Relaxation Tribometry: A Generic Method to Identify the Nature of Contact Forces
TTribology Letters manuscript No. (will be inserted by the editor)
Relaxation tribometry: a generic method to identify thenature of contact forces
Alain Le Bot · Julien Scheibert · Artem A. Vasko · Oleg M. Braun
Received: date / Accepted: date
Abstract
Recent years have witnessed the develop-ment of so-called relaxation tribometers, the free os-cillation of which is altered by the presence of frictionalstresses within the contact. So far, analysis of such oscil-lations has been restricted to the shape of their decay-ing envelope, to identify in particular solid or viscousfriction components. Here, we present a more generalexpression of the forces possibly acting within the con-tact, and retain six possible, physically relevant terms.Two of them, which had never been proposed in thecontext of relaxation tribometry, only affect the oscil-lation frequency, not the amplitude of the signal. Wedemonstrate that each of those six terms has a uniquesignature in the time-evolution of the oscillation, whichallows efficient identification of their respective weightsin any experimental signal. We illustrate our method-ology on a PDMS sphere/glass plate torsional contact.
Keywords
Relaxation tribometer · Damped oscil-lations · Amplitude decay curve · Frequency shift · Nonlinear contact forces · Two-times averaging method
The energy dissipated during relative motion of solidsurfaces in contact corresponds to the work of the fric-tion force. It is therefore appealing to measure frictionforces without force sensors, just from the energy decay
A. Le Bot · J. ScheibertUniv Lyon, Ecole Centrale de Lyon, ENISE, ENTPE, CNRS,Laboratoire de Tribologie et Dynamique des Syst`emes LTDSUMR5513, F-69134, Ecully, FranceE-mail: [email protected]. Vasko · O.M. BraunInstitute of Physics, National Academy of Sciences ofUkraine, 46 prospect Nauki, Kiev, 03028, Ukraine that they induce in a frictional system. This is preciselywhat the so-called relaxation tribometry is about. Thebasic idea is to place a tribological interface in an oscil-lator, provide the latter with a certain amount of initialmechanical energy, and let it oscillate and relax back toits equilibrium position. The time-rate of such a relax-ation informs about the amplitude of the friction force,while the envelope of the decaying oscillation charac-terizes the type of dissipative mechanism involved.The idea of measuring a viscous damping coeffi-cient by monitoring the decay of vibration of an os-cillator dates back to Rayleigh, who described the log-arithmic decrement technique in his famous treatise [1,p.46]. But relaxation tribometry actually starts withthe remark that a velocity-independent friction coeffi-cient can also be measured from the time-decay of theenvelope of the vibration [2,3]. When both friction andviscous dissipation are present, the solution of the gov-erning equation of an oscillator and its amplitude de-cay curve was found by Markho [4]. This led Feeny andal. [5,6] to extend the decrement method to measure si-multaneously viscous and friction coefficients, while Wuand al. [7] proposed to apply the method to nonlinearviscous damping. Rigaud et al. [8] performed similar si-multaneous measurements, not from the amplitude butfrom the energy decay, on contacts lubricated by water-glycerol solutions.Recently, renewed interest for relaxation tribometryhas emerged as a unique tool to measure low forcesefficiently and accurately [9,10,11]. The reason is thatthe smaller the friction force, the smaller the decrement,the more measurable oscillations before rest and thusthe more data available to estimate the force.Moreover, the method is so accurate that non-conven-tional behaviours, neither purely frictional nor purelyviscous, could be detected. On the one hand, nonlin- a r X i v : . [ phy s i c s . c l a ss - ph ] A p r Alain Le Bot et al. ear dissipative forces have been identified and mea-sured [12]. On the other hand, a progressive shift ofthe oscillation frequency has been observed with a ro-tational tribometer [13]. Both observations confirm thattribological interfaces are complex [14] and thus cannotbe fully described with a simple combination of viscousand friction coefficients. A sliding contact, in transla-tion or torsion, is in particular made of a multitudeof micro-contacts [15,16], possibly implying differentmaterials [17]. It can also be divided into coexistingslip and stick zones [18,19]. The question thus arisesof which nonlinear forces can actually be detected byrelaxation tribometry.In this paper, we propose a generic description ofthe type of signals recovered using relaxation tribome-try. By applying systematically the two-times averagingmethod already used in [8], we describe the character-istic signatures of six relevant contact forces, in termsnot only of the decay of their amplitude, but also of thefrequency shift.
The principle of a relaxation tribometer consists in ob-serving the free vibration of an oscillator equipped witha sample rubbing on a surface. The decay of vibration,due to friction and other sources of dissipation, holds in-formation on contact forces acting on the sample. Thus,a simple measurement of the motion allows to extractthe friction force without using a force probe. Note thatrelaxation tribometry, in essence, probes the transient(as opposed to steady sliding) response of a frictionalinterface over the characteristic time scale of the os-cillator’s period. Two types of tribometer have beenreported in the literature [8,13] depending on the kine-matics followed by the rubbing sample: translation ortorsion (see Fig. 1). θ samplesurface P θ samplesurface mass m x P inertia I Fig. 1
Left: principle of a torsional pendulum with asphere/plane contact at the extremity of the rotation axis.Right: principle of a translational oscillator with a sphere incontact with a plane and attached to the moving mass.
In this section, we will argue on the basis of a tor-sional relaxation tribometer (Fig. 1, left), with no lossof generality. For a translational relaxation tribometer(Fig. 1, right), one should simply replace the angle θ with the dimensionless position mω x/P , and the an-gular speed ˙ θ with the dimensionless speed mω ˙ x/P where m is the moving mass, ω the natural frequencyand P the normal load.In a torsional relaxation tribometer, an axisymmet-ric sample is pressed against a surface by a load P and issubmitted to torsional oscillations. The governing equa-tion is:¨ θ + ω θ = − ω f ( θ, ˙ θ ) (1)where ω is the contactless natural frequency of the os-cillator and f ( θ, ˙ θ ) a dimensionless force induced by thesample rubbing on the surface and/or by a non-ideal be-haviour of the pendulum. Here f = − M/ ( Iω ) where M is the external torque on and I the total inertia ofthe pendulum. If we had considered a translational tri-bometer, then f = − T /P where T is the transverseforce. The origin θ = 0 is conventionally fixed at equi-librium. By construction, the pendulum is symmetric.The force f is odd with respect to parity and time re-versal, so that: f ( − θ, − ˙ θ ) = − f ( θ, ˙ θ ) . (2)The reactive force f may still take a wide variety offorms, two special cases being of particular interest.On the one hand, when the force is a dissipative re-action of the contact (typically friction), it is usuallyassumed to depend on the sliding speed ˙ θ only, and noton the position θ and Eq. (2) imposes f ( − ˙ θ ) = − f ( ˙ θ ).For instance, a linear viscous force f ( ˙ θ ) ∝ ˙ θ as wellas any drag force f ( ˙ θ ) ∝ ˙ θ n with n odd, satisfies thiscondition. Another example is the solid friction force f ( ˙ θ ) ∝ sgn( ˙ θ ), which is discontinuous at zero. The gen-eral form including these examples and matching thedissipative condition − f ( ˙ θ ) ˙ θ < f ( ˙ θ ) = λ sgn( ˙ θ ) + 2 ζ ˙ θ/ω + δ ˙ θ sgn( ˙ θ ) /ω + o ( ˙ θ ), where λ , ζ and δ are positivedimensionless constants.On the other hand, when f is conservative, it derivesfrom a potential and consequently does not depend on˙ θ . Since f is an odd function of θ (by Eq. 2) and con-tinuous at zero, a pseudo-polynomial series expansiongives f ( θ ) = αθ + νθ sgn( θ ) + (cid:15)θ + o ( θ ) where theconstants may be positive or negative.Limiting ourselves to the above orders (2 in ˙ θ and3 in θ ), we are therefore left with six terms to study:a constant friction force f = λ sgn( ˙ θ ), a linear viscousforce f = 2 ζ ˙ θ/ω , a quadratic dissipative force f = δ ˙ θ sgn( ˙ θ ) /ω , a linear elastic force f = αθ , a quadratic elaxation tribometry: a generic method to identify the nature of contact forces 3 elastic force f = νθ sgn( θ ), and a cubic elastic force f = (cid:15)θ .For weakly nonlinear systems – when f is small –the solution of Eq. (1) may be approximated by: θ ( t ) = a ( t ) cos [ ω t + φ ( t )] (3)where a ( t ) and φ ( t ) are slowly varying functions, i.e. over times much larger that 2 π/ω . The magnitude a ( t )and phase φ ( t ) will now be obtained by applying thetwo-times averaging method [20,21,22] (see Appendixfor a brief review of the method).In practice, the force f acting on the pendulum isnot known. One has to determine it by observing thebehaviour of the pendulum. Thus, the time evolutionof the magnitude, a ( t ), and of the phase, φ ( t ), will con-stitute the only available information to identify therelevant terms in the right-hand side of Eq. (1).2.1 Decaying envelopeThe magnitude a ( t ) constitutes the envelope of vibra-tion, the decay of which indicates the energy lost in thesample. When the six forces are present simultaneouslyand are small, the two-times averaging method applies.The time derivative of the magnitude, ˙ a , is given in Ap-pendix, Eq. (17) where the right-hand side is obtainedby summing all terms (cid:104) h sin ϕ (cid:105) given in Table 1. Theresult is:˙ a = − π ω λ − aω ζ − a π ω δ. (4)This is an ordinary differential equation of first orderon a . Let us remark that the constants α , ν , and (cid:15) donot appear.As observed in [2], a pure constant friction force f = λ sgn( ˙ θ ) imposes a linear decreasing of the successivelocal maxima. The equation of the envelope is obtainedby integrating (4) with ζ = δ = 0: a ( t ) = a (0) − λπ ω t. (5)An interesting consequence of Eq. (5) is that since a ( t ) ≥
0, the vibration always stops after a finite durationequal to a (0) π/ λω . This is illustrated in Fig. 2, topfor initial angle θ (0) = 1 and speed ˙ θ (0) = 0.A pure linear viscous force f = 2 ζ ˙ θ/ω imposes anexponentially decreasing magnitude [1]. The equationof the envelope is from (4) with λ = δ = 0: a ( t ) = a (0) exp( − ζω t ) . (6)Thus the pendulum vibrates forever (Fig. 2, middle). Time ω t A ng l e θ Fig. 2
Time evolution of angle θ for various forces f . Top:case of constant friction force λ sgn( ˙ θ ) for λ = 0 .
03. Mid-dle: case of linear viscous force 2 ζ ˙ θ/ω for ζ = 0 .
05. Bottom:case of quadratic dissipative force δ ˙ θ sgn( ˙ θ ) /ω for δ = 0 . θ (0) = 1, ˙ θ (0) = 0. Broken line: envelope by Eqs. (5),(6), and (7). The case of a pure quadratic dissipative force of type f = δ ˙ θ sgn( ˙ θ ) /ω is more original, and to our knowl-edge has only been investigated in [12]. The integrationof (4) with λ = ζ = 0 gives: a ( t ) = 1 (cid:104) a (0) + δ π ω t (cid:105) (7)This result is illustrated in Fig. 2, bottom.The three other terms, linear elastic force f = αθ ,quadratic elastic force f = νθ sgn( θ ) and cubic elas-tic force f = (cid:15)θ , are conservative. Consequently, thecorresponding magnitude a ( t ) = a (0) is constant overlarge time scales. This is consistent with the fact that α , ν , and (cid:15) do not appear in Eq. (4).2.2 Varying frequencyThe time evolution of the phase φ ( t ) is a typical non-linear effect. The instantaneous frequency is given by: ω = dd t [ ω t + φ ( t )] = ω + ˙ φ ( t ) (8)where ˙ φ is the time-derivative of φ . Alain Le Bot et al.
Magnitude a [rad] F r equen cy ω / ω Fig. 3
Evolution of frequency with amplitude for a purequadratic dissipative force δ ˙ θ sgn( ˙ θ ) /ω . Symbols: numericalsolution to Eq. (1) with initial conditions θ (0) = 1, ˙ θ (0) = 0(one value per half-period) with δ =0.05 ( (cid:66) ), 0.1 ( (cid:52) ), 0.3 ( (cid:53) )or 0.8 ( ♦ ). Broken line: Eq. (9). There exists analytical results for the value of theangular frequency in the presence of some of the forcesconsidered here. The case of a linear elastic force f = αθ is rather trivial since one can write Eq. (1) as ¨ θ + ω (1+ α ) θ = 0, so that the angular frequency of the oscillationhas a constant value ω √ α . If a viscous dampingforce f = 2 ζ ˙ θ/ω is added, the frequency of the oscilla-tion remains constant during the oscillation, but withthe value ω ∞ = ω √ α (cid:112) − ζ . It has been shownin [5] that this conclusion remains true if a constantfriction force f = λ sgn( ˙ θ ) is added.For the other three forces, we solved the evolutionof the angular frequency using the two-times averagingmethod. Equation (18) of Appendix gives a ˙ φ as a linearcombination of all terms (cid:104) h cos ϕ (cid:105) given in Table 1. Thefrequency ω = ω + ˙ φ is then: ω = ω (cid:18) α a π ν + 3 a (cid:15) (cid:19) . (9)Note that, according to the two-times averaging method,Eqs. (4) and (9) correspond to a linear approxima-tion in all small terms λ , ζ , δ , α , ν and (cid:15) . In partic-ular, to this degree of approximation, (cid:112) − ζ (cid:39) √ α (cid:39) α/
2, so that ω ∞ (cid:39) ω (1 + α/ λ , ζ , and δ do not ap-pear in Eq. (9). In particular, although highly nonlin-ear, the dissipative quadratic force (amplitude δ ) doesnot induce any time variation of the frequency (Fig. 3).The situation is different for elastic forces. In thecase of a pure quadratic elastic force f = νθ sgn( θ ),Eq. (9) gives ∆ω = ω − ω ∞ = 4 νaω / (3 π ). Thus, aquadratic elastic force exhibits a linear variation of fre-quency versus magnitude of envelope. The sign of ν Magnitude a [rad] F r equen cy ω / ω Fig. 4
Evolution of frequency with amplitude for a purequadratic elastic force νθ sgn( θ ). Symbols: numerical solu-tion to Eq. (1) with initial conditions θ (0) = 1, ˙ θ (0) = 0 (onevalue per half-period) with ζ = 0 .
05 and ν =-0.5 ( (cid:66) ), -0.2 ( (cid:52) ),0.2 ( (cid:53) ), 0.9 ( ♦ ). Broken lines: Eq. (9). Magnitude a [rad] F r equen cy ω / ω Fig. 5
Evolution of frequency with amplitude for a pure cu-bic elastic force (cid:15)θ ). Symbols: numerical solution to Eq. (1)with initial conditions θ (0) = 1, ˙ θ (0) = 0 (one value per half-period) with ζ = 0 .
05 and (cid:15) =-0.9 ( (cid:66) ), -0.1 ( (cid:52) ), 0.5 ( (cid:53) ), 1.5( ♦ ). Broken lines: Eq. (9). controls the type of frequency evolution: an experimentexhibiting an increasing frequency with increasing time, i.e. with decreasing amplitude of the oscillation corre-sponds to ν <
0. Conversely, if the period of the pendu-lum increases with time, then ν >
0. Those results aredemonstrated on Fig. 4, which successfully comparesEq. (9) with simulation results.Finally, the case of a pure cubic elastic force, whichcorresponds to the well known Duffing oscillator, is alsosolved by Eq. (9) [20]. The frequency shift is now givenby ∆ω = ω − ω ∞ = 3 (cid:15)a /
8, where the variation of elaxation tribometry: a generic method to identify the nature of contact forces 5 frequency is quadratic in a . As for the quadratic elas-tic force, the sign of (cid:15) may be determined by observ-ing whether the pendulum experiences an increasing( (cid:15) >
0) or decreasing ( (cid:15) <
0) period as time increases.Again, those results are found in good agreement withsimulation results (Fig. 5) although small discrepanciesare visible when (cid:15) is not small enough.2.3 Identification of force strengthTo get effective values of the six coefficients λ , ζ , δ , α , ν ,and (cid:15) , we start from the two curves a ( t ) and ω ( t ), bothassumed to be known with a sufficient precision. Themethod to extract a ( t ) and ω ( t ) from the full measure-ment of θ ( t ) is by no way important at this stage. Weadmit that the magnitude a , its time-derivative ˙ a , andthe frequency ω are known at a finite number of times(see Fig. 6). Let a i = a ( t i ), ˙ a i = ˙ a ( t i ) and ω i = ω ( t i )be these values, for i = 1 , . . . , n . a i π/ω i a i . t i time Fig. 6
Estimation of magnitude a i , slope of magnitude ˙ a i ,and frequency ω i at time t i . Let us first determine λ , ζ , and δ . By Eq. (4):˙ a i = − π ω λ − a i ω ζ − a i π ω δ i = 1 , , . . . . (10)Assembling these equations in a matrix form, we get: ω − π − a − a π . . . . . . . . . − π − a n − a n π λζδ = ˙ a . . . ˙ a n . (11)This equation constitutes an overdetermined system oflinear equations. The solution in the least mean squaresense is given by: λζδ = 1 ω (cid:0) L T L (cid:1) − L T ˙ a . . . ˙ a n , (12)where L denotes the matrix in the left-hand side ofEq. (11) and L T its transpose.The determination of ν and (cid:15) is done following thesame approach. By Eq. (9), and remembering that ω ∞ (cid:39) ω (1 + α/ ω a π a . . . a n π a n (cid:18) ν(cid:15) (cid:19) = ∆ω . . .∆ω n . (13) where ∆ω i = ω i − ω ∞ is the frequency shift. The solu-tion in the least mean square sense is given by: (cid:18) ν(cid:15) (cid:19) = 1 ω (cid:0) M T M (cid:1) − M T ∆ω . . .∆ω n , (14)where M denotes the matrix of Eq. (13).The five coefficients λ , ζ , δ , ν and (cid:15) are determinedby Eqs. (12), (14). The coefficient α is determined as α = ω ∞ /ω − f = γh ( θ, ˙ θ/ω ) may easily be included inthe analysis by calculating its signature on the enve-lope and frequency shift. The terms ω γ (cid:104) h sin ϕ (cid:105) and ω γ (cid:104) h cos ϕ (cid:105) (Appendix) must respectively be added tothe right-hand sides of Eqs. (4) and (9). If the forcemodifies the envelope, the matrix L of Eq. (11) willcontain a fourth column whose entries are (cid:104) h sin ϕ (cid:105) i ,estimated at all time t i and γ will appear as a sup-plementary unknown. If it modifies the frequency shift,then (cid:104) h cos ϕ (cid:105) i at time t i constitutes a third columnof M and γ appears as unknown. More generally, aforce may have a signature on both the envelope andfrequency shift. This raises the question of the unic-ity of the signature of a given force on envelope andfrequency. In general, there is no such unicity. For in-stance, both forces ˙ θ sgn( ˙ θ ) and θ sgn( ˙ θ ) have the ex-act same mean values (cid:104) h sin ϕ (cid:105) and (cid:104) h cos ϕ (cid:105) , makingthem indistinguishable by the method presented here. We illustrate here the method presented above on theexample of a measurement performed with the torsionalmagnetic levitation tribometer described in [13]. Thenatural moment of inertia of the pendulum is I =1 .
24 kg · mm . A mass m = 0 .
90 g has been added at adistance l = 10 . I = I + ml = 1 .
33 kg · mm .A spherical cap of radius R = 1 .
17 mm is attached atthe pendulum’s tip, made of Sylgard 184 PolyDiMethyl-Siloxane (PDMS), prepared as in [16,23]. The PDMSsphere is set into contact against a glass plate undernormal load P = 15 . θ (solid line).The coefficients λ , ζ , δ , ν and (cid:15) have been calculatedby applying Eqs. (12) and (14), where the values of Alain Le Bot et al.
Time [s] -0.3-0.2-0.100.10.20.3 θ [ r ad ] Measurement [ λ , ζ , δ ][ λ , ζ , δ , ν , (cid:6) ]
20 20.5 21 21.5 22-0.04-0.0200.020.04
Fig. 7
Time evolution of θ for a PDMS-sphere/glass-platetorsional contact. Solid line: measurement. Dash-dot line: so-lution to Eq. (1) with λ , ζ , and δ . Dashed line: with λ , ζ , δ , ν , and (cid:15) . a i , ˙ a i , and ω i have been assessed at 76 instants fromthe full measurement of θ ( t ). In practice, each of 78half-periods is first fitted using a half-sine function ofamplitude b i , with an extremum reached at time t i .We then compute a i = ( b i +1 + b i − ) /
2, ˙ a i = ( b i +1 − b i − ) / ( t i +1 − t i − ) and ω i = 2 π/ ( t i +1 − t i − ). The an-gular frequency ω ∞ = 9 .
91 rad/s is taken as the lastcomputed value of ω i , where the influence of nonlin-ear forces is the least. The obtained values are λ = − . ζ = 0 . δ = 0 . ν = − .
40, and (cid:15) =0 .
30. In Fig. 7, the dash-dotted line corresponds to thenumerical solution of Eq. (1) with the three dissipativeforces only, f = λ sgn( ˙ θ )+2 ζ ˙ θ/ω + δ ˙ θ sgn( ˙ θ ) /ω , whilethe dashed line corresponds to the solution with all fiveforces (the effect of α is already accounted for in ω ∞ ), f = λ sgn( ˙ θ ) + 2 ζ ˙ θ/ω + δ ˙ θ sgn( ˙ θ ) /ω + νθ sgn( θ ) + (cid:15)θ .It is clear that both curves, based on either three or fiveforces, well capture the envelope of the measurement.However, the curve for three forces (dash-dotted line)rapidly shows a phase shift and is even out-of-phase af-ter few periods. The curve for five forces (dashed line)does not have this shortcoming and is in perfect agree-ment with the measurements over the full time window.This result highlights that the presence of nonlinearelastic forces is essential to explain the frequency shiftobserved on the tribometer. α can be estimated from theknowledge of ω and ω ∞ . With the magnetic levitation θ [rad] λ , ζ θ , δ θ × -3 λ ζθδθ Fig. 8
Relative importance of solid friction force ( λ ), viscousforce ( ζ ), and quadratic dissipative force ( δ ), as a function ofthe angle, for the experiment shown in Fig. 7. tribometer used here, accessing ω is not direct, be-cause the pendulum cannot be operated in the absenceof a mechanical contact. To estimate ω , we thus per-formed an experiment on a contact which is expectedto be submitted to a small frictional torque. We chose acontact between steel and graphite (SG), for which themeasured coefficient of friction µ = 0 .
05 is the small-est among all cases investigated with this rotationaltribometer. For a normal load P = 16 . ω ∞ , SG =8.87 rad/s, which is an upper limit for ω , ifthe steel/graphite contact had a vanishing frictionaltorque. Assuming that ω = ω ∞ , SG , we infer that thePDMS/glass contact torsional stiffness is estimated by K c ≈ I ( ω ∞ , PDMS − ω ) = 25 . − N · m. The torsionalstiffness of a non-slipping elastic sphere of radius a is K c = 16 Ga / G is the shear modulus of theelastic material [24]. We measured a = 0 .
20 mm bydirect visualization through the glass and with G =0 .
53 MPa [16], it yields K c ≈ . − N.m, whichis in reasonable agreement with the value found fromthe analysis of the oscillation. This agreement suggeststhat the change in final frequency is due to the finitetorsional elasticity of the elastomer contact.Once the coefficients of all forces are known, it is in-teresting to assess the relative weights of those forces inthe signal. Those weights depend not only on the non-dimensional coefficients, but also on the current ampli-tude of θ or ˙ θ . Approximating ω (cid:39) ω , one can estimatethe weight of each of the five considered forces by: λ ,2 ζθ , δθ , 4 νθ/ (3 π ) and 3 (cid:15)θ /
8. As an example, in Fig. 8,the weights of the three forces affecting the amplitudein the PDMS/glass experiment of Fig. 7 are shown. For elaxation tribometry: a generic method to identify the nature of contact forces 7 θ [rad] θ ν / ( π ) , θ (cid:4) / θν /(3 π )3 θ (cid:4) /8 Fig. 9
Relative importance of quadratic ( ν ) and cubic ( (cid:15) )elastic forces on the phase shift, as a function of the angle,for the experiment shown in Fig. 7. the experimental values of the coefficients and the ex-perimental range of angles, one can see that the viscousforce dominates and is thus responsible of most of thedissipation in the pendulum. Such a dominance doesnot necessarily indicate that the interfacial friction isstrongly viscous (and indeed, [25,26] found negligiblevelocity-dependence of friction at similar PDMS/glassinterfaces). Actually, this viscous force term presum-ably also combines viscous dissipation in the air aroundthe pendulum, in the viscoelastic bulk of the PDMS,and in the magnetic device. In comparison, the con-tribution of a solid-friction-like term ( λ -term) is foundcompletely negligible. The obtained value is even nega-tive, which likely suggests that the uncertainty on λ islarger than its value.Figure 9 shows the relative importance of the twoconservative forces, νθ sgn( θ ) and (cid:15)θ , in the same ex-periment. The quadratic elastic force νθ sgn( θ ) clearlydominates and imposes the phase shift in the pendu-lum. This nonlinear force may be due to a combinationof a nonlinear stiffness of the torsional contact, a nonlin-ear elastic behaviour law of the PDMS and a nonlinearrestoring force due to the magnetic device.It is interesting to compare the value of (cid:15) found withthat expected if the pendulum was a pure pendulum un-der gravity. In that case, the second order approxima-tion of sin θ , which enters the exact equation of motionof the pendulum, is sin θ (cid:39) θ − θ /
6. This means thatthe expected value of (cid:15) would be − /
6. The fact thatwe find a value about twice bigger in magnitude and ofopposite sign indicates that apart from gravity, there isa stronger cubic elastic force in the system, presumablydue to the magnetic levitation device. θ [rad] ω [ r ad / s ] MeasurementEq. (9)
Fig. 10
Evolution of angular frequency, ω , versus magnitudeof oscillation, a , for the experiment shown in Fig. 7. Also of interest is the instantaneous frequency ω ver-sus the magnitude a of oscillation (Fig. 10), for themeasured curve θ ( t ) of Fig. 7. ω ( a ) is almost linear, asshown in Fig. 4. This observation is fully consistent withthe dominance of the ν term compared to the (cid:15) term,already demonstrated in Fig. 9. Furthermore, the nega-tive slope of ω ( a ) proves that ν <
0. Note that a similar,affine-like behaviour of ω ( a ) had already been obtained,for the same tribometer, on a steel/glass contact [13],suggesting that a similar quadratic elastic force was alsoimportant in that case.4.2 General commentsThe analysis in terms of the weights of the various forcesreveals several generic features. Concerning the decay ofamplitude, the quadratic term δθ will always dominateat large amplitude, at the beginning of an experiment.In particular, it will be larger than the viscous termfor amplitudes a > ζ/δ (except of course if this valueexceeds π , the maximum possible initial angle). For ourexperiment of Figs. 7 and 8, it would correspond toamplitudes above about 0.5.Another general results is that, at small amplitudes,the constant λ -term will always dominate. This meansthat in any situation in which λ (cid:54) = 0, the oscillationwill vanish at a finite time. A rough estimate of thisarrest time can be obtained as follows. At small ampli-tudes, the λ -term will dominate when the amplitude a will become smaller that a λ = λ/ (2 ζ ) (equating λ and2 ζa ). After that instant, assuming that λ is the onlyterm important for the amplitude decay, one can useEq. (5) to estimate the remaining oscillation time tobe πa λ / (2 λω ) = π/ (4 ζω ). So, after the amplitude a λ Alain Le Bot et al. is reached, the oscillation will cease after a time of theorder of π/ (4 ζω ).Similarly, concerning the frequency shift, the (cid:15) -term,which is quadratic in amplitude, will always dominatethe ν -term for amplitudes a > ν/ (9 π(cid:15) ). For our ex-periment of Figs. 7 and 9, it would correspond to am-plitudes above about 4.76. This value is larger than π , the largest possible initial angle, which means that,in our experimental case, event for the largest ampli-tudes, ω ( a ) − ω will always remain dominantly linear.In general, the linear ν -term will always dominate atsmall amplitudes, when the angular measurement maybecome less accurate, due to discretizaton effects. Ourconclusion is that, in this regime, a linear extrapolationof ω ( a ) for small a is a priori a correct approximationof the behaviour of the system.More generally, friction being an hysteretic phenome-non, it cannot always be described only as a functionof the instantaneous angle and angular velocity, as as-sumed in Eq. (1). In transient regimes, like the oscilla-tions considered in relaxation tribometry, a more com-plete description of the interface should incorporate oneor several state variables, as in the rate-and-state (RS)friction framework [27,28]. Although most RS modelsuse a typical contact-related time as a state variable [29,30,31,32], here we believe that the most relevant statevariable would be related to the oscillating history ofthe contact. When the contact is brought to its initialangle, only a central circular part of radius c of the con-tact has remained in a stuck state, while its peripheryhas already been slipping [18]. When the contact hascompleted its first half-cycle of oscillation and is backto vanishing angular velocity, it has an angle the abso-lute value of which is smaller than the initial one, andthus the stick radius is now c > c . As a consequence,when the oscillator goes back to a vanishing angle, theannulus between c and c in the contact region hasstored a shear strain state which is different from thatof the first contact, and which will survive all along thesubsequent decaying oscillation of the contact. Such ascenario occurs over and over at each half-cycle, build-ing up a complex, onion-ring-like shear strain field, thedescription of which involves knowledge of the series of c i reached at all half-cycles. Incorporating such a com-plex state variable in the analysis of relaxation tribom-etry data is an interesting future challenge. It will likelyrequire extension of studies limited to the first loadingof elastic contacts [24,18,19], to decaying oscillations. We have shown that relaxation tribometry is not lim-ited to the measurement of constant friction and vis- cous coefficients as is usually done using the decrementmethod. More complex dissipative forces but also elas-tic forces can be unambiguously identified and quanti-fied using the general procedure proposed in this study.The key is to exploit the two-times averaging methodto analyse not only the time-evolution of the vibrationdecay, but also that of the frequency shift. The mag-nitudes of forces are then solutions of a linear system,although the forces are themselves nonlinear. This pro-cedure, which has been applied to six relevant types ofcontact forces, can easily be extended to any other de-sired nonlinear force, to identify its characteristic sig-nature, both on the amplitude and frequency. Thoseresults suggest that relaxation tribology has a vast, butstill insufficiently exploited potential, both fundamental(identification of the forces at play) and applied (quan-tification of those forces).
A two-times averaging method
The right-hand side of Eq. (1) is written f = (cid:15)h ( θ, θ (cid:48) ) where (cid:15) << θ (cid:48) = ˙ θ/ω denotes the derivative of θ with respectto dimensionless time. We seek the solution of the form θ = a cos ϕ , where ϕ = ω t + φ and a ( t ) and φ ( t ) are slowly varyingfunctions. Substituting in Eq. (1) gives:¨ a cos ϕ − a ( ω + ˙ φ ) sin ϕ − a ¨ φ sin ϕ − a ˙ φω cos ϕ − a ˙ φ cos ϕ = − (cid:15)ω h. (15)Considering that ¨ a , ˙ a ˙ φ , ¨ φ , and ˙ φ are second order terms in (cid:15) and can thus be neglected in Eq. (15):2˙ aω sin ϕ + 2 a ˙ φω cos ϕ = (cid:15)ω h. (16)We must now develop h at order zero in (cid:15) since the left-handside if of order one in (cid:15) . At order 0, θ = a cos ϕ and ˙ θ/ω = − a sin ϕ therefore h = h ( a cos ϕ, − a sin ϕ ). Then substitutingin Eq. (16) and averaging over a time-period (with ˙ a and ˙ φ constant) gives the so-called averaged equations:˙ a = (cid:15)ω π (cid:90) π h ( a cos ϕ, − a sin ϕ ) sin ϕ d ϕ = ω (cid:15) (cid:104) h ( ϕ ) sin ϕ (cid:105) (17) a ˙ φ = (cid:15)ω π (cid:90) π h ( a cos ϕ, − a sin ϕ ) cos ϕ d ϕ = ω (cid:15) (cid:104) h ( ϕ ) cos ϕ (cid:105) (18)where (cid:104)·(cid:105) denotes mean value over 2 π . These are two first-order ordinary differential equations on a and φ .For instance, consider the case of a quadratic dissipativeforce f = (cid:15) ˙ θ sgn( ˙ θ ) /ω . Then h ( θ, θ (cid:48) ) = θ (cid:48) sgn( θ (cid:48) ) and h ( ϕ ) = − a cos ϕ sgn(sin ϕ ). By averaging: (cid:104) h sin ϕ (cid:105) = − a π , (19) (cid:104) h cos ϕ (cid:105) = 0 . (20)The two differential equations are therefore ˙ a = − a (cid:15)ω / π and ˙ φ = 0. After integration, a ( t ) = (cid:2) a − + 4 (cid:15)ω t/ π (cid:3) − (Eq. 7) and φ ( t ) = φ , a and φ being the initial valuesof a and φ .A second interesting example is f = (cid:15)θ sgn( ˙ θ ) for which h ( θ, θ (cid:48) ) = θ sgn( θ (cid:48) ) and h ( ϕ ) = − a cos ϕ sgn(sin ϕ ). By av-eraging: (cid:104) h sin ϕ (cid:105) = − a π , (21) (cid:104) h cos ϕ (cid:105) = 0 , (22)elaxation tribometry: a generic method to identify the nature of contact forces 9 Table 1
Averaged values of h of Eqs. (17) and (18) for thesix considered forces h sgn( θ (cid:48) ) θ (cid:48) θ (cid:48) sgn θ (cid:48) θ θ sgn θ θ (cid:104) h sin ϕ (cid:105) − π − a − a π (cid:104) h cos ϕ (cid:105) a a π a which gives the same signature ˙ a ∝ a as in Eq. (19).Similar results for all considered contact forces are sum-marized in Table 1. Acknowledgements
We thank J. Perret-Liaudet, E. Rigaud,and O.A. Marchenko for fruitful discussions and critical com-ments. This work was supported by CNRS-Ukraine PICSGrant No. 7422.
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