DYNAMIC CONTACT PROBLEM OF ROLLING ELASTIC WHEELS

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1 DYNMIC CONTCT PROBLEM OF ROLLING ELSTIC WHEELS Dénes Tkács nd Gábor Stépán Deprtment of pplied Mechnics Budpest Uniersity of Technology nd Economics Budpest, H-151, Hungry BSTRCT: The lterl ibrtion of towed wheels, clled the shimmy, is well-known stbility problem in ehicle system dynmics. One of the resons of the ppernce of shimmy is the elsticity of the pneumtic tyre tht is used widely on ehicles. Our model is goerned by coupled system of prtil differentil eqution nd n integrodifferentil eqution. The chrcteristic eqution of this liner system is determined nd the relent eigenlues re clculted numericlly. The bstrct eigenectors belonging to these eigenlues represent the corresponding stnding we solutions. The results re confirmed by simultion nd by eperiments, too. 1. INTRODUCTION Shimmy is the nme of the lterl ibrtion of towed wheels. Shimmy cn be obsered on nose gers of irplnes, on motorcycles, on trucks nd on rticulted buses. lthough, the phenomenon hs been well known for long time, the modelling of shimmy motion still rises new mechnicl nd mthemticl problems. The mechnicl models, which cn be found in the literture, re considering the elsticity of the suspension [1] or the elsticity of the tyre []. Becuse the mjority of ehicles re rolling on pneumtic tires, the elstic tyre model hs become more importnt. This study inestigtes low degree-of-freedom model tht ws nlyzed s time delyed system, for emple in [3], where the liner stbility chrt cn lso be found. The model in question is simple elstic wheel towed by rigid cster through the rigid king pin. The tyre touches the ground t contct surfce. The lterl deformtion of the contct region is modeled by contct line in the centre of the contct re. The system is goerned by prtil differentil eqution (PDE) originted in the kinemticl constrint of rolling, nd by n integro-differentil eqution (IDE), which is deried from the bsic lw of dynmics. In this study, the chrcteristic eqution of the liner system is clculted nd the relent eigenlues re determined by n pproprite numericl

2 method. The stnding wes, tht re the eigenfunctions of the PDE, re clculted nd so the deformtion shpes of the contct line re lso presented t the eperimentlly lidted Double-Hopf bifurction point.. MECHNICL MODEL The model in question is tken from [3]. s shown in the figure, the elstic wheel is towed by the cster of length l on the stedy ground with constnt elocity (see Figure 1). The bsolute coordinte system ( ξ, η, ζ ) is fied to the ground. During rolling, the tyre dheres to the ground in the contct re. This re is modelled s contct line of length. In this wy, the deformtion of the tyre cn be modelled by the lterl displcement q (, t) of this contct line reltie to the plne of the wheel. The coordinte describes the position of the contct points long the contct line, nd t stnds for the time. The lterl stiffness nd dmping fctor distributed long the contct line re c [N/m ] nd k [Ns/m ], respectiely. The mss moment of inerti of the oerll system with respect to the is z t the rticultion point () is θ [kgm ]. z η ξ q(,t) y P ψ(t) c k l Figure 1. Model of the towed tyre.1. Eqution of motion The eqution of the motion of the undmped system is θψ () t = c ( l ) q(,)d t. (1) If rolling is considered, the lterl deformtion long the contct line is determined by the PDE: qt (, ) = sin ψ + ( l ) ψ + q ( t, ) ( cos ψ qt (, ) ψ ), ()

3 with the boundry condition: where [, ] nd t [ t, 0 ) differentition with respect to the spce coordinte. qt (, ) = 0, (3), dot refers to time deritie, prime refers to 3. CHRCTERISTIC EQUTION If we considered smll mplitude ibrtions, the equtions of the system cn be linerized t ψ = 0. So the liner system is described by: c ψ () t = ( l ) q(,)d t θ, (4) qt (, ) = ψ( t) + ( l ) ψ ( t) + q (, t), (5) where [, ], t [ t, 0 ), nd qt (, ) = 0. If we use the stndrd eponentil tril solution λt λt ψ() t = P e, qt (, ) = Q ( ) e (6) in (4) nd (5), we obtin: λt c λt λ Pe = ( l ) Q( ) e d θ, (7) λt λt λt λt λq( e ) = Pe + ( l λ ) Pe + Q ( ) e. (8) fter some lgebric mnipultion, (8) hs the form λ λ Q ( ) Q( ) = P ( l) P, (9) which is non-homogeneous liner ordinry differentil eqution respected to Q() with the boundry condition Qt (, ) = 0. The solution of (9) cn be determined by the sum of the homogeneous solution nd the prticulr solution: Q ( ) = QH + QP, (10) where the homogeneous prt cn be found s ( ) µ QH = K e. (11) If we write tht into the homogeneous differentil eqution, we obtin: λ µ K = 0. (1) For non-triil lues of K, we get the solution µ = λ, nd with this, the homogenous solution ssumes the form ( ) ( λ ) QH = K e. (13)

4 Becuse the right side of (9) hs polynomil form, the prticulr solution cn be found in the form of QP = + b. (14) If we write this tril solution into (9), we get λ λ ( b) P ( l) P (15) Seprte the coefficients of the eqution with respect to the power of : 0 : λ λ b P l P (16) 1 : λ λ P (17) from which the prmeters of (14) cn be determined, nmely = P nd b= Pl. So the prticulr solution is QP = P+ Pl. (18) From (10), the generl solution of (9) is ( λ ) Q ( ) = K e P+ Pl, (19) where K nd P re determined by the initil nd boundry conditions. Substitute this solution into (7) nd pply the boundry condition to obtin the net system of equtions: c ( λ ) λ P+ ( l ) ( K e P+ Pl) d= 0 θ, (0) ( λ ) K e P Pl + = 0, (1) which cn be written in mtri form: ( λ ) e ( l ) K c ( λ ) c ( l ) e d λ ( l ) d P = 0. + θ θ () The integrls in the mtri cn be clculted by prtil nd polynomil integrtion, respectiely. This results the new mtri eqution form ( λ ) e ( l ) K c ( ) λ ( ) λ c = e l e l λ l P 0. (3) θ λ λ λ θ 3 Becuse this mtri eqution must he non-triil solutions for K nd P, the determinnt of the coefficient mtri hs to be zero. This leds to the trnscendentl chrcteristic eqution:

5 λ + l + ( l ) l + e l+ + = 0. (4) c c λ θ 3 θ λ λ λ 3. STBILITY NLYSIS The stbility nlysis of this mechnicl model cn be found in [3,4], for the sme chrcteristic eqution obtined from dely differentil eqution model. The stbility chrt is determined by the D-subdiision method nd stbility criterion deried in [5]. The results re shown in Figure, the stble regions re shded. The prmeters re the dimensionless cster length L = l/ nd the dimensionless towing speed V = /( α), where c 1 α = l + 3 s (5) θ is the nturl ngulr frequency of the system bout the rticultion point () if the towing speed is zero. s it is shown in the picture t the intersection of two stbility boundries there re two independent self-ecited ibrtion frequencies (ω) in the system. ω/α L Log scle V Figure. Stbility chrt V

6 4. STNDING WVES 4.1 Eigenlues The roots of the chrcteristic eqution (4) cn be determined nlyticlly if l =, nmely, when the chrcteristic eqution simplifies to c λ + l + = 0, (6) θ 3 nd the eigenlues of the system re: c λ1, =± iα =± i 3 θ. (7) In the generl cse, the system hs infinitely mny eigenlues, but they cn be found by pproprite numericl methods only. To determine some relent stnding wes of the system, we clculte the relent eigenlues only. 4. Eigenfunctions From (1), we he ( ) λ K ( ) i i = Pi l e, i = 1,,... (8) nd the corresponding eigenfunctions he the form λi Qi( ) = Pi ( l ) ( l ) e. (9) Becuse the sum of the function hs to be rel, it cn be proed tht the coefficients of the comple conjugte eigenlues he to be conjugte pirs, too. The deformtion q (, t) is gien in the form of n infinite series: i= 1 λit λit ( i i ) qt (, ) = Q( ) e + Q( ) e, (30) which leds to: qt (, ) = Re( Pi)( l ) ( l ) Re( Pi)cos Im( λi) i= 1 Re( λi ) Im( Pi)sin Im( λi) e cos( Im( λi) t) Im( Pi)( l ) ( l ) Im( Pi)cos Im( λi) Re( λi ) Re( λi ) t Re( Pi)sin Im( λi) e sin ( Im( λi ) t) e, (31)

7 where λ i denote the comple eigenlues with positie imginry prt only. 5. EXMPLE If we choose relistic prmeter lues obtined from our eperiments, we cn clculte the deformed shpe of the tyre. These prmeters re: l = 0.011[m], = [m], c= [N/m ], (3) θ = [kgm ], = 0.15 [m/s]. t these prmeter lues, there is Double-Hopf bifurction point on the stbility chrt (see Figure ), nd qusi-periodic ibrtion ws detected in the eperiments [6]. If we plot the Eucliden norm of the rel nd the imginry prts of the chrcteristic eqution, we cn determine the loction of the eigenlues (see Figure 3). The relent eigenlues he nerly zero rel prt. The self-ecited ibrtions re originted from these eigenlues, while the other eigenlues he reltiely lrge negtie rel prt, nd the corresponding oscilltions will die wy quickly. The numericl lues of the relent comple conjugte eigenlues re determined: λ1 = i , λ = i (33) The pproprite prmeters P i re coming from the initil conditions, tht is, from the initil shpe q (,0) of the contct line. This time we simply choose: P1 = i0.01, P = i0.01. (34) Im(λ) 0 10 λ λ λ 1 λ Re(λ) Figure 3. Norm of the chrcteristic eqution The corresponding stnding wes Q 1, ( ) re shown in Figure 4 nd the time eolution of the deformed contct line qt (, ) is represented in Figure 5.

8 Q () Q() t = 0 [s] q(,t) Q 1 () Figure 4. Stnding wes t = 0.05 [s] t = 0.5 [s] t = 0.15 [s] Figure 5. Deformtion of the tyre 6. CONCLUSION The inestigtion of the deformtion shpes of this tyre model cn be ery helpful to understnd the nonliner effects of sliding t rer prt of the contct line. This ws obsered on the eperimentl rig (see [6]). This nonlinerity cn eplin the differences between the eperimentl results nd the boe liner theory with respect to the qusiperiodic oscilltions. 6. CKNOLEDGEMENT The uthors gretly cknowledge the finncil support of this work proided by the Hungrin Ntionl Science Foundtion under grnt no. T REFERENCES 1. Gábor Stépán: Chotic motion of wheels. Vehicle System Dynmics, 0(6), pp (1991).. H. Pcejk: Modelling of the pneumtic tyre nd its impct on ehicle dynmic behiour. Technicl report, Technicl Uniersity of Delft, The Netherlnds, (1988). 3. Gábor Stépán: Dely, nonliner oscilltions nd shimmying wheels. F.C.Moon (ed.), pplictions of Nonliner nd Chotic Dynmics in Mechnics, (Kluwer cdemic Publisher, Dordrecht, 1998), pp Dénes Tkács: Dynmics of rolling elstic wheels, MSc thesis, Budpest Uniersity of Technology nd Economics, Hungry, 005. Mechnicl Engineering MSc. 5. Gábor Stépán: Retrded Dynmicl Systems, Longmn, London, Dénes Tkács, Gábor Stépán: Eperimentl study of qusi-periodic oscilltions of towed wheels, In Proceedings of Fifth Conference on Mechnicl Engineering, 006. Budpest, Hungry