DE-Vol. 49, Friction-Induced Vibration, Chatter, Squeal, and ChaosASME 1992

NUMERICAL MODELING OF FRICTION-INDUCEDVIBRATIONS AND DYNAMIC INSTABILITIES

W. W. TworzydloComputational Mechanics Company, Inc.

Austin, Texas

E. B. Becker and J. T. OdenTexas Institute for Computational Mechanics

. University of Texas. AustinAustin. Texas

Abstract

A numerical study of dynamic instabilities and vibrations of mechanical systemswith friction is presented. Of particular interest are friction-induced vibrations. self-excited oscillations and stick-slip motion.

:\ typical pin-on-disk apparatus is modeled as the assembly of rigid bodies withelastic connections. An extended version of the Oden-Martins friction model is usedto represent properties of the interface.

The mechanical model of Ihe frictional system is the basis for numerical analysisof dYllamic instabilities caused by friction arid of self-excited oscillations. Couplingbetween rotationaJ and normal modes is the primary mechanism of resulting self-excited oscillations. Tllese oscillations combine with high-frequency stick-slip motionto produce significant reduction of the apparent kinetic coefficient of friction.

As a particular study model. a pin.on.disk experimental setup has been selected...\ good llualitati\'(! and quantitative correlation of numl'rical <Lndexperimental resultsis observed.

1 Introduction

Cont<\ct <\nd friction phenomena ha\'c been of interest to researcherssince tile historical lI'orks of :\lI1ontons (1] and Coulomb [17). Ex-''''I'inll'ntal results relalpd to friction and corresponding theorips'! 'n'lon,," hpfol'p fhp ..aI'''' I'Hins art' "nll1l11al'ized in tl.., hook-

,.' l30wden and Tabor (11.12]. As it g(~IH:ral rule, all aspects ofIridional behavior lI'ere attributed to specific prope~t.ies of thecontacting surfaces. In particular. t.he difference between staticand kinetic friction lI'as assumed to be an intrinsic property ofthe frictional interface.

It wasn't until recently that the importance of the dynamiccharacteristics of the testing apparati in the results of frictionalexperiments was explicitly 1I0ted. Although discrepancies be·tween the results of frictional experiments performed on differentapparati were noted by various researchers (see, e.g., Burwell andRabinowicz (16], Bowden and Tabor [12]), the first clear statementof the importance of the dynamic characteristics and vibrationsof the appratus is dnc to Tolstoi [58) and Tolstoi, Borisova andGrigol'Ova 159J. These researchers investigated kinetic friction ex-perimentally iu the pl'esencp (and absence) of vibrations. Theyconcluded that the intO/lice coeflicicnt of friction does not explic-itly depend on sliding \'(·Iocity. and thai. the difference betllw'll, .... "tlil/(/'fllt slatic ane! kill<,tic co"lIki"III.' of friction is t he (,Oll"'~

13

lju<:ncl' \Ii' nli(l"o:;cale \'ihraliolls accompanying frictional sliding.This observation lI'as confirmed by experiments of other reo

searchers. in\'estigating the inflnence of vibrations on the staticand kinetic coefficients of friction-Weic [68]. Pohlman and Lehf-feldt [.151. Godfrey [21] or more recently Broniec and Lenkiewicz(1.1). Aronov. D·Souza. [~alpakjian and Shareef (5.6] and Chiou,Kato and Ahc' [19].

Observations and ideas presented by Tolstoi led Oden andflfartins [421 to a new approach in t.he analysis of dynamic fric-tion. They considered a relatively simple constitutive model ofthe interface, with a power law normal response and the coefficientof friction independent of the velocity. This model was combinedwith an analysis of motion, in particular of normal vibrationsof the slider, to give an apparent kinetic coefficient of frictiondifferent-in general-from the interface coefficient of friction. Intheir numerical studies. Oden and :\'ia.rtins [42] obtained a goodqu;tlitati\'(: modeling of general experimental obsen·ations. Thisapproach lI'as later sl.udieJ by Oden el. all43} and by TII'orzydloaud Becker [GO]. wlto applicd it to Ilumerical modeling in the re-I .. ,·t iOIlof s!.at.ic friction hI' viural.ions. The results obtained wen'

in vcry good agreement lI'ith experimental observations of Tolstoi[58].

In this paper we investigate further possibilit.ies of the appli-cation of the Odcn/l\lartills lilt/del to realistic modeling of tl)l'

phenomena of dynamic friction. Our analysis focuses on the typ-ical pin-on-disk experimental apparatus, which is representativeof a broad class of mechanical sliding systems. In particular, westudy self-excited oscillations of the slider and basic mechanismsleading to these oscillations, as well as parameters of the sys-tem affecting their occurrence. The general objective is to obtainqualitative and quantitative agreement between numerical pre-dictions and experimental observations. As a particular objectof our computations we have selected the representative pin-on-disk apparatus investigated experimentally by Aronov, D'Souza.Kalpakjian and Shareef [6] and later by Dweib and D'Souza [24)and D'Souza and Dweib (23).

The paper consists of two general parts. In the first part (Sec-tions 2 through 4) we present a brief discussion of the most impor-tant phenomena affecting the nature of dynamic friction and thevalue of the kinetic coefficient of friction. We also present numer-ical model used in our simulations. In the second part (Sections 5through 8) we present a study of the frictional behavior of a pin-on-disk appratus. 'vVestudy the mechanism of self-excited oscil·lations and analyze the most important parameters of the systemwhich affect this occurrence (stiffness, damping, angle of attack).Then, by means of transient analysis. we study the motion of thesystem in the self-excited zone and the resulting reduction of thekinetic coefficient of friction.

The theory, numerical approach and observations presented inthis work can be generalized to a variety of mechanical systemswith friction.

2 Remarks on the Nature of DynamicFriction

In this section lI'e review the most important phenomena whichaffect the nature of dynamic friction, various forms of frictionalsliding and the value of the kinetic coefficient of friction. Wt'classify each phenomenon analyzed into one of two groups:

A. phenomena resulting primarily from properties of the inter-face, and

B. phenomena associated with the dynamic characteristic ofthe system as a whole.

Our review is based primarily on representative experimental worksand numerical experiments. Some remarks anticipate results pre-sented later in this paper. All of the phenomena considered inthis section are listed in Table I, which also contains a brief de-scription and evaluation of each.

2.1 Dynamic Instability of the System

Dynamic instability of the system is a typical phenomenon ofgroup B. The general idea is that if the system under consider-ation is dynamically unstable. then any perturbation of the so-called steady sliding equilibrinIll position canses propagation ofoscillations and occllrrence of high amplitude vibrations of thesy~ll·ms. These are called self-excited oscillations and are per-

14

ceived as frictional noise 01' squeal.Different types of these vibrations have been studied experi-

mentally by several investigators, including Soom and Kim (54,55],Ko and Brockley (36), Shushan (10). Aronov, D'Souza, Kalpak-jann.Shareef [6]. Dweib and D'Souza (241, and D'Suoza and Dweib(23). A numerical study of this phenomenon was recently pre-sented by aden and Martins (42]. who estimated the frictionalstability of a rigid block sliding on a rigid surface by the analysisof eigenvalues of the linearized equations of motion of the system.This line of research is continued and extended in this paper. Inparticular. we study the mechanism of instability which is due tothe coupling between normal and rotational vibrations of the sliderin the presence of friction.

The detailed analysis of dynamic instability and self-excitedoscillations will be presented in Section 7. Here we mention thatonce the self-excited oscillations have occurred, they can be thecause of other phenomena. such as reduction of the coefficient offriction. stick-slip motion or normal jumps of the slider.

2.2 Rate and Time Dependence of the StaticCoefficient of Friction

The time or rate dependence of the static coefficient of friction isa typical phenomenon of group A. Time dependence of the staticfriction was first described by researchers who believed (sec. e.g.,Sampson et al. (5I). Habinowicz [46.-17]. Brockley and Davis (13))that at zero time of stick the static coefficient of friction wasequivalent to the kinetic coefficienl. but then its value increasedwith the time of stick.

This growth of the coefficient of friction was explained by anincrease of the real contact area due to viscoplastic deformationsof asperities under the static normal load. Various formulas ex-pmssing the Iillle cllilnge of IL, weJ'l~propused. The one in bestagreement lI'ith the cxperimcllt.al observations was proposed byKato, Sato. and Matsubayachi [3:l) in the form:

(2.1 )

where:

time of static contact.

IL, . the static coefficient of friction.

µo . the static coellicient of friction at zero time of stick.

IL,,,,, - the asymptotic value of the static coefficient of frictionafter long time of static contact .

..,. TIl - parameters.

However. more recent experimental works of Johannes. Green.and Brockley (30) and Richardson. Nolle [49] showed that thephellomena at.tributed to the time dependence of the static fric-tioll should rather bt' illlcrpret'.J as rate dependence of the staticcoefficient of friction. From specially dcsiglled experiments theseauthors illferred that the filial va.llle of the static coefficient of fric-t ion ol'p"nds primarily 011 the rate of application of the tangenl ial

..

!!\b Phenomenon Descriplion Grollp Fleason model Result When active Impor·tance

Dynamic Oscillalions grow coupling between -sell exciled When '''0. l"yinstability of until norlllal norlllal and rola- oscillalions

1 B ·slick·slip ++Ihe system jumps occur lional oscillalions

II kind

Rale dependen: I' is slIIalier lor plaslic increase 01 beginning 01 slip2a ce of sialic higher ralOs 01 AlB apprO<ldland con· Slick·slip

in absence 01 nOl·applicalion 01 laCIarea: I kind ++friction langenli;)1 lorce inilial microdispl. lIlal oscillalions

..~---~._.,,~...__ .. . -. .. . . .--.. ~-, ,., . .......Time dependen· JI increases wilh creep increase 01 Slick-slip sialic conlacl

2b ce of stalic timo 01 stalic A Ihe contact area: no slip or normalI kind -friclion conlact diflusioll oscillalions

Dissipalion energy is dissipated a) plaslic dissipalion ·damping 013 during normal during oscillalions AlB bl viSCOElasticand normalosc. whenever nOfmal ++

due 10viscoptaslic viscoplaslic ·decrease of oscillalions occuroscillalions delormalions damping coe/. 01 hie.

.growth of oscillal. -decrease ofNormal jumps Iwo surfaces in unSlalJlccases cae/. olltic. ·unslable cases

4 of a slider lose conlact B -illlerlocking 01 as· ·excitation 01 ·beginning 01 slip ++per. lllUt.'{Jin.01 stip oscillations aller Slick

Load dependen- ·breaking 11110ugh changes 01

5 ce 01 Ihe coel· II changes wilh the oxide layer apparent II whenever normal +nOllnalload A ·ptastic failure al wilen oscilla· oscillations occur

(icient 01 fricl. very high loads lions occur

Dynamic inler· I,"perlections 01 ·excitationlocking of bolll surfaces AlB

rOIl~lhnl.!ss01 01 normlll when macroscopic++6 impinge on each a SUIlace oscill,Uions sliding occurs

,imperfeclions oilIer . lill lorce

('aIJJ.: I: P!.t'noml·lIa alrccting lhe lIi1tllre of dynamic friction.

III

load til. Thc t:-'I':"al I' .. 1'''r~lI~ Ij, "111'1'" nhlaitll'll h.\· Hichanl"ollNolle: [49] is pn:;'t·:lll:d ill I-'i!;n...· I. Th:.; 1'111'\'': (iln 1)(' dcsc.:ribct'I,y t:qna';oll (:l.11. prol·id,·d thai tI ... rat,· or applil'i,tion of tilt'tangcntiallo;\tl is ('OilStani alld t hill lilt,\'(, ar(' no int.errllPtions tothi~ loading.

Load Rate Coeff <D

1.

.8

µ$ .6-µ $$

.4

_2

0I

-310

·210

.,10 1.0

(SeC" )

'0

:Iic<i eljuation). 111 the prcscntation of thc experimental 1I'0rk thc;ullllOrs did nol suggt'St any explanation of thc mechanism of theIlhscl'I'ed pht'nOlllcnon, which is of a somewhat differcnt naturcthan that of time dependencc of stalic friction. It Sl'Cms thatt Ill' most naturnl explanation is the \'iscoplastic growth of thearca of junct.ions due to combincd tangential and normal loads.described within the purely plastic range by the early theory ofjUlIction growth of BOII'den and Tabor [12]. Since the process is ofa \'iscoplastic nature, onc can expect a smallcr junction growth,and consequently a smaller coefficient of friction. at high rates ofthe tangential load.

In the context of kinctic friction. which is of primary interest inthis work. rate dependence of thc static coefficient of friction canhI:' important in stick-slip motion. In this case rate dependencecan strongly affect the \'alue of tangcntial force, at which the slipOCCl1l's. This. in turn. can influence the nature of motion of theslider. value of apparent kinetic coefficient of friction. etc.

'I';

Figu re 1: Rate depcndcnc(' or the coefficicnt of fric-tioll-Richardson. Nollc (49].

2.3 Dissipation of Ellel'gy During Normal Os-cillations

For the more gencral cascs of loading the authors suggestedreformulation of equation (2.1) in t.erms of loading rate as the;lIdependent variahle (they did not propose lit/' form of this mod-

15

The normal oscillations of th .. slider are uSllally accompanied bysCI'ere deformations of sl1l'facc asperities during which dissipationof energy occurs dill' to:

• visco-e1asto·plastic deformations of asperities and wear de·bris, and

z [µ]

These jumps (see Fig. 2) at the very beginning of the slidingphase are significantly diffcrent from the normal oscillations oc-CUlTing dlll'ing steady sliding. which usually do not cause loss ofcontact between the two surfaces. These jumps are important inthat they providc strong initial impulse for the normal oscillationsof the slider and thercby activate most of the other features of thedynamic friction.

0.0

t [sec]

t [secl150'0.0

Figure 2: Normal jumps of the slider- Tolstoi [58).

! x [µ]

o.

t 100

1600

• viscous properties of lubricants.

The main dynamic effect produced by this dissipation is thedamping of normal oscillations of the slider. Moreover, since mostof the dissipation occurs when two bodies approach each other,the oscillations produce a slight rising of the slider, which in turnresults in reduction of the real contact area a.nd of the coefficientof friction. This effect was observed experimentally by Tolstoi

158J.The viscoplastic deformation of aspel;ties is of complex na-

ture and therefore difficult to model exactly in the phenomeno-logical constitutive model of the interface. However, for the sakeof general modeling of vibrations of the slider the damping effectproduced by this deformation can be represented by a nonlineardamping term in the constitutive law of the interface (see Huntand Crossley (29J, Oden and Martins (42], and Tworzydlo andBecker [60)). The corresponding model will be presented in sec-tion 3_1 of this paper.

It should be pointed out that modeling of dissipation of en-ergy on the interface by the damping term is only a rough ap-proximation of real phenomena. Moreover, due to the simplicityof this model and the lack of reliable experimental data, estimat-ing values of coefficients of damping is very difficult. Results ofimpact experiments (lIulIL Crossley (29)) which invoh'e a com-pletely different range of normal velocities are not applicable tothc case of vibrations accompanying frictional sliding. Numcricalexperiments oriented toward cstimatioll of thesc coefficients wercpresented ill our pre\'ious work [GOI.

2.5 Load Dependence of the Coefficient ofFriction

2.4 Normal Jumps of a Slider

Normal jumps of a slider represent a very specific feature of dy-namic friction. They can occur in two typical situations:

• in the case of high-amplitude self-excited oscillations .

• at the very beginning of the sliding after the static contactof two surfaces (slip after stick).

Load dcpendence of the static coefficient of friction, apart from itssignificance in the purcly static friction. also produces certain ad-ditional effects in the case of dynamic friction or, more generally,whenever normal oscillations occur. It can bc inferrcd from theilpproach-friction forcc curve and confirmed by numerical a!lalvsi~of motion of the slider that. under constant mean normal load,the load dependcnce of µ, is the reason for the change in theapparent kinetic coefficient of friction in the presence of normaloscillations. In particular.

Jumps of the first kind, which are easily observed in manyexperiments on frictional vibrations (see, e.g., Ko and Brockley136], Kato, et. al. [31.32). and Aronov, et at. [5,6)) are caused bythe growth of amplitudes of normal oscillations in the dynamicallyunstable cases.

Jumps of the second kind were observed in the experiments ofGrigol'ova, Tolstoi [28). Tolstoi (58), and Tolstoi, Borisova, andGrigorova (59]. They investigated stick-slip motion with pre-cise analysis of normal displacements and observed that -forwardmovements of the slider inva7-iably occu7-red in strict synchronismwith its upward jumps. n

• if µ, decreases with the normal load, then in the presenceof normal oscillations the apparent coefficient of friction de-creases (as compared with the static friction),

• if It. increases with the normal load. the macroscopic coeffi-cient of friction increases in presence of normal oscillations(and in the absence of other effects).

Taking into acount the load dependence of the coefficient offriction in practical applications is somewhat difficult sincc the10MI dependence of µ. has a different nature for different mate-

16

..

rials and types of surfaces. Even for thc same material the dataobtained by different researchers call differ significantly and thereis no agreement even with regard to general nature of load depen-dence. For example, for steel or iron surfaces some researchers(Wilson [62], Bowden and Tabor (12] and other sources referredin this b<>?k) observed a slow decrease ~f the coefficicnt of fric-tion at light loads, rapid fall at moderate loads and then almostconstant value (Fig. 3).

illlerest in this paper, therc is no need for precise modeling ofexcitation due to imperfections of the surface and it suffices tomodel only the initial impulse, which triggers further self-excitedvibrations of the slider.

3 FOrlllUlation of the Boundary-ValueProblem With Friction

Figure 3: Load dependence of the coefficient of friction.

2.6 Dynamic Interlocking of Imperfections

On the contrary. other authors suggest a slight increase ofthe coefficient of friction at light and moderate loads and a dropat high loads (Sl'C. e.g .. Tolstoi [·j::;l. Nolle and Richardson [41],Buckley (l5l. B;LY and Wendheim [8], I3ronicc and Lenkiewiczll-ill. In the computational part of this work we qualitativelyillustrate the inOuence of the load dependence of friction on thevalue of the dynamic coefficient of friction.

In a general situation of two deformable bodies in long-distancesliding contact this leads to a rather complex problem. with a ver-sion of Lagrangian-Eulerian description necessary to effectivelyrepresent the deformation. The details oi this formulation arepresented in our previous IVork [43]. A somewhat simpler formu-lation for a single elastic body in contact with a rigid surface ispresented in references [.12.60].

The numerical study presented in this paper is focused ona simplified numcrical model consisiting of several rigid bodieswith clastic and viscous connections (see Section 4). The rigidmodel can formally be obtaincd as a special case of a generalformulation prescnted in (43). This rather lengthy derivation willnot be discussed here and we will only present the final form ofnonlinear equations of motion for thc rigid body model (Section-!). III thc remainder of this section we disclISS the most importantcomponent of tht: model. which is tlw contact and friction law ontil<' intcrfal:e.

Figurc 4: Elastic bodies ill frictional contact.

The primary objcct of our intercst is a typical pin-on-disk appa-ratus. which is a good representative of a broad class of mechan·ical sliding systems. The computational model of such a setupshould. in general. be three-dimensional. However, if one is in·tercsted primarily in the motion of the pin and of the disk in the':icinity of the pin. the behavior of the system can be representedby a two-dimensional model formulated in a plane tangent to thecylindrical surface defined by the rotational motion of the disk.Thc model is that of an elastic body A in contact with anotherelastic body 8. which is mO\'ing with a prescribed velocity if (Fig.~).

r+3

10I z

::'0

I~O

Load [gm]

I-2

10 ::.C

\

~

c2.q

0...t 1. ~............. 1.00

:::: o. ~.. .0u

0

The dynamic interlocking of imperfections is a key factor in fric-tional sliding. This is not only because the interlocking stronglyaffects the frictional resistance of the interface, but also becauseit provides excitation of normal I'ibrations of the slider. It is thissecond role that we explore.

The first rescarcher II'ho had clearly shown the existence andimportance of normal motion of a slider was Tolstoi [58]. Thenthis phenomenon was studied cxperimentally by Antoniou, et al.[4], Soom and Kim [54.55) and Arotlol', et al. (6]. The explanationof the source of these vibrations as an interlocking of imperfectionsof the surface and the propositions of modeling this interlockinglI'ere presl'nted bv Anand and Soom [21. I.anchun et al. (37) andSoom and Chen (5:3). Using the sinusoidal and quasi-randommodel of waviness of the surface and a simple model of the sliderthey simulated the dynamic effects observed in experiments.

It is important to observe that in the case of self-excited os·cillations the interlocking of imperfections provides the initial im-pulse for oscillations, but then vibrations reach a much higherlevel than those forced by waviness of the surface only (as in so-called steady sliding). Therefore. in this case. which is of primary

17

3.1 Constitutive Relations for the Interface

so that if mT = mN, this model represents the Coulomb frictionlaw with µ = ~. In terms of normal stress on the interface, theequation (3.4) can be rccast in the form

The constitutive equations of the interface applied in this workare based on the aden/Martins model and consist of two basicequations:

• normal interface law.

• friction law.

(3.5)

whcre it was assumed that Ii = 0 (static friction). One can easilyobserve that there are at least two ways of modeling load depen-dence of the coefficient of friction:

The normal response law is of the form

(3,1)

1. by considering IllT =1= I7lN. However, in this case, µ at loadsdecreasing to zero will either decrease to zero (for mT <mN) or grow to infinity (for nlr < m.v).

where UN is a normal stress, a is a normal approach (penetration)on the interface and Ii is its time derivative. This constitutive lawcorresponds satisfactorily to experimental observations at moder-ate loads as reviewed by Back, Burdekin and Cowley [7).

The first term in this equation represents elastic response ofthe interface, while the second damping term models dissipationof energy during normal oscillations. As noted above. the natureof this dissipation is fairly complex and the term in (3.1) is onlysupposed to model the overall damping effect produced by the vis-cous properties of surface asperities, contaminants and lubricants.The coefficients bN, IN arc usually assumed to be constant - seeHunt and Crossley (29) and aden and Martins [42]. However.since the energy is dissipated mostly in the approaching phase ofnormal motion, it may be morc realistic to assllme bN dcpendentof thc phase of normal motion, e.g .. in the simple form:

2. by assuming lilT = inN and CT dependent on normal lo~(or penetration). This approach seems to be more generaland will be used further in this work.

(3.6)

if I~I> (

$,(() = J (2 -I~I)~1 sgn ~

is the regularization function.

where

Another important observation is that within this model thevalue of friction force is /lol uniqnely defined in terms of displace-luents (in particular for zero slidin~ velocitv). In ordpr to ami.:numerical difficulties resulting from that fact, the friction law isusually regularized [40.42.43,60]. One of the possible regularizedforms of the friction law is (for two-dimensional problems):

(3.2)

Damping terms of this form were studied in our prcviolls IVork[601· 4 Discretization

The second constituti\'e equation of the interface is the frictionlaw of the form: 4.1 Rigid Body Model

(3.3)if a ~ 0 then leTTI ~ cT(a)mT and

if a < 0 then eTT = 0 For the particula.r pin-on-disk appara.tus considered here, we uscda relatively simple dynamic model consisting of three rigid bodiesconnected with elastic springs and linear dashpots. The applica-t.ion of ~lIch a modd is iustifipd in St'ction 5.

The model is prcsented in Fig. 5 and represents the experi-mental setup showll iu Fig. 6. It consists of two rigid bodi~s Aand C and a rigid frame F. Body A corrcsponds to the pin. bodyC represents the hcavy block on which the pin was mounted. andthe rigid frame F represents the moving disk. The dimensions ofeach body are length L, height Ii, and thickness b (not all of thesedimellsions are important in computations). Each of the bodieshas three ?cgrces of freedom: displacements of the center of mass(ix. U~ and counterclockwise rotation O. Frame F and body Caresupported by elastic sprillgs of stiffnesses 1\';, J(~, I\t, t1 = F. C.and body A is connected with body C by means of three springsl\';-c .l\;-c. ]\'o4-C. For each of these springs there may exist acorresponding dash pot. Note that although the disk is modeled

(3.-1)µ = CT a(mT-mN)

CN

where d is a sliding velocity calculated as a time derivative ofsliding distance, and index T refers to the direction tangentialto the contact surface. The friction force is a function of thenormal approach of the two surfaces which. in tum, depends onthe normal force. The actual value of the static coefficient offriction can be expressed in the form:

leTTI = cr(a)mT => d = -).eTTt (). > 0)

18

(4.2)

wh~re uAX, Y, t) is the displacement of any point in the body,Ur(t),Uy(t),O(t) are discrete displacements defined in the center

of mass, and .x' Y is the reference position of the center of mass(of the given rigid body). Approximation of velocities and ac-celerations can be obtained by appropriate time differentiation ofthe above formula,

4.2 Semidiscrete Equations of Motion

Substituting discretization formulas (4.1) into the variational equa-tion ano requirinµ; that it be sat.isfied for every combination of te~tfunctions [40.42.43,60]. we obtain semidiscrete nonlinear equa-tions of motion:

M R(t) + (C + C)R(t) + (K + K)R(t)+

P (R(t). R(t») + J (R(t). R(t») = F(t)

withR(O) = itJ

(4.3)R(O}

Figure .j: A rigid body model.In the above formula:

R, R. R the column vectors of discrete displacements. veloci·ties and accelerations.

.,s a rigid body, it differs from the standard rigid surface modpl,I,ecallse it is supported elastically and can move.

The initial configuration of our model is presented ill Fig. 5.

!l1. D, K mass, damping and stiffness matrices,

C, K damping and stiffness matrices corresponding to ad·ditional inertia terms resulting from the prescribedmotion of the disk (representcd by a frame F),

F consistent load vedor

P, J vector of consistent forces due to normal and frictionalresponse of the interface. respectively.

For the discrete rigid hody model, these \'ectors and matrices aredefined below:

(4,5)

(4.6)

(4.4)

M = rMA.MA.j-'I .... ,JcJ

The stiffness matrix K is:Figure 6: Schematic of an experimental pin-on·disk apparatus.

In order to consider different angles of attack, we permit a cer-tain prescribed rotation 0 of the framc F. This rotation is as-sumed to be moderate in the sense, that cos '8 :::::1 and sin 0 :::::0.Since the original pin uscd in the expcriment was of the circularcross-section, we allow variable distribution of thickness of thepin ~(s),s E [-LA/2.LA/21 (this is important in calculations ofresponse of the interface).

The approximation of displacements for each rigid body canbe written in the form:

Ur(X'y,t) = Ur{t) + (cosO(t) -I)(X- .~) +sinO(t)(Y- }~')

1\;<-C 0 0 0 0 0 _1\:-c 0 _ DI\;<-c

0 /\':~_c.: 0 0 0 0 0 _1\;,-r: 0

0 0 l\e~-c 0 0 0 0 0 }' •• c- \.~

0 0 0 1\; 0 0 0 0 00 0 0 0 I\{ 0 0 0 0 r-l.~·0 0 0 0 0 1\:- 0 0 0

f"-C 0 0 0 0 0 I;,~ 0 DI\,'-c- '.0 _l\';~-c 0 0 0 0 0 1\", 0_DI':;<-c 0 -1':,'--' 0 0 0 Dl':;'-c 0 I':..

where D = dA-C is the distance between centers of mass of pin

u 0

II.(X.}, t) = Uy(t) - siIlO(I)(X - X) + (cosO(t) - l){Y - Yl.J.1)and block and 1\77 = 1';; + f\';-C!\'88 = 1\; + f{;-c, f{gg =f\f + gf}-c + f{:-c D2.

19

(4.8)

Since the disk is rigid, the additional stiffness matrix K IS

zero and the additional damping matrix has only two non-zeroelements C56 = 2U MF, C~6 = 2U MFOF. The two vectors P andJ, representing normal and frictional response of the interface are

of the form:

P = {PrA,pvA,PoA,P!,P{,Pf,O,o,o}

J = {J;,J;,Jt,J[,J{,J[,O,O,O}

where the consecntive components can be expressed in the form

KN -ap

- aRJ<T = aJ

aReN _ ap

(4.14)

-8ii.

eT aJail

with the use of formulas presented in section 4.

4.4 Remarks on Methods of Solving The Equa-tions of Motion

In the following sections. we apply the model developed in thefirst pal't of the paper to the analysis of frictional behavior (inparticular, stability) of a typical pin-on-disk setup.

As the particular object of our analysis, we have chosen theapparatus employed by Aronov, D'Souza. Kalpakjian and Sharref(6) in their experiment on frictional vibrations. Our choicc wasbased on the thorough documentation of both experimental setupand results, the relativc simplicity of the apparatus as well ason the interesting results regarding self-excited oscillations andevolution of the kinetic coefficient of friction at increasing loads.Note that the experimental investigations of reference (6] werefurthcr continued by Dwcib and D'Souza [241 and D'Souza andDwcib (23].

The primary goal of our analysis is the further understandingof the nature of dynamic friction. In particular we aim to:

- identify alltl study the most important parameters af-fecting the occurrence of self-excited oscillations.

- study the nature of motion of the system in the sclf-excited zone.

which was obtained by the Householder method. The eigenval-ues and eigenvectors of this systelll an:. due to nonsynunetry ofmatrices, complex quantities.

5 Presentation of Numerical Exam-ples

The frictional behavior of the sliding system is determined by thetransient numerical solution of the system of equations of motion(4.2) or. in certain cases, via analysis of natural frequencies oflinearized equations (4.13).

The primary method of transient analysis was Newmark'smethod with adaptive time increment. as apr1iPd previollsly hy

Oden and Martins [42). In selected cases, primarily for verifica-tion purposes. the standard Gears or Adams integrators lI'ere alsoapplied (general purpose library routines were used). The naturalfrequencies of the linearized system (4.13) were calculated via thesolution of eigenproblem:

M,,\2+(c+c+eN+eT)"\+(J<+K+KN+J(T) =0 (4.15)

(4.9)

(4.12)

(4.11)

-P(I I = x.y,O

JAo

pAo

where

pA:r

and the sliding velocity on the interface is given by the formula:

M R+(C+C+eN +eT)R+(K +K +KN +KT)(R-Rn) = tlF(4.11)

where R - Rn is the infinitesimal increment of displacements,M, e,C,K have the same meaning as in (4.2) and matricesKN. KT, eN, eT can be calculated as derivatives at Rn:

The semidiscrete equations of motion, presented in the previ-ous section, are-due to nonlinearities in the response of theinterface--generally nonlinear. However, for the sake of analysisof infinitesimally small vibrations around a prescribed configura-tion of the system, defined by Rn = 0, Rn = 0, one can use thelinearized version of (4.2):

4.3 Incremental Equations of Motion

Jr = -J(' i= x,y,O

In the above formulas s spans the "bottom" of the pin and r~is the line of actual contact between pin and disk, determined bya(s) > O. The approach a can be calculated from the formula:

a(s) = U{ - U: - sOA + sFOF (4.10)

20

Figure 7: Variation of the kinematic coefficient of friction fOlincreasing normal loads - after Aranov ct. aJ. [6].

J. Region of stcady-state friction with very small vibrationscaused by interlocking of imperfections on the interface

Normal force (NEWTONS;

,.200

IV

. i; ii :

, I ,

120 160I

80I40

o"1o

•.6

'0 .2c

'":§~ .1()

.e .3

~

- model the dilfercnce hetw<',(~11static and kinetic coeffi-cients of friction,

- identify further directions of development of models offriction and relevant <:omputational methods.

Althongh the experiment perfol'l1It'd hy Aronov, et al. [6] wasvery well documented, certain oata nccessary in numerical mod-eling are missing and were not measured at the time of the expel"iment. In these cases we used relevant information presented inother works (e.g. constitutive constants for the interface). More-over we often "perturbedn the data from the basic data set, usu-ally in order to better illustrate specific features of the frictionalbehavior of the system under consideration.

The analysis consists of several stages. First we analyze stabil-ity of frictional sliding at a wide range of loads. The stability andoccurrence of self-excited oscillations is estimated by means ofeigenvalue analysis of linearized equations of motion (4.13), andthe results are confirmed by transient analysis of full nonlinE'arequations of motion (-I.:!). This transient analysis also forms thebasis for modeling the kinctic cocfficient of friction. In particular.the drop of friction force in the ZOlW of self-excited oscillationswas observed.

Finally, in Section S we extend our analysis to angles of attackother than zero. This analysis is heyond the actual program ofthe experimental work [61.

II. Region of nonlinear friction with small vibrations

5.1 Summary of Experimental ResultsThe detailed description of the cxperimental apparatus is pre·sented in the paper by Monov, D'Sonza, I\alpakjian and Sharref[61. Here we will briefly present the basic components and a sim-plified sketch of the setup (Fig. 6).

The apparatus consists of a rot.atin~ iron disk (1), 200lllm indiameter, and strel pin (2) of diamclt:r !imlll. The pin is mountedon an elastic arm (3). fixed at one en (I. A heavy, rigid block (4)was fixed at the same end as the pin. The pin was not really fixedto the supporting arm-a force transducer was sandwiched betweenthe piu and supporting <lrnt, so this t:Onnection did actually havesome compliance.

During the experimcnt, the normal load was exerted on thepin by means of moment applied at the fixed end of the supportingarm. The iron disk was rotating at constant angular velocity, suchthat the average sliding spPed of tlw pin was equal [1 = 46cm/s.

The basic part. of the pxpprimenl. consisted of analysis of t.heevolution of the kinetic coefficient of friction and occurrence ofself-excited oscillations at normal loads increasing from 0 to 200N. The experimental results are shown in 7, which is a simplifiedversion of Fig. 3 by Aronov. et al. [61. The figure presents a plotof the kinetic coefficient of friction versus the normal load on thepin. There are four regions on this plot (for dctailed descriptionrpfer to [6]):

II I. Transient region with periodic bursts of self-excited oscilla-tions accompanied by a drop of tht! apparent kinetic coeffi·cient of friction

IV. Region of self-excited oscillations. i.e. high-amplitude os-cillations perceil'cd as a frictional noise. In this region thekinetic coefficicnt of friction was lower than in other zones.

It is of importancc that the coefficicnt of friction plotted onFig. 7 was not measured on the interface. but was estimated bymeasurements of displaccment of the end of the supporting arm.Therefore it can be considered to be an apparent or macroscopiccoefficient of friction.

6 Analysis of the Stability of FrictionalSliding

The occurrence of self-excited oscillations is the consequence ofdynamic instability of the system. that is. the situation in whicha perturbation of steady sliding causes growth of oscillations andoccurrence of high-amplitude vibrations. This observation wasmade by Arollov et al. 161. D'Souza d al. (23,24], and modelednumerically for a simple test model hy Oden and Martins [421·In our previous work «1:1] we identified t.he mechanism of self-executed oscillations as the dynamic coupling between normal androtational degrees of freedom. A similar conclusion was derivedfrom experimental observations by D'Sollza and Dweib [23]. Thegeneral nature of this mechanism can he dcscrihed as follows:

21

The self-excited oscillations occur when the naturalfrequencies of normal and rotational vibrations of theslider (in contact with a slide line) are relatively closeto one another. Then the existence of friction causescoupling between rotational and normal modes and, asa consequence, frequency coalescence and propagationof self-excited oscillations.

In light of this observation, the crucial parameters in the anal·ysis of stability and self-excited oscillations are all parametersaffecting normal and rotational vibrations of the system.

For the apparatus under consideration, the frequency of rota-tional oscillations of the slider depends primarily on the torsionalstiffness of the supporting arm. On the other hand, frequencies ofnormal vibrations depend strongly on the stiffness of all elementsbetween the slider (rigid block in Fig. 6) and the disk. Sincethe frictional interface is generally quite stiff, the compliance ofthe pin and of the connection between pin and block becomesextremely important. In the computational model presented inFig. 5, linear springs were introduced between body A (pin) ilnobody C (block). These springs represent the compliance of thepin and other elements between the pin and the block. such asforce transduces.

6.1 Basic Computational Model

The basic data corresponding to the parameters of the experi-mental apparatus are (notation corresponds to that of Section4):

1. Body A (pin):L = 0.5 cm (circular cross sect ion)JJ = 1.5 cmAt = 2.2972 . 10-3 kgJ, = 4.6663· 10-4 kg

2. Body C (block):MC = 4.5 kgJf = 96.0 kg cm 2 (6.1)dA-C = 3.085 cmFC = 845.6.102 kg S -2\r

FC = 773.9· 103 kg s -2\u

FC = 432.5.105 kg S -2 cm-2\6

3. Body F (disk):HF = ScmM; = 10.0 kgMF = 24.0 kgJl = BIGF F FI\'r = l\u = A, = BIG

Parameters not listed above (such as the dimcnsions of theblock) were of no importancc in the computation. In the abovedata BIG represents a big number so that the support of the disk(shaft, bearings) is considered. in the basic data set, to be rigid.This was assumed here due to the lack of experimental data, butin further analysis this assumption will be relaxed.

The material constants for the interface were calculated fromthe table presented by Back, Burdekin and Cowley [7] for thesurface finish corresponding to that of disk and pin. These data

22

are:

cN = 1.25. 103 kg cm 3.5 5-2

cr = 0.3125.103 kg cm 3.58-2

mT = 2.5 (6.2)

mN = 2.5

bN = iN = 0

and they correspond to the coefficient of friction µ = 0.25 (seeequation (3.4)). The stiffness of connection between pin and blockwill be defined in the subsequent sections. In the basic set of datathere is no damping in the system whatsoever.

6.2 Estimation of Stability via Eigenvalue AnalYSls

6.2.1 The Algorithm

In the analysis of stability of frictional sliding we followed thegeneral procedure used by Oden and tvlartins [42]. This procedureconsists of two stages:

I. Assume that there are no vibrations of the slider (steadysliding) and find the equillibrium position in this state.

2. perturb this equilibriulll position and analyze t.he dynamicstability of motion uf the system.

Solution of the first stage is equivalent to solving the equationof motion (4.2) under the constraint. that velocities and accelera-tions are zero (except for prescribed velocity of the disk U). Then(4.2) reduces to the quasistatic nonlinear equation:

(K+K)+P(O,R)+J(O,R)=F (6.3)

which can be solved by the Newton method. The solution of thisequation corresponds to the steady slidinl!; equilibrium position.

This situation is. of course, idealized, since interlocking of thesurface asperities provides permanent small perturbations of thisequilibium position. If the system is dynamically stable, vibra-tions caused by this interlocking do not propogate but stay at alow level. perceived as a steady sliding (zone I in Fig.7). If thesystem is, however. dynamically unstahle. after perturbation ofthe steady sliding equilibrium position the oscillations grow andreach high values corresponding to zone IV in Fig. 7.

The stability of motion of the slider after perturbation of thesteady sliding equilibrium position can be estimated by the tran-sient analysis of equation (4.2) or. in a simplified version. byanal·ysis of the ~tability of linearized equations of motion (4.13).

The laUeI' can be reduced 1.0 the solution of the eigenvalue

I>roblem:

500

Normal Force [NJ

,.:,f;

: i: I

I

III

K •2.047 ..6

I ~.,-. K"'':;

4.0e- i 5.0e- 7 y

no fric:ion

µ- ",-._-

3.0e-7

selt-exclted oscillations

2.0e,71.0e-;

(angular frequencyof a Slider}

.0

10

inSl3t1ililY :oneat ~=:s

Figure 9: Zone of self·excited oscillations at various values ofl\'y~-C.

Note that at I\y~-C = 2.057x 106 kgs-2 the self-excited oscilla-tions occur at loads greater than 185N, a case which correspondswell to experimental observations. This value of J\:-c was there-fore included in the basic data set for the model of experimentalsetup. The detailed plot of frequencies of normal and rotationalvibrations in this casc is presented in Fig. 10.

The width of thc instability zone depends strongly on the cocC-licient of friction. and pcrturbation of It from 0.25 to 0.30 reduceslite lIurmal force corresponding to the onset of sclf-excited oscil-lations to 120N.

100

Figure S: Estimation of instability via analysis of natural frequen-cies.

(r)

I"coi "0 21.000

1;.900_

,..oJ;.•oJ;.~J;.=~J

I; . .eoo.1

;.J,.~J

I FN:.:00 . . . . . . . . . ,f.

.:00 . .2iJO .400 .~~ .:00 1.:l00 :.ZOO ~.4:0 ~.soo 1.aco 2.000 1:0 ~

2. Re( Ai) < 0 for i = 1 ... 11 - oue can expect stable oscilla-tious which are gradually damped out,

1. Re(A;) = 0 for i = I ... n - one can expect stable oscilla-tions about the static equilibrium position,

3, 3i s.t. Re(,\;) > 0 - one can expect unstable, growing oscil-lations.

6.2.2 Dynamic instability of frictional sliding

The key factor in occurrence of self -excited oscillations of theslider (understood Iwre as a pin-block asscmbly) is the freqnencyof its normal vibrations when the pin is in contact with a disk.The self-excited oscillations occur when this frcqucncy is closeto the frequency of rotational oscillations which, for the modelnnder consideration. is determined by thc stiffness of supportingarm /If and is almost constant under all loads (about 126 Hz).The normal stiffness of the system depends on the stiffness of thesupporting arm /\';. stiffnes of the interface, and of the springA':-c. Thc last valuc represents the compliance of the pin andthe connection between the pin and the block, which could notbe explicitly determined from thc experimental data. One of thegoals of this \\'ork is to study its influence on the occurrence ofself-excited oscillations.

We begin by showing how, for given parameters of the system,the range of the instability zone is established. As a representa-tive example. consider the basic data set presented in Section 6.1with the normal stiffness J\;-c = 2.4 X 106 kg S-2 and othersprings assumed temporarily to be rigid: I<:-c = K:c = BIG.If one considers this system without friction, then the dependenceof normal and rotational frequencies on the normal load is repre·sented by dashed lincs on Fig. S.

These t\\'o curves interscct at the value of the normal loadFN = 1O.5N. Introdnction of friction (µ = 0.25) into the systemcauses coupling between normal and rotational modes and coales-cence of corresponding frequencies in the vicinity of the intersec-tion point (solid lines in Fig. 8). The corresponding eigenvalueshave positive real parts. so this zone is the region of dynamicinstability and, as a consequence, self-excited oscillations. Thewidth of this instability zone increases with thc increasing valueof the coefficient of friction,

The above analysis was performed for various stiffnesses J\':-cand the range of the unstable zone was established. This zone isprescnted in logorithmic scale in Fig. 9.

The analysis of eigcnvalues establishes the stability of the lin-earized small oscillations about the equilibrium position. Notethat this linear stability analysis gives only the initial estimate ofthe real behal'ior of the model, \\'hich is essentially nonlinear.

Due to the existence of damping and asymmetry of the milt rix(K + KN + K1') thc eigcnvalucs of this problem are complex.The dynamic stability of the linearized system is determined bythe values of real parts of these eigenvalues. In general, threesituations are possiblc:

23

8.100

co (angular frequency2 of. slider)

8.Z00J'10

(6.5)

CO (angular frequency)

1.200~""0 5

This numerical study leads to the conclusion that horizontal androtational stiffucsses of a pin-to-block counection have virtuallyuo effect on natural frequencies of vibrations of the slider (pin-block assembly) and, therefore. on its frictional stability. How.ever, at certain values of stiffness Kt-C the natural normal androtational frequencies of the pin itself are close enough to caufieself-excited, high frequency oscillations of the pin. For example,for general data given in section 6.1 and:

025

µ=.25self·excrteaoscillations

-----------

self· excited oscillations

no friction

with friction

µ= .30

---------- µ=.25

8.0001 Ola==7.900

7.900

7.7001 COy

7.600

7.~

7.400

Figure 10: Self·excited oscillations of the slider a:f{y~-C = 2.057 X lO"kgs-2.

This sensitivity of the range of self-excited vibrations to thefrictional properties of the interface can explain the existencc. inthe experimcntal observations. of the region of transient frictionwith periodic bursts of sclf-excited oscillations (zone III in Fig.7). According to experimental researchers. in this zone "the meanvalue of friction force increases with time at a constant normalload" and "when the mcan friction force reached a sufficientlyhigh value, a temporary burst of self-excited vibrations wouldoccur and the friction force would infitalltly fall to a lower value~(Aronov et aI. (6)).

This observation is in perfect agrccment with the sensitivityof the range of the instability zone to the coefficient of friction.presented in Fig. 10.

Normal Force IN]

8LOCK+PIN

Normal Force (N]

self·erClled O$CIUauons01 lne wnol& stider

~y~~".",."~,,,oo,.~""

.000 .~o .700 1.0!:O :..0400 1.7~O 2.':'00 2 .• ~O a.aoo 3.150 3.500 -"0

. 7S0

.6'0

.::00

.920

l. 060

8.:'001_102

7.900

7.700

c.~o2.0001.~01.i..~.~ 1.000.~7.ZOL

.000

6.2.3 Other Modes of Dynamic Instability

The coupling between normal and rotational oscillations of theslider is a basic mechanism of self-excited oscillations ohscr\'t'din the experiment of Aronov, ct aI. 161. However, even for sncha simple mechanical system a question arises: is there a po.~si·bility of other modes of self-excited oscillations and what are thepUJ'umeters affecting them?

To answcr this question we perturbed selected parameters ofthe system which, in our opinion, could affect dynamic stabilityof the frictional system. In particular, stndied were:

- stiffnesses l\'x~-cand J{O··I-C of a pin-to-block connec-tion.

- stiffness of the disk support.

Figure II: Simultaneous self·excited oscillations of the block andof the pin.

the plot of rotational and normal frcquencies of the pin and theblock are presented in Fig. II.

In this case sell-excited oscillationfi of the pin and block occurindependently and, at a ccrtain range of normal loads, there areactually two self-excited modes of differcnt frequencies. For morecomplex systems. this observation can be generalized to predictthe whole spectrum of sclf-excited modes. each of them corre-sponding to differcnt noises generated by the systcm.

In the analysis prcsented so tar, the disk was assumed to befixed in the axial direction (due t.o lack of relevant data). How·ever. in practical systems. the shaft and bearings have a certaincompliance, so that thc disk can also vibrate during rotat.ion.

24

!i

j·1iI

~.a~o

no friction

µ=.:!S

Normal Force [N]

----~----

self·e:xciledo~lJatlons

C1\ (SLIDER)

analysis). It is evident that oscillations of t he system are actuallysclf-excited and grow as a function of time ,.the same happens torotational oscillations, not prcsented htm~). This behavior com·plies with predictions of the eigenvallle' analysis presented in Fig.10. Thc behavior of the systelll at fnlly developed oscillations willhe allalyzed in the next section.

A more detailed analysis. not prescnt<.-d here, suggests that asufficiently large perturbation can. under certain circumstances.cause self-excited oscillations of the systcms which, according tolinearized analysis. are stable. Thcrefore the actual nonlinearinstability zone may extclId slightly beyond the limits estimatedby linearized analysis.

Figure 12: Self·excited oscillations dne to coupling of rotations ofthe slider with normal vihrations of till' disk.

Therefore thc question arises: Call sl'if-excited o.<ciilations occurdue to a coupling between tite 1II0tioll of the slider and the disk,for example, the rotational vibrations of the slider and the normal(i.e., along the shaft) oscillations of the disk? The answer to thisquestions is positive. In order to illustrate possible situations ofthis type we perturbed the normal sWfness .of the supporting arm(in order to avoid coupling between rotational and normal mod~sof the slider) and assumed compliant disk support. In particnlar,for the data:

6.3 Transient Analysis of Self-Excited Oscil-lations

(6.6)

we obtained graphs of frequencies of normal vibrations of thedisk and rotational vibrations of the slider presented in Fig. 12and the zone of self-excited oscilJatiolls similar to previously pre-sented examples (but now both the disk and slider vibrate). Thisobservation can most probably be generalized to other Illulti-component sliding systems and suggests thc existence of compli-cated modes of self-excited oscillations, with dynamic couplingbet\\'ccn different clements of the system.

The above examples illustrat.e L1le sensitivity of occurrenc.eof self-excited viibrations to various parametcrs of the system.The significance of many of these parameters can be int.uitivelyanticipated by considering their innuence on the frequencit>s ofnormal and rotational vibrations.

The eigclll'ahlt' analysis estimates the dynamic stability of in-finitesimal linearizcd oscillations about thc steady sliding equi·librium position. A more realistic modeling of the behavior offrictional systcms can be ohtained by thc transient solution of anonlineiU' equation (4.2) after a sm,11Ipt~rturbation of the equilib-rium position. Such computations were performed and confirmpredictions of the eigenvalue analysis.

For example, Fig. 13 presents a history of the normal velocityof the block ohtained for the basic Sl't presented in Section 6.1.with the following additional parameters:

]'A-C = 2.05i kg S-1\y

F:v = 200 N

Pv = 0.01 cm/s

e = 0.01 cm/s

ill = 10-(; sec

II, = (1.:25

(6.7)

1.600

l.leO.

WAE 1_'- ....,.-

.100

-1.600.

-2.000

Here. p, is the small initial perblllhation of the normal ve-locity of the block. e is a regularization paramet.er in the frictionlaw and ~t is an initial time step (automatically adapted during Figure 13: Onset of self-excited oscill,ltions of the slider.

25

In this section a transient analysis of motion of the system afterthe onset of self-excited oscillations will be performed. The goalhere is to model changes of the apparent kinetic coefficient offriction in the presence of self-excited oscillations, similar to thosl'observed in the experiment (Fig. 7, zone IV).

The transient analysis was performed using the Newmark method,with the adaptive time stepping procedure presented by Od('nand Martins 142). Although the Newmark's algorithm is uncon-ditionally stable, the basic time step ilt = 5.0l0-6see was smallenough to model all frequencies of the system, in particular, high-frequency vibrations of the pin: (order of 15 kHz).

As the first example we analyzed the motion of the modelpresented in section 6.1 with only two different parameters:

Figure 14: Temporary drop of the killemalic friction force in theself-executed zone.

flUE 1-)...----, I"

0.800 0.900 1.000

0.100.

0.000.,_-. __ ,-,--. +1- -- ,- _, -1-

e.ooo 0.100 0.200 0.300 0.,100 u!.lOO o.60u 0.100

Modeling of the Kinetic Coefficientof Friction

7

J1d = 0.01 em

(7. I)t .000.1 _10~2

ilt = 5.0 x 10-6 sec,

Here J1d is the perturbation of displaccments U:,U~from a steadysliding equilibrium position and other parameters were explainedbefore. This set of data. which will he l'ef('l'red to as the basic.one, corresponds to the zone of self-excited oscillations (zone IVin Fig. 7).

The calculated history of horizontal displacements of the slider(which reflect the apparent friction force) is presented in Fig. 14.The oscillations of the system grow and. after about .3 sec., thefriction force temporarily drops. This reduction of the frictionforce occurs when the sliding of the pin turns into the micro-scopic stick-slip motion in phase with the rotational vibration ofthe slider. However, for this particular model the reduction offriction force is only temporary, which is lIot in agreement withexperimental observations. The main reason is the lack of damp-ing in the model. This allows strong high frequcncy vibrationsof the pin itself. which interfere with the stick phase of the mi·croscopic stick-slip motion (thc stick contact is disturbed). Thebehavio:' of the system. however, changes significantly after intro-duction of a small damping into the system, in particular into thepin-to-block connection.

This change is illustrated in Fig. 15. which presents horizontaldisplact'ments of the block after illt.rodn('in.~ dalllflin~ into t.he pin-block connection. The actual coefficients were equal to:

C'I-C = 100 kg 5-1r

CA-C = 130 1.:95-1 (7.2)~

Cg1-C = 100 1.:g e1ll S-1

0.100.

0.100.

"••. 1S_0.500

0.500.

0.300.

0.:00.

0.100.

W.IEI'Io.ooo·I_~r-_I __ •.~ ,._..--. _..--._0.000 0.100 0.200 0.)00 0.4Ot1 o.~ o,t;OO 0.100 0.000 0

Figure 15: Reduction of the apparent friction force due tosplf-"xrited oscillatiolls.

0.800

0.700

0.600.

0.500.

o. ~oo

O.JOO+

0.200.

O,IDO.

Figure 16: lIistory of the sliding \'c\o('ity of th .. tip of the pin.

and give damping close to critical damping for the pin. bnt almostimperceptible (1% of critical) for the slidcr as a whole, whichconsists of a pin and heavy block. Physically. the existence ofsllch a small damping is realistic and well justified.

In this Ci1.~e.afkr the self-excited oscillations reach certainlevels, the \'allle of t.he apparent kinetic: ('oefficient of friction dropsbelow tht~ steady sliding v..hI<'. 'I'll" rllulion of the tip of the pinhas the form of sticl;-slip ill phase with the rotational vibrationsof the slider. This Gln!>t' S<'<:II in Fig. Ii.which presents the time

26

O.OOO+t--,----r----r-_.-_'_-r0.000 0.100 0.200 o)OQ 0.00 O~o O.GO

," .....-....-

0.100.

0.700.

- .. --:-:-: . ':9

1.000.1*'0 <4

0.500.

....:L ~CO .

'.:CO ~,;:a : ....:':l :.1:~ :,.C:l :.::0 '.!·:C -:: 'L!~~ :,;.:C

0.200.

D.""

0.• 00.

O.eoo.

0.600

with the necessary coefficients based 011 oLlwr sources.In the first example we introduced damping on the interface by

specifying in the interface constitutil'e la\\' (:3.1) bN_ = 0, b.\·+ =3.0 x 1012kg 011-1 s-J with In = 2.5. The values of coefficientbN were estimated according to remarks presented in our previ-ous work [60]. At the relative normal \'elocities on the interfaceattained during self-excitcd oscillation (order of 1.0 cm/s) theyproduce a damping force of thc order of 10% of the elastic re-sponse of the interface. The plot of calculated history of tan-gential displacements of the slider (which reflect the measuredapparent fricion force) is presented in Fig. 19. As the oscillationsgrow, the apparent coefficient of friction decreases and the finalreduction of friction is more significant than in the case with nointerface damping (15% as compared to 11%).

Additional reductioll of friction force was also obtained whenthe coefficient of friction was assumed to be dependent on a nor-lllalload. It was assumed that the coefficient of friction (definedin equation (3.5)) changes linearly from 0.2:\ at zero normal forceto 0.25 at normal load 200 N. For such data. the time history ofthe frictioll force is pr(?scnted in Fig. 20. Again, the reduction ofthe wdficicnt of friction is more significant than in the basic casepresented in Fig. 15. nalllely 21% as compared to 11%.

Figure 17: Evolution of the instantaneous friction force at the tipof the pill.

Figure I:): Drop of apparent friction force in the case of ~stronger"instability.

• occurrences of Slick-slip motion of the tip of the pin, whichtl'nds to altl'r th" roliltiOllill fr('(I'IC'lIey of the slidpr (as inexamples considered here),

• occurrence of normal jumps of the slider, which tends toalter its normal frequency.

Which of these mechanisms is actually activated depends on themechanical characteristics of the system and sliding \'elocity. If ithappens to be normal jumps of a slider, there is no miscroscopicstick-slip and, consequently, no perceptible drop of the apparentcoefficient of friction (Fig. 5 in Aronov, et al. [6]).

In the analysis presented so far, the model of the interface wasrelatively simple, with no interface damping and with the coeffi-cient of friction independent of the normal load. As mentionedin Section 2, these phenomena can. in the presence of vibrations.introduce additional changes to the coefficicnt of friction. For thesystem under consideration, neither the load dependence of thestatic coefficient of friction nor the amount of damping on the in-terface were kno\\'n. \\le present here a few illustrative exampl~.

history of the relatil'e sliding velocity on the frictional interface.The corresponding drop of opposed friction force can be clearlyseen in this history plot of instantaneous friction force at the tipof the pin (Fig. 17).

These results are in perfect qualitative agreement with ex-perimental observations in the zone of self-excited oscillations.However, the actual calculated reduction'of the mean value of thekinetic coefficient of friction with respect to steady state value isabout 10%, which is much smaller than the experimental resultsof Fig. 7. The explanation of this discrepancy is the fact thatthe actual value of the coefficient of friction in the self-excitedzone is extremely sensitive to the mechanical characteristics ofthe system. For example, for a slightly different stiffness of thesupporting arm there occurred no reduction of the kinetic coeffi-cient of friction in the self-excited zone (see Fig. 5 in Aronov etal. [fill.

Our numerical experiments indicate that the actual drop ofthe kinetic coefficient of friction is related to the relative lengthof the micro-stick phase: the longer the stick, the stronger thereduction of the mean friction force. The duration of the stickphase depends in turn on the "strength ~ of instability. which canbe measured by the magnitude of real parts of eigenvalues inequation (6.-1). This fact is confirmed by the example in whichthe stiffness I\'~4-c was changed to the value 2.4 x 106 kg s-2.

This cha.nge made the dynamic instability "stronger" than in theprevious example. In this case the stick phase of microscopicstick-sli p motion \\'as relal.h·ely lon/)I.'I' ;lIId. I:Onsequently, the re-duction of the 'Ippal'cnt kinetic cocllicient of friction was moresignilicant than in the previous example! (compare Figs. 15 andI:)).

It is important to ohser\'e that. in general, there seem to betwo basic factors which limit the unstable growth of oscillationsin the sclf-excited zone. Since t.he instahility is due to couplingbclwt.'ClI normal and rotational modes of the system, the growthof oscillations can be limited by mechanisms that alter one of theabove frequencies. These mechanisms can be, in particular:

27

0000

0.100

0 ....

1110-2

U Ie"'1'_1

8 Influence of the Angle of Attack onthe Stability of Frictional Sliding

figure 19: Reduction of the apparcnt friction in the casc of non-symmetric damping on the interface.

0.500-1--;; .....19

In the preceding sections we have analyzed the behavior of thesystem under the assumption that the angle of attack was equalto zero, which is to say that the surface of the pin is parallel to

the surface of the disk prior to the application of loads. In theexperiment, this condition was assured by bringing the pin intocontact with a revolving disk at a small normal load and allowingit to run for a distance of approximately 12-15 km. This pre-sliding was crucial for the reproductivity of results. The angles ofattack different from zero were not analyzed experimentally.

In the numerical analysis, we also considered various anglesof attack, ranging from _10° to +100

, where the positive anglecorresponds to the clockwise rotation of the slider in Fig. 5 (inactual computation the prescribed rotation was actually imposedon the disk). These angles are within the range of applied theory.in which it was assumed that sin 0;:::; 0 and cos 0;:::; I. At variousangles of attack, the eigenvalue analysis was used to calculateforces corresponding to the onset of the self-excited oscillationand the stability and instability zones were defined for anglcs ofattack ranging from -10° to +I0° and normal forces up to 500 N.

The results obtaineo are presented in fig. 21. which also

TIME Is)-r-O1.000

,0 ....

T0.000

,0.100

~T0.500 0.6C1O

T0._T0.300

T0._0.100

U Ic"'l._,1,1)00

0,700.

0.&00

0.100

0.0000.000

0.000

0.>00

0.100

0.""

0._

0 ....

0.100.

0.:100

sooIIt.IE 1<1

0.100 a,lOG 0.)00 0.400 o.;o.;-;'~.i.oo 0,:.00 \.~.0 ....

0.000

0,100

Figurl: 20: Reduction of the apparent friction in the case ofload-dependent coefficient of friction.

'00

UNSTAIlL.E

.. 300

Figure 21: Dependence of the range of instability zones on theanglc of attack.

sholl'S schematically th,~ configuration of the slider in each ofthe presented zones. Generally the self-excited oscillations oc-cur when the point of contact is ahead of the center of the massof the slider. so that the slider "stumbles". More precisely, insuch a configuration the presence of friction on the interface isthe reason for counterclockwise rotation of the slider, which leadsto the growth of normal force due to inertia of the slider. This in1.111'11 illcreases the friction force and causcs further rotation. This

The above examples show that various parametcrs of the in-terface and the system as a whole affect the final value of thecoefficient of friction in the presence of self-excited oscillations.It can be expeded that a more accurate model of the system, amore realistic model of frictional interface and exact estimates ofall involved parameters should provide better quantitative agree-ment between numerical and experimental results.

The above analysis focused on the modeling of the coefficientof friction only in the unstable zone, where the coupling betweennormal and rotational vibrations was the dominant source of vi-brations. Analysis of the kinetic coefficient of friction in steadysliding will require proper modeling of vibrations due to the in-terlocking of surface asperities, which are the cause of vibrationsin this zone. This problem will not be addressed here.

·6 -5 ' •. J .2 ·1

a

10

28

In this paper, a numerical study of dynamic friction phcnomena,in particular of self-excited oscillatiolls and stick-slip motion. waspresented.

A good qualitative agrcement of numerical and experimentalresult.:; confirms the importance of the mechanical charactcristicsof a system in frictional sliding and the feasibility of the presentednumerical approach. Although wc have achieved quite a goodquantitative correspondence wit.h sclected experimental results,we believe that much more precisc mathematical and numericalmodels will be required to achieve this agreement for a generalclasss of problems.

Identifying the mechanism of self-excited oscillations as thedynamic coupling of normal and rotational oscillations providesinsight into the mechanism of self-excited vibrations and guidancein avoiding thcm in the design of sliding systems.

Although the model analyzed was relatively simple, the ob·servations and methods used in this work can be applied to theanalysis of more complex systems. possibly discretized by the fi-nite element method. In general. a variety of self-excited modes('all be expected in more complex mechanical systems.

coupling between rotational normal motion is the mechanism ofSt~lf-excited oscillations,

On the other hand, if the point of contact is behind the centerof the mass of the slider, the sliding is steady. The reason is thatin this situation the rotation of the slider due to friction causesdecrease of normal and friction forces, so there is no couplingbetween normal and rotational modes,

An especially interesting zone is in the vicinity of the zero an-gie of attack. If this angle is exactly zero, the onset of self-excitedoscillations corresponds to the normal force 185N. However, thisvalue changcs drastically when the slider is rotated, so that atpositive angles of attack greater than 0.0050

, the sliding is steadyat all analyzed loads. This sensitivity to the angle of attack issomewhat stronger than one would intuitively expect in this case(although no corresponding experimental evidence was found).I1owe\'er. the present analysis is bas<.-don a simplified estimate ofthe stability of linearized infinitesimal rotations about the equilib.rium position. As stated previously, if the fully nonlinear systemis analyzed, the instability zone may extend somewhat beyondboundaries established bY'eigenvaluc analysis. In order to verifythis possibility, we performcd full transient analyses for selectedV,"lIt'S of angle of attack in the vicinity of ti = O. These analysesrel'ealed that the instability zone ncar ti = 0 actually extends tovalues of the angle of attack up to 0.05° (one order higher thanfrom Ihe eigcnvalue analysis). This \'alue appears to be morerl'alistic than the r<.'Sultof linearized study.

The study prcsented in this scction confil'llls well known scnsi·I.ivity of stability of frictional sliding 10 the angle of attack. whichis Oill' of thc rcasons for the poor reproductivity of the results ofvarious frictiollal expcrimcnts.

9 Conclusions

29

It should be pointed out that the model of the interface (ageneralization of the Oden/~Iartins model) was relatively simple,with a purely elastic normall'esponse and linear damping termreprc:;cnting dissipation effects on the interface. In future anal-yses. thc mol'l~ general viscoplastic model of interface should beapplied (possibly based on a micromecllanics analysis), Thesemodels should also be combined with appropriate descriptions oflubricants, which reduce the coefficient of friction on the interfacealld introduce a considerable amount of damping.

The introductory analysis of the kinetic coefficient of frictionin the self-excited wne confirms the observation that the differ·ellce between static and kinetic friction is not an intrinsic prop-t:rty of the interface, but also depends on the overall natme ofthe motion of the systems, in part.icular normal vibrations of theslider.

In the context of numerical mooplin!! of frictional vibrationsfor complex mechanical systems it is important to note that theIl'i\nsient analysis of the motion of the slider is extremely compu-tdtionally expensi\'e. because the macroscopic unstable motion ofthe slider (observed in the scale of scconds) results from a com·plicated nature of high frequency I'ibrations of all components ofthe system. Therefore hundreds of thousands of time steps arerequired in order to model e\'en short periods of motion of theslider. For realistic models with higher numbers of degrees offreedom (like Finite Element ~Iodcls), a more efficicnt approachto the time integration will be necessary. otherwise prohibitivelylarge computation times can be expected.

In SUllun:lI'Y, further progress in Ihe modeling of dynamic fric-tion phenomena will require:

- development of nell' phenomenological models of thefrictional interfaces (based on a micro-asperity analy-sis and consistent with experimental results),

- further refinement of numerical techniques used in mod-eling the motion of the sliding systems.

development of a solid mathematical basis for problemsconsidered and numcrical methods applied,

study of thc importancc of lubricants in suppressingself-excited oscillations and stick-slip motion,

flll'ther comparison of numerical results with experi·mClIts, preferably lI'ith joint design of both numericaland experimental aspects.

Acknowledgements

Sponsorship of this effort by the National Science Foundationunder Grant No. MSlvI 84·1614:3 and in part by the Air ForceOffice of Scientific Reesearch under contract F49620-86-C-005l isgratefnlly ackllowledged.

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32