THE DYNAMICS OF A THREE-WHEELED NARROW-TRACK TILTING VEHICLE

Transcription

1 F28SC32 THE DYNAMICS OF A THREE-WHEELED NARROW-TRACK TILTING VEHICLE 1 Beote, Johan *, 1 van Poelgeest, Auguste, 1 Daling, Jos, 1 Edge, Kevin, 1 Plue, Andew 1 Univesit o Bath, United Kingdo KEYWORDS dnaics, hdaulics, odelling, thee-wheele, contol sstes ABSTRACT Thee-wheeled tilting vehicles have an potential advantages. The cobine the coot and saet o a ca with the uel econo, size and anoeuvabilit o a otoccle. One ajo challenge is to ull undestand and account o the unusual dnaics o this tpe o vehicle. This pape sets out the equations o otion o a vehicle with single ont and twin ea wheels and copaes the with those o a ca and a otoccle. The atheatical odel shows that it is necessa to intoduce a tilt-dependent ea wheel stee te to obtain balanced handling chaacteistics. A vehicle with a hdaulicall actuated tilting sste is consideed as a case stud. As the tilt actuatos act against the non-tilting engine odule, this has signiicant eects on the coneing dnaics. Finall, the dieence between the lateal acceleation and the hdaulic esponse tie is shown to be a ke issue. INTRODUCTION In toda s societ, the need o oe envionentall iendl tanspotation is constantl inceasing. In paticula the deand o sae, clean and individual tanspotation is gowing. The CLEVER ca was developed as pat o an EU unded consotiu at the Univesit o Bath between 22 and 25 to ill this gap in the aket. The design objectives wee to design an envionentall iendl, sae ode o tanspot that would take up as little space as possible on the oad. The designes opted o a naow tack thee-wheeled vehicle that would seat a dive and one passenge in tande. This coniguation signiicantl educed the weight and ontal aea o the vehicle, esulting in a oad ootpint siila to that o a otoccle. The naow wheelbase, howeve, eant that the oll-ove stabilit o the vehicle was ve liited. This poble was ovecoe b intoducing a hdaulic tilt echanis that would lean the dive cabin into a cone like a otoccle. The cobination o thee-wheels and tilting abilit eant that the vehicle showed ve dieent dnaic chaacteistics o those o a otoccle o a ca. The dieence in dnaics between a ca and a thee-wheeled vehicle have been analsed b Huston et al. (1). Howeve, this publication did not cove tilting thee-wheeles. On the topic o tilting naow tack vehicles a nube o papes have been published. Kanopp and Fang (2) wee the ist to suggest that, to pevent oll-ove, the vehicle should lean into the tun in the sae wa a otocclist leans. Subsequentl, Kanopp and Hibbad (3) discussed the optiu lean angle equied when tilting a vehicle. Kanopp and Hibbad continued in this aea o eseach and descibed the dnaics o a naow tilting vehicle (4) and the ethods to achieve the tilting position (5). The two ethods descibed wee Diect Tilt Contol (DTC) and Stee Tilt Contol (STC). The oe consisted o a contol loop which calculated the optiu tilt angle equied o the speed and stee angle, and an actuato that pushed the vehicle to this

2 tilting position. STC contained a siila contol loop, but it estiated the countestee equied to achieve the optiu tilt angle, and steeed the vehicle accodingl. With egads to saet, the CLEVER designes peeed DTC to STC, the latte equiing a stee-b-wie sste and theeoe cutting the diect link between the dive and the diectional contol. It should be noted that the authos o papes (2) to (5) onl consideed the tilt contol. None o the publications consideed the actuation in detail, and in paticula, the eect the actuation sste would have on the assebl it was acting against. The hdaulic actuatos on the CLEVER vehicle ae ounted on the non-tilting ea unit, so when the ae diving the cabin to the balanced position, the ae also acting against the suspension o the ea unit causing it to oll as well. I the toque o the actuatos becoes lage enough, the ea unit could oll ove. This pape discusses the vaious anoalies o the dnaics. It stats with a copaison o the equations o otion o a ca, a otoccle and a thee-wheeled tilting vehicle. It then looks close at one paticula dnaic eect, nael ea stee. Following on is a discussion o the eect the hdaulics have on the sste, in paticula the eect o the actuatos on the ea suspension and also the esponse tie o the hdaulic actuation sste. NOMENCLATURE vehicle ass F ea wheel lateal oce ont axle ass C ont te stiness ea axle ass C γ ont te cabe stiness I z vehicle aw inetia C ea te stiness a lateal acceleation K δ ea stee gain V x owad velocit ont te slip angle V lateal velocit ea te slip angle ψ aw angle δ vehicle stee angle a c.g to ont te δ ont stee angle b c.g. to ea te δ ea stee angle l vehicle length γ ont te cabe angle R tun adius g gavit F ont wheel lateal oce θ tilt angle EQUATIONS OF MOTION Ca The equations o otion o a ca ae geneall deived o the biccle odel shown in Figue 1. This odel assues that the dnaic behaviou o the two ont and ea wheels can be odelled b one single wheel poducing twice the lateal oce. Using this odel the equations o otion can be deived as shown in equations [1]and [2]. The side oces F and F ae caused b the te slip angles and. The esulting oce is dependent on the te stiness C, which vaies pe te, with the diving conditions, and an oe actos. This elationship is shown in equation [3]. The slip angles ae a esult o the dieence in the diection the te is pointing and the diection the vehicle is going. The equations that descibe this phenoenon ae shown in equation [4].

3 Figue 1: Biccle odel ( a Vxψ) = F F + & + [1] I ψ& & = af bf [2] z F = C F C = [3] = δ V + aψ& actan Vx V bψ& = actan Vx [4] Motoccle Because the equations o otion o a ca ae usuall deived o a biccle odel as shown in the section above, the otoccle equations ae not too dieent o those o a ca. The ain dieence is the eect the oll otion o a otoccle has on its dnaics. This can ist be seen in the side oce equation [5]. The side oce is also dependent on the cabe angle γ, also noted as the tilt angle θ, and the elated te stiness. The slip angles and ae geneated as in equation [4]. A otoccle ide ust lean the vehicle into a cone in ode to balance the leaning oent caused b lateal acceleation and the oent caused b the gavitational oce. This oll angle is descibed in equation [6]. F = C + C γ γ F = C + C γ γ [5] a a θ = actan g [6] g

4 Thee wheeled tilting vehicle The equations o otion o a thee-wheeled tilting vehicle ae a cobination o those o a ca and a otoccle. The tilting ont te ields side oces o both the sideslip and the cabe, wheeas the ea tes onl ield side oces o thei espective slip angles as shown in equation [7]. The geneation o the slip angles is siila to equation [4] o the ont te. Howeve, the ea tes o a thee-wheeled tilting vehicle can stee too and an additional stee te is added to the second pat o equation [8]. This ea stee te is deived in the next section. F = C + C γ γ F = [7] = δ V + aψ& actan Vx = δ V bψ& actan Vx [8] HANDLING CHARACTERISTICS OF A THREE-WHEELED TILTING VEHICLE Stead State Chaacteistics Using equations [5] and [6] o the biccle odel descibed peviousl and esticting stee and slip angles to ail sall values, the stee angle equied to negotiate a cuve with adius R is given b equation [9]. At low speed, and cancel each othe out and the stee angle δ is equal to l/r. This is eeed to as the Ackeann angle. At highe speeds, te slip inceases and the steeing angle deviates o this idealised condition. The slip angles ae witten with espect to the lateal acceleation as shown in equation [1]. Equation [1] is substituted into equation [8], the 1/R te is eplaced with a /V 2, and the cabe angle is eplaced b the tilt angle o equation [6]. This then ields the new equation o a stead state steeing adius, equation [11]. l δ = + ( ) [9] R = a C C γ θ a = [1] δ = l a V 2 + g a C C g γ a a [11] Using tpical te data o a 12/7R17 ont otoccle te and 195/65R15 ea ca tes (7), gaphs o stee and slip angles vs. lateal acceleation can be plotted o constant adius tuns, Figue 2. These show that the vehicle would consideabl ovestee. This is due to the ont te geneating the ajoit o the coneing oce though cabe. The ont slip, ea slip and cabe angles eain the sae o all tun adii because the equations ae lineaised. The lineaization eans that the angle appoxiations becoe less accuate as the lateal acceleation inceases. Siilal, the cabe angle is highe than 45 o at 1g coneing because the tan unction in equation [6] is lineaised.

5 Figue 2: Coneing chaacteistics (7) To copensate o the signiicant ovestee, the ea wheels would have to stee into the cone as the cabin leans. This ea stee is epesented as a gain o the tilt angle [12]. Substituting the latte into equation [9] then ields the equation o the stee angle o a given tun [13]. Substituting in the slip angles o [1] and dieentiating with espect to the lateal acceleation ields what is known as the ovestee estiation [14]. Fo this ovestee estiation, the ea stee gain K δ can be calculated [15]. δ = K δθ [12] l δ = + ( ) + δ R [13] dδ da = l V 2 g C + C g γ K δ + g [14] K δ g γ = [15] g C C Rea Stee and Tilt Axis Inclination In the pevious section it was shown that ea wheels need to stee as the vehicle olls in ode to achieve neutal handling chaacteistics. This can be achieved though the inclination o the tilt axis. When consideing the inetia tensos o both the ont and the ea unit, the will both take the o o equation [16]. In this case the pitch inetia I 2 is also the inetia about the pinciple axis. Now it can be seen that a oll acceleation o the cabin will ield a aw oent. I xx I zx I 2 I xz I zz [16]

6 Figue 3: CLEVER and its tilt axis Figue 4: Eect o tilt axis inclination (7) I the tilt axis is paallel to the gound, the aw oent will be diectl tanseed to the ea unit and the will aw togethe in the sae diection and with the sae acceleation. Now, i the tilt axis is inclined, the oent tanse will not be diect and the ea will aw eithe oe o less than the ont assebl depending on the inclination. Figue 3 shows a positive tilt axis inclination. This esults in a aw ate o the ea which is geate than the aw ate o the ont, so the ea will stee into the tun. I the tilt axis is negative, the ea will stee awa o the tun. Figue 4 shows the eect o vaious tilt axis inclination angles; it shows how uch the ea wheels will stee o a given tilt angle. The bold line shows the necessa inclination to satis equation [15]. CONTROL In ode to stabilise the vehicle duing coneing, CLEVER has an active diect tilt contol sste. The hdaulic cicuit was designed to contol the position o the tilting pat o the vehicle with two single acting linea hdaulic actuatos. When pessuised, these clindes contol the lean angle o the tilting cabin b otating it with espect to the upight ea odule. A popotional diectional contol valve with a closed cente position odulates the low to the actuatos, contolling thei position and locking the clindes when no coand is given. Having a neutal steeing esponse allows a easonabl accuate estiate o the lateal acceleation to be ade using vehicle speed and stee angle. The contolle cuentl ipleented is a siple Popotional contolle with closed loop eedback o a tansduce easuing actuato displaceent. SIMULATION A non-linea odel o CLEVER was ceated using the MATLAB/SIMULINK package in ode to investigate the unique dnaics o the vehicle. One o the ke aeas o concen elating to the stabilit o the vehicle is the link between the static ea odule and the tilting cabin. Due to the DTC ethod o contolling the lean angle, additional oces act on the ea odule as a esult o the actuatos. The oce acting on the ea odule ceates a oll oent that acts in such a wa as to give the inne wheel the tendenc to lit o the gound. This situation aises when a apid steeing input is ade which esults in a high actuato toque deand. The elativel sall inetia o the ea odule copaed to that o the tilting cabin can lead to the lit o o the inne wheel and loss o contol o the vehicle.

7 a (/s 2 ) a ackeann a eal deand tilt angle absolute tilt angle tie (s) Figue 5: Siulated lateal acceleation and tilt angle tilt angle (deg) Toque (N) actuato toque 2 inne wheel load tie (s) Figue 6: Siulated actuato toque and inne wheel load Foce (N) Figue 5 and 6 illustate the vehicle s esponse to a ap-tpe input in the steeing angle while diving at a constant velocit o 3kph. Figue 5 shows the calculated (Ackeann) lateal acceleation and the equivalent tilt angle equied to bing the ont odule o the vehicle in equilibiu. The delaed vehicle s lateal acceleation esponse due to the geneation o slip and cabe in the tes can also be seen. Futheoe, the tilt angle esponse o the ont odule as a esult o the actuato toque is shown. Figue 6 illustates the equied actuato toque o the anoeuve and the load on the inne wheel. It can be seen that actuato toque esults in a signiicant dop in the load on the inne wheel. Once the load on the inne wheel eaches zeo, the vehicle will oll ove, showing that this paticula anoeuve is close to the vehicle s capabilities in tes o dnaic esponse. Figue 5 illustates the avoued contolle esponse, whee the tilt angle equied to balance the ont odule is achieved at the sae tie as the lateal acceleation is geneated. B slowing down the esponse o the contolle, the actuato toque deand is educed along with the isk o oll-ove. It is howeve ipotant that the vehicle eaches the equied tilt angle beoe the lateal acceleation builds up, because at highe lateal acceleations this would esult in inceased weight being tanseed to the oute wheel and ultiatel the olling o the vehicle. Also, the esponse o the vehicle would eel sluggish, which could esult in the dive copensating b additional steeing deand. Notable is also the act that the absolute tilt angle does not each the deand tilt angle. This is due to the easueent o the cabin tilt angle being elative to the ea odule, which olls out o the cone with the incease in lateal acceleation. DISCUSSION The siulation has shown that a cobination o speed and apid steeing inputs can cause the vehicle to oll ove as a esult o the lage toque equieent o the anoeuve. The actuato applied such a lage oce to the ea unit that the esulting oll acceleation was lage enough to oll the unit out o the bend. The siulation also evealed the iniu equied esponse speed o the hdaulic actuation sste. The Ackeann lateal acceleation, its esulting desied tilt angle, the non-linea lateal acceleation estiate, and the esulting tilt angle o the actuation sste wee copaed. The tie dela between the calculated Ackeann lateal acceleation and the eal lateal acceleation estiate deived o the non-linea odel allowed the hdaulic actuation sste to espond beoe the vehicle s lateal acceleation built up, educing the

8 actuato toque deand and the isk o the ea odule olling ove. The contolle gain had to be set so that the actuation sste esponse ell between the Ackeann lateal dnaics tilt deand and the actual tilt angle equied to balance the vehicle accoding to the non-linea lateal acceleation odel. This eant that the vehicle was in act oveleaning duing tansient anoeuves. This was ound to be a positive esponse, as the gavitational coponent o the ont cabin would help it each the desied angle, educing the toque equied and hence the likelihood o oll-ove. CONCLUSIONS An investigation o the dnaics o thee-wheeled tilting vehicles has been pesented. This investigation showed that the equations o otion o a thee-wheeled tilting vehicle ae a cobination o those o a ca and a otoccle. The ost ipotant dieence between the dnaics o a thee-wheeled tilting vehicle and that o a ca o otoccle was the steeing action o the non-tilting unit. It was shown that the ea wheels had to stee into the cone in ode to achieve neutal steeing dnaics. Also, the ea wheel stee had to be elated to the aount o lean. To achieve this task, it was evealed that the tilt axis had to be inclined with espect to the oad. In addition the chaacteistics and pobles o the hdaulic tilt contol sste wee highlighted. The actuation sste could cause the ea unit to oll out o the bend when appling lage toques in esponse to ast steeing inputs. The contolle set up was such that the cabin would stat leaning into the cone just beoe the lateal acceleation built up. It was citical to slow down the esponse tie o the actuatos to avoid lage toques, while at the sae tie peventing the lateal acceleation o the ont cabin o building up. ACKNOWLEDGEMENTS The authos would also like to thank D M Bake and D B Dew o the wok the caied out on the subject peceding this stud. REFERENCES (1) Huston, J.C., Gaves, B.J., Johnson, D.B., Thee wheeled vehicle dnaics, Societ o Autootive Enginees, 82137, 1982 (2) Kanopp, D., Fang, C., Siple odel o steeing-contolled banking vehicles, ASME Dnaics Sstes and Contol Division (DSC), 44, pp , 1992 (3) Kanopp, D., Hibbad, R., Optiu oll angle behavio o tilting gound vehicles, ASME Dnaics Sstes and Contol Division (DSC), 44, pp , 1992 (4) Hibbad, R., Kanopp, D., The dnaics o sall, elativel tall and naow tilting gound vehicles, ASME Dnaics Sstes and Contol Division (DSC), 52, pp , 1993 (5) Hibbad, R., Kanopp, D., Methods o contolling the lean angle o tilting vehicles, ASME Dnaics Sstes and Contol Division (DSC), 52, pp , 1993 (6) Pacejka, H.B., Bakke, E., Magic oula te odel, Vehicle Sste Dnaics, 21 (suppl), pp 1-18, 1993 (7) Bake, M, Chassis design and dnaics o a tilting thee wheeled vehicle, PhD thesis, Univesit o Bath, Bath, UK, 26