Modeling and Active Damping of StructuralVibrations in Machine ToolsbyAmir Hossein Hadi HosseinabadiB.Sc., Sharif University of Technology, Iran, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2013? Amir Hossein Hadi Hosseinabadi 2013AbstractFeed drives of High Speed Machine (HSM) tools deliver fast motions for rapid positioningof tool or workpiece. The inertial forces generated by acceleration and deceleration of largemachine tool components excite structural modes of the machine tools and cause residual vi-brations. Unless avoided, the vibrations lead to poor surface finish and instability of the drive?scontrol loop. In this thesis, structural flexibilities are represented by linear and torsional springsand dampers to develop a mathematical model of the feed drive dynamics. The model includesthe contribution of structural vibrations in measuring table position by a linear encoder. Anidentification algorithm is introduced to facilitate the estimation of rigid body and structuraldynamics in frequency domain. The identified mathematical model is used to mimic the realmachine in simulations with the purpose of analyzing the interaction between structural dy-namics and a high bandwidth adaptive sliding mode controller.Meanwhile, efficiency of finite element modeling approaches in predicting this interactionprior to the physical production is investigated by replacing the machine dynamics by a FEMbased model. The mathematical model is used to design a Kalman Filter which estimatesthe table?s acceleration by taking double digital derivative of the encoder signal. The table?sacceleration is used to modify the control loop to minimize the effect of undesired structuralvibrations. It is shown that the vibrations can be actively damped, and the bandwidth of thedrive can be increased. The increase in the servo loop bandwidth provides smoother motionand improves the tracking performance significantly.iiPrefaceThis dissertation is original intellectual property of the author, Amir Hossein Hadi Hossein-abadi, under supervision of professor Yusuf Altintas. This work has been completed in theManufacturing Automation Laboratory at the University of British Columbia. The results pre-sented in Chapters 3 to 5 are going to be submitted for publications. All figures and tablesfound in this thesis are used with permission from applicable sources.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Modeling the Dynamics of Flexible Machine Tools . . . . . . . . . . . . . . . 52.3 Identification of Machine Tool Dynamics . . . . . . . . . . . . . . . . . . . . 92.4 Vibration Reduction in Production Machines . . . . . . . . . . . . . . . . . . 112.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Mathematical Modeling of Structural Dynamics . . . . . . . . . . . . . . . . . . 17ivTable of Contents3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Machine Tool Feed Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Mathematical Modeling of Feed Drives . . . . . . . . . . . . . . . . . . . . . 193.3.1 Rigid Body Model of the Ball-Screw Drive . . . . . . . . . . . . . . 203.3.2 Flexible Ball-Screw Model . . . . . . . . . . . . . . . . . . . . . . . 223.3.3 Feed Drive In a Flexible Machine Tool . . . . . . . . . . . . . . . . . 253.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Experimental Identification of Feed Drive Dynamics . . . . . . . . . . . . . . . 344.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Identification of Rigid Body Dynamics (Gr(s)) . . . . . . . . . . . . . . . . . 344.3 Identification of Structural Flexibilities . . . . . . . . . . . . . . . . . . . . . 404.4 Modal Parameters (? ,?n) Estimation . . . . . . . . . . . . . . . . . . . . . . 424.4.1 The Peak Picking Method . . . . . . . . . . . . . . . . . . . . . . . . 434.4.2 The Rational Fraction Polynomial (RFP) Method . . . . . . . . . . . . 454.5 Experimental Identification Results . . . . . . . . . . . . . . . . . . . . . . . 494.5.1 Ball-Screw Test Bed . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5.2 Flexible Machine Tool - FADAL VMC2216 . . . . . . . . . . . . . . 594.6 Validation of the Identified Model for the Flexible Machine Tool . . . . . . . . 664.7 Structural Dynamics in Finite Element Models . . . . . . . . . . . . . . . . . 694.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 Drive-Based Vibration Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Structural Dynamics in Control Loop . . . . . . . . . . . . . . . . . . . . . . 745.3 Vibration Reduction with Acceleration Feedback . . . . . . . . . . . . . . . . 775.3.1 Cascade Control Structure . . . . . . . . . . . . . . . . . . . . . . . 78vTable of Contents5.3.2 Adaptive Sliding Mode Controller . . . . . . . . . . . . . . . . . . . . 815.4 Kalman Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.5 Experimental Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 885.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95AppendicesA ZOH Equivalent of The T.F in Equation 4.6 . . . . . . . . . . . . . . . . . . . . 103B Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106B.1 Construction of Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . 106B.2 Proof of Equation 4.51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107viList of Tables4.1 Identified parameters of the flexible ball-screw drive . . . . . . . . . . . . . . . 554.2 Identified parameters of structural transfer function of the feed drive (G f (s) [m/m],see Equation (4.69)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Identified parameters of G?f (s) [m/N.m](see Equation (4.75)). . . . . . . . . . . 725.1 Comparison of damping ratios before and after active damping . . . . . . . . . 84viiList of Figures1.1 Frequency content of trajectory generation algorithms with infinite, constantand continuous jerk profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Hybrid model of machine tool [1]. . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Comparison of modeling strategies . . . . . . . . . . . . . . . . . . . . . . . 92.3 Electromagnetic active damper . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Vibration reduction approaches in production machines . . . . . . . . . . . . . 163.1 Ball-screw feed drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Linear drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Schematic diagram of a C-frame milling machine . . . . . . . . . . . . . . . . 203.4 Rigid ball-screw mounted on a rigid bed . . . . . . . . . . . . . . . . . . . . . 203.5 Ball-screw feed drive - UBC Mechatronics Laboratory . . . . . . . . . . . . . 223.6 Flexible ball-screw mounted on a rigid bed . . . . . . . . . . . . . . . . . . . . 233.7 FADAL VMC2216 three axis machining center . . . . . . . . . . . . . . . . . 263.8 Flexible ball-screw model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.9 Bending mode of flexible machine tool structure . . . . . . . . . . . . . . . . . 283.10 Bending mode of flexible machine tool structure . . . . . . . . . . . . . . . . . 294.1 Least square identification reference command . . . . . . . . . . . . . . . . . 394.2 Identification of G f Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . 43viiiList of Figures4.3 Single degree of freedom oscillator (Mass,Spring, and Damper System) . . . . 434.4 Transfer function of a SDOF system represented by its real and imaginary parts 444.5 Magnitude plot of frequency response function for a system with two closelyspaced modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 Estimated inertia (Je) and viscous damping coefficient (Be) by unbiased leastsquares technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.7 The experimental FRF between motor torque and table displacement measuredby linear and rotary encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.8 Contribution of the ball-screw torsional flexibility in the experimental FRFsmeasured by linear and rotary encoders . . . . . . . . . . . . . . . . . . . . . 524.9 Real and imaginary plots of the experimental FRF for G f t and G f m . . . . . . . 534.10 Magnitude and phase plot of experimental and reconstructedG f t( j?)G f m( j?) . . . . . . 544.11 Experimental and reconstructed FRF of the transfer function between motortorque (Tm) and table position capture by linear encoder (xt) (see Equation(4.63)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.12 Experimental and reconstructed FRF of the transfer function between motortorque (Tm) and table position capture by rotary encoder (rg?m) (see Equation(4.64)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.13 Direct FRFs measured at ball-screw table . . . . . . . . . . . . . . . . . . . . 574.14 The hammers used for direct FRF measurements at ball-screw table . . . . . . 584.15 Experimental FRFs of the feed drive in x direction for FADAL VMC2216 . . . 604.16 Comparison of the experimental FRF between table acceleration and inputvoltage to the amplifier measured by linear encoder (red line) and accelerome-ter (blue line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.17 Experimental acceleration FRF - Comparison of using two accelerometers (blueline) and linear encoder (red line) . . . . . . . . . . . . . . . . . . . . . . . . . 62ixList of Figures4.18 Estimation of modal parameter with RFP method for the closely spaced col-umn bending and torsion modes. . . . . . . . . . . . . . . . . . . . . . . . . . 634.19 Experimental FRF of the feed drive structure (G f (s), see Equation (4.69)). . . . 654.20 Experimental and reconstructed FRF of the open loop transfer function (Ge(s),see Equation (4.69)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.21 The control loop block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 674.22 Reference trajectory with trapezoidal velocity profile . . . . . . . . . . . . . . 684.23 Comparison of simulated and experimental machine acceleration . . . . . . . . 684.24 Finite Element model of the column bending mode for the machine tool withstructural flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.25 Comparison of Fitted and Simulated FRF of structural dynamics obtained fromFEM (G?f (s), see Equation 4.75) . . . . . . . . . . . . . . . . . . . . . . . . . 715.1 Reference trajectory with cubic acceleration profile . . . . . . . . . . . . . . . 755.2 Comparison of simulation and experimental results for the ball screw drivecontrolled by a high bandwidth sliding mode controller without active damping. 765.3 The simulated table acceleration based on the FE model - high bandwidth slid-ing mode controller without active damping. . . . . . . . . . . . . . . . . . . . 775.4 A single degree of freedom oscillator mounted on the ball-screw table. . . . . . 785.5 Cascade control structure for the ball-screw drive . . . . . . . . . . . . . . . . 795.6 Velocity loop of ball screw drives with single degree of freedom oscillator andactive damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.7 Velocity loop of ball screw drives with single degree of freedom oscillator andactive damping - The indirect velocity loop is replaced by its DC gain (Kv) . . . 805.8 Control loop for the linearized adaptive sliding mode controller . . . . . . . . . 82xList of Figures5.9 Velocity loop of the linearized adaptive sliding mode controller with activedamping network included in the direct velocity loop . . . . . . . . . . . . . . 825.10 Bode plots of different velocity loops . . . . . . . . . . . . . . . . . . . . . . . 835.11 Bode plots of the position loop with and without active damping . . . . . . . . 835.12 Block diagram of sliding mode controller with and without active damping ofvibration mode with a natural frequency ?n . . . . . . . . . . . . . . . . . . . 895.13 Comparison of experimental results for the machine tool feed drive with andwithout active damping (AD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.14 Comparison of the control command (u) with and without active damping (AD) 90xiNomenclatureRoman Symbolsx?T Machine table velocity measured by the tacho-generator coupled to the motor shaft[mm/s]d? Disturbance compensator in the adaptive sliding mode controller [V]y? Output vector estimated by the Kalman Filterz? State vector observed by the Kalman Filteru? Quantization error in the commanded voltage to the servo amplifierx? Quantization error in the table position measured by the linear encoderAd State (or system) matrix of the discrete time state space representation of the plantBd Input matrix of the discrete time state space representation of the plantBe Equivalent viscous damping coefficient reflected on motor shaft [kg.m2/s]Bt Viscous damping coefficient between machine table and guide [kg/s]Cd Output matrix of the discrete time state space representation of the plantct Viscous damping coefficient of nut, bearing and coupling [kg.m2/s]xiiNomenclatureccb Bending viscous damping coefficient of the machine column and bed clampingjoint to the ground [kg.m2/s]cct Torsional viscous damping coefficient of the machine column and bed clampingjoint to the ground [kg.m2/s]cls Axial viscous damping coefficient of the ball-screw [kg/s]cts Axial viscous damping coefficient of the ball-screw [kg.m2/s]Dd Feed-through (or feed-forward) matrix of the discrete time state space representa-tion of the plantFcb The reaction force between machine column and the ground [N]Fmc The axial force between column and motor [N]Ftn The linear force between nut and machine table [N]Ge Transfer function between table displacement measured with linear encoder andapplied torque at the ball-screwG f Contribution of structural flexibilities in the transfer function between motor torqueand table displacement measured by linear encoderGr The Transfer function associated with the rigid body dynamicsGt Transfer function between motor torque and displacement of machine tableGbe Transfer function between motor torque and vibration of the guidway on the bed atthe point where encoder optical head is mounted.hp Pitch length of the ball-screw [m]xiiiNomenclatureJe Equivalent inertia reflected on motor shaft [kg.m2]J f Jacobian matrix in Gauss-Newton curve fitting algorithmJm Inertia of the nut, ball-screw coupling and motor shaft observed at the position[kg.m2]Jn Inertia of the nut, ball-screw coupling and motor shaft observed at nut position[kg.m2]Jcb Column, spindle and bed inertia in bending [Kg.m2]Jct Column, spindle and bed inertia in torsion [Kg.m2]ka Amplifier gain [A/V]kI Gain of the integrator in the position controller loop of the linearized adaptive slid-ing mode controller [1/s2]kp Gain of the position feedback loop in the linearized adaptive sliding mode controller[1/s]kt Motor torque constant [N.m/A]ku Input voltage gain in rigid body identification [V/V]Kv DC gain of the indirect velocity loop [(mm/s)/(mm/s)]kv Gain of the velocity controller in the linearized adaptive sliding mode controller[V/(mm/s)]kcb Bending stiffness of the machine column and bed clamping joint to the ground[kg.m2/s2]xivNomenclaturekct Torsional stiffness of the machine column and bed clamping joint to the ground[kg.m2/s2]k f f Gain of the feed-forward inertia compensator in the linearized adaptive slidingmode controller [s]kls The axial stiffness of ball-screw [kg/s2]kts Torsional stiffness of the ball-screw [kg.m2/s2]L Gain matrix in the Kalman Filter state estimatorM Innovation gain matrix in the Kalman Filter output observermc Column, spindle and bed mass [kg]mm Motor mass [Kg]mt Table mass [Kg]Q Process noise matrix in Kalman filter designR Measurement noise matrix in Kalman filter designrg Nut transformation ratio [m/rad]s Laplace variableTm Motor torque applied to the ball-screw at motor shaft [N.m]Tn Load torque applied to the ball-screw at nut position [N.m]Ts Sampling period [sec]Tcb The reaction moment between machine column and the ground [N.m]xvNomenclatureTct The reaction moment between machine column and the ground in column torsion[N.m]x Table position captured by linear encoder [m]xb Displacement of the middle point of the guide on the bed [m]xc Displacement of machine column COG [m]xm Displacement of the motor [m]xn Displacement of machine table due to axial deformation of ball-screw [m]xr The reference position command for the machine table [mm]xt Total Displacement of machine table where encoder scale is mounted [m]xbe Total displacement of bed at the point where encoder optical head is mounted [m]Y Measured velocity vector in identification of rigid body dynamics based on ULStechniquey Output vector of the discrete time state space representation of the plantz State vector of the discrete time state space representation of the plantGreek Symbols? Vector of unknowns in Gauss-Newton curve fitting algorithm? Modal overlap factor? Poisson?s ratio?ni Natural Frequency of mode number i [rad/s]xviNomenclature?(s) Transfer function matrix between motor torque and displacements at nut, motor,and the column? Sliding surface of the adaptive sliding mode controller [mm/s]?m Angular displacement of ball-screw at motor shaft [rad]?n Angular displacement of ball-screw at nut [rad]?cb Column bending vibration [rad]?ct Column Torsion vibration [rad]?i Damping ratio of mode number iAcronymsADD Active Damping DeviceARMAX Auto-Regressive Moving Average with eXogenous inputARX Auto-Regressive models with eXogenous inputsASMC Adaptive Sliding Mode ControllerCNC Computer Numerical ControlFE Finite ElementFEM Finite Element MethodFFT Fast Fourier TransformFRF Frequency Response FunctionHiLS Hardware in Loop SimulationxviiNomenclatureHSM High Speed MachiningLQG Linear Quadratic GaussianOE Output ErrorPCE Partial Continuous EstimationPID Proportional - Integrator - Derivative controllerRFP Rational Fraction PolynomialSDOF Single Degree of FreedomSUT System Under TestTDE Total Discrete EstimationULS Unbiased Least SquaresZPETC Zero Phase Error Tracking ControllerxviiiAcknowledgmentsIn the battle for success, everybody faces hard times, and no one can prosper without the helpof an experienced and knowledgeable leader. In my way towards masters degree, professorYusuf Altintas was not only a research supervisor but a passionate mentor whose personalityhas been a source of inspiration for the rest of my life.I would like to present my sincere gratitude to my colleagues in Manufacturing AutomationLaboratory who provided a friendly and constructive environment during the last two years andmade this period unforgettable for me.It was not possible to bear the hardships of living thousands of miles away from my familywithout the help of my supportive friends, especially my dear roommates Moein and Sina ofwhom I am deeply thankful.Last but not least, I would like to express my sincere appreciation to my beloved parentsand siblings for being a great source of support and encouragement to me and for their patienceregarding my absence.This research was sponsored by NSERC and Pratt & Whitney Canada as part of IndustrialResearch Chair in Virtual High Performance MachiningAMIR HOSSEIN HADI H.The University of British ColumbiaNovember 2013xixTHIS THESIS IS DEDICATED TO:MY BELOVED PARENTS MOHAMMADREZA AND FATEMEHAND MY DEAR SIBLINGS ALIREZA, AHMADREZA AND NEGINChapter 1IntroductionThe growing demand for increased productivity has directed aerospace, automotive and dieand mold industries towards machining technologies which provide the capability of manufac-turing products in a rapid pace with high accuracy and good surface finish. Therefore, HighSpeed Machining (HSM) has received lots of attention from industrial sectors and provokedresearchers to develop new techniques to reduce the cycle time while enhancing the quality ofthe final product.Three or five axis computer numerical controlled (CNC) machines equipped with fast feeddrives and powerful spindles are used to perform operations at cutting speeds up to 1000[m/min]. Tangential feedrate along the desired trajectory is a function of spindle speed, num-ber of cutting tool?s flutes and feed per tooth; therefore, increase in the cutting speed requireshigher feedrate to maintain an acceptable chip load. To achieve this, feed drives which canreach up to 1 [g] acceleration with feed speeds up to 40 [m/min] are utilized.Accurate manufacturing of the parts with complex geometries, such as turbine blades, re-quires the cutting tool to travel along complicated paths in 3-D space. For this purpose, pathplanning algorithms must be implemented to decouple the desired tool-path into the positioncommands for each axis. Moreover, motion control strategies must be implemented to reducetracking error between the instantaneous commanded and actual axis position. The discrep-ancy in tracking the reference command occurs because of the disturbance loads and limitedbandwidth of servo drives. In multi-axis machining operations, tracking errors in different axiscause a geometrical deviation between the desired and the traversed tool-path, which is called1Chapter 1. Introductioncontouring error. If the contour error is large, the product tolerance will be violated. Moreover,machine tools exhibit structural flexibilities which cause tracking and contouring error profilesto possess low and high frequency components.The low eigen-frequencies correspond to large components (i.e. column, tool changer andspindle housing), and are excited by the inertial forces generated through large accelerationsand decelerations in high speed machine tools. The higher frequency modes originate fromlighter components (i.e. tool and spindle) and are excited because of the metal cutting loads.As Figure 1.1 shows, discontinuities in the position, velocity, acceleration and jerk profilescommanded at discrete time intervals, introduce high frequency content to the inertial forceand cause transient vibrations. These vibrations not only degrade the quality of final productbut also damage the machine itself and reduce the productivity. If the inertial vibrations are feltby the control loop through encoder readings, the servo drive can become unstable and vibrateat the same frequency of structural modes. In general, there are three approaches to avoid theinertial vibrations.The vibrations are notch filtered to prevent their transmission to control loop; therefore,the bandwidth of the drive servo is reduced. This classical approach reduces the high speedcontouring ability of the machine, because the lowering of the bandwidth leads to increasedaxis tracking errors that are proportional to contouring feed speeds.Alternatively, the position commands can be pre-shaped to avoid the excitation of naturalmodes of the machine. However, input shaping brings phase delay to axis commands whichlead to severe distortion of the tool paths in multi-axis operations. Because this approach is anoffline feed-forward technique, it cannot reduce the disturbance induced vibrations. Moreover,large variations of machine dynamics can degrade its performance.It is more common to damp the natural vibrations through active damping network withinthe servo control loop. If the frequency of the vibration is beyond the bandwidth of the drive,external actuators dedicated to damping are added to the system. However, such measures2Chapter 1. Introductionincrease the complexity and the cost of the machine tool. It is more ideal to use the existingservo motor and sensors of the drive to actively damp the vibrations. Trapezoidal Vel. Trapezoidal Acc. Cubic Acc.[mm][mm/s][mm/s2 ][mm/s3 ]Vel.PosAcc.|Acc(?)|JerkTime [sec] Time [sec] Time [sec]Frequency [Hz] Frequency [Hz] Frequency [Hz]Frequency [Hz] Frequency [Hz] Frequency [Hz]0050100200-20002000-2000054-5 0 0.5 1.0005000100005101505100550 100 0 50 100 0 50 10020 40 60 80 100 20 40 60 80 100 20 40 60 80 1000 0.5 1.0 0 0.5 1.0x 10Figure 1.1: Frequency content of trajectory generation algorithms with infinite, constant andcontinuous jerk profiles.This thesis presents modeling, identification and active damping of low frequency machinetool modes for CNC feed drives. Chapter 2 presents a brief overview of the existing literatureon modeling, identification and vibration reduction in machine tools. The fundamental modes3Chapter 1. Introductionof the machine tool column and ball screw drive are modeled, and the relative transfer functionbetween the table and guide is developed analytically in Chapter 3. Chapter 4 is dedicated toidentify the parameters of the transfer function from experimental excitation of servo motorand linear encoder output directly. It is shown that typically bending and torsional modesof the machine tool column structure, as well as the axial vibration mode of the ball screwinterfere with the servo controller most. In Chapter 5, the experiments and simulations showthat the controller becomes highly oscillatory with high vibration amplitudes unless they aredamped. The proposed model is experimentally validated and the axial mode of the table driveis successfully damped with the acceleration feedback derived from linear encoder signals. Ifthe machine tool is at the design stage, the same transfer function can be predicted using FiniteElement model and tested in virtual environment for improved machine tool design. Chapter 6contains the conclusion remarkes and proposed future research directions.4Chapter 2Literature Review2.1 OverviewThis chapter is dedicated to present an overview on state of the art in the field of modelingthe structural vibrations in machine tools and summarizes the existing literature on differentvibration reduction approaches. Therefore, it is structured as follows: Section 2.2 presentsan overview on various modeling approaches to explain dynamics of flexible machine tools.Section 2.3 covers the existing literature on identification techniques to estimates the unknownparameters of the governing transfer function. An overview of existing literature on suppress-ing the oscillations in machine tools is presented in Section 2.4 and finally the chapter is sum-marized in Section 2.5.2.2 Modeling the Dynamics of Flexible Machine ToolsVarious modeling strategies exist to explain the dynamics of feed drives in machine tools.These schemes can be as simple as considering only the rigid body dynamics and neglectingall structural flexibilities, up to complex Finite Element (FE) models with up to millions ofelements and a large number of degrees of freedom. Complex models provide higher accuracyat the expense of heavy computational load. Therefore, selection of the modeling technique ishighly dependent on the modeling objective. For example, heavy computational loads of FEMmakes them non-beneficial for controller design and simulation purposes. However, they are52.2. Modeling the Dynamics of Flexible Machine Toolsquite efficient for topological optimization of machine tool structures.The traditional approach for modeling the dynamics of feed drive systems is to developlumped rigid models based on equivalent inertia (Je) and viscous damping coefficient (Be)reflected on motor shaft [2, 3, 4]. In this model the drive train is considered as a rigid com-bination of links, joints, and couplings which transfer the motion from motor shaft to tableposition, and all structural flexibilities are neglected. The simplest way of including structuralflexibilities in modeling, is called modular approach in which the machine tool componentsare considered as point inertias connected with springs and dampers. Altintas et al. [5] mod-eled the ball-screw drive as a torsional spring connecting two rigid bodies and implementedthe procedure presented by Dadalaue et al. [6] to obtain its torsional stiffness. Chen et al.[7] used springs to model the axial, torsional, and table-guideway stiffnesses of a ball-screwdrive. They showed that in high acceleration motions, where large inertial forces are present,the compliance of the mechanical elements in a ball-screw drive system leads to significantvibrations and elastic deformations which degrade positioning accuracy of the table. Lee etal. [8] modeled the flexibilities of ball-screw drive and machine tool bed as linear springs anddampers. They proposed a procedure for tuning the parameters in a servo control system toavoid closed loop instability due to structural vibrations. A similar procedure for modelingmachine tool flexibilities by assuming its components as lumped masses connected with linearor torsional springs and dampers is presented in [9, 10, 11].Great amount of research has been dedicated to modeling machine tools by using FiniteElement Methods (FEM). FEM is a powerful tool for modeling complex geometries and canhandle problems with varying boundary conditions. Van Brussel et al. [12], Bianchi et al. [13]and Schafers et al. [14] developed full finite element models of machine tools. These modelsare capable of predicting machine tool behavior despite the change in boundary conditions dueto different configuration of machine tool components. However, they have thousands of de-grees of freedom leading to large matrices which impose high computational effort. Therefore,62.2. Modeling the Dynamics of Flexible Machine Toolsin all cases, some model reduction algorithms is proposed to obtain a lower order model havingless computational load and more suitable for control and simulation purposes.Another approach for generating a mathematical model for an element (e.g. ball-screw) isto develop its distributed parameter representation. However, machine tool comprises numer-ous components; therefore, implementation of distributed parameter approach would becomevery difficult in practice because many partial differential equations needs to be incorporatedand different boundary conditions must be associated for each part. To solve this problem,researchers have resorted to hybrid models. In hybrid models some components (e.g. massivebeams, bed plates, and foundation supports) are modeled as lumped point-wise mass, spring,and frictional assemblies, whereas distributed parameter formulation or finite element modelsis developed for components with large length/diameter and slenderness ratios which are moreflexible, such as ball-screw drives [15].Research efforts are mostly concentrated on modeling ball-screw dynamics. Whalley etal. [15] and Pislaru et al. [16] considered torsional and axial flexibilities of ball-screw drivesand employed hybrid, distributed-lumped approach to the modeling of the x-axis dynamics ofa milling machine. Varanasi et al. [17] developed distributed-parameter model of the lead-screw drive system by writing second order wave equations in the frequency domain for thelongitudinal and torsional dynamics. They implemented Galerkin approach to obtain a loworder model. Hybrid finite element models are widely used by researchers to model torsional,axial and lateral flexibilities of ball-screw drives in machine tools. Erkorkmaz et al. [18], Zhouet al. [19], and Kamalzadeh et al. [20] implemented finite element beam formulations to modeltorsional and axial flexibilities of ball-screw drives. Zaeh et al. [21], and Okwudire et al. [22]included lateral dynamics in their models as well as torsional and axial modes. These hybridFE models are capable of predicting the ball-screw dynamics despite different locations of thetable and nut along its stroke. In other words, they handle the varying boundary conditions.72.2. Modeling the Dynamics of Flexible Machine ToolsFigure 2.1: Hybrid model of machine tool [1].Vesely [1] implemented modular approach to represent dynamics of ball-screw drive con-sidering its torsional and axial flexibilities; he also employed finite element methods to modelmachine frame (Figure 2.1). Coupling these two models resulted in a hybrid model whichincludes ball-screw and frame structural dynamics.Moreno-Castaneda et al. [23, 24] used transmission line modeling, quite similar to dis-tributed parameter approach, to make an analogy between elements in transmission line andball-screw feed drives. The developed model was based on linear elements even though thereal feed drive system has non-linearities such as friction and backlash. Pislaru et al. [25] ex-tended this work and presented a dynamic model of a non-linear control systems; they utilizedtransmission line modeling approach to model friction and backlash.Figure 2.2 compares complexity, accuracy, and computational cost of different modelingtechniques reviewed in this section. In Chapter 3, the modular approach is used, because of itssimplicity, to write governing equations and provide a better understanding on how structuralvibrations affect the feed drive transfer function in a C-frame milling machine. Linear andtorsional springs are used to model the column bending as well as the axial and torsional82.3. Identification of Machine Tool Dynamicsmodes of ball-screw. Moreover, results of a full finite element model are also presented todemonstrate how FE methods are beneficial in predicting the effect of structural vibrations incontrol loop.Lumped Parameters Model Modular Approach Hybrid ModelsFinite Element ModelsDistributed ParametersModelComplexity and accuracy IncreasesComputational Load DecreasesModeling StrategiesFigure 2.2: Comparison of modeling strategies2.3 Identification of Machine Tool DynamicsAccurately identified transfer functions are essential for designing complex control laws whichfulfill desired performance specifications. Several strategies are proposed in literature for accu-rate identification of transfer functions from plant input (i.e position command, voltage, torque)to desired output (i.e. position, velocity, or acceleration). These approaches can generally bedivided into two main groups. The first group includes procedures applicable to identify openloop transfer function and the second one deals with closed loop identification.The traditional procedure for open loop identification of rigid body dynamics is calledAd-Hoc technique [26]. In this method the main effort is to force the simulated response tobe as identical as possible with experimental results in time domain. Despite its simplicity,this method provides good estimates of the equivalent inertia, viscous damping coefficient andcoulomb friction parameters, and is therefore favorable. Ad-Hoc technique is off-line, timeconsuming and needs an initial estimation of parameters which are to be identified.Least Squares Estimation [27] is another approach widely used for identification of transferfunctions. This is based on obtaining the discrete time (z-domain) equivalent of the transfer92.3. Identification of Machine Tool Dynamicsfunction in continuous time (Laplace) domain and implementation of least squares optimiza-tion to minimize a desired cost function. The cost function can be defined in frequency or timedomain. This approach is off-line, but more accurate than Ad-Hoc technique.Recursive Least Squares identification [28] is an on-line algorithm which updates the iden-tified parameters at each sampling period based on measurements prior to that sampling time.This approach is usually used in conjunction with adaptive controllers where plant parameters(e.g workpiece mass) change with time.Erkorkmaz et al. [3] proposed Unbiased Least Squares (ULS) identification technique tofind equivalent inertia, viscous damping coefficient and coulomb friction in positive and neg-ative directions. Identification of these parameters is based on open-loop response of machinetool feed drive to a step shaped voltage command. This method is off-line too but more ac-curate than both Ad-Hoc and Least Squares techniques because it takes account for coulombfriction in finding the unknown parameters. The ULS technique is fast and identifies all theparameters together. In other words, there is no need to do separate experiments to obtaincoulomb friction specifications.The literature covered so far are mostly used for identification of rigid body dynamics. Toinclude structural flexibilities in the mathematical model of the open loop, usually frequencyresponse function (FRF) curve fitting is used. Casquero et al. [29] supplied a periodic Chirpsignal in open-loop and took Fast Fourier Transform (FFT) of input and output signals to obtainthe first resonance mode. Erkorkmaz et al. [30] used sine sweeping technique to measureFRF and implemented curve fitting techniques to obtain the transfer function from ball-screwactuating torque to linear position of the table.The aforementioned techniques are concerned with open loop identification. However, insome cases, where it is not possible to conduct open loop experiments (e.g commercial CNCmachines) or closed loop transfer function is required to design controllers (ZPETC [31]), weneed to identify the closed loop transfer function. Tung et al. [32] assumed ARX, ARMAX,102.4. Vibration Reduction in Production Machinesand OE structures for the closed loop system. They implemented least squares, constrainedleast squares, and most likelihood estimator to identify the transfer function parameters indiscrete time domain, and used the identified model to design feed-forward tracking controller.Yang et al. [33] considered the System Under Test (SUT) as a gray box and adopted ARXstructure. They proposed two approaches for plant identification: Total Discrete Estimation(TDE) and Partial Continuous Estimation (PCE). They showed that the identification based onTDE approach is more precise. They also evaluated the effect of sampling time on accuracy ofidentified parameters and demonstrated that a faster sampling rate does not essentially resultin better identification results. In other words, the proper selection of sampling time is affectedby servo drive time constant.The machine tool used in this project is controlled by a D-Space based open architectureCNC; therefore, it is possible to control each axis in open loop. Since the open loop identi-fication methods bypass the complex control laws, they provide more accurate mathematicalrepresentations of feed drives dynamics than close loop identification algorithms. Hence, inChapter 3, the procedure of developing an analytical model of feed drive dynamics based on fit-ting a transfer function to experimentally obtained FRF is explained. The described procedureis implemented in Chapter 4 for experimental identification.2.4 Vibration Reduction in Production MachinesVibration of flexible components in manipulators and production machines, such as roboticarms and machine tool feed drives, has received strong attention during past decades. Sev-eral techniques have been proposed and numerous instruments were developed to reduce theundesired vibrations. A literature review shows that the vibration reduction approaches cangenerally be categorized into two main streams: avoidance and suppression.Vibration avoidance techniques are concerned with modification of supplied command to112.4. Vibration Reduction in Production Machinesthe drives to reduce excitation of structural dynamics. Trajectories with discontinuities in ve-locity, acceleration, and jerk profiles have relatively wider frequency content which excite ma-chine?s natural frequencies and lead to inertial vibrations. Trajectories having limited jerk[34]and continuous jerk [35] profiles are presented in literature to develop smooth reference com-mands. These trajectories exhibit lower frequency content at high frequency region comparedto those with acceleration limited profiles; however, they still cause inertial vibrations in ma-chines with low frequency structural modes. Pre-filtering the motion commands prior to begiven to the machine with low-pass or notch filters has been quite advantageous for vibra-tion avoidance by removing the frequency content of reference command which excites thestructural modes [5]. However, the real time filters introduce delays and, in the case of lowfrequency modes, they limit the bandwidth of servo loop which reduces the productivity.Singer and Seering [36] introduced command shaping as a method which can be applied toany trajectory to remove desired harmonics of the given command to the machine. In this ap-proach, a sequence of pre-designed impulses, known as command shaper, is convoluted by thereference command, and the result is a modified trajectory which does not excite the modes ofinterest. They also presented different procedures for designing the impulses sequence whichcan provide a robust vibration avoidance, when slight shift in natural frequency or dampingratio exists. Although command shaping is highly efficient in vibration reduction, it adds con-siderable delay, distorts the given trajectory and results in geometrical contour errors [37]. Al-tintas and Khoshdarregi [38] utilized input shaping to reduce excitation of natural frequencies,and implemented a contouring error compensation algorithm to reduce trajectory distortionand improve contouring performance. It must be noted that, all these vibration avoidance tech-niques only prevent excitation of the structural modes due to supplied reference command, andare not beneficial in reducing vibrations caused by disturbances during machining operations.On the other hand, vibration suppression techniques known in literature as active damp-ing are concerned with minimizing generated vibrations which are sensible through feedback122.4. Vibration Reduction in Production Machinessensors mounted on the machine. In other words, in active damping, a sensor (i.e. accelerom-eter) is added to the machine to measure vibrations in real time; an external actuator or ma-chine servo drive itself is used in a feed back loop to generate suppressing force based on themeasured vibrations. Therefore, efficiency of active damping is limited by the bandwidth ofactuators and possibility of measuring vibration through feedback sensor.(a) ADD attached to the spindleADDControl CommandFlexible BeamVibration Feedback Sensor(b) ADD attached to a flexible beamFigure 2.3: Electromagnetic active damperThe external actuators can be electromagnetic, piezoelectric or hydraulic [5]. They can bedirectly attached to the flexible structure. However, incorporating external active dampers indesign of CNC machines can be difficult and more costly. Figure 2.3 shows a schematic dia-gram of an industrial active damping device (ADD) attached to a flexible structure. In machinetools industry, external actuators are usually used to improve chatter stability by enhancingstructural damping which allows larger maximum depth of cut for a stable cutting process[39].Drive-based vibration reduction is achieved by using the machine tool feed drives or servomotors in joints of the robotic arms. For this purpose, the control loop is modified to increase132.4. Vibration Reduction in Production Machinesdamping ratio for target modes. The damping enhancement is usually achieved through addinganother feedback loop to increase the damping of the mode of interest while the outer looptakes care of accurate positioning of manipulator arm. The existing literature concerning howto design the active damping loop is very rich. Banavar and Dominic [40] considered a Lin-ear Quadratic Gaussian (LQG) controller to provide adequate damping to the flexible modesand implemented an H? controller in the outer loop to satisfy robustness toward unstructuredperturbations. Symens et al. [41] used a gain-scheduling controller to reduce vibrations ofthe first structural mode in a pick and place machine, natural frequencies of which change fordifferent lengths of the handling arm. Verscheure et al. [42] incorporated an H? controller inthe internal loop to force a pick and place machine track a reference acceleration and lead-lagcontroller for the position loop to follow the reference position command.Mahmood et al. [43] proposed a resonant controller structure in the internal loop to in-crease the damping for the flexible structure, and cascaded an integrator in the outer loop toeliminate steady state error in positioning of a flexible manipulator. However, the resonantcontroller is not robust with respect to variation of the system natural frequency. Dietmair andVerl [44] came up with the idea of compensating the phase delay of flexible structure at its nat-ural frequencies in the internal loop through designing a damping network with incorporatingphase compensators and band-pass filters. The simplicity of designing the active damping net-work without imposing significant computational load has made this approach favorable. Thismethod can easily be applied to any flexible structure, and does not need a precise knowledgeof the dynamic behavior of the machine. Moreover, it is relatively robust compared to non-modeled dynamics and frequency shift. Kenneth NG [26] implemented this approach to dampout the bending mode vibrations of a flexible beam, clamped to a fast linear driven motor, byusing acceleration feedback with a gain of unity at its natural frequency.In the field of machine tools, Chen and Tlusty [45] compared the functionality of us-ing accelerometric feed back and an external tuned damper for vibration reduction of feed142.4. Vibration Reduction in Production Machinesdrives. They also cascaded a feed-forward controller to improve tracking performance. Re-sults showed that using accelerometric feedback for suppression of machine table?s vibrationsis quite efficient and also improves chatter stability in cutting process. However, due to thefeed-forward controller, it was not robust to dynamic variations of drive train. Pritschow et al.[46] used Ferrari sensor to feedback the angular acceleration of the ball-screw drive and dampthe vibrations.Adaptive sliding mode controllers are used to effectively compensate and damp out thetorsional and axial modes of ball-screw drives by Kamalzadeh and Erkorkmaz in [20] and [30]resulting in controllers with significant enhanced bandwidth. Altintas and Okwudire [47] pro-posed a disturbance adaptive sliding mode controller to enhance dynamic stiffness of a directdriven feed drive against cutting force and increased the servo loop bandwidth. Erkorkmaz andKamalzadeh [18] suppressed the ball-screw?s torsional mode by giving a canceling signal pro-portional to its twist angle measured by subtracting the outputs of two rotary encoders placedat either ends of the ball-screw.The aforementioned methods are based on nominal frequency of vibration modes and donot include dynamic variations which, if large enough, can cause destabilization of control loopand saturation of actuators. To include dynamic variation of feed drives, Van Brussel [12] de-signed an H? controller based on nominal position of the machine tool with incorporating theinformation about dynamic variations in model uncertainties. The developed controller showeda better performance in suppressing vibrations but demonstrated a worse disturbance rejectioncompared to a reference PID controller designed based on rigid body dynamics. Hanifzade-gan and Nagamune [48] proposed a switching gain scheduling controller to reduce vibrationsand achieve performance robustness despite varying position-dependent stiffness of ball-screwdrive.Figure 2.4 presents an overview of vibration reduction techniques explained in this section.Regarding to the simplicity and relative robustness of the active damping approach proposed152.5. Summaryby Dietmair and Verl [44], this method is implemented in Chapter 5 to design the phase com-pensator in the internal feedback loop. The added feedback loop increases damping of the axialmode of ball-screw and reduces the vibrations significantly.Vibration Reduction In Production MachinesAvoidanceSmooth Trajectory Generation Set PointFilteringCommand Shaping External Active DamperDrive-Based Vibration ReductionSuppressionFigure 2.4: Vibration reduction approaches in production machines2.5 SummaryA concise overview of existing literature in the field of modeling, identification, and vibrationreduction in production machines are presented in this chapter. It is known that the modularapproach is the simplest way of modeling structural flexibilities for controller design and sim-ulation purposes. It was highlighted that open loop identification result in a better estimationof system parameters due to intrinsic bypassing of the complex control laws. Finally, vibra-tion reduction approaches in production machines were investigated under two main groups ofavoidance and suppression.In this thesis, the mathematical model, developed based on the modular approach, for aC-frame machine tool is identified by utilizing frequency domain curve fitting algorithms. Theidentified model is used to reduce machine vibrations by modifying the servo control loop.16Chapter 3Mathematical Modeling of StructuralDynamics3.1 OverviewThe previous chapters, mentioned that inertial loads excite structural flexibilities and lead toresidual vibrations. This chapter focuses on developing a mathematical model which explainshow feedback sensors mounted on the machine can pick up these vibrations. In developing themodel, the fundamental modes of the machine tool column and ball-screw drive are considered.These modes have low eigen-frequencies and can be excited by inertial loads. The chapter isorganized as follows: Section 3.2 introduces and compares ball-screw and linear driven feeddrives which are widely used in industry. Section 3.3 presents the mathematical modelingprocedure and the chapter is summarized in Section 3.4.3.2 Machine Tool Feed DrivesFeed drives are a part of the control loop used for positioning the cutting tool and workpiece tothe desired location in the machine tools [5]. These electromechanical actuators are classifiedas linear motors and ball-screw feed drives. Ball-screw feed drives (Figure 3.1) are an assemblyof the coupling, ball-screw, and nut to convert the rotational motion generated by servo-motorto linear displacement at machine table. In some cases, a torque reduction gear set is placed173.2. Machine Tool Feed Drivesbetween the servo-motor and the coupling. Each rotation of the ball-screw is converted toa pitch length linear travel of machine table through nut interface. The existence of severalcomponents in the drive train makes the driven inertia by rotary motors larger than that forlinear drives.Servo MotorCouplingBall-ScrewLinear - GuideTableNutSupport BearingsFigure 3.1: Ball-screw feed driveThe linear motor structure (Figure 3.2) is quite similar to an electric motor which its statorand rotor are unrolled. Therefore, the interaction between magnetic field of stator and rotorgenerates a linear force causing the armature, serving as machine table, to move linearly.Armature(Winding Not Visible) Magnet Plate(Usually Stationary)Armature Travels Along Magnet PathMagnets Are Fixedto Magnet PlateFigure 3.2: Linear driveThe existence of nut and gear set in ball-screw feed drives mitigate the effect of cuttingforce and inertial changes on servo loop performance. It also makes them suitable for movinglarge and heavy workpieces without a significant increase in motor size. Therefore, the ball-183.3. Mathematical Modeling of Feed Drivesscrew?s inherent ?gear reduction? in addition to their lower cost have made them favorablein heavy-duty commercial CNC machines. However, the large inertia reflected on motor shaftlimits their speed, acceleration and achievable bandwidth. Because of the flexibility of the longball-screw shaft and wear in mechanical drive components, their motion delivery is not perfect[49]. In the cases of light workpieces and small mass variations linear drives are preferredbecause of their higher speed and acceleration capability.Despite the widespread usage of ball-screw drives, they have several drawbacks: position-ing of machine table at different locations changes supporting condition and causes ball-screwdrives to have varying dynamics. Therefore, if the controller is designed based on identifiedparameters for one position (e.g table in the middle), the closed loop performance would notbe optimal for operation of the machine in the whole travel length. Researchers have consid-ered these dynamic variations in controller design and employed robust control approaches totackle this problem [41]. Another concern about ball-screw drives is a non-linear phenomenacalled backlash which is due to the clearance in joints; however, this can be minimized byusing preloaded balls in nut interface.3.3 Mathematical Modeling of Feed DrivesThe most common way of modeling the drive train in machine tools is to neglect the structuralflexibilities and consider all of the elements as lumped masses connected with rigid links. Pri-marily, this approach is used to develop a rigid body model of a ball-screw drive. The modularapproach is utilized to incorporate the ball-screw and column flexibilities in the mathematicalmodel of a C-frame machine tool as shown in Figure 3.3.193.3. Mathematical Modeling of Feed DrivesColumnSpindleMilling ToolBedTableServo MotorBall - screwFigure 3.3: Schematic diagram of a C-frame milling machine3.3.1 Rigid Body Model of the Ball-Screw DriveFigure 3.4 shows schematic diagram of a ball-screw drive:WorkpieceMachine TableMotorRigid Ball-ScrewzxCouplingFigure 3.4: Rigid ball-screw mounted on a rigid bedIn rigid body modeling, all the flexibilities in the column, bed, ball-screw, coupling andbearings are neglected and they are assumed as rigid links and joints. Therefore, the angulardisplacement at motor shaft (?m) and nut (?n) are the same. The angular displacement isconverted to linear motion at machine table (xt) through the screw-nut interface. For each203.3. Mathematical Modeling of Feed Drivesrotation of the screw, machine table moves for a screw pitch length (hp). The machine tabledisplacement (xt) is a function of angular motion at ball-screw and nut interface (?m) as:xt = rg?m; rg =hp2pi (3.1)where rg is the nut transformation ratio. The actuating torque applied by servo motor (Tm)drives the equivalent inertia (Je) and opposes the viscous friction torque (Bed?mdt ) reflected atthe motor shaft. Therefore, the angular displacement at motor shaft can be modeled as:Tm?Bed?mdt= Jed2?mdt2(3.2)combining Equations 3.1 and 3.2 and taking Laplace transform we have:Tm(s)?Bergdxtdt=Jergs2xt(s) (3.3)by organizing Equation (3.3), the transfer function between actuating torque and table lineardisplacement is:Gt(s)[ mN.m]=xt(s)Tm(s)=rgs(Jes+Be)(3.4)in CNC machines, when linear encoder is used for position feedback measurements, the opti-cal head of linear encoder is usually mounted on machine bed and the scale is placed on themachine table. Therefore, the linear encoder outputs the relative displacement between ma-chine table and machine bed. In rigid-body modeling, it is assumed that the whole drive trainis mounted on a rigid base, machine bed, and its displacement is zero. As a result, the encoderoutput is the same as absolute displacement of the machine table with respect to a stationaryreference outside the machine. Hence, the transfer function between the linear encoder outputand the actuating torque (Ge(s)) equals to Gt(s).213.3. Mathematical Modeling of Feed Drives3.3.2 Flexible Ball-Screw ModelBall-screw drives used as machine tool feed drives exhibit axial and torsional flexibilities. Inball-screws with large pitch length and light weight table, the torsional deformation causesshrinkage and elongation in the axial direction attributed to Poisson?s ratio (?). This variationin length will be reflected as an axial displacement on the machine table. Therefore, there is acoupling between axial and torsional modes.Figure 3.5: Ball-screw feed drive - UBC Mechatronics LaboratoryThe torsional flexibility of a single axis ball-screw shown in Figure 3.5 (hereinafter knownas ball-screw test bed) is modeled as a torsional (kts) spring with its associated damping (cts)connecting the inertias reflected at the motor (Jm) and the nut (Jn) as shown in Figure 3.6. Thewhole assembly is assumed to be mounted on a rigid bed. This setup consists of a ball-screwwith a pitch length of 20 [mm] constrained axially by a thrust bearing at the closer end to themotor; a radial ball bearing is used to constrain the other end. It is driven by a brush-lessDC motor connected to the ball screw through a bellow-type coupling. A linear encoder with0.05 [?m] resolution attached to the table, and a rotary encoder with 0.1 [?m] mounted on themotor shaft are used for table position measurement. The test bed can have 1 [g] acceleration223.3. Mathematical Modeling of Feed Drivesand maximum velocity of 27 [m/min].zxBall-ScrewServo motorCouplingNutTableTable:Feed Drive:Ball-Screw Torsional Mode:Figure 3.6: Flexible ball-screw mounted on a rigid bedThe linear force (Ftn) applied from the nut moves the machine table mass (mt) on the guide-way. Ignoring external disturbance loads, and considering viscous damping (Bt) in the guide,the table dynamics is expressed as:Ftn?Bt x?t = mt x?t ; xt = rg?n (3.5)The reaction force between machine table and the nut (Ftn) is transmitted to the ball screwhaving transmission ratio of (rg) as a load torque (Tn):Tn = Ftn? rg; rg =hp2pi (3.6)Considering the load torque (Tn) acting where the nut is located, the ball-screw vibrations233.3. Mathematical Modeling of Feed Drivesat nut interface (?n) can be modeled as:?Tn? kts (?n??m)? cts(??n? ??m)= Jn??n (3.7)The electrical servo motor creates torque (Tm), which rotates the motor shaft, coupling, andball-screw. Therefore, the ball-screw vibrations observed at motor shaft can be expressed as:Tm? ct ??m? kts (?m??n)? cts(??m? ??n)= Jm??m (3.8)where ?m is the angular displacement of the ball-screw at motor shaft and ct represents thetorsional viscous damping coefficient due to the viscous friction in the nut, bearings and thecoupling. Taking Laplace transform of Equations (3.5) to (3.8), organizing them in terms ofdisplacements (?n, ?m), and motor torque (Tm) yields the following matrix equation:????11 ?(ctss+ kts)?(ctss+ kts) ?22???????n?m???=???01???Tm (3.9)?11 =[Jn +mtr2g]s2 +[cts +Btr2g]s+ kts?22 = Jms2 +(cts + ct)s+ ktsBy solving Equation 3.9, the transfer function between machine table displacement (xt),vibration of the ball-screw at motor shaft (?m) and motor torque (Tm) can be obtained as:Gt(s)[ mN.m]=xt(s)Tm(s)=rg?n(s)Tm(s)=rg (ctss+ kts)s(Jes+Be)(s2 +2??ns+?2n )(3.10)Gm(s)[ mN.m]=rg?m(s)Tm(s)=rg([Jm +mtr2g]s2 +[cts +Btr2g]s+ kts)s(Jes+Be)(s2 +2??ns+?2n )(3.11)where Je,and Be are the equivalent inertia and viscous damping constants. ? and ?n represent243.3. Mathematical Modeling of Feed Drivesthe damping coefficient and natural frequency of the ball-screw torsion mode. Je, Be, ? , and?n can be evaluated from Equation (3.9).Similar to the case of rigid body modeling in 3.3.1, because of the assumption on rigidityof the bed, the transfer function between linear encoder output and actuating torque (Ge(s))equals to Gt(s). The transfer functions in Equations 3.10 and 3.11 can be written as:Gt(s) = Gr(s)?G f t(s); Gm(s) = Gr(s)?G f m(s) (3.12)where the rigid body motion is governed by transfer function:Gr(s)[ mN.m]=rgs(Jes+Be)and the structural dynamics part is dominated by one mode as:G f t(s)[mm]=(?ts+?t)(s2 +2??ns+?2n ); G f m(s)[mm]= K f m(s2 +?ms+?m)(s2 +2??ns+?2n )(3.13)where K f m,?t , ?t , ?m, and ?m are functions of the variables used in Equations 3.10 and 3.11.These formulations will be used in identification of structural dynamics in Chapter 4.3.3.3 Feed Drive In a Flexible Machine ToolA mathematical model for the x axis feed drive of a three axis vertical machining center(FADAL VMC2216) shown in Figure 3.7 is developed. The model includes dynamics of themodes which are transmitted as axial vibrations of the horizontal (x) drive to the linear encoder.These modes are identified as axial vibrations of the ball screw, and bending and torsional vi-brations of the C-framed column. The torsional mode of ball-screw is neglected as its naturalfrequency is far beyond the bandwidth of servo loop.253.3. Mathematical Modeling of Feed DrivesFigure 3.7: FADAL VMC2216 three axis machining centerA brush-less DC motor is used to apply torque to the ball-screw with 10 [mm] pitch lengthso that the table can have maximum acceleration of 2.5 [m/s2] and reach maximum velocityof 25.4 [m/min]. Linear and rotary encoders with 1 [?m] resolution are used to measure tableposition. While the linear scale is mounted on the moving table, the encoder head is mountedon the stationary guide. The motor torque is transmitted to the ball screw with a flexiblecoupling that has a torsional viscous damping constant of ct and angular displacement ?m.(seeFigure 3.8):Jm??m = Tm?Tn? ct ??m (3.14)where Jm is the total inertia of coupling, motor shaft, ball-screw and nut reflected at the motorshaft. The axial load ( Ftn ) felt at the nut ? ball screw is transmitted as the load torque (Tn) asexplained before in Equation 3.6.The total axial displacement of the table (xt) is the summation of the axial vibration of ballscrew reflected at the nut (xn) and the angular displacement of the ball screw (?m) translated as263.3. Mathematical Modeling of Feed Drivesthe axial displacement due to the nut interface:xt = xn + rg?m (3.15)zxBall-ScrewServo motorCouplingNutTableTable:Feed Drive:Ball-Screw Axial Mode:Figure 3.8: Flexible ball-screw modelThe axial force (Ftn) applied from the nut moves the machine table mass (mt) on the guide-way of the bed. Ignoring external disturbance loads, and considering viscous damping (Bt) inthe guide, the table dynamic is expressed as:mt x?t = Ftn?Bt (x?t? x?b) (3.16)where xt?xb is the relative vibration between the table (xt) and the bed (xb). The ball-screw ismodeled as an axial spring (kls) with its associated damping coefficient (cls) as shown in Figure273.3. Mathematical Modeling of Feed Drives3.8. The axial vibrations (xn) of the ball screw can be considered as:?Ftn? kls (xn? xm)? cls (x?n? x?m) = mnx?n (3.17)where xm is the axial vibration of the motor. The axial vibrations of the ball screw and motorexerts reaction force (Fmc) on the motor ? bed mounting bracket:Fmc? kls (xm? xn)? cls(x?m? x?n) = mmx?m (3.18)FmcFcbTcb FcbCOGRbb RbbRmb RmbRgb Rgbzx? ?cbxmb xmbxbbxcxbbxcFmcFigure 3.9: Bending mode of flexible machine tool structureThe machine tool?s column is made of cast iron box and exhibits bending and torsionmodes. Its bending mode is modeled as a rigid mass (mc) connected to the ground with aspring (kcb) and viscous damping element (ccb) as shown in Figure 3.9. The bending vibrations283.3. Mathematical Modeling of Feed Drivesof the column (xc) at its center of gravity (COG) is modeled as:mcx?c = Fcb +Bt (x?t? x?b)?Fmc (3.19)where Fcb is the reaction force between the machine column and the ground, and Bt is theviscous friction constant between the bed and table. The (xcb) of the column creates reactionmoment (Tcb) which is absorbed by the bending stiffness and damping elements at the groundconnection:Tcb = kcb?cb + ccb??cb (3.20)The moment equilibrium at the center of gravity can be expressed as:FmcRmb?FcbRgb?Bt (x?t? x?b)Rbb?Tcb = Jcb??cb (3.21)where Rmb, Rgb, and Rbb are the moment arms as shown in Figure 3.9. The column?s torsionalmode is modeled by a rigid mass (mc) connected to the ground with a torsional spring (ktc) andviscous damping element (ctc) as shown in Figure 3.10.FmcFmcTctCOGRmtRbtRetyx?FcbxbtxetxmtFcbFigure 3.10: Bending mode of flexible machine tool structure293.3. Mathematical Modeling of Feed DrivesThe moment equilibrium around the center of gravity in x-y plane is:?Bt (x?t? x?b)Rbt?Tct +FmcRmt = Jct ??ct (3.22)where Tct is the reaction torque exerted on the column?s center of gravity from the ground dueto its torsional vibration (?ct):Tct = kct?ct + cct ??ct (3.23)The bending ( ?cb) and torsional (?ct) deflections of column causes linear displacements atthe bed (xb), motor shaft (xm) and column?s center of gravity (xc) which can be evaluated bysuperposing the vibrations as: ???????????????xb = xbb + xbtxm = xmb + xmtxc = xcb + xct(3.24)where (xbb,xmb,xcb) and (xbt ,xmt ,xct) are contributed by the bending (b) and torsion (t) ofthe column at the top surface of the bed (b), at the motor shaft (m) and at the center of gravityof column (c), respectively. The torsional vibrations of machine column are assumed to occurat its center of gravity; hence, xct = 0. The linear displacements are estimated from the angularbending and torsional displacements from Figure 3.9 and Figure 3.10 as:xbb =(Rgb?Rbb)?cb; xbt =?Rbt?ct (3.25)xmb =(Rgb?Rmb)?cb; xmt =?Rmt?ct (3.26)xc = Rgb?cb (3.27)where Rmt and Rbt are the moment arms as shown in Figure 3.10. Taking Laplace transform303.3. Mathematical Modeling of Feed Drivesof Equations 3.14 to 3.27, organizing them between output displacements (xn, ?m, ?cb, ?ct),motor torque (Tm) yields the following:?(s)?????????xn(s)?m(s)?cb(s)?ct(s)?????????=?????????0100??????????Tm (3.28)where the symmetric transfer function matrix (?(s)) is:?(s) =??????????11 ?12 ?13 ?14?12 ?22 ?23 ?24?13 ?23 ?33 ?34?14 ?24 ?34 ?44?????????(3.29)?11 = [mn +mt ]s2 +[cls +Bt ]s+ kls?12 = [mts2 +Bts]rg?13 = [clss+ kls](Rmb?Rgb)+Bt(Rbb?Rgb)s?14 = [clss+ kls]Rmt +BtRbt?22 =[Jm +mtr2g]s2 +[ct +Btr2g]s?23 = Bt(Rbb?Rgb)rgs?24 = BtRbtrgs?33 =[Jcb +mcR2gb +mm(Rgb?Rmb)2]s2 +[ccb + cls(Rgb?Rmb)2+Bt(Rgb?Rbb)2]s+[kcb + kls(Rgb?Rbb)2]?34 =[mms2 + clss+ kls](Rmb?Rgb)Rmt +BtRbt(Rbb?Rgb)s313.3. Mathematical Modeling of Feed Drives?44 =[Jct +mmR2mt]s2 +[cct + clsR2mt +BtR2bt]s+[kct + klsR2mt]The transfer functions between the total displacement of table (xt), total displacement ofbed at the point where encoder optical head is mounted (xbe) and motor torque (Tm) can beevaluated from Equation (3.28) as:Gt(s)[ mN.m]=xt(s)Tm(s)=xn(s)+ rg?m(s)Tm(s)=?6i=0 pi(s)sis(Jes+Be)?3j=1(s2 +2? j?n js+?2n j) (3.30)Gbe(s)[ mN.m]=xbe(s)Tm(s)=(Rgb?Rbb)?cb(s)?Ret?ctTm(s)=?4i=0 qi(s)s4s(Jes+Be)?3j=1(s2 +2? j?n js+?2n j) (3.31)where Je and Be are the equivalent inertia and viscous damping constants, and ? j, ?n j arethe damping coefficient and the natural frequency of the mode i. A linear encoder is usedto measure the relative displacement between machine table (xt) and bed (xbe) in x direction(x = xt ? xbe). Its location is shown by a filled triangle in Figures 3.9 and 3.10 The transferfunction between the relative table position (x) and motor torque (Tm) can be obtained as:Ge(s)[ mN.m]= Gt(s)?Gbe(s) =x(s)Tm(s)=xt(s)? xbe(s)Tm(s)= Gr(s)G f (s) (3.32)where the rigid body motion is governed by transfer function:Gr(s)[ mN.m]=rgs(Jes+Be)(3.33)323.4. Summaryand the structural dynamics part is dominated by three modes as:G f (s)[mm]=?6i=0 li(s)si?3i=1(s2 +2?i?nis+?2ni) (3.34)The objective is to predict and control the relative vibrations (x) between the table and bed,which are felt by the position feedback sensor, the linear encoder.3.4 SummaryThis chapter presented the modeling of feed drives by taking structured flexibilities into ac-count. The flexibilities contribute to low frequency vibrations of machine tools, and they areused to obtain the transfer function between motor torque and table displacement. For a lightweight table, driven with a ball-screw, the coupled torsional-axial flexibility of the ball-screwis considered. A similar approach is extended to incorporate column flexibilities as torsionalsprings in modeling the feed drive of a typical C-frame CNC machine. It is shown that thetransfer function between motor torque and table displacement is the product of Gr(s) for rigidbody dynamics and G f (s) for structural flexibilities. Because the machine components areconnected through joints and couplings, flexibilities of the column are reflected on machine ta-ble as residual vibrations. These vibrations are observable through feedback sensors providedthat they have sufficient resolution.33Chapter 4Experimental Identification of Feed DriveDynamics4.1 OverviewThis chapter concentrates on identification of the mathematical models developed in Chapter3 for the ball-screw test bed and the vertical machining center (FADAL). Sections 4.2 to 4.4present the identification procedure of the transfer functions which explain the rigid body andstructural dynamics. Experimental identification results for the ball-screw test bed and theflexible machine tool (FADAL) are covered in Section 4.5. The identified mathematical modelfor FADAL is validated in Section 4.6 through simulating its dynamics in control loop andcomparing the results with experiment. Section 4.7 evaluates the efficiency of finite elementmethods in predicting machine dynamics by comparing the experimentally measured FRF withthe simulated FRF obtained from FEM. This chapter is summarized in Section 4.8.4.2 Identification of Rigid Body Dynamics (Gr(s))Erkorkmaz et al. [3] proposed the Unbiased Least Squares approach to identify rigid bodydynamics of machine tool feed drives. This approach is based on the rigid body model of theball-screw drive developed in Section 3.3.1.344.2. Identification of Rigid Body Dynamics (Gr(s))x(s) =rgs(Jes+Be)Tm(s)Including the effect of disturbance torque (Td) reflected on the motor shaft due to the fric-tion and cutting forces, Equation (3.4) is modified as:x(s) =rgs(Jes+Be)[Tm(s)?Td(s)] (4.1)The control signal input (u[V ]) is used to excite the axis dynamics. The applied torque tothe ball-screw drive (Tm) can be obtained by multiplying the input voltage by amplifier andmotor constants (ka[A/V ],kt [N.m/A]) as:Tm(s) = kaktu(s) (4.2)Having the amplifier gain (ka) and motor constant (kt) known, the disturbance torque (Td) isregarded as a disturbance equivalent voltage (d) to facilitate its estimation. Therefore, Equation(4.1) can be written in terms of the commanded (u) and disturbance (d) voltages as:x(s) =kaktrgs(Jes+Be)[u(s)?d(s)] (4.3)The machine table velocity measured by digital differentiation of the linear encoder signalcan be expressed in terms of control input (u) and equivalent input disturbance (d) as:v(s) = sx(s) =kaktrgJes+Be[u(s)?d(s)] (4.4)Considering Pv =KaKtrgJe, and Qv =?BeJe, the derivative of table velocity in Laplace domain354.2. Identification of Rigid Body Dynamics (Gr(s))can be obtained by rewriting Equation (4.4) in the following form:sv(s) = Pv [u(s)?d(s)]+Qvv(s) (4.5)Equations (4.3) to (4.5) can be expressed in state space notation as:???x?(t)v?(t)???=???0 10 Qv???? ?? ?Ac???x(t)v(t)???+???0 0Pv ?Pv???? ?? ?Bc???u(t)d(t)??? (4.6)The input voltage to the motor is commanded through a D/A converter; the table positionand velocity are measured by sampling the encoder signal at discrete time intervals (Ts). Hence,Equation (4.6) expressed in discrete-time domain with a zero-order hold at the input stage is:???x(k+1)v(k+1)???= Ad???x(k)v(k)???+Bd???u(k)d(k)???where Ad = eAcTs , Bd =Ts?0eAc?d? .Bc (4.7)Following the calculations presented in Appendix A, Ad and Bd are as follows:Ad =???1 ?eQvTS0 eQvTs??? , Bd =PvQv(eQvTS?1)????1 11 ?1??? (4.8)By defining Qvd = eQvTs and Pvd =PvQv(eQvTS?1), Equation (4.7) results in the followingrecursive equation:v(k) = Qvdv(k?1)+Pvd (u(k?1)?d(k?1)) (4.9)364.2. Identification of Rigid Body Dynamics (Gr(s))In identification process, the only disturbance acting on the feed drive is Coulomb friction(d f ) in the guid-way, nut and bearings. The traditional approach to model Coulomb frictionis assigning two constant values to the applied disturbance in positive and negative directions.Therefore, d f can be expressed as follows:d f (v(k)) =???????????????d+f if v(k) > 00 if v(k) = 0d?f if v(k) < 0(4.10)The friction model in Equation (4.10) can be reformulated as:d f (v(k)) = PV (v(k)) .d+f +NV (v(k)) .d?f (4.11)???????PV = Positive Velocity = 12? (v(k)) .(1+? (v(k)))NV = Negative Velocity =?12? (vt(k)) .(1?? (v(k)))(4.12)where ? is the sign function.? (v(k)) =???????????????1 if v(k) > 00 if v(k) = 0?1 if v(k) < 0(4.13)By incorporating Equation (4.11) into Equation (4.9), we have:v(k) = Qvdv(k?1)+Pvdu(k?1)?PvdPV (v(k?1))d+f ?PvdNV (v(k?1))d?f (4.14)374.2. Identification of Rigid Body Dynamics (Gr(s))or in matrix form:v(k)????Y (k,1)=[v(k?1) u(k?1) ?PV (v(k?1)) ?NV (v(k?1))]? ?? ??(K,1)?????????QvdPvdPvdd f+Pvdd?f?????????? ?? ??(4.15)Writing equation (4.15) for each sampling time, results in the following matrix equation:?????????v(2)v(3)...v(N)?????????? ?? ?Y= ??????????QvdPvdPvdd f+Pvdd?f?????????? ?? ??+?????????e(2)e(3)...e(N)?????????? ?? ?EWhere N is the total number of sampled data, and?=?????????v(1) u(1) ?PV (v(1)) ?NV (v(1))v(2) u(2) ?PV (v(2)) ?NV (v(2))............v(N?1) u(N?1) ?PV (v(N?1)) ?NV (v(N?1))?????????(4.16)Solving equation (4.16) by using least squares technique to minimize the error vector (E)gives the unknown matrix (?) as:?=(?T?)?1?TY (4.17)The first and second element of? are Qvd and Pvd , respectively. d+f and d?f can be obtained384.2. Identification of Rigid Body Dynamics (Gr(s))by dividing the third and fourth components by Kvd respectively. Therefore, identificationbased on the ULS technique gives all the parameters engaged in rigid body modeling andthere is no need to conduct a separate experiments to find Coulomb friction parameters. Theequivalent inertia and viscous damping coefficient can be obtained by using equations below:Je =(Qvd?1)ktkaTsPvd ln(Qvd)(4.18)Be =(1?Qvd)ktkaPvd(4.19)To identify the parameters, Erkorkmaz suggested a pulse-shaped voltage command (Figure4.1) to be commanded to the amplifier in open-loop while having the servo drive in torque con-trol mode. The amplitude of the supplied voltage must be scaled with different factors, ku, toprovide sufficient excitation to overcome static friction and the parameters should be estimatedfor each ku. Here care must be taken in selecting ku to avoid saturation of the amplifier. Theestimated parameters (Je and Be) for the largest value of ku, which does not saturate the drives,must be considered as the identified parameters.0-2.0-1.5-1.0-0.50.01.01.52.00.50.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6InputSignal[V]VoltageTime [sec]Figure 4.1: Least square identification reference command394.3. Identification of Structural Flexibilities4.3 Identification of Structural FlexibilitiesThe structural dynamics of the drive in the machine tool with structural flexibilities (G f (s)) isidentified by sending sweep sinusoidal waves to the amplifier at a frequency range of modes.The frequency response function (FRF) of the system is measured with a Fourier analyzer. Themeasured FRF (Ge (s = j?)) is divided by rigid body dynamics (Gr( j?)) at each frequencyto isolate the structural dynamics (G f (s = j?)). If the linear encoder resolution is not high,the low frequency modes with large masses cannot be measured accurately. In such cases, oneaccelerometer is placed at the bed where the encoder reading head is mounted, and another isplaced on the table while exciting the motor. The FRFs from the two locations are subtractedand divided by ( j?)2 to find(Ge ( j?) = xt( j?)?xbe( j?)Tm( j?)), which is equivalent to linear encoderbased measurements. For the machine tool with column and ball-screw flexibility (see section3.3.3), the numerator of the transfer function (Equation (3.34)) is reorganized as the product ofsecond order polynomials to suit the identification algorithm:G f (s) = K f?3i=1(s2 +ais+bi)?3i=1(s2 +2?i?nis+?2ni) (4.20)in which four parameters need to be identified (ai,bi,?i, ?ni) for each mode. The natural fre-quency (?ni) and damping ratio (?i) are estimated from experimental FRFs either using PeakPicking (PP) algorithm [50] for well-separated modes, or Rational Fraction Polynomials (RFP)method [51] for closely spaced modes. These two algorithms are explained in section 4.4. Hav-ing damping ratios and natural frequencies identified, the denominator of G f (s) is known; andonly the numerator coefficients are left as unknowns. Replacing s with ( j?) in Equation (4.20),magnitude of G f (s) becomes:??G f ( j?)??= K f??????ki=1((bi??2)2 +a2i)?ki=1((?2ni??2)2+(2?i?ni?)2) (4.21)404.3. Identification of Structural Flexibilitieswhere the unknowns are stored in a parameter vector as:{?}=[K f ,a1,a2,a3,b1,b2,b3]T(4.22)The parameters are estimated using non-linear least squares method. Starting with an initialguess, ? 0, Gauss-Newton iteration is used to identify the unknown parameters:{?}s+1 = {?}s?([J f]T [J f])?1 [J f]Tf (? s) (4.23)where the Jacobin matrix (J f ) is:[J f]=[? f?K f? f?a1? f?a2? f?a3? f?b1? f?b2? f?b3](4.24)with the partial derivative terms:? f?K f=??????ki=1((bi??2)2 +a2i)?ki=1((?2ni??2)2+(2?i?ni?)2) (4.25)? f?a j=??????ki=1,i 6= j((bi??2)2 +a2i)?ki=1((?2ni??2)2+(2?i?ni?)2) ?a j?((b j??2)2+a2j) (4.26)? f?b j=??????ki=1,i 6= j((bi??2)2 +a2i)?ki=1((?2ni??2)2+(2?i?ni?)2) ?(b j??2)?((b j??2)2+a2j) (4.27)In non-linear least squares problems, when dealing with a large number of unknowns,the iteration convergence highly depends on an appropriate initial guess of the unknowns. Tofacilitate reconstruction of experimental FRF and find the appropriate initial guess, curve fittingis done in several steps. In the first step (m = 1), iterations start with considering only the first414.4. Modal Parameters (? ,?n) Estimationmode and neglecting higher modes dynamics. As G f (s) must have DC gain of 1, the unknownparameters are initialized as:K1f = 1; a11 = 0; b11 = (?n1)2 (4.28)where the superscript indicates the step number (m = 1). Outputs of the first step are stored inK1f , a11, and b11 and they are used as the initial guess in the second step (m = 2) where dynamicsof the second mode are taken into account. Therefore, the initial guesses for the parameters tobe identified in the second step are made as:K2f = K1f ; a21 = a11; b21 = b11 (4.29)a22 = 0; b22 =(?n1?n2)2K1f b21(4.30)A similar procedure is followed in the third step (m = 3) as:K3f = K2f ; a31 = a21; b31 = b21; a32 = a22; b32 = b22 (4.31)a33 = 0; b33 =(?n1?n2?n3)2K2f b31b32(4.32)The flowchart shown in Figure 4.2 summarizes the steps for the identification of G f (s).4.4 Modal Parameters (? ,?n) EstimationThe existing literature for modal parameter estimation is rich and a wide range of approachesin time domain and frequency domain has been developed. In this section, the Peak-Picking(PP) algorithm and Rational Fraction Polynomial (RFP) approach are briefly explained.424.4. Modal Parameters (? ,?n) EstimationNumber of Modes to be IdentifiedMaking The Initial GuessYesNoNoYesGauss - Newton Iteration(Frequency Range Includes modes)Save The Parameter after stepFigure 4.2: Identification of G f Flowchart4.4.1 The Peak Picking MethodFor a single degree of freedom mass-spring and damper system (Figure 4.3) with mass of m,stiffness of k, and damping coefficient of c, the transfer function between applied force to themass and its vibration response has the following structure:G(s) =xsFs=?2n/ks2 +2??ns+?2n; Where ? = c2?km,?n =?km(4.33)Figure 4.3: Single degree of freedom oscillator (Mass,Spring, and Damper System)434.4. Modal Parameters (? ,?n) EstimationThe frequency response function of the single degree of freedom oscillator can be obtainedby replacing the Laplace operator (s) with ( j?) in Equation (4.33) as:G( j?) = ?2n/k(?2n ??2)+ j (2??n?)(4.34)The real and imaginary parts of Equation (4.34) are extracted through multiplying its nu-merator and denominator by(?2n ??2)? j (2??n?):Re [G( j?)] =?2n(?2n ??2)/k(?2n ??2)2 +(2??n?)2(4.35)Im [G( j?)] = ?2n (2??n?)/k(?2n ??2)2 +(2??n?)2(4.36)FRF plot of the single degree of freedom oscillator in terms of its real and imaginary partsis depicted in Figure 4.4. FrequencyFrequencyReal ImaginaryPartPartFigure 4.4: Transfer function of a SDOF system represented by its real and imaginary parts444.4. Modal Parameters (? ,?n) EstimationAs this Figure 4.4 indicates, the real part shows two extremums at the frequencies of(?1 = ?n (1?? )) and (?2 = ?n (1+? )); the imaginary plot has a peak at the natural fre-quency (?n). As a result, for a single degree of freedom oscillator, it is possible to identifyits natural frequency (?n) and damping ratio (? ) by reading the frequencies associated withthe extremums of real and imaginary plots of its experimental FRF and using the followingequation.? = ??2?n; ?? = ?2??1 (4.37)The explained procedure is called Peak Picking approach to identify modal parameters of aflexible system. This is the simplest method for identifying the modal parameters of a flexiblestructure in frequency domain. It is also quite efficient for flexible systems with well-separatedmodes where dynamics of one mode is not affected with dynamics of the neighboring mode.More details on this approach are presented in [52].4.4.2 The Rational Fraction Polynomial (RFP) MethodWhen two modes are closely spaced, each of the modes affects the neighboring mode fre-quency response; this interference makes modal parameter estimation more difficult. There-fore, when the flexible system exhibits closely spaced modes, more advanced methods must beutilized for more accurate estimation of modal parameters.A plenty of studies are published in literature to define a numerical indicator for modalinterference [53, 54]. Modal Overlap Factor (?) is referred to the ratio between the half-powermodal bandwidth and the average modal spacing [54]. Considering Figure 4.5, for two closelyspaced modes, the modal overlap factor for each mode (?i) is defined as:?i =? fi?F(4.38)454.4. Modal Parameters (? ,?n) EstimationWhen ? is larger than 30%, modes are coupled and accurate estimation of modal param-eters requires more advanced identification techniques than Peak-Picking algorithm, such asRational Fraction Polynomials (RFP).Frequency [Hz]Magnitude [dB]Figure 4.5: Magnitude plot of frequency response function for a system with two closelyspaced modesRational fraction form denotes a fraction which has two polynomials in its numerator anddenominator, and the order of denominator (n) is independent of the order of numerator (m)(see Equation (4.39)). The polynomial in denominator is called characteristic equation, and itsroots represent poles of the system.H( j?) = ?mk=0 aksk?nk=0 bksk ; s = j? (4.39)The main effort is finding coefficients of the numerator (ak) and denominator (bk) suchthat the reconstructed FRF (?mk=0 aksk?nk=0 bksk ) makes a good fit with the experimentally measured FRF(H( j?)). The curve fitting problem can be expressed as minimizing the error criterion (J)defined as follows:ei =m?k=0akski ?H( j?i)? ?? ?hin?k=0bkski ; si = j?i (4.40)464.4. Modal Parameters (? ,?n) EstimationJ =L?i=1e2i (4.41)The polynomials in numerator and denominator of Equation 4.39, and therefore the exper-imental FRF (H ( j?)) have Hermitian symmetry about the origin of the frequency axis. Inother words, their real part are even functions and their imaginary part are odd functions. AsRichardson and Formenti [51] proposed, Equations 4.39 and 4.40 can be expressed in terms oforthogonal polynomials (?i, j and ?i, j) as:H( j?) = ?mk=0 ck?i,k?nk=0 dk?i,k; dn = 1 (4.42)ei =m?k=0ck?i,k?hin?k=0dk?i,k (4.43)where the orthogonality condition is defined as:?Li=?L?i,k?i, j =???????0 k 6= j1 k = jand ?Li=?L?i,k|hi|2?i, j =???????0 k 6= j1 k = j(4.44)L indicates the number of frequencies in which the experimental FRF (H ( j?i)) is mea-sured. For negative frequencies (i = ?L, . . . ,?1) the experimental FRF (hi) is constructed byusing its Hermitian symmetry property. Equation 4.43 can be expressed in matrix form as:{E}= [P]{C}? [T ]{D}?{W} (4.45)where{E}=??????e?L...eL??????; [P] =????????L,0 ??L,1 ? ? ? ??L,m....... . ....?L,0 ?L,1 . . . ?L,m??????(4.46)474.4. Modal Parameters (? ,?n) Estimation[T ] =??????h?L??L,0 h?L??L,1 ? ? ? h?L??L,n?1....... . ....hL?L,0 hL?L,1 . . . hL?L,n?1??????, {W}=??????h?L??L,n...hL?L,n??????(4.47){C}=[c0 c1 . . . cm]T{D}=[d0 d1 . . . dn?1]T(4.48)This new formulation overcomes some numerical analysis problems from which the oldleast squares techniques suffer. The procedure of calculating the orthogonal polynomials ateach frequency (?i, j and ?i, j) is presented in Appendix B. The numerator and denominatorcoefficients (ck and dk) of the transfer function expressed in terms of orthogonal polynomials(see Equation 4.42) are obtained through least square minimization of the error vector ({E})as: ???????{D}=?[I? [X ]T [X ]]?1[X ]T {H}{C}= {H}? [X ]{D}(4.49)where[X ] =?Re([P?]T [T ]){H}= Re([P?]T{W})(4.50)The star sign (*) indicates the complex conjugate matrix. Once these coefficients areknown, the coefficients of the denominator of the transfer function in terms of Laplace variable(bk in Equation 4.39) can easily be recovered as:bk =n?i=kdi pik; pik =???????????????0 k < 0 or k > i1D0i = k = 01Dipi?1k?1 +Vi?1 pi?2k 0? k ? i(4.51)The proof of Equation 4.51 is presented in Appendix B. Having these coefficients, it is possibleto reconstruct the characteristic equation in its ordinary form. Damping ratio (? ) and natural484.5. Experimental Identification Resultsfrequency (?n) of a specific mode which is considered in curve fitting can be calculated fromthe roots of the identified characteristic equation. Besides modal analysis, this technique canbe utilized for identification of poles, zeros and resonances of combined electromechanicalservo-systems.4.5 Experimental Identification ResultsConsidering the mathematical models developed in sections 3.3.2 and 3.3.3, the experimentalresults of the identified transfer functions for the ball-screw test bed and the flexible machinetool (FADAL VMC2216) are presented in sections 4.5.1 and 4.5.2, respectively.4.5.1 Ball-Screw Test BedThe amplifier gain (ka), motor torque constant (kt), and ball-screw pitch length (hp) for theball-screw test bed are given in manufacturer?s catalogue as:ka = 0.887[AV]; kt = 0.72[N.mA]; hp = 20 [mm] (4.52)From Equation (3.1), the nut transformation ratio (rg) is:rg =hp2pi =20?10?3 [m]2pi [rad] = 0.0032[ mrad](4.53)The estimated inertia (Je) and viscous damping coefficient (Be) for different gain values (ku)are presented in Figure 4.6. For small values of ku, the motor does not provide enough power toovercome sticking friction in the guide-way; hence, the identification procedure overestimatesthe parameters of interest. By increasing ku, the given voltage to servo drive increases anda higher torque is applied at the ball-screw. Therefore, machine table moves with a higher494.5. Experimental Identification Resultsvelocity; a more rapid motion of the table mitigates the effect of nonlinear friction around zerovelocity on parameter identification.678910111213x 10-4x 10-20.5 1 1.5 2 2.51.52.02.53.0Ku [V/V]0.5 1 1.5 2 2.5Ku [V/V]B e [Kg.m2 /s]Je [Kg.m2 ]Figure 4.6: Estimated inertia (Je) and viscous damping coefficient (Be) by unbiased leastsquares techniqueAccording to manufacturer?s catalogue, the saturation limit for the amplifier is 5 [V ]; hence,ku cannot be larger than 2.5 [VV ]. The estimated parameters for ku = 2.5 are considered as theidentified inertia and viscous damping coefficients reflected on the motor shaft as:Je = 6.9703?10?4 [Kg.m2]; Be = 0.0156[kg.m2/s](4.54)Considering Equation (3.4), the rigid body transfer function between motor torque (Tm) andmachine table displacement (xt) can be obtained as:Gr[ mN.m]=rgs(Jes+Be)=0.0032s(6.9703?10?3s+0.0156)(4.55)504.5. Experimental Identification ResultsTo identify the contribution of structural dynamics (G f t and G f m in Equation (3.13)) intable vibration (xt), the experimental FRFs between motor torque (Tm) and machine table dis-placement measured through linear and rotary encoders are reconstructed by curve fitting. Theexperimental FRFs are obtained through sweeping sinusoidal voltage commands from 10 -300 [Hz]. Table position is measured directly by using the linear encoder. The rotary encodergives table position based on angular displacement of the ball-screw at motor shaft (?m); there-fore, it outputs rg?m. Experimental FRFs and bode plot of the rigid body transfer function(kaktGr ( j?)) are depicted in Figure 4.7.101 102-200204060Linear EncoderRotary Encoder101 102 -400 -300 -200 -1000Phase[Degrees]Magnitude[dB]Frequency [Hz]Linear EncoderRotary EncoderRigid =Body FRFRigid Body FRFFigure 4.7: The experimental FRF between motor torque and table displacement measured bylinear and rotary encodersFigure 4.7 shows that the ball-screw drive has a torsional mode around 220 [Hz] whichis clearly captured by linear and rotary encoders. The rigid body transfer function FRF co-incides with the experimental FRF in low frequency region (1-100 [Hz]). This means that,in low frequency region, the rigid body transfer function can successfully explain ball-screwdynamics.514.5. Experimental Identification ResultsFigure 4.8 presents the contribution of ball-screw flexibility (G f t and G f m) in linear scaleobtained through dividing experimental FRF by rigid body FRF at each frequency in the rangeof 10-300 [Hz]. From the flexible ball-screw modeling in section (3.3.2), the governing equa-tions for G f t and G f m have the following structure:G f t(s)[mm]=(?ts+?t)(s2 +2??ns+?2n ); G f m(s)[mm]= K f m(s2 +?ms+?m)(s2 +2??ns+?2n )50 100 150 200 250 300015101550 100 150 200 250 300-300-200-1000100200Phase[Degrees]Magnitude[m/m]Frequency [Hz]ExperimentalExperimentalExperimentalExperimentalFigure 4.8: Contribution of the ball-screw torsional flexibility in the experimental FRFs mea-sured by linear and rotary encodersBecause there is no other mode closely located to the ball-screw torsional mode, peakpicking algorithm is used to identify the natural frequency (?n) and damping ratio (? ) fromreal and imaginary plots of G f t and G f m shown in Figure 4.9. As explained in Section 4.4, theimaginary plot shows a peak at the natural frequency (?n = 223.91 [Hz]). The damping ratio(? ) is calculated as:? = ?2??12?n=229?2122?223.91= 0.038 (4.56)524.5. Experimental Identification Resultswhere ?1 and ?2 are shown in Figure 4.9.50 100 150 200 250 300-10-5051050 100 150 200 250 300 -800 -600 -400 -2000200400Real[m/m]Imaginary[m/m]Frequency [Hz]ExperimentalExperimentalExperimentalExperimentalFigure 4.9: Real and imaginary plots of the experimental FRF for G f t and G f mConsidering Figure 4.7, the experimental FRF measured by the rotary encoder Gm( j?)shows an anti-resonant mode around 180 [Hz]. This sharp attenuation in bode plot means thatG f m possesses imaginary zeros in its numerator. Therefore, the FRF obtained by dividing G f tby G f m at each frequency , which is shown in Figure 4.10, exhibits a mode around 180 [Hz]and has the following structure:G f tG f m=1K f m?ts+?ts2 +?ms+?m(4.57)Its modal parameters (? ?and ? ?n) are estimated by peak picking algorithm as:? ? = 0.02574; ? ?n = 180[Hz] (4.58)The denominator coefficients (?m and ?m) in Equation (4.57), can be expressed in terms of534.5. Experimental Identification Resultsthe identified modal parameters (? ?and ? ?n) as:?m =(? ?n)2= 1.279?106; ?m = 2???n = 58.22 (4.59)According to Figure 4.8, G f t and G f m have a gain of 1 in low frequency region. Hence:K f m?m = ?2n =? K f m =?2n?m=(223.91?2?pi)21.279?106= 1.5475 (4.60)?t = ?2n = 1.979?106 (4.61)The only unknown parameter is ?t . This parameter is identified by least squares minimiza-tion of the error in phase plot, between experimental and reconstructed FRF ofG f t( j?)G f m( j?) , shownin Figure 4.10, as:?t =?53.952 (4.62)50 100 150 200 250 300051015202550 100 150 200 250 300-200-150-100-500Phase[Degrees]Magnitude[m/m]Frequency [Hz]ExperimentalReconstructedExperimentalReconstructedFigure 4.10: Magnitude and phase plot of experimental and reconstructedG f t( j?)G f m( j?)544.5. Experimental Identification ResultsThe identified parameters are summarized in Table 4.1.Table 4.1: Identified parameters of the flexible ball-screw driveJe = 6.9703?10?4[kg.m2]Be = 0.0156[kg.m2s]rg = 0.0032[mrad]ka = 0.887[AV]kt = 0.72[N.mA]K f m = 1.5475? = 0.038 ?n = 1406.9[rads]?t =?53.952?m = 58.22 ?t = 1.979?106 ?m = 1.279?106The identified transfer functions between motor torque (Tm) and table displacement (xt)measured by linear and rotary (rg?m) encoders are:Gt =xt(s)Tm(s)=0.0032(53.952s+1.979?106)s(6.9703?10?4s+0.0156)(s2 +106.92s+1.979?106)(4.63)Gm =rg?m(s)Tm(s)= 1.5475?0.0032(s2 +58.22s+1.279?106)s(6.9703?10?4s+0.0156)(s2 +106.92s+1.979?106)(4.64)101 102-130-120-110-100-90-80-70-60101 102 -400 -350 -300 -250 -200 -150Phase[Degrees]Magnitude[dB]Frequency [Hz]ExperimentalExperimentalReconstructedReconstructedFigure 4.11: Experimental and reconstructed FRF of the transfer function between motortorque (Tm) and table position capture by linear encoder (xt) (see Equation (4.63))554.5. Experimental Identification Results101 10 2-140-120-100-80-60101 10 2 -250 -200 -150 -100 -500Phase[Degrees]Magnitude[dB]Frequency [Hz]ExperimentalReconstructedExperimentalReconstructedFigure 4.12: Experimental and reconstructed FRF of the transfer function between motortorque (Tm) and table position capture by rotary encoder (rg?m) (see Equation (4.64))Bode plots of the identified transfer functions shown in Figures 4.11 and 4.12 provide agood fit with the experimentally measured FRFs in magnitude plot. The small discrepancyin the phase plot in high frequency region is due to the existence of non-linear friction in theball-screw test bed which works as an additional damping. This additional damping changesthe phase plot and increases the time delay. The effect of sticking friction is considerablein high frequency region where machine table vibration amplitude is less than 3 micro-meters.Moreover, FEM modeling of the ball-screw test bed [22] shows that, beside the torsional mode,the experimental setup exhibits lateral flexibilities which is not included in developing themathematical model.Direct FRF at the ball-screw table is also measured by using a hammer instrumented witha force sensor and mounting an accelerometer on the table. These FRFs are presented inFigure 4.13. When the small hammer (HA5 in Figure 4.14) with the aluminum tip is used, ashift in measured natural frequency and an overestimation of damping ratio is observed. This564.5. Experimental Identification Resultsdiscrepancy is due to non-linearities such as clearance in joints and nonlinear friction in theguide-way [55].0 100 200 300 400 500 60000.10.20.30.40.50 100 200 300 400 500 600-200-1000100200HA6 - Soft Plastic TipHA5 - Rubber TipHA6 - Hard PlasticTipHA5 - Plastic TipHA5 - Aluminium T ipPhase[Degrees]Magnitude[(m/s2)/N]Frequency [Hz]Figure 4.13: Direct FRFs measured at ball-screw tableLampaert et al [56] studied the effect of nonlinear friction on frequency response functionmeasurements. They concluded that the nonlinear friction in the pre-sliding region works asadditional springs connected to the table which increase the stiffness and cause over estimationof the natural frequency. Halverson and Brown [55] emphasized on the effect of clearanceand nonlinear damping in the joints on FRF measurements by impact testing. They pointedout that if excitation is close to a joint which has clearance, or the damping is a function ofrelative displacement, the apparent damping in the measured frequency response will be higherthan the actual value as a result of large relative motion at that location. They also proposedpre-loading the system to take up clearances in joints and bearings.Excitation of the structure with a larger force (harder hammer hit) removes the clearanceinstantly and breaks the pre-sliding friction bonds in the guide-way; therefore, a more in-574.5. Experimental Identification Resultstense excitation of structure, which is possible by using softer tips, mitigates the effect ofnon-linearities (but does not remove them) and leads to a more accurate measurement of thenatural frequency and damping ratio.Figure 4.14: The hammers used for direct FRF measurements at ball-screw tableFigure 4.13 shows that by using softer tips (plastic and rubber) and a larger hammer (HA6)the measured natural frequency and damping ratio converge to those obtained by giving puresinusoidal waves and using the FFT analyzer. In summary, systems which are an assemblyof several components exhibit non-linearities due to the friction and clearance in joints andbearings; therefore, excitation of their structure with pure random (e.g white noise) or sinesignal yields the most accurate analysis of their frequency response.584.5. Experimental Identification Results4.5.2 Flexible Machine Tool - FADAL VMC2216The amplifier gain (ka), motor torque constant (kt), and ball-screw pitch length (hp) for feeddrive of the FADAL VMC 2216 for which the mathematical model is developed in Chapter 3,are given in manufacturer?s catalogue as:ka = 0.4769[AV]; kt = 6.5723[N.mA]; hp = 10 [mm] (4.65)From Equation (3.1), the nut transformation ratio (rg) is:rg =hp2pi =10?10?3 [m]2pi = 0.0016[ mrad](4.66)The equivalent inertia (Je) and viscous damping coefficient (Be) reflected on the motor shaftare identified as [3]:Je = 7.95?10?3 [kg.m2]; Be = 0.0265[kg.m2s](4.67)Considering Equation (3.4), the rigid body transfer function between motor torque (Tm) andmachine table displacement (xt) can be obtained as:Gr[ mN.m]=rgs(Jes+Be)=0.0016s(7.95?10?3s+0.0265)(4.68)Similar to the ball-screw test bed, the sine sweeping approach in the frequency range of2-500 [Hz], is employed to measure the x-axis feed drive FRFs between supplied voltage (u)and the table position read by linear and rotary encoders. Experimental FRFs are depicted inFigure 4.15. The FRF measurements based on the linear encoder output indicates dominantmodes at 49.01, 61.91, and 97.88 Hz which correspond to bending of the column, torsion ofthe column and axial mode of the ball screw, respectively. The modes are verified by analyzing594.5. Experimental Identification Resultsthe mode shapes of the machine from experimental measurements and Finite Element studypresented by Law et la in [57]. FRF measurements based on the rotary encoder output showsonly one mode around 380 [Hz]; this mode is the torsional mode of ball-screw drive capturedbest by the rotary encoder. Its frequency is located far beyond the bandwidth of control loop(40-80 [Hz]); therefore this mode is not included in the mathematical modeling (see section3.3.3) and identification.101 102-160-140-120-100-80-60-40101 102 -400 -300 -200 -1000100Linear EncoderLinear EncoderRotary EncoderRotary EncoderRigid =Body FRFRigid =Body FRFPhase[Degrees]Magnitude[dB]Frequency [Hz]Figure 4.15: Experimental FRFs of the feed drive in x direction for FADAL VMC2216In FRF measurement, an accelerometer is mounted on the table beside the encoders output.Figure 4.16 compares the measured acceleration FRFs in linear scale by using accelerometerand by taking double derivative of linear encoder output. This figure shows a discrepancybetween obtained FRFs based on accelerometer and linear encoder measurements; the modearound 49 Hz is the most flexible one when accelerometer output is used to obtain accelerationFRF; however, the acceleration FRF based on linear encoder reveals that the mode around 97[Hz] is the most flexible mode.604.5. Experimental Identification Results50 100 150 200 250 300 350 400 450 50000.51.01.52.050 100 150 200 250 300 350 400 450 500-600-500-400-300-200-100Phase[Degrees]Magnitude[(m/s2)/V]Frequency [Hz]Linear EncoderLinear EncoderAccelerometerAccelerometerFigure 4.16: Comparison of the experimental FRF between table acceleration and input volt-age to the amplifier measured by linear encoder (red line) and accelerometer (blue line)As explained before, linear encoder outputs the relative displacement between machinetable and bed. On the other hand, the accelerometer measures absolute acceleration of table inx direction. The mode shape at a specific frequency could be in a way which results in relativedisplacement of the machine table and bed to be different from linear displacement of machinetable with respect to a stationary origin outside the machine. In such a case, we expect to see adiscrepancy in measured FRFs. In the range of 30 to 60 [Hz], the machine table and bed movein the same direction since the vibrations are associated with the column bending and torsionmodes; therefore, linear encoder output is less than absolute vibration of the table. Between60 and 100 [Hz], vibrations are dominated by the axial mode of the ball-screw drive. Hence,the machine table and bed move in the opposite directions, leading the output of linear encoderto be larger than the table?s absolute vibration amplitude. For further investigation, anotherexperiment is conducted; the accelerometer (1) is attached to the bed where the optical head isplaced and the accelerometer (2) is attached to the machine table where the scale is located. As614.5. Experimental Identification Resultsclaimed before, subtracting the FRF obtained through the accelerometer (1) from that measuredby the accelerometer (2) must lead to an FRF similar to that measured by using linear encoder.Figure 4.17 shows that this expectation is fulfilled and the obtained FRFs match.50 100 150 200 250 300 350 400 450 50001.00.51.52.0 50 100 150 200 250 300 350 400 450 500-500-400-300-200-100Two AccelerometersLinear EncoderPhase[Degrees]Magnitude[(m/s2)/V]Frequency [Hz]Figure 4.17: Experimental acceleration FRF - Comparison of using two accelerometers (blueline) and linear encoder (red line)According to the mathematical modeling of the feed drive for machine tool with structuralflexibilities presented in Section 3.3.3, the transfer function between motor torque (Tm) andtable position measured by linear encoder (x) has the following structure:Ge(s) =x(s)Tm(s)=rgs(Jes+Be)? ?? ?Gr(s)?K f?3i=1(s2 +ais+bi)?3i=1(s2 +2?i?nis+?2ni)? ?? ?G f (s)(4.69)So far, the rigid body transfer function (Gr(s)) is identified. The total FRF is divided byrigid body dynamics (Gr( j?)) at the frequency domain to obtain the FRF of the structuraldynamics (G f ( j?)).624.5. Experimental Identification Results45 50 55 60 65 45 50 55 60 6545 50 55 60 65 45 50 55 60 65Frequency [Hz]Magnitude[m/m]Magnitude[m/m]Magnitude[m/m]Magnitude[m/m]Frequency [Hz]Frequency [Hz]Frequency [Hz]RFPExperimental11.21.41.61.811.21.41.61.811.21.41.61.811.21.41.61.8RFPExperimentalRFPExperimentalRFPExperimentalFigure 4.18: Estimation of modal parameter with RFP method for the closely spaced columnbending and torsion modes.The contribution of structural dynamics (G f (s)) is estimated by implementing the iden-tification algorithm presented in Section 4.3. Experimentally meausred FRF exhibits threedominant modes in the frequency range of 2-150 [Hz]. These three modes are at 49, 61, and97 [Hz] and correspond to bending of the column, torsion of the column and axial mode of theball screw, respectively. They are considered in the modeling procedure in Section 3.3.3.Because the first two modes are closely-spaced, Peak-Picking method fails to estimatetheir modal parameters properly and Rational Fraction Polynomial (RFP) approach is utilized(Figure 4.18). For this purpose, in RFP modal parameter estimation over the frequency rangeof (45-65 [Hz]), a 4th order polynomial is chosen for the denominator (n = 4) to include the634.5. Experimental Identification Resultsdynamics of column bending and torsion modes. As proposed by Richardson and Formenti[51], the order of numerator is increased in 4 steps to provide a more accurate estimationof natural frequencies and damping ratios. As Figure 4.18 shows, by increasing the orderof numerator (m), the RFP technique compensates the effect of non-modeled dynamics andprovides a better fit. The axial mode of the ball-screw is relatively far from the first two modesand its modal parameters are estimated by Peak-Picking algorithm (similar to the flexible ball-screw drive) as ?3 = 0.1 and ?n3 = 96.84 [Hz].Having the modal parameters estimated, the denominator of G f (s) is known. To identifythe unknown coefficients of its numerator, the nonlinear curve fitting approach shown in Figure4.2 is followed. The estimated modal parameters from RFP and Peak-Picking techniques arefine-tuned in each step to improve the fitting accuracy.The identified transfer function parameters from linear encoder (position) and servo com-mand measurements are given in Table 4.2.Table 4.2: Identified parameters of structural transfer function of the feed drive (G f (s) [m/m],see Equation (4.69)).K f = 0.492a1 = 60.65 b1 = 105.62?103 ?1 = 0.083 ?n1 = 307.94a2 = 88.36 b2 = 150.23?103 ?2 = 0.060 ?n2 = 388.93a3 = 272.5 b3 = 694.56?103 ?3 = 0.091 ?n3 = 615Figure 4.19 compares the experimental and reconstructed FRF of G f ( j?) over the fre-quency range of interest (2-150 [Hz]). The identified open loop FRF of the system whichcontains both rigid body (Gr (s)) and structural dynamics (G f ( j?)) are superimposed on themeasured FRF in Figure 4.20. This figure indicates that the reconstructed FRF matches theexperimental FRF in both magnitude and phase plots.644.5. Experimental Identification Results20 40 60 80 100 120 14000.51.01.52.02.53.020 40 60 80 100 120 140-150-100-50050Phase[Degrees]Magnitude[m/m]Frequency [Hz]ReconstructedStructuralReconstructedStructuralExperimental StructuralExperimental StructuralFRFFRFFRFFRFFigure 4.19: Experimental FRF of the feed drive structure (G f (s), see Equation (4.69)).101 102-140-120-100-80-60-40101 102 -150 -100 -50050Fitted FRFExperimental FRFFitted FRFExperimental FRFPhase[Degrees]Magnitude[dB]Frequency [Hz]Figure 4.20: Experimental and reconstructed FRF of the open loop transfer function (Ge(s),see Equation (4.69)).654.6. Validation of the Identified Model for the Flexible Machine Tool4.6 Validation of the Identified Model for the FlexibleMachine ToolTo validate the identified mathematical model for the flexible machine tool, its dynamics aresimulated in control loop and the simulation and experimental results are compared. An Adap-tive Sliding Mode Controller (ASMC) is implemented to control feed drive motion along areference trajectory. The sliding mode controller is designed based on rigid body dynamics(Gr(s) in Equation 4.68) an has the following structure [58]:usmc = Ks? +me [x?r +? (x?r? x?T )]+bex?T + d? (4.70)me =Jekaktrg; be =BekaktrgSliding Surface: ? = (x?r? x?T )+? (xr? x)Disturbance Observer: d? = ?? t0??dt? =???????????????0(d? < d?&? < 0)0(d? > d+&? > 0)1 OtherwiseIn Equation 4.70, x?T indicates the table velocity measured by the tacho-generator coupledto the motor shaft, x is the table position measured by the linear encoder and xr represents thereference position commanded from the trajectory generation module in CNC. The nonlinearsliding mode controller is tuned to have a bandwidth of 30 [Hz] with parameters ? = 200[rad/s], ks = 0.1 [V/(mm/s)], ? = 30 [V/mm]. The compensated disturbance is limited tovoltage equivalent of (d?,d+) = (?0.3,+0.3) [V]. The reflected mass and viscous dampingcoefficients are calculated from the identified rigid body parameters as me = 1.5625 and be =664.6. Validation of the Identified Model for the Flexible Machine Tool5.3125, respectively. Figure 4.21 shows the block diagram of the servo loop used for thesimulation and experiments. Adaptive Sliding ModeControllerAmplifier Gain[V]Motor Constant[A] [N.m]Flexible Machine Tool[V]-+++++ ++Figure 4.21: The control loop block diagramIn experimental implementation, the reference velocity (x?r) and acceleration (x?r) are ob-tained by numerical differentiation of the reference position command (xr). The integrationterm in controller disturbance compensation is computed numerically through Euler techniquewhere the control interval (Ts) is set to 1 [ms]:x?r = 1Ts [xr (k)? xr (k?1)] x?r (k) =1Ts[x?r (k)? x?r (k?1)] (4.71)d? (k) = d? (k?1)+Ts??? (k) (4.72)The given trajectory is a back and forth motion with trapezoidal velocity profile (Figure4.22) to excite structural dynamics properly, and is designed to attain maximum accelerationand deceleration of 2 [m/s], and feedrate of 0.3 [m/s].674.6. Validation of the Identified Model for the Flexible Machine Tool050100-20002000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2000 -10000100020000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time [sec]Pos[mm]ACCVel[mm/s2 ][mm/s]Figure 4.22: Reference trajectory with trapezoidal velocity profile0 0.2 0.4 0.6 0.8 1 1.2?500005000Table Acc [mm/s2 ]0 0.2 0.4 0.6 0.8 1 1.2?0.0500.05Time [sec]Tracking Error [mm]20 40 60 80 100 120 140 160 180 2000102030Frequency [Hz]|Acc(f)|Fast Fourier Transform of the Acceleration SignalSimulationSimulationSimulationExperimentExperimentExperimentTime [sec]Figure 4.23: Comparison of simulated and experimental machine acceleration684.7. Structural Dynamics in Finite Element ModelsFigure 4.23, compares the simulated and experimental table acceleration measured by thelinear encoder. The second plot in Figure 4.23 presents the FFT transform of the measuredacceleration in the low frequency range of (1-200 [Hz]). As it shows, with the medium band-width adaptive sliding mode controller (? = 200 [rad/s]), the supplied trajectory excites thelow frequency modes (49, 61, and 97 [Hz]) and the ball-screw axial mode (97 [Hz]) is moreflexible than column bending and torsion modes (49 and 61 [Hz]).4.7 Structural Dynamics in Finite Element ModelsFinite element models are proven to be quite efficient in analyzing structural dynamics aheadof physical production for the purpose of improving the designed machine structure. In thissection, based on a Finite Element model developed in ANSYS? software for the same ma-chine [57], the simulated FRF between motor torque and table displacement measured by linearencoder is extracted and compared with experimental results presented in Section 4.5.2.The finite element model does not include the conversion of motor torque to axial forcebetween the machine table and the nut. This axial force which is the main source of vibrationsin real machine, is imitated in FE by exerting an external axial force in negative direction at theball-screw where the nut is mounted. The FRFs between this axial force and axial vibrationat the nodes where encoder scale (node A in Figure 4.24) and its optical head (node B inFigure 4.24) are mounted in the real machine are subtracted. The contribution of structuralvibration in the frequency response function between the torque applied at the ball-screw andthe table vibration captured by the linear encoder can be approximated by multiplying theresultant FRF by the transfer function between motor torque and axial force (1/rg). This FRFwhich is depicted in Figure 4.25 reveals the existence two low frequency modes around 39 [Hz]and 130 [Hz], which are associated with the column bending and ball-screw axial flexibilities,respectively, and lots of higher frequency modes.694.7. Structural Dynamics in Finite Element ModelsNode A(Where Encoder ScaleIs Mounted)Node B(Where Encoder Optical Head Is Mounted)Figure 4.24: Finite Element model of the column bending mode for the machine tool withstructural flexibilityAlthough the FRF obtained by finite element modeling can successfully predict existenceof the column bending mode and dominancy of the ball-screw axial mode in machine tableresponse, there is still a large discrepancy between the simulated and experimental FRFs. Thediscrepancies can be seen in frequency and amplitude of the predicted modes; however, theyare acceptable as long as the simulated and experimental mode shapes are the same. The mainreason for the observed differences is that a typical machine tool includes many types of joints(bolted connections, connections between the guide-block and rail, between ball-screw andnut, and the bearing supports), each with different characteristics that have different effects onthe overall machine dynamic response. Therefore, if an accurate FRF prediction is sought, thejoints? dynamics effects also must be taken into account [59]. Moreover, in developing the704.7. Structural Dynamics in Finite Element Modelsfinite element model, it is assumed that all the modes have the same damping ratio (?i) of 5%,however, experimental results show that different modes manifest different damping ratios.0 50 100 150 200 250 300 350 400 450 50000.51.01.52.02.53.0 x 10-6 0 50 100 150 200 250 300 350 400 450 500 -400 -300 -200 -1000100200Phase[Degrees]Magnitude[m/N.m]Frequency [Hz]Fitted FRFFitted FRFSimulated FRF (FEM)Simulated FRF (FEM)Figure 4.25: Comparison of Fitted and Simulated FRF of structural dynamics obtained fromFEM (G?f (s), see Equation 4.75)Following a similar procedure presented in the modeling section (see Section 3.3.3) withonly considering the column bending and the ball-screw axial modes, the transfer functionbetween motor torque and table position capture by linear encoder has the following structure:Ge(s) =x(s)Tm(s)=rgs(Jes+Be)? ?? ?Gr(s)?K f?2i=1(s2 +ais+bi)?2i=1(s2 +2?i?nis+?2ni)? ?? ?G f (s)(4.73)Because the column torsion mode is eliminated, the numerator and denominator of G f (s)are of 4th order. As stated before, the developed finite element model is not capable of mod-eling the rigid body motion of the machine table; therefore, the FRF depicted with blue colorin Figure 4.25 only represents the contribution of structural vibrations on the machine table714.8. Summaryresponse. The overall machine table response including the rigid body and structural dynam-ics can be obtained by superposing the structural vibrations (G?f (s)) on top of the rigid bodymotion as (Gr(s)) and as:Ge (s) = Gr (s)+G?f (s) (4.74)Since G?f (s) represents the dynamics of two modes (column bending and ball-screw axialmodes), in summation form, we have:G?f (s)[ mN.m]=r1s2 +2?1?n1s+?2n1+r2s2 +2?2?n2s+?2n2=K f(s2 +?s+?)?2i=1(s2 +2?i?nis+?2ni) (4.75)Table 4.3: Identified parameters of G?f (s) [m/N.m](see Equation (4.75)).K f =?0.3075 ? = 24.8 ? = 6.038?104?1 = 0.05 ?n1 = 249.9 ?2 = 0.08 ?n2 = 817.09Table 4.3 summarizes the parameters of G?f (s) identified by following a similar procedurepresented in Section 4.3. Comparison of the fitted FRF (G?f ( j?)) with the simulated FRFobtained from finite element modeling is shown in Figure 4.25.4.8 SummaryIn the ball-screw test bed, the direct table frequency response to disturbance force and its in-direct frequency response to the motor torque are measured by conducting impact test andsweeping sinusoidal voltage supplied to the motor, respectively. Comparison of the experi-mental FRFs reveal that the sine sweeping technique leads to more accurate measurement ofthe natural frequency and the damping ratio of the ball-screw torsional mode than hammer test.724.8. SummaryThe feed drive rigid body dynamics is identified by ULS technique and its structural dynamicsis estimated by Peak Picking algorithm and curve fitting in frequency domain.For the vertical machining center, the experimental FRF showed many low frequencymodes among which the column bending and torsion and the ball-screw axial mode were dom-inant. The machine table dynamics is identified in a similar way to the ball-screw test bed.However, proximity of the modes required more advanced techniques for estimation of modalparameter; therefore, Rational Fraction Polynomials was utilized. The identified mathematicalmodel is validated by simulating feed drive dynamics in control loop and comparing the resultswith experiment.At the end, the simulated FRF obtained from finite element modeling of the machine struc-ture was compared with experimentally measured FRF. Despite the observed discrepancies, itwas shown that the finite element model can efficiently predict the machine mode shapes anddominancy of the ball-screw axial mode in the machine table response.73Chapter 5Drive-Based Vibration Reduction5.1 OverviewThis chapter addresses the effect of structural vibrations on the control loop of CNC machinetools. The mathematical model of the flexible machine tool, identified in Chapter 4, is usedin Section 5.2 to simulate the performance of the servo loop with a high bandwidth slidingmode controller. Section 5.3 is focused on modification of the controller structure by using themachine table?s acceleration in the feedback loop to attenuate the undesired effects of structuraldynamics. Efficiency of the active vibration control technique is validated experimentally inSection 5.5 followed by the summary of the chapter in Section 5.6.5.2 Structural Dynamics in Control LoopDesigning high bandwidth controllers is essential for accurate positioning of feed drives, anddeveloping new techniques for smooth trajectory generation mitigates the excitation of flexibledynamics in machine tools. In this section, the identified mathematical model of the verticalmachining center (Ge(s) in Equation (4.69)) is simulated in a control loop to evaluate theeffect of structural dynamics on feed drive motion when rapid positioning is demanded. Forthis purpose the Adaptive Sliding Mode Controller (ASMC) introduced in Section 4.6 is tunedto have a high bandwidth (~80 [Hz]) with parameters ? = 600 [rad/s], ks = 0.3 [V/(mm/s)] and? = 50 [V/mm]. The disturbance is limited to voltage equivalent of (d+,d?) = (+0.7,?0.7)745.2. Structural Dynamics in Control Loop[V] and the reflected mass and viscous damping coefficients (me and be ) are the same asSection 4.6.usmc = Ks? +me [x?r +? (x?r? x?T )]+bex?T + d?where ? , me, be, and d? are introduced in Equation 4.70. The trajectory generation of the CNCis set to cubic acceleration with a saturation limit of 2 [m/s] and feedrate of 0.3 [m/s] as shownin Figure 5.1. The control interval is set to 1 [ms]. 050100-2000200-2000-10000100020000 0.2 0.4 0.6 0.8 1 1.2Time [sec]Pos[mm]0 0.2 0.4 0.6 0.8 1 1.20 0.2 0.4 0.6 0.8 1 1.2ACCVel[mm/s2 ][mm/s]Figure 5.1: Reference trajectory with cubic acceleration profileThe control system is simulated by including both the rigid body and the structural dynam-ics of the drive (Equation 4.69) to mimic the real machine tool drive controller. The simulatedand experimentally measured acceleration of the table are shown in Figure 5.2. The frequencyspectrum shows that the axial mode (97.88 [Hz]) of the ball screw is heavily excited by thehigh bandwidth controller. The tracking errors have amplitude of 50 [?m] which leave unac-ceptable vibration marks on the finish surface. In addition, the controller attempts to counteract755.2. Structural Dynamics in Control Loopthe vibrations felt by the position feedback (linear encoder), and sends high frequency controlcommands with large amplitudes which saturate the amplifier. Although the high bandwidthcontrollers are desirable for accurate tracking of multi-axis tool paths, the increased frequencycontent of the controller input excites the machine tool vibrations. The bandwidth of the drivehad to be reduced to less than 30 [Hz] (? = 200 [rad/s] ) in order to avoid excitation of thevibrations. The structural vibrations must either be avoided through shaping position com-mands ahead of the control loop, or be actively damped in the control loop if a high bandwidthcontroller is desired to be used on high feed machine tools.-2-1012 x 10340 50 100 150 200 250 300 350 400 450 5000100020003000Frequency [Hz]Fast Fourier Transform of the Acceleration Signal0 0.2 0.4 0.6 0.8 1.0 1.2 -0.1 -0.0500.050.1Time [sec]TrackingError [mm]|Acc(f)|Table Acc [mm/s2 ]SimulationExperimentSimulationSimulationExperimentExperiment0 0.2 0.4 0.6 0.8 1.0 1.2Time [sec]Figure 5.2: Comparison of simulation and experimental results for the ball screw drive con-trolled by a high bandwidth sliding mode controller without active damping.Figure 5.3 presents the simulated table acceleration when machine dynamics in controlloop is replaced by the FRF obtained based on the finite element model (Equation (4.73)). Thefast Fourier transform of the acceleration signal shows that, similar to Figure 5.2, the axial765.3. Vibration Reduction with Acceleration Feedbackmode of the ball-screw is heavily excited and caused saturation of the amplifier output current.Therefore, the model developed based on the finite element analysis of the machine structureis capable of predicting the effect of structural flexibilities in control loop ahead of physicalproduction and can be used to improve the design.-2-1012 x 1040500010000 0.1 0.0500.050.1TrackingError [mm]|Acc(f)|Table Acc [mm/s2 ]0 50 100 150 200 250 300 350 400 450 500Frequency [Hz]Fast Fourier Transform of the Signal Acceleration0 0.2 0.4 0.6 0.8 1.0 1.2Time [sec]0 0.2 0.4 0.6 0.8 1.0 1.2Time [sec]Figure 5.3: The simulated table acceleration based on the FE model - high bandwidth slidingmode controller without active damping.5.3 Vibration Reduction with Acceleration FeedbackDrive based vibration reduction proposed by Dietmair and Verl [44] is an effective, and at thesame time low cost method of increasing damping ratio for a specific mode in production ma-chines and robotic arms. This method increases the damping ratio by modifying the velocityloop, when there is a cascade control structure where position and velocity loops can be dis-tinguished. Section 5.3.1 provides an overview of the theoretical basis of using acceleration775.3. Vibration Reduction with Acceleration Feedbackfeedback for active vibration reduction for the cascade control structure. Section 5.3.2 extendsthis approach to the case of using nonlinear sliding mode controller in servo loop.5.3.1 Cascade Control StructureA thin bar, used as a single degree of freedom (SDOF) oscillator, which is mounted on a ball-screw drive?s table is shown in Figure 5.4.SDOF OscillatorMachine TableMotorRigid Ball-ScrewzxCouplingAFigure 5.4: A single degree of freedom oscillator mounted on the ball-screw table.This bar exhibits bending flexibilities and its dynamics can be expressed as:xAx'?2ns2 +2??ns+?2n(5.1)where xA represents the vibrations at the oscillator?s tip point and x indicates the ball-screw?stable displacement. The inertial forces generated by the rapid positioning of the machine tablecause the flexible beam to vibrate. Drive based vibration reduction concerns the algorithmswhich enhance the damping ratio of a target mode and lead to faster vibration attenuation ofthe flexible beam.Figure 5.5 presents the control loop block diagram of the ball-screw drive when a cascadecontrol law is implemented.785.3. Vibration Reduction with Acceleration FeedbackPosition LoopPosition ControllerVelocityControllerServo DriveIndirect Velocity LoopSDOF OscillatorDirect OpenVelocity LoopFigure 5.5: Cascade control structure for the ball-screw driveThe indirect velocity loop uses the motor?s angular velocity measured from a tacho-generatoror rotary encoder mounted on the motor shaft [5] to ensure a fast system response in trackingthe velocity command. The position loop uses the linear or rotary encoder position feed backfor accurate positioning of the machine table (x).VelocityControllerServo DriveIndirect Velocity LoopSDOF OscillatorDirect Velocity LoopFigure 5.6: Velocity loop of ball screw drives with single degree of freedom oscillator andactive dampingAn easy to implement approach of suppressing the oscillator?s vibrations, is to include itsdynamics in the control loop of the ball-screw drive; for this purpose, 90? phase shift withunity gain at the natural frequency (?0) is made in the direct velocity loop as shown in Figure795.3. Vibration Reduction with Acceleration Feedback5.6. The direct velocity loop uses the velocity measured at the oscillator?s tip point (x?A) forfeedback. This velocity can be measured by using a laser vibrometer or an accelerometerattached to the point A in Figure 5.4.In considering the dynamics of the SDOF oscillator, provided that its vibrations occur ata frequency (?0) within the bandwidth of the indirect velocity loop, this loop can be approxi-mated by its DC gain (Kv) as shown in Figure 5.7.SDOF OscillatorDirect Velocity LoopFigure 5.7: Velocity loop of ball screw drives with single degree of freedom oscillator andactive damping - The indirect velocity loop is replaced by its DC gain (Kv)The closed loop transfer function of the direct velocity loop (Gv(s)) obtained from Figure5.7 is:Gv(s) =x?A(s)x?r(s)=Kv?2ns2 +2(? +Kv/2)?ns+?2n(5.2)Equation 5.2 reveals that the damping property is enhanced. For systems with multiple modes,an active damping network must be designed in which each mode is isolated by cascading aband-pass filter prior to the phase-shifter, summation of the phase shifter outputs builds thefeedback to the velocity loop. The bandpass filters limit the effect of the phase compensatorwithin a frequency range.805.3. Vibration Reduction with Acceleration Feedback5.3.2 Adaptive Sliding Mode ControllerThe simulation and experimental results presented in Section 5.2 showed that, although asmooth trajectory with cubic acceleration profile is supplied to the servo loop, utilizing a highbandwidth controller causes inertial forces to excite the structural flexibilities. The vibrationsinduced by the inertial forces lead to heavy control effort which saturates the servo loop?s am-plifier. To solve the saturation problem and reduce these vibrations, the table?s accelerationmeasured by taking double derivative of the linear encoder signal is used in the direct velocityloop to increase the damping ratio of the problematic mode (Axial mode of the ball-screw).The controller used in Section 5.2 is an adaptive sliding mode controllers which, due tothe adaptation in the disturbance compensation term (d?), has a nonlinear control structure.Therefore, in the first step, the controller must be restructured in a way that it is possible todistinguish the direct velocity loop, the indirect velocity loop and the position control loop.For this purpose, by considering ? in Equation 4.70 as a constant, the control law is linearizedas:usmc = Ks [(x?r? x?T )+? (xr? x)]+me [x?r +? (x?r? x?T )]+bex?T +??[(xr? x)+?? t0(xr? x)dt]=kv[(x?r? x?T )+ kp [xr? x]+ k f f x?r + kI? t0(xr? x)dt]+bex?T (5.3)wherekv = (Ks +me? ) ; kp =Ks? +??Ks +me?; k f f =meKs +me?; kI =???Ks +me?(5.4)Figure 5.8 depicts the closed loop block diagram with the linearized adaptive sliding modecontroller from which it is possible to extract the direct and indirect velocity loops and the po-sition loop. As this figure shows, the control law in the indirect velocity loop is a proportionalcontroller of gain kv which is independent of ? .815.3. Vibration Reduction with Acceleration FeedbackIndirect Velocity LoopOpen Direct Velocity LoopPosition LoopTable Velocity Capturedby the Tachometer Table Position Measuredby the Linear EncoderFigure 5.8: Control loop for the linearized adaptive sliding mode controllerThe velocity loop of the adaptive sliding mode controller has been modified to include adamping network as presented in Figure 5.9.Indirect Velocity LoopDirect Velocity LoopFigure 5.9: Velocity loop of the linearized adaptive sliding mode controller with active damp-ing network included in the direct velocity loopSimilar to the SDOF oscillator, the phase shifter is a signal differentiator which is normal-ized at the natural frequency of the ball-screw axial mode (?n = 97 [Hz]). In this case, althoughthe flexible machine tool has several modes, the damping network includes no bandpass filters,because the modes are closely spaced and incorporating band-pass filters for each mode leadsto sharp phase shifts in the phase plot; these sharp phase shifts make the control loop highlysusceptible to non-modeled dynamics and non-linearities in the system, which renders the in-effectiveness of the active damping network. Therefore, the most flexible mode felt by the825.3. Vibration Reduction with Acceleration Feedbacklinear encoder, which is the ball-screw axial mode, is considered for vibration reduction basedon the servo drive.100 101 102-20-15-10-50510Magnitude [dB]100 101 102 -200 -150 -100 -500Frequency [Hz]Phase [Degrees]-3 [dB]103 [Hz]Closed Direct Velocity LoopOpen Direct Velocity LoopOpen Direct Velocity LoopIndirect Velocity LoopClosed Direct Velocity LoopFigure 5.10: Bode plots of different velocity loops100 101 102?15?10?50510Frequency [Hz]Magnitude [dB]100 101 102?150?100?50050Frequency [Hz]Phase [Degrees]AD OFFAD OFFAD ONAD ONFigure 5.11: Bode plots of the position loop with and without active damping835.3. Vibration Reduction with Acceleration FeedbackBode plots of the indirect and direct velocity loops for the adaptive sliding mode controllerare compared in Figure 5.10. Figure 5.11 compares the bode plots for the position loop withand without active damping.Table 5.1: Comparison of damping ratios before and after active dampingMode 1 (49 [Hz]) Mode 2 (61 [Hz]) Mode 3 (97 [Hz])Without AD ?1 = 0.083 ?2 = 0.060 ?3 = 0.091With AD ? ?1 = 0.10 ??2 = 0.07 ??3 = 0.17Percentage of Increase 20.48% 16.67% 86.61%The frequency of the ball-screw axial mode is around 97 [Hz] which lies within the band-width of the indirect velocity loop (~103 [Hz]); therefore, modifying the control loop must beefficient for drive-based vibration reduction, and as Figure 5.10 shows the damping ratio ofthe axial mode of the ball-screw drive is increased by adding the damping network. As Table5.1 indicates, the damping ratios for all three modes are increased, even though the dampingnetwork was designed based on the ball-screw?s vibrations only.The phase shifter included in the direct velocity loop is an s transfer function which takesderivative of the velocity signal and gives the acceleration as an output. Therefore, instead ofusing the measured velocity and the phase shifter, it is possible to normalize the accelerationsignal at the natural frequency (?n) in the augmented feedback loop. The table?s accelera-tion can be estimated from the linear encoder output. However, taking double derivative ofthe encoder output introduces a heavy noise to the acceleration signal. This noisy signal isnot appropriate for the feedback loop, it can saturate the amplifier and overheat the motor.Employing a low pass filter is the initial attempt to minimize the noise but it adds delay tothe feedback loop which deteriorates the active damping network performance. Designing aKalman filter based on the identified mathematical model is beneficial in removing the noisewithout sacrificing the vibration control performance.845.4. Kalman Filter Design5.4 Kalman Filter DesignThe Kalman Filter algorithm uses the identified model (Equation 4.69) to estimate the states ofa linear system based on the measurements which contain sensor noise and other inaccuracies.Ge(s) =x(s)Tm(s)=rgs(Jes+Be)?K f?3i=1(s2 +ais+bi)?3i=1(s2 +2?i?nis+?2ni)The state space representation of the identified mathematical model in discrete time domainobtained by using the Control Toolbox in MATLAB? software is:?????z(k+1) = Adz(k)+Bdu(k)y(k) =Cdz(k)+Ddu(k)(5.5)where u(k) is control input to the servo amplifier. The output vector (y(k) = [x(k) x?(k)]T )contains measured table position (x) and acceleration (x?). The matrices Ad , Bd , Cd and Dd forthe experimental machine are derived as:Ad =??????????????????????0.5311 ?0.5523 ?0.2160 ?0.3436 ?0.1119 ?0.2497 ?0.0263 00.8239 0.7016 ?0.1015 ?0.1900 ?0.0556 ?0.1403 ?0.0148 00.2316 0.4599 0.9836 ?0.0335 ?0.0093 ?0.0249 ?0.0026 00.0411 0.1243 0.5100 0.9956 ?0.0012 ?0.0033 ?0.0003 00.0027 0.108 0.0654 0.2558 1 ?0.0002 0 00.0001 0.0007 0.0056 0.0328 0.2560 1 0 00 0 0 0.0003 0.0041 0.0320 1 00 0 0 0 0 0 0.001 1??????????????????????855.4. Kalman Filter DesignBd =[0.0064 0.0036 0.0006 0.0001 0 0 0 0]T, Dd =???00.303???Cd =???0 0 0 0.0001 0.0001 0.0004 0.0012 0.74157.9394 13.2433 8.2565 13.0299 5.8278 11.9945 ?1.2357 0???The control input (u(k)) is given to the servo drive as u?(k) after being quantized by D/Aconvertor with the resolution of ?u. The table position (x(k)) is also measured as xm afterbeing quantized by the linear encoder with resolution of ?x. The estimated acceleration x?m (k)is different from the actual table acceleration (x?(k)) because of the quantization error in positionmeasurement (x?). ???????????????u?(k) = u(k)+ u?(k)xm (k) = x(k)+ x?(k)x?m (k) = x?(k)+ ??x(k)(5.6)In experimental implementation, Equation 5.5 can be written in a more accurate way as:???????????z(k+1) = Adz(k)+Bdu(k)+Bd u?(k)???xm (k)x?m (k)???=Cdz(k)+Ddu(k)+Dd u?(k)+???x?(k)??x(k)???(5.7)In Kalman Filter design, u? is called the process noise which is introduced at the systeminput and???x?(k)??x(k)??? is called measurement noise which happens at system output and is causedby poor measurement sensors. The process noise (u?(k)) lies in (??u/2, ?u/2) and the positionquantization error (x?(k)) is in the range of (??x/2, ?x/2). They are assumed to have uniformdistributions with zero mean values [30].865.4. Kalman Filter DesignE (u?(k)) = 0, E (x?(k)) = 0 (5.8)Ru =(?u)212, Rx =(?x)212(5.9)The voltage output of the 16 bits D/A convertor is in the range of (?10,+10) [V], hence itsresolution is ?u = 20/216 = 0.305176?10?3 [V]. The linear encoder resolution is ?x = 10?6[m]. Therefore, from Equations 5.9, the quantization error covariance values are calculated as:Ru = 7.7610?10?9 [V 2]; Rx = 8.3333?10?8 [mm2](5.10)To reach a satisfactory performance for the Kalman Filter, Rx? is tuned to be 50? 10?3[mm2/s2]. The process noise (Q) and measurement noise (R) covariance matrices are definedas:Q = Ru; R =???Rx 00 Rx???? (5.11)The Kalman filter, composed of the state estimator (z?(k+1|k)) and filtered position andacceleration (y?(k|k) = [x?(k) ??x(k)]T ), has the following structure [60]:?????z?(k+1|k) = Ad z?(k|k?1)+Bdu(k)+L(y(k)?Cd z?(k|k?1)?Ddu(k))y?(k|k) =Cd (I?MCd) z?(k|k?1)+(I?CdM)Ddu(k)+CdMy(k)(5.12)The gain matrix (L) and innovation gain matrix (M) are calculated in terms of the noise covari-ance data (Q , R) as:L =(APCT +DdQBTd)(CPCT +R+BdQBTd)?1(5.13)M = PCT(CPCT +R+BdQBTd)?1(5.14)875.5. Experimental Implementationwhere P is solution to the corresponding algebraic Riccati equation. Having the covariancematrices of the process noise and measurement noise computed off-line, the L and M matricesare evaluated for the experimental machine as follows:L =???0.0035 0.0010 0.0055 0.0001 ?0.0041 ?0.0188 0.1787 0.02170 0 0 0 0 0 0 0???TM =???0.0021 ?0.0039 ?0.0042 0.0026 ?0.0045 ?0.0177 0.1793 0.02150 0 0 0 0 0 0 0???T5.5 Experimental ImplementationThe adaptive sliding mode controller has been modified to include a damping network as pre-sented in Figure 5.12. The noise in the estimation of acceleration from double digital differen-tiation of the encoder signal is minimized by the Kalman Filter designed in Section 5.4.The inertial vibrations are actively damped by the following acceleration feedback addedto sliding mode controller command:u(k) = usmc (k)? kv??x(k?1)?n(5.15)where kv is the gain of indirect velocity loop of adaptive sliding mode controller. The slidingmode controller parameters were set the same as in the case without active damping network(see Section 5.2), and the active damping is set to damp the axial mode (?n= 611 [rad/s] =97.88 [Hz]) of the ball screw. The servo performances with and without active damping arecompared in Figure 5.13. The vibration mode is completely damped out, and the amplitudesof tracking error oscillations (i.e. vibrations) are reduced from 50 [?m] to 5 [?m].As Figure 5.14 shows, when active damping is applied the saturation problem with the885.5. Experimental Implementationcontroller is solved.KalmanFilterAdaptive Sliding Mode Controller [m][m/s]Flexible Machine ToolAmplifier Gain Motor Constant[V] [A] [N.m]Feed-back Loop for Active Damping[m/s] [m/s2][m/s2][V]Figure 5.12: Block diagram of sliding mode controller with and without active damping ofvibration mode with a natural frequency ?n895.5. Experimental Implementation-2-1012 x 104Time [sec]0 50 100 150 200 250 300 350 400 450 5000300020001000Frequency [Hz]0 0.2 0.4 0.6 0.8 1 1.2 -0.1 -0.0500.050.1Time [sec]Tracking Error [mm]|Acc(f)|TableACC[mm/s2 ]AD OnAD Off0 0.2 0.4 0.6 0.8 1 1.2Fast Fourier Transform of the Acceleration SignalFigure 5.13: Comparison of experimental results for the machine tool feed drive with andwithout active damping (AD)0 0.5 1 1.5 2 2.5051015Time [Sec]Control Signal [V]Saturation Limits = AD OFFAD ONFigure 5.14: Comparison of the control command (u) with and without active damping (AD)905.6. Summary5.6 SummaryThe simulation and experimental results showed that although a smooth trajectory was com-manded to the servo loop, structural modes are excited. The residual vibrations, which weresensible through the feedback sensors, resulted in saturation of the servo amplifier and cease-less vibration of the machine tool structure. It was shown that using the table?s acceleration,which is normalized at the natural frequency of the ball-screw axial mode, increased the damp-ing property of the problematic mode and caused faster attenuation of the structural vibrations.This acceleration was estimated by passing the double derivative of the linear encoder sig-nal through a Kalman filter. Experimental result confirmed the efficiency of active vibrationcontrol technique in stabilizing the control loop and solving the saturation problem.91Chapter 6Conclusion and Future WorkIn this thesis, a mathematical model of the horizontal feed drive (x-axis) of a C-frame CNCmachining center has been developed. The mathematical model has been intended to explainthe interaction between the control loop and structural dynamics. The model includes thefundamental low frequency modes of the machine tool column and ball screw drive to findthe transfer function between the motor torque applied at the ball-screw and table positionmeasured by the linear encoder. In the modeling procedure, the ball-screw axial and columnbending and torsional modes have been represented by lumped inertias connected with lin-ear and torsional springs and dampers. These modes are excited by inertial forces generatedthrough acceleration and deceleration of large machine tool components.The mathematical model was identified by fitting the obtained transfer function to the ex-perimental FRF, which is measured by sine sweeping technique, in frequency domain. Theexperimental FRF showed that although the column bending mode is dominant in the ma-chine table response, the ball-screw?s axial mode gains dominancy when the relative vibrationbetween the machine table and the bed, which is the linear encoder output, is of interest.By utilizing the identified model, simulations were conducted to analyze the interactionbetween a high bandwidth adaptive sliding mode controller and the structural dynamics. Thesimulation results , which were verified by experiments, showed that the axial mode of the ballscrew was heavily excited by the high bandwidth controller, although a smooth trajectory wassupplied to the servo loop. The controller attempted to counteract against the vibrations felt bythe linear encoder; therefore, it sent high frequency control commands with large amplitudes92Chapter 6. Conclusion and Future Workwhich saturated the amplifier. Saturation of the amplifier led to incessant vibration of themachine table and drastically increased the tracking error.A similar procedure was applied on the simulated FRF which was obtained from the finiteelement modeling of the machine tool structure. The FE model could predict structural modeshapes and dominancy of the ball-screw axial mode in machine table response which causedsaturation of the control signal.Considering the experimental setup, a Kalman Filter was used to estimate the accelerationof the machine table from double digital differentiation of the encoder signal. The measuredacceleration was added in the feedback loop for active control of the undesired vibrations. Themodified control loop with enhanced damping resulted in faster attenuation of the ball-screwvibrations, stabilized the servo loop and solved the saturation problem.In conclusion:1. Structural flexibilities not only affect the surface finish of the final product, they alsointerfere with the control loop provided that the feedback sensors can feel the vibrations.The sensed vibrations can impose heavy control efforts to the control system whichsaturates the actuators in the servo-loop and destabilizes the closed loop system.2. FE softwares are proven to be efficient in predicting the role of structural dynamics oncutting process in machine tools. They are typically used for analysis and topologyoptimization of the machine tool structure. They are also beneficial in modeling thecontrol-structure interaction and can help control engineers in choosing the best locationsfor the feedback sensors and improve their design.3. Active control of machine tool?s vibrations by designing an active damping network in-creases the damping property of the structural modes significantly. It helps in enhancingthe servo loop bandwidth, and improves the closed loop performance. Design of theactive damping network can be performed independently from the controller design and93Chapter 6. Conclusion and Future Workit is only beneficial when the vibrations are captured by the feedback sensors. However,for multi-mode systems with closely spaced modes, design of a proper active dampingnetwork is cumbersome.4. The increased damping due to vibration control can increase the minimum depth ofcut for a stable process in stability lobes diagram. Therefore, the vibration control canincrease the productivity significantly.This research can be continued to improve damping of the machine tools. A mathematicalmodel which includes structural dynamics can be used for Hardware in Loop Simulations(HiLS) of machine dynamics to visually depict the machine mode shapes during the cuttingprocess and the surface quality of the finished product. The Active vibration control techniqueimplemented in this thesis can be extended to the case of three or five axis CNC machines todamp structural vibrations and improve the tracking and contouring performances; moreover,it can be combined with input shaping to minimize the excitation of structural flexibilities. Fi-nally, It is essential to investigate the effect of active damping on the cutting process; therefore,future work can be concentrated on evaluating the effect of feed drives? vibration control onincreasing the stiffness and damping in the FRF between tool-tip and table top. The increaseddamping property can enhance the productivity of cutting process by increasing the absoluteminimum depth of cut for a stable process.94Bibliography[1] J. Vesely and M. Sulitka, ?Machine tool virtual model,? Modern Machinery (MM) ScienceJournal, pp. 146?151, 2009.[2] D. Ford, ?A CAD-CAE aid to design of a machine tool system,? 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(A.2)As presented in equation (4.7), the zero order hold equivalent of the above state-spacerepresentation can be obtained as follows:???xt(k+1)vt(k+1)???= Ad???xt(k)vt(k)???+Bd???u(k)d(k)???where Ad = eAcTs , Bd =Ts?0eAc?d? .Bc (A.3)103Appendix A. ZOH Equivalent of The T.F in Equation 4.6In the above equation, AcTs =???0 Ts0 QvTs???. To find eAcTs , first we need to calculate eigenvaluesof AcTs. Therefore, we need to solve equation below:det(AcTs?? .I2?2) = 0? det????? Ts0 QvTs?????= 0 (A.4)?? (QvTs?? ) = 0? ?1 = 0, ?2 = QvTs (A.5)The next step is to find the eignevectors associated with each eigenvalue. Therefore, for?1 = 0 we have:AcTs???a1b1???= ?1???a1b1???=???00???????0 Ts0 QvTs??????a1b1???=???00??? (A.6)????????b1Ts = 0b1PvTs = 0? b1 = 0 (A.7)From equation (A.7) we notice that b1must be zero and a1 can have any value. In this casewe select a1 = 1. Hence, the eignevector associated with ?1 = 0 is v1 =???10???. For ?2 = PV Tswe have:AcTs???a1b1???= ?2???a2b2???=???QvTs a2QvTs b2???????0 Ts0 QvTs??????a2b2???=???QvTs a2QvTs b2??? (A.8)????????b2Ts = QvTs a2b2QvTs = QvTs b2? b2 = Qv a2 (A.9)104Appendix A. ZOH Equivalent of The T.F in Equation 4.6If we choose a2 = 1, the associated eignevector for ?2 = QvTs will be v2 =???1Qv???. Havingeigen values and eigenvectors:eAcTs =??1eDAcTs?where ?=[v1 v2], DAcTs =????1 00 ?2??? , eDAcTs =???e?1 00 e?2??? (A.10)Following the procedures presented above, we have:Ad =???1 ?eQvTS0 eQvTs??? (A.11)Having Ad , from equation (A.3), Bd can be obtained as follows:Bd =Ts?0eAc?d? .Bc =???Ts ? 1QV(eQvTS?1)0 1QV(eQvTs?1)??????0 0Pv ?Pv???? Bd =???? PvQv(eQvTS?1) PvQv(eQvTS?1)PvQv(eQvTS?1)? PvQv(eQvTS?1)??? (A.12)105Appendix BOrthogonal PolynomialsB.1 Construction of Orthogonal PolynomialsThe simplified Fosythe method presented by Richardson and Formenti [51] to construct thecomplex orthogonal polynomials (?i,k and ?i,k) used in modal parameter estimation throughRFP method are summarized in Equations (B.1) to (B.5):?i,k = ( j)kRi,k (B.1)Ri,?1 = 0;Ri,0 =1D0;Ri,k =Si,kDk(B.2)Si,k = ?iRi,k?1?Vk?1Ri,k?2;k = 2,3, . . . ; i =?L, . . . ,?1,1, . . . ,L (B.3)V0 = 0; Vk?1 =L?i=1?iRi,k?1Ri,k?2qi (B.4)D0 =????L?i=?Lqi; Dk =????L?i=?L(si,k)2qi (B.5)qi is the weighting function and equals to hi for ?i,k. To obtain the orthogonal polynomialsassociated with the numerator (?i,k), the weighting function (qi) must be chosen equal to 1 forall frequencies in Equations (B.2) to (B.5).106B.2. Proof of Equation 4.51B.2 Proof of Equation 4.51The transfer function used in modal parameter estimation through RFP method which is ex-pressed in terms of orthogonal polynomials in Equation 4.42 has the following characteristicequation:Ch.Eq:n?k=0dk?i,k (B.6)Replacing ?i,k by ( j)k Ri,k from Equation (B.1) and expanding Equation (B.6) we have:n?k=0dk?i,k = d0 ( j)0 Ri,0? ?? ?P0( j?i)+d1 ( j)1 Ri,1? ?? ?P1( j?i)+ ? ? ?+dn?1 ( j)n?1 Ri,n?1? ?? ?Pn?1( j?i)+dn ( j)n Ri,n? ?? ?Pn( j?i)(B.7)where Pn ( j?) is the nth order complex orthogonal polynomial. From Equation (B.7), bk, whichis the coefficient of sk in ordinary expression of characteristic equation, can be expressed interms of pik which is the coefficient of sk in the orthogonal polynomial of ith order when it isconsidered in its ordinary form (written in terms of s, s2, s3,. . ., si) as:bn = dn pnnbn?1 = dn pnn?1 +dn?1 pn?1n?1bn?2 = dn pnn?2 +dn?1 pn?1n?2 +dn?2 pn?2n?2...b1 = dn pn1 +dn?1 pn?11 + ? ? ?+d2 p21 +d1 p11b0 = dn pn0 +dn?1 pn?10 + ? ? ?+d1 p10 +d0 p00(B.8)=? bk =m?i=kdi pik (B.9)Considering Equations (B.7), (B.2), and (B.3), we have:107B.2. Proof of Equation 4.51P0 ( j?i) = ( j)0 Ri,0 = 1D0P1 ( j?i) = ( j)1 Ri,1 = ( j)1[?iRi,0D1]= ( j?i)D1[( j)0 R0 ( j?i)]= ( j?i)D1 [P0 ( j?i)]P2 ( j?i) = ( j)2 Ri,2 = ( j)2[?iRi,1?V1Ri,0D2]= ( j?i)D2[( j)1 Ri,1 +V1 ( j)0 Ri,0]= ? ? ?? ? ?= ( j?i)D2 [P1 ( j?i)+V1P0 ( j?i)]P3 ( j?i) = ( j)3 Ri,3 = ( j)3[?iRi,2?V2Ri,1D3]= ( j?i)D3[( j)2 Ri,2 +V2 ( j)1 Ri,1]= ? ? ?? ? ?= ( j?i)D3 [P2 ( j?i)+V2P1 ( j?i)]...Pn ( j?i) = ( j)n Ri,n = ( j)n[?iRi,n?1?Vn?1Ri,n?2D3]= ? ? ?? ? ?= ( j?i)Dn[( j)n?1 Ri,n?1 +V2 ( j)n?2 Ri,n?2]= ( j?i)Dn [Pn?1 ( j?i)+Vn?1Pn?2 ( j?i)](B.10)Replacing ( j?i) with s, the complex orthogonal polynomials (Pi ( j?)) are represented interms of Laplace variable (s) as:P0 (s) = 1D0Pi (s) = 1Di (sPi?1 (s)+Vi?1Pi?2 (s)) ; i = 1, . . . ,n(B.11)From Equation (B.11), the coefficient associated with sk in the complex orthogonal poly-nomial of ith order which is represented by pik in Equation (B.9) can be expressed as:pik =???????????????0 k < 0 or k > i1D0i = k = 01Dipi?1k?1 +Vi?1 pi?2k 0? k ? i(B.12)108
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Modeling and active damping of structural vibrations in machine tools Amir Hossein, Hadi Hosseinabadi 2013
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Title | Modeling and active damping of structural vibrations in machine tools |
Creator |
Amir Hossein, Hadi Hosseinabadi |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | Feed drives of High Speed Machine (HSM) tools deliver fast motions for rapid positioning of tool or work-piece. The inertial forces generated by acceleration and deceleration of large machine tool components excite structural modes of the machine tools and cause residual vibrations. Unless avoided, the vibrations lead to poor surface finish and instability of the drive's control loop. In this thesis, structural flexibilities are represented by linear and torsional springs and dampers to develop a mathematical model of the feed drive dynamics. The model includes the contribution of structural vibrations in measuring table position by a linear encoder. An identification algorithm is introduced to facilitate the estimation of rigid body and structural dynamics in frequency domain. The identified mathematical model is used to mimic the real machine in simulations with the purpose of analyzing the interaction between structural dynamics and a high bandwidth adaptive sliding mode controller. Meanwhile, efficiency of finite element modeling approaches in predicting this interaction prior to the physical production is investigated by replacing the machine dynamics by a FEM based model. The mathematical model is used to design a Kalman Filter which estimates the table's acceleration by taking double digital derivative of the encoder signal. The table's acceleration is used to modify the control loop to minimize the effect of undesired structural vibrations. It is shown that the vibrations can be actively damped, and the bandwidth of the drive can be increased. The increase in the servo loop bandwidth provides smoother motion and improves the tracking performance significantly. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-11-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0165661 |
URI | http://hdl.handle.net/2429/45494 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2014-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
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