NONLINEAR MODEL OF THE VEHICLE HYDROPNEUMATIC SUSPENSION UNIT
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1 Nuber, olue II, Deceber NONLINEAR MODEL OF THE EHICLE HYDROPNEUMATIC UPENION UNIT Libor Kuk ury: The lue reers oel of he hyroneuic susension uni n is verificion is escribe in he er. The relions effecive surfce n volue of he rubber bellows ir sring versus osiion n ressure n Bernoulli equion re use in he heicl oel of hyrulic r. The oil volue coression chnges re consiere negligible. The ibic se equion for he gs is use. The gs flow hrough he hroling injecor is consiere boh in uner n overcriicl coniions. The sic n ynic chrcerisics of he whole uni n is rs were esure. Key wors: hyroneuic susension uni, rubber bellows ir sring, heicl oel. INTRODUCTION The hyroneuic susension uni (fig. consiss of hyrulic n neuic rs. I is ossible o sere he hyrulic r for he esureen (,. Fig. Hyroneuic susension uni Ing. Libor Kuk, Ph.D., Universiy of Prubice, Fculy of Elecricl Engineering n Inforics, Deren of Process Conrol, náěsí Čs. legií 6, Prubice, Tel.: , Fx: , E-il: libor.kuk@uce.c Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 79
2 Nuber, olue II, Deceber The uni (see fig. consiss of: neuic rubber bellows ir sring (, seel chber (, neuic cuor n chnnels (, n, lower n uer rubber bellows sring (6 n 7, silicon oil fille, fixe chnnel (8, hyrulic cuor n chnnels (9,, seel ro ( n lerl guince syse of uer rubber bellows sring (.. NONLINEAR MATHEMATICAL MODEL. Hyrulic r The hyrulic r hs he chrcer of clssic er, bu i hs sll sring chrcer s well. Hyrulic r consiss of wo rubber bellows srings, which re connece wih ech oher hrough hroling bore n rllel hyrulic chnnels wih cuor. Fig. chee of he hyrulic r of he uni (, lower n uer rubber bellows srings, fixe oin of esureen equien, hroling bore, servo vlve The relions beween osiion n ressure of one rubber bellows sring n he effecive surfce were esure using lborory urose-buil esing equien. The relion volue versus osiion n ressure ws roxie wih funcion (. The volue roxiion funcion ws chosen in he following for (, e, ( where is volue, is osiion, is ressure, u o re coefficiens. The coefficiens n were eerine exerienlly fro he esureen on he sseble hyrulic r. The exlici exression = (, fro ( is e. ( The ressure insie e.g. lower rubber bellows sring s funcion of ie is escribe by Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 8
3 Nuber, olue II, Deceber. ( = / is oil volue flow ino he rubber bellows sring. = f (, [c ].8.6. [MP]. - - [c] Fig. D visuliion of he funcion = (, 8 rojecion of = f (, 6 [c ] [c] Fig. Projecion of cross secion curves of he funcion = (,, curves reer is ressure =. u o. MP Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 8
4 Nuber, olue II, Deceber Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 8 Le us coue he erivives / n / fro ( n subsiue he ino ( e e ( ( ( ( ( Le us esigne he ressures insie he lower n he uer bellows n ifference beween he, n = resecively. The following roxiion of he soluion of Bernoulli's equion ws use o coue h h e sign( R, ( R h is hyrulic resisnce n h is coefficien which ffecs he erivive /( for =. The force hef hef h F F F, (6 hef n hef re effecive surfces of he uer n lower bellows srings. The roxiion of he effecive surfce of bellows sring ws chosen in he following for e hef, (. (7 Fig. D visuliion of he roxiion hef = hef (, n esureens of n effecive surfce. Pneuic r The neuic r of he susension uni consiss of rubber bellows srings, seel chber n cuor hroling eleen (see fig. 6. The equion of ibic chnges in ir is kons, (8 where is ressure, is ensiy n =.. The ifferenil for of (8 is
5 Nuber, olue II, Deceber. (9 Fro (9 ( n. ( Mss conservion lw is kons ( n he ifferenil for of his lw is, ( is ss flow. Fig. 6 chee of he neuic r of he uni ( bellows ir sring, seel chber, fixe oin of esureen equien, hroling injecor, loss coefficien PT, servo vlve, loss coefficien P Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 8
6 Nuber, olue II, Deceber Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 8 The oificions of equion ( for boh rs of neuic uni re ( ( n (. ( Fro ( n (, (6 resecively. (7 Bernoulli equion is use for he ibic gs flow hrough he hroling injecor kons v v i i i i i i, (8 i, v i, i n i, v i, i re vribles, which escribe he flow in fron of n behin he hroling injecor. ince v i << v i, he relion beween flow velociies is i i i i i v. (9 For i, i n i, i fro (8 i i i i. ( The ss flow hrough he hroling injecor is i i v, ( is he secionl re of he hroling injecor. Afer subsiuion of (9 n ( ino ( n using he loss coefficien is i i i i i i. ( This equion is vli for uner-criicl (linr flow, in he cse of ir i ens for i / i.8. Le us esigne kri / i =.8. For over-criicl (urbulen flow, for i / i <.8 is i kri i kri i i. (
7 Nuber, olue II, Deceber The resuling force is F F F F G. ( The force F of he rubber bellows ir sring (fig. 7 is F. ( ef The eenences of effecive surfce ef n volue on osiion (see fig. 8 were roxie wih fifh egrees olynoils. Fig. 7 chee of he rubber bellows ir sring Ruben D -7 ef = f(, = f( ef ( ( 6 ef [c ] [c ] [c] Fig. 8 Effecive surfce n volue of he use bellows sring Ruben D -7 ( ef ef n Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 8
8 Nuber, olue II, Deceber. IMULATION The siulion oel ws cree in MATLAB iulink. Plos of n xil force F( cquire by nuericl siulion re in figs. 9 u o, he frequencies of he hronic signl were.6,. n. H, he liues were, n. Fig. 9 Corison of he oel n exerienl for frequency.6 H n liue (xil force F = F( Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 86
9 Nuber, olue II, Deceber 6 exerien oel F( F [kn] [c] Fig. Corison of he oel n exerienl for frequency.6 H n liue (xil force F = F( Fig. Corison of he oel n exerienl for frequency. H n liue (xil force F = F( Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 87
10 Nuber, olue II, Deceber 7 6 exerien oel F( F [kn] [c] Fig. Corison of he oel n exerienl for frequency. H n liue (xil force F = F( Fig. Corison of he oel n exerienl for frequency. H n liue (xil force F = F( Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 88
11 Nuber, olue II, Deceber 6 exerien oel F( F [kn] [c] Fig. Corison of he oel n exerienl for frequency. H n liue (xil force F = F( CONLUION The eveloen of hyroneuic susension uni (wih rubber bellows srings is escribe in he er. The siulion oel ws cree in MATLAB iulink. This oel ches he ynics of he hyroneuic uni uner consierion relively well. Prosecive licion of his uni is ruck bck xle susension. The sring n ing chrcerisics cn be chnge in he cse of sei-cive, or cive conrol. The conrol hs no been esigne ye. ACKNOWLEDGMENT The work hs been suore by he funs No. GFEI/ of he Universiy of Prubice, Cech Reublic. This suor is very grefully cknowlege. REFERENCE ( ŠKLÍBA, J.; BARBORA, J.; CIRKL, D. Hyroneuický člen s rlelní ření hyrulického neuického luení. In Proceeings of Inercion n Feebcks. Prgue: Insiue of Theroechnics A CR,.. 6. IBN ( BARBORA, J.; JANEČEK, B.; KUPKA, L.; ZŮBEK, T. Hyroneuic usension Rubber-bellows Uni. In Proceeings of Colloquiu Euroech on ei-acive ibrion uression [CD ROM]. Prgue: CTU,.. ( BAUER, W. Hyroneuic usensions yses. s e. Berlin Heielberg New York: ringer,. IBN Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 89
12 Nuber, olue II, Deceber ( HANUŠ, B.; OLEHLA, M.; MODRLÁK, O. Číslicová regulce echnologických rocesů: lgoriy, eicko-fyikální nlý, ienifikce, ce. Brno: BUT,. IBN 8--6-X. ( ETFÁLOÁ, M.; TŘEDA, I. Technická ynik lynů. Liberec: TU,. (6 KUPKA L.: yhonocení růběhů růokových chrkerisik elekroneuického servovenilu EF. [Reserch reor.] Liberec: TU,. Kuk: Nonliner Moel of he vehicle hyroneuic usension Uni 9
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