CONSTRUCTIVE CHARACTERISTICS AND MATHEMATICAL MODELLING OF MECHANIC-HIDRAULIC NETWORKS FOR COMPENSATING THE DYNAMICS OF ASSYMETRIC HYDRAULIC MOTORS

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1 Scientific Bulletin of the Politehnic Univerity of Timior Trnction on Mechnic Specil iue The 6 th Interntionl Conference on Hydrulic Mchinery nd Hydrodynmic Timior, Romni, October -, 004 CONSTRUCTIVE CHRCTERISTICS ND MTHEMTICL MODELLING OF MECHNIC-HIDRULIC NETWORKS FOR COMPENSTING THE DYNMICS OF SSYMETRIC HYDRULIC MOTORS Ion CRISTIN, Profeor* Economic Engineering nd Mnufcturing Sytem Deprtment Trnilvni Univerity of Brşov *Correponding uthor: Str. Mihi Vitezul 5, 50074, Brşov, Romni, Tel.: (+40) 4773, Fx: (+40) 4773, Emil: icritin@unitbv.ro BSTRCT The pper preent the contructive chrcteritic of the dynmic compenting mechnic-hydrulic network of ymmetric hydrulic motor. The bic ide i tht the ymmetric hydrulic motor require contruction tructure including ymmetricl element. The ymmetry inide the mechnic-hydrulic network reult from the ymmetry of the hydrulic motor, i.e. from the rtio between the ueful urfce of the ymmetric hydrulic motor. Further on, the pper give mthemticl model nd perform n nlyi of the hock borbing cylinder nd of network coniting of pneumtic-hydrulic ccumultor plced onto hydrulic motor. The chrcter of high-p filter of the dynmic compenting mechnic-hydrulic network i highlighted. KEYWORDS Differentil hydrulic cylinder, network for compenting the dynmic, mthemticl modelling, numericl imultion.. INTRODUCTION The trnient condition which eprte the ttionry running condition of hydrulic motor i ccompnied by preure ocilltion which influence the dynmic behviour of the motor. Of prticulr importnce i the mximum pek (over djutment) of the preure wve nd the preure ocilltion fding time (tbiliztion time). The dynmic of hydrulic motor cn be improved if mechnic-hydrulic compention network re plced on the motor. The contruction of thee dynmic compention network i influenced by the ymmetricl or ymmetricl chrcter of the hydrulic motor they re plced on. The mot widely ued dynmic compention mechnic-hydrulic network for the ymmetricl hydrulic motor re [, 4]: permnent lek (hydrulic reitnce t by-ping) between the hydrulic motor chmber, dmper cylinder between the chmber of the hydrulic motor, trnient lek between the chmber of the motor, trnient lek towrd the exterior, pneumtic-hydrulic ccumultor plced on the motor, hydrulic cpcitie plced on the motor. The condition for mintining the ttionry poition of the ymmetric hydrulic motor determine uneven preure between the two chmber of the motor. Conequently, the hydrulic reitnce by cutting off the chmber of the motor cn no longer be ued with the ymmetricl motor.. CONSTRUCTIVE PRTICULRITIES OF THE DYNMICS COMPENSTION MECHNIC-HYDRULIC NETWORKS OF SYMMETRICL HYDRULIC MOTORS The ymmetricl hydrulic motor need mechnichydrulic network with ymmetricl element in their tructure. The ymmetry within the mechnichydrulic network i determined by the ymmetry of the hydrulic motor nd by the rtion of the ueful re of the hydrulic motor, repectively. The contructive prticulritie will be pecified for the following type of ymmetricl motor dynmic compention mechnic-hydrulic network [3]: dmper cylinder, trnient lek between the chmber of the motor, trnient lek towrd the exterior, pneumtic-hydrulic ccumultor plced on the motor, hydrulic cpcitie plced on the motor. 349

2 In Figure the dmper cylinder, connected in erie with the reitor R h, i plced on the ymmetric hydrulic motor. Inide the dmper cylinder there i mll differentil piton centred on two compreion pring. The ymmetric chrcter of the hydrulic motor impoe the ue of differentil dmping piton. The rtio of it ueful re correpond to the M M rtion of the ueful re of the hydrulic motor. Thi i llowed by the hydrulic pool feturing ctive edge nd zero coverge Y 0 = 0. The opening of the pool occur only in trnient condition when the preure from the two end differ. Fig.. Dmper cylinder plced on the motor The dmping piton eprte the two preure inide the chmber of the hydrulic motor nd borb the preure wve building in the motor in the initil phe. The preure reitor, together with the hydrulic cpcity, repreented by the dmper cylinder, filter the preure ocilltion of certin frequency. The me hydrulic reitor hydrulic dmper cylinder embly cn be found in the mechnic-hydrulic network how in Figure nd 3 where the network re plced in imilr wy on the hydrulic motor. The functionl role remin the me. The network in Figure chieve the ttenution of the preure ocilltion inide the hydrulic motor due to the trnient lek between the chmber of the hydrulic motor. The preure wve i trnmitted from the chmber where the preure build up to the chmber where the preure drop. Fig.3. Trnient lek towrd the exterior The network with trnient lek towrd the exterior, preented in Figure 3, work in imilr wy. Thi network ttenute the preure ocilltion inide the hydrulic motor by evcuting to the exterior the preure pek occurring inide the motor chmber. In thi ce, the hydrulic pool connect the chmber where the preure build up wve occur to the tnk in trnient condition. Fig.. Trnient lek betweeen the chmber Fig.4. ccumultor plced on the motor The network in Figure 4 nd 5 conit of pneumtic - hydrulic ccumultor nd pure hydrulic cpcitie connected in erie witiltering hydrulic reitor R h. The ymmetricl chrcter of the network cn be noticed in thi ce, too. The pure hydrulic cpcitie re repreented by volume nd plced on the hydrulic motor chmber (Figure 5). Thi hydrulic motor dynmic compention network offer the dvntge of implicity to the expene of reduced dynmic performnce. 350

3 n importnt remrk here would be tht the replcement of the hydrulic reitor R h with n inductor permit prior djutment of the network filtering frequency. Fig.5. Hydrulic cpcitie plced on the motor 3. MTHEMTICL MODELLING OF THE SYMMETRIC HYDRULIC MOTOR DYNMICS COMPENSTION NETWORKS Further on the dmper cylinder the network coniting pneumtic - hydrulic ccumultor plced on the motor. 3.. Mthemticl modelling of the dmper cylinder Figure 6 preent the dmper cylinder in n opertionl phe, when n increing preure wve develop in chmber. The dmper cylinder, including m centred on two compreion pring, contitute mechnic-hydrulic cpcity which, together with the hydrulic reitnce of the reitor form mechnic-hydrulic filter high-p, equivlent to the filter in the electric circuit. The input vlue i the P Le chrge equivlent differentil preure of the ymmetricl hydrulic motor while the output vlue i the flow rte Q ic which enter the network. Fig.6. Dmper cylinder The mthemticl modelling of thi compention mechnic-hydrulic i chieved by ccepting the following clcultion umption: between the input-output preure vrition there i the following reltionhip: P B = K P () the reference poition of the dmper piton correpond to the poition for which the following reltionhip exit between the volume of the cylinder chmber: V = () 0 K V0 the rtio between the ueful re of the dmper cylinder correpond to the rtio between the ueful re of the ymmetricl hydrulic motor: = K (3) the hydrulic filtering reitor i conidered of the lminr type, the m of the piton i neglected, the lek flow rte reulting from the clernce between the piton nd the dmper cylinder i neglected, the influence of the network pipe i neglected. The functioning of the network i decribed by the continuity eqution of the flow rte written for the two chmber of the dmper cylinder, the expreion of the flow rte through the hydrulic filtering reitnce nd the dynmic equilibrium eqution of the dmper piton: V 0 Qic = v + P E V0 Q = + ec v PB E Q ic (4) = ( P P ) (5) R ( K P PB ) = m v + c v + Ke v dt (6) where Q ec i the output oil flow exiting from the compention network, P i the preure downtrem the filtering reitor of hydrulic reitnce R, c i the vicou friction coefficient of the dmper piton, v = x i the diplcement peed of the dmper piton, x i the diplcement of the dmper piton from the reference poition nd K e i the eltic contnt of the centring pring of the dmper piton, m - the m of the dmper piton. Following proceing of reltion (4), (5) nd (6) the dynmic network compention i decribed in the time field by the following eqution ytem: C m H R v + c ic + Qic Q v + K e = v in complex, repectively: + C v dt = P H PLe Le R Q ic (7) 35

4 [ K P ( ) K Q ( )] T v( ) = Le ic ζ + + ω ω (8) K3 T Qic ( ) = v( ) + K4 PLe ( ) T + T + K = ;K = R e ;K = ;K 3 4 = R Ke T = ;T = CH R ; ω = ; ζ = 0.5 K m where C Rc= i the hydrulic cpcity of the Ke dmper cylinder. ; K c e m The following remrk cn be highlighted: in the firt tge preure pek occur nd then the repone i chrcteritic to n element (D). ince the dmper cylinder i of mll dimenion compred to the hydrulic motor, it m nd the vicou dmping coefficient cn be, generlly, neglected. In thi ce the network behviour i decribed by the following trnfer function which correpond to high p filter. TF Qic( ) K F PL e( ) (9 TF + The expreion of the mplifiction coefficient nd the time contnt of the network: e H K e CH + K K = (0.) R ( K C + ) K R R Rh f T = CH Rh f + (0.b) K K indicte the wy their vlue re influenced on the ground of the modifiction of the contructive prmeter of the network. e e Fig.7. The block digrm of the network The block digrm (Figure 7), repreented on the ground of eqution ytem (8) highlight the internl connexion nd the intermedite vrible v, the externl connexion with the ymmetricl hydrulic motor, repectively, (by men of the input differentil preure P Le nd the flow entering the compention network Q output vlue). ic The repone to tep ignl of mechnic-hydrulic network hving d f = 0.3 [ mm ] nd being chrcterized by prmeter K F = 3e- [ m 5 N ], T F = 8e- [ ], ( f = [Hz] ), obtined by numericl imultion t preure tep P Le = 0 [br] i given in the Figure below [3]. 3.. Mthemticl modelling of the pneumtic-hydrulic ccumultor plced on the hydrulic motor The hydrulic reitor in the tructure of thi network hve the role to borb the preure ocilltion which could propgte from the pneumtic-hydrulic ccumultor towrd the hydrulic motor, the compreible g inide the ccumultor cting like pneumtic pring. To rpidly borb the preure ocilltion in the hydrulic motor, direction vlve (not hown in the drwing) re plced in prllel with the hydrulic reitor to llow the ccumultor chrging flow to by p the hydrulic reitor. Qic [lpm] RcM_ ( Ple = 0 br ) x Qic PLe t [] Fig. 8. The repone to tep ignl Fig. 9. ccumultor plced on the hydrulic motor The input vlue in the compention network i repreented by the differentil preure equivlent to the lod P Le of the hydrulic motor while the output vlue i flow Q i c which exit the circulr chmber

5 of the hydrulic motor nd enter the ccumultor plced there onto. The mthemticl modelling of compention network coniting of two pneumtic-hydrulic ccumultor i bed on the following clcultion umption: the m of the rubber element (bellow hped, for intnce) eprting the oil from the g in the ccumultor i neglected, thi being conidered piton with infiniteiml m nd hving the urfce equl to,, repectively, their rtio being identicl with the rtio of the ueful re of the motor: = K () with the filtering hydrulic reitnce, repectively: R = K R () h the trnformtion deriving from the chrging dichrging of the ccumultor re conidered polytropic, being chrcteried by polytropic coefficient n, between the preure vrition in the two brnche of the compention network there re the following interdependence reltion: PB = K P (3) P = K P the following oil nd g rtion correpond to the reference poition of the two ccumultor: Vu = K Vu (4) Vg = K Vg the dynmic of ech ccumultor i conidered to be linerly decribed by expreion chrcteried by the elticity equivlent module of the g inide the ccumultor E g = np0. The expreion of the flow rte through the filtering hydrulic reitor nd the flow continuity eqution, written for the oil nd g volume inide the ccumultor, re follow: h R Qe = P P K R QiB = P PB Q e = K z + CHu P QiB = z + CHu P 0 = K z + CH g P 0 = z + CH g P (5) where R = Rh, Q ec i the outlet flow from the ccumultor plced on the circulr chmber of the motor nd enter thi chmber, z nd z - the diplcement of the oil nd g eprtion urfce in the two ccumultor in reltion with their repective reference poition, C Hu nd C Hu - the hydrulic cpcitie of the oil inide ech ccumultor, nd C H g, C H g re the pneumtic cpcitie of the g inide ech ccumultor. Upon completion of the proceing of the eqution ytem bove, the dynmic of the compention network contituted of two ccumultor i decribed in the time field, by the following eqution ytem: nd in complex: ( K + ) R Qe = PLe P Qe = z + CHue Pe = CH ge Pe e [ P ( ) P ( )] Qe( ) = K Le e Pe ( ) = Qe( ) K z T z( ) = K3 Pe ( ) ( ) (6) (7) where Pe = K P P, z = z = z, nd C Hue şi C H ge re the oil nd g equivlent cpcitie inide the two pneumtic-hydrulic ccumultor. Fig.0. The Block digrm of the network The Block digrm in Figure 0, drwn ccording to eqution (7) llow u to deduct the finl trnfer function: Qe ( ) TF H0( ) = = K F (8) P ( ) T + K F = ( K Le + )R ( K + )Rh f ( CH ue + CH g e ) TF = which repreent high-p filter with time contnt T F. ; F 353

6 The repone to tep ignl (Figure ) of mechnichydrulic network [3], with: V = 0.5 [l], d = [mm], f nd chrcterized by prmeter K F = 3.6e-0 [ m 5 N ], T F =.7e- [], f = 60 [H], obtined by numericl imultion gint preure tep P Le = 0 [br], highlight the trnient vrition of flow Q ic which enter the ccumultor plced on the circulr chmber of the ymmetric hydrulic motor nd of the diplcement z of the z rubber bellow. Qe [lpm] Qe PLe RcM_5 ( PLe = 0 br ) t [] Fig.. The repone to tep ignl z The trnient condition dipper fter 0.08 econd nd the repone i determined by vlue of K F nd T F, hence by the functionl contructive prmeter of the network. 4. CONCLUSIONS The numericl imultion confirmed the fct tht the mechnic-hydrulic network hown herein ct high p filter. Their ymmetry correpond to the ymmetry of the hydrulic motor they re plced onto nd their time contnt depend on the functionl contructive prmeter. REFERENCES. Ctnă Il. (979) Contribution to the nlyi of tbility nd to the ynthei of the electro-hydrulic ytem for quick procee (in romnin). Ph. D. thei, Politehnic Univerity of Buchret. Ctnă Il. (995) Electro-hydrulic utomted ytem nlyi nd ynthei. (in romnin). Politehnic Univerity of Buchret 3. Critin I. (998) Reerch on the ynthei nd nlyi by numericl imultion of ervoytem tbiliing the longitudinl fcing feed (in romnin). Ph. D. thei, Trnilvni Univerity of Brşov 4. Guillon M. (97) L erviment hydrulique et electrohydrulique. Dunod, Pri 5. Herbert E.M. (967) Hydrulik Control Sytem. John Wiley, New York 354