ACTIVE DAMPING OF MACHINE TOOLS WITH MAGNETIC ACTUATORS by Fan Chen B.A., Tsinghua University, China, 2008 M.A.Sc., National University of Defense Technology, China, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2014 © Fan Chen, 2014 ii Abstract Flexible parts and tools are often found in machining operations, such as boring large cylinders, or turning long and slender shafts. The excessive flexibility of such tool or shaft may cause static deflection, forced vibration, and even chatter vibrations, which result in poor surface finish, tool breakage, and even damage to the machine, and thus become the main constraints in achieving higher productivity. This thesis presents an active damping solution to such problems, by using a novel three degrees of freedom linear magnetic actuator, which can increase the damping and stiffness of flexible structures in machining. The actuator is comprised of four identical magnetic actuating units; the magnetic force output of each actuating unit is linearized with regard to the input current by biasing magnets. Fiber optic sensors are integrated into the actuator to measure the displacements of the structure during machining. The magnetic actuator is used for three purposes: active damping of boring bar, increasing its static stiffness, and monitoring cutting forces based on the control current signals and fiber optic displacement sensor signals. The active damping is achieved by controlling the magnetic force as a function of measured vibrations. Three different types of controllers (loop shaping controller, Derivative-Integral controller, and H∞ controllers) have been developed to actively damp the displacements of a flexible boring bar during machining tests. The actuator can deliver 248 N force up to 850 Hz, and 107 N force up to 2000 Hz which is limited by the current amplifier used in the experimental setup. The cutting force is estimated through a Kalman filter, which was experimentally verified to be effective up to 550 Hz. Both the dynamic stiffness and static stiffness of the boring bar iii have been increased considerably with the designed magnetic actuator, leading to a significant increase in the chatter-free material removal rates. Although the proposed magnetic actuator is demonstrated for active damping of a slender boring bar in the thesis, the proposed magnetic actuator principle can be applied to suppress vibrations of rotating shafts, long boring bars and flexible structures in machine tools and other machineries. iv Preface Parts of this thesis have been published or submitted for publications. The publications reported work carried out during my Ph.D. research under the supervision of Dr. Yusuf Altintas and Dr. Xiaodong Lu. The relative contributions of the authors in the publications are clarified in this section. The content of Chapter 3 have been published in two journal articles: (1) Lu X., Chen F., Altintas Y., 2014, Magnetic actuator for active damping of boring bars, CIRP Annals – Manufacturing Technology, vol 63, 369-372 [1]. The magnetic actuator design and fabrication is conducted by me under supervision of Dr. Lu. I designed the velocity feedback controller, carried out the experiments, and wrote most of the manuscript. Dr. Altintas contributed significantly in writing the paper. (2) Chen F., Lu X., Altintas Y., 2014, A novel magnetic actuator design for active damping of machining tools, International Journal of Machine Tools and Manufacture, vol 85, 58-69 [2]. I developed the magnetic actuator, designed the loop shaping controller, carried out the impact modal analysis and cutting tests, and wrote the paper. Dr. Lu and Dr. Altintas gave me some helpful advice on writing the paper. Parts of Chapter 4 have been submitted for review [3]. I measured the plant model and identified the modal parameters. I developed the H∞ active damping controllers, carried out the experiments, and wrote the paper. M. Hanifzadegan gave me some helpful advice on the H∞ controllers design and writing the paper. Dr. Altintas contributed significantly in writing the paper. v Table of Contents Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ...........................................................................................................................v List of Tables ................................................................................................................................ ix List of Figures .................................................................................................................................x List of Symbols ........................................................................................................................... xiv List of Abbreviations ...................................................................................................................xx Acknowledgements .................................................................................................................... xxi Dedication .................................................................................................................................. xxii Chapter 1: Introduction ................................................................................................................1 Chapter 2: Literature Review .......................................................................................................8 2.1 Overview ......................................................................................................................... 8 2.2 Spindle speed selection ................................................................................................... 8 2.3 Spindle speed variation ................................................................................................... 9 2.4 Passive damping methods ............................................................................................. 10 2.5 Active damping methods .............................................................................................. 12 2.6 Summary ....................................................................................................................... 18 Chapter 3: Magnetic Actuator Design .......................................................................................21 3.1 Overview ....................................................................................................................... 21 3.2 Magnetic configuration design of the actuator ............................................................. 21 3.3 Working principle of the actuator ................................................................................. 22 vi 3.4 Lumped parameter force/torque analysis ...................................................................... 24 3.5 Comparison with regular magnetic bearings ................................................................ 31 3.6 Material selection and actuator parameters design ....................................................... 33 3.7 Finite element analysis of the actuator performance .................................................... 34 3.8 Fabrication of the magnetic actuator............................................................................. 37 3.9 Summary ....................................................................................................................... 40 Chapter 4: Active Damping of Machine Tools ..........................................................................41 4.1 Overview ....................................................................................................................... 41 4.2 Experimental setup and model identification ............................................................... 41 4.2.1 Experimental setup.................................................................................................... 41 4.2.2 Dynamics measurements of the designed magnetic actuator ................................... 43 4.2.3 Model identification of the active damping setup ..................................................... 46 4.3 Controller design ........................................................................................................... 51 4.3.1 Loop shaping controller design ................................................................................. 52 4.3.2 Derivative-Integral (DI) controller design ................................................................ 55 4.3.3 H∞ controllers design ................................................................................................ 56 4.3.3.1 Control configurations ...................................................................................... 56 4.3.3.2 Guideline for tuning the weighting functions ................................................... 58 4.3.3.3 State space model of the generalized plant ....................................................... 59 4.3.3.4 Controller design ............................................................................................... 61 4.4 Experimental results...................................................................................................... 64 4.4.1 Impact response tests ................................................................................................ 64 4.4.2 Static stiffness measurements ................................................................................... 67 vii 4.4.3 Cutting tests .............................................................................................................. 69 4.5 Summary ....................................................................................................................... 73 Chapter 5: Cutting Force Estimation ........................................................................................75 5.1 Overview ....................................................................................................................... 75 5.2 Methodology ................................................................................................................. 76 5.3 Experimental setup and model identification ............................................................... 77 5.3.1 Experimental setup.................................................................................................... 77 5.3.2 Model identification .................................................................................................. 78 5.4 Kalman filter design ...................................................................................................... 81 5.5 Experimental results...................................................................................................... 85 5.5.1 Frequency domain test .............................................................................................. 85 5.5.2 Time domain test....................................................................................................... 87 5.6 Summary ....................................................................................................................... 90 Chapter 6: Conclusions and Future Research Directions ........................................................91 6.1 Conclusions ................................................................................................................... 91 6.2 Future research directions ............................................................................................. 93 Bibliography .................................................................................................................................95 Appendices ..................................................................................................................................100 Appendix A System Matrices of the Plant and Weighting Functions .................................... 100 Appendix B Summary of H∞ Controller Synthesis with LMI Method ................................... 105 Appendix C System Matrices of the Force Estimation System .............................................. 110 Appendix D Operation instruction for the magnetic actuator ................................................. 112 D.1 Function of the magnetic actuator........................................................................... 112 viii D.2 Installation of the magnetic actuator ....................................................................... 112 D.3 Calibration of the fiber optic sensors ...................................................................... 117 ix List of Tables Table 3.1 Magnetic actuator design parameters ........................................................................... 34 Table 4.1 Estimated modal parameters of the system. .................................................................. 48 Table 4.2 Weighting function parameters ..................................................................................... 62 Table 4.3 Controller parameters ................................................................................................... 63 Table 4.4 Tool tip dynamic stiffness for different control cases .................................................. 67 Table 4.5 Measured static stiffness at the boring bar tip .............................................................. 68 Table 5.1 Identified modal parameters ......................................................................................... 79 Table 5.2 Frequency responses from the shaker input voltage (shaV ) to the output force ( cyF ) ... 88 x List of Figures Figure 1.1 Schematic diagram of deep hole boring and long shaft turning .................................... 1 Figure 1.2 Comparison of surface finish with and without chatter................................................. 2 Figure 1.3 Dynamics of boring process .......................................................................................... 3 Figure 1.4 Block diagram of the cutting mechanics in boring process ........................................... 5 Figure 1.5 Schematic diagram of the proposed experimental setup ............................................... 6 Figure 3.1 Electromagnetic configuration of the designed actuator ............................................. 22 Figure 3.2 Working principle of the magnetic actuating unit ....................................................... 23 Figure 3.3 Current directions and flux paths for force generation ................................................ 24 Figure 3.4 Magnetic circuit model for the biasing flux ................................................................ 25 Figure 3.5 Magnetic circuit model for excitation flux .................................................................. 27 Figure 3.6 Schematic drawing for the torque analysis .................................................................. 30 Figure 3.7 Schematic drawing of a one degree of freedom existing magnetic bearing ................ 32 Figure 3.8 Finite element mesh of the designed actuator using FEMM ....................................... 35 Figure 3.9 Force current relation of the designed magnetic actuator. .......................................... 36 Figure 3.10 Torque current relation of the designed magnetic actuator ....................................... 37 Figure 3.11 Magnetic actuator. (a) Solid model; (b) Fabricated stator assemblies and armature assembly. ....................................................................................................................................... 39 Figure 3.12. Armature assembly clamped onto a boring bar. (a) Solid model; (b) Fabricated assembly. ....................................................................................................................................... 39 Figure 3.13. Active damping setup. (a) Solid model; (b) Fabricated setup. ................................. 40 Figure 4.1 Experimental setup for active damping of boring bar ................................................. 42 xi Figure 4.2 Data acquisition setup.................................................................................................. 43 Figure 4.3 Open loop FRFs from the amplifier command currents (xI , yI, I) to the armature displacement ( x ). ......................................................................................................................... 44 Figure 4.4 Open loop FRFs from the amplifier command currents (xI , yI, I) to the armature displacement ( y ). ......................................................................................................................... 45 Figure 4.5 Open loop FRFs from the amplifier command currents (xI , yI, I) to the armature displacement ( ). ......................................................................................................................... 45 Figure 4.6 Diagram of the active damping setup .......................................................................... 46 Figure 4.7 Measured and curve fitted FRFs of the plant system. (a) IxtxG; (b) Ia xxG; (c) IxxG; (d) FtxG;(e) FaxG; (f) FxG . .................................................................................................................. 48 Figure 4.8 Mode shape analysis of the boring bar assembly. ....................................................... 50 Figure 4.9 Control system block diagram in the x axis ................................................................ 52 Figure 4.10 FRFs of the loop shaping controllers and the NLTs for different axes ..................... 54 Figure 4.11 FRFs of the DI controller and the NLT in the x axis ................................................. 56 Figure 4.12 System block diagrams for controller design ............................................................ 57 Figure 4.13 Schematic diagrams for selection of the weighting functions. (a) 1( )Wx and xFT in log scale; (b) 1( )WI and IFT in log scale. .................................................................................. 59 Figure 4.14 Tool tip FRFs with loop shaping controllers in different number of axes ................ 64 Figure 4.15 Measured FRFs from the cutting force (Fc) to the tool tip displacement (xt) with active damping. (a) Full scale FRFs; (b) Zoomed-in FRFs for the first mode; (c) Zoomed-in FRFs for the second mode; (d) Zoomed-in FRFs for the third mode. .......................................... 66 xii Figure 4.16 Measured FRFs from the cutting force (Fc) to the control current (Ix) with active damping. (a) Full scale FRFs; (b) Zoomed-in FRFs for the third mode ....................................... 67 Figure 4.17 Setup for static stiffness measurement ...................................................................... 68 Figure 4.18 Chatter stability measurements without active damping ........................................... 70 Figure 4.19 Chatter stability measurements with active damping. ............................................... 71 Figure 4.20 Setup for surface roughness measurement ................................................................ 72 Figure 4.21 Surface finish and surface roughness of the workpiece. Machined at 1500 rev/min spindle speed, 0.2 mm/rev feedrate, with and without active damping. ....................................... 73 Figure 5.1 Kalman filter design configurations. ........................................................................... 76 Figure 5.2 Experimental setup for cutting force estimation ......................................................... 78 Figure 5.3 Measured and curve fitted FRFs for yFG , yIG and yFT .............................................. 79 Figure 5.4 Simulated FRFs from the cutting force (cyF ) to the estimated force ( cˆyF ) ................. 85 Figure 5.5 Measured FRFs from the excitation force (cyF ) to the estimated force ( cˆyF ) ............. 86 Figure 5.6 Measured and estimated force at 100 Hz .................................................................... 89 Figure 5.7 Measured and estimated force at 200 Hz .................................................................... 89 Figure B.1 Reconstructed system block diagram ....................................................................... 106 Figure D.12Installation of the clamping fixture 1 to the turret ................................................... 112 Figure D.23Installation of the boring bar and armature assembly to the clamping fixture 1 ..... 113 Figure D.34Installation of the clamping fixture 2 ...................................................................... 113 Figure D.45Gaps between safety bumper and the clamping fixture 2 ........................................ 114 Figure D.56Stator installation ..................................................................................................... 115 Figure D.67Adjusting the air gaps .............................................................................................. 115 xiii Figure D.78Actuator setup view when the stators are installed.................................................. 116 Figure D.89Fully assembled magnetic actuator.......................................................................... 117 Figure D.91Fiber optic sensor calibration .................................................................................. 118 xiv List of Symbols ad Depth of cut Approach angle of the insert m, c, k Modal mass, damping coefficient and modal stiffness of the boring bar xt Tool tip displacement Fc Cutting force in the x direction n Modal frequency of the boring bar Damping ratio of the boring bar dh Regenerative chip thickness T Spindle period cK Cutting force coefficient 0h Static chip thickness lima Critically stable depth of cut x, y, θ Armature displacement in the x, y, θ direction x0 Air gap when armature is centered y0 Air gap between the permanent magnet and the armature core L The length of the permanent magnet B1, B2 Total flux on the upper and lower sides of the armature 1B , 2B Biasing flux on the upper and lower sides of the armature B Excitation flux xv 1R , 2R Reluctance of the upper and lower air gaps aR Reluctance of the air gap between armature and permanent magnet pmR Permanent magnet reluctance A Stator pole area pmA Permanent magnet pole area 0 Vacuum permeability Flux flowing into the armature pm Total flux generated by the permanent magnet rB Remanence of the permanent magnet N Total number of coil turns for each magnetic actuating unit I Excitation current through each coil turn e Flux vector of all excitation sources uF Magnetic force generated by each actuating unit iK , sK Magnetic coefficients for each actuating unit _x totalF , _y totalF Total actuating forces in the x and y directions _xi totalK , _yi totalK Total magnetic coefficients related to the input current in the x and y axes _xs totalK , _ys totalK Total magnetic coefficients related to the armature displacements in the x and y axes T Torque generated by a single actuating unit xvi 0l , 1l Distances from the armature center line to the left and right edges of the stator pole t Thickness of the magnetic cores _ totalT Total torque generated by the actuator _i totalK Total magnetic coefficient related to input current in the θ axis _s totalK Total magnetic coefficient related to armature displacement in the θ axis exI , eyI , eI Excitation currents in the x, y and θ directions through each coil turn 1i , 2i Input currents to the regular magnetic bearing coils ax Air gap of regular magnetic bearing MBF Magnetic force output of regular magnetic bearing mbk Magnetic coefficient for regular magnetic bearing xI , yI, I Amplifier command currents in the x, y and θ directions i I jG Open loop frequency response from amplifier output current jI ( , ,j x y ) to armature displacement ( , , )i i x y xa, a Boring bar displacement and acceleration at the accelerometer location IxtxG,Ia xxG Open loop transfer functions from the input current (Ix) to displacements (xt) and (xa) x FtG,x FaG,xFG Open loop transfer functions between displacements (xt, xa, x) and the cutting force (Fc) It xxk,Ftxk,Ia xxk Modal stiffnesses xvii Faxk,Ixxk,xFk Modal stiffnesses LPxC , LPyC,LPC Loop shaping controllers for x, y and θ axes DIxC DI controller in the x axis IFT Closed-loop transfer function from cutting force ( cF ) to control current (xI ) xFT Closed-loop frequency response from cutting force ( cF ) to armature displacement (x) WI , Wx Weighting functions xq State vector for the plant pA , pB , pC , pD State space matrices of the plant WIq , Wxq State vectors for the weighting functions WI , Wx WIA , WIB , WIC , WID State space matrices of the weighting function WI WxA , WxB , WxC , WxD State space matrices of the weighting function Wx Iz , xz Weighting function outputs pK,1f , 2f , 3f , 4f Tuning parameters of the weighting functions K H∞ optimal controller ia , ib Controller parameters dnk Dynamic stiffness at the peaks Rt Peak to peak height of the surface finish Ra Average roughness xviii cˆF , cˆyF Estimated cutting force cyF Cutting force in the y direction yFG , yFT Open-loop and closed-loop FRFs from the cutting force ( cyF ) to the armature displacement (y) c , o , qi Modal frequencies c , o , qi Damping ratios _yF ck,_yF ok , _yI ik Modal stiffnesses iq , iu State vector and input vector of the force estimation system, ( 1, 2i ) iA , iB , iC System matrices of the force estimation system, ( 1, 2i ) W Observability matrix iw Process noise ( 1, 2i ) v Measurement noise iK Kalman filter gain matrix, ( 1, 2i ) eiq Expanded state vector ˆeiq Estimate of the expanded state vector eiq iQ , R Noise covariance matrices cf Frequency of the cutting force gF Amplitude of the square wave force shaV Shaker input voltage signal xix ga , Magnitude and phase of the frequency response from shaV to cyF xx List of Abbreviations DI Derivative-Integral SSV Spindle speed variation VSSM Variable spindle speed machining CSSM constant spindle speed machining TVD Tuned viscoelastic damper LMS Least mean squares IMC Internal Model Control LQR Linear quadratic regulator LR Inductor-resister AMB Active magnetic bearing FRF Frequency response function FEM Finite element method FEMM Finite element method magnetics SMC Soft magnetic composites LMI Linear matrix inequalities xxi Acknowledgements I would like to express my sincerest gratitude to my supervisor, Prof. Yusuf Altintas, for his guidance and support through all my PhD years. Prof. Altintas is a world-class famous professor. Although he is very busy, he can always manage to discuss with his students about their projects, care about their professional and personal development. I really appreciate that he gave me the opportunity to join his talented and professional research team, and get world-class research experience. I would also like to thank my co-supervisor Dr. Xiaodong Lu for his supervision and guidance through my PhD life. At the beginning of my PhD, I did not have a solid foundation in mechatronics. He gave me lots of helpful advice in the magnetic actuator design. I learnt many practical engineering skills from him. Besides, the most important ability I learnt from him was thinking and working independently. He taught me how to find my interest and my value in my career. I would like to offer my enduring gratitude to my parents. I really thank them for their support in my life and my research. Special thanks are owed to my lovely girl friend, Siyuan Xiong. The last one and half years in my PhD life is very stressful but productive. I spent lots of late nights and weekends working in the lab, and finally finished my research and thesis. Without the support and understanding of Siyuan, it would be very hard for me to accomplish all these work. I would like to thank all my colleagues in Precision Mechatronics Lab and Manufacturing Automation Lab at the University of British Columbia. They are hard working, friendly, and always willing to help others. I really enjoy my research life with them. In the end, I would like to acknowledge NSERC CANRIMT for their support. xxii Dedication Dedicated to: My parents and Siyuan Xiong! 1 Chapter 1: Introduction Flexible workpiece and tools are often found in the boring of large cylinders and turning of long and slender shafts, as shown in Figure 1.1. Y axis driveZ axis driveSpindleWorkpieceSteady restBoring barSteady rest Clamping turretxyzX axis driveZ axis driveSpindle WorkpieceSteady restxyzTurretBoring a deep holeTurning a long flexible shaft Figure 1.1 Schematic diagram of deep hole boring and long shaft turning However, because of the excessive flexibility of the workpieces or tools, static deflection, forced and chatter vibrations become the main constraints in production. Excessive static deflections may violate the dimensional tolerance of the workpiece, and vibrations may lead to poor surface finish, short tool life and chipping of the tool [4]. Forced vibrations are usually 2 caused by interrupted cutting or runout. Chatter, which is a self-excited unstable vibration, is mainly induced by the regenerative effect of the cutting process, and is the biggest limitation in achieving higher material removal rates and acceptable surface finish quality. Figure 1.2 shows the surface finish of two workpieces from my cutting tests in boring operation. When the cutting process is stable, vibration will attenuate, which gives a smooth surface finish. However, once chatter happens, vibration will grow up, which leads to a poor surface finish. Chatter No chatterPoor surface Smooth surface Figure 1.2 Comparison of surface finish with and without chatter The regenerative chatter mechanism was first analyzed by Tlusty [5] and Tobias [6]. They observed that the dynamic chip thickness is affected by the vibration waves left on the surfaces from the previous cut (i.e., outer modulation) and the current cut (i.e., inner modulation). The modulated chip thickness affects the dynamic cutting force, which in turn changes the amplitude of the vibrations. Depending on the phase relation between the inner and outer modulations, the 3 amplitude of the vibrations in the cutting process may be amplified, which will lead to regenerative chatter. The stability of chatter vibrations has been studied by lots of researchers [5]–[18]. Since the cutting mechanics in turning and boring operations are quite similar, boring process is used as an example to briefly demonstrate the chatter mechanisms here for clarity using the theory proposed by Tlusty [5]. A boring bar can be modeled as a single degree of freedom, mass-spring-damper system in x direction, as shown in Figure 1.3. In the figure, ad is the depth of cut; is the approach angle of the insert; m, c, and k are the modal mass, damping coefficient and modal stiffness of the boring bar, respectively; xt is the tool tip displacement; and Fc is the cutting force in the x direction. kcdaBoring barWorkpieceZXY mcF txBoring bar model Figure 1.3 Dynamics of boring process The dynamic model of boring process can be expressed as: 4 22( ) 2 ( ) ( ) ( )nn n cx t x t x t F tk (1.1) where n ( n km ) is the modal frequency of the boring bar; and (2 ncm ) is the damping ratio. The transfer function (G(s)) between the cutting force (Fc) and the tool tip displacement (x) can be calculated as: 22 2( )( )( ) 2nc n nx s kG sF s s s (1.2) The regenerative chip thickness (dh ) is modeled as: ( ( ) ( ))cosdh x t T x t (1.3) where T is the spindle period. The cutting force is a function of chip thickness, expressed as: 0( )c c d dF t K a h h (1.4) where cK is the cutting force coefficient; and 0h is the static chip thickness. Because of the regenerative chip thickness (dh ), the cutting process becomes a closed-loop system, as shown in Figure 1.4. The regenerative effect is considered as a time delay ( sTe ) in 5 the block diagram. As it can be seen, the depth of cut (ad) will affect the gain of the closed-loop system. Thus, the system may become unstable when the depth of cut (ad) keeps going up. ( )G sxc dK a Tex( )x t Tos0hd cF Figure 1.4 Block diagram of the cutting mechanics in boring process According to [9], the critically stable depth of cut (lima ) is proportional to the dynamic stiffness of the boring bar. Therefore, in order to minimize the forced and chatter vibrations, as well as the static deflections, the dynamic and static stiffnesses of the flexible workpieces and tools must be improved. Steady rests are usually used in industry to support large boring bars and long shafts during machining (Figure 1.1) to enhance their dynamic and static stiffness. However, the use of steady rests may cause over constrain problems, since the shafts and the boring bars are already clamped into the spindle and the clamping turret, respectively. Therefore, the shaft may be twisted or even be damaged during machining; and the boring bar may statically deflect, leading to dimensional errors on the diameter of the holes. The objective of this thesis is to reduce the static deflection and improve chatter stability of flexible shafts and large boring bars by increasing both static and dynamic stiffnesses. A non-contact three degrees of freedom linear magnetic actuator is designed and built to actively damp 6 the vibrations and enhance the static stiffness of the flexible shafts and boring bars. Since turning and boring operations have similar cutting mechanics, the developed magnetic actuator is only tested on a long boring bar in this thesis for demonstration of the proposed concept. Based on the Hardinge CNC lathe in the Manufacturing Automation Lab at the University of British Columbia, an experimental setup is proposed to investigate the developed actuator and active damping methods, as schematically shown in Figure 1.5. The armature of the magnetic actuator is clamped to the boring bar; and the stator of the magnetic actuator is installed on clamping fixture 1, which is clamped to the turret of the CNC lathe. The boring bar is cantilevered to clamping fixture 2. Displacement sensors are installed into the magnetic actuator to measure the armature displacements for feedback control. Different types of controllers, including loop shaping controller, Derivative-Integral (DI) controller, and H∞ controllers are designed for active damping of the boring bar vibrations. Besides, cutting force is predicted from the armature displacements and the control currents simultaneously. TurretSpi dlZXYDisplacement sensorControl systemStator of the magnetic actuatorClamping fixture 1WorkpieceArmature of the magnetic actuatorClamping fixture 2Boring barθMagnetic actuator Figure 1.5 Schematic diagram of the proposed experimental setup 7 The thesis structure is organized as follows: Chapter 2 presents a detailed literature review on the different chatter suppression methods. Chapter 3 proposes a novel three degrees of freedom linear magnetic actuator design. The designed actuator can be actuated in two radial directions, and the torsional direction. It is composed of four identical magnetic actuating units. Each actuating unit has a linear force output with respect to the input current. The magnetic configuration and working principle of the designed magnetic actuator is explained. Both lumped parameter method and Finite element method are used to analysis the force/torque generation of the designed magnetic actuator. The material selection, design and fabrication of the magnetic actuator are also explained. In Chapter 4, the installation of the proposed magnetic actuator on a CNC lathe is shown to demonstrate its active damping performance on a flexible boring bar. The frequency response functions of the active damping setup between the system inputs and outputs are measured; and the corresponding modal parameters are identified. Loop shaping controllers are first designed for all the three actuating directions to test the effectiveness of the proposed magnetic actuator. Then, a simpler DI controller is implemented to increase both the static and dynamic stiffness of the boring bar at its tip. Afterward, four different H∞ control strategies are presented to damp the multiple modes of the boring bar. Impact model analysis and cutting tests are carried out to test the active damping performance of the different controllers. Chapter 5 presents two cutting force estimation method. The cutting force is estimated from the armature displacements and the control current using a Kalman filter. The principle of the force estimation algorithm is explained; and its performance is tested experimentally. Concluding remarks and future research directions are discussed in Chapter 6. 8 Chapter 2: Literature Review 2.1 Overview This chapter reviews the relevant work in the literature. Section 2.2 and 2.3 review the spindle speed selection and spindle speed variation methods for chatter suppression. Passive and active damping methods are discussed in section 2.4 and 2.5. Concluding remarks are given in section 2.6. 2.2 Spindle speed selection In spindle speed selection methods, when chatter is detected online, suitable spindle speed and depth of cut are selected based on stability lobe diagram to stabilize chatter. This idea was first put forward by Weck et al. [19]. They proposed an online method for chatter avoidance in face milling operations. Smith and Tlusty [20][21] proposed a spindle speed control algorithm, which is based on Weck’s method but does not require the stability lobe data. The spindle speed is selected to match the tooth passing frequency as the integer divisions of chatter frequency in order to penetrate into the stability pockets of the machine. The spindle speed converges to the most stable region of the stability lobe diagram within two or three iterations. However in both approaches, chatter is detected when it has already developed at a fully grown stage. Dijk [22] proposed an automatic in-process chatter avoidance approach, in which chatter can be detected in premature stage so that no visible marks on the workpiece are present. These strategies work satisfactorily in the high spindle speed regions, where there are well separated lobes. However, for turning and boring process, the spindle speed is usually low, and there is almost no lobe. In this case, spindle speed selection methods may not work well. 9 2.3 Spindle speed variation The goal of spindle speed variation (SSV) method is to disturb the regenerative mechanism of chatter by continuously varying the spindle speed [23]. Speed can be varied sinusoidally [24] or randomly [25]. Jayaram et al. [26] developed an analytical model to predict the chatter stability of variable spindle speed machining (VSSM). Based on the analytical model, they selected the VSSM parameters that can provide a maximum increase in the stability of the machining process compared to constant spindle speed machining (CSSM). Al-Regib et al. [27] investigated the effect of sinusoidal SSV amplitude ratio on stable and unstable cutting process based on energy analysis. They computed the work done by the regenerative force for different amplitude ratios. Their results showed that applying sinusoidal SSV to an unstable cutting process always reduced the work done by the regenerative force comparing to the CSSM case, consequently, increasing the stability of the system. However, applying sinusoidal SSV to a stable cutting process reduced the system stability and might cause chatter, because the regenerative force dissipated less energy comparing to the CSSM case and even delivered energy under certain amplitude ratios. In SSV methods, the frequency and amplitude of the SSV signal are the most important parameters that need to be optimized to achieve a good chatter suppression performance. However, the frequency and amplitude of SSV are limited by the spindle drive system and spindle inertia. Therefore, SSV method may not provide good chatter suppression performance for relatively high spindle speeds. 10 2.4 Passive damping methods Various passive dampers have been designed to improve the dynamic performance of the structures. Passive damping methods require the attachment of a mass-spring-damper system to the structure with an identical frequency which needs to be damped. Rivin and Kang [28] proposed an analytical approach to design vibration absorbers. Their detailed and comprehensive experimental study demonstrated significant performance improvements using their design procedure. Tarng et al. [29] used tuned vibration absorber to reduce the magnitude of the negative real part of the frequency response function (FRF) of the cutting tool in turning operation. The vibration absorber was selected to have a large damping ratio; and the natural frequency of the vibration absorber was tuned to be equal to the natural frequency of the cutting tool. Six times increase in the maximum chatter free depth of cut was achieved from their experimental results. Sims [30] proposed an analytical method for optimally tuning the parameters of vibration absorbers to push the negative real part of the main structure’s FRF. Miguelez et al. [31] considered the parameters of passive dynamic absorbers into the chatter stability model, and the absorber parameters were determined to maximize the minimum values of the chatter stability lobes. Yang et al. [32] presented an optimal tuning method for multiple tuned mass dampers to increase chatter stability. The mass of the dampers are selected to be identical, while the stiffnesses and damping ratios of the dampers are optimized using numerical method to minimize the magnitude of the negative real part of the FRF at the tool-workpiece interface. Moradi et al. [33] studied the influence of the absorber location along the boring bar. The position of the absorber was selected to minimize the tool tip deflection of the boring bar. 11 Besides passive vibration absorbers, Evita et al. [34] designed a friction damper to suppress high frequency (above 5000 Hz) vibrations of boring tools. The friction damper consists of a piece of mass attached to the main vibrating structure with permanent magnet. When the chatter reaches certain threshold amplitude, the damper starts sliding. Therefore, the friction introduced by the damper dissipates the vibration energy and prevent chatter from growing beyond the threshold amplitude. However, since chatter has already happened, the surface finish of the workpiece has been damaged by the chatter vibration, although the vibration amplitude is constrained to the threshold value. Rashid et al. [35] developed a tuned viscoelastic damper (TVD) for vibration suppression of workpiece held on a palletized workholding system in milling operation. The TVD parameters were first calculated with the identified system FRF. Then, the TVD parameters were optimized by applying the analytical model of the TVD directly to the experimentally measured FRF of the system. Daghini et al. [36] used composite material for the tool holder design to improve the system damping capacity. Hydrostatic clamping system was used to maintain a high level of static stiffness; and ten times improvement of the damping ratio was reported from their research work. Houck et al. [37] presented a tuned tool holder design approach, in which the natural frequency of the tool holder was designed to match the natural frequency of the clamped-free boring bar. The resulting boring-bar-holder assembly achieved 69% reduction in the magnitude of the tool tip FRF comparing to the clamped-free boring bar. Generally, passive dampers are relatively cost effective and easy to implement. Also, they never destabilize the system. However, the practically achievable amount of damping is rather limited. Furthermore, passive dampers are not robust to the changing machining conditions. The 12 natural frequencies of the system may differ in each application; and the tuned passive dampers need to be re-manufactured for each mode. 2.5 Active damping methods The active methods are generally able to achieve more damping comparing to the passive methods, since extra energy is applied to the system. Also, it allows damping of several modes simultaneously by adjusting the controller parameters of the actuators. Tanaka et al. [38] installed eight piezo actuators into a boring bar for active damping. An accelerometer was used to measure the boring bar vibration at the tool tip; and velocity feedback controller is implemented to actively damp the boring bar vibrations. Besides, the influence of the actuator positions on the chatter stability was studied. The optimum position of the piezo actuators was found to be at a length of 0.3Lb from the boring bar clamping location, where Lb is the length of the boring bar. Redmond et al. [39] installed four piezo actuators inside a boring bar, and implemented acceleration feedback control to damp the boring bar vibrations in both lateral directions. Andrén et al. [40] presented a boring bar structure, in which a piezo actuator was embedded into the boring bar; and an accelerometer was place at 25 mm away from the tool tip to measure the boring bar vibration. Three different control algorithms were implemented for active damping, including a PID controller, a feedback filtered-x least mean squares (LMS) algorithm, and an Internal Model Control (IMC) controller based on the filtered-x LMS algorithm. Experimental results showed that the latter two algorithms can achieve better active damping performance than the PID controller. 13 Tewani et al. [41] studied the chatter stability of a boring bar which has an embedded active dynamic absorber. A piezo actuator was used to move the absorber mass to generate an inertial force that counteracted the disturbance acting on the boring bar. Two different control strategies, including acceleration and velocity feedback control, and linear quadratic regulator (LQR) control, were considered for the active dynamic absorber. Their experimental results showed that substantial increase in the chatter-free width of cut was obtained for a boring bar with an active dynamic absorber over a plain boring bar and a boring bar with a passive absorber. Marra et al. [42] implemented the active dynamic absorber developed by Tewani et al. [41] on a boring bar with H∞ optimal control, and achieved greater reduction in the magnitude of the peaks of the tool FRF comparing to the case with LQR control. Matsubara et al. [43] presented a vibration suppression method for a boring bar using embedded piezo actuators with an inductor-resister (LR) circuit, which acted as a dynamic vibration absorber. The parameters of the LR circuit were optimally tuned so that the magnitude of the real part of the FRF from the cutting force to the tool tip displacement was minimized to suppress the regenerative chatter. However, in all the above active damping methods, piezo actuators were installed inside the boring bar for chatter suppression. Therefore, their size and output were limited by the boring bar diameter to achieve a better active damping performance. Besides, it was not efficient to apply the actuating force in the axial direction of the boring bar to damp its vibration in the lateral direction. Different from the active methods mentioned above, Pratt and Nayfeh [44] installed two Terfenol-D actuators outside the boring bar for active damping. Therefore, the actuator size and output were not limited by the boring bar size anymore. Moreover, since the directions of 14 actuating force and boring bar vibration were co-parallel in their proposed setup structure, vibrations can be damped more effectively. A survey of active damping of spindle vibrations is presented by Abele et al. [45]. A set of piezo actuators has been installed behind the outer rings of the spindle bearings for active damping with various control strategies [45]. Pan and Su [46] used a piezo actuator to suppress chatter in turning operation. The piezo actuator was placed between the tool holder and tool post to manipulate the displacement at the tool tip. A continuous time dynamic backlash-like hysteresis model was employed to describe the hysteresis in the force output of the piezo actuator; and adaptive controller was implemented to suppress the chatter vibration caused by the hysteresis and time delay in the cutting process. Ast et al. [47] implemented a piezo actuator into a machine tool with parallel kinematics. The piezo actuator was connected to the main cantilever of the machine tool to damp the vibrations at the tool center point. Parus et al. [48] attached a piezo actuator to a flexible base to damp the vibrations of a flexible workpiece in milling process. Linear Quadratic Gaussian algorithm was implemented in the control system for active damping; and a significant increase in the critical depth of cut was reported from their research. For all the active methods discussed above, piezo and Terfenol-D based actuators were used for vibration suppression. However, both piezo and Terfenol-D actuators have hysteresis, which needs to be modeled and compensated during the controller design process, therefore, complicating the design of the linear controllers. 15 Besides piezo and Terfenol-D actuators, other different types of actuators, like electro hydraulic actuators and electromagnetic actuators, were also used for chatter suppression in the previous studies. An electro hydraulic actuator was equipped on a lathe by Nachtigal [49] for active chatter suppression in turning process. Strain gages were used to measure the cutting force, while a force feedback controller was developed to control the chatter vibration. Shiraishi et al. [50] integrated a stepping motor into the drive system of a lathe to control the tool position. The relative motion between the workpiece and the cutting tool was measured with an eddy current sensor; and an optimal state feedback controller was implemented to control the chatter vibration. Choudhury and Sharath [51] installed an electromagnetic actuator between the tool post and tool to control the tool position. An optical sensor was used to measure the relative displacement between the tool and workpiece. A simple gain feedback controller was implemented to control the electromagnetic actuator to neutralize the relative vibration between the tool and the workpiece caused by the cutting process. Ganguli [52] implemented two electromagnetic inertial actuators in a milling demonstrator with collocated sensor and actuator configuration for active damping. A velocity feedback control strategy was implemented to control the actuators so that they behaved like active mass dampers. Besides, Ganguli discussed the collocated and non-collocated control configuration for active damping. A collocated sensor and actuator configuration is always preferable, since it is characterized by an alternating pole zero configuration of the plant transfer function in the complex plane. For a non-collocated sensor and actuator configuration, moving the sensor away from the actuator causes a migration of the zeros along the imaginary axis towards ±j∞. Thus, 16 the interlacing property of the poles and the zeros is no longer guaranteed [52]. When the distance between the sensor and the actuator is beyond a certain value, a non-collocated system always leads to a non-minimum phase system and is almost impossible to be controlled [52]. Munoa et al. [53] developed a two axes inertial actuator for chatter suppression in a ram type travelling column milling machine. Different control laws, including direct acceleration feedback control, direct velocity feedback control, direct position feedback control, and delayed position feedback control, were compared for chatter suppression. The comparison demonstrated that the direct velocity feedback control is the most effective strategy. The actuators presented in [49]–[51] were used to control the relative vibration between the tool and workpiece. However, the dynamic stiffness of the flexible workpiece was not improved. Therefore, if the vibration amplitude of the workpiece was larger than the stroke of the actuator, these methods will not be effective. The inertial actuators presented in [52] and [53] were used to improve the dynamic stiffness of the flexible parts in machine tools. However the inertial actuators cannot be attached to flexible rotating parts and tools, e.g. long engine shafts in turning process and large rotating boring bars in boring operation. Active magnetic bearing (AMB) is a very promising technology for machining applications. It has less hysteresis effect comparing to piezo and Terfenol-D based actuators. Also, it can be designed to have a large load capacity. These advantages make it a better choice for active damping devices in machining. Besides, AMB uses magnetic forces to firmly hold the rotor and maintain its separation from the machine’s stationary components [54]. Therefore, it can be used to damp vibrations in both stationary and rotating parts and tools. AMB has been implemented in turning and milling process for active control of chatter by some researchers in the past [55][56]. 17 Chen and Knospe [55] implemented AMB in turning operation for active chatter control. Their experimental setup consisted of an AMB on an actuator platform which was connected to the tool platform via a leaf spring. Feedback linearization was used to linearize the AMB force output with respect to the input current. The regenerative effect of chatter was modeled as a time delay in the plant model. They developed three different µ-synthesis based chatter control strategies, including: speed-independent control, speed-specified control, and speed-interval control. Due to limitations on their lathe’s spindle speed range, cutting tests were carried out only for PID controller and speed-independent controller. Experiments demonstrated a 63% improvement over PID controller in achievable chatter-free chip width with the speed-independent controller. However, for the other two controllers, large order of Padé approximation had to be used to estimate the time delay in the low spindle speed. Therefore, the controller orders were too large to be implemented, even after model reduction. Dijk [56] developed an active chatter control methodology for high-speed milling process with AMB, which can guarantee chatter-free cutting operations in a priori defined range of process parameters such as spindle speed and depth of cut. The dynamic model of the cutting process was considered into the plant model during the controller design. The stability requirement for a pre-defined range of process parameters was cast into a robust stability problem. Hereto, the most important process parameters (depth of cut and spindle speed) were treated as uncertainties to guarantee the robust stability in the priori specified range of these process parameters. The control problem was solved via µ-synthesis using D-K-iteration. Simulation result demonstrated that the chatter stability boundary was locally shaped to stabilize the desired range of working points. However, experimental results showed that the obtainable increase in the material removal rate was limited. The reasons can be explained as follows: 1) relatively large 18 uncertainties were included in the controller design, which reduced achievable controller performance; 2) no uncertainty related to the cutting force modeling error was considered during the controller design. Therefore, the designed robust controller may not be able to guarantee the closed-loop stability for the real cutting process, due to the unconsidered modeling error. However, if the uncertainties in the cutting force model are also considered into the controller design, µ-synthesis may not be able to find a satisfying controller to handle so many uncertainties. Controlling chatter with delayed feedback loop makes the controller design challenging. Instead, active damping of critical modes of the flexible parts and tools may be more practical, since the increased dynamic stiffness reduces the chatter vibrations and increases the stability limits. Moreover, the AMBs presented in [55] and [56] have nonlinear relation between the magnetic force and the command current. Therefore, the magnetic force has to be linearized first before it can be effectively used for active damping. 2.6 Summary This chapter discusses about different chatter suppression methods published in the literature, including spindle speed selection, SSV, passive damping, and active damping. In spindle speed selection method, suitable spindle speed and depth of cut are selected based on the stability lobe diagram to stabilize chatter. It works well in milling process where there are well separated chatter stability lobes. However, in turning and boring process, it does not work well, since the spindle speed is low and there is almost no lobe. In SSV methods, chatter suppression is achieved by continuously varying the spindle speed to disturb the regenerative mechanism of chatter. However, the frequency and amplitude of SSV are limited by the spindle drive motor’s 19 torque limit and spindle inertia. Therefore, SSV method may not provide good chatter suppression performance for relatively high spindle speeds. Passive damping methods require the attachment of a mass-spring-damper system to the boring bar with an identical frequency which needs to be damped. The parameters of the passive dampers can be optimized using different methods to achieve a good vibration suppression performance. Generally, passive dampers have lower costs and easy to use, but the achievable amount of damping is quite limited. Also, they are not robust to the changing machining conditions. Active damping methods require actuators, sensors and control laws for chatter suppression. They are able to achieve more damping comparing to the passive dampers, and can damp multiple modes simultaneously. Different actuators have been presented in the literature for active damping. The most commonly used actuators include piezo actuator, Terfenol-D actuator, inertia actuator, and AMB. Piezo and Terfenol-D actuators have large load capacity with short stroke. However, the nonlinear hysteresis in both piezo and Terfenol-D actuators complicates the controller design. Inertia actuators are quite convenient and effective for active damping of large stationary parts and tools. But it cannot improve the dynamics of flexible rotating parts and tools. AMBs can have large load capacity and little hysteresis, which is good for active damping. However, the past designs do not have linear relation between the magnetic force and the command current. Therefore, the magnetic force has to be linearized first before it can be effectively used. This thesis presents a new, noncontact linear magnetic actuator instrumented with fiber optic displacement sensors for active damping of machine tools. The magnetic force is linearized with respect to the command current; and it can be delivered in two radial and the torsional directions for active damping of the flexible modes. The proposed novel actuator is installed on a CNC lathe and tested in boring operation with a slender long boring bar. Different control strategies 20 are implemented and compared in this thesis in order to achieve a good active damping performance. Significant increase in the tool tip dynamic stiffness and absolute chatter-free depth of cut has been proven through impact modal tests and cutting experiments. 21 Chapter 3: Magnetic Actuator Design1 3.1 Overview This chapter presents the design principle of a novel three degrees of freedom linear magnetic actuator. The actuator is composed of four identical magnetic actuating units. Each actuating unit has a linear force output with respect to the input current. The proposed actuator concept is designed to damp vibrations of very large slender shafts in turning process, and very long boring bars which are used in boring large diesel engine cylinder walls or similar parts. The content of this chapter is arranged as follows. Section 3.2 gives the magnetic configuration design of the actuator. Section 3.3 explains the working principle of the designed actuator in details. Section 3.4 presents the lumped parameter analysis of the force and torque generation. The designed magnetic actuator is compared with regular magnetic bearings in Section 3.5. Section 3.6 shows the material selection and designed parameters of the actuator. The performance of the designed magnetic actuator is analyzed using finite element method (FEM) in Section 3.7. Section 3.8 shows the solid model of the magnetic actuator and discusses about the fabrication of the armature and stator assemblies. Concluding remarks are given in the last section. 3.2 Magnetic configuration design of the actuator The magnetic configuration of the designed actuator is shown in Figure 3.1. It has four identical magnetic actuating units. Each magnetic actuating unit is comprised of two excitation coil 1 Parts of Chapter 3 has been published in two journal articles: (1) X. Lu, F. Chen, and Y. Altintas, “Magnetic actuator for active damping of boring bars,” CIRP Annals - Manufacture Technology, vol. 63, no. 1, pp. 369–372, 2014. (2) F. Chen, X. Lu, and Y. Altintas, “A novel magnetic actuator design for active damping of machining tools,” International Journal of Machine Tools and Manufacture, vol. 85, pp. 58–69, 2014. 22 windings, one permanent magnet, two stator side cores, one stator middle core, and an armature core. Both the armature and stator cores are made of soft magnetic material. The holes on the armature core are used for assembling. The force generated by each magnetic actuating unit is linearized with respect to the excitation current using the electromagnetic structure design strategy [57]. The designed magnetic actuator can be moved in x, y and θ directions. Stator side coreCoil windingPermenant magnetArmature corexyθStator midlle core Figure 3.1 Electromagnetic configuration of the designed actuator 3.3 Working principle of the actuator Figure 3.2 shows the flux paths of the magnetic actuating unit. The air gaps on the lower and upper sides of the armature are denoted as x0+x and x0-x respectively, where x0 is the air gap when the armature is centered; and x is the armature displacement taken as positive for upward movement. The length of the permanent magnet is denoted as L; and y0 is the air gap between the permanent magnet and the armature core. The permanent magnet plus the excitation coils generate a total flux B1 on the upper side of the armature and a total flux B2 on the lower side. The reference directions of both total fluxes are the same as those of the displacement x. These 23 fluxes contain both biasing fluxes and generated by the permanent magnet, and excitation flux generated by the coil windings. If the actual current in the two coil windings is in the reference current direction shown in Figure 3.2, the biasing and excitation fluxes will add together on the armature top surface and subtract on the bottom surface. The normal force on the top surface will be larger than that on the bottom; and a net force in the +x direction will be generated. When the excitation current reverses its direction, the net force direction will change accordingly. By properly driving the current directions for each magnetic actuating unit, actuating force/torque can be generated on the armature in the x, y and θ directions, as shown in Figure 3.3. In the figure, xF , yF , and F represent for magnetic force generated in the x, y and θ directions, respectively. NSx1BBiasing flux,Excitation flux,B2BBiasing flux,Air gap, x0-xFlux density, 1 1B =B+BAir gap, x0+x Flux density, 2 2=B-BLAir gap, y0 Figure 3.2 Working principle of the magnetic actuating unit 24 xyθNSNSFyFyNS N SFxFxN SNSNSNSFθFθFθFθ(b) (c)(a) Figure 3.3 Current directions and flux paths for force generation (a) x direction; (b) y direction; (c) θ direction. 3.4 Lumped parameter force/torque analysis In this analysis, the permeability of the chosen soft magnetic material for the stators and armature is assumed to be infinite compared to the permeability of air. Thus, the main magnetic reluctance in the actuator is that of the air gaps. 25 pmpmR 1R2Ra1B A2B ANS Figure 3.4 Magnetic circuit model for the biasing flux Figure 3.4 shows a section of the magnetic actuating unit and the magnetic circuit model containing only the components related to the biasing flux generation. The reluctances of the system can be expressed as: 010x xR A (3.1) 020x xR A (3.2) 00apmyR A (3.3) 0pmpmLR A (3.4) where 1R and 2R are the reluctances across the air gaps between the armature and the stator. aR is the reluctance of the air gap between the armature and the permanent magnet. pmR is the 26 permanent magnet reluctance. A is the stator pole area. pmA is the permanent magnet pole area. 0 is the vacuum permeability. is the flux flowing into the armature. The total flux generated by the permanent magnet is: pm r pmB A (3.5) where rB is the remanence of the permanent magnet. According to the magnetic circuit model: 21 20 01 2 02 2pmpm pmpmpm aR LR R A A xR R L y xR R A A x (3.6) Since the actuator is designed to damp the vibration of the armature and hold it at its zero position, x will be much smaller than 0x . We can neglect the fourth part in the denominator of Equation (3.6), resulting in: 0 02pmpmLAL y xA (3.7) The fluxes need to satisfy the following equations from Gauss’ law: 27 1 21 1 2 2B A B AB AR B AR (3.8) whose solution yields to: 010020x xB Bxx xB Bx (3.9) where 2B A. 1R2RBABA2NI2NI-+-+ Figure 3.5 Magnetic circuit model for excitation flux Figure 3.5 shows a section of the magnetic actuating unit and its circuit model including only the components related to the excitation flux generation. From the magnetic circuit model: 28 1 2( )2 2NI NI BA R R (3.10) where N is the total number of coil turns for each magnetic unit, and I is the input excitation current through each coil turn. The excitation flux density can be calculated as: 002NIB x (3.11) The total flux density on both the upper and bottom surfaces of the armature is the superposition of the biasing flux and the excitation flux, giving 0 01 10 00 02 20 022NI x xB B B Bx xNI x xB B B Bx x (3.12) According to [58], the magnetic force generated by the magnetic actuating unit can be calculated from the system stored energy function ( , )eW x : ( , ) | eu eF W xx (3.13) 29 where e is the flux vector of all the excitation sources. Under the assumption of infinite permeability of the soft magnetic material, energy can only be stored in the air gaps. Therefore, the total stored energy in the system can be calculated as: 22 21 1 2 21 1 1( , ) 2 2 2e aW x B A R B A R R (3.14) 12eB AB A (3.15) Thus, the magnetic force generated by each actuating unit can be calculated as: 2 21 2020 0 0 ( )22ui sAF B BBAN ABI xx xK I K x (3.16) where 0/iK BAN x , 2 0 02 /sK AB x . Equation (3.16) shows that the magnetic force uF is a linear combination of the excitation current I and the armature displacement x . Since two actuating units work in parallel to generate the magnetic force in the x and y directions, we therefore have: _ _ ___ _ __2( )2( )x total i ex s xi total ex xs totaly total i ey s yi total ey ys totalF K I K x K I K xF K I K y K I K y (3.17) 30 where _ _ 2xi total yi total iK K K , _ _ 2xs total ys total sK K K , _x totalF and _y totalF are the total actuating forces in the x and y directions. exI and eyI are the x and y directions excitation currents through each coil turn. xyθl 0l1ldlT Figure 3.6 Schematic drawing for the torque analysis Figure 3.6 shows the schematic drawing for the torque analysis. In the figure, is the armature rotation angle; T is the torque generated by a single actuating unit; and 0l and 1l are the distances from the armature center line to the left and right edges of the stator pole, respectively. If the stator pole area is divided into multiple small sections with width dl and distance l to the armature center line, the magnetic force dF applied on each section can be expressed as: 31 i seK tdl K tdldF I lA A (3.18) where t is the thickness of the magnetic cores; and eI is the θ direction excitation current through each coil turn. Then, the torque generated in the θ direction by each actuating unit can be calculated as: 102 21 0 1 1 0 0( ) ( ) 2 3li seli sei sK KK tdl K tdlT I l lA AK l l K l l l lI (3.19) The total torque generated by the actuator is: _ _ _4( )total i e s i total e s totalT K I K K I K (3.20) where _ 4i total iK K , _ 4s total sK K . 3.5 Comparison with regular magnetic bearings Figure 3.7 shows a schematic drawing of a simple one degree of freedom existing magnetic bearing system. The input currents to the magnetic bearing coils are 1i and 2i ; ax is the air gap when the armature is centered; x is the armature displacement; and MBF is the magnetic force. 32 The nonlinear relation between the magnetic force (MBF ), input currents ( 1i , 2i ), and armature displacement ( x ) can be described by the following nonlinear relationship 2 21 22 2( ) ( )MB mb a ai iF k x x x x (3.21) where mbk is the magnetic coefficient. Because of this nonlinearity, the magnetic force created by regular magnetic bearings must be linearized before it can be effectively used. 1i2i ax xax xxStatorAr ature MBFCoils Figure 3.7 Schematic drawing of a one degree of freedom existing magnetic bearing Comparing to regular magnetic bearings, the main advantage of the proposed actuator is that the magnetic force/torque has a linear relation with the excitation current and the armature displacement as shown in (3.17) and (3.20). This advantage can significantly simplify the controller design process for magnetic actuators. 33 3.6 Material selection and actuator parameters design In general, there are two classes of materials that are often used in making magnetic cores. One is laminated steel, and the other is soft magnetic composites (SMC). Both of them can significantly reduce the eddy current generation in the magnetic cores, and therefore dramatically limits the heat generation in the actuator. Laminated steel usually has much higher permeability than SMC. However, it is very difficult and costly to form complex core shapes. On the other hand, although SMC has lower permeability, it is much easier for fabrication and fast prototype test. Therefore, in this thesis one of the SMC materials, Somaloy 500, is selected to fabricate the magnetic cores of the designed actuator. Besides, the rare earth material Nd2Fe14B with grade 42SH is selected to make the permanent magnet. The objective of the magnetic actuator design is to provide at least 350 N force and 40 N·m torque outputs for 1 A DC excitation current input. The design parameters of the magnetic actuator are first selected based on the lumped parameter analysis explained in Section 3.4, and then refined with finite element analysis. The final design parameters of the magnetic actuator are given in Table 3.1. 34 Table 3.1 Magnetic actuator design parameters Parameters Values Air gap, 0x 0.7 mm Stator pole area, A 1056 mm2 Permanent magnet pole area, pmA 966 mm2 Length of permanent magnet, L 25.4 mm Air gap between armature and permanent magnet, 0y 3 mm Remanence of the permanent magnet, rB 1.3 T Number of coil turns for each stator, N 552 turns Thickness of the magnetic cores, t 48mm Distance, 0l 58.8 mm Distance, 1l 80.8 mm 3.7 Finite element analysis of the actuator performance The lumped parameter analysis is sufficient to guide the actuator design. But it does not lead to a very accurate performance calculation. Nonlinearities and nonidealities such as magnetic saturation, finite permeability and flux leakage, always exist in the real device. Therefore, FEM is used to evaluate the force and torque generation of the designed magnetic actuator. A Finite Element Method Magnetics (FEMM) software package [59] is used to carry out the FEM analysis and the B-H curve of soft magnetic material Somaloy 500 [60] is used in the analysis. Since the armature moves only around its zero position during active damping, the output performance of the actuator is evaluated only at zero armature displacement for different input currents. 35 Excitation coilsStator corePermanent magnetArmature coreBoundary Figure 3.8 Finite element mesh of the designed actuator using FEMM The finite element mesh of the designed actuator using FEMM is shown in Figure 3.8. The results of the FEM analysis are compared with the analytical analysis (based on (3.17) and (3.20)) as shown in Figure 3.9 and Figure 3.10. From the FEM analysis, both the force and torque generated by the designed magnetic actuator have a linear relationship with the input current. By implementing linear regression to the FEM results, the magnetic coefficients are identified as: _ _ _ _xi total FEM yi total FEMK K 370.76 N/A, _ _i total FEMK =49.98 N·m/A, while the calculated analytical magnetic coefficients are _ _ _ _xi total ana yi total anaK K 875.81 N/A, _ _i total anaK=122.26 N·m/A. The difference between the FEM results and the analytical results are caused by 36 the nonlinearities of the real actuator, primarily finite permeability of the core material and leakage flux paths in the real device. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2000-1500-1000-5000500100015002000Current [A]Force [N] FEM resultLinear regression resultAnalytical result_ _370.76 N/Axi total FEMK _ _875.81 N/Axi total anaK Figure 3.9 Force current relation of the designed magnetic actuator. (Same in both x and y directions.) 37 -2 -1 0 1 2-300-200-1000100200300Current [A]Torque [N*m] FEM resultLinear regression resultAnalytical result_ _49.98 N m/Ai total FEMK_ _122.26 N m/Ai total anaK Figure 3.10 Torque current relation of the designed magnetic actuator 3.8 Fabrication of the magnetic actuator Since the parameters of the magnetic actuator have been designed, the solid model of the actuator assembly is developed; and the armature and stator assemblies have been fabricated, as shown in Figure 3.11. The stator assembly includes the stator cores, permanent magnet, excitation coils and the stator housing. 20/34 SPNSN served litz wire is used to wind the coils. In order to reduce the inductance of the actuator, the excitation coils are wound and connected in such a way: four sub coils, each has 23 turns × 3 turns, are wound and connected in parallel to make one excitation coil winding. When powering the actuator with the current amplifiers, the two coil windings of each magnetic unit are also connected in parallel. However, because of this coil connection method, the amplitudes of the equivalent excitation currents in each coil turn (exI , eyI,eI ) are 38 one-eighth of the amplitudes of the amplifier command currents (xI , yI, I). After winding the coils on the stator cores, the permanent magnet and the stator cores are epoxied to the stator housing; and all the gaps in the stator assembly are filled with epoxy to make the structure more rigid. The armature assembly is composed of an armature core and two clamping fixtures, as shown in Figure 3.12. Fixture 1 is used to provide the target surfaces for displacement sensors, while fixture 2 is used to clamp the armature core to fixture 1. The armature core and the clamping fixtures are bolted together; and all the gaps in the armature assembly are filled with epoxy so that it can behave like a rigid part. The fabricated armature assembly is clamped to a slender and long boring bar using a trantorque bushing, as shown in Figure 3.12. The weight of the boring bar is about 1 kg, and the weight of the armature assembly is about 3 kg. It should be noted that this is a prototype setup in a laboratory used to demonstrate the effectiveness of the new magnetic actuator. The proposed actuator concept is designed to damp vibrations and increase the static stiffness of very large slender shafts in a turning process and very long boring bars which are used in boring large diesel engine cylinder walls or similar parts. The boring bar size that we can use in our prototype setup is limited by our machine tool dimensions. For real application in very large machine tools, the weight of the armature assembly will be much less than the weight of the large and long boring bars and shafts. The active damping setup has also been designed and fabricated as shown in Figure 3.13. Both the stators and the boring bar are clamped to the turret of a CNC lathe through the clamping fixture. Three fiber optic displacement sensors are installed on the magnetic actuator to measure the armature displacement in the x, y and θ directions. The operation instructions of the designed magnetic actuator are given in Appendix D. 39 Figure 3.11 Magnetic actuator. (a) Solid model; (b) Fabricated stator assemblies and armature assembly. Figure 3.12. Armature assembly clamped onto a boring bar. (a) Solid model; (b) Fabricated assembly. 40 Figure 3.13. Active damping setup. (a) Solid model; (b) Fabricated setup. 3.9 Summary In this chapter, a novel three degrees of freedom linear magnetic actuator, which can have actuation in the radial (x, y) directions and torsional (θ) direction, is presented. The designed magnetic actuator is composed of four identical actuating units; and each actuating unit is linearized with magnetic configuration design strategy. Lumped parameter models are developed to analyze the force and torque generation of the designed magnetic actuator; and finite element analysis is carried out to further analyze the performance of the actuator. The FEM analysis results show that the designed actuator can output 370.76 N force and 49.98 N·m torque for 1 A DC excitation current input. Finally, the actuator is fabricated and ready to use in the active damping tests. 41 Chapter 4: Active Damping of Machine Tools2 4.1 Overview The designed magnetic actuator is mounted on the turret of a CNC lathe to demonstrate its active damping performance for long and flexible structures in machine tools. It is tested on a long and slender boring bar, and three different types of controllers, including loop shaping controller, DI controller and H∞ controllers, are developed to actively damp the boring bar vibrations. The active damping performance is tested through both impact response tests and cutting experiments. The content of this chapter is arranged as follows. Section 4.2 explains the experimental setup and model identification of the active damping setup. Three different types of controllers, including loop shaping controller, DI controller, and H∞ controllers, are designed in Section 4.3. The experimental results of impact response tests, static stiffness tests, and cutting tests are given and discussed in Section 4.4. Concluding remarks are given in Section 4.5. 4.2 Experimental setup and model identification 4.2.1 Experimental setup The fabricated magnetic actuator is mounted on the turret of a CNC lathe as shown in Figure 4.1. Three fiber optic sensors are used to measure the actuator armature displacements in the radial (x, y) directions and the torsional (θ) direction. Also, an accelerometer is attached to the boring bar slightly away from the tool tip to measure the boring bar vibration and avoid interference with the cutting process. A microphone is used to measure the sound signals during the cutting tests. 2 Parts of Chapter 4 have been submitted for review. 42 The armature of the actuator is clamped to a 305 mm long Valenite A16T-SCLPR4 boring bar using a trantorque bushing. The boring bar is cantilevered to the clamping parts (Figure 4.1), which carry the stators of the actuator. The workpiece used in the cutting tests is an aluminum 6061 T6 cylinder with 150 mm outer diameter and 40 mm inner diameter, which will be expanded by the boring operation. The distance between the boring bar tip and the actuator is 145 mm. Four Varedan LA-415s linear power amplifiers each with an 8 kHz current loop bandwidth are used to drive the magnetic actuator. The amplifier can provide maximum continuous current of 5 A, which gives a maximum 0.625 A excitation current in each coil turn. However, above 850 Hz, the maximum current that can be provided in each coil is limited by the maximum output voltage of the amplifier and the actuator inductance, instead of the amplifier current limit; and it decreases when the current frequency goes up. Therefore, the proposed actuator can deliver maximum 248 N force in two radial (x, y) directions and maximum 34 N·m torque in torsional (θ) direction up to 850 Hz. The force and torque reduces to 107 N and 14.5 N·m at 2000Hz. Hence, the actuator is capable of damping a wide range of structure modes. SpindleWorkpieceXZYMagnetic actuatorTurretFiber optic sensorsAccelerometerBoring barθClamping partsMicrophone Figure 4.1 Experimental setup for active damping of boring bar 43 The data acquisition setup is shown in Figure 4.2. An impact hammer from Dytran Instruments is used to give the blows to the boring bar tip for impact modal analysis. An I/O box from National Instruments and CutPro software developed by Manufacturing Automation Lab at the University of British Columbia are used to collect all the data from different sensors during both impact modal tests and cutting tests. Impulse hammerI/O boxCutPro software Figure 4.2 Data acquisition setup 4.2.2 Dynamics measurements of the designed magnetic actuator The objectives of measuring the dynamics of the magnetic actuator are as follows: a) determining whether there is coupling between the different axes or not; b) Providing a plant measurement for the loop shaping controller design and DI controller design in Sections 4.3.1 and 4.3.2, respectively. The open-loop FRFs from the amplifier command currents (xI , yI, I) to the actuator armature displacements ( x , y , ) are measured using a sine sweep, as shown in Figures 4.3 – 4.5. In the figures, ( , , , , , )i I jG i x y j x y represents the open loop FRF from the amplifier 44 command current jI to the armature displacement i . From the measurements, it can be seen that the dynamic stiffness of the cross frequency responses (,i I jG i j, , ,i x y ; , ,j x y ) is much stiffer than that of the direct frequency responses(ii IG, , ,i x y ). Thus, it can be assumed that the mover of the actuator (including the armature assembly and the boring bar) has uncoupled dynamics in each axis; hence, the actuator can be controlled individually in each axis. 10010110210310-810-710-610-510-4Frequency [Hz]Magnitude [m/A]100101102103-600-400-2000Frequency [Hz]Phase [deg]xIyG Ix xxIxGxIyI Figure 4.3 Open loop FRFs from the amplifier command currents (xI , yI, I) to the armature displacement ( x ). 45 yIxGyI yGyI10010110210310-710-610-510-4Frequency [Hz]Magnitude [m/A]100101102103-300-200-1000Frequency [Hz]Phase [deg]yIxGyI yGyI Figure 4.4 Open loop FRFs from the amplifier command currents (xI , yI, I) to the armature displacement ( y ). IyI10010110210310-610-410-2100Frequency [Hz]Magnitude [deg/A]100101102103-800-6004-2000Frequency [Hz]Phase [deg]Frequency [Hz]Magnitude [deg/A]Frequency [Hz]Phase [deg]IxGIyI Figure 4.5 Open loop FRFs from the amplifier command currents (xI , yI, I) to the armature displacement ( ). 46 4.2.3 Model identification of the active damping setup The design of the loop shaping controller and the DI controller does not necessarily require an identified plant model. However, H∞ controllers are model based controllers. The accuracy of the identified plant model plays a very important role in achieving a good control performance. Therefore, the objective of the model identification in this section is to provide a well identified plant model for the H∞ controllers design in Section 4.3.3. Based on the dynamics measurements in Section 4.2.2, the boring bar is considered to have uncoupled dynamics in the two radial ( ) and the torsional (θ) directions. The horizontal and the torsional (θ) deflections have negligible influence on the regenerative chip thickness, but the dynamic chip thickness is directly affected by the axial ( ) and the radial ( ) deflections in a boring process [4]. Since the axial stiffness of the boring bar is high, the chatter is caused by the bending modes in the direction which need to be actively damped by the controller. ZXYBoring barθMagnetic actuatorFc xItx ,ax axArmature Figure 4.6 Diagram of the active damping setup 47 The diagram of the boring bar and actuator setup, i.e. plant, is given in Figure 4.6. The control current (Ix) is applied to the magnetic actuator via amplifiers, which leads to magnetic force at the armature that tries to manipulate the boring bar vibrations. The cutting force ( ) acts as a disturbance input. The outputs of the plant are the tool tip displacement (xt), armature displacement (x), boring bar displacement (xa) and acceleration (a) at the accelerometer location. The open loop transfer functions from the command current (Ix) to displacements (xt) and (xa) are denoted by IxtxG and Ia xxG, respectively. The FRFs are measured by giving an impulse current signal to the current amplifiers and measuring the displacement (xt) with a fiber optic sensor while measuring the acceleration (a) with the accelerometer. The open loop transfer functions between the displacements (xt, xa, x) and the cutting force (Fc) are denoted by x FtG, x FaG, and xFG , respectively. The FRFs are measured with impact hammer tests. The measured and curve-fitted FRFs of the system are given in Figure 4.7, while the estimated modal parameters are given in Table 4.1. It can be seen that the actuator is able to excite mainly the first two modes of the boring bar from its armature location which is 145 mm away from the tool tip. The third mode can hardly be excited by the actuator which makes active damping of the third mode challenging. 48 / [m/A]txxI/ [m/A]axxI/ [m/A]xxI/ [m/N]tcxF/ [m/N]acxF/ [m/N]cxF IxtxG(a)Ia xxG(b)IxxG(c)FtxG(d)FaxG(e)FxG(f) Figure 4.7 Measured and curve fitted FRFs of the plant system. (a) IxtxG; (b) Ia xxG; (c) IxxG; (d) FtxG;(e) FaxG; (f) FxG . Table 4.1 Estimated modal parameters of the system. Mode # nf [Hz] [%] xtx Ik [A/m] xaIxk[A/m] xIxk [A/m] Ftxk [N/m] Faxk[N/m] xFk [N/m] 1 174 1.20 -1.74×105 -1.92×105 -4.38×105 1.62×106 1.98×106 4.38×106 2 614 0.63 1.05×106 1.70×106 -1.02×108 1.72×107 2.89×107 -4.50×108 3 1130 0.18 3.99×107 7.23×107 -1.90×108 2.10×107 3.85×107 -1.24×108 The FRFs between the cutting force and the boring bar displacements show that the first bending mode of the boring bar at 174Hz is the most flexible mode, while the third bending 49 mode at 1130 Hz is also quite flexible. However, the second bending mode of the boring bar at 614 Hz can hardly be seen from the measured FRFs of FxG (Figure 4.7 (f)) and IxxG (Figure 4.7 (c)). In order to explore the reason for this, the mode shapes of all the three modes are measured with impact hammer tests (Figure 4.8). An accelerometer is placed at the tool tip to measure the boring bar vibrations, while hammer blows are given at different locations on the boring bar. The measured mode shapes are given in Figure 4.8. It can be seen that the fiber optic sensor is located at a neutral node of the second bending mode of the boring bar assembly. Therefore, the boring bar vibration at the second modal frequency is too small to be detected by the embedded fiber optic sensor. In real active damping applications, the actuator should not be placed at a neutral node of the structural mode that we want to damp. In our experimental setup, the actuator is located at a neutral node of the second bending mode of the boring bar by coincidence. Since we want to make the boring bar overhang as long as possible to have a good demonstration of the target cases (large and long boring bars/shafts), and the 145mm overhang of the boring bar (Figure 4.1) is the longest we can reach for our setup, limited by our machine tool dimension. 50 AccelerometerFiber optic sensor locationActuator armature assemblyTool tip20 mm40 mm60 mm80 mm100 mmDistance from tool tip [mm]AmplitudeMode shape at 174 HzMode shape at 614 HzMode shape at 1130 Hz135 mm155 mm175 mm195 mm225 mm50 100 150 20000.5150 100 150 200-0.200.20.450 100 150 200-1012 Figure 4.8 Mode shape analysis of the boring bar assembly. The transfer function relations between the inputs cF , Ix and outputs xt, xa, and x are determined from their estimated modal parameters as: 51 ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )t x I x x F ca x I x x F cxI x xF cxt ta x axx s G s I s G s F sx s G s I s G s F sx s G s I s G s F s (4.1) with the transfer functions 22 23 31 13 31 1( )2( ) ( )( ) ( )( ) ; ( )( ) ( )( ) ( )( ) ( )( ) ; ( )( ) ( )( )( )( )( )nipii ni nipi pit tx I x Fi ix I c Fpi pia ax I x Fi ix I c FpixIx Ixt tt ta x aa axxi ixi ixx xx xxG ss sG s G sx s x sG s G sI s k F s kG s G sx s x sG s G sI s k F s kG sx sG sI s k 1,3 1,3( )( ); ( )( )pixFi ic Fi ixG sx sG sF s k (4.2) where 2ni nif and i are the modal frequency (rad/s) and damping ratio, while It xixk, Ft ixk, Ia xixk, Fiaxk, Ixixk, and xFik are the modal stiffnesses of the mode i . Note that the second bending mode cannot be detected with the embedded fiber optic sensor because the sensor is at its neutral point. As a result, the second mode is neglected in xIxG and xFG . 4.3 Controller design The ultimate goal for active damping of the boring bar is to damp the forced and chatter vibrations at the tool tip (xt), and increase the maximum chatter-free depth of cut with the designed magnetic actuator. Since the critical depth of cut for chatter stability is proportional to the dynamic stiffness of the structure, the main objective of the controller design is to improve the boring bar dynamic stiffness of all the dominant modes within the limits of the current amplifiers. 52 4.3.1 Loop shaping controller design In this section, loop shaping controllers are designed for each axis not only to damp the boring bar vibration, but also to prove that the actuator can be experimentally controlled in all the x, y and θ axes; although only the x axis control contributes to the chatter suppression. Since the boring bar has uncoupled dynamics in the x, y and θ axes, a controller is designed separately for each axis; and the x axis is used as an example to show the control configuration, as given in Figure 4.9. Because the tool tip displacement xt is in the cutting zone, the displacement of the armature (x) is used for feedback control. The armature displacement caused by the cutting force (Fc) is considered as a disturbance which needs to be damped with the controller. xref is the reference signal for the controller, which is set to zero during active damping. xI xPlantref ( )C scF Figure 4.9 Control system block diagram in the x axis The loop shaping controller for each axis is designed as given below: 53 2 752 71163.67 1 11.88 1.488 102 260.5 /15( ) 3.47 10 115 1131 1.488 1012 260.5 15LPxss sC sss s s (4.3) 2 742 71123.40 1 31.67 1.488 102 196.4 / 20( ) 8.02 10 120 3016 1.488 1012 196.4 20LPyss sC sss s s (4.4) 1229.34 1 2 365/15( ) 226.54 115 12 365 15LPsC sss (4.5) where LPxC , LPyC,and LPC are the controllers for the x, y and θ axes, respectively. The designed loop shaping controllers are composed of two different parts: a PI controller and a lead-lag compensator. The PI controller is used to achieve zero steady state error and increase the static stiffness. The lead-lag compensator is added to ensure sufficient phase margin for stability and improve the damping property of the boring bar. Since the second bending mode of the boring bar at 614 Hz cannot be detected by the embedded fiber optic sensor because it is located at the neutral node of the mode, a notch filter is added to the x and y axes controllers to remove the boring bar vibration components around 614 Hz. The frequency response of the designed controllers and the negative loop transmission (NLT) of the control system in each axis are measured as shown in Figure 4.10. From the measurements, we can see that the system has a loop transmission unity gain crossover frequency of 260.5 Hz in the x axis, with 51.7° phase margin. The crossover frequency for the y axis is 196.4 Hz, and the phase margin is 55.9°. Besides, the control system in the θ axis has a crossover frequency of 365 Hz, with 57.1° phase margin. 54 100101102103104105106107Frequency [Hz]Magnitude100101102103-1000100200Frequency [Hz]Phase [deg]10010110210310-210-1100101Frequency [Hz]Magnitude100101102103-400-2000Frequency [Hz]Phase [deg]100101102103103104105106Frequency [Hz]Magnitude100101102103-1000100200Frequency [Hz]Phase [deg]10010110210310-210-1100101Frequency [Hz]Magnitude100101102103-200-10001Frequency [Hz]Phase [deg]0 1 2 3101102103r [ z]Magnitude100101102103-1000100Frequency [Hz]Phase [deg]0 1 2 310-1100101102r [ z]Magnitude100101102103-200-1000100Frequency [Hz]Phase [deg]Loop shaping controller, x axis NLT, x axisLoop shaping controller, y axis NLT, y axisLoop shaping controller, θ axis NLT, θ axis Figure 4.10 FRFs of the loop shaping controllers and the NLTs for different axes 55 4.3.2 Derivative-Integral (DI) controller design Besides the loop shaping controller, an even simpler DI controller is designed in this section for active damping of the boring bar. Since vibrations in the x axis is the most crucial in chatter suppression, the DI controller is only designed in x axis to improve both the dynamic and static stiffness of the boring bar. The control configuration is the same as the one shown in Figure 4.9; and the transfer function of the designed DI controller is given below: 6 2 72 75 10 11.88 1.488 10( ) 200 1131 1.488 10DIxs sC s s s s s (4.6) The derivative part of the DI controller improves the damping property of the boring bar, while the integrator increases its static stiffness. A notch filter is added to the controller to remove the boring bar vibration around 614 Hz since the fiber optic sensor is located at a neutral node of the second bending mode. The FRFs of the designed controller DIxC and the NLT of the control system are measured as given in Figure 4.11. It can be seen that the control system has a unity gain crossover frequency of 256.9 Hz with 66.5° phase margin. 56 DI controller, x axis NLT, x axis100101102103103104105106107Frequency [Hz]Magnitude100101102103-1000100200Frequency [Hz]Phase [deg]10010110210310-210-1100101Frequency [Hz]Magnitude100101102103-400-2000Frequency [Hz]Phase [deg] Figure 4.11 FRFs of the DI controller and the NLT in the x axis 4.3.3 H∞ controllers design Besides the loop shaping controller and the DI controller, an H∞ optimal control method is discussed in this section for the active damping controller design because of its advantage in achieving stabilization with optimized frequency response shaping performance. 4.3.3.1 Control configurations Four different control configurations are proposed in this section to damp the tool tip vibrations, and the system block diagrams are given in Figure 4.12. For all the cases, weighting function WI is used to penalize the closed-loop FRF from the cutting force (Fc) to the control current (Ix) to avoid saturation of the current amplifiers. Iz and xz are the weighting function outputs, and ( )K s is the controller which will be synthesized using the H∞ optimization technique. Case 1: The armature displacement (x) is used as a feedback signal for the controller design, and the weighing function Wx is used to penalize the closed-loop FRF from the cutting force (Fc) to the armature displacement (x) which indirectly reduces the vibration at the tool tip (xt). 57 Case 2: The armature displacement (x) is also used for feedback control, but the weighting function Wx is used to directly penalize the closed-loop FRF from the cutting force (Fc) to the tool tip displacement (xt). Case 3: The boring bar acceleration (a) at the accelerometer location (Figure 4.1) is used for feedback control, while the weighting function Wx is used to shape the closed-loop FRF from the cutting force (Fc) to the boring bar displacement (xa) at the accelerometer location. Similar to Case 1, this method reduces the tool tip vibrations indirectly. Case 4: The feedback signal is still the boring bar acceleration (a), but the weighting function Wx is used to directly penalize the closed-loop FRF from the cutting force (Fc) to the tool tip displacement (xt). xIcFx( )WI sIzxz( )K sGeneralized plantP1xCase 1cFx( )WI sIz( )K sGeneralized plantP2xtxcF ( )WI sIz( )K sGeneralized plantP3aaxa cF ( )WI sIz( )K sGeneralized plantP4a atxCase 2Case 3Case 4( )s( )x s ( )x s( )x sxzxzxzxIxIxI Figure 4.12 System block diagrams for controller design 58 The combination of the plant (Pj, j=1, 2, 3, 4) and the weighting functions (WI and Wx ) creates a generalized plant which will be used for the controller synthesis. 4.3.3.2 Guideline for tuning the weighting functions The weighting function structures used in this thesis for tuning WI and Wx are given below: st1221 1 nd22 221 1 rd3242 212(1 order)1212 2( ), ( ) (2 order) 12 212 2 2(3 order)212 2pppsfKsfs sf fWI s Wx s Ks sf fs sf f s fKs fs sf f (4.7) where pK, 1f , 2f , 3f and 4f are the tuning parameters of the weighting functions. The weighting functions are tuned so that the peaks in the closed-loop FRF from the cutting force ( ) to the tool tip displacement (xt) are attenuated as much as possible within the current limits of the amplifiers. Here, a sample guideline for case 1 is provided for the parameter tuning as an example. Figure 4.13 illustrates how the gain plots of the weighting functions 1( )Wx and 1( )WI affect the FRFs of the closed-loop transfer functions from Fc to x, denoted by xFT , and from Fc to Ix, 59 denoted by IFT . In order to reduce the peaks of xFT at the natural frequency locations, the values of the tuning parameters can be adjusted to lower the amplitude of 1( )Wx around the peak frequency regions of xFT . Similarly, the amplitude of 1( )WI can be decreased around the peak frequency regions of IFT to penalize the peaks of IFT . If the 1st order weighting function structure cannot provide a good control performance, the 2nd order or 3rd order weighting function structures can be used in the tuning process. However, a higher amplitude of IFT is normally required to further damp the FRF of xFT . If the amplitude of IFT is too high, it may saturate the current amplifiers during the impact hammer tests and the cutting tests, which needs to be avoided when tuning the weighting functions. ( )xFT j1( )Wx j1pK12 f22 fAmplitude [m/N] ( )IFT j1( )WI jAmplitude [A/N](a) (b)[rad/s] [rad/s]1pK12 f22 f Figure 4.13 Schematic diagrams for selection of the weighting functions. (a) 1( )Wx and xFT in log scale; (b) 1( )WI and IFT in log scale. 4.3.3.3 State space model of the generalized plant In order to synthesize the controller ( )K s using H∞ optimization technique, the state space model of the generalized plant is developed in this section. 60 Since the outputs and the order of the plant are different for the proposed four cases, the state space model of the plant is developed separately for each case, given as: Case j (j=1, 2, 3, 4): :x pj x pjjv pj x pjq A q B uPy C q D u (4.8) ( j=1)( j=2),( j=3)( j=4)tcv axtxxxFu y xIaxa (4.9) where xq is the state vector. Similarly, the state space equations of the weighting functions can also be developed from their transfer functions as: : WI WI WI WI xI WI WI WI xq A q B IWI z C q D I (4.10) (Case1)(Case2): ,(Case3)(Case4)tWx Wx Wx Wx xxax Wx Wx Wx xtxxq A q B uWx uxz C q D ux (4.11) 61 where WIq and Wxq are the state vectors. The derivation processes of the state space models and the system matrices are given in Appendix A. The state space equation describing the generalized plant for each case can be derived from Equations (4.8 - 4.11) as: Case j (j=1, 2, 3, 4): 0 00 00:0 000 0x xpj pjWI WI WI WWx nj Wx Wx njWx WxpjxI WI Wx Wx nj Wx WI Wx njmj mjWxq qA Bq A q B uB C A B Dq qGqz C Dz D C C q D D uC Dv q (4.12) (Cases j=1, 2)(Cases j=3, 4)xv a (4.13) where [0 ]W WIB B , [0 ]W WID D . The matrices njC , mjC , njD and mjD are given in Appendix A. 4.3.3.4 Controller design Having the generalized plant, the controller K is synthesized for each case so that the closed-loop system is internally stable and the H∞ norm of the closed-loop transfer function (zFT ) from cF to z ( TI xz z z ) is minimized. This H∞ optimal control problem can be solved using the linear matrix inequalities (LMI) algorithm presented in [61] with the available MATLAB command 62 hinfsyn.m [62]. A brief summary of how to synthesize the H∞ controller with the LMI method is given in Appendix B. The weighting function parameters that are used in the controller design are given in Table 4.2. Table 4.2 Weighting function parameters Weighting functions Parameters Case 1 Case 2 Case 3 Case 4 WI order 1st order 2nd order 1st order 1st order pK 10 N/A 14 N/A 30 N/A 40 N/A 1f 500 Hz 500 Hz 600 Hz 600 Hz 2f 2000 Hz 1600 Hz 2000 Hz 2000 Hz Wx order 3rd order 3rd order 2nd order 2nd order pK 2.4×106 N/m 2.0×106 N/m 1.0×106 N/m 1.0×106 N/m 1f 160 Hz 200 Hz 200 Hz 160 Hz 2f 540 Hz 540 Hz 735 Hz 550 Hz 3f 1 Hz 2 Hz 4f 0.01 Hz 0.25 Hz The transfer function of the synthesized H∞ optimal controller is obtained as: 00( )k iiik iiib sK sa s (4.14) where k is the order of the controller, and the controller parameters ia , ib (i=0, 1, 2, …, k) for each case are given in Table 4.3. The controller order is 8, 15, 9 and 15 for cases 1, 2, 3 and 4, respectively. It is a summation of the plant order and the order of the weighting functions. In 63 Equation (4.1), the order of transfer functions ( )tx s , ( )ax s and ( )x s are 6, 6, and 4, respectively. Since only ( )x s is needed for case 1, while only ( )ax s is required for case 3, the plant order in cases 1 and 3 is much lower, i.e. 4th order and 6th order respectively, therefore leading to a lower order controller. On the other hand, because both ( )tx s and ( )x s are considered in case 2, while both ( )tx s and ( )ax s are required in case 4 to model the plant, the plant order in cases 2 and 4 becomes much higher, i.e. 10th order and 12th order respectively, therefore leading to a higher order controller. Table 4.3 Controller parameters bi Case 1 Case 2 Case 3 Case 4 ai Case 1 Case 2 Case 3 Case 4 b0 1.812×1032 3.219×1055 2.889×1029 -2.79×1050 a0 1.353×1025 2.417×1048 4.004×1028 1.869×1050 b1 -4.707×1031 -6.513×1054 4.89×1026 1.147×1047 a1 2.169×1026 2.42×1049 3.493×1027 4.242×1048 b2 1.148×1028 -5.574×1051 4.043×1023 -3.911×1044 a2 1.658×1023 4.139×1046 1.85×1025 1.519×1046 b3 -1.106×1025 -9.818×1048 -1.375×1018 -4.057×1040 a3 8.398×1019 5.626×1043 1.179×1022 1.597×1043 b4 5.591×1021 -7.558×1045 -3.537×1015 -2.105×1038 a4 2.441×1016 5.531×1040 6.256×1018 2.006×1040 b5 1.918×1018 -4.685×1042 -7.075×1012 -1.468×1035 a5 4.865×1012 3.785×1037 2.056×1015 1.343×1037 b6 8.52×1014 -2.92×1039 -1.031×109 -6.988×1031 a6 5.468×108 1.951×1034 4.749×1011 7.263×1033 b7 5.894×1010 -1.017×1036 -2.427×105 -2.699×1028 a7 2.77×104 7.493×1030 8.619×107 2.841×1030 b8 0 -4.284×1032 -16.31 -6.982×1024 a8 1 2.265×1027 9717 9.221×1026 b9 0 -1.005×1029 0 -1.792×1021 a9 0 5.452×1023 1 2.395×1023 b10 0 -2.686×1025 0 -2.704×1017 a10 0 1.059×1020 0 5.113×1019 b11 0 -3.287×1021 0 -4.902×1013 a11 0 1.7×1016 0 9.15×1015 b12 0 -5.039×1017 0 -4.457×109 a12 0 2.128×1012 0 1.249×1012 b13 0 -3.282×1013 0 -4.555×105 a13 0 2.259×108 0 1.579×108 b14 0 -2.388×109 0 -26.27 a14 0 1.536×104 0 1.079×104 b15 0 0 0 0 a15 0 1 0 1 When the controllers are implemented, a notch filter is added to the controllers in cases 1 and 2 to remove the vibrations around 614 Hz which are not measurable by the fiber optic sensor. 64 4.4 Experimental results 4.4.1 Impact response tests The designed controllers are implemented using the dSPACE 1103 real time system. Impact modal tests were carried out to measure the FRFs from the cutting force ( ) to the displacement at the boring bar tip (xt) for each controller case. First, the measured FRFs with loop shaping controllers in different number of axes are compared as shown in Figure 4.14. The results show that the boring bar dynamics in the x axis does not change when the controllers in the y and θ axes are turned on, comparing to the case with active control in the x axis only. This further proves that the boring bar has uncoupled dynamics in x, y and θ axes. 200 400 600 800 1000 1200 14000246810 x axis controlx, y axes controlx, y, axes controlFrequency [Hz]/ [μm/N@tcxF1130 1135 11406789 610 615 620345 Figure 4.14 Tool tip FRFs with loop shaping controllers in different number of axes Then, the measured FRFs and the tool tip dynamic stiffness at the peaks are compared for different controller cases, as shown in Figure 4.15 and Table 4.4, respectively. From the measurements, the first bending mode at 174 Hz is completely damped by all the controllers. The second mode at 614 Hz is not damped in all the control cases but H∞ control cases 3 and 4 due to 65 the fiber optic sensor location and notch filter. In H∞ control cases 3 and 4, the second mode is completely damped, since the accelerometer is attached close to the tool tip which can sense the boring bar vibrations caused by all modes. For the third mode at 1130 Hz, the dynamic stiffness is increased considerably for all the control cases. H∞ control case 1 achieves better active damping performance over the other control cases. The simple loop shaping controller and DI controller cannot provide as good active damping performance as the H∞ controller in cases 1 and 2. Due to the non-collocated sensor actuator configuration, the open loop FRF Ia xxG (Figure 4.7 (b)), which plays an important role in the plant model in H∞ control cases 3 and 4, has a right half plane zero. Therefore, the plant becomes a non-minimum phase system, which limits the achievable active damping performance in H∞ control cases 3 and 4. Considering all the three modes, H∞ control case 1 gives the highest minimum dynamic stiffness at the tool tip, and therefore provides the best active damping performance. The closed-loop FRFs from the cutting force (Fc) to the control current (Ix) are also measured by giving a hammer blow at the tool tip and measuring the current from the amplifiers. The measurements are shown in Figure 4.16. It can be seen that the third mode at 1130 Hz is much harder to be damped comparing to the first two modes, since it requires much higher control currents. The results are quite noisy in the control cases in which armature displacement (x) is used as a feedback signal. This is because that the large noise level in the displacement measurements (x) is amplified by the controller and leads to noisy control currents. From Figure 4.16, it can be seen that 1 N dynamic cutting force around the third modal frequency of the boring bar demands more than 1 A control current for all the control cases. When the cutting process is stable, the dynamic cutting force will be small due to the low vibration level; therefore the control current will also be small. However, if chatter happens around the third modal 66 frequency of the controlled boring bar, the large level of vibrations will lead to very high control currents which can easily saturate the amplifiers. Although it is impossible to damp the third mode completely due to the current limit, its dynamic stiffness is still increased from 0.073 [N/µm] to 0.168 [N/µm] with H∞ controller case 1. 1120 1125 1130 1135 114002468101214Frequency [Hz]xt/Fc [m/N] 605 610 615 620 625012345Frequency [Hz]xt/Fc [m/N] 170 172 174 176 178 180051052025Frequency [Hz]xt/Fc [m/N] No controlH∞ control Case 2(a) (b)(c)(d)/ [μm/N@ctxF/ [μm/N@ctxF/ [μm/N@ctxF/ [μm/N@ctxFH∞ control Case 3H∞ control Case 4H∞ control Case 1Loop shaping controlDI controlNo controlH∞ control Case 2H∞ control Case 3H∞ control Case 4H∞ control Case 1Loop shaping controlDI controlNo controlH∞ control Case 2H∞ control Case 3H∞ control Case 4H∞ control Case 1Loop shaping controlDI control500 1000 1500051015202530Frequency [Hz]xt/Fc [m/N] Figure 4.15 Measured FRFs from the cutting force (Fc) to the tool tip displacement (xt) with active damping. (a) Full scale FRFs; (b) Zoomed-in FRFs for the first mode; (c) Zoomed-in FRFs for the second mode; (d) Zoomed-in FRFs for the third mode. 67 Table 4.4 Tool tip dynamic stiffness for different control cases Cases Dynamic stiffness at the peaks dnk [N/µm] Minimum dynamic stiffness dnk [N/µm] 1st mode 2nd mode 3rd mode No control 0.039 0.195 0.073 0.039 Loop shaping control 0.560 0.190 0.112 0.112 DI control 0.657 0.190 0.129 0.129 H∞ control Case 1 0.589 0.212 0.168 0.168 H∞ control Case 2 0.352 0.219 0.149 0.149 H∞ control Case 3 0.220 1.821 0.108 0.108 H∞ control Case 4 0.272 3.096 0.120 0.120 1125 1130 1135 114000.511.52Frequency [Hz]I / Fc [A/N] / [A/N]xcIF(a) (b)H∞ control Case 2H∞ control Case 3H∞ control Case 4H∞ control Case 1Loop shaping controlDI control/ [A/N]xcIF200 400 600 800 000 120000.511.522.5Frequency [Hz]I / Fc [A/N] Figure 4.16 Measured FRFs from the cutting force (Fc) to the control current (Ix) with active damping. (a) Full scale FRFs; (b) Zoomed-in FRFs for the third mode 4.4.2 Static stiffness measurements The tool tip static stiffness of the boring bar in the x direction is measured with a force gauge and a fiber optic sensor, as shown in Figure 4.17. Without control, the static stiffness is only 0.67 N/µm. When the active damping controller is turned on, the static stiffness is increased by about 4 times for all the control cases except for H∞ control cases 3 and 4. Because the acceleration signals do not contain any DC information of the boring bar displacement, acceleration feedback control methods in H∞ control cases 3 and 4 cannot improve the static stiffness of the boring bar. 68 The loop shaping controller and the DI controller have an infinite DC gain, because of the integrator, while the H∞ controllers in cases 1 and 2 are tuned to have a very high DC gain as well. Therefore, the boring bar has almost infinite static stiffness at the actuator location in these control cases, as long as the control current is not saturated. Thus, these control cases have nearly the same tool tip static stiffness as the case when the boring bar is rigidly clamped at the actuator location. Fiber optic sensorForce g u e Figure 4.17 Setup for static stiffness measurement Table 4.5 Measured static stiffness at the boring bar tip Cases Static stiffness [N/µm] No control 0.67 Loop shaping control 2.86 DI control 2.84 H∞ control Case 1 2.67 H∞ control Case 2 2.69 H∞ control Case 3 0.67 H∞ control Case 4 0.67 69 4.4.3 Cutting tests The absolute, chatter-free depth of cut is proportional to the minimum dynamic stiffness of the boring bar. Since H∞ control case 1 has the highest minimum dynamic stiffness, its controller has been used to conduct a series of cutting tests to demonstrate the viability of active damping to improve the chatter-free depth of cuts. The diameter of the hole was increased from 40 mm to 50 mm with the tool holder described in Section 4.2.1. The workpiece material is Al6061-T6. The feedrate was 0.2 mm/rev; and the tool has a 0.8 mm nose radius with a 95 degree approach angle (insert number: CPGT432). The measured chatter stability charts without and with active damping are compared in Figure 4.18 and Figure 4.19. The presence of chatter is determined from the surface finish and the power spectrum of the measured accelerometer signals. When there is chatter, the surface finish is rough and the power spectrum has large amplitude peaks at the modal frequencies as shown in Figure 4.18 and Figure 4.19. Without active damping, the maximum depth of cut is only 0.03 mm due to the high flexibility of the boring bar; and the chatter stability is limited by the first bending mode of the boring bar at 174 Hz, which has the lowest dynamic stiffness. The cutting tests were repeated with an actively controlled boring bar. The maximum depth of cut has been increased from 0.03mm to 0.13 mm, which is improved by about 4 times compared to the case without control. Since the first bending mode is damped with active control, the third bending mode of the actively damped boring bar becomes the new limit for chatter stability. At a 0.15 mm depth of cut, the third mode of the controlled boring bar causes chatter and saturates the current amplifiers. The nonlinear saturation significantly limits the control efforts on not only the third mode but also the first mode of the boring bar, and causes both modes to be out of control. Therefore, the unstable vibration caused by the first mode of the boring bar starts growing and finally becomes dominant due to the higher open-loop flexibility. 70 1000 1200 1400 160000.050.10.15Spindle speed [rev/min]Depth of cut [mm] stablechatterUnstable cutting0.04 mm depth of cut, 1500 rev/min spindle speedStable cutting0.03 mm depth of cut, 1500 rev/min spindle speedChatter stability diagramMeasured boring bar accelerationPower spectrum of boring bar acceleration13 14 15 16-100-50050100150Time [s]Magnitude [g]14 15 16-100-50050100150Time [s]Magnitude [g]0 500 1000 1500 2000 2500051015Frequency [Hz]Amplitude0 500 1 00 1500 2000 2500051015Frequency [Hz]Amplitude Figure 4.18 Chatter stability measurements without active damping 71 1000 1200 1400 160000.050.10.15Spindle speed [rev/min]Depth of cut [mm] stablechatterUnstable cutting0.15 mm depth of cut, 1500 rev/min spindle speedStable cutting0.13 mm depth of cut, 1500 rev/min spindle speedChatter stability diagramMeasured boring bar accelerationPower spectrum of boring bar acceleration0 500 1000 1500 2000 2500051015Frequency [Hz]Amplitude13 14 15-150-100-50050100150Time [s]Magnitude [g]10 11 12 13-150-100-50050100150Time [s]Magnitude [g]0 500 1000 1500 2000 2500051015Frequency [Hz]Amplitude Figure 4.19 Chatter stability measurements with active damping. Figure 4.21 shows the surface finish and measured surface roughness of the machined workpiece. The surface roughness is measured using a surface roughness tester from Mitutoyo, 72 as shown in Figure 4.20. The workpiece is machined at a 1500 rev/min spindle speed with a 0.04 mm depth of cut. For the case without control, because of chatter, the surface finish of the workpiece is rough with a 35.9 μm peak to peak height (Rt) and an average roughness (Ra) of 6.95 μm. When active damping is turned on, the quality of the surface finish is raised significantly (Ra =0.70 μm, Rt =3.94 μm). Surface finish at a 0.13 mm depth of cut is also measured with active damping, as shown in Figure 4.21. It can be seen that the cutting process is stable and the workpiece has a good surface finish with Ra =0.88 µm and Rt =5.39 µm. The surface roughness obtained with control at 0.13 mm depth of cut is even better than that of the stable but uncontrolled cutting at the 0.03 mm small depth of cut (Ra=1.56 µm, Rt=9.0 µm). WorkpieceSurface roughness tester Figure 4.20 Setup for surface roughness measurement 73 [μm][μm]Ra = 0.88 μmRa = 6.95 μm[μm][μm][μm][μm]Ra = 0.70 μmRt = 35.9 μmRt = 3.94 μmRt = 5.39 μm0.04 mm depth of cut without active damping0.04 mm depth of cut with active damping0.13 mm depth of cut with active dampingRa = 1.56 μm[μm][μm]Rt = 9.0 μm0.03 mm depth of cut without active damping2000 3000 4000 5000 6000-20-10010202000 3000 4000 5000 6000-20-10010201000 2000 30 0 4000-20-10010201000 2000 3000 4000-20-1001020 Figure 4.21 Surface finish and surface roughness of the workpiece. Machined at 1500 rev/min spindle speed, 0.2 mm/rev feedrate, with and without active damping. 4.5 Summary The designed magnetic actuator is mounted on a CNC lathe; and its active damping performance is tested on a flexible and slender boring bar. The boring bar vibrations are measured either close to the tool tip via an added accelerometer, or using an existing non-contact, fiber optic displacement sensor within the actuator which is away from the tool tip. Three different types of controllers, including loop shaping controller, DI controller, and H∞ controllers, are designed for active damping. Besides, four different control configurations are proposed for designing the H∞ controllers. The FRF from the cutting force to the tool tip displacement is measured for each control case; and the results show that the first mode of the boring bar is totally damped, while 74 the third mode is also significantly damped for all the control cases. H∞ control case 1 gives the best active damping performance since it provides the highest minimum dynamic stiffness. The simple loop shaping controller and DI controller cannot damp the third bending mode of the boring bar as much as the H∞ controllers in cases 1 and 2 do. Moreover, when an accelerometer is attached close to the tool tip as a feedback sensor, the system becomes a non-minimum phase system due to the non-collocated sensor actuator configuration, which limits the active damping performance of the H∞ control cases 3 and 4. Besides the dynamic stiffness, the static stiffness of the boring bar at its tip is also improved considerably with the designed active damping controllers. Also, cutting tests were carried out with H∞ control case 1; and the chatter-free depth of cut and the surface finish are improved significantly compared to the case without active damping. 75 Chapter 5: Cutting Force Estimation 5.1 Overview Cutting force measurement plays an important role in monitoring and controlling the machining operations during production, as well as optimizing the cutting process. The most common method to measure the cutting force is using force dynamometers. Although dynamometers provide good force measurements, they are more suitable for laboratory use and have limited applications on production machines due to the limitation of workpiece or tool size and mounting constraints [63]. Therefore, developing a cutting force sensing mechanism which is integrated into the machine tool structures or cutting tools is a desired solution [63, 64]. The developed magnetic actuator is used to estimate the cutting forces from the armature displacement and the control current. The cutting force is estimated only from the armature displacement measurements in one method; while both the measured armature displacement and the control current are used in an alternate method. Both methods are tested on a boring bar which is clamped to the turret of a CNC lathe. Experimental results show that the estimated forces agree with the measured forces. The chapter is arranged as follows. Section 5.2 explains the principle of the proposed force estimation methods. Section 5.3 describes the experimental setup and model identification. Kalman filter is designed in Section 5.4 to estimate the cutting force while compensating the dynamics of the force sensing system. Experimental results are discussed in Section 5.5, while the concluding remarks are given in the last section. 76 5.2 Methodology The basic principle of cutting force estimation is to obtain the approximate cutting force by using the information provided from the identified system model and measurements collected from the inherently available sensors and states in the actuator. When the cutting force is quasi static with low frequency content, the force can be correlated to the measured static deflection of the boring bar which is measured with the embedded fiber optic displacement sensor at the armature. However, when the cutting force is dynamic with periodic components, the dynamics of the boring bar’s structural dynamics will distort the estimated forces from the sensor measurements, which is the focus of this chapter. Kalman filter, which is a type of optimal observer, is used to estimate the dynamic cutting forces. Two different design configurations are proposed for the cutting force estimation as shown in Figure 5.1. In the figure, d ( ,d x y ) represents the measured armature displacement in the x or y direction; dI is the control current in the d direction; and cˆF is the estimated cutting force. cF dKalman filterˆcFKalman filterdI Method 1Method 2dFTdFGdIcF ˆcFddI Figure 5.1 Kalman filter design configurations. 77 In method 1, the closed-loop transfer function from the cutting force (Fc) to the armature displacement (d), denoted as dFT , is considered in the Kalman filter design; and the cutting force is estimated from the armature displacement (d). In method 2, the open-loop transfer functions from the cutting force (Fc) to the armature displacement (d), denoted as dFG , and from the control current (Id) to the armature displacement (d), denoted as dIG , are considered in the Kalman filter design; and the cutting force is estimated from both the armature displacement (d) and the control current (Id). 5.3 Experimental setup and model identification 5.3.1 Experimental setup The experimental setup used for cutting force estimation is shown in Figure 5.2. Since it is difficult to install a dynamometer into the proposed active damping setup to measure the cutting force directly for comparison, a shaker (Wilcoxon Research, model F4/F7) is used to simulate the effect of the cutting force; and a Kistler force sensor (type 9712B500) is attached to the boring bar to measure the simulated cutting force. The proposed force estimation algorithms are demonstrated in the y axis. The shaker is hung to a crane, and a stinger is used to connect the shaker to the force sensor; while the fiber optic sensor of the actuator is used to measure the armature displacement in the y axis. The control current signal is read directly from the dSPACE 1103 controller board. 78 ShakerStingerForce sensorFiber optic sensorYXZ Figure 5.2 Experimental setup for cutting force estimation 5.3.2 Model identification A simple derivative controller, with derivative gain 50, is implemented in the y axis for demonstration. The open-loop and closed-loop FRFs (yFG and yFT ) from the cutting force ( cyF ) to the armature displacement (y) are measured by sending sinusoidal signals to the shaker, and recording the corresponding excitation force (cyF ) and armature displacements (y). The open-loop FRF (yIG ) from the control current ( yI) to the armature displacement ( y ) is measured by sending sinusoidal signals to the current amplifiers, and measuring the armature displacement (y) from the fiber optic sensor. The measured FRFs are shown in Figure 5.3, and the identified modal parameters are given in Table 5.1. Due to high dynamic stiffness of the boring bar and low excitation force of the shaker, the armature displacement is buried in the sensor noise when the frequency is higher than 600 Hz. Therefore, only the frequency region less than 600 Hz is considered in this chapter for modeling and cutting force estimation. 79 / [m/A]yyI/ [m/N]cyyF:yFT:IyG:FyG/ [m/N]cyyF100 200 300 400 500 60000.511.52Frequency [Hz]y/Fcy Measured FRFCurve fitted FRF100 200 300 400 500 6000246Frequency [Hz]y/Fcy Measured FRFCurve fitted FRF100 200 300 400 500 600051015Frequency [Hz]y/Icy Measured FRFCurve fitted FRF Figure 5.3 Measured and curve fitted FRFs for yFG , yIG and yFT Table 5.1 Identified modal parameters Transfer function Modes Frequency [Hz] Damping Ratio [%] Stiffness [N/m] yFT Mode 1 198 9.0 2.9 × 106 yFG Mode 1 188 3.4 2.6 × 106 yIG Mode 1 51 3.0 -2.0 × 106 Mode 2 67 8.0 -4.0 × 105 Mode 3 98 3.0 -9.0 × 106 Mode 4 138 8.0 -8.0 × 105 Mode 5 273 5.0 4.0 × 106 Mode 6 450 5.0 -9.5 × 105 80 From Figure 5.3, it can be seen that the FRF of yIG has multiple modes below 600 Hz, which is quite different from the FRFs of yFG and yFT . This is because the installed shaker system changed the boring bar dynamics when the shaker is not working as an excitation source. When the shaker is removed, the measured FRF of yIG has similar shape as the FRFs of yFG and yFT . The transfer functions between the inputs Fcy, Iy and the output y are determined from the estimated system modal parameters for each method as: Method 1: yF cyy T F (5.1) Method 2: yF cy yI yy G F G I (5.2) with the transfer functions: 2_2 2/( ) 2c yF cyFc c ckT s s s (5.3) 2_2 2/( ) 2o yF oyFo o okG s s s (5.4) 262 21 _( )( ) , ( ) 2qi qiyI qii yI i qi qi qiG sG s G sk s s (5.5) 81 where c , o and qi are the modal frequencies (rad/s); c , o and qi are the damping ratios; and _yF ck, _yF ok , and _yI ik are the modal stiffnesses of the transfer functions. The subscript i in the modal parameters represents for mode i. 5.4 Kalman filter design The objective of the Kalman filter design is to reconstruct the cutting force (cyF ) from the identified system model and available system input and output signals. The transfer functions given in Equations 5.1 and 5.2 are mapped into the following state space form: Method i ( i =1, 2): i i i i ii iq A q B uy C q , , =1 , =2 cyTicy yF iuF I i (5.6) where iq and iu are the state vector and input vector, respectively; while iA , iB , and iC are the system matrices, which are given in Appendix C. The observability matrix, W, is found to be full rank for both methods, which guarantees the observability of the system: 21ii ii ini iCC AW C AC A (5.7) 82 where n is the number of the system states. Since the Kalman filter only estimates the state vector (iq ) and system output ( y ), the cutting force (cyF ) is assumed to be piece-wise constant and is considered as an additional unknown state by expanding Equation 5.6 as: Method 1: (5.8) Method 2: (5.9) where iw ( 1, 2i ) is the process noise and v is the measurement noise. The value of the matrices FB , IB , 1G , 2G are given in Appendix C. With the expanded model, the cutting force (cyF ) can be estimated through a Kalman filter as: 83 Method 1: 11 1 1 11 1 1 1 11 1 1 1 11 1 1 1ˆ ˆ ˆˆ ˆˆˆ ˆ , with 0 1e e ee e e ee e ecy F e F nq A q K y yA q K y C qA K C q K yF C q C (5.10) Method 2: 22 2 2 22 2 2 2 22 2 2 2 22 2 2 1ˆ ˆ ˆˆ ˆˆˆ ˆ , with 0 1e e e I ye e I y e ee e e I ycy F e F nq A q B I K y yA q B I K y C qA K C q B I K yF C q C (5.11) where iK ( 1, 2i ) is the Kalman filter gain matrix; in is the state number of iq ; ˆeiq is the estimate of the expanded state vector eiq ; yˆ is the estimate of the system output; and cˆyF is the estimated cutting force. Assume the process and measurement noise are uncorrelated zero-mean white noise signals with covariance matrices Ti i iE w w Q , TE vv R , and 0TiE w v , the Kalman filter gain matrix (iK ) is obtained by minimizing the state estimation error ( ˆi ei eiq q , 1, 2i ) covariance matrix, TP E , as: 1Ti i eiK PC R , 1, 2i (5.12) 84 The minimized state estimation error covariance matrix can be evaluated by solving the following Riccati equation [65, 66]: 1T T Ti ei i i ei i i i i ei ei iP A P PA GQG PC R C P , 1, 2i (5.13) where the covariance matrices of the process and measurement noise are tuned to be 30R , 231 10Q , and 232 1.2 10Q . Therefore, the Kalman filter matrices are obtained as: 5 5 101 9.59 10 3.43 10 5.77 10 TK (5.14) 6 5 15 15 15216 16 17 17 1516 16 16 15 10[1.09 10 3.64 10 5.29 10 4.91 10 5.57 109.63 10 8.43 10 2.71 10 2.40 10 2.60 104.71 10 5.35 10 8.28 10 3.11 10 6.32 10 ]TK (5.15) The simulated FRFs from the cutting force (cyF ) to the estimated force ( cˆyF ) with both methods are given in Figure 5.4. It can be seen that both methods have a 550 Hz force estimation bandwidth. 85 ˆ/ [/]cycyFFNN100 200 300 400 500 600Frequency [Hz]Phase [deg]100 200 300 400 500 600Frequency [Hz]101102-2 0-1000Frequency [Hz]Phase [deg] 10110210-1100101Frequency [Hz]Magnitude [N/N] Method 1Method 2Method 1Method 2Phase [deg]Magnitude [N/N]1010 Figure 5.4 Simulated FRFs from the cutting force (cyF ) to the estimated force ( cˆyF ) 5.5 Experimental results 5.5.1 Frequency domain test In order to evaluate the performance of the proposed force estimation methods in frequency domain, the FRFs from the excitation force (cyF ) to the estimated force ( cˆyF ) with both methods are measured experimentally by generating sinusoidal excitation signals to the shaker and recording the measured and estimated forces. The results are shown in Figure 5.5. 86 10210-210-1100101Frequency [Hz]Fe/Fcy 102-200-150-100-50050Frequency [Hz]Phase [deg] Method 1Method 2Method 1Method 2ˆ/ [/]cycyFFNN100 200 300 400 500 600Frequency [Hz] [ ]Fe/Fcy [ ]Phase [deg] Phase [deg]100 200 300 400 500 600Frequency [Hz] Figure 5.5 Measured FRFs from the excitation force (cyF ) to the estimated force ( cˆyF ) Comparing Figure 5.5 with Figure 5.4, it can be seen that the experimental results well match the simulation results. In the FRF with method 2, there are two peaks at 188 Hz and 450 Hz, which are the natural frequencies of yFG and yIG , respectively. Method 1 gives a smoother FRF compared to method 2. One possible reason for this is that method 1 requires only one transfer function (yFT ), while method 2 needs two transfer functions ( yFG and yIG ). Therefore, method 2 has more uncertainties due to the identification errors and measurement errors which may reduce the estimation performance. 87 5.5.2 Time domain test In order to test the force estimation performance of the proposed methods in time domain, both sinusoidal and square wave forces are generated by the shaker for testing. Since the shaker is not able to output DC force, the first three partial sums of the Fourier series for the square wave are used to estimate the square wave signal. Therefore, the simulated square wave force can be expressed as: 1 1( ) sin(2 ) sin(2 (3 ) ) sin(2 (5 ) )4 3 5cy g c c cF t F f t f t f t (5.16) where cf is the cutting force frequency; gF is the amplitude of the square wave force, and t is the time. When generating the square wave force cyF , the shaker dynamics has to be considered when determining the input voltage of the shaker (shaV ), and be cancelled in the output force ( cyF ). The frequency responses from the shaker input voltage (shaV ) to the output force ( cyF ) at certain frequencies are measured by sending sinusoidal signals to the shaker, and recording the force sensor output, as given in Table 5.2. Thus, the shaker input voltage signal can be determined as: 1 1sin(2 (3 ) (3 )) sin(2 (5 ) (5 ))sin(2 ( )) 3 5( )4 ( ) (3 ) (5 )c c c cc csha gg c g c g cf t f f t ff t fV t Fa f a f a f (5.17) 88 where ga and are the magnitude and phase of the frequency response from shaV to cyF , respectively. Table 5.2 Frequency responses from the shaker input voltage (shaV ) to the output force ( cyF ) Frequency, f [Hz] Magnitude, ga [N/V] Phase, [rad] 100 10.020 -1.726 200 1.140 -3.883 300 2.813 -3.264 500 9.170 -6.013 600 0.780 -6.185 1000 2.385 -6.252 The measured and estimated forces at 100 Hz and 200 Hz are compared as shown in Figure 5.6 and Figure 5.7. It can be seen that the estimated forces match the measured force quite well at 100 Hz. At 200 Hz, since it is quite close to the resonance frequency of the boring bar, the force measurement is distorted by the boring bar vibration. However the proposed force estimation methods can still bring the force to the level provided by the force sensor measurement. 89 0.17 0.18 0.19 0.2 0.21-10-50510Time [s]Cutting force [N] Measured forceEstimated force with method 1Estimated force with method 2Sinusoidal force tests0.17 0.18 0.19 0.2-10-50510Time [s]Cutting force [N] Measured forceEstimated force with method 1Estimated force with method 2Square wave force tests Figure 5.6 Measured and estimated force at 100 Hz 0.145 0.15 0.155 0.16-3-2-10123Time [s]Cutting force [N] Measured forceEstimated force with method 1Estimated force with method 20.12 0.125 0.13 0.13 4-3-2-10123Time [s]Cutting force [N] Measured forceEstimated force with method 1Estimated force with method 2Sinusoidal force tests Square wave force tests Figure 5.7 Measured and estimated force at 200 Hz 90 5.6 Summary In this chapter, two Kalman filter based force estimation methods are proposed. Method 1 estimates the cutting force from the measured armature displacement, while method 2 estimates the cutting force from both the control current and the armature displacement. The cutting force is treated as an unknown system state when designing the Kalman filters. Both methods have a force estimating bandwidth of about 550 Hz; and the estimated forces match the measured force. Compared to method 2, method 1, which uses only armature displacement measurements, provides a better and smoother FRF from the measured force (cyF ) to the estimated force ( cˆyF ). While method 1 only needs one transfer function (yFT ), method 2 requires both transfer functions (yFG , yIG ). Hence it has more uncertainties which lead to a poor estimating performance. 91 Chapter 6: Conclusions and Future Research Directions 6.1 Conclusions For flexible parts and tools, static deflection, chatter and forced vibrations are the main constraints in achieving higher productivity, and may result in poor surface finish, tool breakage, and even damage to the machine. A novel three degrees of freedom linear magnetic actuator, which has actuation in both radial directions (x, y) and torsional direction (θ), is developed for active damping of machine tool vibrations. The designed magnetic actuator is composed of four identical actuating units; and each actuating unit is linearized with magnetic configuration design strategy. Three fiber optic displacement sensors are installed into the magnetic actuator to measure the armature displacements in the x, y and θ directions. The designed magnetic actuator is mounted on a CNC lathe; and its active damping performance is tested on a flexible and long boring bar. The boring bar vibrations are measured either close to the tool tip via an added accelerometer, or away from the tool tip using the embedded fiber optic displacement sensor. Three different types of controllers (loop shaping controller, DI controller, and H∞ controllers) are designed for active damping. The FRF from the cutting force to the tool tip displacement is measured using impact modal test for each control case. The experimental results show that both the dynamic and static stiffnesses of the boring bar are increased significantly with the designed controllers. H∞ controller case 1, which uses the fiber optic displacement sensor signals for feedback control and uses weighting functions to penalize the closed-loop FRF from the cutting force to the armature displacement, achieves a better active damping performance comparing to the other controllers implemented, since it provides a higher minimum dynamic stiffness. Cutting tests were carried out with H∞ controller 92 case 1. The chatter-free depth of cut is improved from 0.03 mm (without active damping) to 0.13 mm (with active damping). The surface finish of the workpiece is significantly improved as well. Besides active damping of vibrations, the developed magnetic actuator can also be used to estimate the cutting force from the armature displacement and the control current. Based on Kalman filters, two force estimation methods are proposed: (1) the cutting force is estimated only from the armature displacement measurements; (2) both the measured armature displacement and the control current are used to estimate the cutting force. Both methods are tested on a boring bar, which is clamped to the turret of a CNC lathe. Experimental results show that the estimated forces match the measured forces. Method 1, which estimates the cutting force only from the armature displacement, gives a better and smoother FRF from the cutting force to the estimated force. Method 1 requires only one transfer function in the Kalman filter design; while method 2 needs two transfer functions, therefore, having more uncertainties which lead to a poor estimation performance. The thesis contributions are summarized as follows: A novel non-contact linear magnetic actuator is developed for active damping of flexible parts in turning process and slender cutting tools in boring operation. The designed magnetic actuator has three degrees of freedom in the radial x, y and the torsional θ directions. The magnetic force is linearized with respect to the input current using biasing magnets. An H∞ optimal control configuration, which uses the armature displacement signals for feedback control and uses weighting functions to penalize the closed-loop FRF from the cutting force to the armature displacement, is proposed for active damping controller 93 design. With the designed controller, the material removal rate is improved by about four times compared to the case without active damping. A Kalman filter based method is proposed to estimate the cutting force from the actuator armature displacements, and its effectiveness has been experimentally verified. This thesis presents a novel three degrees of freedom linear magnetic actuator which can improve both the dynamic and static stiffnesses of the flexible parts and tools, and estimating the cutting force simultaneously. 6.2 Future research directions Soft magnetic composites (SMC) material is used in this thesis to fabricate the magnetic cores of the designed magnetic actuator. However, the permeability of SMC is much lower than the laminated electrical steel, which may lead to low magnetic force capability and large size of the magnetic actuator. Therefore, laminated silicon steel can be used instead of SMC to fabricate the magnetic cores, so that the maximum magnetic force output can be improved, and the actuator size can be reduced. The dynamics of the rotating shaft is changing during the turning operation, since the mass is slowly decreasing. Thus, in order to damp the vibrations of the rotating shafts with the proposed magnetic actuator, adaptive controllers can be designed so that the controller parameters will vary during the machining to adapt to the changing dynamics of the shafts. Alternatively, the uncertainty in the shaft dynamics can be estimated in advance; and robust control strategies, e.g. µ synthesis, can be implemented to guarantee the robust stability over the pre-defined uncertainty range. 94 The proposed force estimation methods work for structures which have time-invariant dynamics. However, for the rotating shaft case in turning operation, the system dynamics is time-varying. Therefore, to estimate the cutting force accurately, online system identification and adaptive Kalman filter design can be combined together. In this way, the system model can be updated in real time; and the covariance matrices Q and R can be estimated simultaneously. In addition, the bandwidth and performance of the proposed force estimation methods is limited by the resolution of the fiber optic displacement sensor which is about 1 µm. In order to increase the force estimation bandwidth and range, displacement sensors with higher resolutions should be used. 95 Bibliography [1] X. Lu, F. Chen, and Y. Altintas, “Magnetic actuator for active damping of boring bars,” CIRP Ann. - Manuf. Technol., vol. 63, no. 1, pp. 369–372, 2014. [2] F. Chen, X. Lu, and Y. Altintas, “A novel magnetic actuator design for active damping of machining tools,” Int. J. Mach. Tools Manuf., vol. 85, pp. 58–69, 2014. [3] F. Chen, M. Hanifzadegan, Y. Altintas, and X. Lu, “Active damping of boring bar vibration with magnetic actuator,” Submitted for reveiw, 2014. [4] F. Atabey, I. Lazoglu, and Y. Altintas, “Mechanics of boring processes — Part I,” Int. J. Mach. Tools Manuf., vol. 43, pp. 463–476, 2003. [5] J. Tlusty and M. 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[56] N. van Dijk, “Active chatter control in high-speed milling process,” PhD dissertation, Eindhoven University of Technology, 2011. [57] X. D. Lu and D. L. Trumper, “Ultrafast Tool Servos for Diamond Turning,” Ann. CIRP, vol. 54, no. 1, pp. 383–388, 2005. [58] X.D. Lu, Electromagnetically-Driven Ultra-Fast Tool Servos for Diamond Turning. PhD dissertation, Massachusetts Institute of Technology, 2005. [59] D. C. Meeker, “Finite ElementMethod Magnetics Version 4.2,” (2009). [online] Available: http://femm.info. [60] I. Usman, M. Paone, and K. Smeds, “Radially Biased Axial Magnetic Bearings/Motors for Precision Rotary-Axial Spindles,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 3, pp. 411–420, Jun. 2011. [61] P. Gahinet, P., Apkarian, “A linear matrix inequality approach to H1control,” Int J. Robust Nonlinear Control, vol. 4, no. 4, pp. 421–448, 1994. [62] I. The MathWorks, “Documentation,” (2012). [Online]. Available: http://www.mathworks.com. [63] Y. Altintas and S. S. Park, “Dynamic Compensation of Spindle-Integrated Force Sensors,” CIRP Ann. - Manuf. Technol., vol. 53, no. I, pp. 305–508, 2004. [64] G. Tantussi, M. Technology, and M. Beghini, “A Sensor-Integrated Tool for Cutting Force Monitoring,” Ann. CIRP, vol. 46, pp. 49–52, 1997. [65] R. G. Brown, Introduction to Random Signals and Applied Kalman Filtering. Wiley, New York, 1997. [66] J. E. Potter, “Matrix Quadratic Solutions,” SIAM J. Appl. Math., vol. 14, pp. 496–501, 1966. 100 Appendices Appendix A System Matrices of the Plant and Weighting Functions The state space realization of the plant in Equation (4.8) is summarized here, taking Case 1 as an example. Denote: 1 11 12 21 11 11 2 21 1 1/ ( ) / ( )( ) ( ) ( ) 2n xI x n xF cp px cxI xF n nxxk I s k F sG Gx s I s F sk k s s (A.1) 3 33 32 23 13 33 2 23 3 3/ ( ) / ( )( ) ( ) ( ) 2n xI x n xF cp px cxI xF n nxxk I s k F sG Gx s I s F sk k s s (A.2) where 1x and 3x are the armature displacements caused by the first and third bending modes, respectively. The corresponding differential equations of the system become: 1 12 22 1 11 1 1 1 1 1( ) 2 ( ) ( ) ( ) ( )n nn n x cxI xFxx t x t x t I t F tk k (A.3) 3 32 22 3 33 3 3 3 3 3( ) 2 ( ) ( ) ( ) ( )n nn n x cxI xFxx t x t x t I t F tk k (A.4) By selecting the state vector 1 1 3 3 Txq x x x x , the state space model of the plant is obtained for Case 1 as: 101 1 13 32 21 121 1 12 2 23 3 3 3 3111112 0 00 01 0 0 00 0 20 0 1 0:0 00 1 0 1 0 0n nxF xIn ncx xxn n n nxF xIcxxxxppppABDCk kFq qIk kPFx qI (A.5) where 1pA , 1pB , 1pC , and 1pD are the state space matrices. In the generalized plant model (4.12), 1 1 1n m pC C C , 1 1 1n m pD D D . The state space matrices of the plant for Cases 2, 3 and 4 are evaluated similarly. Case 2: 2 22 2 2 21 1 2 20 , , ,0n t n np p p pp p m mA B C DA B C DA B (A.6) 21 1 122 2 223 3 32 0 0 0 01 0 0 0 0 00 0 2 0 00 0 1 0 0 00 0 0 0 20 0 0 0 1 0n nn nnn nA (A.7) 102 1 12 23 32 21 12 22 222 23 30 0, 0 , 0 1 0 1 0 10 00 0tn nx F x In nx F x It n t tn nx F x Ixt txtxt tk kk kB C C Ck k (A.8) 2 10m pC C , 2n tD D , 0 0tD , 2 1m pD D (A.9) Case 3: 1 12 23 32 21 12 22 23 33 3 3 33 32 23 30 0, , ,0 00 0n nx F x In nn nx F x Ip n p p pm mn nx F x Ia a xa a xa a xk kC Dk kA A B C DC Dk k (A.10) 3 0 1 0 1 0 1nC (A.11) 2 2 23 1 1 1 2 2 2 3 3 32 2 2m n n n n n nC (A.12) 1 2 3 1 2 32 2 2 2 2 21 2 3 1 2 333 0 0n n n n n nmx F x F x F x I x I x Ina a a a x a x a xDk k k k k kD (A.13) 103 Case 4: 4 44 4 4 43 4 40 , , ,0tn n np p p ppn m mBA C DA B C DBA (A.14) 4 0n tC C , 4 30m mC C , 4n tD D , 4 3m mD D (A.15) The system matrices of the weighting functions are: WI: 1st order: 22 2121, 1 ,1 ,WI WI pWI WI pA B KC D K (A.16) 2nd order: 22 2 212 22 21 2 1 21 122 21, ,01 0WI WI WI pWI p pA B D KC K K (A.17) 104 3rd order: 2 22 4 2 2 4 4 22121 3 2 412 221 2 1 3 2 412 223 1 4 21222211 0 0 , 0 ,0 1 0 0WI WI WI ppWI ppTA B D KKC KK (A.18) where 1 12 f , 2 22 f , 3 32 f , 4 42 f . Wx: 1st order: 22 2121, 1 ,1 ,Wx Wx pWx Wx pA B KC D K (A.19) 2nd order: 22 2 212 22 21 2 1 21 122 21, ,01 0Wx Wx Wx pWx p pA B D KC K K (A.20) 105 3rd order: 2 22 4 2 2 4 4 22121 3 2 412 221 2 1 3 2 412 223 1 4 21222211 0 0 , 0 ,0 1 0 0Wx Wx Wx ppWx ppTA B D KKC KK (A.21) Appendix B Summary of H∞ Controller Synthesis with LMI Method A brief summary of how to synthesize the H∞ optimal controller using LMI method [61] is given. Case 1 is used as an example for demonstration. The state space model of the generalized plant in Equation (4.12) can be reconstructed as: 1 211 11 122 21 22:cxpcxFq Aq B BIGFC D DzqIC D Dx (B.1) 106 where 110 00 00pWIWx n WxAA AB C A , 1 32 21 31 0 0 0 0Tn nxF xFB k k , 1 32 21 32 0 0 0Tn nWIxI xIx xB Bk k , 110 00WIWx n WxCC D C C , 2 1 0 0mC C , 11 0 0 TD , 12 0 TWID D , 21 0D , 22 0D . Assume the controller K has a transfer function as: 1( ) ( )K K K KK s D C sI A B (B.2) where KA , KB , KC , KD are the state space matrices of controller K. The system block diagram given in Figure 4.12 (Case 1) can be reconstructed with the state space matrices of the generalized plant and the controller, as shown in Figure B.1. xcF xzK KK KA BC D 1 21 11 122 21 22A B BC D DC D D Figure B.1 Reconstructed system block diagram 107 Then, the state space model for the closed-loop transfer function from cF to z ( zFT ) can be obtained as: 2 2 2 1 2 212 211 12 2 12 11 12 21K K Kck kK K KK K K ckclclclclBADCq qA B D C B C B B D DFq qB C A B Dqz C D D C D C D D D D Fq (B.3) whereclA, clB, clC, clD are the state space matrices of zFT . Gathering all controller parameters into a single variable gives: K KK KA BC D (B.4) By introducing the following definitions: 10 0 0 1221 12 2210; ; 0 ;0 0 000; ;0000 ;i ii iBAA B C CIBB CCID D DD (B.5) the closed-loop matrices clA, clB, clC, clD can be rewritten as [61]: 108 0 0 20 1 11 1 2; ;;cl i i cl i icl i i cl i iA A B C B B B DC C D C D D D D (B.6) According to the bounded real lemma for continuous time systems [61], the H∞ suboptimal constraints can be turned into an LMI problem: H∞ suboptimal constraints: 1( ) , ( is stable)cl cl cl cl clD C sI A B A (B.7) where is a pre-defined positive scalar. LMI problem: There exists a symmetric positive definite solution clX to the LMI: 120T Tcl cl cl cl cl cl clT Tcl cl clcl clA X X A X B CB X I DC D I (B.8) Then, the controller parameters K KK KA BC D can be solved by four steps [61]: i) Solve the following LMIs for symmetric matrices R and S: 109 1 11 1 1 111 110 000 0T TTR RT TT TARA R ARC BN NC RA I C RC DI IB D I (B.9) 1 11 1 1 111 110 000 0T T TTS ST T TA SA S A SB CN NB SA I B SB DI IC D I (B.10) 0R II S (B.11) where RN and SN denotes the basis of the null spaces of 2 12,T TB D and 2 21,C D , respectively. ii) Find full-column-rank matrices M, N so that: TMN I RS (B.12) iii) Find a unique solution clX of the linear equation: 0 0clT TS I I RXN M (B.13) iv) Substituting the obtained clX into Equation (B.8) and solve the inequality for the controller parameters K KK KA BC D . By using a standard γ-iteration technique, the optimal value of γ and the corresponding controller parameters of the H∞ optimal controller can be obtained. 110 Appendix C System Matrices of the Force Estimation System The system matrices in Equation 5.6 are given as follows: Method 1: 22_1 1 12, , 0 11 00cc c cyF ckA B C (C.1) Method 2: 22 _12 _ 22 _ 322 _ 42 _ 52 _ 60 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0oAAAAAAAA , 2 F IB B B (C.2) _1_ 2_ 3F_ 4_ 5_ 6FoFFFFFFBBBBBBBB , _1_ 2_ 3_ 4_ 5_ 6IoIIIIIIIBBBBBBBB , 22 _12 _ 22 _ 322 _ 42 _ 52 _ 6oCCCCCCCCT (C.3) 2221 0o o ooA , 2_0oyF oFo kB , 00IoB , 2 0 1oC (C.4) 22 _21 0qi qi qiiA , _00F iB , 2_ _0qiI i yI iB k , 2 _ 0 1iC , ( 1,2,...,6)i (C.5) 111 The process noise matrices 1G and 2G in Equations 5.8 and 5.9 are given as: 2 1101G , 14 1201G (C.6) 112 Appendix D Operation instruction for the magnetic actuator D.1 Function of the magnetic actuator The magnetic actuator (Figure 3.13) is designed for active damping of flexible parts and tools, such as long boring bars and large flexible shafts. The clamping fixtures are designed to fit the turret of Hardinge CNC lathe in the Manufacturing Automation Laboratory at the University of British Columbia. The up and down stators are powered together to generate the magnetic force in the y direction; the left and right stators are powered together to generate the magnetic force in the x direction; all the four stators are powered to generate the magnetic torque in the θ direction. D.2 Installation of the magnetic actuator The instructions for installing the magnetic actuator are as follows: 1) Install the clamping fixture 1 to the turret of the lathe, as shown in Figure D.1. Clamping fixture 1Turret Figure D.12Installation of the clamping fixture 1 to the turret 113 2) Install the boring bar and armature assembly to the clamping fixture 1 using trantorque bushings, as shown in Figure D.2. Adjust the boring bar orientation and clamping length to the one you want, and then tighten trantorque bushings in the clamping fixture 1. Clamping fixture 1TurretArmature assemblyTrantorque bushingBoring bar Figure D.23Installation of the boring bar and armature assembly to the clamping fixture 1 3) Install the clamping fixture 2 to the clamping fixture 1, as shown in Figure D.3. Clamping fixture 1Clamping fixture 2 Figure D.34Installation of the clamping fixture 2 114 The gaps (Figure D.4) between the safety bumper and the clamping fixture 2 are expected to be 0.25mm. However, the real heights of the gaps can vary from 0.15mm to 0.35mm depending on the installation. Plastic shims from Artus shim stock can be used to maintain the air gaps while tightening the trantorque bushing in the armature center hole. The heights of the gaps should be as even as possible. After tightening the trantorque bushing, the plastic shims should be removed. The front surface of the safety bumper should not exceed the front surface of the clamping fixture 2 after tightening the trantorque bushing. This is to protect the magnetic cores of the stators from touching the safety bumper during the stator installation process. GapsClamping ficture 2Safety bumper Figure D.45Gaps between safety bumper and the clamping fixture 2 4) Install the stators one by one, as shown in Figure D.5. When installing the stator to the clamping fixture 2, an aluminum plate is placed in the slot of the stator between the permanent magnet and the armature core to protect the stator from sucking to the armature and causing damage. The stator is carefully slidden toward the clamping fixture 2 along the aluminum plate by hand. 115 Aluminum plateStator Figure D.56Stator installation Adjust the air gaps between the stator poles and armature poles with fixture 4, and the gap between the armature core and permanent magnet with fixture 3, as shown in Figure D.6. Use plastic shims to maintain the height of the air gaps. When the air gaps are adjusted to the expected value (around 0.7 mm), tighten the clamping screws of the stator assembly. Then, remove the aluminum plate, plastic shims, and the fixtures 3 and 4. Fixture 3Fixture 4Plastic shim Figure D.67Adjusting the air gaps 116 When all the four stators are installed, it will look like the one shown in Figure D.7. LRUD Figure D.78Actuator setup view when the stators are installed 5) Install the fiber optic displacement sensors to the stators, so the fully assembled magnetic actuator will look like the one shown in Figure D.8. 117 Figure D.89Fully assembled magnetic actuator D.3 Calibration of the fiber optic sensors To calibrate the sensitivity of the fiber optic sensors, put an accelerometer, the sensitivity of which is known, on the armature in line with the fiber optic sensor to be calibrated, as shown in Figure D.9. Give a hammer blow to the boring bar in the sensing direction, and measure the FRFs from the hammer force to the armature displacement using both fiber optic sensor and accelerometer. Afterward, keep adjusting the sensitivity of the fiber optic sensor in CutPro and measuring the FRFs until the FRFs with both sensors are matched. The final sensitivity of the fiber optic sensor is the one we want. 118 xyzθAccelerometer locations for calibration Figure D.910Fiber optic sensor calibration
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Active damping of machine tools with magnetic actuators Chen, Fan 2014
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Title | Active damping of machine tools with magnetic actuators |
Creator |
Chen, Fan |
Publisher | University of British Columbia |
Date Issued | 2014 |
Description | Flexible parts and tools are often found in machining operations, such as boring large cylinders, or turning long and slender shafts. The excessive flexibility of such tool or shaft may cause static deflection, forced vibration, and even chatter vibrations, which result in poor surface finish, tool breakage, and even damage to the machine, and thus become the main constraints in achieving higher productivity. This thesis presents an active damping solution to such problems, by using a novel three degrees of freedom linear magnetic actuator, which can increase the damping and stiffness of flexible structures in machining. The actuator is comprised of four identical magnetic actuating units; the magnetic force output of each actuating unit is linearized with regard to the input current by biasing magnets. Fiber optic sensors are integrated into the actuator to measure the displacements of the structure during machining. The magnetic actuator is used for three purposes: active damping of boring bar, increasing its static stiffness, and monitoring cutting forces based on the control current signals and fiber optic displacement sensor signals. The active damping is achieved by controlling the magnetic force as a function of measured vibrations. Three different types of controllers (loop shaping controller, Derivative-Integral controller, and H∞ controllers) have been developed to actively damp the displacements of a flexible boring bar during machining tests. The actuator can deliver 248 N force up to 850 Hz, and 107 N force up to 2000 Hz which is limited by the current amplifier used in the experimental setup. The cutting force is estimated through a Kalman filter, which was experimentally verified to be effective up to 550 Hz. Both the dynamic stiffness and static stiffness of the boring bar have been increased considerably with the designed magnetic actuator, leading to a significant increase in the chatter-free material removal rates. Although the proposed magnetic actuator is demonstrated for active damping of a slender boring bar in the thesis, the proposed magnetic actuator principle can be applied to suppress vibrations of rotating shafts, long boring bars and flexible structures in machine tools and other machineries. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2014-12-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
IsShownAt | 10.14288/1.0165577 |
URI | http://hdl.handle.net/2429/51547 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2015-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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