one exposure and is included in reduced form at the back of the book. ... restricted access and for the procedures to be followed in applying for this restriction. The ... which describes satisfactorily the behavior of the friction force In this thesis, two friction ...... As the net external force grows to break sticking, the adherence.
INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted.
Thus, some thesis and
dissertation copies are in typewriter face, while others may be from any type of computer printer.
The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order.
Bell & Howell Information and Learning 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 800-521-0600
NOTE TO USERS
The original manuscript received by UMI contains broken or light print. All efforts were made to acquire the highest quality manuscript from the author or school. Page(s) were microfilmed as received.
This reproduction is the best copy available
UMI
University of New Brunswick HARRIET IRVING LIBRARY
This is to authorize the Dean of Graduate Studies to deposit two copies of my thesis/report in the University Library on the following conditions:
(DELETE one of the following conditions)
(a)
The author agrees that the deposited copies of this thesis/report may be made available to users at the discretion of the University of New Brunswick
OR (b)
The authcplSgkqcs thaUbe^raositedpepks of this ttasisfoport may^&made available to uscrc'oiflywith hMTItirwitten^pcrmisSion for fli£j>ejkxl enaiqg
After that date, it is agreed that the thesis/report may be made available to users at the discretion of the University of New Brunswick*
J2.
Date
_________ i___________________ ______________________
Signature of Author 0 and u>n/ 0)o ...................................................................................... 73 Figure 5.5 (a-d) Dynamic responses o f a system with relatively low a>e/o)0 and high (on/ n/ w ,,...................................................................................75
Figure 5.7 Normalized sticking velocity error distributions
76
Figure 5.8 Graph o f In In (I R) vs In Error
76
Figure 5.9 Probability o f normalized sticking velocity error (P = 100 - R)
76
Figure 6.1 Two-dimensional path trajectories of the sample cases
84
Figure 6.2 (a-d) Displacement and velocity responses o f a two-dimensional system with relatively high g)„x, and low coI)V, cocx and
.........................85
Figure 6.2 (e-h) Acceleration and friction force responses o f a two-dimensional system with relatively high oo„v and low co„v, io.x and oocv
86
Figure 6.3 (a-d) Displacement and velocity responses of a two-dimensional system with relatively high natural and excitation frequencies
87
Figure 6.3 (e-h) Acceleration and friction force responses o f a two-dimensional system with relatively high natural and excitation frequencies
88
Figure 6.4 (a-d) Displacement and velocity responses of a two-dimensional system with u)ox and co((Valmost the same, and oocx is much higher than to.,
89
Figure 6.4 (e-h) Acceleration and friction force responses o f a two-dimensional system with co,)Xand a)„v almost the same, and u)cx is much higher than
90
Figure 6.5 (a-d) Displacement and velocity responses of a two-dimensional system in the case when O-FBFM detects one more sticking than the other friction m odels......................................................................... 91 Figure 6.5 (e-h) Acceleration and friction force responses o f a two-dimensional system in the case when O-FBFM detects one more sticking than the other friction models
92
Figure 6.6 (a-d) Displacement and velocity responses o f a two-dimensional system with (j0ox and a)ov almost the same, and u)cv is much higher than
93
Figure 6.6 (e-h) Acceleration and friction force responses of a two-dimensional system with co()Xand
almost the same, and co^. is much higher than 10^
94
Figure 6.7 Normalized sticking velocity error distributions
95
Figure 6.8 Graph o f In In f l R) vs In Error
95
Figure 6.9 Probability o f normalized sticking velocity error (P = 100 - R)
95
CH A PTER 1 INTRODUCTION
1.1 G eneral
In many mechanical systems, friction is the primary source o f oscillations, often leading to unstable behavior o f system components. Friction-induced vibrations are an inhibiting factor for increased accuracy and efficiency, and they can cause excessive wear o f components, surface damage, fatigue failure, and noise. These problems are encountered in many industrial applications including bearings, disc brake systems, electric motor drives, robot joints, wheel/rail mass transit systems, machine and tool/work-piece systems (Ibrahim. 1992) Understanding the dynamics o f stick-slip and its possible elimination are important, especially for applications requiring high precision motion (Armstrong-Helouvrv. 1992) In shell-and-tube heat exchanger systems the friction forces between the tubes and their supports, which are caused by tlow-induced vibrations o f the tubes, play a very important role in the fretting wear o f the tubes (Antunes et al., 1988 and Tan and Rogers, 1996) The nature o f dynamic friction forces developed between bodies in contact is extremely complex and affected by a long list o f factors: the constitution o f the interface, the time scales and the frequency o f the contact, the response o f the interface to normal forces, inertia and thermal effects, roughness o f the contacting surfaces, history o f loadings, wear and general failure o f the interface materials, the presence o f lubricants, and so on. Thus, dynamic friction is not a single phenomena but a collection o f many complex mechanical and chemical phenomena entwined together whose features can not be grasped through isolated simple experiments.
The absence o f a universally accepted friction model, which satisfactorily describes the strongly nonlinear behavior o f the friction force in the vicinity o f zero velocity, makes computer simulations o f dynamic system motions with friction difficult For rigid bodies with dry friction, the classical Coulomb law o f friction, which evolved from early studies in past centuries, is usually employed in many engineering contact problems because o f the simplicity o f its form. However, Coulomb friction results in a sharp discontinuity in the frictional force when the sliding velocity reverses direction. This discontinuity causes numerical slidingdirection chatter, which causes frictional force chatter, and results in many difficulties in numerical simulation. To get smooth and chatter-free simulations o f stick-slip motions, the friction model should have good criteria for the detection o f sticking and have friction force functions which satisfactorily describe the behavior o f the friction force. In this thesis, two friction models are proposed based on the ideas ofT an and Rogers (1996) and Antunes et al (1988) to improve the limitations o f the respective models. The models have their own means o f detecting whether the system is slipping or sticking. Piecewise continuous analytical and numerical solutions are developed for both original and newly developed friction models, considering a dynamic system with variable normal and tangential forces. Various computer simulations are made to test and compare the results o f the proposed friction models with the Tan and Rogers’ and Antunes et al. friction models. Finally, extensive parameter sensitivity studies o f the friction models are carried out.
1.2 Application The motivation behind the present work is the need to develop mathematical models and numerical algorithms to predict the sticking and slipping responses o f bodies sliding on surfaces. O f particular interest is the dynamic interaction between heat exchanger tubes and support surfaces. Tubes in shell-and-tube heat exchangers have small clearances between with their supports and are subjected to vibration due to How turbulence The nature o f the contact between the tubes and their supports is intermittent, i.e.. either the contacting surfaces are apart or physically in contact depending on the amplitude o f vibration o f the tubes The physical contact can be impact or rubbing, and causes tube wear The extent of wear damage in heat exchangers depends on four factors ( Fisher et al.. 1989) These are: tube-to-support dynamic interaction, material combination, contact configuration, and environmental factors. Because o f the difficulties in measuring the dynamic interactions in real heat exchangers, which are required for the analysis o f tube wear damage. Rogers and Pick (1976) introduced a finite element computer code called VIBIC (Vibration o f Beams with Intermittent Contact) as an analytical tool to estimate this interaction. The current version o f VIBIC is 7 0 and has two components, namely FREMOD and VIBSIM. FREMOD (for natural frequencies and modal shapes) discretizes the tube using beam finite elements, and calculates the undamped natural frequencies and mode shapes using the RSG (EISPACK) eigen-solver. The second component, which is VIBSIM (for vibration and impact simulation), is used to simulate the tube response to external sinusoidal or random force excitation, predicts tube motion and tube-to-support dynamic interaction, and computes
the wear work rate (normal forces multiplied by sliding velocity) based on the tube response history VIBSIM uses modal superposition o f the free modes o f vibration to reduce the number o f equations, and fourth-order Runge-Kutta numerical integration to solve the resulting set o f modal equations in a time marching fashion Tan and Rogers (1996) and Zhou and Rogers (1997) introduced a friction model into VIBIC that can detect and simulate stick/slip motion o f the tubes The aim o f the present work is to improve the current friction model. This organization o f this thesis is as follows: Chapter 2 reviews the complicated nature o f friction, and discusses the problem in computer simulation o f stick-slip motion The algorithms o f three existing dynamic friction models and their analytical and numerical solutions are provided in chapter 3 Chapter 4 introduces two improved dynamic friction models. The models are tested for one-dimensional and two-dimensional dynamic system motions with stick-slip, and the limitations o f the existing friction models are discussed. Parameter sensitivity studies on one-dimensional and two-dimensional dynamic system motions are presented in chapters 5 and 6, respectively These studies are made by running 1000 simulations, in each case, with wide range o f input parameters to answer two basic
questions. These are: I ) Are the force-balance and spring-damper friction models effective in capturing sticking for a wide range o f input parameters. 2) Which model has the best performance? Finally, chapter 7 presents the conclusion o f the work done in this thesis, and indicates areas that need further work. References which are quoted by other researchers are listed as the secondary references. The complete analytical solution o f a sliding system, simulation o f a dynamic system motion using numerical solution, and parameter sensitivity data for one-dimensional and two-dimensional dynamic system motions are given in appendices A. B and C, respectively.
CHAPTER 2 BACKGROUND
2.1 Classical laws of dry sliding friction When two metallic bodies in contact are subjected to applied forces which tend to produce relative sliding motion, friction induced stresses develop on the interface that tend to oppose that motion. According to Moore ( 1975), the classical laws o f friction, as they evolved from early studies in the past centuries, are the following:
|/*'i £ n;V i The friction force, F. (at the onset o f sliding and during sliding) is proportional to the normal contact force, N. Two values o f the coefficient o f friction, (i. are quoted : the coefficient o f static friction, |i„ which applies to the onset o f sliding and the coefficient o f kinetic friction. uk . which applies during sliding motion. ii. The coefficient of friction is independent o f the apparent area o f contact iii. The static coefficient is greater than the kinetic coefficient iv The coefficient of kinetic friction is independent o f the sliding velocity v. When tangential motion occurs, the friction force acts in the same direction o f the relative velocity but in opposite sense, f
= -ji.iV r r / ! r„i
The first two laws, usually known as the Amontons' laws o f friction, are generally observed to hold for gross motions o f rigid bodies. However, deviations from the first law have been observed at various circumstances (Bowden and Tabor. 1964. and Bay and Wanheim, 1976). Since there is a microslip regime during acceleration from a rest or near-rest condition before macroscopic sliding takes place, the statement that the friction force on the onset o f sliding is equal to the (static) coefficient o f friction times the normal force is valid only in a macroscopic sense. The third law derives from the classical experiments (Euler. 1748) if a body rests in equilibrium on an inclined plane and the inclination o f the plane is slowly increased up to the angle at which the sliding initiates (this angle is known as the angle o f static friction
0 =
tan'‘uN), it is often observed that, when the motion starts, it starts abruptly and the body acquires a large velocity in a short time. The fourth law is now known to be invalid, (eg .. Kragelskii. 1965) However, for many purposes in which only a limited velocity' range is o f interest, the kinetic friction coefficient may be taken to be a constant independent o f the sliding velocity ( Rabinow icz. 1965) The fifth law was confirmed by experiment (Rabinowicz. 1965)
2.2 The origins of friction
2.2.1 The early theories on the origin o f friction In early theories, frictional behaviour was explained in terms o f the surface roughness and the interlocking o f the surface asperities. It was thought that the sliding o f two contacting
bodies involved the riding o f rigid asperities o f one surface over the other If the average asperity angle is 0 , the coefficient o f friction would be equal to tan 0 and consequently it would be independent o f the load or the size o f the contacting surfaces The assumption that the asperities on one surface could traverse the gap between asperities on the other surface provided an explanation for the fact that the kinetic friction is smaller than the static one The main weakness o f this early work on friction (Amontons. 1609 . Euler 1748 and Coulomb. 1785) is that the models proposed are non-dissipative (Tabor. 19 8 1)
2.2.2 The adhesion-plowing theory The adhesion-plowing theory (Bowden and Tabor. 1964) has been the most widely accepted in recent decades among the researchers. The interfacial friction between metallic bodies is attributed essentially to two causes the formation and shearing o f metallic junctions between the surface asperities and the plastic deformation o f the softer surface by hard asperities As a consequence, the friction coefficient can be given as the sum o f two components resulting from the each o f the above effects.
where ua results from the adhesion (welding) and uP results from the plastic deformation (plowing). The theory verifies the third classical law the static friction is often greater than the kinetic one because the strength o f the junctions would increase with the time o f stationary contact.
Many researchers believe that the strength o f contact increases with the time o f stationary contact, and they give expressions relating the static coefficient o f friction with the time o f stationary contact (Kato and Matsubayashi 1970, Derjaguin. Push and Tolstoi. 1957. and Rabinowicz, 1958). However more recent experimental work (Johannes, Green, and Brockley, 1973, and Richardson and Nolle, 1976) showed that the phenomena attributed to the time dependance o f the static friction should rather be interpreted as rate dependance o f the static coefficient o f friction. The explanation for this is
small junction growth (small
coefficient o f friction) is expected at high rates o f tangential loading
2.2.3 Recent theories Rignev and Hirth (1979) developed a new model to explain the origin o f friction during the steady-state sliding ofmetais. with the assumptions that the frictional force arises from plastic deformation in the near-surface region and that most o f the deformation work is confined to the highly deformed region o f the micro-structure, w hich is developed during the initial break-in stage. They equate the work o f the friction force to the work o f deforming plastically the subsurface volume and obtain an expression for the frictional coefficient Kulmann-VVilsdorf (1981) proposed a modification o f the adhesion theory o f friction to include the interlocking effect due to the micro-roughness superimposed on the hills and tops Suh and Sin (1981) suggested that the coefficient o f friction between sliding surfaces is due to the various combined effects o f an asperities deformation component. ult, a component, up from plowing by wear particles and hard surface asperities and a component. uu . from adhesion between flat surfaces. The relative contribution o f these components depends on the
condition o f the sliding surfaces, which is affected by the history o f sliding, the specific materials used, the surface topography and the environment The anisotropy o f the contacting surfaces also influences the tangential stiffness at zero displacement, the length o f the micro slip zone and the energy dissipated in the contact (Olofsson and Hagman, 1997)
2.3 Dynamic friction models
2.3.1 Concerns on discontinuity of Coulomb law For rigid bodies with dry friction, the classical Coulomb law o f friction is usually employed. However, Coulomb friction results in a sharp discontinuity in the frictional force when the sliding velocity reverses direction. This discontinuity causes numerical slidingdirection chatter, which causes frictional force chatter, and results in many difficulties in numerical solution. To overcome the difficulty arising from Coulomb friction, a simple velocity-limited friction model (VLFM) is used for some engineering problems (e g.. Rogers and Pick. 1976) The model uses a linear force-velocity function to avoid the discontinuity when the relative sliding velocity is close to zero. A smooth velocity-limited friction model (S VLFM) was given by Oden and Martins (1985) and is used by many researchers In this model a quadratic function is adopted in the region where the sliding velocity is near zero in order to make the frictional force vary continuously and smoothly. The schematic diagrams o f these velocitylimited friction models are shown in figure 2 .1 There are two common characteristics o f these models. First, these models need a limiting velocity value. V . to smooth or to improve the discontinuity of friction force in the velocity range near zero The selection o f the limiting
velocity value is very important, however, there is no common formula for estimating the limiting velocities Second, at low velocities, these models allow the moving bodv to accelerate even when the external forces on the body are less than the peak static force They, therefore, cannot be expected to accurately predict the effects associated with sticking phenomena. Meijaard (1997) approaches the problem with a variable time step size and interpolation around zero velocity until 'nearly exact zero velocity' position is obtained On the next time step the initial values, the time step size and the system equations will be adjusted, and solved as a new system The problem with this method is that it needs high computation time
Figure 2.1 Schematic diagrams o f the velocity-limited friction models
2.3.2 Concerns on the behaviour of dynamic friction during sliding It was not until recently that the importance o f the dynamic characteristics o f the testing apparatus on the results o f frictional experiments was explicitly noted The first clear statement o f the importance o f the dynamic characteristic and vibrations o f the apparatus is done by Tolstoi (1967). Tolstoi (1967) and Tolstoi. Borisova and Girigorova (1971) investigated kinetic friction experimentally in the presence (and absence) o f normal vibrations They conclude that the coefficient o f friction does not explicitly depend on sliding velocity, and the difference between the apparent static and kinetic coefficient friction is the consequence o f microscale vibrations accompanying frictional sliding
The dynamic
interlocking o f imperfections is responsible for providing excitation o f normal vibrations in addition to strongly affecting the frictional resistance o f the interface Observations and ideas presented by Tolstoi led Oden and Martin (1985) to a new approach in the analysis o f dynamic friction. They considered a relatively simple constitutive model o f the interface with two key ingredients One o f the key ingredients is a normal contact constitutive law which takes into account the normal deformability o f the interface asperities. This constitutive law is characterized by a relationship between the normal stress and the normal approach, and was first developed by Back, Burdekin and Cowley (1974 ) for finite element computation o f static contact problems. The other key ingredient is the friction law which is a function o f the normal approach o f the two surfaces which, in turn, depends on the normal force. Some o f the points that Oden and Martin provide about their model are (a) The preliminary micro-displacements which occur before gross-sliding are disregarded. (b) No distinction is made between static and kinetic friction. (They believe that the apparent decrease o f measurable coefficients o f friction are the result o f dynamic instabilities which are
a consequence o f the normal vibration.) and (c) Variations o f coefficient o f friction with velocity are not considered. The difficulties and limitations o f single-valued friction (assuming constant load and steady state motion) have been indicated by Bell and Burdekin (1969-70) and Rabinowicz (1965) Multi-valued friction coefficients, different, for example, during the acceleration and deceleration phases o f a velocity oscillation are often observed This was addressed by Hess and Soom (1990) who showed that this behavior could be modeled by a constant time delay (determined empirically) between a change in sliding velocity and the corresponding change in friction However, the time delay model does not apply to general dynamic changes in velocity or load. In the Rice and Ruina (1983) model in its simplest form, i e . for unidirectional sliding and constant normal stress, the friction coefficient is a function o f the sliding velocity and a scalar state variable. Their model is termed a "State variable friction model" For a system with lubricated line contact and dynamic load. Polycarpou and Soom ( 1995c), developed a friction model by combining a two-dimensional quasi-steady friction model and a normal dynamics model to estimate the time-varying friction coefficient (or force). This model consists o f the important parameters, such as the sliding velocity, the instantaneous separation o f the sliding bodies normal to the sliding direction, the normal load and lubricant properties. The two-dimensional quasi-steadv friction model. Polycarpou and Soom (1995a), is an empirical model which is based on unsteady friction experiments carried out under constant normal loads and under time-varying sliding velocity The model explicitly includes normal motion, load and sliding velocity The normal dynamics model. Polvcarpou and Soom (1995b), is used to linearize the nonlinear normal dynamics. It is
represented by the contact interface model and two springs, which represents the bulk deformations o f the sliding bodies. The contact interface model, itself, is represented by a spring and a damper in parallel. The contact stiffness and damper depend on the normal load, the sliding velocity, and the roughness o f the two contacting surfaces This model is used to compute normal contact forces and normal displacements. It is combined with the friction model (two-dimensional friction model). Bouissou et al (1997) observed from their experiment that normal load and surface roughness have strong influence on the distance and the duration o f slip, the maximum velocity during sliding, and the relative displacement and/or deformation during sticking.
2.3.3 Concerns on stick-slip motion Stick-slip vibrations are self-sustained oscillations induced by dry friction Stick-slip determines the lowest performance bounds o f a machine, the lowest sustainable speed and the shortest governable motion ( Armstrong-Hélouvry. 1992) Since the friction characteristic consists of two qualitative different parts (kinetic and static frictions) with a nonsmooth transition, the resulting motion also shows a nonsmooth behavior Thus, stick-slip systems belong to the class of nonsmooth systems, like systems with stops, impacts, backlash or hysteresis Stick-slip motion behavior depends on the normal vibrations o f the slider, and dynamic characteristics o f the system (Tworzydlo, Becker and Oden. 1992) Normal vibrations are excited by dynamic interlocking o f imperfections (Tolstoi, 1967) Dynamic characteristics o f the system affects the instability o f the system, which means, propagation o f oscillations and occurrence ofhigh amplitude vibrations, under the circumstances when the system oscillations draw energy from friction (Swayze and Akay. 1992).
Popp (1992) modeled a simplified friction-induced self-sustained vibration system for a violin string, using a mass-spring system running on a moving belt The energy How from the belt to the oscillator is controlled by the friction characteristics In his model, sticking occurs when the relative velocity between the slider and the belt is greater than or equal to zero, and the net external force is less than the limiting static friction force During sticking the relative velocity is assumed to be zero, which means that the mass moves at the same velocity as the belt. Sliding occurs when the net external force is greater than the maximum static friction force, and the relative velocity is less than zero McMillan (1997) proposed a friction model, which is a function o f velocity and acceleration, to predict the dynamic responses o f a block running on a moving conveyor belt The criteria for capturing and leaving sticking are the same as Popp's. Pfeiffer and Glocker (1992) analyze unsteady processes of an impact drilling machine and the peg-in-hole problem (problem in assembly process) using time varying constraints, which are contact constraints, friction constraints and impact constraints. The criteria for transitions from sliding to sticking are the same as Popp's, whereas the conditions for the transitions from sticking to sliding are the relative acceleration should be non-zero and the net external force is greater than or equal to the maximum static friction force. However, the above models did not clearly define the friction characteristics during micro-slipping and micro-sticking states. An experiment by Polycarpou and Soom (1992) on the transitions between sticking and slipping o f lubricated line contacts showed that during the transition to sticking, the tangential contact stiffness changes abruptly from zero to a finite value, and then continues to decrease. Thereafter, the micro relative velocity increases and the tangential force rises
until, just before reaching its peak value, gross slip begins During stick/slip and slip/stick transitions the measured normal vibrations are higher They suggested that the instantaneous separation o f the sliding bodies, normal to the sliding direction, is an essential element in determining the tangential contact stiffness during sticking and the extent o f the microslip and microstick regimes during transitions. They also observed that the microstick part o f the slip to stick (zero velocity) transition takes place over a much shorter distance than the corresponding stick to slip transition.
2.3.4 Computer simulation algorithms Computer simulations o f stick-slip friction in mechanical dynamic systems are difficult because of the strongly nonlinear behavior in the vicinity o f zero velocity ArmstrongHelouvry (1992) used the Stribeck friction curve equation, which is developed by Hess and Soom (1990), for the analysis o f stick-slip behavior o f fluid lubricated machines in the low velocity region. The Stribeck friction curve (Stribeck, 1902) is a friction verses velocity curve for lubricated systems. Kamopp (1985) developed a model with a small-velocity window, surrounding zero velocity. Outside the velocity window (slip region), the friction force is an arbitrary function o f velocity Inside the velocity window (stick region), the friction force is determined by other forces in the system in such a way that the velocity remains constant until the breakaway value o f force is reached, i.e., the friction force takes on the force required to keep zero acceleration. The system enters the sticking region when the velocity is in the velocity window and the net external force is less than or equal to the limiting static friction force The
system leaves the sticking region when the net external force exceeds the breakaway value o f friction force. This model is implemented for many practical actuator and mechanism problems. Based on Kamopp’s one-dimensional model. Tan and Rogers (1996) developed a two-dimensional friction model that can be used for the simulations o f mdof systems with friction acting on a surface. Applications include a model for the simulation o f stick-slip motion o f a four-degree-of-freedom ball system on a surface (Tan and Rogers. 1998) and a finite element model o f a heat exchanger tube on a loose circular support surface (Tan and Rogers. 1996). Antunes et al. (1988) developed a spring-damper friction model (Figure 2 2). which expresses sticking frictional forces in terms o f an "adherence stiffness (K,) and damper (C ,)" To detect the sticking state, there must be a change in sign o f the tangential velocity, w hich shows the existence o f zero velocity between the last two steps, and the adherence friction force must be smaller than the sliding friction force The adherence friction force is composed of adherence stiffness and viscous damping forces.-A.'. [.V(/) - A'J - (
AY/)
The
adherence stiffness force is given by the product o f an adherence stiffness constant and the contact distance from the nominal adherence surface location. XM.. which is the value o f displacement corresponding to the "zero" value o f tangential velocity A viscous damping force is incorporated in order to damp out any numerical residual velocity If the two criteria are satisfied, then the system will be assumed to be in sticking state and the fictitious adherence spring and damper will be active. These adherence forces pull the mass back slowly to the sticking position. As the net external force grows to break sticking, the adherence spring and damping forces grow as well. When the adherence friction force is greater than or
equal to the sliding friction force, the system leaves sticking The simulations o f mass, spring. Coulomb damped systems (Antunes et al„ 1988, McGinn. 1990, Tan and Rogers, 1995) showed that the algorithm is capable o f detecting the stick/slip phenomena for straight-line motions.
X = Xst V = 0
X =0
X = X (0 ►
Figure 2.2 Schematic diagram o f Antunes et al friction model
In the Antunes et al. model, a single coefficient o f friction is used following the view o f Oden and Martins (1985), who believe that the variation o f coefficient o f friction is due to variation in normal oscillations rather than velocity The constants (adherence stiffness and viscous damping) are chosen such that the system will be over damped, and consequently, the residual adherence oscillations will be damp out before rebound occurs During sticking, the micro-displacements are attributed to the local deformations o f asperities o f the two bodies in contact, and are assumed to have a linear-elastic relationship with the frictional force.
CHAPTER 3 ANALYTICAL AND NUMERICAL SOLUTIONS FOR COULOMB, TAN AND ROGERS, AND ANTUNES et al. FRICTION MODELS
In many engineering problems, dynamic force balance equations are involved, which are differential in nature. Their solutions may be determined analytically or numerically The analytical solution is a (unction that satisfies both the differential equation and also certain initial conditions. However, the analytical method is limited to special kinds o f equations. Den Hartog (1931) provided the analytical solution for a single degree-of-freedom (sdof) system with a constant normal force and combined viscous and Coulomb damping, excited by a harmonic force. On the other hand, numerical integration has almost no limitations to the type o f equations, and the solutions are obtained as a tabulation o f values using the previous information, such as the initial condition or previous evaluation, not as a functional relationship. In this chapter, the analytical and numerical solutions for a sdof, mass-spring-damper dynamic system, with variable normal force and excitation force will be discussed briefly For simplicity the tangential excitation force is harmonic (cosine function o f time) and the normal force is a combination o f harmonic and constant forces. The harmonic normal force represents the variation o f the normal force due to external forces and surface roughness o f the contacting surfaces, which is one o f the important factors in the determination o f dynamic friction (Hess and Soom, 1991).
Figure 3 .1 Schematic diagram o f the sdof system
Figure 3 .2 Free body diagram
For the systems shown in Figures 3 .1 and 3.2, the general equation o f motion is given as: MY + Fe - F
= F
+ F,
(3 - 1 )
where: /•' is the viscous damping force, I \ is the spring force. l\. is the excitation force, and /•’, is the friction force Except for /•,, these forces are represented by:
/•; = r.v, f'e = Pcos(ioj),
/•; = a t , l\. - .V = (;V. ►,V cos(w /))
2)
where/V, < /V. and the frequencies, co,and sinco t ►/-." a) cosaw - (i
- Xp)
(3-18)
where A'„ and A'„ are the initial conditions.
Summary for the analytical solution of a dynamic system with Coulomb friction ►
The steps that should be taken to determine the instantaneous displacement and velocity of the system at any time o f sliding are: 1. Update the time, t ,by adding the time step to the previous time 2. Calculate the values o f Y, W, Ÿ. W, A'p . and A ,using equation (3-14) 3. Use the current .-1 and B values, and calculate the displacement and velocity using equations (3-15) and (3-16), respectively
►
The steps that should be taken to determine constants A and B. whenever the system starts sliding from sticking or when its sliding velocity sign changes, are 1. Update t,t to the elapsed time, and reset t to zero 2. Determine the values o f constants D ". l\ \ (/'*. //**. and / using equations (3-9), (3-10), (3-11), (3-12), and (3-13), respectively 3 Determine the values o f Y. W. Y. tt'. A’.,. and A ,using equation (3-14) 4. Determine the values o f constants A and B using equations (3-17) and (3-18)
In this work the most popular numerical integration method, the fourth-order RungeKutta method, is used for the numerical computation. It is a self-starting and single-step method that uses only one step back information to predict the current state, and results in good accuracy. The integration is performed at a finite time step, h, which will determine the accuracy o f the solution. The error per step is o f order h 4 (Gerald and Wheatley, 1994) The fourth-order Runge-Kutta method requires six equation evaluations to complete a single time step for a second order differential equation. The steps that should be followed in the implementation o f this method are: Step 1: The system dynamic equation o f motion, equation (3-1). should be written in the form:
(3-19)
.V - / (/, .V. X)
From Figure 3.2, the function /con sists o f the system damping force, spring force, excitation force and friction force.
X = f ( i , X X) = -Î- [ - F . - F * M
- F, ]
(3-20)
For the Coulomb model, equation (3-19) will be: X = / ( / , X, X) = — [ - O f - K X - Pcostojt - t j M -(/V uCOSG)n( / - / ) - .V )u , .V/S'Z/I.V)]
(3-21)
Step 2 : The function/needs to be evaluated four times for each full time step, as follows (Thomson,! 993): /► .'I = Jf ( t rt', X n', X n') h
=/ ( ' „ "
v \
f fit h A =1(1« + y
y
7 ~
hy
/ . = /J( /x n * h,* X /? > hX n - —
" y fx) h‘f y h r\ " — U V, ' -Jz)
(3-22)
.V" - ///',) '
where /rt is the current time o f integration with known inputs.V, and A', (displacement and velocity, respectively), and / , is the evaluation o f A'^ at fri, /, and / , are estimations o f A' j_ at / hXn - !Lo -v „ , - .v, . t L y . z f , : /, o
- /,)
(3-23) ,3 .2 4 ,
From a step by step calculation, the system dynamic responses and motion state are obtained The state o f the system (sticking or slipping) is determined at the end o f each full time step calculation.
The Tan and Rogers model (1996) accounts for both sticking and sliding phenomena, unlike the Coulomb model which cannot capture sticking since zero sliding velocity cannot be obtained in numerical computation except as an initial condition. Tan and Rogers made two separate friction force definitions for each phenomena. The first one is the sliding friction force, which is given by the same definition as the Coulomb kinetic friction force. The second one is the sticking friction force, which is determined by making a force balance and is similar to Kamopp’s method (1985). A rigid body is sticking when it has no motion in a direction tangential to the contacting surfaces, which means its velocity and acceleration are zero Since it is impossible to get exact “zero’ velocity in a numerical computation, a small velocity window is assigned, ¡A] < A',,
( - A', < A'< A '), a velocity inside this region is assumed
to be zero The algorithm developed by Tan and Rogers to decide whether the system is sticking or slipping is summarized as follows: ! If ¡A"] > A', then the system is sliding, and the friction force is given by Ff = -
sign(X)
(3-25)
2. If ¡Al < A, ^ calculate the net force (Fnet) acting on the system excluding the friction force. Fne, = Pcosto.(/ + /„) - (M X - ( X - KX)
(3-26)
2.1 If |F„J > N//s then the system is sliding, and the friction force is given by equation (3-25). 2.2 If |F„J < N//s then the system is sticking, and the friction force is given by:
F. = - F , = - Pcosio (/
/ ) - iM X - ( ’X ' fCX)
(3-27)
Since .V is very small and X is zero, due to the force balance. /•. can be closely approximated bv /•', = -Pcoscojf + t ) + KX
(3-28)
3.2.1 Analytical and numerical solutions for sliding state with Tan and Rogers model During sliding, the friction force is given by equation (3-25), and therefore, the dynamic system equation o f motion, will be given by equation (3-7). which is the same equation of motion as the Coulomb model. Therefore, the dynamic characteristics o f motion, displacement and velocity, o f the system are given analytically and numerically by equations (3-15) and (3-16), and equations (3-23) and (3-24), respectively
3.2.2 Analytical and numerical solutions for sticking state with Tan and Rogers model At the very beginning o f sticking, the time / is set to zero and t„ is updated to the elapsed time. During sticking, the friction force is given by equation (3-28) It balances the net force and creates an equilibrium state, which means the system will have a very small constant velocity and zero acceleration. Therefore, analytically, the system in Figure 3 1 will have the following characteristics o f motion: .V = 0
(3-29)
.V = Constant
(3-30)
and X =X
- Xt
(3-31)
where .V, is the position where the system starts sticking (i.e.. enters the velocity window)
Numerically, the system characteristics o f motion can be analyzed with the same method and procedures described in section 3.1.2. Equations (3-20) and (3-22) are zero due to the force balance made by the friction force, equation (3-27) Consequently, equation (3-23) and (3-24) will be reduced to: = X, + hXn - i “ -'C
(3-32) (3-33)
3.3 Antunes et al. Spring Damper Friction Model (AABG-SDFM)
The Antunes et al. model (Antunes, Axisa, Beaufils and Guilbaud. 1988) uses the Coulomb kinetic friction force and an adherence friction force during sliding and sticking, respectively. The adherence friction force is defined in terms o f an "adherence stiffness" and "adherence damping". The model detects sticking if and only if the following two criteria are simultaneously satisfied: the sliding velocity changes its sign and the sticking friction force is smaller than the sliding friction force. It leaves sticking when the adherence friction force is greater than the Coulomb kinetic friction force. This model uses a single coefficient o f friction (u ), which is the kinetic coefficient o f friction. The model is based on the notion that stick/slip motion is not necessarily the result o f a decreasing o f u with velocity, but rather a consequence o f the coupling o f the normal force and relative tangential motion o f the contacting bodies.
The preceding arguments for the Antunes et al model are summarized bv the following algorithm, which is applied at each time step /„ I
Recall state for step /„.,.
2.
If state (/„.,) = sliding, then examine the possibility o f adherence (zero velocity between /„., and /„ ) by calculating: ( ' = X (tJ
AY/,, ,)
(3-34)
2 1 If ( ' > 0, then the system is sliding and the friction force is given by /■',(/„) =
v/tf//[AY/„)|
(3-35)
2.2 I f f ’ < 0, then estimate displacement A', by interpolation between AY/,,.. ) and A (/J. and follow procedure step (3). 3
If state (/„.,) = adherence, then calculate /-, (/„) and /•'. the maximum friction force l’,Vn) = -K, [-'ï/„) - .Vf] - ( ’, AY/,,)
(3.36)
and '• = 'V,' >
(W 7 ,
3 1 If F > j F, (/„)|, then adherence is maintained and /• . (/,,) is accepted; 3 2 If F < | F, (/J|, then sliding occurs and F. (/„) is given by equation (3-35)
The subscripts// and //-/ represent the present and previous time steps, respectively. A', is the position where the ‘zero velocity’ is assumed to be. and K. and ( '. are the adherence stiffness and damping constants, respectively. Their values are chosen so that the system will be over damped, and consequently, the residual adherence oscillations will be damped out before sliding recurs.
3.3.1 Analytical and numerical solutions for sliding state with Antunes et al. model During sliding, the friction force is given by equation (3-35), which is identical to the friction force used in the Coulomb and Tan and Rogers models. Therefore, the dynamic equation o f motion, and the analytical and numerical solutions for the dynamic characteristics o f motion, are given by equation (3-7), equations (3-15) and (3-16), and equations (3-23) and (3-24), respectively.
3.3.2 Analytical and numerical solutions for sticking state with Antunes et al. model During sticking, the friction force is given by equation (3-36), and the dynamic equation o f motion, equation (3-3), will have the form,
M X + C X + K X = Pcoso3e(t - r j - K, (A' - A'vf) - C, X
(3-38)
Rearranging equation (3-38) gives;
MX
C X -i- K X = P cosojj/ ~ ( )
Kt X r
(3-39)
where two o f the dynamic characteristics o f the system, damper and spring, have been redefined as C = C + C'( and K = K * Kr
3.3.2.1 Analytical solution during sticking with Antunes et al. model The system equation o f motion during sticking, equation (3-39), is a nonhomogeneous second-order differential equation. Therefore, the general solution. A'', should include both
complementary, X ’ , and particular, Xp , solutions. The system is supposed to be over damped so that the residual oscillations will be damped out before rebound occurs. Consequently, the complementary soIution,.V.’, for equation (3-39) will have the form derived in appendix A, equation (A- 8 ), and given as:
Xc' = A e ( i J 2)2 - (2C gW o) ( ); ] ( ! ) > /
(3-45)
XX,
/ =
(3-46) K
Finally, the complete analytical solution o f the dynamic system during sticking, equation (3-39), is given as the sum of the complementary and particular solutions, equations (3-40) and (3-43), respectively.
■V = X : * X :
V = A' c' v ’’ ">l>' - B e
(3-47)
' - D coso) / +■ E sinco / * /
(3-48)
This equation can be simplified by defining the following terms y
_ t/ i->, • «■».,*
W =e V' = (-(g ),
* d>j ) e' v > '
= ( -;o ),, -
(3-49) w ' - -(Co)„ +
^ ’
' ,/>l = ~((o>, *
X p' = D coso) ei - E sinoo / - / X p = - D o)e sino)e / * E o) coso) /
Thus, the displacement and velocity o f the system, during sticking, are given by:
X ' = À Y ' ►B W ’ ►x ;
(3-50)
X ‘ = À Ÿ ’ * B W ' * X'p
(3-51)
The constants À and B are determined from the initial conditions, which are the displacement {X't ) and velocity ( X ’ ) o f the system when it begins sticking from sliding. At this instant the tim e,/, will be reset to zero, and therefore W Ay = /) + l , X
= l , K ’ = - ( à ) ( -i- ù , , W ' = - ( ( û , * to,).
= E gj,, and equations (3-50) and (3 -5 1) will be solved simultaneously for
constants A and B using Cramer's rule (McCracken and Dorn, 1964)
( x ; - x ; w ' - ç c ’0 - x p’ ) A = ---------p — --------. W - Y
(3-52)
_ ( X ’ -X/,) - (A;;-A-;)K(3-53) W
- Y'
Summary for the analytical solution o f a dynamic system during sticking using Antunes et al. model ►
The steps that should be taken to determine the instantaneous displacement and velocity o f the system at any time o f sticking are: 1. Update the time, t, by adding the time step to the previous time, t 2. Calculate the values o f Y ‘, W \ Y , W , X ' and
using equation (3-49)
3. Use the current A and B values, and calculate the displacement and velocity using equations (3-50) and (3 -5 1), respectively.
►
The steps that should be taken to determine constants A and H. whenever the system begins sticking from sliding, are: 1 Update /„ to the elapsed time, and / is reset to zero 2. Determine the values o f constants P . hi and / using equations (3-44). (3-45) and (3-46), respectively. 3. Determine the values o f Y ', W '. Y . W . A',', and A',, using equation (3-49) 4. Determine the values o f constants À and H using equations (3-52) and (3-53)
3.3.2.2 Numerical solution during sticking with Antunes et al. model Numerically, the system characteristics o f motion during sticking, displacement and velocity, are given by the same method and procedures described in section 3 1 2. except that equation (3-21) will be replaced by equation (3-54) From equation (3-39) A’ = /(A ', A', /) = — [ - ( ’V - KX - /'costo (/ * / ) - K. ,Vf|
M
(3-54)
3.3.3 Relationship between adherence stiffness. Kr and damping. C, In the Antunes et al. model, the nature o f the system motion during sticking is determined by the values of adherence stiffness and damping. Basically, the adherence stiffness and damping are implemented to minimize and damp out small oscillations. Therefore, their values should be chosen so that the system will have over-damped motion The following analysis is made to find a relationship between the two values, i.e.. K. and (
CC -----——
where ( \ = 2A / \K /\ f
_ r ^ _ r L_ 2JM(K - K.)
(3-55)
K. = K a
c =
c / i + o
since K - Kt / o .
KIM -
(3-56)
2A'/(o)V/l r a
Kt / M o , and defining w, - ^ K, !M.
to
-
va
r v/1 - a
2Mv>t \f(I - o) / o
o) Consequently, the relationship for over damped motion ( (
c
For example if a = 100, stickinu.
(I + O) _ _C_
°
when
2 \ / oj,
(3-57)
I ) will be.
(3-58)
Ja
> 1.005 - 0 1( is required for over damped motion during
CHAPTER 4 IMPROVED FRICTION MODELS AND SYSTEM SIMULATIONS
The difficulty with simulations o f dynamic system motions with friction is mainly in the explicit determination o f the behaviour o f the friction force for different states o f motion. The tw o main areas that need special attention in dealing with friction are: when the velocity changes sign and when the system searches for sticking. In the first case, the external forces are usually large and the friction force changes direction so rapidly that there is a discontinuity in the friction force and acceleration functions. In the second case, when the system approaches sticking after oscillating for some time, the velocity as well as the friction force change their sign so frequently that numerical chattering results. Therefore, there must be criteria that can detect the system 's requirements and implement a friction force function that can satisfy these requirements In this chapter the algorithms o f tw o improved friction models are discussed, followed by simulations o f one- and two-dimensional dynamic system motions with friction using the existing and newly developed friction models. These simulations are intended to show and com pare their performance with the existing friction models' (CFM. O-FBFM , A ABG-SDFM )
4.1 Improved Friction Models A new friction model is required to improve, and if possible eliminate, the limitations o f the CFM, O-FBFM and AABG-SDFM, and to get smooth and chatter-free simulations o f dynamic system motions. In this thesis tw o improved friction models are presented.
4.1.1 Revised Force Balance Friction Model (R-FBFM) The R-FBFM is one o f the proposed models, which is based on the original forcebalance friction model (O-FBFM), but uses sliding velocity sign changes rather than a velocity window as one o f the criteria for the detection o f sticking. As the system approaches sticking, the sliding velocity goes to zero and changes sign, therefore sticking will not be missed and low velocity non-sticking cases will be avoided. During sticking the friction force is determined by making a force balance. This model gives a smaller constant sticking velocity, and consequently smaller sticking displacement, compared to the ordinary O-FBFM . and also does not need a predetermined velocity window The algorithm is 1
Recall state for step
2
I f .1state
= sliding => calculate (' - X ( t j
X(f„ ,)
2 I If C > 0, then the system is sliding and the friction force is given by
(4-1) 2.2 If C < 0 => calculate the net external force excluding the friction force - P c o s(io jn) - [M X d J - I ’Xd") *■ ALVi/j] 2.2.1 If \FnJ t n)\ > M /„)n,then the system is sliding, and the friction force is given by equation (4-1). 2.2.2 If |/*'w/(/„)| ^ M /„)fivthen the system is sticking, and the friction force is given by: = -Costco/,,) - AlV(/„)
(4-2)
The analytical solutions o f a system during both sliding and sticking are the same as ORBFM 's since R-FBFM is also a force balance model.
The TR-SDFM is the second proposed friction model and is based on the AABGSDFM. The model’s criteria for detecting sticking has three com ponents which must be satisfied simultaneously. These are: ►
There should he a change in velocity sign. This shows the existence o f zero velocity between the last two steps.
►
The net external force should he less than the maximum friction force. This avoids capturing sticking at changes o f velocity sign with high net external force and avoids detection o f momentary sticking.
►
The adherence friction fo rce must he less than the maximum friction force. This is because during sticking, the friction force is given by the adherence friction force and cannot be larger than the limiting friction force (which is the static friction force).
After the existence o f sticking is confirmed by the above conditions, the model activates the adherence stiffness and damping and goes a few time steps back until the sliding friction force (Coulomb kinetic friction force) and the adherence friction force (adherence spring and dam per force) are roughly equivalent. For the case with a very short sliding region between the current and the previous sticking states, the backing up will be terminated if the model enters into the previous sticking region before the required equivalence o f friction forces is achieved. The algorithm then re-approaches the zero velocity position with the adherence spring and dam per in place. The result is a very smooth transition from sliding to sticking.
From the detection o f sticking until the velocity attains its lowest value, the sticking friction force value is usually dominated by the adherence damping force com ponent Afterwards the adherence spring force becomes dominant until the system breaks sticking During the sticking period the sliding velocity and acceleration continuously decrease and approach zero values. When the adherence friction force is greater than the maximum friction force, the sticking state is broken and sliding is initiated The TR-SDFM algorithm is 1
Recall state for step
2
I ( state (/„./) = sliding => calculate ( ’ = X(tn)
X(tn ,)
2 . 1 If ( ’ > 0, then the system is sliding, and the friction force is given by: (4_3)
2.2 If ( ’ 1 0, then calculate the net system force and the maximum friction force
= /jcos(gw ,) - [MX(tn) * C.V(/„) - KXd ,)\ F = M /„ )n v 2 .2.1 If |/ \ J > F. then the system is sliding and the friction force is given by
equation (4-3) 2.2.2 If |/- J
< /•', then
2 2.2.1 Estimate displacement .V, by interpolation between A'(/„.,.) and A'(/,) 2.2.2.2 Calculate the adherence friction force. /- ,(/„): F , UJ = ~K, [AT/,) - A'J - (
X( t J
(4- 4 )
2.2.2.3 If IFo'(t nf' )I > F. then the svstem is slidinu —and the friction force is given by equation (4-3) 2 .2 .2 .4 If F j t j l z F. then the system is sticking, and now search for
the start-up sticking friction force:
2.2.2 4.1 If |/\,(/_,)! >
/•;
or
^ s tic k in g . then /•, (/„) = / .,(/,.)
2.2.2.4.2 If \FJ/„./)) < \F, (/„.,)! => step back (// = //-/), obtain new /•',,(/„) from equation (4-4). and check for the start up adherence friction force using condition 2 2 2 4 I 3
I(sta te (/„.,) = adherence, then calculate /•', (/,,) and F
|/ \ (/„) |, then adherence is maintained and /•'. (/.,) is accepted. 3 2 If F < |/ ’* (/„) |, then sliding occurs and F, (/„) is given by equation (4-3) This algorithm is also given in the flow chart shown in Figure 4 I The analytical solutions o f the svstem during both sliding and sticking are the same as A ABG -SDFM 's since the TR-SDFM also uses an adherence spring and damper during sticking, and Coulomb kinetic friction force during sliding. At the very first step the model tests whether the system is initially sticking or sliding using the equilibrium position as the sticking position: F. - -AT. (A'
- A',) - ( \ . V .
4.2 Transition from sticking to slipping The transition from sticking to slipping occurs only if the magnitude o f the net force is greater than the maximum friction force, which is the criterion o f all friction models discussed above for leaving sticking. Initially there is a micro-slip region where the magnitude o f the sliding velocity is very small and its direction is not reliable It is therefore assumed that the svstem mass has to accelerate in the direction o f the net force regardless o f its previous velocity direction. In the micro-slip region the friction force acts in opposite direction to the net external force, which means the direction o f the friction force is determined by the net
Calculate c = x „ .X*
< —
----->
Sliding
Calculate adherence friction force W O)
Sticking Calculate static friction force (F) < --------
FKO = -N ^ ksig n (X ,^< H Sliding
Calculate net force 7 ^ ( 'Vl1
Calculate static friction force (F)
Calculate Estimate > adherence sticking friction force position X(l Fo(t„)
C/3
S F | Sticking
Calculate P IÙ l igure 4 I TR-SDFM algorithm flow chart 42
external force direction, not by the direction o f the sliding velocity This continues until the sliding velocity develops enough and gets its right sense. The following algorithm is used to determine the right friction force direction o f one-dimensional dynamic system motion. For the case o f two-dimensional dynamic system motion, the direction o f the friction force is in opposite direction to the net force until the magnitude o f the sliding velocity is greater than 0 I mm/s This value is chosen after comparing different simulations with various ranges o f velocity values 1
Recall state for step
2
If state (/„.,) = sticking, and N\ns < | ( / j | , then state (/,,) = micro-slip, and the friction force is given by:
/•;(/„) 3
v/i'/zf/-(/,,)]
It'slate (/„./) = micro-slip = > calculate S(t„) =
(4-6) .V(/„) ] (to determine if the
sliding velocity and the net external force are in the same direction) 3 1 If S(/„) < 0, then state (i„) = micro-slip, and the friction force is given by equation (4-6) 3 2 If S(/„) > 0, then state (/„) = sliding, and the friction force is given bv /•',(/„) =
4
v/tf//[.V(/,,)]
(4_7)
if state (tn_j) = sliding, then the friction force is given by equation (4-7)
4.3 Simulation of Dynamic System Motions
4.3.1 Introduction Simulations o f one- and two-dimensional dynamic systems motions with friction are presented in this section, using the algorithms and solutions discussed in chap ter 3 and sections 4.1.1 and 4.1.2. The simulations are carried out by utilizing Lahey Fortran-90
programs to get the outputs, and Axum-5 0 to analyze the dynamic responses o f the various models. The time steps are chosen such that they will be less than one thousandth o f the period o f the highest frequency, among the natural, excitation and normal force frequencies In this thesis, the original AABG-SDFM is slightly modified with the following two objectives: a) To com pare the model with the other friction models regardless o f the type and values o f coefficients o f friction, the modified model uses both static and kinetic coefficients o f friction. Various simulations with AABG-SDFM confirmed that the capability o fth e model is not affected by using tw o distinct values o f coefficient o f friction b ) To distinguish between 'tru e ' sticking and momentary sticking states. Due to the very small time step used in the present work, the original AABG-SDFM detects momentarily sticking when the system 's sliding velocity changes direction even though the net external force is higher than the maximum friction force. To avoid this momentary sticking, the model is modified in such a wav that it captures sticking only if the net external force is less than the maximum friction force. For each considered friction model and test case, two independent simulations are made using analytical and numerical solutions. However, in this section, the dynamic response figures are from the analytical solutions. The equivalent simulations using the numerical solutions based on the fourth-order-Runge-K utta method, and the relative errors that can be generated by using numerical solutions (for one sample case), are shown in appendix B. In this thesis, the dynamic responses and state o f motion o f the systems are represented by a solid line for TR-SDFM , dotted line for .AABG-SDFM. dashed line for RFBFM, long dashed line for O-FBFM and dash-double-dot-dash line for CFM
The schematic diagram o f the dynamic system considered for this test case is shown in Figure 3.1, and the param eters and the initial conditions are described in Table 4 I This simulation is intended to show the problems o f the CFM in simulations o f dynamic system motion near zero velocity, and also to discuss the dynamic responses o f the models during sticking. The system is subjected to both sinusoidal tangential and normal forces, in addition to constant normal, spring and damping forces. To increase the chance o f getting more stick/slip types o f motion, which is a challenging type o f motion for the models to simulate, a normal force with a high amplitude and frequency is used This variable normal force makes the friction force change fast and leads to stick/slip motion The velocity window for the O-FBFM is less than 0 1% o f the maximum velocity The sticking velocity o f the R-FBFM for any sticking period is the smallest velocity recorded near zero velocity, i.e., just before or just after the velocity changes sign. An adherence stiffness o f 10000 times the system stiffness is used for both the AABG-SDFM and TR-SDFM after comparing different simulations with various ranges o f stiffness values The analytical time history responses o f the displacements, v elocities, accelerations, and friction forces o f the system using all five models are shown in Figure 4 2 (a-d) To magnify the differences and limitations o f the force-balance and spring-dam per friction models during sticking, detail figures ( Figure 4.3, details A-D). are provided The state o f motion o f the system is displayed at the bottom o f each dynamic response figure with a rectangular stepfunction, where the upper value represents sliding and the lower value sticking
NAM E
SY M B O L
VALLES
I NITS
VIODELS
M ass
m
1
ku
ALL
Stiffness
K
400
N ill
ALL
Damper
C
1
\ s 111
\l.l.
Damping ratio
"3
o 025
-
ALL
Static coefficient o f friction
Ms
0 4
-
ALL
Dynamic coefficient o f friction
Mk
0 23
-
ALL
0 1
m
ALL
Initial displacem ent Initial velocity
.V
0
ni/s
ALL
Natural frequency
to,,
20
rad; s
ALL
Constant normal force
N
9 s
N
ALL
A m plitude o f v ariable normal force
N„
>
\
ALL
Am plitude o f excitation force
P
N
ALL
Frequency o f v ariable normal force
„
15
rati, s
ALL
Frequency o f excitation force
(O
10
rad. s
ALL
Tim e step
At
> I4E - 4
Simulation duration time
r
V elocity w indow
V
Adherence stiffness
K,
\L L >
\L L
0 1
mm s
O -FB FM
4000
V mm
\A B G - & T R -SD FM
A dherence dam ping ratio
'il
15
-
A A B G -& T R -S D F M
Table 4 I T he input parameters o f the one-dim ensional system .
Figure 4.2 (a-d) Dynamic responses of friction models with input parameters given in Table 4.1
. ►R* *.t
H*8FM
WfyFM ■»/»BO- 'X F M ;dfw
M B G -S O F M | ’ R-SD FM I
•
j 0C J
*s*o r
*150K •■*» r •*t*2r
•iisa h
•jjta ;?20
>*efw
=*-PBFU AA6G -S0FM ,
'R-SOFM
3 9»8
T92D
0 9 22
192*
392«
1 323
T930
J F3f1,1
**»: :df» *»
xfm
'1332
Figure 4 3 Detailed dynamic responses o f friction models during the first sticking occurrence
As shown in Figure 4.2 (a-d), the slipping portions o f the simulations ofall considered models agree well since all o f them use Coulomb kinetic friction. However, when the system searches for sticking, the sign o f the velocity response o f the CFM rapidly changes, as shown in Figure 4 2(b), causing considerable numerical chattering with large amplitude oscillation in the acceleration and friction force responses. Figures 4 2(c) and (d). respectively Therefore, it can be concluded that the CFM cannot simulate stick/slip motion, where the sliding velocity is close to zero. By comparison the simulation results o f the force-balance and spring-damper friction models do not chatter as that o f CFM (Figure 4 2 (a-d)) However, their responses during sticking have slight differences as they have different criteria and friction force tiinctions as shown in Figure 4 3(a-d) During sticking, the O-FBFM gives zero acceleration (due to the force balance). and a constant small sliding velocity for each sticking period. Therefore, the system is actually in motion with a constant low velocity during each sticking period, rather than slowing down and stopping. Each sticking state has a different constant low sliding velocity Their differences depend on the size o f the velocity window If the velocity window is large, the sticking velocity will be large which will give a considerable displacement during sticking, and which results in early breaking o f sticking states. If the velocity window is small, the O-FBFM algorithm might not detect sticking as the velocity solution may jum p over the window Therefore, its performance mainly depends on the size o f the velocity window, which consequently results in limitations in approaching zero' velocity
During sticking, the R-FBFM also has a constant sticking velocity However, the RFBFM avoids the requirement o f a predetermined velocity window by taking the smallest velocity recorded near zero velocity, i.e., before or after the velocity changes sign, as the sticking velocity As shown in Figure 4.3, detail B, the sticking velocity o f the R-FBFM is much lower than that ofO -FB FM . Consequently, the micro displacement during sticking is small com pared to O-FBFM displacement (Figure 4.3. detail A) The acceleration and friction force behavior o f the tw o models are nearly the same, i.e., step changes to constant values (Figure 4 3, details C and D, respectively) In the AABG-SDFM, the sticking friction force has a sharp peak (Figure 4 3. detail D) in the opposite direction to the first sticking velocity direction ( Figure 4 3. detail B ). which is because the fictitious adherence spring and damper are set to be active just after the velocity has changed its sign. The adherence damping force dominates over the adherence spring force. It accelerates the mass back and overshoots zero velocity Figure 4 3. detail C. shows the sharp acceleration peak corresponding to the initial sticking friction force This situation affects the consequent smooth approach o f the system to zero velocity ( Figure 4 3. detail B) When the sticking velocity finally approaches to zero, the adherence spring dominates Despite the acceleration peaks, the displacement during sticking is negligible as shown in Figure 4.3, detail A. As shown in Figure 4.3. details A-D. the TR-SDFM gives a smooth and chatter-free simulation o f dynamic system motion. The sticking velocity continuously decreases and approaches to zero (Figure 4.3, detail B), unlike the O-FBFM and R-FBFM. which do not improve or change the sticking velocity. The displacement during sticking is much less than a micro-metre (Figure 4.3, detail A). The acceleration and friction force functions (Figure
4 3, details C and D, respectively) do not have sharp peaks unlike the AABG-SDFM . but rather they have smooth transitions from sliding to sticking. The sticking example shown in Figure 4.3 involved backing up and repeating only tw o time steps As a further check the first simulation case is repeated using numerical solutions (fourth-order Runge-Kutta numerical method). The system 's dynamic responses and relative errors, which are produced by using fourth-order Runge-Kutta method, are shown in appendix B, Figure B -l and B-2, respectively The relative error is defined as the difference between the analytical and numerical values divided by the analytical value, and multiplied by 100% The relative errors during sliding are negligible. During sticking, the force-balance friction models reduce the system equation o f motion to a very simple expression (constant velocity motion), and therefore their relative errors are insignificant The spring-dam per friction models' relative errors are higher during the transition from sliding to sticking, and also during sticking. The AABG-SDFM has the highest relative errors in all dynamic responses. Its maximum relative displacement error is -2.5E-5% . velocity error 2 0 1%. acceleration error 20%. and friction force error - 10%
4.3.4 Simulation of two-dimensional dynamic system motion The ultimate goal o f this work is to provide a more stable and good friction model that can detect and simulate stick-slip motion o f a dynamic system on a two-dimensional sliding surface. A two-dimensional dynamic system with a mass on the x-v plane, and springs and dampers connected in x and y directions is shown in Figure 4.4 The excitation forces drive the mass to move on the plane and the normal force produces the friction forces.
In the study o f two-dimensional motion, the sliding friction force is divided into X and Y components. The m agnitudes and directions o f the com ponents are determined by the instantaneous sliding velocity unit vector, which consequently results in coupling o f the X and Y equation o f motion as follows.
A/A' - ( ' X - K X - [Po cosco / - P J - [.V c o s w / * A' Ju. vA~- r M Y + C Y +■ K Y - [Po coso)d / + P ] - [¿V cosco /
-
^
va
- r
where P. is the constant part o f the excitation force, and P„ is the amplitude o f the variable excitation force part. This coupling o f the system equations requires the formulation o f a new analytical solution function for every time step by resetting the time t to zero and using the latest displacement and velocity as the initial values.
The algorithms and solutions o f the tw o force-balance and the two spring-damper friction models, which are discussed in chapter 3 and section 4 I. are extended to simulate two-dimensional dynamic system motions. The R-FBFM and the spring-dam per friction models require change o f sign o f tangential sliding velocity in order to detect sticking However, in the two-dimensional case the simultaneous change o f signs o f both \ and v velocity com ponents is very unlikely. Basically, the change o f sign criterion only indicates that the sliding velocity is close to zero and calls for the second criterion (comparison o f the net external force magnitude and the maximum friction force) to check for sticking Since both velocity com ponents may not change their sign simultaneously, the second criterion should check for sticking whenever either o f the velocity com ponents changes sign With this concept all the friction models which require sign change to check for sticking (R-FBFM. AABG-SDFM and TR-SDFM ) are able to simulate two-dimensional dynamic system motion. Simulation o f the two-dimensional dynamic system shown in Figure 4 4 with input system param eters given in Table 4 2. which is used by Tan and Rogers (1996). is made using the tw o force-balance and two spring-damper friction models The analytical time history responses o f the displacements, velocities, accelerations, and friction forces o f the svstem in the X- and Y-direction are shown side by side in Figure 4 5 The present AABG-SDFM results are an improvement on those obtained by Tan and Rogers (1996). Their method for detecting sticking for this model was unsuccesstiil for motions on a two-dimensional surface.
As Figure 4.5 (a) - (d) show, both the x and y displacement and velocity com ponents are positive, which means the mass is sliding in the first quadrant o f the x-y plane with no return. The state o f motion o f the system has a cyclic pattern which is: slide - stop for relatively long period o f time - again slide - stop for a short period o f time - repeat the cycle During sticking, the displacement responses o f all friction models are supposed to be nearly constant. However, the sticking velocities o f the O-FBFM are relatively large and results in significant displacement during sticking. Figure 4.5 (a) and (b). The acceleration and friction force com ponents o f the AABG-SDFM (Figure 4.5 (e) - (h)) spike when it detects sticking, which were observed in the one-dimensional motion simulation case as well ( Figure 4.3. detail C and D). R-FBFM has small constant sticking velocities and the displacements during sticking are very small compared to the O-FBFM The dynamic responses o f the TR-SDFM are smooth and well behaved (no peaks). In spite o f the O-FBFM and AABG-SDFM limitations mentioned earlier, their dynamic responses agree very well with the R -FB FM 's and TR-SD FM 's. Therefore, they might be safely used for simulation o f two-dimensional dynamic system motion. M ore exhaustive evaluations o f the friction models are provided by the sensitivity studies given in the next tw o chapters.
NAME
SYMBOL
M ass
m
L NITS
MODELS
kg
ALL
N/m
ALL
N s, m
ALL
0 02 9 . o 0 4 !
-
ALL.
o 40
-
ALL
0 I ). 0 ( )
ill
ALL
Initial velocity
0 0 0 1 . 0.001
Ill/S
ALL
Natural frequency
1 1 55. 8 15
rad/s
ALL
Stiffness
Kn . K,
Damper
C ,. c ,
Dam ping ratio
r r 'S\ * "9\
C oefficient o f friction Initial displacem ent
• Y„
VALLES ** 40 0 . 200
Constant normal force
N.
20
N
ALL
Am plitude o f variable normal force
N„
4
N
ALL
C onstant excitation force
P , P,
5. 4
N
ALL.
Am plitude o f excitation force
P ‘ .*\- P * o\
2. 1
\
ALL
Frequency o f variable normal force
50
rad/s
ALL
Frequency o f excitation force
10. 20
rad/ s
ALL
0 0001
s
ALL
s
ALL
Tim e step
■It
Sim ulation duration time
T
V elocity w indow
V
«) i) 1
min >
n -r c F M
K,
40 0 0
N mm
A.ABG- A: TR-SDFM
15
-
A A B G -& TR-SDFM
A dherence stiffness A dherence dam ping ratio
f
'at
Table 4.2 The input parameters o f the two-dimensional system
On
¿0 00« I
K 3
V l . l l t l 4 l It'll Ht V I l U i l l M . . . . ,
0*FBFM R-FBFM AABG-SOFM TH-SOFM
!
J i i t i i M M . i » «i A t t t l r i «ii.SOFM i ------------ TR-SDFM
CHAPTER 5 PARAMETER SENSITIVITY STUDY OF ONE-DIMENSIONAL SYSTEM
5.1 Introduction The following param eter sensitivity study was undertaken to answer tw o basic questions:
1) Are the force-balance and spring-damper friction models effective in capturing
sticking for a wide range o f input parameters? 2) Which model gives the lowest sticking velocity (or which model has the best performance)? The schematic diagram o f the dynamic system considered for the experimental study is shown in Figure 3.1. The experiment was carried out by; first, randomly generating unique input values from the uniformly distributed parameter domains shown in Table 5 I (at the end o f this chapter); second, computing derived input variables which are shown in Table 5 2 using the param eter relations given in Figure 5.0; third, specifying other necessary input variables such as time step, simulation duration, initial conditions, velocity window ( for OFBFM), and adherence properties (for AABG-SDFM and TR-SDFM ); fourth, simulating dynamic responses o f the system using the algorithms and solutions discussed in chapters 3 and 4; and fifth, collecting measures o f performance. This procedure was repeated for 1000 simulation cases. The smallest sticking velocities and sticking frequency counts attained by the four friction models were recorded and studied. Determination o f all the required input variables using definite param eter relations ensured unique input system parameters and a unique dynamic response for each simulation case. Table 5 .2 shows that the derived input quantities have a wide range o f values.
Y Po
Figure 5.0 Input parameter relationships for the sensitivity study
O ther important simulation data were determined from the above param eters as follows: a) The time step is one thousandth o f the shortest period o f the three frequencies, i.e.. among natural, tangential or normal force excitation frequencies; b) The simulation duration is four times the transient decay time o f the damped system with no friction ( T = 4 / (u>J. In other words, it is a duration until the amplitude o f the transient response falls to roughly 2% o f its initial value. This is determ ined from the fact that the inclusion o f friction will accelerate occurrence o f a sufficient sticking count before the end o f time T. H ow ever in some cases there will be a lot o f sticking in the prespecified period T which is due to the nature o f the system parameters or the period T itself is too long due to relatively small values o f damping ratio and natural frequency In these cases, the simulation is term inated when there has been five periods o f sticking;
is stretched to a position where it can overcome the tangential excitation force and the friction force,
( KX. > P.. + ( N.. +
,
d) The velocity window for the O-FBFM is given as an input param eter Its value is automatically regulated by the amplitudes o f the transient and steady state velocity responses due to the tangential force and friction force. The size o f the velocity window is 0 025 percent o f the com posite average velocity shown below
where Xd = ¡¡A2 + B 2 (1 + ( a ) J , .V, = \ j D2 - A'; . and
Xrl = \ j d z - H 2
The expressions for A and B, D, £ , G, and H are given by Equations (3-17 and 3-18) and (3-9 to 3-13). This velocity window was found to be suitable after comparing different simulations with various velocity factors. When the velocity window is large, there is a higher sticking frequency count and a higher sticking velocity value, and when the velocity window size is small, the algorithm misses some occurrences o f sticking. e) For the AABG-SDFM and TR-SDFM friction models, the adherence spring stiffness (K ,) and damping constant (C,) are determined by the following relations and ( \ = 2sj M K t (,, where K is the system spring stiffness (derived parameter), \ f is the mass (randomly generated parameter), o is the stiffness ratio o f the adherence spring to the system spring, and (, is the adherence damping ratio. The same stiffness ratio and damping ratio are used for all simulation cases, and their values are 10000 and 1 5. respectively This means, each simulation has a unique adherence stiffness and damping constant values since K and \ f vary from case to case. Stiffness ratios o f 100 and 1000 were tried, but both cases give higher sticking velocity errors compared to a stiffness ratio o f 10000
The four models (O-FBFM, R-FBFM, AABG-SDFM and T R -SD FM ) are com pared to each other with respect to their capabilities in approaching zero velocity and the number o f sticking states detected for the same simulation duration. The first sticking occurrences are used for comparison since the subsequent responses and sticking velocities will be affected by the enforced sticking criteria during the preceding sticking. This can be illustrated as follows: during the first sticking period, the O-FBFM will have a higher sticking velocity than R-FBFM, and consequently will leave sticking earlier. There is no guarantee that the models will capture the next sticking state around the same time This is true to a lesser extent for the AABG-SDFM and TR-SDFM friction models as well. The deviation o f the smallest sticking velocity value from zero is taken as an error since theoretically the system is supposed to have zero velocity during sticking (no motion) For the AABG-SDFM and TR-SDFM friction models, which gradually reduce the velocity while sticking, the smallest (final) sticking velocities values are used for comparison To relate the velocity errors with the input system parameters, they are normalized by dividing by 0 I percent o f the composite average velocity (referred to above) This dimensionless quantitv is used for comparison o f the models' performance.
5.2 Observations O f the 1000 simulation cases the R-FBFM. AABG-SDFM and TR-SDFM friction models found only 268 cases with sticking, while the O-FBFM detected 344 cases In each o f the additional 76 cases the O-FBFM detected only one sticking state This happened for systems with parameters o f high natural frequency and high to .
to,, frequency ratio To
illustrate this situation. Figure 5 l(a-d) shows the displacement, velocity, acceleration and friction force responses o f the system using the four friction models The system attained a small amplitude and high frequency steady-state vibration after about 0 55 second o f the simulation duration. Only the O-FBFM detected a brief sticking occurrence at the time o f transition from damped to steady-state motion. It was also observed that the distributions o f all input param eters o f the 268 simulation cases except the natural frequency are fairly uniform as expected. The input values o f the natural frequency ranged from 6.2 to 304 4 rad/s However the mean value was 78 7 rad/s For this skewed distribution, 64 % o f the natural frequencies w ere below the mean value and one third o f the cases had a natural frequency between 15 - 45 rad/s This result shows that natural frequency has a large effect on the occurrence o f stick-slip motions Although this result looks interesting, further analysis on the effects o f individual input parameters on the dynamic responses is not considered in the present study The extreme values, means and standard deviations o f the input parameters are shown in Table 5 3 Although the R-FBFM captured the same number o f sticking cases as the AABGSDFM and TR-SDFM friction models, it recorded one more sticking period for six simulation cases, which is not as serious as missing sticking. A sample case is shown in Figure 5 2(a-d). where the two spring-damper friction models show a continuous sticking period, whereas, the sticking periods o f the two force-balance friction models are interrupted by a small slip region which consequently results in one more sticking record. This is because the O-FBFM and RFBFM are passive during sticking, i.e.. they do not do anything to reduce their relatively large
sticking velocities, and consequently, the net forces grow with the sticking friction forces (due to the force balance), and eventually break sticking (Figure 5 2 (b)) The AABG-SDFM and TR-SDFM friction models recorded the same sticking frequency count for all simulation cases The TR-SDFM backs up only 2 time steps for 50 percent and 3 time steps for 25 percent o f the sticking cases, while the minimum back up is 1 and the maximum is 7 This confirms that the TR-SDFM does not have much extra computational cost compared to the AABG-SDFM To see the effect o f the simulation duration, it was doubled and the experiment was repeated. The results show that the number o f sticking cases increased by only 11 Therefore, the suggested simulation duration and number o f simulation cases are sufficient. M oreover, another 1000 simulation cases were run using a different seed value The R-FBFM and the tw o spring-damper friction models detected sticking for 2 9 1 cases while the O-FBFM captured 366 sticking cases. These values are almost the same number o f sticking cases that were obtained with the original seed. As a further check the solutions were also run using fourth-order Runge-K utta numerical integration. In all simulation cases, the dynamic responses, sticking velocities and sticking frequency counts o f systems using the numerical solutions agreed very well with those obtained by using the piece-wise continuous analytical solutions
5.3 Four Example Cases Four simulation cases are presented as samples o f the sensitivity experim ent. The samples have been selected such that they represent simulations o f dynamic systems with various extrem e combinations o f frequency ratios oo./ g>0 and u>„/ a>0 The values o f coc/co„, u>n / to,,, P„ /K and N„ / Nc are given in Table 5.4. The analytical tim e history responses o f displacement, velocity, acceleration and friction force are shown in Figures 5 3-6
The first simulation sample. Figure 5 3, has relatively high
and low o>n / u)„
frequency ratios. After a simulation time o f 0.7 second, the system exhibits stick-slip motion with roughly equal durations o f sticking and slipping. Since the variable normal force is very small, the friction force during sliding is almost constant. Figure 5 3 (d) The sticking friction forces grow to -16.2 N before leaving sticking, where as the sliding friction force is - II 2 N, this is because the static coefficient o f friction is 1.45 times o f the kinetic coefficient o f friction in this example. The second and third simulation samples. Figures 5 4 and 5 5, respectively, have relatively high u>n/ co„ frequency ratio and normal force amplitude ratio N„ 1 N The w, /w,, frequency ratios are 3.73 and 0.41, respectively, which are relatively high and low In the second sample the first sticking is detected at about 1 7 second, and the sticking durations are much longer than the slipping durations. In the third sample the models detect the first sticking at about 0.6 second, and the sliding durations are longer than the sticking durations except the first sticking period. In both samples, the AABG-SDFM gives sharp peaks in the accelerations and friction forces function (Figure 5.4 (c-d) and 5 5 (c-d), respectively) when it d etects sticking. In the last sample. Figure 5.6, both coVco,, and o)n/ o>„ frequency ratios are relatively low. As shown in Figure 5.6 (d), the sliding friction force functions are continually changing with time since the normal forces amplitude ratio N„ / Nwis high (0 9 1 8 ) The acceleration and friction force curves (Figure 5.6 (c) and (d), respectively) o f the AABG-SDFM also have small sharp peaks when sticking is detected.
The normalized sticking velocity error distributions ( histogram s) o f the 268 simulation cases, which had sticking at least once, are shown in Figure 5 7 The frequency counts o f the distributions decrease with increasing normalized sticking velocity error, except for the OFBFM which is in the other way around. These distributions can
be approxim ated by
exponential functions and can be treated statistically by Weibull distributions As shown in Figure 5.8 the normalized sticking velocity error distributions have satisfied the requirement o f the Weibull distribution, which demands that the In In (I R) verses In Error plot to be a straight line (Shigley and Mischke, 1989) Ii (reliability) is the value o f the cumulative density function complementary to unity Error is the difference between the normalized sticking velocity error and a guaranteed normalized sticking velocity value (x „ ). which is zero for R-FBFM, .AABG-SDFM and TR-SDFM , and 0 3758 for O-FBFM The other tw o Weibull parameters are the characteristic value, 0, and the shape parameter, b. Their corresponding values are determined using Figure 5 8 The characteristic value found from ///(0 - x„) which is the abscissa point corresponding to a reliability (R) o f 0 368 or In In ( / R) = 0. The shape parameter is simply the slope o f the fitted straight line The values o f these three Weibull parameters (x„, 0. and b) for the four friction models are given in Table 5.5. The mean and the standard deviation o f the Weibull distribution are given as ( Shigley and Mischke, 1989): H - x, - (0 - x j T(1 - Mb)
a = (0 - r , ) [ m
- 2Ih) - T2( I - I/A)]| :
where T is the gam ma function. Using these equations and the Weibull param eters (Table
5 5), the mean and the standard deviations o f the normalized sticking velocity errors o f the four friction models are calculated. As a summary. Table 5 6 shows the descriptive statistics o f the normalized sticking velocity errors. This experimental study indicates that the TR-SDFM friction model has the smallest mean, standard deviation, minimum and maximum normalized sticking velocity error values o f all the friction models. The AABG-SDFM has the second smallest sticking velocity error values, and followed by R-FBFM and O-FBFM, in order The mean normalized sticking velocities o f the O-FBFM, R-FBFM, and AABG-SDFM are 30 3. 8 5 and 2.7 times, respectively, that o f the TR-SDFM friction model. Its mean sticking velocity error is only 6 millionths o f the composite average system velocity The expression for reliability is given by (Shiglev and Mischke,
1989)
R(x) = exp(-[(.r - x j t (0 - . r j f ) where x is the normalized sticking velocity error. Figure 5.9 shows the normalized sticking velocity error probability curves o f the four friction models, which are given by P( \ ) = 100 [1 - R(x)]. in percent The probability curves o f the two spring-damper friction models are steeper than the two force-balance models, and consequently they give much smaller normalized sticking velocity errors for the same percent o f probability (certainty). For example, the maximum expected sticking velocity errors o f the O-FBFM, R-FBFM, AABG-SDFM and TR-SDFM friction models with a 95 % confidence are: 0.256, 0.170, 0.076 and 0.028 in order. The last o f these represents only 28 millionths o f the com posite average system velocity which confirms the effectiveness o f the TR-SDFM model in reducing the sticking velocity nearly to zero
Minimum
Maximum
VI (kg)
10
50
co,, (rad/s)
2n:
1o-
0 0|
in n
0 2
() 4
to. / to,,
0
>o
«»/«*>.!
0
50
P„ / K (mm)
0
10 0
N„ / N.
0
10
LU/ !-k
10
2 'I
Table 5 I Ranges for randomly generated input parameters which are used for one-dimensional parameter sensitivity study
Minimum
Maximum
K. ( N/ m)
39 o
-}>) ;4 si 111
C (N s/m )
0 125
15”
P. (N)
0
49.>5 o
to. (rad/s)
0
500r
ton (rad/s)
0
500-
N (N )
9 81
49 i)5
N„l N)
0
4 U i >5
u,
0 15
M-III
Table 5 2 Derived input parameter ranges for one-dimensional parameter sensitivity study
Variable
Vlean
Std Dev
Minimum
Maximum
VI (kg)
2 96
1 14
1 01
4 95
ti)„ (rad/s)
78.7
70 0
62
304 4
r'S
0 028
0.011
0 010
0 049
Mi
0.2S8
0 070
n 151
0 400
< £L cx * tw 8. o 3» (IQ
S. n
t UQ o Q. » •3 £ o cr >< O
3 BJ 3 Q. ?o i T) DO CL.
C ft
IMihoti fc*n*iNt
----- o>*eFM ............. RFOFM AABO-SOFM -----------
TII'SOFM
Normalized sticking velocity error
s„cW.n8'etocUy
error
histogram toV
the
, 1 M on"aV'zeli
In Error
Figure 5.9 Probability o f normalized sticking velocity error ( P = 100
w ootf I*1 h*
“ «« -
(rad/s)
C ,, cy
(rad/s) (rad/s)
Table 6 1 Ranges for randomly generated input parameters which are used for two-dimensional parameter sensitivity study
Minimum
Maximum
(N/mm)
0.039
1973 920
Cx , Cv (Ns/m)
0.125
314 159
PC,, PCy(N)
0
1973 920
Pov P o ,(N )
0
2950 880
NC(N)
9 81
49 05
N„ (N)
0
49 05
0.15
0.80
Table 6.2 Derived input param eter ranges for two-dimensional param eter sensitivity study
03^ (rad/s)
Sample Figure
Simulation case number
o)„ (rad/s)
toov (rad/s)
6.2
12
479 10
65 16
38 05
61 96
6.3
257
507.36
319 46
247 56
100 78
6.4
541
304.11
355 65
101 01
13 08
6.5
773
543.47
22.71
156 69
206 37
6.6
907
63.81
58.30
7 48
94 69
(rad/s)
Table 6.3 Some o f input parameters o f the five two-dimensional example cases
0
Model
b
O-FBFM
0.2751
2.35E-OI
-2.7020
R-FBFM
0
3 48E-02
7 734E-0I
AABG-SDFM
0
2.33E-03
5 2S5E-01
TR-SDFM
0
2.17E-03
5 I80E-01
Table 6.4 Weibull parameters for the normalized sticking velocity error distributions tor two-dimensional cases
Model
Mean
Std. Dev.
Minimum
Vlaximum
O-FBFM
2 .180E-1
4.820E-2
4.819E-2
2 499E-1
R-FBFM
4.043E-2
5.287E-2
4.047E-4
2.387E-1
AABG-SDFM
4.225E-3
8.756E-3
4.263E-6
9 045E-2
TR-SDFM
4.073E-3
8.677E-3
2.365E-6
4.922E-2
Table 6.5 Errors summary for 183 two-dimensional cases
(C)
D iiplactracnt ia X -direction tm ni
Figure 6.1 Two-dimensional path trajectories o f four sample cases Motions start in upper right-hand corner
OFBFU
R-FBFM AABG-SOFM TR-SOFM 06 04
0 2
7-0 0 1-0 2
; -o 4
: -0Q -oa -i o -t 2
02
03
r u m at
04
Q5
O -FB PM R -F B F M AABO-SOFM t r .s o f m
vM
n
U
u
o oo
00 3 a 04 a o a o o a
a to a
12a
rmnn
t4 a i a o
140.20022 a 2 4
IÍOPN ?a. 3 &
fa
Timm i
B Ï CL O e> « ? 8 to a o s cm D.
f l
it
oo
vo
p 2
ë c/) _ - 8
9e lc/> bñ* O B P
co 3
-
Í
0
s: CL1
rt* 3 9 § 9* SC s § £ ^ 5 Jg SS n
3
s.
D tip U ic n rn i
ih
V
4hcoy almost the same, and gj.n is much higher than co.,
O-FBFM R-FBFM AABO-SDFM TR-SOFM
07K i 35 0 3
=^
■*
* = •0 5 1
-0 7
9 •t ! t 3 •I 5
ooo a
020040. 0*0o a a
to a
12o
i4 ate o ta a
2002202*
Figure 6.5 (a-d) Displacement and velocity responses o f a two-dimensional system in the case when O-FBFM detects one m ore sticking than the other friction m odels
•hi O -FB FM R -F B F M A A B O -S D F M T R-SD FM
Q 0 0 0 0 0 2 0 0 4 0 06 QOS 0 t o 0 12 0 14 0 T6 0 18 0 20 0 2 2 0 24 rx rv t • i
Figure 6.5 (e-h) Acceleration and friction force responses o f a two-dimensional system in the case when O-FBFM d e te a s one m ore sticking than the other friction models
/ D ' ' coso),/ + E " sinw ,/
(A-22)
where
D'’ =
P [coso,/od - (oWu),)2) + sinai/^2^0),/g>o)] [d - (oWw/ ) 2 + ( 2CgWg>, )2](o>/
P [cosu /p C w A 0«,) - sinco/J I - (aWwj2)| [(I - ( o i j u j 2)2 - (2C w /u , r ] w > /
(A-23)
(A-24)
The particular solution Xp, is developed by considering only the harmonic normal force from the right hand side terms o f equation (A-5):
M X + CX + K X = - N cos(Jn(f + t ) \ i k sifpi(X)
(A-25)
Since equation (A-25) has similar structures as that o f equation (A -11). the particular solution
Xp, will have the same form as that o f Xp,, and therefore, all the steps that have been taken on the latter equation to arrive at equation (A-22) will be applied on equation (A -25) to get an expression for the particular solution Xp, as follow:
A' , = ( ¡ " c o s io j * H "sinoy
(A-26)
where
(}■ =
H" =
Nu (!,. [c o s o w ( 1 - (w /w ,,)2) - s in w /p C w /w ,) ] signiX) ------------------------------- —-------------------— ------------------[(1 - (co/to,)-)- - (2Ca>ya)„ )*Jw>/ N |i, [cosw / (2Qix>Jw) - sin w / ( I - ( to /c o ) 2)] sign(X) --------------------------------------------------------;------------------[(1 - ( o W o ) /) 2 - (2Cco,/w, )2]co;V/
(A-27)
(A-28)
Particular solution Xp, is developed by considering only the constant normal force from the right hand side term s o f equation (A-5):
MX + CX + KX = - N jik sign(X)
(A-29)
As long as X does not change sign, the right hand side term o f equation (A-29) has a constant magnitude with a sign opposite to the sliding velocity direction Therefore, the particular solution Xp, will have the form:
-V, = /
(A-30)
and its derivatives:
XpS = °-
=0
(A-31)
Substituting equations (A-30) and (A -3 1) in equation ( A-29) and solving for I yields:
I = -
N.\i,
sign(X)
(A-28)
The com plete particular solution o f equation (A-5), Xp, is the sum o f the three independent particular solutions, i.e.,
XP = Xri + XP2 + - \ s
(A-33)
Substituting equations (A-22), (A-26) and (A-30) for Xp, , Xp, and Xp„ respectively, gives:
X p - D " c o s u )e i +
'sinew + •.
, i
i l , i ‘.
i / i / .
J . 4
/* I / . ’. S»/
V. / 4-4
il
I'>‘ M h>
/
.
i »-v
f/ 1 / f V .
■
>/
"
ir,
"
it o .
1 '
o
O O ll-t
. 1 4 *
O »/ _' / . **
■■ b
1 /
- ^. s
1 «■/ /
/
■>
...1
’
/>
/
! 4
J 4 ' / 4 ‘>I
/
,,
o
■-
,
, , - >/
.¡O /l» ,o
, , 1 , IH . J . '!
• ■1. n i
/
.1
*• /
' !
• /
‘
/ , t ?+
,
\
i» , h i h t n l i , , , 4
4
. » ‘ >1 '■ ' 4 ‘ h , 1
,
. ‘ 4 ' I I *
' >, ' >■>1 1
!
ì ? n !t . < / . 14*1
4 >,
i ■*+. "
•
i-UH
,1 H I U h i *
o O}
.1 ^ /
•'/
1 .
•>' ? 4 * » n
1
■, ■ 1
..*/
o 7
...
/
,,
O .'4
-•
/
-
,i , n
f .i
.1 , , 1. I l i ' r > .
- » . / -t 1'
.
t i >i
■*/ 1 . 4
1
W'
I H H I
1 / .
- • ¿1 .
/
I .
l . l k
a s i:
* - -
.
.
■
■
>
II
li
\
OltS
\l
II..
/
Ik II,' lin
I bi II.-
l ' nk \.» \ i
Í N f'A
I
.
Ni l i
Is/'
//W
H % A'
.S
N s
/ /i7v*
!
I-
* *
'
(
V m i - #■
/
l ' * ‘f ‘4 > l ' I H ' l ' l - l>
’H ' ‘ f ~ t l
‘ t‘
/ •- ¿ 4 i* I I « /
f
/ ■
, 'I /
( ■ , , ’ t I - • ,
>i > i ’ i,
I,,,, n
*
t .< >
t - t’
>'
{,
I
/ • ■>! ' / I ’ ’ / f
mi i
,,,
I
, ■I . " l i j t ' . ‘ l ’ I I
,,
f. f - I ■ • I H> I >
........... .............................. I ' l l
- , *■ ' . •< * f-
,v '
I I I t-K' I " ~i ‘I ' I >
,
*
--L. ■I
V 'V
II
•. r I / / < < *
; ; I r I r. ' . i' i' •' ' *■
- , , , */
, / * *
,,,, II f
,
,,I'I>
I
- i f - I O I ' l ' !•
f»-■ f - ‘. /
>. {
/ , .1 : f
f l ' l ’l ' I ' r
i
II ' I ' I '
s'
i ,*
'■ !
I I
>i m u i
••>.*/,f - xi ' ti I'
- I.
'
/ II ■
■
. 1 . 1 ' , '
/
mrt ’
i
y
• ■*
.
/*-'/•
, *■ t
f
.*
'
•
* i,
. # y.;
• -, b ’ ^
'■
> !•
/
r 1S /
•
'
.....................
i
* / *. .
/
,
-
.
............
I I '
1
' ♦
1 /
> 1 1 '
-,
..................... .
r-
.
s'
f
,,
. ‘
. i t
1,
>■ "
** ’
............
1•
1 /
'■
f ' *
/ , * 1‘ /
.•**,.
/
.
/ M i i /-
. f •/, "
- m / m i.
t-
•/
/
l o u r *
c
t I
I K + ff
i ' / */*■■ . + >*■>>‘>t/"/ >• f "bf-u
/: /'
/
V'
V
/
•#./(. M
.....................................'
.V '
i
*/ /
// "
.V W / /
• - f S i -/ /,
so.
/ s /
*. »
/
' S / '
.
/f
*/f I It
f/ *
"/
. */ '■ /
■■ /
t
"* /
• /f .
^I
.
I ( ' '
I
I ri i i.
Í.
•».
V.■ " f 1 i // '
* 'I ' f i
■s t
‘ ' . W*
•
!•' •>
/ k.
1S-/ «
'. «. /»
*'■■
/K'*
/. «. / . /
■ 1111 ’ / • .»■
"
•>!
*•
Y/
/
».
/ .
I*
• •-/
• •■ "H
/V
vi/o
w .
»•/*/-»
«.
-•‘ ■ i / i n ;
/.s-,/ S
/ l l ' t ' li I J ’. •
! ¡ n f
V '" '
l ¿*>¡ *•! ’I
t - ’ i ' ■ f - s ’, 1 / i
! / < II H '• * H
''/ 'i f » ;
/
t ' f i m ‘n i i i / • f
t
.ss'/
S- . / s ’ f / í ' M * . M . /)K
. / o / * , / ' . , , , , , , ,,
f II il H ' II
I ft-f
f
It-l ( u n a
,
n n * •1 ' 1
r-
\ . n n / .*«. «' »-, < i > ; n i . ' * / ‘. n i ’i ' h
jK/f' .
i ’
+ ,,< ' , *( s'/
/. t-1
■
/'
i -w f f -
r
r
n:
f O ; * /»
y
/ ■;
!•
n
/'/(f i
*
f "
j n ‘.
*-/,t n i r
* S ■ i l 1,
• M / M ,,
. -,
■
/'/
II
" t- • I I
/•
1S 1
S1,'
1
/
< t / I ' ,1
“i i “ i '
o.-.-o
/ ' » " *.»
/ :
t ,,
/. f
, f / - 11
•- s ' *■ *
/
/ " /
t-,
»./'*» M
t-Si
~ l ■■ I
/ t t- f t , ; ■ !
,•
I /
¡ n + n
■•>.■.
/ .
Kf v i . ‘
* «
s - f o f » , /
n /
■/.■»■ /**,',
/
.V '
u mu
i
'
i mi l
l' M
1
,--VM f"./ /' / /
M
n *»./ r-M«
n1f- n ; In n •. s’" n
.
/• 1
n/
■*■1 11 *
/
■ ‘•7
.*•- 1 n ......... • ,V n
l^f/l 1»
inn n1' n nnfu i' s’s' ■ nn1 n *■ ; : n*ns ' s'f i•n : ,,t nn i n S’»-' n ......
•. “ i h'C ti fJ
0/ 1.'
/t « '* 1
-« . " 1
>
./ -, ,
# ■7
.1
„
* ♦" .'I/.
. /
-, '
s ■« /
/
.,/
■» *>,
/
- ,
•
n
/.» .
.< , i f
,»
?. .
.
■" .
.
1 :
’
1
,
'
'
i
*
'
■> ■'
■' 4 „
.
i
'■
>> : i
.,/,»
> ’ 1 . ’ .s '•
- i f *
•
t ■
.
/
, 1 II]
k
-,
1
.
I» IÍ-V '■ •
i ' n
*
i /*• 7 / -
i
U n
>1 4 >t, t
■'.*» .
I I .
l h /
1 .
1 H >>,,
.s •’
lli>
.S
./ < O M u
■
1 >> S i/.*i
S --4
-
lit»
>4 1 “>
. .r.-*.
k
/!
s
t
/
■'
!
'
•
'
OIIS
\l
lin
/.
Ik
Ur Un Un Un
t'n K
\n
I\lk
I
/■ IKK
R
\ R
I RKS
xn I RK
> r.
; S. ; Ï. ■»
; í
f. :
m
II tt
M
J 1 i >1 i
J ' lu ni
V
II U
V
l l . , H' l
/
V
. . . »„■i ,
4?o.tu
X/*
OS . l o
^ «« ¡ -
il
•
■i i
1 +i h l
ill
. U4 i
7J
/. ' - i '
n> : ' a
i > | I ( M I >i>
m i ' * ;
Il ¡(»«>
,
.* Il . i ' i it i 1 :
»*(». »• >
............. I. N ¿-
il
mi..f ;.
. . ». M1J « „.
>*.■>
!•>
• .......... .. ... .
..................
■-
- - U - .
' *- *
ili'
0 . J M>
" •/ » / '
b»i*b
■ U n î'lM lï.i
i
i i. » . * H i j o i / *
i
I»
'l;
u i n i ¡ril.H . iu
■
ri I* » .'.* » í >U i
i
*1 i w , u > > :
'
«» 1» ¡ i >«W .- «
*
• H U t l t . í. l « -
,1 '» « ,* * » > .'
i » i h ». » r
i.'
t
v« ¿ : ¿ 11
J '
(»
i» l i ' i W i {.• » » ’
■>
1) 1»1**1 i
‘ > ‘ i + 'i 'r >«■
-
I I Ift .lf
,m » . r v j - i i ' i t *
i
■ M
11
-
/ ...
1
• m »< *»•»». / .
...
,1 ,* . . » ! , • >
. . .. .
■ ... - .
*
'
*1/
¡
» »
-
. ’
. . . . .
'
...................... *-■
.1 l * » .* M . - «
. i
;
iiiu m
i
) '« » ;
i
■i i »i »k ■ » u i « ;
i
*
• M »lM ¿ ¡ 1 i i.-
i
-
•.......... ....
»
-i .» ,* » .,i/.i
i
I M . » . 'V M , - i
>
.............
1
■ .. * i * » i j . ' . i *. .*
111, k »i i
•
’ »«» i ;
-*
, . . » . ( » ; i ; • - .i '
i i»
11,
i,« .. . o «
• ----------- --
•
.
•
............... ... .
,
.
-.
. - . -■
.
■ * » ■ -..
»-
• ...
. . ---------------
..
.....
-
■ -----------
....
i» «»!»:»*». * i « J l
I» ' l i l l H J >+ M> >
.
. . .
■
>U,I~ '
............. -■
............... ...
* '
. -
i
...
■
. .
••
•. > ••
- -
- .
■
i
■
.i
... ■
4
1» ,» !» (
i
i
'
l H H H J i f ',4 i
I I >«». f ;
4»»1 M
...........
.. i» ..' •
>
« *.« /
•i . * I U
.1 1 > «
...............
"
a i i 1» » « . ;
I l M il i i
; / ,
l i MH
• t-
„
,
J 1
n i » » ) < /« ;.
1 ’ »
I l 1 „• '
..r .H i.- l a • í«»* , v .i* *
It l i l i l í * 1« « f
H ,.,." ’,
I l I h t i l 1i l !
11
, . i » » , ' '.» ,» ; ;
«. i «i»- ; i - i. - *
V
•»
■ / ( .< * ' ' ;
, i .». » .*!«.
; ■ ' « im i
«•
‘ *r>r> i
1
*
. . »i
'
nil
«'
'
-r„.K. !
HI * - . 1
.
.............. ..
*'
» i f * ' í ¿ i « >ia) h
, »• i
'»•»#»' l i / *«'
■
»i.
' 'ti'
i
_
»
■
Í ! /t*"f
}
f*
i
'
'
M
*■. . . **■
/ n i
- - : «■ 11• i
.
i.
» i . *>
*
-• . i■
• i
-• i
*•> i
?'*t O
■ / ' . .................................... ... ■
s* >nn
* i.i
i
•>¡ r i - } i i i i > i l »i i ^
........ .
'
■ , . . , * • «.
¡ ,!
li
*
..........................
s*
, ,* ,
,
I i t / i 'i m i 'i 1
.................................
c
*»
--
-•
. / í i) , fn / «
-
*>. n i s * w - . *vi .
J
f
.,*• ,
,*• t , i n m « .| i «i
!
*
■
'
...............................
« . . . i r ' »h* * K
!
1-
. J »■•••/ /«' «'
• i.-L
-
•
:.H
........
\
M
- « « * • / ; »i j i»
t » . * r>¿ / « i n il
t r._ .......................
* • , . * ( . < ,*,* |< i
,-o>...-i,.,.|.
t
•- : - < * ' i i .* •>
/
*
n
•
i
w, *i . n . **.«■ h
»'Mi.
f
*
............
1
». v»; * i i . fc* ».
