Indian Journal of Fibre & Textile ResearchVol. 23, December 1998, pp. 201-208

Theory of stick-slip effect in friction

R P Nachane, G F S Hussain & K R Krishna Iyer

Central Institute for Research on Cotton Technology, Adenwala Road, Matunga, Mumbai 400019, India

Received 6 October 1997; accepted 12 December 1997

Friction in textile materials exhibits stick-slip effect in a pronounced way. Theoretical relationshipshave been derived to explain this stick-slip effect. These relationships give the velocity of the fibre padholder, its displacement and the frictional force, measured as functions of time. The relationships clearlyshow that the extent of stick-slip effect depends on the difference between the static and dynamicfrictional forces, and on the experimental parameters. The results also indicate why and when thephenomenon of stick-slip effect is observed/not observed. The experimental results have been found toagree well with those obtained by calculations based on the relationships derived.

Keywords: Dynamic friction, Frictional force, Static friction, Stick-slip effect

1 IntroductionFriction between the surfaces of solids is a very

complex phenomenon. It depends not only on thephysical and chemical properties of the materialsinvolved but also on the mechanical finish of thecontacting surfaces. No theory can, therefore, beformulated to explain the frictional behaviour quitesatisfactorily. For this reason, the study of frictionhas always been based on the experimentalevidence. From the available data, the general lawsof empirical nature can be stated. This was firstdone by Amonton I. It has been observed that therelative motion between the sliding metallic objectsis very complex and jerky' though it appears to besmooth. The relative motion consists of momentarypauses between the positions of rapid slip.

In the case of textile materials, it becomes evenmore difficult to explain the frictional behaviour'r",This is mainly due to high variability among thespecimens in their surface characteristics. Also,their mechanical properties are complex due to theviscoelastic nature of the material. The staticcharges generated on the specimen surface add todifficulties in obtaining consistent experimentalresults.

Several researchers have studied the frictionalproperties or different types of fibres, yams andfabrics3

-27

. Some of these researchers haveobserved that the frictional behaviour betweencertain materials is of the stick-slip type9,17,18,23,24,27.

The friction curve represents cyclic increase anddecrease in frictional force, the decrease beingalmost instantaneous. Morton and Hearle24 havepresented a simple model to explain the occurrenceof stick-slip effect.

Bowden and Taborl derived an equation ofmotion for a surface which is sliding or slippingunder the elastic constraint on the surface of a testspecimen. The constraint in their experiment was atorsion wire. The equation put forward by them is :

d2SM--+KS=F

dt2

where S is the displacement of the measuringsurface (i.e. slider) from its equilibrium position;M, the mass of the sliding system; K, the springconstant of the elastic constraining mechanism; andF, the frictional force at the surface on which themass is sliding. This equation is based on theassumption that the kinetic friction between thetwo surfaces is Coulombic, i.e. the frictional forceremains constant during the sliding. Lyons andScheier" modified this equation by assuming thatthe resistance to the motion of the slider during theslip phase has Coulombic as well as viscouscomponent. The viscous resistance decreases indirect proportion as the velocity of slipping. Thetotal frictional resistance then becomes [F-G(dS/dt)] instead of F in the above equation. Here,G is a constant. The modified equation is :

202 INDIAN J. FIBRE TEXT. RES., DECEMBER 1998

Md2

S +GdS +KS=Fdt2 dt

Derjaguin et al.29 considered a sudden drop ofstatic friction to a lower kinetic friction at the startof slip, followed by a constant kinetic friction.Hunt et al.30 proposed that friction depends onacceleration as well as velocity. But Brockley etal" demonstrated that the error involved incomputing slider amplitude with respect to drivervelocity, by assuming constant kinetic frictioninstead of a velocity dependent friction, is quitenegligible. Banerjee" assumed a steady statenonlinear velocity-dependent function to calculatecritical velocity. Critical velocity means velocity ofthe driven surface at which stick-slip motionchanges to smooth sliding. Banerjee assumedkinetic frictional force of the type:

dx (dx)2F. -a.-+~ -o dt dt

where x is the displacement of the slider withrespect to a fixed frame of reference; and Fo, a andB, the constants for the particular experimental set-up. Driving mechanism is a spring and a dashpot asshown in Fig.I. The equation of motion obtainedby him is:

d1x [fix ] fixm-+c --v +k(x-J-1-xo)+Fo-a-df ~ ~

+ p(:J =0

where k is the spring constant; xo, the initialposition of the mass m at rest; v, the drivervelocity; and c, the coefficient of damping. Heshowed that at high damping levels, as the drivervelocity is increased, the slider velocity approachesthe driver velocity and beyond a certain drivervelocity there is no stick-slip motion.

Cockerham and Symmons" assumed adiscontinuous friction model which consists of anegative damping action for the slip acceleration

>' ;>

Fig.1 - Driving mechanism consisting of a spring and adash pot [m - mass of the slider, v- constant drive velocity,k - stiffness of the spring, c - coefficient of damping of thedamper and x - displacement of the slider from a fixed point]

phase and a constant frictional resistance for theslip deceleration phase.

Except for Lyons and Scheier", all otherresearchers have studied the theory of stick-slipphenomenon in metal-on-metal. The type of kineticfriction assumed by them is highly complicated,but at the same time not based on any physicalexplanation as to why the friction should behave inthe way they have assumed. We feel that theobservation by Brockley et at.3! that "the errorinvolved in computing slider amplitude withrespect to driver velocity, by assuming constantkinetic friction instead of a velocity-dependentfriction, is quite negligible" should be given moreweightage than assuming highly complicatedkinetic frictional force.

It is an accepted fact that friction depends onmany variables like the substances in contact, thesurface asperities, the relative velocity between thesurfaces, the normal reaction between them, thelocalised fusion that might be taking place at thepoint of contact between the two surfaces, etc. Inviscoelastic materials like textile fibres, the timefor which the substances are in contact would alsomatter as the asperities in contact would getdistorted more and more with time. It seems thatthe task of predicting the precise nature of frictionis almost impossible.

The stick-slip motion observed in textile fibreshas very high amplitude. To study it theoretically,it is therefore assumed in this work that there aretwo distinct frictional forces between the fibremasses. One is the static frictional force when thereis no relative motion between the fibre pads and theother is the dynamic frictional force when the padsmove relative to each other. This assumption is inline with that made by Bowden and Tabor',Derjaguin'", and some other workers". Forsimplicity, we have assumed that the dynamicfrictional force is constant and does not changewith velocity of the moving fibre pad. This simpleassumption . is good enough to explain theexperimental results not only qualitatively but alsoquantitatively.

2 TheoryConsider a specimen holder of mass m (with

specimen on its lower face) placed on a surfacealso lined with another specimen (may be of thesame material). Let a string be attached to thespecimen holder, as shown in Fig.2. The other end

, NACHANE et al.: THEORY OF STICK-SLIP EFFECT IN FRICTION 203

of the string is attached to a strain gauge whichsenses tension in the string. Let the strain gauge bemoving with a constant velocity v. This is normallythe experimental set-up in a friction test.

Initially, the tension in the string is zero. As thestrain gauge starts moving, t1K string getsextended. This causes the development of tension Tin the string. But due to the normal reaction mg (gis acceleration due to gravity) at the lower surfaceon the specimen holder, a frictional force Fdevelops between the two contact surfaces. Thisforce just balances the tension in the string whichextends at a constant rate v. Let K be the stringconstant, i.e. the force required for unit extensionof the string. This tension would increase at aconstant rate Kv if it is assumed that theload-extension curve for the string is linear.

As this built-up tension in the string getsbalanced by the frictional force, there is no netforce on the specimen holder. Therefore, it remainsstationary. A stage is later reached when the stringtension becomes equal to the maximum frictionalforce that can develop between the two surfaces.This force is called as static frictional force F; Themoment the tension T becomes greater than F;there is a net force acting on the holder which startsmoving. And once the motion begins, the frictiondrops to Fd called the dynamic frictional force. Thenet force F acting on the holder is given by :F=T-Fd (J)

To determine the tension in the string at anyinstant after the holder has started moving, let usstart 'counting the time from the moment the holderbegins to move. Tension in the string at t=0 would,therefore, be equal to F" the static frictional forcebetween 'the two surfaces in contact. There wouldalready be some extension m the stringcorresponding to the tension F; In time I, the straingauge would have traversed a distance vt. In themean time, the holder would have moved by somedistance, say x. The additional extension (e) of thestring during this interval will be e = VI - x. Thus,tension in the string at any time I after the motionhas started would be :T=K (vt - xy+F; ... (2)

Substituting the above value of T in Eq.( 1) andrearranging the terms, the equation of motion forthe specimen holder can be written as :

N

T

F

Fig.2 - Forces acting on a fibre holder. The top of the lowersurface is some textile material. The bottom of the movablefibre holder is a pad of parallelised fibres held firmly at theleading end. [Mg - weight of the specimen holder,N - normal reaction acting on the specimen holder,T - tension in the string and F - frictional force between thetwo surfaces in contact]

d2x.K Kvt F.-Fd--+-x = --+--=---=-dt? m m m

... (3)

Solution of this differential equation would givex as a function of I. Substituting this value of x inEq.(2) one would be able to get tension T as afunction of time. T is what is actually sensed by thestrain gauge. To evaluate the constants ofintegration, the following initial conditions wereused:

dxx == 0 when I = 0; and - = 0 when I = 0

dlThe solution of the differential equation gives thefollowing relationship:

F-Fd ~" Ix=- sK cos[v'(Klm)tj]'-v(mIK)

F-Fvsin[~(Klm)t]+ s d+vt ... (4)

KThe velocity of the specimen holder (u) is given bythe following relationship:

dx [[;7.(F-' '] F, -Fd . ctrr:!/"-=I'-VCOS v\l\./m)( +-r=== sm[v(Klm)t}

dt .J(mK) ... (5)Once the holder starts moving, it will continue

to move with the velocity u given by Eq.(5). As canbe seen from the equation, II is a function of timeand of the experimental constants for a givenparticular case. If u becomes zero at any time I=/:; O.the object comes to rest at that instant. It cannotstart moving again till tension in the string becomesequal to Fs, the static frictional force at this newplane of contact.

Let us find out the condition. required for theobject to come to rest and the time =t, at which the

204 INDIAN 1. FIBRE TEXT. RES., DECEMBER 1998

object comes to rest. Substituting u=O in Eq.(5), thefollowing condition is obtained:

tan [~(K / m) ts J = _ F.- Fd• 2 ~(mK)v

This condition is satisfied only for time t, such that

~(K / m)~ > 2: i.e.rc-n ~(m/ K)2 2

Thus, for t s 1[~(m / K), the holder cannotcome to rest. To find out ts> the followingrelationship was taken:~(K I m)~=!!.- + 0

2 2This gives

C\ .Jv ~(niK)]~=tan l---F.-~

As the specimen holder starts moving and gainsin velocity, the rate of extension of the string getsreduced. But it is still extending and rising intension. When the velocity of the holder equals v,there is no net increase in extension and tension ofthe string. For a velocity greater than the cross headspeed, the string extension and tension arediminished. At some time, the string tensionbecomes equal to Fd• This corresponds to themaximum velocity of the holder. The holder movesfurther with this velocity, reducing the stringtension still further. Tension T is now less than Fd•

Thus, there is a net retardation force that tends toreduce the velocity of the specimen holder. A stagecomes when the velocity of the holder becomesequal to the crosshead speed. Here, the reduction inextension of the string stops. Tension at this timecorresponds to the minimum tension observed inthe string. As the velocity reduces still further,extension in the string starts giving rise to increasein tension. Ultimately the holder comes to rest andthe string tension starts following its load-extensioncurve till the tension equals F; Thus, the peaks andtroughs observed in the frictional test exhibitingstick-slip effect correspond to the times when thevelocity of the holder (u) is equal to the crossheadspeed v.

Now let us find out the time when u=v.Substituting v for u in Eq.(5), we get

tan [~(K / m) tmax] = tan [ ~(K/m) tmin]

= v.J{mK)F. -Fd

... (6)

... (7)

. . . (8)

where tmax is the time when the velocity u isincreasing; and tmin, the time when the velocity u isdecreasing. To find out the relationship betweentime tmax and tmin at which u=v, the followingrelationship was used:tan[~(K I m )tmaJ = tan [11" + ~(K I m ) tmax 1Therefore,tmin - tmax = 11" ~(m I K) ... (9)

Velocity u changes cyclically with a period21[~(m I K). It becomes equal to the crossheadspeed v at time Imax corresponding to maximumtension. At time tmin, the velocity of the object isagain equal to v when u is decreasing, i.e. tmin

corresponds to the minimum tension. Afterreaching the minimum tension the holder comes torest in time tmax.

Let us now consider the variation in the stringtension with respect to time t. On substituting thevalue of x in Eq.(2) from Eq.(4) and simplifyingand rearranging the terms, we getT = Fd + (F, - Fd)COS[~(K I m)t]

+Y~(mK)sin[~(Klm)t] ... (10)To arrive at the maximum tension (Tmax), substitutet by tmax• Thus, Eq.(10) becomes

Tmax = Fd +(F, - Fd)COS[~(K Im)tmaxl

+ Y~(mK)sin[~(K I m)tmax]

Substituting for cas [~(K I m) tm.J and sin [~(K i m) tmaxl

from Eq.(8), we getTmax = Fd + [(F, - Fd)2 + y2mKf5 ... (11)

Similarly, to find the minimum tension (Tmin)'

substitute t by tmin and use the value of tmin fromEq.(9). On simplifying, we getTmin = Fd -[(F, - Fd)2 + y2mKf5 ... (12)

3 Materials and MethodsIn two specially designed clamps, the fibre tufts

prepared from a cotton sample were held as shownin Fig. 3. One clamp was placed' over the other insuch a way that the fibre beards pointed in oppositedirections. The lower clamp was held stationaryand the upper clamp was attached to the load cellof the Instron tensile tester by a string going rounda frictionless pulley. As the Instron crossheadmoved up, tension developed in the string. Thistension was recorded with respect to time. A stringmade of nylon filaments was used in the study .Different crosshead speeds were employed.

NACHANE et al.: THEORY OF STICK-SLIP EFFECT IN FRICTION 205

4 Results and DiscussionIt may be seen from a representative stick-slip

curve (Fig. 4) that the time interval during whichthe actual slippage of the sliding specimen holdertakes place is of the order of a tenth of a second. Itwas, therefore, not possible to determineexperimentally the exact time period over whichanyone single slippage occurred for any crossheadspeed. Between the two slips, however, there wasno movement of the holder for a long interval oftime depending on the crosshead speed and theaverage difference between the static and dynamicfrictional forces.

Table 1 shows the experimentally observedvalues of Fs-Fd, total time for one period of slipand stick together and the distance travelled perslip by the holder. For crosshead speed greater than100 mm/min, no stick-slip effect was observed,

(0 ) ( b)Fig.3 - Fibre holder with cotton fibres held in it: (a) sideview, and (b) top view

while for crosshead speed below 1 mm/min, thefriction at the pulley transforming the vertical pullinto horizontal tension in the nylon string renderedthe measurement of string tension somewhatinaccurate. Therefore, the crosshead speeds rangingfrom 1 mm/min to 100 mm/min only were used inthe present study.

Tension in the string with reference to time wasrecorded for a period corresponding to the totaldisplacement of the holder by 8 mm. The holderwas of mass 30 g. During the stick period when thestring extended at the rate of crosshead speed, itwas observed that for the period just after the slipcompleted and the holder had come to rest, the rateof increase in the tension was faster but it sloweddown as time passed. This effect was prominent forslower crosshead speed as compared to faster ones.Also, for more drop in tension during a slip, thisnon-linearity in the change in string tension wasobserved to be more. This is due to the occurrencfof inverse relaxation in the nylon string when thereis drop in its tension34

,3s. String constant K of thestring was 40,000 dynes/em (400 mN/cm).

~8o..------------------...,

~ 140z\oj

••120

100

.800~-----~12----2~4---~~---~~~-~60

TIME, S

Fig.4 - A typical curve showing stick-slip effect (crossheadspeed=lOmmlmin)

Table I- Comparison of theoretical and experimental values for nylon string

Experimental CalculatedExperimental Total time for Av.distance Time per slip Distancevalue ofF. - Fd one slip+stick, s travelled per s travelled per

dynes slip, em slip, cm

1225 0.374 0.0817 0.0942 0.07731550 1.060 0.0882 0.0860 0.07892300 3.375 0.1125 0.0844 0.11142400 7.250 0.1210 0.0852 0.11811800 12.100 0.1010 0.0902 0.09912400 39.000 0.1300 0.0864 0.12102140 69.800 0.1160 0.0893 0.1154

Crossheadspeed

mm/min

10050201052I

206 INDIAN 1. FIBRE TEXT. RES., DECEMBER 1998

Using all these experimental values, the velocityof the holder during slip as a function of time wascalculated by using Eq.(S). Tables 2-4 give thechange in velocity of the holder for three of theseven crosshead speeds studied, viz. 100 mm/min,10 mm/min and I mm/min, till the velocity justbecame negative. Physical meaning of the negativevelocity is that the holder comes to rest at that time.A similar trend is observed at the other crossheadspeeds. It is observed from the tables that the timetaken by the holder to reach crosshead velocity isless than a thousandth of a second and the totaltime per slip is less than one tenth of a second.Also, the maximum velocity attained is very highas compared to the crosshead speed. Thesecalculations show why it is not possible todetermine the. time per slip experimentally.

Tmax and Tmin values corresponding to maximumtension and minimum tension as calculated by Eqs(11) and (12) respectively (see Tables) are almostequal to F; and 2Fd-Fs. Hence, assuming the peakas the static frictional force, and the mean of thepeak and trough as the dynamic frictional force inthese experiments is quite reasonable. Thedisplacement values in Tables 2-4 were calculatedby using Eq. (4). The calculated values of totaldisplacement per slip are given in Table 1. Thesevalues are quite in agreement with the valuesdetermined experimentally.

Since cotton is a highly variable material, it isdifficult to say anything about the relation betweenthe rate of dragging and Fs-Fd between the cottonsurfaces. But as a general rule", it is known that thedifference between static friction and dynamicfriction decreases at higher speeds. Assuming asmaller difference of 300 dynes between F, and Fs.Fd to be 17900 dynes and using m and Kcorresponding to the present experimental set-up,for a crosshead speed of 200 mm/min, variation intension in the string has been calculated withrespect to time by using Eq.( I0). This is shown inFig.5. It may be seen from the Fig.5 that onecomplete cycle of stick-slip takes place over aperiod of less than 0.2 second. Variation in tensionabout the mean value is about 2% of the meanvalue. Fig.S(b) shows how slip and stick are almostequal, giving rise to a smooth sinusoidal variationin tension of the string. For any strain gauge,generally, it is difficult to measure such smallvariations in tension over such short periods like0.2 second. Therefore, it would measure only an

Table 2 - Velocity of the object, its displacement and tensiondeveloped in the string for crosshead velocity 0.1667cm/s

[Dynamic frictional force (Fd)=17,885 dynes]

Times

0.00000.00020.00200.00400.03000.05000.07000.09000.09420.0944

Velocity (u)cm/s0.00000.00820.08200.16451.08531.29390.93000.17860.0057

-0.0024

Displacement (x) Tension (7)em dynes0.0000 19110.00.0000 19111.30.0001 19120.00.0003 19123.50.0176 18610.10.0423 17757.60.0654 16969.70.0769 16646.60.0773 16659.10.0773 16660.4

Table 3 - Velocity of the object, its displacement and tensiondeveloped in the string for crosshead.velocity 0.01667cm/s

[Dynamic frictional force (Fd)=17,220 dynes]

Time Velocity (u) Displacement (x) Tension (7)s cm/s em dynes

0.0000 0.0000 0.0000 19620.00.0002 0.0160 0.0000 19620.10.0010 0.0800 0.0000 19619.00.0040 0.3190 0.0006 19596.40.0100 0.7830 0.0040 19463.90.0300 1.9452 0.0325 18301.20.0600 1.7443 0.0943 15779.20.0800 0.4164 0.1170 14861.40.0852 0.0026 0.1181 14820.00.0853 -0.0054 0.1181 14820.1

Table 4 - Velocity of the object, its displacement and tensiondeveloped in the string for crosshead velocity 0.001667cm/s

[Dynamic frictional force (Fd)=17,980 dynes]

Time Velocity (u) Displacement (x) Tension (7)s cm/s em dynes

0.000000.000040.004000.010000.030000.060000.080000.089000.089340.08936

0.00000.00290.28440.69881.76511.74160.65490.02480.0005

-0.0009

0.00000.00000.00060.00350.0292o.oan0.11230.11540.11540.1154

21120.021120.021099.120989.520036.317881.316954.116840.116840.016840.0

average value of tension for such short intervals oftime. The tension curve would appear to be smoothand would correspond to the dynamic tensionbetween the fibres at those specific positions of thespecimen holder. Thus, for static and dynamicfrictions of approximately 20000 dynes with adifference of about 300 dynes between them, onewould observe almost smooth movement of thespecimen holder at a crosshead (driver) speed of

200

1!50

100

!50zE,.z 00

!184...182

180

118

176

NACHANE et al.: THEORY OF STICK-SLIP EFFECT IN FRICTION 207

0·15 1·0 I·15 2·0

174~ ~ ~ ~~

1·36 1·46 1·156 1·66TIME, S

Fig.5 - (a) Variation of tension calculated by using Eq.12 fora crosshead speed of 200 mmlmin, Fd=17,900 dynes andF,-Fd=300dynes. At this fast speed, one cycle of stick-sliptakes place in less than 0.2 s. Normally, no strain gauge is ableto measure this variation in such a short time(b) Magnification of a portion of the curve

200 mmlmin, as the experimental set-up will not beable to show stick-slip effect. What thesecalculations mean is that for small differences in F,and Fd and high driver velocities, the stick-slipeffect may not be observed due to the experimentallimitations, though the actual motion may be jerky,i.e. consisting of stick and slip parts.

The values of F, and Fd are not constant over theentire length of the surface. However, it wasobserved that the variation in F, and Fd was small,thus making it possible to apply the equations usingtheir average values. It may be seen from Table Ithat the theoretically calculated and experimentallyobserved values of the distance travelled per slipagree quite well.

The relationships derived here show that theextent of stick-slip effect is dependent not only onthe difference between the static and dynamicfrictional force but also on other experimentalvariables like m, K and v. In the presentexperimental study, variation of valone has been

attempted. However, similar studies can be madeby varying the one or more ofthese parameters.

It is not possible to predict the exact frictionalbehaviour even between two metal surfaces.Friction is a highly complex phenomenondepending on various factors. It depends on thematerials involved, i.e. the adhesive force betweenthe molecules of the materials. Frictional forcedoes not depend on the actual area of contact aslong as the latter is not very small or very large.When the surfaces are rough, the pressure overcontact spots will be large indeed. Localised fusionof materials would occur necessitating a higherforce to separate the rough surfaces. On the otherhand, when the two surfaces are very smooth, therewould be a large number of molecules in onesurface in the vicinity of molecules in the othersurface, thus increasing the total adhesive force. Infact, for highly smooth surfaces, it has beenobserved that the frictional force can be as high asten times the normal reaction.

Normally, the surfaces of a material are coveredwith adherent oxides and gas molecules includingwater molecules. Therefore, the frictional forcebetween the surfaces of different materials may nottruely represent the material property. It maydepend on the level of oxidation and on the amountof adhesive gas or water vapour present on thesurface. In the case of textile materials, oxidationdoes not occur at room temperature, but adhesionof gas molecules and particularly water moleculesdoes affect the friction considerably.

In the context of present experiments, it may besaid that the asperities in contact will undergodeformation. This deformation is viscoelastic innature. The adhesive force between these surfacesin actual contact determines the static frictionalforce. Once the relative motion between thesurfaces begins, the deformation of the asperities isreduced because the time of contact betweenasperities is reduced. The adhesive force betweenthe two surfaces in relative motion is thereforediminished, giving rise to a lower value fordynamic friction as compared to the static friction.When the crosshead velocity is high, the time overwhich the fibre holder remains stationary betweenthe two consecutive slips is quite low. For example,it will be of the order of tenth of a second when thecrosshead speed is 200 mm/min. This being tooshort an interval, would not permit viscoelasticdeformation to occur and therefore only a small,

208 INDIAN 1. FIBRE TEXT. RES., DECEMBER 1998

often undetectable difference could be expectedbetween the static and dynamic frictional forces.

5 ConclusionsThe theoretical relationships developed in this

study explain the observed frictional behaviour intextile materials. The occurrence of stick-slip effectis explained. Application of the theory to someexperimental data shows good agreement betweenthe calculated and observed values. Stick-slipeffect depends on various factors like thedifference between static and dynamic friction,mass of the fibre pad holder, elastic constant of thestring used for pulling the specimen holder and thevelocity with which the string is pulled.

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