Theo retical and experim enta l investig ation s of creep gro an in auto motive dis k brakes vo rgel egt von M.Sc. Xin gwe i Zh ao geb. in Hu nan, Chi n a v on der Fakultät V - Verkehrs - und Masc hinen sy stem e der Technischen Universität Berlin zur Erlangung des akadem ischen Grades Dokto r der Ingen ieurw issensch af ten - Dr. - Ing. - gene hm igte Dis sertation Prom otionsa ussch uss: Vorsitz e nder : Prof. Dr. rer. nat. Wolfgang H. Müller Gutachter: Prof. Dr.- I ng. U tz v on W agner Gutachter: Prof. Dr.-Ing. Hartm u t Hetzler Tag der wissenschaftlichen Aussprache: 21. Dezember 201 7 Berlin 201 8 1 Acknow ledgments The m ost sincer e tha nks to m y super visor, Prof. Dr. - In g. Utz v on W agner , wit h- out h is thou ghtf ul guida nce a nd e nth usiast ic suppo rt thi s stu dy wou ld not ha ve bee n com pleted on ti me . I a m gr eatl y grate ful to Dr. - Ing. N ils Grä bner for pr o- vidi ng in va luab le di scus sion s and c om m ents. I am deeply than kfu l to Prof. Dr. - Ing. H artm ut He tzler for his f ine c o m me nts. L ast but c er tainly not lea st, I w oul d like to de vote m y heartfe lt tha nks to m y wife L u Qian f or al l the be st sh e has done fo r me . 2 Contents 1 I ntroduction ..................................................................................................................................... 3 1.1 Backg r ound and m otivation ................................................................................................ 3 1.1.1 N VH prob lem s in vehi cl e b rakes ................................................................................. 3 1.1.2 Friction l a w and stick- slip m otion ............................................................................... 4 1.1.3 C reep groan ................................................................................................................. 6 1.2 Obje ctiv e of the w ork .......................................................................................................... 7 1.3 Outline of the w ork .............................................................................................................. 7 2 Exper im ental I nvestig at ion s of Creep G roan .................................................................................. 9 2.1 Test rig with an idea lized b r ak e ........................................................................................... 9 2.2 Test rig with a rea l brak e ................................................................................................... 15 2.3 S um ma r y ........................................................................................................................... 19 3 T heo ret ic al Inve st igations of Creep G roan on the Test Rig with an Idealiz ed Brake ................... 20 3.1 Min im al mo del w ith Cou lom b’s frict ion law .................................................................... 20 3.1.1 O ne DOF m odel with Coul om b’s friction law .......................................................... 20 3.1.2 Two DOFs m odel with Coulom b’s fri ction l aw ........................................................ 25 3.2 Min im al m odel with the b ri stle fri ct ion law ...................................................................... 29 3.2.1 O ne DOF mod el with th e b ris tle f rict ion law ............................................................ 29 3.2.2 Two DO Fs m odel with the brist le friction law .......................................................... 37 3.3 S um ma r y ........................................................................................................................... 43 4 Com pa rison study of exper im ent al and theore tical results on the idealiz ed brake ........................ 44 4.1 Param eter ident ifica tion for th e sy stem with C oul omb ’s fr ictio n law ............................... 44 4.2 Param eter ident ifica tion for th e sy stem with th e bris tle fr ict ion law ................................. 46 4.3 Friction observ er ................................................................................................................ 54 4.4 S um ma r y ........................................................................................................................... 56 5 Theore tica l and exp erim ental analy sis of creep groan on the tes t rig with a real brak e ................ 57 5.1 Mode ling of the tes t rig with a real b rak e .......................................................................... 57 5.2 A reduced- orde r m odel for the sim ul ation of c reep groan ................................................ 59 5.3 Transfer function identifica tion thr ough m odal analysis ................................................... 61 5.4 Com pa rison study of exper im ent al and sim ulation results ................................................ 64 5.5 S um ma r y ........................................................................................................................... 73 6 Counte r m easures again st cr eep g roan ........................................................................................... 74 6.1 Suppression o f creep groa n through a a ctive pad .............................................................. 74 6.2 Passiv e method ag ainst cre ep g roan .................................................................................. 80 6.3 Suppression o f creep groa n through an optimal brake te chnique ...................................... 82 6.4 S um ma r y ........................................................................................................................... 87 7 Conclusions and Fu ture Wo rk ....................................................................................................... 89 Bibliography .......................................................................................................................................... 91 3 1 Introduction 1.1 Back ground and m otivat ion The brak e is one of the m ost im portant safe ty and p erformance c om ponents of v e hicles. On the one hand, the developm ent of b rakes has fo cused on the increase of b r aking power an d reliability . On the other han d, the r efinem ent of v ehicle acoust ics and co mfort of v ehicl e design ha ve increase d the rela- tive contribu tion of br ake noise, v ibration and ha r shness (NVH). A co nsum er m a y believe that th e NVH pro b lem i s sy mptomati c of a defect ive br ake and f ile a war r an ty c l aim , ev en thoug h the b rake is function ing as desig ned in all ot h er par ts [1] , [2] . Th e entire au tom otive indu stry can at tes t t o N V H repair s ofte n dom inating w arranty c laims a t afterm arket servic e cent er and dea lership s. Abendro t h and Wernitz deno te that m any m a kers of material s for br ake pad s spen d up to 50% of their engine ering budg ets on N V H i ssues [3] . Concentrati ng on NVH pe rformance can be dr awn back to the early 1990s, engineers focus e d m uc h of the ir attention on e l im inating high - frequency sque als. There is a wea l th of literatu re on autom otive di sk brak e squeal. Rev iews c onducted in the las t 30 years p rovide a com pre- hensiv e source of info r m ation [4] . In contrast, creep groan received m uch les s at tention. How ever, this type of vibration rece ives a growing inter est fro m the automo tive indu stry becaus e it prim ari ly aff ects the comfort. Lik e other brake noise problem s such as brake squea l, creep groan m a y bring complaints of custom ers, wh ich ev entually cau ses warranty claims and resu l ts in r efin ement c osts to t he indust r y . 1.1.1 NVH pro blems in veh icle brake s I n general, br ake vibrat ion and /or noi se can be clas sified into m any categ ories based on th e occu rring frequencies, s uc h as judder, creep groan, m oa n, and sq ue al [5] , [ 6] . As shown i n T abl e 1 . 1, in the low- frequency range (0 - 1000 H z) there are in general thre e different types o f structur al vibra tions, nam ed judder, creep g roa n, and m oan, while squeal occu rs in the hig h- frequency r ang e ( >1000 Hz ). • Brake sque al is a high f r equency vibration noi s e, wh ich is in gene r al highe r tha n 1000 Hz. I t is c ause d by the f lut ter ins tab ility [7]-[17] . Sq ueal is a fr iction - induced self - exc ite d osc illa tion . When an un stable equ ilibrium s olution exists in the sy stem, the sy st em oscillates with increas ing am plit ude from the eq ui lib rium solu tion and reach es a lim it cycle (LC) . The lim it cycle osci lla- tion ca n gene r ate sound, and th e bra ke di sk e mi t s the s ound. • Moan s hows th e same exc itat ion m echan isms as low frequency squeal. I t i s a sel f exc ited v i bra- tion , wh ere the equ ilib rium solutio n of the system becom es unstable due to the f riction coup l ing of vi br a tion m odes . During m oan, t he h ar m oni c v i bration of brak e carrier and axle can n or m ally be observed [18 ] , [19]. • Judde r is cause d by period ic fea tures on t he rotor sur face th at resu lt in cy clic brak e t orques. T he ty pica l feature of this type of noise is t hat its frequenc y is a multiple o f the rotor spe ed of rotation [20]-[26]. • Creep g roan is a low fr equency vibrat ion no ise caus ed by the stick - slip - effect, which descr ibes the brake p ad’s tota l or pa r tial, a lternating adhesion and sliding on the disk. T he phenom enon m ay take p lace wh enev er the br ake is s lowly rele ased w hi le the car s tarts m oving from a sta ti o- 4 nary state , which is a freq uen t problem of vehicle s wit h autom atic t ransm issions due to the con t i- nuous driving torque on th e drive sh aft [2 7] - [33]. Brake noise Frequency re gi o n Occur rence brake pre ssure Occur rence speed T ri gger i n g source Judder [2 0] - [26] Around 10 Hz Low brak e pressu re Proportiona l to speed Fo r ce d vi br at i on Moan [18], [1 9] 100- 1000 Hz Low brak e pressu re Hig h speed Self - excited ( flu tter insta bili ty ) C reep g roan [27]-[33] 0- 500 Hz Hi gh brake pressur e Low speed Self - excited ( stick - slip ) Squeal [7] -[17] >1000 Hz Low brak e pressu re Hig h speed Self - excited ( flu tter insta bili ty ) Table 1.1: N V H p roblem s in vehicl e brake 1.1.2 Friction law and stick -slip motion Tribology i s the study of adhes i on, fric tion, lubricati on a nd wear o f surfaces in relativ e m otion. It rem ains im portant to day as it was in anci ent times, ari sing in the f i elds o f physics, chem i stry, g eology , biology and engineering [34 ] . Fr ictio n is a c lassi cal f ield th at track s back to Leonardo da Vinci, G ui l- laume Am ontons, and Charles August in de Coulom b [ 3 5] . Am ont ons po inted out t hat t he fr icti on force is propor tional to nor m al load but does n ot depend on the a rea of the ap parent cont act surfac e . Coulom b proposed a model where the friction force is opp osite to the dire ction of velocity w ith a m agnitude proportional to the norm a l force. Besides, it d escribe s a static force a t zero sliding veloci t y to be l ar ger than a kin eti c force at finit e s lid ing veloci ties [36]. Measu rements o f the contac t surf ace of rocks show that the f riction force i s proportiona l to true con tact area, wh ich is typica lly m uch less than the appar ent contac t su rface [3 7] , [38 ] . B y m easu r ing the velocity dependence of friction, St ribeck found that friction de c reases w ith inc reasing v elocity in ce r tain v el ocity reg imes. T his phenom enon is called the Str i be ck effe ct [39] . Stick - slip v ibra tio n is c hara cteriz ed by a saw to oth d isplacem ent - tim e ev olution [40 ] w hi ch h as cl earl y defined stick and slip reg ions. I t appears i n every day life a s we ll as in eng ineering system s, such as t he sound of b owed ins truments, c reaking door s , ratt ling joints o f a robot, c reep g roan of brak e sys tems, and ch attering machine t ools [ 41] . T he st ick - slip m otion can in the si m plest usua l way modeled as a 1- degree - of- freedom ( DOF ) sy stem coupled with Coul om b ’ s friction law. Wh en dry friction is m odel ed as Cou l om b’s friction law, se l f- sustain ed stick - sl ip motion m ay occur [4 2] - [46]. Coulom b’s fri ction m odel is norm ally describe d by piecew ise d i ffer entiab l e eq uations , with swit ching be tween the stick and slip regi on . T he w ay to solv e the se pi ece wi se di ff ere nt iabl e equations is stu d ied in l itera tur e [47]-[53]. Popp e t at. em ployed a point- m apping approach to ca lculat e the stick - slip lim it cy cle [4 7] , whil e Leine et al. studied the shooting m ethod to find the limit cycle by solving a two- point boundary -v alue problem [4 8] . Another method to 5 solve t he piecew ise d iffer enti able eq uations i s so ca lled “sw itch m ethod”. Durin g the “switch m ethod” the num eri cal c alculation st arts fro m an initial sta t e wi th a dif ferent ial equ ation. A fter each tim e step it is inspec t ed if poss ible sw it ch cond itions wi t h in this ti me step are sa tisfied. I f the switc h conditions ar e sat isfie d , a new integ ratio n proce ss is s tarted with a m odified set o f di ffe ren tial eq uat ions and i ts in itia l conditions are the s t ate at the swi tching point [49] . B a sed on the studies o f flo ws for non - sm ooth dy- nam ic system s , the switching c onditions o f a non - smooth sy stem can be calcu lat e d by f inding the separation bounda r ies of flows [ 50]- [53] . Due to t he non - s m ooth character istic s of the st ick - slip m o- tion, chaotic m oti on can o c cur as we ll as m ulti - pe riodic soluti ons. Bilinea r and nonlinear dy nam ic m odels hav e been consi dered to exp lain such fri ction phenomen a as sti ck - sli p, chat ter and chaos by I brahim [54] , [ 55 ] . G alvanett o and Knudsen d escribe d an even t map of a t wo D OF s mechanic al sy s- tem under self - sus tained os c illations in duced by dry friction, and the par ameter depende nt bifurca tion behavior is analy zed by the defi ned m appings [56]. Pop p et al. prop osed tha t the lim it cycles of stick - slip vibrat ions can b e broken up by a harmonic dis turbance, a nd th e bifu rcation be havior and th e cha os of a sti ck - slip system under exte rnal ex citation are studied f or differen t system param eters [41]. St o rc k et al . proposed that the f ricti on w ill be redu ced in pres ence o f ultraso nic v i b ration [57]. Som e sc holars stated th at the st atic fri ction coef ficien t can be str ong ly reduced u nder a no rm al or lateral m echanica l osc illa tio n [58]- [61]. With furth er inv est iga tio n s of fr icti on law , it is we ll k now n that t he p he nom ena such as pre - slid ing , rate depen dence, and hy steresis hav e been ob ser v ed e xperim entally and are reproduced only by dy- nam ic models [6 2] - [64]. A s a res ul t, the sim ple classical static m odels of C oulomb, Stribeck , etc. have giv en way g ra dually to m ore sophisticated, dy namical m odel s with due atten tion to p r esiding hyster e- sis and tim e - lag ef fects. D ahl devel oped a sim ple dynam ic fr icti on model with one st ate in the late 1960s , w hi ch is wid ely use d to sim ulate aerospa ce system s [ 65] , [ 66] . Ho we ver , t hi s m odel doe s not capture t h e Strib eck effec t and t hus canno t pred i ct s ti ck - slip motion. Later, the LuGre m odel is an extensio n of the D ahl m odel that ca n de scrib e the Strib eck effec t, sti ck - slip ef f ect, and hy steres is [67]- [71]. Canudas d e Wi t et al. [71] analy zed the s t ick - slip experiment us ing the LuG re friction m ode l, whose gross fea tures of th e be havio r are sim i lar to those obta ined with Coulo mb’s model, but the transi tions are capture d by dynamics for the LuGr e friction m odel. Li et a l. st ud i ed t he bif ur c at ion and chaos in friction - induced vibration th rough the LuGr e friction m odel [ 72 ] . Thanks to the av ailability of m easur em ent techniques a nd equipm ent s such as sc anning prob e m icr o- scopy , laser inter ferom etry, and th e surf ace force ap paratus, i t is p ossible to m easure f r icti on a t the nanosc ale [73 ] . Ma te et a l. first introd uce d th e frict i o n force m icroscop e (FF M) in 198 7 . I t becom es possible to observ e the atom i c friction processes for a tip sliding ov er graphite [74], [75] . Later, To- m anek et al. propo sed an accur at e d escr iption o f t he microscop ic me chanism of energy dissipation in the fr ictio n forc e microsco pe [ 76] . On th e atom ic sca l e, severa l ex perim ents con firm th at the friction force on t h e nanom eter scale exhibits a sawtoo th behavior, comm only k nown as atom i c stick - slip [77] . As a res ult, th e Pran dtl - T om linson model is p roposed to exp lain the frictional stick - sli p m otion on the atom ic scal e [78] , [ 79] . Soc o li uc el at . con fi rm ed tha t the atom ic s tick - slip m oti on ca n be e lim ina te d under a norm al mechanical osci llation [8 0] , [8 1] . A comparison of different frict ion laws is shown in Table 1.2. 6 Na me Equations Descr ibed fr ictio n charact eris tics Coulom b f riction law [ 36] [ ] sgn( ), s l ip , , stick d ss FF v F FF = ∈− Stick - slip Dahl m odel [ 65] 0 0 c dz v vz dt F Fz σ σ = − = Hy steresis LuGre m odel [ 67] ( ) ( ) 0 01 () ( ) exp / ( ) d sd s v dz vz dt g v gv F F F v v F z z fv α σ σσ = − = +− − = ++ Stribeck effect Hy steresis Stick - slip Pr a ndt l - Tomlinson model [78] 10 sin( ) d mz F z F z σλ = −− Atom stick - slip Table 1.2 : D iffe ren t fric tion laws 1.1.3 Creep groan I t is well a ccept ed th at cree p g roan is caused by the stick - slip - effect [ 27] -[30], [ 82] -[89]. S o me wo r ks focus on the e xperim e ntal study of c reep groan . J ang et al. invest igated creep g r oan pro pensity of different f riction m aterials, and p roposed tha t the creep g r oan can b e elim i nated by employing the friction m aterial w ith les s diff erence b et we en the stat ic and dy namic frictio n coeffici ent [82] . Fuadi et al. studied a fundamenta l m ec hanism for creep groan g ener ation by adopting a c aliper - slid e r exp er i- m ental m odel, whe re a m ap that show s the n ecessary con dition fo r avoidi ng creep groa n was introduced [83], [8 4] . Other literature wo r ks on t he fundam e ntal mechan ism of cr eep g roan nois e generat i on a s wel l as the corresponding suppression m ethods. Brec ht e t al. [29], [30] m easured the v ibration char acteristi cs o f creep groan and stud i ed the stick- sli p lim i t cycle of creep g roan. Jung et al. m easured th e interio r noise in the ev ent of cre ep groan n oise by using a c hassis dynam o m eter, and th e way t o redu ce cre ep g roan noise w as stud ied experim entally [85] . Va dari et a l. stated th e stiffness a s one of the m ost imp ortant param eters in brake cr eep g roan gener ation [ 86] , and Donley et a l. through exp erim ental observ at ions dem onstrated th at the struc ture of the McPherso n’s su spension system is th e key t o damp creep g roan [87] . Zhang a t el. s t udied the conditions leadi ng to creep groan nois e through road tests , as w ell as object i v e charac t eri stics of creep groan noise [ 88] . Ne i s e t al. c om bi ned in - v ehicle tes ts and laborat o- ry - scal ed tribom eter tests to seek the condit ions of cr eep g roan no ise occu r rence s [ 8 9] . Bett ella et al. focused on t he transm is sio n of the v i bration f rom the brake com ponent regions to the cockpi t during creep groan, and show ed t hat t he airbo rne tra nsm i ssion can be neg lected com par ed to stru cture - borne path [90] . Gau terin et al. studi ed creep groan of di sk brakes of ca rs and exp lained how creep g roan origina t es an d perm itted th e assessm ent of co rrective m easures [91]. With respect to modeling, t he stick - slip - effect has i n gener al been m odeled by correspo nding dy namic system s coupled with Cou lomb’s friction law. C rowther et a l. inv estigat e d the b rake creep g roan p ro b- 7 lem by f orm ulati ng the issues in t e rm s of two dy namic sub - s ystems coupled w ith C oulomb’s frict ion la w [9 2] , [93] . Heg de et al. fo cused on the nonl ine ar dyn am ic transien t analy sis of b rake vibra t ion with a m ulti - body dynamic simulation [94 ] . Br ech t et al. [29] , [30] stud ied the sti ck - slip lim it cycl e of creep groan with 3 DOFs m inim al mo del co upl ing w ith Co u lom b’s fric tion la w as w ell a s wi th fin ite elem ent m odel. H owever, the m odel using Coulom b’s friction law c annot ex plai n sev eral effec ts du r- ing creep groan. He tzler et al. [9 5] - [99] presen ted an analyt ical investig ation on stabi lity and bifurcation behav ior of a friction oscillator, where a f riction model is described by Coulomb’ s fr iction law coupled w ith the Stribe ck effect. Som e works of the author referring to creep g roan ha ve been published in [3 1] -[33]. A com parison study of the creep gro an models wi th Coulom b’s frict ion law and t he br i stle fric tion law is prop os ed in [31]. Furtherm ore, a 1 DO F model i n [32] and a 2 DO F s model in [ 33] with th e brist le fric tion law are invest i g ated to descr ibe the fundam ental mech anism of c reep groan. T he sim ulation r esults are co m- pared with the experim ental resu lts m easured in t he test rig with an ideal ized b rake in [ 31] -[33], and in [33] the ex perim ental resu l ts in a t est ri g with a real bra ke is al so an alyz ed. 1.2 Objectiv e of t he w ork Motivated by the a forem entioned observat ions, the m ain objective of this thesis is to study t he fund a- m ental m echanism of creep g roan on brake sy stem s, as well as the suppre ssion m et hod s o f creep g roan. To be sp ecific, the tasks of thi s thesi s are s tated as follow s: • Desig n and set up test rig s with an idealiz ed brak e a nd a real brake. C reep groan should be g e ne r- ated in both tes t rigs. Th e fundam ental m echanism of creep groan should be investig ated b ased o n the ex per imen t al r es ult s . • I nvestig ation o f m inimal model s of creep g roan, so th at the f rictio n induced stick - sli p m otion will be carefully s tudied for a deeper unde rstanding of creep g roan. • I nvestig ation o f th e multip le deg r ee s of f reedom m odel to descr ibe creep groan of a real b rake. A reduced- or der model wi ll be studie d to im pr ove calc ulati on e fficiency . • Searc h for m ethods ag ainst creep gro an . The feasib ility and effectiv eness of the methods shou ld be confirm ed t hrough exp erim ents. 1.3 Outline of the work The org ani z ation of this thesis is descr ibed as f oll o ws . According to the g eneral introduction on c r eep groan an d NVH i ssues o f brak e system given in C hapter 1, C hapter 2 fi rst presen ts the construct i on of te st rig s with an idealiz ed brake and with a rea l brake. Then , the experim ental resu l ts dur ing c r eep groan from both t est rig s are shown to g ive a first im pression o f creep groa n. I n Chapter 3, based on the exper imental r esults, m i nim al m ode l s of creep groan for an ide alized brak e are investig ated. Both Coulom b ’ s friction l aw and th e bristle f r ict i on law ar e em ployed to de scribe the friction force. By coupling the minim al model with differ ent friction laws, the stab ility of the equi l i- brium solu tio n as well as t h e stick - slip lim it cycle of th e nonlinear system is studie d. According to that, the syst em has di fferen t par ameter regio ns with d iffere nt t y pes of m otion. 8 Chapter 4 focuse s on the st udy of the par ameter iden tification for differe nt fricti on models. After p a- ram eter iden tifica ti o n, the theoretic al and experim ental r esult s are c ompare d with each ot her quantit atively. Exper imen tal resu lts co nfirm the existen ce of three di fferen t param eter regions. I n add i- tion, a f riction fo r ce obs erv er is desi g ne d and the observed friction force is compared with the simulated f riction force. The study of creep groan o n the test rig with a real bra ke is presented in Chapte r 5. A large number o f degrees o f freedom m odel and a correspon di ng reduc ed - or der model are set up t o describe th e brake system . Num erical sim ulation show s that th e reduce d - order m ode l can describe creep groan wi th high calculation e fficiency and lim i ted error. Counterm ea sures against c reep groan a re finally discussed in C hapter 6 . A pad, w hich cont ains piez o- ceram ic stap le actu ators, i s successfu lly used to supp ress creep g roan by exciti ng a high frequency oscill ation of the system . Besides , the risk of the generation o f creep groan can also b e reduc ed by increasing the dam pi ng of the shaft. Another m et hod to sho rt en the tim e of creep g roan is an optim al brake tech nique, throu gh th at t h e system can leave the regions w ith cr eep groan rapidly. By i ntegrating this optim al brake technique into an a nti - lock b raking system (AB S) , t he ABS can perform the optim al braking process through a sim pl e control loop t o a void c reep g roan. Chapte r 7 con clude s the the si s and discu sses the f uture scope. This work was perform ed at the Cha ir of Me chatron ics and Mach ine Dynam ics (MMD ) TU Berlin and fund ed by the China Scho l arship C ouncil (CSC). The author is de e pl y th a nkf ul t o Dr . - In g . Tors ten Trey de (ZF TRW) f or helpf ul comm ents . 9 2 Experi m ental Inv estig ation s of Cre ep Groa n I n this chapt er, the t est r igs and th e experim ental r esult s are presen ted to gi ve a first im pression of creep g roan. In o rder to u nderstand cr eep groan st ep by step, a tes t rig with an idealized b rake was designed and as sem bled at MMD TU B erlin. Su bsequ ently, a t est rig with a real brake was set up as a com parison study w ith th a t of an ideal ized b rake. Th e st i ck - slip lim i t cycl e can be m easured i n b oth tes t rig s. Accord i ng to the measurem ent s, the mechanism of cr eep groan w i ll be explained in this chap- ter. 2.1 Test r ig wit h an ideal ized b rak e Creep g roan of b rake sy stem s is a wel l - known low fre quency v ibration noise caused by the stick - slip - effect, wh ich des cribes the brak e pad ’s to tal o r par t ia l, altern ating adh esion and slid i ng on the disk [27]-[30], [ 83] -[94] . The phenom enon may take place whenev er light pr e ssure is exer ted by the driver on the br ake ped al and so me forces are a cting on the veh icle, s uch as an idling engine throug h an au- tomatic transm is sion, or g ravity due to the v ehicle on a slope. The structure a front - wheel dri v e vehicl e is shown in F ig. 2. 1, where a suspensio n is hung in a chassis , and a brake system is attached to the suspen sion . A dr ive shaft d riven by m otor is c onnec t ed to the disk, while a whee l i s c o nnec t ed t o t he ot h er si de of the disk with wheel bol t s. I f a car is an autom atic car, the motor - di sk- pad sub- s ystem is the prim e part r elev ant to creep groan . Fig. 2.1: Struc ture of a fron t - wheel driv e v ehic le in fron t view I n order to concentrate on the inves tigation of t he frictiona l contact, a t est r ig wit h an idealiz ed brak e has been designed and assem ble d. T he intent ion of designing this set - up is to concentrate on the pad- disk con tact. Th erefor e, the brake d isk, the drive sha f t, the brake p ads, th e brak e caliper, and the br ake carrie r are tak en out from a real vehicl e to cons titute the te st rig ; b ut the di ffic u lt - to -model pa rts , such as the com plicated str ucture of th e carri er and the rubber coate d pins on the c arrier, ar e replaced by an idealiz ed carri er, which consists of tw o L - shaped stee l plates. This ide alized carrier has hig h stif fnes s in t he in - p lane dire ction and low stiffne ss in the out -of- pl ane direction. To m ake sure that the stiff ness Suspension Wheel Brak e disk Brak e pads with c alipe r Dr i ve Chass is P r i me part of creep g roan 10 of the ca r rier h as compara ble order of stiffness similar to the rea l brak e carrier, a finite el eme nt ( FE) analysi s i s perform ed t o calcula te the stiffne ss of a real car ri er in the in - pl ane di rectio n . In th e FE analysi s, the screw h ole o f t he cal iper is set as f ixed po sition, wh ile a late ral force is acting on the b oth pads. Fig. 2. 2 show s the FE analysis res ults, wh ere the s tiffn ess in the in - plane direction is 7 8.21 10 N/m ⋅ in the piston s i de and 7 2.33 10 N/m ⋅ in the oth er side. T hose tw o st eel plates are des i g n ed according to the calcu lated sti ffness . Fi g. 2.2: FE an alysis of t he brak e carrier An AC m otor coupled with a reduc t ion g e ar box is us ed as dr i ve, w hich can pro vide a low r evo l uti on speed w i th h igh m oment. A shaft is assem bled between the reduct ion gear bo x and a brak e disk. B rake pads are fixed on t he long edges of the s teel plate, a nd the short edg es are fixe d on the fram e. T he brake caliper is hung on th e long edg es of t he steel pl at es, and prov ides the b rake pressur e with a h y- draulic system . T he CA D m ode l of the te st rig is sk etched in Fig. 2.3. T he test ri g h as som e advantages fo r the expe rimental inv es tigation of creep groan, such as : 1. The t est r ig is sim ilar with th e brak e system of a real ca r , s ince th e pad s, the disk, the sha ft and the cal iper of t h e test rig co m e f rom a real car ; 2 . The t est rig has sim ple st ructure of the com ponents an d the ref o re th e param eters a re easy t o ide n- tify and there is less un cert ainty i n dynam ics; 3 . The t est r ig is ea sy to equi p with d ifferent t y pes of s ensors. I n order to observe c r eep g r oan in the test r i g, senso rs have been fitted in su itable positions , shown i n Fig 2 .4. An accelerom eter is atta ched to t he car r ier, which m easures th e accele ration of the p ad. A turning ang le transm i tter is connected to th e disk by which the absolute angle and ang ular vel ocity of the disk can be measured . T he p ressu re of the brak e can be read from a pressure m eter. Strain sen sor s are place d in the m iddle of the drive shaft. Once the shaft has a tors ional deform a tion, the chang e of the el ectr ical resis tance of the s t rain sensor can be m easured by a Wh eatst one brid ge, which is related to the strain. T he c alibratio n of the st rain sensor is pe rformed by a sta tic measu re m ent and its relat i o n- ship to t he torsional angle θ ∆ is ident ifi ed [100]. I nf ormat ion on sensors is given in Table 2.1. Fixed posit ion Fixed posit ion Fric tio n fo rce Fixed posit ion Fixed posit ion Fric tio n fo rce 11 Fi g. 2.3: CAD draw i ng of the te st rig with an i deal ized brake Fig. 2.4: Test rig with an idealiz ed brake and th e corres ponding sensor s Be f ore a ppl yi ng a n experim enta l investigat ion, modal analy s is is perform ed [101]. T he eige n fr eq ue n- c ies and correspon ding e igenm odes are presente d in Table 2 .2 [100]. T he d isk ha s eig enfr eq uencie s at 1400 Hz, 1871 Hz and 3148 Hz (with f ree -free bound ar y condition ), while the L- shaped s t eel pl at e h as eigenfrequenc ies at 16 Hz, 174 Hz and 300 Hz (wit h fix -free boundary condit ion) . T he t orsi o na l ei ge n- fr e que n cy of the shaft is at 36.5 H z (w ith f ix - free boun dar y cond i tion ). Since c r ee p groan is lower t han 500 Hz, i t will be more related to the c omponents with lowe r eing enfrequenc ies, such as the L - shaped steel p late an d the shaf t. D is k C aliper P ad B rake car rier F r a me S tr ai n gau ge A ccelerom eter T urn i ng a n gle tran smitte r 12 Sensor Paramet ers Accelerometer PCB 4507 ICP accelerometers, frequ ency range 0.3 Hz – 6 kHz , se nsit ivity 101.2 mV /g T urning a n gle tra n s- mitter Pow er supply 5 V DC , max i mum sam ple rate 40 k Hz, res oluti on 14 bi t D ata acq uisition mo d ul e Fea turi ng four 24 - bit s imul taneous l y sampl ed A /D ch a nnels, m axi mum sample rate 80 kHz, and an alog ue a nti - alia sing filte r Table 2.1: I nformation of sensors 1400 Hz 1871 Hz 3148 Hz Dis k 16 Hz 174 Hz 300 Hz L- shaped stee l pla te Shaft 36.5 Hz Table 2.2. E igenfrequen cies and eigenm odes of the com ponents [100] The exp eriment is p erformed at th e speed 0.2 rad/s w ith br ake pressu re 9 bar. T he m easured tor si o nal angle and acceler ation are disp layed in Fig. 2.5 witho ut creep g roan, where onl y sm al l vi br ation can be observed in the to rsional ang le and accelera tion sig nals. Fig. 2.6 (a) and (b) s ho w the torsiona l angle of the shaf t θ ∆ m easured from the strain senso rs on the dr ive shaf t and i ts tim e deri v ativ e θ ∆ (to rsio nal vel o ci ty) d ur in g cr ee p gr oa n . In the expe r iment al re sults, th e typ ica l sti ck - sli p m oti on can be observed : in the stick region, θ ∆ is approxim at ely constant and θ ∆ increa ses linear ly; in the s lip region, θ ∆ changes with t ime. T he accele ration of the brak e pad is g i ven in F ig. 2. 6 (c) meas ured from the acc ele- rom eter on t he brake c arrier. Th er e i s a larg e im pulse w hen the sy st em is s hif ti ng from stick t o s lip, which is cause d by sudden change of the friction. 13 Fig. 2.5: To rsional angle of t he sh af t (a) and accelerat ion of the p ad (b) withou t creep g roan The frequen cy spectr a of th e t orsional v elocity and the accele ration during creep g r oan are exhibi ted in Fig. 2. 7 by pursuing Discrete Fast Fourier Tran sform ation (DFFT). T he s pectrum of the t orsiona l an- gle show s a sing l e pe ak at 36.25 H z, which is the f requency of t he stick - slip motion. I t is noti ced th at this f requency is close to the eigen fr eq uency of the shaft, wh ich im plies tha t the shaft m ay be the pr i- m ary part related to creep g roan. T he spectrum of t he pad osc illation show s a lot of peak s, which ar e m uch higher than the frequency of the stick - sl ip m otion, suc h as 146 H z, 256 Hz a nd 292 Hz . These frequenc ies are app roached to the ei g enfrequ encies of the L - shaped s teel plate, w hich ar e heard f elt by the human during creep g ro an. T h e m easured frequ encies are l ower th an the fi rst eig enfrequen cy of the disk, so that it is possib le to cons ider the disk a s a rig i d body during the study of creep groan . Torsional angle [rad] Ti me [s] (a) Ti me [s] (b) Accelera tion [m /s 2 ] 14 Fig. 2.6 : Torsional ang le (a) and torsional v e locity (b) of the sh aft and a cce l erat i on ( c) o f br a ke pad during creep g roan Ti me [s] (a) Torsional angle [rad] Torsional ve locity [rad/s] Tim e [ s] (b) Accelera tion [m /s 2 ] Tim e [s] (c) Frequency [Hz] (a) DFFT (veloci ty) [ rad/s] 15 Fig. 2.7: Freque ncy spec tra o f to rsio na l v elo ci ty of the shaft ( a) and acceler ation of t he pa d (b) Fig. 2. 8 show s th e m easured stick - slip limit cycles w it h the pressure at 5, 7, 9 a nd 11 bar, where its horizontal axis is θ ∆ and th e vertic al ax i s is θ ∆ . I t is c lear tha t th e a m plitude of the lim it cy cle i n- creases with the pressure. Experim ental results confir m that the stick - slip happe ns during c reep gro an . As a res ult , the friction force i s sw itched between the static and dynam ic friction forces , and the v ibr a- tions i n the br ake pad and t he brake ca rrier a re exc i ted by the varie d fr i ction force . Cre e p groa n is t he result ing v i brations th at can be h ear d or fe lt by humans. Fig. 2.8 : S tick - slip lim it cycles u nder d ifferent b rake p ressure 2.2 Test r ig wit h a r eal b rak e As a com pari so n study , a test rig with a re al brak e wa s assem bled, whi ch co nsti tutes the br ake disk , the brak e pads, the shaft, the cal iper, th e carrier and the suspen sion sy stem. Th e brake car r ier i s as- sembled on the suspension , w ith two brak e pads fix ed in it. The struc ture o f the t est r ig w ith a real brake is shown in Fig. 2. 9. S ensors a re assem bled i n the test rig. A 3D m ot ion and deform ation sensor is used to m easure the dynam ic motion of the pad, the cal iper and the car rier. The 3D m ot ion and def or m ation sensor is a non- contact and mater ial - indepe ndent m easuring sy stem based on digital i m age correla tion. I t offers a stab le solu tion fo r po int - b ased analy ses of test ob j ects of j ust a few m i llim eters i n size [102] . T he t o r- sional ang le of the d r ive sh aft is m easured by the strai n sensors, and th e acceler ations o f t he p ad can b e m easured by accele r om eters. Fig. 2 .1 0 sh ows t he po sit ions of t he s ensors. Torsional ang le [ra d] Torsional ve locity [rad/s] Frequency [Hz] (b) DFF T(accele ratio n) [m/s 2 ] )LJ 7HVWULJ ZLWKDUHD OEUDNH )LJ'P RWLRQDQGG HIRUPDWL RQV HQVRUVWUD LQVH QVRUDQG DFFH OHURP HWHUV 7KH H[SHULP HQW ZDV SHUIRU PHG XQGHU WKH VSHHG DW UDGV ZLWK E U D N H S U H V V X U H D W E D U 7 K H U H V X O W V IURP ' PRWLRQ DQG GHIRUP DWLRQ VHQVRU DUH VKRZQ LQ )LJ Z KLFK LV P HDVXUHG WRJHWKHU ZLWK 'U *Ul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i g. 2.11: In - plane d isplace ments of the p ad, the ca liper and the ca rrier during creep g roan , the arrows shows t he rel ative disp lace m ent vector referen ce to the st atic po sition. Fig. 2.1 2 (a ) a nd ( b) shows the to r sional a ngle θ ∆ of the shaft measu red f rom the st rain se nso rs an d the to rsio nal ang le v elocity θ ∆ , which i s the t ime der ivativ e of θ ∆ . Com pared to the measu rements from the tes t rig wit h an ide alized b rak e, th e sim ilar sti ck - slip motion can b e observ ed at the rea l brak e. The acce leration o f the brak e pad is g i ven i n Fig . 2.1 2 (c ) m easured from the acce lerom eter on the brake pad. Differing from the m easured acceleration at the idealiz ed brake shown in Fig. 2.6 (c), there are double im pul se signals in one s tick - slip period , which are the impulse s i gna ls from sti ck to sl ip transi ti on and from slip to stick trans ition . The fr equency spectr a of the signals during creep g roan are exhibited in Fig. 2.13 by pursuing Di sc r ete Fast Fo urier Tr a nsf or mat i on ( D FFT ) . T he spect rum of the t orsional a ngular ve locity shows d ouble peaks , w here the dom inant frequency is at 30 Hz and the second frequency appears at 60 Hz . T he double peaks ind icate the nonlinea r characte ristic of t he stick - sl ip m otion. T he spect rum of the ac cele- ration of the pa d is shown in Fig. 2.1 3 (b). It s first three peak f requencies (30 Hz , 60 Hz and 90 Hz) are th e fre quency of the stick - sli p m otion and its doub le and trip le f requenc ies. T he hig her frequen cies such as 152 H z, 243 Hz , 366 Hz and 518 Hz are the e xcited vibration of the carrier by the varied f ri c- tion. I t is obvious t ha t the real b rake system has m ore hig h frequencie s of peak th an the ideal i zed brak e. Fig. 2.1 4 shows the exp eri m ental limit cycl es with pressu re at 4, 6, 8 and 10 ba r. Com paring Fi g. 2 .8 with Fig. 2.14, sim ilar s tick - slip limit cy cles can b e observ ed in bo th set -ups . Besides, b oth re s ul ts show that the size of the l im it cycle increase s with the brak e pressure. On the other hand , the ex ited vibrations of pa ds ha ve different f requencies in d i ffere nt test rigs as shown i n Fig. 2.7 (b) and F i g. 2.13 (b). I f a real v ehicle is co nsidered , the vibra ti on of a chassi s can als o be ex cited by t he sudden ly changed frict i on during cr eep g roan. Displacem ent s during s lip Displacem ent s during st ick 18 Fig. 2.12 : Torsional angle (a) and t orsion al velo city (b) of t he shaft and a cce leration of b rake pad du r- ing creep g roan Torsional ve locity [rad/s] Accelera tion [m /s 2 ] Ti me [s] (b) Ti me [s] (c) Torsional angle [rad ] Ti me [s] (a) Frequency [H z] (a) DFFT( veloci ty) [rad /s] 19 Fig. 2.13: F requency spec tra of tors ional velocity (a) and accele ration of the pad ( b) Fig. 2.14: S tick- slip lim i t cy cles under dif f erent brake pres sure 2.3 Summary I n thi s chapter, the des ign of a test r ig w ith an idea liz ed brak e and a tes t rig w ith a real b rake has be en present ed . In both test rigs, creep groan wa s m easured under low velocity with high brake pressure (>4 bar). Com pared the m easured sig nal s to the eigenf requ encies of each com ponent, it implies that cr eep groan m ay relate to the shaft a nd brake ca rrier. Dur ing creep g roan, stick - slip lim it cycles ( ar o und 30 Hz) as we ll as the vibration of t he carri er (between 0~500 Hz ) are m e asured thr ough the stra in senso r and a ccele rom et er. T h e stic k - slip limit cycles from both test rig s are qu al itat ively simila r wi th ea ch other, but the exited v ibr at ions of pads have d ifferent freq uen cies in diffe rent te st rigs. Based on t he m easurements, models con sidering the m ot ion of th e shaft and the ca r rier, a s well as fric tio n laws including the stick- slip e ffect w ill be stu die d in the following chapters . Frequency [H z] (b) DFF T(accele ratio n) [m/s 2 ] Torsional ve locity [rad/s] Torsional ang le [ra d] 20 3 Theoretical Inv estiga tions of Creep Groan o n the Tes t Rig with an I dealized B rake I n this chapter, different minim al models wi ll be proposed to s tudy creep g roan. Com pare d to finite element m e thods with la rge num ber s of deg rees of fr e edom , nonlinear m odel s w ith a low num ber of de gr ees of fre edom are more conveni ent to study the basic excit ation m echanism of creep g roan . At first, m odels with Cou l om b’s friction law are inv esti gated and the stick - slip lim it cyc le c an be sim u- lated. Subsequ ently, m ode ls with the bris tle friction law are propose d to im prove the m odeling. Diff erent m odels w ill be compared with each othe r . Part s of the r esult s i n thi s section h ave already been d escrib ed in [31] -[33]. 3.1 Minim al m odel with Cou lom b’s fr ict ion la w 3.1.1 One DOF model w ith Coulo mb’s friction la w Fig. 3.1: Mode l of the test rig with an ideal ized brak e with 1 DOF [31] The m odel o f the te st rig with a n ide al ize d te st rig is sh own in Fig . 3.1. In th is te st rig , th e disk is co n- sidered as a ri gid b ody and c onnect ed to the m otor by the drive s haft. The pads are fixed on the frame through the carr ier. As the disk can pe rform only rotation but no wobb ling m ot ion, the number of pa ds in the sub s equent m odel is reduced f rom two to one w it hout any in fluence on the qualita tive behav ior of the syst em. The drive s haft is consid ered as a ro tational sp ring with stiffness k θ and damping d θ . I n t his test r ig, the st iffne ss of t he d rive sh aft is m uch lower th an the s tiffness of the brak e carrier. Therefore, only the v ibration of the disk is considered for the sim ulation of creep g roan firstly. The equation of m ot ion fo r this one degree of freedom s y stem is described by ( ) ( ) 00 , R I d k t Fr θθ θθ θ + −Ω + −Ω = − (3.1) Replac e 0 t θ Ω− by θ ∆ , R I d k Fr θθ θ θθ ∆ + ∆+ ∆ = (3.2) 0 t Ω R F 0 t θθ ∆= Ω− , kd θθ N θ θ ∆ θ 0 t Ω 21 where I i s the m oment of i nertia o f the di sk; r is the r adius of the point whe re the pad ac ts; k θ and d θ are the stiffness and dam ping of the drive shaft; θ ∆ is the to rsiona l angle o f the driv e shaf t supe r- posed to the d isk rotation with ang le θ , θ ∆ is equal to 0 t θ Ω− , 0 Ω is the ro tating speed of m otor which assum e d as a cons tant, and R F is the friction forc e in the con t act betwe en the disk and pad. I n Coulomb’s friction law , if the cont act surface s are at rest re l ativ e t o each other, the friction for ce work s as the st atic friction force, whi ch prevent any relat ive m otion up unti l a threshold force . T hi s thresho ld force is ch aract eri zed b y the norm al force tim es the st atic f riction co efficien t s µ [103]. If t h e conta ct sur faces are in rela tive m otion , the frictio n co eff ic ien t is th e dy nam ic fric tio n coe ffic ien t d µ . s µ is no rma lly la rger th an d µ . d µ can be d escri bed a s a constant or a v elocity depend ent term [ 4 3], [58], [95], [97], [104]. For the sak e of s implif icati on, d µ is at firs t assum ed to be co nst an t. The co n- tact fo rce in the stick and s lip reg ions is t herefo r e g iven as ( ) [ ] 00 0 sgn , if (slip) , , , if (stic k ) Rd s R ss s FN r r N d k F NN N d k θθ θθ µ θ θµ θ θ µµ θ µ θ θ = Ω −∆ Ω ≠ ∆ ∪ ≤ ∆ + ∆ ∈− Ω = ∆ ∩ > ∆ + ∆ (3.3) where sgn m eans sign func t ion, N is norm al force. No te that the force law f r om Eq. ( 3.3) is m ultiv a- lued , a fact that s eem s t o contrad i ct a unique d eterm ination of the forc es betw een the st ick and slip re gi o n s . Ho we ver , if th e force law is com bin ed with som e equations of m ot ion of a dy namical system , uniqu ene ss can b e gu arant eed in m any cases [51], [103] . By com bining the friction law with the d y- nam i cs of the b rake sy stem, the dyn am ic equat i on i n the slip reg i on ( 0 0 rr θ Ω −∆ > ) is 0 , fo r ds I dk N r N dk θθ θθ θ θθ µ θ µ θθ ∆ + ∆ + ∆ = Ω ≠ ∆ ∪ ≤ ∆+ ∆ (3.4) while the dy na m i c equation in the s tick region i s 00 0 , for s t Nd k θθ θθ θ µ θ θ ∆ = ∆ +Ω Ω = ∆ ∩ > ∆ + ∆ (3.5) The sy stem is then a p iecew ise line ar sy st em . I t can be analy t ical l y solved in e ach region . On t he on e hand, the solution in the sl ip region is ( ) ( ) ( ) ( ) 01 0 2 0 21 0 1 2 0 0 exp ( - ) cos ( ) sin ( ) , 2 cos ( ) sin ( ) exp ( ) , 2 22 l l le l ll d t t C tt C tt I d dd C C tt C C tt tt I II θ θθ θ θθ θ θθ θ θ ω ωθ θω ω ω ω ∆ = − − + − +∆ ∆= − − − − − − − wher e 2 02 1 0 1 1 /, = + . 22 ,, e d le l dd k k Nr I I C I C C θθ θ θθ θ ωµ θ θθ θ ω ∆ ∆∆ ∆ = −= − = (3.6) where t is current tim e, 0 l t is the time when the system just ente r s the s lip region, e θ ∆ is the equil i- brium position, and 0 l θ ∆ and 0 l θ ∆ ar e the initia l tors ional a ng le an d ang le ve lo city at tim e 0 l t . On the other han d, the so lut ion i n the s tick r egion is 22 00 0 0 () , , t tt θθ θ ∆ =Ω − +∆ ∆= Ω (3.7) where 0 l t is the t im e when the sy st em j ust en ters the sti ck reg ion, and 0 l θ ∆ is the initia l to rsio na l a n- gle at tim e 0 l t . Eqs. ( 3.6) and (3.7 ) shows that a uni que so lution exi sts when init ial cond ition s are gi ven . I f the brak e system f alls in to the at tracto r of the s t ick region, it s motion is described by Eq . ( 3.7). I n con trast, its m otion is desc ri bed by Eq. ( 3.6) in the slip reg ion. I f the system is alway s switche d be- tween those tw o so lutions, a st ick - slip lim it cycle can be o bserv ed as shown i n Fig. 3.2 , with the fr i cti o n l aw s ho wn i n Fi g. 3.3 . I n the stick region, the d isk adhe res to the pads , lea di ng to th e increas ed torsional angle of t he drive shaft, a nd en er gy is stored in the driv e shaft. In the slip reg ion, the disk is separat ed w i th th e pads and m ov es under a dynam ic fric tio n . I n t his reg ion, the k inematic en ergy co n- verts into heat. There is a non - s m ooth behavior from sl ip to stick or from stick to slip . Fig. 3.2 : S tick - s lip m otion It s hould be noted that the period of the stick- sli p m otion T p is larg er than T n , w here T n is the harmonic pe rio d of the system with out stic k - slip (dashed line i n Fig . 3.2) . T he period of stick - slip motion is equal to the stick time p lus the s lip t im e , p st sl TTT = + (3.8) Assum ing that the d am ping of the system is ignored, the am plitude o f t he stick - slip lim it cy cle max θ ∆ are approxim ately calcula t ed as 2 2 2 2 22 2 22 0 max 0 0 ma x 2 0 () , () sd sd Nr Nr θ ω µµ θ µµ ω Ω ∆= Ω + − ∆= + − , (3.9) where max θ ∆ is the am plitude of the s tick - slip vibration, max max eq θ θθ ∆ = ∆ −∆ . T hen, the slip time sl T can be ca lculated a s 0 00 max 2 arcsin , sl T π ωω θ Ω = + ∆ (3.10) and the stic k tim e st T can be e xpressed as Ω 0 θ ∆ θ ∆ θ ∆ t S tick S lip S tick r e gi on S lip re gi on T p T n max θ ∆ max θ ∆ 23 0 2( ) . sd st Nr T k θ µµ − = Ω (3.11) As a re sult , the period o f st ick -slip m ot ion is giv en as 0 00 0 max 2( ) 2 arcsin . sd p Nr T k θ µµ π ωω θ −Ω = ++ Ω ∆ (3.12) Therefore, som e c onclus ions based on the sim ple model can be d rawn as: • A stick - slip lim i t cycle can be obs erved du ring the st i ck - slip motion; • T he per iod of the stick - s lip m ot ion is larger than th e harmonic periods of the system wit hout stick - slip; • T he pe riod of the stick - slip motion increase s with in creasing normal force or d ecreasing m otor speed; • T he amp litu de o f the s tick - slip v ibratio n incr ease s with the norm al f orce or the m otor speed. The aim of the rem aining part of this sectio n is to find out the ex istenc e condit ions of stick - slip m o- tion s. In nonlinear d ynam ics, a l im it cycle is an isolated period i c solut ion in a self - excit ed system [104] . Th ere fore , the exi stence of a stick - slip lim it cyc le can be judged by: if a system leaves the sti ck region and can return t o the attracto r of the stick region ag ain under isolated env ironmen t , the system has a stab le stick - slip m otion . Since Eqs. (3.6 ) and (3 .7) are the analy tic al s olu ti on of the p iecew ise differ entiab le Eqs. (3.4) an d (3.5), subs tituting an i niti al condit ion in the stick reg i on int o E q. (3.6) and (3.7), its solu tion, whe ther a st ick - slip lim it cy cle o r equil ibrium solu tion, c an be d i rectly obtained. Fig. 3.3: Fric tion coeffic ient of Coulom b’ s fri c- tion law , where s µ is assum ed l arger tha n d µ , and d µ is const ant , v is the r elative v elocity Fig. 3.4: E xistence condi t ion of the s t ick - slip m otion, regio n I : witho ut stic k - slip lim it cy cle b u t with a sta b le equ ilib rium solution, region II : with stable st ick - slip lim it cycl e and stab le eq uilib rium solu tion When the dam ping d θ is larg er than 0, the system has alw ays a stab l e equil ibrium solution in the slip region according t o Lyapun ov stability theory [104] . I n contrast, t he conditions for the existence o f the stick - slip lim it cycle depen ds on the p arameters such as N and 0 Ω . With var ied pa ram eters, the system s µ d µ II I 0 Ω N 0 0 v µ 24 has two regions as sket ched in the Fig . 3.4, i.e. a r egion wi th stable s t ick - slip lim it c y cl e and equ i l i- brium solution , nam ed as region II ; a region with out the st ick - slip l imit cy cle bu t wi th a sta ble equilibrium solution , nam ed as reg ion I . It shou ld be noti ced that in reg i on II the sy stem has two stabl e solu tions so tha t the o ccurrence a nd absenc e of th e stick - sl ip m otion depends on it s initial cond ition. I t is in te res tin g to k now under wh ich in itial cond itio n s creep g roan w ill occu r in re gion I I . T h ere is a boundary between the attractor s of the equilibri um soluti on and t hat of the lim i t cy cl e. I n or der to find this boundary , a cri t ical poi nt is def ined as ,0 ,0 0 , b eb θ θθ ∆= ∆ ∆= Ω as shown in Fig. 3.5 m arked wi th the red point. I f a trajectory is low er than this cr itical p oi nt, th is tra jectory cannot reach the stick region anym ore and g oe s to the equilib rium so lution . The ref ore, it is poss ible to f ind a traje cto ry w hich can just pass thro ugh th e crit i cal point , and this tra ject ory is noth ing els e but the boundary between t wo attracto rs. When the initial condition is outside the cri tic al traje ctory , the solution is the lim it cycle solution. Otherwis e , the resulting solution is th e equili brium solution. Fig. 3.5: The boun dary betw een the attracto rs of the equi l ibri um solution and st ick - s lip l im it cycle . T he r ed p oin t i s the cr iti cal poi nt d efi ned as ,0 ,0 0 , b eb θ θθ ∆= ∆ ∆= Ω . I f a given tr aject ory is l owe r than thi s point, th e system can not r each the st ick reg ion (show n with green curve) . T he r e d da sh li ne shows the bou ndary b etween the attractors . I n shoul d be noticed th at the system in the slip region h as a linea r behavio r , and only a unique trajecto- ry can p ass throug h this critica l poin t. As a re sult, i t is possib l e to u se negativ e time integ r ation t o ca l cul ate t h is trajecto r y bas ed on t he crit ical point (su bst itu ting neg tt = − t int o E q. ( 3. 6) ) . T h e cr iti c al tra jec tory is ca lcu late d as ( ) neg 1 neg 2 neg neg 2 1 neg 1 2 neg exp cos( ) sin( ) , 2 exp cos( ) sin( ) , 22 2 b d de b d dd d d tC t C t I dd d t C C t CC t II I θ θθ θ θ ω ωθ θ ω ωω ω ∆ = − + − +∆ ∆= − − − − − with (3.13) θ ∆ θ ∆ 25 ( ) 0 neg 2 neg 0 1 11 , C = , [0 0 , ], , , i bi d tt t t t C θ ω ++ Ω = −∈ = ∆= Ω where the ca lcu la ted the cri tical tr a ject ory is shown in Fig. 3.5 wit h th e r ed dash li ne. Thi s tra jec to ry is the boundary be tween th e attractor s of the equ ilibrium solution and st ick- slip lim it cy cle. 3.1.2 Tw o DOFs model w ith Coulomb’s friction l aw I f both the vibration s of the disk and the pad a re cons ide red in th e m odel, the te st rig wi th a n idea lize d brake can b e tre ated as t wo sub - system coupled by the friction forc e . There are r ot at in g pa rt s i .e. di sk- shaft sub - system, and non - rotat ing part s i.e . pad - car ri er sub - system . T wo pads ar e cons idered as one rigid body conne cting to the base frame with a spring , since the two pad s m ove sim ultaneously during creep groan. The disk is treated as a r igid body connec t ing to the m otor with a torsional sp ring. I f the system is in the sl i p reg ion, the dy namics o f t he t wo sub - system s can be de scribe d as ( ) ( ) 0 0 sgn , sgn , d xx d I d k Nr r r x m xd xk x N r r x θθ θ θ θµ θ µθ ∆ ∆ ∆ Ω −∆ − + + = Ω −∆ − + += (3.14) where m is th e m ass of th e pad; x k and x d are the stiffne ss and damp ing of the carrier; x is the di s- plac eme nt in v ertic al di rec tio n o f th e pad, th e m ode l is g i ven in Fig . 3.6. Fig. 3.6: Mode l of the te st rig with an idealiz ed brak e with 2 DOF s In contrast, the pad and d isk are adhered w i th eac h oth er in the stick reg ion, the d ynamic equation is ( ) 2 00 /, ,. xx m I r xd xk x k d xr t r xr r θθ θθ θθ + + + = − ∆− ∆ = Ω −∆ = Ω −∆ (3.15) The sy stem is in th e stick reg ion when th e follow ing condit ions are f ul fi lled : The di sk a nd p ad h a ve no relativ e motion; meanwhi le the m aximal stat i c fr i ction force is l arge enou gh t o mak e t he pad and d isk to adher e, stick condition: 0 . s xx r x N m xd xk x θµ −= ∩ ≥ + + (3.16) Otherw i se, the sy st em is in the s lip reg ion. 0 t Ω R F , kd θθ N x θ 26 The sy stem is a piecew ise l inear system . It can be sep arately s olved in each regio n. In the s lip reg ion, the motion of the shaf t an d the pad are ( ) ( ) ( ) ( ) 01 0 2 0 21 0 12 0 0 exp ( - ) c os ( ) sin ( ) 2 cos ( ) sin ( ) exp ( ) 2 22 l l le l ll d t t C tt C tt I d dd C C tt C C tt tt I II θ θθ θ θθ θ θθ θ θ ω ωθ θω ω ω ω ∆ = − − + − +∆ ∆= − − − − − − − ( ) ( ) ( ) ( ) 01 0 2 0 2 1 0 12 0 0 exp ( - ) cos ( ) sin ( ) 2 cos ( ) - sin ( ) exp ( ) 2 22 x l xl xl d x xx x xl x xl l d x t t D tt D tt x m d dd x D D tt D D tt tt m mm ωω ω ωω ω = − − + − + = − −− − − − (3.17) wher e 2 02 1 0 1 1 /, = + , 22 ,, e d le l dd k k Nr I I C I C C θθ θ θθ θ ωµ θ θθ θ ω ∆ ∆∆ ∆ = −= − = 1 0 2 02 1 1 / , , =+ , 22 , xx x dx l ee l x x k xN D dd k xD x mm xD m ωµ ω = −= = − (3.18) where 0 l t is the tim e when the sy stem just ente rs in the slip region, t is t he current time, and 0 l θ ∆ , 0 l θ ∆ are th e in iti al to rsi ona l ang le and to rsio na l veloc ity o f the shaft at 0 l t , 0 l x and 0 l x ar e t he ini tial displac ement and v el o city of the pad at 0 l t . On the other han d, t he so lution i n the s t ick r egion can b e calculat ed by num erical integ ration ( ) 00 0 2 00 0 0 1 () , / () ( ) , () ( ) , /, () ( ) , ll l xx tt ll tt t l t xt d x k x k d m Ir xt x d x xt x d x xr td θθ θθ ττ ττ θ θ θτ τ θ = − − − + ∆+ ∆ + = += + ∆ = Ω− ∆ = ∆ +∆ ∫∫ ∫ (3.19) where 0 l t is the tim e when the sy st em j ust enters i n t he stick region, t is the cur rent tim e, and 0 l θ ∆ , 0 l x , 0 l x are the in itia l stat e vari ables at ti me 0 l t . 27 Fig. 3.7 : Stick - slip lim it cycle o f the shaf t (a) and the pad (b) Fig. 3.8 : S im ula tio n to rs ion al angle θ ∆ (a ) and tors iona l velo city θ ∆ (b) of the disk Fig. 3.9 : Acc e leration of the pad (a) and f riction fo rce ( b) Num er ical simulation is ca rried out to study t he stick - s lip lim it cycle of th e syste m. The param et ers o f the syst em are arbitr arily chosen by the au thor and g iv en in T ab le 3.1. A stick - sl ip lim it cycle can b e observed in the phase d iagr am θ ∆ as a function of θ ∆ , shown in Fig . 3.7 . In ord er to d i ffer the stick region from t he slip region, the red line deno t es the stick re gion and blue line den ot es the slip reg i on. One can se e t h at t her e ar e no n - s mooth r egi o n s between the stick motion and sl ip m oti on in the simul a- tion resul ts [103], [104] . The simula t ed torsio nal angle θ ∆ an d tor si o nal ve l ocit y θ ∆ are exhibit ed in Ti me [s] (b) R F [N] Ti me [s] (a) θ ∆ [ rad ] θ ∆ [rad /s] Ti me [s] (b) Tim e [ s] (a) x [m /s 2 ] θ ∆ [ rad ] (a) θ ∆ [rad /s] x [ m] (b) x [m /s] 28 Fig. 3.8 . The sim ulated acceleratio n of the pad x and fri ction force R F are shown in Fig. 3. 9 . T he advan tages o f t h e m odel w ith Co ulom b’s friction l aw ar e th at the st ick and slip reg ions can b e sepa- rated cl early , and the s tick and slip m otion can be analy zed separate ly. The disa dv antage is that the non- smooth pr oblem appear s between the stick and slip regions. The n on -smooth nature of Coulom b’s m odel m a kes the d i ffi culty of t h e num erical integ ration. T herefore, instead of C oulom b’ s friction l aw, the bri stle f riction law will b e stu di ed in the l ater section to i m prove the sim ul ation resul t s. Paramet ers Values Paramet ers Values I 1kgm 2 m 0. 1kg k θ 100 Nm k x 1000 N/m d θ 1 Nms d x 10 Ns/ m s µ 0.3 r 1 m d µ 0.35 N 5 N 0 Ω 0.01 rad / s Table 3 .1: Par ameters o f th e system I f the damping of the drive shaft is incr eased to 0.4 Nms and the damping of the carrier i s de crease d to 0.01 Ns/m, different thing s will happen as shown in Fig. 3. 10. Fig. 3. 10 (a) shows the frict ion force and (b) shows the stick - slip lim it cyc le of the p ad. It is n ot ice d tha t th e s tick - s lip l im it cyc le i s re late d to the dam ping of the syste m . If the damping of the shaft i s low , a stick - slip lim it cy cle is o bserv ed in the di sk- s haft sub - system . In contrast, if the damping of the ca rrier is low , the stic k - slip lim it cy cle is obtained in the pad - carr ier sub - system . Fig. 3.10 : F rictio n fo rce (a) , stick - slip limit cy cle of th e pad (b) with 0.4 Nms d θ = , 0.01 Ns/m x d = Ti me [s] (a) R F [N] x [m] (b) x [m /s] 29 3.2 Minim al m odel with the bri stl e fri ction la w Even thoug h the model with C oulomb’s friction law c a n help us to unde r stand the m echanism of the stick - slip motio n, it i s t oo sim ple to exp lain sev er al effects o f cree p groan in a bra ke sy stem. In ord er to mak e our sim ul ation c l ose to t he experimenta l result s, the bristle f riction law i s c hosen f or model ing creep groan in this s ec tion [67]-[70]. Fig. 3.11 : Mode l of th e bra ke tes t rig with an ideal ized brake w ith th e br istl e fri ction l aw 3.2.1 One DOF model w ith the bristle friction la w The m odel of th e test rig i s given in F i g. 3.11. T he d isk is conside r ed as a rigid body and co nnected with the m otor by the d rive sha ft, w hile t he driv e shaft is co nsidered as a rotation al spr i ng . Th is model and correspon ding sim ulat ion resul ts com pared with t he experim ent al resul ts ha ve been published in [32]. At first, onl y the vibration of the disk is con side red i n the m odel . The equation of m oti on of the one deg ree of freedom system is described by , R I d k Fr θθ θ θθ ∆ + ∆+ ∆ = (3.20) wher e I is the m o m ent of inertia o f the disk ; r is the r adius of the po int wh ere th e pad ac ts; k θ and d θ are the st iffne ss and dam ping o f the drive shaf t; ∆θ is the t ors ional angle of the drive shaft supe rposed to the disk rota tion with an gle θ , θ ∆ is equ al to 0 t θ Ω− , 0 Ω is the rotating s peed of m otor , and R F is the f riction force in t he co ntact b et we en t h e disk and pad. The br istle fr ictio n law , which is propos ed by Canudas d e Wi t e t al. [6 7] -[70], is used t o calcu late R F . This theo ry is b as ed on the im agina tio n th at two rig id bod ies are in co n tac t th roug h v i sco - el astic br i s- tle sur faces. Wh en a tangen tial forc e i s ap plied, t he b ristles w ill de flec t like sp ring s which giv e rise to the fr iction force. If the force i s suff icien t ly l ar g e t o make som e of the bris t les d eflect th en sl ip occu rs. The dynam ic friction force can be expr ess ed as 01 , R F zz σσ = + (3.21) whe r e z is the averag e deflect ion of the bri stles, 0 σ is their sti ffness, 1 σ is their damping, and z is a nonlinear function of θ ∆ and z 01 , ,, , s ds v µµ σ σ r θ 0 t Ω R F 0 t θθ ∆= Ω− , kd θθ N x θ x 30 00 0 ( , ) | |. z z z rr rr g φθ θ θ = ∆ = Ω −∆ − Ω −∆ (3.22) Here 0 g is giv en as 0 0 0 () 1 ( ) exp , d sd s r g NN v α θ µ µµ σ Ω −∆ = +− − (3.23) where v s is t he Str i be ck ve l oc it y, N is the b rak e normal force, s µ is the sta tic fric tion co ef ficie nt, d µ is the dy namic fr iction coeff icien t , α i s an empiri cal pa rameter w hich can be m easured in experim ents. The value α = 1 is sug gested for th e dry contact wh ile α = 2 is prefer r ed for the lubri ca tion contact [105]. The friction coeffic ients for α = 1 an d α = 2 are giv en in Fig. 3.12. Fig. 3.12 : F ricti on co effic ient of α =1 an d α =2 , v i s t h e rel ativ e veloci t y This brist l e fric tion law ca n descr ibe pre - slid ing and hy steresis ch aracte ristics of frict ion. The c o m- plete dynam ic equations c an be wr i tten as a set of firs t order diff erential equa tions T 01 00 0 ( ,) ( ,) , ( , ) | |, , dk rr zz I I II z z rr rr g z z θθ θ θ θ σ σφ θ φθ θ θ φθ θ θ = ∆ − ∆− ∆+ + ∆ ∆ ∆∆ ∆ = = Ω −∆ − − Ω∆ Y Y (3.24) where Y is the v ector of s tate v ariables of t he sy stem. The equ ili brium s olut ion o f Eq. (3.2 4 ) i s e x- pressed a s T 00 0 0, 0 , ( ) exp / , ( ) e x p / . eq eq eq eq eq d s d eq d s d ss z rr r NN k z NN vv αα θ θ θ µ µµ µ µµ σ = = ∆ ΩΩ ∆ = +− − = +− − YY Linearizing Eq. ( 3.24 ) about its equi librium pos ition 31 0 11 , 01 0 ( ,) ( ,) . ( ,) ( ,) 0 kd r rr zz I II I I z zz z θθ σ σσ φθ φ θ θ φθ φ θ θ = ∂∆ ∂∆ = − −+ + ∂ ∂∆ ∂∆ ∂∆ ∂ ∂∆ A A YY (3.25) I n t he case o f α = 1 ( ) ( ) 0 2 0 0 0 0 ( ( ,) (, ) ) , ( )e , ( exp / ) xp / sd ss s dd ds s d z r r z z NN r v v r v µµ µµ µ µ φθ φ µ σ µ θ θ −Ω −+ Ω ∂∆ = Ω −Ω ∂∆ − ∂∆ = ∂ +− and in the ca se of α = 2 ( ) ( ) 32 0 2 22 2 0 00 22 2 0 2 (/ , ex () ( ,) , exp ) ( ,) () p/ sd sd d d sd ss s r v rv r r z z z N v N µµ φθ θ µµ µ φθ µ µµ σ Ω +Ω −Ω + − ∂∆ = ∂∆ − ∂∆ = ∂ − − Ω where A is the co rr espond ing system matrix. The stab ility of the equilibrium solution can be ob ta i ned by ana lyz ing the eigenv alues of A . If t he real pa rts of al l eig envalues ar e neg ative, th e equilib rium solutio n i s asy mptotica lly stab le. I f any of the rea l p arts of e igenv alues is posi tiv e, t he eq uilib rium solutio n is unst able, and the so lution w ill sh ow incr easing am plitudes. On the other hand, the lim it cycle of E q. ( 3.24) ca n be d et erm i ned by transi ent anal ysis using num erical in tegration. Fig. 3.1 3 : Three reg ions and the pos sible phase plots w ith variation of driving speed (a) α =1, (b) α = 2, the red line is t he m aximum real part of eigenv a lues of th e equilib rium solution, and t he blu e line giv es the amplitude of the s tick - slip lim it cy cle. I n the following, possible parameter reg ions of solut i ons of the sy stem a re discussed qualita tively. According to the ex iste nce conditi ons of the st ick - slip lim it cy cle an d th e stabi lity o f the eq uilib rium solutio n, the system shows three dif f e rent reg ions w ith dif ferent types of so lutions. When the sy stem Re( ) λ + + III II I 0 Ω (a) LC A 0 - + 0 - + II I III 0 Ω (b) Re( ) λ LC A 0 - - II 0 32 has no stick - slip lim it cycl e and has a st able equ ilibrium soluti on , cre ep groan cannot occur in thi s param eter r egion w hich i s label ed as region I . When t he system has a s table stick - sli p lim it cycle an d stable equ ilibrium solutio n (and an uns table lim it cycle so l ution in betw een), the syst em may have creep groan or not, whic h depend s on its initial condi tions. This region is labele d as region II . When the system has a stabl e stick - slip li m it cycle and an unstable equilibrium solut ion, creep g r oan will alway s occur and this r egion is la beled as r egion II I. Fig . 3.1 3 exhib its t he three r egions quali tatively and the po ssibl e phas e plot s with th e varia tion of th e motor speed, w here th e r ed line is the m axim um real part of eig envalues of the equil ibrium solu tion, and th e blue line giv es the a mplitude of the s t ick- slip lim it cy cle . With th e in crease of d r iving speed, the sy stem for α = 1 g oes throug h r egion II I, II and I , respectively , while the system f or α = 2 goes throug h region II , III , II and I, respectively. I f the driving spee d and b rak e pressure a r e b oth v aried, th e three p aram eter regi ons I - I II are quali t a- tiv ely sk etch ed in F ig. 3. 1 4. The d istr ibutio n of para meter region s are nam ed as map of cr e ep gr o an, since it can show the condi tion of gene rating creep g r oan. The boundary between regions II and III is defined as a curve, where the largest real p art of the e igenvalues o f the sy stem matrix A is equal to 0. The way to determine the boundary between r egions I and II i s giv en as follow s: For vary ing brak e pressure and speed, the s olutio n of the non linear sy st em can be calcul ated by a n umerical t ime integr a- tion of Eq . (3. 24) wit h in itial cond itio ns in the sti ck reg ion. I f the so lutio n is s till th e sti ck - slip lim it cycle af ter a w hile, th e stic k - slip lim it cy cle is con side red to exi st a nd be s t able. In additio n, its equ i l i- brium solu tion is a sym ptotical ly stab le and the system is in region II . If the solution converges to the equilibrium solution, a stable stick - s lip l im it cycle is cons idere d not to ex ist, i. e. the sy stem i s in r e- gi on I . By varying the brake pressure and dri vin g speed, the regions I and II, as well as the b oundary between them can be determ ine d. Fig. 3.1 4 : Map of creep g roan , (a ) α = 1, (b) α = 2 , r egion I : the syst em has no stick - slip lim it cycl e and has a stabl e equ ilibriu m solution ; region I I : the sy stem has a stab l e sti ck - slip lim it cy cle and s table equilib r ium solution; reg ion II I : the system has a stabl e stick - slip limit cycle and an unstable equi l i- brium solution There are som e differen ces betwe en the c ases α = 1 and α = 2. For the case α = 2, the system i s unde r reg ion II with v ery lo w v eloc ity, whi ch is not the ca se when α = 1. T his differe nce is shown in Fig. 3.14 . The stable equ ilibrium solution and the stick - s lip lim it cy cle so lution in regio ns I to III are ske t c he d i n Fi g. 3 . 1 5 fo r α = 1 and in Fig. 3.1 6 for α = 2. Simula tion resu lts prov e that the stabl e equ i- librium solution and the stick - s lip lim it c yc le solu tion exi st sim ultan eou sly in reg ion I I . In region I, ev en th oug h the in itia l con dition is in the st ick reg ion, th e s y stem return s th e equ ilibr ium soluti on a fter fin ite tim e . In reg ion III , even though the i nitia l cond ition is i n th e equi librium solution, the am pl itude of the sy stem increase s until it rea c hes the st ick - slip li m it cycle. The lim it cycles are s i m i lar with e ach (a) (b) 33 other in the cases of α = 1 and α = 2. Accor ding to t he experim enta l r esults show n in the ch apter 5, α = 1 is a m ore reasonable cho ice for our test rig . For the late r modeling, only α = 1 is considered. The m ap of creep groan in dicates the n ecessary condi tions for g ener ating cr eep groan, which can be used to evalu ate the brak e system with respect to creep g roan. If the a r ea o f reg ions I I and II I is larg e, the brak e system has a h igh prob abili ty to show cre ep g roan, and vice v ersa. Com pa red to the system with Coulomb’s fric tion law, the system using the bristle friction law has an additional region, i.e. reg ion III . Besides, si n ce the pr e - sliding effe ct includ es in the brist le fr ictio n law , there is no sudden change b etween the st ick reg i on and sli p region any more. Fig. 3.15: Eq uil ibrium solu tion an d th e stic k - slip l imit cycle so lutio n α = 1 Fig. 3.16: Eq uil ibrium solu tion an d th e stic k - slip l imit cycle so lutio n α = 2 I n the following pa r t of this se ction, th e bifurc ation beha vi o r of the sys t em wi ll be st udi e d. F i g. 3 .1 4 already shows the bifu rcation beh avior, and its pro perty wil l be further studied by analy tic al method i n this part. I n order to sim plify the analysis, it i s assum ed that 0 σ is approxim at e to infinite and z is approxim ate to 0, Eqs. ( 3.20)- (3.23 ) can be written as in s l ip r egion and stick reg ion separate l y . I n the slip region, t h e dy nam i c equ ation i s giv en as 0 () exp , b bb d s r I d k rN N v θθ θ θ θθ µ µ Ω −∆ ∆+∆+ ∆ = + ∆ − with . sd µµ µ ∆= − (3.26) I n the stick region, the disk will stick with the pad and the friction f orce between [ ] , ss NN µµ −+ will balance the force of the torsiona l spring. θ ∆ [ rad] θ ∆ [rad ] θ ∆ [rad ] 0 = 0.05 rad s Ω 0 = 0.03 rad s Ω 0 = 0.015 rad s Ω θ ∆ [rad ] [ ] rad s θ ∆ θ ∆ [rad ] θ ∆ [rad ] 0 = 0.05 rad s Ω 0 = 0.03 rad s Ω 0 = 0.01 rad s Ω [ ] rad s θ ∆ [ ] rad s θ ∆ [ ] rad s θ ∆ [ ] rad s θ ∆ [ ] rad s θ ∆ 34 At f irst, the ana lytic al stud y is g iven in th e slip reg i on . A similar case has been studied by Hetz ler et al. in [95] with the same m ethod . By introducing the coo rdinate transform at ion ( ) ( ) 0 1 exp / ds rN r v k θ θθ µ µ ∆ = ∆ + + ∆ −Ω 00 () exp exp . ss dk r r rN rN II I v I v θθ θ µµ θ θθ Ω −∆ Ω ∆∆ ∆+ ∆+ ∆ = − − − (3.27) Rewriting Eq. (3. 27 ) into ( ) 2 00 2 ex p , D θ ω θ ω θ γ βθ γ ∆+ ∆ = − ∆+ ∆ − with 2 0 0 0 , , ex p , . 2 ss kd r rN r D II I v v θθ µ ωγ β ω Ω ∆ = = = −= (3.28) T hen , wit h the assum ptions 2 0 1 D ω and 1 γ , the state v ariab les ca n be described by sine or cosi n e functions ap prox imately 00 22 00 sin ( ) cos ( ) sin ( ) A AS A AC A AS θ ϕϕ θ ω ϕω ϕ θ ω ϕω ϕ ∆= = ∆= = ∆= − = − (3.29) where S denotes sine func t ion and C deno tes cosine function . Eq. (3.29 ) can be interpre t ed as the transfo rm ation to po lar coo rdinate f rom state spa ce . Subst ituting Eq. (3.29) into E q. (3.28) ( ) 22 00 0 2 () e x p () . D AC AC θ ω θ ω ϕ γ βω ϕ γ ∆+ ∆ = − + − (3.30) The right- hand si de of the e quation can be written a s ( ) 2 00 2 () e x p () . f D AC AC ω ϕ γ βω ϕ γ = −+ − (3.31) According to the slowly changed am pl itude and phase m et hod [104], the am plitud e is obtained by averag ing f over one p eriod ( ) 2 0 22 00 1 cos 2 1 exp ( ) , A fd D A C AC C π ϕϕ π ω γ βω ϕ π ′ = = −+ − ∫ (3.32) where . denotes the averag e of a function ov er one period. I n order to calcu l ate A ′ , Eq. (3.32 ) i s rewritten as a series represe ntation of the exponent ial functions, yielding 35 2 22 0 0 0 1 2 21 21 2 21 0 0 1 1 2 2! . ( 2 1) ! 2 n nn n n k kk k k AC A D A C Cd C n C DA A k π βω γ ω ϕπ ππ βω ωγ π ∞ = −− ∞ − = ′ = −+ − = −+ − ∑ ∫ ∑ (3.33) As a re sult, a constant A can be cal cul at ed w ith A ’ =0 2 21 21 2 21 0 0 1 0. ( 2 1) ! 2 k kk k k C DA A k βω ωγ π −− ∞ − = −+ = − ∑ (3.34) I t is obvious that A = 0 is one of th e solutions in the sy stem, which is the eq uilibrium sol ution. Another approxim ate solu tion , wh ic h is th e lim it cy cle solu tion , has bee n calc ulated by H etzler e t al . i n [ 95] by chosen k as 1 and 2 (3 r d order approximation) . Her e , it is p os sib le t o get a better app roxim ate result with h igh orders appro ximation 2 3 34 5 56 2 21 21 00 0 2 3 5 21 0 0 ... 0. 2 12 240 (2 1 ) ! 2 k kk k CC C C D AAA A A k βω βω βω βω ωγ γ γ γ ππ π π −− − − + + + ++ = − (3.35) I n order to distinguish the stick - s lip lim it cycle , the no ze r o s ol ut i on of Eq. (3.3 5 ) is call ed sli p limit cycle, si nce it occur s in sl ip reg i on. H etzl er et a l. [9 5] provided that this s ol ut ion is an unsta ble lim it cycle. I f stick - slip occu rs in the system , by assuming the system wit h low dam ping, the am pl itude to the stick - slip lim it cy cle can be calcul a t ed as ( ) 2 2 0 max 0 () . sd I N rd k θ θ θ µµ Ω ∆ = + − −Ω (3.36) Fig. 3.1 7 shows the e qu ilibrium solut ion, s tick - slip li m it cycle and slip lim it cy cle w ith diff eren t or der approxim ation . I n this figure, regions I , II and III are calcula ted by numerica l integ ration . The boun- dary between reg ions II and III is coincided with the H opf - bifurcat ion. With increasing of t he speed, the am plitud e of s tick - slip lim it cy cle wi ll go ac ros s th at o f th e sli p lim it cy cle . By in c rea si n g the or der of approximation, the c ross point will be ap pr oxi mat e to the boundary between regions II a nd I. Ther e- fore, it is pos sible to us e t his cro ss point to determine the boundary betw e en reg ion s II and I . I n re gion I I, the slip limit cy cle is nothing else than t he boundary between the a ttractors of the stic k - slip limit cycle and the eq ui librium sol ution, show n in Fig. 3.18 (a). When the initial condition lie s ins ide of the sl ip lim it cyc le, the sy stem goes to an eq u ilib rium solu tion. When the in iti al cond ition lies outs ide th e slip lim it cycl e, the amp li tude of the sy stem will incr ease un til it re aches the stick reg ion [95]. 36 Fig. 3.1 7 : Amplitude of st ic k - slip lim it cycl e and s lip lim it cycle with 3 rd, 5th , 7th , 9t h and 11 th or der approxim ation , the red points are the boundarie s between region s I, II and III calculated by the num e r- ica l in teg ra tion Fig. 3.18 : (a) E quilibrium solution, slip lim it cycle sol ution and stick - slip lim it cy cle so lu tion in r e- gion II . ( b) The sy stem h as no stick - slip lim it cycle wh en the am pli tud e of s l ip l i mit cy cle i s larg er than that of the s tick - slip li mit cy cle. At a criti cal speed, the amplitude of sti ck - slip lim it cyc le and tha t o f the s lip lim it cycle go across with each othe r . I t is intere sting to k now the phy sical meani ng of this c ross point. Whe n the stick - slip lim it cycle is la rge r than the s lip lim it cy cle, the sy stem w ill alw ays h a ve stick - slip m otion i f th e init ial c o n- dition is gi ven in the stick r egion. Therefo re, th e stic k - slip l imit cy cle l arger than the slip lim it cyc le is a condition th at a sy stem has a stick - slip lim it cycle . Fig. 3.1 8 (b) shows th at the s tick - slip wil l not happen when t he amplitude of t he slip limit cy cle is lar ger th an th at o f the sti ck - sl ip li mi t cycle. Th ere- fore, i t is pos sible to use t his cross po int to app roximate the boundary between reg i ons I and II . I f the order of approxim ation is chosen as 5, the am pl itude of t he slip l imit cycl e can b e c alcul ated through Eq. (3.35) 3 3 6 62 5 5 2 0 0 00 0 55 0 4 16 256 384 2 . 192 D A β ωγ β ω γ β ω γ β ωγ ω β ωγ −+ − − = (3.37) 0 Ω [ ra d/ s] A [ ra d] θ ∆ [ rad ] (a) θ ∆ [ ra d/ s ] θ ∆ [ rad ] (b) θ ∆ [ ra d/ s ] 37 Making the ampli tude of t he sl ip limit cy cl e to be equ al to th at of th e sti ck - slip li m it cycle, i . e. s ubs t i- tuting Eq. (3.3 6 ) into Eq. ( 3.3 7 ), it is ob tained tha t ( ) 2 2 55 33 6 6 2 55 0 2 0 0 0 0 00 0 2 () 4. 192 16 256 384 2 sd N rd I D kk θ θθ µµ β ω γ β ωγ β ω γ β ω γ β ωγ ω − −Ω Ω + = −+ − − (3.38) By s ol vi n g this equation, the bounda ry between reg ions I a nd II can be approxim ately solved. T he m ap of creep g r oan obtained w ith analytical m et hod is gi ve n i n Fi g. 3.19 ( a). On the other hand, t he m ap can also b e solved by n um erical integ ration as shown in Fig . 3.19 (b) . The e rro r between the an al yti cal and the numerical m aps increa ses wit h th e br a ke f or ce . In the fo llow ing parts of this thesi s, al l m aps are obtained th rough num eri cal analy sis to g uarantee the accu racy. Fig. 3.19 : Map of c reep groan ca lculated by t he analyti cal m ethod (a), and the num erical method (b) 3.2.2 Tw o DOFs model w ith the bristle friction l aw I n order to study the noise generation o f creep groan, t he vibr ation of the pa d should also be cons i- dered . The brake system consis ts of rotating par t s i .e. disk - shaft sub - sy stem and non- rot ating part s i . e. pad- carrier sub - system . T his model and corr esponding si m ulati on re su lts compared with the exper i- m ental results are al so d escr i bed in [3 3] . The dynam ic s of the two sub- system s can b e described by , . R xx R I d k Fr m xd xk x F θθ θ θθ ∆ ∆∆ + += + += (3.39) where m is the m ass of the pad; k x and d x are the stiffne ss and dam ping of the carrier; x is the di s- placem ent in vertical di rection o f the pad. Suppose that two r ig id bod ies are in contact wit h t he elasti c bristle su rf aces [ 67] and t he friction fo rce F R is generated by the deform at ion of the bristle. Th e dynamic friction force R F can in general be expresse d as 01 , R F zz σσ = + (3.40) and z is a nonlinear func t ion of θ ∆ , x and z N [N] III II I [ ] 0 s rad/ Ω (a) N [N] III II I [ ] 0 s rad/ Ω (b) 38 00 0 ( , , ) | |, z z y z r rx r rx g φθ θ θ = ∆ = Ω −∆ − − Ω −∆ − (3.41) wher e g 0 is a sc al e wh ich includ es the Str ibeck ef fect . Here g 0 is given as 0 0 0 () 1 ( ) exp . d sd s rx g NN v θ µ µµ σ Ω −∆ − = +− − (3.42) The com plete dynam i c equations c an be writt en as a se t of first or der differen tial e quations T 01 01 00 0 ( , ,) 11 ( , ,) ( , ,) ( , ,) | , , , | xx x dk rr z xz I I II dk x x z xz m mm m xz xx z x z r rx r rx g z θθ θ θ θ σ σφ θ σ σφ θ φθ θθ φθ θ θ = ∆ − ∆− ∆+ + ∆ −−+ + ∆ ∆ ∆∆ ∆ = Ω −∆ − − Ω −∆ = − Y Y (3.43) where Y is t he v ector o f sta te variab les of the sy stem. Num erical simu lation is carri ed out to study the stick - slip limit cy cl e o f the syst em . T he sy stem para- m eters are a rbitra rily chosen by the autho r and giv en in T able 3.2 . A st ick - slip lim it cycl e can be observed in the to rsional ang le in Fig. 3.20 . Th is limit cy cle is qual i tativ ely s im ilar to t he l im it cycl e shown in Fig . 3.10 but without non- smooth part , so that it can descr i be the expe rimen tal results b etter . The pre - sli ding effe ct betw een the stick region and s lip region can b e observed in the simu lated resu lt . The sim ulated tor sio na l an gle and torsional velocity a re exhibited in Fig . 3.21 . Besid es, the acc elera- t io n of th e pa d an d t he fr iction force are sh own in F i g. 3.22 . In t he st ic k r e gi on, th e fr iction forc e increas es lin early , while in the slip reg ion the f riction force is almost const ant. Paramet ers Values Paramet ers Values I 1kgm 2 m 0. 1kg k θ 100 Nm k x 1000 N/m d θ 1 Nms d x 10 Ns/ m s µ 0.3 v s 0.005 m/ s d µ 0.35 σ 0 50000 N/m r 1 m σ 1 10 0 Ns/ m N 5 N 0 Ω 0.01 rad / s Table 3 .2: Par ameters o f th e system 39 Fig. 3.20 : L i m i t cycle o f the dr ive shaf t (a), and t he p ad (b) Fig. 3.21 : Torsional ang le θ ∆ (a) and t orsion al veloci t y of the s haft θ ∆ (b) Fig. 3.22 : Acc e leration o f the pad (a) and f r iction forc e (b) during the stick - s lip m otio n I n the fo llow ing, th e s tab ili ty of the sy stem is a nal yzed . The equilibrium pos ition of Eq. (3.4 3 ) is e x- pressed a s Ti me [s] (a) θ ∆ [ rad ] θ ∆ [ ra d/ s ] Ti me [s] (b) Ti me [s] (b) R F [N] Ti me [s] (a) x [ m/ s 2 ] θ ∆ [ rad ] (a) θ ∆ [ ra d/ s ] x [m ] (b) x [ m/ s ] 40 ( ) ( ) ( ) 0 00 T / // 0 00 , ( ) /, () / , () / . s ss eq eq eq eq rv eq d s d rv rv eq d s d x eq d s d xz rN N e k x NN e k z NN e θ θ θ µ µµ µ µµ µ µµ σ −Ω −Ω −Ω = ∆ ∆= + − = +− = +− Y (3.44) Linearizing Eq. ( 3.43 ) under its equi librium posi tion 0 1 11 0 1 11 , 0 1 00 0 ( , ,) ( , ,) ( , ,) 0 0 0 01 0 ( , ,) ( , ,) ( , ,) 0 (, , ) (, , ) (, 00 xx kd r r rr xz xz xz I II I x I I z kd xz xz xz m m mm x m m z xz xz x x θθ σ σ σσ φθ φ θ φθ θ σ σ σσ φ θ φ θ φθ θ φθ φ θ φθ θ = = ∂∆ ∂∆ ∂∆ − −+ + ∂∂ ∂∆ ∂∆ ∂∆ ∂∆ − −+ + ∂∂ ∂∆ ∂∆ ∂∆ ∂∆ ∂ ∂∆ Y AY A . ,) z z ∂ (3.45) w ith 00 0 00 2 / / 00 / () ( , ,) ( , ,) ( , ,) , ( )e () ,, () () ss s rr d sd sd vv ss ss s d dd rv dd xz xz x r xz z rr v N ve N e µµ µµ µµ µ µµ µ µ µµ φ θ φθ θ σ φθ ΩΩ −Ω ∂∆ ∂∆ = = ∂ ∂∆ − − Ω −Ω −+ − + Ω ∂∆ = ∂ +− where A is the cor respond ing system m at rix. The s tabil ity of t he equ ilib rium solu tion can be ana l y zed by the eig envalues of A . I f the real parts o f all eigenv alues are neg ative, the eq uilibrium solution is asym ptotically stab le. If any of t he real p arts of eig envalues is positiv e, the e qui librium s olution is unstable, and the solut i on w ill show increasing am pli tudes. The st ic k - slip lim it cycle of Eq. (3.4 3 ) can be determined by transien t analysis us ing num er ical integration. T hree di f feren t r eg ions with differen t types of solut ions are sh own i n Fi g. 3 . 23 . The white region is region I , where the sy st em has no stick - slip l imit c ycle a nd has a s table equi libr ium s olut ion . T he green reg ion repres ents reg ion II , w he re the sy stem has a stab le stick - slip lim it cycle a nd a stable equ i- librium solut i on. T he yel l o w reg ion den otes region III , wher e the system has a stable st i ck - slip l im it cycle and an un s table equ i librium s olution. 41 Fig. 3.23 : Map of c reep groan Two sim ulations are given t o expla in the m ap of creep groan . At f irst, t he norm al force is i ncrea sed from 0 N to 10 N under a c onstant driv i ng speed. S imulation resul ts are shown i n F ig. 3. 24 (a ) , whe re the blue line presen t s the vibration o f the torsiona l angle ( eq θ θθ ∆ = ∆ −∆ ) and the red line denote s the norm al force. It can b e seen th at the stick - slip motion occurs wh e n the norm al brak e force is l arge r than a crit ical force N c 23 . The reason is that the stick - slip lim it cy cl e is the only stab le so lut ion wh en the syst em is in reg ion III . T he am plitude o f the s tic k - slip vibration incre ases w ith the brake forc e. Afte r that, the norm al brake force is de creas ed from 10 N t o 0 N. The st ick - slip v ibrat ion wi ll a lway s exis t unt il th e n or ma l brake fo rce is low e r than an other critical for ce N c 12 . The stic k - slip v ibra tion wi ll vanish s ince the s tick - slip solution does not exist in reg ion I. N c 23 is larger t han N c 12 , this can con firm the ex is te n ce of re gi on I I . The c ritica l forces obtai ned by the s imulation with the c ontinuously ch an ged norm al forc e are the sam e with the c ritical f orces read from the map i n Fig . 3. 23 . By apply ing shor t time Fourier transform ation ( using Ham m ing window with window s i ze 6.2s ) to the simulated torsion- al angle, the frequ ency of t he stick -slip motion decre ases with t he brake fo r ce, sh own in Fig . 3.24 (b ). T hen , the driving speed is increased slowl y under a constant brak e normal force. The simulation r e- sults show in Fig. 3.25. A t the beginning, the syst em is under low speed and in region I II , the stick - slip lim it cy cle is the only stabl e solution of the system . The stick - slip m oti on ex ists until t he speed is lar g- er than a critical sp eed Ω c 12 . When the speed is larger than Ω c 12 , the stic k - slip motion does not exist and the system wil l return to th e equilibrium s olution (in region I ) . After that, the speed is decre ased slow ly . The stick - sl ip moti on appears once agai n whe n the speed is lower tha n a critical spe ed Ω c 23 , which i s sm aller than Ω c 12 . By apply ing short time Fouri er transform at ion to th e simulated t orsional angle, it is c lear th at the fre quency of the st i ck - slip motion increases with the speed, and app roaches to the eigenfreque ncy of t he s ystem before the c ritical sp eed Ω c 12 i s rea ch ed. III II I 0 Ω [rad /s] N [N] 42 Fig. 3.2 4 : The stic k - slip vibration wi t h inc reasing a nd decreas ing norm a l brak e forc e (a ) and t he ir sh ort tim e Four ie r tra ns fo rm ation usi ng Ham m ing window w ith window size 6.3s , where b r ight co lor m eans the frequency with large am pl itude (b) Fig. 3.2 5 : The stick - sli p vibration with increasing and decr eas ing motor speed (a) and the ir sho rt tim e Fourier transform ation using Ha mm ing window with window size 6.3s ( b) Ti me [s] (b) Frequency [Hz] Ti me [s] (b) Frequency [Hz] Ti me [s] (a) θ ∆ [ rad ] Ti me [s] (a) θ ∆ [ rad ] Ω 0 [rad /s] N [N] 43 3.3 Summary I n this chapter, different friction l a ws i.e. Coul omb’s friction law and t he brist le friction law are use d to sim ulate cre ep groan o f the te st rig w ith an idealized brake. The disk is consid ered as a rigid body connected to the m ot or wit h a rotat ional spring. T he two pads ar e tre ated a s o ne r i gi d bo d y connecte d to the ba sed fram e thr ough a linear spring. The st ick - slip lim it cy cle is f irstly obta ined by coupl ing th e system model with Coulo m b’s fr iction l aw, in w hich the stat ic friction coef ficien t is larger than the dynam ic friction coe fficient. How ever, the n o n- sm ooth characterist ic app ear s be tw ee n t he s ti ck a n d slip reg ions due to the swit c h function i n C oulom b’ s friction l aw. As an im pr ovem ent of the m odeling, the br istl e fric tio n la w, inc lud ing the Str ibe ck velocit y, pre - sli din g effec t and hy steresis, is use d t o describ e t h e frict ion betwee n the con tac t surfa ces. According to t he s tability of the equilibrium sol uti on and existence of the s tick - slip lim it cy cle, th e system with Coulom b’ s friction ha s two pa r am eters reg ions. A reg i on with a stable equil ibrium sol u- tion and no s tick - slip lim it cy cle nam ed as reg io n I and a r egion wi th a s tab le equ ilibr ium solu tion an d a stick - sl ip lim it cycle named as region II . In contrast, the m ode l with the bristle f riction l aw has three parameter r egions. Excep t for reg i ons I and II, there i s an add itional reg ion with uns t able equi l ibrium solu tion a nd a s table stick - slip limit cycle, wh ich i s nam ed as region II I . Table 3.3 exhib i ts the lim i t cycles and param eter regio ns of th e sys t em with diffe r ent f riction law s. Since di ff eren t friction law s lead to d ifferent dy namic chara ct eri stics , one shou ld be carefu l to choose the suit able fr ictio n law to describ e creep groa n of brak e system . Fri ctio n law Coulom b’s model Bristle m ode l ( Swi tch fu n c tion ) ( α = 1 ) ( α = 2 ) Fric tion co efficients Stick - slip lim it cy cle Map of cr e e p gr oa n Table 3.3: Com parison of different m odels θ ∆ θ ∆ θ ∆ θ ∆ θ ∆ θ ∆ 0 Ω N 0 Ω N 0 Ω N v µ v µ v µ 0 0 0 0 0 0 44 4 Co m pa rison st udy o f experimen tal and theore tical results o n the idealized bra ke I n thi s chapter , th e param eters of the m odels with dif f erent f r ict ion law s wil l be iden tif ied . Then, the simulation w ill b e qu an tita tiv ely compared with the experim ental resul ts , which a re measured from the tes t rig with an id ealiz ed br ake . A fr iction ob serv er will be des i gned at the end, to show a n alter nativ e way to obt ain the friction force . 4.1 Param eter id enti ficati on for t he system with C oulom b’s fr ictio n l aw Considering the two DOFs m odel with Coulom b’s friction l a w, the dy namic equations are giv en in Eq s . (3. 14) and ( 3.15). Du e to the sim ple struc ture of the te st rig with an id ealiz ed b ra ke , i ts par am e- ters ar e easy t o id entify com pared to the tes t rig with a rea l brake. T he unknow n param eters of t he te st rig with a n ideal i zed br ake includ e the m ass of the d is k and pads, the s t iffness, the dam ping of th e shaft and t he c arri er, as we l l as the stat ic and dy namic f rictio n coe fficients. The m ass of the d isk and the brak e pads is m easured by a weighing dev i ce. T he s tiffness and d a m pi ng of the dr ive shaft and t he carrie r can be m easured t hroug h modal analy sis. The m easured param eters are g iv en in Table 4.1. Paramet ers Values Paramet ers Values I 0.2025 kgm 2 k x 6 9.87 10 N/m ⋅ k θ 4 1.036 10 Nm ⋅ d x 3 3 10 N s/m ⋅ d θ 2 Nms r 0.15 m Table 4. 1 : Par ameters of t he sy stem Howev er, the st at ic and dynam ic fri ction coe f ficien t s cannot be directly m easured. There is a sim ple way to ident if y those p aram eters from the measur ed sti ck - slip limit cy cle. Assu me that the sy stem has low dam ping ra tio, th e sha pe of the l imit cycl e will be app roxim ately sym metrical. A s a r esult, th e displac ement dur ing the stick region is approxim at e t o 2 ( )/ sd Nr k θ µµ − , and the sy mm etri cal a xis of the l imit cy cle lies in / d Nr k θ µ , shown in Fig. 4.1. Therefore, i t is p oss ible to m e a sur e the d isp l ac e- m ent of th e s tick re g ion in the limit cy cle and ca l cu late the correspon ding static a nd dy nam i c friction coeffi cient s 21 21 ( ) / ( 2 ), ( ) / ( 2 ), sd d k Nr k Nr θ θ µµ θ θ µ θθ − = ∆ −∆ = ∆ +∆ (4.1) where 1 θ ∆ is the to rsiona l ang le when the sy stem just enter s the sti ck reg ion, 2 θ ∆ is the tor sional angle w hen the sys t em leaves the s tick r egion . 45 Fi g. 4.1: I dentification of th e fric tion c ha ra cte rist ic acc ording to th e l im it cycle , 1 θ ∆ is the to rsio nal angle when the sys tem j ust ente r s the stick region, 2 θ ∆ is t he tors i ona l angle w hen t he system leaves the sti ck reg i on According to Eq. (4.1 ), d µ is equal to 0.325, sd µµ − is equal to 0.0125. W i th th e param eters to hand, the torsiona l ang l e, torsion al veloc ity as we l l as t he friction forc e can b e calcul ated th r ough Eqs. (3.14) and (3.15). Fi g. 4.2 shows the m eas ured and simulated tors ional ang l e and to rsional veloc ity of th e drive shaft , wh ere the simulat ed resul ts are p resent ed with the red line and the experim ental resul ts ar e plotted with the blue line . The sim ulated acce leration of th e br a ke pa d is presen ted in Fig. 4.3 (a) . A large im pulse appears when th e system converts to the s l ip reg i on from t he s t ick region due to the sudden change of t he frictio n force as shown i n Fig . 4.3 (b). Fig. 4.2: Experim e ntal and si m ul ation torsional angle (a) and torsional velo city (b) of t he shaft θ ∆ [ ra d/ s ] θ ∆ [rad ] Ti me [s] (b) θ ∆ [ rad ] θ ∆ [ ra d/ s ] Ti me [s] (a) 46 Fig. 4.3: Simulated acceler ation of t he pad (a) and f riction force (b) 4.2 Param eter id enti ficati on for t he system with t he b r istle fr iction law Par ts o f the exper i m ental resu lts in th is section hav e already been de scribe d in [ 32], [ 3 3] . I n orde r to describ e t h e dyn am ic beh av iors of the fr ictio n fo rce, the bris tle f ric tio n la w is used to describe the friction contac t between the pads and di sk. Fiv e parameters are em pl oyed to describe the friction for ce, in term of the sta tic fr icti on co eff icien t, the d ynam ic fric tion c oef ficie nt, the Stribe ck veloc ity, th e contact stiffnes s and the co nt act dam ping . S in ce t he y are diffi cult t o m easure, it i s poss ib le to estim ate them through som e parameter id entification t echniqu es. An ob j ective f unction i s identified by com pa r- ing the ex per imen tal re sul ts with the sim ulat i on r esul ts, which i s defined as 2 22 2 exp, sim , 0 exp, sim , 11 ( ) () + () , nn i ii ii ii WX e θθ ω θθ = = = = ∆ −∆ ∆ −∆ ∑∑ (4.2) where i is th e su bscrip t of the sam ple poin ts f rom 1 to n , n is maxim al sam ple number, exp, i θ ∆ and sim , i θ ∆ are the measured and sim ul ated tors ional angle s, exp, i θ ∆ and sim , i θ ∆ are the corr esp ond ing to r- sional v elocities , 2 0 / kI θ ω = . T he o bjec ti ve f un ct ion a ppr oa chi ng the m inim um v alue m eans that the sim ulation r esult s hav e the best fit to the expe ri m ental resul ts. The sim , i θ ∆ is calcu lated by Eq. (3.43 ) with g iven initia l con di tion s in the st ick reg i on exp, 0 exp, 0 0 exp, 0 0 exp, 0 0 0 0 0 , , , 0, . x kk x xz rk r θθ θθ θθ θθ σ ∆∆ ∆= ∆ ∆= ∆ = = = (4.3) Classical optim i zation m ethods (such as gra di e nt - based algorithm s ) are norm ally suitab l e for th e pa- ram eter ident ificat i on of a linear m odel [106 ]. The bristle fric tion m odel has hi gh nonlin ear charact eris tics and its opti mization problem is no t conv ex. Therefore, a binary genetic algor i thm is used to find the m i nimum value of the ob jective func tion W(X ) . A genetic alg orithm is a metaheu ristic inspired by the pro c ess o f natural selection, which is commonly used t o gene rate hig h - quality solu- tion s to opt im iz ation problem s by relying on bio - insp ir ed opera t o rs such as m utation, crossov er an d select ion. The g enetic algor ithm has the adv antage to fin d the global m i nimum of non- convex optim i- zation problem s. The algorithm i s given in the MATLAB genetic algor ithm toolbox with it s param eters show n in Table 4.2. After 200 g ene rations, the ob jective func tion appr oaches the m ini m um value, and the optim ized p aram eters are p resent ed in Tab le 4. 3 . Fi g. 4.4 shows the v alues of param e- x [ m/ s 2 ] Ti me [s] (a) R F [N] Ti me [s] (b) 47 ters and t he v alue of th e ob j ectiv e funct i on ov er each g eneration. I n order to p resen t the p aram eters with different physi cal unit s in one figure, they hav e be en divid ed by their m a xim um optimal boun- dary. Once the param eters are obtained, t he exp erimental and sim ul ation res ults are c om pared with ea c h ot he r. F i g. 4 . 5 shows the m easured and s imulated t orsional ang l e and t ors ional velo city of the drive sha ft , whi le the simula ted acceler at ion of the brak e pads and the simulated fric tion force are presented in Fig . 4.6 . When the system conv erts to th e slip reg i on f rom th e stic k r eg ion , a la rge i m- pulse app ears. Mean wh ile, the sim ulated fri ction f orce chang es sudden ly. Paramet ers Values Paramet ers Values Maximum generations num ber 200 Generation g ap 0.8 Num ber s of individual 10 Mutation probab i lity 0.7 Table 4 .2: Par ameters o f th e gen etic a lgorith m Paramet ers Values Paramet ers Values d µ 0.325 σ 1 5 3.84 10 Ns/m ⋅ sd µµ − 0.009 v s 0.025 m/ s σ 0 8 3.97 10 N /m ⋅ Table 4 .3: Par ameters o f th e fric tion law Fi g. 4.4 : Param ter valu es (a) and the v alue o f W(X ) (b) in each g enerat i on Fig . 4.7 exhibits the s im ulated s tick - slip l im it cycle by Coulomb’s friction law com par ed t o exper i- m ental resul ts , while Fig. 4.8 exhibits the s im ulated stick - slip lim it cyc le by the b rist le fr ictio n law com pared t o experim e nt al resu lts . Bot h simula ted lim it cy cles a re coin ci d ed wi th the m easured limit cycle. I t is clear tha t bo th fricti on laws hav e the ab ility to d escribe the st ick - slip lim it cy cle [ 6 7] , [ 68 ]. How ever, the br i stle frictio n law can descr ibe the pr e - sliding ef fec t between the st ick and sl ip regions. I n the lat er discus sion , i t c an be found that the un stable equ ilibrium solu tion of the system unde r low speed w ill show up i f the b ristle fr iction law is us ed. Generation (a) Generation (b) W(X) [ ra d/ s ] V alu es 48 Fi g. 4.5: Torsiona l angle (a ) and torsional velocity (b) of the sh aft Fi g. 4.6: Accelera tion of th e brake pads (a), and the si mulated friction force (b ). Fig. 4. 7 : Sim ulated sti ck - slip lim it cyc le by Co u- lomb’s frictio n law com par ed to experiments Fi g. 4.8: Sim ul ated stick - slip l im it cycle by t he bristle friction l aw com par ed t o exper iments Wit h the ide ntif ied param eters of the sy stem , the m ap of creep groan can be calculated th rough the m ethod proposed in section 3.2, shown in F i g. 4.9, w hose horizont al axis is t he driving speed and the vertic al axis is t he brake p re ssure w hich is pr oportio nal to the b rake norm al f orce. Ac cording to t he analysi s of sec tion 3. 2, t he system has th ree pa rameter r egions if the bri stl e fr ictio n l aw i s use d . R e- gion I means tha t the system has a stable e quilibri um solution b ut n o st ic k - slip li mit cy cle. R eg ion I I m eans that the system has a stab le s tick - slip lim it cy cl e a nd equili brium solu tion (a nd an unstable li m i t Ti me [s] (b) θ ∆ [ rad ] θ ∆ [ ra d/ s ] Ti me [s] (a) Ti me [s] (a) x [ m/ s 2 ] Ti me [s] (b) R F [N] θ ∆ [ ra d/ s ] θ ∆ [ rad ] θ ∆ [ ra d/ s ] θ ∆ [ rad ] 49 cy cle s olutio n in b etwe en) . Region I II m eans that the sy stem has a stable st ick - s lip lim it cyc le and an unstable equ ilibrium solution. This m ap can be us ed to st udy creep g roan du ring the acce lerating and dece leratin g process es. If t h e operati on starts w ith pa ram eters in reg i on II I, creep groan m ust occur, as the limit cy cl e i s the on ly stab le solu tion. I f then t h e parameter reg ion of type II is entered fr om r eg ion III (i .e. with c r eep g roan) , creep g roan will p roceed u ntil re g ion I is en ter ed . This is usu ally the case w hen t he vehicle i s acc ele- rating. Differ ent things w ill happen if the vehicle is decelera ting. I n region I the sil ent so lution without creep g roan is the o nly stab l e so lutio n. The system will s tay in t he att r acto r of the silent solut i on wh en entering reg i on II from region I and no creep groan occurs. C reep groan will the n occu r unle ss region III i s e ntered. Fig. 4.9: Map of creep g r oan w ith ide nt ifie d par ame ter s , the r ed po ints repre sent the m easured bou n- dary points betwee n re gio ns II and III, w hile the blue points rep resent the m easured bou ndary points between region s I and II Experim e nts are carried out to confi rm the existence of the different c ritical conditions of creep g r oan during accelera ting and decelerating of a v ehicle , and the approach is shown in Fig. 4.9 w ith the s oli d lin e . The brak e pressure is constant (8 bar) and the dri vi ng speed of the motor is varied. Correspond ing experim ental results a re sh own in Fig . 4.10. The r ed line de s cribes t he dr iving s peed, wh i le th e bl ue lin e presen ts t he to rsional vibration ang le θ ∆ ( eq θ θθ ∆ = ∆ −∆ ) of the drive shaft. The torsional v i- bration of the driv e shaft becom es large w hen cree p g roan app ears . At fir st, the speed i s slowly increas ed from 0 rad/s to 0.7 rad/s, the sy stem has creep groan at low sp eed, bu t creep g r oan disa p- pears w hen the speed is hi gher tha n a critical speed Ω c12 (0 .62 rad/s , mark ed with a red p oint in Fig . 4.9 ). Af ter tha t, the s peed is sl owly decrea sed from 0.7 rad/s. Creep g roan d oesn’ t oc cur at high speed and app ears w hen th e speed is lowe r than anoth er criti cal pressu re Ω c23 (0.42 rad/ s, mark ed with a blue point in Fig . 4.9). Therefo re, the bot h critical speeds du ring acc elerating and decelerat ing are found out , and the measured critical speeds a re near the simulated boundary of r e gio n s. Th e dif f ere nce b etw een these t wo l i m i t speed s in fact proo fs th e exis t en ce o f reg ion II with b oth stabl e equ ilibr ium solu tio n and lim it cycle. III II I Ω 0 [rad /s ] p [bar] V ari ed s pee d Varied pressu re Ω c 23 Ω c 12 p c 23 p c 12 50 Fig. 4.11 shows t he equ ilibrium solution and the lim i t cy c le of the br ake system with diffe r en t speeds in the phase plot with θ ∆ as a function of θ ∆ . When the driv ing spee d i s higher t han Ω c 12 , only the equilibrium sol ution can be measured bot h in the acce l erating and dec elerating p rocess, meaning that the system i s in region I and cr eep g r oan canno t occur. When the speed is l ow er than Ω c23 , cr eep g r oan is m easured in bo th proce sses, m eaning that the sy stem is in region I II and creep groan alw ays occu rs in t his region. When the speed is higher than Ω c23 but lower than Ω c12 , creep groan is m e asured in th e accele r ating process, b ut no t in the dec elerating proce ss, m eaning that the sy stem is in region II . In the decelerating process, the sy st em begins with the equilibrium solution and stay s in the attractor of the stable e quilib rium so lution. In the accel erati ng proces s, the sy stem begins with th e stick - slip sta te, a nd the system is in the attrac tor of the s table l imit cy cle. Therefore, experim ent s dem onst rated the exi s- tence of the three reg ions. Fig. 4.10: Tor si on al v ibration a ngl e with variation o f the speed of the m otor , the red and blue po ints represe nt the m easured bo undary spee ds Fig. 4.11: Th e equilibrium sol ution and the stick - s lip limit cycle with di fferent spe eds, (a) in region I , (b) in region I I with coexistence of the s table equili brium solut ion and limit cycle, ( c) in reg ion III Another possib ility to show t he existen ce of the th ree reg ions is to keep the constant driv i ng speed but vary the brake pr essure. H ere, the alternative approach is shown i n Fig . 4.9 w ith t he dotted l ine, wh ere the driv i ng speed o f the m otor is co nstant ( 0.2 r ad /s) and the b r ak e pressu re va r i es . Corresp onding experim ental results a re sh own in Fig . 4.12. The red line des c ribes t he brak e pressure, w hile t he bl ue lin e present s the to rsio nal v ibrat ion angle θ ∆ of th e dr i ve s ha ft . At first , the brak e pressure is slowly increas ed from 0 bar to 5 bar, t he system has no creep g roan at low pre ssure, b ut creep groan occu rs when the p ressure i s h igher t ha n a critic al pressu re 23 c p (2.9 bar, m arked with a red point in Fig. 4.9 ) . After that, the press ure is slowly decr eased from 5 bar to 0 bar. Creep g r oan occ urs at high pressu re Ti me [s] θ ∆ [ rad ] Ω 0 [rad /s] θ ∆ [ ra d/ s ] θ ∆ [ rad ] (a) θ ∆ [ ra d/ s ] θ ∆ [ rad ] (b) θ ∆ [ ra d/ s ] θ ∆ [ rad ] (c) 0 0.65 ra d/s Ω= 0 0.45 ra d/s Ω= 0 0.3 rad/s Ω= 51 and d isappea rs when the pressure i s l ower than another criti cal pres sure 12 c p (2.3 bar , mar ked wi t h a blue point in Fig. 4.9 ). I t should be no ticed th at the m e asured critica l pressu re ar e ar ound the simulated boundary between region s. The di fference b etwe en these t wo cr i tic al p ressur e value s al s o proofs t he exis tence o f reg ion II with both stable equ ilibr ium sol ution a nd lim it cy cle. Fig. 4.13 shows th e equil i- brium solution and the limit cycle of the br ake system under different pr essu re in the phas e plo t wit h θ ∆ as a function of θ ∆ . When the pre ssure is low er than 12 c p , only the equi librium soluti on can be m easured i n both the pressure - incre asing and pres sure - d ecreas ing proce ss es , m eaning that the sy st em is in region I . When the pressure is hig her than 23 c p , creep groan is m easured in both p rocesses, m ea n- ing that t he sy stem is in reg i on III . When the pre ssur e i s low er tha n 23 c p but higher than 12 c p , creep groan is m easured in the p ressure - decreasing p rocess b ut not in the pressure - increasing process, m ea n- ing that the system i s in region I I. In the p ressure - inc reasing proces s, the system is i n the attract or of the stable equi librium s olution, so that creep g r oan doesn’ t appear. I n cont rast, the syst em is in the attracto r of the st able l im it cy cl e and creep g roan occu rs. Fig. 4.12: Torsio nal vibra tion angle w i th v ariation o f the brake p ressu r e , the red and b lue points represe nt the m easured bo undary pressu re Fig. 4.13: T he equilibrium sol ution and the stick - slip li mit c yc le with d iffe rent p re ssure , (a ) in reg ion I , (b) in region I I with coexistence of the s table equili brium solut ion and limit cycle, ( c) in reg ion III During the p ressure - increasing proces s, the am plitude of th e torsion al ang l e incr eases w ith the b rake pressure, shown in Fig . 4.14 (a). By apply ing short ti me Fourier transform ation (SFF T ) ( using Ha m- m ing window w ith window siz e 0.5s ) of θ ∆ , one can s ee that the frequency of the stick - slip m otion decreas es with inc reasing o f t he b rak e pressure , s how n in Fig. 4.14 (b ) . Meanwh i le, the am plitude of the accel erati on of the pad x increa ses with t he brake p ressure, sh own i n Fig . 4.14 ( c). The reason i s tha t the ac cele ration of the pad is proportiona l to the differ ence betw een sta tic a nd dy namic frictio n forc es , as well as the brake pres sure. The SFFT of x is p resented in Fi g. 4.14 (d) . I f the brak e press ure Tim e [ s] θ ∆ [ rad ] P [b ar] θ ∆ [ ra d/ s ] θ ∆ [ rad ] (a) θ ∆ [ rad ] (b) θ ∆ [ rad ] (c) θ ∆ [ ra d/ s ] θ ∆ [ ra d/ s ] 2 bar p = 2.5 bar p = 3.5 bar p = 52 is large enough, som e high f requency v ibr ation of the carrie r can be excited. For the low brake pre s- sure, the frequen cy of peak appears at 200 Hz , while an additional peak f r e que ncy at 900 Hz appears when th e brake pres sure is larg er than 6 bar. Fig. 4.14: Measu red signal s d ur in g pr e ss ur e - increas ing process, (a) v ibration of the torsiona l angl e θ ∆ , (b) SFFT of θ ∆ , (c ) accel erat ion o f the pad x , ( d) SF FT of x . Ti me [s] (c) x [ m/ s 2 ] Ti me [s] (d) Frequency [Hz] Ti me [s] (a) θ ∆ [ rad ] Ti me [s] (b) Frequency [Hz] p [b ar] p [b ar] 53 Similarly , the am plitude of th e torsional ang le incr eases with the speed, show n i n Fig. 4.1 5 (a). T he frequency of the sti ck - sl ip m otion increa ses wi th the d ri ving speed, and ap proa ches the eig enfrequency of the shaft a t high speed, expressed in Fig. 4.15 (b). Fig. 4.15 (c) show s that the amplitud e of the accele r ation o f the pad, whi ch is le ss inf luenced by the speed, sin ce the b rake press ure is const ant. T he SFFT of x is presented in Fig . 4.15 (d) . The speed has less influenc e on the vibrati on frequen cy of t he pad. Fig. 4.15 : Me asured s ignal s dur ing acce lerating process , (a ) vibration of the torsiona l angle θ ∆ , (b) SFFT of θ ∆ , (c) acc el era ti on of th e pad x , (d) SFFT of x . Ti me [s] (a) θ ∆ [ rad ] Ti me [s] (b) Frequency [Hz] Ti me [s] (c) x [ m/ s 2 ] Ti me (d) Frequency [Hz] Ω 0 [ ra d/ s ] Ω 0 [ ra d/ s ] 54 4.3 Friction obser v er I t is possib le to calcu late the friction a ccording to the abov em entioned model. How ever, a lot of work is required for m odeling and pa rameter id entificat ion . Ther efore, in this section , an o bserv er is d e- signed to estimate the fricti on fo rce dir ectly from the m easured sig nals. I n t he brak e sy stem, the fri ction f orce acts on the b r ak e disk and leads to a t or si ona l def or mat i on o f t he driv e shaf t. I f the fr ictio n f orce i s co nsi der ed a s a n inp ut of the d isk - s haft sub - sy stem, and the tor si o nal an gl e and a ngle velocity of the drive sh aft a re treated as the state v ariable s , the dynam ic system is T 01 , , 0 R F kd r I II θθ θ θ θ θ θθ ∆ ∆ ∆ = + −− ∆ ∆ = ∆ X (4.4) where R F is the input f riction force, and X ar e t he stat e vari abl es . Since the sta te var iables θ ∆ and θ ∆ are know n , and R F chang es slow ly com pared to the sa mple freq uency , it is po ssi ble to des ign a Proportiona l- I ntegral observ e r to estimate the unknow n input force R F [107], [108] . The observ er is gi ven as ( ) [ ] 1 2 T T 0 ˆ ˆ ˆ , ˆ 00 ˆ 01 ˆ ˆ ˆ , , 0 /, // R R F F rI kI d I θθ θθ = +− = ∆∆ = = −− X XL AB XX L XA B (4.5) where 1 L and 2 L is the observ er matrixes, ˆ θ ∆ and ˆ θ ∆ are th e estim at ed torsion al angle an d torsio n- al v eloc ity. I f the abov e equations is reform ed s o that X is as the input and ˆ X and ˆ R F are th e st at e variabl es, then 11 22 ˆ ˆ . ˆ 0 ˆ R R F F − = + − X AL B X L X LL (4.6) Af ter the transf ormation, t he unknown in put is considered as a state v ariable. The erro r dyn am ics of the o bser ve r is 1 2 ˆ , 0 R R F F − = ∆− ∆ e AL B X L (4.7) where ˆ = − eX X and ˆ R RR F FF ∆= − . I f the matrix 1 2 0 − − AL B L has onl y eigenvalue s with negativ e real par t and the unk nown input changes slow ly (i.e. quasi - static), the estim ated errors conv erge asymptotica lly to z ero, and ˆ RR FF ≈ after finite time. A ccording to t he conv ergent critical, obs erver m atrixes are chos en as 3 1 3 10 0 0 10 1/s = L and 77 2 10 N/s 10 N = L . F i g. 4.16 shows t he flo w 55 chart of th e designed obser ver. T he f ri cti on force can be observ ed th rough the m easured state vari ables. Fig. 4.17 exhibi ts the observ ed f riction force . After som e t im e ˆ R F becomes st able a nd is appro xi m ate to the real fr icti on for c e. I t is easy to dis ting uish th e st at ic f ricti on for ce fr om the dy nam ic fric tion force, whe r e the dy nam i c friction fo rce approach es a constan t whi l e th e static friction fo rce varies between the m inimal and m axi mum values (marked with blue co lor in Fig . 4.17 (b)). Com pa red to the m odeling approach , des i gni ng a fri ct i on o bs er ve r re qui r es onl y a li n ear di sk - shaft model, so th at th e parameters of the fricti on law don’t need to be iden tified . How ever, a disadv antage is tha t the m ea- surem ent no ise w ill str on gly infl uence the accu racy of the obse r v ed fric tion fo r ce. Fig. 4.16: F low chart of the friction force observer Fig. 4.17: O bserv ed frictio n force ( a) and it enla rged detail s (b), w here the s tick r eg ion mark ed with blue color At t he end, the calculated friction fo rce s from differen t models are com pa red wi th the obse rved fri c- tion force in T able 4.4 . The obse rved friction fo r ce is si m ilar t o the friction calculated by the model with the b rist le fr ictio n law . I t prov es tha t the b ristle fric tion l aw is m ore sui tab le to describ e t h e co n- tac t inte rface between the disk and the pads. I n contrast, Coulom b’s friction law cannot desc r ibe the fu ll dynam ics of the friction. ∫ A 1 L 2 L B X ˆ X ˆ X ˆ R F − ˆ − XX ∫ Ti me [s] (a) ˆ R F [N] Ti me [s] (b) ˆ R F [N] 56 Modeled friction by Coulom b ’s fr icti on l aw Modeled friction by the bristle fr icti on l aw Observ ed fricti on Table 4.4: Sim ula ted f riction force a nd t he ob s erved fri ction force 4.4 Summary I n thi s chapter, param ete r identi fication m ethods ar e proposed for the models w i th different friction laws. For the m odel with Coulomb’s friction law , its static and dynam ic friction coeffic ients can be identified by analyz i ng the g eo metric s hape of t he st ick - slip l im it cycl e. Fo r th e m odel w ith th e b ris tle friction law, their par amete rs, inclu di ng the sta t ic f r ict ion co efficien t, the dyn am ic friction coefficien t, the Str i be ck v elocity, th e con tact stiffne ss, and the co ntact dam ping, are estimat ed by com par i n g the sim ulated and m easured r esu lts. A g enetic alg orithm is assi sted t o find the op timized pa rameter set. The model with the brist le friction law in d ica ted t hat th e brake sy stem has thre e reg i ons of pa r am eters according to the stabi lity of the equilibr i um solutio n and the ex istenc e of th e s tick - slip lim it cy cle . T wo possib le m ethods are employ e d to confirm the three pa rameter r egions. On e way is to k eep the brake pressure con s tant and v ar y the driving speed; another w a y is to k eep the driving speed c ons tant and v ary the b rake pressu re. Both experim ents are carri ed ou t and th e t h eoret ical an d experim ental results hav e good agreem ent with eac h other. I n the region I I , both stable equi librium soluti on and the stick - slip limit cy cl e can be measured w ith d iff er ent ini tial conditions. At the end, a Proportion al- I ntegral observ er i s des igned to observe the fr iction force from the mea s- ured θ ∆ and θ ∆ . Com pa red to the m ode ling m et hod, des igning a f riction obse r ver r e qui r es onl y a linear disk - shaft model, so that the param et ers o f the f riction law d on’t need to be ident ified . Ho we ver , it has a disa dvantage t hat the m easurement noise w ill str on gly i nfluence the acc uracy of the ob served force. B y com paring the ca l cu lat ed frict ion force with the obs erved frict ion force, it is found that the observed f riction for ce is mo r e sim ilar to the fric tio n c alc u late d b y the mod el w ith th e bri stle fric tion law than that with Cou l om b ’ s fric tion law . It pr ov es that th e br istle f rict io n law is m ore suit able to describ e t h e frict ion interfa ce of the brak e system . Ti me [s] R F [N] Ti me [s] R F [N] Ti me [s] ˆ R F [N] 57 5 Theoretical a nd e xperi ment al ana ly sis of creep groa n o n the test rig w ith a rea l bra ke I n this chapte r, a dyn am ic model of the te st rig w i th a real b rake is studi ed. Different from t he test rig with an ideal ized br ake, the tes t rig with a real b r ak e consist s of a com plex bra ke carrier w i th a su s- pension system . Therefore, a mo del with larg e number of deg rees of freedom is r equired to des cribe the dy namic sy st e m . With respect to the modeling , i t is hard to identify t he st iffness and damping of each com ponent of the test rig due to their com pl ex co nstruction. O n the othe r ha nd, it needs plenty of calcula tion t i m e f or the tra nsition analy sis of a non linear system with l arg e num b er of deg rees of f r ee- dom . In order to solv e t hese issu es, Butlin and Woodh ouse use the transfer function m odel to de scribe the fr icti on - induced vibrati on [109] . I nspired by this work , a redu ced - order m odel expressed by tran s- fer functions is stud ied in this chapt er . Af t er param eter iden tifica t ion o f the reduced - order model, sim ulation resu lts w ill b e com pare d with the experimental results. Som e results of experimenta l inve s- tigat ions are already descri bed in [3 3] . 5.1 Model ing of t he te st rig wit h a real bra ke The re al brak e system can be decomposed in to thre e sub - systems, in terms of the ro t ating par t s , the non- rotating part s and the friction in terface. The rota ting part s i nc l ude the disk, the drive sha ft and the m otor. T he non - rota ting part s inc lu de the brak e pads, the c arrier, the c aliper, and the suspen sion sy s- t e m. During modeling, the suspensio n is considered as a rigid body and hung on the basic fram e with a set of m ass - spring system s. According to exper imental ana l y sis , both pads m ove si mu ltan eously du r- ing creep groan. T hen, two pads ar e trea ted as one rigid body which connec t s to the suspensio n through a se t of mass - spri n g sy stems. The disk is assem bled in the d r ive sh aft, w hich is supported b y a bearing assem bled in the suspension. The shaft is driv en by the m ot or with a reduction gear box. The structu re of t he b rake t est ri g and its dy nam ic model are sk etched in Fig. 5 .1. Fi g. 5.1: Structure of the tes t rig of real b rake and i ts dynam i c m odel The dynam ic e quation of th e non -rotating parts is giv en as , M X+D X +K X =F (5.1) K g , kd θθ x 1 x L x N θ x 1 x L x N , kd θθ K g θ 0 t Ω 0 t Ω 58 wher e [ ] [ ] [ ] T 12 1 2 T 11 1 1 1 1 ... , diag( ... ), , 0 ... 0 , 0 00 ... ... 0 0 0 ... ... 0 , 0 0 ... ... 00 0 NN R LL N NN x x x mm m F kk k kk k kk β + − − = = = = − − = + − − XM D KF K whe r e M is th e mass ma t r ix , D and K are t he dampin g and s tiffness matrixes , wh i c h are symm etric m atrices , F R is the frict ion force , x 1 , x L and x N are the displacem e nt of the pad, th e suspension an d the fram e in ver tical d irec tion, respectiv ely . I t is assum e d that D is a proportional m o dal dam pi ng propo r- tion al to K w ith a fac t or β . The rotating par t s c on si st of the disk and th e drive sha ft. I t s dynam i c equation is w r itten as , R I d k Fr θθ θ θθ ∆ + ∆+ ∆ = (5.2) where r is the radius of t he p oint w her e t h e pad a cts , k θ and d θ are the stiffne ss and dam ping of the drive sh aft , ∆θ is th e tors i on al ang le of the dr ive shaf t . According to the bristl e friction law, the friction forc e between the conta ct surface of the pads and disk is 01 , R F zz σσ = + (5.3) and z is a nonlinear func t ion of v and z 0 ( , ) | |. z z vz v v g φ = = − (5.4) and ( ) ( ) 0 0 1 ( ) exp / , d sd s g N N vv µ µµ σ = +− − (5.5) where 0 g is a sc alar incl u din g th e Stri b ec k eff ect , v is the relative velo city between the pads and disk . I t is imp ortant to de fine v . Since the disk is rotati ng around the b earing fi xed in the su spen sion , t he disk has no re lative transi tion m otion referring to t he suspen sion. T hen , the r el at i v e veloci ty betw een the pad and di sk is ( ) 01 , L v r r xx θ = Ω −∆ + − (5.6) where 1 x is th e ve loc ity o f th e pads i n v ertic al di rectio n , L x is the ve loc ity of the susp ension in v erti c- al dir ection , 0 Ω is the d riving speed of the m otor , the rota ting angle of the disk is θ , and the torsiona l angle of the shaft is θ ∆ with 0 θθ ∆ = Ω− . The r elations hip in Eq. (5.6) is also t rue for a real v ehicle , i.e. it is pos sib le to iso la te the b rak e - suspe nsion sy stem fr om the fram e or t he ch assis for t he study of the sti ck - slip m otion , and complex com ponent s such as the cha ssis can be at first i gn ored. 59 Substituting Eq s . ( 5.3 )- ( 5.6) into (5. 1) and (5.2), th e in t egrated dy na m i cs of the te st rig c an be wr i tten as fir st o rder differ ential equat ions ( ) ( ) -1 -1 -1 01 01 0 . R LL dk r F I II z r r xx r r xx g z θθ θ θ θθ θ θθ ∆ ∆ − ∆− ∆+ ∆ −− + Ω −∆ − = Ω∆ − − − −− X X MD X MK X M F X (5.7) 5.2 A reduced - o rder m odel f or the s imu lati on of c ree p gro an I n practice, it is very hard to m easure the dam ping and stiffne ss m at rix D and K . Instead, it is possible to m e asure the t ransfer funct ion of the non - rotating part s by modal analysis. Therefore, 1 L x − can be expressed thr ou gh a rationa l transfer funct ion ( ) 1, x xL F Hj ω − as ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1, 21 2 2 21 2 2 2 1, 2 21 11 , ... s , kg ... L L x xL F R LL LL x xL F LL L x xx H jF a j a j aj Hj j bj b ω ωω ω ω ωω −− −− −− − − − = −= + ++ = + ++ (5.8) whe re the in put of th e tran sfer f unctio n is the f rictio n fo rce, the ou tput is th e rel ativ e v elo city of the pad referring to the suspen sion, ω repres ents t he angula r frequency with unit [rad/s] , j is t he im ag i- na r y un it wit h t he pr op er ty 2 1 j = − [107]. T he transfer function is t he ra tio of two poly nom ials. The orders of the numerator polynom i als are less than the order s of the denomina tor polynomials. The poles of the transfer functi on c orrespo nd to values of the ω - v ariabl e for whi ch the denom inator pol y- nom ial i s zero. The zeros of the transf er function cor respond to values o f the ω - v ar iable for w hich t he nom inator polynom ial is zero. T he transfe r fun ction can also be w rit ten in the p arti al transfe r function, w hich is expressed a s th e sum of independent secon d or der transform func tion s. Since D is pr opor tional to K , the partial t ransf er function has a stand ar d form as ( ) ( ) ( ) ( ) ( ) 1, 2 11 , LL i x xL F i ii ii j H j Hj j dj k αω ωω ωω − = = = = ++ ∑∑ (5.9) where ( ) i Hj ω is a m ode of the sy stem. Th e transfer function ca n be obta ined from the FE m odel , m ulti - body dy na mic model, or m easured by modal analysis on the tes t rig . Sinc e the or i g inal transf er function m ay have lots of m odes, it is necessary to reduce t hem to gain a re duced - or der model . For purposes of potential modes reduction, the H 2 norm of ( ) i Hj ω is cal culate d , and t he m odes wit h ( ) 2 i Hj ωε > is def ined as the rela tive im portance modes [110]. 2 . is H 2 norm of a stable cont i- nuous- time sy stem and m e asures t he steady - stat e pow er of the output respon se, which is given by 60 ( ) 2 1 T race ( ) ( ) 2 H i ii H j Hj Hj d ω ω ωω π ∞ −∞ = ∫ . ε is a thresho l d de t erm ined by the designer . By ignoring rela tive unim port ant m odes , the reduce d- order t ransfe r funct ion can b e w ri tten as ( ) ( ) ( ) ( ) ( ) ( ) 1, 1 , 1, 2 1 R L i x x LF x LF x x LF i ii j Hj H j Hj j dj k αω ωω ω ωω −− − = = −∆ = ++ ∑ , (5.10) where L R < L , ( ) 1, x xL F Hj ω − consists of t he r elativ e important m odes , ( ) 1, x xL F Hj ω − ∆ is th e rel ativ e unimportant m odes. In order to simplify the calculation, ( ) 1, x xL F Hj ω − is instead of ( ) 1, x xL F Hj ω − t o calc ulate the fric tion fo rce . Fo r the rota ting par ts, Eq. (5.2) converts to a t ransform func tion as ( ) ( ) ( ) ( ) , 2 , FR R rj H jF F Ij d j k θ θθ ω θω ωω ∆= = ++ (5.11) where , F H θ is a transfe r function from t he inpu t friction force to the to rsional veloc ity of th e shaft . Then, t he com plete equat i o ns of the reduced - orde r m odel are given as ( ) ( ) , 01 1, 1 01 01 0 ( ), ( , , ) | |. F x xL F L pL L Hj zz Hj x z z r rx r rx g z θ ω θ σσ ω φθ θ θ − − ∆ = + ∆ =Ω −∆ − − Ω −∆ − = X (5.12) With two transfer func tions to hand, the calculat ion procedure is g iven as fol low s with i ts flow chart i n Fig. 5.2: 1 . Giv e the in itial re lativ e velo city 1 L x − and th e tors iona l v eloc ity θ ∆ , 2. C alcu late the frictio n fo rce by the bri stle f rict ion la w with g ive n 1 L x − and θ ∆ , 3. Let the calcula ted friction forc e feedback to the transfe r function ( ) 1, x xL F Hj ω − and ( ) , F Hj θ ω , so tha t th e 1 L x − and θ ∆ for the nex t tim e st ep can be calc ulated, 4. Repeat step s 1 to 3 ite rativ ely to obtain the fr iction force during creep g roan. Com pa red to the original m odel, the reduce d - order model has some adv a ntages. On the on e hand, the calc ulat ion tim e of the redu ced - order m odel is muc h les s tha n t ha t of ori gi n al mo de l ; o n the ot her ha nd , only the sy stem parameters are requi red in the reduced - order m ode l, which can be easily identif ied by m odal analy sis. 61 Fi g. 5.2: Flow cha rt of the ca lcu l at ion procedure 5.3 Tran sfer fu nction id entif icati on thr ou gh m odal an alysi s I n thi s section, m odal analy si s is perform ed to identify the tra nsfe r function bet ween the fric tion force and the vel ocity of the pa d related to the suspension. In practice, the acceleration of the pad instead of its v eloc ity is m easured , bu t the velo city can be obtain ed by doing tim e integratio n of the ac c elerat ion. The frequency r esponse funct ion (FRF) is really the trans fer function ev aluated along the frequ ency axis [111] , wh ich is rewr itten as ( ) ( ) ( ) 0 1 0 () . m k k k n nk k k aj H j bj ω ω ωω = − = = + ∑ ∑ (5.13) The way to find th e unkno wn para m e ters a k and b k i s equ iv alent to solv e the curve fitting prob lem such that the error b et we en the analy t ical expression a nd the frequen cy respon se m easurement is m i nim i zed over a chosen f requency ra nge [111]. Th e error betwe en t he ana lytic al FRF and m easured FRF is 0 1 0 () , () () m k k k ii n nk k k aj eh j bj ω ωω = − = = − + ∑ ∑ (5.14) where h i is the freque ncy respon se m easurem ent at ω i , which is a complex num ber. Mul tiply i ng 0 () n k k k bj ω = ∑ in both sides of E q. ( 5.14) 11 0 00 () () () () () . n mn kn k kn i i ki i ki i ki i k kk e e bj j a j h bj j ωω ω ωω −− = = = = += − + ∑ ∑∑ (5.15) Then, th e error v ector ca n b e expres sed as [ ] T 1 ... L ee = E . The erro r vector can be rew ritten in a m ore compact form by expa nding ea ch of the summ ations , −− E = PA TB W (5.16) wher e Bris tle m odel R F R F R F R F 1 L x − θ ∆ 1, x xL F H − , F H θ 62 2 21 1 1 1 1 11 11 11 2 21 2 2 2 222 22 22 2 21 1 () . . . () () (). . . () 1 () . . . () () () . . . () , ... ... ... ... .... ... ... ... ... .... 1 () . . . () () () . . . () mn mn mn L L L L LL LL LL j j j h hj hj hj j j j h hj hj hj j j j h hj hj hj ωω ω ω ω ω ωω ω ω ω ω ωω ω ω ω ω − − − = = PT [ ] [ ] T 01 T 01 1 T 11 2 2 ... , ... , ( ) ( ) ... ( ) . m n nn n LL aa a bb b hj h j h j ωω ω − = = = A B W (5.17) The squ ar ed error criter ion is the s quared norm of the e rror v ect o r ( ) ( ) ( ) 2 *T 2 T *T T *T *T T *T T * T *T 2R e 2R e 2R e , J = = = ++ − −− E EE A P PA B T TB W W AP T B AP W B T W (5.18) where 2 E is the Eucl i d ian norm . Once the c riterion func tion has a single minim um value, its deriv a- tives with resp ect to the v ariables A and B should be zero in the m inimum point. Therefore, the following equations exist [ 111] ( ) ( ) ( ) ( ) * ** ** * 2 2 Re 2 Re 0 , 2 2 Re 2 Re 0. TT TT J J ∂ = −− = ∂ ∂ = −− = ∂ P PA P TB P W A T TB T PA T W B (5.19) The above equ ations c an be written in a com pact for m ( ) ( ) ( ) ( ) ** * ** * Re Re . Re Re TT TT − = − P P PT P W A B TP T T T W (5.20) I n this equation, P , T and W are k nown variab l es, and A and B are unk nown variable s. T he param eter vectors A and B can b e obt a ined by solv ing the l inear equations (5.20 ) . Howev er, the m atrixes P an d T are comm only in il l - condi tioning a nd hence Eq. ( 5.20 ) i s diff icult to solv e. A m ethod to s olv e thi s issue i s prop ose d in [111] , where orthogonal polynom ial s are used to r em ov e the i ll -condition. I n orde r to obtain the frequency response function o f the test rig, m odal analysis is perfo r m ed , shown in Fig. 5.3. A m odal hammer is knocked on the br ake carrier when t he disk is rotating under constant speed and brake press ure w itho ut sti ck - slip. The input force () in Ft can be m easured by a force sensor . I n the meantim e, t wo accel erom et ers , which are adh ere d to the b rake pad a nd near the bearing o f the suspen sion, m easure the excit ed acceler ations . 63 Fi g. 5.3 : M oda l analy sis of the tes t rig with a re al bra k e The frequen cy spectr a of the v elocity on the pad 1 () xt and the v elocity near the b earing () L xt are presented in Fig. 5. 4 with t he red line and the b lue line, resp ectiv el y. As a resu lt, the tran sf er fun ct i on 1, x xN F H − and 1, x xN F H − can be identif i ed as ( ) ( ) ( ) ( ) 1 1, , () () ,, () () L x xN F xL xN F in in xt x t HH Ft Ft −− = = (5.21) where ( ) . denotes Four i er tra nsform . It should be no ted that in Fig . 5. 4 the frequency spectr a of 1 () xt and () L xt coincide with each o ther at frequency 27.5 H z. After ana lysis, this frequ ency is the eigenfrequency of the susp ens ion sy stem, so that the suspension and b r ake sy s tem m ove s ynchronou s- l y. Si nce the motion of the suspensio n will not in f luen ce the fr iction fo rce be tw een the pad s and di sk , this frequency shoul d be n ot conside red in th e cal culation of t he s tick - slip motion . The tran sfe r fun c- tion bet w een the relative velocity and the f riction force 1, x xL F H − is expr essed as ( ) ( ) 1 1, () () . () L x xL F in xt x t H Ft − − = (5.22) Fi g. 5.5 shows th e measure d FRF of 1, x xL F H − with th e blu e lin e. O b v iously, there is no peak fr equen cy at 27.5 Hz. T he tr ansfer funct ion is i den tified by a na ly zin g the m easured FRF according to Eq. (5.20) . T he FRF of the estim ated transfe r function with 12 m odes is shown in Fig. 5.5 (a) wi th the re d lin e . It is worth to d ecl are tha t with 12 m odes the estim ated FR F has g ood ag r eem ent with the m easured FRF . I n general, the m ore m odes are used, the bette r resu lt can b e obtained. H oweve r , a l arge num ber of m odes of a system m ay l ead to an ove r - f itting problem, and m a y bring som e diff ic ulties for the num e r- ical calcu lation. There fore, only 4 important m odes are chosen to describe the non- rotating p art s . The FRF of the estimated transfer fun ction w ith 4 m odes is shown in F i g. 5. 5 (b) with the red line . Co m- pared to Fig. 5. 5 (a), the er ror between the est im ated and the m easured FR F becom es lar g er. It i s we ll understood that the cal culation ef ficien cy can be inc reased by sacrificing som e calculation accu racy by using the redu ced - order m ode l. H owev er, since the erro r between those m odels is lim ited, it i s mor e Modal ham m er Accelero me t er on pad Accelero me t er on suspension 1 x L x 64 sensibl e t o u se the re duced - order model to sa v e ca lcu lation time . The transfer fun ction of the re duced- order m odel 1, x xL F H − is giv en as 2 52 6 1 28 , 27 0.01028 ( ) s 0.07617( ) s () () k g () () k 150 3.234 10 359.3 6.434 10 144.9 1.432 1 g 0.1334( ) s 0.06125 ( ) s . () () 0 543.7 1.47 kg ( ) ( ) kg 5 10 x xL F jj H jj j j jj jj j j ωω ωω ω ω ωω ωω ω ω − + +⋅ + + ⋅ + = + ⋅ + + + +⋅ + (5.23) Fi g. 5.4 : Frequ ency spectr a of the v elocit ies of the pad and suspension Fig . 5. 5 : Measur ed FRF an d estim ated FRF, (a) appro xi m ated with 1 2 DOFs, (b) approx imated wi th 4 DOFs 5.4 Com parison stud y of exper im ental and s imulat ion resu lts The param eters of th e test rig with a rea l brak e should be identified at firs t . T he way to ide ntif y th e transf er fun ction 1, x xL F H − has been presen ted in section 5.3 and is giv e n in Eq . (5.2 3) . T he t r ans fer Frequency [H z] DFFT(vel ocity) [m/s] Frequency [Hz] (a) FRF [s/kg] Frequency [Hz] (b) FRF [s/kg] 65 function of , F H θ is easy t o m easure, since on ly one m ode is considere d. The static f r ict i on coe ffic ient, the dyn am ic friction c oefficien t, the St r ibeck velocity , the contact s t iffn ess and conta ct dam ping can be identified accord ing to the process p roposed in sect i on 4.2. As a resu lt, the param et ers of Eq. (5.12) are giv en in Table 5.1. It should be not i ced that the p arameters in Table 5.1 a re different wi t h the p a- ram eters in Tab le 4.3 , since differen t brak e com ponent s are used in t he test rig wit h a real brak e as that of th e idea li zed brak e. Paramet ers Values , F H θ ( ) ( ) ( ) 2 3 0.15 s kgm 0.225 1.8 9.38 10 j jj ω ωω + +⋅ d µ 0.335 sd µµ − 0.03 σ 0 8 8.88 10 N/m ⋅ σ 1 5 8.88 10 Ns/m ⋅ v s 0.0547 m /s Table 5.1: Param et ers of the frictio n law With par am et ers to han d, t he simulation resu l ts are e xpressed in Fig s. 5. 6- 5.9 . T he driving speed is 0.31 rad/s and th e brak e pressure is at 8 bar. F ig. 5.6 sh ows the m easured an d sim ulated to rsional ang le and torsio nal v elocity of t he d rive shaft, while th e sim ulated accele ration o f th e brake p ads and the simulated fric tion force are pres ented in Fig. 5.7 . Wh en the system converts to the sli p region f r om the stick region, th e fric tion fo rce has a sudd en chang e, so that the accele r atio n shows a la rge im pulse . Th e sam e signal can a lso be ob served in the m easurement s as shown in Fig. 2.12. How ever, the m easured accele r ation also exh ibits a n i m pul se be tween the s tic k an d s lip re gio n, w hile th e sim ulati on resul ts show only a week impulse. In Fig . 5.8, t he frequency spectrum of the torsional v elocity of the shaft shows a dom inant frequency at 30 Hz and a second frequency at 60 Hz , which is the frequency of th e stick - sl ip m otion . The freq uency spect rum of the acce leration of th e pad shows a lot of frequenc ies of peak , also at m uch hi gher f requencies than the freque ncy of the stick - slip m otion , t his v ibration is the driver heard or felt during c reep g roan . F ig. 5. 9 exhib its the simula ted and m easured st ick - slip lim it cycle in the phase plo t with θ ∆ as a function of θ ∆ , where the simulation resul ts have g ood a gre e- m ent with ex per im ental re sult s. Just as t h e tes t rig with an ideal i zed brak e, both tes t rig s can be des cribed by the sim ilar m odels , whi ch are two linea r system s coupled by a n onl i ne ar friction law. Therefore, the limit cycle obtained f rom those m odels are qu alitativ el y sam e. How ever, the test rig w i th a re a l br a ke is much mo r e complex than th at of the id ealized bra ke , so tha t a m ode l w ith mor e DOFs is required to desc ribe the vibration of the ca r rier. T he q uest ion w i ll be i f the prop osed m odel still work s for a real v ehicle. The answ er seem s to be yes , becau se the st i ck - slip m otion is relat ed to t he velo c ity o f th e p ad ref erring to the s u s- pension, i.e. the st ick - slip m otion is on ly rela ted to the iso lated br ak e - su spension sub - system. Therefore, the s tick - slip motio n will be the sam e no m at ter whethe r the susp ens i on is assem bl ed in the fram e or a ssembled in t he chass i s o f a veh icle. However , even though the stick - slip m otion i s u n- 66 chang ed, the g enerated cr eep groan will b e different if t he chass is of a vehi cl e is c on sid er ed , si nce the vibration of t he c ha ssis ca n also b e excited by the var ied friction fo rce. As a conc lusion, the proposed m odel in Eq. (5.12) also work s for a real vehicle . T he stick - slip motion only r e lat es t o t he br a ke - suspension s ub - system , but creep groan nois e wil l be influenced by t he oth er components of a veh i cl e, such a s the ch assis, the axle and so on. Fi g. 5.6: Exp erim ental and simulat ed tors iona l ang le ( a) an d tors ional v eloc ity ( b) of the sh af t Fi g. 5.7: S im ulated acc eleratio n o f the pad (a) and the sim ulated friction forc e (b) Fi g. 5.8: Frequency spe ctr a of torsiona l ve loc it y of the shaf t (a ) and acceleration o f the pad (b) θ ∆ [ rad ] Ti me [s] (a) θ ∆ [ ra d/ s ] Ti me [s] (b) DFFT ( ) x [ m/ s 2 ] Frequency [Hz] (a) DFFT ( ) θ ∆ [ ra d/ s ] Frequency [Hz] (b) R F [N] Ti me [s] (b) x [ m/ s 2 ] Ti me [s] (a) 67 I n orde r to study the existence condition of creep g roan, the s tability of the solution is p roposed a s follows. At f irst, the transf er funct ion 1, x xL F H − is conver t ed to t he stat e space func ti o n [ ] ( ) [ ] ( ) [ ] [ ] ( ) 1 1 TT , , diag .... ... , diag .... ... , .... ... , ... ... , 1 , 2, ...., . R pp pp pp p R L L ip i p i i p i pi R F xx d k xi L α = ++ = = = = = = = ∑ I XD XK X B DK BX (5.24) Then, the equiv a lent posi tion of Eq. ( 5.12) i s giv en as ( ) ( ) ( ) 0 00 / // 1 ,, 0 ( ) / , 0, () , 0 , () / . s ss rv eq d s d eq rv rv p eq d s d p p eq eq d s d rN N e k NN e z NN e θ θ µ µµ θ µ µµ µ µµ σ −Ω −Ω −Ω − ∆= + − ∆= = +− = = +− X KX (5.25) The stability of the equilibrium solution of Eq. (5.12) is obtained by analyz ing the eigenv alue of the linear iz ed system under its equi librium pos ition T , , pp z θθ = = ∆∆ Y AY Y XX 11 11 1 01 11 1 1 1 01 01 0 ( , ,) ( , ,) ( , ,) (, , ) (, , ) (, RR R R R RR RR RR R LL p pp L p L L LL LL LL p pp L p p pp p p z zz kd rr r I II I I z zz θθ φθ φθ φθ σσ σσ θ φθ φθ φθ σ σ σσ θ ×× × ××× × × × = ∂∆ ∂∆ ∂∆ − −+ + ∂∂ ∂∆ ∂∆ ∂∆ ∂∆ − −+ + ∂ ∂∆ A 00 X XX 0 X 000 I 0 X XX 0 B K DB B X 1 ,) ( , ,) ( , ,) ( , ,) 0 R p pp L p z z z zz z φθ φθ φθ θ × ∂ ∂∆ ∂∆ ∂∆ ∂∂ ∂∆ X XX 0 X (5.26) with 0 0 00 2 0 / 00 / 00 / / () , () () () ( , ,) ( , ,) , ( )e ( , ,) 0...0 0 ( ...0 ) , )( s s s s p r dd p rv dd p rr p dd d sd v ss s sd sd vv ss ss d r ve rr ve ve z z r z NN z φθ θ φθ σ φ µµ µµ θ µ µ µµ µµ µµ µµ µ µµ µ Ω −Ω ΩΩ −Ω −+ Ω −Ω −Ω − ∂∆ = ∂∆ ∂∆ − = ∂ +− ∂∆ = ∂ −+ −+ X X X X where A is the cor respond ing system m at rix. The s tabil ity of t he equ ilib rium solu tion can be an aly zed by studying t he eigenvalu es of A . I f the real par t s o f al l eigenv al u es are neg ative, the equi librium s olu- tion is asy m ptot ically stable. I f any o f th e rea l p arts of the e igenvalues is posi tiv e, the equili brium solu tion is unstable, and th e s olut ion will show increasing am plitudes. The s tick - s lip lim it cycl e o f the test rig with a real brake ca n be ca lcu l ate d b y performing numerica l integ ration b ased on Eq. (5.12) . If 68 the sy stem has n o sti ck - sli p l im it cycle, the system reaches its eq uilib rium solu t ion wi th the init ial condition in th e stick regio n. Otherwise, t he st ic k - slip lim it cyc le is r eac hed . Fi g. 5.9: Sim ulated and m easured s tick - slip lim it cy cle Accord ing to th e condition s for the exis tence o f the stick - slip lim it cy cl e and the sta b ility of the equ ili- brium solut ion, the sy s tem show s three different regions w ith di fferent ty pe s of solutions . Fig. 5. 10 shows the simulated m ap of creep groan with the estim ated param et ers. It is sim ilar to th e ma p of creep groan for t he idealized br a ke shown in Fig . 4.9. Fi g. 5.10 : S imula ted m ap of cr eep g roan of the real br ake Exper im ents are car ried ou t t o con firm the existence of t he thr ee regions, w here the driv i ng s peed of the motor is c onstant (0.31 rad/s) an d the brake pressur e is varied. Corresponding exper imen tal result s are shown in Fig. 5.11. T he red line desc ribes the brak e pressure, while the blue line shows the to r- siona l vib ration angle θ ∆ of th e drive sh aft. When t he bra ke pressure i n creas e slow ly from 0 bar to 5 bar, the system has no cree p groan at l ow pr essure, bu t creep g roan occu rs whe n the pr essure is high er than a critica l p ressure p c 23 (2 bar). Afte r that, the pre ssure is slowly decreased from 5 bar t o 0 bar . Creep groan occurs a t high pr essure and disapp ears w hen the pressu r e is lower th an another cr itic al θ ∆ [ rad ] θ ∆ [ ra d/ s ] Ω 0 [ rad / s ] p [bar] 69 pressur e p c 12 (1.2 bar). T he di ff eren ce betwee n crit ical p ressure i n fact p roofs t he existence of thre e regions o f the syst em. Fig. 5.12 shows the equilibrium s olution and t he lim it cycle of the bra ke system with dif feren t pressu re in the pha s e plo t wit h θ ∆ as a function o f θ ∆ . When the pressure is lower than p c 12 , onl y the equil i- brium solution can b e measured both in the p ressur e - increasing and pressu re - d ecreas ing p rocesse s, m eaning that the system is in reg ion I and cr eep groan cannot o ccur. When the pressu re is hig her tha n p c 23 , creep g roan is m easured i n b oth pro cesses, m eaning tha t the sy stem is in region I II and creep groan alw ays o ccurs i n th is reg i on. When the pressur e i s lower than p c 23 but hi gher tha n p c 12 , cr eep groan i s measur ed in the pres sure - decreasing process, bu t not in the pre ssure - increasing proc ess, me a n ing that t he sy stem is in region I I. Fi g. 5.11: Tors ional vibrat ion angle w ith varied b rake pressure , t he red and b lue points represen t the m eas ured boundary pr essure betw een the three region s. Fig. 5.12: Eq uil ibrium solution an d stick- sl ip lim it cycle under diff erent pressur e I n the rest part of this section, a n experim e ntal m e thod is propos ed to id en tify the m ap o f cree p gro an of the brak e system . The deta iled st eps are giv en as follows. A t a constant sp eed 0, ci Ω , the brake pre s- sure is in creased from 0 bar. Th e system is in its eq uilibr ium solutio n at l ow brak e pressu re. If th e pressur e is hig her than a critical v alue 23, ci p , c ree p groan w ill oc cur. At this mom ent, the speed as we l l as t he brake pres sure is recorded as a bo undary point 0 , 23 , ci c i p Ω in the v eloc ity - pressure map wit h a tr iangl e . On ce cr eep g roan occur s in t he syst em, the b rake p ressure is s l owl y decreased un til th e disappea rance of c reep gr oan. Meanwh ile, t he b oundary poin t is recorded a s 0, 1 2, ci c i p Ω in the veloci t y - pressure m ap with a cycle . Ti me [s] θ ∆ [ rad ] p [b ar] θ ∆ [ rad ] θ ∆ [rad/s] θ ∆ [ rad ] θ ∆ [rad/s] θ ∆ [ rad ] θ ∆ [rad/s] 1 b a r p = 1.8 bar p = 12 bar p = 70 Furtherm ore, the driving speed is ch anged to 0, 1 ci + Ω and find th e corresp onding boundary po ints 0 ,1 2 3 ,1 ci c i p ++ Ω as well as 0, 1 1 2, 1 ci c i p ++ Ω . Repeating t his process, the boundary poi nts unde r di f- ferent speed s can be m easured. I t is pos sible to f i nd a poly nomial reg ressi on that fits the da t a set 0, 1 2, , 1 , 2, ..., ci c i pi N Ω= . This curve i s t h e boun dary between reg ions I and II. I t is also poss ible t o find a poly nom ial re gre ss io n tha t fits th e dat a se t 0 , 23 , , 1 , 2, ..., ci c i pi N Ω= . Thi s curve i s t h e bou n- dary b etween regions I I and III. As a result, the b oun dar ies of reg ions I, II and III can be obtained by experim ental analysis, sho wn in Fi g. 5. 13 (Similar re sults have been publ ished in [33]) . Com pa r e d t o the s imula ted map of creep groan in Fig . 5.10 , the m easured m ap shows the sim ilar beha v ior wi th the simulated one. Fi g. 5.13 : Map of cre ep groan id entified by ex per i ments , the triangle s represen t the m easured bou n- dary points between r egions II and I II , while the c ycles represe nt the m easured boundary points between region I and II ; t he red line and the blue li ne are the poly nomial reg ression curv es of the m easured boundary po ints Du ri n g t he pr es sur e - increa sing proces s, the v ibration amplitude of the d rive sh aft increas es with t he brake pressure, shown in Fi g. 5. 14 ( a). By doing SFF T of θ ∆ ( using H amm i ng window with window size 0.5s ) as shown in Fig. 5. 14 (b), one can see that the frequ ency of the st i ck - sl ip m oti on d ecreases with in c rea sing o f t he brak e pressure. Me anwhile, th e amplitudes of t he ac celeration of the pad 1 x increas es w ith th e brak e pressu re, shown in Fig. 5. 14 (c). T he SFF T of 1 x is shown in Fig. 5. 14 ( d) , and t he frequency of 1 x chang es with t he brake pre ssure. Sim ilarly , during the accelera ting proces s, the vib ration am pl itude of t he dr ive s haft is in crease d wi th the speed, shown in F i g. 5. 15 (a). The frequ ency of th e stick - slip motion increase s with the sp eed , and approach es t he eigenf reque ncy of the dis k - shaft sub - system , s hown in Fig. 5.1 5 (b). In Fig. 5.1 5 (c), the am plitude of t he accel eration of th e pad x is almost not influenced by t he speed. The SFFT of x is presented in Fig . 5.15 (d), and t he speed has alm ost no influen ce on t he v ibra tio n frequency of the p ad . III II I Ω 0 [ rad / s ] p [bar] 0, 1 2, ci c i p Ω 0 , 23 , ci c i p Ω 71 Fig. 5. 14 : Mea sured sig nal s during pressur e - increasin g p rocess, (a ) vibr ation of the torsional angle θ ∆ wi t h inc r ea si ng b ra ke pr essu r e, ( b) S FFT of θ ∆ , (c) accel eration of t he pa d with increasi ng brake pres sur e, (d ) SFF T of x . Ti me [s] (a) θ ∆ [ rad ] p [bar] p [bar] Ti me [s] (b) Frequency [ Hz ] Ti me [s] (c) 1 x [ m/ s 2 ] Ti me [s] (d) Frequency [ Hz ] 72 Fig. 5.15 : Measu red signal s during acce lerating proce ss , (a) v ibration of th e to rsion a l ang le θ ∆ with increasing spee d, (b) SFF T of θ ∆ , (c) accelera tion of the pad with increasing speed, (d) SFF T of x . Ω 0 [rad /s ] Ti me [s] (b) Frequency [ Hz ] Ti me [s] (d) Frequency [ Hz ] Ω 0 [rad /s ] Ti me [s] (c) x [ m/ s 2 ] Ti me [s] (a) θ ∆ [ rad ] 73 5.5 Summary I n t his ch apter, t heo ret ical and exp erim ental inv estigation s of creep gro an on a test rig wi t h a rea l brake are carried out. Acc ording to the analysi s of the dy na m i c model, it is possible to isolat e the br a ke - suspensio n system from the frame, the ax le, and the ch assis for the study of the sti ck - sl ip motion. Therefore, com pl ex com ponents such as the ch assi s can b e a t fir st igno red. I n order t o an alyz e creep groan in such tes t rig effici ently , a reduced - order model is proposed by ig- nor ing the relativ e unimpor tant m odes. The advantage s of the reduced - order model ar e that i t requi res only sy stem param eters and it h as hi gh com puta tional e fficiency. Moda l analy sis is carr ied out t o i d entify param eters of the test rig w i th a real brak e. Base d on the ide n- tified transfer func tion, the simulated resul ts are compa r ed with the exper imental resul t s qu antit ativ ely. Both resu lts h av e good ag reem ent with e ach other. It is confirm ed that th e reduc ed - order m odel is of effic iency to describe c reep groan of the b rake system . Furtherm ore, a map of creep g roan is obtained by stabi lity analysis of the equi librium solution and stick - slip l im i t cycle. Additionally, thi s map is measured thro ugh a n experim ental method. The m ea s- ured and calc ulated maps s how the sim ilar beh avior . 74 6 Countermeasures again st creep gro an After under standing the m echanism of creep groan, differ ent m e thods ag ainst cree p groan are dis- cussed in this c hapter. A t f irst, a p ad, wh ich con tains piezoce ramic stap le a ctuato r s [112] , is used to suppress creep groan. Th e n, damping m ateri al s a re add ed between the shaf t and di sk to suppr ess creep gro a n . Furt herm or e, during the accelera t ing process , a skillfu l driv er can e m ploy the ca dence braking technique to decr ease the tim e of cr eep groan. I nspired by the optim al braking techniqu e , a control loop is designed and integrated into an anti - lock brak ing sy stem to p revent c r eep groan. 6.1 Suppressio n of cree p groan through a active pad I t is a w ell - k nown effect th at vib ration can influe nc e f ricti ona l conta cts s ee e.g. [47], [ 5 7] -[61], [80] , [81] . There f ore, a pa d, which con tains pi ezoceram ic staple a ctuato rs, i s appli ed to e liminate creep groan o f the brak e system . Von Wag ner et al. ha ve su ccessfu l ly u s ed such “sm art pad s ” f or the a ctive suppression o f br ake sque al via o p tim al cont rol [11 2], [ 113] . As an extension of th is work , sim ilar activ e pads are u sed to sup press creep groan in thi s thesis . I n this case, c reep g roan can b e elim inated by gi vi ng a n external vibrat i on at a cons t ant f requ ency v ia the act ive pad . At the beginning of this section, the Prandtl - T omlinson m odel i s studied to und erstand the friction al m echanism on the atom ic scale [78] . The Prandtl - To m linson m odel describ es a sha r p tip sc ann ing a corrug ated surfa ce at a constan t n orm al forc e in nanotribo logy, and the friction f orce is g enerated by the torsional bending of t he can tilever wh ich the t ip is m ounted on. The way t o calcul ate the s tatic friction o n the atom ic scale is giv en in [80] . A ssum ing that the co rrugated su rface has s inusoidal su r- face potentia l with am plitude E 0 an d pe ri odi c it y a , the potenti al en er gy V int of the system is ( ) 2 0 int 1 cos 2 , 22 t tt s Ex V kx x a π = − +− (6.1) where the pull ing spring with stiffness t k is extended be tween the posi tion of the tip t x and the m o v- ing tip s x . I f the tip m oves slow ly, it a lway s resi des in a m inim u m of the e ffe ctiv e po ten tial. Th e condition of th e m inimum of the effect ive potent ial is int /0 t Vx ∂ ∂= [80] int 0 si n 2 ( ). t tt s t VE x kx x xa a π π ∂ = +− ∂ (6.2) The pulling force F L is equ al to the fo rce in th e spring if ignoring the dam ping and acceleration of the tip. With () L tt s F kx x = − , it fo llows that 0 sin 2 . t L Ex F aa π π = − (6.3) T he c or r uga ti on o f t he sur fa c e pot e nt ial E 0 is l inear ly rela ted to th e m axim u m la teral force F L , the m axi m um of the abso lute value of the force F L is f ound a t /4 t xa = . I t is obtain ed 75 0 , L E F a π = (6.4) where F L is considered as the s tatic friction f or ce, whi c h is the m ini m um force to let the tip jump ou t the energy barrier of heig ht 0 / Ea π . Th e s tatic fr ictio n co eff ici ent is therefo re given as 0 ,0 , s N E F a π µ = (6.5) where F N i s the n ormal fo rce. I f a m echanical vibr ation f is a dd ed i n t he out -of- plane direction of the di s k, the oscill ation caus es t h e variat ion of the norm al forc e F N and the energ y corrugation E 0 . It is a ssume d that the c orrugation en e r- gy ch anges w i th tim e as 0 ( ) (1 c o s 2 ) E t E ft απ = + when the norm a l oscillatio n acts on the lateral tip m otion, where α is a sc ale assoc i ated to the am plitude of the v ibration . I f / s f xa >> , t he ti p r eac hes the minim um corrugation 0 (1 ) E α − ma n y t im es when the tip slowly m oves on the sur f ace [81] . It m eans that the surface po tential ene rgy u nder m echanical v ibration c an be express ed as 0 (1 ) E α − , then 0 (1 ) . L E F a πα − = (6.6) As a re sult, the sta t ic f rict ion coe fficient un der m echanica l vibra t ion i s expressed as ,0 (1 ) . ss µµ α = − (6.7) Obv iously , the sta tic fric tion c oeff icien t is de crease d with in crea sin g α . The dynam i c coeffic i en t will not chan ge un til the tip sep ar ates fr om the surfac e. Fig. 6.1: Pra ndt l -Tom linson m odel [78] A pad w ith i ntegra t ed piez oceram ics m anufacture d at MMD TU Berlin p rovide s hi gh frequen cy exc i- tation in the out - of - plane dir ect ion, which is sim ilar to the pad used in [ 112] . Th is pad cont ains t wo piezoel ectric l ay ers with an elect rode l ayer between the m , shown in F i g. 6.2. A signal genera tor can produce a sine form voltage with frequency from 1 Hz t o 100 kH z. A signal amplifier is uti lized for the volt age am plificatio n. Then th e excitation v oltage i s a x t x s k t F L F N 76 0 sin 2 , p U U ft π = (6.8) where U p i s t he am plitude of th e voltage, an d f 0 is its frequency . T here are numerous pu blic ation s on activ e inf luencin g fric tiona l co n tac ts [47] , [57]- [ 61], [ 80] , [81] , but author is to the best of his know ledge not aware of the prior usag e of such activ e pads for the sup- pression o f creep groan so far. I n order to suppress ion of creep groan, t he a ctiv e pad is assem bled to the carr ier as shown in Fig. 6.3 ( a). A cc ording to the abov e analysis, it is pos sible to de cr ease the st ati c fr i cti o n co ef fi ci ent b y gi vin g a hi gh fr eq ue nc y mec ha ni ca l vi bra ti o n in t he o ut -of- plane direct ion via the activ e pad. Creep g roan can theref ore be eliminate d when the stati c frict ion co efficien t i s less than som e critical v al ue. Ex perim ents are pe rform ed to con f irm this statem ent. T h e frequency of the exte r- nal var i ation is chosen in such a way tha t , o n the one hand, the frequen cy shou ld be one of eigenfr eque ncies o f t h e cali per to m aximiz e t he am plitude of the extern al vibratio n . O n th e o the r ha n d, it should be higher than the hum an l imits of hearing t o avoid additional n oise. As a result, the freque n- cy 20 k H z is chosen whi ch fulfills bot h requirem ent s. Fig. 6.2: Cons truction of th e ac tive pad Fig. 6.3: Locat ion of the p iezoceramic actuato r , (a) a ctiv e pad in the id ealiz ed b rak e, (b) activ e carr i- er, (c) active pad in the real brake The spe ed of the m otor is set a s 0.2 rad/s and the brak e pressure is given as 6 bar. U nder thi s conditio n the syst em is in region I II and cr eep g roan wi ll alwa y s occur witho ut ext ernal excitation . T h e me a- surem ent is started whe n a st able stick - slip lim it cy cle is observed in th e syst em. I n Fi g. 6.4, the vibration of t he s haft θ ∆ is e x hibi ted wi th a b lue l ine, w hile th e d rivin g v oltag e U p is show n as a red Brak e plate Piezoele ctric lay er Elec trode Piezoele ctric lay er Br a ke pad (a) (b) (c) Piezoact uator Piezoact uator Piezoact uator 77 lin e. I t is notice d that th e a mplitu de of the creep g roan decreases with the incre ase of the voltage. Onc e the vol tage is larg er than a nother critica l value U c 1 , cre e p gr oa n is elim inated. The re ason i s t h at th e system shifts into reg i on I due to the d ecre ase of the static fric tion coef f ici ent . This exp erim ent co n- firms that cre ep groan can be e lim i nated by adding an externa l vibrat ion in the ou t- of -plane direction. Furtherm ore, if the vo l tage is dec reased an d low er t han a cri tica l valu e U c 2 , creep g roan appears ag ain. The r eason i s that th e syst em return s to reg ion I II due to t he inc rease o f the st atic fric tion c oe ffic ient. U c 2 is lower th an U c 1 , which also con firms the exi st en ce of r egion II wit h both stable equi librium sol u- tion an d lim it cy cle . Fig. 6.5 shows the equilibr ium sol ution and st ick - sl ip lim it cycle of th e system . If th e dri vi n g vol t age is lower than U c 2 , t he sy stem i s in region II I and the stick - slip lim it cy cle is th e o nly stable solution of the system . If the driv ing voltag e is higher than U c 1 , the system is in reg ion I and the stick - slip lim it cycle do es not ex ist. If t he dr i vi ng vo l ta ge is bet we en U c 1 and U c 2 , t he sy s tem is in region I I and the occurrence o r absence of creep groa n depends on i t s initial condition. Therefore , the stick - slip lim it cycle can be ob serv ed in th e vo ltag e - i ncreasing proces s in r egion II , whil e the equilibrium sol uti on ca n be observed in the voltage- decreas ing process. Fig. 6.4: Su p pr es sion of creep g roan by adding a n external excitation in the out-of- pl ane direction of the disk, the blue point s a re the crit ical v oltage tha t creep groan disapp ears, while the red point s are the c ritica l vo ltag e that creep g roan appear s again. Fig . 6.5: E qu ilib rium solut ion an d l im it cycle w ith d if ferent am plitud e of v oltage , (a) sy stem in r e- gion II I, (b) system i n region I I, (c) system in region I Furth ermo re , a m echanical v ibration added in the in - plane d irection of th e di s k will be consid ered. I n this ca se, the o scillation wi ll c hang e t he posi tion of x t . According to the Ref. [58 ] , Eq. (6.3) becom es Ti me [s] θ ∆ [ rad ] U p [V] θ ∆ [ rad ] (a) θ ∆ [rad/s] θ ∆ [ rad ] (b) θ ∆ [rad/s] θ ∆ [ rad ] (c) θ ∆ [rad/s] 0 V p U = 60 V p U = 80 V p U = 78 ( ) ( ) 0 0 2 sin sin 2 , Lt E F x ft aa π π βπ = −+ (6.9) where β is a scal ar assoc iated to the am plitude of t he in - plane v i bration. I ntegrat ing Eq. (6.9) with a period 0 1/ Tf = ( ) ( ) ( ) ( ) ( ) ( ) 1/ 0 0 1/ 1/ 00 00 2 sin sin 2 22 sin c os si n 2 c os sin sin 2 . f Lt ff tt E F x ft dt aa EE x ft dt x f t dt aa a a π π βπ ππ ππ βπ βπ = −+ = −− ∫ ∫∫ (6.10) T he mean v alue of the sec ond term i s equ al to 0, si n ce it is a n odd function. The mean val u e o f t he first term can be calc ulated with the help of the Besse l -function [58] 0 0 2 si n ( ), Lt E F xJ aa π π β = − (6.11) where J 0 is the Bess el funct ion of the fir st kind. The m axi m um value of Eq. (6.11) is 00 00 max 2 s i n () () , Lt EE F xJ J aa a ππ π ββ = −= (6.12) where F L i s t h e static friction for ce. As a r esult, the sta tic friction coef f ici ent is 0 ,0 () . ss J µ βµ = (6.13) There fore, th e static fr iction coeffi cient is decre ased w ith in cr easing β . I n orde r to con firm the abov e analysi s, a piezoce r am ic ac tuator is ass em bled under t he short ed ge of the L- sha ped s teel p late which consti t utes a n a ctive carrier, shown in Fig. 6.3 (b ). The v ol tage o f the piez oceram ic actuator i s giv en as 0 sin 2 p U U ft π = . After test , it is found that creep g roan can be eliminated when t he fr equency of the vol t a ge is 8 kHz , which is one of the eig enfrequenc ies of the carri er. The operati on conditio n is at the brake pre ssure 10 bar w i th the speed 0.3 rad/s. Expe rimental results are p r esente d i n Fig . 6.6. At the beginning, the system is in the reg i on II I and creep groan occ ur s with out externa l excitat ion. I t is o b- vious that cre ep groan is el iminated w hen the piezoceram ic actuator t urns on. On ce the p i ez oceram i c actuato r turn s off, creep groan app ears ag ai n. I t should be noted that the activ e carrier has disa d van ta ge s comp ared to the ac tiv e pad : The vi br at i on source is f ar from the contact su rface ; it is hard to ad d a piezoceram ic actuato r in a real b rake car rie r due to its c om pa ct struc ture. 79 Fig. 6.6: Supp ression of cr eep g roan by an ex terna l exc itatio n i n th e in -plane direct ion of the disk Therefore, the a c tiv e pad is em ployed in t he tes t rig with a r eal brake again st creep g r o an. Once this techniqu e is succe ssfully perform ed in the tes t rig wi th a real br ake, it can be conside red to use in a real v ehi c le. An ac tive pad is assem bled in the p iston side o f the cali p er as show n in Fig. 6.4 (c). T he speed of the motor is set as 0. 2 rad/s and th e brake pressur e i s g iven as 6 b ar. Un der this con dition the system is in region III and creep groan occu r s withou t external e xcitation. I n the first experiment, the am plit ude o f t h e vo lt age i s ke pt constan t an d its freq uency is ch anged . The volt a ge of pi ezoceram ic actuator is set as 7 0 V and v ar ies from 10 k H z to 20 kHz . Exper im ent result s ar e exhibit ed in Fig . 6.7, where the re d line pr esents the frequency of the v ol tag e and the blue line shows t he v i bration o f th e tors ion al ang le of the shaft . Th e vibrat ion am pl itude of the sha ft varies w ith the frequency and has th e m inimum value at 15.8 k Hz and 16.5 kHz , wh ich ar e eigenfreque nc ies of the cal i pe r . The external vibration has the m aximum amplitude under those f requencies, so t hat the s tat ic frict i on co efficient has the mi n i mu m va l u e. I n the second experim ent, the frequency of the v oltag e i s set as 15.8 kH z, and the am plitud e of th e v oltag e is slow ly inc rease d from 0 V to 100 V. I n Fi g. 6.8, the stick - slip v ibrat ion θ ∆ is exhib ited with a blu e line, and t he am plitude of the v olt ag e is show n as a red line. T he system has creep groa n when the voltage equals to 0. The vibration amplitude of the shaft decreas es wi th incr easing the am pl i- tude of the v oltage. On ce t he voltag e is la rger than a critica l value, creep groa n is el imin ate d. I f th e am plit ude of the vol tage is decreas ed , creep groan oc curs agai n. For a conv entional ac t ive noise cance llation m ethod, a feedb ack con trol loop is required [112] . In order to guarantee the s tabi lity of feedback cont ro l as well as good control perfo rm ance of the control loop, lots of work is requ i red and the corresponding hardware is g eneral ly expen sive. I n contrast, the propos ed act i v e pad syst em is easier and cheap er. Fig. 6.7: Supp ression of cr eep g roan by an ex terna l exc itatio n with va r ie d vo l tage frequency Ti me [s] θ ∆ [ rad ] U p [V] Ti me [s] θ ∆ [ rad ] f 0 [H z] 80 Fig. 6.8 : Supp r ession of cre ep groan by an e xte rnal e x cita tion with va ri ed vol ta ge amplitude 6.2 Pas sive m eth od ag ainst c re ep gr oan Ano ther feasible method ag ainst creep groan is to increas e t he dam ping of the shaft. C ompared to the activ e m et hod, th e passiv e method doesn ’t require ext ernal a ctuato r s and sensor s. When the dam ping of the driv e shaf t increas es, energ y dissipation in the slip reg ion becomes so la r g e, that the sy stem m ay not return t o the stick r egion. As a resul t, the s tick - s lip motion c a nnot r epeat an d creep groan can no t occur in t he sy s tem. O n the other hand, i f damping of the carr ier i s increa sed , t he vi b rat i o n of t he p ad ma y be reduced , but the stick - slip mo tion on t he di s k - shaft sub- sy st em is not influenced. T he map of creep groan w i ll not chang e. Therefo re, on ly the cou ntermeasu r e w i th inc rea si ng t he damping of t he shaft i s discu ssed here. Fig. 6. 9 : Sim ulated m ap of cree p groan of the system with ( - ) and withou t (- -) damping m ater ial s. T he rhombus rep resent th e m easured critical points of t h e sy stem withou t dam ping m aterials , the cycles represe nt the m easured c ritical poin ts of the sy stem with dam ping material s T heo ret ic al ana l ysis is b y t he model proposed in Eq. (3.42) w ith its param eters in Tab le 4.1 and T a ble 4.3. Fig. 6. 9 shows the m ap of the param eter region s with and without damping material s , wher e the dotted line repres ents the reg ion boundaries o f the orig ina l system ( 2 N ms d θ = ), w hil e the solid line denotes the re gi on bounda ries of the system wit h da mping m ateri al s ( 2.5 Nm s d θ = ). Comp ar e d to ma p of the o r igin al system , the a rea of regions I I and III is decre ased w ith t he inc r eas e of da mpi n g of the driv e shaft, i. e. dam ping m at erial s reduce the risk of cr eep groan g eneration. θ ∆ [ rad ] U p [V] Tim e [ s] Ω 0 [rad /s ] p [bar] 81 Fig. 6.10: Add ing dam pi ng m a terials betwe en the d isk and th e driv en shaft Fig. 6.11 : T he critic al spe ed of o rigina l the syst em (a), and that of the sy st em with damping material s (b) , the rhom bus repres ent the m easured criti cal point s of the syst em without dam pi ng material s , the cycles r eprese nt the m easured cr itical po ints of t he sy stem with dam ping m at erial s I n pract ice, d amping mater ial s ar e added be t ween th e shaft a nd brak e disk as s hown in F i g. 6.10, and experim ents are c arried ou t to t estify the above theo retica l analy si s . The driv ing speed is varied u nder a constant brak e pr essure ( 8 bar). The m easured results are shown i n Fig. 6.11, wher e the b lue line denotes th e vibration o f t h e shaft and th e r ed line pr esents th e motor speed. F or the orig inal b ra ke system , the critica l speed b etween reg i ons I and II is 0.62 rad /s and the c ritical sp eed between r egion s II and III i s 0.42 rad/s, which are m arked in Fig. 6.9 with the rhom bus . For the system wit h dam ping m aterials , b oth th e critical speeds a re decreas ed. The critica l sp eed betw een region I and II becom es 0.48 rad/s and the cr itical speed betwee n r egion s II and III i s decreased to 0.35 rad /s, which are m arked in Fig. 6. 9 wi th cir cle s . The ex pe rim ents ind icate tha t b oth cri tical speeds ar e de crea sed b y Tim e [ s] (a) θ ∆ [ rad ] Ti me [s] (b) θ ∆ [ rad ] Dam ping m ater ial s Ω 0 [rad /s] Ω 0 [rad /s] 82 adding dam pi ng m ater ial s bet ween the sha f t and the d isk. I n pr actice, the shaft can be des i gned wi t h a large dam ping coe fficient. It is poss ible t o incre ase the th ickness of the sha ft so tha t the dam ping of the shaf t incr eases. 6.3 Suppressio n of cree p groan th rough an opt imal br a ke techni que I n t his sec tion, the occurre nce of cre ep g roan in drive pro cesses will be stud ied with the assi stanc e of the map of cre ep groan . Instead of redesig ning the com ponents of a vehicle , anot h er way to influenc e creep gro an is t o co nt rol t he vel o ci t y of t he w he el and b r ake p r es sur e during driv e process es . F or t he sake of sim plification, a two w heel vehicle m odel is used for the simu lat ion . The dynam ics of a v e- hic le is gi ven a s 2 0 1 () ( () ) () , r pi i R M t r F t Tr A p t r µ = Ω +Ω = − ∑ (6.14) where M is the m ass of th e vehicle, r is the radius of th e wheel, r F is the r esist ance f orce such as w ind force, 0 T is the drive m oment provided by the m ot or, () pt is the brak e pressure, A p is the are a of the pressur e surface , µ is th e fric tion c oe ffic ient, R is the ra dius o f th e fr iction conta ct to th e cen ter o f th e disk, () t Ω is the ang ul ar speed o f whee ls , wh ich is c alcul ated as 0 0 () ( ) . t td ττ Ω = Ω +Ω ∫ (6.15) I n t he ac celer ating process, i t is assu m e d that the dr iver re leases the brake ped al slowly , wh ile t he d r i ve mo me n t and the r esi stance for ce keep consta nt. A s a resul t, the brak e pre ssure an d the corre s- ponding v elocity is gi ven as 00 2 0 0 00 () , 11 () 2 , rp p p t p at RR t Tr F p A t A a t Mr r Mr r µµ = − Ω = − − + +Ω (6.16) where p 0 is the in it ial brak e pressure, 0 a is the s lop of t he linear decrea s i ng brake pressure. I f this process is plotted in the map of creep g roan , the tim e of creep groan c an be eas ily estim ated with th e assistance of the m a p. As shown i n Fig . 6.12 (a ), the system is at fi rst in reg i on II I and creep groan occurs since t he stick - s lip limit c y cle is the only stable solution o f the system , where the red line denotes the system with c reep groan . With in cre asing spee d, creep g roan disap pears once the system enters region I , where the blue line denotes the sy s tem w ithout creep groan. Fig . 6.12 ( b) shows the brake pressu re and the ang ular speed of the w heel. I n t he acce lerating process , creep groan appea rs at the beginning and di sap pear s at 4.7 seconds. I n the decelerat ing proc ess, it is assum e d that the dr i ver p resses the brake ped al s lowly , and the br a ke pressure is incr eased with a slop 0 a . Then, the brake pr essure and the cor respond ing v el ocity of the vehicle i s g iven as 83 00 2 0 0 00 () , 11 () 2 . rp p p t p at RR t Tr F p A t A a t Mr r Mr r µµ = + Ω = − − − +Ω (6.17) The dece lerat ing proce ss is plotted in the m a p of cr eep gr oa n as shown in Fig. 6.13 (a). The system is at first in r egions I and creep g roan doesn ’t appear . Creep groan app ears when t he system reaches to re gion III since the equil i brium solution o f the system becomes uns table . Fig . 6.13 (b) show s the br a ke pressure and the angular s peed of th e wheel. I n the de ce l erat ing p roces s, creep groan appea r s at 7 second s and d isapp ears at 9 secon ds. From the sim ulations, it is obv i ous that the tim e of creep groan in the acceler ating proces s is long er than the time of th e decele r ating process. I n t he de celera ting pr ocess, the i nit i al condition i s near the equilibrium sol ution, and cr eep groan happens wh en t he equi librium s olution is uns t abl e . I n the accele- rating proces s, the in i tial c ondition is ne ar the st ick - slip limit cycl e, and cre ep groan disappe ars whe n the st ick - slip lim it cy cle do esn’t exis t. Th is can e xplain why creep g roan is m ore serious in the accele- rating process t han in the decel erating process , which ag rees with dr iving experience. I n practice , creep groan no rmally happ ens in the acce lerating process. I n order to reduce t he t ime of creep g roa n, an optim al accelerat i ng process is proposed. I nstead of linear dec r easing t he br a ke pressure , t he cadence br a king techniqu e is in trodu ced to preven t creep groan. Cade nce braking m eans that the driv er releas es the b r ak es for a sho rt tim e and brake s a gai n . When the d river rel eases the brakes a t low speed , th e system leave s regi ons III an d II r apidly. After that, even the brak e pressur e i n crease s agai n and e nter s region I I , creep groan doesn’ t appear s ince its initial cond ition is nea r its equilibri um solution. Sim ulation results are shown in Fig. 6.14. I n this op- timal acc elerating process, creep g roan appears at the beg i nning and disappe a rs at 0.8 seconds, whic h is much shorter than the tim e of cre ep groan under the lin ear acce l erat i ng process des c ribed in Eq. (6.16) ( with 4.7 se conds). Fig. 6.12: Creep g r oan in accel erating p rocess 0 Ω [ ra d/ s ] (a) p [bar] Ti me [s] (b) III II I 0 Ω [ ra d/ s ] p [bar] 84 Fig. 6.13: Creep g r oan in decel erating process Fig. 6.14 : Cr eep groan in optim al acce l erat i ng process An anti - l o ck b r aki n g sys t em ( ABS) is an a utomobil e safe t y system that all ows t he w heels on a vehicle to keep traction cont act prev e nting the wheels from loc king up. It can im itate the caden ce brak ing that w as p racti ced by skillf ul dr ivers [114]. An ABS can apply cadenc e braking 15 tim es per se cond, whil e hum an can do j u st 1 or 2 ti m es per secon d. Beca use of this, the whee l s o f cars equipp ed w ith the A BS wo rk m uch bet ter than the hum an. Typ ica lly , an ABS inc ludes a cen tral e lect roni c con tro l un it (E CU) , wheel sp eed sen sors, a nd hy draulic v al v es within th e brake hy draulics. T he spee d sensor s det er mine the accel erat ion or de celer ation of ea ch whee l; the ECU constantly m onitors the rotational spe ed of each whe el an d con trol s the valv es to avoid whe el lock. I f a wheel rota t es sig nifica ntly slow er than the others, the ECU c an j ud ge t he impending wheel lock and actua t e the v alve to re duce hydrau lic p res- sure of the brak e at th e af fecte d whe el . As a resul t, t he b ra ki ng pressu re on that wheel is reduced and the wh eel th en turns faste r. C onv ersely, i f the EC U detects a wheel tu rning sig nificant ly fas ter than th e others, t he b r ake pressu re to the wh eel is in creas ed so the b rak i ng pressure is reap p lied, slowing down the wheel. This process is r epe ated c ontinuously to pr ev ent the wheel blo ck. I n the following par ts, sim u lation an alysi s will be proposed to con fi r m that an ABS can be del ivered to prevent creep g roan of a v ehicle, i f caden ce braking is applied by t h e ABS dur ing the oc cur r en ce of creep groan. 0 Ω [ ra d/ s ] (a) Ti me [s] (b) 0 Ω [ ra d/ s ] (a) Ti me [s] (b) III II I III II I p [bar] p [bar] 0 Ω [ ra d/ s ] p [bar] 0 Ω [ ra d/ s ] p [bar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¦ ZKHUH $%6 S LVWKH FRQWUROOH GEUDN HSUHVVXUHE\ WKH$%6, WVG\QDP LFVLVJ LYHQDV ZLWKFUH HSJURDQ ZLWKRXWFUH HSJURDQDQG RWKHUZLVH $%6 L GH $%6 L $%6 L LQ $%6 L G $%6 G $%6 L SE S SE S S S S S ° ® ° ¯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f an AB S is used in the bra ke system , the brake pre ssur e can be controlled t o prev ent the vehi cle fro m creep groan. The acc elera ting process with the a ssista nce of t he ABS is p lo tted in the m ap of creep gro a n as show n in the Fig . 6.1 7 (a). T his p rocess can be described as that th e brak e pressu re is d e- crease d o nce th e sti ck - slip m otion occu rs, whi le th e brak e pressure returns to the designed pressure when the sti ck - s lip m otion disa ppea r s. As a resu lt, the speed an d the pressur e of the vehic l e wi ll fo l- low the bo undary betwe en regions II and II I . The angular veloci ty of disk is show n in Fig. 6.1 7 ( b), where th e sti ck reg i on is m arked w i th red color . Fig. 6.1 7 (c) exhib its the accele ratio n of t he pad. Com pa red to the simulation resu lts without ABS, cr eep g roan o f t he brak e system w i th the ABS o c- curs only i n short time interv als , i .e., cr eep groan can be success f u lly elim inated throug h the ABS . Fig. 6.1 6 : Acce lerating process w ithou t t he ass istance of an AB S, (a) acce leratin g process pl otted in the map of cree p groan , (b) ang ular veloci ty of di sk, (c) acce l erat i on of the p ad Tim e [ s] (b) θ [ ra d/ s ] III II I 0 Ω [ ra d/ s ] (a) p [bar] Tim e [ s] (c) x [ m/ s 2 ] 87 Fig. 6.1 7 : Acc elera ting p rocess w ith th e ass ist an ce o f an ABS, (a) acceler ating proce ss plott ed i n the ma p of creep groan , (b) angular ve loc ity of d isk, (c ) acce lera tion o f th e pad 6.4 Summary I n this chapter, diffe rent m ethods are proposed to suppres s creep groan, in terms of adding pie zo c e- ram ics actua tor , in creasing the dam ping of the drive sh aft, and by an opti mal b ra kin g tech nique. At f i rst, theoretica l and experim ental investigat ions are carried out to con firm that creep groan can be elim inated b y gi v ing a high fr equency m echanical vibration in the out - of - plane dir ect ion of the disk. The hig h frequency vibration is provided by an active pad, w hich contains two piez oe lec tric layers with an ele ctrode layer be t ween them . The stat ic fricti on coef ficien t is dec reased by giving a hig h frequency v i bration, and c reep groan c an be elim inated when th e stati c friction coe ff ici e nt i s l ess t han a crit ical v alue. I f a piez oceram ic actua tor i s assem bled betw een the short edges of the L- shaped steel pla tes and the frame, a hig h frequency mechanica l v ibration can be prov i ded in the i n - p l an e dir ec ti on of the disk, and Creep g r oan can als o be elim inated. Furtherm ore, t he activ e pad is ass embled in the test rig wi t h a real brak e, and c reep g roan of the real b rake is su ppressed by the ac t ive pa d successfu ll y . 0 Ω [ ra d/ s ] (a) p [bar] Tim e [ s] (b) θ [ ra d/ s ] Tim e [ s] (c) x [ m/ s 2 ] III II I Desig ned path Real p ath 88 Ano ther feasib le m ethod ag ainst cre ep groan is t o incre as e the d amping of the shaft. By ad di ng d am p- ing m aterial s between the drive s haft and disk , energy dissipa tion dur i ng th e slip region becom es large. As a resul t, the system cann ot re turn to the st ick reg i on again so that c r eep gr oan i s suppressed. E xp e- rim enta l and simulation res ults ind i cate t h at the area of r eg ions II and III d ecreases with the inc rease of the dam pi ng of the shaft, i .e. the re is less the r i sk of cr eep g roan gene ration . With the as si stan ce of th e m ap of creep groan , cre e p groa n in accelerat i ng and decel erat i ng process es is s tud ied . It is found out that creep g roan is m ore serious i n the acce lerating proc ess than in t h e dece- lerating process , which ag r ees with driving experi e nce. After tha t, an optim al brake technique (cadence b raking) is propos ed to minim ize the time of creep groan. Fur thermore, a n ABS can perform the cadence b raking through a simple control l oop to avoid creep g roan just lik e a sk illful dri ver . S i- m ulation resu l ts i ndicate th at creep g roan can be eliminated with the as sistance o f the A BS . 89 7 Conclusions and Futur e Work The main o bject ive of th i s thes is is to study the funda mental m echanism of cree p groan theoret i ca lly and experim ent ally. I n Chapt er 1, br ief introduc tions on m aj or developm e nt of b rake noise, v ibration and harshn ess, stick - slip motion and creep groan are presented. Due to t he increasing com fort claims of custom ers, th e study on lo w frequ ency creep groan has b een re ceiving i n creas ing att en tion curren tl y . I n order to understand cre ep g r oan step by st ep, test ri gs concentrating on cre ep groan are presented in Chapter 2. At first, a test rig with a n idealized brake is built to study c reep groan concentrated on fri c- tion contac t. Subsequ ently , a test rig with a re al brake, which is m ore sim ilar to a real veh i cle, i s set up to study creep groan. Exp erimental results from bot h test rigs are presen ted and compared with each other. I n Chapter 3, di fferent frict ion laws i.e. Coul om b’s friction law and t he bristle fri ction law are used t o describ e creep groan of the tes t rig with a n ide al i zed bra ke . The s t ick - slip lim it cycle is firs tly obta ined by coupling the syst em model with Coulom b’ s friction law, in wh ich the static friction coe fficient is larger than t he dy namic friction coefficient. H owever, t he non - s moot h c har a cter is ti c ap pe ars b et ween the stick and slip reg ions due to the switch function in Coulom b’s friction law. In contrast, the br istl e fr icti on la w , contain ing t he Strib eck effe ct , pr e - sliding ef fect and h yste r esis , c a n overcom e t his issue . Acc ord ing to the s tabil ity of th e e qu ilib riu m soluti on an d the sti ck - s lip lim it cy cle, th e sys tem with Coulom b’s friction law h as two param eters regions, i. e. a region wi th a stab le equilibrium solution bu t no stick - slip lim it cy cle n am ed as reg ion I ; and a reg ion wit h bo th stab le eq u ilibr ium solu tion an d stick - slip lim it cy cle named as region II . In contrast, the m odel with the brist le friction law has thre e parameter region s . Beside s reg ions I and I I, there is an additi onal region w i th an unstable equ ilibrium solutio n and a s table st ick - slip l im it cycle, nam ed as reg i on II I. S ince diff erent fri ction l aw s lead to differ ent dy namic chara cteris tics, on e should ch oose the suitabl e fric tion l aw carefu lly to descr ibe creep g roan of brak e system s. Chapter 4 focuses on the s tudy on the param eter identifica tion of the friction law . If Coulom b’s fri c- tion law is em ployed, the unknow n parameters relate d to the contact surfac es are static and dy namic friction coefficients. They can be identified by analyz ing the geo m etric shape o f the s tick - slip lim it cycle . I f the br istle fric tion law is em ployed, the unk nown parameters rela ted to the contact sur f aces contain th e sta tic fri ction coeff icien t, the dyna m ic fric tion coef ficien t, the Stri bec k v elocity , th e c o n- tact stif f ne ss and the co ntact dam ping. As a resu lt, a g eneti c a lgori thm is used to e s tima te thos e param eters by compar i ng the s imulation results w ith the exp erimen tal resu lts. Co m pared to the m odel with Coulom b’ s friction l aw , the model w ith the bris tle friction l aw can de scribe the pre - sli d ing eff ect between the stick and slip regions. B esi de s , by changing t he speed of the m ot or under constan t brake pressur e, three p ar am eter reg ions nam el y region s I, II and III can be detected experim entally . This dem onstrates tha t the mo del with bri stle frict ion law is more reasona ble to descr ibe cree p g r oan. At the end of Chapter 4, a Propo rti onal- I ntegral obs erver is designed to observ e the friction force from the m easured θ ∆ and θ ∆ . Compared to the modeling method, the friction ob server require s only a line ar disk- shaft m odel. Howev er, it has a d isadv antage t ha t the m easurem ent noise w ill st r on gl y i nf l ue nce the accu racy of t he o bserv ed force. The ob se rv ed f rictio n i s s im ilar to the ca lc ula ted frictio n by th e m odel with the b rist le fr icti on law. 90 The theor etica l and ex peri m ental studie s of creep g roan on the te st rig with a real br ake are p resented in Chap t er 5. Com pared to the idealized b r ak e, the te st rig with a real b rake con tains a real brake ca rr i- er and a n additio nal suspen sion sy stem. A m odel with lar ge num ber of degrees of fre edom is firstly set up to d escr ibe t he brake sy stem . By anal y zing the dyn am ic m odel, it is pos sible to isola t e th e brake suspen sion sy stem from the fram e, the ax le and t he chass is for the stu dy of th e stick - sl ip motion. T her efo re , com plex com ponent s suc h as the ch assis ca n be at fi rst ignored fo r the fr iction force calcu- lation. Subs equently , a red uced - order m ode l is pr oposed to im pr ove the calcula tion efficiency . The advan tages of th e r ed uce - or der model are tha t it requires only s ystem para m et ers inst ead of the phy s i- cal pa r am eter s an d it has high co m putationa l efficien cy. With t he ident i fied param eters to hand , t he sim ulated st ick - slip limit cycle i s com pared to the expe rimental on e. Both result s hav e good agreem ent with each other , wh ich con firms tha t t h e reduced - order m odel is of eff iciency to desc ribe creep g roan of the brak e system . Furth erm ore, a m ap of creep g roan is ob tained by stabi lity analysis o f t h e equil i- brium so lution and stick - s lip lim it cycle. T h is m ap is com pared wit h t he m ap measured t hr oug h experim ents. The m easured and ca lculated m aps show the sim ilar behav ior. I n Chapter 6, som e suppression m et hods of creep g roan are f inally im plemented on the theor etical m odel as well as on th e test rig s. At firs t, an activ e pad, containing two piez oelect ric lay ers with an electrode l ayer betwee n them , successfully e liminates creep g r oan by providing a high frequency m e- chanical v ibration in the out - of - p lane direc tion of t he d isk. The high frequen cy vibrat ion d ecreas es the static f r i ction coe fficient, a nd thus elim i nates creep g roan. Later, by adding a high frequency vibration in the in - plane direction, creep g r oan is also elim i nated. Anothe r feasible method against creep groan is to incr ease th e damping of t h e shaf t . I t i s co nfirm e d that by increasing th e dam ping of the shaf t, the area o f the reg ions I I and I II d ecreases, i .e. ther e is l es s risk t o gen erat e cre ep groan. A t last, a method to shor ten the t im e of cre ep g roan by us i ng an o ptimal b rake techn ique is p resented. The sy stem can leave the regions II I and II rapidly by preceding t he optimal brake t echniq ue. I f t his optimal b rake techn ique is integ rated in a n ABS , t he ABS can p erform the optim al brak ing proc es s th rough a sim pl e control loop to avoid creep groan just l ik e a sk ill ful dri ver . Sim ulat ion resul ts indi cate tha t creep g roan can be el iminated w i th th e assist ance of the ABS. I n general, this t hes is atte mpts to bu i ld a d etail m odel to desc ribe creep groan on the br ake system . Methods for supp ress ion o f creep g roan are s ug gested based o n the m odel . So me future work i s ou t- lin ed as fo llow s. • Theore t ica l and experim ental invest igations of cr eep g roan on a real vehicle c an be stud ied and com pared with the r esults from the test rigs. • The ef ficie ncy o f the a ctiv e pad n eeds to b e t ested under rea l drive cond i tions. I t is im portant to know the influence of the external hig h frequency exci t ation on a r eal vehic le. • Experim ent al investigation s on the suppression of c reep groan by a n anti - lock braking sy stem can be further stud ied. Since an A BS is a sa fety r elevan t sy stem , is it sens ible to u se it to solve th e com fort problem? This qu estion sho uld be fur ther disc ussed. 91 Bibliogra phy [1] Ki n kai d , N. M.; O ’R eilly , O. M. and Papadopoulos P. : Autom otive di sk brak e squeal. Journal of Sound and V ibration , 26 7(1 ), 105- 166, 2003. [2] AbuBak a r , A. 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