T.G.V disk brake squeal : understanding and modeling

Transcription

1 .G.V disk brake squeal : udersadig ad modelig 1 X. Lorag, 2 F. Margiocchi ad 3 O. Chiello SNCF, Iovaive & Research Deparme, PSF, 45 rue de Lodres, 75379, Paris, Frace (1) el: +33 (0) , xavier.lorag@scf.fr (2) el: +33 (0) , florece.margiocchi@scf.fr (3) INRES, 25 av. F. Mierrad, Bro cedex, Frace el: +33 (0) , Fax: +33 (0) , olivier.chiello@ires.fr Iroducio Brakes are oe of he mos impora safey compoes i rai operaig codiio bu he brake squeal geeraed by disk brakes is a everyday source of discomfor for he passegers boh iside ad ouside he rai i saios. his is he case i paricular for GV. he developme of a refied mechaical modelig of he pheomeo was carried ou i order o udersad he mechaism of squeal geeraio. Priciples of soluios ad oise reducio assessme hrough he developed models akig io accou he brakig oise ad he safey will be ow possible. he paper deals wih fricio iduced vibraios ad especially wih GV disk brake squeal. he paper is divided i hree pars. I begis wih he sraegy o model such fricio iduced vibraios ad is applicaio o a 3D es case. I he secod par, a experimeal ivesigaio of he pheomeo is preseed boh i real codiios ad o a fixed esig pla. he hird par is devoed o he model of he brake mechaism. A fiie eleme approach is used o model he pheomeo via he preseed sraegy ad is compared wih he experimeal resuls. he firs par of he paper is devoed o he sraegy used o model he geeral problem of selfexcied vibraios of wo elasic bodies i fricioal coac. Uilaeral coac codiios wih Coulomb fricio ad cosa fricio coefficie are cosidered. I order o predic he occurrece of self-excied vibraios, a classical sabiliy aalysis is performed, which cosiss o compuig he complex modes associaed o he liearized problem. A commo ierpreaio of he sabiliy aalysis is ha frequecies of usable complex modes correspod o squeal frequecies. o check his assumpio, he behavior of he soluio far from he slidig equilibrium is deermied by usig a o liear rasie aalysis. Moreover, a expasio of he rasie soluio o he complex modes provided by he sabiliy aalysis helps us o highligh he role of he usable modes ad he vibraory field i geeral. he ifluece of iiial codiios of he rasie aalysis is aalyzed. I he secod par, a complee experimeal ivesigaio is preseed. I order o have a good represeaiviy of he pheomeo, acousic measuremes have bee doe i he saio o various kid of GV rais. o go ahead io deeper udersadig, oe brake sysem was equipped whe he rai is ruig. he vibraio specrum i oe poi of he disk was explored via a sigle poi laser doppler vibromeer. o ge more iformaio ha a vibraio specrum a a poi, a fixed esig pla was used o coiue he ivesigaio. he moivaio o ge he eire vibraio field is he ideificaio of he exied modes of he disk durig he brakig phase. he scaig laser doppler vibromery echique helped us o highligh he wave propagaio alog he disk which are prediced umerically. For a give frequecy, whe a roaig wave is deeced, i meas ha he field is a cosequece of a usable mode a he same frequecy. he hird par of he paper focuses o he applicaio of he approach o a Fiie Eleme model of he GV disk brake sysem. he umerical model is firs validaed wih a classical modal aalysis whe he disk is i free codiio. he modal dampig facors are measured. he complex eigevalue problem is compued (sabiliy aalysis) ad shows ha for a give fricio coefficie (f = 0.4) he slidig equilibrium is usable eve akig io accou he maerial dampig. 1

2 A. Mechaical sraegy o model brake squeal 1. Formulaio of he problem he mechaism of he simplified disc brake sysem represeed o figure 1 is cosidered. he roaio speed Ω of he disc is assumed o be cosa ad sufficiely small so ha he gyroscopic erms ad he volume forces iduced by roaio may be egleced. A euleria descripio is adoped. Uilaeral coac wih Coulomb fricio codiios are ake io accou. By usig a fiie eleme mehod, he oliear dyamics problem may be wrie i a discree form as follows (see deails i referece [3]): [ M ]{ U&& } + [ C]{ U& } + [ K ]{ U} = { F} + [ P ] { r } + [ P ] { r } { r } = ({ r } α ([ P ]{ U} { g })) { r } = Pro ({ r } α ([ P ]{ U& } + { V} ) R C Pro 0 (1) where [M], [C] ad [K] deoe he mass, Rayleigh dampig ad siffess marices whereas {U} ad {F} represe he vecors of odal displacemes ad exeral odal forces. I addiio, {r } ad {r } deoe he vecors of ormal ad ageial reacios forces a he coac odes whereas [P ] ad [P ] are proecio marices o he ormal ad ageial relaive displacemes bewee he disc ad he pads a he coac odes. C is he Coulomb coe ad coefficies α ad α mus be posiive. Fially, {g 0 } is he vecor of iiial gaps whereas {V} is he vecors of imposed slidig velociies due o he disc roaio speed Ω a he coac odes. 2. Sabiliy ad rasie aalysis Liear sabiliy aalysis Sysem (1) is a se of o-liear differeial equaios characerised by a slidig equilibrium. Cosiderig small regular perurbaios ha does o break he coac (bilaeral coac), he fricioal forces may be liearised ad he evoluio of he perurbaios {U} verifies (see [2]): ([ M ] + f [ M ]){ U&& } + ([ C] + f [ C ] + f [ C ]){ U& f f e } + ([ K] + f [ K f ]){ U } = [ P ] { r } [ P ]{ U } = 0 where [M f ], [C f ], [K f ] ad [C e ] are o symmerical marices provided by he liearisaio of he fricioal forces, f is he fricio coefficie ad {r } deoe he vecor of perurbed ormal reacios forces a he coac odes. By elimiaig he bilaeral coac cosrais, a o symmerical liear sysem of equaios is obaied ad he sabiliy of he equilibrium may he be deduced from a complex eigevalue aalysis of he sysem, providig complex modes ad complex eigevalue λ i. A complex mode i is usable if Re(λ i )>0, which may happe sice he sysem is o symmeric. A modal growh rae ζ i =Re(λ i )/Im(λ i ) may also be defied (physically equivale o a egaive modal dampig). No liear rasie aalysis I addiio o he sabiliy aalysis, a implici umerical resoluio of he sysem of equaios (1) may be performed. A ime discreisaio mehod is used here, resig o a former work of M. Jea ad J.J. Moreau (see [3]). he θ-mehod allows oe o avoid umerical problems a he ime of a impac. Ideed, he isabiliy of he slidig equilibrium may lead o srogly oliear eves like a separaio followed by a shock, as well as sick/slip rasiios. I order o iroduce a ielasic shock law, oe uses he modified versio of he θ-mehod (see [4]). (2) 2

3 Relaios bewee sabiliy ad rasie aalyses I order o udersad he role of he usable modes i he o liear par of he rasie behaviour, i is proposed o expad he umerical rasie soluio o he basis of he complex modes. akig io accou he o symmerical characer of he marices ivolved i he problem (2), i is ecessary o iroduce he complex modes provided by rasposed liearised problem. Ideed, he geeralisaio of he usual orhogoaliy codiios i he case of symmerical marices leads o bi-orhogoaliy codiios, ad give by : { L i } [ A]{ Φ } = 0 i (3) where {φ } are he complex modes of he direc problem (2) ad {L i } he complex modes of he rasposed problem of (2). [A] is he mass marix i sae space variable (see [5]). his bi-orhogoaliy propery allows oe o compue he evoluio of he complex ampliude of he mode, β (), from he rasie perurbaio wrie i sae-space variables {α()}: { L } [ A]{ α ( ) } { L } [ A]{ Φ } β ( ) = (4) wih {α()} provided by umerical resoluio of sysem (1). he coribuio of he mode o he perurbed displaceme ad velociy fields ca he compued from β (). Fially, he variaio of he oal perurbed eergy of a mode ca be compued from hese coribuios. 3. Applicaio o a simplified disc brake sysem I his par, he simplified disc brake sysem of figure 1 is cosidered. he geomeric ad physical characerisics of he srucures are give i able 1. he oher parameers are f=0.35, Ω=2.5 rad/s ad δ= m. A sabiliy aalysis is performed i he [0 15kHz] frequecy rage. Amog he 100 compued complex modes, hree modes are foud usable, called M1 (8583 Hz, ζ=0.05 %), M2 (9288 Hz, ζ=0.13 %) ad M3 (10130 Hz, ζ=0.17 %). he rasie soluio is also calculaed wih θ = 0.5 ad ime sep Δ = s. I order o sudy he ifluece of he iiial codiios o he rasie soluio, 4 cases are cosidered (A o D) for which he iiial coribuios of he usable modes are differe. he coribuios of he usable modes o he oal perurbed eergy E M1 (), E M2 () ad E M3 () are represeed o he figure 2 for he 4 cases. he differe figures show ha he sabilised soluio is o depede o he iiial codiios. I may be observed ha his sabilised soluio is made up of 2 of he 3 usable modes (M1 ad M2). hese compuaios show ha he sabiliy aalysis is able o predic he proe-squeal modes bu ha a rasie aalysis is ecessary o predic which modes remai i he sabilised soluio. I also highlighs he possible coexisece of several usable modes i he self-susaied vibraios. 3

4 Disc Pads Youg s modulus E Pa Pa Poisso s raio ν Desiy ρ 7850 kg.m kg.m 3 Dampig param. α 0 s s 1 Dampig param. β s s Ex. diameer 0.6 m I. diameer 0.2 m 0.16 m hickess 0.04 m 0.04 m ab 1. Physical ad geomeric characerisics Pad P δ z Disc D Disc - Pads δ Fig 1. Simplified disc brake sysem Case A Case B Case C Case D Fig 2. Evoluio of modal coribuio o oal perurbed eergy E () [J] for he differe cases as a fucio of ime [s]. E M1 (), E M2 (), O E M3 (). 4

5 B. Experimeal ivesigaio o a GV brake sysem his par preses he aalysis of experimeal daa comig from a saisical sudy i saios, o-board measuremes ad ess o bech i laboraory. 1. Measuremes i saios I he firs experimeal sep of he proec, he oise of 41 GV comig io Sai-Pierre des Corps ad Avigo-GV saios has bee performed wih a microphoe locaed o he plaform, a 1m from he rai. From hese recordigs, oise levels L ad maximum Aeq, 125 ms specra have bee calculaed for 173 squealig bogies i brakig operaio. he measured levels are coaied bewee 85 ad 106 db(a), wih a rae of 89%, superior a 90 db(a). he mai squealig frequecies are represeed o figure 3. Despie he dispersio of hese measured frequecies, 3 groups may be clearly disiguished. he firs group correspods o frequecies coaied bewee 1000 ad 5000 Hz. hese frequecies vary a lo ad heir coribuios are ofe secodary. he secod group is represeed by oly oe frequecy aroud 6600 Hz. I has a sigifica coribuio o he global oise level. he hird group is composed of he frequecies coaied bewee 8000 ad Hz, which domiae he specrum. Figure 3.-Squeal frequecies measured i saios 2. I-board measuremes I a secod sep, more deailed measuremes have bee carried ou o rollig sock i corolled codiios. Differe liigs ad discs have bee isrumeed wih a microphoe i he close field, a laser vibromeer aimig a he disc surface ad some hermocouples ad acceleromeers o he liigs. he mai characerisics of he squeal oise measured i saio have bee foud agai. he effec of several parameers has bee sudied as he ruig direcio, he brakig pressure or he vehicle speed. Oe of he mai resuls is he relaioship bewee he oise level ad he vibraio level a he disc surface. Ideed, i appears ha above 5000 Hz (groups 2 ad 3), mos of he squealig frequecies correspod o he vibraio frequecies (see figure 4). For hese groups, he squeal oise is maily radiaed by he disc i axial vibraios (ou-of-plae). 5

6 Fig 4. Experimeal power Specrum of he ormal velociy [db] (below, ref 1 [m/s]) ad acousic pressure [db] (above, ref 20 [μpa]) 3. Laboraory measuremes From hese measuremes, cosidered like refereces, he represeaiveess of some brakig beches has bee esed. I appears ha o bech is able o reproduce exacly he pheomeo observed o he rai. I srogly depeds o he liig ad he frequecy of ieres. Geerally speakig, oly he squealig frequecies of he hird group (above 8000 Hz) are well reproduced. he 6600 Hz frequecy ever appears ad he frequecies less ha 5000 Hz are oo variables. Some experimes are i progress o explai hese differeces. However, he bes es bech has bee used o characerize he vibraios field o he disc by laser measuremes durig brakig (see figure 5). A frequecies above 7000 Hz, he measuremes have show ha he disc vibraes maily axially. he measured vibraios fields do o correspod o saioary flexio modes bu raher flexio waves, which may be ierpreed as flexio modes roaig alog he circumferece of he disc (see figure 6). From 7000 o Hz, he ideified modes have o odal circles ad odal diameers varyig from 9 o 15. 6

7 Figure 5 : Vibraio field ivesigaios wih laser measuremes Figure 6 : Vibraio field a differe frequecies he mai compoes of he brake sysem (disc ad liigs) have also bee esed i free codiios (e.g. wihou fricioal coac). he resuls of hese experimeal modal aalyses have bee used o improve he fiie eleme modellig of he compoes. C. Sabiliy aalysis of a GV brake sysem I his par, a fiie eleme model of he GV brake sysem (cf. fig 3) is cosidered. Firs, he modes of he disc i free codiios have bee calculaed ad have bee classified accordig o he direcio of he domia deformaios. I paricular, i-plae circumfereial modes C-m, i plae radial modes R-m ad ou-of-plae axial modes A-m may be disiguished where ad m deoe respecively he umber of odal circles ad odal diameers. A sabiliy aalysis has bee performed. More ha 800 complex modes have bee calculaed o reach a upper limi frequecy of abou 14 khz. he correspodig growh facors are 7

8 represeed o figure 5. wo kids of modes may be disiguished: he pad modes, for which he disc vibraios are very small, ad he disc modes for which he vibraios of he disc are domiaig. he correspodig mode shapes ad frequecies of he disc modes are close o he modes of he disc i free codiios bu roae alog he disc. Mos of hese modes are axial modes wihou odal circles ad wih oe odal circle (see fig 3). Aoher mode is raher a iplae mode (C0-2) bu wih some axial compoes. Some experimeal resuls have bee obaied from a GV i brakig operaio a abou 10 km/h. he acousic pressure a oe ceimere from he disc ad he axial vibraory velociy a a poi o he disc surface have bee measured durig brakig. Figure 4 shows ha he disc is resposible for he emied oise ad ha he vibraios are composed of 8 high frequecies from 5000 o Hz. Excep he axial modes wih oe odal circle, all he usable modes are close o he experimeal squeal frequecies. Mode C0-2 : 6684 Hz - ζ=0.02% Mode A1-6 : 9494 Hz - ζ=0.12% Mode A0-10 : Hz - ζ=0.23% Fig 3. Some usable modes of he GV brake F.E. model Fig 5. Growh raes ζ [%] of complex modes as a fucio of frequecy [Hz] (O : disc mode : Pad modes) Coclusio I his paper, he modellig sraegy of disc brake squeal has bee sudied. I order o ivesigae he relaios bewee sabiliy ad rasie classical aalyses, a ew mehod has bee proposed, which cosiss o expadig he rasie vibraory field o he complex modes provided by he sabiliy aalysis. his mehod has bee esed o a simplified disc brake model 8

9 for various iiial codiios. Resuls have show ha he sabilised soluio is made up of wo coexisig usable modes ad ha his soluio is he same for differe iiial codiios. A sabiliy sudy has also bee performed o a fiie eleme GV brake sysem ad compared wih vibraio measuremes. I has bee foud ha he frequecies of he usable disc modes correspod o mos of he vibraio frequecies. he experimeal ivesigaios performed o GV disc brakes have bee preseed. he experimeal daa comig from a prior saisical sudy i saios, o-board measuremes ad ess o bech i laboraory has bee aalyzed. Refereces [1] F. Moiro ad Q.S. Nguye, Brake squeal: a problem of fluer isabiliy of he seady slidig soluio? Arch. Mech. 52, 2000, pp [2] F. Moiro, Eude de la sabilié d'u équilibre e présece de froeme de coulomb. PhD hesis, Ecole polyechique, Palaiseau, Frace, [3] M. Jea, he o-smooh coac dyamics mehod, Compu. Mehods Appl. Mech. Eg., 177, 1999, pp [4] D. Vola, E. Pra, M. Jea ad M. Raous, Cosise ime discreizaio for a dyamical fricioal coac problem ad complemeariy echiques. REEF Volume 7, [5] E. Balmès, Srucural Dyamics oolbox, 2006, [6] X. Lorag, Isabilié des srucures e coac froa : applicaio au crisseme des freis à disque de GV. PhD hesis, Ecole polyechique, Palaiseau, Frace,