Stick-slip motion in a driven two-nonsinusoidal Remoissenet–Peyrard potential

Physica D 191 (2004) 31–48

Stick-slip motion in a driven two-nonsinusoidalRemoissenet–Peyrard potential

G. Djuidje Kenmoe∗, A. Kenfack Jiotsa, T.C. KofanéDépartement de Physique, Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I,

BP 812, Yaoundé, Cameroon

Received 14 January 2002; received in revised form 27 October 2003; accepted 31 October 2003Communicated by E. Ott

Abstract

Frictional stick-slip dynamics is studied theoretically and numerically in a model of a particle interacting with two de-formable potentials, one of which is externally driven. We focus our attention on a class of parameterized one-site potentialURP(φ, r) whose shape can be varied as a function of parameterr and which has the sine-Gordon shape as the particular case.Periodic stick-slip, erratic and intermittent motions, characterized by force fluctuations, and sliding above the critical velocityare the three regimes that are identified in the motion of the driven plate. The onset of chaos is studied through an analysisof the phase space, and a computation of the Lyapunov exponent. This study strongly suggests that stick-slip dynamics ischaracterized by chaotic behavior of the top plate and the embedded molecular system, and is strongly dependent on thedeformable parameterr.© 2003 Published by Elsevier B.V.

Keywords:Stick-slip motion; Remoissenet–Peyrard potential; Lyapunov exponent

1. Introduction

The concept of friction in general, and the prob-lem of stick-slip motion in particular, at microscopicto macroscopic length scales, is central to many areasof physics, from friction and lubrication of materials[1–3] to earthquake models and avalanches[4–6]. Inmany cases of current interest is the study of the fric-tion which arises from slippage at solid–solid[7,8],fluid–fluid, fluid–solid interfaces and contact points[9–11].

More recently, frictional forces in an electrolyticenvironment[12], and the behavior of an elastically

∗ Corresponding author. Fax:+237-222-13-20.E-mail address:[email protected] (G. Djuidje Kenmoe).

supported mass with three translational degrees offreedom that can make contact with a rigid Coulombfriction support [13] have been studied. Recent de-velopments in the physics of granular matter haveillustrated that the dissipative nature of the interac-tions between grains can result in a variety of differ-ent phenomena[14–17]. The interest in this problemcomes from the large range of physical applicationsthat are related to it. One can note, for example, mag-netic storage and recording systems (computer diskheads), miniature motors, and many aerospace com-ponents.

The investigation of the friction has taken five prin-cipal directions. The first approach is to use first-principles calculations[18]. The second approach isto use scanning probe microscopes, such as the atomic

0167-2789/$ – see front matter © 2003 Published by Elsevier B.V.doi:10.1016/j.physd.2003.10.012

32 G. Djuidje Kenmoe et al. / Physica D 191 (2004) 31–48

force microscope[19,20], and the friction force mi-croscope[21]. The third approach is the molecular-dynamics simulations[9–11]. The fourth approach isto use theoretical models based on both single- andtwo-block models[4,22]. The one-dimensional uni-form Burridge–Knopoff model of earthquake fault isthe well-known example[4]. Many other models havebeen proposed including discrete cellular automata[23,24], lattice models[25–33]. The last approach isthe experimental studies[34–41].

From these approaches, important new features havebeen outlined. For instance, it has been demonstratedthat:

(1) Friction can depend dramatically on the localchemical and atomic nature of the surfaces incontact, and can be extremely sensitive to mono-layers or submonolayers of adsorbed atoms ormolecules[42].

(2) The stick-slip motion is based on the well-knownobservation that the frictional force on static ob-jets is larger than on sliding ones[3].

(3) During the stick-slip motion, the system makesperiodic transitions between a static (solid-like)state and a kinetic (liquid-like) state. These peri-odic transitions are characterized as the conven-tional static–kinetic transitions[3].

(4) In the ultra-low friction, a second dynamic stateduring stick-slip motion has been identified,which is fundamentally different from conven-tional static–kinetic transitions[36].

(5) The origin of stick-slip motion is thermodynamicinstability of the sliding, rather than a dynamicinstability [43].

(6) For a large class of sliding systems, the ratiobetween the kinetic and static friction coefficientsis equal to 1/2[27].

(7) Stick-slip models evolve to the self-organizedcritical state[32].

(8) When the friction is not velocity weakening butvelocity strengthening, different self-organizedbehavior is obtained[35,44–46].

(9) Generally, electronic contributions to sliding fric-tion are weak; only in cases with significant elec-tronic coupling between molecule and substrate

does electronic friction dominate phononic fric-tion [47].

(10) Depending on the initial conditions, the systemexhibit chaotic spatiotemporal behavior[22,48].

Generally, the interactions between adsorbed atomsare known to play an important role in many sur-face phenomena, such as catalysis, surface diffusion,desorption and phase transitions in adsorbate layers.Conventionally, one distinguishes between direct andindirect interactions between adsorbates. The directinteractions are those that could be present also be-tween atoms and molecules in the gas phase, for exam-ple covalent bonding[49], van der Waals interaction[50,51] and dipole–dipole interaction[52]. The indi-rect interactions are mediated through the substrate.This category includes for instance the elastic interac-tions between the displacement fields due to the ad-sorption of large molecules[53]. It has been shownthat the indirect interactions also depend very sensi-tively upon the adsorbate and substrate geometry ofthe system considered[54–57]. For real physical sys-tems, such as in rock–rock[58] and metal–metal[1]frictions, the account of various disturbances and of amore complex character of atomic interactions cannotbe satisfactorily described by substrate potentials withconstant parameters, for which much work has beenalready done in the context of their dynamical behav-ior. Thus, for instance the periodic sinusoidal potentialfails completely to reproduce the correct disturbanceof sinusoidal shape of substrate periodic potential inreal adsystem situation.

Presently, we want to investigate the dynamicsof a particle interacting with two nonlinear periodicsubstrate potentialsURP(φ, r), one of which is ex-ternally driven, and whose shape can be varied con-tinuously as a function of a parameterr and whichhas the sine-Gordon potential as a particular case[59]. This potential, which plays an important role inone-dimensional atomic chains, is produced by theinteraction of an adatom with substrate atoms, wherethe parameterr could account for the temperature orpressure dependence, or for the geometry of the sur-face of the metallic surface. Quite in contrast, to ourknowledge, the use of deformable potentials[59–64]

G. Djuidje Kenmoe et al. / Physica D 191 (2004) 31–48 33

has been largely neglected in the study of systemsthat can exhibit chaotic behavior[65–72].

The main purpose of this article is to go beyond theabove restrictive assumption concerning the periodicsinusoidal potential[59], and to present a more gen-eral theory for a particle interacting with two periodicpotentials, in the general case where the substrate po-tential relieves are nonsinusoidal, with particular ref-erence to the effect of the shape parameterr. Section 2defines the model and describes briefly the Lagrangianformalism, from which we obtain our basic set ofunderdamped equation of the field displacements. InSection 3, using the adiabatic approximation for theparticle and the plates, one can understand dynamicalprocesses in the deformable oscillator model. In orderto give a physical significance to the results which weobtained under the adiabatic approximation, we car-ried out a direct numerical integration of the coupledexact equations of motion describing the slow modefor the top plate and the fast mode for the particle. Agood agreement between both methods has been ob-served. Next, positive values of the largest Lyapunovexponent have been obtained, which represent the dy-namical chaos states. InSection 4, we study the dy-namical behavior of the top plate and of the particle asthe driving velocity of the stage is varied. The powerspectra of the spring force has been also calculated.Section 5concludes the paper.

2. Model description and Lagrangian formalism

The purpose of this section is to introduce our modelschematized inFig. 1. It includes two rigid plates anda single particle of mass m embedded between them.The top plate of massM is pulled by a linear springwith force constantK connected to a stage movingwith a constant velocityV . The displacement of theparticle is characterized by the variablex, while the

Fig. 1. Schematic sketch of a model geometry.

variableX denotes the displacement of the top plate.The lower plate is assumed to be fixed. To model theinteraction between the particle and each of the plates,we consider a general class of nonsinusoidal oscil-lators where the Remoissenet–Peyrard (RP) potentialURP(φ, r) given by[59]

URP(φ, r) = U0(1 − r)21 − cos(2πφ/b)

1 + r2 + 2r cos(2πφ/b)(1)

has constant amplitude and is 2π-periodic inφ, whereφ denotes the displacement field;U0 is a constantwhich measures the amplitude of the nonsinusoidalRP potential (1), whileb is the period of the potential.The shape of the potential is defined by the parameterr (|r| < 1). Forr > 0, the nonsinusoidal RP potential(1) has flat bottoms separated by thin barriers, whilefor r < 0, it has the shape of sharp wells separatedby flat wide barriers. The potential (1) has been in-troduced before by RP[59] in the context of solitontheories[73].

For r = 0, the nonsinusoidal RP potential (1) re-duces to the sinusoidal potential which is the famil-iar sine-Gordon (SG) potential. The SG potential hasbeen used to model the driven pendulum[74], chargedensity waves in the presence of time-varying electricfield [75], charged particles in viscous medium in asinusoidal potential with a time-varying electric field[76], Josephson junction[77–79] and phase-lockedloops[77].

The nonsinusoidal RP potential has been used tocalculate the pinning energy of kinks due to the dis-creteness of substrate lattices[80–82], as a model forreconstructive surface growth[79], and to describe thecomplicated exchange-mediated diffusion mechanism[83]. The nonsinusoidal RP potential has been alsoused to calculate the diffusion coefficient of adsor-bates in metallic substrates[84], the nucleation rateof kink–antikink pairs at low temperatures and in thelimit of strong damping[85], the total density of kinkand antikink at low temperatures[86]. The deformablespin model Hamiltonian has been introduced[87],while two classes of topological solitons have beencalculated in the model of long-range interatomicinteractions with nonsinusoidal RP potential[88].Nanopterons have been recently found in the nonlinear


Klein–Gordon chain using the slightly modified non-sinusoidal RP potential[89]. The low-temperaturethermodynamics[90], the pinning potential and thepinning barrier of kinks due to the discreteness of lat-tices[91] for deformable systems have been derived.

The substrate potential (1) can also be calculatedfrom first principles[92]. The shape parameterr canbe determined directly from the measured values of theactivation energy for diffusion of an isolated adatom[93], and the frequency of adatom oscillations parallelto the surface[94]. For instance, an estimate for theH/W adsystem (hydrogen atoms adsorbed on a tung-sten surface) yieldsr ≈ −0.3 [81,95].

From the Lagrangian for nonconservative systems,the coupled equations of motion for the top plate andthe particle can be written in dimensionless form as

Y + α2(Y − Vτ)+ εγ(Y − y)

− ε

2π(1 − r21)

2 sin 2π(y − Y)

[1 + r21 + 2r1 cos 2π(y − Y)]2= 0,

(2a)

y + γ(2y − Y )

+ 1

2π(1 − r21)

2 sin 2π(y − Y)

[1 + r21 + 2r1 cos 2π(y − Y)]2

+ 1

2π(1 − r22)

2 sin 2πy

[1 + r22 + 2r2 cos 2πy]2= 0, (2b)

whereY = X/b, andy = x/b are the coordinates ofthe plate and particle, respectively, in units of the pe-riod of the potentialb, τ = ω0t is the dimensionlesstime, ε = m/M the ratio of particle and plate masses,α = Ω/ω0 the ratio of frequencies of the free oscilla-tions of the top plate and the particle,Ω = √

K/M.To simplify the investigation, we have assumed

b1 = b2 = b, µ1 = µ2 = µ,

U01 = U02 = U, (3a)

where 0< ε 1. The frequencyωi of the small os-cillations of the particle in the minima of the potentialis connected to the shape parameterri by

ω2i = ω2

0

(1 − ri

1 + ri

)2

, i = 1,2, (3b)

γ = µ/mω0 is a dimensionless friction constant whichaccounts for dissipation due to phonons and/or other

oscillations, andV = v/bω0 is the dimensionless topplate velocity,ω0 = (2π/b)

√u/m the frequency of

the small oscillations of the particle in the minima ofRP potential whereri = 0. Let us introduce, as an ex-ample, the following set of parameters correspondingto the underdamped system, namely[22]: α = 0.02,γ = 0.1, ε = 0.125.

3. Adiabatic approximation

In this section, the coupled equations are derivedthat support two modes: the slow mode for the topplate and the fast mode for the particle, respectively.In the first step, we assume the adiabatic approxima-tion which is very convenient from an analytical pointof view since there is a separation of time scales be-tween the particle and the plates. This approximationis strictly accurate only for the characteristic frequencyof the large scale plate motion that is much smallerthan the characteristic frequency of the particle oscil-lations and the natural frequency, and for the mass ofthe particlem that is smaller than the mass of the topplateM. In the framework of these assumptions, andassuming that, forEq. (2b), the plate moves with aconstant velocityY = V , Eqs. (2a) and (2b)become

Y = V, (4a)

y + γ(2y − V)

+ 1

2π(1 − r21)

2 sin 2π(y − Vτ)

[1 + r21 + 2r1 cos 2π(y − Vτ)]2

+ 1

2π(1 − r22)

2 sin 2πy

[1 + r22 + 2r2 cos 2πy]2= 0. (4b)

In a second step, we solve numericallyEqs. (4a) and(4b), and explore the behavior of the particle. In thecaser1 = r2 = 0, the system of equations (4) candescribe a dissipative parametrically driven pendulumand a dissipative motion of a particle in two waves.

3.1. Numerical analysis of the motion of theparticle

In order to understand dynamical processes inour deformable oscillator model, the system of


Fig. 2. Particle trajectories for selected values of the plate velocities, forr1 = 0.5 andr2 = 0.9. (f) Magnification of enclosed part of (e).(a) V = 0.05; (b)V = 0.15; (c) V = 0.38; (d) V = 0.39; (e)V = 0.45.

equations (4) have been integrated numerically witha fourth-order Runge–Kutta scheme with the timestep equal to 0.055. The results of this numericalintegration are shown inFig. 2(a)–(f), which we ob-tained forr1 = 0.5 andr2 = 0.9. Depending on thevalue of the plate velocities namelyV = 0.05, 0.15,0.38, 0.39 and 0.45, the dynamical behavior of theparticle can be classified in one of the following fivecategories:

(i) For V = 0.05, the particle clings to one of theplates as indicated inFig. 2(a). In this case, thetime-averaged velocity of the particle is equal toV or 0.

(ii) For the case ofFig. 2(b) whereV = 0.15, weobserve that the particle undergoes fast oscilla-tions with the periodV−1, around the trajectoryy = (1/2)Vτ. Here, the time-averaged velocityof the particle is equal to (1/2)V.

(iii) With appropriate velocityV = 0.38, as inFig. 2(c), the particle jumps between the twoplates being trapped by each of them fortime much longer thanV−1. Here also, thetime-averaged velocity of the particle is equal to(1/2)V.

(iv) A further increase in the plate velocity namelyV = 0.39, yields another changes in the dynamicof the system where the motion gradually evolves


until it reaches a critical velocityVc = 0.3847for which the particle is trapped by one of theplates, for example, the top plate as shown inFig. 2(d). The time-averaged velocity of the par-ticle is equal toV.

(v) For a still higherV value of 0.45, the particle iscompletely trapped by the lower plate and oscil-lates within one cell of the corrugated RP poten-tial URP(φ, r). These oscillations are visible inthe zoom ofFig. 2(e) (shown inFig. 2(f)). Thetime-averaged velocity of the particle is closeto 0.

We would like to mention that the above scenariolooks very similar to the one that Rozman et al.described for ther1 = r2 = 0 parameters of de-formability, that is, for the sinusoidal potential[22].However, as we will show below, the above scenariois not standard for all parameters of deformability.Thus, in the next section, we discuss numerically theeffects of parameters on the dynamical behavior ofthe particle motion.

The phase space plot ofy(t) versusy(t) is a veryefficient tool in the study of dynamic systems, be-cause it presents the global behavior of these systems.Fig. 3(a) and (d) shows various examples of phaseportraits of the particle in the underdamped regime asthe plate velocity increases. The results presented havebeen calculated for the case ofr1 = 0.5 andr2 = 0.9.Fig. 3(a) and (d), which we obtained forV = 0.05 andV = 0.45, respectively, indicates that the motion ofthe particle is quasi-periodic.Fig. 3(b) and (c), whichshow the results obtained forV = 0.15 andV = 0.38,respectively, exhibit chaotic behavior.

To end this section, it is interesting to compare theglobal behavior described here and that which weobtained on the approximate system under the adi-abatic approximation and the exact equations. Thus,we carry out a direct numerical integration of theexact equations (2) by using a standard fourth-orderRunge–Kutta method with the time step equal to0.055 and with the set of fixed initial conditions(y, Y, y, Y ) = (0,0,0, V). Let us mention that con-trary to the approximate system, the exact equationsdiffer from the approximate ones in that terms in

the equation of motion for the top plate involve theratio of particle mass to plate massε, and the ratioof free oscillations of the top plate to the particleα.These two parameters will have significant influenceon the numerical solutions of these exact nonlineardifferential equations. Plotting on the same graphthe approximate (solid line) and exact (dashed line)results, we can easily do the comparison. For thesecalculations, we have been able to reproduce as indi-cated inFig. 3(a) the exact result in good agreementwith the adiabatic approximation, which we obtainedfor ε = 0.125 andα = 0.97, with the values ofr1,r2, V andγ entered here being those in the adiabaticapproximation. Withε = 0.125 andα = 0.03, weobtained an excellent agreement between both theresults as depicted inFig. 3(d). But when we changedthe value ofα, and maintained other parameters con-stant, the figures obtained were totally different. Itwould take very long to study all the possible caseswhich can occur inEq. (2).

3.2. Influence of parameters of deformability on thetype of the particle motion

In respect of the previous results, we define thethree mean trajectories as follows: types TMI, TMIIand TMIII for the time-averaged velocityVm, equalto 0, (1/2)V, and V, respectively. These differenttypes also correspond to the ratioVm/V equal to 0,1/2, 1. For convenience, we introduce this quantityRv = 2(Vm/V) − 1, in order to recognize differenttypes of mean trajectory for a given velocity, forwhich −1, 0, and 1 correspond to types I, II and III,respectively. Following the variation ofV, we ob-served that, to each couple of parameters (r1, r2) cor-responds a particular variation of the type of motion.We have tried to group them which resulted in fivephases:

(1) Phase I obtained forr1 = 0.3 and r2 = −0.9(seeFig. 4(a)), corresponds to type TMI in the allvelocities interval.

(2) In phase II which we obtained forr1 = 0.05 andr2 = −0.5 (seeFig. 4(b)), type TMIII succeeds toTMI at a certain velocityVp, and comes back toTMI just after at the next value ofV.


Fig. 3. Phase space representation of the particle trajectories in the underdamped regime for selected values of the plate velocities and forr1 = 0.5 andr2 = 0.9. y and y are presented in dimensionless units. Dashed lines are obtained by the exact equations for: (a)α = 0.97and ε = 0.125; (d)α = 0.03 andε = 0.125.

(3) At phase III obtained forr1 = 0.3 andr2 = −0.3(seeFig. 4(c)), phase I appears after a close in-terval [V1, V2] giving type II. We can also givethis phase the following name: intermittent phase

due to the existence of the intermittent stick-slipmotion in this phase. Its particularity lies in thefact that it is the only phase in which the criticalvelocity can be found.


Fig. 4. The quantityRv plotted againstV, for fixed values of the couple shape parameters (r1, r2). One can see how things hang, arelinked, together. The particular caser1 = r2 corresponds to (c). (a)r1 = 0.3 and r2 = −0.9; (b) r1 = 0.05 andr2 = −0.5; (c) r1 = 0.3and r2 = −0.3; (d) r1 = −0.7 andr2 = 0.1; (e) r1 = −0.7 andr2 = −0.85.

(4) Phase IV obtained forr1 = −0.7 andr2 = 0.1(seeFig. 4(d)), Fig. 4(e) corresponds to the pas-sage of type TMIII to type TMI, above a certainvelocity.

(5) In the last phase, that is phase V obtained forr1 =−0.7 andr2 = −0.85 (seeFig. 4(e)), the trajectoryis not predictable in a certain intervalJ of V. Wemove to TMI towards TMIII for a certain velocity,and after follows the reverse phenomenon, i.e. toTMIII towards TMI at the next value; and so onin this interval.

Note that, in each of phases II–IV, windows of phaseV can occur. When, varying parametersr1 andr2 from−1 to 1, respectively, we carry out our simulation,

depending on the sign ofr1, the following two resultshave been obtained:

(a) Negative values of the parameter r1. For the neg-ative value of r1, the phases mentioned abovesucceed themselves as follows: phase I precededphase II which makes place for the third phasewhen the parameterr2 is in the vicinity ofr1. Afterthat arrives phase IV, leading once more to phaseIII while r2 approaches the vicinity of |r1|. The in-tensity of the intermittency will decrease and leadus to phase I forr2 tending to 1. We noticed thatthis intensity of intermittent motion is highest forr2 ∼ r1. For this reason, we are not astonished ofthe ‘precocious’ appearance of the intermittency


as r1 and r2 tend to−1. We also note that, be-tween phases III and IV, intervenes a very smallinterval of r2 corresponding to phase II.

(b) Positive values of the parameter r1. Concerningthe parameterr1 > 0, the different phases followin this order I, II and III. Contrary to the previoussign ofr1, phase II occupies a more larger band of

Fig. 5. The quantityRv is plotted vs.V and r2 for: (a) r1 = −0.69; (b) r1 = −0.9; (c) r1 = r2 = r; (d) graph of the variation of theintensity of the intermittency for negative values ofr1; (e) graph of the variation of the intermittency for positive values ofr1. The dashedline corresponds to the particular caser1 = r2 = 0.

r2, and begin to coexist with the third one sincer2

is in vicinity of −r1. Its intensity decreases withthe increasing ofr2 and the coexistence of bothphases gains more and more on intermittence. Thiscoexistence makes place for phase III asr2 is closeto r1. This intermittency persists, but decreasesin intensity, until r2 tends to 1. The intermittent


stick-slip motion is once again very intense whenr2 ∼ r1. It is useful to notice the absence of phaseIV in this case since the intermittency occurs.

It is also important to note that the critical velocity,while we are presently in phase III, does not varysignificantly with r2 for any sign ofr1. We have theintensity of intermittency in the interval ofV giv-ing the intermittent stick-slip motion, i.e.Rv closeto 0. Fig. 5(a) and (b) presents the two cases men-tioned above of sign ofr1. As we can see in thesefigures, the top and bottom platforms denote typesTMIII and TMI, and the intermediate level is typeTMII. A particular case is presented inFig. 5(c) usingr1 = r2 = r. One can observe a sort of street denotingthe existence of the intermittency for any value ofr.In Fig. 5(d) and (e), we have a schematic represen-tation of the variation of the intensity of the inter-mittency.

3.3. Lyapunov exponent

The value of the Lyapunov exponent is responsiblefor exponential divergence or convergence of nearbytrajectories and, here we shall investigate only thevalue of this Lyapunov exponentλ. This largest ex-ponent plays a very important role because it is pre-cisely this exponent that determines the motion char-acter for the majority of the trajectories of the system.The states withλ > 0 represent the dynamical chaosstates and the states withλ ≤ 0 represent the regu-lar (periodic) states.Figs. 6 and 7show the Lyapunovexponent of our system calculated usingEq. (4), fordifferent values of the couple parameters of deforma-bility ( r1, r2). Fig. 6(a)–(d) which we obtained forr1 = 0.5 andr2 = 0.9 presents the evolution of theLyapunov exponent for four values of the plate veloc-ities corresponding to different dynamical behavior ofthe particle. The positive values of the largest Lya-punov exponent obtained forV = 0.15 andV = 0.38,for which the time-averaged velocity of the particle isequal to (1/2)V, indicate that the motion of the particleis essentially chaotic. This is illustrated inFig. 6(b)and (c). However, inFig. 6(a) and (d), the Lyapunovexponents obtained forV = 0.05 andV = 0.45 are

Fig. 6. Lyapunov exponentλ(t) vs. time for selected values of theplate velocities, forr1 = 0.5 andr2 = 0.9.

negative, which implies that the motion of the particleis periodic.

We have also computed the time-averaged largestLyapunov exponent as a function ofV for some phases.This evolution is presented inFig. 7(a)–(d). Each fig-ure presents the time-averaged largest Lyapunov ex-ponent and the corresponding phase. For example, inFig. 7(a) obtained forr1 = 0.3 andr2 = −0.9, whichcorresponds to phase I, the time-averaged largest Lya-punov exponent equals zero, and the motion of theparticle is regular. InFig. 7(b) obtained forr1 =0.3 and r2 = 0.4, corresponding to phase III, reg-ular and irregular behavior are presented. Phase IVwhich has been obtained forr1 = −0.7 and r2 =0.1 is completely regular (seeFig. 7(c)) because thetime-averaged largest Lyapunov exponents are essen-tially negative in all the velocity intervals. InFig. 7(d)obtained forr1 = −0.7 andr2 = −0.8, correspondingto phase V, the particle behavior is essentially chaotic.Finally, chaotic domains can be clearly distinguished


when representing the variation of the time-averagedlargest Lyapunov exponent as function ofV andr2 forr1 = −0.6 andr1 = 0, respectively (seeFig. 7(e) and(f)), and also as a function ofr1 and r2 for V = 0.2(seeFig. 7(g)).

Fig. 7. Lyapunov exponentλ(t) vs. the dimensionless velocity of the plateV: (a) r1 = 0.3 and r2 = −0.9; (b) r1 = 0.3 and r2 = 0.4;(c) r1 = −0.7 and r2 = 0.1; (d) r1 = −0.7 and r2 = −0.8. Time-averaged largest Lyapunov exponent as functions ofV and r2 for: (e)r1 = −0.6; (f) r1 = 0; (g) time-averaged largest Lyapunov exponent as functions ofr1 and r2 for V = 0.2.

4. Force fluctuations and influence of the shapeparameters

We now wish to explore the motion of the top plateas well as the effects of the shape parameters on the


Fig. 7. (Continued).

particle–plate interaction force by examining the com-plete coupled equations of motion for the top plate andthe particle,Eqs. (2a) and (2b).

4.1. Influence of the frictional force on the type ofthe particle motion

The study of the influence of the frictional forceon the particle motion is facilitated by considering thesolutions ofEq. (4b), that is,y(t, Y ), that depend para-metrically onY and also on the shape parametersr1

andr2. Substituting these solutions intoEq. (2a)yieldsthe equation describing the plate in the nonadiabaticapproximation

Y − εF(τ, Y, Y)+ α2(Y − vτ) = 0, (5)

where the particle—plate interaction force

F(τ, Y, Y)= −γ(Y − y)+ 1

2π(1 − r21)

2

× sin 2π(y − Y)

[1 + r21 + 2r1 cos 2π(y − Y)]2(6)


contains fast-oscillating components. Following[22],we averageEqs. (5) and (6)over the fast oscillations,and obtain an equation for the slow-oscillating com-ponent of the spring length

L(τ) = Y(τ)− vτ, L− εϕ(L+ v)+ α2L = 0,

where the time-averaged forceϕ(Y) = 〈F(τ, Y, Y)〉depends only on the velocity of the plate, and presentsthe effective friction for the top plate motion.

Alternatively to the slow-oscillating component ofthe spring length, one may characterize the effec-tive friction which is the velocity dependence of thetime-averaged force and which takes into accountthe component of the frictional force as well as thedissipative contribution. The results of the numericalcalculations are presented inFig. 8(a)–(e), where we

Fig. 8. Frictional forces acting on the top plate as a function of plate velocityV: (a) r1 = r2 = 0; (b) r1 = 0.6, r2 = −0.9; (c) r1 = 0.2,r2 = −0.9; (d) r1 = 0.2, r2 = −0.2; (e) r1 = −0.6, r2 = 0.4.

have shown the frictional forces acting on the topplate as a function of the plate velocityV for fivevalues of the couple parameters of deformability (r1,r2). Fig. 8(a) which we obtained forr1 = r2 = 0 isalso shown for comparison, where the lower curve isthe dissipative contribution and the upper curve is thenet force. As the figures indicate, we have identifiedthree characteristic behaviors of particle trajectories.

At V V ∗, the particle is trapped by one of theplates. AtV < V ∗, we observed a regular stick-slipmotion of the particle. AtV > V ∗, the particle alwaysclings to one of the plates.

Following the variation of the deformability param-etersr1 andr2, the structure of forces shows the fivephases of the particle motion presented inSection 3.2(seeFig. 8(b)–(e)).


Fig. 9. Time series of the spring force corresponding to four different values of the stage velocitiesV with r1 = 0.5, r2 = 0.4: (a) 0.05;(b) 0.1; (c) 0.35; (d) 0.37.

4.2. Numerical analysis of the motion of the topplate

We are first looking for the dynamical behavior ofthe top plate within velocity values of interest, leadingto three different dynamical regimes obtained forr1 =0.5 andr2 = 0.4 and for four different values of thestage velocity, namelyV = 0.05, 0.1, 0.35, and 0.37,respectively:

(a) The stick-slip motion. Initially at rest, the top platebegins to slide when the spring force acting onthe top plate exceeds the static frictional forceFS ,which is equal to 2π(U0/b)[(1 − r2i )/(1 + r2i )]

2.The spring force decreases, when the plate velocityis Y > V , until it reaches some value where themotion stops and then the process repeats. Thisphenomenon is produced at low velocities and isillustrated inFig. 9(a) and (b) where we have takenV = 0.05 andV = 0.1, respectively.

Fig. 10. Time series of the spring force corresponding to four different values of stage velocitiesV with r1 = −0.9; r2 = 0.5: (a) 0.05;(b) 0.1; (c) 0.35; (d) 0.85.

(b) The intermittent stick-slip motion whereV = 0.35is characterized by spring force fluctuation. Here,the amplitude of the spring force strongly dependson the stage of the velocityV and the oscillationsof the spring force are faster than in the formerregime. This is shown inFig. 9(c).

(c) The sliding motion for whichV = 0.37 andwhich occurs above a critical driving veloc-ity V = 0.3539 in the neighborhood of thatfound in the model of adiabatic approximation[8]. The spring force is practically constant (seeFig. 9(d)).

As shown inFig. 9(a)–(d), these regimes are closelylinked to the spring force fluctuations which the am-plitude decreases asV increases. These results havebeen also obtained by Rozman et al. for the sinusoidalpotential[22], for which r1 = 0 andr2 = 0. The nextsubsection presents the modifications brought out bythe parameters of deformability.


Fig. 11. Plot of the average amplitude of the spring force fluctuations vs.V and r2 for: (a) r1 = −0.8; (b) r1 = −0.5.

4.3. Influence of the shape parameters on the springforce

In the last regime (the sliding motion), keeping fixedthe values of the deformability parametersr1 and r2,

Fig. 12. Plot of the average amplitude of the spring force fluctuations vs. (r1, r2) for: (a) V = 0.05; (b) V = 0.25.

from −1 to +1, respectively, we obtained the follow-ing results:

• Firstly, we notice that these three regimes can onlybe observed for an intervalJ belonging to℘ = ] −


Fig. 13. The power spectrum of the spring force fluctuations forV = 0.75, r1 = 0.15 andr2 = 0.95.

1; 1[ of the parameters of deformability. This inter-val J is nearby [−0.5; 0.5].

• Secondly, taking only one of the parameters out ofthe intervalJ, the spring force, begins once againto fluctuate far after the critical velocity, at aboveV ≥ 0.6. It is very important to note that we cannotconsider again this model as the first one for highvelocities.

• The last point is the case where both parameters aretaken in℘ but far from J. In this case, the forcefluctuates at any velocities.

For the last two cases mentioned above, the ampli-tude of fluctuations of the spring force are not largecompared to those obtained withr1 andr2 taken in theintervalJ. This is illustrated inFig. 10(a) and (b) ob-tained forr1 = −0.9 andr2 = 0.5, and for four differ-ent values of the stage velocity, namelyV = 0.05, 0.1,0.35, and 0.85. But at high velocities, time-averagedamplitude is almost the same.

We have defined the time-averaged amplitude ofoscillations of the spring force as the difference ofmean maximal and mean minimal amplitudes. Thisallows us to present graphically the results mentionedabove. We have, inFig. 11(a) and (b), obtained forr1 = −0.8 andr1 = −0.5, respectively, the variationsof the difference of time-averaged amplitude of oscil-lations of the spring force as functions ofV and r2,and inFig. 12(a) and (b) as functions ofr2 andr1 forV = 0.05 andV = 0.25, respectively.

The power spectrum of the spring force fluctuations,shown inFig. 13, computed forV = 0.75 (i.e.V >Vc), r1 = 0.15 andr2 = 0.95, which corresponds tothe sliding regime, has the highest peak centered at

155. The presence of other frequency peaks indicatesthat in the sliding regime, the spring force performsmicroscopic oscillations with a period of the orderV−1

[22].

5. Conclusion

In this work, we have been able to describe fric-tional phenomena in a nonlinear model that containsa deformable nonsinusoidal substrate potential, whichis different from the well-known sinusoidal potentialwhich has a constant profile. It should be noticed thatour basic coupled equations describing the slow modefor the top plate and the fast mode for the particle arerather general and could be considered as some gen-eralization of the Rozman et al. model of a particleinteracting with two sinusoidal potentials. Moreover,some effects mentioned in the present work are alsorather general. For instance, for the shape parametervalues such asr1 = 0.5 andr2 = 0.9, and dependingon the value of the plate velocities, the motions of theparticle, such as periodic stick-slip, erratic, intermit-tent stick-slip and sliding motions, are very similar tothe ones described by Rozman et al. for the sinusoidalpotential (r1 = 0 andr2 = 0). We have also investi-gated the influence of the parameters of deformabilityon the particle motion. For negative as well as positivevalues of the deformability parameters, the intermit-tent stick-slip motions are very intense whenr1 ∼ r2.

In order to evaluate our results which we obtainedin the adiabatic approximation, we made use of thefourth-order Runge–Kutta scheme to integrate the ex-act equations of motion. By carefully considering the


values of the parametersε andα, namelyε = 0.125andα = 0.97, andε = 0.125 andα = 0.113, respec-tively, good agreement between the adiabatic approx-imation and the exact equations was found. We havealso characterized the effective friction which is thevelocity dependence of the time-averaged force andwhich takes into account materials that undergo struc-tural changes under some external conditions. As a fi-nal remark, we want to stress that we have provided aquite thorough description of the features of the mech-anism of frictional phenomena in a deformable poten-tial introduced by Remoissenet and Peyrard. We hopethat these theoretical considerations will be of help indealing with more complex situations. In forthcomingstudies, intensive investigations will be carried out onEq. (2)and particular attention will be allocated to in-vestigations on the effective frictional force.

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Documents

Stick-slip motion in a driven two-nonsinusoidal Remoissenet–Peyrard potential