APS/123-QED
Modeling of dry sliding friction dynamics : from heuristic models to physically motivated models and back Farid Al-Bender,∗ Vincent Lampaert, and Jan Swevers Katholieke Universiteit Leuven Mechanical Engineering Department Celestijnenlaan 300B B3001 Heverlee (Belgium) (Dated: March 25, 2004)
Abstract After giving an overview of the different approaches found in the literature to model dry friction force dynamics, this paper presents a generic friction model based on physical mechanisms involved in the interaction of a large population of surface asperities and discusses the resulting macroscopic friction behavior. The latter includes the hysteretic characteristic of friction in the pre-sliding regime, the velocity weakening and strengthening in gross-sliding regime, the frictional lag and the stick-slip behavior. Out of the generic model, which is shown to be a good, but rather computationally intensive simulation tool, a simpler heuristic model, which we call the Generalized Maxwell-slip (GMS) friction, is deduced. This model is appropriate for quick simulation and control purposes being easy to implement and to identify. Both of the generic and heuristic model structures are compared, through simulations, with each other and with experimental data. PACS numbers:
∗
Electronic address:
[email protected]
1
The issue of friction characterization and modelling has been steadily gaining in importance over the last number of decades. However, despite persistent and painstaking efforts of researchers over a long period of time, no satisfactory, comprehensive, practicable friction model, i.e., a model showing all the experimentally observed aspects of friction force dynamics in one formulation, has been achieved yet. Friction comprises multi-scale processes requiring multiscale theories. Such theories are however not fully developed and the importance of each theory at each level is not fully understood. Therefore most of the friction models that have been proposed to date are heuristically/empirically motivated; based on limited observations and interpretations. In this sense, the resulting models are in most of the cases only valid for that specific scope of test/observation conditions, e.g. the level and type of excitation used to obtain the data sets. On the other hand, development of simulation models and, where possible, predictive theories, at scales from atomic, through continuum, to useful engineering models, is an important endeavor. This paper presents an example of such model, where we start from the generic mechanisms behind friction to construct a model that explains observed macroscopic friction behavior. Once there are effective physically based models at different scales, future work on the inter-relationships among models should be useful in furthering our understanding of emergent, collective frictional properties. Moreover, a predictive physical model of this type could help us derive simple (reduced) heuristic models useful for control purposes.
I.
INTRODUCTION
Considering friction as a mechanical system, a close examination of the sliding process reveals two friction regimes: the pre-sliding regime and the gross-sliding regime. In the presliding regime the adhesive forces (at asperity contacts) are dominant such that the friction force appears to be primarily a function of displacement (rather than velocity). This is so because the asperity junctions deform elasto-plastically thus behaving as nonlinear hysteretic springs. As the displacement increases, more and more junctions will break (and less will have time to form) resulting eventually in gross-sliding. The sliding regime is characterized 2
by a continuous process of asperity junction formation and breaking such that the friction force becomes predominantly a function of the velocity, in addition to the other states of the system [1]. The transition from pre-sliding to gross-sliding is a sort of criticality that depends on many factors, e.g. the displacement rate and acceleration. Besides the field of tribology, where the origin of friction is one of the main topics, the modelling and understanding of friction dynamics is important in several other domains. (i) In the machining and assembly industry, the demand for high-accuracy positioning systems and tracking systems is increasing. The increasing interest and research in the class of controlled mechanical systems with friction is mainly caused by the increasing demand for these systems. Friction can severely deteriorate the performance; it can cause increasing tracking errors, larger settling times, hunting and stick-slip phenomena. (ii) In the domain of structural dynamics the damping of large space structures is important. This damping is mainly caused by the energy dissipation due to the pre-sliding displacement in mechanical joints, the so-called structural damping. (iii) Geomechanics studies the origin of earthquakes, which are unstable stick-slip phenomena due to the interaction of friction between tectonic plates. In short, friction is one of the main players in spatially distributed systems. We have chosen to divide the approaches to model friction into two classes. The first, namely, the physics motivated approach starts at the micro-level of asperity interaction, with assumed physical phenomena governing the interaction forces and an interaction scenario, and synthesizes a mechanical system that integrates them. The resulting model could then simulate and analyze the macroscopically observed friction force dynamics. Thus while the basic mechanisms responsible for friction may remain phenomenological, i.e. empirical, the resulting macroscopic behavior will be a sort of ”emergent” behavior that might qualitatively vary with the choice of the parameters of the micro-process. This approach of modelling is popular in the domain of tribology [2],[3], atomic friction [4] and geomechanics [5]. The purpose of this approach is to gain insight in the mechanisms determining the friction behavior. The second class of models, namely the heuristic/empirical ones work the other way round, i.e. looking at the process from without, by asking the question: which dynamical model can best account for (or fit) observed macroscopic frictional behavior, regardless of what the actual physics of the problem might be. The purpose of these models is then to capture as much as possible friction properties in one formulation with a minimum set of parameters, which have to be determined. Owing to their relative simplicity, those resulting 3
models are appropriate for quick simulation or for control purposes. Let us finally note that when those heuristic models would have attained a high degree of complexity, they could approach, in terms of predictive power, the physics based ones. However, they would remain, in essence, heuristic since they do not predict the behavior from the ”internal” physics of the problem, but rather from an ”external” system consideration. The underlying aim of this paper is to lay a link between those two approaches; in particular, to deduce a simple heuristic formulation from the results of a generic physics-based model, and to compare the two with experimental observation. The next section highlights the state of the art of these two approaches in more detail. Section III outlines a generic model for dry friction force dynamics [2] based on the first approach of modelling and section IV views the resulting macroscopic friction behavior using this novel friction model. Section V briefly discusses the formulation of a heuristic friction model, called the GMS model [6], useful for control purposes. This model is an empirically motivated model based on the frictional macroscopic behavior of the generic model. The friction dynamics resulting from both model formulations correspond to the experimentally observed friction behavior, as is shown in section VI. Finally, section VII poses some open questions which have to be dealt with in the near future.
II.
MODELLING OF FRICTION IN THE DIFFERENT DOMAINS
To recapitulate, friction models found in literature can be divided roughly into two approaches: (i) the models based on the physics behind friction phenomena and (ii) models which are based on heuristic considerations of experimentally observed data through which a best fit is sought.
A.
Physics motivated friction models
The different physics motivated friction models proposed in literature can be divided into three classes according to the scale of approach: (i) at asperity scale (common in the control domain), (ii) at tectonic plate scale (common in the geophysics community), and (iii) at atomic/molecular scale (common in nanotechnology). Despite the wide range of scales, there is some similarity between the approaches, which emanates from the self-similar nature
4
of the friction process. Historically, Coulomb was perhaps the first researcher looking into friction at asperity scale. Realizing that surfaces were not ideally flat and were formed by asperities, he proposed that the interlocking asperities could be a source of friction force. Recognizing the presence of asperities and the existence of interfacial forces, Bowden and Tabor [7] proposed that adhesion occurs at asperity surfaces and that shearing occurs upon translational motion. This model explains a number of effects such as the disparity between true area of contact and apparent area of contact and the relation between friction force and load, since the asperities and thus the true area of contact change with load. However, their models remained restricted to the basic mechanisms of friction, rather than the dynamics of asperity interactions. During a friction modeling seminar in 1987, Francis Moon suggested that a friction contact can be treated as the contact between bristles of a brush and another surface. His talk started a revival of Coulomb’s idea of asperities in the 90’s [8–10]. Based on this idea, Haessig and Friedland [8] developed a bristle model which tries to simulate friction in the pre-sliding as well as in the sliding regime. The physical paradigm underlying the bristle model is a pair of facing surfaces with deformable bristles extending from each; the friction interaction occurring at the bristle tips. In the geophysics community, at tectonic scale, Brace and Byerlee suggested in 1966 that earthquakes might be caused by stick-slip instabilities in the relative sliding of tectonic plates [11]. The use of computational simulation of earthquake activity as a strategy for understanding seismicity begins with a paper written by Burridge and Knopoff in 1967 [5]. They proposed that a vertical strike-slip fault in the Earth’s crust can be modelled as a one-dimensional string of sliding blocks with mass m connected to one another by springs with stiffness kc (see figure 1). In this model, the build-up of tectonic forces is represented by the externally-driven motion of a ’loader plate’ that is connected to the blocks via springs kt . The main feature of the local friction function Ff , of the relative velocity at each block, is that it must decrease more or less suddenly as the velocity deviates from zero such that the system is in a condition of dynamic instability during the shock. Many variations on the Burridge-Knopoff model have been proposed which simulate not only the earthquake but also the aftershocks. Since the invention of the Atomic Force Microscope, the research on the physics behind friction force at atomic scale has intensified. The classical molecular/atomic models are 5
v
kt
kc m
FIG. 1: The Burridge-Knopoff model.
described extensively in a number of review articles and books [12, 13]. The basic outline of the different models/methods is straightforward. One begins by defining the interaction periodic potential VE , such as the Lennard-Jones potential, sine-wave potentials, simple ideal springs, etc. This produces forces on the individual particles, typically atoms or molecules, whose dynamics will be followed. Next the geometry and boundary conditions are specified, and initial coordinates and velocities are given to each particle. Then the equations of motion for the particles are integrated numerically, stepping forward by discrete steps [12]. The underlying basic models used to simulate wearless friction in low dimension systems are the Tomlinson model [14], the Frenkel-Kontorova model [15–20], and a combination of both: the Frenkel-Kontorowa-Tomlinson model [21]. These models have in fact a great similarity with the Knopoff-Burridge model where the masses correspond with atoms or molecules. The main difference is that the phenomenological friction law for Ff is replaced by a kind of atomic realization of such a ’friction’ law [22], resulting in the Frenkel-KontorovaTomlinson model. The original models are special cases of this model: (i) Tomlinson’s atomic model [14] has no springs which interconnect the atoms (kc = 0), and (ii) the Frenkel-Kontorova model has no springs to the upper surface (kt = 0) except the first spring. Tomlinson advances a dissipation mechanism of wearless (sliding) friction based on one surface ’plucking’ the atoms of the other surface thus causing them to vibrate and dissipate energy through the bulk material. The underlying idea is in fact also applicable to asperity interaction during sliding, i.e., on a scale that is larger than the atomic. Two additional aspects must be added in this case however, namely ’adhesion’ and ’creep’. Baumberger [3] and his group proposed another heuristic friction model at atomic/molecular level. It is based on two main ingredients: (i) a phenomenological contact age and (ii) the Brownian motion of an effective creeping volume in a pinning potential, the strength of which increases with age. At this stage, however, the nature of the pinned units 6
and the mechanism of de-pinning are still unknown. Finally, the physics motivated model outlined in this paper (section III) belongs also to this category.
B.
Empirically motivated, heuristic friction models
Empirically motivated models are especially useful in the control domain. The requirements for such models are threefold: (i) they should be as simple as possible in order for one to be able to use them online, (ii) but complex enough to describe the frictional properties, which are relevant for control, and (iii) the number of parameters should be as small as possible and those should be easy to identify. During history, numerous friction models have been developed for control, varying from simple static models to more complex ones. The most elementary models are the so-called classical friction models which describe the friction force as a function of the velocity only (e.g. Coulomb friction model). Armstrong [23] made a first attempt to model also the dynamic aspects of friction behavior by introducing a time lag and an extra equation in the case of pre-sliding displacement. However, the transition from one equation describing the sliding regime behavior to another describing the pre-sliding regime behavior is not obvious and was not incorporated into the model. In the late 1960s, the Dahl model [24] introduced the start of the development of dynamic friction models based on internal states. The Dahl model captures only an approximation of the pre-sliding behavior, but it was the basis for more elaborated models like the BlimanSorine model [25] and the LuGre model [9]. These models are now called ’integrated’ friction models because they integrate a variety of phenomena into one formulation. The LuGre model is still a very popular model in the sense that it is well suited for use in theoretical calculations and easy to implement. More elaborated models are the elasto-plastic model [26], the Leuven model [10, 27] and the Generalized Maxwell-slip friction model [6]. This last model will be discussed in more detail in section V. In the geomechanics community, Rice and Ruina [28] developed a general empirical velocity-state dependent friction-dynamics law based on experiments where there is a suddenly imposed step change in the velocity. Such experiments suggest that the friction force response to this velocity step be a combination of an instantaneous increase with the velocity 7
coupled with a first-order-like decay with the evolving state. Analysis of this class reveals that all the empirically motivated friction models can be written in one generalized form. The friction force Ff is a generalized function of internal states, z, and the external states (i.e., the velocity v and the position x of the moving object): Ff = F(z, v, x),
(1)
and the dynamics of the internal states z can be written as a first order differential equation of a general form: dz = G(z, v, x), dt
(2)
with G(.) a general nonlinear function. The task of the empirically motivated friction modeling is now to find suitable expressions for the generalized functions F(.) and G(.), such that the model correspond to the measured friction behaviors. Two essential, limiting criteria (that are properties of friction) may be readily applied to simplify the task. Firstly, for constant velocities, i.e., in steady sliding, the friction force is function only of the velocity s(v). This friction behavior imposes a first condition on the general functions: if G(z, v, x) = 0 and v = cst
then F(z, v, x) = s(v)
(3)
A second condition is afforded by the frictional behavior in pre-sliding regime. The friction force is then, for sufficiently small displacements, a hysteretic function with nonlocal memory of the position: Ff = F(z, v, x) = Fh (x).
(4)
A faithful friction model has to fulfill these two conditions and, moreover, has to simulate the other frictional properties (frictional lag, stiction, stick-slip, breakaway force, etc.) properly, as will be seen later.
III.
A GENERIC MODEL BASED ON PHYSICAL PHENOMENA
This section outlines a generic friction model based on physical phenomena, which the authors have recently developed (see [2] for more detail). The starting point of the developed model is two multi-scale rough surfaces, belonging to two solid bodies, in contact with each other without sliding and under a static compressive force, Fn . Since the surfaces are not 8
smooth, contact will occur only at discrete points which sustain the total compressive force. The spatial randomness (or other stochastic or fractal distribution) of the contact points corresponds to the particular nature of surface roughness, whereas the different sizes of contact spots are due to the multiple scales of roughness. For a given load, the size of spots depends on surface roughness and the mechanical properties of the contacting bodies. It is evident that even for an isotropic surface the shapes of the contact spots are not isotropic and can be quite irregular and complex. Consider the two surfaces now to slide relative to each other. To do so, one will experience a tangential (frictional) force Ff . This force will be the result of ’adhesive’ forces and ’deformation’ forces arising from asperity interactions. ’Adhesive’ refers to the tangential surface forces (resisting tangential movement) arising from a variety of sources, when the two surfaces are at a certain proximity of each other. Thus, not only metallic adhesion (junction shear strength) is considered, but any other, short or long range forces, such as van der Waals, can lie at source of ’adhesion’ [29, 30]. ’Deformation’ will refer to hysteresis losses, of rate-independent nature [7, 27, 31] in the bulk of the materials as a consequence of geometrical deformation of asperities. This takes place during interlocking of asperities and persists till after they have passed over each other, by vibration (structural) damping to a halt. A third important mechanism, which is more directly responsible for the friction dynamics is ’creep’ of the contact in the normal direction. This is best imagined as follows: two asperities that are loaded against each other will tend to ’sink’ into one another over time, leading to higher tangential forces upon sliding. This concludes the first category of ingredients of the model; i.e. the phenomenological friction mechanisms. Obviously, these three mechanisms may be interconnected with each other to the degree that it would be difficult to draw a clear borderline between the spheres of influence of each. We assume a priori the existence of these mechanisms, consider them abstractly, and lump adhesion and creep expediently in one parameter: a time increasing ’local adhesion coefficient’, while deformation is accounted for by loss of stored (elastic and inertial) energy of the deforming asperities. Thus, by varying the parameter responsible for each mechanism, we can gain insight into its effect on the global friction behavior. It is the dynamic interaction of these three mechanisms with the asperity mass and stiffness that is responsible for the richly varied and rather complex behavior of the friction force. This interaction is mainly brought about by the continuously changing contact topography, in time and relative displacement. We call this, category of ingredients therefore ’the as9
perity interaction scenario’, which will be seen to be equally significant in determining the friction force dynamics. This scenario only considers the vertical distance between asperities of both surfaces as a function of the relative horizontal translation between the two objects while the vertical distance between the two objects remains constant. In the literature dealing with the contact modelling of two rough elastic surfaces the model is often reduced to that of the contact of one elastic surface, having the equivalent (or composite) roughness and elasticity of both surfaces, with a rigid perfectly flat second surface [7, 32, 33]. While such a reduction is acceptable, when considering more or less stationary contact (e.g. in presliding displacement), it falls totally short of revealing the dynamics of the non-stationary case (i.e., sliding), since it masks the interlocking of asperities and gives a false picture of the continuously changing relative topography and hence of the resulting tangential forces. We must mention here that the actual quantification of the two sets of ingredients for a given contact may not always be obvious. However, since we are concerned with the qualitative, formal aspects of friction force dynamics, we consider this issue to be outside the scope of this paper. Having identified the ingredients, the next step is to formulate a spatially distributed model by applying them to a large population of individual (idealized) asperities that vary randomly in the time of sliding. This is carried out as follows. The contact surfaces of two blocks rubbing against each other can be represented, after transformation, as follows. One (upper) flexible surface, containing all the possible equivalent asperities, each with its own equivalent stiffness, mass and shape interacts with a rigid, ’shaped’ lower surface. This representation allows determining whether or not an equivalent asperity will be active or inactive. Figure 2(A) depicts the life cycle of one such equivalent asperity, where it is assumed that the upper surface is moving from left to right with respect to the fixed lower surface. Topographical characteristics are assigned to both surfaces. The equivalent characteristics of the two interacting asperities (namely stiffness, mass, compression and adhesion) are lumped into one point (•). This point is initially moving freely (i), until it touches the lower rigid surface (ii), after sticking (iia) and slipping (iib) over the lower profile it breaks completely loose from the lower profile (iii). Doing so, the asperity is assumed to dissipate all its elastic energy through internal hysteresis losses. Figure 2(B)depicts the tangential, hysteretic deformations that the asperity undergoes. In case (ii) the asperity is said to be in an active state, for the other cases the asperity is said to be 10
(i)
(ii)
(iii) spring force
asperity base
h
(iib)
x
(iii)
kn
kt 2
(iia)
spring extension
(B)
kt 2 d
ζ ξ
(A)
w αw asperity tip
FIG. 2: (A) Life cycle of an average equivalent asperity contact. (B) Deformation, hysteretic behavior of the asperity.
inactive. (This may be reminiscent of the Tomlinson-Prandtl atomic model, except that it accounts for creep, adhesion and load-carrying, which prove essential in revealing friction force dynamics). In this formulation, the possible vibration of an asperity during contact are ignored, deeming it for the time being to be of secondary importance. From the moment the asperity becomes active, it will begin to follow the profile of the lower surface, by deforming normally ζ and tangentially ξ, resulting in a normal and tangential force. The normal force Fn (t) is given by kn ζ(t) and the tangential force is given by: Ft (t) = kt ξ(t), where kn and kt equal the normal and tangential stiffness of the equivalent asperity. The maximal tangential force an asperity can sustain, before slipping, equals the adhesion force: Fµ (t) = µ(t)kn ζ(t), where the expedient local adhesion coefficient µ(t) is function of the contact time owing to normal creep. This behavior can be deduced from the static friction versus dwell-time relation. Coulomb in 1785 [34] was probably the first to explore this time dependence. As a rule, static friction Fs grows with dwell time. An overview of the various formulae for the time-dependence is given by Gitis and Volpe [29]. 11
Thus, depending on the relative values of ζ, µ, kn and kt , the asperity tip (•) will initially stick (case iia in figure 2(B)) to the lower profile then slip on the profile (micro-slip) and, finally, break completely loose from the profile. Assuming, without loss of generality, linear tangential (and normal) elasticity of the top asperity, the force increases initially linearly with the displacement until slip occurs (in the mean time µ keeps increasing with contact time). When the asperity tip has fully traversed the bottom surface, it will break loose, vibrate (tangentially and normally) and thereby dissipate (part of) its ’elastic’ energy, by internal hysteresis, until it comes to rest or comes in contact with the next bottom profile. For low sliding speeds and relatively large separation of consecutive asperities all of the tangential elastic energy is lost. Besides the deformation and adhesion of the equivalent asperity the mass can contribute also to the friction force. When the equivalent asperity comes into contact with the surface profile, the asperity will undergo an impulse, resulting in an energy equal to mv 2 /2, with v the relative velocity of the equivalent asperity. To investigate the different types of friction behavior the generic model can simulate at macroscopic level, more than a thousand of equivalent asperities are simulated. The contact is modelled as that taking place between one flat object covered by a population of mutually independent elastic point asperities with randomly chosen constant heights h, masses m , normal stiffness kn , and tangential stiffness kt , and a rigid surface having a square wave profile (dashed line of Figure 2(A)), for each asperity, with a constant height but randomly chosen wavelengths w (that are, however, fixed during the life time of a single asperity contact). The stochastic distribution functions of the aforementioned parameters depend on the actual contact considered and will generally influence the resulting friction dynamics. The mean ratio between the characteristic length of inactivity and activity is given by a topography factor α. The shape of the local adhesion coefficient is chosen, for subsequent simulations, to be an exponential saturation function (though other forms are also possible) and is equal for all the asperities. As may be apparent in figure 2, the asperity will tilt so that its load carrying capacity will also vary in function of the relative position of its tip. This will lead to a normal force dynamics (the lift-up effect) that will, however, not be considered further in this paper owing to its lesser relevance, (for more detail see [2]).
12
IV.
DIFFERENT TYPES OF FRICTION BEHAVIOR
This section overviews the behavior of the generic model for different well-established characteristics of dry friction: (i) the pre-sliding hysteresis characteristic (friction force versus position), (ii) velocity weakening and strengthening (friction force versus steady state velocity), (iii) the dynamical behavior of the friction force in the velocity during gross-sliding, and (iv) the transition between pre-sliding and gross-sliding. The parameters of the generic model that we used to simulate the macroscopic behavior have been chosen on purpose to be rather unrealistic, in order to exaggerate their effects for better visualization. The results are all expressed in normalized dimensionless units. We are mainly interested, at this point, in the qualitative behavior of the friction. The real scale of the figures, i.e., the quantitative behavior, will depend on the studied material combination and surface topography.
A.
Pre-sliding behavior
At very small displacements, i.e., in the pre-sliding regime, different researchers found a hysteretic displacement-dependent friction force [27, 35–37]. Figure 3 shows the results obtained by the generic model. The position signal is chosen such that there is an inner loop within the outer hysteresis loop. The resulting friction-position curve is rate-independent; i.e., the friction-position curve is independent of the speed of the applied position signal. When an inner loop is closed, the curve of the outer loop is followed again, proving the nonlocal memory of the hysteresis. The shape of the hysteresis function is determined by the distribution of the asperity heights, the tangential stiffness, and the normal stiffness. It may be worthy of mention here that this hysteresis behavior arises, in the present simulation, primarily from micro-slip, i.e. the breaking of the adhesive contacts, just as in the Maxwell-slip model (see section V). The contribution of deformation losses (i.e. hysteresis losses in the bulk materials) depends on the relative value of this part as compared to the adhesive part, as well as on the tangential stiffness of the asperities, which governs the extent of deformation before slip.
13
0.5
5
friction force [−]
10
position [−]
1
0
−0.5
−1
0
2
4 6 time [−]
8
0
−5
−10
10
0
2
4 6 time [−]
8
10
0.8
1
friction force [−]
10
5
0
−5
−10 −1
−0.8
−0.6
−0.4
−0.2
0 position [−]
0.2
0.4
0.6
FIG. 3: Simulation results of the generic model in pre-sliding regime. The two upper figures show the imposed displacement and the resulting force. The lower figure shows the corresponding (nonlocal-memory) hysteresis loops obtained by plotting the force as a function of the displacement. B.
Velocity weakening and strengthening
When the asperity junctions are constantly being created and broken, the frictional interface is in the gross-sliding regime. Two main properties are observed here. The first property to be studied is the steady state friction force for different steady state sliding velocities, or the so-called Stribeck effect. The actual form of the friction-velocity curve is determined by different process parameters, namely, the normal creep (i.e., the time evolution of adhesion), the surface topography and the asperity parameters, i.e. tangential stiffness and mass [2]. Figure 4 shows the steady state curves for different mean tangential stiffness values, with the asperity mass being neglegted. All friction force curves show velocity weakening, namely decreasing friction force for increasing steady state velocity. Decreasing friction force for increasing velocity is a consequence of the rising local adhesion coefficient as a function of the contact time. Rabinowicz [38] pointed out an important correlation between the coefficient of static friction plotted as a function of the rest (or dwell) time, and the coefficient of steady sliding 14
8 7 friction force [−]
3 6 5
2
4 3
1
2 1
0
1
2 3 velocity [−]
4
5
FIG. 4: Velocity weakening in steady state sliding, as simulated by the generic model. The influence of the tangential stiffness of the asperities on the behavior is shown as an example: cases 1, 2 and 3 are for decreasing mean tangential stiffness.
(or kinetic) friction plotted as a function of the sliding speed. This has, much later, also been confirmed by Baumberger [3]. Taking the static friction vs. dwell time behavior to be a measure of the local adhesion coefficient-time characteristics, this correlation can be seen as the following transformation: µ µs
D0 tstick
¶ = CFf (v) , µd (v),
with µs and µd being the static and the sliding (=kinetic or dynamic) friction coefficients, respectively. The (process characteristic) number D0 is termed the ’creep-length’ and thus D0 /v can be seen as the time needed to surpass an average asperity contact; hence, the ’creep-time’. Figure 5 depicts such correlation, as simulated by our generic model, compared with experimental results. Note that two sets of data are plotted on each sub-figure, namely, µd (or CFf ) as function of v and µs (or the local adhesion coefficient µ(t)) as function of D0 /tstick . Figure 5(a) shows the two simulated curves (the friction force with dots and the local adhesion coefficient curve with crosses) using an exponential local adhesion coefficient curve, whereas figure 5(b) shows the two simulated curves using a logarithmic local adhesion coefficient curve. Figure 5(c) shows some experimentally measured curves by Baumberger [3] (correlating the static and the sliding friction coefficients). In all cases, both the static and 15
3
µ*(t*
5
f
10
C F*(v*)
f
stick
stick
)
)
µ*(t*
4
C F*(v*)
15 5
2 a
b
1 −1 10
0
v*
0
10 D /t*
0 stick
c −2
0
10
v*
10 D /t*
0 stick
FIG. 5: Correlation of the local adhesion coefficient, as function of time, with the sliding coefficient of friction as a function of the velocity. Figures a and b show the simulation results using the generic model with an exponential and a logarithmic local adhesion coefficient curve, respectively. The dots correspond to the normalized friction force (∝ friction coefficient) as a function of the normalized velocity, the crosses correspond to the normalized adhesion coefficient as a function of rest (dwell) time. The full line corresponds to a fitted logarithmic law of the form A + B log(v). Figure c shows experimental results by Baumberger (reproduced from [3]).
the sliding (kinetic) values of the coefficient of friction lie approximately on the same curve. Figures b and c represent a special case in that this curve is a straight line for several orders of magnitude of the abscissa. Until now the mass of the equivalent asperity has been assumed to be negligible. When the asperity masses are taken into account, the friction increases with the velocity for high velocities, the phenomenon being called velocity strengthening. Figure 6 shows the splittingup of a steady state friction curve (curve 1) in its different components: a component due to the deformation of the asperities (curve 2), a component due to the adhesion forces (curve 3) and a component due to the impact of the asperity masses (curve 4). The first two components are decreasing functions of the velocity, whereas the last component is an increasing function of the velocity. Based on only measured curves like curve 1 of figure 6, it is impossible to distinguish (i.e. differentiate) the different components. The importance of each different component depends on the surface topography and the dynamic asperity properties (mass and stiffness). Let us note finally that the same type of friction force vs. velocity behavior is known in the case of lubricated sliding friction, namely the Stribeck effect, where the velocity weakening arises from the build up of hydrodynamic pressure and the velocity strengthening 16
is attributed to the viscous shear of the lubricating film. This ”viscous” friction component is mostly assumed to be proportional to the velocity (σ2 v, with σ2 the viscous friction coefficient). 7
1
friction force [−]
6 4
5 4 3 2
3
1
2
0
0
1
2 3 velocity [−]
4
5
FIG. 6: The steady state friction-velocity curve (curve 1) and its different components: a part due to the deformation of the asperities (curve 2), a part due to the adhesion forces (curve 3) and a part due to impact of the asperity masses (curve 4).
C.
Frictional lag
Frictional lag, also called hysteresis in the velocity (erroneously) or frictional memory, is a well-known effect in lubricated friction [39], where the physical process giving rise to frictional lag appears to relate to the time required to modify the lubricant film thickness. The same effect has been observed in dry friction experiments [3] where no form of lubrication is used. Figure 7 shows the friction force as a function of the velocity using the generic model. The applied velocity signal is sinusoidal, of different frequencies, plus a constant offset in order to ensure a unidirectional signal at all times. The simulations show that the friction force is higher for acceleration than for deceleration, i.e., that it circles around the steady-state friction curve. Let us note that the departure from the steady state curve becomes more pronounced and qualitatively different at high accelerations. Rice and Ruina [28] developed a general heuristic velocity and state dependent friction law based on experiments where there is a suddenly imposed step change in the velocity. Such 17
6 a acceleration friction force [−]
5
b c
4
e
3 2
d
deceleration 0
1
2 3 velocity [−]
4
FIG. 7: The friction force as a function of the velocity for non-steady state velocities. The large dots represent the steady state friction curve. Frictional lag behavior is shown by the loops a, b, c, d and e, corresponding to (increasing, dimensionless) frequencies of 0.1, 0.25, 0.5, 1 and 5 respectively.
experiments suggest that the friction force response to this velocity step be a combination of an instantaneous increase with the velocity coupled with a first-order-like decay with the evolving state. The first factor, namely that the friction force increases simultaneously with the suddenly imposed increase of velocity, corresponds to the asperity inertia (or the ’viscous’) effect. The second factor results in a non-linear lag response. The generic model is able to simulate this behavior (figure 8(b)) and show that the time lag can be approximated as a first order system. However, the time constant depends not only on the velocity step, but also on the initial steady state velocity. Figure 8(a) shows the response for the same velocity steps but with negligible asperity masses. The frictional lag effect is independent of the asperity masses. The asperity mass effect can be seen as an instantaneous static velocity-dependent effect on the friction force. This explains why the mass is neglected for the study of the frictional lag (see figure 7) and the study of the transitional behavior (see below).
18
6
6 a
b 5
4
Ff [−]
∆ F1
*
F*f [−]
5
3
1
∆ F2
3 *
v*=1 2
∆F
4
*
v =2.5 10
15
20
v =2.5
v =1 2
25
τ [−]
v*=0.5
*
*
v =0.5
10
15
20
25
τ [−]
FIG. 8: The normalized friction force as a function of the normalized time for two different velocity step sizes (a velocity step equal to +1.5 and a step equal to −2). Figure a shows the response without the asperity mass interaction. Figure b shows the response with asperity mass interaction. ∆F1 corresponds to the evolving friction change and ∆F2 corresponds to the instantaneous friction change for a velocity step. D.
Transition behavior
Another important aspect of friction is the behavior at transition from pre-sliding regime to sliding regime and vice versa, which most continuum models, such as the rate-state model, are not able to deal with. This transition behavior can be investigated by applying a periodic (e.g. sinusoidal) displacement signal to the system. Figure 9 shows the friction force as a function of a sinusoidal velocity excitation. The dots on figure 9 represent the friction force for constant velocity. Not only frictional lag is observed, but more importantly, the crossover at velocity reversals, passing through pre-sliding. When the velocity goes through zero (1), the direction of motion is reversed and all the active asperities will be relaxed and reloaded in the new direction of displacement. The friction force will increase until it reaches a maximum value (2), called the breakaway force, lying above the steady state curve. From the moment of breakaway, the object is in gross-sliding and the friction force will be attracted to the steady state curve. As in the case of frictional lag, the steady state curve lies between the curves of acceleration (2 − 3) and deceleration (3 − 4). The results obtained by the generic model have the same qualitative behavior as experimental results reported in
19
[37, 40]. Note that the great diversity observed in this type of behavior depends not only on 6
2
4
2
friction force [−]
4 4
3
2 0 −2 1 −4 1
−6 −5
0 velocity [−]
5
FIG. 9: The friction force as a function of the velocity under a sinusoidal excitation using a low (full line) and a high (dashed line) frequency. The dots correspond to the velocity weakening curve.
the process parameters but also on the velocity and its time evolution. This is yet another topic that is of great importance in friction modeling and on which other existing models have very little, if anything at all, to say.
E.
Stick-slip (limit cycle) behavior
x(t) Ff
k M
V0
FIG. 10: Schematic set-up to investigate the stick-slip phenomenon.
Figure 10 schematically shows the commonly used apparatus. It is described by a block in contact with a fixed surface. The block is connected via a spring and damper to a system which imposes a constant desired velocity v0 . The stick-slip oscillation is not only influenced by the nature of the surfaces in contact (the friction behavior), but also by the dynamics of 20
the system (inertia, stiffness, damping, ...) of the experimental apparatus, and the desired sliding velocity. The negative slope in the friction-velocity curve is a necessary, but not sufficient, condition for steady sliding instability. Figure 11 shows the results for a mass object using the generic friction model to simulate the friction interaction. The stick and slip phases are clearly visible in figure 11(a) and the friction force as a function of time has the same behavior as experimentally observed [37]. The object sticks until the frictional force equals the breakaway force. The block begins to slide and the increasing velocity results in a decreasing friction force. At the same time the applied (spring) force decreases also because the desired velocity is smaller than the real velocity of the block. The block accelerates until the friction force equals the spring force. From that moment on, the block decelerates, resulting in a higher friction force which maintains this process until the block sticks again. During the stick period an oscillating behavior in the pre-sliding regime occurs. The frequency of this oscillation is determined by the mass of the moving object and the total stiffness of the spring and the contact stiffness in pre-sliding regime. The damping of the oscillation is determined by the energy loss at the asperity contacts (given by the enclosed surface of the hysteresis curve, i.e., structural damping). Figure 12(a) shows the time history of the states during the stick-slip. The time, as angular coordinate, is scaled by the averaged period of one stick-slip cycle. This diagram contains the time, position and velocity information of the limit cycle in one graph. Figure 12(b) is the corresponding phase portrait of the limit cycle.
V.
A REDUCED FRICTION MODEL APPROPRIATE FOR SIMULATION AND
CONTROL PURPOSES
The generic model, though capable of simulating very fine nuances of the friction force dynamics, is too complex to implement for general simulation, identification and control of systems with friction. In this sense, heuristic models, when they are well constructed, have the advantage of capturing the basic characteristics of behavior in simple, tractable formulations. Thus, based on the results obtained from the generic friction model and on the general formulation of heuristic friction models (Eq. 1 and Eq. 2), the authors have been able to formulate a novel reduced heuristic friction model appropriate for quick simulation 21
30
a
25
25
20
20
15
15
10
10
5
5
0
0
0
5
10
15
20
25 time [−]
30
35
40
45
50
5
10
15
20
25 time [−]
30
35
40
45
50
velocity [−]
position [−]
30
10
force [−]
b 5
0
−5 0
FIG. 11: Stick-slip behavior. Figure a shows the desired position (dotted line), the real position (full line) and the real velocity (dashed line) as a function of time. Figure b shows the applied force to the mass (dotted line) and the friction force (full line) as a function of time.
and control purposes. It has been christened the Generalized Maxwell-slip friction model [6]. Since the (classical) Maxwell-slip model has proved to be the most convenient way to model pre-sliding, it was logical to use it as the starting point for the new formulation. The developed model is based explicitly on three friction properties: (i) the steady state friction curve s(v) for constant velocities (Eq. 3), (ii) the hysteresis function with non-local memory in the pre-sliding regime (Eq. 4) and (iii) the frictional lag behavior in the sliding regime. The developed model is in fact a parallel connection of different elementary massless blockspring models, having all the same input, namely the velocity v (see figure 13). The output of the model is the summation of all the individual friction forces Fi acting on the blocks. The dynamic behavior of each elementary model is determined by the following rules (ki , νi and C, are constant parameters):
22
10 8
a
b
6 velocity [−]
velocity [−]
5
0 position error
−5 5
−2 −4 −5
0 −5
2 0
5 0
4
−5
0 5 position error [−]
10
FIG. 12: Figure a: The time history of the states during stick-slip, plotted in cylindrical coordinates, with displacement as the radial axis, velocity as the longitudinal axis, and time as the circumferential coordinate. Figure b: The phase portrait corresponding to figure a.
k1 if stick F
if slip
υ i s i(v) υ i Fs
{
=
i,max
(for GMS)
F1
(for MS)
Fi υ i C (1− ) (for GMS) dF i υ i s i(v) = dt 0 (for MS)
{
ki
Fi
v kN
FN FIG. 13: Representation of the (Generalized) Maxwell-slip friction model. The maximum force Fi a massless block can sustain is constant in case of the the classical Maxwell-slip (MS) model. In case of the generalized Maxwell-slip (GMS) model, Fi varies depending on the sticking or slipping of the element.
• If the elementary model sticks the state equation is given by: dFi = ki v, dt and the model remains sticking until Fi = νi si (v).
23
(5)
• If the elementary model slips the state equation is given by: µ ¶ dFi Fi = sgn(v)νi C 1 − , dt νi si (v)
(6)
and the model remains slipping until the velocity goes through zero. In practice, however, the model will be very difficult to identify experimentally if we do not take the si (v)’s to be the same for all elements (si (v) equals s(v) for all i). In the latter case, one must require that the following compatibility condition be satisfied: N X
νi = 1.
i=1
The friction force is given as the summation of the outputs of the N elementary state models plus an extra viscous term to account for the viscous friction: Ff (t) =
N X
Fi (t) + Fv (v),
i=1
where Fv (v) can for example be set equal to σ2 v(t). This model is capable of simulating the pre-sliding and sliding regime accurately. In the pre-sliding regime and for small velocities, P the function νi si (v) can be approximated by the static force Fs . The GMS friction model is then reduced to a Maxwell-slip model [41, 42], which is capable of simulating the hysteresis characteristic with nonlocal memory, and satisfying the second condition (Eq. 4). In case of full gross-sliding (i.e., all the elementary models are slipping), the friction force dynamics can be written as: µ ¶ dFf Ff = sgn(v)C 1 − , dt s(v)
(7)
satisfying the first condition (Eq. 3). The model simulates also properly the frictional lag behavior and the transitional behavior from pre-sliding to sliding and vice versa. Used in combination with a mass with or without spring, the total system can simulate the non-drifting property, the breakaway force, the stick-slip behavior and hunting [6]. (See also next section). The identification of the GMS friction model is relatively easily achieved by performing three dedicated experiments. In a first experiment different constant velocity signals are imposed on the moving mass. By measuring corresponding friction force, the steady state 24
curve s(v) can be identified using curve fitting techniques. A second experiment imposes a small constantly increasing position signal to the object. By plotting the friction force as a function of the displacement, the parameters νi and ki can be estimated using curve fitting techniques. A last experiment imposes a non-steady-state positive velocity signal to the object. For this signal the object is always in sliding regime and equation (7) is valid. By measuring the velocity, the friction force and the estimated steady state curve s(v), the attraction parameter C can be estimated using least-squares techniques.
VI.
COMPARISON BETWEEN MODELS AND EXPERIMENTAL OBSERVA-
TIONS
Having presented a generic, physics-based model, on the one hand, and a heuristically derived model, on the other, it may be pertinent to ask: to what extent are both models capable of capturing true, experimentally observed friction behavior? This question contains a problematic, which will be first discussed, then some comparisons will be presented. First, owing to the fact that both models contain a (large) number of unknown parameters, (which so far cannot be determined from the contact description and specification alone,) a direct comparison with experimental observation is, generally, not possible. The unknown parameters have first to be identified, using appropriate techniques, on the experimentally obtained data. In this way, the presented models may be viewed as ”model structures” with unknown parameters, whose effectiveness, i.e. the correctness of their structures, may be gauged by their ability to simulate the qualitative behavior of experimentally observed friction force (as well as whole system’s) dynamics as has been described in the two previous sections. Second, we must bear in mind that a systematic experimental identification of friction is generally not possible. This is so because the presence of friction, with its highly non-linear character, in mechanical systems, makes it very difficult to control them effectively. (In fact, one of the prime motivations of this research is to understand friction better in order to be able to control it.) Thus far, experimental friction data are obtained as follows. We apply a certain desired trajectory to the system comprising frictional contact, (the system will generally only follow it with a varying degree of fidelity, depending on the effectiveness of the control scheme), and measure the instantaneous, real displacement and friction force. 25
The latter are however measured, in the time, with a very high degree of accuracy, see [37]. With the foregoing remarks in mind, three representative comparisons (in addition to the one already presented in Fig. 5) will be presented and discussed in the following. Figure 14 depicts the pre-sliding hysteresis behavior as measured experimentally (Fig. 14(a)) and as simulated by each model (Figs. 14(b) and 14(c)), with ad hoc tuning of the unknown parameters. One external and two internal loops are shown to demonstrate the nonlocal-memory character of the hysteresis. Obviously, the unknown parameters could be so tuned that better quantitative agreement (in this case the form of the hysteresis loop) is
10
10
5
5
5
0
−5
−10
−1
−0.5
0
0.5
1
friction force [−]
10
friction force [−]
friction force [N]
achieved, but this subject is outside the scope of this paper.
0
−5
−10 −1
−0.5
displacement [µm]
(a)Measured on tribometer.
0
0.5
displacement [−]
(b)Simulated: generic model.
1
0
−5
−10
−1
−0.5 0 0.5 displacement [−]
1
(c)Simulated: GMS model.
FIG. 14: Pre-sliding hysteresis behavior: measurement and simulation.
Figure 15 compares the results for periodic sliding, showing the frictional lag phenomenon. These results have been obtained in the following way. First, a sinusoidal velocity command signal was input to the tribometer (dashed line of Fig. 15(a), left). The resulting (actual) velocity signal (solid line of Fig. 15(a), left) follows that command except for two intervals, which correspond to transition from and to pre-sliding. This is precisely the problem, which we pointed to in the opening paragraphs of this section. Finally, the measured true velocity is plotted against the measured friction force (Fig. 15(a), right). In order to effect a reasonable comparison with the models, the essential parts of the tribometer system (including the controller) are now simulated, in Matlab software, using each model in turn to simulate the friction contact. Again, the tuning of the model parameters was carried in an ad hoc manner, since we are mainly looking for qualitative behavior. Figures 15(b) and 15(c) 26
show the obtained results using the generic and the GMS models respectively. The overall qualitative agreement in the general trends (as well as some details) of the behavior is strikingly good. 150
5
friction force[N]
velocity [ µm/s]
100 50 0 −50
−5
−100 −150
0
0
1
2
3
4
−150 −100
5
time [s]
−50
0
50
velocity [ µm/s]
100
150
(a)Measured on tribometer. Left, sliding velocity. Right, friction force vs. velocity. 60
1.5 1
friction force [N]
velocity [µm/s]
40 20 0
0.5 0
−0.5
−20 −40
−1
−1.5 −60
0
1
2
3
4
−60
5
−40
−20
0
20
40
60
velocity [ µm/s]
time [s]
(b)Simulated with the generic model. Left, sliding velocity. Right, friction force vs. velocity. 10
600
friction force [−]
velocity [−]
400 200 0 −200
5 0 −5
−400
−10
−600 0
2
4
−500
6
time [−]
0 velocity [−]
500
(c)Simulated with the GMS model. Left, sliding velocity. Right, friction force vs. velocity.
FIG. 15: Periodic sliding behavior: measurement and simulation.
A final comparison is afforded by Fig.16 concerning stick-slip behavior. Again, the starting point is the experiment. The sliding mass is commanded to follow a ramp (constant velocity) input, subject to a low-gain (= low stiffness) controller. This resulted in stick-slip 27
motion (Fig.16(a)), which is quite complex in behavioral detail. In particular, the spring force (broken line in right figures), which is erroneously often used in experiments to gauge the friction force, is not equal to the total force (solid line). There are qualitative and quantitative differences owing to the presence of high frequency inertia forces of the moving mass during stick-slip vibrations. Simulation of the whole system using the generic and the GMS models, Figs.16(b) and 16(c) respectively, yields results, which to all intents and purposes resemble the experimental ones. We believe that these simulations are the first of their kind, which reveal the intricate details of the stick-slip phenomenon, especially the high frequency, damped vibrations superimposed on the basic motion. In conclusion, on the basis of the foregoing evidence, we believe that the presented models provide a fairly accurate description of friction force dynamics, at least in its broad lines. The ultimate test of the presented models would be to tune their parameters for a large set of experimental friction data, covering different regimes, which are however obtained from the same contact configuration. The error residues, resulting from the fitting optimization process, should then attain their minimal values for the same set of model parameters, regardless of the friction regime. This is however an elaborate subject, which (together with other issues) is left for the future, (see following section).
VII.
FURTHER COMMENTS
Obviously, both of the presented friction models, the generic physics motivated friction model and the Generalized Maxwell-slip friction model, are far from perfect and are still open to further developments. (i)A study of the dependency of the parameters on each other for one asperity and the interdependency of the asperities. Using other values and distribution functions for the parameters of the generic model may result in qualitatively different macroscopic friction phenomena. In the first instance, the coupling of stiffness, mass and geometrical properties of real surfaces may be considered, based on stochastic/fractal analysis; (likewise, the consideration of interaction of neighboring asperities). However the mechanisms and the interaction between the different mechanisms will remain the same as described. The possibility to transform two real surfaces into equivalent asperities has not been considered either. The local load-carrying function and the rigid surface profile are physically continuous functions, 28
24 500
22 400
force [N]
position [µm]
20
300
18
200
16
100
14
0 0.2
0.4
0.6
12 0.2
0.8
0.4
0.6
0.8
time [s]
time [s]
(a)Measured on tribometer. Left, displacement in time. Right, Force in time (dashed line is the
30
8
25
6
20
4
force [−]
position [−]
spring force; full line is total force).
15
2
10
0
5
−2 −4
0
10
20 30 time [−]
40
50
10
20 30 time [−]
40
50
(b)Simulated with the generic model. Left, displacement in time (dotted line is the command signal). Right, Force in time (dotted line is the spring force; full line is total force). 8
500
6 4 force [N]
position [µm]
400 300 200
2 0 −2 −4
100
−6 0
0
100
200 300 time [s]
400
0
500
100
200 300 time [s]
400
500
(c)Simulated with the GMS model. Left, displacement in time (dashed line is the command signal). Right, Force in time (dashed line is the spring force; full line is total force).
FIG. 16: Stick-slip behavior: measurement and simulation.
but we do not expect that the simplifications we used for them would greatly influence the 29
behavior of the model. Finally, there should be a relationship between the compression of the asperities and the tangential/normal stiffness of the asperities (because the contact area is related to the compression), which can also be included in in more elaborate formulation. (ii) An extension of the model to add the normal degree of freedom. The considered generic friction model assumes that the separation between the two objects remains constant. The discussed friction behaviors in section IV are only valid if the normal capacity load remains constant (e.g. curve 1 in figure 4). If the normal force varies, the influence of the slider dynamics (inertia) on the friction force dynamics has to be taken into account [43]. This increases the complexity of the friction problem enormously by adding a second d.o.f. (iii) Extension to the problem of lubricated friction. At very low velocities, the lubricant is not able to build-up a fluid film by hydrodynamic effects. Therefore the dominating mechanism will by solid-to-solid friction. Due to the lubrication the developed model has to be extended taking the hydrodynamic effects into account. When the normal degree of freedom is also incorporated into the model a lift-up of the moving object will be possible, resulting in a smooth transition from solid-to-solid effects (i.e., dry sliding effects) to purely hydrodynamic effects. (iv) the extension of the generic friction model to lubricated friction and the normal degree of freedom, can also be used to extend the GMS friction model such that it can deal also with these phenomena. (v) Self-organized criticality. The functional form of the dynamic spring-mass interactions between the local spots is still unclear. However this is not insignificant since collective phenomena such as the onset of sliding and stick-slip can depend upon these types of interactions. Recently there has been some interest in studying this as a self-organized critical phenomenon [44]. The onset of sliding friction can be pictured as follows. When an attempt is made to slide one surface against another, the force on a contact spot can be released and distributed among neighboring spots. The forces may turn out to be limited to a small region or become large enough so that the whole surface starts sliding. During this process, the interface evolves into a self organized critical system insensitive to the details of the distribution of initial disorder. This type of analysis has been used to provide a physical interpretation of the Guttenberg-Richter relation between earthquake magnitude and its frequency. We have not yet investigated the possibility of this phenomenon occurring in our generic model. 30
VIII.
CONCLUSIONS
This paper discussed the two different approaches to model friction dynamics namely physics motivated models and heuristic or empirically motivated models, laid the link between them, and presented (the authors’) novel versions of each type. The physical mechanisms behind the generic dry friction model are explained and the different frictional types of behavior are illustrated using the generic dry friction model: i.e., the hysteretic friction force as a function of the displacement in pre-sliding regime, the velocity weakening and strengthening steady-state friction curve in gross-sliding regime, the lift-up and lift-down effect, the frictional lag property, the transition behavior from pre-sliding to gross-sliding and the modelling of stick-slip. This paper also discusses briefly a novel empirically motivated friction model, called the Generalized Maxwell-slip friction model, which has been derived from the generic dry friction model. The resulting friction dynamics correspond accurately with the experimentally measured friction behavior and the parameters of the novel model are easy to identify. Finally, possible refinement and extensions of the models have been outlined.
Acknowledgments
This paper presents research results of the K.U.Leuven’s Concerted Research Action GOA/99/04. The Volkswagenstiftung grant I/76938 is also gratefully acknowledged by the first author. The scientific responsibility is assumed by its authors.
31
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34
IX.
FIGURE CAPTIONS
1. The Burridge-Knopoff model. 2. (A) Life cycle of an average equivalent asperity contact. (B) Deformation, hysteretic behavior of the asperity. 3. Simulation results of the generic model in pre-sliding regime. The two upper figures show the imposed displacement and the resulting force. The lower figure shows the corresponding (nonlocal-memory) hysteresis loops obtained by plotting the force as a function of the displacement. 4. Velocity weakening in steady state sliding, as simulated by the generic model. The influence of the tangential stiffness of the asperities on the behavior is shown as an example: cases 1, 2 and 3 are for decreasing mean tangential stiffness. 5. Correlation of the local adhesion coefficient, as function of time, with the sliding coefficient of friction as a function of the velocity. Figures a and b show the simulation results using the generic model with an exponential and a logarithmic local adhesion coefficient curve, respectively. The dots correspond to the normalized friction force (∝ friction coefficient) as a function of the normalized velocity, the crosses correspond to the normalized adhesion coefficient as a function of rest (dwell) time. The full line corresponds to a fitted logarithmic law of the form A + B log(v). Figure c shows experimental results by Baumberger (reproduced from [3]). 6. The steady state friction-velocity curve (curve 1) and its different components: a part due to the deformation of the asperities (curve 2), a part due to the adhesion forces (curve 3) and a part due to impact of the asperity masses (curve 4). 7. The friction force as a function of the velocity for non-steady state velocities. The large dots represent the steady state friction curve. Frictional lag behavior is shown by the loops a, b, c, d and e, corresponding to (increasing, dimensionless) frequencies of 0.1, 0.25, 0.5, 1 and 5 respectively. 8. The normalized friction force as a function of the normalized time for two different velocity step sizes (a velocity step equal to +1.5 and a step equal to −2). Figure a 35
shows the response without the asperity mass interaction. Figure b shows the response with asperity mass interaction. ∆F1 corresponds to the evolving friction change and ∆F2 corresponds to the instantaneous friction change for a velocity step. 9. The friction force as a function of the velocity under a sinusoidal excitation using a low (full line) and a high (dashed line) frequency. The dots correspond to the velocity weakening curve. 10. Schematic set-up to investigate the stick-slip phenomenon. 11. Figure a shows the desired position (dotted line), the real position (full line) and the real velocity (dashed line) as a function of time. Figure b shows the applied force to the mass (dotted line) and the friction force (full line) as a function of time. 12. Figure a: The time history of the states during stick-slip, plotted in cylindrical coordinates, with displacement as the radial axis, velocity as the longitudinal axis, and time as the circumferential coordinate. Figure b: The phase portrait corresponding to figure a. 13. Representation of the (Generalized) Maxwell-slip friction model. The maximum force Fi a massless block can sustain is constant in case of the the classical Maxwell-slip (MS) model. In case of the generalized Maxwell-slip (GMS) model, Fi varies depending on the sticking or slipping of the element. 14. Pre-sliding hysteresis behavior: measurement and simulation. (a) Measured on tribometer. (b) Simulated: generic model. (c) Simulated: GMS model. 15. Periodic sliding behavior: measurement and simulation. (a) Measured on tribometer. Left, sliding velocity. Right, friction force vs. velocity. (b) Simulated with the generic model. Left, sliding velocity. Right, friction force vs. velocity. (c) Simulated with the GMS model. Left, sliding velocity. Right, friction force vs. velocity. 16. Stick-slip behavior: measurement and simulation. (a) Measured on tribometer. Left, displacement in time. Right, Force in time (dashed line is the spring force; full line is total force). (b) Simulated with the generic model. Left, displacement in time (dotted line is the command signal). Right, Force in time (dotted line is the spring force; 36
full line is total force).(c) Simulated with the GMS model. Left, displacement in time (dashed line is the command signal). Right, Force in time (dashed line is the spring force; full line is total force).
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