Damping Optimization in Simplified and Realistic Disc Brakes

57 Pages·2.227 MB·English

by  Hagedorn, Peter, Jekel, Dominic, Sampaio, Rubens, Wehner, Jan-Hendrik

SPRINGER BRIEFS IN
APPLIED SCIENCES AND TECHNOLOGY
Jan-Hendrik Wehner · Dominic Jekel
Rubens Sampaio · Peter Hagedorn
Damping
Optimization in
Simplified and
Realistic Disc
Brakes
SpringerBriefs in Applied Sciences
and Technology
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Jan-Hendrik Wehner Dominic Jekel
(cid:129)
Rubens Sampaio Peter Hagedorn
(cid:129)
Damping Optimization
fi
in Simpli ed and Realistic
Disc Brakes
123
Jan-Hendrik Wehner RubensSampaio
Weinheim Department ofMechanical Engineering
Germany Pontifical Catholic University ofRio
RiodeJaneiro
Dominic Jekel Brazil
Dynamics andVibrations Group
Technical University of Darmstadt PeterHagedorn
Darmstadt, Hessen Dynamics andVibration Group
Germany Technical University of Darmstadt
Darmstadt, Hessen
Germany
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefs inApplied SciencesandTechnology
ISBN978-3-319-62712-0 ISBN978-3-319-62713-7 (eBook)
DOI 10.1007/978-3-319-62713-7
LibraryofCongressControlNumber:2017946042
©TheAuthor(s)2018
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Acknowledgements
The support through DFG HA 1060/55-1, Ingenieurgesellschaft für technische
Software mbH (INTES), and Dr.-Ing. h.c. F. Porsche AG is gratefully
acknowledged.
v
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
References.. .... .... .... ..... .... .... .... .... .... ..... .... 2
2 Theoretical Background .. ..... .... .... .... .... .... ..... .... 3
2.1 Linearization of Nonlinear Equations of Motion.. .... ..... .... 3
2.2 Time-Invariant MDGKN-Systems. .... .... .... .... ..... .... 4
2.3 First-Order Systems .. ..... .... .... .... .... .... ..... .... 5
2.4 Time-Periodic Systems and FLOQUET Theory .... .... ..... .... 5
2.5 Optimization of Damping... .... .... .... .... .... ..... .... 7
2.5.1 Time-Invariant Systems... .... .... .... .... ..... .... 7
2.5.2 Time-Periodic Systems ... .... .... .... .... ..... .... 7
2.6 Linear Damping Models.... .... .... .... .... .... ..... .... 8
2.6.1 COULOMB Damping .. .... .... .... .... .... ..... .... 9
2.6.2 Viscous Damping ... .... .... .... .... .... ..... .... 10
2.6.3 Structural Damping.. .... .... .... .... .... ..... .... 10
2.7 Modal Reduction .... ..... .... .... .... .... .... ..... .... 12
2.8 Brake Squeal ... .... ..... .... .... .... .... .... ..... .... 13
References.. .... .... .... ..... .... .... .... .... .... ..... .... 13
3 Optimization of a Minimal Model of Disc Brake ... .... ..... .... 17
3.1 Equations of Motion.. ..... .... .... .... .... .... ..... .... 17
3.2 Optimization Technique .... .... .... .... .... .... ..... .... 19
3.3 Optimization Results.. ..... .... .... .... .... .... ..... .... 20
3.3.1 Time-Invariant Model.... .... .... .... .... ..... .... 20
3.3.2 Time-Periodic Model .... .... .... .... .... ..... .... 24
3.3.3 Discussion.... ..... .... .... .... .... .... ..... .... 26
3.4 Traps and Shortcomings of CEA . .... .... .... .... ..... .... 26
References.. .... .... .... ..... .... .... .... .... .... ..... .... 30
vii
viii Contents
4 Optimization of Finite Element Models of Disc Brakes .. ..... .... 31
4.1 Theoretical Background .... .... .... .... .... .... ..... .... 31
4.2 Low-Degree-of-Freedom Model.. .... .... .... .... ..... .... 36
4.2.1 Optimization Results. .... .... .... .... .... ..... .... 36
4.2.2 Discussion.... ..... .... .... .... .... .... ..... .... 39
4.3 High-Dimensional Industrial Model ... .... .... .... ..... .... 40
4.3.1 Optimization Results. .... .... .... .... .... ..... .... 42
4.3.2 Discussion.... ..... .... .... .... .... .... ..... .... 47
References.. .... .... .... ..... .... .... .... .... .... ..... .... 47
5 Conclusion. .... .... .... ..... .... .... .... .... .... ..... .... 49
Chapter 1
Introduction
Inmechanicalengineering,self-excitedvibrationsareusuallyunwantedandsome-
timesdangerous.Afamousexampleisthesquealingnoiseofautomotivediscbrake
systems.Thephysicaloriginofthisphenomenonisduetofrictionforcesinthecon-
tactinterfacebetweenthebrakepadsandtherotatingbrakediscforcingthesystemto
barelyvisiblebutaudiblehighfrequencyvibrations[1].However,maybecontraryto
thepublicopinion,brakesquealisnoindicatorofafaultybrake.Evenifthesquealing
noise is a nuisance, the brake system may be in a perfect technical condition with
thebrakingperformanceensured[2].Nevertheless,theautomotiveindustryaimsto
avoid the occurrence of high-frequency vibrations for comfort reasons. Therefore,
finiteelement(FE)modelsareusedtostudybrakesunderdifferentaspectsincluding
the influence of damping on squeal noise [3, 4]. These models may have several
hundred thousand or even millions of degrees of freedom (DOF) and are about to
reflecttherealityinareasonablemanner.
It is well known that the structure of the damping matrix plays an important
role in self-excited mechanical systems [5, 6]. Since the physics of damping is
the most unknown part in large finite element models, damping is often assumed
to be Rayleigh damping, i.e. D = αM + βK. Although this type of modeling
the damping matrix is not done on physical grounds but keeps the eigenvectors of
the MDK-system real, experimental tests identify parameters α and β describing
thedampingaccordingtodifferentsituations.However,sincearealisticautomotive
brakesystemcontainsvariouscomponentsanddampingorigins,Rayleighdamping
isinsufficienttoidentifytheeffectsofdampingforeachcomponentindividuallyand
stabilityanalysescarriedoutwiththisapproacharequestionable[7].
In this paper, a simple minimal model of disc brake with two DOF is used to
gaininsightintothedifferentphysicaloriginsofdamping.Furthermore,FEmodels
ofsimplifiedandrealisticdiscbrakesderivedbythecommercialsoftwarepackage
Permasareinvestigatedaimingtooptimizethedampingpropertieswithregardto
anequilibriumpositionwhichisasstableaspossiblesubjecttosensibleconstraints.
©TheAuthors2018 1
J.-H.Wehneretal.,DampingOptimizationinSimplifiedandRealistic
DiscBrakes,SpringerBriefsinAppliedSciencesandTechnology,
DOI10.1007/978-3-319-62713-7_1
2 1 Introduction
InPermas,dampingcanbemodeledbydifferenttypesoflineardampingandeach
component can be treated individually. This, for example, enables assessing the
efficiencyofanti-squealshims,whichareoftenappliedinautomotivebrakes[8, 9].
Inaddition,itispossibletoidentifythosecomponentsbeingmoreorlessworthwhile
to be damped in order to avoid brake squeal. Even components where reducing
damping has a stabilizing effect may be found, which sometimes can be observed
inexperimentaltests[10].Consequently,itisnotalwayspurposefultoadddamping
butinsomecasesitmustbereducedtooptimizethestabilitybehavior,whichmay
becounterintuitivefromanengineer’sperspective.
References
1. Kinkaid,N.M.,O’Reilly,O.M.,Papadopoulos,P.:Automotivediscbrakesqueal.J.SoundVib.
267(1),105–166(2003)
2. Breuer,B.,Bill,K.H.(eds.):Bremsenhandbuch:Grundlagen,Komponenten,Systeme,Fahr-
dynamik.ATZ/MTZ-Fachbuch,4thedn.SpringerVieweg,Wiesbaden(2013)
3. Fritz,G.,Sinou,J.-J.,Duffal,J.-M.,Jézéquel,L.:Effectsofdampingonbrakesquealcoales-
cencepatterns—applicationonafiniteelementmodel.Mech.Res.Commun.34(2),181–190
(2007)
4. Fritz,G.,Sinou,J.-J.,Duffal,J.-M.,Jézéquel,L.:Investigationoftherelationshipbetween
dampingandmode-couplingpatternsincaseofbrakesqueal.J.SoundVib.307(3–5),591–609
(2007)
5. Hagedorn, P., Heffel, E., Lancaster, P., Müller, P.C., Kapuria, S.: Some recent results on
MDGKN-systems.Z.Angew.Math.Mech.95(7),695–702(2015)
6. Jekel,D.,Hagedorn,P.:StabilityofweaklydampedMDGKN-systems:theroleofvelocity
proportionalterms.Z.Angew.Math.Mech.(1–8)(2017)
7. Hagedorn,P.,Eckstein,M.,Heffel,E.,Wagner,A.:Self-excitedvibrationsanddampingin
circulatorysystems.J.Appl.Mech.81(10),101009–1–9(2014)
8. Festjens,H.,Gaël,C.,Franck,R.,Jean-Luc,D.,Remy,L.:Effectivenessofmultilayervis-
coelasticinsulatorstopreventoccurrencesofbrakesqueal:anumericalstudy.Appl.Acoust.
73(11),1121–1128(2012)
9. Kang,J.:Finiteelementmodellingfortheinvestigationofin-planemodesanddampingshims
indiscbrakesqueal.J.SoundVib.331(9),2190–2202(2012)
10. Massi,F.,Giannini,O.:Effectofdampingonthepropensityofsquealinstability:anexperi-
mentalinvestigation.J.Acoust.Soc.Am.123(4),2017–2023(2008)