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Stick-slip friction and rotational vibration under large contact areas and uneven distributed loads Baleri, Mudlagiri 2000

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Stick-Slip Friction and Rotational Vibration under Lar Contact Areas and Uneven Distributed Loads by Mudlagiri Baled  A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Mechanical Engineering)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A September 2000 © Mudlagiri Baled, 2000  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of department or by his or her representatives. It is understood copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Mudlagiri Baleri)  Department of Mechanical Engineering The University of British Columbia Vancouver, Canada  Date  q ^  QtLp-UmsbeA,, &D6c>  ABSTRACT The proposed research is aimed at the analysis of stick-slip frictional phenomenon when large contact areas are involved and, due to the physical structure of the system, high uneven contact loads are generated. The study will look at both mechanistic and tribological aspects of the problem. The main objective of the research is to achieve a better understanding of the phenomenon based on experimental observations and to develop a mathematical representation under realistic conditions. A non-linear finite element model has been developed using A N S Y S to obtain an understanding of the contact pressure distribution at the interface of contact of large surfaces due to the applied loads. A mathematical model of this distribution has been derived based on the results of the A N S Y S solution. The friction torque acting at the contact surface has then been estimated by integrating elemental friction torque over the entire plate area. The LuGre friction model for the localized contacts has been extended for large contact areas using an integration approach. The simulations have been carried out using Simulink to understand the dynamic behavior of the integrated LuGre model as well as to assess the effect of various system parameters on the stick-slip vibrations. A test rig has been designed and fabricated to re-construct the realistic situations that occur in large robots and excavators for example. The instrumentation, signal processing, data acquisition and analysis have subsequently been carried out. The experiments have been conducted to investigate the effect of various parameters such as total load, loading configuration, speed, and spring stiffness on the stick-slip vibrations. The simulations using proposed integrated LuGre model are found to be in good agreement with the experimental results. The experiments have revealed that with all other parameters being same, the stick-slip vibrations are more likely to occur under concentrated loading as compared to distributed loading for the same total load.  n  T A B L E OF CONTENTS ABSTRACT  ii  T A B L E O FC O N T E N T S  iii  N O M E N C L A T U R E  vi  L I S T O FT A B L E S  ix  LIST O FF I G U R E S  x  xiii  A C K N O W L E D G E M E N T S  CHAPTER 1  1  INTRODUCTION A N DO V E R V I E W  1.1  Introduction  1  1.2  Terminology  1.3  Historical background  3  1.4  Friction  5  '.  1.4.1  Static  1.4.2  Kinetic friction  1.5  2  friction  The effects of tribosystem variables on frictional response  5 7 8  1.5.1  Effects of surface finish  8  1.5.2  Effects of load and contact pressure  9  1.5.3  Effects of sliding velocity  9  1.5.4  Effects of temperature  9  1.5.5  Effects of surface films and chemical environments  10  1.5.6  Combined effects of several variables  11  1.6  Stick-slip vibrations  11  1.6.1  Stick-slip vibration for large surface contact area  13  1.6.2  Undesirable effects of stick-slip motion  14  1.6.3  Factors causing stick-slip vibrations  15  1.6.4  General remedies  15  1.7  Discussion  CHAPTER 2  CONTACT  16 PRESSURE DISTRIBUTION A T T H EINTERFACE  17  2.1  The topography of contact  17  2.2  Experimental apparatus  18  iii  2.3 Problem description  19  2.4 Simple linear model using ANSYS  20  2.5 Non4inear model using contact elements  21  2.6 Post-processing and analysis of the results  23  2.7 Development of an empirical model for the distribution of the contact pressure  25  2.8 Calculation of the friction torque  27  2.9 Discussion  29  CHAPTER 3  MODELING AND SIMULATION  3.1 Parameters to be considered for a stick-slip friction model  30  30  3.1.1  Pre-sliding displacement  30  3.1.2  Stribeck effect  31  3.1.3  Frictional memory  32  3.1.4  R i s i n g static friction  33  3.2  Stick-slip models  34  3.2.1  Classical friction (Coulomb) model  35  3.2.2  K a m o p p model  35  3.2.3  The seven-parameter friction model  37  3.2.4  L u G r e model  38  3.3 Proposed approach for large contact areas  40  3.4  42  Simulation of dynamical model behavior  3.4.1  Pre-sliding displacement  43  3.4.2  Frictional lag  44  3.4.3  V a r y i n g break-away torque  45  3.4.4  Stick-slip motion  46  3.5 Discussion CHAPTER 4  EXPERIMENTS AND RESULTS  51 52  4.1 Friction test system  52  4.2 Instrumentation  52  4.3 Signal conditioning  53  4.4 LabVIEW interface for data acquisition and analysis  53  4.5  54  Surface preparation  4.6 Pre-tests  56  4.6.1  Test procedure  57  4.6.2  Results o f the pre-tests  57  4.7 Stick-slip tests  61 iv  4.7.1  Surface conditions for experiments  61  4.7.2  Experimental procedure  61  4.7.3  Experimental results  78  4.8  Surface measurements  78  4.9  Analysis of the results  80  4.10  Discussion  CHAPTER 5  83 84  CONCLUSIONS A N D FUTURE DIRECTIVES  5.1  Summary  84  5.2  Conclusions  85  5.3  Scope for future work  85  REFERENCES  87  APPENDIX IANSYS  90  E L E M E N T S  A P P E N D I X II A N S Y S  94  BATCH FILE  A P P E N D I X III  MATLAB  A P P E N D I X IV  CALIBRATION DATA  P R O G R A M  ,  97 103  V  NOMENCLATURE C  constant  Ci, C2 constants c  damping coefficient other than inter-surface friction  g  Stribeck friction function  E  Young's modulus of elasticity  F  Coulomb friction force  Ff  instaneous friction force  F  magnitude of the Stribeck friction force  F  magnitude of the Stribeck friction after a long time at rest (with a slow application of  c  s  yo0  force) F  SM  magnitude of the Stribeck friction at the end of the previous sliding period  F  viscous friction force  J  mass moment of inertia of the vibrating disc  k  spring stiffness of the elastic suspension  k  tangential stiffness of the static contact  m  mass of the system  Pi  minimum contact pressure due to self-weight of the plate  P  maximum contact pressure due to the applied load  R  radius at the point of contact with the spring  v  t  m n  max  R  average surface roughness value  R  inner radius of the plate  Ri  radius at the point of application of load  R  outer radius of the plate  R  v  maximum peak to peak surface roughness value  r  x  x co-ordinate distance from the load application point  r  y co-ordinate distance from the load application point  a  t  0  y  Tf T  v  frictional  torque applied on the disc  integrated viscous frictional torque  vi  t  momentary time  t  dwell time, time at zero velocity  to,  infinite time  t  time of stationary contact in a stick episode  VQ  driver velocity  V  critical driver speed above which stick-slip motion ceases  V, v  relative velocity  v, x  characteristic velocity of the Stribeck friction  W  applied normal load  x  momentary sample displacement  XQ  time averaged sample displacement  Xmax  maximum sample displacement  x  minimum sample displacement  x  stick-slip amplitude  x+s  displacement at the end of a stick episode  Xs  displacement at the beginning of a stick episode  2  s  C  R  s  s  min  s  x  absolute sample velocity  B, y  constants  0  momentary angular position for rotary system  0o  time-averaged angular position for the rotary system  0  average asperity angle  0/  frictional angle  0  momentary angular velocity for rotary system  0  momentary angular acceleration for rotary system  a  H  Coulomb friction coefficient  Hk  kinetic friction coefficient  u  static friction coefficient  ju oc  static coefficient of friction for very long time of contact  fx  viscous friction coefficient  v  Poisson's ratio  p  density of the plate material  c  s  S  v  vii  a  variance of the normal distribution  <To  stiffness of the system for LuGre model  ai  damping coefficient for LuGre model  x  time constant of frictional memory  co  absolute angular velocity of the vibrating disc  a>o  angular velocity of the driving disc  co  relative angular velocity  L  r  viii  LIST OF TABLES Table 2.1 Parameter values used in A N S Y S model  20  Table 2.2 The comparison of the values from A N S Y S and mathematical model  26  Table 3.1 Model parameter values used in all simulations  42  Table 3.2 System parameter values used in simulation of stick-slip motion  46  Table 3.3 The simulation data for the variation the load and the spring stiffness  49  Table 3.4 The simulation data for the variation of the driving speed and the loading radius. 50 Table 4.1 Pre-test data for different loading configurations  58  Table 4.2 Pretest data for the variation of the normal load at radius = 84 mm  59  Table 4.3 Pretest data for the variation of the normal load at radius = 59 mm  60  Table 4.4 Stick slip data for the variation of the normal load at 0.22 rad/sec  63  Table 4.5 Stick slip data for the variation of the normal load at 0.28 rad/sec  64  Table 4.6 Stick slip data for the variation of the normal load at 0.33 rad/sec  65  Table 4.7 Stick slip data for W=90 N ; R=84 mm; k=3.0377 N/mm  66  Table 4.8 Stick slip data for W=90 N ; R=84 mm; k=9.1864 N/mm  67  Table 4.9 Stick slip data for W=l 80 N ; R=84 mm; k=3.0377 N/mm  68  Table 4.10 Stick slip data for W=l 80 N ; R=84 mm; k=9.1864 N/mm  69  Table 4.11 Stick slip data for W=270 N ; R=84 mm; k=3.0377 N/mm  70  Table 4.12 Stick slip data for W=270 N ; R=84 mm; k=9.1864 N/mm  71  Table 4.13 Stick slip data for W=90 N ; R=59 mm; k=3.0377 N/mm  72  Table 4.14 Stick slip data for W=90 N ; R=59 mm; k=9.1864 N/mm  73  Table 4.15 Stick slip data for W=l80 N ; R=59 mm; k=3.0377 N/mm  74  Table 4.16 Stick slip data for W=l 80 N ; R=59 mm; k=9.1864 N/mm  75  Table 4.17 Stick slip data for W=270 N ; R=59 mm; k=3.0377 N/mm  76  Table 4.18 Stick slip data for W=270 N ; R=59 mm; k=9.1864 N/mm  77  Table 4.19 Surface roughness values for the different locations of the plates  80  ix  LIST OF FIGURES Fig 1.1 Plot of static friction coefficient as a function of time of stick  7  Fig 1.2 Plot of kinetic friction coefficient as a function of sliding speed  8  Fig 1.3 Spring and dash-pot representation of stick-slip  12  Fig 1.4 Schematic illustration of displacement and velocity during stick-slip motion  13  Fig 1.5 Stick-slip phenomenon for large contact areas  14  Fig 2.1 Laboratory experimental set-up  18  Fig 2.2 Exploded view of load application points  19  Fig 2.3 Contour plot for the contact pressure distribution (90 N load at 59 mm)  20  Fig 2.4 The geometric appearance of the model using A N S Y S  22  Fig 2.5 Radial and Circumferencial contact pressure distribution plot (load at 59 mm and 45°) 23 Fig 2.6 Contour plot of pressure distribution for the non-linear model (90 N at 59 mm)  24  Fig 2.7 Model for the contact pressure distribution at the interface  26  Fig 2.8 Distribution of the contact pressure over the whole plate (90 N at 59 mm)  27  Fig 2.9 Calculation of friction torque at the interface between the plates  28  Fig 3.1 Asperity deformation under applied force and pre-sliding displacement  31  Fig 3.2 The Stribeck effect  31  Fig 3.3 Typical friction-speed time shift [25]  33  Fig 3.4 Commonly used friction models  34  Fig 3.5 Classical friction model  35  Fig 3.6 Karnopp's model [30]  36  Fig 3.7 LuGre model showing the contact between bristles  38  Fig 3.8 Schematic representation of the plate subjected to stick-slip  40  Fig 3.9 Sub-function for calculating the friction torque acting over the entire plate  42  Fig 3.10 Pre-sliding displacement as described by the model  43  Fig 3.11 Hysteresis in friction torque with varying velocity  44  Fig 3.12 Relation between break-away torque and the rate of increase of applied torque  45  Fig 3.13 Displacement, velocity and friction torque waveforms for the stick-slip motion  47  x  Fig 3.14 Contact pressure distribution for concentrated and distributed loading  48  Fig 3.15 The simulation plots for the variation of the load and the spring stiffness  49  Fig 3.16 The simulation plots for the variation of the driving speed and the loading radius.. 50 Fig 4.1 Initial surface roughness testing results  54  Fig 4.2 The front panel for the pre-test data analysis  57  Fig 4.3 The variation of spring force with load at radius = 84 mm  59  Fig 4.4 The variation of spring force with load at radius = 59 mm  60  Fig 4.5 The front panel for the stick-slip test data analysis  62  Fig 4.6 Plot of variation of the normal load at 0.22 rad/sec  63  Fig 4.7 Plot of variation of the normal load at 0.28 rad/sec  64  Fig 4.8 Plot of variation of the normal load at 0.33 rad/sec  65  Fig 4.9 Stick-slip plot for W=90 N ; R=84 mm; k=3.0377 N/mm  66  Fig 4.10 Stick-slip plot for W=90 N ; R=84 mm; k=9.1864 N/mm  67  Fig 4.11 Stick-slip plot for W=180 N ; R=84 mm; k=3.0377 N/mm  68  Fig 4.12 Stick-slip plot for W=l 80 N ; R=84 mm; k=9.1864 N/mm  69  Fig 4.13 Stick-slip plot for W=270 N ; R=84 mm; k=3.0377 N/mm  70  Fig 4.14 Stick-slip plot for W=270 N ; R=84 mm; k=9.1864 N/mm  71  Fig 4.15 Stick-slip plot for W=90 N ; R=59 mm; k=3.0377 N/mm  72  Fig 4.16 Stick-slip plot for W=90 N ; R=59 mm; k=9.1864 N/mm  73  Fig 4.17 Stick-slip plot for W=180 N ; R=59 mm; k=3.0377 N/mm  74  Fig 4.18 Stick-slip plot for W=180 N ; R=59 mm; k=9.1864 N/mm  75  Fig 4.19 Stick-slip plot for W=270 N ; R=59 mm; k=3.0377 N/mm  76  Fig 4.20 Stick-slip plot for W=270 N ; R=59 mm; k=9.1864 N/mm  77  Fig 4.21 Surface texture at different radial locations  79  Fig 4.22 Surface roughness measurements at different locations of the plate  80  Fig 4.23 Comparison of results for the variation in normal load  81  Fig 4.24 Comparison of results for the variation in driving speed  82  Fig A-I.l Element geometry for SOLID95  90  Fig A-I.2 Element geometry for T A R G E 170  91  Fig A-I.3 Element geometry for CONTA174  92  Fig A - I V . l The calibration charts for L V D T and driving disc speed  103  xi  Fig A-IV.2 The calibration charts for springs  104  Fig A-IV.3 The measurement chart for static coefficient of friction  105  xn  ACKNOWLEDGEMENTS I extend my sincere gratitude to my supervisor, Dr. F. Sassani for opening the doors to this unique opportunity for me and for providing me with invaluable guidance and insightful ideas. I also wish to thank Dr. Pak Ko for his assistance and support throughout the project. I would like to recognize the help of staff and faculty in the Mechanical Engineering Department. I am also grateful to the Faculty of Graduate Studies and the sponsors of the University Graduate Fellowship provided during the course of my degree. The support from Natural Sciences and Engineering Research Council of Canada under the Grant number EQPEQ 205290 is highly appreciated. I would like to recognize the help of graduate students Henning Keiner and Suresha Udupi for helping me walk through A N S Y S modeling. I am thankful to Dr. Cyrille DecesPetit for his invaluable help extended during instrumentation and LabVIEW programming. I am grateful to the people in Tribology Laboratory, N R C and the Manufacturing Automation Laboratory for their facilities, which enabled the completion of this project. I also wish to thank everyone in the Process Automation and Robotics laboratory for providing me with a happy environment to work in. M y deepest love and gratitude is felt for my parents and family for their never ending encouragement and support. This would have been impossible without them. Last but not the least, I would like to thank all my friends who are directly or indirectly involved in making this dream come true.  xin  C H A P T E R 1 INTRODUCTION AND OVERVIEW 1.1  Introduction Friction is the resistance to motion that exists when a solid object is moved  tangentially with respect to the surface of another that it touches, or when an attempt is made to produce such a motion. In a modern automobile, 20% of the power is wasted in overcoming friction; in an airplane piston engine, 10%; and in a modern turbojet 1.5 to 2% is wasted. Since the friction losses amount to 0.5% of the GNP of industrialized countries [37], reduction of friction through improved design, through the use of more suitable contacting materials, or through the application of better lubricating substances is an important consideration for modern technology. It must not be overlooked, however, that many processes of everyday life are dependent for their effectiveness on the presence of friction at high levels. Hence, when required, the provision of sufficiently large friction is also a task of great importance. Although the foregoing two categories comprise the two main frictional requirements, that of lowering friction when unwanted or of maintaining it at a sufficiently high level when required, there is a third problem of some importance, that of maintaining friction constant within narrow limits. A fourth problem of considerable importance in many practical applications is of controlling and reducing friction induced vibrations. Friction induced vibrations can occur at the bearing surfaces of many moving machines and mechanisms. It is often the result of non-linear friction characteristics at the interface that induce vibration. About 40% of the industrial frictional problems involve friction-induced vibration. For instance, joints and guideways of robots during load carrying and positioning experience this form of friction. Another example of the highly undesirable presence of this effect is in continuous machining operations as the induced vibrations give rise to chatter marks on the workpiece. Stick-slip sliding is a severe form of friction-induced vibration in dynamic modeling and causes significant problems in the control of computer controlled mechanical systems. It also leads to uneven and often a step form of loading on components that could result in subsequent wear in various forms and finally failure.  1  1.2  Terminology Before proceeding, several definitions are provided that will be used quite frequently  in the following pages: Tribology: Literally, the study of rubbing; the name given to the modern study of friction, lubrication and wear of rubbing surfaces. Static Friction (Stiction): The torque (force) necessary to initiate motion from rest. It is often greater than kinetic friction for metal contacts. Coulomb friction: A friction component that is independent of the magnitude of the velocity. Viscous Friction: A friction component that is proportional to velocity and, in particular, goes to zero at zero velocity. Break-away: The transition from rest (static friction) to motion (kinetic friction). Break-away Torque (Force): The amount of torque (force) required to overcome static friction. Break-away Distance: The distance traveled before break-away; that is, the distance over which static friction operates, a consequence of the materials used and forces applied. Dahl Friction or the Dahl Effect: A friction phenomenon that arises from the elastic deformation of bonding sites between two surfaces, which are locked in static friction. The Dahl effect causes a sliding junction to behave as a linear spring for small displacements. Stribeck Friction or the Stribeck Effect: A friction phenomenon that arises from the use of fluid lubrication and gives rise to decreasing friction with increasing velocity at low velocity. Negative Viscous Friction: Decreasing friction with increasing velocity. Stribeck friction is an example of negative viscous friction.  2  Hertzian contacts: Contact between elastic curved bodies where the region of contact is small compared with the dimensions of the bodies, resulting in very high local stresses when pressed together by external loads. 1.3  Historical background The friction effects are those that arise from the tangential forces transmitted across  the interface of contact when solid surfaces are pressed together by a normal force. Friction is usually classified as a branch of physics or mechanical engineering. The wear phenomenon consists of the removal of material from the surfaces of one of the contacting bodies, as a result of interaction with the other contacting body and is often considered to be part of material science and mechanical engineering. Adhesion, the ability of contacting bodies to withstand tensile forces after being pressed together, is the third of the interaction phenomena. It seldom occurs to any marked extent and has been much less investigated than others. Lubrication, the study of substances that affect friction and wear, comes under the heading of chemical and mechanical engineering, whereas adhesion probably belongs more to physics. These various surface interaction phenomena are closely related and hence the literature is widely scattered. At this point it may be of value to give a very brief survey of the development of our knowledge about friction [37]. In 16 century, Leonardo da Vinci ascribed the first value (0.25) of the coefficient of th  proportionality between the frictional force and the weight of the sliding body. The period extends over almost 200 years until late 17 century when Amontons rediscovered these th  aspects. He noticed that the proportionality value is approximately equal to 1/3 and is invariable for the materials tested, provided the pressure is constant. At a later time, Bulffingeri (1727) assigned the value of 0.3. During this period many of the attempts to explain frictional behavior involved surface roughness and the interlocking of surface asperities. Both Amontons and Coulomb considered that sliding involved riding of rigid asperities of one surface over the other. If the average asperity angle is 6 , the coefficient of friction /u, is equal to tan 6 and does not a  a  depend on the load or the size of the bodies.  3  Coulomb, in the late 18 century, carried out systematic scientific investigations on the friction on pulleys, capstans and slipways. He was able to distinguish between static friction, the force required to start sliding, and kinetic friction, the force required to maintain it. He showed that the kinetic friction could be appreciably lower than static friction. He also observed that kinetic friction is nearly independent of the sliding speed. These activities led to the enunciation of the law of dependence of the frictional force through the intermedium of the sliding friction coefficient, together with its consequence, the dependence of the friction coefficient on the two forces. Early  investigators Amontons (1699), Coulomb (1785), and Morin (1833)  hypothesized that friction is due to the interlocking of mechanical protuberances or asperities on the surfaces of the contacting materials, and in this way, they were able to explain why the friction force is proportional to the load, and independent of the contact area. It is clear that they had seriously considered an alternative explanation, that friction is due to adhesive forces between the contacting surfaces. However, they rejected this Adhesion hypothesis because it implied that friction is proportional to the contact area, which is contrary to the experimental evidence. Three different groups of researchers, namely Holm (1938), Ernst and Merchant (1940), and Bowden and Tabor (1942) cleared up the difficulties with adhesion hypothesis. These researchers pointed out that there is a crucial difference between the apparent and the real areas of contact and it is the real area alone, which determines the magnitude of the friction force. Since the real area could be shown to be proportional to the load and independent of the apparent area, the adhesion hypothesis was now able to explain the experimental result that the friction force is independent of the (apparent) surface area. In the late 1960s, the British government was persuaded by the Jost report (1966) that much waste of the resources (about 515 million pounds sterling per year) occurred because of ignorance of mechanical surface interaction phenomena. A coherent program of education and research was launched to remedy this situation and the word "Tribology" was coined to describe this program.  4  1.4  Friction Friction is expressed in quantitative terms as a force, being the force exerted by either  of two contacting bodies to oppose relative tangential displacement of the other. The three quantitative relations are as follows: 1. The friction force (F) is proportional to the normal force (W),  F=juW  1.1  This relationship enables us to define a coefficient of friction pi. Alternatively it is often convenient to express this law in terms of a constant angle of repose, or frictional angle Of defined by, tandf=ju  1.2  It can be shown that Of is the angle of an inclined plane such that any object, regardless of its weight, placed on the plane will remain stationary, but that i f the angle is increased by any amount, the object will slide down. 2. The friction force is independent of the apparent area of contact. Thus the large and the small objects have the same coefficients of friction. 3. The friction force is independent of the sliding velocity v (due to Coulomb). This implies that the force required to initiate the sliding will be the same as that of the force to maintain sliding at any specified velocity. It is well known that the friction force required to start sliding is usually greater than the force required to maintain sliding, and this has given rise to notion that there are two coefficients of friction - static (for surfaces at rest) and kinetic (for surfaces in motion). 1.4.1  Static friction  The static coefficient of friction is often considered to be a function of time constant. Rabinowicz [37] proposed that the shear force at the asperities built up as a result of imposed tangential micro-displacement and this indicated that the static friction coefficient would build up as a result of the time of stationary contact. There are various proposed expressions to describe the changes in static friction with time of stationary contact (t ), s  1) Howe et al. [27] suggested that a relation of the form,  5  Ms =Mk+(Msoo-Mk)(l-e  )  Cts  1.3  where, Pk  kinetic coefficient of friction;  ju oo  static coefficient of friction for very long time of contact;  S  c  2)  a constant.  Another possible curve for the time dependence of the static coefficient of friction was presented by Derjaguin et al. [ 17] Ci t „ Ms = Mk + — C +t 2  1  s  where, fik  kinetic coefficient of friction;  Ci ,C2 are constants. 3)  Rabinowicz [37] and Brockley et al. [12] proposed a general relation in the form, Ms ~Mk = Yh  1.5  where, fik  kinetic coefficient of friction;  y, f3  are constants.  These expressions have the general shape as shown in Fig 1.1. As it can be seen from the figure, the static coefficient of friction increases with the increase of time of contact until it reaches a saturation value. This saturation time depends on the materials that are in contact and may vary from fraction of a second to several days. However, experiments by Johannes et al. [28] and Richardson and Nolle [38] have demonstrated that at constant high loading rates, the coefficient of static friction is constant and independent of contact time. For intermediate loading rates, the coefficient of static friction decreases with increasing loading rates. Thus under these conditions the time of stationary contact is not controlling the friction behavior and it appears that the rate of tangential loading is the principal factor. But the mechanism responsible for the rate dependence on the static friction has not been properly understood.  6  Time of Stick (Q  Fig 1.1 Plot of static friction coefficient as a function of time of stick  1.4.2  Kinetic friction Kinetic friction or the frictional force during sliding may be divided into three  categories: •  Dry friction between clean surfaces.  •  Fluid friction between surfaces separated by thick lubricant film, such as in the case of bearings, often determined by the hydrodynamic properties of the fluid.  •  Boundary friction between surfaces separated by a thin surface film. Friction force often varies during the process of sliding, and it is considered to be  velocity dependent as experimentally studied by Ko et al. [34]. They showed that the existence of the type of friction induced vibration is critically dependent on the particular shape of the dynamic friction curve. The kinetic friction coefficient for lubricated metal-tometal contact often has a negative slope at slow sliding speeds and a positive slope at high sliding speeds. A typical plot of coefficient of kinetic friction vs. sliding velocity appears as shown in Fig 1.2.  7  Fig 1.2 Plot of kinetic friction coefficient as a function of sliding speed  1.5  The effects of tribosystem variables on frictional response  1.5.1  Effects o f surface finish  There is often a direct relationship between the friction forces generated in a system and the initial surface texture, which includes the roughness, waviness, and lay of the moving parts. If the contact pressure is low, or if the contacting materials are hard, initial surface features may be preserved for an extended period of sliding and therefore they will play a role in determining friction. If, on the other hand, contact pressure is high or if one or both contacting materials is relatively soft, the initial surface finish asperities will quickly be obliterated and it will have a minimal effect on sliding friction. Thus, friction can affect wear, which in turn affects the surface characteristics which affect friction, and the interdependence of friction on wear becomes a recursive one. In summary [9], the morphology of the initial surface finish can have an important role in friction if: •  The contact pressure is small enough to avoid loss of the original geometric features by wear.  8  •  The wear resistance of the sliding materials is high enough to preserve the surface features even after prolonged sliding takes place.  •  In the presence of liquids, the lubrication regime is affected by surface roughness.  •  The surface microgeometry has characteristics that enable it to trap loose particles that would otherwise affect friction.  1.5.2  Effects of load and contact pressure When surfaces are placed together under a normal load, the asperities will deform  first elastically and then plastically to support the load. When the normal pressure being applied is exactly balanced by the upward force distributed over the expanding contact area, the growth of junctions will stop. However, the fact that friction coefficient decreases with increasing load in some cases does not guarantee that Hertzian elastic behavior is the cause. There are other possible causes. For example, higher load increases wear, which in turn causes more surface roughening or generates a layer of shear-strength modifying wear debris in the interface. The dependency of friction coefficient on contact pressure and load cannot be generalized because it is determined by both the materials and the tribosystem. 1.5.3  Effects of sliding velocity Under some conditions increasing the velocity increases the wear rate, which in turn  roughens the surface and raises the plowing and cutting contributions to friction. Continuing to increase the velocity produces surface softening and the friction decreases again. Therefore, before assuming that the friction always decreases as velocity increases, it is wise to consider the specific materials and systems involved. In many engineering applications, however the velocity is neither steady nor periodic. It therefore becomes important to study the behavior of practical friction systems to understand not only how friction is related to velocity, but also to changes in velocity. 1.5.4  Effects of temperature Thermal energy in a sliding interface affects the properties of the materials in the  vicinity of the interface. If the properties of the materials are altered sufficiently by heating, then the resistance to sliding offered by those materials will be changed. The main effects of temperature on friction are summarized as follows [9]:  9  •  The shear strength of interfacial materials is dependent on temperature.  •  The viscosity of liquid and solid lubricants is dependent on temperature.  •  The tendency of the surfaces of materials to react with their surrounding environment to form films or tarnishes is dependent on temperature.  •  The tendency of formulated liquid lubricants to change chemical composition is dependent on temperature exposure.  •  The wear processes in materials, which affect surface roughness and tractional characteristics, change with temperature.  •  The ability of a metal or alloy to work-harden or otherwise alter its structure and properties depends on temperature.  •  The tendency of certain materials to transfer to the rubbing partner may depend on temperature.  •  The ability of a surface to adsorb contaminants from the surrounding environment is affected by temperature. In view of these, the dependence of the friction of a given system of contacting  materials on temperature is not straightforward. Therefore, it is not unusual to observe irregular behavior of friction as a function of temperature over wide range of temperatures. However, in this work, the experimental system operates at very low speeds (in the order of 1.6 rad/sec) and reasonably low loads (up to 300 N resulting in maximum contact pressure of 3.5 N/mm ), the temperature effects are considered to be negligible. 2  1.5.5  Effects of surface films a n d chemical environments  Any substance interposed between sliding surfaces has the potential to affect friction. Under some circumstances, the contact conditions may cause that substance to be penetrated or wiped away quickly and its effects will then be suppressed. On the other hand, it is often observed that surface films and those chemical environments that cause such films to be formed and replenished can have a major effect on frictional behavior. Surface oxides and tarnishes can significantly affect both the static and kinetic friction between solids. Relative humidity has been shown [9] to influence the frictional behavior of a wide variety of materials, but the mechanistic details of its influence vary from one material to  10  another. Increasing relative humidity can either increase or decrease friction, depending on the material combination involved. 1.5.6  Combined effects of several variables The effects of several variables on frictional behavior have been discussed separately.  During frictional contact, several frictional processes could be occurring simultaneously, and, as mentioned earlier, it is difficult to change just one variable at a time when conducting a friction experiment. This interrelationship among various factors makes laboratory studies on the effects of only one variable difficult to design and interpret correctly. Therefore, investigating the conjoint effects of multiple variables on either static or kinetic frictional behavior remains a challenging and very important area for future friction research from both a practical and fundamental standpoint. 1.6  Stick-slip vibrations Stick-slip is caused by the interaction between the dynamic behavior of the frictional  force and dynamics of the system and is characterized by a periodic cycle of motion and arrest. This phenomenon occurs often between slowly moving bodies in dry or boundary lubricated contact. In a majority of sliding systems as that shown in Fig 1.3, the sliding friction has a static friction component, which has to be overcome by the driver acting through the compliant element. When the static friction is surpassed, the mass begins to move; once the mass begins to move the sliding friction may change. If the friction force becomes smaller, then the mass accelerates backwards until the spring force becomes less than the friction force. At this point the mass will decelerate until it comes to rest and the cycle repeats. Stick-slip due to the relative motion between the machine members exerts significant influence on system dynamic behavior giving rise to undesirable self-excited oscillations. This is called stick-slip instability and is characterized by a uniform motion (stick phase) followed by a non-uniform motion (slip phase).  11  Fig 1.3 Spring and dash-pot representation of stick-slip  The stick-slip phenomenon was first observed by Wells in 1929 while trying to measure kinetic coefficient of friction at low speeds and proposed that the motion could occur only if the static coefficient were larger than the dynamic coefficient of friction. Thomas in 1930 was first to demonstrate stick-slip, who used graphical and analytical techniques to solve pertinent differential equations. Blok in 1940 outlined the correct criteria for the onset of stick-slip by linearizing the friction-velocity curve that was followed by the work of Eliasberg in 1951. Rabinowicz [37] undertook the study of stick-slip process in 1957 and developed a conceptual model. Stick-slip motion as a whole can be treated in terms of a general motion trajectory x(t), where during stick episodes, the slider velocity equals that of the driving component. In Fig 1.4, the relative motion of a slip episode after stick phase begins at t=0 at a displacement of x+ , which is larger than x , the time averaged position. Inertia carries the slider further s  ()  upwards for a short interval to x , where its absolute velocity vanishes and then becomes max  increasingly negative. The maximum relative velocity is reached at the average position xo and then diminishes. After reaching a zero absolute velocity at x , the slider speeds up to min  the driver velocity VQ and the slider sticks to the driver at a displacement of x. < XQ. The s  observed stick-slip amplitude is thus x = x+ - x. . s  s  s  12  V=V„ (V =0) R  Fig 1.4 Schematic illustration of displacement and velocity during stick-slip motion  1.6.1  Stick-slip vibration for large surface contact area Many empirical models have been developed for small contact areas, i.e. localized  contacts where every point on the mass moves with the same velocity as that of the slider. But, very little attention has been given in the literature to the friction phenomenon for large contact areas involving uneven distributed loads. It is only recently that Tseng and Wickert [42] published a paper on stability analysis of a friction loaded disc. Fig 1.5 shows the schematic representation of rotational discs in contact with uneven distributed loads. The lower disc rotates continuously at a constant speed. The upper disc, which is in frictional contact with the lower disc, is constrained by means of a spring. When the lower disc starts rotating at a constant speed, the upper disc goes with it due to the static friction. As soon as the spring torque exceeds the friction torque, the upper disc starts to slip and in the event due to the drop in friction torque, it exhibits stick-slip vibrations.  13  Vibrating Disc  Driving Disc  Fig 1.5 Stick-slip phenomenon for large contact areas  The phenomenon for large contact areas (especially  rotary contacts)  is very  complicated due to the fact that the tangential velocity at every point on the surface varies with the radial distance. A l s o , the distribution o f normal load over the surface would vary due to the uneven distribution o f the loads. The elastic contact depression can result due to high local loads. Thus every point on the surface w i l l have different combination o f normal load and different linear velocity. If all these points were to behave  on their own  independently, one point may exhibit stick-slip and the other may not, depending on all these various parameters. But now, because o f the fact that all these points are kinematically constrained on to the same disc, they have to obey the overall behavior o f the disc. Thus the modeling and the analysis becomes complicated. This research mainly aims at developing a better understanding o f the phenomenon under such circumstances and obtaining an insight into it.  1.6.2  Undesirable effects of stick-slip motion The following are examples o f some undesirable effects resulting from stick-slip  motion when large contacts are involved:  14  •  Initial jerky motion in excavators carrying large loads causing negative effects on functionality and performance.  •  Vibrations in large robotic manipulators leading to positioning and tracking errors.  •  Unstable motion in machine tool slides and the resulting vibrations of the tool affecting the quality of the machined surface.  •  Detraction in the accuracy and reliability of Mechanisms and measuring devices.  1.6.3  Factors causing stick-slip vibrations  The basic causes of the stick-slip friction vibration have been studied by Rabinowicz [37], Derjaguin et al. [17], Brockley et al. [11] and Gao et al. [22]. The following could be the probable reasons for the onset of stick-slip: •  The most accepted cause for stick-slip is that static friction coefficient being greater than kinetic friction coefficient or kinetic friction coefficient dropping rapidly at low speeds.  •  Stick-slip motion does not necessarily occur even i f static friction is greater than kinetic friction, its occurrence and the severity when it occurs can be significantly influenced by sliding speed, spring stiffness and system damping.  1.6.4  G e n e r a l remedies  Different compensation techniques have been tried for elimination of stick-slip vibrations at a given velocity by various researchers like Rabinowicz [37], Derjaguin et al. [17], Arnov et al. [8], Gao et al. [22]. They established different criteria for the elimination of stick-slip at a given velocity, some of which are as follows: •  Using a large stiffness of the elastic suspension i.e. having a spring of large stiffness. However, increasing the stiffness may not be always economical and there is a limit beyond which the stiffness of the system can not be increased.  •  Reducing the difference between forces of static and dynamic friction, which may be achieved by using an adequate lubricant.  •  Increasing the degree of the damping parameter, but this may not be always desirable or possible to implement.  15  •  Having a positive friction-velocity characteristic at the sliding velocity employed. Certain materials having higher friction coefficient at increased sliding velocity have been found to reduce the stick-slip.  •  In addition, the compensation techniques using PID control [19], Accelerated Evolutionary Programming [32] and Neural Networks [39] have also been employed by various researchers to compensate for stick-slip vibrations. The stick-slip amplitude decreases with increasing driving speed and there exists a  critical speed beyond which no stick-slip motion occurs as observed by Brockley et al. [11]. This critical speed increases with increasing load, leading to increasing prevalence of stickslip motion at higher loads. 1.7  Discussion Since each of these compensation techniques has its own limitations, i f a better  understanding of the phenomenon is achieved, it can be used to model the stick-slip vibrations. This mathematical model can be later used to control the stick-slip vibrations under different conditions. Hence the current research is aimed at obtaining an insight into the phenomenon for large contact areas involving uneven distributed loads.  16  C H A P T E R 2 CONTACT PRESSURE DISTRIBUTION AT T H E INTERFACE 2.1  The topography of contact In a B B C radio program, tribology pioneer F.P. Bowden observed that "putting two  solids together is like turning Switzerland upside down and standing on Austria - the area of intimate contact will be small". Crystalline surfaces, even apparently smooth surfaces, are microscopically rough. The protuberant features are called asperities and the true contact occurs at points where asperities come together. The accurate representation and deflections between rough bodies in motion is of significant importance in tribological applications. In 1881, Hertz considered the contact between two smooth elastic spheres and cylinders. It has long been realized that surfaces are rough on microscopic scale, and that this causes the real area of contact to be extremely small compared to the nominal area. The calculation of the area of contact, or even the prediction of how this varies with load, is very difficult. Early attempts to do this by applying the Hertzian theory of contact between spheres to individual contacts met with two difficulties: the area of contact depends on the radius of asperity, which is not usually known; and the predicted variation of area with load proved to be incorrect. Both these obstacles were removed when Holm [35] introduced the idea that although the overall stresses are in the elastic range, the local stresses at the contact spots are much higher so that the elastic limit will be exceeded and the contact will yield plastically. Each contact can then be visualized as a small hardness indentation, so that the mean contact pressure will be equal to the hardness and effectively independent of the load and the contact geometry. This has been a very fruitful concept, based on which Bowden and Tabor [10], developed their adhesion theory for friction. Bowden and Tabor [10] noted that the real area of contact for rough surfaces is much smaller than the area predicted by Hertz. Earlier investigators considered the asperities to deform either elastically [23] or plastically [10]. However, in the contact region, some asperities will deform elastically and others plastically.  17  For the present project, since the contact occurs over a large area, the analytical approach using asperities concept would be very complicated. Moreover, the literature available for analytical approach for the contact pressure distribution for large areas is scarce. Hence finite element modeling using A N S Y S is employed to calculate the contact pressure distribution between the circular plates. 2.2  Experimental apparatus The test rig consists primarily of a rotating disc driven by a variable speed DC motor  and a slider disc attached to an elastic system as shown in Fig 2.1. The facing surfaces of the discs are lapped to provide good contact between the two. Normal loads can be applied using dead weights via pulleys at four different load points. This loading configuration helps to isolate the applied load from the vibrating mass. The position of these loading brackets can be moved towards or away from the center thereby changing the point of the load application.  Fig 2.1 Laboratory experimental set-up  18  2.3  Problem description The schematic representation of the plates used in A N S Y S model consists of a  circular plate that rests on top of another circular plate similar to the one shown in Fig 1.5. The plates can freely rotate about the central ball bearing and the four loads can be applied 90° apart at a specified radial distance. The loads are applied using deadweights with the help of roller bearing and a pivoted bracket whose exploded view is shown in Fig 2.2. Fixed Pulley Fixed Hinge  Loading Bracket  Load on the plate  Roller Bearing  One of the four load application points Plate  Fig 2.2 Exploded view of load application points The realistic representation of the experimental set-up is done by modeling the plates that are in contact with each other to their real dimensions and material properties (for steel), which are as shown in Table 2.1. As contact problems are highly non-linear and require significant computer resources to solve, only a quarter of the circular plate has been modeled by taking advantage of the symmetry. Most contact problems need to account for the friction and these friction models and laws are non-linear.  19  Parameter  Top Plate  Bottom Plate  Inner Radius, Rj (mm)  8  8  Outer radius, R (mm)  100  100  Thickness, t (mm)  6  32  0  Young's modulus, E (N/mm )  210 X 10  Material density, p (Kg/mm )  7.8 X 10"  7.8 X 10"  Poisson's ratio, v  0.3  0.3  Coefficient of friction, n  0.192  0.192  2  3  210 X 10  3  6  3  6  Table 2.1 Parameter values used in A N S Y S model  2.4  Simple linear model using ANSYS A finite element model in A N S Y S [45] has been developed to obtain the distribution  of the contact pressure at the interface between the plates due to the applied loads. In modeling, as a first approximation, the top plate was assumed to be supported at all points on its lower surface, thereby considering the bottom plate to be perfectly rigid.  Fig 2.3 Contour plot for the contact pressure distribution (90 N load at 59 mm)  20  The normal pressure distribution across the surface of contact was found to be concentrated only at the point of application of load and diminished rapidly away from the point. Thus the pressure distribution can be visualized as peaks and valleys whose contour plot is shown in Fig 2.3. Although, the areas away from the load application point showed a positive value for the stress, physically it is likely that the plate is slightly separated and thereby there is no contact between the two surfaces in these areas. The drawback of this model is that in reality, the bottom plate is not very rigid, instead it is elastic. Subsequently, this limitation was overcome in the next step of modeling by using contact elements, which falls into the category of non-linear modeling. 2.5  Non-linear model using contact elements The plate segments are modeled by placing the top plate over the bottom one and  modeling the contact elements at the interface of contact. As both plates are elastic in nature, the contact problem is considered to be flexible-to-flexible surface contact. The brief description of various elements used in A N S Y S model has been provided in A P P E N D I X I. Both plates are modeled with 20 node, SOLID95 element with 3 degrees of freedom/node. A contact pair has been created between the lower face of the top plate and the upper face of the bottom plate. The upper face of the bottom plate is considered as the target surface for the contact and has been modeled with TARGE 170 target element. The element 8 node, CONTA174, higher order quadrilateral, flexible contact element is attached to the lower face of the top plate. Both contact and target will deform under the influence of load, i.e. they are not rigid. The mesh discretization with SMART SIZE 6 option has been incorporated near the point of application of load to achieve finer element size closer to the load and better approximation in calculating stress distribution. This has resulted in the total number of 11076 nodes with 31698 degrees of freedom. In reality, the load is applied using a roller bearing of 5 mm width and it is reasonable to assume that the load would be distributed over a line. Hence the specified amount of load is applied over a line, at a given radial distance, on the upper surface of the top plate. The boundary conditions for the plates include constraining the displacement at the cylindrical wall of the center hole of the inner bearing surfaces, in all directions except in the direction of application of load. In addition, the lower surface of the bottom plate is constrained for  21  displacement in the direction of load as well, thereby making it perfectly rigid. The resulting model is shown in Fig 2.4. The solution for the above model is obtained by considering large deformation effects, which enable the individual elements to go in and out of contact throughout the analysis. The loading is accomplished in 10 sub-steps to facilitate convergence of the solution.  AN  Fig 2.4 The geometric appearance of the model using A N S Y S  Limitations of this model: •  The estimated contact pressure distribution is for the stationary plate, but when the plate vibrates due to stick-slip phenomenon, the load distribution change. However, due to the limited capabilities of the A N S Y S analysis package available, there is no attempt to accommodate the stick-slip phenomenon in the analysis.  •  Also, the modeling does not take into account the elastic depression of the asperities due to high-localized loads.  22  2.6  Post-processing and analysis of the results A section plane has been created at the contact surface in the solution domain to  achieve a better understanding of the contact pressure distribution between the two plates and to obtain a contour plot of nodal solution. To visualize the variation of the contact pressure, paths have been defined along the radial and the circumferencial directions of the circular plate and the corresponding 2D distributions have been plotted as shown in Fig 2.5. The contour plot is shown in Fig 2.6, which corresponds to A N S Y S solution for the model carried out for a load of 90 N applied at a radial distance of 59 mm. It can be clearly seen that the maximum contact pressure is at the load application point (at 59 mm and 45°). The effect of the applied load distributed over the line is clearly reflected by the flattened peak.  Radial Contact Pressure Distribution plot 3.5 - W = 45 N W = 90 N W= 180 N W = 270 N  3  e  I H  f  25  i  2  1  I  (ft  I  Q.  I  15  I  1b  j  o  o 0.5  10  20  30  i  •  ^/  40  50  -•-  ...  \ \ Y. \ i  60  70  -80  90  0 Circumferential Contact Pressure Distribution plot  30  40 50 Angular orientation (in degree)  60  70  80  90  Fig 2.5 Radial and Circumferencial contact pressure distribution plot (load at 59 mm and 45°)  23  AN NODAL SOLUTION STEP=1 SUB =6 TIME=1 CONTPRES (AVG) DHX =.936E-03 SMX =1.053 ZV =i *DIST =55 *XF =50 *YF =50 *ZF =-13 SECT ION .100E-03 .122311 .244522 .366733 . 486944 .611156 .733367 .855578 .977789 1. 1  Fig 2.6 Contour plot of pressure distribution for the non-linear model (90 N at 59 mm)  To facilitate the repetition of the analysis for different combinations of loads and load application points, a batch file of the above A N S Y S model has been written as shown in APPENDIX II. By simply changing the corresponding parameters of this batch file, solutions were obtained for several combinations. Based on these solution sets, a generalized empirical model for the contact pressure distribution across the entire interface of contact has been derived. The contact pressure distribution is observed to be approximately normal around the load application point and found to decrease rapidly away from this point. The applied load has negligible contribution to the contact pressure distribution away from the load application point. The weight of the plate would contribute a contact pressure uniformly distributed across the interface of contact. Hence the contact pressure varies from a maximum value (at the point of application of load) due to the applied load plus weight of the plate, to a minimum value (away from this point) contributed essentially by the weight of the plate  24  only. The results of the solution carried out at various loads and the radial distances are tabulated as shown in Table 2.2. The minimum value of the contact pressure, away from the load application point has been found to be independent of the applied load and hence inferred to have been caused by the weight of the plate alone. The applied load has no influence on the nature of the distribution except for the maximum value of the contact pressure, which is proportional to the applied load. The variation in the radial distance of the point of application of load was found to have no significant effect on the distribution of the contact pressure. 2.7  Development of an empirical model for the distribution of the contact pressure Based on the results revealed by the finite element modeling using A N S Y S , a  generalized empirical model for the contact pressure distribution across the entire interface of the plate has been developed as shown in Fig 2.7. The inputs to the model are the total "applied load and the radius of the load application point from the center of the circular plate. The maximum value of the contact pressure has been calculated in terms of the load applied and the minimum value in terms of the self-weight of the plate. The contact pressure distribution is found to be normal around the load application point. The variance of this distribution has been calculated to make the total normal load causing this pressure distribution to be equal to the sum of the applied load and the weight of the plate.  p(ne)=p +(p mm  2.1  max  v where, Pmi„  minimum contact pressure due to self-weight of the plate = Weight of the plate/Area of the plate surface;  Pmox  maximum contact pressure due to the applied load  = Factor X (Applied load + Self-weight)/Area of the plate surface; r  x co-ordinate distance from the load application point;  r  y co-ordinate distance from the load application point;  x  v  variance of the normal distribution.  25  Pmax  Pniin  x Fig 2.7 Model for the contact pressure distribution at the interface  Load  Radius  From A N S Y S solution  W(N)  R(mm)  Pmax (N/mm )  Pmin (N/mm )  Pmax (N/mm )  P  45  59  0.5267  4.60 X 10"  4  0.5262  4.59 X 10"  90  59  1.0529  4.60 X 10"  1.0524  4.59 X 10"  180  59  2.1042  4.60 X 10~  2.1048  4.59 X 10"  270  59  3.1559  4.60 X 10"  4  3.1571  4.59 X 10"  45  80  0.52668  4.60 X 10"  0.5262  4.59 X 10"  90  80  1.0531  4.60 X 10"  1.0524  4.59 X 10"  180  80  2.1063  4.60 X 10"  2.1048  4.59 X 10"  270  80  3.1613  4.60 X 10"  3.1571  4.59 X 10"  From mathematical model 2  4  4  4  4  4  4  2  (N/mm ) 2  m m  4  4  4  4  4  4  4  4  Table 2.2 The comparison of the values from ANSYS and mathematical model  26  The factor of applied load that contributes to the maximum contact pressure calculated is found to be 365 and the variance of the normal distribution, a has been computed to be equal to 5.3292. These values have been computed using non-linear curve fitting techniques to make the total normal load resulting from this contact pressure distribution to be equal to the sum of the applied load and the weight of the plate. As a result, the distribution of the contact pressure across the interface for a single load of 90 N at a radial distance of 59 mm appears as shown in Fig 2.8. The values are found to be in good agreement with the results extracted from the ANSYS model, Table 2.2.  Pressure Distribution at the contact surface  Fig 2.8 Distribution of the contact pressure over the whole plate (90 N at 59 mm)  2.8  Calculation of the friction torque The total frictional torque between the two plates is calculated by integrating the  torque acting over a small elemental area over the entire area of the plate. Consider an elemental area as shown in Fig 2.9 at a radial distance of r, of width dr and angular width dO. The coefficient of friction at this elemental area, ft, depends on the relative speed between the two plates and can be obtained by one of the point contact models that will be explained later  27  in Sec 3.2. Since the relative speed depends on the radial distance from the center, the coefficient of friction is a non-linear function of the radial distance. Thus, dT = ju{r) P{r,6)r  = ju{r)P(r,d)r  2  dd dr r  ,  dr dd  2.2  Fig 2.9 Calculation of friction torque at the interface between the plates  This torque is integrated over the entire area to obtain the total frictional torque applied on the disc. lit „ T= \ j ju(r)p(r,0)r 0 Rj R  2  dr dd  2.3  This double integration has to be carried out over the entire area of the plate of 100 mm radius at every instant of time. The coefficient of friction is a function of the radial distance from the center while the contact pressure varies as a function of both radial distance and angular orientation. As evidenced from the A N S Y S results, the contact pressure rapidly diminishes to the minimum value away from the load application point and hence the friction torque is a function of radial distance alone except in the vicinity of the load application point. Thus it is sufficient to perform the double integration only in the immediate neighborhood of the load while the single integration along the radial direction is performed across the entire interface of the plate. In doing so, the circumferencial strip is first integrated  28  along the radius and then double integration is performed only in the neighborhood of the load application point, thereby minimizing the number of computations to be performed. The Gaussian numerical integration algorithms are used and the Matlab code for the calculation of the friction torque is given in APPENDIX III. This total friction torque acts as the driving torque for the top plate and would be used in the simulation of the stick-slip vibration as shown later in C H A P T E R 3. 2.9  Discussion The analysis of the variation of the friction torque across the interface is not  straightforward as it involves the non-linear expressions for both coefficient of friction as well as contact pressure. As we move radially away from the center of the plate (along a radial line passing through the load application point), the friction torque slowly increases from zero to a maximum value at the load application point and then decreases rapidly to a minimum followed by a slow increase towards the periphery. The contact pressure model does not take into account the elastic depression of the asperities due to indentation effects at the load application point. Tribology has not yet been able to offer a theoretically motivated model for such a phenomenon. A further study in this direction is necessary i f these effects have to be taken into consideration.  29  C H A P T E R 3 MODELING AND SIMULATION 3.1  Parameters to be considered for a stick-slip friction model The stick-slip phenomenon is governed by the following four factors: •  The mass-spring dynamics of the system;  •  Velocity dependent friction characteristics;  •  The interdependence of the static friction and dwell time;  •  The time lag between a change in system state and the corresponding change in friction.  The friction model incorporates Coulomb friction, stiction and viscous friction with frictional memory and rising static friction. Both viscous and Coulomb friction produce a force against the motion, but the former is proportional to the speed while the latter depends on the sign of the velocity. Since these two parameters are well known in tribology, four other parameters of importance will be discussed in more detail. 3.1.1  Pre-sliding displacement Pre-sliding displacement is a consequence of elastic deformation of the surface  asperities where contact and sliding occur and significantly influence friction forces during velocity reversals. It is well known that contacts are compliant in both the normal and tangential directions [29]. Dahl [16] concluded from his experimental observations on small rotations of ball bearing that for small displacement, a junction in static friction behaves like a spring as shown in Fig 3.1. This pre-sliding displacement is approximately a linear function of the applied force up to a critical force, at which breakaway occurs. When a force, F (t) is applied, the u  asperities will deform, but recover when the force is removed just like a spring. Pre-sliding displacement is pivotal in correctly predicting displacements while sticking, including velocity reversals. The tangential force is governed by, F {t) = k x u  t  3.1  30  where, F (t)  applied tangential force;  k  the tangential stiffness of the static contact;  u  t  x  the displacement away from the static position. Force 5 Static Friction  z  Part A Idealized Asperity Junctions PartB  /////  Break-Away PartB  /  /  /  /  /  Fig 3.1 Asperity deformation under applied force and pre-sliding displacement 3.1.2  Stribeck effect The Stribeck effect is shown in Fig 3.2. Because of the Stribeck effect, the friction  force decreases with increase of relative velocity within the low-velocity sliding region. The phenomenon is due to partial fluid lubrication in solid-to-solid contact and may be one of the reasons for stick-slip motion [25]. Armstrong [1] showed that Stribeck effect contributes to, and perhaps dominates, stick-slip.  X  Fig 3.2 The Stribeck effect  31  Stribeck effect plays an important role in correctly predicting the initial conditions leading to stick-slip and can be mathematically represented as,  sgn(i)  F(i) =  3.2  1+  where, Stribeck friction force; characteristic velocity of the Stribeck friction. 3.1.3  Frictional memory The Stribeck curve shows the dependence of friction upon the relative velocity. If  there is a change in velocity, one might presume the corresponding change in friction force to occur simultaneously. In fact a time delay or a phase shift has been observed experimentally between a change in sliding velocity and a corresponding change in friction by Rabinowicz [37], Hess and Soom [25] and Armstrong [3]. Thus the frictional memory is a delay between a change in sliding velocity and the subsequent change in friction. Hess and Soom [25] carefully measured this shift and found it in the range of 3 to 9 ms in a range of load and lubrication conditions. Physically, frictional memory is the result of state in the interface, which must adjust to the new sliding condition before the friction force will attain its new value. Frictional memory is required to be included in the model for correctly predicting interaction of stiffness and stick-slip amplitude. This phenomenon is dominant in the lower velocity region and can be included in Equation 3.2 as,  F(x) = F  v  ( -A. X 1+  -  \\  2  sg:n(i)  3.3  JJ where, TL  represents a frictional memory (times range between 3 to 9 ms).  32  14.0  F (N) s  3.0 0.1 X (m/sec)  0.0  05  0.6 t(sec) Fig 3.3 Typical friction-speed time shift [25]  3.1.4  Rising static friction The static friction rises as a junction spends more time at rest, and this rising static  friction has a pronounced influence on the continued occurrence of stick-limit cycle. Physically, it can be attributed to the junction growth of asperities between the surfaces in contact. This is also due to the existence of cold-welding effect when contacting asperities come within molecular distances of each other. Increase in junction area with increase in time of stick is due to the plastic behavior of contacting asperities as they are deformed in a way analogous to creep. Derjaguin et al. [17], Rabinowicz [37] investigated the influence of this phenomenon and showed that inclusion of the rising static friction is needed to correctly predict interaction of velocity and stick-slip amplitude. The static friction force increases with increase of dwell time (Y^), and the relationship between the static friction force and dwell time is given by,  t +r  3.4  2  where, F t,  magnitude of the Stribeck friction;  F  magnitude of the Stribeck friction at the end of the previous sliding period;  Si  SM  33  F„ s  magnitude of the Stribeck friction at k = °°\  y  temporal parameter of the rising static friction.  3.2  Stick-slip models In Fig 3.4 common friction models are shown: in Fig 3.4(a), a coulomb + viscous  friction model; in Fig 3.4(b), a static + coulomb + viscous friction model; and Fig 3.4(c), a static + coulomb + viscous friction model with negative viscous friction at low velocity. - Friction force  Frictioriiforce  1i  .——,  .. Slope due to • Viscous friction.  Velocity  Veiociiv  Level of Coulomb'friction Static friction* (b) Static *>Soulomb + viscoussmodel  :(a) Coulomb + viscous model Friction;force  StribeckefTect  Velocity  •  ici Static ' Coulomb - Stribeck model  ., .'  Fig 3.4 Commonly used friction models Coulomb friction is independent of velocity and is always present. Viscous friction is in proportion to the velocity and is present in fluid lubricated junctions, such as machines lubricated by grease or oil. In many situations, the force required to "break-away", that is to commence motion from rest, is greater than that required to sustain motion, as suggested by the static friction model of Fig 3.4(b). Detailed observation of friction suggests that the drop from static friction does not occur instantly and that Fig 3.4(c) is often appropriate.  34  3.2.1  Classical friction (Coulomb) model  The discontinuity at the origin in the model represented by Fig 3.4(c) is practically unrealistic and unacceptable for the purposes of simulation. One approach in overcoming this discontinuity has been to approximate this by a linear region as shown in Fig 3.5, which results in the classical model.  F  H  ^  / -F . H  Fig 3.5 Classical friction model But, the problem is that it allows the body to accelerate even though the external forces on the body are less than the peak stiction force FH- SO, this model can not be expected to accurately predict limit cycling or other effects associated with the sticking phenomenon. Also, the very steep slope at zero velocity can result in very short integration time steps and numerical difficulties. 3.2.2  K a r n o p p model  To overcome the aforementioned difficulties in the classical model at very low velocities, Karnopp  [30] developed an alternative approach in which the order of the  dynamic system is effectively reduced at every instant that the relative velocity is zero. Also, the position constraint of the zero motion is incorporated into the system of differential equations.  35  As shown in the Fig 3.6, when the relative velocity is within the small band ±D  R  the  system "sticks" and while sticking the friction force exactly cancels the driving force. When the driving force exceeds the break-free force FH, the body accelerates, and after a short time the magnitude of V will exceed D and the model will switch from stick to slip mode. R  V  Fig 3.6 Karnopp's model [30]  The disadvantages of the above model are: •  Difficulty incorporating into a simulation model because it requires the derivation of separate sets of equations for each possible slipping and sticking condition.  •  With Karnopp's approach, the complexity of the friction model increases as the complexity of the system increases. The number of sets of equations depends on the details of each particular application.  •  Also, this model is application dependent. The main advantage of Karnopp's model is that it is computationally efficient, as it  requires minimum computer time.  36  3.2.3  The seven-parameter friction model Theoretically motivated models for the components of friction are not yet available,  but a variety of empirically motivated forms has been presented. One of these is the sevenparameter model proposed by Armstrong  [4] that incorporates all the seven friction  parameters explained before, where the friction force is given by: Not sliding (pre-sliding F (x)  displacement)  3.5  = -k x  f  t  Sliding (Coulomb + viscous + Stribeck friction  with frictional  F +F \x\+F {y,t )-  Ff{ ,t):  c  X  v  s  1+  y  3.6  5gn(x)  2  f •(. _ \ \ i(t-T )  memory)  2  L  S  X  Where, the rising static friction (friction level at breakaway), is given as, Fs  h) = F ,a + (Fs,oo ~ F , ) s  s a  3.7  7^7-  where, Ff  instaneous friction force;  F (*)  Coulomb friction force;  F (*)  viscous friction force;  F  magnitude of the Stribeck friction (frictional force at breakaway is F + F );  F  magnitude of the Stribeck friction at the end of the previous sliding period;  c  v  s  c  Si0  F  SiO0  s  (*) magnitude of the Stribeck friction after a long time at rest (with a slow application of force);  k (*) t  tangential stiffness of the static contact;  (*) characteristic velocity of the Stribeck friction; TL(*)  time constant of frictional memory;  y(*)  temporal parameter of the rising static friction;  tj  dwell time, time at zero velocity;  (*)  marks friction model parameters, other variables are state variables.  37  The magnitudes of the seven friction parameters will naturally depend upon the mechanism and lubrication, but the typical values may be offered as in [7]. The procedure for identification of these parameters has been explained by Armstrong et al. [7]. 3.2.4  L u G r e model The LuGre model (stands for Lund and Grenoble) is a dynamic friction model  presented by Canudas et al. [13], which is inspired by the bristle interpretation of friction discussed in [24], in combination with lubrication effects. Surfaces are very irregular at the microscopic level, therefore two surfaces make contact only at a number of asperities. This is visualized as two rigid bodies that make contact through elastic bristles. When the tangential force is applied, the bristles will deflect like springs, which give rise to friction force.  Sliding Body  i  Bonded Bristles  Stationary Surface  Fig 3.7 LuGre model showing the contact between bristles  If the force is sufficiently large some of the bristles deflect so much that they will slip. This phenomenon is highly random due to the irregular forms of the surfaces. Haessig and Friedland [24] proposed a bristle model where the random behavior was captured and a simpler reset-integrator model, which describes the aggregated behavior of the bristles, was developed. This model is based on the average behavior of the bristles as shown in Fig 3.7. The average deflection of the bristles is denoted by z and is modeled by, dz  |v|  dt  g(v)  where, v is the relative velocity between the two surfaces. 38  The first term gives a deflection that is proportional to the integral of the relative velocity. The second term asserts that the deflection z approaches the value,  55=n^( )=^( ) g ( )  z  v  |v|  v  s  n  3.9  v  in steady state, i.e., when v is constant. The function g is positive and depends on many factors such as material properties, lubrication, and temperature. Direction dependent behavior can therefore be captured. For typical bearing friction, g(v) will decrease monotonically from g(0) when v increases. This corresponds to Stribeck effect. The friction force generated from the bending of the bristles is described as, dz  ^  F=<7QZ + <T — X  3.10  dt  where, OQ is the stiffness and 0 7 is the damping coefficient. A term proportional to the relative velocity could be added to the friction force to account for viscous friction so that, dz + CTY —  F-CTQZ  + CT V  3.11  2  dt  The model in Eq 3.9 and Eq 3.11 is characterized by function g and the parameters a , 0  (7/ and 02. The function ao g(v) + 02 v can be determined by measuring the steady state friction force when the velocity is held constant. A parameterization of g that has been proposed to describe the Stribeck effect is,  °M=F {F -Fcy^ lv^ C+  3.12  S  where, F the Coulomb friction level, F the level of stiction force, and v the Stribeck C  S  s  velocity. With this description the model is characterized by six parameters 00 , 0 7 , 02 , F , F and v . C  S  s  Thus for steady-state motion, the relation between velocity and friction is given by, ss  F  (  V  g(v)sgn(v)+  ) = C 0  CT V 2  = F sgn(v) + (F, - F > - ( C  c  v / v  *)  2  sgn(v)-rcr v 2  Thus the model can describe arbitrary steady-state friction characteristics. It supports the hysteretic behavior due to frictional lag, spring-like behavior in stiction and gives a  39  break-away force depending on the rate of change of applied force. A l l these phenomena are unified into a first order nonlinear differential equation. A n identification procedure for the LuGre model is described in [14]. The method is based on a series of experiments in different friction regimes. Experiments in stick region are used to identify GO and G\. Steady state slip experiments are used to identify v , F and 02s  c  Finally F is identified from breakaway experiments. This model has been experimentally s  proven to represent the richer behavior in terms of capturing the friction phenomena and is strongly favored by Gafvert [20] and Kelly et al. [31]. 3.3  Proposed approach for large contact areas The proposed approach for friction modeling of the dynamic motion for the system  under consideration involves extending the LuGre dynamic model for large contact areas. It considers a circular plate subjected to stick-slip motion by being frictionally driven at a constant speed by another plate and constrained by a spring. The schematic representation of the interfacial frictional phenomenon in an elemental area is shown in Fig 3.8.  Plate Fig 3.8 Schematic representation of the plate subjected to stick-slip  The equation of motion for the vibrating plate is given by, je\t)+ce\t)+kR 0{t)=T {ca ) 2  f  r  3.14  40  where, J  mass moment of inertia of the disc;  c  damping coefficient other than inter-surface friction;  k  spring stiffness of the elastic suspension;  R  radius of the vibrating disc at contact with the spring;  coo  angular velocity of the driving plate;  co  relative angular velocity = a>o- 0{t).  r  In Eq. 3.14, the expression on the right hand side represents the total frictional torque acting on the plate being exerted by the driving plate. Since, this is a non-linear function of relative speed, a> , an analytical solution is difficult to obtain. r  Instead, the total torque acting over the entire area is obtained by integrating this frictional torque over the whole area of the disc as explained earlier in Sec 2.8. The friction force acting on this elemental area can be obtained by one of the point contact models explained in Sec 3.2. The integrated torque for the entire area of the plate is obtained using the following set of equations based on the LuGre model for the localized contacts as explained in Sec 3.2.4 and have been incorporated into simulation as shown in Fig 3.9. TfM  = cr z + (7 ^ 0  dz — = a),, dt  l  °0  + T  3.15  v  3.16  z  r  2K R  1  gM=  0  J  0  J_  R;  \2  (  >  ^  5  P(r,9)r drd9 2  3.17  J  In R  1J 0  T = j  j  0  Rj  v  jU P{r,d)r drde 2  v  3.18  The simulation of stick-slip motion of the disc for different system parameters has been carried out using Simulink [44]. A s-function in Matlab has been written as shown in APPENDIX III to calculate the total friction torque between the two discs by integrating the elemental torque over the entire area of the disc as explained in Sec 2.8. The input to this s41  function is the relative angular speed and it outputs the total friction torque at the interface as shown in Fig 3.9. si  d >  -N  sfunjugre S-Function  dz/m  SO-  Demux  hKZD Torque  •SO  Fv  Fig 3.9 Sub-function for calculating the friction torque acting over the entire plate 3.4  Simulation of dynamical model behavior As a preliminary assessment of the model, the dynamic behavior of the model is  investigated in some typical cases. In all the simulations the parameter values that are shown in Table 3.1 have been used. The bristle stiffness, ao and damping coefficient, 07 have been chosen to give the pre-sliding displacement of the same magnitude as reported in various experiments. The static friction coefficient, /u , has been measured to be 0.1919 as explained 0  in APPENDIX IV. The coulomb friction, JU , viscous friction coefficient, //,,, and the Stribeck C  velocity, v , are estimated based on the non-linear optimization techniques outlined in [14] s  and are also of the same magnitude as given in [2] and [13]. Parameter  Description  Value  I nit  OQ  Bristle stiffness  10000  Nm/rad  Damping coefficient of bristle  100  Nm sec/rad  Mo  Friction coefficient at co = 0  0.1919  Mc  Coulomb friction coefficient  0.1019  Hv  Viscous friction coefficient  0.0408  V  Stribeck velocity  0.001  s  r  1  m/sec  Table 3.1 Model parameter values used in all simulations The different behaviors shown in the following subsections cannot be attributed to a single parameter but rather to the behavior of the nonlinear differential equation and the shape of the function g. The pre-sliding displacement and the varying break-away torque are  42  due to the dynamics. The shape of the Stribeck function, g, together with the dynamics give rise to the type of hysteresis observed due to frictional lag. 3.4.1  Pre-sliding displacement  If a torque less than the break-away torque is applied to the two surfaces in contact, there will be a displacement. A simulation has been performed to investigate i f the integrated model captures this phenomenon.  Applied torque friction torque • vs Angular displacement  y ET 0  y* r  -2  Angular displacement (in rad)  x 1D"  Fig 3.10 Pre-sliding displacement as described by the model  A n external torque was applied to the disc subjected to friction. The applied torque was slowly ramped to 2.5 Nm, which is 83% of the break-away torque. The torque was kept  43  constant for a while and later ramped down to the value -2.5 Nm, where it was kept constant and then ramped up to 2.5 Nm again. The results of the simulation are shown in Fig 3.10, where the friction torque is shown as a function of angular displacement. The behavior shown agrees qualitatively with the experimental results in [13]. 3.4.2  Frictional lag Hess and Soom [25] studied the dynamic behavior of friction when velocity is varied  during unidirectional motion and showed that there is hysteresis in the relation between friction and velocity. The friction torque is lower for decreasing velocities than for increasing velocities. The hysteresis loop becomes wider at higher rate of velocity changes. Hess and Soom explained their experimental results by a pure time delay in relation between velocity and the friction torque.  Friction torque vs Angular speed  Friction Mbdel  28"  w - 25 rad/sec w = 10 rad/sec w = 1 rad/sec  27  I  26  ;  25  """•••-C^S i  24  % ^>.* te  1  !  ,  2 3 ;  22: 2'1  •  2 •  j  I 9 •  n nna  0.01  0.012  0.014 Angular speed (in rad/sec)  0.016  0 018  0 02  Fig 3.11 Hysteresis in friction torque with varying velocity  44  Fig 3.11 shows a simulation of the Hess-Soom experiment using the integrated friction model. The input to the integrated friction model was the velocity, which was changed sinusoidally around an equilibrium point. The resulting friction torque is given as a function of velocity in Fig 3.11 and the proposed integrated model clearly exhibits hysteresis. The width of the hysteresis loop also increases with the frequency of velocity change and the model thus captures the hysteretic behavior of real friction described in [25].  3.4.3  Varying break-away torque  vs  Angular displacement  Applied torque  Applied torque  n  1  1  1  1  J  I  I  '50  60  70  1-  3.2 i  ;  .2.8  * 26  I  *  24  :  *  i  2.2 2h J  10  30 - -'  L  '-r"40  * : :1  • ^ ' ' T o ' q u e . application  * r a , e  L . _ _ . ^  L  80  100  (N-m/sec)  Fig 3.12 Relation between break-away torque and the rate of increase of applied torque  The break-away torque experiments can be investigated through experiments with stick-slip motion. In earlier experiments it was found the dwell-time during sticking and the rate of increase of applied torque are always inter-related and hence the effects of these  45  factors can not be separated. The break-away experiments were therefore redesigned so that time in stiction and the rate of increase of applied torque could be varied independently. The results showed that the break-away torque did depend on the rate of increase of the torque but not on the dwell-time. Simulations have been performed using the integrated model to determine the break-away torque for different rates of torque application. Since the model is dynamic, a varying break-away torque can be expected. A torque applied to the disc was ramped at different rates and the friction torque when the disc started to slide was determined. Note that since the model behavior in the stiction is essentially that of a spring, there will be microscopic motion, i.e. velocity different from zero, as soon as a torque is applied. The break-away torque was therefore determined at the time when a sharp increase in the velocity could be observed. Fig 3.12 shows the torque at the break-away as a function of the rate of applied torque. The results agree qualitatively with [38]. 3.4.4  Stick-slip motion The simulation of the stick-slip motion for the plates under consideration has been  carried out for the system parameters shown in Table 3.2. These system parameters have either been measured or calculated. Parameter  Description  Value  I nit  J  Mass moment of inertia of the disc  0.01X26  Kg-m-  k  Stiffness of the spring  3.0377 X 10  c  Damping parameter  0.0  N-sec/rad  R  Radius at the contact with the spring  0.121  m  Ri  Radius at the load application point  0.059  m  W  Self-weight of the disc  14.331  N  w  Applied load  90  N  CO()  angular speed of the driving disc  0.2  rad/sec  s  3  N/m  Table 3.2 System parameter values used in simulation of stick-slip motion The simulation starts with the disc originally at rest, and the displacement and the spring torque increase linearly. When the spring torque equals the breakaway torque, the plate starts to slip and the friction decreases rapidly due to the Stribeck effect. In the event  46  spring torque decreases and the plate slows down. The friction torque increases because of Stribeck effect and the motion stops. The phenomenon then repeats itself. In Fig 3.13 shown are the position, the velocity and the friction torque. A highly irregular behavior of the friction torque around the region where the disc stops can be observed.  Friction torque  th (t) = Ax+Bu y =Cx+Du  w(t)  Displacement  State-Space Model for the System  Demux  ->  wO wO  Velocity  Friction Model Displacement waveform 0 07  A  0 06  new  7  /  0.03  1  fill  -f. —  J  /  0! -0M  / •-/  )  A  /! i  /  0 04  0.01  Velocity waveform  05  /  t:  73 IfgSBpii:.!:..  /  . ...]...1  1 i/  I  i . -cs J-  1i..  O  ;  I  cn  1  -I -1.5  VI 0;2::  0.4  -1  SB  O.B 06 Ti •ne On «SPC)  1  1.2  1  0.2  -2  0.4  0.6 0.8 Time [in sec)  1,2  Friction torque waveform  25 IT  t  15  cr  2  1  C  ^  !  2  ±  .  :  111111 -0 5  j  { /j  i  / i  /  /  i /  r  i 0B 0.8 Time fin serf  1  1  12  Fig 3.13 Displacement, velocity and friction torque waveforms for the stick-slip motion  47  The simulations have been carried out for different combinations of the driving speed, applied load, loading radius and the following conclusions have been drawn: •  The increasing applied load W, increases the stick-slip amplitude and reduces the frequency of vibrations (Fig 3.15).  •  The increase in the spring stiffness of the system k, resulted in the decrease of stick-slip amplitude and increase in frequency (Fig 3.15).  •  The increase in the speed of the driving disc COQ, resulted in decreased stick-slip amplitude and increased frequency of vibrations (Fig 3.16).  •  The increased radius of load application point Ri, caused the increase in stick-slip amplitude and reduction of frequency (Fig 3.16). The effect of the loading configurations viz. concentrated and distributed loading for  the same amount of total load has been assessed. This is achieved using either a single load at a given radial distance or dividing it into two equal loads at the same specified radial distance 90° apart or replacing it with four equal loads at 90° apart. Simulations have revealed no change in the behavior of the stick-slip vibrations for different loading configurations. This is quite logical as the contact pressure distribution in the vicinity of the load (the peak of the distribution) gets divided up into equal smaller distributions at different locations, but at the same radial distance. In other words, if all these different individual peaks were superimposed over one another, the resultant distribution would be the one caused by the single load as shown in Fig 3.14. This results in same value of the friction torque irrespective of the loading configuration.  (a) Distributed loading  (b) Concentrated loading  Fig 3.14 Contact pressure distribution for concentrated and distributed loading  48  Table 3.3 The simulation results for the variation of load and spring stiffness Spring stiffness Load (N/mm)  (N)  Angular displacement (rad) Amplitude Maximum Minimum (rad)  22.5 45 90 135 180 225 270 22.5 45 90 135 , ; , 180 225 270  k=3.0377  k=9.1864  0.0196 0.0334 0.0610 0.0880 0.1155 0.1430 0.1700 0.0068 • 0.0102 0.0172 0.0241 0.031 1 0.0380 0.0450  -0.0034 -0.0042 -0.0055 -0.0060 -0.0054 -0.0040 -0.0020 -0.0014 -0.0006 0.0007 0.0019 0.0032 0.0045 0.0060  o  "O  Q.  E  CO  Q.  2  O  '•5  CO  (Hz)  0.0230 0.0376 0.0665 0.0940 0.1209 0.1470 0.1720 • 0.0082 0.0108 0.0165 . . . , 0.0222 0.0279 0.0335 0.0390  Stick-slip amplitude plot c  Frequency  5.495 3.984 2.577 1.88 1.499 1.255 1.092 12.195 10.81 8.475 6.849 . 5.747 4.95 4.386  • k = 3.1 N/mm  0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00  A k = 9.2 N/mm  50  100  150  200  250  300  N o r m a l L o a d (in N)  Stick-slip frequency plot  • k = 3.1 N/mm  14  >.  u c d) cr ._ o> N A: X  Q.  10  8 6  In  4  o  2  CO  A k = 9.2 N/mm  12  0 50  100  150  200  250  300  N o r m a l L o a d (in N)  Fig 3.15 The simulation plots for the variation of the load and the spring stiffness  49  Table 3.4 The simulation results for the variation of the driving speed and loading radius Applied loac = 90 N ; Spring stiffness = 3.0377 N/mm Radial location (mm)  Disc Speed (rad/sec)  Rl= 40  ... .  Rl=59  Rl=80  Angular displacement (rad) Amplitude Maximum Minimum (rad)  0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5  0.0470 0.0455 0.0441 0.0432 0.0427 >d:o6'49 0.0610 . . 0.0578 0.0555 0.053.7 0.0835 0.0763 0.0709 0.0668 0.0632  -0.0100 -0.0075 -0.0054 -0.0041 -0.0034 -0.0110 -0.0055 -0.0015 0.0015 0.0037 -0.0085 -0.0005 0.0058 0.0106 0.0145  0.08 0.06  a.  0.04  <B  ui •  •  m  A  •  2  1.58 3.058 4.376 5.411 6.192 1.222 2.577 3.831 5 5.952 1.017 2.247 3.559 4.95 5.988  • RI = 40 mm A RI = 59 mm • RI = 80 mm  0.10  3 o. _ E -a  (H/)  0.0570 0.0530 0.0495 0.0473 0.0461 0.0759 0.0665 0.0593 0.0540 0.0500 0.0920 0.0768 0.0651 0.0562 0.0487  Stick-slip amplitude plot  •a  Frequency  A  •  n  J  I  t  0.4  0.5  0.02 0.00  0.1  D  0.2  0.3  0.6  D i s c S p e e d (in r a d / s e c )  • RI = 40 mm RI = 59 mm H RI = 80 mm  Stick-slip frequency plot  A c  =>.  6  g.  5  ^  4  U  -Q.  N 1  3  W o  (/> 0.1  0.2  0.3  0.4  0.5  0.6  D i s c S p e e d (in r a d / s e c )  Fig 3.16 The simulation plots for the variation of the driving speed and the loading radius  50  3.5  Discussion The major focus of stick-slip phenomenon for large contact areas with unevenly  distributed loads is on how the frictional condition of each elemental area affects the overall behavior of the disc. In other words, some of these small areas would have gone into stickslip mode i f they were individual isolated entities. Because of the fact that all of them are kinematically constrained on to the same disc, it is the cumulative frictional effect that decides the overall behavior of the disc. Since both the friction coefficient and the contact pressure occur non-linearly in the expression for friction torque, it is very difficult to derive a generalized condition for the occurrence of stick-slip at any location on the disc. First of all, it is to be kept in mind that the combination of the contact pressure and the linear speed at any point (with all other parameters being same) on the disc determine the possibility of the occurrence of stick-slip. For a given driving speed of the disc, the linear speed at any point on the disc is directly proportional to the radial distance from the center. As we move radially outwards from the point of load application, the contact pressure decreases whereas the linear speed increases. So, there is little chance that the infinitesimal areas in this region would be in stick-slip mode, unless the driving speed of the disc is very low. Similarly, as we move radially inwards from the load application point, both contact pressure and the linear speed decrease and based on these values, some points may tend to exhibit stick-slip whereas others may not. In the vicinity of the load application point, the contact pressure is relatively high whereas the linear speed is moderate and thus this region is most likely to be in stick-slip mode and it would overcome the effect of other points. If the contact pressure were more widely distributed, then probably every individual small area would have had a contribution of its own. But, since the contact pressure due to the applied load is very dominant in the immediate vicinity of the load application point, it is logical to conclude that the conditions at this location play a pivotal role in the occurrence of the stick-slip vibrations. Thus the combinations of the applied load, the driving speed and the radial distance at the load application point essentially determine the chances of occurrence of stick-slip vibrations.  51  C H A P T E R 4 EXPERIMENTS AND RESULTS 4.1  Friction test system A test rig has been designed and fabricated to re-construct the realistic loading  scenarios in the laboratory environment. A brief explanation of the apparatus has been provided earlier in Sec 2.2. The test rig consists primarily of a rotating disc driven by a variable speed DC motor (1/8 HP) and a slider disc attached to an elastic system as shown in Fig 2.1. The facing surfaces of the discs are lapped to provide good contact between the two. Normal loads can be applied using dead weights via pulleys at several load points. The position of these loading brackets can be moved towards or away from the center thereby changing the point of the load application. The rotational speed of the disc is controlled by a series of gears and pulleys with rubber belts. The latter, a crucial design aspect for friction test devices helps to isolate extraneous vibrations from the motor and gearbox. A cantilever beam made of 17-4 Chromium-Nickel stainless steel (T-630), heat-treated to condition H 900 was used as a spring to represent the stiffness of the system. Friction forces between the slider and the disc surface are monitored by a series of transducers. A n accelerometer and a L V D T are used to measure the dynamic transient motion patterns of the slider. The speed of the driving disc is measured using a tachogenerator. The signal conditioning is carried out using a charge amplifier and various signals are then acquired using a data acquisition card to analyze the data on a desktop PC with the help of LabVIEW software [43]. 4.2  Instrumentation The current research in part employs the instrumentation and measurement techniques  used by Ko et al. [33] with computer interface using Data acquisition card and LabVIEW. •  Displacement - Sangamoto L V D T to measure the displacement of the spring. DCR100 with linear stroke of ±100 mm.  •  Acceleration - KISTLER accelerometer. Type 8628B50 Trimount PiezoBeam accelerometer with acceleration range of ± 50g.  •  Speed - Pittman tachogenerator for measuring the speed of the driving disc.  52  9414H204 with 24 VDC, 100 CPR. •  Dual Mode Charge Amplifier - KISTLER Model 5004  •  Data Acquisition card - by National Instruments. PCI-6024E low-cost multifunction I/O board for OC1200 kS/s, 12 bit, 16SE/8DI inputs, 2 AO, 8DIO.  • 4.3  Data analysis software - LabVIEW for Win N T by National Instruments. Signal conditioning The signal acquired using accelerometer is conditioned using a dual mode amplifier  with a gain of 10. The L V D T is powered using a constant D C supply of 12V and is calibrated to read the output in linear distance. Since the signal contained high frequency noise, it was passed through a first order analog filter of 50 Hz, constructed using a simple R C circuit. Similarly the output signal from tachogenerator containing high frequency noise was passed through a first order analog filter of 50 Hz. The signal is accordingly calibrated to read the speed of the driving disc. The calibration data has been provided in A P P E N D I X IV. These conditioned signals are then read using the data acquisition system consisting of B N C board and data acquisition card and further analysis is carried out on desktop PC using LabVIEW. 4.4  LabVIEW interface for data acquisition and analysis An application has been developed in LabVIEW data analysis software [43] for signal  processing and analysis. The LabVIEW interface consists of two parts: •  Front Panel: It displays the various graphs and the conditioned signals acquired such as acceleration, speed, force, etc. It is also a mode of user interaction with the program as it allows the user inputs like calibration constants, user operated buttons to run or stop the program.  •  Block Diagram: This contains the graphical program for reading, processing and analyzing the acquired signals. The signals acquired from various channels of data acquisition card are first read,  then multiplied with calibration constants to convert into appropriate units and finally stored in different arrays. Then these signals are digitally filtered using a median filter to obtain a  53  cleaner signal. The velocity is obtained by differentiating the displacement signal. This was preferred over integrating an acceleration signal, as the noise to signal ratio was small. The mean value of the speed of the driving disc is displayed at every instant the program runs. A provision has also been made to store the data in a spreadsheet file whenever the user prompts to stop the data acquisition. A separate program has been written in LabVIEW to read the stored data from the spreadsheet file in the following order; time, acceleration, velocity, displacement, spring force and driving speed. The program calculates the maximum and minimum spring force, average amplitude of the spring force and the average frequency and the average speed of the driving disc. 4.5  Surface preparation Careful surface preparation is a very crucial aspect in any friction experiment to  obtain reliable results. The contacting surfaces of the plates were lapped in order to get a reasonably good contact between the two. But, since the contact wasn't uniform, the plates were set for self-lapping with a lapping compound (jeweler's rouge) between them. The surfaces were cleaned with isopropyl alcohol and the contact was then found to be more uniform. When these plates were tested for dry stick-slip test, several circumferencial grooves were found to be formed on the surfaces. On measuring the surface conditions using a surface profilometer, a roughness of about 35 pm was revealed as shown in the Fig 4.1. 30  S 20 O  -20 L 0  6  16  18  Fig 4.1 Initial surface roughness testing results  54  In the plot shown, the stylus of the profilometer was moved radially over a length of 16 mm near the edge of the disc and the groove formed was recorded. This means that the dry friction tests after lapping could be detrimental to the surface and hence a small amount of lubrication was applied between the surfaces to avoid wear and surface damage. Thus, careful surface preparation is very important between the tests. Even after careful surface preparation, inconsistency in the readings was encountered i.e. the amplitude and frequency of the force was not consistent over one rotation of the disc. It was thought that this might be due to the unevenness in the flatness of the surface. When a dial gauge was placed on the surface of the disc and rotated slowly, 4/1000 of an inch th  displacement was observed. However, when the disc was rotated by 180° and again the dial gauge readings were taken, it was concluded that the inconsistency in readings was not due to the out of flatness of the surface, rather it was due to the misalignment of the driving shaft and bearings. This should not affect the friction behavior as long as the two surfaces are in good contact with each other. These variations in the readings were repeatable over every cycle (cyclical), and are quite common in many tribological experiments. Hence it was decided that the entire disc would be divided into four localized regions, which would give fairly uniform readings, and the experiments would just concentrate on these areas. Later, these areas were examined under the optical microscope to view the tribological surface texture. The stick-slip for large contact areas (or the phenomenon for any contacts for that matter) depends on the following interfacial conditions: •  How long the surfaces have been in contact.  •  The total time the system has been running (this determines the amount of debris formed).  •  The condition of the lubricant in the interface, such as age, contamination, temperature etc.  •  The oxide film formed between the surfaces.  •  Temperature of operation (although the temperature changes are very small). Due to the above-mentioned reasons, it is difficult to obtain consistent results in such  kinds of friction experiments for large contact areas. Hence, comparison between one test and a test conducted a month later becomes difficult. Also, it is difficult to establish the same  55  result for the same given physical parameters over a period of time because the interfacial conditions won't remain the same. However, the trend of the results is repeatable. Hence the current research is aimed at how the various parameters like loading, stiffness, speed etc affect the stick-slip phenomenon for large contact areas. Nevertheless, attempts were made to keep the experimental condition as identical as possible. Terminology used in the tests: Multi-point  Loading: Four equal loads (W/4) applied at 90° intervals at the same radial distance.  Two-point Loading: Two equal loads (W/2) applied adjacent to each other at 90 intervals at the same radial distance. Single-point Loading: A single load (W) applied at a given radial distance, where, W is the total applied normal load. It was kept in mind that the plate itself weighs 22.5 N and this load, due to the mass of the plate is distributed uniformly over the entire area. Whereas the distribution of the externally applied load is more concentrated at the load application point and decreases rapidly away from this point as depicted by finite element model for the contact pressure distribution explained in CHAPTER 2. 4.6  Pre-tests The initial sets of tests were conducted to assess the effect of normal load on the  friction force behavior with lubrication. For these sets of tests, fairly large quantity of motor oil (Tellus-32) was used for lubrication to ensure hydrodynamic or full-fluid regime. At higher loads, the fluid was found to seep out from the contact interface. The speed of the rotation of the driving plate was slowly increased from zero to 15.5 rpm (1.65 rad/sec) and then decreased slowly back to zero. A sample experimental display is shown in Fig 4.2, where the variation of spring force with time and the speed of the driving plate has been shown for increasing and decreasing directions of the speed. No stick-slip vibration was observed and the results were fairly consistent and provided good repeatability.  56  Spring Force (N)!v*.Tiroe^el  S0377  | N/mm  J2lTT0~[  Sprog Force iU] vs Speed (fpmll  M  [0.0301 "| rad  Fig 4.2 The front panel for the pre-test data analysis 4.6.1  Test procedure Pre-tests have been conducted to understand the behavior of the lubricated contacts  taking the following aspects into consideration: •  The single point normal load was varied from 22.5 N to 225 N in steps of 45 N using dead weights.  •  The tests were conducted for clockwise and counterclockwise direction of rotation of the driving plate.  •  The effect of loading configuration was tested using multi-point, two-point and singlepoint loading to analyze the effect of concentrated and distributed loads at the contact interface for the total load of 90 N .  •  The effect of spring stiffness on the behavior of the system was tested using two springs having stiffness of 3.0377 N/mm and 9.1864 N/mm.  4.6.2  Results of the pre-tests The following conclusions have been drawn from the results of the experiments.  •  The variation of spring force with the speed of the driving plate is approximately linear and the spring force increases with the increase in the speed of rotation of the driving plate. This behavior is expected as the friction force increases with the increase in velocity in the hydrodynamic lubrication regime.  57  •  A small hysteresis was observed in the spring force variation between increasing (from rest to maximum speed) and decreasing (from maximum speed back to rest) direction of the driving plate speed as shown in Fig 4.2. This could be due to the reversal in the direction of various (frictional) forces involved and can be explained in part by considering the frictional memory as explained in Sec 3.1.3.  •  As it can be seen from Table 4.1, the spring force experienced due to multi-point loading was found to be slightly lower than single-point loading for most cases. This shows that the friction force experienced would be higher in case of the concentrated loading as compared to the distributed loading.  •  The spring force was found to increase with the increase in total normal load as shown in Fig 4.3 and Fig 4.4 although this was not linear.  •  The change in radial distance of the load application point did not appear to have any considerable effect on the spring force. However, only two radial positions could be realized at 84 mm and 59 mm due to space limitations of the instrumentation.  Table 4.1 Pre-test data for different loading configurations Loading type  Spring force (N) Spring 1 Spring 2  Deflection (rad) Spring 1 Spring 2  Slope (N/rpm) Spring 1 Spring 2  "btal load = 90 N ; Loading Radius = 84 mm; Clockwise rotation 5.6 5.5 0.0182 Multipoint 0.0061 0.3884 5.7 5.6 Two-point 0.0185 0.0062 0.3915 5.75 5.55 0.0187 Single-point 0.0061 0.3956 Tota load = 90 N ; Loading Rac ius = 84 mm; Countereloe cwise rotation 5.6 6.05 0.0182 Multipoint 0.0067 0.3912 5.65 6 0.0184 Two-point 0.0066 0.3954 5.7 6.1 0.0185 Single-point 0.0067 0.3998 "btal load = 90 N ; Loading Radius = 59 mm; Clockwise rotation 5.85 5.7 0.0190 Multipoint 0.0063 0.3806 5.9 5.75 0.0192 Two-point 0.0063 0.3854 5.95 5.8 Single-point 0.0193 0.0064 0.3889 Tota load = 90 N ; Loading Rac ius = 59 mm; Countercloccwise rotation 6.4 6.25 0.0208 Multipoint 0.0069 0.3831 6.45 6.1 Two-point 0.0210 0.0067 0.3854 6.5 6.15 0.0211 Single-point 0.0068 0.3866  0.3786 0.3845 0.3803 0.4022 0.3989 0.4075 0,3602 0.3644 0.3673 0.3789 0.3726 0.3795  58  Table 4.2 Pretest data for the variation of the normal load at radius - 84 mm  Load(\)  22.5 45 90 135 180 225 22.5 45 90 135 180 225  Spring force (N) Deflection (rad) Slope (N/rpm) Spring 1 Spring 2 Spring 1 Spring 2 Spring 1 Spring 2 Radial distance = 84 mm; Clockwise rotation; 3.9 4.4 0.0127 0.0048 0.262 0.2695 4.3 4.6 0.0140 0.0051 0.284 0.3113 5.9 5.7 0.0192 0.0063 0.3906 0.3826 6.5 7.2 0.0211 0.0079 0.4054 0.5033 9.5 8.65 0.0309 0.0095 0.5975 0.5482 12.3 11.6 0.0400 0.0128 0.7762 0.7124 Radial distance = 84 mm; Counterclockwise rotation; 4.4 4.2 0.0143 0.0046 0.262 0.2695 4.5 4.6 0.0146 0.0051 0.284 0.3113 5.95 6 0.0193 0.0066 0.3906 0.3826 6.8 6.9 0.0221 0.0076 0.4054 0.5033 9.2 9.1 0.0299 0.0100 0.5975 0.5482 12.1 11.8 0.0384 0.0133 0.7762 0.7124  Clockwise rotation of the disc 14 12 10  o u.  • Spring 1 • Spring 2  ii  6 4  CL  1  8  n  2 0 50  100  150  200  250  Total Load (in N)  Counterclockwise rotation of the disc  A Spring 1 • Spring 2  100  150  200  250  Total Load (in N)  Fig 4.3 The variation of spring force with load at radius = 84 mm  59  Table 4.3 Pretest data for the variation of the normal load at radius = 59 mm I.oad(N) Spring force (\) Deflection (rad) Slope (N/rpm) Sprinu 1 Spring 2 Spring 1 Spring 2 Spring 1 Spring 2 Radial distance = 59 mm; Clockwise rotation; 3.95 4.35 22.5 0.0128 0.0048 0.262 0.2695 4.25 4.5 45 0.0138 0.0050 0.284 0.3113 5.9 5.75 0.0192 90 0.0063 0.3906 0.3826 135 6.35 7.05 0.0206 0.0078 0.4054 0.5033 9.4 8.5 180 0.0306 0.0094 0.5975 0.5482 12.1 225 0.0393 0.0127 11.5 0.7762 0.7124 Radial distance = 59 mm; Counterclockwise rotation; 0.0141 22.5 4.35 4.25 0.0047 0.262 0.2695 4.4 4.5 0.0143 45 0.0050 0.284 0.3113 6 6.1 0.0195 90 0.0067 0.3906 0.3826 6.7 6.9 135 0.0218 0.0076 0.4054 0.5033 9.1 9 0.0296 180 0.0099 0.5975 0.5482 12 0.0387 225 0.0132 0.7762 11.9 0.7124  Clockwise rotation of the disc  A Spring 1 • Spring 2  100  150  250  Total Load (in N)  Counterclockwise rotation of the disc  A Spring 1 • Spring 2  100  150  250  Total Load (in N)  Fig 4.4 The variation of spring force with load at radius = 59 mm  4.7  Stick-slip tests The actual stick-slip tests followed the pre-tests. The second set of experiments was  conducted under lightly lubricated conditions (boundary lubrication regime) using motor oil (Tellus-32). Under these conditions, stick-slip vibrations were observed in the low speed regime. The amplitude of vibrations decreased with increasing speed and eventually disappeared. 4.7.1  Surface conditions for experiments  The surface conditions for the tests conducted were as follows: The surfaces were lapped with 1500-grit paper (Grade A0814W) and cleaned with varsol followed by isopropyl alcohol. A small amount of lubricant (Motor oil, Tellus-32) was applied to ensure a boundary lubrication regime between the surfaces. They were run for a period of about 5000 revolutions to allow a layer of wear debris to be formed at the interface.  Pre-tests had  revealed that this would lead to a more stable (settled) interfacial condition. 4.7.2  Experimental procedure  The stick-slip tests were conducted taking the following aspects into consideration: •  The single point normal load was varied from 22.5 N to 225 N in steps of 45 N using dead weights to determine the effect of normal load on stick-slip phenomenon. These tests were conducted at a specified constant driving speed and were repeated for three different speeds.  •  The effect of loading configuration was tested using multi-point, two-point and singlepoint loading to analyze the effect of concentrated and distributed loads at the contact interface for the total load of 90 N , 180 N and 270 N .  •  The tests were repeated for a range of speeds varying from 0.07 rad/sec to 0.5 rad/sec in steps of 0.1 rad/sec.  •  Two springs having stiffness 3.0377 N/mm and 9.1864 N/mm were used to investigate the effect of spring stiffness on the behavior of the system.  •  Finally, the radial distance of the point of load application was changed by moving the loading brackets radially from the center of the plate. Only two radial positions (i.e. 84  61  mm and 59 mm) could be realized due to the constraints caused by the instrumentation and physical configuration of the loading brackets. The front panel in LabVIEW that is used for the data analysis of the stick-slip results is shown in Fig 4.5. The front panel displays the acceleration, the displacement, the velocity and the spring force waveforms in addition to the average driving speed. Different calculated parameters such as maximum and minimum displacement, average stick-slip amplitude and frequency, average spring force are also displayed. JAngular^1spT15nSrt (reBp^Srna|seG)]  lAtiquiar,acceleraton-(rad/secfl v s Time (sec)|  ;  r 30 OO-!  Ir-  10.000.00•10 00-  -2000-1 -3000 IZ~  : -50.00-1 -60 0D ' -70.00-1 0.00  •  J 1.500 2.000. 2500 3.000'3500 4.C  ~3\n 0000 0.500 1.000 1.500 2.D00 2.500 3.000 3.500 4 000 4.500 5 000 5 500 B000|  5.000 8.500 6.000!  Spnnq Force (N) vs Time  lAnqular Velocity (rad/sec) vs Time(sec)j • OJflfl-l  •=."1  -1.400-j -1.6QQ -1 800  !  H2000' P  t  ' j  <M»FolSloSFi»^2jiD0 Z5'oo-.3.do6'i3.5'o?'4.o'o0 Toialilioadl  1 _aao.oi:-jiJ N  4.500 5dOO?5,5lpO-6.00Oji';  JMax Displacement! 101036 | rad  30377  3  "Sack-slip Amplitude] SSoWlrad Average  Spring stiftnessj I  , 0000 0.500 1.000 1 500 2 000-2 500,3 000 _3500 J000 :4 500  N/mm  rad  Forceamplitudej  R7.2102 I N  1  •A^rage"DiscSpeed| 10.1063 [ rad/sec Average Frequency! lO 9132;-1 Hz  Fig 4.5 The front panel for the stick-slip test data analysis  62  Table 4.4 The stick slip data for the variation of the normal load at 0.22 rad/sec 1 mal Load  Angular Displacement (rad) Maximum  (N>  Minimum  Amplitude  Spring Force  (rad)  Frequency  (N)  Radial distance =84mm; Disc speed = 0.22 rad/sec; Spring 1 = 3.0377 N / m m 22.5  0.0247  -0.0072  0.027  7.5924  4.0635  45  0.0398  -0.0083  0.0431  12.1317  2.9259  90  0.0648  -0.0020  0.0628  16.5825  2.2323  135  0.1154  -0.0259  0.1004  33.9573  1.2367  180  0.1397  -0.0176  0.1332  37.5794  1.0918  225  0.1608  -0.0130  0.1519  22.5  0.0058  -0.0010  0.0039  3.2552  11.2567  45  0.0090  0.0005  0.0053  4.6061  10.0855  90  0.0192  -0.0012  0.0157  13.3885  5.7782  135  0.0305  -0.0030  0.0278  23.7569  3.7397  180  0.0423  -0.0033  0.0405  34.6136  2.7569  225  0.0593  -0.0132  0.066  56.2622  1.856  42.7666 Radial distance =84mm; Disc speed = 0.22 rad/sec; Spring 2 = 9.1864 N / m m  0.972  S t i c k - s l i p A m p l i t u d e plot  • Spring 1 A Spring 2  100  150  200  250  Normal Load (in N)  S t i c k - s l i p F r e q u e n c y plot  12 10  • Spring 1  3  o  e  A Spring 2  2  50  100  150  200  250  Normal Load (in N)  Fig 4.6 Plot of variation of the normal load at 0.22 rad/sec  63  Table 4.5 The stick slip data for the variation of the normal load at 0.28 rad/sec Amplitude Spring Force Angular Displacement (rad) Frequency Maximum Minimum (rad) (II/) (N> Radial distance =84mm; Disc speed = 0.28 rad/sec; Spring 1 = 3.0377 N/mm 22.5 0.0270 -0.0095 0.0287 8.0753 5.1238 0.0394 45 -0.0060 0.0387 10.912 4.1409 90 0.0661 -0.0017 0.0576 16.214 3.1227 135 0.1039 -0.0151 0.0825 30.2989 2.0053 180 0.1251 -0.0032 0.1159 32.6719 1.8548 0.1625 225 0.0016 0.1425 40.1319 1.4744 Radial distance =84mm; Disc speed = 0.28 rad/sec; Spring 2 = 9.1864 N/mm 22.5 0.0080 -0.0027 0.0063 5.4963 10.9758 45 0.0107 -0.0011 0.0078 6.6212 10.3966 90 0.0217 -0.0022 0.0183 15.5747 6.8269 135 0.0293 -0.0020 0.0253 21.5609 5.6547 0.0404 180 -0.0025 0.0382 27.4366 4.7619 225 0.0569 -0.0127 0.0621 52.8757 2.8752  Total Load <\»  Stick-slip Amplitude plot  0.16 TJ 0.14 ra .= 0.12 0.1  • Spring 1 A Spring 2  a 0.08 E ™ 0.06 |  0.04  m 002 0  50  100  150  200  250  Normal Load (in N)  Stick-slip Frequency plot  12 10  • Spring 1  3  CO  A Spring 2  50  100  150  200  Normal Load (in N)  Fig 4.7 Plot of variation of the normal load at 0.28 rad/sec  250  Table 4.6 The stick slip data for the variation of the normal load at 0.33 rad/sec Total Load  Amplitude Angular Displacement (rud) Spring Force Frequency Maximum Minimum (rad) (Hz) (N) Radial distance =84mm; Disc speed = 0.33 rad/sec; Spring 1 = 3.0377 N/mm 22.5 45 0.0385 -0.0043 0.0358 10.0808 4.9491 90 0.0635 -0.0021 0.0527 14.8349 3.8286 135 0.1074 -0.0138 0.0825 28.8736 2.3675 180 0.1258 -0.0024 0.1129 31.851 2.1568 0.1564 225 0.0055 0.1343 37.8297 1.8575 Radial distance =84mm; Disc speed = 0.33 rad/sec; Spring 2 = 9.1864 N/mm 22.5 0.0115 45 -0.0010 0.0073 6.2311 11.0064 90 0.0208 -0.0020 0.0154 13.0716 8.0328 135 0.0294 -0.0009 0.0243 20.6425 6.2784 0.0383 180 -0.0026 0.0359 26.2857 5.381 225 0.0598 -0.0087 0.0596 50.773 3.2644  (N)  Stick-slip Amplitude plot  0.16 =0 0.14 co °0.1 0.08 12  •a U E  • Spring 1 A Spring 2  ™ 0.06 0.04 S 0.02  1  50  100  150  200  250  Normal Load (in N)  Stick-slip Frequency plot  12 10  (Spring 1 i Spring 2  55  2  50  100  150  200  250  Normal Load (in N)  Fig 4.8 Plot of variation of the normal load at 0.33 rad/sec  65  Table 4.7 Stick slip data for W=90 N ; R=84 mm; k=3.0377 N/mm Loading I ype Disc Speed Angular displ accmciit (rad) Amplitude Spring Force Frequency (rad* sec I Maximum Minimum (rad) (11/) <N> Two-point 0.1043 0.0624 -0.0004 0.0576 17.7796 1.3287 Two-point 0.208 0.0590 0.0030 0.0495 15.2272 2.6311 Two-point 0.3032 0.0569 0.0052 0.0452 13.9243 3.7917 Two-point 0.4031 0.0570 0.0063 0.0434 13.3757 4.6652 Multi-point 0.1046 0.0572 0.0063 0.048 14.8106 1.5488 Multi-point 0.2044 0.0573 0.0067 0.0451 13.8716 2.8151 Multi-point 0.3092 0.0570 0.0077 0.0435 " 13.41.63 3.8382 Multi-point 0.4 ' No stick-slip observed Single-point 0.1037 0.0708 -0.0175 0.0688 21.2089 0.9936 Single-point 0.2034 0.0699 -0.0107 0.0613 18.8735 1.9628 Single-point 0.3068 0.0639 -0.0028 0.0534 16.4538 3.4682 Single-point 0.4043 0.0543 0.0097 0.0392 12.0793 5.0753 Stick-slip Amplitude plot ad)  0.08 •  k> c  tud  01  0.07  • Two-point loading  0.06 0.05  a. 0.04  -  0.03 ra a.  -  E  A Multi-point loading  To 0.02  Ji  9Single-point loading  ~ 0.01 CO  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Stick-slip Frequency plot > Two-point loading  & Multi-point loading  i Single-point loading 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Fig 4.9 Stick-slip plot for W=90 N ; R=84 mm; k=3.0377 N/mm  66  Table 4.8 Stick slip data for W=90 N ; R=84 mm; k=9.1864 N/mm Loading Type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency (rad/sec) Maximum Minimum (rad) (Mz) <N) 0.0734 0.0252 Two-point -0.0069 0.0278 25.8995 1.6659 Two-point 0.1066 0.0209 -0.0024 0.0198 18.3923 2.9519 0.0184 Two-point 0.2047 0.0017 0.0114 10.6481 6.9671 Two-point 0.3066 0.0163 0.0033 0.0083 7.6705 10.0089 Multi-point 0.0734 0.021.1 -0.0027 0:0202 18.7937 2.1591 0.1052 0.0199 Multi-point -0.0012 ' 0.0186 17.2478 . 3.1421 Multi-point 0.2062 . 0.0164 0.0039 0.007 6.5445 .8.9257 0.3 Multi-point No stick-slip observed Single-point 0.0758 0.0256 -0.0087 0.0288 26.8248 1.487 Single-point 0.1059 0.0224 -0.0032 0.0213 19.833 2.6499 Single-point 0.2068 0.0200 -0.0013 0.0167 15.5297 5.7409 Single-point 0.3067 0.0163 0.0040 0.0074 6.8229 10.3173  rad)  Stick-slip Amplitude plot  c —o TJ 3  :  Q.  E ra Q. W  Jc o  CO  0.035 0.03  • Two-point loading  0.025 0.02  & Multi-point loading  0.015 0.01 0.005 0-  I Single-point loading  0.05  0.1  0.15  0.2  0.25  0.3  0.35  Disc Speed (in rad/sec)  Stick-slip Frequency plot  12 • Two-point loading  10  A  8 A Multi-point loading  6 4 0  1 Single-point loading  4  2 0.05  0.1  0.15  0.2  0.25  0.3  0.35  Disc Speed (in rad/sec)  Fig 4.10 Stick-slip plot for W=90 N ; R=84 mm; k=9.1864 N/mm  67  able 4.9 Stick slip data for W=180 N ; R=84 mm; k=3.0377 N/mm Loading Type Disc Speed (rad/sec)  Angular displacement (rad) Amplitude Maximum  Minimum  (rad)  Spring Force Frequency <N)  Two-point  0.1083  0.1096  0.0154  0.0884  27.2102  Two-point  0.2044  0.1079  0.0206  0.0807  24.8767  1.7217  Two-point  0.3065  0.1043  0.0221  0.0727  22.3944  2.5623  0.0996  0.0344  0.0574  17.6673  3.7717  0.1009  0.0296  0.0674  • 20.7587  1.1059  . 0.0334  0.0581  17.9167  2.2731  0.0400  0.049  15.0573  3.5212 0.6181  Two-point  0.4061  Multi-point  0.1.027  Multi-point  0.2073  0.0965  Multi-point  0.3066  0.0921  Multi-point  0.4  ,  ,  0.9132  N o stick-slip observed  Single-point  0.106  0.1370  -0.0093  0.1215  37.4302  Single-point  0.2071  0.1263  0.0016  0.1001  30.8428  1.3365  Single-point  0.3042  0.1197  0.0116  0.0874  26.9261  2.0714  Single-point  0.4109  0.1174  0.0173  0.0737  22.7043  2.8841  S t i c k - s l i p A m p l i t u d e plot 0.14  • Two-point loading =•  0.1  co  •a |  0.08  A Multi-point loading  Q.  i  0 0 6  f 0.04  1 Single-point loading  •£ 0.02 CO  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  D i s c S p e e d (in r a d / s e c )  S t i c k - s l i p F r e q u e n c y plot 4  • Two-point loading  £ 3.5 c — 3 >. £ 2.5 a> 2  A  3-  A Multi-point loading  2 «  1  £  0.5  • Single-point loading  CO  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  D i s c S p e e d (in r a d / s e c )  Fig 4.11 Stick-slip plot for W=180 N ; R=84 mm; k=3.0377 N/mm  68  Table 4.10 Stick slip data for W = 1 8 0 N ; R=84 m m ; k=9.1864 N / m m Loading Type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency (rad/sec) Maximum Minimum (rad) (ll/> (M Two-point 0.0779 0.0502 -0.0105 0.0524 48.7775 0.9651 Two-point 0.1073 0.0461 -0.0069 0.0464 43.2477 1.4359 Two-point 0.2077 0.0388 -0.0018 0.0348 32.3745 3.3192 Two-point 0.3068 0.0364 0.0021 0.0273 25.4419 5.1112 0.0741 Multi-point 0.0352 , 0.0027 0.0284 26.5044 1.643 Multi-point 0.1052 • , 0.0348 0.0043 0.0256 ' 23.9589 2.4444 Multi-point 0.208 0.0321 0.0052 0.0203 18.9035 4.9994 Multi-point 0.3 N o stick-slip observed Single-point 0.0723 0.0498 -0.0192 0.0555 51.6829 0.7782 Single-point 0.1048 0.0511 -0.0146 0.051 47.5147 1.1668 Single-point 0.2051 0.0440 -0.0060 0.0426 39.669 2.6554 Single-point 0.3067 0.0042 0.0042 0.025 23.28 5.3263 Stick-slip Amplitude plot 0.06 •o  C O 0.05 ku- 0.04 <  • Two-point loading  _c  ~u 3  Q. 0.03  A Multi-point loading  E cs  Q. 0.02 Jc  O 55  0.01  1 Single-point loading 0.1  0.05  0.15  0.2  0.25  0.3  0.35  Disc Speed (in rad/sec)  Stick-slip Frequency plot • Two-point loading  k Multi-point loading A  A  0.05  i 0.1  1 Single-point loading 0.15  0.2  0.25  0.3  0.35  Disc Speed (in rad/sec)  F i g 4.12 Stick-slip plot for W = 1 8 0 N ; R=84 m m ; k=9.1864 N / m m  69  Table 4.11 Stick slip data for W=270 N ; R=84 mm; k=3.0377 N/mm Loading Type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency ^(ra'd/ see I Maximum Minimum (radl (II/) (N) Two-point 0.1074 0.1581 0.0416 0.1058 32.5944 0.7418 Two-point 0.2072 0.1544 0.0445 0.0965 29.7377 1.4758 Two-point 0.3051 0.1413 0.0602 0.0738 22.6831 2.5038 Two-point 0.4076 0.1358 0.0613 0.0659 20.3021 3.3292 Two-point 0.5039 0.1419 0.0708 0.0524 16.1305 4.4961 0.1049 Multi-point .0.1407 0.0583 0.0802 .24.7431 0.9657 Multi-point 0.2066 0.1376 0.0649 0.0716 22.0299 1.9029 Multi-point 0.3052 0.1361 . 0.0668 0.0653 20.1117 2.7041 Multi-point 0.408 0.1269 0.0735 0.0466 . 14.3797 4.2447 Multi-point 0.5 No stick-slip observed 0.1084 Single-point 0.1857 0.0093 0.1639 50.449 0.4867 Single-point 0.1774 0.2053 0.0181 0.149 45.8813 0.9822 Single-point 0.3081 0.1727 0.0270 0.1313 40.5081 1.5407 Single-point 0.4117 0.1625 0.0323 0.1195 36.8371 2.1061 Single-point 0.506 0.1600 0.0420 0.1078 33.193 2.6958 Stick-slip Amplitude plot 0.18 to  -i  0.16  • Two-point loading  _c 0.14 o 0.12 3  0.1 •  & Multi-point loading  o.  E 0.08 to Q.  0.06 •  to  A  . i 0.04 • o tn 0.02 0  I Single-point loading 0.1  0.2  0.3  0.4  0.5  0.6  Disc Speed (in rad/sec)  Stick-slip Frequency plot 5 £  4.5  c  4  • Two-point loading  A  > 3.5  u  5 |  a *  co  ii  3  A Multi-point loading  2.5  2  A  1.5  I Single-point loading  1  5> 0.5 -I 0 0.1  0.2  0.3  0.4  0.5  0.6  Disc Speed (in rad/sec)  Fig 4.13 Stick-slip plot for W=270 N ; R=84 mm; k=3.0377 N/mm  70  Table 4.12 Stick slip data for W=270 N ; R=84 mm; k=9.1864 N/mm Loading Typo Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency (rad/sec) Maximum Minimum (rad) <N) Two-point 0.0736 0.0627 -0.0031 0.0581 54.0756 0.8045 Two-point 0.1045 0.0591 -0.0032 0.0571 53.0651 1.154 Two-point 0.2021 0.0592 -0.0023 0.0555 51.7224 2.1221 Two-point 0.3076 0.0575 -0.0004 0.052 48.3325 3.1391 Two-point 0.407 0.0567 0.0025 0.0494 45.9344 4.0688 Multi-point 0.075 •0.0537 0.0026 0.0463 \ 43.2658 1.0311 Multi-point 0.1071 • 0.0543 0.0070 0.0404 37.6198 1.6061 Multi-point 0.2067 <• 0.0452 0.0136 0.0266 24.7278 3.9164 Multi-point 0.3086 0.0427 0.0176 0.0216 20.1611 6.0472 0.4 Multi-point No stick-slip observed Single-point 0.0761 0.0625 -0.0083 0.0657 61.1949 0.7382 0.1062 Single-point 0.0632 -0.0087 0.0634 58.9427 1.0392 Single-point 0.206 0.0642 -0.0055 0.0618 57.5572 1.9575 Single-point 0.3068 0.0597 -0.0026 0.0548 51.0154 3.0495 Single-point 0.4123 0.0577 -0.0005 0.0506 47.1779 4.0586 Stick-slip Amplitude plot 0.07  • Two-point loading =• 0.05 o  u  I  0.04  A Multi-point loading  Q.  | 0.03 re  f  0.02  I Single-point loading  0.01 cn  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Stick-slip Frequency plot 7 -! N I  • Two-point loading  6  oi 4  A Multi-point loading  O"  I 3 a 1 2  = CO  A  4  1  I Single-point loading  •  0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Fig 4.14 Stick-slip plot for W=270 N ; R=84 mm; k=9.1864 N/mm  71  able 4.13 Stick slip data for W=90 N; R=59 mm; k=3.0377 N/mm Loading Type Disc Speed \ngular displacement (rad) Amplitude Spring Force Frequency (rad/sec) Maximum Minimum (rad) (II/) (N) Two-point 0.1028 0.0683 -0.0037 0.0646 19.8419 1.1522 Two-point 0.2075 0.0591 -0.0017 0.0546 16.8033 2.4046 0.3052 Two-point 0.0569 -0.0005 0.05 15.3918 3.4753 Two-point 0.4068 0.0579 0.0014 0.0469 14.4249 4.2972 Multi-point 0.1025 0.0640 -0.0068 0.063 19.3842 1.2392 0.2047 Multi-point 0.0536 0.0030 0.0476 14.6535 2.7242 Multi-point 0.3043 0.0520 0.0057 0.0416 12.8434 3.9782 0.4 Multi-point No,stick-slip observed Single-point 0.1045 0.0678 -0.0123 0.0683 21.0372 1.0584 Single-point 0.2047 0.0664 -0.0080 0.0659 20.313 2.0763 Single-point 0.3069 0.0648 -0.0058 0.0624 19.2034 2.9311 0.4047 Single-point 0.0655 -0.0028 0.0605 18.6157 3.6558  Stick-slip Amplitude plot  • Two-point loading  \ Multi-point loading  I Single-point loading 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Stick-slip Frequency plot  N" c  &  5 4.5 4  • Two-point loading  35  i Multi-point loading  cr 2.5 o £ 2 1.5 1 0.5 0  I Single-point loading 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Fig 4.15 Stick-slip plot for W=90 N; R=59 mm; k=3.0377 N/mm  72  able 4.14 Stick slip data for W=90 N ; R=59 mm; k=9.1864 N / m m Loading Type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency Minimum (rad/sec) Maximum (rad) (Hz) (N) 0.072 0 0250 Two-point -0.0050 0.0256 23.6353 1.6966 0.1019 0.0227 -0.0044 Two-point 0.0235 21.9507 2.4758 0.0204 Two-point 0.2036 -0.0019 0.0175 16.2607 5.2934 0.0202 Two-point 0.3053 -0.0011 0.0158 14.7829 7.1847 Multi-point 0.0756 : 0:0220 -0:0.041 0.0229 ' 21.2487 1.9619 Multi-point 0.1056 0.0220 -0.0026 0.0195 ... 18.2284 2.9157 0.0189 Multi-point 0.2071 -0.0002 0.0144 13.3852 6.1058 Multi-point 0.3 No stick-slip observed '. . Single-point 0.0709 0.0250 -0.0071 0.0286 26.5571 1.418 0.0234 Single-point 0.1097 -0.0053 0.024 22.3098 2.4747 Single-point 0.2009 0.0222 -0.0026 0.0193 18.0953 4.462 0.3022 Single-point 0.0210 -0.0018 0.0166 15.5079 6.5847 Stick-slip Amplitude plot 0.035  • Two-point loading  A Multi-point loading  I Single-point loading 0.15  0.2  0.3  0.35  Disc Speed (in rad/sec)  Stick-slip Frequency plot • Two-point loading  £ 7 C  A  — 6 >. 8 5 o>  A Multi-point loading  s I co  A mm  3  2  I Single-point loading  •5 1  in  0.05  0.1  0.15  0.2  0.25  0.3  0.35  Disc Speed (in rad/sec)  Fig 4.16 Stick-slip plot for W=90 N ; R=59 mm; k=9.1864 N / m m  73  Table 4.15 Stick slip data for W=180 N ; R=59 mm; k=3.0377 N/mm Loading Type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency (rad/sec) .Maximum Minimum (rad) (II/) (N) 0.1054 Two-point 0.0918 0.0152 0.0738 22.6878 1.0354 Two-point 0.2053 0.0904 0.0192 0.0659 20.3308 2.0545 Two-point 0.3039 0.0892 0.0217 0.0617 18.6503 2.9868 Two-point 0.4093 0.0822 0.0285 0.0497 15.3099 4.0658 Multi-point , 0.1036 0.0911 0.0145 0.073 22.4688 1.0736 . 0.2074 Multi-point 0.0899 .0.0195 0.0655 20.1814 2.1139 0.305 Multi-point 0.0820 0.0284 0.0501 ' 15.4238 3.5036 0.4 Multi-point , No stick-slip observed 0.1032 Single-point 0.1079 0.0008 0.096 29.5811 0.7597 0.2034 Single-point 0.0968 0.0157 0.0782 24.1132 1.729 Single-point 0.3026 0.0889 0.0228 0.0609 18.7447 2.8993 0.4002 Single-point 0.0819 0.0282 0.0487 14.986 4.1381 Stick-slip Amplitude plot 0.12  •a  re  0.1  • Two-point loading  •  c o •a 0.08  m  3  Q.  E  m  0.06  A Multi-point loading  ••  (0  a. 0.04 Ji To o  0.02  1 Single-point loading  5) 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Stick-slip Frequency plot 4.5  • Two-point loading  A  si. 3.5 3 2.5 cr  2 S"  A Multi-point loading  0k  2 1-5  i Single-point loading  1  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Fig 4.17 Stick-slip plot for W=180 N ; R=59 mm; k=3.0377 N/mm  74  Table 4.16 Stick slip data for W=180 N ; R=59 mm; k=9.1864 N/mm Loading Type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency (rad/sec) Maximum Minimum (rad) (M/) (N) Two-point 0.0424 0.0719 -0.0134 0.0503 46.8041 0.9194 Two-point 0.1065 0.0402 -0.0057 0.0397 36.9414 1.686 Two-point 0.2048 0.0346 0.0021 0.0274 25.6815 3.8238 Two-point 0.3037 0.0295 0.0055 0.0168 15.682 6.8665 Multi-point 0.0751 0.0314 0.0000 0.0275 25.6286 1.6876 Multi-point 0.1073 0.0299 • 0.0028 0.0239 22.2177 2.56 Multi-point 0.205 0.0284 0.0042 0.0195 18.1432 4.9228 Multi-point .0.3 No stick-slip observed Single-point 0.0738 0.0451 -0.0148 0.0541 50.4167 0.8609 Single-point 0.1044 0.0436 -0.0122 0.0487 45.3505 1.3286 Single-point 0.2057 0.0342 -0.0033 0.0311 28.9659 3.3514 Single-point 0.3056 0.0301 0.0028 0.0221 20.6373 5.7779 Stick-slip Amplitude plot 0.06  Tj" C O 0.05 c o 0.04 3 TJ Q. 0 . 0 3 E C O Q. 0 . 0 2 Jc0.01 O c7 >  • Two-point loading  A Multi-point loading  i Single-point loading  0 0.05  0.1  0.15  0.2  0.25  0.3  0.35  Disc Speed (in rad/sec)  Stick-slip Frequency plot 8  ?  c  7  • Two-point loading  ••=• 6 >.  g o 5  A  A Multi-point loading  & 4 o  S  3  A  I Single-point loading  $ 1 cn  0 0.05  0.1  0.15  0.2  0.25  0.3  0.35  Disc Speed (in rad/sec)  Fig 4.18 Stick-slip plot for W=l 80 N ; R=59 mm; k=9.1864 N/mm  75  Table 4.17 Stick slip data for W=270 N ; R=59 mm; k=3.0377 N/mm Loading type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency Maximum "(rad/sec) Minimum (rad) (M/) (N) Two-point 0.1074 0.1421 0.0256 0.0983 30.2364 0.7689 Two-point 0.2072 0.1384 0.0285 0.0917 28.2354 1.5467 Two-point 0.3051 0.1253 0.0442 0.0701 21.5642 2.6789 Two-point 0.4076 0.1198 0.0453 0.0652 20.0564 3.4578 Two-point 0.5039 0.1259 0.0548 0.0517 15.8947 4.5879 Multi-point 0.1049 0.1247 .0.0423 • 0.078 . 23.9854 - 1.0589 Multi-point 0.2066 0.1216 . 0.0489 0.0711 ' 21.8745 2.1045 Multi-point 0.3052 0.1201 0.0508 0.0636 19.5648 , 2.9364 Multi-point 0.408 0.1109 0.0575 0.0455 14.0012 ' - 4.3657 Multi-point 0.5 No stick-slip observed 0.1084 Single-point 0.1697 -0.0067 0.154 47.3698 0.4572 Single-point 0.2053 0.1614 0.0021 0.1406 43.2654 0.9269 Single-point 0.3081 0.1567 0.0110 0.1268 39.0124 1.4878 Single-point 0.4117 0.1465 0.0163 0.1144 35.2147 2.0171 Single-point 0.506 0.1440 0.0260 0.1055 32.4587 2.6381  Stick-slip Amplitude plot 0.18 0.16  • Two-point loading  0.14 0.12 0.1  0.08  A Multi-point loading  A  0.06 0.04  I Single-point loading  0.02  0 0.1  0.2  0.3  0.4  0.5  0.6  Disc Speed (in rad/sec)  Stick-slip Frequency plot 5  N -  .E  4.5  A  • Two-point loading  4  ^ 3.5  o  3 | 2.5 £ 2 S  A Multi-point loading  A 55  I Single-point loading  0.5 o 0.1  0.2  0.3  0.4  0.5  0.6  Disc Speed (in rad/sec)  Fig 4.19 Stick-slip plot for W=270 N ; R=59 mm; k=3.0377 N/mm  76  Table 4.18 Stick slip data for W=270 N ; R=59 mm; k=9.1864 N/mm Loading Type Disc Speed Angular displacement (rad) Amplitude Spring Force Frequency (rad/sec) Maximum Minimum (rad) (II/) (N) Two-point 0.0736 0.0577 -0.0081 0.0538 50.2345 0.9457 Two-point 0.1045 0.0541 -0.0082 0.0532 49.7554 1.257 Two-point 0.2021 0.0542 -0.0073 0.0523 48.8621 2.3247 Two-point 0.3076 0.0525 -0.0054 0.0504 47.1024 3.2478 Two-point 0.407 0.0517 -0.0025 0.0462 43.2178 4.1654 Multi-point 0.075 0.0487 -0.0024 0.0461 43.1245 1.0457 Multi-point 0.1071 0.0493 0.0020 0.0399 37.2548 . 1.699 Multi-point 0.2067 0.0402 0.0086 0.0258 24.1234 '4.0157 Multi-point 0.3086 0.0377 0.0126 0.0213 . 19.8953 6:2147 0.4 Multi-point • ' No stick-slip observed Single-point 0.0761 0.0575 -0.0133 0.0622 58.1245 0.8795 0.1062 Single-point 0.0582 -0.0137 0.0602 56.2357 1.2547 Single-point 0.206 0.0592 -0.0105 0.058 54.2398 2.1547 Single-point 0.3068 0.0547 -0.0076 0.0528 49.3478 3.2569 Single-point 0.4123 0.0527 -0.0055 0.0485 45.3214 4.234 :  Stick-slip Amplitude plot 0.07 ra 0.06  • Two-point loading  =• 0.05 o TJ | 0.04  A  A  a  i  A Multi-point loading  0 0 3  f  002  I Single-point loading  •H 0.01 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Stick-slip Frequency plot 7 -  N  X  C  • Two-point loading  6  > 5  o c  <J> 4  i Multi-point loading  c3r  <u 3  LL  Q.  0> 2 _i o  55  I Single-point loading  1 0 0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Fig 4.20 Stick-slip plot for W=270 N ; R=59 mm; k=9.1864 N/mm  77  4.7.3  Experimental results The stick-slip tests for large contact areas yielded the following general results and  trends: •  Increasing the total load increases the stick-slip amplitude and decreases the frequency, although not linearly, as depicted by Fig 4.6, Fig 4.7 and Fig 4.8.  •  The stick-slip amplitude was the largest for single-point loading followed by two-point loading and then multi-point loading for the same total load as seen from Fig 4.9 to Fig 4.20. Similarly, the stick-slip frequency was the lowest for single-point loading followed by two-point loading and then multi-point loading for the same total load.  •  The stick-slip amplitude was found to decrease with the increase in the speed of the driving disc as seen from Fig 4.9 to Fig 4.20. Critical speed at which the stick-slip vibrations disappear was found to be higher for single-point loading as compared to twopoint and multi-point loading.  •  Increasing the spring stiffness resulted in decreased stick-slip amplitude and increased frequency as seen from Fig 4.6, Fig 4.7 and Fig 4.8.  4.8  Surface  measurements  Different areas on the contacting surfaces were examined with an optical microscope to view the tribological surface texture. Also, the surface roughness readings were taken using profilometer to obtain numerical values of the surface properties. The surface measurements were recorded at the following locations along the radius of the circular plates: •  load application points (i.e. at radial distances of 84 mm and 59 mm)  •  near the edge of the disc (radial distance of 100 mm)  •  close to center (radial distance of 35 mm) The area captured using the microscope at these locations was 5 mm X 5 mm. This  was carried out for both upper and lower plates and along four radial locations 90° apart. The pictures clearly showed the groove that was formed initially near the edge of the disc and the surface texture was more or less uniform at other locations for both upper and lower plates. Photos taken at one such location at four different radial distances are shown in Fig 4.21.  78  (c) 59 mm  (d) 35 mm  Fig 4.21 Surface texture at different radial locations  Similarly, the surface roughness readings were taken at these marked locations using a stylus-type measurement instrument, Surftest SV-502 with a resolution of 0.005 pm. The stylus of the profilometer was moved radially at these four locations with a sampling length of 8 mm with a speed of 1.0 mm/sec and the data was recorded. The groove towards the edge can clearly be seen as shown in Fig 4.22 and hence the surface roughness measured was more towards the edge. The roughness value towards the center was comparatively less and the results concluded reasonably good contact between the two surfaces. The surface roughness values for the different locations of the plates have been listed in Table 4.19. The groove that was formed initially should not affect the friction characteristics as this results only in a negligible reduction in the contact area.  79  Fig 4.22 Surface roughness measurements at different locations of the plate  Radial Distance (mm) 100 84 59 35 75-100 50-75 25-50  fop Plate Regions (um) Quadrants 1 II III 0.315 0.221 0.533 0.069 0.081 0.081 0.09 0.061 0.066 0.0624 0.067 0.065 0.2 0.2 0.19 0.14 0.13 0.13 0.13 0.11 0.13  IV 0.247 0.08 0.056 0.071 0.37 0.14 0.15  Bottom Plate Regions (pm) Mirror image quadrants I 11 III IV 0.467 0.355 0.409 0.387 0.101 0.081 0.084 0.09 0.09 0.08 0.07 0.07 0.09 0.09 0.08 0.09 0.54 0.5 0.61 0.57 0.17 0.17 0.16 0.19 0.11 0.13 0.11 0.13  Table 4.19 Surface roughness values for the different locations of the plates 4.9  Analysis  of the  results  A direct comparison between the results of simulation and that of the experiments has been made and the effect of various parameters on stick-slip vibrations and the probable factors affecting the above stated behavior are explained below: •  Total normal load: It was found that an increase in total normal load would increase the stick-slip amplitude and decrease the frequency as expected, although not linearly. This is 80  due to the reason that the friction torque at the interface of contact increases with the increase in the total normal load. A comparison of simulation and experimental results has been made in Fig 4.23 and are found to reasonably agree with each other. The loads higher than 270 N could not be realized experimentally as they tended to cause scoring on the contact surfaces. Stick-slip amplitude plot (for k = 3.1 N/mm; Wo = 0.2 rad/sec)  • Simulation , Experimental  0.16 0.14 0) 0.12 •D 3 0.10 0.08 2 0.06 a Tn 0.04 • o 0.02 «-< CO 0.00  Isi  50  100  150  200  250  N o r m a l L o a d (in N)  Fig 4.23 Comparison of results for the variation in normal load Loading configuration: The stick-slip amplitude was the largest for single-point loading followed by two-point loading and then multi-point loading for the same total load. Similarly, the stick-slip frequency was the lowest for single-point loading followed by two-point loading and then multi-point loading for the same total load. This implies that for this to be possible, there has to be a higher value of frictional torque for the former case as compared to the other two cases. Physically, one single load leads to stronger interlocking whereas distributing the same load over several locations will lead to weaker interlockings thereby a greater chance of break-away. The simulations using proposed integrated LuGre model could not predict this behavior simply because these effects were not included in the model. Speed of the driving plate: A comparison between the simulation and experiments made in Fig 4.24 shows that the increase in the speed of the driving plate reduces the stick-slip amplitude. The stick-slip vibrations disappear after a certain critical speed has been reached. Physically, this occurs because with the increase in driving speed, the rate of applied torque increases, which lowers the static friction at break-away. Critical speed at which the stick-slip vibrations disappear is higher for single-point loading as compared to  81  two-point and multi-point loading. This can be explained by the same reasoning that there is a higher tendency for stick-slip with concentrated loading as compared to distributed loading for the same total load. The accurate estimation of this critical speed could not be achieved due to the reason that the stick-slip vibrations are not entirely uniform over one complete rotation of the plate. Hence the speed at which the stick-slip vibrations disappear is not a good indication of the critical speed. Stick-slip amplitude plot (for W = 90N; RI = ip amp litude (in rad) 1  to  A Experimental  •  0.08  CO  • Simulation  59 mm)  0.07  ^A  0.06  •A  0.05 0.04 0.03 0.02 0.01 0.00 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45  Disc Speed (in rad/sec)  Fig 4.24 Comparison of results for the variation in driving speed •  Stiffness in the system: The increase in spring stiffness of the system reduces the stickslip amplitude and increases the frequency as seen from both simulation and experiments. Physically, this is due to the fact that with a stiffer system the rate of increase of applied force increases and the breakaway force depends on the rate of increase of force. Thus the increased elastic stiffness of the system reduces the tendency for the stick-slip amplitude and might even completely eliminate the stick-slip. However, it is not always desirable to increase the stiffness of the system, as there lies a limit up to which this can be achieved.  •  Radial distance of loading points: As predicted by simulations in Fig 3.16 (page 50),  the increased radial distance of the loading points should ideally increase the stick-slip amplitude. This is due to the fact that with the increase in radial distance, the friction torque at the load application point increases and consequently there is higher tendency for stick-slip vibrations to occur. However, for the tests conducted at two radial distances, the interfacial conditions were not the same and therefore, they cannot be directly compared. Only two radial positions (i.e. 84 mm and 59 mm) could be realized due to the constraints caused by the instrumentation and physical configuration of the loading 82  brackets. Since very little could be concluded from the variation in radial distance of loading points, no further tests were attempted under this purview. 4.10  Discussion  A direct comparison between the results of simulation and experiments in Fig 4.23 and Fig 4.24 reveals that they are in good agreement with each other for the variation in parameters. However, the simulation using integrated LuGre model could not demonstrate the effect of loading configurations on the behavior of the stick-slip vibrations. Experiments have shown that there is a higher tendency for stick-slip vibrations to occur under concentrated loading as opposed to the distributed loading for the same total load. But the simulation results make no distinction between these loading cases. This can be attributed to the greater inter-locking of the asperities at the load application area as a result of higher applied load. The proposed integrated LuGre model does not take these effects into consideration. A further understanding of this phenomenon is necessary i f these effects are to be accounted for.  83  C H A P T E R 5 CONCLUSIONS AND FUTURE DIRECTIVES 5.1  Summary  The current research was aimed at gaining a better understanding of the stick-slip phenomenon for large contact areas involving uneven distributed loads. The proposed approach involved accomplishing this objective by: •  Observing and formulating the effects of stick-slip in the presence of other system components (i.e. the drive train) so that realistic interactions are included.  •  Achieving  a better understanding  of the phenomenon  based  on experimental  observations. •  Developing a mathematical representation that can be used in dynamic modeling and control of stick-slip phenomenon under realistic conditions.  Each of these criteria has been addressed in the pursuit of obtaining an insight into the phenomenon under such circumstances. The contact pressure distribution at the interface between the plates was obtained by means of a non-linear finite element model using A N S Y S . The analysis showed that the contact pressure is more dominant in the vicinity of the point of load application and decreased rapidly away from this point. Based on the results obtained, a mathematical model was developed and the contact pressure distribution was found to be normal around the load application point. The friction torque acting between two plates was then calculated by integrating the elemental torque over the whole area of the plate. Different friction models for the localized contacts were presented and the importance of friction parameters was discussed. The LuGre model for the localized contacts was integrated for large contact areas and the dynamic behavior of the model was examined. The simulations were carried out using Simulink for different combinations of the system parameters. The phenomenon was well simulated to assess effect of various system parameters on the behavior of the stick-slip vibrations. A test rig was fabricated to re-construct the realistic situations in the laboratory environment. The instrumentation, data acquisition, signal processing and analysis using  84  LabVIEW were subsequently carried out. Experiments were conducted and the surface conditions were defined for each set of experiments. The effect of various parameters, on the stick-slip vibrations was investigated. The surface roughness measurement results revealed a reasonably uniform contact between the two surfaces at the interface. A n insight into the phenomenon was achieved by analyzing the behavior of the system under the effect of various parameters. 5.2  Conclusions  The experiments revealed that the stick-slip vibrations for large contact areas were influenced by the loading conditions. It was concluded from both simulation as well as the experiments for large contact areas with uneven distributed loads that, the stick-slip vibrations are more dominant at: •  Higher applied loads;  •  Lower spring stiffness;  •  Lower driving speeds;  •  Larger loading radius. More importantly, the experiments showed that there is a higher tendency for the  stick-slip vibrations to occur under concentrated loading as compared to the distributed loading for the same total load. But, the simulation results using the integral-model made no distinction between these cases. Although this can be attributed to the greater inter-locking of the asperities at the load application area, this could not be included in the model due to the lack of the knowledge about how this affects the friction torque. The experimental results were found to be in reasonably good agreement with the behavior predicted by simulation using proposed integrated LuGre model. Hence the proposed integral-model can be readily used for simulation, design and control applications for large contact areas. Although limited in the range of contact pressure, material and surface preparation, the present results give insight into some previously uncertain aspects of stick-slip friction behavior for large contact areas. 5.3  Scope for future work  The double integration function for friction torque calculation has been carried out using an s-function written in Matlab. The current algorithm makes use of numerical 85  Gaussian integration rule. As a result of this the calculation and the simulation are slowed down and hence not very efficient for control applications. This limitation could be overcome by writing a c-mex file instead of s-function, thereby making the formulation more suitable for real-time control applications. The current integrated model has could not simulate the observed experimental behavior for different loading configurations, simply because indentation effects of the asperities were not modeled.  With the inclusion of this factor, it would be possible to  incorporate the plowing effect of the asperities due to applied load as a result of which the magnitude of the contact pressure would be higher in these regions. A further investigation in this direction is necessary so that the modeling would be complete. The proposed integrated LuGre model has simulated the stick-slip vibrations for the system under study. But this is not yet generalized for user specified applications. However, it can be extended to obtain the generalized model in terms of load distribution, loading configuration, etc. Thus it can be made available for easy implementation for specific applications.  86  REFERENCES [I]  Armstrong-Helouvry B. (1990). Stick-slip arising from Stribeck friction. Proc. 1990 Inter. Conf. on Robotics and Automation,  Cincinnaati, IEEE, pp. 1377-1382.  [2]  Armstrong-Helouvry B. (1991). Control of Machines with Friction, Academic Publishers  Boston : Kluwer  [3]  Armstrong-Helouvry B. (1992). Frictional lag and stick-slip. Proc. 1992 Inter. Conf. on Robotics and Automation, Nice, IEEE, pp. 1448-1453.  [4]  Armstrong-Helouvry  B. (1993). Stick-slip and control in low-speed motion, IEEE  transactions on Automatic Control, 38(10), pp. 1483-1496.  [5]  Armstrong-Helouvry B., P. Dupont (1993). Friction modeling for controls, and Compensation Techniques for Servos with friction, Proc. 1993 American Control Conference, AACC, San Francisco, pp. 1905-1915.  [6]  Armstrong-Helouvry B . (1994). Dynamic friction in control of robots.  IEEE  transactions on Automatic Control, pp. 1202-1207.  [7]  Armstrong-Helouvry B., P. Dupont, C. Canudas (1994). A survey of Models, Analysis tools and Compensation Methods for the Control of Machines with Friction. Automatical n7 (1994), pp. 1083-1138.  [8]  Aronov V., A . F. D'souza, S. Kalpakjian, I. Shareef (1984). Interactions among Friction, Wear, and System Stiffness, Transactions of ASME, Part 1,2 and 3. Vol. 106, pp. 54-69.  [9]  Blau P. J. (1996). Friction Science and Technology. New York : M . Dekker  [10] Bowden, F. P. and D. Tabor (1956). The Friction and the Lubrication Wiley and Sons , N Y .  of solids. John  [II] Brockley C. A . , Cameron R. and Potter A . F. (1967). Friction induced vibration, J. of Lubrication Technology, 89(2), pp. 101-108. [12] Brockley C. A . and Davis H . R. (1968). The time dependence of static friction, J. of Lubrication Technology, 90(1), pp. 35-41. [13] Canudas de Wit C , H . Olsson, K . J. Astrom and P. Lischinsky (1995). A new model for control of systems with friction. IEEE Trans, on Automatic Control, 40(3); pp. 419-425. [14] Canudas de Wit C , P. Lischinsky (1997). Adaptive friction compensation with partially known dynamic friction model, Int. Journal of Adaptive Control and Signal  Processing,  Vol. 11, pp. 65-80. [15] Capone G., V . D'Augostino, S. D. Valle and D. Guida (1993). Influence of the variation between static and kinetic friction on stick-slip stability. Wear, 161, pp. 121-126. 87  [16] Dahl P.R.(1968). A solid friction Corporation, E l Segundo, C A .  model, TOR-158(3107-18), The Aerospace  [17] Derjaguin B. V., V . E. Push and D. M . Tolstoi (1956). A theory of stick-slip sliding of solids,/, of Technical Physics (Moscow), 6.  [18] De Silva, C.W. (1989). Control Sensors and Actuators. Prentice Hall. [19] Dupont P. E. (1994). Avoiding stick-slip through PD control, IEEE Trans, on Automatic Control, 39(5), pp. 1059-1097. [20] Gafvert M . (1997). Comparisons of two dynamic friction models. Proc. of the 1997 Int. Conf. on Control Applications. Hartford, IEEE, pp. 386-391. [21] Gao C , D. K. Wilsdorf and D. D. Makel (1993). Fundamentals of stick-slip, Wear, 162164 (1993) pp. 1139-1149. [22] Gao C , D. K . Wilsdorf and D. D. Makel (1994). The dynamic analysis of stick-slip motion, Wear, 173 (1994) pp. 1-12. [23] Greenwood, J. A . and J. H . Williamson (1966). The contact of two nominally flat rough surfaces. Proc. of the Royal Society (London), Vol. A230, pp. 531-548. [24] Haessig D. A. and B. Friedland (1991). On the modeling and simulation of friction, J. of Dynamic Systems, Measurement and Control, 113(3), 354-362.  [25] Hess D. P. and A . Soom (1990). Friction at a lubricated line contact operating at oscillating sliding velocities, J. ofTribology, 112(1), pp. 147-152. [26] Hess D. P. and A . Soom (1991). Normal vibrations and friction under harmonic loads, /. ofTribology, 113(1), pp. 80-92. [27] Howe P.G., D. P. Benton, and I. E. Puddington (1955). London-Van der Waals attractive forces between glass surfaces, Canadian Journal of Chemistry, Vol. 33, pp. 1375. [28] Johannes V . I., M . A . Green and C. A . Brockley (1973). The role of the rate of application of tangential force in determining the static friction coefficient. Wear 24, pp. 381-385. [29] Johnson, K. L. (1987). Contact Mechanics. Cambridge University Press, Cambridge. [30] Karnopp D. (1985). Computer simulation of stick-slip friction in mechanical dynamic systems, ASME J. of Dynamic Systems, Measurement and Control, 107(1), pp. 100-103.  [31] Kelly R., J. Llemas (1999). Determination of viscous and Coulomb friction using velocity responses to torque ramp inputs, Proc. of IEEE Int. Conference on Robotics and Automation, pp 1740-1745.  88  [32] K i m J-H., H - K . Chae, J-Y. Jeon and S-W. Lee (1996). Identification and control of systems with friction using Accelerated Evolutionary Programming. IEEE-ControlSystems-Magazine, v 16 n 4 Aug 1996, pp. 38-47 [33] Ko P. L. and C. A . Brockley (1970). The measurement of friction and friction induced vibration, ASME J. of Lubrication Technology, 92(4), pp. 543-549. [34] Ko P. L. and C. A . Brockley (1970). Quasi-harmonic friction induced vibration, ASME J. of Lubrication  Technology, Oct. 1970, pp. 550-556.  [35] Lance B.J. and F. Sadeghi (1993). The normal approach and stick-slip phenomena at the interface of two rough bodies. [36] Ludema, K . C. and R. G. Bayer (1991). Tribological Designers. A S T M .  modeling for  Mechanical  [37] Rabinowicz, E. (1965). Friction and wear of materials. John Wiley and Sons , N Y [38] Richardson R. S. H . and H . Nolle (1976). Surface friction under time-dependent loads, Wear, 37, pp. 87-101 [39] Seidl D. R., T. L. Reineking and R. D. Lorenz (1992). Use of Neural Networks to identify and compensate for friction in precision, position controlled mechanisms. [40] Singer, I. L. and H . M . Pollock (1992). Fundamentals of friction: microscopic processes. Kluwer Academic. [41] Timoshenko S. and J.N. Goodier (1951). Theory of Elasticity. Company.  macroscopic and  McGraw-Hill Book  [42] Tseng J. G. and J. A . Wickert (1998). Nonconservative Stability of a Friction Loaded Disc. Transactions of ASME. Vol. 120, pp. 922-929. [43] LabVIEW User Manual, National Instruments Inc. [44] Matlab and Simulink User manual, Mathworks Inc. [45] A N S Y S 5.5 Element reference, Modeling and Structural analysis guide, A N S Y S Inc.  89  APPENDIX I  A N S Y S ELEMENTS  A brief description of various elements used in modeling the discs using A N S Y S in CHAPTER 2 is as follows [45]:  SOLID95  3-D  20-Node  Structural Solid  SOLID95 is a higher order version of the 3-D 8-node solid element. It can tolerate irregular shapes without as much loss of accuracy. SOLID95 elements have compatible displacement shapes and are well suited to model curved boundaries. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y and z directions. The element may have any spatial orientation. The element has plasticity, large deflection and quadratic shape functions which offer greater accuracy in displacement and stress approximation capabilities as compared to their 8-noded counterparts. The geometry, nodal locations, coordinate system and the stress output for this element are shown in Fig A-1.1.  Fig A - I . l Element geometry for SOLID95  90  TARGE170  3-D  Target Segment  TARGE170 is used to represent various 3-D "target" surfaces for the associated contact elements. The contact elements themselves overlay the solid elements describing the boundary of a deformable body and are potentially in contact with the target surface, defined by T A R G E 170. This target surface is discretized by a set of target segment elements (TARGE 170) contact and is paired with its associated contact surface via a shared real constant set. One can impose any translational or rotational displacement on the target segment element. It is also possible to impose forces and moments on target elements. The geometry and the node locations are shown in Fig A-I.2.  Y  Fig A-I.2 Element geometry for T A R G E 170  For rigid target surfaces, these elements can easily model complex target shapes. For flexible targets, these elements will overlay the solid elements describing the boundary of the deformable target body. The target surface is modeled through a set of target segments, typically, several target segments comprise one target surface. The target surface can either be rigid or deformable. For modeling rigid-flexible contact, the rigid surface must be represented by a target surface. For flexible-flexible contact, one of the deformable surfaces must be overlayed by a target surface.  91  For any target surface definition, the node ordering of the target segment element is critical for proper contact. The nodes must be ordered so that the outward normal to the target surface is defined by the right hand rule. Therefore, for general target segments, the outward normal by the right hand rule is consistent to the external normal.  CONTA174  3-D  8-Node  Surface-to-Surface C o n t a c t  CONTA174 is used to represent contact and sliding between 3-D "target" surfaces (TARGE 170) and a deformable surface, defined by this element. This element has three degrees of freedom at each node: translations in nodal x, y and z directions. This element is located on the surface of 3-D solid or shell elements with midside nodes (eg. SOLID95). It has the same geometric characteristics as the solid or shell element face with which it is connected. Contact occurs when the element surface penetrates one of the target segment elements (TARGE 170) on a specified target surface. The geometry and the node locations are shown in Fig A-I.3. >Associated Target Surfaces  Fig A-I.3 Element geometry for CONTA174 The element is defined by eight nodes. It can degenerate to a six-node element depending on the shape of the underlying solid or shell elements. The positive normal is  92  given by the right-hand rule going around the nodes of the element and is identical to the external normal direction of the underlying solid or shell element surface. The target element must always be on its outward normal direction. The target and associated contact surfaces are identified via a shared real constant set. This real constant set includes all real constants for both target and contact elements.  93  A P P E N D I X II ! Batch  file  A N S Y S BATCH FILE  f o r t h e ANSYS model f o r the  contact  pressure  ! Defining W = 90 Ro = 100 Ri = 8 R l = 59 tl = 6 t 2 = 32  t h e g e o m e t r y and t h e l o a d ! A p p l i e d L o a d ( i n N) ! O u t e r r a d i u s o f t h e p l a t e ( i n mm) ! I n n e r r a d i u s o f t h e p l a t e ( i n mm) ! R a d i u s o f t h e l o a d i n g p o i n t ( i n mm) ! T h i c k n e s s o f t h e t o p p l a t e ( i n mm) ! T h i c k n e s s o f t h e b o t t o m p l a t e ( i n mm)  ! Defining /prep7  the  element types  et,1,82 et,2,95 et,3,170,1 et,4,174 keyopt,4,12,1  Element Element Contact Contact  f o r the  distribution  plates  type f o r top p l a t e type f o r bottom p l a t e element type f o r t a r g e t element type f o r top p l a t e  r,1,,,,,le-3 ! Defining  material  properties  f o r the  plates  uimp,1,ex,nuxy,dens,210e3,0.3,7.8e-6 uimp,1,mu,,,0.192 ! Meshing the p l a t e s along csys, 0 k,101,0,0,0 k, 1 0 2 , 0 , 0 , t l k,103,0,0,-t2 numstr,line,101 1,102,101 1,101,103 lesize,101,1.5,,,1/2 lesize,102,8,,,2 numstr,line, 1  the  ! G e n e r a t i n g t h e c y l i n d e r by cyl4,0,0,Ro,0,R1,45 cyl4,0,0,Ro,45,Rl,90 cyl4,0,0,Rl,0,Ri,45 cyl4,0,0,R1,45,Ri,90 aglue,all  thickness  g l u i n g the  four  quarters  ! D e f i n i n g the f o r c e a p p l i c a t i o n p o i n t csys,1 ForceKP=KP(Rl,45,0) kscon,ForceKP,1,0 ! Defining  the  smart, m e s h i n g a t t r i b u t e s  94  lesize,2,5,,,1/5 lesize,19,5,,,5 lesize,3,5,,,1/5 lesize,18,5,,,5 type, 1 amesh,all ! Extruding the plates along the thickness csys, 0 agen,2,all,,,0,0,-t2 asel,s,loc,z,0 vdrag,all,,,,,,101 asel,s,loc,z,-t2 vdrag,all,,,,,,102 aclear,all numcmp,node numcmp,elem csys, 1 nrotat,all ! Forming t h e contact csys, 0 vsel,s,loc,z,-t2,0 aslv,s asel,r,loc,z,0 nsla,s,1 type,3 real, 1 esurf  between  the plates using  the contact  elements  v s e l - , s, l o c , z , 0 , t l aslv,s asel,r,loc,z,0 nsla,s,1 type,4 real, 1 esurf allsel finish /solu ! D e f i n i n g t h e degree of freedoms csys, 1 nsel,s,loc,z,-t2 d, a l l , u z , 0 nsel,s,loc,x,Ri d,all,ux,0 d,all,uy,0 nsel,s,loc,y,0 nsel,a,loc,y,90 d,all,uy,0 allsel  f o r nodes  95  ! A p p l y i n g t h e d i s t r i b u t e d l o a d o v e r the l i n e ForceN=NODE(Rl,45,tl) f,ForceN,f z,-W/4 Nl=NODE(Rl-l,45,tl) f,N1,fz,-W/4 N2=NODE(Rl+l,45,tl) f,N2,fz,-W/4 N3=N0DE(Rl-2,45,tl) f,N3,fz,-W/8 N4=N0DE(Rl+2,45,tl) f,N4,fz,-W/8 I A p p l y i n g t h e g r a v i t y l o a d due t o t h e s e l f - w e i g h t acel,0,0,9.81 ! Non-linear solution ANTYPE,0 NLGEOM,1 NROPT,AUTO, , LUMPM,0 EQSLV, , ,0, PREC,0 PIVCHECK,1 SSTIF PSTRES TOFFST,0, OUTRES,ALL,2, TIME,1 AUTOTS,-1 NSUBST,10, , ,1 KBC, 0  options  96  A P P E N D I X III function  MATLAB PROGRAM  [sys,xO,str,ts]  = sfun_lugre(t,x,u,flag,friction,load)  % % % % % %  SFUN_LUGRE M - f i l e S - f u n c t i o n t o c a l c u l a t e t h e t o t a l f r i c t i o n t o r q u e by i n t e g r a t i n g t h e elemental torque over the whole area of t h e p l a t e The i n p u t s t o t h e f u n c t i o n c o n t a i n two p a r a m e t e r s e t s o f f r i c t i o n and l o a d c o n t a i n i n g : f r i c t i o n = [muO;muc;muv;vs] parameters d e f i n i n g f r i c t i o n curve load = [W;Rl;config;order] parameters of loading d e t a i l s  % %  The g e n e r a l 2 p i Ro I  -'6  %  equation  of m - f i l e S-function  syntax i s :  I  | | {mu (wr, r ) *P ( r , t h e t a ) *r' 2 } d r d t h e t a N  %  0 Ri  % % %  The o u t p u t v e c t o r c o n t a i n s t h e i n t e g r a t e d S t r i b e c k t o r q u e a n d t h e v i s c o u s f r i c t i o n torque. Which a r e l a t e r used t o c a l c u l a t e t h e T o t a l F r i c t i o n T o r q u e g e n e r a t e d a t t h e i n t e r f a c e o f t h e two p l a t e s  %  Copyright  switch  (c) 1 9 9 9 - 2 0 0 0 by The M u d l a g i r i B a l e r i ,  All. Rights  Reserved  flag,  % % % % % % % % % 15 % % % % % % % %  % Initialization % %%%%%%%%%%%%%%%%%% % I n i t i a l i z e t h e s t a t e s , sample times, c a s e 0, [sys,xO,str,ts]=mdlInitializeSizes;  and s t a t e o r d e r i n g  %%%%%%%%%%% % Outputs % %%%%%%%%%%% % Return t h e outputs of the S - f u n c t i o n b l o c k , c a s e 3, sys=mdlOutputs(t,x,u,friction,load); % %%%%%%%%%%%%%%%%%% % Unharidled f l a g s % g. a a Q. a o, a Q. a. 9. a u. q. a t> 9. a. a cu •o t> r> 'o -6 -6 a -b '6 o o 'o -& -6 0 ^> '& -o o  % T h e r e a r e no t e r m i n a t i o n t a s k s ( f l a g = 9 ) t o b e h a n d l e d . % A l s o , t h e r e a r e no c o n t i n u o u s o r d i s c r e t e s t a t e s , % s o f l a g s 1,2, a n d 4 a r e n o t u s e d , s o r e t u r n a n e m p t y u % matrix c a s e { 1, 2, 4, 9 } sys= [ ] ;  % %%%%%%%%%%%%%%%%%%% % Unexpected f l a g s % %%%%%%%%%%%%%%%%%%%% % R e t u r n an e r r o r message f o r unhandlecl otherwise  flag  values,  strings,  error(['Unhand!ed  flag =  , num2str(flag)]);  end  % mdllnitializeSizes % Return the s i z e s , i n i t i a l function.  function  % % % % % %  c o n d i t i o n s , and sample times  f o r t h e S-  [sys,xO,str,ts]=mdlInitializeSizes  c a l l simsizes f o r a sizes structure, f i l l sizes array.  i t i n andconvert  i t toa  Mote t h a t i n t h i s example, t h e v a l u e s a r e h a r d coded. This i s not a recommended p r a c t i c e a s t h e c h a r a c t e r i s t i c s o f t h e b l o c k a r e t y p i c a l l y defined by the S-function parameters.  sizes = simsizes; sizes.NumContStates sizes.NumDiscStates sizes.NumOutputs sizes.Numlnputs sizes.DirFeedthrough sizes.NumSampleTimes  = = = = = =  0; 0; 2; 1; 1; 1;  % a t l e a s t one sample time  i s needed  sys = s i m s i z e s ( s i z e s ) ;  ialize  the i n i t i a l  conditions  • ; % % s t r i s a l w a y s a n empty m a t r i x 'o  str  = [] ;  % initialize  the a r r a y o f sample  times  'o  ts  =[0  0];  return; % end m d l l n i t i a l i z e S i z e s Q. Tl  % mdlOutputs % Return the block  -  outputs.  function sys=mdlOutputs(t,x,u,friction,load) F=load(l); % A p p l i e d l o a d (N) Rl=load(2); % L o a d i n g r a d i u s (m)  98  config = load(3); % Loading configuration o r d e r = r o u n d ( l o a d ( 4 ) ) ; % no. o f Gauss p o i n t s Ro=0.100;  % outer radius  Ri=0.008;  % i n n e r r a d i u s (m)  t=0.006;  (m)  % plate thickness  A=pi*(Ro 2-Ri 2); A  A  type  (m)  % a r e a o f c / s (m2)  Fs=7800*A*t*9.81; % S e l f - w e i g h t (N) Pmin=Fs/A; % Minimum c o n t a c t p r e s s u r e (N/m.2) i f Rl>0.080 rx=0.1-Rl; else rx=0.02; end switch config % S i n g l e - p o i n t l o a d i n g case c a s e {1} Pmax=350*F/A; % Maximum c o n t a c t p r e s s u r e (N/:rn2) others=[Pmin;Pmax;Rl]; [gvl,Tvl]=intGauss (Ri,Ro,order,friction,others,u); [gv2,Tv2]=dblIntGauss ( R l - r x , R l + r x , - r x , r x , o r d e r , f r i c t i o n , o t h e r s , u ) ; sys = [gvl+gv2,Tvl+Tv2]; % Two-point l o a d i n g case c a s e {2} Pmax=350*F/(2*A); % Maximum c o n t a c t others=[Pmin;Pmax;Rl];  pressure  (N/m2)  [gvl,Tvl]=intGauss (Ri,Ro,order,friction,others,u); [gv2,Tv2]=dblIntGauss ( R l - r x , R l + r x , - r x , r x , o r d e r , f r i c t i o n , o t h e r s , u ) ; sys = [gvl+2*gv2,Tvl+2*Tv2]; % M u l t i - p o i n t loading case case{4} Pmax=350*F/(4*A); % Maximum c o n t a c t others=[Pmin,• Pmax;Rl] ;  pressure  (N/m2)  [gvl,Tvl]=intGauss (Ri,Ro,order,friction,others,u); [gv2,Tv2]=dbl!ntGauss ( R l - r x , R l + r x , - r x , r x , o r d e r , f r i c t i o n , o t h e r s , u ) ; sys  = [gvl+4*gv2,Tvl+4*Tv2];  end return; % end mdlOutputs function  [yl,y2]  = intGauss  (xmin,xmax,n,friction,others,wr)  'o  % % % % % % % % % % %  y = intGauss  (xmin,xmax,n,friction,others,wr)  intGauss - n u m e r i c a l l y i n t e g r a t e s a f u n c t i o n f u n from a t o b u s i n g Gauss-Legendre quadrature w i t h n gauss p o i n t s i t c a l c u l a t e s t h e S t r i b e c k . and v i s c o u s f r i c t i o n t o r q u e on t h e e l e m e n t a c i r c u m f e r e n c i a l s t r i p c a u s e d due t o t h e Pmin... i . e . c o n t a c t p r e s s u r e due t o t h e s e l f - w e i g h t . T h e i n p u t i s t h e r a d i a l d i s t a n c e from t h e center and t h e f u n c t i o n outputs the S t r i b e c k and v i s c o u s friction t o r q u e . The c o e f f . o f f r i c t i o n i s a f u n c t i o n o f r a d i a l d i s t a n c e a n d t h contact p r e s s u r e i s constant and independent of the r a d i a l d i s t a n c e . This elemental torque i s then i n t e g r a t e d along the r a d i a l distance  % to obtain if  nargin  u [vc,x] [x,k] w x yl y2  = =  the  < 6,  total n = 20;  friction  t o r q u e c a u s e d by  self-weight  alone,  end  = ( l : n - l ) . / s q r t ( ( 2 * ( l : n - l ) ) ."2-1) ; = eig(diag(u,-1)+diag(u,1)); = sort(diag(x)); = 2*vc(l,k)'."2; = (xmax+xmin)12 + ( x m a x - x m i n ) 1 2 . * (xmax-xmin)12 (xmax-xmin)12  the  * sum(w .* * sum(w .*  x;  GV1(x,friction,others,wr)); TV1(x,friction,others,wr));  return; function gvl=GVl(r,friction,others,wr) % gvl=GVl(r,friction,others,wr) % This f u n c t i o n c a l c u l a t e s the s t r i b e c k f u n c t i o n at a s p e c i f i e d r a d i a l % l o c a t i o n f o r g i v e n f r i c t i o n p a r a m e t e r s , l o a d p a r a m e t e r s and a n g u l a r % speed.  Pmin=others(1) ; mu0=friction(1); muc=friction(2); vs=friction(4); vr=wr.*r; % r e l a t i v e l i n e a r v e l o c i t y (m/sec) g=muc+(muO-muc)*exp(-((vr/vs)."2)); gvl=(2*pi*Pmin*g).*(r."2); % Stribeck f r i c t i o n torque (N-m) return; function Tl=TVl(r,friction,others,wr) % Tl.=TVl ( r , f r i c t i o n , o t h e r s , wr) % This f u n c t i o n c a l c u l a t e s the v i s c o u s f r i c t i o n f u n c t i o n a t a s p e c i f i e d % r a d i a l l o c a t i o n f o r g i v e n f r i c t i o n p a r a m e t e r s , l o a d p a r a m e t e r s and % a n g u l a r speed. Pmin=others(1); muv=friction(3); vr=wr.*r; Tl=2*pi*Pmin*(muv.*vr).*(r."2); return; function % y =  [yl,y2]  = dbllntGauss  % r e l a t i v e l i n e a r v e l o c i t y (m/sec) % viscous f r i c t i o n torque (N-m)  (xmin,xmax,ymin,ymax,n,friction,others,wr)  dbllntGauss(xmin,xmax,ymin,ymax,n,friction,others,wr)  o  % % % % % % %  N u m e r i c a l l y d o u b l e i n t e g r a t e s a f u n c t i o n f u n o f two v a r i a b l e s u s i n g Gauss-Legendre quadrature w i t h n gauss p o i n t s . I f no number o f g a u s s p o i n t s i s s p e c i f i e d , n w i l l b e s e t t o 20. x i s the outer, y the i n n e r v a r i a b l e I t c a l c u l a t e s t h e S t r i b e c k a n d v i s c o u s f r i c t i o n t o r q u e on an elemental a r e a due t o F... i . e . t h e c o n t a c t p r e s s u r e due t o t h e a p p l i e d l o a d The i n p u t i s t h e x a n d y d i s t a n c e s o f t h e e l e m e n t a n d t h e f u n c t i o n  100  % % % % % % % %  o u t p u t s t h e S t r i b e c k and v i s c o u s f r i c t i o n t o r q u e . The c o e f f i c i e n t o f f r i c t i o n i s a f u n c t i o n o f r a d i a l d i s t a n c e a n d t h e contact, p r e s s u r e i s a f u n c t i o n of t h e r a d i a l and a n g u l a r d i s t a n c e . T h i s e l e m e n t a l t o r q u e i s t h e n i n t e g r a t e d o v e r t h e e n t i r e area, t o o b t a i n , t h e t o t a l f r i c t i o n t o r q u e c a u s e d by t h e a p p l i e d l o a d a l o n e . However, t h i s has been c a l c u l a t e d o n l y i n t h e v i c i n i t y of t h e l o a d a p p l i c a t i o n p o i n t a s t h e c o n t a c t p r e s s u r e d i m i n i s h e s r a p i d l y away from t h i s p o i n t .  if  n a r g i n < 6,  u [vc, x] [x,k] w  n = 20;  end  = (l:n-l)./sqrt((2*(l:n-l)).~2-l); = eig(diag(u,-1)+diag(u,1)); = sort(diag(x))• = 2*vc(1,k)'."2;  x_bar = (xmax+xmin)12 + (xmax-xmin)12 .* x; y _ b a r = (ymax+ymin)12 + (ymax-ymin)12 . * x; 11 = 0; 12 = 0; for  i=l:n 11=11+ s u m ( w ( i ) * 12=12+ s u m ( w ( i ) *  w w  .* .*  GV2(x_bar(i),y_bar,friction,others,wr)); TV2(x_bar(i),y_bar,friction,others,wr));  end yl=(xmax-xmin)*(ymax-ymin)/4 * I I ; y 2 = ( x m a x - x m i n ) * ( y m a x - y m i n ) / 4 * 12; return; function gv2=GV2(x,y,friction,others,wr) % gv2=GV2(x,y,friction,others,wr) % T h i s f u n c t i o n c a l c u l a t e s t h e S t r i b e c k f u n c t i o n a t a s p e c i f i e d x and y % l o c a t i o n f o r g i v e n f r i c t i o n p a r a m e t e r s , l o a d p a r a m e t e r s and a n g u l a r % speed.  Pmin=others(1); P m a x = o t h e r s (2) ; Rl=others(3 ) ; mu0=friction(1); muc=friction(2); vs=friction(4); rsx=5.3292*le-03; rsy=5.3292*le-03;  % x - v a r i a n c e of d i s t r i b u t i o n % y - v a r i a n c e of d i s t r i b u t i o n  (m) (m)  r=sqrt(x."2+y.^2); % r a d i a l d i s t a n c e (m) vr=wr.*r; % r e l a t i v e l i n e a r v e l o c i t y (m/sec) g=muc+(muO-muc)*exp(-((vr/vs).~2)); Rx=abs(Rl-x); % x - d i s t a n c e f r o m l o a d i n g p o i n t (m) Ry=abs(y); % y - d i s t a n c e f r o m l o a d i n g p o i n t (m) P= (Pmax-Pmin) * e x p (- ( ( R x / r s x ) . "2+ ( R y / r s y ) . "2 ) ) ; % c o n t a c t p r e s s u r e gv2=(g.*P).*r; % S t r i b e c k f r i c t i o n torque (N-m) return;  (N/'m  function T2=TV2(x,y,friction,others,wr) % T2=TV2(x,y,friction,others,wr) % This f u n c t i o n c a l c u l a t e s the viscous f r i c t i o n % and y l o c a t i o n , f o r g i v e n f r i c t i o n parameters, % a n g u l a r speed. Rl=others(3); P m i n = o t h e r s (1) ; P m a x = o t h e r s (2) ; muv=friction(3); rsx=5.3292*le-03; rsy=5.3292*le-03; r=sqrt(x."2+y."2); vr=wr.*r; Rx=abs(Rl-x); Ry=abs(y); P=(Pmax-Pmin)*exp(-((Rx/rsx) T2=(muv.*vr).*(P.*r); return;  function at a specified l o a d parameters and  x  % x - v a r i a n c e o f d i s t r i b u t i o n (rrt) % y-variance of d i s t r i b u t i o n (m) % radial distance % relative linear x-- d i s t a n c e f r o m % y - d i s t a n c e from "2+(Ry/rsy) . 2 ) ) ; % viscous friction A  (m.) v e l o c i t y (m/sec) l o a d i n g p o i n t , (m) l o a d i n g p o i n t (m) % contact pressure torque (N-m)  (N/'m2)  102  APPENDIX iv  CALIBRATION DATA  103  C\l <N  CO  Q c  D)  a. a. CO CO 1  1  ST  .2  (0 :  JD  -  "5 U O)  C ':'  a  CO  3 o BJ  ra O  a E  0)  E  Q  re cr c ;<0 Dl  Q  O  E CM  CO  E o o  CC  d§ ~i  CD  '< "o CD O  c re u5 lb re  CC  CM  O  03  CD  ^  CM  O  ( s q | U|) peon  CO  ice  104  Measurement of static coefficient of friction:  The static coefficient of friction between the two plates has been measured at different locations. This was done using evenly distributed loads on the disc and the tangential force required to initiate the motion was measured by increasing this applied load at regular intervals. The ratio of the total friction torque to the applied tangential torque gives the static coefficient of friction.  Data for measurement of Static friction coefficient Quadrant II Quadrant 1 Applied Load (lb) Friction force (lb) Coef. of friction Applied Load (lb) Friction force (lb) Coef. of friction 0.1838 15.2 5.35 15.2 5.1 0.1913 0.1885 25.2 25.2 8.35 8.85 0.1985 0.1946 35.2 35.2 11.85 0.1908 11.6 0.1922 45.2 45.2 14.85 15.1 0.1952 0.1898  |Avq friction coeff. 0.1919  0.1940  Static Friction Coefficient  16 14 12 10 8 6 4 2 0  - • — Location A • - - Locarion B  10  20  30  40  50  Deflection (in inches)  Fig A-IV.3 The measurement chart for static coefficient of friction  105  

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