N/Aspace, and t is the thickness of the piezo rings. By evaluating the required C0from Eq.(3.9), the required thickness and cross-section area of the piezo rings areevaluated from Eq.(3.10).The selected hard piezoelectric material, which has a high mechanical qualityfactor and low dielectric loss, are comparable to PZT-8, and its properties are listedin Table 3.1. The selected thicknesses of piezoelectric rings are 4.5 mm in z and 5.5mm in X and Y directions with all having 40 mm outer and 15 mm inner diametersin order to match the designed capacitance. Table 3.2 lists the parameters used inthe simulations with selected piezo rings.Table 3.2: Lumped parametersC0 [nF] Cm [nF] Lm [H] Rm[Ω]Simulation in X 6.58 0.438 0.376 100Experiment in X 7.4 0.366 0.454 325Simulation in Z 4.71 0.312 0.588 50Experiment in Z 5.5 0.327 0.561 113The impedance frequency response functions (FRF), which are used to designand check resonance frequencies, of the electromechanical system in three direc-tions have been predicted from the equivalent circuit and validated experimentallywith an impedance analyzer (Keysight E4980AL). Figure 3.4 shows the impedanceFRFs of the ultrasonic vibration actuator clamping an 8 mm dummy tool along Xand Z. The impedance FRFs of X and Y actuators are close to each other within50 Hz due to symmetry. The resonance frequencies of the fabricated actuator weremeasured to be 17.45 kHz in X and Y , and 16.51 kHz in Z as planned. The pa-rameters including Cm, Rm, Lm and C0 can be realized through the impedance FRF231.65 1.7 1.75 1.8 1.85Frequency [Hz] 10 410 210 310 410 5|Z|x []1.65 1.7 1.75 1.8 1.85Frequency [Hz] 10 4-100-50050100Phase[deg]experimental resultssimulated1.6 1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 1.78 1.8Frequency [Hz] 10 410 010 210 410 6|Z|z []1.6 1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 1.78 1.8Frequency [Hz] 10 4-100-50050100Phase[deg]experimental resultssimulateda)b)Figure 3.4: Impedance frequency response functions of piezoelectric actua-tor: a) X actuator; b) Z actuator.241st longitudinal modezxyzxy3rd bending modew(z)neutral node of X vibrationneutral node of Z vibrationclamping fixturea)b)u(z)0.000e+0002.253e-0014.505e-0016.758e-0019.011e-0011.126e+0001.352e+0001.577e+0001.802e+0002.027e+0002.253e+0002.478e+0002.703e+000AMPRES0.000e+0003.537e-0017.074e-0011.061e+0001.415e+0001.768e+0002.122e+0002.476e+0002.830e+0003.183e+0003.537e+0003.891e+0004.244e+000AMPRESFigure 3.5: Mode shapes of the 3DOF vibration actuator in FE simulation: a)mode shape of X vibration; b) mode shape of Z well, and the values are listed in Table 3.2. The electromechanical couplingcoefficients for both actuators were measured to be about KM = Finite Element Model of the ActuatorThe finite element (FE) analysis of the 3DOF ultrasonic vibration actuator hasbeen carried out to verify natural frequencies, mode shapes and neutral nodes ofthe modes to locate the clamping locations of the actuator with the tool holderbody. A solid model of the actuator with estimated material properties determinedfrom the lumped parameter model was used under free-free boundary conditionsin the analysis. The mode shapes of the actuator were predicted using commercialsoftware (COMSOL Multiphysics) as shown in Figure 3.5 for the radial (X ,Y ) andaxial (Z) directions.25The neutral nodes of the mode shapes, where the modal displacements are zero,are shown in Figure 3.5. The neutral node is a circular band for Z vibration, andtwo neutral points per X and Y directions exist along the axis of the actuator. Theactuator is clamped to the housing with four leaf springs at the neutral points ofthe third bending mode in order to minimize the transmission of vibrations to thetool holder body while not affecting the values of natural frequencies. The nodesof X/Y and Z vibrations have been tuned to a small region by adjusting the shapesof the front mass and the back mass. The orientations of half-ring piezos for X andY vibrations align to the leaf springs because of excitation directions. The naturalfrequencies of the actuator with a carbide tool having 50 mm length (35 mm stickout of the collet) and 8 mm diameter have been predicted as 17.5 kHz in X , 17.4kHz in Y and 16.6 kHz in Z, respectively.3.5 Prototype ManufacturingA prototype ultrasonic vibration tool holder has been fabricated to fit CAT40 spin-dle as shown in Figure 3.6. Figure 3.6b) shows the ultrasonic vibration actuatorinside the holder housing. An 8 mm carbide dummy tool with a 35 mm length outof the collet was clamped in order to measure the impedance FRFs shown in Figure3.4. A two-flute end mill with 8 mm diameter was inserted into the actuator with acollet as shown in Figure 3.6a), and the stick out of the tool was adjusted to matchthe natural frequencies.The flexibility FRFs of the actuator assembly with the end mill have been mea-sured by impact modal tests (shown in Figure 3.7). The natural frequencies of theactuator assembly are measured to be 16.5 kHz in Z and 17.5 kHz in X and Y di-rections which agree with the predicted values during the design. If the stick outlength must be different from the design value in production, the natural frequen-cies will shift. However, the resonance tracking controller presented in Chapter4 identifies the resonance frequency on-line and update the excitation frequencydynamically to keep the amplitudes at the desired level.The maximum dynamic flexibilities of the system are 0.06 µm/N in radial and0.09 µm/N in axial directions respectively to excite the ultrasonic vibrations. Thestatic flexibility of radial direction is measured as 0.04 µm/N from the impact tests,26a)b)slip ringshousingleaf springcutting toolback masspiezo stacks front massFigure 3.6: Prototype of 3DOF ultrasonic vibration tool holder: a) fabricatedultrasonic vibration tool holder; b) 3DOF ultrasonic vibration actuator.which is about twice of the flexibility (0.02 µm/N) of a normal tool holder clamp-ing the same cutting tool through an ER25 collet. The main reason for the staticstiffness loss is contributed by the leaf springs, which are the clamping mechanismbetween the ultrasonic vibration actuator and the housing. The dynamic stiffnessof the tool holder with actuator is 960 kN/m, which is sufficient to resist againstchatter in targeted machining of CFRP at a limit depth of cut of 2.5 mm. The re-duction in the stiffness limits the application of ultrasonic vibration-assisted tools271 1.2 1.4 1.6 1.8 2 2.2f[Hz] 10400.511.522.53|X/F| [m/N]10 -70.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2f[Hz] 10400.|Z\F| [m/N]10 -7a)b)fnz=16.5 kHzfnx=17.5 kHzFigure 3.7: Structural frequency response functions: a) FRF along X; b) FRFalong heavy cuts, hence they are designed for finishing operations to obtain a goodsurface finish.Slip rings are used to transmit the power to the rotating ultrasonic vibrationtool holder. Four rings, which assemble together as a rotor, are required to connectpositive electrodes for all three groups of piezos (X , Y and Z) and the ground. Thedistance between two adjacent rings is determined by the maximum supply voltageof piezo actuators, which is 200 V, and it is designed to be 10 mm according to the283DOF vibration tool holderSlip ringCarbon brushFigure 3.8: Slip rings assemblyspecifications from the vendor. The maximum current on the rings is 5 A. Fourcarbon brushes, which contact the rings with preload, are installed to the CNC ma-chine statically to deliver power as shown in Figure 3.8. The maximum rotationspeed of the slip rings is 3500 rev/min to prevent sparks. The slip rings limit themaximum spindle speed of ultrasonic vibration-assisted milling or drilling oper-ation using the proposed holder. Contactless power transmissions such as rotarytransformer are preferred in the future improvement to expand applications of the3DOF ultrasonic vibration tool holder.29dSPACE PWMgeneratoractive filtercurrent sensing amplifierconditioning circuit ultrasonic vibration actuatorsensing resistorfx+-voltage sensingamplifierIpower amplifierVVCADACV0ADCsVpp=200VFigure 3.9: Electronic system for X vibration.3.6 InstrumentationThe instrumentation for the 3DOF ultrasonic vibration tool holder has been de-signed and implemented. The instrumentation system per channel includes: 1) aninput conditioner to produce appropriate signals for the voltage amplifier from thedigital computer; 2) a voltage stage as a power amplifier to drive the three piezovibration actuators; 3) a current sensor for feedback control. A digital real-timecomputer governs all analog instruments to control and monitor the 3DOF ultra-sonic vibration tool during cutting. Figure 3.9 shows the instrumentation of Xpiezo actuator as an example.The first objective of the instrumentation system is to prepare signals for thevoltage power amplifiers. In order to excite the ultrasonic vibration, three sinu-soidal voltages are generated as:Vx =Vx0 sin(2pi fxt)Vy =Vy0 sin(2pi fxt+θin)Vz =Vz0 sin(2pi fzt) (3.11)where fx, fz are the excitation frequencies in (X ,Y ) and Z directions, respectively,and θin is the phase difference between the input voltages of X and Y actuators.Vx and Vy are synchronized and share the same excitation frequency to generateelliptical vibrations.30A dSPACE microLab is used as the digital real-time computer, which can runup to 50 kHz as a sampling rate. Figure 3.9 shows the instrumentation of X actuatoras an example. The periodic square wave signals with fx or fz are modulatedby a waveform generator embedded in the digital controller. The square wave isconverted to a sinusoidal wave using an active RC, fourth-order low pass filterin order to eliminate high-frequency harmonics. Following the filter, a voltage-controlled operation amplifier (Texas Instruments VCA822) is used to adjust theamplitude of the input sinusoidal signal to the desired value (Vx0, Vy0 and Vz0) foreach channel. The gain of the VCA is adjusted by an analog continuous voltageproduced by the digital controller through a digital-analog converter (DAC). Theinput phase difference (θin) is adjusted by a phase-shift integrated circuit using apotentiometer manually.A voltage stage is used to amplify the signal from the VCA to drive the piezoactuator. The supply voltage of the actuator is designed to be 200 V peak-to-peakvoltage. The bandwidth of the voltage stage needs to reach high frequency toexcite the selected vibration modes of the actuator without voltage loss. MP118from APEX , which is a linear power amplifier, is selected as the power amplifierin the voltage stage. It has maximum supply voltage of 200 V with 10 A continuousoutput current, and the voltage closed loop bandwidth is up to 100 kHz with thegain of 10.The actuator can produce the ultrasonic vibration for each axis independentlyusing the power amplifiers and the input conditioner, and Figure 3.10 shows theresults of preliminary experiments for all three directions. A laser vibrometer(Polytec CLV-2534) was used to measure the vibrations in the experiments. Theinput voltage peak-to-peak amplitude was adjusted to be 200 V. A 8 mm diame-ter carbide dummy cutting tool was clamped in the holder with 30 mm overhang.The ultrasonic vibration actuator was resonated at 17.5 kHz in X- and Y -directionswith measured peak-to-peak displacement amplitudes of 20 µm in X and 18 µmin Y , respectively. The Z-axis was excited at 16.5 kHz with a measured vibrationamplitude of 25 µm.It is important to drive the piezo actuator at its resonance frequencies to ob-tain enough amplitudes during cutting. The full details of the control system arepresented in Chapter 4. Briefly, the phase of impedance, which can be character-310 0.5 1 1.5 2 2.5t [s] 10-4-20-1001020x [m]0 0.5 1 1.5 2 2.5t [s] 10-4-20-1001020y [m]0 0.5 1 1.5 2 2.5t [s] 10-4-20-1001020z [m]Figure 3.10: Vibrations outputs from the laser vibrometer for X, Y and Z.ized from the current flowing through the piezoelectric actuator and the voltageoutput from the power amplifier, has been used to track the resonance frequency.The current flowing through each piezo actuator must be measured to obtain thephase. According to this requirement, a current sensing circuit is designed andtested. In the 3DOF ultrasonic vibration tool holder, three piezo actuators, one foreach vibration freedom, are pressed by a central bolt and share the same groundwhen receiving supply voltages. Therefore, to measure the current individuallyfor each channel, the current sensing resistor is placed in front of the piezo ac-tuator, as shown in Figure 3.9. Thus, the common-mode voltage is high up tohundreds of volts when measuring the voltage drop across the sensing resistor. Ahigh common-mode voltage difference op amplifier (Texas Instruments INA117)is selected to obtain the current signal from the sensing resistor. The current sig-nal is collected by an analog-to-digital converter (ADC) in the digital controller asfeedback. On the other hand, the supply voltage signal is fed back using a step-32down voltage amplifier to calculate the phase in order to track the resonance duringcutting.3.7 SummaryA 3-DOF ultrasonic vibration tool holder design to be used in milling and drillingoperations was presented in this chapter. The proposed design includes a piezoelec-tric actuator, which can be activated in three directions to expand its application tovarious machining operations in a single tool holder. While the radial vibrationsare delivered by resonating the third bending mode, the axial vibrations are createdby exciting the first axial mode of the actuator assembly at around 17 kHz witha peak-to-peak amplitude range of 20-25 µm. The system has a feature of main-taining a phase shift between X and Y vibrations, which can be used in deliveringelliptical vibrations. The instrumentation system was also developed to drive theproposed piezoelectric actuator.33Chapter 4Sensorless Control of 3DOFUltrasonic Vibration Tool Holder4.1 OverviewThe piezoelectric actuator of the vibration tool holder works at natural frequenciesto enlarge the amplitudes for high-frequency vibration. The natural frequencies ofultrasonic vibration actuators vary under the varying cutting force disturbances dur-ing machining. Consequently, the amplitudes of delivered vibrations deviate fromthe desired reference values. As opposed to using vibration sensors, this chapterpresents a sensorless digital control system for the proposed 3DOF ultrasonic vi-bration tool holder using a digital controller platform. The chapter starts with thefundamental principles of resonance tracking and amplitude control in section 4.2and a dual-loop control system in section 4.3. The crosstalk compensation betweenthe two axes is analyzed and discussed in section 4.5. Section 4.6 shows the millingand drilling experiments with the proposed control system.4.2 Principle of Frequency and Amplitude ControlEach axis of the 3DOF ultrasonic vibration actuator can work independently us-ing the same sensorless control strategy. The resonance frequency is tracked andthe desired vibration amplitude is delivered in a closed-loop manner. A linearized34V(t)Cd vt+-It=ψxI(t)Electro-meachnical SystemCircuitFilter∏ PA ψV0fV=V0sin(2̟ft)Ft=ψVmkbFext xFigure 4.1: Linear lumped model of the ultrasonic actuator.lumped parameter model is proposed to estimate the vibration characteristics fromthe measured supply voltage and driving current signals without having to use ded-icated vibration sensors.The piezoelectric ultrasonic vibration actuator is represented by a lumped lin-ear model as presented in section 3.3 for each axis. The model is also used to trackthe varying natural frequencies under cutting loads and estimate the velocities tocontrol the amplitudes. The piezo actuator can excite the selected structural modesof the holder to generate ultrasonic vibration as mentioned in Chapter 3. The natu-ral frequency of the selected vibration mode shifts under the cutting load, but thisshift is not wide enough to excite the neighboring modes of the system. Conse-quently, a single degree of freedom (SDOF) mechanical system corresponding toselected excitation mode is used to model the piezo actuator during operation. Theoverall tool holder is lumped into the mechanical subsystem of the piezoelectricactuator stack for each vibrating axis, and the resulting electromechanical systemmodel is shown in Figure 4.1. The actuator is excited to deliver motion by applyinga sinusoidal voltage at its resonance frequency on the terminals of the piezoelec-tric element. The capacitive component (Cd) represents the electrical subsystemof the piezoelectric stack, while the mass, spring, damper elements represent thecombined mechanical subsystems of the piezo stack, and the tool holder deliversthe vibration in each direction as shown in Figure 4.1).The model is presented for X axis as an example. The constitutive linear elec-35tromechanical relationship of the piezoelectric actuator is defined as [28]:Ft = ψV (4.1)It = ψ x˙ (4.2)where Ft is the force transduced by the electrical energy generated by the supplyvoltage (V ) on the actuator with a force factor or transforming gain ψ . It is thetransforming current to generate the vibration, and x˙ is the vibration velocity at thecutting tool tip. The force generated by the actuator (Ft), and the external cuttingforce (Fext) are applied on the actuator to excite the structural mode with mass (m),stiffness (k) and damping constant (b) causing tool displacement (x):mx¨+bx˙+ kx= Fext +Ft (4.3)The overall driving current (I) from the power amplifier is the sum of the cur-rent flowing through capacitance (Cd) and the transforming current (It):I =CdV˙ + It (4.4)The transfer function between the transforming current (It) and supply voltage(V ) can be evaluated by taking the Laplace transforms of (4.1) to (4.3) and lettingFext = 0 :Ga(s) =It(s)V (s)=ψ2sms2+bs+ k(4.5)The frequency response function Ga(s= jω) (4.5) can be obtained as:Ga( jω) =ψ2sms2+bs+ k∣∣∣∣s= jω=1kjψ2ω2nωω2n −ω2+ j2ζωωn(4.6)where the natural frequency (ωn =√k/m), damping ratio (ζ = b/(2√mk)) andstiffness (k) are evaluated from the modal parameters. When the input voltage fre-quency (ω) matches the natural frequency of the mode (ωn), the phase of the FRFgiven in (4.6) becomes zero (ϕ =∠Ga( jωn) = 0). In the proposed control system,the phase between transforming current and supply voltage (ϕ =∠Ga( jω)) is used36to track the resonance.The velocity of vibration at tool tip can be derived from (4.4) and (4.2) as:x˙(t) =1ψIt =1ψ[I(t)−Cd·V (t)](4.7)where driving current (I) and voltage (V ) can be measured from the power ampli-fier, and the capacitance (Cd) of the piezo is measured with an LCR meter (KeysightE4980L). The transforming gain (ψ) in each direction is defined in (3.6). It is pro-posed to estimate the amplitude of vibrations from the driving current and supplyvoltage measurements during cutting by utilizing the model given in (4.7).4.3 Dual-loop Control SystemThe dual loop control system is used to track the vibration resonance frequency andamplitude for each vibrating axis as shown in Figure 4.2. The inputs to the con-troller are the desired phase (ϕr) between the transforming current (It) and supplyvoltage (V ) to deliver vibration amplitude linearly, and the desired vibration ampli-tude at the targeted excitation frequency. The first loop tracks the resonance of toolstructure by using the phase between supply voltage and transforming current, andthe phase is tracked based on the frequency response function defined in (4.6). Thesecond loop tracks the reference vibration amplitude by using estimations based on(4.7).The vibration observer shown in Figure 4.2) is implemented in real time toestimate the amplitude of transforming current (It) and the phase between supplyvoltage V and current It . The transforming current (It) is evaluated from Eq.(4.4)using the measured current I and voltage V , i.e. It = I−CdV˙ where derivativeV˙ is digitally evaluated. Two digital phase detectors (PD1 and PD2) are used tocalculate the phase difference between It and V , and the digital amplitude detector(AD) tracks the amplitude of the transforming current (It) estimated by the observer.The current It is used to control the amplitude of vibration.Various detection methods have been used to track the phase between the over-all driving current (I) and supply voltage (V ) in the literature, mostly using elec-tronic circuits. Jung et al. [39] used a phase lock loop (PLL) circuit as a phase37C1 sine wavegeneratorpiezo actuatorV=V0cos(2̟ft)ddtCddifferentiator-phase detectorPI controllerPD1PD2--rf0Δf fI(t)V(t)It(t)VIC2|x|rψ|It|rADamplitude detector|It|-V0vibration observervoltage Kalman filtercurrent Kalman filterphase tracking loopamplitude control loopReference phaseReference velocity amplitudeFigure 4.2: Dual-loop ultrasonic vibration control systemdetector to track the resonance during elliptical vibration cutting. Dow et al. [20]utilized a lock-in amplifier to detect the phase of impedance for resonance trackingduring nanocoining processes. Furuya et al. [23] designed a circuit with a differ-ence amplifier to detect the phase between supply voltage and vibration velocitywhose accuracy is dependent on the physical tuning of the capacitor to match thecapacitance of piezo components. As for the detection of amplitude, Juang andGu [37] designed a circuit to recognize the amplitude of operation current for atwo-phase ultrasonic motor, which also relied on a well-tuned capacitor to removethe influence of piezo capacitance. The existing circuit based methods can trackthe small variations in the natural frequency if the same tool geometry is usedin the holder. However, when various tools are used, the variation in the naturalfrequency becomes too wide to be tracked by analog circuit based systems. Theproposed, digital dual loop amplitude and phase detector adapts itself to the chang-ing structural dynamics of the tool holder, which is essential in regular machiningapplications in production. Though there is a limitation of the adaptation becausethe large changing dynamics shifts neutral nodal positions, which can increase thedamping during the cutting process and then reduce vibration amplitudes.38The dual loop vibration observer consists of two sets of dedicated Kalman fil-ters; one detects the amplitude and phase of supply voltage on the piezo stack asshown in Figure 3.2 altered by the conditioning circuit, and the other detects thevarying amplitude and phase of transforming current disturbed by the cutting load.The measured periodic supply voltage (V ) and transforming current (It) withamplitude Ak and phase ϕk are defined as:s(tk) = Ak cos(ωtk+ϕk)+ v(tk)→ s(tk) =V (tk) or s(tk) = It(tk) (4.8)where tk = kT is the discrete time at interval k with a sampling period T . The ex-citation frequency is ω = 2pi f , and v(tk) is the measurement noise. Since the sam-pling frequency is just above Nyquist frequency, it is too low to track the amplitudeAk and phase ϕk accurately. Kalman filter is used to estimate them accurately usingthe following two state variables defined as:qk =[q1,kq2,k]=[Ak cos(ωtk+ϕk)Ak sin(ωtk+ϕk)](4.9)At the consecutive discrete time step tk+1 = tk+T , the states vary due to the processnoise wk: [q1,k+1q2,k+1]=[Ak cos(ω(tk+T )+ϕk)Ak sin(ω(tk+T )+ϕk)]+[w1,kw2,k](4.10)or by using the trigonometric identities:[q1,k+1q2,k+1]︸ ︷︷ ︸qk+1=[cos(ωT ) −sin(ωT )sin(ωT ) cos(ωT )]︸ ︷︷ ︸Φ[q1,kq2,k]︸ ︷︷ ︸qk+[w1,kw2,k]︸ ︷︷ ︸wk(4.11)where Φ is the state matrix. The measurement equation defined in Eq.(4.8) for thestates with noise vk becomes as follows:sk =[1 0]︸ ︷︷ ︸H[q1,kq2,k]+ vk (4.12)where H is the output matrix of the model. Combining Eq.(4.11) and Eq.(4.12)39leads to the completed state space model for the periodic voltage (V ) and current(It) signal (sk) .The state vector (qk) is estimated with the following Kalman filter [10]:qˆ−k =Φqˆk−1P−k =ΦPk−1ΦT +QKk = P−k HT(HP−k HT +R)−1qˆk = qˆ−k +Kk(sk−Hqˆ−k)Pk = (I−KkH)P−k(4.13)where Pk[2× 2] is the error covariance matrix and the superscript (·)− denotesprior estimate. Q = QI = E[wwT ] is the 2× 2 diagonal process noise covariancematrix; R is the covariance of the measurement noise E[vvT ], where E[·] is theexpectation operator, and it is found by taking the covariance of measured currentand voltage signals in air cutting. The process noise covariance is assumed tobe Q = λR, where λ is tuned as λ = 10−6. In the ultrasonic vibration controlsystem, the angular excitation frequency of the periodic signal (ω) is generated bythe controller (C1) of phase tracking loop, and it is a known value when estimatingthe states. However, the frequency (ω) varies during machining due to time varyingcutting forces, hence the state matrix Φ, so as the Kalman gain matrix Kk[2×1] isupdated recursively.The amplitude (Aˆk) and the phase (ϕˆk) of the signal shown by PD and ADblocks in Figure 4.2 are estimated at each sampling interval (k) as:Aˆk =√qˆ21,k+ qˆ22,k (4.14)ϕˆk = arctan(qˆ2,kqˆ1,k)−ωtk (4.15)The phase difference (ϕ) between the transforming current (It) and supply volt-age (V ) (see Eq.(4.6)) is tracked from their estimated phases from the Kalman fil-ters (Eq.(4.15) ) as:ϕ = ϕˆIt − ϕˆV (4.16)which is as a feedback to track the reference input phase (ϕr) in the dual loop con-40troller. A digital differentiator is embedded in the vibration observer to calculatethe derivative of supply voltage (V˙ (k)) for the transforming current (see Figure4.2). The differentiator is smoothed by a Kalman filter to reduce the effect of noisein digital differentiation at a sampling frequency which is just above two times ofthe natural frequencies of the actuator. The transforming current (It) is calculated inreal time by the differentiated supply voltage using Eq.(4.4) (It(k) = I(k)−CdV˙ (k))before detecting the phase and amplitude by Kalman filters.In the phase tracking loop as shown in Figure 4.2, a PI controller (C1) is de-signed to update the excitation frequency ( f ) with a feed forward input ( f0), whichis measured natural frequency of the tool holder with impact modal tests ahead ofcutting operation. The feedback of this loop is the estimated phase between supplyvoltage and transforming current derived by Eq.(4.16). The PI controller is definedas follows:C1(s) =∆ f∆ϕ= Kp(1+ωIs) (4.17)where Kp and ωI are the proportional gain and the integrator corner frequency,respectively.In the vibration amplitude control loop, the referenced vibration velocity am-plitude (|x˙|r) is converted to the referenced transforming current amplitude (|It |r =ψ|x˙|r) where ψ is the transforming gain in Eq.(4.7). The amplitude of transform-ing current (|It |) is tracked by another PI controller (C2(s) in Figure 4.2), as New-comb et al. [60] suggested that the piezoelectric actuator performs more linearly bycontrolling the current in comparison to the voltage control. PI controllers are im-plemented with anti-windup configuration to remove the steady state errors whilepreventing the integration wind-up when the outputs are saturated.The sine wave generator in Figure 4.2 consists of a digital controller, a condi-tioning circuit and a power amplifier per channel as presented in section 3.6 (seeFigure 3.9) to generate the supply voltage to the piezo actuator. The output signalis a sinusoidal function defined as follows:V (t) =V0 cos(2pi f t) (4.18)whereV0 is the supply voltage amplitude produced by the vibration amplitude con-41troller (C2(s)), and f is the excitation frequency calculated by the phase trackingcontroller (C1(s)).4.4 Verification of Control SystemThe dual control system has been implemented, and multiple tests have been con-ducted to verify its performance. Both closed loops are designed to have at least60◦ phase margins to guarantee the stability of the system with the highest band-width possible within the physical limits of the conditioning circuit. The cornerfrequency of integrator is set as tenth of the gain crossover frequency (ωI =ωc/10)to minimize the steady state errors. The measured frequency response functions ofthe dual loop controller are shown in Figure 4.3, where the phase and amplitudetracking loops have 130 Hz and 140 Hz bandwidths, respectively. The vibration es-timation is based on a linear relationship between the vibration velocity and trans-forming current in the implemented system. The effect of nonlinearity of the piezoactuator on the transforming gain (ψ) is analyzed using the experimental setupshown in Figure 4.4.The proposed dual loop controller is used by setting the desired phase (ϕr)between the transforming current and supply voltage and varying the referencedtransforming current amplitude (|It |r). The transforming gain (ψ) is found from theratio of the current (|It |) and the vibration velocity measured with a laser vibrometer(Polytec CLV-2534). The experiments were repeated at several reference phases(−45◦ ≤ ϕr ≤ 45◦), and the identified sample transforming gains for X axis areshown in Figure 4.5. The stationary excitation frequency ( f0) was matched withthe unloaded resonance frequency of the actuator which was 15.43 kHz for X axis.The transforming gain for each reference phase is slightly different, but it is mostlinear at the referenced phase of ϕr = −30◦, where the goodness of linear fit is0.9996, without saturating the power amplifier. Consequently, the reference phasebetween the transforming current and supply voltage is fixed at ϕr = −30◦ in thecontroller setup for the ultrasonic actuator in machining applications. Since thetool length and diameter will change at various operations, the transforming gainis calibrated for each cutting tool clamped to the holder.The performance of controller has been tested in turning aluminum alloy AL42f [Hz]phase  [deg/deg]f [Hz]velocity |x| [(m/s)/(m/s)]100 101 10210-1100101100 101 10210-11001013 dB-3 dB3 dB-3 dBa)b)Figure 4.3: Closed-loop frequency response functions: a) phase trackingloop; b) vibration velocity tracking loop.7050 workpiece attached to the spindle, while a turning tool was clamped to thestationary actuator as shown in Figure 4.4. The transforming gain was measured tobe ψ = 0.1754 A/(m/s) for the tool at a fixed reference phase φr =−30◦, see Fig-ure 4.5. The measured vibration velocity with laser vibrometer is compared againstthe estimated velocity in Figure 4.6a). The referenced vibration velocity amplitudewas set at 0.6626 m/s which was closely tracked by the controller during air cutting,43Spindle Laser beamStationary ultrasonic vibration toolFigure 4.4: Turning setup with a laser vibrometerbut the vibrometer measurements deviated up to around 1.0 m/s when the tool cutsthe material. However, the laser vibrometer does not only measure the ultrasonicvibrations delivered by the actuator but also the vibrations of tool holder clampedon the flexible cantilevered plate which vibrates at its lower natural frequency dur-ing cutting. The natural frequency of the flexible fixture, which is 175 Hz, wasmeasured with impact modal tests with an impulse hammer (Dytran 5800B4) andan accelerometer (Dytran 3225F) at a sampling rate of 50 kHz. The vibrations ofthe actuator and plate fixture are separated in frequency domain as shown in Figure4.6 b). The amplitude of the actuator vibration velocity was 0.6413 m/s at the ultra-sonic vibration excitation frequency of 15.47 kHz during cutting, which was closeto the reference velocity of 0.6626 m/s. The plate vibrations contribute 0.1963 m/sat its natural frequency of 175Hz. The controller also maintained the referencephase at φr =−30◦ (see Figure 4.6 c)) and adjusted the excitation frequency fromunloaded natural frequency of actuator at f0 = 15.43 kHz to the loaded natural fre-440 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 [A]°,  ψ =0.1586 A/(m/s)=30°, ψ =0.1679 A/(m/s)=0°, ψ =0.1710 A/(m/s)=-30°, ψ =0.1754 A/(m/s)=-45°, ψ =0.1843 A/(m/s)|It|measured |x| [m/s]Figure 4.5: Measured transforming gains (ψ) as a function of reference track-ing phase (ϕr).quency of f = 15.47 kHz (Figure 4.6 d)) during cutting under the measured cuttingforce of 55 N (see Figure 4.6 e)).The sensitivity of transforming gain to different cutting loads was also testedby changing the feed rate in turning tests between 0.01 mm/rev and 0.1 mm/rev,which corresponded to cutting force range of 12 N to 145 N. The transforminggain changed from ψ = 0.1754 A/(m/s) in air cutting to ψ = 0.1548 A/(m/s) (seeFigure 4.7) at the highest cutting load of 145 N obtained at 0.1 mm/rev feed rate.The variation in vibration velocity caused by the transforming gain fluctuation un-der the cutting load around 150 N is within 12%, which is acceptable in ultrasonicvibration-assisted milling and drilling of targeted CFRP materials. The transform-ing gain can either be calibrated at the operational cutting load or the deviationcan be neglected in ultra-precision machining applications where the disturbancecutting force is negligibly small.458.5 9 9.5 10 10.5-101Velocity [m/s]measurementestimation8.5 9 9.5 10 10.5-31-30.5-30-29.5-29 [deg]8.5 9 9.5 10 10.515.4215.4415.4615.4815.5f ext [kHz]8.5 9 9.5 10 10.5Time [s]020406080Ft [N]air cutting cutting0 5 10 15 20 25Frequency [kHz] [m/s](15.47, 0.6413)a)b)c)d)e)(0.175, 0.1963)Figure 4.6: Single-axis ultrasonic vibration during turning: a) estimated andmeasured ultrasonic vibration velocity; b) FFT of vibration velocity dur-ing cutting; c) phase between supply voltage and transforming current;d) excitation frequency generated by phase tracking controller; e) turn-ing force along tangential direction. The spindle speed of turning is1500 RPM, the feed rate is 0.05 mm/rev, and the width of cut is 1.5mm.460 0.02 0.04 0.06 0.08 0.1Feed rate [mm/rev] gain  [A/(m/s)]measurements during cutting=0.1754 [A/(m/s)]Figure 4.7: Transforming gains with different feed rates. The turning spindlespeed is 1500 RPM, and the width of cutting is 1.5 mm.4.5 Two-axis Ultrasonic Vibration ControlThe actuator’s two-axis (X ,Y ) ultrasonic vibration generation capability has beentested by delivering elliptical vibration loci for milling applications based on thedual-loop control system proposed in section 4.3. The vibrations in X and Y di-rections are excited by the supply voltages having amplitudes V0x and V0y at theidentical excitation frequency ( f ) as,Vx(t) =V0x cos(2pi f t)Vy(t) =V0y cos(2pi f t+θin)}(4.19)where V0x and V0y are the amplitudes of supply voltages for X and Y vibrationsrespectively, θin is the input phase difference between two supply voltages and it isadjusted through a potentionmeter.The frequency response functions (FRFs) in X and Y directions of the toolholder are measured by impact modal tests using a miniature impulse hammer(PCB 086C80) and a laser vibrometer. A sample FRF measurement for 8 mmdiameter end mill with four flutes are shown in Figure 4.8. The measured naturalfrequencies are fx = 16.58 kHz and fy = 16.61 kHz which differ only by about 31Hz due to symmetric tool geometry.4713 14 15 16 17 18 19 20f [kHz]|FRF| [m/N]xyfX=16.61 kHzfy=16.58 kHzFigure 4.8: Measured FRFs along X and Y when clamping an 8 mm millingtool with 4 flutesThere is crosstalk arising between X and Y directions of the actuator. Thecrosstalk is removed from the sensorless estimation of the vibration velocities tocontrol the elliptical vibration locus using the observer shown in Figure 4.9. Thesupply voltage (Vx,Vy) and driving current (Ix,Iy) for each axis are collected in realtime. The vibration observers designed in section 4.3 are used to estimate thetransforming currents (Itx.Ity) flowing through piezo stacks (X ,Y ) and their phases(ϕx,ϕy). Since the phase controller is used only to track the resonance frequency,and the actuators in X and Y directions have almost identical natural frequencies,only the phase in one direction (ϕx) is closed loop controlled. The resulting res-onance frequency ( f ) is then used to excite both X and Y actuators. The phasecontroller for y axis is deactivated for applications with elliptical vibrations.The relationship between transforming currents (Itx and Ity) and the vibrationvelocities (x˙ and y˙) with crosstalk is expressed as:[x˙y˙]=[1ψxx1ψyx1ψxy1ψyy][ItxIty](4.20)where (ψxx, ψyy) are direct and (ψyx, ψxy) are crosstalk transforming gains, re-48X VibrationoberserverY Vibrationoberserver1/Ψxx1/Ψxy1/Ψyx1/ΨyyADxADyPDxPDycrosstalk observerIx(t)Vx(t)Iy(t)Vy(t)фx|x||y|θxy -x(t)y(t)(to X phase tracking controller)Itx(t)Ity(t)θxθyFigure 4.9: Two-axis ultrasonic vibration crosstalk observerspectively. The estimated vibration velocity amplitude considers the crosstalk, andit is fed back to closed loop controller for on-line compensation. The transform-ing gains are calibrated from the transforming current and the vibration velocitymeasured with a laser vibrometer. It is observed that the crosstalk has range of10−30% depending on tool geometries.The phase difference (θxy) between the vibrations in two directions are eval-uated from the estimated phases (θˆx and θˆy) in the crosstalk observer shown inFigure 4.9 as:θxy = θˆx− θˆy (4.21)The input supply voltage phase θin in (4.19) is tuned using a potentiometer untilthe controller delivers the desired phase θxy between the vibrations in X and Y di-rections. For example, a circular locus is generated by θxy = 90◦. The proposedtwo-axis ultrasonic vibration control system has been verified by generating a cir-cular (θxy = 90◦) and elliptical (θxy = 120◦) loci when the ultrasonic vibration toolholder was stationary as shown in Figure 4.10. The estimations from the proposedcrosstalk observer can track the referenced elliptical loci. The estimated tool tiploci match with the laser vibrometer measurements with a maximum error of 13%49as shown in Figure 4.10.-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5[m/s]-0.5-0.4-0.3-0.2-[m/s]-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5[m/s]-0.5-0.4-0.3-0.2-[m/s] measurementreferenceestimationy yxxFigure 4.10: Elliptical loci generated by two-axis ultrasonic vibration con-troller with phase differences of a) θxy = 90◦, b) θxy = 120◦. Trackedreferenced vibration velocities are set to : ˙|x|r = ˙|y|r = 0.45 m/s.4.6 Cutting ExperimentsThe sensorless control system for the 3DOF ultrasonic vibration tool holder hasbeen tested in several milling and drilling operations on a CNC machining centershown in Figure 1.4. A stationary table dynamometer (Kistler 9256C1) with 3 kHzbandwidth was used to measure the cutting forces in three directions. The band-width of dynamometer measurements is much lower than the ultrasonic vibrationfrequencies (16-17 kHz) generated by the tool holder, hence the cutting forces canbe measured only at tooth passing frequencies and their harmonics under 3 kHz.Sample drilling and milling test results are presented to illustrate the performanceof ultrasonic actuator controller performance during machining.4.6.1 Drilling testsA twist drill with 6 mm diameter was used in drilling aluminum alloy AL 7050while exciting the actuator in Z axis at its 17.95 kHz natural frequency. The trans-forming gain was identified as ψz = 0.1333 (m/s)/A. The referenced vibration ve-locity amplitude was set to |z˙| = 1.5 m/s. Figure 4.11 shows amplitudes of vibra-500 0.5 1 1.5Time [s]|z| [m/s]controlled vibrationuncontrolled vibrationreferenceFigure 4.11: Amplitudes of vibration velocities in drilling tests. The spindlespeed is 1500 RPM, and the feed rate is 0.1 mm/rev. The workpiecematerial is aluminum alloy 7050.tion velocities. In the first test, the actuator’s Z axis was excited at a fixed inputfrequency (17.95 kHz) while the ultrasonic vibration controller was off. The vibra-tion velocity during cutting was attenuated from the desired 1.5 m/s to about 1.32m/s with oscillations caused by periodic cutting forces at spindle periods. When thecontroller was turned on, the vibration velocity was tracked at the desired velocity(1.5 m/s) with small amplitude oscillations due to disturbance force compensationof the system as shown in Figure 4.11.The effect of ultrasonic vibrations on the cutting forces is also demonstratedin Figure 4.12, where the drilling forces were reduced by about 30% relative tothe normal drilling without vibration assistance and reduced by 10% relative tothe drilling with uncontrolled ultrasonic vibration. When the frequency and ampli-tude of the ultrasonic vibration are delivered without the controller, the vibrationvelocity amplitude is attenuated which affects the drilling force amplitude.4.6.2 Milling testsAn 8 mm diameter end mill with 4 flutes was used in up milling of aluminum al-loy AL 7050 material. Measured FRFs of the end mill set up is shown in Figure510 1 2 3 4 5 6 7 8 9time [s]-50050100150200250Thrust force Fz [N]controlled vibrationuncontrolled vibrationnormal drillingFigure 4.12: Thrust forces along axial direction during drilling. The spindlespeed is 1500 RPM, and the feed rate is 0.1 mm/rev. The workpiecematerial is aluminum alloy 7050.4.8 with the natural frequencies 16.61 kHz and 16.58 kHz, in X and Y directions,respectively. The transforming gains were ψxx = 0.1359 A/(m/s), ψyy = 0.1373A/(m/s), ψyx = 1.400 A/(m/s) and ψxy = 0.7003 A/(m/s). The two-axis ultrasonicvibration control system was tested to generate elliptical locus in xy plane as shownin Figure 4.13. The elliptical vibration with a circular locus was excited with thesensorless closed loop controller in period I, where the reference amplitudes of vi-bration velocities were set to 0.45 m/s in both X andY directions with a phase angledifference of θxy = 90◦ to generate a circular path. The controller was switched offin period II, and vibrations were excited at a fixed frequency and a constant supplyvoltage, which resulted in a distorted shape deviated from the circle. Without thecontroller, the θxy = 90◦ phase difference to generate the circular path cannot bedelivered. In period III, the vibration actuator was turned off completely which ledto the increased cutting forces. The cutting force amplitudes were reduced by about30% in Period I with a controller relative to Period III where the ultrasonic vibra-tion actuator was off, and the cutting forces were reduced by about 10% relative toPeriod II where uncontrolled ultrasonic vibrations were excited.524.7 SummaryThis chapter presented a sensorless closed loop control system for a 3DOF ultra-sonic vibration tool holder. A linearized lumped parameter model of the piezoelec-tric actuator for each axis is used to track the resonance and to estimate the vibra-tion amplitudes during cutting. A Kalman filter based observer is used to estimatethe vibration amplitude and phase from the measured driving currents and supplyvoltages without using vibration sensors. The proposed closed loop control sys-tem has been demonstrated to maintain the vibration amplitude at the desired levelwhile adjusting the excitation frequency to match the varying resonance frequencyof the actuator during machining. The proposed method ensures the delivery ofdifferent vibration amplitudes in each direction with the desired phase, which al-lows circular or elliptical vibration paths during machining. The system has beendemonstrated in turning and drilling with unidirectional vibrations, as well in two-axis milling with circular vibration loci. The ultrasonic vibrations with a sensorlesscontroller reduced the cutting forces by about 30% compared to the conventionalcutting in sample drilling and milling tests.5330 40 50 60 70 80 90 100-120-80-40040 Fx[N]30 40 50 60 70 80 90 100t[s]-60-40-20020 Fy[N]Period I-0.5 0 0.5-0.500.5-0.5 0 0.500.5-0.5a)b) c)Period II Period IIIperiod I locusperiod II locusFigure 4.13: Milling tests with 3DOF ultrasonic vibration tool holder: a)Feed (Fx) and normal (Fy) cutting forces during up milling of Al7050.b) circular vibration locus during milling generated by the sensorlesscontrol system; c) vibration locus during milling at a fixed excitationfrequency and supply voltage without controller. Cutting conditions:Spindle speed=1000 rev/min, feed rate=0.1 mm/rev, depth of cut=1.5mm, the width of cut=2 mm. The workpiece material is aluminumalloy 7050.54Chapter 5Dynamics of Vibration-AssistedMilling5.1 OverviewTo understand the vibration assistance and to use the proposed 3DOF ultrasonic vi-bration tool holder properly, this chapter presents the dynamics of elliptical vibration-assisted milling and predicts the chatter stability. Section 5.2 models the dynamicsof elliptical vibration-assisted milling process by investigating the uncut chip thick-ness, the process time delay, tool-workpiece separations, and the process dampingeffect. Section 5.3 presents the stability prediction of vibration-assisted milling us-ing the semi-discrete method, followed by simulations and milling experiments insection 5.4 for two materials.5.2 Milling Process with Elliptical Vibration5.2.1 Chip generation with elliptical vibration assistanceThe elliptical vibrations are delivered in radial and tangential directions by thepiezo actuators embedded in the tool holder at its resonance frequency which de-pends on the diameter and length of the end mill (i.e. fe ≈ 16− 17kHz). Thevibrations along the tangential and radial directions are excited at the same ultra-55sonic vibration frequency fe but with an adjustable phase shift of θ to generate anelliptical path (Figure 5.1a)). The generated vibrations by tooth j in the tangential(a j(t)) and radial (b j(t)) directions are:a j(t) = A0 cos[ωet+( j−1) 2piN]b j(t) = B0 cos[ωet+( j−1) 2piN +θ] } (5.1)where N is the number of flutes on the end mill, ωe = 2pi fe [rad/s] is the ultrasonicexcitation frequency, and A0 and B0 are vibration amplitudes which can be adjustedbetween 5-10 µm by using the closed loop controller developed in Chapter 4. Thevibration locus at the tool tip is an ellipse with harmonically varying amplitudeand speed. The tooth spacing angle is φp = 2pi/N for end mill with uniform pitchcutters. The maximum velocity of the tangential vibration (da j(t)/dt) is higherthan the nominal cutting speed (Vs) in order to force the cutting edge to producehigh-frequency intermittent contact with the workpiece surface. The tangentialvibration (a j(t)) leads the radial vibration (b j(t)), i.e. θ > 0. As shown in Figure5.1 b), the tool first moves into the workpiece at point A, and moves away radiallytowards B to lose the contact with the cut surface. The cutting edge also losesthe contact with the chip because the motion in direction u has a velocity in theopposite direction to the nominal cutting speed Vs (see point B in Figure 5.1b)).The immersion angle φ j(t) at tooth j is influenced by both the spindle speed (Ω[rad/s]) and the tangential vibrations (a j(t)) delivered by the ultrasonic actuator:φ j(t) =Ωt+a j(t)R+( j−1)φp (5.2)where R is the radius of the milling tool.The cutting edge motion is governed by the rigid body rotation and linear feedof the tool; self excited, regenerative vibrations between the tool and workpiecewhich are contributed by the dynamic flexibilities of their structures; and the ultra-sonic vibrations imposed at a constant frequency with fixed amplitudes (A0,B0) intangential and radial directions by the piezo actuator. The dynamic chip thickness(h j(t)) removed by the tooth j at time instant t is measured in the radial directionwith the following three components (Figure 5.2):56C1XYj(t)RWorkpieceuvΩVibration locuskxcxkycyVsMilling toolA0B0a)BVsuvworkpieceAVsuvworkpieceΩmill tooth in cutb)Ωmill tooth out of cutcutting tooth patha(t)b(t)Figure 5.1: Elliptical vibration-assisted milling process.h j(t) = g j(t) [h0, j(t)+∆b j(t)+∆v j(t)] (5.3)The quasi-static chip thickness (h0, j(t)) removed by the rigid body motion of millingtool at the instantaneous immersion angle φ j(t) ish0, j(t) = csinφ j(t). (5.4)where c is the feed rate in [m/rev/tooth]. The regenerative chip generated by struc-tural vibrations (v j) between the present and previous in-cut teeth can be expressedas:∆v j(t) =− [v j(t)− v j(t− τ j)] (5.5)where τ j is the time delay between the present and the previous teeth which arein contact with the work material. If two adjacent teeth are in cut, the time delay(τ j) will be the tooth passing period (T = 2piΩ/N). The structural vibrations con-tributed by the machine and the workpiece are modeled in feed (x) and normal (y)directions and projected to the cutting edge ( j) location in the radial direction as:v j(t) = x(t)sinφ j(t)+ y(t)cosφ j(t) (5.6)57Ω cutterhj(t)bq(t-τj)+vj(t-τj)bj(t)+vj(t)h0,jφjuvtooth path ABvj(t)bj(t)Figure 5.2: Dynamic chip thickness.The chip thickness produced by the ultrasonic vibrations along the radial di-rection is modeled as the difference between the marks left by the present tooth jat time t (b j(t)) and the previous in-cut tooth q in the radial direction with τ j delay( bq(t− τ j)):∆b j(t) =− [b j(t)−bq(t− τ j)] (5.7)The excited radial vibration (b j(t)) changes the dynamic chip thickness, but it hasa fixed frequency and amplitude which contributes to the forced vibrations only.However, the tool loses its contact with the cut surface when the amplitude of theultrasonic vibration becomes larger than the chip thickness.The tool-workpiece contact is indicated by the step function g j(t) in Eq.(5.3)as {g j(t) = 1, tool is in contactg j(t) = 0, tool is not in contact(5.8)Tool loses the contact with the workpiece under several conditions. When the toothis out of the immersion as in regular milling,{g1, j(t) = 1,φst ≤ φ j ≤ φexg1, j(t) = 0,φ j < φst or φ j > φex(5.9)58where φst and φex are the entry and exit angles of the cutter relative to the part,respectively.5.2.2 Time delay in elliptical vibration-assisted millingThe time delay (τ j) in Eq.(5.7) may not always be equal to the tooth passing perioddepending on the instantaneous chip thickness and ultrasonic vibration amplitudeof the tooth. By ignoring the regenerative term (∆v j) in the resultant chip thicknessin Eq.(5.3), the chip thickness becomes:hr0, j = csinφ j(t)−b j(t)+bq(t− τ j) (5.10)which can be zero or negative for small feed rates (c) or the immersion angles (φ j).Hence the tool disengages from the workpiece along the radial direction when:{g2, j(t) = 1,hr0, j(t)≥ 0g2, j(t) = 0,hr0, j(t)< 0(5.11)The time delay (τ j) is determined by numerically searching the previous in-cuttooth at the present immersion angle φ j(t) in time domain as described in Algo-rithm 1, where ti is the time at discrete interval i, n is the number of tooth passingperiods to search last tooth in cut, and q is the index of the tooth. The index qis derived based on the present tooth j and the number of teeth N as described inAlgorithm 1 Lines 4-7. The searching procedure starts from n = 1, which corre-sponds to the last tooth present at the immersion angle φ j(ti). If the last tooth isnot in cut, where g2,q(tq) = 0, then the tooth in the previous two cycles is examineduntil a tooth contacts the material (i.e. g2,q(tq) = 1). Once the time of last in-cuttooth (tq) is determined, the time delay is evaluated as:τ j(ti) = ti− tq (5.12)Figure 5.3 shows two simulations at the same spindle speed but with differentfeed rates. If the feed rate is much larger than the amplitude of the radial vibrationsuch as the conditions given in Figure 5.3 a), the tooth will only jump out of theworkpiece along radial direction around the entry angle for up milling and around59Algorithm 1: Time delay solverInput : φ j(ti)Output: τ j(ti)1 foreach ti do2 foreach j do3 n= 14 q= mod ( j−n,N)5 if q= 0 then6 q= N7 end8 φq(tq) = φq(ti)− 2piN9 obtain tq from φq10 while g2,q(tq) = 0 do11 n= n+112 q= mod ( j−n,N)13 if q= 0 then14 q= N15 end16 φq(tq) = φq(tq)− 2piN n17 obtain tq from φq18 end19 τ j(ti) = ti− tq20 h j(ti) = cτ sinφ j(ti)+bq(tq)+b j(ti)21 end22 endthe exit angle for down milling where sinφ j is small. The time delay will be equalto the tooth passing period (T ) except at the entry region for up milling and the exitregion for down milling. If the feed rate per tooth is comparable to the amplitude ofradial vibration such as in Figure 5.3b), multiple tool-workpiece separations willoccur within the immersion but the time delay will be integer multiples of toothpassing periods. In particular, when the tooth passing period (T ) is selected basedon the excitation vibration period as:ωeNΩ=TTe= k (5.13)601080 1260 1440 [deg]0.511.522.5/T1st tooth2nd tooth1080 1260 1440 [deg]0.511.522.5/T1st tooth2nd tootha) b)Figure 5.3: Time delay for two cases: a) c= 0.1 [mm/rev/tooth]; b) c= 0.01[mm/rev/tooth]. The spindle speed is 1600 [rev/min], and the mill hastwo teeth in a slot cutting. The radial vibration amplitude B0 = 10 [µm].where k is an integer number and Te = 2pi/ωe is the the ultrasonic excitation vi-bration period, the time delay is a constant and equals to one tooth passing period(τ j = T ). Hence the selection of tooth passing frequencies (i.e. spindle speed timesnumber of teeth) as the integer division of ultrasonic vibration frequency cancelsits contribution (∆b j(t) = 0) to the dynamic chip thickness in Eq.(5.3).5.2.3 Tangential tool-workpiece separationThe resultant tangential speed (Vr, j) of the tooth j is the summation of cutting speedand tangential vibration velocity:Vr, j(t) =Vs+ a˙ j(t) =Vs−A0ωe sin[ωet+( j−1) 2piN](5.14)The cutting edge periodically disengages from the cut surface when the actua-tor imposed oscillating tangential vibration velocity (da j(t)/dt) exceeds the nomi-nal cutting speed (Vs). The amplitude of the vibration velocity is always set greaterthan the nominal cutting velocity (Vs < A0ωe) to produce periodically intermittentcontacts along the tangential direction. Figure 5.4 shows one cycle of the tool-workpiece separation. When the harmonically varying vibration velocity (a˙ j(t))becomes negative, and the resultant velocity at the cutting edge (Vr, j) reaches zero(Vr, j = 0, t = tA in Figure 5.4), and the tooth begins to leave the workpiece. WhentA < t < tB, the cutting edge travels back until the resultant velocity (Vr, j) reaches61tttAtBtCt=tAtA<t<tBVr,j<0tB<t<tCVr,j>0 Vr,j>0t≥tCVr,j=0Vr,jsr,jtAtBtCsmaxVsVstFigure 5.4: Tangential resultant velocity at tool again at time tB in Figure 5.4. Then, the tangential vibration velocity increasesthe resultant cutting speed and reaches the nominal cutting speed (VS) at t = tC.The resultant displacement (sr, j) generated at the tool tip by the nominal cuttingspeed Vs and tangential vibration a j(t) is:sr, j =Vst+A0 cos[ωet+( j−1) 2piN](5.15)The resultant displacement (sr, j) oscillates about the nominal displacement (Vst)due to the tangential vibration. As shown in Figure 5.4 for one cycle separation,when the tool leaves the workpiece at tA, the resultant displacement (sr, j) reachesa local maximum point (s = smax). As the tool travels back after t = tA, the re-sultant displacement is decreased until t = tB because of the negative tangential62velocity. The resultant displacement rises after tB when the resultant tangential ve-locity becomes positive, and then it reaches the local maximum point smax againat t = tC. Meanwhile, the cutting tool touches the workpiece. The local maximum(smax) keeps updating until the resultant velocity reaches zero again after tC .Thetool contact along the tangential direction is considered in the step function as:{g3, j(t) = 1,sr, j(t)≥ smaxg3, j(t) = 0, else(5.16)The resultant displacement (sr, j) is recorded, and the local maximum point (smax) iscalculated and updated using the recorded displacements at each sampling periodin the simulation.The overall tool-workpiece contact (g j(t)) is determined by combining thethree types of tool-workpiece separations as:g j(t) = g1, j(t)g2, j(t)g3, j(t) (5.17)where g1, j, g2, j and g3, j are defined in (5.9), (5.11) and (5.16) respectively.5.2.4 Process damping effectContacts between the round cutting edge of cutting tool and the wavy workpiecesurface produce process damping in dynamic cutting [35]. In the elliptical vibration-assisted cutting, there exists additional ploughing process due to the radial vibra-tions as explained in section 5.2.1, which change the process damping. The processdamping forces are modeled as a function of elastic contact volume between thecutting tool and workpiece [2], and the instantaneous process damping forces alongthe radial and tangential directions are defined as:F pdr, j (t) = KspVpdj (t) ; Fpdt, j (t) = µFpdr, j (5.18)where Ksp is the material dependent indentation coefficient which can be identifiedfrom cutting experiments, and µ is the Coulomb friction constant between cuttingtool and workpiece. V pdj is the dynamically indented volume (Vpdj ) for tooth j, andit can be evaluated from the cross-section area (Apdj in Figure 5.5) and depth of cut63LSPβeSPγReγFrpdFtpdVca(t)b(t)uvLWorkpieceCutting toolFigure 5.5: Process damping effectw, i.e. V pdj = wApdj . The area (Apdj ) indented by the flank of tool with an equivalentwear length (L) at cutting velocity (Vr) is expressed as [2]:Apdj =L22Vrr˙ j (5.19)where r˙ j is the velocity along the radial direction for tooth j. The equivalent wearlength (L) can be modeled as (see Figure 5.5):L= Re [sinβe+ sinγ+(cosγ− cosβe)/ tanγ] (5.20)where Re is the edge radius of the cutting tool, γ is the clearance angle of the tool,and βe is the angle to define the separation point (SP) as shown in Figure 5.5.If the process is conventional milling and there is no vibration assistance, theseparation angle (βe) is assumed to be a constant so as the equivalent wear length(L). However, when the elliptical vibrations are present, the separation point dy-namically changes so as the separation angle (βe) and equivalent wear length (L).The dynamic separation angle for tooth j can be evaluated from the geometry givenin Figure 5.5 as:βe(t) = arccos[Re cosβ0+b j(t)Re](5.21)where β0 is the separation angle without elliptical vibration assistance, and b j(t) is64the excited vibration displacement along the radial direction defined in Eq.(5.1).The radial velocity (r˙ j) is contributed by the excited ultrasonic vibration (b j(t))and the regenerative vibration (v j(t)) as:r˙ j(t) = v˙ j(t)+ b˙ j(t) (5.22)The instantaneous indented area (Apdj ) in (5.19) can be evaluated from (5.22). Theultrasonic vibration (b j(t)) is a forced vibration with the known amplitude andfrequency, thus it doesn’t contribute to the chatter stability of milling process.5.3 Chatter Stability Analysis5.3.1 Dynamic milling forceThe quasi-static components (csinφ j(t)+∆b j(t)) of the dynamic chip thickness(Eq.(5.3)) cause only forced vibrations and do not affect the chatter stability of theprocess, hence they are dropped. Only regenerative term with the step functiong j(t) which reflects the tool disengagements contributed by the ultrasonic vibra-tions is considered as follows:hw, j(t) = g j(t) [−(v j(t)+ v j(t− τ j)] = g j(t)[sinφ j cosφ j][x(t)− x(t− τ j)y(t)− y(t− τ j)](5.23)where φ j is the immersion angle of tooth j, and [x(t),y(t)]T are the structural vi-brations in feed and normal directions, respectively. Dynamic tangential (Ft, j) andradial (Fr, j)cutting forces generated by tooth j are expressed as:[Ft, jFr, j]= Ktwhw, j(t)[1Kr]+g j(t)Ceq[µ1]v˙ j (5.24)where w is the axial depth of cut and Ceq = L2w/(2Vr) is the equivalent processdamping coefficient. The milling forces contributed by N number of teeth are65projected to feed and normal directions:[FxFy]=N∑j=1[−cosφ j −sinφ jsinφ j −cosφ j][Ft, jFr, j](5.25)By substituting Eqs.(5.23) and (5.24) into Eq.(5.25), and the dynamic millingforces in xy coordinates are expressed as:[FxFy]=12KtwN∑j=1g j(t){[αxx, j αxy, jαyx, j αyy, j][x(t)− x(t− τ j)y(t)− y(t− τ j)]+Ceq[βxx, j βxy, jβyx. j βyy, j][x˙(t)y˙(t)]}(5.26)where the directional factors (αxx, j, αxy, j, αyx, j, αyy, j) and process damping coeffi-cients (βxx, j,βxy, j,βxy, j,βyy, j) are:A j(t) =[αxx, j αxy, jαyx, j αyy, j]=[−cosφ j sinφ j−Kr sin2 φ j −cos2 φ j−Kr sinφ j cosφ j−sin2 φ j−Kr cosφ j sinφ j sinφ j cosφ j−Kr cos2 φ j]B j(t) =Ceq[βxx, j βxy, jβyx. j βyy, j]=Ceq[−µ cosφ j sinφ j− sin2 φ j −µ cos2 φ j− sinφ j cosφ j−µ sin2 φ j− cosφ j sinφ j µ sinφ j cosφ j− cos2 φ j](5.27)which are periodic and g j(t) is time varying as the spindle rotates (φ j=Ωt) becauseit is dependent on ultrasonic vibrations (a j,b j) and feed rate (c).5.3.2 Stability of milling with ultrasonic vibrationsThe milling dynamics with multiple modes are expressed in modal coordinatespace as:¨ˆx+C˙ˆx+Dxˆ = UTF (5.28)66where U is the mode shape matrix, C=2ζ1ωn,1 · · · 0.... . ....0 · · · 2ζnωn,n, D=ω2n,1 · · · 0.... . ....0 · · · ω2n,n,ωn,k and ζk are the natural frequency and modal damping ratio for mode k, and n isthe number of dominant modes of the spindle-holder-tool assembly. F=[Fx Fy]Tis the dynamic milling forces acting on the milling tool. The modal displacements( xˆ =[xˆ1 xˆ2 · · · xˆn]T) can be transformed to physical vibrations with:[xy]= Uxˆ (5.29)Substituting Eq.(5.26) into Eq.(5.28), the system is represented by a set of de-layed differential equations with periodic coefficients:¨ˆx+C˙ˆx+Dxˆ = δN∑j=1g j(t)UTA j(t)U [xˆ(t)− xˆ(t− τ j)]+UTBU˙ˆx (5.30)where δ = 12Kta and B=∑Nj=1 B j. By letting q(t) =[xˆ(t)T ˙ˆx(t)T]T, Eq. (5.30)is rearranged into a set of first-order equations as [32]:q˙(t) = Lq(t)+N∑j=1R jq(t− τ j) (5.31)where L(t) =[0 Iδ ∑Nj=1 g j(t)UTA j(t)U−D UTBU−C]andR j(t) =[0 0−δg j(t)UTA j(t)U 0]. The matrices L and R j are time-varying due tothe time-varying factors A j and B j.The Semi-discrete method proposed by Insperger and Stepan[32] is used tosolve the stability of elliptical vibration-assisted milling process governed by Eq.(5.31).The simulation time length (τm) is determined by the maximum time delay, whereτm = max(τ j(t)). Let ∆t be the sampling period, and the simulation period is di-vided into m number of time discrete intervals, i.e. τm = m∆t. At current time67t = ti, for tooth j, the time delay can be expressed in as:li, j =τ j(ti)+∆t/2∆t(5.32)Let q(ti) = qi and the state variable (q) at t = ti+1 is:qi+1 = eLi∆tqi+N∑j=112(eLi∆t − I)L−1Ri, j (qi−li, j+1+qi−li, j) (5.33)where Li = L(ti) and Ri, j =R j(ti). The solution requires the states between qi andqi−m, and the state function in the discrete time domain is expressed as follows:Qi+1 = HiQi (5.34)where Qi =[qTi qTi−1 .. qTi−m+1 qTi−m]T, and Hi is defined as:Hi =eLi∆t 0 ... 0 0I 0 ... 0 00 I ... 0 0....... . .......0 0 ... I 02n(m+1)×2n(m+1)+N∑j=1[0 ... 12(eLi∆t − I)L−1Ri, j 12 (eLi∆t − I)L−1Ri, j ... 00 ... 0 0 ... 0]2n(m+1)×2n(m+1)(5.35)Thus, within a simulation period τm, the transition equation of the states becomes:Qm = HQ0 = Hm · · ·H2H1Q0 (5.36)According to the Floquet theory, the system in (5.36) is stable if all the eigenvaluesof H are less than unity, and the critical stability corresponds to unity eigenvalues.68ultrasonic vibration tool holdermicrophoneworkpieceFigure 5.6: 3DOF ultrasonic vibration tool holder on CNC machine5.4 Simulations and ExperimentsThe stability of milling with elliptical vibration assistance is simulated using thesemi-discrete time domain method as presented in Section 5.3 and compared againstexperimental measurements using the setup shown in Figure 5.6. The helical endmill had 2 flutes with 6 mm diameter and 25 mm overhang. The edge radius of thecutting tool was 15 µm with a clearance angle of 5◦.The frequency response function (FRF) of the tool at its tip was measuredthrough impulse tests using a small (Dytran 5800B2) and miniature (PCB086E80)hammers to cover both low (< 2,000 Hz) and high frequency (< 20,000 Hz) modesof the end mill. A laser Doppler vibrometer (Polytec CLV-2534) was used to cap-ture the displacements. The measured FRFs in feed (x) and normal (y) directionsare given in Figure 5.7 with the curve fitted modal parameters listed in Table 5.1.The actuator’s third bending mode at fe= 16.35 kHz is used to excite the ultrasonicelliptical vibrations with A0 = 10µm in tangential and B0 = 5 µm in radial direc-tions with a phase difference of θ = 90◦. The stability of the system is analyzed for692 4 6 8 10 12 14 16 18 20f [Hz] 103024681012| FRF | [m/N]10-6XY3rd bending mode of the vibration actuator 1st bending mode of the vibration actuatorholder housing bending modeFigure 5.7: Frequency response functions of the milling toolTable 5.1: Modal parametersDirection ωn[Hz] ζ [%] k [N/m]X 369.85 3.16 7.709e61146.36 1.14 8.464e6Y 411.21 1.68 7.056e61069.21 1.55 5.8242e6the first bending dominant modes of the holder (369 Hz,411 Hz) and the end mill(1146 Hz,1069 Hz) since the higher modes beyond 4 kHz are process damped out[35].Aluminum alloy 7050 and AISI 1045 steel alloys have been used as workpiecematerials with the experimentally identified cutting force coefficients (Kt ,Kr) andcomputed indentation (Ksp) coefficients [21]. The separation angle of plastic chipflow under the edge radius is set to βe = 50◦ [35]. The predicted stability lobes andsample measurements during half immersion up milling of Al 7050 are shown inFigure 5.8. The minimum stable depth of cut increases from 1.3 mm to 2.00 mm upto the spindle speed of 2300 rev/min under ultrasonic vibrations. The effect of ul-trasonic vibrations on the stability diminishes at higher speeds because the cutting70800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000spindle speed [rev/min]0.511.522.533.544.5axial depth of cut [m]10-3lobes with elliptical vibrationlobes without vibration assistanceunstable regardless vibration assistancestable regardless vibration assistancestable with vibration assistance but unstable without vibration assistanceAFigure 5.8: Stability lobes for half immersion up milling of Aluminum AL7050 with a feed rate of c = 0.1 mm/rev/tooth. Tool: Two fluted, 6mm diameter helical end mill with a helix angle of 30◦. The modalparameters are given in Table 5.1. Cutting coefficients: Kt = 770 MPa,Kr = 0.218,Ksp = 10000 N/mm3.speed becomes higher than the vibration velocity, hence the intermittent contact be-tween the tool and workpiece is lost. A sample sound measurements correspondingto point A in Figure 5.8 at the axial depth of cut of 1.5 mm and spindle speed of1624 rev/min are shown in Figure 5.9. When the ultrasonic vibration was turnedon in the beginning of the cut, the process was stable, the surface was smooth andthe sound spectrum was dominated by the actuator’s excitation frequency at 16.37kHz, which was a forced vibration. When the vibration actuator was turned off, thechatter developed quickly at the first bending mode of the tool at 1107 Hz whichled to poor surface finish. The presence of chatter was decided by checking theamplitude and side bands which occur at tooth passing frequency (54 Hz) away[55]. Similar phenomenon was observed in all marked experimental results.The predicted stability lobes for half immersion down milling of AISI 1045steel is shown in Figure 5.10. The minimum depth of cut was increased from0.2 mm to 0.35 mm under ultrasonic vibrations. Similar to milling Al 7050, theeffect of ultrasonic vibrations diminished again after 2300 rev/min because thetool does not lose the contact with the workpiece beyond this cutting speed. A710 2 4 6 8 10 12Time [s]-0.01-0.00500.0050.01Micphone [V]ultrasonic vibration offultrasonic vibration on16.37kHza)b) c)5 10 15 20[Hz] 10300.511.522.53[V]10-4800 900 1000 1100 1200 1300 1400[Hz]0.511.522.53[V]10-41107 Hz1053 Hz1162 HzfT=54Hz1107 Hz5 10 15 20[Hz] 10300.[V]10-3800 900 1000 1100 1200 1300 1400[Hz][V]10-41090 Hz 1353Hz 16.37kHz Figure 5.9: Microphone outputs in half-immersion up milling of AL 7050with a two-tooth cutter. Spindle speed: 1624 rev/min; axial depth ofcut: 1.5 mm. a) Microphone data in time domain; b)FFT of microphoneoutput with ultrasonic vibration assistance; c)FFT of microphone outputwhen ultrasonic vibration turned off.sample of sound measurements at 1770 rev/min with 0.4 mm depth of cut (B inFigure 5.10) is given in Figure 5.11. When the ultrasonic vibration was turned on,the vibration was dominated by the frequency of actuator’s ultrasonic excitation721000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000spindle speed [rev/min] depth of cut [m]10-3lobes with elliptical vibrationlobes without vibration assistanceunstable regardless vibration assistancestable regardless vibration assistancestable with vibration assistance but unstable without vibration assistanceBFigure 5.10: Stability lobes for half immersion down milling of Steel AISI1045 with a feed rate of c = 0.1 mm/rev/tooth. Tool: Two fluted, 6mm diameter helical end mill with a helix angle of 30◦. The modalparameters are given in Table 5.1. Cutting coefficients: Kt = 3850MPa, Kr = 0.19,Ksp = 47000 N/mm3.frequency (16.1 kHz). When the actuator was turned off, the system chattered atthe holder’s bending mode of 393 Hz with having side-bands at 59 Hz away.5.5 SummaryThe chatter stability model of the synchronized elliptical vibration-assisted millingprocess has been presented in this chapter. It is shown that the chatter is dominatedat low-frequency modes of the end mill at low speeds, and the excitation of actua-tor’s ultrasonic mode contributes to forced vibrations while improving the chatterstability. The ultrasonic vibrations can improve the chatter stability only if thetool has periodic contacts with the workpiece at the ultrasonic vibration frequency,which occurs if the tangential vibration velocity is higher than the nominal cuttingspeed of the tool or the vibration amplitude is greater than the static chip thicknessgoverned by the feed rate and immersion. The process must be designed to selectthe feed rate, cutting speed, and the excited vibration amplitudes to avoid chatterwithout saturating the amplifier of the piezoelectric actuator. The proposed modelhas been experimentally validated in milling aluminum alloy 7050 and AISI 1045steel alloys.730 2 4 6 8 10 12Time [s]-0.03-0.02-0.0100.010.020.03Micphone [V]ultrasonic vibration offultrasonic vibration ona)b) c)5 10 15 20[Hz] 103012345[V]10-4100 200 300 400 500 600[Hz]00.511.522.5[V]10-4393 Hz334 Hz452 HzfT=59Hz5 10 15 20[Hz] 10300.511.522.533.5[V]10-3100 200 300 400 500 600[Hz]00.511.5[V]10-416.1kHz59 Hz118 HzFigure 5.11: Microphone outputs in half-immersion down milling of AISI1045 steel with a two-tooth cutter. Spindle speed: 1770 rev/min; axialdepth of cut: 0.4 mm. a) Microphone data in time domain; b)FFTof microphone output with ultrasonic vibration assistance; c)FFT ofmicrophone output when ultrasonic vibration turned off.74Chapter 6Vibration Assistance on ChipFormation of Ti-6Al-4V6.1 OverviewThe high-frequency tool-workpiece separation due to ultrasonic vibration assis-tance changes the strain rate and heat transfer conditions in the primary shear zone,further affect the chip formation mechanism. This chapter selects a widely useddifficult-to-cut titanium alloy, Ti-6Al-4V, as the target to investigate the effects ofultrasonic vibration on chip formation. Experimental observations are presentedfirst in section 6.2, followed by a plastic flow model in the primary zone to explainthe observations (section 6.3). Simulations and experimental results are presentedin section 6.4 to verify the proposed plastic flow model.6.2 Experimental ObservationsOrthogonal cutting experiments of Ti-6Al-4V with ultrasonic vibration assistancewere carried out, with the turning setup shown in Figure 6.1. The Ti-6Al-4V work-piece clamped into the spindle was a tube with 6 mm diameter and 1mm thickness.A turning insert was mounted in a stationary ultrasonic vibration actuator, and thetool edge, which was wider than the tube thickness, was oriented along Y-axis toachieve the orthogonal cutting setup. In the cutting experiments, the cutting tool75Spindle Stationary vibration toolTi-6Al-4V tubeKistler 9256C1DynamometerX YZFigure 6.1: Turning experimental setup with an ultrasonic vibration cuttingtool.was fed to the workpiece along Z-axis, and the ultrasonic vibration was generatedalong X-axis (tangential direction) in the cutting experiment, as shown in Figure6.1. The material of the turning insert is uncoated tungsten carbide, and the rakeangle (α) is 5◦. The cutting tool was fixed on a tool holder driven by a stackingpiezoelectric actuator. The bending vibration mode of the tool holder structure isexcited at 15.5 kHz, and the vibration assistance at the turning insert is along thetangential direction at the cutting point. The piezo vibration actuator can producevibrations at the amplitude of 10 µm. Multiple cutting experiments with differentuncut chip thicknesses and cutting speeds listed in Table 6.1 were conducted. Foreach condition, both conventional and vibration-assisted cutting experiments wereconducted. The cutting forces for each experiment were measured by a dynamome-ter (Kistler 9256C1) as shown in Figure 6.1. The bandwidth of the dynamometeris 2 kHz, which is much lower than the excitation frequency of the ultrasonic vi-bration.The machined chips under each cutting condition with and without vibration76Table 6.1: Conditions of orthogonal cutting experimentsNo. Vc (m/min) Uncut chip thickness (mm) Vibrations1 40 0.15 no2 40 0.15 yes3 50 0.15 no4 50 0.15 yes5 40 0.2 no6 40 0.2 yes7 50 0.2 no8 50 0.2 yesassistance were collected and polished. They were etched by Kroll’s reagent to re-veal the microstructure, and then were examined by scanning electron microscopy(SEM) imaging (Hitachi SU3500). Figure 6.2 shows the SEM images of the chipscorresponding to 40 m/min cutting speed and 0.15 mm uncut chip thickness, atthe magnifications of 200X, 500X, and 3000X, respectively. When no vibrationassistance is applied, the chip segmentations were formed in a saw-tooth shape, asshown in Figures 6.2 a) - c). Adiabatic shear bands are clearly observed betweentwo neighboring segmented chips, including highly distorted grain structures (seeFigure 6.2c)) compared to other locations. The width of the shear band is about6 - 8 µm from the SEM measurement. Figures 6.2d) - f) show the morphologyand microstructure of the chip in ultrasonic vibration-assisted cutting. The chipsformed with ultrasonic vibration assistance also has segmentations (Figure 6.2d)),however, compared to the no-vibration result, there is no obvious shear band for-mation, and no severe distorted grain structures were observed between the seg-mented chips (see Figures 6.2 e) and f)). In addition, the pitch length of the chipsegmentation with ultrasonic vibration (about 130 µm) is larger than that in con-ventional cutting (about 90 µm) by comparing Figures 6.2a) and d). Therefore,the experimental results demonstrate that the ultrasonic vibration assistance playsa significant role on the shear band formation and chip segmentation in cuttingTi-6Al-4V. Figure 6.3 shows the SEM micrographs of the machined chips cor-77adiabatic shearbandsadiabatic shearbandsadiabatic shearbanda) b) c)d) e) f)90μm130μmFigure 6.2: SEM images of segmented chips: a)-c) conventional cutting withmagnifications of 200X, 500X, 3000X respectively; d)-f) ultrasonicvibration-assisted cutting with magnifications of 200X, 500X, 3000Xrespectively.responding to tests No. 3-8 in Table 6.1, and similar observations on the effectof vibration assistance on adiabatic shear bands were obtained at different cuttingconditions.The studies in [18, 44] report that the chip segmentations in cutting Ti-6Al-4V at high cutting speed (larger than 30 m/min) is caused by heat accumulationwithin the primary shear zone to generate shear bands. Thus, in this chapter, it ishypothesized that the disappearance of adiabatic shear bands caused by the ultra-sonic vibration assistance is due to less heat accumulation, which results in lowertemperature in the primary shear zone. Therefore, ultrasonic vibration assistanceis able to improve the machinability of Ti-6Al-4V with lower temperature as theexperimental observations suggest. Furthermore, the tool life is expected to beelongated without reducing productivity. However, no physical model of this cut-ting process was reported in the literature. Section 6.3 is to model the stress andtemperature variations as well as plastic shear flow in primary shear zone in cut-ting of Ti-6Al-4V with ultrasonic vibration assistance, in order to quantitatively78no vibration assistanced=0.15 mm, Vc=50 m/min d=0.20 mm, Vc=40 m/min d=0.20 mm, Vc=50 m/mina) b) c)d) e) f)adiabatic shear bandadiabatic shear bandadiabatic shear bandwith vibration assistanceFigure 6.3: SEM images of segmented chips under cutting conditions in Ta-ble 6.1 with 200X and 3000X magnifications for each case. a) test No.3;b) test No.5; c) test No.7; d) test No.4; e) test No.6; f) test No.8.determine the effect of vibration assistance on the chip formation mechanism.6.3 Plastic Flow in Primary Shear ZoneThe process model focuses on the material deformation in primary shear zone,which determines the chip segmentation, further to explain the observations in sec-tion 6.2. The model is based on plasticity theory to characterize the material flowin forming the chip. The heat transfer in the shear zone is considered to determine79the effect of temperature increasing on the plastic flow. In the literature, multi-ple researchers contributed to modeling the chip formation analytically. Burns andDavies established governing equations to solve the shear stress and temperaturein the primary shear zone[11]. Bai et al. studied the chip formation of Ti-6Al-4Vanalytically using governing equations in the primary shear zone and the tool-chipinterface, and they predicted the pitch of segmented chip caused by adiabatic shearbands[7]. Ning et al. modeled the cutting process of ultra-fine-grained titanium an-alytically using Johnson-Cook constitutive relationship[64]. Ning and Liang pre-dicted the temperature in the primary shear zone and tool-chip interface by using amaterial constitutive model[63].The vibration assistance changes the cutting tool kinematics, therefore influ-ences the stress and temperature variations in the primary shear zone. In thismodel, the governing equations of shear stress and temperature including the effectof vibration assistance are established, and the material flow is predicted from thematerial constitutive law. By solving the governing and the constitutive equations,the shear stress and the temperature in the primary shear zone are simulated in timedomain.When vibration assistance is applied along the tangential direction (X-axis inFigure 6.1) in orthogonal cutting, the cutting speed changes periodically. The as-sisted ultrasonic vibration is a sinusoidal function at the excitation frequency ( f ),then the cutting tool displacement (x(t)) due to the assisted vibration is defined as:x(t) = x0 cos(2pi f t) (6.1)where x0 is the amplitude of the ultrasonic vibration. The overall cutting speed(V ) is the summation of original cutting speed along tangential direction (Vc) andvibration velocity (Vf ), defined as:V (t) =Vc+Vf (t) =Vc−V0 sin(2pi f t) (6.2)where V0 is the amplitude of the vibration velocity defined as V0 = 2pi f x0.With vibration assistance, the workpiece and the cutting tool contact intermit-tently ifV0 >Vc. Otherwise, the tool keeps in contact with the workpiece with vary-80αcungtoolc)σταφVcungtoola)primary shear zoneσταφVcungtoolb)Figure 6.4: Orthogonal cutting with ultrasonic vibration along tangential di-rection: a) tool-workpiece in-contact period with cutting motion; b) chipelastic recovery period; c) tool-workpiece separation speeds. In this study, we focus on the intermittent contact condition, thus threestages are discussed regarding the tool-workpiece interaction, with the schematicsshown in Figure 6.4: a) the cutting tool is in contact with the workpiece materialwhen the relative displacement between the tool and the workpiece is zero and therelative speed (V ) between them is positive, that the cutting motion occurs; b) cut-ting tool retracts whenV < 0, but the tool and the workpiece are still in contact dueto the elastic recovery of the chip; c) cutting tool is separated from the workpiecematerial when the relative displacement between tool and workpiece is less thanzero.Stage a): Cutting tool in contact with workpiece with cutting motionUnder this condition, the workpiece material has plastic deformation in primaryshear zone due to the cutting tool motion, and forms the machined chip. Figure6.5 shows the deformation zone in 2-D orthogonal cutting process. The primaryshear zone is assumed to have a small thickness (h). Meanwhile, it is assumed thatthe shear stress and temperature are distributed uniformly along the shear plane.According to Burns and Davies [11], although the shear stress in the tool-chipcontact interface causes plastic deformation of the chip material, the normal stressapplied from the rake face to the chip generates local elastic deformation of thechip material because the back side of the machined chip is free surface, shown inFigure 6.5.Based on the force equilibrium principle of the chip, the resultant force applied81dσταφVchLFfrcu ngtoolprimary shear zoneV2V1FnFselastic deformation zoneFigure 6.5: Deformation zone in 2D orthogonal the chip in the shear plane direction is zero, that is:Fn cos(φ −α)−µFn sin(φ −α) = Fs (6.3)where Fn is the normal force on the rake face, Fs is the shear force in the primaryshear zone, µ is the average friction coefficient between the tool and the workpiece,φ is the shear angle, and α is the rake angle of the cutting tool. Based on equation(6.3), the relationship between the average normal stress along the chip contactregion and the shear stress in the primary shear zone is expressed as:[σ cos(φ −α)−µσ sin(φ −α)]Lb= τb dsinφ(6.4)where σ is the average normal stress on the contact length L, τ is the shear stressinside the primary shear zone, d is the uncut chip thickness, and b is the width ofcut.When chip segmentation occurs, the chip has a displacement ∆u perpendicularto the tool rake face, causing unloading on the cutting tool. The displacement (∆u)82causes a normal strain in the local elastic deformation region (the green area inFigure 6.5), expressed as ε = ∆u/(d/sinφ cos(φ −α)).The chip segmentation causes time-varying chip displacement ∆u. As a re-sult, the time-differentiation of normal stress due to the unloading from the chipsegmentation is:σ˙ = Esinφd cos(φ −α)∆u˙ (6.5)where E is the elastic modulus of the chip material.Combined with (6.4), the vari-ation of shear stress with respect to time is expressed as:τ˙ =ELsin2 φd2[1−µ tan(φ −α)]∆u˙ (6.6)where ∆u˙ is the speed difference between the tool and the segmented chip alongthe direction perpendicular to tool rake face, which is:∆u˙= (V1−V2)cos(φ −α) (6.7)where V1 and V2 are the tool and formed chip speeds along the shear plane, respec-tively.The shear speed due to cutting tool motion V1 is a function of cutting speed V ,shear angle φ , and tool rake angle α , expressed as:V1 =V cosαcos(φ −α) (6.8)Meanwhile, the chip flow speed corresponding to chip segmentation V2 is re-lated to the actual shear strain rate of the plastic flow in the primary shear zone,defined as:V2 = γ˙ph (6.9)where γ˙p is plastic shear strain rate of material in primary shear zone due to thechip segmentation. Substitute (6.8) and (6.9) into (6.7), and based on the cuttingspeed equation (6.2), the shear stress variation with time in (6.6) is updated as:τ˙ =ELVc sin2 φd2[1−µ tan(φ −α)]cosα[1− V0Vcsin(2pi f t)− γ˙pγ˙0](6.10)83where γ˙0 = Vc cosα/cos(φ −α)/h refers to the shear strain rate in the primaryshear zone corresponding to the original cutting speed without vibration assistance(Vc). In equation (6.10), γ˙0 is a constant determined by the tool geometry and theoriginal cutting speed, while γ˙p is a time-varying parameter associated with thechip segmentation.Heat is also generated in primary shear zone during the cutting motion due tothe plastic deformation of the workpiece material. By taking half of the primaryshear zone as a controlled volume, the energy conservation condition is satisfied inthe volume [11]. The overall stored energy is balanced by the heat generation, theconvection of mass inflow and the heat transferred by conduction. The fundamentalenergy conservative relationship within the controlled volume is defined as follows:E˙st = ∆E˙conv+∆E˙cond+ E˙gen (6.11)where Est is the stored energy inside the controlled volume, ∆Econv is the heatconvection variation due to the mass flow rate related to cutting speed inside theprimary shear zone, ∆Econd is the heat conduction, and heat generation (Egen) isfrom strain energy caused by material deformation of the workpiece material.The temperature at both boundaries of the primary shear zone is assumed asroom temperature (T0), and the temperature at the centerline of the primary shearzone is T . Therefore, the governing equation of temperature variation with respectto time is derived through (6.11) as:ρcT˙ = ρc2(T0−T )hV sinφ +4kT0−Th2+ τγ˙p (6.12)where k is the thermal conductivity, ρ is the mass density, and c is the specific heatcapacity of the workpiece material.The plastic strain rate of the material in the primary shear zone is influencedby the shear stress and the temperature, and it determines the plastic flow for chipformation. Since material plastic deformation occurs in the primary shear zone,Johnson-Cook (J-C) constitutive model which is used to describe the flow stressunder large strain rate and high temperature is used in this model, described as84follows[45]:τ =[τA+ τB(γp√3)n][1+C ln(γ˙pγ˙re f)][1−(T −T0Tm−T0)m](6.13)where γp is the shear strain in the plastic region, which is given by the average valueof the plastic strain in this model, γ˙p is the plastic strain rate, τA, τB, Tm,C, γ˙re f , n, mare material property related constants. To simplify the proposed model so that thegoverning and constitutive equations are analytically solvable, in the simulations,the average plastic strain (γp) is used for the strain hardening term based on con-stant material flow assumption, and it is obtained by γp= cosα/(sinφ cos(φ−α)),where φ is experimentally calibrated shear angle from average thickness of themachined chip. By letting τ0 = τA+ τB(γp/√3), the strain rate of the material isobtained from the Johnson-Cook constitutive law as a function of stress, strain andtemperature, expressed as:γ˙p = γ˙re f exp 1C ττ011−(T−T0Tm−T0)m −1 (6.14)The equations (6.10), (6.12) and (6.14) describe the variations of shear stress,temperature and shear strain rate of the workpiece material in the primary shearzone. They form the coupled governing equations in describing the mechanism ofsegmented chip formation during cutting motion with vibration assistance in thetangential direction.Stage b): Tool retraction when tool and chip are in contact (elastic recovery)As the overall cutting tool velocity including the vibration assistance changesdirection, the tool retracts and moves away from the workpiece material. Since thechip is deformed elastically at the tool-chip contact area as shown in Figure 6.5,it will bounce back when the cutting tool retracts, until complete elastic recoveryis achieved. During this period, the cutting tool and the chip are still in contact,therefore sharing the same velocity, as shown in Figure 6.4 b). The normal stress(σ ) in the tool-chip contact area reduces based on the cutting tool′s displacement,expressed as (6.15), and the shear stress (τ) in the primary shear zone is calculated85from the chip equilibrium condition by (6.16):σ = E1cos(φ −α) ·∆usep−u(t)d/sinφ(6.15)τ = σ [cos(φ −α)−µ sin(φ −α)] sinφd(6.16)where ∆usep is the relative displacement between the tool and the chip when thecutting tool retraction just starts to occur, and V (t) is zero at this moment; u(t) isthe displacement of the cutting tool during retraction period, and it is calculated bythe integration of the cutting tool velocity.During the tool retraction period, no further plastic deformation occurs in theprimary shear zone, therefore, the heat generation and mass inflow components in(6.12) become zero. However, heat conduction still occurs, which transfers the heatout of the primary shear zone. As a result, the temperature variation is governedby:T˙ =4kρcT0−Th2(6.17)Stage c): Tool-chip separation periodAfter the complete elastic recovery of the chip material, the cutting tool is sep-arate from the workpiece material, shown in Figure 6.4 c). There is no mechanicalloading on the workpiece material from the cutting tool, hence the shear stress inthe primary zone is zero:τ = 0 (6.18)Only heat conduction occurs in the primary shear zone, and it is also governedby (6.17) in the separation period. Based on the governing equations in each tool-workpiece interaction period, the stress and temperature variations in the primaryshear zone can be solved in time domain using finite difference method, with thesimulations and experimental validations presented in the next section.86Table 6.2: Ti-6Al-4V material propertiesParameter Value UnitMass density ρ 4430 kg/m3Young’s modulus E 105 GPaSpecific heat conductivity k 16 W/(m·K)Melting temperature Tm 1610 ◦C6.4 Simulations and ExperimentsTime-domain simulations are performed to predict the stress and temperature varia-tions in the primary shear zone in cutting of Ti-6Al-4V with and without ultrasonicvibration assistance.As discussed in section 6.3, Johnson-Cook constitutive law is used in thismodel to determine the relationship between the strain rate, shear stress, shearstrain, and temperature. Different methods in determining the Johnson-Cook con-stitutive parameters have been reported in the literature. Split-Hopkinson pressurebar tests were widely used to experimentally calibrate the parameters under dif-ferent strain, strain rate and temperature of the material [45, 77]. In addition,inverse identification of J-C parameters from orthogonal cutting tests was inves-tigated [1, 90]. Numerical approaches were also used to adjust J-C constants to beused in cutting process with better accuracy [91]. Moreover, analytical methodsfor obtaining J-C constants in cutting simulations were reported in the literature.Ozel and Zeren identified the J-C parameters based on Oxley’s analytical cuttingprocess model[65]. Recently, Ning and Liang developed an inverse identificationmethod to obtain J-C constants based on chip formation model and an iterativegradient searching algorithm [62].The mechanical properties of Ti-6Al-4V are listed in Table 6.2, which are usedin solving the governing equations from the developed model. The parameters ofJohnson-Cook constitutive property for Ti-6Al-4V are obtained from [45], whichuses SHPB tests to calibrate the parameters, with the material constants ( τA, τB,C,m, n) defined in J-C equation (6.13) listed in Table 6.3.87Table 6.3: Ti-6Al-4V Johnson-Cook parametersτA (MPa) τB (MPa) C n m γ˙re f724.7 683.1 0.035 0.47 1.0 10−5In the simulations, the rake angle (α) of the cutting tool is 5◦. The thicknessof the primary shear zone is assumed as 1/10 of the uncut chip thickness, andthe contact length is assumed as twice of the uncut chip thickness[4]. The frictioncoefficient between the cutting tool and and the machined chip is assumed to be 0.3[12]. The shear angle (φ ) was calibrated through orthogonal cutting experimentsby measuring the average chip thickness under different cutting conditions listedin Table 6.1, and the shear angle was calculated by assuming constant materialflow using (6.19), and was used to determine the average strain in J-C constitutiveequation (6.13)[4].φ = arctanr cosα1− r sinα (6.19)where r is the compression ratio of the chip defined as r = d/dc, dc is the averagethickness of the machined chip. Based on the measured chip compression ratio,the average shear angle φ is calculated as 40◦.The proposed physical model predicts the shear stress and temperature varia-tions in the primary shear zone given the cutting speed and the uncut chip thicknessvalues. Figure 6.6 shows the comparison of the simulated shear stress and temper-ature without and with ultrasonic vibration assistance for two seconds to show thesteady state response from 1.9 s. When the cutting speed is 40 m/min and the uncutchip thickness reaches 0.15 mm under no vibration assistance condition, the shearstrain rate increases and the heat accumulation is localized in the primary shearzone. The adiabatic shear bands are formed to generate the segmented chip, asshown in the SEM image in Figure 6.2 a). In the simulations, the cutting motionoccurs in the whole time period, therefore, the governing equations (6.10), (6.12)combining with the constitutive relationship (6.14) are solved with constant cut-ting speed, and the results show periodic variations of shear stress and temperature88in Figure 6.6 a), which corresponds to the shear band formation. The maximumtemperature within the primary shear zone reaches above 800 ◦C. The oscillationfrequency of the shear stress and temperature is about 27 kHz from the simulationresults.In the simulations of ultrasonic vibration-assisted cutting, the assisted vibrationparameters are based on the experimental conditions corresponding to Figure 6.2d)-f) where the frequency is 15.5 kHz, and the amplitude is 10 µm. Intermittenttool-chip contact and speed variations are considered in the simulations, with theresults of shear stress and temperature variations shown in Figure 6.6 b). Oscilla-tions of shear stress and temperature still exist, however, the oscillation frequencyis determined by the excitation frequency of the ultrasonic vibration, which is 15.5kHz. Furthermore, the shear stress drops to zero periodically due to the tool-chipseparation, and the maximum of temperature in the primary shear zone is about300 - 350 ◦C, which is much lower than the temperature in cutting without vibra-tion assistance. In vibration-assisted cutting, when the tool is separated from theworkpiece, there is no heat generation from the plastic deformation of the mate-rial, and the heat is transferred out of primary shear zone due to conduction. Fur-thermore, when the tool and the workpiece start to be in contact, the shear stressincreases from zero, and the tool-workpiece contact period is smaller comparedto continuous contact without vibration assistance. Thus, the temperature is lowercompared to conventional cutting, which shows the benefit of vibration assistance.This proves the SEM results in Figure 6.2b) that, the application of ultrasonic vi-bration assistance suppresses the generation of adiabatic shear zones between thechip segments.For more quantitative verifications of the proposed chip flow model, the pitchlengths of the chip segmentations between the predictions and the experimentalmeasurements are compared. From the simulations, the relative displacement ofthe chip to the cutting tool is derived by integrating the speed difference betweenthe chip and the cutting tool defined in (6.7).The relative displacement (∆u) repre-sents the chip thickness in the primary shear zone in the cutting process, and it iscalculated by:∆u=∫ t0Vc cosαcos(φ −α)dt−h∫ t0γ˙pdt (6.20)891900 1900.1 1900.2 1900.3 1900.4 1900.5 Time [ms]10001500200025003000Shear stress [MPa]1900 1900.1 1900.2 1900.3 1900.4 1900.5Time [ms]2004006008001000 Temperature 1900 1900.1 1900.2 1900.3 1900.4 1900.5 Time [ms]01000200030004000Shear stress [MPa]1900 1900.1 1900.2 1900.3 1900.4 1900.5Time [ms]0100200300400 Temperature [°C][°C]no vibration with ultrasonic vibrationa) b)Figure 6.6: Steady-state shear stress and temperature in the primary shearzone of cutting with Vc = 40 m/min and d = 0.15 mm. a) without ul-trasonic vibration assistance; b) with ultrasonic vibration assistance atf = 15.5 kHz and 10 µm amplitude.Therefore, the chip geometry can be predicted in spatial domain. The pitchlength of the chip segmentations, which is the distance between the two adjacentpeak values of ∆u, is then calculated from the material speed along the tool rakeface direction and the oscillation frequencies of the shear stress. In orthogonalcutting experiments, the chips from every cutting condition were collected, andthe pitches were measured using an optical microscope. The measured and pre-dicted pitch lengths of the machined chips under different cutting conditions arepresented and compared in Figure 6.7. It is found that the simulated results matchwith the experimental values within 20% errors for both without and with vibrationassistance situations. The error might be from the difference of the material prop-erties used in the simulations and the experiments, and the assumptions of primaryshear zone thickness used in the model. Without ultrasonic vibration assistance,9038 40 42 44 46 48 50 52Vc [m/min]50100150200pitch length [µm]d=0.15mm,no vibrationd=0.15mm,with vibrationd=0.2mm,no vibrationd=0.2mm,with vibrationd=0.15mm,no vibration (simulation)d=0.15mm,with vibration (simulation)d=0.2mm,no vibration (simulation)d=0.2mm,with vibration (simulation)Figure 6.7: Comparison of pitch lengths between simulations and experi-ments with different cutting speeds and uncut chip thicknesses (d).the pitch length increases as the uncut chip thickness increases for different cuttingspeeds (50 m/min and 40 m/min). In contrast, when applying ultrasonic vibration,the pitch lengths with different uncut chip thicknesses (d = 0.2 mm and d = 0.15mm) at the same cutting speed (50 m/min or 40 m/min) are close to each other, asshown in Figure 6.7, where the pitch length is around 130 µm at Vc = 40 m/min,and around 140 µm at Vc = 50 m/min. Thus, it is concluded that the segmenta-tion pitch is only determined by the cutting speed and vibration frequency whenultrasonic vibration assistance is applied, and it does not depend on the uncut chipthickness. This verifies the simulation results in Figure 6.6, that the frequency ofstress and temperature variations is determined by the excitation frequency of theapplied ultrasonic vibration assistance.In addition, the cutting forces from the simulations and the experiments arecompared.The cutting forces along tangential and feed directions were measuredusing a small-size dynamometer (Kistler 9256C1) with 2 kHz bandwidth. Thedynamometer is not able to measure the instantaneous cutting forces due to thebandwidth limit. Therefore, only average cutting forces are evaluated in experi-ments. The simulated tangential cutting force (Ft) is determined by the shear stress91in the primary shear zone as follows:Fs = τbdsinφ(6.21)Ft =Fs cos(βa−α)cos(φ +βa−α) (6.22)where φ is the shear angle, and βa is the friction angle derived from the frictioncoefficient on rake face (βa = arctanµ). From the experiments, the average tan-gential cutting forces (Ft) in the orthogonal cutting process were measured by thedynamometer along X-direction shown in Figure 6.1.The edge radius of the turning insert is measured to be 10 µm. Since it issmaller than 10% of the uncut chip thickness value, the effect of ploughing forcedue to the tool round edge is neglected. Figure 6.8 shows the comparison of the av-erage cutting forces with different uncut chip thicknesses and cutting speeds givenin Table 6.1. The results show that the ultrasonic vibration reduces the averagecutting force in the primary zone, which is due to the intermittent tool-workpiececontact. About 15% difference exists between the predicted and the measuredvalues, which may be due to the inaccurate material property parameters and theassumptions of shear zone thickness in the chip flow model. When the ultrasonicvibration is applied, the simulated cutting forces are smaller than the measured val-ues for different uncut chip thicknesses (d = 0.15 mm and d = 0.2 mm) and cuttingspeeds (Vc = 40 m/min and Vc = 50 m/min). The reason for larger measurementvalues is that the chip flow model neglects the friction between the bottom of thetool edge and the workpiece along the tangential direction during cutting motionand tool retraction, while the friction also contributes to the average cutting forcesalong the tangential direction in the experiments, which leads to larger values thanthe simulated results.6.5 SummaryThe effect of ultrasonic vibration assistance on the chip formation mechanism incutting Ti-6Al-4V is investigated in this chapter. Examinations of the machinedchips using SEM show that the adiabatic shear bands disappear in the segmented9240 50Vc [m/min]0100200300400500600700800cutting force [N]40 50Vc [m/min]020040060080010001200cutting force [N]simulation without vibrationexperiment without vibrationsimulation with vibrationexperiment with vibrationuncut chip thickness: 0.15 mm uncut chip thickness: 0.20 mmFigure 6.8: Comparison of tangential cutting forces between simulations andexperiments with different cutting speeds and uncut chip thicknesses(d). a) d = 0.15 mm; b) d = 0.2 mm.chips when ultrasonic vibration assistance is applied. An analytical plastic flowmodel in the primary shear zone is developed to simulate the shear stress and thetemperature variations associated with the shear banding and chip segmentation.Governing equations on the thermal-mechanical behavior of the material are de-veloped in cutting, tool retraction with chip elastic recovery, and tool-chip sepa-ration periods in vibration-assisted cutting. It is shown that the temperature in theprimary zone shear is much lower compared to that in conventional cutting whenultrasonic vibration assistance is applied in orthogonal cutting of Ti-6Al-4V, dueto periodic tool-workpiece separation which allows heat dissipation and decreaseof shear stress. The predicted pitch lengths of the segmented chips and the averagecutting forces are verified by orthogonal turning experiments with different uncutchip thicknesses and cutting speeds.93Chapter 7Conclusion7.1 ContributionsIn this thesis, a novel 3DOF ultrasonic vibration tool holder was developed, inwhich a piezoelectric resonance actuator is used to excite vibrations along all threeaxes. While radial vibrations are delivered by resonating the third bending mode,axial vibrations are created by exciting the first axial mode of the actuator assemblyat approximately 17 kHz with an amplitude range of 20-25 µm. A power amplifierwith conditioning circuits was also developed to drive and monitor the vibrationcutting tool. The proposed design expands ultrasonic vibration assistance to vari-ous machining operations (milling, drilling, and turning) in a single tool holder.In addition to the design of the vibration tool holder, this thesis presents a sen-sorless closed loop control system to generate the desired vibrations during cutting.A linearized lumped parameter model of the piezoelectric actuator is used for eachaxis to track the resonance and to estimate the vibration amplitude during cutting.A Kalman filter based observer is used to estimate the vibration amplitude andphase from the measured driving currents and supply voltages without the use ofvibration sensors. The proposed closed loop control system has been demonstratedto maintain the vibration amplitude at the desired level while adjusting the exci-tation frequency to match the varying resonance frequency of the actuator duringmachining. The system has been demonstrated in turning and drilling with uni-directional vibrations, as well as in two-axis milling with elliptical vibration loci.94The ultrasonic vibrations applied with the sensorless controller reduced the cut-ting forces by approximately 30% compared with conventional cutting in sampledrilling and milling experiments. The experimental results also showed that thecutting forces of the proposed control system were 10% lower than those for theuncontrolled ultrasonic vibration-assisted cutting.The 3DOF ultrasonic vibration tool holder can deliver spindle-synchronizedelliptical vibrations in the tangential and radial directions to assist in milling oper-ations. The chatter stability must be accurately predicted to select the appropriatespindle speed and depth of cut for the milling process. This thesis proposed a dy-namic model of elliptical vibration-assisted milling, including an analysis of thedynamic chip thickness and the process damping effect. The stability was derivedfrom the proposed dynamics model using the semi-discrete method, and millingexperiments were conducted to verify the predictions. It was concluded that ellip-tical vibrations can suppress chatter under certain cutting conditions.As experimentally demonstrated, the ultrasonic vibration assistance alters thechip formation mechanism in the cutting of Ti-6Al-4V. Adiabatic shear bands aresuppressed due to the high-frequency tool-workpiece separation. An analyticalmodel of plastic flow in the primary shear zone was proposed in this thesis bycombining the governing equations of shear stress and heat transfer. Simulationsbased on the proposed model showed a significant reduction in temperature in theprimary zone when ultrasonic vibration is applied along the tangential cutting di-rection. The model was also verified in orthogonal cutting tests with force mea-surements.7.2 Future Research DirectionsThis thesis proposed a novel 3DOF ultrasonic vibration tool holder with a sensor-less control system. Furthermore, the dynamics of vibration-assisted milling andthe effects on chip formation were investigated. Future work is suggested for thefollowing aspects of this system:• The power transmission of the 3DOF ultrasonic vibration can be improvedto replace the slip rings. A rotary transformer can be designed to wirelesslytransmit power to the tool holder (rotor), thus the spindle can reach much95higher speed (i.e. 20,000 rev/min).• The power amplifier for the piezoelectric actuator can be improved. A switch-type (PWM) power stage can be designed to replace the linear amplifiers inthe current system for higher efficiency, improving the compatibility of thevibration tool holder for industrial applications. In addition, higher supplyvoltages can be applied to broaden the adjustable range of vibration ampli-tudes.• In the presented sensorless control system, the vibration amplitudes are es-timated using a linearized electromechanical model of the piezo actuator,which ignores the nonlinear effects of the piezo material. Therefore, a non-linear electromechanical model can be implemented to improve the trackingaccuracy of the vibration loci.• The proposed digital sensorless control system is adaptive to different ultra-sonic vibration cutting tool designs, but the adaptation is limited in a certainrange of structural design. Therefore, it is valuable to investigate the con-troller performance from the aspect of the changing structural dynamics ofthe tool holder, which is important to industrial applications.• The effects of ultrasonic vibration assistance on chip formation have beeninvestigated in this thesis, but only for vibration along the tangential cut-ting direction. 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