122 Noise Control Eng. J. 53 (4), 2005 Jul–Aug

Analysis of friction-induced vibration leading to “EEK” noise in a dry friction clutch

Pierre Wickramarachia), Rajendra Singha), and George Baileyb)

(Received 2004 March 29; revised 2005 May 09; accepted 2005 August 19)

This article deals with the “EEK” sound that may occur during engagement of a dry friction clutch in vehicles with manual transmissions. Experimental data showed that near the full engagement point of the clutch, the pressure plate suddenly starts to vibrate in the rigid-body wobbling mode with a frequency close to the first natural frequency of the pressure plate and clutch disk cushion. The self-excited vibration exhibits typical signs of a dynamic instability associated with a constant friction coefficient. A linearized lumped-parameter model of the clutch was developed to explain the physical phenomenon. The effect of key parameters on system stability was examined by calculating the complex eigensolutions. Results of the analytical study were in agreement with experimental observations. For instance, it was observed that the instability of the rigid-body wobbling mode was controlled by friction forces. This mode may, however, be also affected by the first bending mode of the pressure plate. A stiffer plate could lead to an improved design with a reduced tendency to produce the “EEK” sound. © 2005 Institute of Noise Control Engineering.

Primary subject classification: 75.6; secondary subject classification: 11.8.2

1 IntroductIon

Asquealingsoundmaybeproducedduringthelastpartoftheengagementofadryfrictionclutchinvehicleswithmanualtransmissions.Thestronglyannoyingsoundofthetransientevent is known as “EEK” noise, which may dominate anoverallperceptionofvehiclesoundquality.Someconsumersmight even prematurely change the clutch in an effort toeliminateorreducethedisturbingnoise. Figure1illustratesatypicalarrangementofapressureplateandaclutchdiskusedtotransmittorquefromtheenginetothetransmissionandultimatelytothevehicle’sdrivenwheels.Depressingtheclutchpedal(notshown)movesathrow-outbearing(notshown)towardadiaphragmand,inturn,causes

the pressure plate to move away from the clutch disk intothedisengagedpositionwheretheengineisuncoupledfromthetransmission.Lettingupontheclutchpedalallowsthepressure plate to squeeze the clutch disk against the flywheel mountedontheengine’scrankshaftintheengagedposition.Acriticalpositionisachievedatthepointof“liftoff”where,inreactiontotheforceappliedbythethrow-outbearing,thepressureplateisjustdisengagedfromtheclutchdisk. Despitemanyexperimentalstudies,thisfriction-induced“EEK”sound(likemanyotherfrictionproblems[1])remainedpoorly understood. One hypothesis was that the “EEK”sound was triggered by a dynamic instability of the sub-systemcomposedoftheclutchpressureplateandtheclutchdiskcushion.Ithadbeenobservedexperimentally[2]that,duringthe“EEK”event,thepressureplatevibratedinarigid-body wobbling (out-of-plane) motion corresponding to thenaturalfrequencyofthepressure-plate/disk-cushionsystem.Experimentalstudiesshowedthatthemodalpropertiesofthepressureplate,asaresultofthefabricationprocess,seemedto play a significant role in the mechanism(s) that cause the “EEK”sound.Toclarifysomeoftheseissues,theanalyticalmodel described in this article was developed and used toidentifykeyparametersaffectingthestabilityofadry-friction-clutchsystem.

2 EXPErIMEntAL MEASurEMEntS And ProBLEM ForMuLAtIon

Twoconditionsaretypicallynecessarybeforethetransient“EEK”sound(Figure2)canbeinduced. First,theenginespeedshouldbewithinthe1500to2500rev/minrange.Thetransmissionspeeddoesnotseemtohavea significant effect. Since the slip speed is relatively high, a constant coefficient of friction µat thethresholdof“EEK”maybeassumed.Theseconditionssuggestthatthe“EEK”sound may not follow the symptoms of classical stick-slipphenomenon[1].

a) Acoustics and Dynamics Laboratory, Center for Automotive Research, The Ohio State University, Columbus, Ohio, 43210, USA; e-mail: pierre@dataphysics.com, singh.3@osu.edu.

b) LuK, Inc., 3401 Old Airport Road, Wooster, Ohio 44691, USA; e-mail: George.Bailey@luk-us.com

Wickramarachi - 1

Cover

Liftoff

Disk

Flywheel

Diaphragm

Pivot rings

Pressure plate

Cam

Tab

Bearing

force

Engaged Disengaged

Fig. 1—Sections of a pressure-plate clutch-disk system in engaged and disengaged stages.

123Noise Control Eng. J. 53 (4), 2005 July–Aug

Second, the clutch pedal motion (or throw-out bearingtravel)hastobesuchthattheclutchisclosetofullengagementattheonsetofthe“EEK”sound.Figure2showsthisparticularposition(wherethe“EEK”soundbeginstooccur)wherethepressureplateandclutchdiskcushionarealreadyincontact,butnotyetfullyengaged.ThetwoverticaldashedlinesinFigure2indicatethetimest

1andt

2attheonsetandcessation

ofthestrong“EEK”soundasnotedbytherapidincreaseanddecreaseofthesoundpressure. Measurementsofthespectraofthe“EEK”soundexhibitadominantfrequencyf

w,relatedtothebearingtravelposition,of

approximately450Hzto500Hz,asshownatvariousrelativetimesinFigure3(a).Thisfrequencyhasbeencorrelatedto

Wickramarachi - 2

Sound pressure

Engine speed

Disengagement

Transmission speed

Bearing travel

“EEK”

threshold

“EEK”

Engagement

Full engagement

Fig. 2—“EEK” sound as measured in the interior of a vehicle as a clutch is being engaged and then disengaged as corre-lated with the variation of engine speed and the position of the clutch throw-out bearing.

Wickramarachi - 3a

1100

1000

900

800

700

600

500

400

300

Elapsed time (s)

Dominant“EEK”frequency ƒw

Freq

uenc

y (H

z)

t2

4 5.1

t1

Wickramarachi - 3b

Frequency (Hz)

Mag

nitu

de (

V)

0 200 400 600 800 1000 1200 1400 1600

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

ƒb

ƒw 2ƒw 3ƒw

Wickramarachi - 3c

Frequency (Hz)

Mag

nitu

de (

V)

0 200 400 600 800 1000 1200 1400 1600

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

ƒw 2ƒw 3ƒw

ƒb

Fig. 3—Spectra of the “EEK” sound; (a) Principal frequency components as the clutch is engaged; (b) Magnitude of the spectral components of “EEK” sound as measured by a spectrum analyzer before the full engagement of the clutch at relative time t

1, where the component at

the fundamental frequency fb of the bending mode was

dominant at approximately 950 Hz; (c) at time t2, where

the component at the fundamental frequency fw of the

wobbling mode was dominant at approximately 500 Hz. [Vertical lines indicate the fundamental, second harmon-ic, and third harmonic of the wobbling-mode frequency in (b) and (c).]

(a)

(b)

(c)

124 Noise Control Eng. J. 53 (4), 2005 Jul–Aug

the wobbling rigid-body motions of the pressure plate (inboth out-of-plane rotations) from dynamometer tests [2]and from a single-degree-of-freedom (SDOF) model thatincludedtheinertiaofthepressureplateandthestiffnessoftheclutchdiaphragmcoupledwiththestiffnessoftheclutchcushion. The pressure plate and clutch disk cushion wereassumedtobethemaincomponentsthatcontrolthe“EEK”phenomenon. This assumption was reinforced by manyempiricalexperimentsthatshowedthatelementsexternaltotheclutch(gears,releasesystem,enginepulsations,etc.)aswellasinternaltotheclutch(diaphragmandcover)donotsignificantly influence the tendency to produce the “EEK” sound[2]. Further,achangeinthepressureplatematerialfromgray-iron(PressurePlateG)tovermiculariron(PressurePlateV)seemedtoreducetheoccurrenceof“EEK”.Suchachangeofmaterialcouldalterthemodalpropertiesofthepressureplate,asfoundinsomebrake-squealstudies[3,4]. ExperimentalmodalstudiesshowedthatapressureplatemadefromPlateVmaterialisessentiallystifferthanapressureplatemadefromPlate G material. Thus, the influence of pressure-plate elastic deformationsneededtobeinvestigated. Fromexperimentalmodalanalyses,thesecondfrequencyfb,between900Hzand1000HzasshowninFigure3(a),could

be linked to the first bending mode of the plate. Figure 3(b) showsthattheprominenceofthisbendingmode(atfrequencyfb)atrelativetimet

1isquicklyreplacedbythedominanceof

thewobblingmode(atfrequencyfwandharmonics)0.3slater

atrelativetimet2inFigure3(c).

Thisarticledevelopsapreliminarysimulationmodelfortheclutchassembly.Atthethresholdofthe“EEK”sound,vibration amplitudes were considered sufficiently small to buildalinearizedmodelthatwouldreproducethetrendsofthe observed dynamic instabilities. An analytical lumpedparameter approach was preferred over the structural finite-elementmethod.Theanalyticalmodelexaminesaself-excitedsystemwithoutanyexternaltorqueexcitations.Thepressureplatewasassignedan initialangle inorder toexamine thegrowthofvibrationamplitudewithtimeanddemonstratetheexistenceofdynamicinstability.

3 AnALytIcAL ModEL

ThelumpedparametermodelofFigure4incorporatestheinertiaofadeformablepressureplatealongwiththeclutchcushion stiffness k

c that was divided into four springs of

stiffnesseskxf,k

xr,k

yf,andk

yr,wheresubscriptsxandyindicate

coordinateaxesandfandrindicatethedirectionofthemotiontowardtheforwardandrearofthepower-transmissionaxis,respectively. Figure 4(a) shows the model for the fixed-base flywheel andtheclutchdisk.The+z-axispointstowardtheengineandisalignedwiththeaxisofthecrankshaft. Theclutchdisk,Figure4(b),wasmodeledasfourmasses,m

f,atsegmentsA,

B,C,andDandconnectedbyfourspringsofstiffnesskf.

In order to consider the nonlinear characteristic of theclutch-diskcushion,thestiffnessesk

yfandk

yrwererespectively

divided and multiplied by a non-dimensional amplitude ratio λ,

asshowninFigure5andinthefollowing,wherethestiffnessofanindividualspringwasassumedtoequalonefourthofthetotalclutchstiffnessk

c.

Assume that the bearing travel is such that the “EEK”thresholdpoint(PinFigure5)isreached.Thisinstantwouldcorrespond to a constant plate liftoff υ = υ

“EEK”. Liftoff

correspondstothedisplacementoftheplatefromitsengagedpositiontoadisengagedposition.Nowsupposethattheplateisalreadyvibratingaboutthex-axiswithamplitudeθ

x0.This

Wickramarachi - 4

kyf

x

z

x

y z

kf

kf

kf

kf

mf

mf

mf

mfzxf

zyr

zxr

zyf

Fixed-base flywheel

Friction surface

(from disk) a)

Pressure plate

f = forward

r = rear

a)b)b)

A

B

C

D

y

mfkf

kf

mf

kf

kf

mf

mf

y z

x

ããã

y

kxr

kxf

x

kyr

zyf

zxf

zxr

zyr

Fig. 4—Schematic of the analytical model for a clutch-disk/pres-sure-plate assembly; (a) two rotational degrees-of-freedom θ

x and θ

y about the x and y axes, respectively,

stiffnesses shown for the forward and rear directions, pressure-plate masses at points A, B, C, and D; (b) four translational degrees-of-freedom in the z direction for the pressure-plate masses m

f coupled by stiffnesses k

f.

125Noise Control Eng. J. 53 (4), 2005 July–Aug

vibrationwillchangethenormalspringforcesatlocationsAandBasshowninFigure5. Consequently, thestiffnessesk

yf and k

yr were obtained by assuming a constant stiffness

slopearoundpointP.ThedifferentspringforcesareusedinEquations(1)and(2).

c

r

4y

kk

λ=

(1)

c

f

4y

kk λ=

(2)

The ratio λ could be used to incorporate an asymmetry into the resulting stiffness matrix. For instance, λ = 0.9 in Equations(1)and(2)allowsasmalldegreeofasymmetry. Friction forces from the clutch disk cushion were alsoincluded by assuming a constant coefficient of friction µduetoahighslipspeed(>700rev/min).Forthesixdegree-of-freedom(DOF)modelofFigure4, twoDOFsdescribe therigid-body“out-of-plane”rotations(wobblingmode)andfouradditionaltranslationalDOFs(normaltothepressureplate)depict the first two nodal-diameter bending modes of the pressureplate.Bychoosinganappropriatelumpedbendingstiffnessesk

f, thisbendingmodewascreated inourmodel

withafrequencyclosetotheobservedfrequencyfb.Setting

oneofthemodalmasses,mfinFigure4(b)tocorrespondtoa

quarterofthetotalplatemasswasbelievedtobeareasonableapproximationofthemodalmass. Next,thefree-bodydiagramsaredrawnasshowninFigure6.Here,eventhoughitisrepresentedasarigidbodyforthesakeofclarity,thepressureplatecanalsoelasticallydeformasexplainedbefore. Thenormalspringforces(N)andthecorrespondingfrictionforces(F)arethereforeexpressedasfollowsalongthexandyaxesandintheforwardandrearwarddirections:

f f f( )x x x yN k z rθ= −

(3)

r r r( )x x x yN k z rθ= +

(4)

f f f( )y y y xN k z rθ= +

(5)

r r r( )y y y xN k z rθ= −

(6)

Wickramarachi - 5

A

C B

Dy

x

θ xo

z

Akc/4λ

λ kc/4B

υEEK υ

kc/

4

P

Fig. 5—Nonlinear model of the stiffness of the clutch-disk cushion as a function of the liftoff position where the “EEK” sound begins to occur.

Wickramarachi - 6

y

z

θ

r

x

z

View on YZ plane

View on XZ plane D

B

A

C

1 2ìï

í

ó

óìïí

ο⊗

•

ο

⊗

•

x

Fyf

Nyf

Fxr F

xf

Nyr

Fyr

θy

Fxr

Nxr

Fyf

Fyr

Fxf

Nxf

Fig. 6—Free-body diagrams for the vibration modes of the pres-sure plate.

f fx xF Nµ= (7)

r rx xF Nµ= (8)

f fy yF Nµ=

(9)

r ry yF Nµ=

(10)

whereristheradiusofthepressureplate,θisarotationaboutthexoryaxis,andµ is the coefficient of friction. Theresultinggoverningequationsforalinear,undampedandunforcedsystemwereobtainedinamatrixformintermsofthemass[M]andstiffness[K]matricesandthegeneralizedcoordinatevector{X}. [ ]{ } [ ]{ } {0}M X K X+ =&&

(11)

where [ ]

T

x y xf xr yf yrX z z z zθ θ= (12)

([ ])x y f f f fM diag I I m m m m= (13)

and

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

2

2 2

2 2

0 2 0

0 0 2

0 2 0

0 0 2

yf yr xf xr xf xr f yf yr f

yf yr xf xr xf f f xr yf yr

xf f xf f f

xr f xr f f

yf f f f yf

yr f f f yr

r k k lr k k r k k r k k

lr k k r k k r k k r k k

rk k k k kKrk k k k k

rk k k k k

rk k k k k

µ

µ

+ + −Γ −Γ + − − − + + − − + −Γ −Γ

− + − − =

+ − − − − +

− − − +

(14)with ( )f f f f

2 2x x

rk l k kµΓ = + + (15)

( )r f f r2 2

x xrk l k kµΓ = − + (16)

l

l

126 Noise Control Eng. J. 53 (4), 2005 Jul–Aug

forthestablemode(representedbytheeigenvectorψ1)andin theoppositedirectionforanunstablemode(denotedbyeigenvectorψ2).Thisresultmeansthattherotationaboutthez-axis(denotedasdynamicmotioninFigure9)actuallyrotatesasafunctionoftimeandinadifferentdirection.Hence,thefrictionforcesmayenhancetheenergyofanunstablemode;

Wickramarachi - 7

UNSTABLE STABLEσ < 0

f 2

f 1

f c

µ ↑

f

σ > 0

Fig. 7—Two eigenvalues of a pressure plate that could be coupled by varying the coefficient of friction µ. The index of dynamic stability σ (real part of the eigenvalues) becomes negative for the stable wobbling mode (dashed line) and positive for the unstable wobbling mode (solid line).

Wickramarachi - 9

0.0015

-0.0015

00

0.05

-0.05 0.05

-0.05

0ψ ψ

Static equilibrium Dynamic motion

Frictionforces

Platerotation

x/d

y/d

z/d 1 2

Wickramarachi - 8

Modes are decoupled

10

5

0

-5

-10

350

340

330

320

310

Plate G Plate V

Rea

l par

t σ

Pressure plate stiffness kf (MN/m)

Freq

uenc

y (H

z)

5 5.5 6 6.5 7 7.5

ωaωb

ωc

a)

b) 5 5.5 6 6.5 7 7.5

Fig. 8—Components of the predicted eigenvalues for the wobbling modes of the pressure plate; (a) Index of dynamic stability σ or real part of the eigenvalues for Plate G and Plate V, (b) natural frequency ω

s of the modes (in hertz), which

exhibits two distinct values ωa and ω

b for high pressure-

plate stiffnesses kf and a single value ω

c for low pressure-

plate stiffnesses.

Fig. 9—Visualization of the complex wobbling modes of the pres-sure plate for a coefficient of friction µ = 0.35. These modes occurred at the natural frequency f

c of the clutch-

disk cushion and illustrate the directions of the traveling waves induced by modes ψ

1 and ψ

2, moving with the axis

of rotation indicated by the dynamic-motion line. Mode ψ

1 is a stable mode associated with waves that move in

the direction of plate rotation; mode ψ2 is an unstable

mode associated with waves that move in the direction of the friction forces. The axes were normalized for this sketch.

( )f f f f

2 2y yrk l k kµΓ = + +

(17)

( )r f f r

2 2y yrk l k kµΓ = − +

(18)

4 StABILIty StudIES And dESIgn guIdELInES

The asymmetric nature of [K] is controlled by thecoefficient of friction µ and amplitude ratio λ and yieldscomplexeigenvalues

~λs = σs±ωs,wheretheimaginarypart

ωsisthedampednaturalfrequency(inrad/s)ofmodeindexsandtherealpartσsisanindexofdynamicstability.Whenσsispositive,theamplitudeofvibrationgrowsexponentiallyintime.Theeffectofµon

~λsisillustratedinFigure7.Two

distinctwobblingmodeswithfrequenciesf1andf

2(inHz)can

be seen,where thedifferencebetween the two frequencieswas found tobe related to the ratioλ for lowvaluesofµ.Nevertheless,asµisincreased,thecouplingbetweenthetwomodesisenhanced.Itresultsintwomodes(onestableandoneunstable)atµ = 0.35 (a typical value for clutch disk cushions orlinings).Essentially,frequenciesf

1andf

2mergetobecome

thesamefrequencyfc.

The effect of the pressure-plate structural stiffness kf

is investigated in Figure 8, which diagrams the naturalfrequenciesandrealpartsofthewobblingmodes.Ask

fis

increased,onepassesfromanunstableregion(forPlateG)toastableregion(forPlateV).Theseresultsareinagreementwithexperimentalobservations[2].Therefore,Young’smodulusdoes have a significant impact on the stability of a pressure-plateclutch-disksystemandconsequentlyonthegenerationofthe“EEK”sound. Figure 9 shows an animation of the complex wobblingmodesthatwasdevelopedtovisualizethephasedifferences.Asthepressureplatewobbles,acircleindicatesthelocationalongtheedgethathasthemaximumpositiveamplitude(inthez-direction).Atµ = 0.35, this circle actually moves along the edge of the plate in the direction of the friction forces

l

l

127Noise Control Eng. J. 53 (4), 2005 July–Aug

conversely,theymaydissipatetheenergyofastablemode.For coefficients of friction less than 0.35, the axis of rotation remains stationary, showing the absence of any couplingbetweenθ

xandθ

y.

Stability diagrams are presented next to summarize theeffectsofpressureplategeometry(innerandouterradiir

iand

ro,andthickness2l ),pressureplatestructuralstiffness(k

f)

anddiskcushioncharacteristics(µandkc).

First, a diagram in terms of ri and r

o is displayed in

Figure 10. Defining δ = 0.5(ro-r

i),theradiiwerevariedfrom

theirbaselinevalueminusδtotheirbaselinevalueplusδ.Theincreasing degrees of instability (described by the real parts σsoftheunstableeigenvalues)arerepresentedgraphicallyandgroupedfromalowtoamediumtoahighdegreeofinstability.Twostraightlinesindicatethebaselineradii.Itwasfoundthatlargerradiiwouldresultinincreasedstability.However,outerradiusr

0 seemed to have a greater influence than the inner

radiusri.Forpracticalpurposes,itwouldbemoreconvenient

tomaintainthesamecontactareabetweenthepressureplateandthedisk.Asmallersurfaceisnotrecommendedsinceitwouldproducemoreheat and reduce the lifeof theclutchdisk.Moreover,theheatcapacityoftheclutchisalsoaffectedbythepressureplatemass.Basedontheseconsiderations,itseemedthatr

0shouldbeincreased,sincea15%increasewould

theoreticallyresultinamorestablesystem.Inaddition,thischangewouldincreasethetorquecapacityoftheclutch. Second, thestabilitydiagramofFigure11shows thatathickerpressureplatewouldresultinreducedstability.Thisisconceivablesinceplate thickness2l controls the torquegeneratedbythefrictionforces.Thistorqueisresponsiblefor

thewobblingmode,becauseitcouplesangulardisplacementθ

xwithangulardisplacementθ

y.Interestingly,a10%decrease

inplatethicknesswouldmakethesystemstable.Aswiththepreviousdesignchangeproposedfortheouterplateradiusr

o,

oneneedstoconsidertheconsequencesofathinnerplateintermsofclutchlifeandperformance.Forinstance,achangeinplatethicknesswouldaffectthestiffnessk

f,whichisrelated

tothebendingmodeofthepressureplate. Figure 12 compares the influence of the coefficient of frictionµonthestiffnessofPlatesGandV. ThepressureplateoftheplateVmaterialwasstableforµ<0.45.FortheplatemadefromPlateGmaterial,decreasingµby10%(from0.35to0.31)wouldmakethesystemstable.Onceagain,thischangemaycauseproblemsintermsofclutchperformancebecause the transmitted torquewouldbe also reduced. Tocompensateforthiseffect,theclampload(normalloadatfullengagement)wouldhavetobeincreasedaccordingly. Finally, thestabilitydiagramofFigure13wasobtainedintermsofpressure-platestiffnessk

fandclutch-diskcushion

stiffnesskc.Thelatterstiffnesswasvarieduptoitsmaximum

value at full engagement. In parallel, the eigenvalueswere computed via an iterative process to determine thecorrespondingvaluesfork

yrandk

yfateachpointontheplot.

Figure10shows the lossof stabilityas theclutch isbeingengaged.Hereitoccursaroundk

c ≈ 26 MN/m for plate G,

whereastheinstabilityregionisavoidedwithplateVatthesame cushion stiffness. Such predictions match empiricalobservations[2]fairlywell.

Wickramarachi - 10

55 65 75 85 95 r i (mm)

r o (

mm

)

130

120

110

100

Low

Med

ium

H

igh STABLE

Fig. 10—Regions of stable and unstable wobbling modes of the pressure plate for a range of the inner radius, r

i, and

outer radius, ro. Plate thickness, structural stiffness, and

coefficient of friction were held constant. The vertical and horizontal lines indicate baseline radii. The indices of dynamic stability (the real part of the eigenvalue for the wobbling modes) were organized in three categories: Low to Medium to High.

Wickramarachi - 11

5 7 9

Plate stiffness kf (MN/m)

Thi

ckne

ss 2l (

mm

)

20

16

12

8

Low

Med

ium

H

igh

STABLE

Plate V

Plate G

Fig. 11—Regions of stable and unstable wobbling modes of the pressure plate for a range of plate thickness, 2l , and plate stiffness, k

f. Baseline inner and outer radii and coef-

ficient of friction were held constant. The horizontal line represents the baseline plate thickness. The indices of dynamic stability (the real part of the eigenvalue for the wobbling modes) were organized in three categories: Low to Medium to High.

128 Noise Control Eng. J. 53 (4), 2005 Jul–Aug

5 concLuSIonS

Theanalyticalstudyreportedinthispaperyieldedmuchinsightintothemechanism(s)of“EEK”noiseindryfrictionclutches. Specifically, the friction-induced vibration of the pressure-platedisk-cushionsystemwascorrelatedwithalossofdynamicstability.Theanalyticalmodelincludesasixdegree-of-freedomlumped-parametersystemthatdescribesrigid-bodywobbling modes and the first elastic deformation mode of the pressureplate.Thecomplexeigenvaluemethodwasutilizedto study the system stability in terms of the coefficient of frictionbetweenthepressureplateandthecushionorliningoftheclutchdisk,pressure-plategeometry,andthestructuralstiffnessofthepressureplate.Thelatterwasfoundtobeakey parameter. By controlling the frequency of the first elastic deformation mode, the coupling with the wobbling modecouldbealtered. This instabilitymechanismand itseffectseemedtocorrelatewellwithempiricalobservations.Futureworkshouldfocusondevelopinganenhancedanalyticalbasis(includingnonlinearmodels)tobetterunderstandthe“EEK”phenomena.

Wickramarachi - 12

5 7 9

Plate stiffness kf (MN/m)

Coe

ffic

ient

of

fric

tion

µ

0.45

0.35

0.25

Low

Med

ium

Hig

h

STABLE

Plate V

Plate G

Fig. 12—Regions of stable and unstable wobbling modes of the pressure plate for a range of the coefficient of friction, µ and plate stiffness, k

f. Baseline inner and outer radii and

plate thickness were held constant. The horizontal line represents the baseline plate thickness. The vertical lines represent the structural stiffness of Plate G and Plate V. The indices of dynamic stability (the real part of the ei-genvalue for the wobbling modes) were organized in three categories: Low to Medium to High.

6 AcKnowLEdgMEntS

WethankToddRookofGoodrichCorp.andtheengineeringteamatLuK,Inc.,fortheirvaluablehelpinsolvingpartsofthe“EEK”mystery.

7 rEFErEncES[1] A. J. McMillan, “A Non-linear Friction Model for Self-excited

Vibrations,”J.SoundVib.205(3),323-335(1997).[2] ReviewofexperimentalstudieswiththeLuK,Inc.engineers(2000-

2001).[3] S.W.Kung,K.B.Dunlap,andR.S.Ballinger,“ComplexEigenvalue

Analysis for Reducing Low Frequency Brake Squeal,” Society ofAutomotiveEngineers,SAEPaper2000-01-0444(2000).

[4] D.J.FeldandD.J.Fehr,“ComplexEigenvalueAnalysisAppliedtoanAircraftBrakeVibrationProblem,”AmericanSocietyofMechanicalEngineers,ASMEDesignEngineeringTechnicalConferences,DE-Vol84-1,Volume3,PartA(1995).

Wickramarachi - 13

5 6 7

Plate stiffness kf (MN/m)

Cus

hion

stif

fnes

s k

c (M

N/m

)

55

45

35

25

15

Low

Med

ium

Hig

h

STABLE

k c at point P

Plate V

Plate G

Fig. 13—Regions of stable and unstable wobbling modes of the pressure plate in an analysis that simulated clutch en-gagement for a range of the stiffness, kc, of the clutch-disk cushion and plate stiffness, k

f. Baseline inner and outer

radii, coefficient of friction, and plate thickness were held constant. The horizontal line represents the clutch-disk stiffness at Point P, as shown in Figure 5. The vertical lines represent the structural stiffness of Plate G and Plate V. The indices of dynamic stability (the real part of the eigenvalue for the wobbling modes) are organized in three categories: Low to Medium to High.