A Simple Model for the Determination of Jackknifing

Transcription

1 A Siple odel or he Deerinaion o Jakkniing Oren asory Florida Alani Universiy Boa Raon, FL 3331 Grainer Thoas IFA Insiu Franais De ehanique Avanee Capus de Cleron, Les Cezeaux BP , Frane Absra Jakkniing is unonrolled braking o a ruk-railer obinaion hih auses he railer roaes relaive o raor and as a resul exends o adjaen lanes, ausing in any ases sever aidens. In his paper a siple o diensional odel is presened. The odel inludes paraeers suh as ruk s and railer s diensions, oeiiens o riion, asses, loaion o he load on he railer e. Based on his odel siulaion progra, hih allo he user o deerine heher or no jakkniing ill our, as rien. Siulaions resuls or e ases are presened. oenlaure L 1 - Disane beeen he raor s ron axle and is GC L - Disane beeen he raor s rear axle and is GC L 3 - Disane beeen he raor s rear axle and he railer s GC L - Disane beeen he railer s GC and is rear axle W - Traor s eigh W - Trailer s eigh (inluding load) - Traor/railer idh b - Laeral loaion o he railer s ener o graviy - Coeiien o riion on he raor s ron ires - Coeiien o riion on he raor s rear ires r - Coeiien o riion on he railer s rear ires - Coeiien o riion on he railer s righ rear ires - Coeiien o riion on he railer s le rear ires I - ass oen o ineria o he railer abou he Z-axis F - Horizonal ore aing on he kingpin P - Verial ore aing on he kingpin - oral ore on he raor s ron axial - oral ore a he raor s rear axial r - oral ore a he railer s rear axial - oral ore a he railer s righ rear heels - oral ore a he railer s le rear heels Inroduion A he presen, he ruk ranspor is one o he ornersones o he eonoy o he Unied Saes. Truking is vial o he US eonoy. The Aerian Truking Assoiaion saes, Truks ove os o Aeria s reigh and are he ransporaion bakbone o he Aerian eonoy he ruking indusry aouns or 81% o he ounry s reigh revenue. [1] This assive lo o produs and aerials an be easily ineuped, hoever, by a variey o poenial aidens espeially sei-railer aidens ha oen our as he ruk and he railer jakknie. Hene, in 000, 57,000 large ruks ha vehiles hih eigh raing are greaer han 10,000 pounds; ere involved in rai rashes in he Unied Saes;,930 ere involved in aal rashes. A oal o 5,11 people died, 1 peren o all he rai aaliies repored in 000, and an addiional 10,000 ere injured in hose rashes. Suh aidens an lose highays or hours, o say nohing o he hrea o he lives o ruk drivers and ar passengers. Jakkniing igh ause a very severe aiden due o he a ha he railer, hih roaes abou he raor, exends o adjaen lanes bloking he inoing or olloing rai. A ull undersanding o his phenoenon is hereore riial or aiden reonsruion and braking syse design. Previous ork on railer/ruk jakkniing inlude: Tiohy V. Fossu and Gilber. Leis presening a dierenial equaion [] hih is a odel or he posiion o a railer relaively o he ab hih is pulling i. Chieh Chen and asayoshi Toizuka ro he Universiy o Caliornia, Berkeley [3] proposed a odeling approah designed o preven jakkniing and urherore redue raking eors o he railer. Soe ork as done in he Universiy o Windsor, Onario, deerining he equaions o oion o raor/railer obinaion. The linearized equaions ere rien in ers o he laeral speed o he ruk, he ya rae o he ruk, and he say angle o he railer. Finally, he sabiliy o he syse had been evaluaed by peroring an eigen value analysis on he arix yielded ro he previous equaions []. A general ehod o deriving dynai odels or any oniguraion o heavy duy vehiles, inluding vehiles ih unis, is presened in appendix [5]. By he ay, oerial

2 heavy vehile researh a he Caliornia Parners or Advaned Transi and Highays (PATH) progra has oused on he developen o onrollers or parially o ully auonoous driving environens. A uli-body dynai odel o a raor sei-railer has been developed using a oerially available dynai analysis soare progra o predi any one o he any ailure odes suh as rollover or jakkniing ha are oon o ariulaed heavy vehiles [6]. A orsase senario generaion algorih able o ind he inpu obinaion hih is os derienal or ruk rollover or jakknie hen he seering and braking inpus o a ruk are liied belo a erain hreshold value, as also presened [7]. any ruk brake produers suh as Radlinski & Assoiaes, In orking on he sae opi also developed ani-jakknie devies [8]. The ai o hose orks has been o igure ou a dynaial odel able o deerine heher he ariulaed ruk ould jakknie or no, hih required he knoledge o a lo o very speii paraeers. This paper presens a siple o diensional odel, hih is an exension o a one diensional sik odel. The odel aoodaes variables, hih ould no be reaed ih one diensional ode, and hereore, hrough siulaions, he ee o hese variables on he jakkniing phenoena an be sudied. These variables inlude raor/railer laeral diensions, loaion o he ener o graviy o he railer (in order o rea load posiion on he railer), oeiien o riion or eah on he ire and oher. As a resul, ore realisi analysis and an be perored hile ainaining he sipliiy o he odel. To-diensional odel The odel assues no oion in he verial direion (Z-direion). Thus, he su o he ores in his direion and he su o he oens abou he Y-axis are zero. Using he diensions o he raor-railer obinaion shon in Figure 1 and he verial ores aing on he raor and he railer are shon in Figures and 3 respeully, he reaions a he heels and he ore a he king pin an be deerined. by: Reeing o Figure he sai equilibriu is given Fz = r W + P = 0 (1) y = r L PL3 = 0 () Solving equaions 1 and yields he noral ores aing on he railer s ires and he verial ore on he king pin: L P = W (3) ( L + L3 ) r = W L 3 ( L3 ) One r and P are knon, and, he reaion ores on he raor, an be easily ound using he sae ehodology. Suaion o he ores aing on he raor in he Z-direion and he oens abou he Y direion (reer o Figure 3): Fz = W P = 0 () y = L PL L1 = 0 (5) Solving equaions and 5 yields he ores aing on he ron and rear ires o he raor: L 1 = W + W (6) ( L + L3 ) L1 + L = W L ( L + L1) To deerine he reaions on he le and righ ires o he railer, and, he sae approah is used. The su o he oens abou Y and X axis should be zero: x = ( ) W ( b ) = 0 (7) y = ( ) L P L3 = 0 (8) Solving equaions 7 and 8 yields: W L3 L3 = ( b ) L3 L3 W L3 L3 = + ( b ) L3 L3 (9) In order o deerine he riion ores aing on he ires, i is assued ha he oeiien o riion, r, is independen o he ire s sliding direion. Figure shos he veloiies o he railer during a severe braking. Aordingly, Figure 5 illusraed he horizonal ores aing o he raor and he railer during braking. The angle β is given by: & θ ( L3 )os( θ ) β = aran( ) X& + & θ ( L + L )sin( θ ) and he angle α is given by: α β + θ 3 L (10) = (1) The ranslaion oion or he raor/railer obinaion, he raor and he railer an be desribed by equaions 13, 1 and 15 respeully:

3 X&& = ( 1 + ( ) ( )os( β )) X& = F (1) X& = F ( )os( β ) (15) + (13) The ore on he kingpin, F, an be deerine by: F 1 ( ) = ( + ) ( )os( β ) (16) The equaion desribing he roaion o he railer is deerined by suing he oens abou he railer s ener o graviy: I && θ = F b os( θ ) L (( b)os( α) L sin( α)) 3 sin( θ ) ( bos( α) + L sin( α) ) (17) W 37 Kg 5176 lb W 8 Kg lb. o.8. o.8. o.8. o.8 r. o.8. o.8 I Kg Slug Case 1: Trailer s ires ih dieren oeiiens o riion In rainy ondiions aer igh auulaes on he side o he road. As a resul, he oeiien o riion o he righ and he le ires ill be dieren. To siulae hese ondiions i is assued, in his siulaion, ha he righ rear railer s ires have loer oeiien o riion han he oher ires. Figure 6 illusraes he angular displaeen o he railer or hree dieren oeiiens o riion. I is lear ha or a oeiien o riion 0. he ruk beoes unsable. Also, as shon in Figure 7, he braking disane i being aeed even hough only one se o ires has loer oeiien o riion. & θ = (θ ) Equaion 17 is a nonlinear equaion in he or ha is solved nuerially ih he iniial ondiions θ ( 0) = θ o, & θ ( 0) = 0 and θ o assues a very sall value. Θ = = = = = = Siulaion Resuls in speial ases The odel desribed above as siulaed using Siulink (an obje oriened prograing ool provided by alab). The progra allos he user o se values o all paraeer shon in he oenlaure. The noinal values used in he siulaion ere adoped ro reerene [3] and are shon in Table. Alhough any roaion o he railer relaive o he raor igh be onsidered as jakkniing, he probabiliy o ollision is higher i he railer peneraes adjaen lanes. Thereore, jakkniing as deerined by evaluaing he value o θ hen he raor/railer oes o sop. I he railer rossed o adjaen lane i is onsidered ha jakkniing oued. Tie [se] Figure 6: Trailer s angle ih several oeiiens = = 0. Table 1: ain values used or he siulaion Value Uni Value Uni L L L L D i s a n e = = 0. = = 0.6

4 Case 3: Lo oeiien on he raor s ron heels. Tie [se] Figure 7: Traor/railer s braking disane. Soe repors lai ha he jakkniing igh our i he oeiien riion on he ron and iddle heels is oo lo sine he ires ould no provide he riion ore required o sop he railer. To exaine his lai he raor s ires oeiien o riion as se o a lo value and he siulaion as perored. Case : The ee o he loaion o railer s ener o graviy Alhough his aspe is rarely sudied in he jakknie siulaion and odeling, i sees naural o hink ha an ariulaed ruk-railer hose railer is no syerially loaded has ore hane o jakknie han anoher one hose railer is perely loaded. Insead o doing siulaions ih several values o he disanes beeen he railer s CG and he heels o kno heir ees on he behavior, he olloing siulaion opares our dieren loadings in hih he ener o graviy is shied righ o le and ron and rear as shon belo: C1 - L 3 = 10,5, L =0,5 & b=6 C - L 3 =10,5, L =0,5 & b= C3 - L 3 =0,5, L =10,5 & b=6 C - L 3 =0,5, L =10,5 & b= Θ Tie [se] Figure 9: Trailer s angle ih loer oeiiens o riion on he raor s ires. Θ C3 C Tie [se] C1 C Figure 8: Trailer s angle ih dieren loadings. Figure 8 indiaes ha in his ase here he oeiien o riion o all ires is he sae, here is no ee o shiing he load ro righ o le (C1 is syeri o C and C3 o C). Obviously, shiing he ener o graviy o he ron o he railer (C3 and C) ill onribue o ouene o jakkniing due o reduion o riion ores on he railer s ires. The resuls shon in Figure 9 are quie surprising. Alhough he braking disane and he braking ie have inreased, he sei-railer sees o reain sable under erain irusanes. Conlusion As a onlusion i is easier o oe o an unsable ase ih he o diensional odel, indeed here are he one diensional unsable ases and he ases here he railer is no syerially loaded or he oeiien o riion on he rear heels are no he sae. The raor/railer ay also beoe unsable i he graviy ener is oved orard or behind and i i is no loaed on he railer syery axis or i he oeiiens o riion o he rear heels are no he sae. Hene, aording o his odel a raor sei-railer ould beoe unsable i i is braking in a non regular surae area ha is o say aer or grease under one o he rear heels or exaple or even i he brakes don ork oely on one heel. C Changing soe oher paraeers suh as he iniial ondiions and pariulay he iniial speed o he raor ay also hange he behavior o he raor railer obinaion. In a hose hanges ay inrease he speed o he reaion and lead he railer o he lii o sabiliy. aybe, he jakknie proble is no as siple as seen beore. During he braking he oeiien o riion ay derease i he heel is loked or i i is heaed. oreover

5 hy should he oeiien o riion be he sae as he heel is skidding orard or on any oher direion. Cona Oren asory an be onaed by pos a ehanial Engineering Deparen, Florida Alani Universiy, Boa Raon, FL 3331, or via eail a [email protected]. Reerenes 1. The Aerian Truking Assoiaion, hp://.rukline.o. Fossu, T. V., and Leis, G., A aheaial odel or railer-ruk jakkniing, SIA Revie Wen-Hou a and Huei Peng, Wors-Case Vehile Evaluaion ehodology Exaples on Truk Rollover/Jakkniing and Aive Ya Conrol Syses, Vehile Syse Dynais, 3 (1999), pp /99/ Rihard W. Radlinski, The anaoy o a raor railer jakknie, Presiden Radlinski & Assoiaes, In, hp://.raibrakes.o 9. K & R. Giek, Forulaire Tehnique, Giek Veag, Ediion rançaise 1990, ISB U.S. Deparen o The U.S. Deparen o Transporaion s, Transporaion Coprehensive Truk Size and Weigh Sudy, Volue II, Issues and Bakground, Publiaion uber: FHWA-PL (Volue II), HPTS/Augus Chieh Chen and asayoshi Toizuka, odeling and Conrol o Ariulaed Vehiles, Universiy o Caliornia, Berkeley, UCB-ITS-PRR-97-, oveber 1997, ISS Dr Brue P. inaker, Universiy o Windsor, Onario, Canada, hp://harger.esx.uindsor.a/~inaker/index.hl 5. eihua Tai, Jeng-Yu Wang, Pushkar Hinge, Chieh Chen, asayoshi Toizuka Laeral Conrol o Heavy Duy Vehiles or Auoaed Highay Syses, Universiy o Caliornia, Berkeley, UCB-ITS-PRR-98-8, February 1998, ISS J. Chrisian Gerdes, Paul Yih, Krishna Sayan, Saey Perorane and Robusness o Heavy Vehile AVCS, Sanord Universiy, January 00

6 , I z Z X L 1 L L 3 L b Figure 1: Traor Sei railer Diensions. P W r Figure : Verial ores aing on he raor. P Z X W Figure 3: Verial ores aing on he railer

7 1 1 L 1 F 1 1 L F L 3 Traor b L Trailer β θ α θ Figure : Horizonal ores aing on he raor and he railer.

8 θ & X & L 3 b θ θ L X & ( L + ) θ & 3 L β α Figure 5: Veloiies on he railer.