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Precision turning of shafts with piezoelectric actuator tool Jun, Martin Byung-Guk 2000

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PRECISION TURNING OF SHAFTS WITH PIEZOELECTRIC ACTUATOR TOOL by Martin Byung-Guk Jun B. A . Sc. (Mechanical Engineering) University of British Columbia A THESIS SUBMITTED IN THE PARTIAL F U L F I L L M E N T OF THE REQUIREMENT FOR THE D E G R E E OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES M E C H A N I C A L ENGINEERING We accept this thesis as conforming to the required standard  The University of British Columbia January 2000 © Martin Byung-Guk Jun, 2000  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood  that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  DEPARTMENT OF M E C H A N I C A L ENGINEERING The University of British Columbia 2324 Main Mall Vancouver, Canada V6T 1Z4 Date:  CTIAIOJ  £  l^o^O  Abstract  There is an increasing demand for high tolerances and good surface finish for most industrial applications of shafts such as the seats of roller bearings or gearwheel compression joints.  A  grinding or ultra-precision machine tool is used to finish these parts in the final production sequence because the conventional turning machine cannot meet the tolerances. This thesis investigates the use of a piezoelectric fast tool servo designed for precision turning of shafts on a conventional turning machine. In doing so, the grinding operation, the final process of precision machining, can be eliminated, which leads to significant savings in time and cost. A piezoelectric fast tool servo previously designed and developed for chatter vibration suppression is used for the investigation. It is mounted on a turret of a standard C N C turning center and is modeled as a second order dynamic system. Sliding mode control or variable structure control is used to control the actuator, taking advantage of its characteristics of robustness against disturbances and parameter uncertainties. The concept of using the piezoelectric actuator to accurately position the tool on a conventional machine and the effectiveness of the control scheme are verified both in simulations and experiments. Cutting experiments are carried out to identify the cutting conditions which produce the required surface quality. The effect of feedrate, cutting speed, tool geometry, and use of cutting fluids are investigated The results show that the surface quality that grinding offers can also be attained with the help of the piezoelectric actuator and proper selection of the cutting parameters.  ii  Table of Contents  Abstract  ii  Contents  iii  List of Figures  iv  List of Tables  v  Acknowledgments  vi  Nomenclature  vii  1 Introduction  1  2 Literature Review  5  2.1  Overview  5  2.2  Piezoelectric Actuator  2.3  Control of Piezoelectric Actuator  2.4  Surface Roughness  12  2.5  Influencing Parameters  15  2.6  Summary  21  .6 8  3 Surface Roughness Model  23  3.1  Geometrical model  23  3.2  Empirical model  28  4 Sliding Mode Control  30  4.1  Introduction  30  4.2  Controller Design  32  4.3  Controller Design with Parameter Adaptation  38  iii  4.4  Summary  42  5 Modeling and Control of Piezo Tool Actuator  44  5.1  Overview  44  5.2  System Setup  44  5.2.1  Tool Positioning Unit  45  5.2.2  Measurement Unit  45  5.2.3  Signal Processing Unit  48  5.3  Modeling of Actuator Assembly  50  5.4  Sliding Mode Control of the System  53  5.4.1  Classical Sliding Mode Control  55  5.4.2  Modified Version with Parameter Adaptation  57  5.4.3  Relations to PID and Pole-Placement  60  5.5  Experimental Results of Controller  61  5.6  Summary  65  6 Experimental Results  67  6.1  Overview  67  6.2  Experimental Setup  67  6.3  Preliminary Selection of Cutting Parameters  69  6.4  Cutting Tests  73  6.4.1  Wet Cutting  73  6.4.2  Dry Cutting  82  6.5  Summary  91  7 Conclusions and Future work  7.1  93  Conclusions  93 iv  7.2  Future Work  94  Bibliography Appendix A  96 Measured data for surface  finish  v  105  List of Figures  1.1  Cutting process of precision machining [10]  4  2.1  Learning parameter estimator and one step ahead control [54]  10  2.2  Surface roughness parameters  14  2.3  Surface roughness parameters characterizing the distribution of roughness  15  2.4  Feed mark caused by feedrate and tool nose radius  16  2.5  Effect of cutting speed on surface roughness in orthogonal cutting due to different types of chip formation  17  2.6  Schematic illustration of grooving wear  20  3.1  Tool tip geometry  24  3.2  Surface roughness geometries (0 < d < d^)  25  3.3  Surface roughness geometries (d^- < d < ds)  26  4.1  Sliding surface on a phase plane  31  4.2  Sliding zone and boundary layer to avoid chattering  37  5.1  Picture of the piezoelectric actuator assembly on the turret of a standard lathe  46  5.2  Schematic diagram of the piezoelectric actuator setup  47  5.3  Toom motion by the piezoelectric actuator  48  5.4  Modeling of piezoelectric actuator assembly  50  5.5  Block diagram of the piezoelectric actuator system  51  5.6  Frequency response of the piezoelectric actuator assembly  52  5.7  Simulated open loop step response  54 vi  5.8  Block diagram of the system with sliding mode controller  57  5.9  Simulated step response of the system with sliding mode controller  58  5.10 Block diagram of the system with modified version of sliding mode controller (parameter adaptation) 61 5.11 Simulated step response of the system with sliding mode controller with parameter adaptation  62  5.12 Step response of open and closed loop systems  64  5.13 Comparison between cuts with and without control  66  6.1  Schematic diagram of workpiece used for the experiments  68  6.2  Stylus-type surface measurement instrument used  69  6.3  Effect of feedrates on surface roughness  72  6.4  The position of the tool and control signal into the piezoelectric actuator of a typical cut  74  Average values of surface roughness R and cutting results)  76  6.5  6.6  6.7  6.8  6.9  a  Rmax  versus cutting speed and feedrate (wet  Average values of surface roughness R and R versus cutting speed and feedrate (wet cutting results)  77  Surface roughness profiles of machined surfaces at different cutting speeds for wet cutting (/ = 0.03 mm / rev)  78  Surface roughness profiles of machined surfaces at different cutting speeds for wet cutting (/ = 0.045 mm / rev)  79  Surface roughness profiles of machined surfaces at different cutting speeds for wet cutting (/ = 0.06 mm / rev)  80  z  q  6.10 Verificaton of the selected cutting conditions  81  6.11 Average values of surface roughness obtained from dry cutting  84  vii  6.12 Average values o f surface roughness obtained from dry cutting  85  6.13 Surface roughness profiles o f machined surfaces at different cutting speeds for dry cutting ( / = 0.04 m m / rev)  86  6.14 Surface roughness profiles o f machined surfaces at different cutting speeds for dry cutting ( / = 0.055 m m / r e v )  87  6.15 Workpiece with rubbing mark  88  6.16 Surface roughness profiles o f machined surfaces at different cutting speeds for dry cutting using the same tool tip for all three cuts ( / = 0.05 m m / rev)  89  6.17 Wear on the tip o f the inserts  90  6.18 Chips produced from wet and dry cutting  90  viii  List of Tables  5.1  Piezoelectric actuator specification  4 5  5.2  LNS 1 8 / 6 0 sensor specifications  4  9  5.3  DSP specifications  4  9  6.1  The peak-to-valley roughness values for different feed rates obtained from E q .  7  0  (6.1) 6.2  Critical cutting speed for disapperance of built-up-edge [ 8 6 ]  7 1  6.3  The cutting conditions tested  7 2  6.4  Optimized cutting conditions  9 2  ix  Acknowledgments  I would like to express my sincere gratitude to my supervisor Dr. Yusuf Altintas for his insightful instructions, practical advices, and constant support throughout the course of my research. I would also like to thank Dr. Ismail Lazoglu, who supervised me in the absence of Dr. Altintas, for his support and understanding. I would like to give special recognition to Dr. Wen-Hong Zhu for his help with the research and for his teachings. Many thanks to my colleagues in the Manufacturing Automation Laboratory U B C for their kindness and assistance. M y deepest gratitude goes to my family for their constant supports and ceaseless prayers. Their encouragements and confidence in me have kept me up to this day. I dedicate this work to them.  "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they display knowledge."  Psalm 19:1-2  Nomenclature  a  ferrite  A  system matrix in continuous-time state space representation  a  number which determines cut-off frequency for low pass filter  B  input matrix in continuous-time state space representation  c  h, b , b ... b  distance from one intersection to the other of a peak in a surface profile  c  predetermined distance below the highest peak of a surface profile  C  matrix which determines the dynamics of sliding surfaces  c  structural damping of the piezoelectric actuator system  eff  effective damping of the piezoelectric actuator system  eff  estimate of C //  Cl  end relief angle  d  depth of cut  D  workpiece diameter  2  3  n  C  C  p  e  , disturbance vector  D  constant disturbance to the actuator  d d ,d~  lower and upper bounds of d  d  disturbance observer  dA, dB  depth of cut limits  f  feedrate  F  applied force to the piezoelectric actuator  +  F  cutting force disturbance  IOA, IA, JAB, IB  feed limits  7  austenite  d  r  G  diagonalized vector which contains the inverse of updating gains a  gain representing the amplifier and the actuator together  xi  minimum convergent rate for sliding surface feedback gain matrix vector which imposes limits on the integral control action stiffness of the piezoelectric actuator system function which imposes limits on the integral action against disturbance control interval counter in discrete time domain derivative gain derivative gain actually used in the computer program integral gain integral gain actually used in the computer program proportional gain proportional gain actually used in the computer program gain of digital to analog converter effective stiffness of the piezoelectric actuator system estimate of K /f e  end cutting edge angle or minor cutting edge angle major cutting edge angle feedback gain sampling length for a surface profile measurement parameter which determines the dynamics of the sliding surface mass of the piezoelectric actuator system effective mass of the piezoelectric actuator system estimate of M ff e  constant parameter vector estimate of the constant parameter vector P lower and upper bounds of the j constant parameter of P th  nose radius  xii  p Ra  parameter updating gain average surface roughness updating gain for j  th  Pi  parameter  Rmax  peak-to-valley surface roughness  R  root-mean-square surface roughness  q  R  ten-point height surface roughness  S  sliding surface vector  s  sliding surface  z  continuous time domain dominant closed-loop poles «3  continuous time domain closed-loop pole  Si  j  T  control period  t*  time that takes for all states to converge to the sliding surface  th  element of S  bearing length ratio at the level of the mean line bearing length ratio U  control vector  u  control input to the piezoelectric actuator  V  cutting speed  V  Lyapunov stability function  V*  critical cutting speed  V  width of normal flank wear at the nose  B  v  width of grooving wear at the nose  W(z)B  reference signal delayed by one cycle  w„  natural frequency of the piezoelectric actuator system  W (z)  reference signal  X  state vector of the system  X  position of the tool tip  0  xiii  X  d  desired vector  Xd  desired position of the tool tip  Y  regressor matrix for parameter vector P  ip  approach angle  C  damping constant of the piezoelectric actuator system  r  z  1 2  discrete time domain dominant closed-loop poles  z  3  discrete time domain closed-loop pole  xiv  AISI  American Iron and Steel Institute  BHN  Brinell Hardness Number  BUE  Built-up-edge  CNC  Computerized Numerical Control  CVD  Chemical Vapor Deposition  D/A  Digital to Analog  DSP  Digital Signal Processor  I/O  Input/Output  LNS  Laser Nano Sensor  PC  Personal Computer  PI  Physik Instrumente  RMS  Root Mean Square  SAE  Society of Automotive Engineers  SISO  Single-Input Single-Output  SMC  Sliding Mode Control  VSC  Variable Structure Control  ZPETC  Zero Phase Error Tracking Control  XV  Chapter 1 Introduction In the field of precision machining, diameter and surface quality tolerance of the workpieces imposed by industrial applications is increasing continuously. Locations of bearings, gearwheels, or pulleys on shafts, for example, bearing mounting surfaces on gas turbine shafts, are considered as a typical field of precision machining applications. Figure 1.1 shows the conventional precision machining processes in which the last stage, after roughing, semi-finishing, and finishing on a lathe, is carried out on a grinding machine. This is because conventional lathes cannot provide dimensional machining accuracy under 10 /xm due to backlash, friction in the guideways, thermal expansion, position dependent static deformation, and servo positioning errors. Typically in precision machining, the required dimensional tolerance is < 1.0 /im, and the surface quality tolerance is  R  a  <  1.0 / i m and  Rm  ax  < 2.5 uxn, where  R  a  and  Rm x a  are average and peak-to-valley surface  roughness. Nevertheless, turning is favorable over grinding in many aspects such as greater hourly production, greater savings in capital investment, less floor requirement, less tooling cost, and more environmentally friendliness [27, 6]. Thus, performing all the operations of precision machining on a single lathe would lead to significant savings in machining cost by keeping the same reference settings and fixture in one machine tool during the complete machining process. The use of a micro-positioning actuator to enhance the positioning accuracy can allow the last stage of precision machining process to be performed on a conventional lathe, thus eliminating the need for the grinding operation. One of the technologies widely used for fine motion actuators is use of piezoelectricity. A piezoelectric actuator was previously developed for active chatter vibration control  1  Chapter 1. Introduction  2  and precision turning at U B C Manufacturing Automation Laboratory [88], and it can be mounted on a standard turret of a conventional lathe. This piezoelectric actuator is used to investigate the feasibility of performing precision turning on a conventional lathe. The feed drive of the machine tool executes the coarse positioning while the piezoelectric actuator performs the fine positioning as shown in Figure 1.1. Surface finish of machined parts is also important since the quality of a machined surface plays a crucial role in its functioning, and significant research has been conducted in this area. Surface finish quality, however, mainly depends on the cutting conditions, tool geometry, and stiffness at the tool contact point. Therefore, an in-depth study and proper selection of cutting conditions is required to obtain the equivalent surface quality that grinding achieves. This thesis provides modeling of the piezoelectric actuator, development of the control law, study on the influences of cutting conditions, and identification of the optimum cutting conditions. The thesis is organized as follows: In Chapter 2, piezoelectric fast tool servos and their applications in precision machining are first reviewed, and then, control techniques for controlling the piezoelectric actuators during a cutting process are investigated.  As disturbances, vibrations, and parameter uncertainties are  associated with a cutting process, control methods provided in the literature to compensate them with the use of piezoelectric actuators are also studied. Furthermore, roughness parameters that assess the quality of a surface are reviewed. The influencing parameters that affect the quality of surface finish during a turning operation are studied, and some geometrical and empirical models to represent the surface roughness of a machined surface are provided. Chapter 3 introduces the application of sliding mode control to the piezoelectric actuator. The mathematical and analytical derivations of the sliding mode control law are presented. First, the  Chapter I.  Introduction  3  classical sliding mode control and its problems are studied; then, the modified version of the sliding mode control using adaptation methods for system parameters is presented. Chapter 4 presents a description of the piezoelectric actuator and its assembly used in this thesis, and the mathematical model of the piezoelectric actuator assembly is presented based on its physical properties. The simulation and experimental results of the piezoelectric actuator system with the selected control schemes are presented. The experimental results with the controller are also compared to the results without any control. Identification of various cutting parameters which yield the required surface finish are provided in Chapter 5. The setup of the experiments is described. The piezoelectric actuator under control is used for the precision turning of the workpiece, and the results are presented. The influence of cutting parameters with and without cutting fluids on the quality of a machined surface is discussed. Chapter 6 concludes the thesis with a summary of contributions and further research work to be done for the project.  Chapter 1. Introduction  Conventional precision maching  Precision machining with piezoelectric actuator Figure 1.1: Cutting process o f precision machining [10]  Chapter 2 Literature Review  2.1  Overview The potential of the performing finishing operation on a turning machine center and eliminat-  ing grinding processes has been of great interest to the industry and researchers, because of the significant reductions in production time and cost that would ensue. Much work has been done to increase the accuracy in turning operations by using high precision actuators. The piezoelectric actuator is one of them and it can be used in various applications. Disturbances due to cutting force, parameter uncertainties, unmodeled dynamics, and vibration are associated with a cutting process.  Therefore, control of the actuator becomes a significant  factor in obtaining desirable positioning accuracy and surface  finish.  Many researchers have  attempted different types of control laws for their actuators. Their methodologies and effectiveness are investigated in this chapter. Surface finish of a machined part is characterized by methods that assess the quality of finished products. Most surface measurement methods are based on profile assessment, and there are more than 50 such profile parameters in use. Some of the standard profile parameters commonly used in industry are introduced, and how they are defined and what kind of information about the surface they provide are reviewed. Surface finish of a machined part is influenced in varying amounts by a number of factors. These factors include feed rate, work material properties, cutting speed, temperature, depth of cut, time of cut, tool nose radius, geometry of the tool, stability of the machine tool, cutting fluids,  5  6  Chapter 2. Literature Review  etc.  In-depth study o f these parameters is essential in order to understand how they affect the  dimensional accuracy and surface quality. M a n y models have been developed in order to determine the surface roughness in terms o f influencing parameters. From a machining point o f view, it is essential that the relationship between the surface roughness and the many process variables be established so that the appropriate cutting conditions can be selected to satisfy the surface finish requirements o f the component to be machined. From the kinematics o f the tool and the workpiece, a geometrical model o f the surface finish can be obtained, and many researchers developed empirical models to represent the actual surface roughness. Both models are discussed in this chapter.  2.2  Piezoelectric Actuator  A piezoelectric material develops an electrical charge when subjected to mechanical stress, or conversely, when an electric field is applied to a piezoelectric material, it causes mechanical stress or movements, the phenomenon known as converse piezoelectric effect.  These effects allow a  piezoelectric material to be useful in many areas as a sensor or actuator.  The advantages o f a  piezoelectric material used in an actuator include [47]: • unlimited resolution and sub-micrometer positioning accuracy • extremely fast frequency response, high bandwidth, and high resonant frequency • high mechanical stiffness • compact size • no friction and no wear Even though a piezoelectric material has drawbacks such as limited stroke, non-linearity, hysteresis, and thermal variation, it provides excellent features for a high precision, sub-micrometer positioning actuator.  7  Chapter 2. Literature Review  Various fast tool servo concepts using the effect of piezoelectricity have been presented in the past. Hara et al. [40] presented a piezo driven fast tool servo with a bandwidth of 2 kHz . The fast tool servo was developed in order to detect the initial contact between the tool and the cutting surface with an accuracy of ± 0.1 over a whole disk.  and make a groove of uniform and submicrometer depth  The micro cutting device consisted of a pair of parallel springs, two piezo  elements and a diamond tool. One piezo element was used for driving the tool and the other for contact detection. The piezo element expanded to 12 /j,m with an applied voltage of 100 V, but the cutting tool expanded only 3.7 /xm due to the reaction of the parallel springs. The stiffness of the device, measured at the top of the tool by applied weight, was about 80 N / /um. Similarly, Rasmussen et al. [81, 82] presented a piezoelectric tool servo with a stroke of 50 /xm for dynamic variable depth of cut machining The piezo actuated cutting tool was designed and developed to test the capability of generating irregular surface waveforms which could compensate for the nonuniform deformation due to clamping forces of fixtures. L i et al. [53] presented a piezoelectric fine positioning system for compensation of guideway and spindle error in precision diamond turning. The piezo tool servo was used in machining a cylindrical rotor which is a part of a gyro to meet the dimensional and form tolerances that are ± 3 /im of the nominal and < 0.3 pm of the roundness error. Okazaki [71] developed a micro-positioning piezo servo to improve the resolution and compensate high-speed movements for ultra-precision diamond turning machines.  It consisted of a  wire-cut parallel spring mechanism, a piezo stack, and a capacitive gauge with the static stiffness of 23 N / /im. He achieved a resolution smaller than 25 nm in depth-of-cut control. Shamoto and Moriwaki [67, 90, 91, 92] designed various piezo actuators to deliver elliptical vibrations to the diamond turning tool to reduce the chip-tool friction.  The elliptical vibration  caused the tool to have a velocity component in the chip flow direction in each cutting cycle after  Chapter 2. Literature  Review  8  it penetrated into the workpiece. Consequently, the friction force between the tool rake face and the chip is effectively reduced by reversing the frictional direction, and the reversed frictional force assists the chip to flow out. From the experiments, the friction force was significantly reduced, and the chip thickness and cutting force reduced substantially as well. Furthermore, formation of burrs was suppressed by the elliptical vibration cutting, and the surface roughness generated was small (0.02 /j,m  R x)ma  K i m et al. [45, 46] developed a piezo actuator to mount on a precision lathe in order to control depth of cut precisely and compensate the waviness on the surface of the workpiece. The piezo actuator was used in machining the face of magnetic disks, which recently have been machined on a precision machine tools using single crystal diamond tools. However, waves occur periodically on the surface of the disk, and this waviness reduces the form accuracy of the disk. By using a piezo actuator with piezoelectric voltage feedback, the authors were able to reduce the maximum surface waviness of 3.3 fim to 0.3 fxm. Most of these applications were for diamond turning where the chip loads are extremely small, and so are the corresponding cutting force disturbances.  Therefore, the rigidity of the actuator  was not a significant issue, and conventional PID controllers were sufficient to handle negligible cutting force disturbances for precision position control.  2.3  Control of Piezoelectric Actuator The precision position control and surface finish depend on compensating static and dynamic  deformations caused by the cutting forces, variations on the cutting process, tool and material, and piezo hysteresis.  Furthermore, disturbances due to cutting force, parameter uncertainties,  unmodeled dynamics, and vibration are associated with cutting process. Therefore, the control of the actuator becomes a significant factor in obtaining desirable positioning accuracy and surface finish.  Chapter 2. Literature Review  9  Rasmussen et al. [81, 82] employed a digital repetitive control to their piezo actuated tool servo and achieved a tool motion error of less than 0.5 /xm. The repetitive controller stores all of its outputs and all of the errors over the past cycles.  The current control is then calculated  based on the output and errors from the previous cycles. This controller is especially suitable for non-circular machining since the trajectory followed by the cutting tool is periodic. If the error is zero, the controller simply sends the same control outputs as the previous one, and i f the error is not zero, the controller updates the previous control output at each point within the cycle. A general discrete time description of a repetitive signal is given as  where W (z) is the original or reference signal, W(z) is the same signal delayed by one cycle, 0  and N represents the length of the cycle in points.  Comparing with results using PID control,  Rusmussen et al. showed that using the repetitive controller resulted in 6.32 times less R M S surface error. L i . et al. [54,55] employed a similar approach as that of Rasmussen et al to control a tool positioning piezo actuator for a diamond turning lathe. They used the controller called the self-tuning regulator which consists of two part as shown in Figure 2.1: learning parameter estimator and one step ahead control. Typically, a self-tuning regulator models a plant with a difference equation, uses a recursive estimator to estimate the parameter vector at each sampling instant, and adjusts its control action accordingly. They used the learning recursive least squares estimator to improve the estimates of the parameters by using the information acquired from previous repetitions. With better estimates of parameters, the self-tuning controller is able to reduce tracking errors over the repetitions of a task. The actuator was controlled to follow a square wave with an amplitude of 1.1 pm and a period of 2.3435 s, and after the 10 repetition, the tracking error was less than 0.1 th  /im.  Repetitive control technique is applicable to mass production where the material and tool  10  Chapter 2. Literature Review  conditions do not change, the process is slow, and the process is repeatable at each set-up. However, this may not be the case in the real production floor. Furthermore, more memory is needed for implementing the controller since the controller needs to store the information of the previous cycles.  / One Step Ahead Control  Plant  Learning Parameter Estimator Parameters  Figure 2.1: Learning parameter estimator and one step ahead control [54]  In order to eliminate the non-linearities in the piezo actuator and to improve the stiffness and response, Okazaki [71] applied two types of closed-loop control systems to control a micropositioning piezo tool servo and compared the results: pole-zero cancellation system with a notch filter and state feedback system with a state observer. For the former system, he used a notch filter to cancel the resonant peak of the piezo tool servo at 950 Hz, and an integrator was substituted by a first order low-pass filter to attain high loop gain and stiffness in the low frequency range. For the state feedback system with a state observer, a velocity signal was generated from the displacement signal using a state observer, and this velocity signal was used to move the oscillating poles of the system to the left in the s-plane so that the resonances are depressed. Okazaki noticed that both controllers were effective in obtaining a flat frequency response and sufficient static stiffness. However, for dynamic stiffness improvement, only the state feedback system with a state observer was effective.  11  Chapter 2. Literature Review  Ro and Abler [83] introduced a general control scheme known as directional damping control. They applied this control scheme to a single point diamond turning machine, but it can also be used to control a piezo actuator.  This scheme is quite effective in reducing vibration without  compromising system bandwidth for a class of linear single-input/single-output (SISO) systems whose single highly underdamped mode is at a frequency well above the system bandwidth. The design of the control scheme involves using a pair of complex zeros placed near the lightly damped poles, not to cancel them, but instead to direct them to a more damped region.  The complex  pair of zeros are placed at such an angle that this angle would give the lightly damped poles the desired angle of departure to minimize sensitivity to disturbance.  Since these lightly damped  poles and zeros would have little effect, the controller can be first designed to produce the desired system bandwidth and steady state characteristics, and then the complex pair of zeros can be added to reduce the vibration. Ro and Abler compared the results with PID control and showed that the peak-to-valley amplitude of vibration decreased from approximately 250 n m to under 100 nm. The control scheme also improved the surface finish of a machined surface on a single point diamond turning machine; peak-to-valley surface finish was reduced from 264 n m to 98 n m . Rubio et al. [84] applied and compared four different modern control techniques with position feedback using a Laser Diode Micheson Interferometer for positioning a piezo actuator for ultraprecision machining of brittle materials: lead-lag filter, PID, PID + feedforward, and fuzzy logic. With the simulation results, they showed the possibility of knowledge-based techniques such as fuzzy logic to control solid state actuators as these techniques do not require detailed mathematical models to formulate the algorithm and have more adaptive capability. The hysteresis of piezoelectric actuators is a phenomenon of energy dissipation associated with the magnitude and rate of input signal and its past history. It is one of the major non-linearities in piezoelectrically driven devices. A standard piezoelectric actuator may exhibit a 10-15% hystere-  Chapter 2. Literature Review  12  sis loop, which represents a phase lag between input and output. However, conventional closed loop controllers can eliminate the effect of hysteresis in regulation control applications [51] with the aid of feedback signal. Nevertheless, Jouaneh and Tian [44] pointed out that without modeling and incorporating the hysteresis in the design of controllers for piezoelectric actuators, the hysteresis would act as an unmodeled phase lag and cause poor control performance in tracking applications. They modeled the hysteresis of a stacked piezo actuator and obtained the linear behaviour under open loop control by applying the estimate of the nonlinear hysteresis behaviour in a feedforward control scheme. Eddy [26] considered the hysteresis of piezoelectric element as a time delay in the transfer function of the overall system. He applied a feedforward state controller with cutting force disturbance observer-compensator to control a piezoelectric actuator and obtained the tool positioning accuracy of < 1 /im. He compared the result with a Zero Phase Error Tracking Controller (ZPETC) and pole-placement control. He concluded that ZPETC's performance is dependent on the cutting force disturbances, which caused transient oscillations during actual cutting. Pole-placement controller resulted in sluggish response i f tuned to have large damping and was not capable of compensating cutting force disturbances quickly.  2.4  Surface Roughness Surface finish of single-point turning operations has traditionally received a great deal of re-  search attention [12, 65, 105] because the quality of a machined surface plays a crucial role in its functioning [107]. The characteristics of machined surfaces not only include how a finished part fits and wears but also include how it looks, reflects light, transmits heat, distributes and holds lubricant, resists fatigue, or accepts coating. Moreover, finish specifications are useful in determining and monitoring the stability of a manufacturing process, where a deteriorating finish may  Chapter 2. Literature Review  13  be interpreted as a sign of material non-homogeneity, progressive tool wear, or even impending catastrophic tool failure [65]. Surface finish of a machined part is characterized by methods that assess the quality of a surface. A surface is, in reality, a three-dimensional structure containing peaks and valleys. However, the conventional surface measurement methods are based on profile assessment with more than 50 profile parameters in use. Figure 2.2 shows some of the surface finish measurement parameters commonly used in industry today [107]. These parameters characterize the height of the surface profile. Average roughness (R ) and root mean square roughness (rms, R ) are the most commonly a  q  specified means of measuring surface finish. By themselves, however, R and R tell very little a  q  about how a surface will function. For example, two workpieces can have completely different functional characteristics yet have the same average roughness value. Nevertheless, they provide a simple value for accept/reject decisions, which accounts for the common use in industry. Geometric average roughness, R is more sensitive to occasional highs and lows and is more useful, but q  roughness average, R offers simpler computation. Rmax is the maximum peak-to-valley rougha  ness height within the sampling length. This parameter is the most sensitive indicator of high peaks and deep scratches.  Ten-point height, R , is the average distance between the five highz  est peaks and the five deepest valleys within the sampling length, measured from a reference line parallel to the mean line but not crossing the roughness profile [36, 98, 107]. There are also parameters characterizing the distribution of roughness material such as t and p  t. m  Bearing length ratio (t ) simulates wear at various cutting depths of a surface. p  It is most  useful for bearing surfaces that must be analyzed and qualified for lubrication and wear properties. t is bearing ratio at a depth of c expressed as a percentage of Rmax below the highest peak. It is p  obtained by establishing a reference line parallel to the mean line at a predetermined distance, c,  Chapter 2. Literature Review  14  Mean line  R (approx.) = a  Rq (approx.) =  ^  +  V2 V3 + V4 + - + Vn +  n  y2 +y2 1  2  +  y2 3  +  y 2+„ + y2 4  n  n max  Mean line  Mean line  Reference line  _(P1  R (approx.) =  +-P2-  +  P5)  "( 1 v  +V  2  + ...+-V )  z  Figure 2.2: Surface roughness parameters  5  15  Chapter 2. Literature Review  below the highest peak of the profile (Figure 2.3). The line intersects the profile, generating one or more subtended lengths (&i, b2...b ) which are distances from one intersection to the other of n  a peak. Bearing length ratio, t , is the ratio of the sum of the subtended lengths to the sampling p  length, L. t is the bearing ratio at the level of the mean line, i.e., c is the distance from the line m  c = 0 to the mean line [36, 107].  — c= 0 Jtp =  0%)  Mean line c = max  ( f = 100%) p  *  o/  t %= D  bi + b —  -  2  -  + b ... + b 3  —  -  n  —  X 100  Figure 2.3: Surface roughness parameters characterizing the distribution of roughness  2.5  Influencing Parameters Machining is a very complicated process where a large number of parameters are involved.  The tool properties (material, shape, geometry, tool life, wear resistance), the cutting parameters (feed rate, cutting speed, depth of cut), the workpiece (material, hardness, geometry), and the machine tool (positioning accuracy, rigidity) have a significant influence on the surface and on the accuracy in size and shape. Thus, it is important to understand the effect of these parameters, so the surface finish can be improved by changing the appropriate cutting conditions. Many researchers  16  Chapter 2. Literature Review  have attempted to isolate each parameter, examine its effect, provide explanations analytically and empirically as to the effect of each parameter. Feed rate and tool nose radius are the main cause for the feed mark of a machined part on a turning center. Figure 2.4.  Each revolution leaves a mark on the surface of the workpiece as shown in  Therefore, from the mechanics of turning it is known that for higher feed rates,  the cutting edge will move more distance for each revolution, thus resulting in higher surface roughness [74, 92, 104, 105, 108, 114]. Conversely, decreasing the feed rate yields better surface finish. However, there is limitation on decreasing the feed rate because below a certain feed rate, the surface starts to deteriorate.  Also, a bigger nose radius has a positive effect on the surface  finish. However, the workpiece material gets more squeezed under the cutting edge and the cross section of cut, and simultaneously the cutting force increases. This may lead to vibrations and a dynamically superimposed roughness.  Cutting edge  Feed rate  Cutting edge after one revolution  v  Tool nose radius  Figure 2.4: Feed mark caused by feedrate and tool nose radius  The cutting forces increase with the depth of cut, and high forces lead to poor surface finish due to structural deformation.  Also, i f the depth of cut is increased beyond a certain value, it  would result in self-excited vibration leaving chatter marks on the machined surface. However,  17  Chapter 2. Literature Review  in the absence of significant structural deformation and chatter, the depth of cut is found to have a negligible effect on the surface quality El-Wardany et al. [29] found that the surface roughness slightly increases as the depth of cut is increase, and it decreases slightly with further increase of the depth of cut during hard turning steel AISI 1552 using ceramic tools. However, the effect is negligible compared to those of other parameters. Takeyama and Ono [42] and Sata [87] obtained similar result, which showed that the depth of cut has little influence on the surface roughness.  Type 2 Chip  Cutting speed, v  Figure 2.5: Effect of cutting speed on surface roughness in orthogonal cutting due to different types of chip formation  Different kinds of chips form at different cutting speeds, and it has been known that the type of chip formation has significant influence on the surface quality. Figure 2.5 shows the regions of which different types of chip are formed [5]. At low cutting speed (In region (a)), discontinuous chips (type 1) form, which results in periodically oscillating force and tool deflections. As a result, a wavy surface finish and tool failure are induced[48]. Discontinuous chips involve the formation of a crack that starts out as a tensile crack at the tool tip but gradually changes to a shear crack as it approaches the free surface at 45 deg [93, 116]. The crack occurs periodically as individual segments are ejected from the tool. This is due to low temperature and rapid cooling at low speeds;  18  Chapter 2. Literature Review  in other words, the material cracks and fractures instead of deforming. This process is also seen with brittle materials [48]. It has been reported that for some materials and cutting conditions, the first stage (a) may not occur [5, 12]. Other reasons for discontinuous chips have also been reported such as random orientation and non-uniform nature of polycrystalline metal and periodic variation of shear angle due to the instabilities in the tool-chip friction and the tool-workpiecemachine tool system [48].  As the cutting speed is increased, the temperature on the tool-chip  interface is sufficient enough for the chip to behave in a ductile manner. Thus, the chip becomes more continuous. However, as the cutting speed is further increased, i.e., region (b), the chip metal becomes ductile and the resulting plastic flow causes strong welds to form between chip and tool [87, 93]. This weld is known as the built-up-edge (BUE). The B U E gradually increases in size and shape until it becomes unstable and leaves with the chip, and this occurs periodically. This cyclic variation in the undeformed chip thickness (type 3) contribute to a substantial increase in surface roughness [116]. When a B U E is present, it is producing the chips rather than the tool, and it grows downward causing the finished surface to be undercut. At sufficiently high cutting speeds (region (c)), the B U E disappears, continuous chips (type 2) start to form, and the surface roughness starts to decrease and approach a steady low value.  Because the chip formation in  this region (type 2) is desirable, many researchers have obtained the critical speed for different materials above which the continuous chip without a B U E occurs [69, 80, 87]. Sata [86] reported the B U E disappears more rapidly with higher rake angles and harder workpiece material. As the cutting speed is increased to avoid the built-up-edge, high temperature becomes another concern. At high temperature, the tool wears more rapidly, work material softens, and the chips snarl and scratch the finished surface [22,112], altogether causing poor surface finish. Konig et al. [49] provided explanation for the temperature build-up for high cutting speed: the heat generated cannot be discharged with the chip due to increased removal rate, so most of the heat flows into the  19  Chapter 2. Literature Review  workpiece and tool. The surface integrity and machining accuracy are related to the temperature as it will cause thermal expansion during machining and metallurgical changes. If the temperature exceeds the 7—a transition temperature (700—900 °C) during machining of hardened bearing steel, frictional martensite is produced which is known as "white" layer on the surface of the workpiece [48, 111]. It is known that this "white"" layer or even slightest structural changes in the outer zone of machine surface may cause damage to gear and roller bearing components [49]. Many researchers have used different types of cooling/lubricating methods to improve the result at high cutting speed. interface.  Conventional methods use cutting fluids discharged to the tool-chip  Cutting fluid acts as a coolant and prevents rewelding by cooling and washing small  particles away [1]. It also acts as a lubricant, reducing the friction on the interface and preventing buildup on the tool nose.  Thus, using cutting fluid results in less tool wear, better surface  finish, and better chip formation. U m et al. [112] investigated the spray cooling method which resulted in lower force, lower surface roughness, and easily breakable chips. Ding and Hong [22] improved the chip breaking in machining low carbon steel by cryogenically precooling the workpiece. Mazurkiewicz and Kubala [61] used a high pressure water jet cooling method to force the fluid into the tool-chip interface since at very high cutting speed there is not enough time for the cutting fluids to penetrate into the interface. Tool wear is also an important factor which would significantly affect the surface quality and dimensional accuracy. Tool wear is defined as a gradual loss of tool material at workpiece and tool contact zone [3]. The moving chip is in contact with rake face where crater wear occurs. As the cutting edge wears, the flank face of the tool starts rubbing against the finished surface leading to flank wear and poor surface finish. Also, concentrated wear in the form of clearly defined grooves may develop, which is localized at the boundaries of the areas of contact between rake face and chip, and between clearance face and workpiece [74, 92, 101, 108]. Figure 2.6 illustrates how  20  Chapter 2. Literature Review  grooves affect the surface roughness.  Several theories have been proposed for the formation of  grooves [93, 102, 117]: • Embedded particles of tool material removed on the previous revolution acts as cutting edges. • Crater wear on the rake face of the tool breaks through the end clearance edge. • This area with grooves may suffer oxidation because the hot workpiece and the tool are exposed continuously to the ambient atmosphere. • Squeezing of metal in the chip-forming process leads to an increase in the height of the feed marks, which may in turn have some effect on groove formation. f  ^max / /  7  Cut without wear  ' Grooving w e a r \ Figure 2.6: Schematic illustration of grooving wear  El-Wardany et al. [29] obtained the result that the surface roughness of case hardened steel AISI 1552 increases almost linearly with increasing cutting time with the same tool. Solaja [102] and Petropoulos [75] obtained similar results except that the finished surface deteriorates more rapidly in the beginning of tool wear. Liu and Mittal [59] reported that residual stresses tend to become tensile with increasing tool wear. It has been reported that negative rake angles yield better surface finish and less wear [29,102]. Also, from a wear consideration, a greater clearance angle is more desirable [102], but increasing  21  Chapter 2. Literature Review  this angle increases the danger of mechanical break-down of the tool because the tool becomes weaker, thus reducing the range of optimum values. Albrecht [1] reported that surface finish improves when the carbon content of steel (hardness) is increased, free-machining additives are added, and the grain size of the material is reduced. Also, residual stress becomes more compressive with increased hardness of steel because the shear angle increases with material hardness [57, 58, 127]. Stability and rigidity of the machine tool and workpiece set-up is a significant issue for surface finish and dimensional accuracy.  Sata [85] compared the machined surfaces on three different  machine: an old, flexible lathe and a rigid, precision lathe. The peak-to-valley roughness values decreased from greater than 20 ^ m with the flexible lathe to around 5 pm with the rigid, precision lathe. The vibration term in the flexible lathe was about 6 jum greater than one produced on a rigid lathe (0.2 — 0.3 fxm). Therefore, in the absence of chatter, the vibration term of roughness depends on the quality of the machine tool used.  2.6  Summary Considerable efforts have been made to control nonlinearities associated with the piezoelectric  actuators, to reject disturbances associated with cutting process, and to compensate inherent vibration and uncertainties in the system. One of the control methods, which is very effective and robust in controlling a nonlinear system and disturbances, is Sliding Mode Control (SMC). Essentially, sliding mode control utilizes a high-speed switching control law to. drive the nonlinear plant's state trajectory onto a specified and user-chosen surface in the state space, and the switching control law maintains the plant's state trajectory on this surface for all subsequent times [20, 118]. However, because control switches around the sliding surface at a very high frequency, some unmodeled dynamics may be excited which results in the chattering phenomenon [8, 20, 99, 118, 123], and due to this phenomenon, SMC has not been widely accepted within the control research community.  Chapter 2. Literature Review  22  Therefore, numerous approaches have been attempted to avoid this chattering phenomenon around the sliding surface. Slotine et al. [100] later introduced the concept of parameter adaptation, which does not require the switching type of control. The switching was not necessary since the effect of parameter uncertainties can be cancelled by applying parameter adaptation. This thesis presents the slightly modified version of Slotine's sliding mode control with parameter adaptation proposed by Zhu et al. [123, 124]. Zhu et al. incorporated parameter upper and lower bounds into the Slotine's controller, so each parameter is updated within its upper and lower bounds independently, thus maintaining the robustness against parameter uncertainties in the system and disturbance. Height parameters  (R , a  R , Rmax, q  and  R) z  provide enough information on the quality of a  surface, and they are the ones that are generally used for surface finish specifications in the industry. Thus in this thesis, only the height parameters are used to determine the quality of a machined surface. For the case that the tool, workpiece, and machine tool are given parameters, only cutting parameters are left to manipulate for the condition of optimum surface finish.  For this thesis,  since the tool, workpiece, and machine tool were given, the optimum range of cutting parameters are presented in which the desirable surface finish is obtained. The geometrical model represents the minimum surface roughness value, so it is used to initially select the range of relevant cutting parameters for acceptable surface finish. Empirical models are used as a comparison.  Chapter 3 Surface Roughness Model Many models have been developed in order to determine the surface roughness in terms of influencing parameters. From a machining point view, it is essential that the relationship between the surface roughness and the many process variables be established so that the appropriate cutting conditions can be selected to satisfy the surface finish requirements of the component to be machined.  3.1  Geometrical model From the kinematics of the tool and the workpiece, a geometrical model of the surface finish  can be obtained.  Because the geometrical model represents the theoretical or ideal model, it  remains valid under the following assumptions [105]: • The roughness is caused by the feed marks only. • Turning is vibration free. • A single-point tool is used to turn the surface. Figure 3.1 shows the geometry of tool tip with the nose radius, r. The parameters on the figure are defined as the following: • K = End cutting edge angle or minor cutting edge angle r  • K' = Major cutting edge angle r  • tp ~ Approach angle r  • d,A,dB = Depth of cut limits • foA, /A, IAB, IB = Feed limits  23  Chapter  3. Surface  Roughness  24  Model  f  B  Figure 3.1: Tool tip geometry dA and d are the limits where the arc of the nose radius ends. Beyond dA and d , the analysis B  B  of the roughness model is approached differently. Similarly, depending on the depth of cut, Rmax is analyzed differently beyond the limits of foA, IA, IAB, and f . B  From the geometry of the tool  profile, dA and d are defined as the following: B  dA = r • (1 — cos/4) d  B  =  0-2)  r • (1 + s h T 0 ) = r • (1 — c o s K )  and the feed limits are defined as the following:  r  r  Chapter 3. Surface Roughness Model  25  A  =  2 • V2 • r • d - d?  f  =  2 • r • sin n!  JAB  =  V% • r • d — d + r • sin A^, + cot K£ • [d — r • (1 — cos «£)]  J B  —  jo  A  (3-3)  2  r • [1 - sin(V; r  :  <)]  ;  sin K'  r  Figure 3.2 and 3.3 show five cases o f turning operations and the roughness for each cut [5]. There are four more cases in which the depth o f cut is larger than d ; B  however, in precision  machining, the depth o f cut is very small, and it is assumed that the depth o f cut is less than d . B  f d  Depth o f Cut: 0 < d < d  Depth o f Cut: 0 < d < d  A  A  Feed L i m i t s : f < foA  Feed L i m i t s : f > foA  Figure 3.2: Surface roughness geometries (0 < d < d ) A  For simplicity, the peak-to-valley height o f the roughness is considered. The following shows the roughness models of the cases indicated in Figure 3.2 and 3.3 [5].  Chapter 3. Surface Roughness Model  Figure 3.3: Surface roughness geometries (d^ <d<  26  ds)  27  Chapter 3. Surface Roughness Model  /4 Rmax  =  R  Rmax  =  RAB  O  . 2_f2 r  = r-^  A  —  (3.4)  J  = r • (1 - cos n' ) + f • cos n' • sin K' - ^2 • f • sin K' - f 3  r  r  2  t  T  where / is the feed rate and r is the nose radius of the tool.  • sin K' 4  T  When the depth of cut is smaller  than d , i.e., case I, the feed is limited by JOA, and the surface roughness model is given by ROA A  in (3.4). If d < dA and / > JOA, case II occurs, in which the feed is too fast and some of the surface are left uncut. For the case III, d^ < d < d , i f the feed is less than /A, and the surface B  roughness model is the same as that in case I: Rmax  =  ROA- If dA < d < d  B  and / > JA, the  surface roughness is given by the equation RAB in (3.4), and the feed is limited by JAB- Similar to case II, if the feed is greater than JAB, partly uncut surface will result, which is shown as case V in Figure 3.3. A number of researchers have sought to test whether the "ideal" or theoretical surface roughness models can be applied in practice. It has been found that above the critical cutting speed, the theoretical or geometrical equations (3.2)-(3.4) can be used in practice [69, 75, 80, 86]. MunozEscalona and Cassier [69] empirically found the mathematical models of the critical cutting speed, V* for AISI 1020 and AISI4140 as the following: AISI 1020 : V* = 31.93 + l . l l r - 48.83/ - 4 3 . 1 7 d  (3.5)  AISI 4140 : V* = 30.56 + 14.52r - 115.33/ - 20.5d The analysis for the geometrical model above corresponds to the condition with little tool wear.  So, some researchers analyzed the geometry of the tool tip with wear and attempted to  develop a mathematical model of surface roughness including the effect of tool wear [102,42,114]. According to Solaja [102], the peak-to-valley surface roughness can be expressed by the following equation when grooving wear exists within the nose circle as shown in Figure 2.6.  Chapter 3. Surface Roughness Model  28  Rmax = (V > - V ) t a n Cl B  B  p  (V&-VJ) D  ( / - 0-065VW/T ( r - t a n CZ )  (3.6)  P  where V is the width o f normal flank wear at the nose, V > the width o f grooving wear at the nose, B  B  Clp the end relief angle, D the work diameter, / the feed, and r the nose radius. Solaja obtained from his experiments a good agreement between the values o f surface roughness measured and calculated from the equation (3.6). However, the geometrical model considers only a few parameters that influence surface finish and does not take into account other dynamics that occur during machining.  Thus, the actual  surface finish or roughness o f the machined surface may be expected to be generally worse than the theoretical or geometrical roughness  3.2  Empirical model A number o f researchers have attempted to establish equations relating the "actual" surface  roughness to the various process variables to reflect the qualitative trends discussed in the previous section and to provide a means o f predicting the surface roughness [23, 28, 29, 52, 65, 69, 105, 109]. Munoz-Escalona [69], by using the multiple linear regression method, developed a general roughness mathematical model for all steel types:  R where R : a  = 124.83 • r " 0  a  2 9 7  •d  0 0 5 6  • /°-  607  • v" 0  138  • B H N-0.445  (3.7)  average roughness (^m), r: tool's nose radius (mm), d: depth o f cut (mm), / : feed  rate (mm / deg), v: cutting speed (m / min), and B H N : Brinell Hardness Number. The exponents indicate whether each variable has favorable or adverse effects on the roughness. El-Wardany et al. [29] developed a log-transformed second order model for hard turning o f steel A I S I 1553 (60 Rc) with ceramic tools. M i t a l and Mehta [65] obtained the non-linear general surface roughness model for three different metals.  However, they assumed that the depth o f cut does not have  Chapter 3. Surface Roughness Model  29  much influence on the surface roughness, so it was not included in the model. Dontamsetti and Fischer [23] added the effect of the initial flank wear to their model.  Sundaram and Lambert  [105] developed a model with six variables: cutting speed, feed rate, depth of cut, time of cut, nose radius, and type of tool, coated or uncoated. Instead of developing a model to predict the average surface roughness values, they developed a model which predicts R M S surface roughness values, R. q  Chapter 4 Sliding Mode Control 4.1  Introduction Sliding Mode Control (SMC), also known as Variable Structure Control (VSC), is a high-  speed switching feedback control, providing an effective and robust means of controlling nonlinear plants. For example, the gains in each feedback path switch between two values according to some rule. This rule allows the controller to appropriately switch feedback gains at a very high frequency in order to drive the nonlinear plant's state trajectory onto a specified and user-chosen surface in the state space [20, 118]. This surface is called the sliding surface as the plant's states slide on it. Thus, V S C involves good design of both the sliding surface which guarantees stability once the states are on the surface, and the rule which enforces all the states to approach the sliding surface in finite time [20, 43, 118, 126]. The sliding surface is normally associated with a stable differential equation, meaning that the states on the sliding surface must be self-stable and self-convergent. The rule or control law is normally developed using the Lyapunov Stability Theorem. Figure 4.1 shows a typical sliding surface for a second-order system on a phase plane, where x and x are system states. Thus, the sliding surface for a second-order system can be generally specified as S = x + cx  where c is a positive constant.  (4.1)  It is clear that S = 0 corresponds to a stable differential Eq.  x + cx = 0 which leads to x —• 0 and x —* 0. Therefore, once the control law drives the states onto the sliding surface, the system is stable.  30  31  Chapter 4. Sliding Mode Control  x  Convergent Trajectories Figure 4.1: Sliding surface on a phase plane For the design of the control law, Lyapunov stability theorem provides a good analysis for its derivation. The Lyapunov stability theorem states that [34], An equilibrium point of a time invariant dynamical system is stable if there exists a continuously dijferentiable scalar function V(x) such that along the system trajectories the following is satisfied  V(x) > 0,  V(0) = 0 dVdx dx dt  (4.2)  If the condition (4.2) is a strict inequality then the system is asymptotically stable.  The Lyapunov function, V(x), can be denned as a function of any parameter to be stabilized. If the Lyapunov function V(x) is denned as a function of the sliding surface, i.e., V(S), and the theorem (4.2) is satisfied, then, the sliding surface will approach to zero. Therefore, the control law which satisfies the theorem (4.2) will guarantee the stability of the system.  The analysis in  the next section explains how the sliding surface and Lyapunov function are defined and how the control law is obtained from them. The problem of the sliding mode control is also discussed.  Chapter 4. Sliding Mode Control  4.2  32  Controller Design Consider a n-order linear system in the state form as X = A X + B U - D  where X e R  n  U e  is the state vector of the system, A e / ^  (m < n) is a control vector, and D e f f  1  n x n  (4.3)  and B 6 p^xm  a r e  m a u  -i  c e S j  denotes the disturbance. The sliding surfaces  should have the same dimension as the control action, and is specified as S = C • (X - X)  (4.4)  d  for tracking applications, where X ^ G R is the desired vector of X , S £ R n  m  of the system states from the sliding surfaces, and C e pjnxn j  s a  m a t r  i . x  denotes the deviation Note that X is n-  dimensional and S is m-dimensional. Therefore, S = 0 only specifies m-dimensional subspace. The remaining n — m subspace has to be self-stable such that X —> X . This is carried out by d  suitable design of matrix C , i.e. matrix C has to be designed such that i) X —> X ^ when S = 0, and ii) (C • B) is symmetrically positive-definite. The suggested control law design by Utkin [113] is as follows U = [S • ( | | X J + ||A|| • ||X|| + ||D||) + r,} • sign(S)  (4.5)  c5 > IKC-B)- !! • ||C||  (4.6)  where <5 is defined as 1  and T) > 0 specifies the minimum convergent rate for S. Many different kinds of controllers have been suggested and developed by numerous researchers in the literature.  In fact, the design of  a control law is an important area of sliding mode control. The following steps show how the control law in (4.5) is developed.  Chapter 4. Sliding Mode Control  33  A Lyapunov function can be chosen as [34]  V =  i  •S  • (C • B ) - • S  T  (4.7)  1  Since ( C • B ) is symmetric, differentiating (4.7) yields V = S  T  • (C • B ) - •S  (4.8)  1  Substituting the E q . (4.3), the time-derivative o f S can be written as S  =  C • ( X - X ) = C • (Xd - A X - B U  =  C X  d  d  - C A X - C B U  + D)  + C D  (4.9)  Therefore, combining (4.8) and (4.9) gives \> = S  T  - ( C - B ) -  1  - ( C - X  d  - C - A - X - C - B - U + C-D)  (4.10)  and the E q . (4.10) can be rewritten as  V  =  S-(C-B)- -C-(X -A-X  =  S  1  d  T  + D)-S -(C-B)- -(C-B)-U T  • (C • B ) " • C • (Xd - A • X + D ) - S 1  1  T  (4.11)  • U  and  V < | | S | | - ||(0 - B ) | | • | | C | | • (HXdll + | | A | | • | | X | | + | | D | | ) - S T  - 1  T  •U  (4.12)  From the E q . (4.12), i f the right hand side is always negative, the Lyapunov stability theorem is satisfied, i.e., V < 0. Therefore, the controller must be designed such that the right hand side o f the E q . (4.12) is always negative, and then, S —• 0 can be guaranteed. B y setting the right hand side o f the E q . (4.12) equal to a negated function which is always positive as the following, the inequality in (4.12) can be always satisfied.  Chapter 4. Sliding Mode Control  34  | | S | | • | | ( C • B ) - i • | | C | | • ( | | X J + ||A|| • | | X | | + | | D | | ) - S r  T  Since | | S | | = S T  S  T  T  • U = - 77 • | | S | | r  (4.13)  • sign(S), the E q . (4.13) becomes  • sign (S) - | | ( C • B )  _ 1  | | • | | C | | • ( | | X | | + | | A | | • | | X | | + ||D||) - S  T  d  • U = - 77 • S  T  • sign(S) (4.14)  Solving the E q . (4.14) for U gives  U = [||(C • B )  | | • | | C | | • ( | | X | | + | | A | | • | | X | | + | | D | | ) 4- 77] • s i g n ( S )  - 1  d  (4.15)  Therefore, the Lyapunov stability theorem is always satisfied i f the control law is designed as  V = [S- (UX.,11 + | | A | | • | | X | | + | | D | | ) + 77] • s i g n ( S )  (4.16)  where S is defined as in (4.6). It can be verified that the control law in (4.16) satisfies the Lyapunov stability theorem. Premultiplying (4.9) by ( C • B )  yields  _ 1  (C •B ) - • S = (C •B ) - • (C •X 1  1  d  - C •A •X + C •D ) - U  (4.17)  Substituting (4.16) into (4.17) yields  (C-B)" -^  =  1  ( C - B ) -  1  - ( C - X  d  - C - A - X + C-D)  -[6 • ( H X J + | | A | | • | | X | | + | | D | | ) +77]-sign (S)  (4.18)  From the E q . (4.18), the derivative o f the Lyapunov function (4.8) can be written as  V  =  S -[(C-B)- -(C-X -C-A-X + C-D) T  1  d  -[6 • (\\X \\ + | | A | | • | | X | | + | | D | | ) + 77]- sign (S)] d  =  S  T  -S  • (C •B ) " • ( C - X d - C - A - X + C - D ) 1  T  • [8 • (\\X \\ + | | A | | • | | X | | + | | D | | ) + 77]- sign (S) d  (4.19)  Chapter 4. Sliding Mode Control  35  but, the first three terms S • (C • B ) T  • C -X , S • ( C • B ) " • C • A • X , and S • (C • B ) " • C • D  _ 1  T  1  T  1  d  can make the following inequalities.  S -(C.B)- .C-X T  1  <  d  ||S -(C-B)- -C-X || T  1  d  < ||s || - ||(c - B ) r  (C B ) "  r  1  - iix^ii  ||S -(C-B)- -C-A-X||  <  | | S | | - ||(C - B)- X |l  1  S  ! ! - lien  <  S -(C-B)- -C-A-X T  1  C D  r  x  r  <  -  - HA||  - ||X||  ||S -(C-B)- -C-D|| r  <  1  | | S | | • ||(C - B ) | | • | | C | | • | | D | | T  _ 1  Therefore, the Eq. ( 4 . 1 9 ) becomes  v < ||s || - ||(c - B ) r  1  - ||C|| - iix.,11  !!  +||S ||.||(C-B)- ||.||C||.||A||.||X|| r  1  + ||S ||.||(C.B)- ||.||CM|D|| T  -S  T  1  • sign (S) • [S • (HXdll + | | A | | • | | X | | + ||D||) + rj]  < IISl^KC-B^II-llCll. -S  T  • sign(S) • [S  •  ( 1 1 X ^ 1 1 + ||A||.||X|| + ||D||)  ( | | X J + ||A|| • ||X|| +  ||D||) + n]  Since S - s i g n ( S ) = | | S | | and<5 > j|(C • B ) - ) ! • | | C | | , T  T  V  <  1  | | S | | - | | ( C . B ) - | | . | | C | | . ( | | X | | + | | A | | - | | X | | + ||D||) r  1  d  -||S || • [||(C • B ) T  _ 1  | | • ||C||  • (HXrfH +  ||A|| • ||X|| +  ||D||) + rj]  < IISl-IKC-Br^l-IICII-ClXill + IIAII^IXII + IIDII) -||S || • ||(C • B ) T  Therefore,  _ 1  | | • ||C|| • ( | | X | | + d  ||A|| • ||X|| + ||D||) -  \\s \\ T  r, •  Chapter 4. Sliding Mode Control  36  V<-»7-||S|li  (4.20)  It can be seen from (4.20) that the control law (4.15) satisfies the Lyapunov stability. For a particular case that (C • B ) is diagonal, premultiplying (4.18) by sign ( S ) • (C • B ) T  yields sign(S)j-Sj<-(C-B) r7  (4.21)  r  where Sj denotes the j  th  element of S and (C • B ) j denotes the j  th  diagonal element of (C • B ) .  Suppose Sj(0) — So, then, Sj < — (C • B ) j • n. Since the slope is negative, S j , with a positive initial value, will approach 0.  Suppose Si(0) = —So, then, Si > ( C • B ) ; • n.  Sj will again  approach 0 because the slope is positive with a negative initial value. Thus, all states will converge to the sliding surfaces within a finite time, i.e. t 6 (0, t*), where t* is specified as f = max 3  f^ T] • (C • B ) j (0)  (4.22)  For example, i f Sj(0) = So, Sj < —(C • B ) j • n. This gives the equation for S j , Sj < - ( C - B ) j -rj-t + So  (4.23)  Therefore, the time for all the states to converge to the sliding surfaces can be determined by setting Sj = 0. So, the Eq. (4.23) becomes 0 < - ( C • B ) j • 77 • t* + So Solving fort*, * <  t  ^°  -v(C-B)j Sliding mode control has an important property: the corresponding motion of the system is independent of changes in the plant parameters and of external disturbances, so it shows strong  Chapter 4. Sliding Mode Control  37  robustness against parameter uncertainties and disturbances [43]. However, because the control switches around the sliding surface at a very high frequency (Figure 4.1), some unmodeled dynamics may be excited, which results in the chattering phenomenon [20, 100, 118, 125]. This phenomenon can be seen in the Eq. (4.16) as well. The control law switches depending on the sign of the sliding surface, S, and this switching occurs at a very high frequency.  Sliding Zone  Figure 4.2: Sliding zone and boundary layer to avoid chattering  Many approaches have been attempted to avoid this chattering around the sliding surface. The sliding mode control of using the sliding zone has been introduced by Young et al. [119] as shown in Figure 4.2(a). Instead of having the controller switch around a single surface, they had their controller switch around a sliding zone, thus reducing the chattering.  However, the  chattering was avoided only when the states were far from zero because the sliding zone narrows near zero. Slotine and Coetsee [101] and Bartolini and Zolezzi [8] improved the Young's solution by applying a boundary layer to the sliding surface as shown in Figure 4.2(b).  Their control  laws enforces the states towards the boundary layer instead of the sliding surface, and when the states are inside the boundary, Slotine and Coetsee used linear high gain feedback control, and  Chapter 4. Sliding Mode Control  38  Bartolini and Zolezzi nonlinear feedback control.  Using the boundary layer control, however,  results in a trade-off between performance and robustness. Slotine et al. [100] later introduced the concept of parameter adaptation, which does not require the switching type of control; the effect of parameter uncertainties can be cancelled by applying parameter adaptation. Zhu et al. [123, 124] modified the concept initially introduced by Slotine.  They incorporated parameter upper and  lower bounds into the controller, so each parameter is updated within its upper and lower bounds independently. In the next section, Zhu's modified version of Slotine's sliding mode control with parameter adaptation is introduced.  4.3  Controller Design with Parameter Adaptation  Parameter adaptation has its significance in updating each parameter in the system within its upper and lower bounds independently so that switching type of control is not necessary. Also, output stability of the system does not require parameter convergence [124]. Assuming that (C • B )  • (C •  _ 1  — C • A • X + C • D) in the Eq. (4.11) is linear in parameters,  i.e.,  (C • B ) - • (C • X 1  d  - C • A • X + C • D) = Y(Xd, X ) • P  (4.24)  where P denotes a constant parameter vector and Y ( X , X ) is the corresponding regressor matrix, d  Slotine and Li [100] then suggested the following control law for parameter adaptation,  U = Y(Xd,X) where K e pj^xn j updated through  s a  P + K  S  f }back gain matrix, and P is the estimate of the parameter P . eec  (4.25) P is  Chapter 4. Sliding Mode Control  39  P = r - - ^ - ^ ; tf = Y - S 1  P Pj  T  = T- -SR-Y -S  (4.26)  =  (4.27)  T  1  Pj-Kj-tpj  J =  K  I  0 0  Pj < Pf Pj > Ff  and <pj < and ^ >  0  0  I 1 otherwise where the subscript j is assigned to the j  th  parameter, Yj denotes the j  th  of the j  th  parameter, Pj > 0 denotes the update gain for the j  th  column of Y ( X d , X ) , P~ and  denote the lower and upper bounds  parameter Pj. Noting that Pj e [Pf, Pf), it follows from (P - P f • [ Y ( X , X ) • S T  d  r  • P] =  Y,l(Pi ~ j) • p  (4.27)  that  • (1 -  <0  ( - 8) 4  2  where T = diag{—, • • • , — , • • • } .  The derivative of P , P is chosen as in (4.26) such that the control law  (4.25)  satisfies the  Lyapunov function which is chosen as 1 V  =2  S  T  • (C • B ) "  1  • S + (P -  P)  T  • T • (P -  P)  (4.29)  The Lyapunov function is chosen as above because there are two variables to be stabilized: S, the sliding surface, and ( P — P ) , the error between the real parameter and the estimate. Any control law which satisfies this Lyapunov function will guarantee the convergence of S and ( P — P ) to zero.  The following steps show how the control law in (4.25) is developed and prove that it  satisfies the Lyapunov function in  (4.29).  Since (C • B ) and V are symmetric and P is a constant parameter vector, differentiating yields  (4.29)  Chapter 4. Sliding Mode Control  40  V = S • (C • B ) " • S - (P - P) T  1  •T •P  T  (4.30)  Substituting (4.9) and (4.26) into (4.30) gives • ( C • B ) " • C • ( X - A • X - B • U + D ) - (P - P )  • T • T " • 3? • Y  V  =  S  V  =  S -(C-B)- -C-(X -A-X + D)-S -U-(P-P) -3?-Y -S  1  T  T  1  T  T  V  =  V  =  d  r  (4.31)  T  r  T  r  T  Because Y is symmetric, S • Y • P = P r  •Y  Adding and subtracting P •Y  T  - S  T  T  •Y  T  T  •S - S • U - P  r  T  •Y  T  T  (4.32)  T  • S. Consequently, the Eq. (4.32) becomes T  • ft • Y  •S + P  T  •  T  •Y  T  •S  (4.33)  • S to the Eq. (4.33) yields,  •S - S • U - P  T  T  S -Y-P-S -U-P -Sft-Y -S + P -K-Y -S  T  •S  d  S - Y - P - S - U - ( P - P f - K - Y  V =P  T  - C • A • X + C • D ) = Y ( X , X ) • P , it follows that  1  P  1  d  Since (C • B ) " • (C • ±  V =  T  d  T  T  • ft • Y  T  •S + P  T  • ft • Y  •S + P  T  T  •Y  T  •S - P  T  • Y •S (4.34) T  Rearranging the terms in the Eq. (4.34) and factoring, V  =  P  V  =  P -(Y -S-5R-Y -S)-P -(Y -S-Sft-Y -S)-S -U + P -Y -S  V  =  (P-P) -(Y -S-SR-Y -S)-S -U  T  T  - Y  - S - P  T  T  T  T  - ^ - Y  T  - S - P  r  T  - Y - S + P r  T  T  T  T  - K - Y  r  T  - S - S  T  T  + P -Y -S  T  T  T  - U + P -Y -S T  r  T  T  (4.35)  T  From the Eq. (4.26), it follows that  r-p  =  T-P  -  r-r -»-Y -s _ 1  T  ft-Y -S T  (4.36)  Chapter 4. Sliding Mode Control  41  Thus, substituting (4.36) into (4.35) V = (P - P ) • ( Y • S - r • P ) - S • U + P r  T  T  T  •Y  T  •S  (4.37)  From the Eq. (4.28), it is known that (P - P ) • ( Y • S - T • P ) < 0 T  T  (4.38)  Therefore, the Eq. (4.37) becomes V < -S  •U + P  T  T  •Y  T  •S  (4.39)  Again, since Y is symmetric, V < -S  •U + S '•Y •P .  T  T  (4.40)  In order for the Lyapunov stability requirement to be satisfied, the right hand side of the Eq. (4.40) must always be less than or equal to 0. Therefore, it can be set to -S -U + S -Y-P = -S -K-S T  T  T  (4.41)  Solving the Eq. (4.41) for U yields. U = Y(X ,X)-P + K-S  (4.42)  d  Notice that this control law, which uses parameter adaptation, does not contain a switching function. Consequently, the chattering phenomenon does not occur. This controller can also be verified that it satisfies Lyapunov stability. From (4.30), it follows that V = S • Y • P - S • U - (P - P ) • T • P T  T  T  (4.43)  Substituting the control law (4.42) into (4.43), the differentiation of the Lyapunov function becomes  42  Chapter 4. Sliding Mode Control  V  = S -Y-P-S -Y-P-S -K-S-(P-P) -r-P T  T  T  T  = S • Y • (P - P) - S • K • S - (P - P) • r • P T  T  T  =' (P - P) • Y • S - S • K • S - (P - P) • r • P T  T  T  T  = (P - P) • [Y • s - r • P] - S • K • s T  T  T  (4.44)  From (4.28), it follows that  V < -S  T  K S  (4-45)  Therefore, the control law satisfies the Lyapunov stability.  4.4  Summary Sliding mode control is basically a feedforward plus feedback control. The feedforward part is  devoted to compensate the system dynamics while the feedback part is to perform output tracking and is tracking error driven.  It has a prevailing property that the corresponding motion of the  system is independent of changes in the plant parameters and of external disturbances [43]. This is owing to the fact that uncertainties in the system are conquered by large, discontinuous control signals. However, the uncertainties must be bounded; sliding mode control uses control signals larger than the bounds of the uncertainties to conquer them. If the uncertainties are unbound, the system has to be modeled again so that the uncertainties are bound. Parameter adaptation on the other hand conquers the uncertainties with parameter estimates, which allow the control signal to be continuous. However, since the parameter estimate is based on the feedback, it may constantly increase or decrease i f error in the feedback builds up and gets accumulated in the estimate. This is why Zhu et al. [123, 124] incorporated upper and lower bounds to the Slotine's sliding mode control with parameter adaptation. The bounds set the limit to the estimate of uncertainties, hence  Chapter 4. Sliding Mode Control  43  it does not constantly increase or decrease. When the estimate hits the bound, it stays at the value, thus making the controller robust against measurement noise as well.  Chapter 5 Modeling and Control of Piezo Tool Actuator 5.1  Overview In this chapter, modeling of the piezoelectric actuator assembly and simulation of the sliding  mode control with parameter adaptation are provided. Firstly, the specifications of each component of the piezoelectric actuator system setup and related parts are given in detail, followed by modeling and simulation results. Also, the simulation results of the control law for traditional discontinuous sliding mode control are compared with those of a modified version using parameter adaptation. Furthermore, using the developed model, the implementation of the control law to the piezoelectric actuator is presented, followed by the positioning accuracy and robustness results.  5.2  System Setup A photograph of the piezoelectric actuator assembly mounted on the turret of a standard C N C  lathe is shown in Figure 5.1. A schematic diagram of the system setup is illustrated in Figure 5.2. The assembly consists of a piezoelectric stack actuator, the tool, its holder, and a pair of flexures that support the tool holder. The piezoelectric stack actuator, held between a rigid block and one of the flexures. It pushes the tool indirectly so that its stroke is amplified approximately 50% by the leverage of the flexure bars. The PC sends the control signal to the amplifier which drives the piezoelectric actuator. The displacement of the tool is measured by an optical laser sensor and is fedback to the PC controller.  44  45  Chapter 5. Modeling and Control of Piezo Tool Actuator  5.2.1  Tool Positioning Unit  The actuator used in this research is a piezoelectric stack actuator from Physik Instrumente, model number P-242.30 [77]. Table 5.1 provides the specification for the actuator: Table 5.1: Piezoelectric actuator specification Attribute  Value  Expansion @1000V  40  Stiffness  325  Electrical Capacitance  650 nF  Resonant Frequency  4.7 k H z  Temperature Expansion  0.65  Weight  725  fim N/ M  m  fim/K g  With the piezoelectric stack actuator between the rigid block and the flexure, the tool holder part is connected to the base with two parallel flexures. The actuator is used to exert force against the flexure bar in order to position the tool, utilizing the flexure bar as a leverage. The flexures provide a theoretical mechanical amplification of one and a half times the motion delivered by the piezoelectric stack actuator. Figure 5.3 shows the motion of the tool driven by the actuator and flexure assembly. In addition, the flexure structure provides the returning force for the piezoelectric element. 5.2.2  Measurement Unit  A Laser Nano Sensor (LNS), model number 18/60, produced by DynaVision [25], is mounted at the back of the actuator assembly block to measure the motion of the tool with respect to the actuator by emitting a laser to the mirror which is attached to the back of the tool holder. The laser is projected onto a reflecting surface via a system of two high-grade lenses from a laser source of a miniature laser diode unit. This system is used in conjunction with a beam splitter, and part of the scattered light spot is reflected on a photo detector in the form of two light beams. When  Chapter 5. Modeling and Control of Piezo Tool Actuator  Figure 5.1: Picture o f the piezoelectric actuator assembly on the turret o f a standard lathe  46  Chapter 5. Modeling and Control of Piezo Tool Actuator  47  Laser  Figure 5.2: Schematic diagram of the piezoelectric actuator setup  Chapter 5. Modeling and Control of Piezo Tool Actuator  48  Figure 5.3: Toom motion by the piezoelectric actuator the object being measured is in focus, the spots of the light beam are projected on the center of the photo detector. When the object is shifted out of the focal point, the light spots also move. The object's movement, with respect to the focal point of the sensor, is calculated by summing the outputs from the photo detectors. In order to perform accurate measurements, a highly reflective surface is required to reflect the laser beam. Table 5.2 shows the specifications of the laser sensor. 5.2.3  Signal Processing Unit  A n Indy F3 processor board with TMS320C32 DSP microcontroller is used for data processing, and DL3-A1 Analog Module for I/O operations.  Table 5.3 shows the specifications of the DSP  board. The sampling frequency used is 10 kHz, and the output signal is sent from a PC through the Indy F3 board and D / A converter to the piezoelectric amplifier which sends the input signal to the actuator. The piezoelectric amplifier is a model P-270 Power Amplifier from Physik Instrumente (PI) of Germany, and it is specifically designed to drive piezoelectric elements. The amplifier can supply 0 - 1000 V of output and is controlled by +0..10 V command input.  49  Chapter 5. Modeling and Control of Piezo Tool Actuator  Table 5.2: L N S 18/60 sensor specifications Attribute  Values  Measuring distance (focal point)  18 mm  Effective range  ±60  Resolution  0.1 /mi  Measuring frequency  50 kHz  Temperature range  fj.m  0-50 °C  Distance (X)  ±7.8 v  Intensity (I)  1.5  max  Vdc  -0.7 v dcin range  Enable (E)  +4.7 v dcout of range  Load (X, I, E)  R > lOkfi 130 mV  Sensitivity Angle between sensor and object  / fim  typ.  90°±1° 100 g  Weight Power supply  ±15 vdc ±5%  Power consumption  max ±200 mA  Laser power  5 mW (780 nm)  Table 5.3: DSP specifications Processor  Indy F3  Bus configuration  ISA (2K of 16-bit Dual Port R A M )  Supplier  Spectrum Signal Processing  Analog Input  4 Channels  Analog Output  2 Channels  Chapter 5. Modeling  5.3  50  and Control of Piezo Tool Actuator  Modeling of Actuator Assembly The mechanical flexure together with the piezoelectric actuator is modeled by a single degree  of freedom system, which consists of the spring constant (K) due to the two flexures, the mass ( M ) of the upper carriage (tool holder), and the structural damping (C). Figure 5.4 shows the dynamic model of the piezoelectric actuator system.  Tool Holder  X(s) Tool Extension  "rVWVS  Force — Disturbance F (s) d  Actuator Force F(s) \ \ \ \ \ \ \ v^A \  \ \ \  \ \  \ \  \ \  \ \  Figure 5.4: Modeling of piezoelectric actuator assembly  Therefore, the tool holder and actuator dynamics are modeled as a second order system as shown in Figure 5.5. The open loop transfer function between the tool position (x[/zm]) and the control input (M[N]) in Laplace domain is x{s) =  KG Ms + Cs + K d  a  2  u{s)  1  F (s)  KG d  d  (5.1)  a  where K [V / N] is the gain of digital to analog converter, G [ N / V] is the gain representing the d  a  amplifier and the actuator together, and F is the cutting force disturbance to the actuator [3]. d  Expressing Eq. (5.1) as a differential equation and rearranging,  Chapter 5. Modeling and Control of Piezo Tool Actuator  M KG d  ..,  s  C KG d  a  .. .  K KG  a  d  . .  51  . ,  1  „. .  (5.2)  KG  a  d  a  or M x(t) + C x(t) + K x(t) eff  eff  (5.3)  = u(t) - d(t)  eff  where M ff, C ff, and K f are the equivalent mass, damping, and stiffness of the system, respece  e  e  f  tively.  Disturbance \Fd u  'a  Control Signal D/A  1  o  Piezo Amplifier  Ms  2  + Cs + K  Piezo Actuator Assemby  Figure 5.5: Block diagram of the piezoelectric actuator system  The values of M ff, C ff, and K ff can be identified from impulse hammer test. e  e  e  Figure  5.6 shows the frequency response of the system in the direction of tool motion obtained from the impulse hammer test. As it is seen, the magnitude and phase plots of the transfer function shows one dominant mode at 600 Hz which represents the natural frequency of the flexure structure. The transfer function between the applied force (F) and the tool tip displacement (x) is expressed by a single degree of freedom system as,  £ W F(s)  i  =  ( 5 4 )  M s* + C s + K eff  eff  K  }  eff  Since the dynamics of the piezoelectric actuator are so high compared to that of the flexure structure, the dynamics of the whole system is dominated by the flexure structure, and it is acceptable to treat the dynamics of the piezoelectric actuator as a gain [26].  52  Chapter 5. Modeling and Control of Piezo Tool Actuator  Frequency Response from Impulse Hammer Test 0.6  400  600  800  1000  1200  1400  1600  1800  2000  I  \  j\ I  200  400  I  600  800  1000  1200  1400  1600  1800  Frequency [Hz]  Figure 5.6: Frequency response of the piezoelectric actuator assembly  2000  53  Chapter 5. Modeling and Control of Piezo Tool Actuator  The estimates of the parameters of the system, M ff, e  C ff, e  and K ff e  are obtained as the  following: M  =  2.6 kg  C  =  1.0 x IO" N s / > m  Keff  =  37 N / > m .  eff  eff  3  and the corresponding natural frequency and damping ratio are oo  n  C  =  600 Hz  = 0.05  Figure 5.7 shows the M A T L A B simulated open loop step response of the piezoelectric actuator system with 5 fim reference input, using the estimates of the system parameters above. As shown, it is quite oscillatory without any control.  5.4  Sliding Mode Control of the System  Many control approaches can be applied to design a controller to control the piezoelectric actuator system, such as sliding mode control, conventional PID control, pole-placement control, and so on. In this section, the comparison between the classical sliding mode control and modified version with parameter adaptation is discussed, and relations of the sliding mode control to PID control and pole-placement control are remarked. Since the actuator is used for regulation tasks, the desired displacement x stays constant. d  Therefore, the sliding surface is designed as S  = X(x - x) + (x - x) d  =  X(x — x) — x d  d  (5.5)  Chapter 5. Modeling and Control of Piezo Tool Actuator  54  5 jam reference input  Li»—  0.05  0.1  0.15  0.2  time (s)  Figure 5.7: Simulated open loop step response  0.25  0.3  55  Chapter 5. Modeling and Control of Piezo Tool Actuator  where A [ l / s ] > 0 is a constant which determines the dynamics of the sliding surface. The reason for the selection of the sliding surface as in Eq. (5.5) is that convergence of the sliding surface to zero (S — > 0) would force the position error and velocity of the output to converge to zero (x —> Xd and x —> 0) so that the tool tip can be maintained at the commanded position. The control input (u) must be chosen such that it guarantees asymptotic convergence of the sliding surface. The Lyapunov stability theorem is used to obtain the control law, and it is assumed that M  =M,  eff  eff  5.4.1  Ceff = C ff, and K e  =  eff  K . eff  Classical Sliding Mode Control  With the sliding surface designed as (5.5), in order to design the control law which guarantees S —> 0, a non-negative Lyapunov function is chosen as (5.6)  V Differentiating the Lyapunov function gives MeffSS  V  (5.7)  Since S = —Xx — x, Eq. (5.7) becomes V  = =  M ffS(-Xx-x) e  -MeffSXx  -  MeffSx  (5.8)  and the acceleration of the tool can be expressed from Eq. (5.3) as, x  1  (u  -  d -  CeffX  -  (5.9)  K ffX) e  It follows from (5.9) that, S = -Xx  1 (-CeffX  -  KeffX  + U-  d)  (5.10)  56  Chapter 5. Modeling and Control of Piezo Tool Actuator  Thus, the derivative of the Lyapunov function becomes, V  — d — C ffX  —M ffSXx  — S(u  =  -M SXx  - Su + Sd + SC ffX  =  S(-M ff\±  <  \S\(\M Xx\  e  e  e  eff  + C ffX  + K ffX  e  e  e  e  +  SK x eff  + d) -  e  + \C ffx\  eff  K ffX)  —  =  + \K ffx\ e  Su  + \d\)-Su  (5.11)  In order to satisfy the Lyapunov stability, the derivative of the Lyapunov function should be always non-positive. Thus, the right hand side of the Eq. (5.11) is set equal to a negated function as the following: |5| ( | M  e / /  A x | + \C x\ eff  + \K x\ eff  + \d\) -Su  =  -r,\S\  where 77 > 0 is the minimum convergent rate for S. The control law is attained by solving for u as u = s i g n ( S ) ( | M A x | + \C x\ e//  eff  + \K x\ eff  + \d\ + n)  (5.12)  The block diagram of the piezoelectric actuator system using the control law (5.12) is shown in Figure 5.8, and the control parameters are chosen as A =  2.0 x 10  4  77 = 100 The effect of the control law is simulated in M A T L A B , and Figure 5.9 shows the simulated step response of the system with 5 u.m reference input. As expected, the zoomed-in graph shows the chattering effect of using sliding mode controller (5.12) because of the switching control.  Chapter 5. Modeling and Control of Piezo Tool Actuator  57  Input + ef  M  +  ef  C  +  eff  K  Piezo Actuator System  s  •  C  eff+  M k ef}  + \x\  +  ,ff  K  Laser Sensor Feedback Figure 5.8: Block diagram of the system with sliding mode controller  5.4.2  Modified Version with Parameter Adaptation  Parameter adaptation [100] is a technique that the parameter incorporated in the controller is updated in order to compensate any uncertainties present in the system.  For the piezoelectric  actuator system, the disturbance is considered as uncertainty, and this disturbance is assumed to be constant.  Denoting d as a constant disturbance and d as the disturbance observer, now there are  two variables to be stabilized: the sliding surface S and the error between the real disturbance and its estimate (d - d). Therefore the Lyapunov function is chosen as  V(t) =  MS  2  eff  +  (d - df  (5.13)  where p > 0 is the parameter updating gain. If the derivative of the Lyapunov function is negative definite, the rate of change of the sliding surface and the disturbance prediction error will decrease, thus resulting in a stable system. The derivative of the disturbance observer, d is designed as [124],  d = pnS  (5.14)  Chapter 5. Modeling and Control of Piezo Tool Actuator  5 jum reference input  4.965  0.18  0.182 0.184 0.186 0.188 0.19 0.192 0.194 0.196 0.198 0.2  time (s)  0  0.05  0.1  0.15 time (s)  0.2  0.25  Figure 5.9: Simulated step response of the system with sliding mode controller  0.3  Chapter 5. Modeling and Control of Piezo Tool Actuator  59  or in discrete time domain, d(k) = d(k-l)+  pnST  (5.15)  where T[sec] is the control period, k is the control interval counter in the discrete time domain, and K is used to impose limits on the integral control action against the disturbance and is defined as, 0 «=<  ifd<dand5<0  0 ifd>dandS> 0 1  (5.16)  otherwise  where dr and d denote the lower and upper bounds of d such that d e [d~, d ). +  +  The lower and  upper bounds can be set manually based on the knowledge of the disturbance. Large uncertainty of the disturbance results in a big difference between the two bounds. Differentiating the Lyapunov function in Eq. (5.13), V = M SS  -  eff  (5.17) P  Substituting Eq. (5.10) and Eq. (5.14) into Eq. (5.17), the derivative of the Lyapunov function then becomes, V  =  -M XSx  + C Sx  + K Sx-Su  =  -M XSx  + C f Sx  + K Sx  <  -M XSx  + C Sx  +K Sx-  eff  eff  eff  eff  e f  eff  +  eff  eff  eff  -Su  Sd-S(d-d)K  + Sd + S(d - d)(l - «)  Su + Sd  (5.18)  since S(d — d)(l — K) < 0 always according to Eq. (5.16). Therefore, the following criteria will ensure asymptotic stability (V(t) < 0), -M ffXSx e  + CeffSx + KeffSx  - Su + Sd = -K S  2  S  where K > 0 is the feedback gain. Solving for u, the control law is obtained as, s  (5.19)  60  Chapter 5. Modeling and Control of Piezo Tool Actuator  U = -M \x  + CeffX + K fX  eff  ef  (5.20)  + d + KS S  where u denotes the control voltage of the piezoelectric actuator. The first four terms in Eq. (5.20) in the right hand side is for feedforward compensation and is for feedback control. In the feedforward control part, —M ffXx  KS S  e  is used to handle M ffX, e  C f fi + K f fx is used to compensate C f fi + K f fx, and d is the output of a disturbance observer e  e  e  e  for estimating the disturbance d. In the feedback control part, K is a control parameter. s  The block diagram of using the modified version of sliding mode control with parameter adaptation is illustrated in Figure 5.10, and after tuning the controller the following control parameters are selected: K  =  1.325 x 1 0  A =  4.302 x 10  2  p =  3.354 x 10  1  s  - 3  [Ns/H [1/s] [N//H  Figure 5.11 shows the simulated step response with 5 urn reference input, and it is zoomed to compare with the simulation result of the classical sliding mode control. As shown, the response is very satisfactory not showing any chattering effect, and the tracking error is less as well. 5.4.3  Relations to PID and Pole-Placement  Since the reference input is always constant for regulation purposes, zeros of the closed-loop transfer function do not have much influence on the system response. Therefore, for unbounded disturbance (d~ = —oo, d  +  the following gains:  = +oo), the control law (5.20) corresponds to a PID controller with  61  Chapter 5. Modeling and Control of Piezo Tool Actuator  J_  s x  d  +  +  m  X  p h  s  +  K  +  +  U  +  M  +  >-f  +  ef  C  +  eff  K  Piezo Actuator System (C -M f) eff  K  e  +  eff  Laser Sensor Feedback  Figure 5.10: Block diagram of the system with modified version of sliding mode controller (parameter adaptation)  K  =  K + MX  K  =  K X + p-  Ki  -  pX = 1.443 x 10  D  P  s  s  eff  - C  = 1.443 x 10~ [N s / p,m] 3  eff  Keff = 2.886 [N / pm] [N/pms]  4  This PID controller would result in the same dynamics since the poles of the closed-loop transfer function are the same as that of the controller (5.20). For the experiment, this PID controller structure is used to control the piezoelectric actuator. The limits of the force disturbance are approximately — 100 and +100 N . However, incorporating these limits to the controller would force the sampling frequency to be decreased due to the extension in the computational time. Thus, the disturbance limits are assumed to be unbounded. It should be noted that when dr — —oo and d  +  = +oo, the control law (5.20) also corresponds  to pole-placement control with two zeros at z\ — 43.0 and z = —2532.5 and three poles at 2  Chapter 5. Modeling  62  and Control of Piezo Tool Actuator  . - - . .K  5.004 5.003 5.002 „ C  5.001  £  5.000  <s a>  •  CO 4.999  a. 4.998 ~ Q  4.997 4.996 4.995  0. 8 0.182 0.184 0.186 0.188 0.19 0.192 0.194 0.196 0.198 0.2  time (s)  0.05  0.1  0.15  time (s)  0.2  0.25  0.3  Figure 5.11: Simulated step response of the system with sliding mode controller with parameter adaptation  63  Chapter 5. Modeling and Control of Piezo Tool Actuator  s  l 2  = —254.7 ± 3582.8i and S3 = —430.2. The system dynamic performance is dominated by the  principal poles s i ^ = —254.7 ± 3582.8i.  5.5  Experimental Results of Controller The displacement of the tool x is measured by Laser Nano Sensor (LNS) as shown in Figure  5.2, and the velocity x is estimated by taking the derivative of the measured displacement from the LNS. However, the evaluation of velocity from discrete position commands and LNS readings may be noisy, so the following simple low pass filter is used to smooth them, x(k) = a x(k - 1) +  [x (k) - x {k - 1)]  c  d  d  (5.21)  where T[s] is the control period and a e [0,1]. c  In the measurement part, one micrometer corresponds to 7.8 x 1 0  Volts, and in the actuator  - 2  part, one Volt corresponds to 10 /jm which is equivalent to 370 N force delivery by the actuator. Hence, the real parameters used in the computer program are  K*  D  = iW(7.8  K*  P  =  K /(7.8  K*  =  10/(7.8 x 10~ x 370) = O05 x 10  P  x  !0~  x 10  2 x  - 2  2  370) = 0 5 x 10~ x 370) =  4  -01 4  The control law was implemented using a real time cyclic executive software designed specifically for high speed intelligent signal processing on a PC with Windows N T operating system [72], and the developed cyclic executive was used as an operating system for DSP. The software was developed in standard C and designed according to open software standards; therefore, it is portable to other DSP platforms. Figure 5.12 shows the step response of the open and closed loop systems with 5 u.m reference input without cutting. The closed loop step response is very similar to the simulated result. The  Chapter 5. Modeling and Control of Piezo Tool Actuator  64  rise time for the closed loop step response is about 0.02 s, the overshoot is less than 2%, and the steady state error is less than 50 n m . Cutting experiments were conducted to evaluate the effectiveness of the controller. The feedrate was 0.05 mm / rev, the cutting speed was 300 m / min, and the depth of cut was 0.1 m m for the cut with and without control. Figure 5.13 shows the cutting results.  Without any con-  trol, because of the cutting force, the tool statically deflects, and the deformation is proportional to the magnitude of the cutting force. However, for the cut with control, there is no change in the position of the tool, meaning that the controller completely compensates the cutting force and generates a robust control performance with high positioning accuracy (±0.1 /xm). The positioning accuracy is maintained within the resolution of the laser sensor which is ± 0 . 1 ^ m .  5.6  Summary  Development of the mathematical model of the piezoelectric actuator and implementation of the classical sliding mode controller and the modified version of sliding mode controller was presented and simulated.  The performance of the piezoelectric actuator with the modified version  was then tested with a step response and compared to the open loop response. The performance is satisfactory, and positioning accuracy of ±0.1 / / m has been achieved, and this can improve if the resolution of the feedback sensor increases. It was also shown that the controller is robust against cutting forces by comparing to the cutting result without any control.  Chapter 5. Modeling and Control of Piezo Tool Actuator  65  Open loop step response (5 urn reference input) 1  0.05  1  1  i  i  0.1  0.15  1  i 0.2  i  i  i  0.25  0.3  0.35  0.3  0.35  time (s) Closed loop step response (5 u.m reference input)  0.05  0.1  0.15  0.2  0.25  time (s) Figure 5.12: Step response of open and closed loop systems  Chapter 5. Modeling and Control of Piezo Tool Actuator  Without Control CD  -I—»  0)  E o o E c g  'w o Q. . O  o  time (s)  With Control CD  •*—»  cu E o i— o E c o to o  Start of cut •-+  End of cut  Q. .  "5 O  time (s)  Figure 5.13: Comparison between cuts with and without control  Chapter 6 Experimental Results  6.1  Overview Achievement of high positioning accuracy (±0.1 pm) by means of the piezoelectric actuator  and an effective controller is presented in the previous chapters. However, surface finish quality of a machined surface mostly depends on the cutting parameters, so it is important to determine adequate cutting parameters which would yield the best surface finish. The required surface roughness values for precision machining are R  a  < 1.0 fim and Rmax < 2.5 fim. Considering  that for precision machining the depth of cut is very small, the optimum range of feed rate and cutting speed for desirable surface finish needs to be attained. are obtained and what the resulting surface finish is.  This chapter presents how they  The preliminary selection of the cutting  parameters are made based on the geometrical model of surface finish and suggestions made in the literature. Then a series of experiments is conducted to examine the effect of each parameter on the quality of the machined surface and to determine the range of each parameter which would result in better surface finish than is required.  6.2  Experimental Setup The turning machine used is the Hardinge SUPERSLANT with Fanuc GN6T C N C Chemical  vapor deposition (CVD) coated carbide inserts from Valenite Inc. with multi-layer coating of TiC, TiCN,  AI2O3,  and TiN are used, and the nose radius of the inserts is 0.4 mm. The tool holder has  the following geometrical information: • Back rake angle = 9 deg  67  68  Chapter 6. Experimental Results  • Side rake angle = — 5 deg • End cutting edge angle = 52 deg • End relief angle = 9 deg • Side cutting edge angle = — 3 deg • Side relief angle = 5 deg The workpiece used for the experiment is low alloy, heat treated steel (SAE 4340) with the composition of 0.80Cr, 1.8Ni, 0.25Mo, and 0.38-0.43C with a hardness of 35-40 HRC. The length of the workpiece is 100 mm, and the diameter is approximately 65 mm. Figure 6.1 shows the workpiece used for the experiments. The cutting fluid used is undyed C I M P E R I A L 1070 with 5% dilution from Milacron company. 5.0 mm  TI  U  9.5 mm  U  U  U  E E  o  iri co  42.0 mm  58.0 mm  Figure 6.1: Schematic diagram of workpiece used for the experiments  The surface roughness is measured with a stylus-type surface measurement instrument, Surftest SV-502 with manual operation column which is manufactured by Mitutoyo (Figure 6.2).  The  Chapter 6. Experimental  69  Results  Figure 6.2: Stylus-type surface measurement instrument used resolution of the measurement instrument is 0.005 /im, and the length of the machined surface measured is 8.0 m m at a speed of 1.0 m m / s.  6.3  Preliminary Selection of Cutting Parameters  The geometrical model of surface roughness discussed in chapter 2 represents the minimum peak-to-valley surface roughness value (Rmax), so the actual Rmax value will always be greater. It is nevertheless a good indicator of the peak-to-valley roughness values at the best cutting conditions, that is, when the effect of other factors deteriorating the surface finish is minimized. Since the depth of cut is always very small ( ^ 5 — 10 //m), the geometrical Rm  ax  value is given by the  following equation: Rmax = r -  -  v  J  (6.1)  where r is the nose radius and / the feed rate. The nose radius of the insert used is 0.4 mm. The following table shows the values of geometrical R x ma  for different feedrates:  Chapter 6. Experimental Results  70  Table 6.1: The peak-to-valley roughness values for different feed rates obtained from Eq. (6.1) Feedrate [mm / rev]  Rmax [^ni]  0.01  0.0313  0.02  0.1250  0.03  0.2813  0.04  0.5003  0.05  0.7820  0.06  1.1266  0.07  1.5342  0.08  2.0050  0.09  2.5393  As shown, the feedrate cannot be greater than 0.08 mm / rev since at 0.09 mm / rev, the value of Rmax exceeds the maximum value required, 2.5 am.  Therefore, feedrates from 0.01 — 0.08  mm / rev would result in peak-to-valley surface roughness values less than 2.5 ^xm at an ideal cutting condition. The cutting speed should be selected above the range where built-up-edge occurs. The disappearance of the built-up-edge mainly depends on the cutting speed; however, it is also affected by other factors such as feedrate, hardness of the workpiece, and cutting fluid. Many researchers have tried to identify the critical cutting speed above which the built-up-edge does not appear. Chandiramani [12] has conducted experiments with cold-drawn resulfurized steel (MXC) and cold-drawn leaded steel (PBC). His results show that at cutting speeds greater than 100 m / min, the surface finish began to improve, meaning that the built-up-edge began to disappear. For smaller feedrates, the disappearance of the built-up-edge is delayed to a higher cutting speed. He also found that for Steel 4340 the critical cutting speed could be as low as 10 m / min. However, depending on the treatment on the material, the value of critical cutting speed may vary. Shaw [93] showed that at the feedrate of 0.05 mm / rev, the built-up-edge disappear at cutting speeds greater than  71  Chapter 6. Experimental Results  100 m / min for AISI 1045. Sata [86] provided the critical cutting speed for three different steels, AISI1015, AISI 1040, and AISI 4140. Table 6.2 shows the critical cutting speeds for the steels at the feedrate of 0.1 mm / rev. Table 6.2: Critical cutting speed for disapperance of built-up-edge [86] Work material  Critical cutting speed [m / min]  AISI1015  280  AISI1040  115  AISI4140  37  As it is seen, for harder materials the built-up-edge disappears at lower cutting speeds. For AISI 4140 which is similar in hardness to SAE 4340, the critical cutting speed increased to 70 m / min at 0.05 mm / rev, and it could increase further for smaller feed rates.  Therefore, the  cutting speed of 100 m / min is chosen as the minimum cutting speed for the experiment in order to make sure that built-up-edge does not form. To see the effect of feedrates, experiments without any cutting fluid were performed for the range of feedrates 0.005 — 0.09 mm / min at a cutting speed 150 m / min and a depth of cut 5 ^ m . The result is shown in Figure 6.3. Ra, Rmax, Rz, and R are average, peak-to-valley, ten-point height, and root mean square q  surface roughness values respectively. These values are obtained from the software, SurfPak 4.10, which analyzes the measurement from the surface measurement instrument Surftest SV-502. As shown in Figure 6.3, the surface finish deteriorates at small feedrates owing to the fact that either a built-up-edge formed or, rather than cutting, the tool edge was pushing and rubbing the material. Furthermore, as expected, the surface roughness increased at higher feedrates. the selected range of feedrates and cutting speeds.  Table 6.3 shows  Chapter 6. Experimental  72  Results  Surface finish at v = 150 m/min -•-Ra  Rmax  Rz  ^-Rq  9 8  I*  <  £ 6 •§•5 (A W  0  c o  -  v i * ^ * = — i — - <  i  0.01  I  0.02  0.03  0.04  0.05  0.06  0.07  0.08  Feed rate [mm/rev]  Figure 6.3: Effect of feedrates on surface roughness  Table 6.3: The cutting conditions tested Range  Feedrate Cutting speed  0.03  -0.06  125 - 2 5 0  mm/rev m/min  Increment  0.005 m m / rev 25 m /  min  0.09  0.1  Chapter 6. Experimental  6.4  Results  73  Cutting Tests The cutting process in Figure 1.1 shows four stages of cutting involved for the precision ma-  chining on a turning center. For the experiment, only the last two stages of the cutting process, i.e., finish and precision finish turning, were performed to determine the surface finish quality of the given cutting conditions. For finish turning, the depth of cut was set by the C N C lathe as 0.1 mm, whereas for precision finish turning, the depth of cut was set by the piezoelectric actuator as 5.0 jum. In practice, the final precision depth of cut is set by the operator who determines it via electronic snap gages. The finish machining is done by positioning the tool 0.1 mm radially into the workpiece, and after turning the workpiece for a short distance (for example, bearing locations), the C N C lathe's radial axis is electronically locked. The tool is then retracted radially with the fast tool servo by 5 (im to provide a clearance between the tool tip and the workpiece surface and brought to the beginning of the workpiece by traversing along the shaft axis. Then, the tool is given a total of 10 fim radial displacement which results in 5 /im effective radial depth of cut. Finally, the precision turning can be done with 5 jum depth of cut. A series of experiments were conducted within the ranges given in the Table 6.3. The length of cut was 9.5 mm for each cutting condition. After cutting the length on the workpiece for each cutting condition, new tool tip is used for the next cut.  Readings of surface roughness values  were recorded at three different locations on the workpiece. The piezoelectric actuator was under control for the duration of cut. Figure 6.4 shows the reading from the laser sensor and the control input to the actuator for a typical cut. 6.4.1  Wet Cutting  Wet cutting results are shown in Figure 6.5 and 6.6, and surface roughness profiles at different cutting speeds for the feedrates of0.03, 0.045, and 0.06 mm / rev are shown in Figure 6.7, 6.8, and  Chapter 6. Experimental  •55  74  Results  4  E  2  Jetting  o  I  c g 55 o  :  2  T~'  0  0.  6  8  10  12  14  10  12  14  time (s) Or  o •0.2 >  "ro -0.4 '(/>  •0.6 c o O •0.8 -1  6  8  time (s)  Figure 6.4: The position o f the tool and control signal into the piezoelectric actuator o f a typical cut  Chapter 6. Experimental  75  Results  6.9 respectively. The values shown in Figure 6.5 and 6.6 are average values of three measurement data. The actual values are shown in Appendix A . For all cutting results, no built-up-edge was observed. As expected from the kinematics of a cutting process, the surface roughness is seen to increase as the feedrate increases. However, it can be noted that in contrast to the geometrical model, at the feedrate of 0.03 mm / rev for low speeds, the surface roughness is higher than the ones obtained at higher feedrates. As the speed increases, the roughness value decreases for 0.03 mm / rev. It can be noted that at 225 and 250 m / min, the surface roughness value for 0.03 mm / rev is the lowest as compared to that of other feedrates, which is expected theoretically.  Shaw and Crowell [93] reported that the cutting temperature  should be sufficiently high that the layer of material adjacent to the tool face be thermally softened to provide low friction and a high shear angle, thus self-lubricating the tool-workpiece and toolchip contact zones. Therefore, at low feedrates, because the temperature is not high enough, the friction on the tool-chip and tool-workpiece interfaces is high, which may have resulted in the poor surface finish. There is also a minimum undeformed chip thickness below which a chip will not be formed, and when this occurs, rubbing takes place. This minimum undeformed chip thickness depends primarily upon the feedrate and also upon the nose radius, the cutting speed, and the stiffness of the system [93]. Therefore, for small feedrates such as 0.03 mm / rev, the cutting speed should be increased so that the undeformed chip thickness is higher than the minimum chip thickness. The effect of increasing the cutting speed at small feedrates (0.03 mm / rev) is well illustrated in Figure 6.5 and 6.6. Many researchers observed that when the built-up-edge is absent, the cutting speed has little effect on the surface roughness [87, 93, 42]. However, at high cutting speeds, though the friction on the contact zone is low, the temperature can partially change the metalurgical property of the material on the surface resulting in poor surface finish. When the temperature is high, the tool  Chapter 6. Experimental Results  76  Ra  0.7  f  0.6  |  0.5  o  1 c  "I 2 8  —•-  0.03 mm/rev  —•-  0.035 mm/rev  -f" f = — A - -f" — • - f~ —0- - f "  0.04 mm/rev 0.045 mm/rev 0.05 mm/rev  —B- - f " 0 . 0 5 5 m m / r e v —A- - f - 0.06 mm/rev  0.4 0.3  0 2  100  125  150  175  200  225  250  275  Cutting speed [m/min]  Rmax co  c  E  —•-  - f - 0.03 mm/rev  —•—A-  -f" 0.035 mm/rev -f" 0.04 mm/rev  —«-  -f = 0.045 mm/rev  —0- - f " 0 . 0 5 m m / r e v —B- - f " 0 . 0 5 5 m m / r e v  £o 4  Ito©  —A- -f = 0.06 mm/rev  c  O) 3  o 1— o o  €  0 100  125  150  175  200  225  250  275  Cutting speed [m/min] Figure 6.5: Average values o f surface roughness R (wet cutting results)  a  and Rmax versus cutting speed and feedrate  Chapter 6. Experimental Results  77  Rz  4.5 (fl C  E  mm/rev  mm/rev  0.045 m m / r e v —e— - f " 0.05 m m / r e v — B - - f " 0.055 m m / r e v  2.5 2 1.5  0)  1  CO  0.5  o 3  —A- - f - 0.06  0.035  —•— f -  3  D) 3  mm/rev  —A—  o  CD  - f = 0.04  a  4  E  c .c  mm/rev  -f"  3.5  w w  - f - 0.03  0 100  125  150  175  200  225  250  275  225  250  275  Cutting speed [m/min]  w  c o o  1 (fl (fl  cu  c  -C  D) D  2 O fi 3  CO  100  125  150  175  200  Cutting speed [m/min]  Figure 6.6: Average values of surface roughness R and R versus cutting speed and feedrate (wet cutting results) z  q  Chapter 6. Experimental  Results  78  Rmax = 5.606 micrometer, f = 0.03 mm/rev, v = 125 m/min  I  i  i  i  i  i  i  i  i  i  I  2  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  4  Rmax = 3.984 micrometer, f = 0.03 mm/rev, v = 125 m/min i  T  CD CD  I 2  I  2.2  I  2.4  i  I  2.6  i  I  2.8  i  I  3  i  I  3.2  i  I  3.4  i  I  3.6  r  I  3.8  I 4  Rmax = 3 . 1 9 9 micrometer, f = 0.03 mm/rev, v = 175 m/min  Rmax = 2.596 micrometer, f = 0.03 mm/rev, v = 2 0 0 m/min CD  |  I 2  1  I  2.2  1  I  2.4  1  I  2.6  1  I  2.8  1  I  3  1  I  3.2  1  I  3.4  r  I  3.6  I  I  3.8  4  L 3.8  4  Rmax = 1.751 micrometer, f = 0.03 mm/rev, v = 2 5 0 m/min  2  2.2  2.4  2.6  2.8  j 3  i 3.2  i 3.4  i 3.6  Length [mm] Figure 6.7: Surface roughness profiles of machined surfaces at different cutting speeds for wet cutting ( / = 0.03 mm / rev)  Chapter 6. Experimental  79  Results  Rmax = 1.305 micrometer, f = 0.045 mm/rev, v = 125 m/min  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 1.309 micrometer, f = 0.045 mm/rev, v = 150 m/min  £  2  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 1.501 micrometer, f = 0.045 mm/rev, v = 175 m/min  'CD  X  2.2  to to  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 1.658 micrometer, f = 0.045 mm/rev, v = 2 0 0 m/min  <D  c  CD  o  0*  o  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 2.655 micrometer, f = 0.045 mm/rev, v = 2 2 5 m/min  t7)  Rmax = 2.821 micrometer, f = 0.045 mm/rev, v = 2 5 0 m/min T  2.6  2.8  3  3.2  3.4  3.6  3.8  Length [mm] Figure 6.8: Surface roughness profiles of machined surfaces at different cutting speeds for wet cutting (/ = 0.045 mm / rev)  Chapter 6. Experimental  80  Results  Rmax = 2.1870 micrometer, f = 0.06 mm/rev, v = 125 m/min  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 2.4714 micrometer, f = 0.06 mm/rev, v = 150 m/min  CD  2  2.2  o  X  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 2.1441 micrometer, f = 0.06 mm/rev, v = 175 m/min  2  2.2  CO CO CD  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 2.6014 micrometer, f = 0.06 mm/rev, v = 2 0 0 m/min  c  o  2  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 2.3073 micrometer, f = 0.06 mm/rev, v = 2 2 5 m/min  t/)  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  3.8  Rmax = 2 . 9 3 9 9 micrometer, f = 0.06 mm/rev, v = 2 5 0 m/min  Figure 6.9: Surface roughness profiles of machined surfaces at different cutting speeds for wet cutting ( / = 0.06 mm / rev)  Chapter 6. Experimental Results  81  Ra  0.3 T  -f" • -f" -f" —hr— 0 - -f" —B- - f "  —  2 0.26 o  •  -  0.035 mm/rev 0.04 mm/rev 0.045 mm/rev 0.05 mm/rev 0.055 mm/rev  E  0.22  in o  c  g or  0.18  cu  I  0.14  3 CO  0.1 100  125  150  175  200  225  Cutting Speed [m/min]  Rmax  —  •  •  2.5  -  -f =  -f"  — 4 - -f = —©- - f =  w c o  —B-- f "  2  0.035 mm/rev 0.04 mm/rev 0.045 mm/rev 0.05 mm/rev 0.055 mm/rev  to CO  c  TO  8  1.5  1  CO  0.5 100  125  150  175  200  Cutting Speed [m/min]  Figure 6.10: Verificaton of the selected cutting conditions  225  Chapter 6. Experimental  82  Results  wears more rapidly which also has detrimental effects on surface finish. Yet, since cutting fluid is used, temperature may not affect tool wear and surface finish that critically. Nevertheless, any inherent vibration due to the flexibility of the machine is more amplified at high cutting speeds. Therefore, except for the feedrate of 0.03 mm / rev, the graph shows a slight increase in surface roughness at cutting speeds greater than 200 m / min. As shown in Figure 6.8, the tool marks are larger at high cutting speeds due to the increased vibration. From Figure 6.5, it can be seen that for the cutting conditions of 0.035 — 0.055 m m / rev and 125 — 200 m / min, R < 0.3 um and Rmax < 2.0 um have been achieved. This result is better a  than the required (R  a  < 1.0 um and Rmax < 2.5 /xm ). Without the accurate positioning of the  tool into the depth of cut by the piezoelectric actuator and robust control of the actuator, this can be obtained only on a grinding or ultra-precision turning machines. Cutting experiments have been repeated at those cutting conditions to verify, and Figure 6.10 shows R and Rmax values at the a  conditions. 6.4.2  Dry Cutting  Dry cutting results (average values) are given in Figure 6.11 and 6.12 for different feedrates, and the surface profiles for the feedrates of 0.04 m m / rev and 0.055 m m / rev are shown in Figure 6.13 and 6.14 respectively. The actual values of measured data are shown in Appendix A. For dry cutting, the surface finish quality is poor until the feedrate is increased to 0.05 m m / rev. This may be due to the increased friction since no lubricant is used to lubricate the tool-chip and tool-workpiece interfaces. In fact, many researchers [69, 93] observed similar result that the surface finish is poor at low feedrates, and it improves until the feedrate is increased to a certain value 0.05 m m / rev for case hardened steel AISI 1552 [69]). Then, as the feedrate is further increased, the surface roughness value follows the theoretical trend: for higher feedrates, the surface roughness increases. Due to the increased friction, the minimum undeformed chip thickness for  Chapter 6. Experimental  Results  83  the tool tip to penetrate into the material becomes higher. When the tool tip cannot penetrate into the workpiece, rubbing takes place. Therefore, though rubbing did not occur at 0.04 m m / rev and 125 m / m i n for wet cutting, rubbing took place for dry cutting at the same condition. When rubbing actually takes place, it is well displayed on the surface o f the workpiece. Figure 6.15 shows the workpiece with the rubbing mark ( / = 0.04 m m / rev and v = 125 m / min), and the surface is dull and non-reflective. A l s o , it is observed that the surface finish is deteriorated at high cutting speeds because o f the increased temperature, which causes rapid tool wear, softening o f material, thermal expansion, and metallurgical changes on the surface o f the workpiece. For dry cutting, because no cutting fluid is used, the increased temperature can exceed the a — 7 transition temperature (700 — 900 ° C ) . A s shown in Figure 6.11 and 6.12, the surface finish starts to worsen as cutting speed is increased. It can be further noted that for higher feedrates, the surface finish starts to deteriorate sooner, that is at lower cutting speeds. For example, at 0.05 m m / rev, the surface finish is good until the cutting speed is at 275 m / m i n , but at 0.055 m m / rev and 0.06 m m / rev, the surface finish is only good until 225 and 150 m / m i n respectively. Desired surface finish is obtained for the following cutting conditions: • feedrate: 0.05 — 0.055 m m / rev • cutting speed: 125 — 275 m / m i n at 0.05 m m / rev and 125 — 200 m / m i n at 0.055 m m / rev For dry cutting, the tool wears faster compared to wet cutting; therefore, coated inserts which are resistant to wear should be used for dry cutting in order to obtain acceptable surface finish. The same type o f inserts are used for both dry and wet cutting, and Figure 6.17 shows the wear on the tool tips after a single cut ( / = 0.05 m m / rev and v = 250 m / min). There is hardly any tool wear on the tool tip from wet cutting; however, tool wear from dry cutting is easily seen even only  Chapter  6. Experimental  Results  84  f = 0.04 mm/rev  225 Cutting speed [mm/rev]  f = 0.045 mm/rev  150  175  200  225  250  275  Cutting speed [mm/rev]  f = 0.05 mm/rev  150  175  200  225  250  275  300  325  Cutting speed [mm/rev]  Figure 6 . 1 1 : Average values of surface roughness obtained from dry cutting  85  Chapter 6. Experimental Results  f = 0.055 mm/rev —•— Ra — m — Rmax —±— Rq —•— Rz  275 Cutting speed [mm/rev]  f = 0.06 mm/rev —•— Ra —m— Rmax — A — Rq —•— Rz  225 Cutting speed [mm/rev]  Figure 6.12: Average values of surface roughness obtained from dry cutting  Chapter 6. Experimental  Results  86  Rmax = 3.568 micrometer, f = 0.04 mm/rev, v = 125 m/min T  Rmax = 1.573 micrometer, f = 0.04 mm/rev, v = 150 m/min T  Rmax = 3.934 micrometer, f = 0.04 mm/rev, v = 175 m/min  Rmax = 4.418 micrometer, f = 0.04 mm/rev, v = 2 0 0 m/min ~r  Figure 6.13: Surface roughness profiles of machined surfaces at different cutting speeds for dry cutting (/ = 0.04 mm / rev)  Chapter 6. Experimental Results  87  Rmax = 1.337 micrometer, f = 0.055 mm/rev, v = 125 m/min  3.2  3.4  3.6  3.8  4.2  4.4  4.6  4.8  Rmax = 1.408 micrometer, f = 0.055 mm/rev, v = 150 m/min T r  3.2  3.4  3.6  3.8  4  4.2  4.4  4.6  4.8  Rmax = 1.436 micrometer, f = 0.055 mm/rev, v = 175 m/min  Rmax = 1.397 micrometer, f = 0.055 mm/rev, v = 200 m/min T  Z3  O CH CD O  00  Rmax = 2.669 micrometer, f = 0.055 mm/rev, v = 2 2 5 m/min  Rmax = 11.99 micrometer, f = 0.055 mm/rev, v = 250 m/min  3.8  4  4.2  Length [mm] Figure 6.14: Surface roughness profiles of machined surfaces at different cutting speeds for dry cutting ( / = 0.055 mm / rev)  Chapter 6. Experimental Results  Figure 6.15: Workpiece with rubbing mark  88  Chapter 6. Experimental  89  Results  1  Rmax = 1.227 micrometer, f = 0.05 mm/rev, v = 125 m/min  Rmax = 3.151 micrometer, f = 0.05 mm/rev, v = 150 m/min  cu o  Rmax = 5.616 micrometer, f = 0.05 mm/rev, v = 175 m/min (fi  4.8  5  5.2  Length [mm]  Figure 6.16: Surface roughness profiles of machined surfaces at different cutting speeds for dry cutting using the same tool tip for all three cuts ( / = 0.05 mm / rev)  Chapter 6. Experimental Results  Wet cutting  90  Dry cutting  Figure 6.17: Wear on the tip of the inserts  Wet cutting  Dry cutting  Figure 6.18: Chips produced from wet and dry cutting  Chapter 6. Experimental Results  91  after a single cut. At low speeds, more tool wear results for dry cutting because of high friction and insufficient material plastification. At high speeds, tool wear is caused by high temperature. Figure 6.16 shows the surface profiles with the effect of tool wear on surface finish. The same tool tip is used to cut all three surfaces in the sequence. At the feedrate of 0.05 mm / rev, the cutting speeds in the range of 125 — 200 m / min do not have much effect on surface finish. Therefore, the poor surface finish shown for the second and third cut can be asssumed to be caused by tool wear. Konig et al. [50] obtained the result that tool life has an upside-down parabolic relationship with cutting speed, and minimum tool wear and maximum tool life occurred near the cutting speed of 150 m / min with 100Cr6 (62 HRC) work material, P C B N inserts, no coolant, feedrate of 0.05 mm / rev, and depth of cut of 0.05 mm. Also, at high speed, problems are associated with regard to the high temperature strength of the cutting material and transformation of the micro-structure in the outer zone of the workpiece, resulting in poor surface finish. Figure 6.18 shows the chips produced from both wet and dry cutting (/ = 0.05 mm / rev and v = 250 m / min). As it is shown, the structure of the chip from dry cutting has been transformed, and the color of the chips changed to dark blue because of high temperature associated on the tool-chip interface. 6.5  Summary This chapter presented the experiments carried out with the piezoelectric actuator. The feed-  back signal from the laser sensor showed accurate positioning result of the tool, and the compensation of the cutting forces were shown to be achieved successfully with the modified version of the sliding mode control using the parameter adaptation scheme. Carefully examining and testing the effect of cutting parameters and using the piezoelectric actuator to control the depth of cut (5 pm), the cutting conditions which produce the required surface finish, /xm, are achieved for both wet and dry cutting (Table 6.4).  R  a  <  1.0 /xm and  Rmax  < 2.5  The positioning accuracy and sur-  92  Chapter 6. Experimental Results  face finish achieved are sufficient to permit the piezoelectric actuator to be used as a substitute for precision grinding operations of short distances such as bearing locations. Table 6.4: Optimized cutting conditions Wet cutting Feedrate Cutting speed  0.035 -0.055 125  -200  mm/rev  m/min  Dry cutting 0.05 -0.055 m m / r e v 125 — 275 m /  m i n at  0.05  m m / rev  125 - 200 m / m i n at 0.055 m m / rev  Chapter 7 Conclusions and Future work 7.1  Conclusions The objective of this thesis is to achieve micro-positioning accuracy of the tool and high quality  of surface finish, which only grinding offers, on a conventional turning machine.  It has been  demonstrated that all the processes of precision machining can be carried out on a turning machine, eliminating the need to use the grinding machine for the final precision finishing process.  The  positioning accuracy of a conventional machine is approximately 10 u.m; therefore, it is not capable of providing the accuracy required (< 1 um).  A piezoelectric actuator, previously designed and  developed, was used to enhance the positioning accuracy of a conventional lathe. Mounted on the turret, it executes the precision positioning while the lathe executes the coarse positioning. The piezoelectric actuator is modeled as a second order dynamic system because the dynamics of the assembly is dominated by the mode of theflexurestructure. Since robust control is necessary to control the piezoelectric actuator during cutting process, sliding mode control is introduced, and problems associated with it are discussed.  A modified version of sliding mode control using  parameter adaptation is finally used to control the actuator. Sliding mode control with parameter adaptation utilizes parameter estimates to compensate for the disturbances and system parameter uncertainties. Using the mathematical model of the actuator assembly, this controller is simulated and tested experimentally, and very robust and accurate positioning of the tool is achieved  ±0.1  Using the piezoelectric actuator to control the tool and set the depth of cut into the workpiece, cutting experiments are carried out to optimize the cutting conditions. Surface finish of the work-  93  Chapter 7. Conclusions and Future work  94  piece is measured on three different locations with a stylus-type surface measurement instrument. The effect of each parameter is carefully studied and explained. The required surface finish quality to emulate a precision finishing operation on a grinding machine is to have an average surface roughness  R  a  <  1.0 //m and peak-to-valley surface roughness  Rmax  < 2.5 /mi. The follow-  ing range of feedrate and cutting speed resulted in surface finishes within the required quality of a finished surface (depth of cut = 5 fj,m). • For cutting with cutting fluids a. feedrate: 0.035 — 0.055 mm / rev b. cutting speed: 125 — 200 m / min • For cutting without cutting fluids a. feedrate: 0.05 — 0.055 mm / rev b. cutting speed: 125 - 275 m / min at 0.05 mm / rev and 125 - 200 m / min at 0.055 mm / rev Therefore, the results indicate that with the help of the piezoelectric actuator, the positioning accuracy of tool and surface finish quality of a machined surface which grinding offers can be achieved on a conventional turning machine.  7.2  Future Work  The natural frequency of the piezoelectric actuator is may be too low, as this frequency can be easily excited during cutting, leading to chatter vibrations. Thus, a stiffer actuator is required to increase the bandwidth of the actuator and actively damp out frequencies that are higher than the natural frequency of the current actuator. Also, the current actuator occupies two tool locations in the turret of the lathe, so a smaller actuator that fits in one tool location and has a solid mounting needs to be considered in order to be used more practically in industry.  Chapter 7. Conclusions and Future work  95  The laser sensor measures the position of the tool tip, not how much of the workpiece is actually cut. Structural deflection and eccentricity on the spindle rotation can cause the workpiece to be undercut or to have increased roundness and cylindricity error. Thus, feedback from the measurement of the workpiece is necessary in order to compensate these errors on-line when structural deflection or eccentricity on the spindle rotation are present, and simultaneously the diameter of the workpiece can be determined. 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Measured data for surface finish  106  0.8 J2 0.7 o o 0.6 to  (0 0  0.03 mm/rev 0.035 mm/rev 0.04 mm/rev • f = 0.045 mm/rev o f = 0.05 mm/rev • f = 0.055 mm/rev A f = 0.06 mm/rev •  Ra •  • •  t  0.5  = 0.4 O)  g 0.3 3 CO  i !  f-  0.1 0 100  I -4  i  a 0.2 •g  125  150  f =  • f =  175  200  225  250  275  Cutting speed [m/min]  U  •  CO  c o  • f = 0.03 mm/rev • f = 0.035 mm/rev A f = 0.04 mm/rev • f = 0.045 mm/rev o f = 0.05 mm/rev • f = 0.055 mm/rev A f - 0.06 mm/rev  Rmax  R • •  •J  l_  o 4  •  'E  CO CO CO  r  c  s:  O) 3  2 ©  CO  &  2  A  1 1  1 3  2  100  125  150  •  175  !  1  1  200  225  250  275  Cutting speed [m/min]  Figure A . l : Measured data of Surface roughness R and R (wet cutting results) A  MAX  versus cutting speed and feedrate  Appendix A. Measured data for surface finish  107  0.03 mm/rev 0.035 mm/rev Af = 0.04 mm/rev • f = 0.045 mm/rev o f = 0.05 mm/rev • f = 0.055 mm/rev A f = 0.06 mm/rev •f=  Rq  •  1 0.9  •  0.8 0.7  S 0-6 cu  c 0.5 o  0.4 0.3  £ 0.2 co 0.1  100  t  1  125  150  i  f=  i A  ! 1 175  200  225  250  275  Cutting s p e e d [m/min]  • f = 0.03 mm/rev • f = 0.035 mm/rev A f = 0.04 mm/rev f = 0.045 mm/rev o f = 0.05 mm/rev • f = 0.055 mm/rev A T = 0.06 mm/rev  Rz  4.5 c o u.  o  4 3.5 3  1  2.5  c  2  o  1.5  (fl (fl cu  3  0  •  cu o  1  CO  0.5  A H  0 100  fm  S  125  150  175  200  225  :  ! 250  275  Cutting s p e e d [m/min]  Figure A.l: Measured data of Surface roughness R and R versus cutting speed and feedrate (wet cutting results) q  z  108  Appendix A. Measured data for surface finish  •f= 0.04 mm/rev •f= 0.045 mm/rev 0.05 mm/rev  Ra  1.2 to  § o  1  •  of = 0.055 mm/rev • f = 0.06 mm/rev  •  • i . 0.8 ID CO  8 c  0.6  i  3  O  0.4  Q:  CD O  •  •  0.2  co  100  125  i 150  •  i  r  175  200  225  250  275  300  Cutting speed [mm/rev]  • f= 0.04 mm/rev • f= 0.045 mm/rev 0.05 mm/rev  Rmax to 0  I  to to  of = 0.055 mm/rev • f = 0.06 mm/rev  °  o 4  0  E3 O)  1 2 s  • •  o  I  co 100  125  150  175  1 200  I  225  •  • 4  250  275  300  Cutting speed [mm/rev]  Figure A. 3: Measured data of Surface roughness R and Rj, versus cutting speed and feedrate (dry cutting results) a  Appendix A. Measured data for surface finish  109  Rq 1.2 to  •  C  O  •  £ 0.8  • f = 0.04 mm/rev • f = 0.045 mm/rev AT = 0.05 mm/rev o f = 0.055 mm/rev • f = 0.06 mm/rev  w to  1 = 0.6 09 3  £ 0.4 © o | 0.2 to 100  8  4  • <  0  2 125  :  i  •  o  j  ft 150  £ 175  200  225  250  275  300  Cutting speed [mm/rev]  • f = 0.04 mm/rev • f = 0.045 mm/rev Af = 0.05 mm/rev o f = 0.055 mm/rev • f = 0.06 mm/rev  Rz to  § 5 u  I  to to  4 3  !  2  B  CO  E O)  I  Q O  .2  1  • 1  * i•  a  o  • •  u  \  t  225  250  u t  Ii  CO  100  125  150  175  200  275  300  Cutting speed [mm/rev]  Figure A.4: Measured data of Surface roughness R and R versus cutting speed and feedrate (dry cutting results) q  z  

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