Determination of Modal Parameters of a Half-Car Fitted with a Hydraulically Interconnected Suspension from Simulated Free decay Responses

Transcription

1 5 th Australasan Congress on Appled Mechancs, ACAM December 007, Brsbane, Australa Determnaton of Modal Parameters of a Half-Car Ftted wth a Hydraulcally Interconnected Suspenson from Smulated Free decay Responses Nong Zhang, Wade Smth, Jeku Jeyakumaran and Wllam Hu Faculty of Engneerng, Unversty of Technology, Sydney Abstract: Ths paper presents an alternatve approach for determnng the vbraton modal parameters, n terms of natural frequences, dampng ratos and modal shapes of a roll-plane half-car ftted wth a general hydraulcally nterconnected suspenson system. The dynamc model of the system, whch conssts of the sprung mass and the HIS and the wheels, s formulated usng the state space representaton approach. The state varables descrbng rgd body motons of the sprung and unsprung masses are heavly coupled wth those descrbng the dynamcs of HIS flud crcuts. A numercal smulaton scheme s developed to obtan the transent and free decay responses of the half car vehcle usng specfc ntal condtons or road nputs. The obtaned results are compared wth those determned from the free vbraton analyss of the system usng the transfer matrx method. Dscussons on the advantages and lmtatons of the presented method are also provded. Keywords: Hydraulcally nterconnected suspensons, mult-body systems, dynamcs of flud crcuts Introducton Fatal crashes due to vehcle rollovers have been frequently reported n recent years around the world. Four Wheel Drve vehcles (4WDs), typcally havng hgher mass centres, are partcularly vulnerable to ths type of accdent, wth over one thrd of 4WD fataltes nvolvng rollover []. The events leadng to vehcle rollovers are complex wth many factors nfluencng the vehcle moton. However, good suspensons can greatly reduce vehcular rollover propensty durng extreme manoeuvres. Recently, research efforts have been focused on advanced suspenson systems, wth a vew to overcomng the nevtable compromse between rde and handlng performance encountered n conventonal suspensons. These advanced suspensons generally nclude varable or adjustable stffness or dampng parameters, most often acheved through actve or sem-actve means [, 3]. However, completely passve hydraulcally nterconnected suspenson (HIS) s also becomng ncreasngly popular for passenger vehcles. In addton to havng the normal functonalty of a conventonal suspenson, a HIS has advantages n provdng addtonal stffness n roll, bounce, ptchng, artculaton, or ther combnatons, dependng on the system confguraton. A typcal HIS often contans several flud crcuts that lnk double-actng cylnders, damper valves and accumulators through ppe, hose or curved fttng elements. Knetc H suspenson s one example [4]. A HIS behaves nonlnearly n relaton to external dsturbances. Therefore, unlke a conventonal suspenson, a lumped mass assumpton does not apply to a vehcle wth a HIS because the hghly pressursed flud wthn the crcuts s a dstrbuted mass system. The vbraton modes of bounce, roll, ptchng, artculaton and wheel hoppng of the vehcle system can not be easly determned usng the conventonal multple lumped mass, s prng and dampng approach. Although rch practcal knowledge on HIS systems has been ganed from expermental studes and applcatons n both racng and passenger vehcles [4-6], only recently a systematc theory and soluton procedure for determnng the dynamc characterstcs of HIS vehcles has been presented by Smth, et al [7] and Zhang, et al [8]. Ths paper presents an alternatve approach for determnng the vbraton modal parameters, n terms of natural frequences, dampng ratos and modal shapes of a roll-plane half-car ftted wth a general hydraulcally nterconnected suspenson system. The sprung mass, suspenson sprngs, wheels and tres are modelled usng the free body dagram method. The ndvdual flud elements, such as lnes, valves and accumulators of the HIS flud crcuts are modelled usng ther own dynamc models. A state vector for the vehcle dynamc system s so defned that t contans all the ndependent dsplacements and veloctes of the sprung mass and wheels, and the pressure and flow at all nodes (between whch are varous hydraulc elements) of the flud crcuts. The dynamc model of the system, whch couples the sprung mass and the HIS and the wheels, s then formulated usng the state space representaton approach (see the detals n reference [8]). The state varables descrbng rgd body moton are heavly co upled wth those descrbng the dynamcs of the HIS flud crcuts. The dynamc

2 model conssts of tme-varant coeffcent matrces, whch accommodate the nonlneartes n the fluds crcuts, and can be used to obtan the transent responses nduced by external dsturbances. A numercal smulaton scheme s developed to obtan the transent and free decay responses of the half car vehcle usng specfc ntal condtons or road nputs. The state varable method s appled to dentfy the modal parameters of the half car system from the smulated tme doman responses. The obtaned results are compared wth those determned from the free vbraton analyss of the system usng the transfer matrx method. Dscussons on the advantages and lmtatons of the presented approach and the free vbraton analyss method are provded. A Half-Car wth a Hydraulcally Interconnected Suspenson (HIS) System In an effort to retan smplcty, whlst stll accountng for flud nterconnectons between wheel statons, a lumped-mass four-degree-of-freedom half-car model, as shown n Fg., s used n ths nvestgaton. The system conssts of lnear tre dampng and sprngng, lnear conventonal suspenson sprngng, and a typcal HIS system. y wl k sl ml chl k hl y v M, I θ k hr HIS System Model chr mr θ k sr y wr Compresson damper valve P Ul P Ll X X X 3 X 6 X 8 V V 3 X 7 Roll damper X 4 X 4 V valves V X 5 X 5 J Hydraulc lne Q X Ntrogen-flled accumulators Q X X 6 Hydraulc lne X 3 X J V Double-actng cylnder V3 X X 7 X 8 P Ur P Lr c tl k tl ktr ctr Rebound damper valve Cylnder pston/rod Fg. Schematc of a half-car wth an HIS Fg. Schematc of a typcal HIS for half-car applcatons From Fg., t can be seen that the shock absorbers of conventonal ndependent suspensons between the sprung and unsprung masses are replaced by a HIS system. A typcal roll-resstant HIS arrangement for ths applcaton conssts of two dentcal hydraulc crcuts as shown n Fg.. The two hydraulc crcuts are coupled wth each other knetcally va two dentcal double actng pstoncylnder absorbers. Each of the hydraulc crcuts comprses three damper valves, a ntrogen-flled daphragm accumulator, and a hydraulc ppelne. HIS crcuts mght typcally nclude addtonal elements, such as hydraulc fttngs and flexble hoses, but they fall beyond the consderaton of ths nvestgaton. Wthn the two hydraulc crcuts, the cylnders are mounted on the car body and the pston rods are fxed on the wheel statons. The dynamc nteracton between the hydraulc system and the sprung and unsprung masses can be descrbed as such: relatve veloctes n the suspenson struts cause flud flows n both crcuts and accompanyng pressure changes n the cylnder chambers, whch leads to new suspenson strut forces beng appled to the sprung and unsprung masses. As a result, vehcle body and wheel motons occur, whch, n turn, affect the hydraulc system. Ths nteracton wll contnue untl the system reaches a new equlbrum. HIS systems can provde greater freedom to ndependently specfy modal stffness and dampng characterstcs. Ideally, HIS system functon s charactersed entrely by mode, though factors such as flud compressblty and frequencydependent hydraulc crcut mpedance cause mperfect system functon. The workng mechansms and features of the half-car HIS system can be found from [8]. 3 Modellng of the Half-Car wth a HIS System The detals of the dervaton of the equatons of moton of the half-car model for free vbraton analyss usng transfer matrx method are presented n [8]. Here a bref descrpton of the dynamc model,.e., the state space representaton of the ntegrated half-car system for the analyss of the transent responses caused by road nputs s provded. For the ntegrated half-car system, the system state vector s defned as

3 X = [ y, y, y, θ, y&, y&, y&, θ&, P, P, P, P,...,] T () wl wr v wl wr v X 8 X X 8' X' whch ncludes the system dsplacement vector Y = [ y, y, y, θ ] T, velocty vector Y &, and wl wr v pressure vector P, whch ncludes the end nodes of the flud crcuts and several nternal nodes dependng on the desred soluton accuracy. Applyng Newtons second law, the equaton of moton of the system s: MY&& + CY& + KY = F = F + F () ext h Concentratng only on transent responses due to external forces, F ext, s caused by the road nput from the two tres and F h s the forces appled to the sprung and unsprung masses from the suspenson flud crcuts resulted from the relatve moton between sprung and unsprung masses. The equaton of moton for the ntegrated system can then be determned n the form: X& = JX + F% (3) ext n whch the system coeffcent matrx s 0 I 0 = J -M K -M C M D - 0 -V A G where M, K, C are the mass, stffness and dampng coeffcent matrces respectvely, see Eq.(); D, V, A are the coeffcent matrces couplng the motons and dynamcs of the rgd body system and the suspenson flud crcuts; G s the coeffcent matrx determnng the dynamcs of the flud crcuts. We should note that coeffcent matrces V and G are tme varant as ther values depend on the system T state and the external force vector F% ext = 0, M Fext, 0 depends on the road nput only. 4 Modellng of the HIS flud crcuts To obtan the soluton of equaton (3), the dynamc state of the rgd body system and that of the flud crcuts can be determned smultaneously usng the frst 8 equatons for the rgd body system and the last n equatons for the flud crcuts respectvely. Ths secton provdes a bref descrpton of the modellng and the numercal soluton scheme of the flud crcut dynamcs. A HIS flud crcut, as shown n Fgure, conssts of key elements such as the cylnder-pston unt, damper valves, straght and curved ppes or fttngs and an accumulator. These elements need to be properly modelled so ther dynamc effects on the ntegrated system are taken nto account. Due to space lmtatons, only the dfferental equatons descrbng the key hydraulc elements are provded. Damper valves - The damper valves play an mportant role n the HIS system performance. The relatonshp between flow rate and pressure drop of the dampers s usually nonlnear. For obtanng the system modal parameters, t s common to assume that the vbraton of the system nvolves small oscllatons about the equlbrum poston. For smplcty, therefore, the flow rates of the damper valves are modelled as lnear wth respect to the pressure drop between ther nlets and outlets, as stated mathematcally below. Q = C ( p p ) (5) dv dv n out where Q dv and C dv stand for flow rate and pressure loss coeffcent of the damper valve respectvely. Ppelnes - The vehcle layout and packagng constrants of the suspenson system requres relatvely long flexble ppelnes. A lumped parameter model s developed by dvdng the flud ppelnes nto several elements. Each flud ppelne of constant dameter s handled as one element. The mean pressure and mean flow n each element s assumed as an arthmetc mean of the pressure and flow rate at both ends of the ppe. The flud flow n the ppe s assumed as one-dmensonal compressble (4)

4 flow to accommodate the water hammer phenomenon. Assumng the pressure losses due to vscosty are proportonal to the mean flow rate, and the magntude of losses s the same as the nerta and pressure forces, the momentum equaton can be wrtten as ρl Q & = ( p p ) k l Q (6) A where the vscous loss k =8/A, s the flud densty, l the ppe length, and A the ppe secton area. The contnuty equaton for the ppelne s wrtten n terms of the mean pressure and flow dfference between the ends of the ppe element as β ( p) p& = ( Q Q ) (7) V where (p), the bulk modulus, s expressed as a functon of mean pressure p. The mean pressure and mean flow rate of the ppe element are gven as the arthmetc mean at both ends of the element,.e., p = ( p + p ) (8) Q = ( Q + Q ) (9) It s noted that Eqs. (6-9) apply to each ppe element. Substtutng Eq. (8) and (9) nto Eq. (6) results n one coupled frst order dfferental equaton governng the pressures at defned nodal ponts of the suspenson flud crcuts. Flow rates and ther frst order dervatves at nodes can be determned n terms of the nodal pressures, pressure change rates and/or boundary condtons. Accumulator - As shown n Fgure, the HIS system features gas-pressurzed hydraulc accumulators to reduce shock pressure loadng due to system nputs. The accumulator conssts of a pressure housng dvded nto two chambers by an elastomerc daphragm. One chamber s flled wth gas and the other wth hydraulc flud. The compressblty of the ol n the accumulator s neglected, as the ol stffness s much greater than that of the ntrogen contaned n the bladder. A drop n the system pressure s accompaned by flow from the accumulator and therefore the accumulator needs to be suffcently large to meet the peak flow demands wthout apprecable drop n the system pressure. The accumulator s modelled by assumng an adabatc process. γ pv = p V = const (0) γ 0 0. The adabatc gas law s used to model the accumulator pressure as a functon of gas volume at a precharged pressure. Takng the partal tme dervatve of Eq. (0) notng that the flow nto the accumulator, Q, s gven by Q=V/t, the pressure gradent of the accumulator can be wrtten as a nonlnear functon of pressure p,.e., γ Qp p p = ( ) V p / γ & () 0 0 Fnte element method based numercal soluton scheme In order to obtan the transent state of the suspenson flud crcuts, a fnte element method based numercal scheme s developed. The long ppe lnes are meshed nto several smaller elements and at each node of each straght nternal ppe, the out flow rate value of the upstream element s equal to that of the n flow rate of the neghbourng down steam ppe element. Note that at a juncton node, net n flow s equal to net out flow, and the pressures (for dfferent flud elements) are the same. Because of these physcal constrants, the flow rates at nternal nodes can be elmnated, or represented by nodal pressures and those at boundary nodes can be determned from the relatve moton of the rgd bodes. Through mathematcal manpulaton, a set of n frst order dfferental equatons, whch descrbes the dynamcs of the suspenson flud crcuts, are obtaned. These equatons are then combned nto the state space equaton for the ntegrated system,.e., Eq. (3).

5 As ths nvestgaton prmarly focuses on the determnaton of the modal parameters of the mult-body system, we try to obtan the free vbratons of the system domnated by sngle vbraton modes by usng specfc road bump nputs. At tme 0, the system s at ts statc equlbrum state and the ntal state can be determned from the statc force balancng equatons. At the next nstant, the dsturbances from specfed road nputs are appled to the system, so the dynamc state of the ntegrated system s determned from the state space system model combnng the dynamcs of the rgd body system and the suspenson flud crcuts usng the Runge-Kutta numercal ntegraton method. 5 Results and Dscusson In ths secton, the developed half-car model of the HIS system s smulated for specfed road nputs n order to determne the frequences, dampng ratos and modal shapes of vbraton modes of bounce, roll and wheel hop. The mechancal and hydraulc system parameters used for the smulaton are presented n Table of the Appendx [8]. The smulaton s performed wth the accumulator pre-charge pressure of 0.5 MPa and system pressure of.0 MPa. Bounce mode To excte the response of the half-car domnated by the bounce mode, a step bump nput, wth ampltude of 0.05m, was appled to the two tres smultaneously. The obtaned transent responses for y wl, y wr, y v, and are plotted n Fg. 3. The results show that the sprung mass has a decayed response, and ts roll s almost zero. The wheels have dentcal vertcal dsplacement. The response s comprsed manly of the bounce mode and n-phase wheel hop mode. Roll m ode To excte the response of the half-car domnated by the roll mode, two specfc step bump nputs, wth ampltude of ±0.05m, were appled to the two tres smultaneously. The obtaned transent responses for y wl, y wr, y v, and are plotted n Fg. 4. The results show that the sprung mass has a clean decayed response n roll and ts bounce s almost zero. The wheels vbrate wth the same ampltudes but out of phase, and they decay very rapdly Z - Tme [s]: HCARM_STP 8.03E E- Z - Tme [s]: HCARM_ROL E- 5.77E-.050E- 6.59E- 3, 4 : y v : 3: y wl 4: y wr Fg. 3 Bounce and wheel hoop responses E E E E Fg. 4 Roll and wheel hoop responses Table Four system natural modes domnated by the half-car mult-body moton Mode & frequency (Hz) State varable Bounce (.05) ± Bounce (.08) ±6.786 Roll (.60) ±4.987 Roll (.43) ±5.8 Dampng rato Left wheel (y wl) 0.098, 47. o 0.099, 39.9 o 0.79, 48. o 0.9, 4.3 o Rght wheel (y wr) 0.098, 47. o 0.099, 39.9 o 0.79, -48. o 0.9, -4.3 o Centre of gravty (y v) Roll angle () Mode & frequency (Hz) State varable Hop (0.8) ±60.54 Hop (0.8) ±60.54 Hop (0.8) -6.6 ±5.5 Hop (.6) ±.94 Dampng rato Left wheel (y wl) Rght wheel (y wr) Centre of gravty (y v) 0.085, 77. o 0.074, Roll angle () , 4.8 o 0.303, 7.5 o The results of free vbraton analyss [8]; the results of the presented transent analyss.

6 Modal parameters of the mult-body system From the smulated transent responses, the modal parameters, n terms of natural frequences, dampng ratos and modal coeffcents, of all four vbraton modes of the mult-body system are determned usng the State Varable Method [9]. All of these results and those obtaned n the free vbraton analyss method [8] are lsted n Table for comparson. From the frst upper-left three columns n Table, for the bounce mode, t s seen that the damped natural frequency obtaned usng the presented method s about 3% hgher than that obtaned from the free vbraton analyss. The newly obtaned dampng rato s about 4% less and the modal coeffcents for the wheels have almost the same magntude but 5% dfference n ther phase referred to the sprung mass. From the frst two upper-rght columns, for the roll mode, the relatve errors between the modal parameters obtaned by the methods are all less than 0%. From the lower part of Table, for wheel hop mode, t s seen that the damped natural frequences and dampng ratos are the same for both methods. The modal coeffcents for the sprung mass have 3% dfference n magntude and 9% dfference n ther phase referred to the wheels. For wheel hoop mode, the relatve errors between the obtaned results are even larger than those for the other modes because ths mode s heavly damped. Dscusson From the results lsted n Table, although large relatve errors occurred for the state varables where the absolute coeffcents are very small (less than 0% of those of the reference state varables), most of the modal coeffcents obtaned wth the two methods compare favourably. Therefore t s reasonable to state that, n general, the modal parameters obtaned from the presented method agree well wth those obtaned from the free vbraton analyss method. Therefore the transent analyss of the half-car system can provde the free decay type of system responses, from whch the modal parameters of the system can be determned wth adequate accuracy. In ths nvestgaton, for smplcty, the half-car system s almost symmetrc from the left wheel to the rght wheel. So the bounce mode and wheel hoop mode can be easly excted by applyng the same step nputs to the tres (see Fgure 3) and the roll mode and wheel hoop mode can be easly excted by applyng two opposte step nputs to the tres (see Fgure 4). But t should be noted that the flud crcuts are not exactly left-rght symmetrc due to the locaton of the accumulators. So the smulated roll response s not zero n Fgure 3 and the sprung mass vertcal response s not zero n Fgure 4. In real applcatons, the left and rght sdes of the half car are not symmetrc and therefore all of the four modes wll contrbute to the response when step nputs are appled to the tres. Ths wll be left for further nvestgaton due to the lmted space. The most obvous lmtaton of the free vbraton method s ts lnearty assumpton, the detals of whch are dscussed n [8]. For the presented method, for smplcty, all of the hydraulc elements except the accumulators are also assumed as lnear. The frcton effects, such as those n the pston rod seals and those between the pston and cylnder are gnored n the smulaton. However, the presented method can be easly modfed to nclude these frctonal effects f so requred. Furthermore, nonlneartes such as the chamber volume couplng nonlnearty, and nonlnear damper valve characterstcs, can also be consdered n the transent analyss. 6 Conclusons Ths paper has presented an alternatve approach for determnng the modal parameters of a vehcle ftted wth a general hydraulcally nterconnected suspenson system. A smplfed lnear roll-plane halfcar model s used as an example to llustrate the applcaton of the proposed methodology. The ndvdual flud elements, such as lnes, valves and accumulators of the HIS flud crcuts are modelled usng a frst prncples approach. The dynamc model of the system, whch couples the sprung mass and the HIS and the wheels, s then formulated usng the state space representaton approach. The state varables descrbng rgd body moton are heav ly coupled wth those descrbng the dynamcs of the HIS flud crcuts. The dynamc model conssts of tme-varant coeffcent matrces, whch accommodate the nonlneartes n the fluds crcuts, and can be used to obtan the transent responses nduced by external dsturbances. A numercal smulaton scheme s developed to obtan the transent responses of the half car vehcle usng specfc road nputs. The smulated free decay responses are then used to determne the modal parameters of the half-car system. The obtaned modal parameters are compared wth those

7 determned from the free vbraton analyss of the system usng the transfer matrx method. It has been found that, n general, the results obtaned from the presented method agree well wth those obtaned from the free vbraton analyss method. Thus the presented transent analyss method can obtan the free decay type of system responses, from whch the modal parameters of the system can be determned wth adequate accuracy. References [] Natonal Hghway Traffc Safety Admnstraton (004), Traffc Safety Facts. [] Sms, N.D. and Stanway, R. (003), Sem-actve vehcle suspenson usng smart flud dampers: a modellng and control study. Internatonal Journal of Vehcle Desgn, 33 (-3) pp [3] Mlanese, M., Novara, C., Gabrell, P. and Tennerello, L. (004), Expermental Modellng of Vertcal Dynamcs of Vehcles Wth Controlled Suspensons. SAE Paper Seres, [4] Wlde, J.R., Heydnger, G.J., Guenther, D.A., Malln, T. and Devensh, A.M. (005), Expermental Evaluaton of Fshhook Maneuver Performance of a Knetc Suspenson System. SAE Techncal Paper Seres, [5] Bhave, S.Y. (99), Effect of connectng the front and rear ar suspensons of a vehcle on the transmssblty of road undulaton nputs. Vehcle System Dynamcs, (4) pp [6] Fontdecaba, J. (00), Integral Suspenson System for Motor Vehcles Based on Passve Components. SAE Techncal Paper Seres, [7] Smth, M.C. and Walker, G.W. (005), Interconnected Vehcle Suspenson. Journal of Automoble Engneerng, 9 (3) pp [8] Zhang, N, Smth, W., Jeyakumaran, J., Free vbraton of a half-car ftted wth a hydraulcally nterconnected suspenson system, under revew. [9] Zhang, N. and Hayama, S. (990), Identfcaton of structural system parameters from tme doman data (Identfcaton of global modal parameters of structural system by mproved state varable method), Intl. Journal of JSME, Seres 3, Vol. 33, No., pp Appendx The parameter values used n ths study were assumed based on avalable data n the publc doman and, to a lesser extent, on realstc estmates made by the authors. Cylnder and roll damper loss coeffcents were chosen to acheve bounce and roll dampng ratos of approxmately 40%. Snce no mechancal frcton or hysteretc dampng has been consdered, the damper loss coeffcents selected here would be larger than those requred n practce. Some parameters of the half-car system are shown n Table. The mean system operatng pressure P of the flud system s MPa. The parameter quanttes used n the study other than those lsted n the table can be found n [8]. Table Propertes of the half-car mult-body system Symbol Value Unts Descrpton M, m 750, 35 kg Sprung and unsprung mass I θ 30 kg/m Sprung mass moment of nerta about roll axs k s, k t 0, 00 kn/m Mechancal suspenson sprng and tre sprng stffness c tj 300 N.s/m Tre dampng µ 0.05 N.s/m Hydraulc ol vscosty β ol 400 MPa Hydraulc ol bulk modulus l 3, l ,.5 m Length of ppe from X to X 3 and from X 6 to X 7 d 7.07 mm Internal ppe dameter V p 00 ml Accumulator pre-charge gas volume P p 5 bar Accumulator pre-charge pressure A U, A L 506.7, 4.4 mm Upper and lower pston areas R V 5e+09, 3.e+09 kg/s.m 4 Loss coeffcent for cylnder valves and accumulator valves