

small (250x250 max)
medium (500x500 max)
Large
Extra Large
large ( > 500x500)
Full Resolution


MODELING THE EFFECTS OF BELT COMPLIANCE, BACKLASH, AND SLIP ON WEB TENSION AND NEW METHODS FOR DECENTRALIZED CONTROL OF WEB PROCESSING LINES By RAMAMURTHY V. DWIVEDULA Bachelor of Engineering Andhra University College of Engineering Visakhapatnam, India 1987 Master of Technology Indian Institute of Technology, Delhi New Delhi, India 1992 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2005 MODELING THE EFFECTS OF BELT COMPLIANCE, BACKLASH, AND SLIP ON WEB TENSION AND NEW METHODS FOR DECENTRALIZED CONTROL OF WEB PROCESSING LINES Thesis Approved: Thesis Adviser, Dr. Prabhakar R. Pagilla Committee Member, Dr. John J. Shelton Committee Member, Dr. Eduardo Misawa Committee Member, Dr. Rafael Fierro Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to express my sincerest appreciation to my major advisor, Dr. Prabhakar R. Pagilla for his intelligent supervision, constructive guidance, inspiration, and friendship. I would like to extend my warmest thanks to my doctoral committee members: Dr. Eduardo A. Misawa, Dr. John J. Shelton, and Dr. Rafael Fierro for their support and suggestions in completion of this research. Their guidance and understanding made the development of this thesis a positive learning experience. I would like to thank my colleagues at Oklahoma State University Yongliang Zhu, Nilesh B. Siraskar, Seshadri Kuppuswamy, Anil Abbaraju, Aravind Seshadri, Ryan Ratliff, Raiza Jafari, Mauro Cimino, and Yu. They are among the finest people I know and are a joy to work with. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Effect of compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Effect of backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Effect of slip on web tension dynamics . . . . . . . . . . . . . . . . . . . . 8 1.4 Decentralized control schemes for web process lines . . . . . . . . . . . . 9 1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Organization of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Modeling and analysis of the belt compliance 13 2.1 Beltpulley transmission system . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Analysis of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Singular perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Resonant frequency due to compliance of belt . . . . . . . . . . . . . . . . 27 2.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Effect of backlash and compliance on the output speed of a gear drive 39 3.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Analysis of backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Backlash model with compliance . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.1 A model of backlash including a compliant shaft . . . . . . . . . . 50 3.3.2 Effect of belt compliance and backlash in gears . . . . . . . . . . . 57 iv 3.4 Error bounds in the presence of backlash . . . . . . . . . . . . . . . . . . . 60 3.4.1 Method of finding a bound on error due to backlash . . . . . . . . . 61 3.4.2 Bound on error due to backlash and a compliant shaft . . . . . . . . 64 3.4.3 Bound on error due to backlash and belt compliance . . . . . . . . 69 3.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Effect of compliance and backlash on web tension 76 4.1 Simulations and experiments on the unwind section of the HSWL . . . . . 79 4.2 Effect of gearbacklash on controlled tension . . . . . . . . . . . . . . . . 88 4.3 Experiments conducted on Rockwell web line . . . . . . . . . . . . . . . . 92 4.3.1 Brief description of Rockwell web line . . . . . . . . . . . . . . . 93 4.3.2 Experiments conducted on the Rockwell web line . . . . . . . . . . 94 4.3.3 Braking input to mitigate the effect of backlash . . . . . . . . . . . 110 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5 Effect of slip on web tension dynamics 118 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Location of slip in the arc of contact . . . . . . . . . . . . . . . . . . . . . 119 5.3 Calculation of the sliparc angle . . . . . . . . . . . . . . . . . . . . . . . 122 5.4 Contact region between the web and the roller: A closer look . . . . . . . . 124 5.5 Propagation of tension waves . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.6 Dynamics of web tension in a free span . . . . . . . . . . . . . . . . . . . 132 5.7 Slippage within the region of wrap . . . . . . . . . . . . . . . . . . . . . . 134 5.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.9 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6 Decentralized Control of Web Process Lines 142 6.1 Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 v 6.1.1 Unwind Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.1.2 Master Speed Section . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.1.3 Process Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.1.4 Rewind Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2 Proposed Decentralized Control Scheme . . . . . . . . . . . . . . . . . . . 147 6.2.1 Equilibrium Control and Reference Velocities . . . . . . . . . . . . 149 6.2.2 Feedback Control Design . . . . . . . . . . . . . . . . . . . . . . . 151 6.3 Decentralized Adaptive Control Scheme . . . . . . . . . . . . . . . . . . . 158 6.4 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7 Summary and future work 170 BIBLIOGRAPHY 173 A The standard singular perturbation method 185 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 A.2 Timescale properties of the standard model . . . . . . . . . . . . . . . . . 186 A.3 Linear timeinvariant systems . . . . . . . . . . . . . . . . . . . . . . . . . 188 B Brief description of experiments conducted on backlash 192 vi LIST OF FIGURES Figure Page 1.1 Schematic of a transmission system . . . . . . . . . . . . . . . . . . . . . 3 1.2 A physical illustration of backlash . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Inputoutput plot for frictioncontrolled backlash . . . . . . . . . . . . . . 6 2.1 Schematic of a belt driven transmission system . . . . . . . . . . . . . . . 15 2.2 Block diagram of belt driven transmission system. . . . . . . . . . . . . . . 19 2.3 Two feedback schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Torquespeed characteristics of motor and controller . . . . . . . . . . . . . 28 2.5 Belt drive with motor in velocity control mode. . . . . . . . . . . . . . . . 28 2.6 Belt drive with motor in torque control . . . . . . . . . . . . . . . . . . . . 30 2.7 Picture of the HSWL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 A picture of the transmission system. . . . . . . . . . . . . . . . . . . . . . 33 2.9 A picture of the transmission system. . . . . . . . . . . . . . . . . . . . . . 34 2.10 Transient response of the motor and load . . . . . . . . . . . . . . . . . . . 35 2.11 Steadystate loadspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.12 FFT of the load speed signal . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Schematic of backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Inputoutput plot for frictioncontrolled backlash . . . . . . . . . . . . . . 40 3.3 A rotary model of meshing spur gears . . . . . . . . . . . . . . . . . . . . 42 3.4 A simple backlash model . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Computing load velocity from the solid velocity . . . . . . . . . . . . . . . 48 3.6 Deviations of load/motor velocities from solid velocity . . . . . . . . . . . 49 vii 3.7 Load/Motor velocity and solid velocity . . . . . . . . . . . . . . . . . . . . 50 3.8 Displacement of the load/motor and center of mass . . . . . . . . . . . . . 51 3.9 Schematic of a gear drive . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.10 rectilinear analog: (a) without backlash, (b) with backlash . . . . . . . . . . 53 3.11 Rectilinear analog where compliance precedes backlash . . . . . . . . . . . 56 3.12 Schematic of a transmission system. . . . . . . . . . . . . . . . . . . . . . 58 3.13 (a) A system without backlash and (b) System with backlash . . . . . . . . 61 3.14 Inputoutput plot of backlash . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.15 Block diagram of a controller for system with backlash . . . . . . . . . . . 66 3.16 ECP Rectilinear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.17 Closedloop experiment with backlash of 1.55 mm . . . . . . . . . . . . . 72 3.18 Closedloop experiment with backlash of 3.56 mm . . . . . . . . . . . . . 73 3.19 Closedloop experiment with backlash of 5.38 mm . . . . . . . . . . . . . 74 4.1 Pictures of the HSWL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Schematic of the HSWL . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Control scheme to regulate web tension and web velocity . . . . . . . . . . 79 4.4 Schematic of the unwind section. Master speed roller, a span following the master speed roller, and locations of two loadcells are also shown. . . . . . 80 4.5 Tension behavior in a span for a change in the reference speed . . . . . . . 82 4.6 Tension behavior in a span for a change in reference speed . . . . . . . . . 84 4.7 Tension behavior in a span for a change in the reference speed . . . . . . . 85 4.8 Tension behavior in a span for a change in the reference speed . . . . . . . 86 4.9 Web speed and tension with a 36 mm wide belt . . . . . . . . . . . . . . . 87 4.10 Web speed and tension with a 24 mm wide belt . . . . . . . . . . . . . . . 88 4.11 Web speed and tension with a 12 mm wide belt . . . . . . . . . . . . . . . 89 4.12 Schematic of Rockwell web line. . . . . . . . . . . . . . . . . . . . . . . . 95 4.13 Drive system in Rockwell web line. . . . . . . . . . . . . . . . . . . . . . 95 viii 4.14 Mean velocity and tension with zero backlash and no disturbance. . . . . . 97 4.15 Tension disturbance introduced. . . . . . . . . . . . . . . . . . . . . . . . 98 4.16 Mean velocity and tension with zero backlash and disturbance applied. . . . 99 4.17 Mean velocity and tension with a backlash of 0.5mm and no disturbance applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.18 Mean velocity and tension with a backlash of 0.5mm and with disturbance applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.19 Standard deviation of webtension. . . . . . . . . . . . . . . . . . . . . . . 102 4.20 Standard deviation of webvelocity. . . . . . . . . . . . . . . . . . . . . . 103 4.21 Frequency content of speed and tension signals. . . . . . . . . . . . . . . . 105 4.22 Frequency content of tension and speed when a disturbance is introduced. . 106 4.23 Frequency content of tension and speed with backlash. . . . . . . . . . . . 107 4.24 Summary of results with and without backlash: disturbance amplification. . 108 4.25 Summary of results with and without backlash: disturbance amplification. . 109 4.26 Schematic of the rewind section with braking input . . . . . . . . . . . . . 111 4.27 Frequency content of tension speed with backlash and braking input . . . . 112 4.28 Summary of results with braking input . . . . . . . . . . . . . . . . . . . . 113 4.29 Summary of results with braking input . . . . . . . . . . . . . . . . . . . . 114 4.30 Summary of comparative results with braking input of 14lbf . . . . . . . . 116 5.1 A schematic showing the nomenclature . . . . . . . . . . . . . . . . . . . 119 5.2 A beltpulley system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Slip arc in the region of contact . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 An element of the belt in the slip arc . . . . . . . . . . . . . . . . . . . . . 124 5.5 Friction force and regimes of contact . . . . . . . . . . . . . . . . . . . . . 125 5.6 Contact of a web around a roller . . . . . . . . . . . . . . . . . . . . . . . 126 5.7 Web considered as elements . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.8 The force distribution in the contact region . . . . . . . . . . . . . . . . . . 127 ix 5.9 The three regions of contact . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.10 Tension wave propagation in elastic medium . . . . . . . . . . . . . . . . . 130 5.11 Control volume considered for deriving web tension dynamics . . . . . . . 133 5.12 (a) Friction and normal forces and velocities at entry and exit when the web slips throughout the contact region.(b) An element in the contact region and forces acting on it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.13 A model of traction between the web and the roller . . . . . . . . . . . . . 136 5.14 Slip under the influence of slowly varying tension. . . . . . . . . . . . . . . 137 5.15 Three regions of contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.16 A free span. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.17 Response of tension t2 for a step change in vr,2 in Figure 5.16 . . . . . . . . 140 6.1 Crosssectional view of unwind roll . . . . . . . . . . . . . . . . . . . . . 145 6.2 Freebody diagram of master speed roller. . . . . . . . . . . . . . . . . . . 146 6.3 Decentralized PI controller: Reference velocity 500 ft/min . . . . . . . . . 157 6.4 Proposed decentralized controller: Reference velocity 500 ft/min . . . . . . 158 6.5 Decentralized PI controller: Reference velocity 1000 ft/min . . . . . . . . . 166 6.6 Decentralized PI controller: Reference velocity 1500 ft/min . . . . . . . . . 167 6.7 Decentralized adaptive controller: Reference velocity 1000 ft/min . . . . . 168 6.8 Decentralized adaptive controller: Reference velocity 1500 ft/min . . . . . 169 A.1 Actuator form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 B.1 Backlash gap in experiments. . . . . . . . . . . . . . . . . . . . . . . . . . 193 x NOMENCLATURE A : Area of crosssection of web bm, bL : Viscous friction coefficients of the motor and the load BR : Speed ratio = R1/R2 E : Young’s modulus of web material FL : Load/disturbance force Fm : Force exerted by motor Fi,f : Frictional force on ith roller Fi,n : Normal force on ith roller Jm, JL : Inertias of the driving and driven gears Jm, JL : Inertias of the motor and the Load Jc0 : Inertia of the coreshaft Jmi : Inertia of the ith drive motor Kb : Stiffness of the belt Km : Motor constant Kp, Ki : Proportional and the integral gains Li : Length of ith span Ri : Radius of rollers/pulley Rb1, Rb2 : Base circle radii of gears Rfull : Outer radius of full unwind roll Rg1, Rg2 : Pitch circle radii of gears Run : Radius of the unwind roll Rc0 : Outer radius of the coreshaft with core on it xi T1, T2 : Torques on the driving and driven ti : Tension in ith span vi : Velocity of web on ith roller/roll vr,i : Peripheral velocity of ith roller/roll V0, V1 : Web speed on rollers w : Width of the web αi : Wrap angle on ith roller δ : Thickness of web ¢ : Half backlashwidth ε : Small parameter = 1/√Kb εi : Strain in ith span θ1, θ2 : Angular displacements of the driving and driven gears θm, θL : Angular displacements of the motor and the load ρ : Density of the web material τL : Load/disturbance torque τm : Torque exerted by motor τmc : Time constant of the motor ωd : Desired angular velocity ωi : Angular velocity of ith roller/roll ωm, ωL : Angular velocities of the motor and the load xii CHAPTER 1 Introduction A web is any material which is manufactured and processed in a continuous, flexible strip form. Examples include paper, plastics, textiles, strip metals, and composites. Web handling refers to the physical mechanics related to the transport and control of web materials through processing machinery. Web processing pervades almost every industry today. It allows us to mass produce a rich variety of products from a continuous strip material. Products that include web processing somewhere in their manufacturing include aircraft, appliances, automobiles, bags, books, diapers, boxes, newspapers, and many more. Web tension and velocity are two key variables that influence the quality of the finished web, and hence the products manufactured from it. Web handling refers to the physical mechanics related to the transport and control of web materials through processing machinery. The primary goal of research in web handling is to define and analyze underlying sciences which govern unwinding, web guiding, web transport, and rewinding in an effort to minimize the defects and losses which may be associated with handling of the web. Web handling systems facilitate transport of the web during its processing which is typically an operation specific to a product; for example, in the case of an aluminum web, the web is brought to a required thickness, cleaned, heattreated, and coated; and in the case of some consumer products, the web is laminated and printed. It is important that the tension in a web span be maintained within a close tolerance band while it is transported at a prescribed velocity through the web processing machine. For example, if the tension in the web changes during printing/perforating processes, the 1 print (perforation) gets skewed. Further, excessive tension variations may cause wrinkles and may even tear the web. Tension control plays a key role in improving the quality of the finished web. It is essential to keep the web in the process at a preset tension, which could change throughout the process by many conditions such as disturbances from uneven rollers and web speed variations. As the demand for higher productivity and better performance from the web processing industry increases, better models for the machinery as well as the web behavior and more accurate control algorithms for the processes must be developed. In specific, the imperfections and nonideal effects inherently present in the machine components must be identified and their characteristics must be analyzed before attempting to address the control schemes that limit/eliminate the deleterious effects. The nonideal effects manifest in web handling systems due to factors such as nonlinear behavior of motors, presence of nonlinear friction, presence of compliant members, presence of backlash in the transmission systems, and imperfect contact between the web and the roller resulting in slippage of web over the rollers. It is important to model these nonideal effects with the objective of synthesizing controllers to improve the performance of the system. Modeling aspects of some such nonideal effects, as discussed below, is considered in this thesis. Besides, advanced control schemes for regulating web tension and velocity are also presented in this thesis. A typical web process line uses transmission systems to couple the drive motors and the driven shafts. Such transmission systems may consist of a beltpulley arrangement, a gearbox, a direct coupling, or a combination of these elements. Figure 1.1 shows a schematic of a transmission system used in the unwind station of a web process line. In such transmission systems, three basic drive characteristics/nonlinearities dominate the empirical observation of machine behavior and these are compliance, backlash, and friction. Of course, “perfect” machines do not portray any of these characteristics/nonlinearities, but the world is certainly not perfect, especially when perfection costs more money. For example, in the transmission system shown in Figure 1.1, all the three drive nonlinearities 2 Tach. web Tweb Unwind Roll Belt Motor Web Bevel Gears V Figure 1.1: Schematic of a transmission system may be present: the belt and/or the shafts may offer the compliance; the gearpair almost invariably has some amount of backlash; and the bearings used at various mounting points may offer considerable friction in addition to the friction inherently present in the drive motor. These characteristics directly affect the controlled variables, which are web tension (Tweb) and web velocity (Vweb). Thus, if a controller is designed to regulate the controlled variables, ignoring the characteristics of the transmission system, the performance of such a controller may be different from expected performance. In particular, if one of the shafts or the belt shown in Figure 1.1 is compliant, it affects the transient response of the system and also may contribute a resonant frequency. Similarly, backlash and friction, when present may cause limit cycles and may even render the closedloop system unstable. This report considers modeling and analysis of the effect of drive characteristics on the performance of closedloop speedcontrol systems and new control schemes to regulate the web tension and velocity. In the following a brief introduction of each of the aspects considered in this report is presented. A detailed introduction and prior work are presented at the beginning of each chapter. 3 1.1 Effect of compliance Compliance is an intrinsic property allowing an object to yield elastically when subjected to a force, and in this sense is synonymous with elasticity. Thus, compliance appears in any machine where there are elastic members that are subjected to external loads. Research on compliance (or elasticity) dates back to some hundreds of years. Galileo Galilei first documented the “resistance” of solids in 1638. Robert Hooke in 1660 discovered the proportionality of stress and strain. Following these studies, several famous physicists and mathematicians, including Claude Navier, Leonhard Euler, Charles Augustin Coulomb, Thomas Young, SimeonDennis Poisson, AugustinLouis Cauchy, Heinrich Rudolf Hertz, and Lord Rayleigh contributed to the research on elasticity. An excellent historical survey of early research on elasticity is given in [1]. Following the classical works, many researchers reported important results that address the effect of compliance on various aspects of machine behavior [2–22]. The classical compliance model, based on Hooke’s Law, relates the strain of a flexible body to the stress induced in the body. It is well known that Hooke’s law is only true for a limited amount of strain, after which permanent plastic deformation will occur. For large strains, often, a softening spring or a hardening spring model [23, pp. 9] or some other nonlinear model is used. In such cases, presence of compliance gives rise to complex nonlinear system of equations [5,19–22] whereas in many industrial drive applications, the effects of compliance can be modeled by linear equations [8–13] using the Hooke’s law. These linear models may be developed along the lines of theMaxwell model or the Kelvin Voigt model as given in [24, Chapter 2]. As already noted (on page 2), compliance arises in the transmission system shown in Figure 1.1 either due to a compliant belt or due to compliant shafts. However, the torsional rigidity of the shafts may be much larger than the linear rigidity of the belt and thus, the strain induced in the belt may be much larger than the strain induced in the shafts for a given force. Chapter 2 proposes a model to include the compliance of the belt in a beltpulley trans 4 mission system and analyzes the closedloop speed control system. Some issues such as the feedback configuration, and natural frequencies which arise due to belt compliance are considered in this chapter. 1.2 Effect of backlash Backlash, in the context of mechanical engineering, means “the play between adjacent movable parts (as in a series of gears) or the jar caused when the parts are put into action1.” Backlash is one of the most important nonlinearities that affect the control strategies implemented in the industrial machines and degrades the overall performance of the machines. In industrial drive systems, backlash occurs in mating gear teeth either due to unavoidable manufacturing tolerances, or often deliberately introduced to avoid other deleterious effects. For example, a pair of spur gears are generally mounted at a center distance slightly larger than the designed center distance to avoid interference/undercutting of the teeth. As a result of this, the width of the tooth of one gear along the pitch circle is slightly less than the recess in the mating gear, thus giving rise to backlash. Presence of backlash, though is advantageous from the point of view of interference/undercutting, causes delays and/or oscillations and consequently gives rise to inaccuracies in the position and velocity of the machine. The effect of backlash can be explained with reference to Figure 1.2. In Figure 1.2, Mm is the mass of the driving member (called “motor” hereafter) and ML is the mass of the driven member (called “load” hereafter). As long as there is contact, as shown in Figure 1.2, the motor is able to move the load. However, if the direction of the motion of the motor were to reverse, contact between the motor and the load is lost and any motion of the motor will not result in a corresponding motion of the load. The classical backlash model, shown in Figure 1.3, considers the displacement of the motor (u) as input to the backlash nonlinearity and displacement of the load (y) as output. 1As given in MerriamWebster’s Dictionary. 5 Motor, Mm contact Right contact a c F u y 2D Left Load, ML Figure 1.2: A physical illustration of backlash G y D −D D −D u A B C D E F Figure 1.3: Inputoutput plot for frictioncontrolled backlash 6 In Figure 1.3, the closed path BCDEFGB represents the hysteresis loop due to the effect of backlash. Notice that along the segments AB, CD, and FG, though the displacement of the motor is changing, the displacement of the load remains the same since contact is lost. Many researchers used this model of backlash to propose control schemes [25–30]. However, in actual practice, the inputoutput plot shown in Figure 1.3 is not realized. In considering the input as a displacement, the classical model ignores the momentum of the load during the periods where there is no contact between the motor and the load. Also, the classical model implicitly assumes that disturbances acting on the load do not affect the backlash characteristics, which is not the case. Chapter 3 discusses some aspects of backlash to overcome these lacunae and presents a model for describing backlash. Friction, in all its manifestations, is an important phenomenon which can be put to use for a positive effect (as in the case of brakes) and at the same time has deleterious effects, causing selfexcited oscillations due to stickslip friction. Friction is a very widely studied aspect [31, pp. 169] dating back to Leonardo da Vinci [32], an extraordinary artist and an extraordinary scientist. Since that time, a number of researchers worked in the field of friction [12, 33–48]. Design of control schemes to compensate for nonlinear friction is not considered in this report. The focus of this report is on the effect of transmission system (compliance and backlash) on the speed and tension control in a web processing system. A specific objective laid down is to find the achievable accuracy of a given speed control system when the compliance and the backlash present in the system are known. The transmission system considered for the purpose of analysis consists of a motor driving a load inertia through a compliant shaft/belt and a gear drive, as shown in Figure 1.1. This kind of transmission system is used on the unwind/rewind station of the High SpeedWeb Line (HSWL) in theWeb Handling Research Center (WHRC) at Oklahoma State University, Stillwater. Such transmission systems are especially used since they use the available floor space most efficiently and also offer certain advantages in terms of installation and maintenance. For example, 7 using a directly coupled transmission system mandates exact collinearity of the motor shaft and the load shaft. Any eccentricity/nonparallelism results in unwanted vibration and wear and tear of the bearings. If a beltdrive is used, any small eccentricity/nonparallelism is absorbed into the compliance of the belt. However, the compliance of the belt and the backlash present in the bevel gears may introduce additional nonlinearities which need to be studied. In this context, it is of importance to know the achievable accuracy when the parameters of the motor/tachometer, controller, belt, and the backlash in the gears are known. Such a bound on the achievable accuracy lets the designer/plant engineer in reconfiguring the system with a different set of parameters. As noted on page 1, the compliance present in the system may be due to the belt in the beltpulley transmission system, or the compliance of the shafts. The effect of the compliance of the shafts is more pronounced especially when the transmission system uses long shafts. Chapter 2 presents a model to include the compliance of the belt into dynamics and analyzes the belt driven transmission system. Chapter 3 presents a model of backlash that includes either the effect of the compliance of the shafts or the compliance of the belt and presents a method to compute the achievable accuracy in a given system with a known backlash. Chapter 4 presents the dynamic model of webtension and webvelocity in the unwind/rewind station to include the effects of compliance and backlash present in the drive system. Results of experiments are presented in respective sections. 1.3 Effect of slip on web tension dynamics Modern manufacturing processes exploit the continuous nature of the basic material in web form by transporting it through and out of the process. In such processes, it is essential to maintain continuity and avoid cracks/breakage in the web. Though tests have been conducted to determine breaking strength of webs, it is found in practice that webbreaks occur even when the web tension is much less than the break tension determined under test conditions. There are two main reasons for web breakage: (i) the cracks could be 8 the result of local stress concentrations. In the event of these stress concentrations, cracks may appear and propagate even at moderate overall web tension (ii) second cause for web breaks could be considerable variation of tension about the mean tension. Fatigue may set in when tension fluctuations are rapid and their amplitude is considerable. In general, the web breakage is probably a result of a combination of these two effects. The local stress concentrations may be avoided by improving the manufacturing processes to reduce the severity and density of irregularities. Such efforts fall under the purview of the design of manufacturing process and are specific to the product being manufactured. On the other hand, controlling web tension within tight tolerance band is a common feature to all manufacturing processes which involve material in web form at some stage of production. Thus, there is a definite need for the study of synthesis of web tension control systems. Before attempting to devise such control systems, it is essential to find out how the tension disturbances occur and how they are propagated through the system. Chapter 5 discusses the effect of slip on web tension dynamics. 1.4 Decentralized control schemes for web process lines A web processing line is a largescale complex interconnected dynamic system with numerous control zones to transport the web while processing it. A web processing line typically consists of an unwind roll, several web spans supported by driven/idle rollers and a rewind roll. In such systems one is interested in designing control input to the unwind motor, the rewind motor, and each of the driven rollers to maintain webtension and webvelocity at prescribed reference values. It might be noted that the physical size of the process line in most cases is very large and the various drive motors and tension/velocity sensors located at various points of interest may be situated far apart. Consequently, it is convenient to improvise control design algorithms that use only information available from tension/velocity sensors nearest to the drive motors, thus allowing decentralization. Chapter 6 considers the decentralized control of web process lines and presents two such schemes applicable to 9 web process lines. 1.5 Contributions The contributions of this report can be summarized as follows. 1. A dynamic model to include the effect of compliance of the belt on the speed control system is proposed. Using this model, a method of setting the proportional and integral gains of the controller is proposed. Using the model developed for the beltpulley transmission system, it is shown that using the feedback signal only from the load side is not desirable. Such a result is counterintuitive. 2. Resonant frequency due to the compliance of the belt is computed for the case where the drive motor is in velocity control mode. Frequency content of the speed signal obtained from experiments closely agrees with the computed resonant frequency. 3. A dynamic model for backlash is proposed to present the effect of backlash on the output speed of a gearpair. This model includes the momentum of the load when contact is lost. In addition, the model considers the compliance of the shaft (or the compliance of the belt) in series with the backlash in the gears. 4. The dynamic model proposed is used to derive an estimate of the upper bound on deviation in the velocity of the load due to the presence of backlash. Results from the experiments agree with the theoretically computed bound. Further, it is shown that, in transmission systems that use beltpulley system and a gearpair, the ratio of driven pulley radius to the driving gear radius needs to be small to minimize the effect of backlash. 5. The dynamics of the rewind section of an experimental platform is developed. The model of backlash is extended to include the effect of backlash on web tension. Simulation study and experimental investigation is conducted to investigate the effect of 10 beltcompliance and backlash on web velocity and tension. 6. It is shown and experimentally verified that the mean tension at rewind station is shifted up when the transmission system for driving the rewind roll has backlash. 7. Experiments were designed on the rewind station of a web process line (Rockwell web process line) to excite the backlash gap. The experimental results show that, the amplitude of tension signal at disturbance frequency is amplified when backlash is present in the transmission system. Further experiments on the same process line show that using a braking input on the rewind shaft is successful in mitigating the effect of backlash on web tension. 8. A scheme to include the effect of slippage of web over a roller is proposed. This scheme shows that, when there is slippage between the web and the roller, the web tension disturbances can travel opposite the direction of web travel as well as in the usual direction of web travel. 9. Decentralized control schemes for web process lines were systematically investigated. Two schemes of decentralized control, viz., a non adaptive scheme and an adaptive scheme, are proposed for regulation of web tension and web velocity. Both schemes are experimentally evaluated and are compared to with existing decentralized PI control scheme. 1.6 Organization of the report The rest of the report is organized as follows. Chapter 2 considers the effect of the compliance of the belt on the speed control system considering the shafts to be rigid. In chapter 3, a dynamic model of the backlash in the gears is proposed. Chapter 4 presents the effect of backlash and compliance on webtension in an unwind/rewind station. Effect of webslippage on the tension is studied in Chapter 5. Chapter 6 presents decentralized control of 11 web process lines. Summary and future work are given in Chapter 7. 12 CHAPTER 2 Modeling and analysis of the belt compliance In all web process lines, the driven rollers and the unwind/winder rolls are driven through a transmission system which may be a direct coupling, a gear driven system, or a beltpulley transmission system. Since the driving and the driven shafts have torsional compliance, torsional oscillations can be expected during power transmission. In the case of a belt drive, the compliance of the belt also contributes to the torsional oscillations. Also, if the transmission system uses gears, backlash between the mating gears is unavoidable. Although backlash is necessary for lubricating the tooth surface and preventing the teeth from getting jammed, it results in reduced stability and creates unwanted vibration. In particular, during high speed operation and intermittent motion requiring change in the direction of rotation, this problem becomes critical [26, 49]. Though there are techniques that can reduce backlash, the production cost is very high compared to the achievable accuracy [40]. Sometimes, a beltpulley arrangement is a better method of transmitting power. For example, coupling the drive motor directly to the process end mandates very accurate collinearity of the axes and takes a considerable amount of time. In such cases a belt driven transmission offers great ease since small inaccuracies can be absorbed into compliance of the belt. However, compliance of the belt brings additional dynamics into the system that need to be studied. There is a large body of literature on the characteristics of belt drives and design of mechanisms using belt drives. Many such works concentrated on mechanism of motion/ power transfer, location and extent of sliparc, nature of frictional contact, efficiency 13 limit of the beltdrive system, and methodology of design/selection of beltdrive components [2–5, 8–13]. While many other papers report control schemes which use either estimation of the transfer function of the system, “fuzzy” control schemes, or a simple, ad hoc model to describe the dynamics included by the compliance of the belt [14–16, and the references therein]. In [14], modeling and control of a beltdrive positioning table is discussed. However, no specific model is reported for including the effect of compliance of the belt; system identification techniques were used to obtain the system dynamics, to be later used in tuning the feedback gains. Similarly in [15], a composite fuzzy controller, consisting of a feedback fuzzy controller and a feedforward acceleration compensator, is proposed to control a belt drive precision positioning table. Again, no model to include the effects of belt compliance was reported in this paper. In [16], a robust motion control algorithm for beltdriven servomechanism is reported. In this paper, the beltstretch dynamics is assumed to contribute a pair of purely imaginary poles to the transfer function of the system. Also, the fact that belt stands as an interconnection from loadside to the motorside is ignored in this paper. Analysis and control of speed drive systems with torsional loads is reported in [6, 7, 17, 18]. In [6, 7], a motor drive system driving an inertial load through a gear and spindle is considered for analysis. Though deadzone and backlash are entirely different (deadzone is a stabilizing influence while backlash is destabilizing), the “backlash” in the gear box is modeled as a “deadzone” and the spindle is modeled as a torsional spring. No specific analysis is presented to demonstrate the effect of the compliance of the spindle. In [17], two application examples in the field of remote handling are presented. In this paper, a quasistatic control scheme is presented to compensate for the compliance in the actuator. Similarly, [18] considers shaft torsional oscillations of an induction machine including saturation and hysteresis in the actuator. Though the dynamics of torque generation are dealt with extensively, the compliance present in the drive train is not modeled completely. Of particular importance in the analysis of the beltdriven transmission system is the computation of resonant frequency due to the compliance of the belt. This topic is 14 J L b L 2R2 L q Kb Jm 2R1 bm m t qm Web Tension, T1 Tight side Slack side Belt Figure 2.1: Schematic of a belt driven transmission system not addressed clearly in existing literature. In this chapter, the effect of the compliance of the belt on the speed control system is studied. Using a simple model to include the effect of compliance of the belt, aspects such as the feedback scheme to be used and choice of the feedback gains, and computation of the resonant frequencies when the motor is in velocity/torque mode are addressed. 2.1 Beltpulley transmission system In many applications, a beltpulley transmission system is a convenient alternative over a gear transmission system. When the center distance between the driving shaft and the driven shaft is too large for use of a gearpair, using a belt to transmit motion/power may be the only practical alternative. Figure 2.1 shows a schematic of the drive system considered. A typical approach in analyzing beltpulley/gear transmission systems is to find an equivalent inertia and equivalent damping as referred to either the motor side or to the load side. Assuming that the speed 15 ratio is BR = R2/R1, the equivalent inertia and damping as referred to the motor side are Jeq,m = Jm + µ R1 R2 ¶2 JL, beq,m = bm + µ R1 R2 ¶2 bL. (2.1) With the equivalent quantities defined in (2.1), the dynamics of the system may be written as Jeq,mθ¨m + beq,mθ˙m = τm. (2.2) The control objective is to design a feedback control law τm such that the closedloop system is stable and the load velocity tracks a given reference signal1 ωdL with a prescribed accuracy. Using the feedback law τm = Kpm(ωdm − ωm) + Kim Z (ωdm − ωm)dτ, (2.3) the characteristic equation of the system may be written as s2 + α1s + α0 = 0 (2.4) where α1 = R2 2bm + R2 1bL + KpmR2 2 R2 1JL + R2 2Jm , α0 = KimR2 2 R2 1JL + R2 2Jm . (2.5) Similarly, equivalent inertia and damping as referred to the load side are Jeq,L = JL + µ R2 R1 ¶2 Jm, beq,L = bL + µ R2 R1 ¶2 bm. (2.6) With equivalent inertia/damping defined in (2.6), and using the control law τm = KpL(ωdL − ωL) + KiL Z (ωdL − ωL)dτ, (2.7) 1the ratio of !dm to !dL is the same as the speed reduction ratio of the beltpulley transmission system. 16 the characteristic equation may be written as s2 + β1s + β0 = 0 (2.8) where β1 = R2 2bm + R2 1bL + KpLR1R2 R2 1JL + R2 2Jm , β0 = KiLR1R2 R2 1JL + R2 2Jm . (2.9) Considering the characteristic equations given by equations (2.4) and (2.8), one may be led to the conclusion that the gains of the PIcontrollers given by equations (2.3) or (2.7) may be chosen to place the poles of the characteristic equations (2.4) or (2.8) appropriately. However, as shown in the subsequent sections, the control law given by (2.3) is preferable over the control law given by (2.7). 2.2 Analysis of the system In this section dynamics of the system including effect of compliance of the belt is derived. Also, it is shown that, when ProportionalIntegral (PI) control law is used, it is not advisable to use feedback only from the loadside (i.e., ωL in Figure 2.1). Notice that for a given direction of rotation of the pulley, the belt has a tight side and a slack side as shown in Figure 2.1. To derive the dynamic equations of the system, a simplifying assumption regarding the power transmission is made. Assume that the transmission of power is taking place on the tight side and the transport of the belt is taking place on the slack side. Under this assumption, the net change in tension on the slack side will be much smaller than that in the tight side and thus may be ignored. The tight side of the belt can then be modeled as a spring with spring constant of Kb. Thus, for given angular displacements θm and θL, net elongation of the tight side of the belt can be written as (R1θm − R2θL). Because of this elongation, the driving pulley experiences a torque of (R1θm − R2θL)KbR1 and the driven pulley experiences a torque of (R1θm − R2θL)KbR2. 17 Thus, dynamics of the system, ignoring the inertias of pulleys, is given by τm = (Jmθ¨m + bmθ˙m) + R1Kb(R1θm − R2θL), (2.10a) R2Kb(R1θm − R2θL) = (JL¨θL + bL θ˙L). (2.10b) Remark 2.2.1 Notice that the dynamics of the beltpulley transmission system given in (2.10) may be easily obtained using the wellknown EulerLagrange equations of motion [50, pp. 129–135]. The kinetic energy of the system shown in Figure 2.1 may be written as K(θ˙m, θ˙L) = 1 2 [Jm θ˙2 m + JL θ˙2 L] (2.11) and the potential energy stored in the belt in the form of strain energy may be written as V (θm, θL) = 1 2 Kb[R1θm − R2θL]2. (2.12) Defining the Lagrangian L = K − V , dynamics of the system may be written as d dt ∂L ∂θ˙j − ∂L θj = τj − bj θ˙j , j = m or L, and τL = 0. (2.13) Explanation leading to (2.10) is elaborately given to emphasize the implicit assumption made in (2.12) that the transmission of power is taking place on the tight side and the transport of belt material is taking place on the slack side. Figure 2.2 shows a block diagram representation of the system given in (2.10). Note that the block diagram given in Figure 2.2 represents the openloop system. The two “loops” appearing in the block diagram represent the interconnections in the (2.10). From the block diagram, we obtain openloop transfer functions G¿m!m(s) , ωm(s) τm(s) = JLs2 + bLs + R2Kb D(s) (2.14a) G¿m!L(s) , ωL(s) τm(s) = R1R2Kb D(s) = G(s) (2.14b) 18 +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Figure 2.2: Block diagram of the belt driven transmission system. BR is the speed ratio, BR = R2/R1. where Jeq = R2 2Jm + R2 1JL, (2.15a) beq = R2 2bm + R2 1JL, (2.15b) D(s) = JmJLs3 + (bLJm + JLbm)s2 + (KbJeq + bmbL)s + Kbbeq. (2.15c) The relative degree of the transfer function given in (2.14a) is one, whereas the relative degree of the transfer function given in (2.14b) is three. For plants with relative degree greater than or equal to three, adaptive control schemes are more complex than for plants with smaller relative degrees [51,52]. Hence, the transfer function given in (2.14a) is more suitable for adaptive schemes than the transfer function given in (2.14b). Also, note that in the case of the belt driven transmission system considered (see Figure 2.1), the feedback signals may be obtained from the motor side (that is ωm) or from the load side (that is ωL) as noted in the previous section. These two cases are shown in Figure 2.3. Let us first consider the case when only ωL is used for feedback with the control law given in (2.7). This control law represents a PIcontroller and is a widely used control law. With the control law given by (2.7), the closedloop transfer function from ωdL to ωL is obtained as ωL(s) ωdL(s) = (R1R2Kb/JmJL)(sKpL + KiL) ψL(s) (2.16) 19 +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Controller Motor + w  dL (a) +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Controller Motor + w  dm (b) Figure 2.3: Two feedback schemes: (a) feedback from load shaft and (b) feedback from motor shaft 20 where ψL(s) = s4 + (bmJL + JmbL) JmJL s3 + (Kb[R2 2Jm + R2 1JL] + bmbL) JmJL s2 + (Kb[R2 2bm + R2 1bL] + R1R2KbKpL) JmJL s + R1R2KbKiL JmJL . (2.17) Notice that the coefficients of s3 and s2 do not depend on the gains, KpL and KiL, of the control law. Thus, it may not be possible to place the poles of the characteristic equation at desired locations. On the other hand, consider the control law given by (2.3). With this law, the closedloop transfer function from ωdm to ωL is obtained as ωL(s) ωdm(s) = (R1R2Kb/JmJL)(sKpm + Kim) ψm(s) (2.18) where Ãm(s) = s4 + c3s3 + c2s2 + c1s + c0, c3 = (bmJL + JmbL + KpmJL) JmJL , c2 = (Kb[R2 2Jm + R2 1JL] + bmbL + KpmbL + KimJL) JmJL , c1 = (Kb[R2 2bm + R2 1bL] + R2 2KbKpm + KimbL) JmJL , c0 = R2 2KbKim JmJL . (2.19) The coefficients of s3 and s2 now depend on the gains of the control law. Thus, we have more flexibility in placing the poles of the characteristic equation. The following theorem establishes the stability of the system when feedback from motor side is used in a PI control law. Theorem 2.2.1 The closedloop system defined by (2.3) and (2.10) is stable and ωm approaches ωdm for all Kpm, Kim > 0. Proof: Substituting (2.3) into (2.10), we obtain dynamics of the closedloop system as Kpm(!dm − !m) + Kim Z (!dm − !m)d¿ = (Jmµ¨m + bmµ˙m) + R1Kb(R1µm − R2µL), (2.20a) R2Kb(R1µm − R2µL) = (JL¨µL + bL µ˙L). (2.20b) 21 Differentiate (2.20) to obtain −Kpmω˙m + Kim(ωdm − ωm) = (Jmω¨m + bmω˙m) + R1Kb(R1ωm − R2ωL), (2.21a) R2Kb(R1ωm − R2ωL) = (JLω¨L + bLω˙ L). (2.21b) Defining errors, em = ωm − ωdm and eL = ωL − (R1/R2)ωdm, (2.21) may be written as −Kpme˙m − Kimem = Jme¨m + bme˙m + R1Kb(R1em − R2eL), (2.22a) R2Kb(R1em − R2eL) = JLe¨L + bLe˙L (2.22b) Choose V (t) = 1 2 £ Jme˙2 m + JLe˙2 L + Kb(R1em − R2eL)2 + Kime2 m ¤ . (2.23) Then, the time derivative of V along the trajectories defined by (2.22) is obtained to be dV (t) dt = −(bm + Kpm)e˙2 m − bLe˙2 L. (2.24) Thus, V (t) is a Lyapunov function and em, eL, e˙m, e˙L ∈ L∞ which implies, from (2.22), that ¨em, ¨eL ∈ L∞. From (2.23) and (2.24), we conclude that because V (t) is bounded from below and is nonincreasing with time, it has a limit [52, Lemma 3.2.3], i.e., limt→∞ V (t) = V∞. Now from (2.24), we have lim t→∞ Z t 0 (bm + Kpm)e˙2 m + bLe˙2 L = V0 − V∞ < ∞ (2.25) Therefore, e˙m, e˙L ∈ L2 and by Barbalat’s Lemma [23], we have e˙m → 0 and e˙L → 0. Thus, ωm and ωL tend to become constants as t → ∞ and from (2.21), we see that ωm → ωdm and ωL → (R1/R2)ωdm. ¥ Theorem 2.2.1 shows that any PI controller with positive proportional/integral gains will stabilize the system when the feedback is from the motor side. However, such a result could not be established for the closedloop system defined by equations (2.7) and (2.10) since a Lyapunov function candidate could not be found for the system. This prompted us 22 to look for other tools which could reveal the stability issues of the system when feedback obtained from the loadside is used in the PI controller. Observing the dynamics of the openloop system given by (2.10), it is noticed that the numerical value of the belt stiffness, Kb, is much larger than other parameters such as radii of pulleys, or the inertias. This situation is reminiscent of the singular perturbation problem [53–56] which addresses the dynamics of systems when one parameter in the dynamics is very small. In the present case, the inverse of the square root of the belt stiffness (1/√Kb) is used as the small parameter. 2.3 Singular perturbation analysis To perform the singular perturbation analysis, the system of equations (2.10) with the control law given by either (2.3) or by (2.7) needs to be expressed in the form x˙ = A11x + A12z, x(t0) = x0 (2.26a) εz˙ = A21x + A22z, z(t0) = z0 (2.26b) where x and z are the states of the slow and the fast subsystems and ε is a small parameter. The elements of matrices Aij may depend on ε. However, to use the singular perturbation method2, the matrix A22 needs to be nonsingular (Please see Remark A.3.1 on page 190 ) at ε = 0. Let us first analyze the system of equations (2.10) with the control law given by (2.3), that is, feedback from the motor shaft. A natural choice of the state variables is to use θm, θ˙m, θL and θ˙L. However, with this choice of the state variables, the matrix A22 becomes singular at ε = 0. To obtain a statespace representation in the standard form, a transformation 2In an effort to make this report selfcontained, a brief review of the singular perturbation method used is given in Appendix A. 23 given by θc , Jmθm + JL(R2/R1)θL Jm + JL (2.27a) θs , θm − (R2/R1)θL (2.27b) is used. The variable θc is a weighted average of angular displacements (θm and θL) referred to the motor side and the variable θs is difference between the angular displacements (θm and θL) referred to the motor side. The idea of the weighted average of the displacements arises naturally in the case of a translatory system wherein θc represents the position of the centroid of the masses. Now, choosing the state variables as x = [θc, θ˙c]⊤ and z = [θs/ε2, θ˙s/ε]⊤, the state space representation of the system is obtained in the form given by (2.26) where A11 = 0 1 f1 f3 , A12 = 0 0 ε2f21 + f22 εf4 , A21 = 0 0 g1 g3 , A22 = 0 1 ε2g21 + g22 εg4 , (2.28) f1 = −Kim/J0, f21 = −KimJL/J2 0 , f22 = (R2 2 − R2 1)/J0, f3 = −(Kpm + bm + bL)/J0, f4 = (bLJm − bmJL − KpmJL)/J2 0 , g1 = −Kim/Jm, g21 = −KimJ2L /(JmJLJ0), 24 g22 = −(R2 1JL + R2 2Jm)/(JmJL), g3 = (bLJm − bmJL − KpmJL)/(JmJL), g4 = −(KpmJ2L + bmJ2L + bLJ2m )/(JmJLJ0), where J0 = Jm + JL, and 1/ε2 = Kb. Notice that det(A22(ε)"=0) = −g22 6= 0, thus satisfying the requirement noted in Remark A.3.1. Characteristic equation for the system given by (2.28) can be factored as (refer to [56, Sec. 2.1–2.3] or the Appendix A for details) ψm(s, ε) ≈ 1 ε2ψms(s, ε)ψmf (p, ε) = 0 (2.29) with ψms(s, ε) , det[sI2 − (A11 − A12L(ε))] (2.30a) ψmf (p, ε) , det[pI2 − (A22 + εL(ε)A12)] (2.30b) where ψms(s, ε) is the characteristic polynomial for the slow subsystem and ψmf (p, ε) is the characteristic polynomial of the fast subsystem exhibited in the highfrequency scale p = εs. The matrix L(ε) is obtained using the iterative scheme given by (A24). Using the matrices given by equations (2.28), the slow and the fast characteristic polynomials are evaluated as ψms(s, ε) ≈ s2 + α1s + α0, (2.31a) ψmf (p, ε) ≈ p2 + α′ 1p + α′ 2 (2.31b) where α1, α0 are defined in (2.5) and α′ 1 = R2 2KpmJL Jm(R2 2Jm + R2 1JL) ε α′ 2 = R2 2JL + R2 1Jm JmJL . (2.32) Equation (2.31) indicates that both the fast and the slow subsystems are stable for all Kpm, Kim > 0. This result is in agreement with Theorem 2.2.1. 25 Similar analysis is performed for the case of feedback from the load shaft, that is, using (2.10) and the control law given by (2.7), to obtain the slow and fast characteristic polynomials as ψls(s, ε) ≈ s2 + β1s + β0 (2.33a) ψlf (p, ε) ≈ p2 − β′ 1p + β′ 0 (2.33b) where β0, β1 are defined in (2.9) and β′ 1 = R2 2bm + R2 1bL + R2 2KpL (R2 2Jm + R2 1JL) ε, β′ 0 = R2 2JL + R2 1Jm JmJL . (2.34) Comparing equations (2.31a) and (2.33a), we notice that the slow subsystems are stable for all Kpm, Kim, KpL, KiL > 0. However, when the feedback from load shaft is used, characteristic polynomial of the fast subsystem given by (2.33b) is unstable for all KpL > 0 and KiL > 0. Also notice that the characteristic polynomials given by equations (2.31b) and (2.33b) are identical when ε = 0. Thus, analyzing the limiting case of an infinitely stiff belt, that is, ε = 0 will not reveal the instability exhibited by (2.33b). The instability exhibited by (2.33b) may be attributed to the fact that the belt connecting the inertias is assumed to be purely elastic, without any damping in it. If we were to assume the existence of damping in the material of the belt, it will only add a positive term to the coefficient of p in equations (2.31b) and (2.33b). In this case, the fast subsystem with characteristic polynomial given by (2.33b) will be stable for some values of KpL and unstable for others. Remark 2.3.1 Notice that the characteristic equations given by equations (2.4), (2.31a) are identical and so are the characteristic equations given by (2.8), (2.33a). That is, the singular perturbation analysis also indicated that an equivalent inertia and an equivalent damping may be defined and these equivalent quantities may be used to place the poles of the characteristic equation by properly choosing the gains of the PI controller. However, the analysis, in addition, revealed that using feedback from only the load side is not preferable since such feedback scheme may make the fast system unstable. 26 Remark 2.3.2 Equation (2.10b) may be used to give an interpretation of the foregoing analysis. Differentiating the (2.10b), we obtain JLω¨L + bLω˙ L + R2 2KbωL = R1R2Kbωm. (2.35) This equation shows that ωL can attain steadystate only when ωm attains steadystate first. Even after ωm attains steadystate, ωL continues to exhibit damped oscillations for some time before it attains steadystate. Thus, by measuring only ωL and using the control law given by the (2.7), we will not be able to say for sure whether the oscillations in ωL are due to fluctuations in motor speed or, the oscillations are indeed damped oscillations. In such a situation, the controller attempts to react to the damped oscillations also, and in this process, changes ωm, which in turn affects ωL because of the dynamics given by the (2.35). This process of correcting the load speed may go on for a very long time, if not forever, depending on the damping present on the load side. Thus, the control law given by (2.7) does not present a desirable situation; when only ωm is observed and the control law given by the (2.3) is used, such a situation does not arise. 2.4 Resonant frequency due to compliance of belt This section presents the transfer functions of the speed control systems when the motor used is in the velocity control mode or speed control mode. Figure 2.4(a) shows the torque speed characteristics of a DC permanent magnet motor. A shunt motor has similar but nonlinear characteristics [57]. By using appropriate feedback and control elements, the characteristics shown in Figures 2.4 (b) and 2.4(c) can be realized. Figure 2.4(b) shows the motor in velocity mode where the input voltage results in a proportional speed of the motor irrespective of the load torque. Similarly, Figure 2.4(c) shows the characteristics of motor in perfect torque control. In this case, the motor produces torque proportional to the input variable at any speed within the operating range of the motor. 27 Torque Control m tm wm = Ke E −Kt tm < < < E1 E2 E 3 E4 Ke E4 K e E3 Ke E2 Ke E1 (a) Uncompensated Motor wm tm KevE4 KevE3 KevE2 KevE1 < < < E1 E2 E 3 E4 (b) Motor under Speed Control wm tm Ket E4 KetE3 KetE2 KetE1 < < < E1 E2 E 3 E4 (c) Motor under w Figure 2.4: Torquespeed characteristics of motor and controller First, consider the motor in velocity control mode as shown in Figure 2.4(b). The block diagram of the system in velocity control mode is shown in Figure 2.5. In this mode, speed of the motor is maintained at the reference value, ωdm, irrespective of the load torque due to the inertia/viscous forces and the forces in the belt. The dotted line showing “feedback path” to the motor in Figure 2.5 represents the interconnection (see equation (2.10)) due to belt dynamics. The effect of this interconnection is taken care of by the motor when the motor is in velocity control mode. Thus, the motor and the controller may be represented by a gain Kev. If perfect speed control shown in Figure 2.4(b) is assumed, it is implied that the current (torque) limit for the motor is not reached [57] and ωdm = ωm. Thus, the + +  J L s + b L 1 K R BR 1 2 b s BR w m wL tL Controller Motor wdm Motor under speed control Interconnection Figure 2.5: Belt drive with motor in velocity control mode. transfer function from ωm to ωL for the block diagram shown in Figure 2.5 may be written 28 as T(s) = ωL(s) ωm(s) = KbR1R2 JLs2 + bLs + KbR2 2 . (2.36) It may be noted that the transfer function given in (2.36) and the transfer function obtained by differentiating (2.10b) are the same. This is because the load torque due to inertia/damping and the torque due to interconnection (belt) are absorbed by the velocity control scheme shown in Figure 2.4(b). Equation (2.36) may be used to predict the transient behavior of the load speed (ωL) for a given belt stiffness (Kb) or to compute the stiffness of a belt to be used to obtain a given transient behavior when the motor is under velocity control. If bL ≈ 0, the transfer function given in (2.36) reduces to T(s) = ωL(s) ωm(s) = (KbR1R2/JL) s2 + ω2n (2.37) where ωn = p KbR2 2/JL, thus indicating a natural frequency at ωn. Also, the sensitivity of the transfer function, T(s), given in (2.36), with respect to the belt stiffness Kb indicates the effect of Kb variations on the transient performance of the system. The sensitivity of T(s) is obtained as ST Kb = ∂T/T ∂Kb/Kb = ∂T ∂Kb · Kb T = s(s + bL JL ) s2 + bL JL s + KbR2 2 JL . (2.38) The sensitivity transfer function given in (2.38) indicates that, when Kb/JL is small, then the load speed, ωL, is very sensitive to variations in Kb. When the motor is under torque control, the load torque due to the inertia, damping, and the interconnection, given in (2.10b) also need to be considered as shown in Figure 2.6. If perfect torque control is assumed, then the torque produced by the motor is equal to the torque reference. In this case, the gains of the PI speed controller also affect the transient performance of the system. The closedloop transfer function of the system is given by G1(s) = ωL(s) ωdm(s) = (R1R2Kb/JmJL)(sKp + Ki) ψm(s) (2.39) 29 +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Torque Control Motor Under Torque Control Motor +  wdm PI Speed Controller Torque reference Figure 2.6: Belt drive with motor in torque control where Ãm(s) = s4 + (bmJL + JmbL + KpJL) JmJL s3 + (Kb[R2 2Jm + R2 1JL] + bmbL + KpbL + KiJL) JmJL s2 + (Kb[R2 2bm + R2 1bL] + R2 2KbKp + KibL) JmJL s + R2 2KbKi JmJL , s4 + ®3s3 + ®2s2 + ®1s + ®0. (2.40) The sensitivity of transfer function given in (2.39) is obtained as SG1 Kb = ∂G1 ∂Kb · Kb G1 = s4 + α3s3 + γ2s2 + γ1s s4 + α3s3 + α2s2 + α1s + α0 (2.41) where αi are defined in (2.40), γ2 = (bmbL+KpbL+KiJL)/(JmJL), and γ1 = (KibL)/(JmJL). Analysis for finding the effect of belt stiffness on the dynamics of the system shown in Figure 2.6 is not as straightforward as it is for the case when the motor is under velocity control. This is because the characteristic polynomial given in (2.40) is of fourth order. Also, in the case of a drive motor in torque control mode, an external speed control loop needs to be used. If this speed control loop is designed to give fast response, a reasonable approximation for the resonant frequency is the value obtained in the case of velocity control mode, which is given by (2.37). 2.5 Experiments Experiments were conducted on the unwind drive system in the High Speed Web Line (HSWL). Figure 2.7 shows a picture of the HSWL. The HSWL consists of an unwind 30 station, a winder station, and two nip stations. The experiments were conducted on the transmission system used in unwind station of the HSWL. Figures 2.8 and 2.9 show closer pictures of the transmission system used in the unwind station of the HSWL. The drive motor is an RPM AC 3phase induction motor rated at 30 HP, under vector control . These type of motors mimic the features of DC motors under torque control mode (that is the torquespeed characteristics as shown in Figure 2.4(c)). The beltpulley transmission system offers a speed reduction of 1:2 and the bevel gear system has unity speed ratio. To mimic a known inertial load due to the unwind roll, four steel discs of known mass (20.45 kg each) are mounted on the unwind shaft. The total inertia of the core shaft and the metal disks is approximately 2.13 kgm2. The drive motor, shown in Figure 2.9 has an encoder connected at one end of the motor shaft to measure the angular velocity of the motor shaft and this encoder is used as feedback element in the speedcontrol loop.. A tachogenerator is mounted on the chuck holding the unwind shaft as shown in Figure 2.8. This tachogenerator is used to measure the speed of the unwind shaft. Web is not threaded in the machine and all the other motors are shutoff except for the unwind motor. Three different belts3 are used in the experiments viz., 8MGT2179212mm, 8MGT2 179224mm, and 8MGT2179236mm. The 36 mm belt has a nominal stiffness of 4.7160× 104 N/m and the stiffnesses of the other belts decrease with the widths, thus 12 mm belt has least stiffness. The driving sprocket has a pitch diameter of 9.023 in and the driven sprocket has a pitch diameter of 4.51 in, thus the speed reduction is approximately equal to 2. Figure 1.1 shows a schematic of the drive system. A series of experiments are conducted to study the performance of the speed control system using different belts by specifying step changes in speed as reference to the motor speed; the motor speed and the load speed are acquired. Figures 2.10 to 2.12 show a representative sample of the experimental results. Figure 2.10 shows the response of the control system when a step input of 200 RPM is specified. The top plot shows the response 3Widths of the belts are 12 mm, 24 mm, and 36 mm respectively 31 Figure 2.7: Picture of the HSWL 32 Figure 2.8: Picture of the unwind transmission system in the HSWL (as seen from operatorend) 33 Figure 2.9: Picture of the unwind transmission system in the HSWL (as seen from gearend) 34 with 12 mm belt, the middle plot shows the response with 24 mm belt and the bottom plot shows the response with 36 mm belt. It is seen that the motor speed and the load speed reach steadystate in less than two seconds. However, with 12 mm belt and 24 mm belt, a “glitch” is seen immediately after the step change in reference occurred. Such a phenomenon is expected and is due to the stretching of the belt (the term R1θm − R2θL in (2.10)). Normally, such a “glitch” does not affect the performance of a speed control system too much. However, in the case of web handling systems, this may cause serious changes in the tension and hence is undesirable. 0 200 400 600 12mm belt RPM Motor Speed Load Speed Reference 0 200 400 600 24mm belt RPM 0 2 4 6 8 10 0 200 400 600 36mm belt Time (sec) RPM Figure 2.10: Transient response of the motor and load Figure 2.11 shows steadystate load speed. It may be noticed that with 12 mm belt, the amplitude of the sustained oscillations is around 5 RPM. This amounts to a linear velocity 35 oscillation to the tune of 0.25m/s at a nominal roll radius of 0.5m. Though such oscillations in themselves form an insignificant percentage of the process speed, their effect on the webtension is significant. If we consider a web material with a nominal stiffness of 100 N/m, the linear velocity oscillations amount to tension variations to the tune of 25 N, which may not be acceptable in many process lines. 270 280 290 12mm belt RPM Load Speed 270 280 290 24mm belt RPM 0 2 4 6 8 10 270 280 290 36mm belt Time (sec) RPM Figure 2.11: Steadystate loadspeed An important consideration in any process line is resonance avoidance. For the unwind drive system, resonant frequencies are computed using the natural frequency given in (2.37). For the 12 mm belt, the natural frequency was evaluated to be approximately 5.5 Hz, for the 24 mm belt, the natural frequency was evaluated to be 7 Hz, and for the 36 mm belt, the natural frequency was 8.6 Hz. Since it is not advisable to verify these nat 36 ural frequencies experimentally by exciting the system at these frequencies, the load speed signal is analyzed using fast fourier transform (FFT). Figure 2.12 shows the FFT content of the load speed signal. Since the natural frequencies for all the three belts are very close to each other, it is difficult to distinguish these from the FFT plots. However, a resonant peak is seen at approximately 5 Hz in the top plot in Figure 2.12. Also, the middle and bottom plots in Figure 2.12 show resonant peaks near 10 Hz, closely corresponding to the theoretically calculated natural frequencies. 0 0.1 0.2 RPM 12 mm Belt Load speed 0 0.1 0.2 RPM 24 mm Belt 2 4 6 8 10 12 14 16 0 0.1 0.2 RPM 36 mm Belt Figure 2.12: FFT of the load speed signal 37 2.6 Summary This chapter considered the effect of compliance of the belt on the speed control system in a web process line. Contrary to the intuitive idea, it is shown that, when the belt in a transmission system is compliant, it is not advisable to use only the load speed as the feedback signal. Also, in the case of unwind/rewind drives, where the load inertia changes with time, a method of tuning the proportional and integral gains of the controller is proposed. Experiments indicate that sustained angular velocity oscillations occur in the load speed even after it attained steadystate. Such oscillations severely affect the web tension. The natural frequency due to the compliance of the belt is computed. Theoretically computed value of the natural frequency agrees with the value estimated from the experiments. 38 CHAPTER 3 Effect of backlash and compliance on the output speed of a gear drive Backlash is one of the most commonly encountered nonlinearities in drive systems employing gears or ballscrews and indicates the play between adjacent moveable parts. Since the action of two mating gears can be represented by the action of one pair of teeth, backlash is commonly represented by the schematic shown in Figure 3.1. When used in the context of mechanical engineering, backlash denotes two salient features as shown in Figure 3.1: (i) a mechanical hysteresis due to the presence of clearance (¢), and (ii) impact phenomena between the surfaces of the masses (Mm and ML). In Figure 3.1, Mm and ML are the masses (inertias) of the driving and driven members, xm and xL are the linear (angular) displacements of the driving and driven members, respectively, from a fixed reference position, and Fm and FL are the driving and load forces (torques). It is a common practice to lump all the mass (inertia) on the driving side into one quantity, Mm, and refer to it as the “motor” and lump all the mass (inertia) on the driven side, and refer to it as the “load”. The classical backlash model considers the schematic shown in Figure 3.1 with input to the backlash as the displacement xm and the output of the backlash as the displacement xL. The inputoutput characteristics of the backlash are represented by Figure 3.2. The slopes of lines GBC and FED are equal to the speed ratio of the gearing in the case of rotary systems. The closed curve BCDEFGB in Figure 3.2 represents mechanical hysteresis due to the presence of clearance ¢. At points B, D, and G in Figure 3.2, the two masses impact and near these points, the inputoutput plot may not be straight but may “oscillate” with a small amplitude. However, impact may be considered to be sufficiently plastic so that 39 D m FL x m xL Mm ML D F Figure 3.1: Schematic of backlash Impact m x L D −D D −D A B C D E F G Impact Impact x Figure 3.2: Inputoutput plot for frictioncontrolled backlash 40 points on these lines lie along a curve bounded by the dotted circles shown, before they resume to lie on the straight lines. The classical backlashmodel resorts to this simplification mainly because in large industrial machines, which operate at steadystate do not reverse direction, impact does not arise except during starting/stopping conditions. Also, in smaller machines, the gear and impact energy are very small. Thus, a plastic impact is considered to be a reasonable assumption. Consequently, all the impacts are assumed to be plastic in this report. Since large industrial machines do not reverse direction many times during their operation, the lines CDE and FGB in Figure 3.2 are ignored and the inputoutput graph of backlash is represented by the curve FEABC, which is the inputoutput graph for deadzone nonlinearity. Though it is a misnomer, often, backlash and deadzone are used synonymously [27, 58]. Notice, however, that input to the backlash is considered to be a displacement of the motor and, the inputoutput plot shown in Figure 3.2 represents quasistatic behavior, at best. The rest of this chapter is organized as follows. Section 3.1 presents a brief review of the existing work on modeling and control of systems containing backlash. Section 3.2 presents a method of analyzing the effect of backlash ignoring the effect of compliance. Analysis of the effect of backlash is presented in Section 3.3 and a bound on velocity error due to presence of backlash is presented in Section 3.4. Experiments are conducted on a tabletop platform to validate the analysis. Results of these experiments are presented in Section 3.5. Summary of this chapter is given in Section 3.6. 3.1 Literature review Research on modeling backlash and its effects dates back to the 1960’s. Much of this research focused on the method of describing functions to investigate limit cycles and deriving stability criteria for systems containing backlash [59–62]. Dubowsky and Freudenstein [63,64] developed a rectilinear model called “impact pair” and presented a dynamic analysis of mechanical systems with clearances wherein it was shown that the compliance can be 41 represented by a linear spring rate, without significantly affecting the dynamic response of the model. Using the Dubowsky’s model, Azar and Crossley [65] studied the dynamic behavior of meshing gears. As a further development, Yang and Sun [49] developed a rotary model for spur gear dynamics and computed the contactspring rate and a time dependent damping for a pair of standard spur gears (pressure angle = 15 or 20 degrees). Figure 3.3 shows a mating pair of gears considered by Yang and Sun. Note that the spring and the D to teeth common normals D P 2a O1 O2 R1 R2 Rb1 q1 T1 z 1 J 1 T2 q2 R b2 z 2 J2 Figure 3.3: A rotary model of meshing spur gears damper shown in Figure 3.3 correspond to the stiffness of the gear tooth in bending and 42 the material damping respectively. In industrial drive systems, the gear teeth have large bending strength and little material damping and thus, for practical purposes the gear tooth may be considered rigid. The “compliance” to be considered in the model typically comes from the shafts on which the driving and driven gears are mounted. Using the models developed, a number of researchers reported control strategies to compensate for the effects of backlash. These control strategies may be grouped into two main categories: (i) strategies for controlling the displacement of the driven member, and (ii) strategies for controlling the velocity of the driven member. A comprehensive survey of various such strategies is reported in [66]. A delayed output feedback controller is proposed for a backlashfree plant in [67] to compensate for the effects of backlash with displacement as its input and output. However, it is not clear as to how the delayed feedback controller stabilizes the system. In the same year, Tao [26] proposed an “adaptive right backlash inverse” for unknown plants with backlash and showed that all closedloop signals are bounded. Similar work on dynamic inversion using neural networks was reported in [68, 69]. Though stability of the system using these inversion schemes is shown through simulations, it is reported in [66] that “the adaptive control seems to yield bad transients during adaptation, while after adaptation, the gain, and hence the bandwidth of the adaptive control system is lower than the gain of the robust linear system”. Prior to these observations, Dean, Surgenor and Iordanou [70] reported a study to experimentally evaluate the inversion scheme presented by Tao in [26] wherein the backlash inverter was found to actually degrade performance in the experiments. Besides, these inversion schemes pertain to position controlled drive systems and are not directly applicable to speed controlled systems. Quantitative design of a class of nonlinear systems with parameter uncertainty was considered by Oldak in [27]. The nonlinearities y = N(x) considered are such that they can be expressed as y = Kx + η(x) where η(x) < M. Several nonlinearities, such as preload, deadzone, quantization, dry friction, and backlash, are shown to belong to this 43 class. Using this idea, Boneh and Yaniv [71], proposed a scheme to reduce the amplitude of limit cycles caused by backlash. Again, the classical backlash model was used in this paper. In contrast to the number of papers published on position control in the presence of backlash, the number of papers published on the speed control is relatively few [29, 30, 58, 72]. The lack of interest, as noted in [66], is either due to the fact that high precision speed control is not required for many systems, but also due to the fundamental difficulties in analyzing speed control of elastic systems with backlash from the load side. A noted exception to this observation are the web handling systems where tight tension control mandates even tighter velocity control. In [72], a nonlinear controller with “soft switching” is proposed. Though improved performance was shown on a large real life drive system, it is not clearly explained how the gains of the lowgain and highgain controllers are computed. A gear torque compensation scheme using a PID speed controller is proposed by Odai [58]. Though it is a novel idea, the PID gains appear to be chosen according to an ad hoc empirical formulae. Warnecke and Jouneh [30] proposed a backlash compensation scheme using an openloop modification of the input trajectory. The proposed velocity compensation method is most efficient only for low operating speeds and large mounting allowance between gears. Extensive literature survey onmodeling and control of industrial speed controlled drives indicates that there is a definite need for a simple model of backlash. Besides, it is of practical importance to know the achievable accuracy in a given drive system with a known backlash. This practical consideration is not addressed in any of the existing literature. Motivated by this practical aspect, Sections 3.3 and 3.4, respectively, present a backlash model and a bound on the achievable accuracy in a given plant with a given backlash. To begin with, Section 3.2 presents a method to evaluate the phase delay and the gain due to the presence of backlash. 44 3.2 Analysis of backlash Consider a simplified system shown in Figure 3.4, where the displacements of masses, xm and xL, are measured from an arbitrary frame of reference. To simplify the analysis, assume that there is no rebound in all collisions and that the masses are free to slide on a smooth, frictionless surface. Notice that at any instant of time, the masses Mm and ML may be either in leftcontact, or rightcontact, or no contact. 2D m x L xm Fcos ( w t ) contact Right contact Left Load, ML Motor, M Figure 3.4: A simple backlash model The dynamics of masses Mm and ML in Figure 3.4 may be written as Mm¨xm = F cos ωt ML¨xL = 0 (3.1) when there is no contact between masses, and Mm¨xm = ±fc + F cos ωt ML¨xL = ∓fc (3.2) when masses are in left/rightcontact where fc is the contact force between masses. It is interesting to note that when the equations given in (3.1) or (3.2) are summed, we obtain Mm¨xm +ML¨xL = F cos ωt. (3.3) 45 Equation (3.3) is free of contact forces and this provides motivation to define a new variable y = Mmxm +MLxL Mm +ML (3.4) and write (3.3) as ¨y = F cos ωt Mm +ML . (3.5) The new variable y defined in (3.4) has a physical significance: it is the position of the center of mass of the system [33, 54]. Also, (3.5) indicates that the center of mass is not affected by the contact forces and moves only under the action of external forces and so the dynamics of y may be considered as the motion of a solid of mass (Mm + ML); the dynamics of y is referred to as solid motion hereafter. It is now possible to view the motions of the masses Mm and ML as the sum of two motions: (i) solid motion and (ii) deviations from solid motion. Define xm , y + zm xL , y + zL (3.6) where zm and zL denote the deviations of motions of Mm and ML from the center of mass. Differentiating (3.4) and using (3.6), Mmz˙m +MLz˙L = 0. (3.7) From (3.7), it can be inferred that the deviations in motions of masses from the center of mass are always proportional to each other and are oppositely directed. Integrating (3.7) results in Mmzm +MLzL = 0 (3.8) where the constant of integration is forced to become zero by choosing the frame of reference such that zm = 0 and zL = 0 at t = 0. Equations (3.4)–(3.8) and the concept of solid motion enable us to view the motion of the mass ML in simplified terms. Since the maximum amount of “delay” introduced into the motion of the mass ML due to backlash is of interest, consider the motion of the motor 46 as it closes the backlash gap, beginning to move from a rightcontact (see Figure 3.4) until it establishes a leftcontact. With the dimensions indicated in Figure 3.4, for the backlash gap to be closed, the following condition has to be satisfied (xm − xL)right − (xm − xL)left = 2¢. (3.9) Using equations (3.6), (3.8), and (3.9), (zL)right − (zL)left = −2¢Mm Mm +ML = −2¢ 1 + ML Mm , −br (3.10) where the subscripts right and left indicate rightcontact and left contact, respectively. Equation (3.10) shows that, when the ratio ML/Mm is very large, the effect of backlash is minimal since the effective backlash is very small and the effective backlash gap is very small and is traversed very quickly. Equation (3.10) offers a conceptually simple way of analyzing the effect of backlash; instead of considering backlash gap b between motor mass and load mass, one can consider the backlash gap br between the solid motion and load mass. Since solid motion is not affected by contact forces, it is sinusoidal when the force exerted by the motor is sinusoidal. In contrast to this, the motor mass dynamics depend on the contact condition as given in equations (3.1) and (3.2). To illustrate the method, simulations were conducted using the numerical values,Mm = 10 kg, Mr = 20 kg, b = 0.05 cm, f = 30, N, and ω = 1 cycles per second. Noting that the solid motion is sinusoidal with the same frequency as that of the load, the velocity of the load mass may be computed as shown in Figure 3.5. Note that, when the backlash is very small, the shaded area in Figure 3.5 will be very small. To add clarity, the shaded area has been enlarged. Starting from the right contact position as shown in Figure 3.4, the load mass and the center of mass of the system travel at the same velocity till a point where the motor mass slows down. This occurs at the peak of the solid motion curve in Figure 3.5. 47 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 time (sec.) velocity Solid Motion Load motion Figure 3.5: Computing load velocity from the solid velocity Each vertical segment in between the solid motion curve and the load motion curve in Figure 3.5 represents the deviation z˙L. The deviation of motor velocity, z˙m, may now be computed using (3.7) and the motor velocity may be constructed using (3.6). Figure 3.6 shows the deviations of the load and motor velocities from the solid velocity. It can be observed that the deviations are either zeros or are oppositely directed at all times. Figure 3.7 shows the velocities of the load and motor mass and the solid velocity in which the peaks/nadirs of the sinusoidal solid velocity curve indicate the points of separation and the points where all the three curves meet indicate sudden equalization of velocities of load and motor due to plastic impact. Figure 3.8 shows the displacements of the load/motor and the center of mass. It is interesting to note that the motor displacement may exhibit two ”peaks”, one peak at the instant where the load and motor masses are separated and the other just after the equalization of velocities. The load velocity curve shown in Figure 3.7 may be used to find an estimate of the gain and the phase delay introduced by backlash. For a given reduced backlash, the load 48 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 time (sec.) velocity Deviation of motor velocity Deviation of load velocity Figure 3.6: Deviations of load/motor velocities from solid velocity velocity for a given amplitude of the force may be computed as shown in Figures 3.5 to 3.7. The amplitude (xL) and the phase (φL) of the load velocity at fundamental frequency (ω) may be obtained from harmonic analysis of the load velocity curve. The ratioM = xL/ym, where ym is the amplitude of solid motion, indicates the gain introduced due to backlash, and φL indicates the phase delay introduced due to backlash. For a given backlash gap b, φL indicates an estimate of the phase lead to be provided by the controller to minimize the effects of backlash. When there is considerable friction on the load side, the analysis is slightly different. In this case, the load velocity does not remain constant upon loss of contact; it gradually decays due to friction till contact is reestablished. The load velocity may be computed in this case also, and a harmonic analysis of the load velocity may be carried out to obtain the gain and phase delay introduced due to backlash, similar to the nofrictioncase. If M is the gain due to backlash at frequency ω and Y (jω) is the frequency response of the system without backlash (i.e., b = 0), then, the condition, Y (jω)M = −1, indicates the possibility 49 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 time (sec.) velocity Motor Load Solid Figure 3.7: Load/Motor velocity and solid velocity of sustained oscillations at frequency ω [23, pp. 280–295]. 3.3 Backlash model with compliance This section considers the transmission system shown in Figure 1.1 and develops a model for backlash. Compliance in the transmission systems may arise due to the elasticity of shafts on which gears are mounted, or due to the belt. Section 3.3.1 presents a model of backlash with compliant shaft and Section 3.3.2 presents a model of backlash with compliant belt. 3.3.1 A model of backlash including a compliant shaft To develop a simplified model, consider the schematic shown in Figure 3.9. In this figure, a load (JL) is driven through a compliant shaft (k is the stiffness) and a pair of gears (radii R1 and R2). Usually, the motor (Jm), is mounted near the driving gear, thus the driving shaft is not very long and so may be assumed to be rigid. 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 5 5.02 5.04 5.06 5.08 5.1 5.12 5.14 time (sec.) displacement Motor Load Solid Figure 3.8: Displacement of the load/motor and center of mass 2R m Tm qg2 qL JL T L q m Driving shaft Driven shaft k 2R 2 1 J Figure 3.9: Schematic of a gear drive 51 To avoid jamming of the gears at high speeds, the gears are mounted with a center distance slightly greater than the designed center distance. This gives rise to clearance between the teeth as shown in Figure 3.3; this clearance is termed “backlash”. To pictorially represent backlash in torsional systems, at least two orthographic views are needed viz., a front view as shown in Figure 3.9 and a side view as shown in Figure 3.3. Also, for studying the effect of a given backlash on the output speed experimentally, one has to assemble the system shown in Figure 3.9 by varying the center distance between the gears. Such experimentation takes a lot of effort and very precise measurement and mounting techniques since the relation between the center distance and amount of backlash gap is not linear. Due to these reasons, often, the rotary system shown in Figure 3.9 is analyzed using a rectilinear analog as shown in Figure 3.10. When a rectilinear analog is used, pictorial representation as well as experimentation is considerably simplified. Further, the model developed using rectilinear analog can be easily converted to the rotary system shown in Figure 3.9 using rectilineartorotary transformation. Figure 3.10(a) and 3.10(b), respectively, show a rectilinear analog of the system with and without backlash. The objective of the analysis is to study how the linearity of the system shown in Figure 3.10(a) gets affected by introducing backlash as shown in Figure 3.10(b). To obtain the equations of motion for the system shown in Figure 3.10(b), first conditions under which contact occurs at points P or Q need to be evaluated. Without loss of generality (see Remark 3.3.1 on page 55), consider the displacements Xm0, Y0, and XL0 as shown in Figure 3.10(b). The free length of the spring when the system is at rest is obtained as Ls0 = XL0 − Y0 − ¢. (3.11) 52 (a) b m bL Fm Mm Xm0 (b) P Q k Reference ML FL XL0 k Reference b m bL Mm ML Fm FL Xm0 XL0 Y0 c 2 c 1 c 1 =c 2 = D Figure 3.10: rectilinear analog: (a) without backlash, (b) with backlash 53 Defining the deviations xm = Xm − Xm0, xL = XL − XL0, y = Y − Y0, (3.12) contact at point P occurs if Xm = Y − ¢, that is, if xm = y − ¢ (3.13) and contact occurs at point Q if Xm = Y + ¢, that is, if xm = y + ¢. (3.14) The length of the spring at any instant of time is obtained as Ls = XL − Y − ¢ = xL + XL0 − y − Y0 − ¢ = (xL − y) + Ls0. (3.15) Thus, when contact occurs at point P, the length of the spring may be obtained from equations (3.13) and (3.15) as LP = (xL − xm − ¢) + Ls0. (3.16) If loss of contact at point P were to occur, it must be either due to mass Mm moving to the right, that is due to increase in xm, or due to mass ML moving to the left, that is due to decrease in xL. In either case, loss of contact at P is occurring due to decrease in (xL−xm) and thus, due to decrease in (xL − xm − ¢). Since the spring tries to regain its original length soon after contact is lost, we say that, LP keeps on changing till either its value is equal to Ls0 or a contact is established, whichever occurs first. From (3.16), we see that if the value of LP tends to change in the direction of Ls0 and (xL − xm − ¢) is decreasing soon after loss of contact, we see that, to begin with (xL − xm − ¢) must be greater than or equal to zero. Thus, (xL − xm − ¢) ≥ 0 (3.17) 54 is the condition for sustained contact at P and (xL − xm − ¢) is the change in the length of the spring. Similarly, when the contact occurs at point Q, the length of spring may be obtained from equations (3.14) and (3.15) as LQ = (xL − xm + ¢) + Ls0. (3.18) If the loss of contact at point Q were to occur, it must be either due to mass Mm moving to the left, that is due to decease in xm, or due to mass ML moving to the right, that is due to increase in xL. In either case, the loss of contact is occurring due to increase in (xL − xm) and thus due to increase in (xL − xm + ¢). Again, given the fact that the spring tries to regain its original length, and the length of the spring when contact exists at Q, given by (3.18), we see that the condition for contact at point Q is (xL − xm + ¢) ≤ 0 (3.19) and (xL − xm + ¢) is the change in the length of the spring. From equations (3.17) and (3.19), we see that, as long as there is no contact, −¢ < (xL − xm) < ¢. (3.20) Remark 3.3.1 At a first glance, it appears that, assuming the system to be centrally located in the backlash gap (that is, ¢ on either side in Figure 3.10 (b)) renders loss of generality. However, since the objective in the “thought experiment” is to see how the behavior of system shown in Figure 3.10(a) deviates from linearity due to the presence of backlash as shown in Figure 3.10(b), such an assumption does not stand as an obstruction. Besides, the same analysis can be carried out assuming the gaps on either side to be c1 = 2¢α and c2 = 2¢(1 − α) for some α ∈ [0, 1]. Remark 3.3.2 It may be noted that the location of the compliance, shown by a spring of stiffness k, with respect to the backlash does not change the contact conditions or the dynamics of the system. Equations (3.17), (3.19), and (3.20) continue to be the conditions 55 of contact at P, contact at Q, and no contact for the backlash configuration shown in Figure 3.11 the compliance precedes backlash. Thus, the dynamic equations developed later apply to the configuration shown in Figure 3.11 also. k Reference Mm P Q c X c 2 1 m0 Y0 c 1 =c 2 = D XL0 b m Fm ML bL FL Figure 3.11: Rectilinear analog where compliance precedes backlash With the deformations of the spring given by equations (3.16), (3.18) and the contact conditions given by (3.17), (3.19), and (3.20), the kinetic energy and the potential energy of the system shown in Figure 3.10(b) may be written as K(x˙m, x˙L) = 1 2 [Mmx˙ 2 m +MLx˙ 2 L] V (xm, xL) = 1 2k(xL − xm − ¢)2 if (3.17) holds 1 2k(xL − xm + ¢)2 if (3.19) holds 0 if (3.20) holds (3.21) With the kinetic energy and potential energy defined in (3.21), the dynamics for the system shown in Figure 3.10(b), ignoring the inertias of the spring and the shaft, may be written as Mmx¨m + bmx˙m + ψ(xm, xL) = Fm, MLx¨L + bLx˙L − ψ(xm, xL) = FL, (3.22) 56 where ψ(xm, xL) = k(xm − xL + ¢) if (3.17) holds, k(xm − xL − ¢) if (3.19) holds, 0 if (3.20) holds. (3.23) 3.3.2 Effect of belt compliance and backlash in gears Section 3.3.1 considered the effect of backlash when one of the shafts in the transmission system is compliant. This section considers the transmission system shown in Figure 1.1 and presents a model that includes the effect belt compliance and backlash in the gears on the output speed of the transmission system. The transmission system shown in Figure 1.1 uses a bevel gear set to transmit motion across the shafts which are at right angles. In such cases, a standard approach is to consider an equivalent spur gear set in place of the bevel gears and perform the analysis of the spur gear set. Figure 3.12 shows a schematic of the transmission system using a beltpulley transmission system and an equivalent spur gear pair. The backlash effect on the output speed of the transmission system shown in Figure 3.12 may be computed using the approach given in Section 3.3.1. Since a pair of mating spur gears rotate in opposite directions, a sign convention is needed to keep track of the angular displacements. The sign convention followed here is that, looking from the load side (that is, from the right hand side of Figure 3.12), θ is considered to be positive in counterclockwise direction and θL is considered to be positive in clockwise direction. Also, reference for angular displacements is taken to be θm = θ = θL = 0 and hence, the deviations in the angular displacements and their absolute values are the same. Further let the freelength of the tight side of the belt in Figure 3.12 be L0. At any instant, the length of the tight side of the belt may be obtained as L = L0 + (R2θ − R1θm). (3.24) A condition for contact at point P is determined by considering the length of the tight side 57 D A q Gear 1 Gear 2 Sprocket 1 L 2R m Jm 2R t L Pitch Circle P Q Enlarged view of A q 1 g1 Belt m q t JL 2R Sprocket 2 2 2Rg2 D Figure 3.12: Schematic of a transmission system using beltpulley arrangement and a gearpair 58 of the belt when contact does exist at point P. First, notice that, for contact at point P, Rg1θ = Rg2θL + ¢ ⇒ θ = 1 Rg1 [Rg2θL + ¢] (3.25) which indicates that a point on the pitch circle of gear 1 has to travel an extra distance of ¢ for contact to be established. Thus, the length of the tight side of the belt during sustained contact at point P may be written as LP = L0 + R2 Rg1 [Rg2θL + ¢] − R1θm , α1θL − R1θm + α2¢ + L0 (3.26) where α1 , R2GR, α2 , R2/Rg1, and GR = Rg2/Rg1. Notice that equation (3.26) is similar to (3.18) except for the coefficients α1, R1, and α2. Contact at point P will be lost either when θ decreases or when θL increases. In either case, (α1θL − R1θm + α2¢) increases. Therefore, upon loss of contact, LP → L0. Coupled with this fact, (α1θL − R1θm + α2¢) increases when contact is lost at point P means that to begin with (α1θL − R1θm + α2¢) ≤ 0. (3.27) Thus, (3.27) gives a condition for contact at point P. The quantity on the left hand side of inequality (3.27) is the change in the length of the tight side of the belt. Similarly, condition for contact at point Q may be written as (α1θL − R1θm − α2¢) ≥ 0. (3.28) The quantity on the left hand side of inequality in (3.28) is change in the length of the tight side of the belt. From equations (3.27) and (3.28), when there is no contact, −α2¢ < α1θL − R1θm < α2¢. (3.29) Thus, the kinetic energy and the potential energy of the system shown in Figure 3.12 may 59 be written as K(x˙m, x˙L) = 1 2 [Jm θ˙2 m + JL θ˙2 L] V (xm, xL) = 1 2Kb(α1θL − R1θm + α2¢)2 if (3.27) holds 1 2Kb(α1θL − R1θm − α2¢))2 if (3.28) holds 0 if (3.29) holds. (3.30) Using the kinetic energy and the potential energy given in (3.30), the dynamics of the system shown in Figure 3.12, ignoring the inertias of the pulleys and the gears, may be written as: Jm¨θm + bm θ˙m + R1ψ(θm, θL) = τm (3.31a) JL¨θL + bL θ˙L − α1ψ(θm, θL) = τL (3.31b) where ψ(θm, θL) = Kb (R1θm − α1θL − α2¢) if (3.27) holds (R1θm − α1θL + α2¢) if (3.28) holds 0 if (3.29) holds. (3.32) 3.4 Error bounds in the presence of backlash This section presents a bound on the error due to the presence of backlash. The idea behind the approach is to consider backlashfree system and see how the presence of backlash affects the dynamics. Section 3.4.1 presents a general idea of obtaining a bound on the error due to the presence of backlash. This idea is used in Section 3.4.2 to present a bound on error due to the presence of backlash and a compliant shaft. Section 3.4.3 presents a bound on error due to the presence of backlash and a compliant belt. 60 3.4.1 Method of finding a bound on error due to backlash This section presents a method of computing the effect of backlash in a system as shown in Figure 3.13(b). Figure 3.13(a) shows two subsystems with transfer functions G1(s) and G2(s) in series and Figure 3.13(b) shows the situation when a backlash nonlinearity, represented by BL is inserted in between them. Assume, for the purpose of illustration, that the inputoutput relation of the backlash nonlinearity BL follows the plot shown in Figure 3.2. (a) G 1 (s) 2 r u y z BL (b) G 1 (s) G 2 (s) r u = y z G (s) Figure 3.13: (a) A system without backlash and (b) System with backlash Suppose that the system G1(s) is represented in statespace form as x˙ 1 = A1x1 + B1r, (3.33a) u = C1x1, (3.33b) and the system G2(s) is represented in statespace form as x˙ 2 = A2x2 + B2y, (3.34a) z = C2x2, (3.34b) 61 with matrices Ai, Bi, and Ci of appropriate dimensions. Fortunately, inputoutput characteristics of backlash nonlinearity shown in Figure 3.2 may be considered as a sum of a linear function and a bounded nonlinear function. With this idea, the output of the backlash may be written as y = u + f(u) (3.35) where the function f(u) is bounded by f(u) ≤ ¢ for all u ∈ R as shown in Figure 3.14. Then, using equations (3.33), (3.34), and (3.35), the statespace representation of the y u −D = + u f(u) (a) (b) (c) u −D u D D D −D Figure 3.14: Inputoutput plot of backlash system shown in Figure 3.13(b) may be obtained as x˙ = Ax + Br + Df(C1x1) (3.36a) z = Cx (3.36b) where A = A1 0 B2C1 A2 , B = B1 0 , (3.37a) C = · 0 C2 ¸ , D = 0 B2 , (3.37b) and x⊤ = [x⊤1 , x⊤2 ]. If the backlash were to be absent (f(u) = 0 and so y = u), as shown 62 in Figure 3.10(a), the same system is described by w˙ = Aw + Br (3.38a) v = Cw (3.38b) where v is the output of the system without backlash. Equations (3.36) and (3.38) are similar except for the extra term Df(C1x1). To analyze the effect of backlash, the output of the system with backlash (that is, z given by (3.36b)) and the output of the system without backlash (that is v given by (3.38b)) need to be compared. Since kz −vk = kC(x−w)k, kx−wk may be evaluated to analyze the effect of backlash. For a given r, the solution of (3.38a) is obtained as w(t) = eAtw0 + Z t 0 eA(t−¿)Br(τ )dτ , φ(r, t,w0) (3.39) where w0 = w(0) is the initial condition. Then, taking the initial condition of (3.36a) as x0 = x(0) = w(0) = w0, solution of (3.36a) may be obtained as x(t) = φ(r, t, x0) + Z t 0 eA(t−¿)Df(C1x1(τ ))dτ. (3.40) Therefore, the required deviation, kx − wk, may be obtained as kz − vk = °°°° Z t 0 eA(t−¿)Df(C1x1(τ ))dτ °°°° ≤ °°°° Z t 0 eA(t−¿)D · ¢dτ °°°° (since kf(u)k ≤ ¢ ∀u ∈ R) ≤ ¢ °°°° Z t 0 eA(t−¿)Ddτ °°°° . (3.41) Equation (3.41) gives an important result: the deviation in the output at any time instant due to backlash is proportional to the halfwidth of the backlash, ¢ and the deviation can be minimized byminimizing the integral appearing in the last line of (3.41). Thus, by choosing the elements of the matrices Ai, Bi, and Ci in equations (3.33) and (3.34), backlash effect on the output can be reduced. In this sense, (3.41) gives a bound on the achievable accuracy when a known backlash is present in a known system. 63 Remark 3.4.1 A physical interpretation of (3.41) is that, to minimize the effect of the backlash, the system dynamics should be able to “close” the backlash gap as quickly as possible. 3.4.2 Bound on error due to backlash and a compliant shaft This section presents a bound on the output error due to backlash and compliant shaft using the idea presented in Section 3.4.1. To use this method, the dynamics of the backlashfree system need to be obtained. To this end, consider the situation without backlash, shown in Figure 3.10(a). When the system is at rest, the displacements of the masses are Xm0 and XL0 from a fixed reference as shown in Figure 3.10(a) and the free length of the spring is Ls0 = XL0 − Xm0. (3.42) When the system is in motion, let XL and Xm be the displacements of the masses from the fixed reference. Defining the deviations xm = Xm − Xm0, xL = XL − XL0, (3.43) the length of the spring at any instant of time may be written as Ls = XL − Xm = xL + XL0 − xm − Xm0 = (xL − xm) + Ls0. (3.44) Thus, the change in the length of the spring is Ls − Ls0 = xL − xm and kinetic energy and potential energy of the system may be written as K(x˙m, x˙L) = 1 2 [Mmx˙ 2 m +MLx˙ 2 L], V (xm, xL) = 1 2 k[x2 m + x2 L]. (3.45) Using the wellknown EulerLagrange equations of motion, given by (2.13), the dynamics of the system shown in Figure 3.10(a) may be obtained as Mmx¨m + bmx˙m + k(xm − xL) = Fm, MLx¨L + bLx˙L − k(xm − xL) = FL. (3.46) 64 Upon rearranging the terms in equations (3.22) and (3.23), the dynamics of the system shown in Figure 3.10(b) may be written as Mmx¨m + bmx˙m + k(xm − xL) − φ(xm, xL) = Fm, MLx¨L + bLx˙L − k(xm − xL) + φ(xm, xL) = FL, (3.47) where φ(xm, xL) = − k¢ if (3.17) holds, k¢ if (3.19) holds, k(xm − xL) if (3.20) holds. (3.48) Notice that equations (3.46) and (3.47) are identical except for the extra term, φ(xm, xL), present in (3.47). And this extra term, because of the condition in (3.20), is bounded by φ(xm, xL) ≤ k¢ for all xm, xL ∈ R. Defining the statevariables zm1 = xm, zm2 = vm = x˙m, zL1 = xL, zL2 = vL = x˙L, and z = [zm1, zm2, zL1, zL2]⊤, a state space representation of the system shown in Figure 3.10(b) is obtained as z˙ = Apz + BpFm + CpFL + βDp(xm, xL) y = Lpz (3.49) where Ap = 0 1 0 0 − k Mm − bm Mm k Mm 0 0 0 0 1 k ML 0 − k ML − bL ML , Bp = 0 1 Mm 0 0 ,Cp = 0 0 0 1 ML , Dp(xm, xL) = 0 Á(xm,xL) Mm 0 −Á(xm,xL) ML , Lp = 0 1 0 0 0 0 0 1 , Lp1 Lp2 (3.50) and β is zero if the backlash gap is zero and unity otherwise. Thus, with β = 0, (3.49) is a state space representation of the system shown in Figure 3.10(a) and with β = 1, (3.49) is a state space representation of the system shown in Figure 3.10(b). 65 Equation (3.49) represents a system with two inputs (the actuating force, Fm and load force, FL) and two outputs (the motor speed, zm2 and the load speed, zL2) as shown in Figure 3.15. e L + G − vr Fm L2 Load speed, z Motor speed, zm2 System Dynamics with Backlash Load force Controller F Figure 3.15: Block diagram of a controller for system with backlash In the control scheme shown in Figure 3.15, the controller, G, uses feedback from the motorside (zm2). If the load velocity is used as feedback, the controller keeps on accelerating/decelerating the motor during the nocontact period since the motor has no “control” over the load during the nocontact period. This might have been the reason for specific lack of interest in using the load side feedback, as noted in [66, Section 3]. Suppose the controller, G, has the statespace representation x˙ c = Acxc + Bce, Fm = Ccxc + Dce. (3.51) Then the statespace representation for the closedloop system shown in Figure 3.15 may be obtained as z˙cl = Aclzcl + CclFL +Wvr + Dcl(xm, xL) zb L2 = Lczcl (3.52) 66 where zcl = [z⊤ x⊤c ]⊤, W = [D⊤ c B⊤ p B⊤ c ]⊤, Lc = [0 0 0 1 0], and Acl = (Ap − BpDcLp1) BpCc −BcLp1 Ac , Ccl = Cp 0 , Dcl(xm, xL) = Dp(xm, xL) 0 . (3.53) The superscript in zb L2 indicates the output in the presence of backlash. If the backlash were to be absent, β = 0 and the statespace representation of the closedloop system may be written as v˙cl = Aclvcl + CclFL +Wvr z0 L2 = Lcvcl (3.54) where the matrices Acl, Ccl, W are given in (3.53) and z0 L2 is the output in the absence of backlash. Equations (3.52) and (3.54) are similar except for the last term in the state equation in (3.52) and the deviation zb L2 − z0 L2 = Lc(zcl − vcl) represents the effect of backlash. For a given reference velocity vr, and the disturbance force FL, solution of the state equation in (3.54) is obtained as vcl(t) = eAcltv0 cl + Z t 0 eAcl(t−¿)[FL(τ ) +Wvr(τ )]dτ , φ(vr, FL, t). (3.55) where vcl(0) = v0 cl is the initial condition. Then, taking the initial condition to be zcl(0) = z0 cl = v0 cl, the solution of the state equation in (3.52) may be written as zcl(t) = φ(vr, FL, t) + Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ (3.56) Thus, the deviation in state variable due to the effect of backlash may be written as kzcl − vclk(t) = °°°° Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ °°°° ≤ k¢ °°°° Z t 0 eAcl(t−¿)D1dτ °°°° , δb (3.57) where D1 = [0 1/Mm 0 − 1/ML 0]⊤. Thus, the deviation in the states of the system at any instant of time is proportional to half backlash width, ¢. 67 Remark 3.4.2 A physical interpretation of (3.57) is that the closedloop system must be able to “close” the backlash gap as fast as possible to minimize the effect of backlash. This is similar to Remark 3.4.1. It is possible to simplify the bound given in (3.57) and express it in terms of the eigenvalues of the matrix Acl. Assume that the eigenvalues of Acl are placed at distinct real values using the controller (3.51). Let these eigenvalues be −λ1, −λ2, · · · , −λn. Since the eigenvalues of Acl are distinct, an orthonormal matrix T exists such that Acl = T¤T−1 and so eAclt = Te¤tT−1. (3.58) Using (3.58), (3.57) is simplified as kzcl − vclk(t) = °°°° Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ °°°° ≤ k¢ °°°° Z t 0 eAcl(t−¿)D1dτ °°°° ≤ k¢ °°°° Z t 0 Te¤(t−¿)T−1D1dτ °°°° ≤ k¢kTk °°°° Z 0 t e¤(t−¿)dτ °°°° kT−1kkD1k (3.59) Since ¤ = diag{−λ1,−λ2, · · · ,−λn}, the integral in the last line of (3.59) is also diagonal and the (i, i)th element of the integral may be written as Z t 0 e−¸i(t−¿)dτ = 1 λi £ 1 − e−¸it¤ (3.60) Thus, the bound in (3.59) is further simplified to kzcl − vclk(t) ≤ k¢kTk °°°° diag{ 1 ¸1 h 1 − e−¸1t i , · · · , 1 ¸n h 1 − e−¸nt i } °°°° kT−1kkD1k (3.61) As t → ∞, the exponentials in (3.61) tend to zero and the bound may be written as kzcl − vclk(t) ≤ k¢kTk °°°° diag{ 1 λ1 , · · · , 1 λn } °°°° kT−1kkD1k ≤ k¢kD1kC1 λmin (3.62) 68 where λmin = min{λ1, · · · , λn} and C1 is the condition number of T. The bound given in (3.62) is considerably simpler to evaluate than the bound given in (3.57). If some eigenvalues of Acl are complex conjugate pairs, the matrix ¤ is block diagonal matrix and the bound may be simplified along the same lines described in equations (3.58)– (3.62). 3.4.3 Bound on error due to backlash and belt compliance This section considers the schematic of the transmission system shown in Figure 3.12 and presents a bound on the error due to the presence of backlash. Defining the statevariables zm1 = θm, zm2 = ωm = θ˙m, zL1 = θL, zL2 = ωL = θ˙L, and z = [zm1, zm2, zL1, zL2]⊤, and using the equations (3.31), (3.32), a statespace representation of the system shown in Figure 3.12 is obtained as1 z˙ = Apz + Bpτm + CpτL + βDp(θm, θL) y = Lpz (3.63) where Ap = 0 1 0 0 −KbR2 1 Jm −bm Jm KbR1®1 Jm 0 0 0 0 1 Kb®1R1 JL 0 −Kb®2 1 JL −bL JL , Bp = 0 1 Jm 0 0 ,Cp = 0 0 0 1 JL , Dp(xm, xL) = 0 −R1Á(µm,µL) Jm 0 ®1Á(µL,µL) JL , Lp = 0 1 0 0 0 0 0 1 , Lp1 Lp2 (3.64) 1the same symbols Ap, Bp etc. are used here and in equation (3.50) to highlight the fact that the dynamic model for the rectilinear analog shown in Figure 3.10 and the dynamic model for the system shown in Figure 3.12 are “analogous” to each other. 69 and β is zero if the backlash gap is zero and unity otherwise and φ(θm, θL) is defined as φ(θm, θL) = Kb − α2¢ if (3.27) holds α2¢ if (3.28) holds (α1θL − R1θm) if (3.29) holds. (3.65) Equation (3.63) represents a system with two inputs (the actuating force, τm and load force, τL) and two outputs (the motor speed, zm2 and the load speed, zL2) as shown in Figure 3.15. Consequently, a bound, as given in (3.57) may be obtained. Comparing the equations (3.20) and (3.29), we see that an additional term α2 = R2/Rg1 multiplies the backlash width ¢ in the case of analysis of the effect of belt and backlash. Due to this term, the bound given in (3.57) is modified to kzcl − vclk(t) = °°°° Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ °°°° ≤ Kbα2¢ °°°° Z t 0 eAcl(t−¿)D1dτ °°°° , δb (3.66) for the case of belt compliance. In equation (3.66), D1 = [0, −R1/Jm, 0, α1/JL, 0]⊤. If α2 is small, the bound δb is also small and so it is advantageous to have R2/Rg1 ≪ 1. 3.5 Experiments Experiments were conducted to verify the bound on the deviation in load speed due to backlash, given in (3.57). Experiments are conducted on an ECP Rectilinear System setup. The setup, shown in Figure 3.16, consists of three masses mounted on carriages which are free to slide. Since the system considered for experimentation (that is shown in Figure 3.10) is a two mass system, only masses 1 and 2 are used. A spring is used to represent the compliance k shown in Figure 3.10. That is the system shown in Figure 3.10 is realized as masses 1 and 2 connected by a spring so that Mm = M1 and ML = M2. Position of each of the masses is measured by a high resolution encoder. 70 Figure 3.16: ECP Rectilinear System Nominal values of the masses are M1=2.28 kg and M2=2.55 kg (these values include the masses of the carriages as well). Nominal stiffness of the spring is k=200 N/m. The damping present at masses, as estimated by a preliminary identification procedure, are bm = bL =0.05 Ns/m. A ProportionalIntegral (PI) controller, using velocity of mass 1 as feedback signal, is implemented to impart a prescribed velocity to mass 1. Positions and velocities of the masses 1 and 2 are acquired firstly without backlash present in the system and then with a known backlash. A description of these experiments is given in Appendix B. From each set of experiments, the difference between the load velocity (velocity of mass 2) with backlash and load velocity without backlash is computed using the experimental data. This difference is then compared with the bound computed using (3.57). Figures 3.17, 3.18, and 3.19 show the results of experiments. In these experiments, a PI controller is used to impart a sinusoidal velocity with amplitude of 10 mm and a 71 frequency of 3 Hz to mass 1 and load position and velocity are measured. The solid line in Figure 3.17 shows the deviation in the load velocity due to presence of backlash obtained from experimental data and the dashed horizontal line shows the bound on the deviation as evaluated from (3.57) using a backlash gap of 1.55 mm. It is noticed that the experimentally evaluated deviation is within the bound. Similarly, Figures 3.18 and 3.19 show the results with backlash gaps of 3.56 mm and 5.38 mm, respectively. These figures show that the deviation due to presence of backlash, as evaluated from experiments is within the bound obtained using (3.57). 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 x 10−3 m/s Deviation in load velocity due to a backlash of 1.55mm Figure 3.17: Closedloop experiment with backlash of 1.55 mm 72 0 1 2 3 4 5 6 7 0 0.005 0.01 0.015 0.02 0.025 m/s Deviation in load velocity due to a backlash of 3.56mm Figure 3.18: Closedloop experiment with backlash of 3.56 mm 73 0 1 2 3 4 5 6 7 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 m/s Deviation in load velocity due to a backlash of 5.38mm Figure 3.19: Closedloop experiment with backlash of 5.38 mm 74 3.6 Summary This chapter presented a model for backlash to include the dynamics of the driven member during loss of contact and to include the effect of disturbing forces on the load. Using the model, an upper bound on the achievable accuracy in a given system using a given controller is obtained. Experiments conducted on a prototype system agree with the theoretically estimated upper bound. 75 CHAPTER 4 Effect of compliance and backlash on web tension As noted in Chapter 1, the two important variables to be regulated in a web process line are the web velocity (Vweb) and the web tension (Tweb). The analysis presented hitherto pertains to only one variable, namely, the web velocity. In specific, Chapter 2 considered the effect of belt compliance ignoring the effect of backlash, Section 3.2 presented the backlash effect on output velocity ignoring the effects of compliance, and Sections 3.3.1 and 3.3.2 considered the analysis of the effect of backlash including the compliance of the shaft/belt. This chapter is intended to motivate further investigation of the effect of compliance and backlash on web tension. To begin with, consider the High Speed Web Line (HSWL) shown in Figure 4.1. A schematic of the HSWL is shown in Figure 4.2. This process line consists of a number of tension control zones with each zone consisting of a number of web spans. Tension/ velocity disturbances, induced in any one span, propagate in the direction of the web traversal and hence affect web tension/velocity in all spans downstream to the span where disturbances originate. To attenuate such disturbances, a driven roller is installed near each section where tight control of tension/velocity is required. The schematic in Figure 4.2 shows two such zones labeled as Nip station 1 and Nip station 2: Nip station 1 has three driven rollers and Nip station 2 has one driven roller. Themotors driving these driven rollers use tension feedback from loadcells and the speed feedback from encoders/tachometers to regulate web tension and velocity. It is common practice to designate one driven roller in the process line, usually the one nearest to the unwind roll, as the master speed roller and use it to only regulate the web 76 Figure 4.1: Pictures of the HSWL 77 Rewind Roll LC LC LC NR NR Unwind Roll Loadcell Roller E Edge Sensor E Displacement Guide, DG DG Rewind Section Unwind Section Master Speed Section Process Section Nip M3 M0 3 v T 2 1 T v T1 v 3 2 Driven Roller M1 M2 Figure 4.2: Schematic of the HSWL transport velocity and the other driven rollers to regulate web tension and web velocity in individual zones. For example, in the HSWL shown in Figures 4.1 and 4.2, the driven roller labeledM1, is used as the master speed roller and hence, the controller for the motor driving this roller uses only speed feedback. All the other controllers, including those for the motors driving the unwind and the winder rolls, use tension feedback as well as velocity feedback. Figure 4.3 shows a control scheme commonly used in regulating web tension and velocity. The control scheme shown in Figure 4.3 uses two feedback loops: an outer tensionloop and an inner speedloop. Each of these loops uses a PIcontroller. As can be seen (from Figure 4.3), the output of the tensionloop controller acts as a vernier correction to the speedloop. It may be observed that there are two PIcontrollers in the feedforward path in the block diagram shown in Figure 4.3: the error in tension passes through both these PIcontrollers whereas the error in speed passes through only one PIcontroller. Thus, the control scheme may be thought of as a combination of PIaction on speederror and a PIDaction on the integral of the tensionerror. The controller for motor driving the master speed roller does not have outer tension loop shown in Figure 4.3. Hence, it is possible to tune PI gains to make the master speed roller very closely follow the reference web speed. Under this condition, it is possible to make 78 x r Vr V0 T1 Reference + − Tension Controller PI ++ − PI Controller Motor/ Controller Dynamics Speed Dynamics Web Loadcell/ Dancer Position Feedback Speed Feedback Speed Reference Tension Speed Correction Torque Reference y T Figure 4.3: Control scheme to regulate web tension and web velocity the assumption that the velocity of web passing over the master speed roller is constant and is equal to the reference velocity [57]. Further, since energy input to the web from driven rollers is much larger than energy dissipated by the idle rollers, the length of web between two driven rollers (or an unwind/ winder roll and the driven roller nearest to it) is considered as a single span, though there may be idle rollers in between. However, these nondriven rollers are sources of tension disturbances because of their inertias in combination with springiness of the web. This chapter attempts to evaluate the effect of belt compliance on the span tension immediately next to the unwind roll under the assumptions mentioned in the preceding two paragraphs. As a first step, simulation study and preliminary experiments are conducted using the belt model presented in Chapter 2. 4.1 Simulations and experiments on the unwind section of the HSWL The unwind section of the process line shown in Figure 4.2 is considered. Figure 4.4 shows a schematic of the unwind section considered for simulation. Themaster speed roller shown in Figure 4.4 sets the web reference speed and the unwind motor sets the span tension. The motor driving the master speed roller (not shown in the figure) is under speed control and it is assumed that the speed of the web leaving the master speed roller is constant at Vr. One source of tension disturbance entering the span immediately next to the unwind roll 79 Backlash un V0 1 2 Kb wm R2 R1 Rg1 wL Rg2 T0 Unwind roll roller T V Loadcell 1 r Master speed roller Unwind motor w Pitch Circle P Q D D Enlarged view of A A Nip Loadcell roller R Figure 4.4: Schematic of the unwind section. Master speed roller, a span following the master speed roller, and locations of two loadcells are also shown. is the wound in tension (T0). To highlight the characteristics of the tension behavior in the presence of belt compliance, a sinusoidal component in T0 is assumed in all the simulations. In the first simulation study, effect of belt compliance and backlash are ignored. Thus, the equivalent inertia as given in equation (2.1) is used in writing the dynamics of the closedloop system. The dynamics of the closedloop system with the control scheme 80 shown in Figure 4.3 may be written as τm = Jeqθ¨m + beqθ˙m − R1 R2 T1Run, (4.1a) JL = Jc0 + KJ (R4 un − R4 c0), (4.1b) Jeq = Jm + µ R1 R2 ¶2 JL, (4.1c) beq = bm + µ R1 R2 ¶2 bL, (4.1d) KJ = ρwπ 2 , (4.1e) ˙R un = − δθ˙L 2π , (4.1f) V0 = Run θ˙L, (4.1g) L1 ˙T 1 = EA(Vr − V0) + V0T0 − VrT1, (4.1h) x = Kpt(Tr − T1) + Kit Z t 0 (Tr − T1)dτ1, (4.1i) y = Kps(x + R2 R1 [Vr − Rwnωm]) + Kis Z t 0 (x + R2 R1 [Vr − Rwnωm])dτ1, (4.1j) τmcτ˙m = −τm + Kmy. (4.1k) Equations (4.1a)–(4.1e) describe the dynamics of rotation of the unwind roll to include the effect of changing radius, changing inertia, and span tension . Equations (4.1f)–(4.1g) specify the rate at which the unwind roll radius is changing and specify the linear speed of the web. Equation (4.1h) is the standard nonlinear model for web tension dynamics as described in [73–76]. Equations (4.1i)–(4.1j) indicate the control law shown in Figure 4.3. And lastly, equation (4.1k) describes the dynamics of the unwind motor. In equations (4.1), the time dependence of the dynamic variables is not shown explicitly. The variables τm, JL, Jeq, Run, T1, θm, θL, V0, x, and y are time dependent variables, while all the others are constants. In practice, a step change in reference is not specified because a sudden change in web velocity will cause a large “surge” in web tension. Instead, any change in the reference (either tension or velocity reference) is allowed to ramp up towards the new value, slowly. 81 0 5 10 15 20 25 30 0 500 1000 1500 Web speed and tension (without compliance of belt) ft/min 0 5 10 15 20 25 30 15 20 25 30 Time (s) lbf Tension Tension Reference Web speed Reference speed Figure 4.5: Tension behavior in a span for a change in the reference speed 82 Figure 4.5 shows the web speed and the tension when the speed reference is rampedup from 200ft/min to 1000ft/min over a period of 5 seconds. Even in this case, the span tension increases drastically above the reference value and takes a long time to reach the reference value. To investigate the effect of belt compliance on web tension, simulations are run to reflect the same changes in the reference speed as was done in Figure 4.5. The dynamics of the system, including belt compliance, is the same as equation (4.1) except that (4.1a) and (4.1b) are modified to (4.2a) and (4.2b) as given below τm = Jmθ¨m + bmθ˙m + KbR1(R1θm − R2θL), (4.2a) KbR2(R1θm − R2θL) + T1Run = Jl ¨θL + bL θ˙L. (4.2b) Figures 4.6–4.8 show the results of the simulation. In each of these plots, the speed reference is changed from 200 ft/min to 1000 ft/min along a slow ramp. The top plot in Figure 4.6 shows the web speed and the reference set by the master speed roller and the bottom plot shows the reference tension and the tension in span immediately next to the unwind roll. A belt of width 36 mm is assumed in this simulation result. It is seen that tension in the span oscillates for a long duration even when the variations in speed are very small. This behavior is in contrast with the simulation result shown in Figure 4.5 where the belt compliance is ignored. Figure 4.7 shows the simulated tension using a smaller value for Kb to correspond to a belt with lesser width (24 mm). Notice that the amplitude of oscillations in web tension increase with decrease in belt width (stiffness). Figure 4.8 shows the tension behavior when a 12 mm belt (lower Kb than previous two choices) is assumed. It can be observed that the amplitude of oscillations in web tension increase even more than those seen in the previous cases (in Figures 4.5, 4.6, and 4.7). To further investigate the effect of compliance on the web tension, experiments were conducted on the unwind section of the HSWL (shown in Figures 4.1 and 4.2) using the three different belts. A schematic of the unwind section considered is shown in Figure 83 0 500 1000 1500 Web speed and tension with 36 mm belt FPM 0 5 10 15 20 25 30 16 18 20 22 24 26 Time lbf Tension Web speed Reference Figure 4.6: Tension behavior in a span for a change in reference speed 84 0 500 1000 1500 Web speed and tension with 24 mm belt FPM 0 5 10 15 20 25 30 16 18 20 22 24 26 Time lbf Tension Web speed Reference Figure 4.7: Tension behavior in a span for a change in the reference speed 85 0 500 1000 1500 Web speed and tension with 12 mm belt FPM 0 5 10 15 20 25 30 16 18 20 22 24 26 Time lbf Tension Web speed Reference Figure 4.8: Tension behavior in a span for a change in the reference speed 86 4.4. The two load cells shown in Figure 4.4 measure the tensions in the spans immediately next to the unwind roll and the master speed roller.
Click tabs to swap between content that is broken into logical sections.
Rating  
Title  Modeling the Effects of Belt Compliance, Backlash, and Slip on Web Tension and New Methods for Decentralized Control of Web Processing Lines 
Date  20051201 
Author  Dwivedula, Ramamurthy V 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  Nonideal behavior of web process lines and new methods for regulation of web tension and web velocity are investigated. The nonideal behavior considered includes the compliance and backlash in the transmission system and slippage of web over a roller. The effect of nonideal elements on the controlled web tension is examined. Dynamic models are developed and analyzed to bring out the effect of nonideal elements on controlled web tension. A method of mitigating the effect of backlash on controlled web tension is proposed. Effect of web slippage over a roller on span tension dynamics is investigated. The web tension and velocity dynamics are systematically laid down. Two decentralized control schemes viz. /, a nonadaptive scheme and an adaptive scheme are proposed for web tension/velocity regulation. Experimental study is conducted to validate the results. The dynamic model of beltpulley transmission system is derived by considering the tight side of the belt as a spring. The singular perturbation analysis reveals that using only the measured load side speed as the feedback signal is not desirable. A method of setting the proportional and integral gains of the speed controller is proposed. Starting from the first principles, a dynamic model of backlash is derived. This model accounts for the momentum of load during no contact. Using the model, a bound on the error due to presence of backlash is derived. The proposed bound agrees with results of experiments. The dynamic model of rewind section of a transmission system containing beltcompliance and backlash is derived. Analysis of the dynamic model shows that the mean rewind tension is shifted when there is backlash in the transmission system. This result agrees with experimental data. Further, the experiments show that providing an additional braking input on the rewind shaft is successful in mitigating the effects of backlash. The dynamics of idle roller, material rolls, and web spans are systematically laid out and a decentralized control scheme for the entire web process line is proposed. A method of calculating the equilibrium inputs and setting the reference tensions is derived. An exponentially stable nonadaptive scheme and an asymptotically stable adaptive scheme are presented. These schemes explicitly account for the varying inertia of the unwind/rewind rolls and compensate for unmatched interconnections. Experimental results show that the proposed methods offer superior web tension regulation as compared to the existing decentralized PIcontrol scheme. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  MODELING THE EFFECTS OF BELT COMPLIANCE, BACKLASH, AND SLIP ON WEB TENSION AND NEW METHODS FOR DECENTRALIZED CONTROL OF WEB PROCESSING LINES By RAMAMURTHY V. DWIVEDULA Bachelor of Engineering Andhra University College of Engineering Visakhapatnam, India 1987 Master of Technology Indian Institute of Technology, Delhi New Delhi, India 1992 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2005 MODELING THE EFFECTS OF BELT COMPLIANCE, BACKLASH, AND SLIP ON WEB TENSION AND NEW METHODS FOR DECENTRALIZED CONTROL OF WEB PROCESSING LINES Thesis Approved: Thesis Adviser, Dr. Prabhakar R. Pagilla Committee Member, Dr. John J. Shelton Committee Member, Dr. Eduardo Misawa Committee Member, Dr. Rafael Fierro Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to express my sincerest appreciation to my major advisor, Dr. Prabhakar R. Pagilla for his intelligent supervision, constructive guidance, inspiration, and friendship. I would like to extend my warmest thanks to my doctoral committee members: Dr. Eduardo A. Misawa, Dr. John J. Shelton, and Dr. Rafael Fierro for their support and suggestions in completion of this research. Their guidance and understanding made the development of this thesis a positive learning experience. I would like to thank my colleagues at Oklahoma State University Yongliang Zhu, Nilesh B. Siraskar, Seshadri Kuppuswamy, Anil Abbaraju, Aravind Seshadri, Ryan Ratliff, Raiza Jafari, Mauro Cimino, and Yu. They are among the finest people I know and are a joy to work with. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Effect of compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Effect of backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Effect of slip on web tension dynamics . . . . . . . . . . . . . . . . . . . . 8 1.4 Decentralized control schemes for web process lines . . . . . . . . . . . . 9 1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Organization of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Modeling and analysis of the belt compliance 13 2.1 Beltpulley transmission system . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Analysis of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Singular perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Resonant frequency due to compliance of belt . . . . . . . . . . . . . . . . 27 2.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Effect of backlash and compliance on the output speed of a gear drive 39 3.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Analysis of backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Backlash model with compliance . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.1 A model of backlash including a compliant shaft . . . . . . . . . . 50 3.3.2 Effect of belt compliance and backlash in gears . . . . . . . . . . . 57 iv 3.4 Error bounds in the presence of backlash . . . . . . . . . . . . . . . . . . . 60 3.4.1 Method of finding a bound on error due to backlash . . . . . . . . . 61 3.4.2 Bound on error due to backlash and a compliant shaft . . . . . . . . 64 3.4.3 Bound on error due to backlash and belt compliance . . . . . . . . 69 3.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Effect of compliance and backlash on web tension 76 4.1 Simulations and experiments on the unwind section of the HSWL . . . . . 79 4.2 Effect of gearbacklash on controlled tension . . . . . . . . . . . . . . . . 88 4.3 Experiments conducted on Rockwell web line . . . . . . . . . . . . . . . . 92 4.3.1 Brief description of Rockwell web line . . . . . . . . . . . . . . . 93 4.3.2 Experiments conducted on the Rockwell web line . . . . . . . . . . 94 4.3.3 Braking input to mitigate the effect of backlash . . . . . . . . . . . 110 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5 Effect of slip on web tension dynamics 118 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Location of slip in the arc of contact . . . . . . . . . . . . . . . . . . . . . 119 5.3 Calculation of the sliparc angle . . . . . . . . . . . . . . . . . . . . . . . 122 5.4 Contact region between the web and the roller: A closer look . . . . . . . . 124 5.5 Propagation of tension waves . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.6 Dynamics of web tension in a free span . . . . . . . . . . . . . . . . . . . 132 5.7 Slippage within the region of wrap . . . . . . . . . . . . . . . . . . . . . . 134 5.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.9 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6 Decentralized Control of Web Process Lines 142 6.1 Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 v 6.1.1 Unwind Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.1.2 Master Speed Section . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.1.3 Process Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.1.4 Rewind Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2 Proposed Decentralized Control Scheme . . . . . . . . . . . . . . . . . . . 147 6.2.1 Equilibrium Control and Reference Velocities . . . . . . . . . . . . 149 6.2.2 Feedback Control Design . . . . . . . . . . . . . . . . . . . . . . . 151 6.3 Decentralized Adaptive Control Scheme . . . . . . . . . . . . . . . . . . . 158 6.4 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7 Summary and future work 170 BIBLIOGRAPHY 173 A The standard singular perturbation method 185 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 A.2 Timescale properties of the standard model . . . . . . . . . . . . . . . . . 186 A.3 Linear timeinvariant systems . . . . . . . . . . . . . . . . . . . . . . . . . 188 B Brief description of experiments conducted on backlash 192 vi LIST OF FIGURES Figure Page 1.1 Schematic of a transmission system . . . . . . . . . . . . . . . . . . . . . 3 1.2 A physical illustration of backlash . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Inputoutput plot for frictioncontrolled backlash . . . . . . . . . . . . . . 6 2.1 Schematic of a belt driven transmission system . . . . . . . . . . . . . . . 15 2.2 Block diagram of belt driven transmission system. . . . . . . . . . . . . . . 19 2.3 Two feedback schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Torquespeed characteristics of motor and controller . . . . . . . . . . . . . 28 2.5 Belt drive with motor in velocity control mode. . . . . . . . . . . . . . . . 28 2.6 Belt drive with motor in torque control . . . . . . . . . . . . . . . . . . . . 30 2.7 Picture of the HSWL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 A picture of the transmission system. . . . . . . . . . . . . . . . . . . . . . 33 2.9 A picture of the transmission system. . . . . . . . . . . . . . . . . . . . . . 34 2.10 Transient response of the motor and load . . . . . . . . . . . . . . . . . . . 35 2.11 Steadystate loadspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.12 FFT of the load speed signal . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Schematic of backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Inputoutput plot for frictioncontrolled backlash . . . . . . . . . . . . . . 40 3.3 A rotary model of meshing spur gears . . . . . . . . . . . . . . . . . . . . 42 3.4 A simple backlash model . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Computing load velocity from the solid velocity . . . . . . . . . . . . . . . 48 3.6 Deviations of load/motor velocities from solid velocity . . . . . . . . . . . 49 vii 3.7 Load/Motor velocity and solid velocity . . . . . . . . . . . . . . . . . . . . 50 3.8 Displacement of the load/motor and center of mass . . . . . . . . . . . . . 51 3.9 Schematic of a gear drive . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.10 rectilinear analog: (a) without backlash, (b) with backlash . . . . . . . . . . 53 3.11 Rectilinear analog where compliance precedes backlash . . . . . . . . . . . 56 3.12 Schematic of a transmission system. . . . . . . . . . . . . . . . . . . . . . 58 3.13 (a) A system without backlash and (b) System with backlash . . . . . . . . 61 3.14 Inputoutput plot of backlash . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.15 Block diagram of a controller for system with backlash . . . . . . . . . . . 66 3.16 ECP Rectilinear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.17 Closedloop experiment with backlash of 1.55 mm . . . . . . . . . . . . . 72 3.18 Closedloop experiment with backlash of 3.56 mm . . . . . . . . . . . . . 73 3.19 Closedloop experiment with backlash of 5.38 mm . . . . . . . . . . . . . 74 4.1 Pictures of the HSWL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Schematic of the HSWL . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Control scheme to regulate web tension and web velocity . . . . . . . . . . 79 4.4 Schematic of the unwind section. Master speed roller, a span following the master speed roller, and locations of two loadcells are also shown. . . . . . 80 4.5 Tension behavior in a span for a change in the reference speed . . . . . . . 82 4.6 Tension behavior in a span for a change in reference speed . . . . . . . . . 84 4.7 Tension behavior in a span for a change in the reference speed . . . . . . . 85 4.8 Tension behavior in a span for a change in the reference speed . . . . . . . 86 4.9 Web speed and tension with a 36 mm wide belt . . . . . . . . . . . . . . . 87 4.10 Web speed and tension with a 24 mm wide belt . . . . . . . . . . . . . . . 88 4.11 Web speed and tension with a 12 mm wide belt . . . . . . . . . . . . . . . 89 4.12 Schematic of Rockwell web line. . . . . . . . . . . . . . . . . . . . . . . . 95 4.13 Drive system in Rockwell web line. . . . . . . . . . . . . . . . . . . . . . 95 viii 4.14 Mean velocity and tension with zero backlash and no disturbance. . . . . . 97 4.15 Tension disturbance introduced. . . . . . . . . . . . . . . . . . . . . . . . 98 4.16 Mean velocity and tension with zero backlash and disturbance applied. . . . 99 4.17 Mean velocity and tension with a backlash of 0.5mm and no disturbance applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.18 Mean velocity and tension with a backlash of 0.5mm and with disturbance applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.19 Standard deviation of webtension. . . . . . . . . . . . . . . . . . . . . . . 102 4.20 Standard deviation of webvelocity. . . . . . . . . . . . . . . . . . . . . . 103 4.21 Frequency content of speed and tension signals. . . . . . . . . . . . . . . . 105 4.22 Frequency content of tension and speed when a disturbance is introduced. . 106 4.23 Frequency content of tension and speed with backlash. . . . . . . . . . . . 107 4.24 Summary of results with and without backlash: disturbance amplification. . 108 4.25 Summary of results with and without backlash: disturbance amplification. . 109 4.26 Schematic of the rewind section with braking input . . . . . . . . . . . . . 111 4.27 Frequency content of tension speed with backlash and braking input . . . . 112 4.28 Summary of results with braking input . . . . . . . . . . . . . . . . . . . . 113 4.29 Summary of results with braking input . . . . . . . . . . . . . . . . . . . . 114 4.30 Summary of comparative results with braking input of 14lbf . . . . . . . . 116 5.1 A schematic showing the nomenclature . . . . . . . . . . . . . . . . . . . 119 5.2 A beltpulley system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Slip arc in the region of contact . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 An element of the belt in the slip arc . . . . . . . . . . . . . . . . . . . . . 124 5.5 Friction force and regimes of contact . . . . . . . . . . . . . . . . . . . . . 125 5.6 Contact of a web around a roller . . . . . . . . . . . . . . . . . . . . . . . 126 5.7 Web considered as elements . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.8 The force distribution in the contact region . . . . . . . . . . . . . . . . . . 127 ix 5.9 The three regions of contact . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.10 Tension wave propagation in elastic medium . . . . . . . . . . . . . . . . . 130 5.11 Control volume considered for deriving web tension dynamics . . . . . . . 133 5.12 (a) Friction and normal forces and velocities at entry and exit when the web slips throughout the contact region.(b) An element in the contact region and forces acting on it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.13 A model of traction between the web and the roller . . . . . . . . . . . . . 136 5.14 Slip under the influence of slowly varying tension. . . . . . . . . . . . . . . 137 5.15 Three regions of contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.16 A free span. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.17 Response of tension t2 for a step change in vr,2 in Figure 5.16 . . . . . . . . 140 6.1 Crosssectional view of unwind roll . . . . . . . . . . . . . . . . . . . . . 145 6.2 Freebody diagram of master speed roller. . . . . . . . . . . . . . . . . . . 146 6.3 Decentralized PI controller: Reference velocity 500 ft/min . . . . . . . . . 157 6.4 Proposed decentralized controller: Reference velocity 500 ft/min . . . . . . 158 6.5 Decentralized PI controller: Reference velocity 1000 ft/min . . . . . . . . . 166 6.6 Decentralized PI controller: Reference velocity 1500 ft/min . . . . . . . . . 167 6.7 Decentralized adaptive controller: Reference velocity 1000 ft/min . . . . . 168 6.8 Decentralized adaptive controller: Reference velocity 1500 ft/min . . . . . 169 A.1 Actuator form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 B.1 Backlash gap in experiments. . . . . . . . . . . . . . . . . . . . . . . . . . 193 x NOMENCLATURE A : Area of crosssection of web bm, bL : Viscous friction coefficients of the motor and the load BR : Speed ratio = R1/R2 E : Young’s modulus of web material FL : Load/disturbance force Fm : Force exerted by motor Fi,f : Frictional force on ith roller Fi,n : Normal force on ith roller Jm, JL : Inertias of the driving and driven gears Jm, JL : Inertias of the motor and the Load Jc0 : Inertia of the coreshaft Jmi : Inertia of the ith drive motor Kb : Stiffness of the belt Km : Motor constant Kp, Ki : Proportional and the integral gains Li : Length of ith span Ri : Radius of rollers/pulley Rb1, Rb2 : Base circle radii of gears Rfull : Outer radius of full unwind roll Rg1, Rg2 : Pitch circle radii of gears Run : Radius of the unwind roll Rc0 : Outer radius of the coreshaft with core on it xi T1, T2 : Torques on the driving and driven ti : Tension in ith span vi : Velocity of web on ith roller/roll vr,i : Peripheral velocity of ith roller/roll V0, V1 : Web speed on rollers w : Width of the web αi : Wrap angle on ith roller δ : Thickness of web ¢ : Half backlashwidth ε : Small parameter = 1/√Kb εi : Strain in ith span θ1, θ2 : Angular displacements of the driving and driven gears θm, θL : Angular displacements of the motor and the load ρ : Density of the web material τL : Load/disturbance torque τm : Torque exerted by motor τmc : Time constant of the motor ωd : Desired angular velocity ωi : Angular velocity of ith roller/roll ωm, ωL : Angular velocities of the motor and the load xii CHAPTER 1 Introduction A web is any material which is manufactured and processed in a continuous, flexible strip form. Examples include paper, plastics, textiles, strip metals, and composites. Web handling refers to the physical mechanics related to the transport and control of web materials through processing machinery. Web processing pervades almost every industry today. It allows us to mass produce a rich variety of products from a continuous strip material. Products that include web processing somewhere in their manufacturing include aircraft, appliances, automobiles, bags, books, diapers, boxes, newspapers, and many more. Web tension and velocity are two key variables that influence the quality of the finished web, and hence the products manufactured from it. Web handling refers to the physical mechanics related to the transport and control of web materials through processing machinery. The primary goal of research in web handling is to define and analyze underlying sciences which govern unwinding, web guiding, web transport, and rewinding in an effort to minimize the defects and losses which may be associated with handling of the web. Web handling systems facilitate transport of the web during its processing which is typically an operation specific to a product; for example, in the case of an aluminum web, the web is brought to a required thickness, cleaned, heattreated, and coated; and in the case of some consumer products, the web is laminated and printed. It is important that the tension in a web span be maintained within a close tolerance band while it is transported at a prescribed velocity through the web processing machine. For example, if the tension in the web changes during printing/perforating processes, the 1 print (perforation) gets skewed. Further, excessive tension variations may cause wrinkles and may even tear the web. Tension control plays a key role in improving the quality of the finished web. It is essential to keep the web in the process at a preset tension, which could change throughout the process by many conditions such as disturbances from uneven rollers and web speed variations. As the demand for higher productivity and better performance from the web processing industry increases, better models for the machinery as well as the web behavior and more accurate control algorithms for the processes must be developed. In specific, the imperfections and nonideal effects inherently present in the machine components must be identified and their characteristics must be analyzed before attempting to address the control schemes that limit/eliminate the deleterious effects. The nonideal effects manifest in web handling systems due to factors such as nonlinear behavior of motors, presence of nonlinear friction, presence of compliant members, presence of backlash in the transmission systems, and imperfect contact between the web and the roller resulting in slippage of web over the rollers. It is important to model these nonideal effects with the objective of synthesizing controllers to improve the performance of the system. Modeling aspects of some such nonideal effects, as discussed below, is considered in this thesis. Besides, advanced control schemes for regulating web tension and velocity are also presented in this thesis. A typical web process line uses transmission systems to couple the drive motors and the driven shafts. Such transmission systems may consist of a beltpulley arrangement, a gearbox, a direct coupling, or a combination of these elements. Figure 1.1 shows a schematic of a transmission system used in the unwind station of a web process line. In such transmission systems, three basic drive characteristics/nonlinearities dominate the empirical observation of machine behavior and these are compliance, backlash, and friction. Of course, “perfect” machines do not portray any of these characteristics/nonlinearities, but the world is certainly not perfect, especially when perfection costs more money. For example, in the transmission system shown in Figure 1.1, all the three drive nonlinearities 2 Tach. web Tweb Unwind Roll Belt Motor Web Bevel Gears V Figure 1.1: Schematic of a transmission system may be present: the belt and/or the shafts may offer the compliance; the gearpair almost invariably has some amount of backlash; and the bearings used at various mounting points may offer considerable friction in addition to the friction inherently present in the drive motor. These characteristics directly affect the controlled variables, which are web tension (Tweb) and web velocity (Vweb). Thus, if a controller is designed to regulate the controlled variables, ignoring the characteristics of the transmission system, the performance of such a controller may be different from expected performance. In particular, if one of the shafts or the belt shown in Figure 1.1 is compliant, it affects the transient response of the system and also may contribute a resonant frequency. Similarly, backlash and friction, when present may cause limit cycles and may even render the closedloop system unstable. This report considers modeling and analysis of the effect of drive characteristics on the performance of closedloop speedcontrol systems and new control schemes to regulate the web tension and velocity. In the following a brief introduction of each of the aspects considered in this report is presented. A detailed introduction and prior work are presented at the beginning of each chapter. 3 1.1 Effect of compliance Compliance is an intrinsic property allowing an object to yield elastically when subjected to a force, and in this sense is synonymous with elasticity. Thus, compliance appears in any machine where there are elastic members that are subjected to external loads. Research on compliance (or elasticity) dates back to some hundreds of years. Galileo Galilei first documented the “resistance” of solids in 1638. Robert Hooke in 1660 discovered the proportionality of stress and strain. Following these studies, several famous physicists and mathematicians, including Claude Navier, Leonhard Euler, Charles Augustin Coulomb, Thomas Young, SimeonDennis Poisson, AugustinLouis Cauchy, Heinrich Rudolf Hertz, and Lord Rayleigh contributed to the research on elasticity. An excellent historical survey of early research on elasticity is given in [1]. Following the classical works, many researchers reported important results that address the effect of compliance on various aspects of machine behavior [2–22]. The classical compliance model, based on Hooke’s Law, relates the strain of a flexible body to the stress induced in the body. It is well known that Hooke’s law is only true for a limited amount of strain, after which permanent plastic deformation will occur. For large strains, often, a softening spring or a hardening spring model [23, pp. 9] or some other nonlinear model is used. In such cases, presence of compliance gives rise to complex nonlinear system of equations [5,19–22] whereas in many industrial drive applications, the effects of compliance can be modeled by linear equations [8–13] using the Hooke’s law. These linear models may be developed along the lines of theMaxwell model or the Kelvin Voigt model as given in [24, Chapter 2]. As already noted (on page 2), compliance arises in the transmission system shown in Figure 1.1 either due to a compliant belt or due to compliant shafts. However, the torsional rigidity of the shafts may be much larger than the linear rigidity of the belt and thus, the strain induced in the belt may be much larger than the strain induced in the shafts for a given force. Chapter 2 proposes a model to include the compliance of the belt in a beltpulley trans 4 mission system and analyzes the closedloop speed control system. Some issues such as the feedback configuration, and natural frequencies which arise due to belt compliance are considered in this chapter. 1.2 Effect of backlash Backlash, in the context of mechanical engineering, means “the play between adjacent movable parts (as in a series of gears) or the jar caused when the parts are put into action1.” Backlash is one of the most important nonlinearities that affect the control strategies implemented in the industrial machines and degrades the overall performance of the machines. In industrial drive systems, backlash occurs in mating gear teeth either due to unavoidable manufacturing tolerances, or often deliberately introduced to avoid other deleterious effects. For example, a pair of spur gears are generally mounted at a center distance slightly larger than the designed center distance to avoid interference/undercutting of the teeth. As a result of this, the width of the tooth of one gear along the pitch circle is slightly less than the recess in the mating gear, thus giving rise to backlash. Presence of backlash, though is advantageous from the point of view of interference/undercutting, causes delays and/or oscillations and consequently gives rise to inaccuracies in the position and velocity of the machine. The effect of backlash can be explained with reference to Figure 1.2. In Figure 1.2, Mm is the mass of the driving member (called “motor” hereafter) and ML is the mass of the driven member (called “load” hereafter). As long as there is contact, as shown in Figure 1.2, the motor is able to move the load. However, if the direction of the motion of the motor were to reverse, contact between the motor and the load is lost and any motion of the motor will not result in a corresponding motion of the load. The classical backlash model, shown in Figure 1.3, considers the displacement of the motor (u) as input to the backlash nonlinearity and displacement of the load (y) as output. 1As given in MerriamWebster’s Dictionary. 5 Motor, Mm contact Right contact a c F u y 2D Left Load, ML Figure 1.2: A physical illustration of backlash G y D −D D −D u A B C D E F Figure 1.3: Inputoutput plot for frictioncontrolled backlash 6 In Figure 1.3, the closed path BCDEFGB represents the hysteresis loop due to the effect of backlash. Notice that along the segments AB, CD, and FG, though the displacement of the motor is changing, the displacement of the load remains the same since contact is lost. Many researchers used this model of backlash to propose control schemes [25–30]. However, in actual practice, the inputoutput plot shown in Figure 1.3 is not realized. In considering the input as a displacement, the classical model ignores the momentum of the load during the periods where there is no contact between the motor and the load. Also, the classical model implicitly assumes that disturbances acting on the load do not affect the backlash characteristics, which is not the case. Chapter 3 discusses some aspects of backlash to overcome these lacunae and presents a model for describing backlash. Friction, in all its manifestations, is an important phenomenon which can be put to use for a positive effect (as in the case of brakes) and at the same time has deleterious effects, causing selfexcited oscillations due to stickslip friction. Friction is a very widely studied aspect [31, pp. 169] dating back to Leonardo da Vinci [32], an extraordinary artist and an extraordinary scientist. Since that time, a number of researchers worked in the field of friction [12, 33–48]. Design of control schemes to compensate for nonlinear friction is not considered in this report. The focus of this report is on the effect of transmission system (compliance and backlash) on the speed and tension control in a web processing system. A specific objective laid down is to find the achievable accuracy of a given speed control system when the compliance and the backlash present in the system are known. The transmission system considered for the purpose of analysis consists of a motor driving a load inertia through a compliant shaft/belt and a gear drive, as shown in Figure 1.1. This kind of transmission system is used on the unwind/rewind station of the High SpeedWeb Line (HSWL) in theWeb Handling Research Center (WHRC) at Oklahoma State University, Stillwater. Such transmission systems are especially used since they use the available floor space most efficiently and also offer certain advantages in terms of installation and maintenance. For example, 7 using a directly coupled transmission system mandates exact collinearity of the motor shaft and the load shaft. Any eccentricity/nonparallelism results in unwanted vibration and wear and tear of the bearings. If a beltdrive is used, any small eccentricity/nonparallelism is absorbed into the compliance of the belt. However, the compliance of the belt and the backlash present in the bevel gears may introduce additional nonlinearities which need to be studied. In this context, it is of importance to know the achievable accuracy when the parameters of the motor/tachometer, controller, belt, and the backlash in the gears are known. Such a bound on the achievable accuracy lets the designer/plant engineer in reconfiguring the system with a different set of parameters. As noted on page 1, the compliance present in the system may be due to the belt in the beltpulley transmission system, or the compliance of the shafts. The effect of the compliance of the shafts is more pronounced especially when the transmission system uses long shafts. Chapter 2 presents a model to include the compliance of the belt into dynamics and analyzes the belt driven transmission system. Chapter 3 presents a model of backlash that includes either the effect of the compliance of the shafts or the compliance of the belt and presents a method to compute the achievable accuracy in a given system with a known backlash. Chapter 4 presents the dynamic model of webtension and webvelocity in the unwind/rewind station to include the effects of compliance and backlash present in the drive system. Results of experiments are presented in respective sections. 1.3 Effect of slip on web tension dynamics Modern manufacturing processes exploit the continuous nature of the basic material in web form by transporting it through and out of the process. In such processes, it is essential to maintain continuity and avoid cracks/breakage in the web. Though tests have been conducted to determine breaking strength of webs, it is found in practice that webbreaks occur even when the web tension is much less than the break tension determined under test conditions. There are two main reasons for web breakage: (i) the cracks could be 8 the result of local stress concentrations. In the event of these stress concentrations, cracks may appear and propagate even at moderate overall web tension (ii) second cause for web breaks could be considerable variation of tension about the mean tension. Fatigue may set in when tension fluctuations are rapid and their amplitude is considerable. In general, the web breakage is probably a result of a combination of these two effects. The local stress concentrations may be avoided by improving the manufacturing processes to reduce the severity and density of irregularities. Such efforts fall under the purview of the design of manufacturing process and are specific to the product being manufactured. On the other hand, controlling web tension within tight tolerance band is a common feature to all manufacturing processes which involve material in web form at some stage of production. Thus, there is a definite need for the study of synthesis of web tension control systems. Before attempting to devise such control systems, it is essential to find out how the tension disturbances occur and how they are propagated through the system. Chapter 5 discusses the effect of slip on web tension dynamics. 1.4 Decentralized control schemes for web process lines A web processing line is a largescale complex interconnected dynamic system with numerous control zones to transport the web while processing it. A web processing line typically consists of an unwind roll, several web spans supported by driven/idle rollers and a rewind roll. In such systems one is interested in designing control input to the unwind motor, the rewind motor, and each of the driven rollers to maintain webtension and webvelocity at prescribed reference values. It might be noted that the physical size of the process line in most cases is very large and the various drive motors and tension/velocity sensors located at various points of interest may be situated far apart. Consequently, it is convenient to improvise control design algorithms that use only information available from tension/velocity sensors nearest to the drive motors, thus allowing decentralization. Chapter 6 considers the decentralized control of web process lines and presents two such schemes applicable to 9 web process lines. 1.5 Contributions The contributions of this report can be summarized as follows. 1. A dynamic model to include the effect of compliance of the belt on the speed control system is proposed. Using this model, a method of setting the proportional and integral gains of the controller is proposed. Using the model developed for the beltpulley transmission system, it is shown that using the feedback signal only from the load side is not desirable. Such a result is counterintuitive. 2. Resonant frequency due to the compliance of the belt is computed for the case where the drive motor is in velocity control mode. Frequency content of the speed signal obtained from experiments closely agrees with the computed resonant frequency. 3. A dynamic model for backlash is proposed to present the effect of backlash on the output speed of a gearpair. This model includes the momentum of the load when contact is lost. In addition, the model considers the compliance of the shaft (or the compliance of the belt) in series with the backlash in the gears. 4. The dynamic model proposed is used to derive an estimate of the upper bound on deviation in the velocity of the load due to the presence of backlash. Results from the experiments agree with the theoretically computed bound. Further, it is shown that, in transmission systems that use beltpulley system and a gearpair, the ratio of driven pulley radius to the driving gear radius needs to be small to minimize the effect of backlash. 5. The dynamics of the rewind section of an experimental platform is developed. The model of backlash is extended to include the effect of backlash on web tension. Simulation study and experimental investigation is conducted to investigate the effect of 10 beltcompliance and backlash on web velocity and tension. 6. It is shown and experimentally verified that the mean tension at rewind station is shifted up when the transmission system for driving the rewind roll has backlash. 7. Experiments were designed on the rewind station of a web process line (Rockwell web process line) to excite the backlash gap. The experimental results show that, the amplitude of tension signal at disturbance frequency is amplified when backlash is present in the transmission system. Further experiments on the same process line show that using a braking input on the rewind shaft is successful in mitigating the effect of backlash on web tension. 8. A scheme to include the effect of slippage of web over a roller is proposed. This scheme shows that, when there is slippage between the web and the roller, the web tension disturbances can travel opposite the direction of web travel as well as in the usual direction of web travel. 9. Decentralized control schemes for web process lines were systematically investigated. Two schemes of decentralized control, viz., a non adaptive scheme and an adaptive scheme, are proposed for regulation of web tension and web velocity. Both schemes are experimentally evaluated and are compared to with existing decentralized PI control scheme. 1.6 Organization of the report The rest of the report is organized as follows. Chapter 2 considers the effect of the compliance of the belt on the speed control system considering the shafts to be rigid. In chapter 3, a dynamic model of the backlash in the gears is proposed. Chapter 4 presents the effect of backlash and compliance on webtension in an unwind/rewind station. Effect of webslippage on the tension is studied in Chapter 5. Chapter 6 presents decentralized control of 11 web process lines. Summary and future work are given in Chapter 7. 12 CHAPTER 2 Modeling and analysis of the belt compliance In all web process lines, the driven rollers and the unwind/winder rolls are driven through a transmission system which may be a direct coupling, a gear driven system, or a beltpulley transmission system. Since the driving and the driven shafts have torsional compliance, torsional oscillations can be expected during power transmission. In the case of a belt drive, the compliance of the belt also contributes to the torsional oscillations. Also, if the transmission system uses gears, backlash between the mating gears is unavoidable. Although backlash is necessary for lubricating the tooth surface and preventing the teeth from getting jammed, it results in reduced stability and creates unwanted vibration. In particular, during high speed operation and intermittent motion requiring change in the direction of rotation, this problem becomes critical [26, 49]. Though there are techniques that can reduce backlash, the production cost is very high compared to the achievable accuracy [40]. Sometimes, a beltpulley arrangement is a better method of transmitting power. For example, coupling the drive motor directly to the process end mandates very accurate collinearity of the axes and takes a considerable amount of time. In such cases a belt driven transmission offers great ease since small inaccuracies can be absorbed into compliance of the belt. However, compliance of the belt brings additional dynamics into the system that need to be studied. There is a large body of literature on the characteristics of belt drives and design of mechanisms using belt drives. Many such works concentrated on mechanism of motion/ power transfer, location and extent of sliparc, nature of frictional contact, efficiency 13 limit of the beltdrive system, and methodology of design/selection of beltdrive components [2–5, 8–13]. While many other papers report control schemes which use either estimation of the transfer function of the system, “fuzzy” control schemes, or a simple, ad hoc model to describe the dynamics included by the compliance of the belt [14–16, and the references therein]. In [14], modeling and control of a beltdrive positioning table is discussed. However, no specific model is reported for including the effect of compliance of the belt; system identification techniques were used to obtain the system dynamics, to be later used in tuning the feedback gains. Similarly in [15], a composite fuzzy controller, consisting of a feedback fuzzy controller and a feedforward acceleration compensator, is proposed to control a belt drive precision positioning table. Again, no model to include the effects of belt compliance was reported in this paper. In [16], a robust motion control algorithm for beltdriven servomechanism is reported. In this paper, the beltstretch dynamics is assumed to contribute a pair of purely imaginary poles to the transfer function of the system. Also, the fact that belt stands as an interconnection from loadside to the motorside is ignored in this paper. Analysis and control of speed drive systems with torsional loads is reported in [6, 7, 17, 18]. In [6, 7], a motor drive system driving an inertial load through a gear and spindle is considered for analysis. Though deadzone and backlash are entirely different (deadzone is a stabilizing influence while backlash is destabilizing), the “backlash” in the gear box is modeled as a “deadzone” and the spindle is modeled as a torsional spring. No specific analysis is presented to demonstrate the effect of the compliance of the spindle. In [17], two application examples in the field of remote handling are presented. In this paper, a quasistatic control scheme is presented to compensate for the compliance in the actuator. Similarly, [18] considers shaft torsional oscillations of an induction machine including saturation and hysteresis in the actuator. Though the dynamics of torque generation are dealt with extensively, the compliance present in the drive train is not modeled completely. Of particular importance in the analysis of the beltdriven transmission system is the computation of resonant frequency due to the compliance of the belt. This topic is 14 J L b L 2R2 L q Kb Jm 2R1 bm m t qm Web Tension, T1 Tight side Slack side Belt Figure 2.1: Schematic of a belt driven transmission system not addressed clearly in existing literature. In this chapter, the effect of the compliance of the belt on the speed control system is studied. Using a simple model to include the effect of compliance of the belt, aspects such as the feedback scheme to be used and choice of the feedback gains, and computation of the resonant frequencies when the motor is in velocity/torque mode are addressed. 2.1 Beltpulley transmission system In many applications, a beltpulley transmission system is a convenient alternative over a gear transmission system. When the center distance between the driving shaft and the driven shaft is too large for use of a gearpair, using a belt to transmit motion/power may be the only practical alternative. Figure 2.1 shows a schematic of the drive system considered. A typical approach in analyzing beltpulley/gear transmission systems is to find an equivalent inertia and equivalent damping as referred to either the motor side or to the load side. Assuming that the speed 15 ratio is BR = R2/R1, the equivalent inertia and damping as referred to the motor side are Jeq,m = Jm + µ R1 R2 ¶2 JL, beq,m = bm + µ R1 R2 ¶2 bL. (2.1) With the equivalent quantities defined in (2.1), the dynamics of the system may be written as Jeq,mθ¨m + beq,mθ˙m = τm. (2.2) The control objective is to design a feedback control law τm such that the closedloop system is stable and the load velocity tracks a given reference signal1 ωdL with a prescribed accuracy. Using the feedback law τm = Kpm(ωdm − ωm) + Kim Z (ωdm − ωm)dτ, (2.3) the characteristic equation of the system may be written as s2 + α1s + α0 = 0 (2.4) where α1 = R2 2bm + R2 1bL + KpmR2 2 R2 1JL + R2 2Jm , α0 = KimR2 2 R2 1JL + R2 2Jm . (2.5) Similarly, equivalent inertia and damping as referred to the load side are Jeq,L = JL + µ R2 R1 ¶2 Jm, beq,L = bL + µ R2 R1 ¶2 bm. (2.6) With equivalent inertia/damping defined in (2.6), and using the control law τm = KpL(ωdL − ωL) + KiL Z (ωdL − ωL)dτ, (2.7) 1the ratio of !dm to !dL is the same as the speed reduction ratio of the beltpulley transmission system. 16 the characteristic equation may be written as s2 + β1s + β0 = 0 (2.8) where β1 = R2 2bm + R2 1bL + KpLR1R2 R2 1JL + R2 2Jm , β0 = KiLR1R2 R2 1JL + R2 2Jm . (2.9) Considering the characteristic equations given by equations (2.4) and (2.8), one may be led to the conclusion that the gains of the PIcontrollers given by equations (2.3) or (2.7) may be chosen to place the poles of the characteristic equations (2.4) or (2.8) appropriately. However, as shown in the subsequent sections, the control law given by (2.3) is preferable over the control law given by (2.7). 2.2 Analysis of the system In this section dynamics of the system including effect of compliance of the belt is derived. Also, it is shown that, when ProportionalIntegral (PI) control law is used, it is not advisable to use feedback only from the loadside (i.e., ωL in Figure 2.1). Notice that for a given direction of rotation of the pulley, the belt has a tight side and a slack side as shown in Figure 2.1. To derive the dynamic equations of the system, a simplifying assumption regarding the power transmission is made. Assume that the transmission of power is taking place on the tight side and the transport of the belt is taking place on the slack side. Under this assumption, the net change in tension on the slack side will be much smaller than that in the tight side and thus may be ignored. The tight side of the belt can then be modeled as a spring with spring constant of Kb. Thus, for given angular displacements θm and θL, net elongation of the tight side of the belt can be written as (R1θm − R2θL). Because of this elongation, the driving pulley experiences a torque of (R1θm − R2θL)KbR1 and the driven pulley experiences a torque of (R1θm − R2θL)KbR2. 17 Thus, dynamics of the system, ignoring the inertias of pulleys, is given by τm = (Jmθ¨m + bmθ˙m) + R1Kb(R1θm − R2θL), (2.10a) R2Kb(R1θm − R2θL) = (JL¨θL + bL θ˙L). (2.10b) Remark 2.2.1 Notice that the dynamics of the beltpulley transmission system given in (2.10) may be easily obtained using the wellknown EulerLagrange equations of motion [50, pp. 129–135]. The kinetic energy of the system shown in Figure 2.1 may be written as K(θ˙m, θ˙L) = 1 2 [Jm θ˙2 m + JL θ˙2 L] (2.11) and the potential energy stored in the belt in the form of strain energy may be written as V (θm, θL) = 1 2 Kb[R1θm − R2θL]2. (2.12) Defining the Lagrangian L = K − V , dynamics of the system may be written as d dt ∂L ∂θ˙j − ∂L θj = τj − bj θ˙j , j = m or L, and τL = 0. (2.13) Explanation leading to (2.10) is elaborately given to emphasize the implicit assumption made in (2.12) that the transmission of power is taking place on the tight side and the transport of belt material is taking place on the slack side. Figure 2.2 shows a block diagram representation of the system given in (2.10). Note that the block diagram given in Figure 2.2 represents the openloop system. The two “loops” appearing in the block diagram represent the interconnections in the (2.10). From the block diagram, we obtain openloop transfer functions G¿m!m(s) , ωm(s) τm(s) = JLs2 + bLs + R2Kb D(s) (2.14a) G¿m!L(s) , ωL(s) τm(s) = R1R2Kb D(s) = G(s) (2.14b) 18 +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Figure 2.2: Block diagram of the belt driven transmission system. BR is the speed ratio, BR = R2/R1. where Jeq = R2 2Jm + R2 1JL, (2.15a) beq = R2 2bm + R2 1JL, (2.15b) D(s) = JmJLs3 + (bLJm + JLbm)s2 + (KbJeq + bmbL)s + Kbbeq. (2.15c) The relative degree of the transfer function given in (2.14a) is one, whereas the relative degree of the transfer function given in (2.14b) is three. For plants with relative degree greater than or equal to three, adaptive control schemes are more complex than for plants with smaller relative degrees [51,52]. Hence, the transfer function given in (2.14a) is more suitable for adaptive schemes than the transfer function given in (2.14b). Also, note that in the case of the belt driven transmission system considered (see Figure 2.1), the feedback signals may be obtained from the motor side (that is ωm) or from the load side (that is ωL) as noted in the previous section. These two cases are shown in Figure 2.3. Let us first consider the case when only ωL is used for feedback with the control law given in (2.7). This control law represents a PIcontroller and is a widely used control law. With the control law given by (2.7), the closedloop transfer function from ωdL to ωL is obtained as ωL(s) ωdL(s) = (R1R2Kb/JmJL)(sKpL + KiL) ψL(s) (2.16) 19 +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Controller Motor + w  dL (a) +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Controller Motor + w  dm (b) Figure 2.3: Two feedback schemes: (a) feedback from load shaft and (b) feedback from motor shaft 20 where ψL(s) = s4 + (bmJL + JmbL) JmJL s3 + (Kb[R2 2Jm + R2 1JL] + bmbL) JmJL s2 + (Kb[R2 2bm + R2 1bL] + R1R2KbKpL) JmJL s + R1R2KbKiL JmJL . (2.17) Notice that the coefficients of s3 and s2 do not depend on the gains, KpL and KiL, of the control law. Thus, it may not be possible to place the poles of the characteristic equation at desired locations. On the other hand, consider the control law given by (2.3). With this law, the closedloop transfer function from ωdm to ωL is obtained as ωL(s) ωdm(s) = (R1R2Kb/JmJL)(sKpm + Kim) ψm(s) (2.18) where Ãm(s) = s4 + c3s3 + c2s2 + c1s + c0, c3 = (bmJL + JmbL + KpmJL) JmJL , c2 = (Kb[R2 2Jm + R2 1JL] + bmbL + KpmbL + KimJL) JmJL , c1 = (Kb[R2 2bm + R2 1bL] + R2 2KbKpm + KimbL) JmJL , c0 = R2 2KbKim JmJL . (2.19) The coefficients of s3 and s2 now depend on the gains of the control law. Thus, we have more flexibility in placing the poles of the characteristic equation. The following theorem establishes the stability of the system when feedback from motor side is used in a PI control law. Theorem 2.2.1 The closedloop system defined by (2.3) and (2.10) is stable and ωm approaches ωdm for all Kpm, Kim > 0. Proof: Substituting (2.3) into (2.10), we obtain dynamics of the closedloop system as Kpm(!dm − !m) + Kim Z (!dm − !m)d¿ = (Jmµ¨m + bmµ˙m) + R1Kb(R1µm − R2µL), (2.20a) R2Kb(R1µm − R2µL) = (JL¨µL + bL µ˙L). (2.20b) 21 Differentiate (2.20) to obtain −Kpmω˙m + Kim(ωdm − ωm) = (Jmω¨m + bmω˙m) + R1Kb(R1ωm − R2ωL), (2.21a) R2Kb(R1ωm − R2ωL) = (JLω¨L + bLω˙ L). (2.21b) Defining errors, em = ωm − ωdm and eL = ωL − (R1/R2)ωdm, (2.21) may be written as −Kpme˙m − Kimem = Jme¨m + bme˙m + R1Kb(R1em − R2eL), (2.22a) R2Kb(R1em − R2eL) = JLe¨L + bLe˙L (2.22b) Choose V (t) = 1 2 £ Jme˙2 m + JLe˙2 L + Kb(R1em − R2eL)2 + Kime2 m ¤ . (2.23) Then, the time derivative of V along the trajectories defined by (2.22) is obtained to be dV (t) dt = −(bm + Kpm)e˙2 m − bLe˙2 L. (2.24) Thus, V (t) is a Lyapunov function and em, eL, e˙m, e˙L ∈ L∞ which implies, from (2.22), that ¨em, ¨eL ∈ L∞. From (2.23) and (2.24), we conclude that because V (t) is bounded from below and is nonincreasing with time, it has a limit [52, Lemma 3.2.3], i.e., limt→∞ V (t) = V∞. Now from (2.24), we have lim t→∞ Z t 0 (bm + Kpm)e˙2 m + bLe˙2 L = V0 − V∞ < ∞ (2.25) Therefore, e˙m, e˙L ∈ L2 and by Barbalat’s Lemma [23], we have e˙m → 0 and e˙L → 0. Thus, ωm and ωL tend to become constants as t → ∞ and from (2.21), we see that ωm → ωdm and ωL → (R1/R2)ωdm. ¥ Theorem 2.2.1 shows that any PI controller with positive proportional/integral gains will stabilize the system when the feedback is from the motor side. However, such a result could not be established for the closedloop system defined by equations (2.7) and (2.10) since a Lyapunov function candidate could not be found for the system. This prompted us 22 to look for other tools which could reveal the stability issues of the system when feedback obtained from the loadside is used in the PI controller. Observing the dynamics of the openloop system given by (2.10), it is noticed that the numerical value of the belt stiffness, Kb, is much larger than other parameters such as radii of pulleys, or the inertias. This situation is reminiscent of the singular perturbation problem [53–56] which addresses the dynamics of systems when one parameter in the dynamics is very small. In the present case, the inverse of the square root of the belt stiffness (1/√Kb) is used as the small parameter. 2.3 Singular perturbation analysis To perform the singular perturbation analysis, the system of equations (2.10) with the control law given by either (2.3) or by (2.7) needs to be expressed in the form x˙ = A11x + A12z, x(t0) = x0 (2.26a) εz˙ = A21x + A22z, z(t0) = z0 (2.26b) where x and z are the states of the slow and the fast subsystems and ε is a small parameter. The elements of matrices Aij may depend on ε. However, to use the singular perturbation method2, the matrix A22 needs to be nonsingular (Please see Remark A.3.1 on page 190 ) at ε = 0. Let us first analyze the system of equations (2.10) with the control law given by (2.3), that is, feedback from the motor shaft. A natural choice of the state variables is to use θm, θ˙m, θL and θ˙L. However, with this choice of the state variables, the matrix A22 becomes singular at ε = 0. To obtain a statespace representation in the standard form, a transformation 2In an effort to make this report selfcontained, a brief review of the singular perturbation method used is given in Appendix A. 23 given by θc , Jmθm + JL(R2/R1)θL Jm + JL (2.27a) θs , θm − (R2/R1)θL (2.27b) is used. The variable θc is a weighted average of angular displacements (θm and θL) referred to the motor side and the variable θs is difference between the angular displacements (θm and θL) referred to the motor side. The idea of the weighted average of the displacements arises naturally in the case of a translatory system wherein θc represents the position of the centroid of the masses. Now, choosing the state variables as x = [θc, θ˙c]⊤ and z = [θs/ε2, θ˙s/ε]⊤, the state space representation of the system is obtained in the form given by (2.26) where A11 = 0 1 f1 f3 , A12 = 0 0 ε2f21 + f22 εf4 , A21 = 0 0 g1 g3 , A22 = 0 1 ε2g21 + g22 εg4 , (2.28) f1 = −Kim/J0, f21 = −KimJL/J2 0 , f22 = (R2 2 − R2 1)/J0, f3 = −(Kpm + bm + bL)/J0, f4 = (bLJm − bmJL − KpmJL)/J2 0 , g1 = −Kim/Jm, g21 = −KimJ2L /(JmJLJ0), 24 g22 = −(R2 1JL + R2 2Jm)/(JmJL), g3 = (bLJm − bmJL − KpmJL)/(JmJL), g4 = −(KpmJ2L + bmJ2L + bLJ2m )/(JmJLJ0), where J0 = Jm + JL, and 1/ε2 = Kb. Notice that det(A22(ε)"=0) = −g22 6= 0, thus satisfying the requirement noted in Remark A.3.1. Characteristic equation for the system given by (2.28) can be factored as (refer to [56, Sec. 2.1–2.3] or the Appendix A for details) ψm(s, ε) ≈ 1 ε2ψms(s, ε)ψmf (p, ε) = 0 (2.29) with ψms(s, ε) , det[sI2 − (A11 − A12L(ε))] (2.30a) ψmf (p, ε) , det[pI2 − (A22 + εL(ε)A12)] (2.30b) where ψms(s, ε) is the characteristic polynomial for the slow subsystem and ψmf (p, ε) is the characteristic polynomial of the fast subsystem exhibited in the highfrequency scale p = εs. The matrix L(ε) is obtained using the iterative scheme given by (A24). Using the matrices given by equations (2.28), the slow and the fast characteristic polynomials are evaluated as ψms(s, ε) ≈ s2 + α1s + α0, (2.31a) ψmf (p, ε) ≈ p2 + α′ 1p + α′ 2 (2.31b) where α1, α0 are defined in (2.5) and α′ 1 = R2 2KpmJL Jm(R2 2Jm + R2 1JL) ε α′ 2 = R2 2JL + R2 1Jm JmJL . (2.32) Equation (2.31) indicates that both the fast and the slow subsystems are stable for all Kpm, Kim > 0. This result is in agreement with Theorem 2.2.1. 25 Similar analysis is performed for the case of feedback from the load shaft, that is, using (2.10) and the control law given by (2.7), to obtain the slow and fast characteristic polynomials as ψls(s, ε) ≈ s2 + β1s + β0 (2.33a) ψlf (p, ε) ≈ p2 − β′ 1p + β′ 0 (2.33b) where β0, β1 are defined in (2.9) and β′ 1 = R2 2bm + R2 1bL + R2 2KpL (R2 2Jm + R2 1JL) ε, β′ 0 = R2 2JL + R2 1Jm JmJL . (2.34) Comparing equations (2.31a) and (2.33a), we notice that the slow subsystems are stable for all Kpm, Kim, KpL, KiL > 0. However, when the feedback from load shaft is used, characteristic polynomial of the fast subsystem given by (2.33b) is unstable for all KpL > 0 and KiL > 0. Also notice that the characteristic polynomials given by equations (2.31b) and (2.33b) are identical when ε = 0. Thus, analyzing the limiting case of an infinitely stiff belt, that is, ε = 0 will not reveal the instability exhibited by (2.33b). The instability exhibited by (2.33b) may be attributed to the fact that the belt connecting the inertias is assumed to be purely elastic, without any damping in it. If we were to assume the existence of damping in the material of the belt, it will only add a positive term to the coefficient of p in equations (2.31b) and (2.33b). In this case, the fast subsystem with characteristic polynomial given by (2.33b) will be stable for some values of KpL and unstable for others. Remark 2.3.1 Notice that the characteristic equations given by equations (2.4), (2.31a) are identical and so are the characteristic equations given by (2.8), (2.33a). That is, the singular perturbation analysis also indicated that an equivalent inertia and an equivalent damping may be defined and these equivalent quantities may be used to place the poles of the characteristic equation by properly choosing the gains of the PI controller. However, the analysis, in addition, revealed that using feedback from only the load side is not preferable since such feedback scheme may make the fast system unstable. 26 Remark 2.3.2 Equation (2.10b) may be used to give an interpretation of the foregoing analysis. Differentiating the (2.10b), we obtain JLω¨L + bLω˙ L + R2 2KbωL = R1R2Kbωm. (2.35) This equation shows that ωL can attain steadystate only when ωm attains steadystate first. Even after ωm attains steadystate, ωL continues to exhibit damped oscillations for some time before it attains steadystate. Thus, by measuring only ωL and using the control law given by the (2.7), we will not be able to say for sure whether the oscillations in ωL are due to fluctuations in motor speed or, the oscillations are indeed damped oscillations. In such a situation, the controller attempts to react to the damped oscillations also, and in this process, changes ωm, which in turn affects ωL because of the dynamics given by the (2.35). This process of correcting the load speed may go on for a very long time, if not forever, depending on the damping present on the load side. Thus, the control law given by (2.7) does not present a desirable situation; when only ωm is observed and the control law given by the (2.3) is used, such a situation does not arise. 2.4 Resonant frequency due to compliance of belt This section presents the transfer functions of the speed control systems when the motor used is in the velocity control mode or speed control mode. Figure 2.4(a) shows the torque speed characteristics of a DC permanent magnet motor. A shunt motor has similar but nonlinear characteristics [57]. By using appropriate feedback and control elements, the characteristics shown in Figures 2.4 (b) and 2.4(c) can be realized. Figure 2.4(b) shows the motor in velocity mode where the input voltage results in a proportional speed of the motor irrespective of the load torque. Similarly, Figure 2.4(c) shows the characteristics of motor in perfect torque control. In this case, the motor produces torque proportional to the input variable at any speed within the operating range of the motor. 27 Torque Control m tm wm = Ke E −Kt tm < < < E1 E2 E 3 E4 Ke E4 K e E3 Ke E2 Ke E1 (a) Uncompensated Motor wm tm KevE4 KevE3 KevE2 KevE1 < < < E1 E2 E 3 E4 (b) Motor under Speed Control wm tm Ket E4 KetE3 KetE2 KetE1 < < < E1 E2 E 3 E4 (c) Motor under w Figure 2.4: Torquespeed characteristics of motor and controller First, consider the motor in velocity control mode as shown in Figure 2.4(b). The block diagram of the system in velocity control mode is shown in Figure 2.5. In this mode, speed of the motor is maintained at the reference value, ωdm, irrespective of the load torque due to the inertia/viscous forces and the forces in the belt. The dotted line showing “feedback path” to the motor in Figure 2.5 represents the interconnection (see equation (2.10)) due to belt dynamics. The effect of this interconnection is taken care of by the motor when the motor is in velocity control mode. Thus, the motor and the controller may be represented by a gain Kev. If perfect speed control shown in Figure 2.4(b) is assumed, it is implied that the current (torque) limit for the motor is not reached [57] and ωdm = ωm. Thus, the + +  J L s + b L 1 K R BR 1 2 b s BR w m wL tL Controller Motor wdm Motor under speed control Interconnection Figure 2.5: Belt drive with motor in velocity control mode. transfer function from ωm to ωL for the block diagram shown in Figure 2.5 may be written 28 as T(s) = ωL(s) ωm(s) = KbR1R2 JLs2 + bLs + KbR2 2 . (2.36) It may be noted that the transfer function given in (2.36) and the transfer function obtained by differentiating (2.10b) are the same. This is because the load torque due to inertia/damping and the torque due to interconnection (belt) are absorbed by the velocity control scheme shown in Figure 2.4(b). Equation (2.36) may be used to predict the transient behavior of the load speed (ωL) for a given belt stiffness (Kb) or to compute the stiffness of a belt to be used to obtain a given transient behavior when the motor is under velocity control. If bL ≈ 0, the transfer function given in (2.36) reduces to T(s) = ωL(s) ωm(s) = (KbR1R2/JL) s2 + ω2n (2.37) where ωn = p KbR2 2/JL, thus indicating a natural frequency at ωn. Also, the sensitivity of the transfer function, T(s), given in (2.36), with respect to the belt stiffness Kb indicates the effect of Kb variations on the transient performance of the system. The sensitivity of T(s) is obtained as ST Kb = ∂T/T ∂Kb/Kb = ∂T ∂Kb · Kb T = s(s + bL JL ) s2 + bL JL s + KbR2 2 JL . (2.38) The sensitivity transfer function given in (2.38) indicates that, when Kb/JL is small, then the load speed, ωL, is very sensitive to variations in Kb. When the motor is under torque control, the load torque due to the inertia, damping, and the interconnection, given in (2.10b) also need to be considered as shown in Figure 2.6. If perfect torque control is assumed, then the torque produced by the motor is equal to the torque reference. In this case, the gains of the PI speed controller also affect the transient performance of the system. The closedloop transfer function of the system is given by G1(s) = ωL(s) ωdm(s) = (R1R2Kb/JmJL)(sKp + Ki) ψm(s) (2.39) 29 +  +  J m s + bm 1 +  J L s + b L 1 K R BR 1 2 b s BR w m wL tm tL Torque Control Motor Under Torque Control Motor +  wdm PI Speed Controller Torque reference Figure 2.6: Belt drive with motor in torque control where Ãm(s) = s4 + (bmJL + JmbL + KpJL) JmJL s3 + (Kb[R2 2Jm + R2 1JL] + bmbL + KpbL + KiJL) JmJL s2 + (Kb[R2 2bm + R2 1bL] + R2 2KbKp + KibL) JmJL s + R2 2KbKi JmJL , s4 + ®3s3 + ®2s2 + ®1s + ®0. (2.40) The sensitivity of transfer function given in (2.39) is obtained as SG1 Kb = ∂G1 ∂Kb · Kb G1 = s4 + α3s3 + γ2s2 + γ1s s4 + α3s3 + α2s2 + α1s + α0 (2.41) where αi are defined in (2.40), γ2 = (bmbL+KpbL+KiJL)/(JmJL), and γ1 = (KibL)/(JmJL). Analysis for finding the effect of belt stiffness on the dynamics of the system shown in Figure 2.6 is not as straightforward as it is for the case when the motor is under velocity control. This is because the characteristic polynomial given in (2.40) is of fourth order. Also, in the case of a drive motor in torque control mode, an external speed control loop needs to be used. If this speed control loop is designed to give fast response, a reasonable approximation for the resonant frequency is the value obtained in the case of velocity control mode, which is given by (2.37). 2.5 Experiments Experiments were conducted on the unwind drive system in the High Speed Web Line (HSWL). Figure 2.7 shows a picture of the HSWL. The HSWL consists of an unwind 30 station, a winder station, and two nip stations. The experiments were conducted on the transmission system used in unwind station of the HSWL. Figures 2.8 and 2.9 show closer pictures of the transmission system used in the unwind station of the HSWL. The drive motor is an RPM AC 3phase induction motor rated at 30 HP, under vector control . These type of motors mimic the features of DC motors under torque control mode (that is the torquespeed characteristics as shown in Figure 2.4(c)). The beltpulley transmission system offers a speed reduction of 1:2 and the bevel gear system has unity speed ratio. To mimic a known inertial load due to the unwind roll, four steel discs of known mass (20.45 kg each) are mounted on the unwind shaft. The total inertia of the core shaft and the metal disks is approximately 2.13 kgm2. The drive motor, shown in Figure 2.9 has an encoder connected at one end of the motor shaft to measure the angular velocity of the motor shaft and this encoder is used as feedback element in the speedcontrol loop.. A tachogenerator is mounted on the chuck holding the unwind shaft as shown in Figure 2.8. This tachogenerator is used to measure the speed of the unwind shaft. Web is not threaded in the machine and all the other motors are shutoff except for the unwind motor. Three different belts3 are used in the experiments viz., 8MGT2179212mm, 8MGT2 179224mm, and 8MGT2179236mm. The 36 mm belt has a nominal stiffness of 4.7160× 104 N/m and the stiffnesses of the other belts decrease with the widths, thus 12 mm belt has least stiffness. The driving sprocket has a pitch diameter of 9.023 in and the driven sprocket has a pitch diameter of 4.51 in, thus the speed reduction is approximately equal to 2. Figure 1.1 shows a schematic of the drive system. A series of experiments are conducted to study the performance of the speed control system using different belts by specifying step changes in speed as reference to the motor speed; the motor speed and the load speed are acquired. Figures 2.10 to 2.12 show a representative sample of the experimental results. Figure 2.10 shows the response of the control system when a step input of 200 RPM is specified. The top plot shows the response 3Widths of the belts are 12 mm, 24 mm, and 36 mm respectively 31 Figure 2.7: Picture of the HSWL 32 Figure 2.8: Picture of the unwind transmission system in the HSWL (as seen from operatorend) 33 Figure 2.9: Picture of the unwind transmission system in the HSWL (as seen from gearend) 34 with 12 mm belt, the middle plot shows the response with 24 mm belt and the bottom plot shows the response with 36 mm belt. It is seen that the motor speed and the load speed reach steadystate in less than two seconds. However, with 12 mm belt and 24 mm belt, a “glitch” is seen immediately after the step change in reference occurred. Such a phenomenon is expected and is due to the stretching of the belt (the term R1θm − R2θL in (2.10)). Normally, such a “glitch” does not affect the performance of a speed control system too much. However, in the case of web handling systems, this may cause serious changes in the tension and hence is undesirable. 0 200 400 600 12mm belt RPM Motor Speed Load Speed Reference 0 200 400 600 24mm belt RPM 0 2 4 6 8 10 0 200 400 600 36mm belt Time (sec) RPM Figure 2.10: Transient response of the motor and load Figure 2.11 shows steadystate load speed. It may be noticed that with 12 mm belt, the amplitude of the sustained oscillations is around 5 RPM. This amounts to a linear velocity 35 oscillation to the tune of 0.25m/s at a nominal roll radius of 0.5m. Though such oscillations in themselves form an insignificant percentage of the process speed, their effect on the webtension is significant. If we consider a web material with a nominal stiffness of 100 N/m, the linear velocity oscillations amount to tension variations to the tune of 25 N, which may not be acceptable in many process lines. 270 280 290 12mm belt RPM Load Speed 270 280 290 24mm belt RPM 0 2 4 6 8 10 270 280 290 36mm belt Time (sec) RPM Figure 2.11: Steadystate loadspeed An important consideration in any process line is resonance avoidance. For the unwind drive system, resonant frequencies are computed using the natural frequency given in (2.37). For the 12 mm belt, the natural frequency was evaluated to be approximately 5.5 Hz, for the 24 mm belt, the natural frequency was evaluated to be 7 Hz, and for the 36 mm belt, the natural frequency was 8.6 Hz. Since it is not advisable to verify these nat 36 ural frequencies experimentally by exciting the system at these frequencies, the load speed signal is analyzed using fast fourier transform (FFT). Figure 2.12 shows the FFT content of the load speed signal. Since the natural frequencies for all the three belts are very close to each other, it is difficult to distinguish these from the FFT plots. However, a resonant peak is seen at approximately 5 Hz in the top plot in Figure 2.12. Also, the middle and bottom plots in Figure 2.12 show resonant peaks near 10 Hz, closely corresponding to the theoretically calculated natural frequencies. 0 0.1 0.2 RPM 12 mm Belt Load speed 0 0.1 0.2 RPM 24 mm Belt 2 4 6 8 10 12 14 16 0 0.1 0.2 RPM 36 mm Belt Figure 2.12: FFT of the load speed signal 37 2.6 Summary This chapter considered the effect of compliance of the belt on the speed control system in a web process line. Contrary to the intuitive idea, it is shown that, when the belt in a transmission system is compliant, it is not advisable to use only the load speed as the feedback signal. Also, in the case of unwind/rewind drives, where the load inertia changes with time, a method of tuning the proportional and integral gains of the controller is proposed. Experiments indicate that sustained angular velocity oscillations occur in the load speed even after it attained steadystate. Such oscillations severely affect the web tension. The natural frequency due to the compliance of the belt is computed. Theoretically computed value of the natural frequency agrees with the value estimated from the experiments. 38 CHAPTER 3 Effect of backlash and compliance on the output speed of a gear drive Backlash is one of the most commonly encountered nonlinearities in drive systems employing gears or ballscrews and indicates the play between adjacent moveable parts. Since the action of two mating gears can be represented by the action of one pair of teeth, backlash is commonly represented by the schematic shown in Figure 3.1. When used in the context of mechanical engineering, backlash denotes two salient features as shown in Figure 3.1: (i) a mechanical hysteresis due to the presence of clearance (¢), and (ii) impact phenomena between the surfaces of the masses (Mm and ML). In Figure 3.1, Mm and ML are the masses (inertias) of the driving and driven members, xm and xL are the linear (angular) displacements of the driving and driven members, respectively, from a fixed reference position, and Fm and FL are the driving and load forces (torques). It is a common practice to lump all the mass (inertia) on the driving side into one quantity, Mm, and refer to it as the “motor” and lump all the mass (inertia) on the driven side, and refer to it as the “load”. The classical backlash model considers the schematic shown in Figure 3.1 with input to the backlash as the displacement xm and the output of the backlash as the displacement xL. The inputoutput characteristics of the backlash are represented by Figure 3.2. The slopes of lines GBC and FED are equal to the speed ratio of the gearing in the case of rotary systems. The closed curve BCDEFGB in Figure 3.2 represents mechanical hysteresis due to the presence of clearance ¢. At points B, D, and G in Figure 3.2, the two masses impact and near these points, the inputoutput plot may not be straight but may “oscillate” with a small amplitude. However, impact may be considered to be sufficiently plastic so that 39 D m FL x m xL Mm ML D F Figure 3.1: Schematic of backlash Impact m x L D −D D −D A B C D E F G Impact Impact x Figure 3.2: Inputoutput plot for frictioncontrolled backlash 40 points on these lines lie along a curve bounded by the dotted circles shown, before they resume to lie on the straight lines. The classical backlashmodel resorts to this simplification mainly because in large industrial machines, which operate at steadystate do not reverse direction, impact does not arise except during starting/stopping conditions. Also, in smaller machines, the gear and impact energy are very small. Thus, a plastic impact is considered to be a reasonable assumption. Consequently, all the impacts are assumed to be plastic in this report. Since large industrial machines do not reverse direction many times during their operation, the lines CDE and FGB in Figure 3.2 are ignored and the inputoutput graph of backlash is represented by the curve FEABC, which is the inputoutput graph for deadzone nonlinearity. Though it is a misnomer, often, backlash and deadzone are used synonymously [27, 58]. Notice, however, that input to the backlash is considered to be a displacement of the motor and, the inputoutput plot shown in Figure 3.2 represents quasistatic behavior, at best. The rest of this chapter is organized as follows. Section 3.1 presents a brief review of the existing work on modeling and control of systems containing backlash. Section 3.2 presents a method of analyzing the effect of backlash ignoring the effect of compliance. Analysis of the effect of backlash is presented in Section 3.3 and a bound on velocity error due to presence of backlash is presented in Section 3.4. Experiments are conducted on a tabletop platform to validate the analysis. Results of these experiments are presented in Section 3.5. Summary of this chapter is given in Section 3.6. 3.1 Literature review Research on modeling backlash and its effects dates back to the 1960’s. Much of this research focused on the method of describing functions to investigate limit cycles and deriving stability criteria for systems containing backlash [59–62]. Dubowsky and Freudenstein [63,64] developed a rectilinear model called “impact pair” and presented a dynamic analysis of mechanical systems with clearances wherein it was shown that the compliance can be 41 represented by a linear spring rate, without significantly affecting the dynamic response of the model. Using the Dubowsky’s model, Azar and Crossley [65] studied the dynamic behavior of meshing gears. As a further development, Yang and Sun [49] developed a rotary model for spur gear dynamics and computed the contactspring rate and a time dependent damping for a pair of standard spur gears (pressure angle = 15 or 20 degrees). Figure 3.3 shows a mating pair of gears considered by Yang and Sun. Note that the spring and the D to teeth common normals D P 2a O1 O2 R1 R2 Rb1 q1 T1 z 1 J 1 T2 q2 R b2 z 2 J2 Figure 3.3: A rotary model of meshing spur gears damper shown in Figure 3.3 correspond to the stiffness of the gear tooth in bending and 42 the material damping respectively. In industrial drive systems, the gear teeth have large bending strength and little material damping and thus, for practical purposes the gear tooth may be considered rigid. The “compliance” to be considered in the model typically comes from the shafts on which the driving and driven gears are mounted. Using the models developed, a number of researchers reported control strategies to compensate for the effects of backlash. These control strategies may be grouped into two main categories: (i) strategies for controlling the displacement of the driven member, and (ii) strategies for controlling the velocity of the driven member. A comprehensive survey of various such strategies is reported in [66]. A delayed output feedback controller is proposed for a backlashfree plant in [67] to compensate for the effects of backlash with displacement as its input and output. However, it is not clear as to how the delayed feedback controller stabilizes the system. In the same year, Tao [26] proposed an “adaptive right backlash inverse” for unknown plants with backlash and showed that all closedloop signals are bounded. Similar work on dynamic inversion using neural networks was reported in [68, 69]. Though stability of the system using these inversion schemes is shown through simulations, it is reported in [66] that “the adaptive control seems to yield bad transients during adaptation, while after adaptation, the gain, and hence the bandwidth of the adaptive control system is lower than the gain of the robust linear system”. Prior to these observations, Dean, Surgenor and Iordanou [70] reported a study to experimentally evaluate the inversion scheme presented by Tao in [26] wherein the backlash inverter was found to actually degrade performance in the experiments. Besides, these inversion schemes pertain to position controlled drive systems and are not directly applicable to speed controlled systems. Quantitative design of a class of nonlinear systems with parameter uncertainty was considered by Oldak in [27]. The nonlinearities y = N(x) considered are such that they can be expressed as y = Kx + η(x) where η(x) < M. Several nonlinearities, such as preload, deadzone, quantization, dry friction, and backlash, are shown to belong to this 43 class. Using this idea, Boneh and Yaniv [71], proposed a scheme to reduce the amplitude of limit cycles caused by backlash. Again, the classical backlash model was used in this paper. In contrast to the number of papers published on position control in the presence of backlash, the number of papers published on the speed control is relatively few [29, 30, 58, 72]. The lack of interest, as noted in [66], is either due to the fact that high precision speed control is not required for many systems, but also due to the fundamental difficulties in analyzing speed control of elastic systems with backlash from the load side. A noted exception to this observation are the web handling systems where tight tension control mandates even tighter velocity control. In [72], a nonlinear controller with “soft switching” is proposed. Though improved performance was shown on a large real life drive system, it is not clearly explained how the gains of the lowgain and highgain controllers are computed. A gear torque compensation scheme using a PID speed controller is proposed by Odai [58]. Though it is a novel idea, the PID gains appear to be chosen according to an ad hoc empirical formulae. Warnecke and Jouneh [30] proposed a backlash compensation scheme using an openloop modification of the input trajectory. The proposed velocity compensation method is most efficient only for low operating speeds and large mounting allowance between gears. Extensive literature survey onmodeling and control of industrial speed controlled drives indicates that there is a definite need for a simple model of backlash. Besides, it is of practical importance to know the achievable accuracy in a given drive system with a known backlash. This practical consideration is not addressed in any of the existing literature. Motivated by this practical aspect, Sections 3.3 and 3.4, respectively, present a backlash model and a bound on the achievable accuracy in a given plant with a given backlash. To begin with, Section 3.2 presents a method to evaluate the phase delay and the gain due to the presence of backlash. 44 3.2 Analysis of backlash Consider a simplified system shown in Figure 3.4, where the displacements of masses, xm and xL, are measured from an arbitrary frame of reference. To simplify the analysis, assume that there is no rebound in all collisions and that the masses are free to slide on a smooth, frictionless surface. Notice that at any instant of time, the masses Mm and ML may be either in leftcontact, or rightcontact, or no contact. 2D m x L xm Fcos ( w t ) contact Right contact Left Load, ML Motor, M Figure 3.4: A simple backlash model The dynamics of masses Mm and ML in Figure 3.4 may be written as Mm¨xm = F cos ωt ML¨xL = 0 (3.1) when there is no contact between masses, and Mm¨xm = ±fc + F cos ωt ML¨xL = ∓fc (3.2) when masses are in left/rightcontact where fc is the contact force between masses. It is interesting to note that when the equations given in (3.1) or (3.2) are summed, we obtain Mm¨xm +ML¨xL = F cos ωt. (3.3) 45 Equation (3.3) is free of contact forces and this provides motivation to define a new variable y = Mmxm +MLxL Mm +ML (3.4) and write (3.3) as ¨y = F cos ωt Mm +ML . (3.5) The new variable y defined in (3.4) has a physical significance: it is the position of the center of mass of the system [33, 54]. Also, (3.5) indicates that the center of mass is not affected by the contact forces and moves only under the action of external forces and so the dynamics of y may be considered as the motion of a solid of mass (Mm + ML); the dynamics of y is referred to as solid motion hereafter. It is now possible to view the motions of the masses Mm and ML as the sum of two motions: (i) solid motion and (ii) deviations from solid motion. Define xm , y + zm xL , y + zL (3.6) where zm and zL denote the deviations of motions of Mm and ML from the center of mass. Differentiating (3.4) and using (3.6), Mmz˙m +MLz˙L = 0. (3.7) From (3.7), it can be inferred that the deviations in motions of masses from the center of mass are always proportional to each other and are oppositely directed. Integrating (3.7) results in Mmzm +MLzL = 0 (3.8) where the constant of integration is forced to become zero by choosing the frame of reference such that zm = 0 and zL = 0 at t = 0. Equations (3.4)–(3.8) and the concept of solid motion enable us to view the motion of the mass ML in simplified terms. Since the maximum amount of “delay” introduced into the motion of the mass ML due to backlash is of interest, consider the motion of the motor 46 as it closes the backlash gap, beginning to move from a rightcontact (see Figure 3.4) until it establishes a leftcontact. With the dimensions indicated in Figure 3.4, for the backlash gap to be closed, the following condition has to be satisfied (xm − xL)right − (xm − xL)left = 2¢. (3.9) Using equations (3.6), (3.8), and (3.9), (zL)right − (zL)left = −2¢Mm Mm +ML = −2¢ 1 + ML Mm , −br (3.10) where the subscripts right and left indicate rightcontact and left contact, respectively. Equation (3.10) shows that, when the ratio ML/Mm is very large, the effect of backlash is minimal since the effective backlash is very small and the effective backlash gap is very small and is traversed very quickly. Equation (3.10) offers a conceptually simple way of analyzing the effect of backlash; instead of considering backlash gap b between motor mass and load mass, one can consider the backlash gap br between the solid motion and load mass. Since solid motion is not affected by contact forces, it is sinusoidal when the force exerted by the motor is sinusoidal. In contrast to this, the motor mass dynamics depend on the contact condition as given in equations (3.1) and (3.2). To illustrate the method, simulations were conducted using the numerical values,Mm = 10 kg, Mr = 20 kg, b = 0.05 cm, f = 30, N, and ω = 1 cycles per second. Noting that the solid motion is sinusoidal with the same frequency as that of the load, the velocity of the load mass may be computed as shown in Figure 3.5. Note that, when the backlash is very small, the shaded area in Figure 3.5 will be very small. To add clarity, the shaded area has been enlarged. Starting from the right contact position as shown in Figure 3.4, the load mass and the center of mass of the system travel at the same velocity till a point where the motor mass slows down. This occurs at the peak of the solid motion curve in Figure 3.5. 47 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 time (sec.) velocity Solid Motion Load motion Figure 3.5: Computing load velocity from the solid velocity Each vertical segment in between the solid motion curve and the load motion curve in Figure 3.5 represents the deviation z˙L. The deviation of motor velocity, z˙m, may now be computed using (3.7) and the motor velocity may be constructed using (3.6). Figure 3.6 shows the deviations of the load and motor velocities from the solid velocity. It can be observed that the deviations are either zeros or are oppositely directed at all times. Figure 3.7 shows the velocities of the load and motor mass and the solid velocity in which the peaks/nadirs of the sinusoidal solid velocity curve indicate the points of separation and the points where all the three curves meet indicate sudden equalization of velocities of load and motor due to plastic impact. Figure 3.8 shows the displacements of the load/motor and the center of mass. It is interesting to note that the motor displacement may exhibit two ”peaks”, one peak at the instant where the load and motor masses are separated and the other just after the equalization of velocities. The load velocity curve shown in Figure 3.7 may be used to find an estimate of the gain and the phase delay introduced by backlash. For a given reduced backlash, the load 48 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 time (sec.) velocity Deviation of motor velocity Deviation of load velocity Figure 3.6: Deviations of load/motor velocities from solid velocity velocity for a given amplitude of the force may be computed as shown in Figures 3.5 to 3.7. The amplitude (xL) and the phase (φL) of the load velocity at fundamental frequency (ω) may be obtained from harmonic analysis of the load velocity curve. The ratioM = xL/ym, where ym is the amplitude of solid motion, indicates the gain introduced due to backlash, and φL indicates the phase delay introduced due to backlash. For a given backlash gap b, φL indicates an estimate of the phase lead to be provided by the controller to minimize the effects of backlash. When there is considerable friction on the load side, the analysis is slightly different. In this case, the load velocity does not remain constant upon loss of contact; it gradually decays due to friction till contact is reestablished. The load velocity may be computed in this case also, and a harmonic analysis of the load velocity may be carried out to obtain the gain and phase delay introduced due to backlash, similar to the nofrictioncase. If M is the gain due to backlash at frequency ω and Y (jω) is the frequency response of the system without backlash (i.e., b = 0), then, the condition, Y (jω)M = −1, indicates the possibility 49 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 time (sec.) velocity Motor Load Solid Figure 3.7: Load/Motor velocity and solid velocity of sustained oscillations at frequency ω [23, pp. 280–295]. 3.3 Backlash model with compliance This section considers the transmission system shown in Figure 1.1 and develops a model for backlash. Compliance in the transmission systems may arise due to the elasticity of shafts on which gears are mounted, or due to the belt. Section 3.3.1 presents a model of backlash with compliant shaft and Section 3.3.2 presents a model of backlash with compliant belt. 3.3.1 A model of backlash including a compliant shaft To develop a simplified model, consider the schematic shown in Figure 3.9. In this figure, a load (JL) is driven through a compliant shaft (k is the stiffness) and a pair of gears (radii R1 and R2). Usually, the motor (Jm), is mounted near the driving gear, thus the driving shaft is not very long and so may be assumed to be rigid. 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 5 5.02 5.04 5.06 5.08 5.1 5.12 5.14 time (sec.) displacement Motor Load Solid Figure 3.8: Displacement of the load/motor and center of mass 2R m Tm qg2 qL JL T L q m Driving shaft Driven shaft k 2R 2 1 J Figure 3.9: Schematic of a gear drive 51 To avoid jamming of the gears at high speeds, the gears are mounted with a center distance slightly greater than the designed center distance. This gives rise to clearance between the teeth as shown in Figure 3.3; this clearance is termed “backlash”. To pictorially represent backlash in torsional systems, at least two orthographic views are needed viz., a front view as shown in Figure 3.9 and a side view as shown in Figure 3.3. Also, for studying the effect of a given backlash on the output speed experimentally, one has to assemble the system shown in Figure 3.9 by varying the center distance between the gears. Such experimentation takes a lot of effort and very precise measurement and mounting techniques since the relation between the center distance and amount of backlash gap is not linear. Due to these reasons, often, the rotary system shown in Figure 3.9 is analyzed using a rectilinear analog as shown in Figure 3.10. When a rectilinear analog is used, pictorial representation as well as experimentation is considerably simplified. Further, the model developed using rectilinear analog can be easily converted to the rotary system shown in Figure 3.9 using rectilineartorotary transformation. Figure 3.10(a) and 3.10(b), respectively, show a rectilinear analog of the system with and without backlash. The objective of the analysis is to study how the linearity of the system shown in Figure 3.10(a) gets affected by introducing backlash as shown in Figure 3.10(b). To obtain the equations of motion for the system shown in Figure 3.10(b), first conditions under which contact occurs at points P or Q need to be evaluated. Without loss of generality (see Remark 3.3.1 on page 55), consider the displacements Xm0, Y0, and XL0 as shown in Figure 3.10(b). The free length of the spring when the system is at rest is obtained as Ls0 = XL0 − Y0 − ¢. (3.11) 52 (a) b m bL Fm Mm Xm0 (b) P Q k Reference ML FL XL0 k Reference b m bL Mm ML Fm FL Xm0 XL0 Y0 c 2 c 1 c 1 =c 2 = D Figure 3.10: rectilinear analog: (a) without backlash, (b) with backlash 53 Defining the deviations xm = Xm − Xm0, xL = XL − XL0, y = Y − Y0, (3.12) contact at point P occurs if Xm = Y − ¢, that is, if xm = y − ¢ (3.13) and contact occurs at point Q if Xm = Y + ¢, that is, if xm = y + ¢. (3.14) The length of the spring at any instant of time is obtained as Ls = XL − Y − ¢ = xL + XL0 − y − Y0 − ¢ = (xL − y) + Ls0. (3.15) Thus, when contact occurs at point P, the length of the spring may be obtained from equations (3.13) and (3.15) as LP = (xL − xm − ¢) + Ls0. (3.16) If loss of contact at point P were to occur, it must be either due to mass Mm moving to the right, that is due to increase in xm, or due to mass ML moving to the left, that is due to decrease in xL. In either case, loss of contact at P is occurring due to decrease in (xL−xm) and thus, due to decrease in (xL − xm − ¢). Since the spring tries to regain its original length soon after contact is lost, we say that, LP keeps on changing till either its value is equal to Ls0 or a contact is established, whichever occurs first. From (3.16), we see that if the value of LP tends to change in the direction of Ls0 and (xL − xm − ¢) is decreasing soon after loss of contact, we see that, to begin with (xL − xm − ¢) must be greater than or equal to zero. Thus, (xL − xm − ¢) ≥ 0 (3.17) 54 is the condition for sustained contact at P and (xL − xm − ¢) is the change in the length of the spring. Similarly, when the contact occurs at point Q, the length of spring may be obtained from equations (3.14) and (3.15) as LQ = (xL − xm + ¢) + Ls0. (3.18) If the loss of contact at point Q were to occur, it must be either due to mass Mm moving to the left, that is due to decease in xm, or due to mass ML moving to the right, that is due to increase in xL. In either case, the loss of contact is occurring due to increase in (xL − xm) and thus due to increase in (xL − xm + ¢). Again, given the fact that the spring tries to regain its original length, and the length of the spring when contact exists at Q, given by (3.18), we see that the condition for contact at point Q is (xL − xm + ¢) ≤ 0 (3.19) and (xL − xm + ¢) is the change in the length of the spring. From equations (3.17) and (3.19), we see that, as long as there is no contact, −¢ < (xL − xm) < ¢. (3.20) Remark 3.3.1 At a first glance, it appears that, assuming the system to be centrally located in the backlash gap (that is, ¢ on either side in Figure 3.10 (b)) renders loss of generality. However, since the objective in the “thought experiment” is to see how the behavior of system shown in Figure 3.10(a) deviates from linearity due to the presence of backlash as shown in Figure 3.10(b), such an assumption does not stand as an obstruction. Besides, the same analysis can be carried out assuming the gaps on either side to be c1 = 2¢α and c2 = 2¢(1 − α) for some α ∈ [0, 1]. Remark 3.3.2 It may be noted that the location of the compliance, shown by a spring of stiffness k, with respect to the backlash does not change the contact conditions or the dynamics of the system. Equations (3.17), (3.19), and (3.20) continue to be the conditions 55 of contact at P, contact at Q, and no contact for the backlash configuration shown in Figure 3.11 the compliance precedes backlash. Thus, the dynamic equations developed later apply to the configuration shown in Figure 3.11 also. k Reference Mm P Q c X c 2 1 m0 Y0 c 1 =c 2 = D XL0 b m Fm ML bL FL Figure 3.11: Rectilinear analog where compliance precedes backlash With the deformations of the spring given by equations (3.16), (3.18) and the contact conditions given by (3.17), (3.19), and (3.20), the kinetic energy and the potential energy of the system shown in Figure 3.10(b) may be written as K(x˙m, x˙L) = 1 2 [Mmx˙ 2 m +MLx˙ 2 L] V (xm, xL) = 1 2k(xL − xm − ¢)2 if (3.17) holds 1 2k(xL − xm + ¢)2 if (3.19) holds 0 if (3.20) holds (3.21) With the kinetic energy and potential energy defined in (3.21), the dynamics for the system shown in Figure 3.10(b), ignoring the inertias of the spring and the shaft, may be written as Mmx¨m + bmx˙m + ψ(xm, xL) = Fm, MLx¨L + bLx˙L − ψ(xm, xL) = FL, (3.22) 56 where ψ(xm, xL) = k(xm − xL + ¢) if (3.17) holds, k(xm − xL − ¢) if (3.19) holds, 0 if (3.20) holds. (3.23) 3.3.2 Effect of belt compliance and backlash in gears Section 3.3.1 considered the effect of backlash when one of the shafts in the transmission system is compliant. This section considers the transmission system shown in Figure 1.1 and presents a model that includes the effect belt compliance and backlash in the gears on the output speed of the transmission system. The transmission system shown in Figure 1.1 uses a bevel gear set to transmit motion across the shafts which are at right angles. In such cases, a standard approach is to consider an equivalent spur gear set in place of the bevel gears and perform the analysis of the spur gear set. Figure 3.12 shows a schematic of the transmission system using a beltpulley transmission system and an equivalent spur gear pair. The backlash effect on the output speed of the transmission system shown in Figure 3.12 may be computed using the approach given in Section 3.3.1. Since a pair of mating spur gears rotate in opposite directions, a sign convention is needed to keep track of the angular displacements. The sign convention followed here is that, looking from the load side (that is, from the right hand side of Figure 3.12), θ is considered to be positive in counterclockwise direction and θL is considered to be positive in clockwise direction. Also, reference for angular displacements is taken to be θm = θ = θL = 0 and hence, the deviations in the angular displacements and their absolute values are the same. Further let the freelength of the tight side of the belt in Figure 3.12 be L0. At any instant, the length of the tight side of the belt may be obtained as L = L0 + (R2θ − R1θm). (3.24) A condition for contact at point P is determined by considering the length of the tight side 57 D A q Gear 1 Gear 2 Sprocket 1 L 2R m Jm 2R t L Pitch Circle P Q Enlarged view of A q 1 g1 Belt m q t JL 2R Sprocket 2 2 2Rg2 D Figure 3.12: Schematic of a transmission system using beltpulley arrangement and a gearpair 58 of the belt when contact does exist at point P. First, notice that, for contact at point P, Rg1θ = Rg2θL + ¢ ⇒ θ = 1 Rg1 [Rg2θL + ¢] (3.25) which indicates that a point on the pitch circle of gear 1 has to travel an extra distance of ¢ for contact to be established. Thus, the length of the tight side of the belt during sustained contact at point P may be written as LP = L0 + R2 Rg1 [Rg2θL + ¢] − R1θm , α1θL − R1θm + α2¢ + L0 (3.26) where α1 , R2GR, α2 , R2/Rg1, and GR = Rg2/Rg1. Notice that equation (3.26) is similar to (3.18) except for the coefficients α1, R1, and α2. Contact at point P will be lost either when θ decreases or when θL increases. In either case, (α1θL − R1θm + α2¢) increases. Therefore, upon loss of contact, LP → L0. Coupled with this fact, (α1θL − R1θm + α2¢) increases when contact is lost at point P means that to begin with (α1θL − R1θm + α2¢) ≤ 0. (3.27) Thus, (3.27) gives a condition for contact at point P. The quantity on the left hand side of inequality (3.27) is the change in the length of the tight side of the belt. Similarly, condition for contact at point Q may be written as (α1θL − R1θm − α2¢) ≥ 0. (3.28) The quantity on the left hand side of inequality in (3.28) is change in the length of the tight side of the belt. From equations (3.27) and (3.28), when there is no contact, −α2¢ < α1θL − R1θm < α2¢. (3.29) Thus, the kinetic energy and the potential energy of the system shown in Figure 3.12 may 59 be written as K(x˙m, x˙L) = 1 2 [Jm θ˙2 m + JL θ˙2 L] V (xm, xL) = 1 2Kb(α1θL − R1θm + α2¢)2 if (3.27) holds 1 2Kb(α1θL − R1θm − α2¢))2 if (3.28) holds 0 if (3.29) holds. (3.30) Using the kinetic energy and the potential energy given in (3.30), the dynamics of the system shown in Figure 3.12, ignoring the inertias of the pulleys and the gears, may be written as: Jm¨θm + bm θ˙m + R1ψ(θm, θL) = τm (3.31a) JL¨θL + bL θ˙L − α1ψ(θm, θL) = τL (3.31b) where ψ(θm, θL) = Kb (R1θm − α1θL − α2¢) if (3.27) holds (R1θm − α1θL + α2¢) if (3.28) holds 0 if (3.29) holds. (3.32) 3.4 Error bounds in the presence of backlash This section presents a bound on the error due to the presence of backlash. The idea behind the approach is to consider backlashfree system and see how the presence of backlash affects the dynamics. Section 3.4.1 presents a general idea of obtaining a bound on the error due to the presence of backlash. This idea is used in Section 3.4.2 to present a bound on error due to the presence of backlash and a compliant shaft. Section 3.4.3 presents a bound on error due to the presence of backlash and a compliant belt. 60 3.4.1 Method of finding a bound on error due to backlash This section presents a method of computing the effect of backlash in a system as shown in Figure 3.13(b). Figure 3.13(a) shows two subsystems with transfer functions G1(s) and G2(s) in series and Figure 3.13(b) shows the situation when a backlash nonlinearity, represented by BL is inserted in between them. Assume, for the purpose of illustration, that the inputoutput relation of the backlash nonlinearity BL follows the plot shown in Figure 3.2. (a) G 1 (s) 2 r u y z BL (b) G 1 (s) G 2 (s) r u = y z G (s) Figure 3.13: (a) A system without backlash and (b) System with backlash Suppose that the system G1(s) is represented in statespace form as x˙ 1 = A1x1 + B1r, (3.33a) u = C1x1, (3.33b) and the system G2(s) is represented in statespace form as x˙ 2 = A2x2 + B2y, (3.34a) z = C2x2, (3.34b) 61 with matrices Ai, Bi, and Ci of appropriate dimensions. Fortunately, inputoutput characteristics of backlash nonlinearity shown in Figure 3.2 may be considered as a sum of a linear function and a bounded nonlinear function. With this idea, the output of the backlash may be written as y = u + f(u) (3.35) where the function f(u) is bounded by f(u) ≤ ¢ for all u ∈ R as shown in Figure 3.14. Then, using equations (3.33), (3.34), and (3.35), the statespace representation of the y u −D = + u f(u) (a) (b) (c) u −D u D D D −D Figure 3.14: Inputoutput plot of backlash system shown in Figure 3.13(b) may be obtained as x˙ = Ax + Br + Df(C1x1) (3.36a) z = Cx (3.36b) where A = A1 0 B2C1 A2 , B = B1 0 , (3.37a) C = · 0 C2 ¸ , D = 0 B2 , (3.37b) and x⊤ = [x⊤1 , x⊤2 ]. If the backlash were to be absent (f(u) = 0 and so y = u), as shown 62 in Figure 3.10(a), the same system is described by w˙ = Aw + Br (3.38a) v = Cw (3.38b) where v is the output of the system without backlash. Equations (3.36) and (3.38) are similar except for the extra term Df(C1x1). To analyze the effect of backlash, the output of the system with backlash (that is, z given by (3.36b)) and the output of the system without backlash (that is v given by (3.38b)) need to be compared. Since kz −vk = kC(x−w)k, kx−wk may be evaluated to analyze the effect of backlash. For a given r, the solution of (3.38a) is obtained as w(t) = eAtw0 + Z t 0 eA(t−¿)Br(τ )dτ , φ(r, t,w0) (3.39) where w0 = w(0) is the initial condition. Then, taking the initial condition of (3.36a) as x0 = x(0) = w(0) = w0, solution of (3.36a) may be obtained as x(t) = φ(r, t, x0) + Z t 0 eA(t−¿)Df(C1x1(τ ))dτ. (3.40) Therefore, the required deviation, kx − wk, may be obtained as kz − vk = °°°° Z t 0 eA(t−¿)Df(C1x1(τ ))dτ °°°° ≤ °°°° Z t 0 eA(t−¿)D · ¢dτ °°°° (since kf(u)k ≤ ¢ ∀u ∈ R) ≤ ¢ °°°° Z t 0 eA(t−¿)Ddτ °°°° . (3.41) Equation (3.41) gives an important result: the deviation in the output at any time instant due to backlash is proportional to the halfwidth of the backlash, ¢ and the deviation can be minimized byminimizing the integral appearing in the last line of (3.41). Thus, by choosing the elements of the matrices Ai, Bi, and Ci in equations (3.33) and (3.34), backlash effect on the output can be reduced. In this sense, (3.41) gives a bound on the achievable accuracy when a known backlash is present in a known system. 63 Remark 3.4.1 A physical interpretation of (3.41) is that, to minimize the effect of the backlash, the system dynamics should be able to “close” the backlash gap as quickly as possible. 3.4.2 Bound on error due to backlash and a compliant shaft This section presents a bound on the output error due to backlash and compliant shaft using the idea presented in Section 3.4.1. To use this method, the dynamics of the backlashfree system need to be obtained. To this end, consider the situation without backlash, shown in Figure 3.10(a). When the system is at rest, the displacements of the masses are Xm0 and XL0 from a fixed reference as shown in Figure 3.10(a) and the free length of the spring is Ls0 = XL0 − Xm0. (3.42) When the system is in motion, let XL and Xm be the displacements of the masses from the fixed reference. Defining the deviations xm = Xm − Xm0, xL = XL − XL0, (3.43) the length of the spring at any instant of time may be written as Ls = XL − Xm = xL + XL0 − xm − Xm0 = (xL − xm) + Ls0. (3.44) Thus, the change in the length of the spring is Ls − Ls0 = xL − xm and kinetic energy and potential energy of the system may be written as K(x˙m, x˙L) = 1 2 [Mmx˙ 2 m +MLx˙ 2 L], V (xm, xL) = 1 2 k[x2 m + x2 L]. (3.45) Using the wellknown EulerLagrange equations of motion, given by (2.13), the dynamics of the system shown in Figure 3.10(a) may be obtained as Mmx¨m + bmx˙m + k(xm − xL) = Fm, MLx¨L + bLx˙L − k(xm − xL) = FL. (3.46) 64 Upon rearranging the terms in equations (3.22) and (3.23), the dynamics of the system shown in Figure 3.10(b) may be written as Mmx¨m + bmx˙m + k(xm − xL) − φ(xm, xL) = Fm, MLx¨L + bLx˙L − k(xm − xL) + φ(xm, xL) = FL, (3.47) where φ(xm, xL) = − k¢ if (3.17) holds, k¢ if (3.19) holds, k(xm − xL) if (3.20) holds. (3.48) Notice that equations (3.46) and (3.47) are identical except for the extra term, φ(xm, xL), present in (3.47). And this extra term, because of the condition in (3.20), is bounded by φ(xm, xL) ≤ k¢ for all xm, xL ∈ R. Defining the statevariables zm1 = xm, zm2 = vm = x˙m, zL1 = xL, zL2 = vL = x˙L, and z = [zm1, zm2, zL1, zL2]⊤, a state space representation of the system shown in Figure 3.10(b) is obtained as z˙ = Apz + BpFm + CpFL + βDp(xm, xL) y = Lpz (3.49) where Ap = 0 1 0 0 − k Mm − bm Mm k Mm 0 0 0 0 1 k ML 0 − k ML − bL ML , Bp = 0 1 Mm 0 0 ,Cp = 0 0 0 1 ML , Dp(xm, xL) = 0 Á(xm,xL) Mm 0 −Á(xm,xL) ML , Lp = 0 1 0 0 0 0 0 1 , Lp1 Lp2 (3.50) and β is zero if the backlash gap is zero and unity otherwise. Thus, with β = 0, (3.49) is a state space representation of the system shown in Figure 3.10(a) and with β = 1, (3.49) is a state space representation of the system shown in Figure 3.10(b). 65 Equation (3.49) represents a system with two inputs (the actuating force, Fm and load force, FL) and two outputs (the motor speed, zm2 and the load speed, zL2) as shown in Figure 3.15. e L + G − vr Fm L2 Load speed, z Motor speed, zm2 System Dynamics with Backlash Load force Controller F Figure 3.15: Block diagram of a controller for system with backlash In the control scheme shown in Figure 3.15, the controller, G, uses feedback from the motorside (zm2). If the load velocity is used as feedback, the controller keeps on accelerating/decelerating the motor during the nocontact period since the motor has no “control” over the load during the nocontact period. This might have been the reason for specific lack of interest in using the load side feedback, as noted in [66, Section 3]. Suppose the controller, G, has the statespace representation x˙ c = Acxc + Bce, Fm = Ccxc + Dce. (3.51) Then the statespace representation for the closedloop system shown in Figure 3.15 may be obtained as z˙cl = Aclzcl + CclFL +Wvr + Dcl(xm, xL) zb L2 = Lczcl (3.52) 66 where zcl = [z⊤ x⊤c ]⊤, W = [D⊤ c B⊤ p B⊤ c ]⊤, Lc = [0 0 0 1 0], and Acl = (Ap − BpDcLp1) BpCc −BcLp1 Ac , Ccl = Cp 0 , Dcl(xm, xL) = Dp(xm, xL) 0 . (3.53) The superscript in zb L2 indicates the output in the presence of backlash. If the backlash were to be absent, β = 0 and the statespace representation of the closedloop system may be written as v˙cl = Aclvcl + CclFL +Wvr z0 L2 = Lcvcl (3.54) where the matrices Acl, Ccl, W are given in (3.53) and z0 L2 is the output in the absence of backlash. Equations (3.52) and (3.54) are similar except for the last term in the state equation in (3.52) and the deviation zb L2 − z0 L2 = Lc(zcl − vcl) represents the effect of backlash. For a given reference velocity vr, and the disturbance force FL, solution of the state equation in (3.54) is obtained as vcl(t) = eAcltv0 cl + Z t 0 eAcl(t−¿)[FL(τ ) +Wvr(τ )]dτ , φ(vr, FL, t). (3.55) where vcl(0) = v0 cl is the initial condition. Then, taking the initial condition to be zcl(0) = z0 cl = v0 cl, the solution of the state equation in (3.52) may be written as zcl(t) = φ(vr, FL, t) + Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ (3.56) Thus, the deviation in state variable due to the effect of backlash may be written as kzcl − vclk(t) = °°°° Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ °°°° ≤ k¢ °°°° Z t 0 eAcl(t−¿)D1dτ °°°° , δb (3.57) where D1 = [0 1/Mm 0 − 1/ML 0]⊤. Thus, the deviation in the states of the system at any instant of time is proportional to half backlash width, ¢. 67 Remark 3.4.2 A physical interpretation of (3.57) is that the closedloop system must be able to “close” the backlash gap as fast as possible to minimize the effect of backlash. This is similar to Remark 3.4.1. It is possible to simplify the bound given in (3.57) and express it in terms of the eigenvalues of the matrix Acl. Assume that the eigenvalues of Acl are placed at distinct real values using the controller (3.51). Let these eigenvalues be −λ1, −λ2, · · · , −λn. Since the eigenvalues of Acl are distinct, an orthonormal matrix T exists such that Acl = T¤T−1 and so eAclt = Te¤tT−1. (3.58) Using (3.58), (3.57) is simplified as kzcl − vclk(t) = °°°° Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ °°°° ≤ k¢ °°°° Z t 0 eAcl(t−¿)D1dτ °°°° ≤ k¢ °°°° Z t 0 Te¤(t−¿)T−1D1dτ °°°° ≤ k¢kTk °°°° Z 0 t e¤(t−¿)dτ °°°° kT−1kkD1k (3.59) Since ¤ = diag{−λ1,−λ2, · · · ,−λn}, the integral in the last line of (3.59) is also diagonal and the (i, i)th element of the integral may be written as Z t 0 e−¸i(t−¿)dτ = 1 λi £ 1 − e−¸it¤ (3.60) Thus, the bound in (3.59) is further simplified to kzcl − vclk(t) ≤ k¢kTk °°°° diag{ 1 ¸1 h 1 − e−¸1t i , · · · , 1 ¸n h 1 − e−¸nt i } °°°° kT−1kkD1k (3.61) As t → ∞, the exponentials in (3.61) tend to zero and the bound may be written as kzcl − vclk(t) ≤ k¢kTk °°°° diag{ 1 λ1 , · · · , 1 λn } °°°° kT−1kkD1k ≤ k¢kD1kC1 λmin (3.62) 68 where λmin = min{λ1, · · · , λn} and C1 is the condition number of T. The bound given in (3.62) is considerably simpler to evaluate than the bound given in (3.57). If some eigenvalues of Acl are complex conjugate pairs, the matrix ¤ is block diagonal matrix and the bound may be simplified along the same lines described in equations (3.58)– (3.62). 3.4.3 Bound on error due to backlash and belt compliance This section considers the schematic of the transmission system shown in Figure 3.12 and presents a bound on the error due to the presence of backlash. Defining the statevariables zm1 = θm, zm2 = ωm = θ˙m, zL1 = θL, zL2 = ωL = θ˙L, and z = [zm1, zm2, zL1, zL2]⊤, and using the equations (3.31), (3.32), a statespace representation of the system shown in Figure 3.12 is obtained as1 z˙ = Apz + Bpτm + CpτL + βDp(θm, θL) y = Lpz (3.63) where Ap = 0 1 0 0 −KbR2 1 Jm −bm Jm KbR1®1 Jm 0 0 0 0 1 Kb®1R1 JL 0 −Kb®2 1 JL −bL JL , Bp = 0 1 Jm 0 0 ,Cp = 0 0 0 1 JL , Dp(xm, xL) = 0 −R1Á(µm,µL) Jm 0 ®1Á(µL,µL) JL , Lp = 0 1 0 0 0 0 0 1 , Lp1 Lp2 (3.64) 1the same symbols Ap, Bp etc. are used here and in equation (3.50) to highlight the fact that the dynamic model for the rectilinear analog shown in Figure 3.10 and the dynamic model for the system shown in Figure 3.12 are “analogous” to each other. 69 and β is zero if the backlash gap is zero and unity otherwise and φ(θm, θL) is defined as φ(θm, θL) = Kb − α2¢ if (3.27) holds α2¢ if (3.28) holds (α1θL − R1θm) if (3.29) holds. (3.65) Equation (3.63) represents a system with two inputs (the actuating force, τm and load force, τL) and two outputs (the motor speed, zm2 and the load speed, zL2) as shown in Figure 3.15. Consequently, a bound, as given in (3.57) may be obtained. Comparing the equations (3.20) and (3.29), we see that an additional term α2 = R2/Rg1 multiplies the backlash width ¢ in the case of analysis of the effect of belt and backlash. Due to this term, the bound given in (3.57) is modified to kzcl − vclk(t) = °°°° Z t 0 eAcl(t−¿)Dcl(xm(τ ), xL(τ ))dτ °°°° ≤ Kbα2¢ °°°° Z t 0 eAcl(t−¿)D1dτ °°°° , δb (3.66) for the case of belt compliance. In equation (3.66), D1 = [0, −R1/Jm, 0, α1/JL, 0]⊤. If α2 is small, the bound δb is also small and so it is advantageous to have R2/Rg1 ≪ 1. 3.5 Experiments Experiments were conducted to verify the bound on the deviation in load speed due to backlash, given in (3.57). Experiments are conducted on an ECP Rectilinear System setup. The setup, shown in Figure 3.16, consists of three masses mounted on carriages which are free to slide. Since the system considered for experimentation (that is shown in Figure 3.10) is a two mass system, only masses 1 and 2 are used. A spring is used to represent the compliance k shown in Figure 3.10. That is the system shown in Figure 3.10 is realized as masses 1 and 2 connected by a spring so that Mm = M1 and ML = M2. Position of each of the masses is measured by a high resolution encoder. 70 Figure 3.16: ECP Rectilinear System Nominal values of the masses are M1=2.28 kg and M2=2.55 kg (these values include the masses of the carriages as well). Nominal stiffness of the spring is k=200 N/m. The damping present at masses, as estimated by a preliminary identification procedure, are bm = bL =0.05 Ns/m. A ProportionalIntegral (PI) controller, using velocity of mass 1 as feedback signal, is implemented to impart a prescribed velocity to mass 1. Positions and velocities of the masses 1 and 2 are acquired firstly without backlash present in the system and then with a known backlash. A description of these experiments is given in Appendix B. From each set of experiments, the difference between the load velocity (velocity of mass 2) with backlash and load velocity without backlash is computed using the experimental data. This difference is then compared with the bound computed using (3.57). Figures 3.17, 3.18, and 3.19 show the results of experiments. In these experiments, a PI controller is used to impart a sinusoidal velocity with amplitude of 10 mm and a 71 frequency of 3 Hz to mass 1 and load position and velocity are measured. The solid line in Figure 3.17 shows the deviation in the load velocity due to presence of backlash obtained from experimental data and the dashed horizontal line shows the bound on the deviation as evaluated from (3.57) using a backlash gap of 1.55 mm. It is noticed that the experimentally evaluated deviation is within the bound. Similarly, Figures 3.18 and 3.19 show the results with backlash gaps of 3.56 mm and 5.38 mm, respectively. These figures show that the deviation due to presence of backlash, as evaluated from experiments is within the bound obtained using (3.57). 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 x 10−3 m/s Deviation in load velocity due to a backlash of 1.55mm Figure 3.17: Closedloop experiment with backlash of 1.55 mm 72 0 1 2 3 4 5 6 7 0 0.005 0.01 0.015 0.02 0.025 m/s Deviation in load velocity due to a backlash of 3.56mm Figure 3.18: Closedloop experiment with backlash of 3.56 mm 73 0 1 2 3 4 5 6 7 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 m/s Deviation in load velocity due to a backlash of 5.38mm Figure 3.19: Closedloop experiment with backlash of 5.38 mm 74 3.6 Summary This chapter presented a model for backlash to include the dynamics of the driven member during loss of contact and to include the effect of disturbing forces on the load. Using the model, an upper bound on the achievable accuracy in a given system using a given controller is obtained. Experiments conducted on a prototype system agree with the theoretically estimated upper bound. 75 CHAPTER 4 Effect of compliance and backlash on web tension As noted in Chapter 1, the two important variables to be regulated in a web process line are the web velocity (Vweb) and the web tension (Tweb). The analysis presented hitherto pertains to only one variable, namely, the web velocity. In specific, Chapter 2 considered the effect of belt compliance ignoring the effect of backlash, Section 3.2 presented the backlash effect on output velocity ignoring the effects of compliance, and Sections 3.3.1 and 3.3.2 considered the analysis of the effect of backlash including the compliance of the shaft/belt. This chapter is intended to motivate further investigation of the effect of compliance and backlash on web tension. To begin with, consider the High Speed Web Line (HSWL) shown in Figure 4.1. A schematic of the HSWL is shown in Figure 4.2. This process line consists of a number of tension control zones with each zone consisting of a number of web spans. Tension/ velocity disturbances, induced in any one span, propagate in the direction of the web traversal and hence affect web tension/velocity in all spans downstream to the span where disturbances originate. To attenuate such disturbances, a driven roller is installed near each section where tight control of tension/velocity is required. The schematic in Figure 4.2 shows two such zones labeled as Nip station 1 and Nip station 2: Nip station 1 has three driven rollers and Nip station 2 has one driven roller. Themotors driving these driven rollers use tension feedback from loadcells and the speed feedback from encoders/tachometers to regulate web tension and velocity. It is common practice to designate one driven roller in the process line, usually the one nearest to the unwind roll, as the master speed roller and use it to only regulate the web 76 Figure 4.1: Pictures of the HSWL 77 Rewind Roll LC LC LC NR NR Unwind Roll Loadcell Roller E Edge Sensor E Displacement Guide, DG DG Rewind Section Unwind Section Master Speed Section Process Section Nip M3 M0 3 v T 2 1 T v T1 v 3 2 Driven Roller M1 M2 Figure 4.2: Schematic of the HSWL transport velocity and the other driven rollers to regulate web tension and web velocity in individual zones. For example, in the HSWL shown in Figures 4.1 and 4.2, the driven roller labeledM1, is used as the master speed roller and hence, the controller for the motor driving this roller uses only speed feedback. All the other controllers, including those for the motors driving the unwind and the winder rolls, use tension feedback as well as velocity feedback. Figure 4.3 shows a control scheme commonly used in regulating web tension and velocity. The control scheme shown in Figure 4.3 uses two feedback loops: an outer tensionloop and an inner speedloop. Each of these loops uses a PIcontroller. As can be seen (from Figure 4.3), the output of the tensionloop controller acts as a vernier correction to the speedloop. It may be observed that there are two PIcontrollers in the feedforward path in the block diagram shown in Figure 4.3: the error in tension passes through both these PIcontrollers whereas the error in speed passes through only one PIcontroller. Thus, the control scheme may be thought of as a combination of PIaction on speederror and a PIDaction on the integral of the tensionerror. The controller for motor driving the master speed roller does not have outer tension loop shown in Figure 4.3. Hence, it is possible to tune PI gains to make the master speed roller very closely follow the reference web speed. Under this condition, it is possible to make 78 x r Vr V0 T1 Reference + − Tension Controller PI ++ − PI Controller Motor/ Controller Dynamics Speed Dynamics Web Loadcell/ Dancer Position Feedback Speed Feedback Speed Reference Tension Speed Correction Torque Reference y T Figure 4.3: Control scheme to regulate web tension and web velocity the assumption that the velocity of web passing over the master speed roller is constant and is equal to the reference velocity [57]. Further, since energy input to the web from driven rollers is much larger than energy dissipated by the idle rollers, the length of web between two driven rollers (or an unwind/ winder roll and the driven roller nearest to it) is considered as a single span, though there may be idle rollers in between. However, these nondriven rollers are sources of tension disturbances because of their inertias in combination with springiness of the web. This chapter attempts to evaluate the effect of belt compliance on the span tension immediately next to the unwind roll under the assumptions mentioned in the preceding two paragraphs. As a first step, simulation study and preliminary experiments are conducted using the belt model presented in Chapter 2. 4.1 Simulations and experiments on the unwind section of the HSWL The unwind section of the process line shown in Figure 4.2 is considered. Figure 4.4 shows a schematic of the unwind section considered for simulation. Themaster speed roller shown in Figure 4.4 sets the web reference speed and the unwind motor sets the span tension. The motor driving the master speed roller (not shown in the figure) is under speed control and it is assumed that the speed of the web leaving the master speed roller is constant at Vr. One source of tension disturbance entering the span immediately next to the unwind roll 79 Backlash un V0 1 2 Kb wm R2 R1 Rg1 wL Rg2 T0 Unwind roll roller T V Loadcell 1 r Master speed roller Unwind motor w Pitch Circle P Q D D Enlarged view of A A Nip Loadcell roller R Figure 4.4: Schematic of the unwind section. Master speed roller, a span following the master speed roller, and locations of two loadcells are also shown. is the wound in tension (T0). To highlight the characteristics of the tension behavior in the presence of belt compliance, a sinusoidal component in T0 is assumed in all the simulations. In the first simulation study, effect of belt compliance and backlash are ignored. Thus, the equivalent inertia as given in equation (2.1) is used in writing the dynamics of the closedloop system. The dynamics of the closedloop system with the control scheme 80 shown in Figure 4.3 may be written as τm = Jeqθ¨m + beqθ˙m − R1 R2 T1Run, (4.1a) JL = Jc0 + KJ (R4 un − R4 c0), (4.1b) Jeq = Jm + µ R1 R2 ¶2 JL, (4.1c) beq = bm + µ R1 R2 ¶2 bL, (4.1d) KJ = ρwπ 2 , (4.1e) ˙R un = − δθ˙L 2π , (4.1f) V0 = Run θ˙L, (4.1g) L1 ˙T 1 = EA(Vr − V0) + V0T0 − VrT1, (4.1h) x = Kpt(Tr − T1) + Kit Z t 0 (Tr − T1)dτ1, (4.1i) y = Kps(x + R2 R1 [Vr − Rwnωm]) + Kis Z t 0 (x + R2 R1 [Vr − Rwnωm])dτ1, (4.1j) τmcτ˙m = −τm + Kmy. (4.1k) Equations (4.1a)–(4.1e) describe the dynamics of rotation of the unwind roll to include the effect of changing radius, changing inertia, and span tension . Equations (4.1f)–(4.1g) specify the rate at which the unwind roll radius is changing and specify the linear speed of the web. Equation (4.1h) is the standard nonlinear model for web tension dynamics as described in [73–76]. Equations (4.1i)–(4.1j) indicate the control law shown in Figure 4.3. And lastly, equation (4.1k) describes the dynamics of the unwind motor. In equations (4.1), the time dependence of the dynamic variables is not shown explicitly. The variables τm, JL, Jeq, Run, T1, θm, θL, V0, x, and y are time dependent variables, while all the others are constants. In practice, a step change in reference is not specified because a sudden change in web velocity will cause a large “surge” in web tension. Instead, any change in the reference (either tension or velocity reference) is allowed to ramp up towards the new value, slowly. 81 0 5 10 15 20 25 30 0 500 1000 1500 Web speed and tension (without compliance of belt) ft/min 0 5 10 15 20 25 30 15 20 25 30 Time (s) lbf Tension Tension Reference Web speed Reference speed Figure 4.5: Tension behavior in a span for a change in the reference speed 82 Figure 4.5 shows the web speed and the tension when the speed reference is rampedup from 200ft/min to 1000ft/min over a period of 5 seconds. Even in this case, the span tension increases drastically above the reference value and takes a long time to reach the reference value. To investigate the effect of belt compliance on web tension, simulations are run to reflect the same changes in the reference speed as was done in Figure 4.5. The dynamics of the system, including belt compliance, is the same as equation (4.1) except that (4.1a) and (4.1b) are modified to (4.2a) and (4.2b) as given below τm = Jmθ¨m + bmθ˙m + KbR1(R1θm − R2θL), (4.2a) KbR2(R1θm − R2θL) + T1Run = Jl ¨θL + bL θ˙L. (4.2b) Figures 4.6–4.8 show the results of the simulation. In each of these plots, the speed reference is changed from 200 ft/min to 1000 ft/min along a slow ramp. The top plot in Figure 4.6 shows the web speed and the reference set by the master speed roller and the bottom plot shows the reference tension and the tension in span immediately next to the unwind roll. A belt of width 36 mm is assumed in this simulation result. It is seen that tension in the span oscillates for a long duration even when the variations in speed are very small. This behavior is in contrast with the simulation result shown in Figure 4.5 where the belt compliance is ignored. Figure 4.7 shows the simulated tension using a smaller value for Kb to correspond to a belt with lesser width (24 mm). Notice that the amplitude of oscillations in web tension increase with decrease in belt width (stiffness). Figure 4.8 shows the tension behavior when a 12 mm belt (lower Kb than previous two choices) is assumed. It can be observed that the amplitude of oscillations in web tension increase even more than those seen in the previous cases (in Figures 4.5, 4.6, and 4.7). To further investigate the effect of compliance on the web tension, experiments were conducted on the unwind section of the HSWL (shown in Figures 4.1 and 4.2) using the three different belts. A schematic of the unwind section considered is shown in Figure 83 0 500 1000 1500 Web speed and tension with 36 mm belt FPM 0 5 10 15 20 25 30 16 18 20 22 24 26 Time lbf Tension Web speed Reference Figure 4.6: Tension behavior in a span for a change in reference speed 84 0 500 1000 1500 Web speed and tension with 24 mm belt FPM 0 5 10 15 20 25 30 16 18 20 22 24 26 Time lbf Tension Web speed Reference Figure 4.7: Tension behavior in a span for a change in the reference speed 85 0 500 1000 1500 Web speed and tension with 12 mm belt FPM 0 5 10 15 20 25 30 16 18 20 22 24 26 Time lbf Tension Web speed Reference Figure 4.8: Tension behavior in a span for a change in the reference speed 86 4.4. The two load cells shown in Figure 4.4 measure the tensions in the spans immediately next to the unwind roll and the master speed roller. 



A 

B 

C 

D 

E 

F 

I 

J 

K 

L 

O 

P 

R 

S 

T 

U 

V 

W 


