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... Inc.
Typicalmodesofbackplatevibrationinthefirstmodeareshownin
Figure7anditsnodallines,indicatedbydashedcurvesinFigure7,are
shownaloneinFigure8.Figures 9and1 0showabackplatebeforeandafter
araisedportion(Figure11)wasaddedtoreducethefrequencyofthenoise
generated.
III.FINITEELEMENTANALYSIS
Murakami,Tsudada,andKitamura[13]reportedonafiniteelementanalysis
ofautomotivediskbrakestocomparethecalculatedresonancefrequencies
withpreviousmeasurementsofbrakesquealonachassisdynamometerand
toassociatethemwithcalculateddeformationofthebrakecomponents.
Secondarylow-frequencysquealfrom2to3kHzandprimaryhigh-frequency
squealfromabout5.5to10.5kHz,asshowninthehistograminFigure12,
correlatedwellwiththeclusteringoffrequenciesfoundforthebrakedisk,
cylinder,pad,andtorquemember,showninFigure13.Calculateddiskand
caliperfrequencymodeswhereverifiedbyholographicinterferometryandby
accelerometermeasurementsinthecaseofthecaliper,orcylinder.
Althoughtheclusteroffrequenciescorrelatedwiththeprimaryand
secondarysquealregions,severalcomponentresonantfrequenciesbetween3
and5 .5kHzdidnotresultinbrakesquealinthisfrequencyrange.This
impliedthatthedrivingforce,thestick-slipphenomenon,didnotexcitethese
frequenciesandthattheywerenotexcitedbystructuralcoupling.Linearized,
seven-degree-of-freedomanalysisofthesystemusingthelumped-parameter
model,asshowninFigure14,gavesolutionsoftheform
u¼Ae
at
sinðNtþfÞð3-1Þ
foreachofthesevenvariablesinwhichexponenta,whichdeterminedthe
regionsofinstabilityintheJarvisandMillsanalysis,wastermedthesqueal
index.Thisanalysisindicatedthatthesquealindexwasrelatedtothenegative
gradientofthefrictioncoefficient,asillustratedinFigure15,whichalso
showstestresultsfortwotestbrakeliningpads.Theseresultsalsocorrelate
withSpurr’sfindingthatthesquealprobabilityincreasedasthefrictionco-
efficientincreased.Thelowsquealindexbetween3. 0and5 .5kHzanditsslope
alsocorrelatewiththehistogramshowninFigure12,whichimpliesthat
negativeslopeofthefrictioncoefficientversusvelocitycurveisoneof
severalsignificantparametersinthegenerationofbrakesqueal.Analysis
oftheinfluenceofthetorquemember,showninFigure16,indicatedthat
itsinfluencewasalsominimuminthevicinityof3kHz,asshowninFigure
17,whichalsoaccountsforthelowsquealamplitudebetween3. 0and
5.5. ... Inc.
ingsurfacessufficientlyforthecomponentstonearlyreassumetheiroriginal
configurationandtheprocesstorepeat.Intheyearsthatfollowed,many
investigationswereconductedtobetterunderstandtheroleofthefactorsthat
affectedbrakeandclutchvibration.Onlyafewofthemanycontributorsto
theliteratureofbrakevibrationwillbediscussedexplicitlyinthissection.This
partiallistingissufficient,however,toportraythegeneraltrendsintheyears
after1935.
AnexperimentalinvestigationbyHollmann[2],reportedin1954,
indicatedthatthetendencyforfrictionalvibrationincreasedwiththecontact
pressureandthatitalsoincreasedwiththetemperatureofthefrictionma-
terialsupto100jCbutthendecreasedrapidlyasthetemperatureroseabove
100jC.Afterstudiesofbrakesquealonrailwayvehicles,Broadbent[3]also
concludedthatbrakevibrationwasdependentonthemechanicallinkageused
andwasassociatedwithafrictioncoefficientthatdecreasedwithincreasing
velocitybetweenshoeandbrake.Healsoobservedthatnochatterwasfound
whenwoodenbrakeswhosefrictioncoefficientincreasedwithincreasedslip
speedwereused.SimilarresultswerefoundbySinclair[4],whoalsoreported
thatthefrequencyofoscillationwasstronglydependentontheequivalent
massandspringconstantofthemechanicalsystemthatheldthebrakelining
inplaceinthesliderblockconfigurationusedinthelaboratorymodel.
Spurr[5]rejectedthenotionthatitwasnecessaryforthefriction
coefficienttodecreasewithincreasingslipspeedinordertohavefrictional
vibration.Hisexperimentswithrailwayblockbrakesonarailwaywheelin-
dicatedthatvibrationwasindependentofthechangeinthefrictioncoefficient
withslipspeedbutthatitwasmorelikelyifthefrictioncoefficientwaslarge.
Thestick-sliptheoryoffrictionappearedtoholdasthedrivingmeans,
however,andSpurralsoobservedthatthefrequencyofoscillationdepended
ontheassociatedmechanicalsystemusedtoforcethebrakeblockagainstthe
wheel.
InthediscussionofSpurr’spaper,F.R.Murrayintroducedwhathas
beencalledthespragtheoryinBritishliterature,involvingasprag(acan-
tileveredbeampressedagainstamovingsurface),asshowninFigure1.A
somewhat ... Inc.
Typicalmodesofbackplatevibrationinthefirstmodeareshownin
Figure7anditsnodallines,indicatedbydashedcurvesinFigure7,are
shownaloneinFigure8.Figures 9and1 0showabackplatebeforeandafter
araisedportion(Figure11)wasaddedtoreducethefrequencyofthenoise
generated.
III.FINITEELEMENTANALYSIS
Murakami,Tsudada,andKitamura[13]reportedonafiniteelementanalysis
ofautomotivediskbrakestocomparethecalculatedresonancefrequencies
withpreviousmeasurementsofbrakesquealonachassisdynamometerand
toassociatethemwithcalculateddeformationofthebrakecomponents.
Secondarylow-frequencysquealfrom2to3kHzandprimaryhigh-frequency
squealfromabout5.5to10.5kHz,asshowninthehistograminFigure12,
correlatedwellwiththeclusteringoffrequenciesfoundforthebrakedisk,
cylinder,pad,andtorquemember,showninFigure13.Calculateddiskand
caliperfrequencymodeswhereverifiedbyholographicinterferometryandby
accelerometermeasurementsinthecaseofthecaliper,orcylinder.
Althoughtheclusteroffrequenciescorrelatedwiththeprimaryand
secondarysquealregions,severalcomponentresonantfrequenciesbetween3
and5 .5kHzdidnotresultinbrakesquealinthisfrequencyrange.This
impliedthatthedrivingforce,thestick-slipphenomenon,didnotexcitethese
frequenciesandthattheywerenotexcitedbystructuralcoupling.Linearized,
seven-degree-of-freedomanalysisofthesystemusingthelumped-parameter
model,asshowninFigure14,gavesolutionsoftheform
u¼Ae
at
sinðNtþfÞð3-1Þ
foreachofthesevenvariablesinwhichexponenta,whichdeterminedthe
regionsofinstabilityintheJarvisandMillsanalysis,wastermedthesqueal
index.Thisanalysisindicatedthatthesquealindexwasrelatedtothenegative
gradientofthefrictioncoefficient,asillustratedinFigure15,whichalso
showstestresultsfortwotestbrakeliningpads.Theseresultsalsocorrelate
withSpurr’sfindingthatthesquealprobabilityincreasedasthefrictionco-
efficientincreased.Thelowsquealindexbetween3. 0and5 .5kHzanditsslope
alsocorrelatewiththehistogramshowninFigure12,whichimpliesthat
negativeslopeofthefrictioncoefficientversusvelocitycurveisoneof
severalsignificantparametersinthegenerationofbrakesqueal.Analysis
oftheinfluenceofthetorquemember,showninFigure16,indicatedthat
itsinfluencewasalsominimuminthevicinityof3kHz,asshowninFigure
17,whichalsoaccountsforthelowsquealamplitudebetween3. 0and
5.5....