STUDY ON CONTACT PATCH MEMORY AND TRAVELLING WAVES OF TYRES

Transcription

1 STUDY ON CONTACT PATCH MEMORY AND TRAVELLING WAVES OF TYRES Dénes Takács 1,Gábor Stépán 2 1 MTA-BME Research Group on Dnamics of Machines and Vehicles 2 Department of Applied Mechanics, Budapest Universit of Technolog and Economics Műegetem rkp. 3. H-1111 Budapest, Hungar address of lead author: takacs@mm.bme.hu In this stud, the self-ecited vibrations of the caster-wheel sstem are analsed. A low degree-of-freedom mechanical model is considered, in which the lateral deformation of the tre is described both in the contact patch and outside it. The simple brush tre model is implemented in order to obtain analtical results b means of minimum number of relevant parameters. The frequencies of the detected self-ecited vibrations are presented against the towing speed. Some intricate shapes of the corresponding tre deformations are presented based on numerical simulations. 1 INTRODUCTION The shimm of towed wheels (e.g. the front wheels of motorccles [1, 2], the nose-gears of air-planes [3, 4, 5]) is a fascinating phenomenon in vehicle dnamics. The safet haard of shimm induced man publications in the field, but the elimination of shimm requires special attention from engineers even nowadas. On the one hand, towed wheels are often parts of comple vehicle sstems, which require the analses of multi-bod sstems like the fuselage of air-planes [4]. On the other hand, the investigation of shimm demands improved and detailed models of tres and tre/road interactions. The memor effect of the tre/ground contact patch was alread recognied in [6]. However, the available mathematical methods did not provide opportunit for engineers at that time to analse dela differential equations. An engineering approimation of the tre lateral deformation in the contact patch helped to describe some behaviour of the caster-wheel sstem [7, 8]. Later, the accurate modelling of the contact patch lateral deformation and the analsis of the corresponding dela differential equations provided new eplanations for some quasi-periodic vibrations [9]. The noise generation of tres has become an important research field in tre dnamics, which makes the developments of different theoretical models of tre/road interaction to be a relevant topic again (see, for eample, [1, 11, 12, 13]). The mechanical models usuall consider the radial and/or the longitudinal deformation of the tre thread elements. However, the lateral vibration of vehicles can also lead to such periodic tre deformations that can generate noise and wear (see [14]). The application of the time delaed tre model in the single track car model [15] also identified parameter domains where small amplitude lateral vibrations ma appear leading to enhanced noise and heat generation. In this stud, a simple mechanical model of a shimming wheel is constructed, in which the lateral deformations are described both inside and outside the contact patch. The delaed contact patch model is combined with a simple tre carcass model. Critical parameter ranges of self-ecited vibrations are determined versus the longitudinal speed of the towed tre with special attention to the vibration frequencies. Tre deformations are illustrated b means of numerical simulations at certain towing speeds. 2 MECHANICAL MODEL The mechanical model is shown in Figure 1. The wheel of elastic tre is attached to the rigid caster of length l. The caster is pulled at the king pin A with the constant towing speed v, R refers to the outer radius of the tre, and it contacts to the ground at a contact patch of length 2a. One of the state variables is the caster angle denoted b ψ(t). The lateral deformations of the tre are described with the help of the lateral deformation q(, t) of the centre line of the contact patch. This will serve as another state variable that is a distributed one along the coordinate attached to the caster. For the sake of simplicit, 1

2 l R v A X Z Y L P R 2a R w(,t) q(-a,t) k R k -a P 2 q(,t) L P 1 a q(a,t) L Figure 1: The mechanical model pure rolling of the tre is considered. Outside the contact patch, along the circumference of the tre, the vibrations of the thread elements are described b the lateral deformation function w(χ, t), which is an etended distributed state variable. The angle χ [,β] (where β =2(π α) and α =arcsin(a/r)) sweeps along the circumference of the tre starting from the trailing edge R and ending at the leading edge L of the contact patch. In this stud, we use the so-called brush tre model, which considers separated tre particles along the circumference of the tre. Here, the threads are modelled b means of continuousl distributed masses supported b springs. These inertial and elastic properties are characteried b ρa [kg/m] and k [N/m 2 ], respectivel. 2.1 Lateral deformation in the contact patch Due to the fact that we consider pure rolling, the tre particles attached to the ground have ero velocities. The position vector of a tre particle P 1 can be written in the (X, Y, Z) ground-fied coordinate sstem as vt (l )cosψ(t) q(, t)sinψ(t) R P1 = (l )sinψ(t)+q(, t)cosψ(t), (1) d for [ a, a]. The kinematic constraint of rolling means that dt R P 1 =, which leads to partial differential equation (PDE) characteriing the lateral deformations in the contact patch (see details [16]). Here we present its linearised form onl: q(, t) =vψ(t)+(l ) ψ(t)+q (, t)v, (2) for [ a, a]. In our stud, dot and prime refer to the partial derivatives with respect to time t and the space coordinates, respectivel. 2.2 Lateral deformation outside the contact patch The equations of motion of the tre particles outside the contact patch are determined with Hamilton s Principle. The Lagrangian can be written as the difference of the kinetic energ and the potential function 2

3 of the spring forces: L = 1 β ( ρav 2 2 P2 (χ, t) kw 2 (χ, t) ) Rdχ, (3) where v P2 (χ, t) refers to the velocit of the tre particle at the position χ: v cos ψ(t) w(χ, t) ψ(t) v cos(α + χ) v P2 (χ, t) = v sin ψ(t) (l + R sin(α + χ)) ψ(t)+ẇ(χ, t)+ v R w (χ, t). (4) v sin(α + χ) Here we assumed that the translational rate of change of the tre particles are equal to the towing speed v. This approimation is acceptable in case of small amplitude vibrations of the towed wheel. According to Hamilton s Principle, the action functional I = L dt is etremal for the realied motion of the sstem, i.e. δi =. This weak form of the equation of motion can be transformed into the form of differential equations b means of the Euler Lagrange equation. Considering (4) in (3), the action reads I = t1 and the Euler-Lagrange equation leads to t β df dw F t ẇ χ F (w, ẇ, w,χ,t)dχdt, (5) F w =. (6) Considering small amplitude vibrations, the tre deformation is characteried b the linearised partial differential equation: ẅ(χ, t)+ 2v R ẇ (χ, t)+ v2 R 2 w (χ, t)+ω 2 c w(χ, t) =(l + R sin(α + χ)) ψ(t)+2v ψ(t) cos(α+χ), (7) for χ [,β]. The natural angular frequenc ω c = k/ρa represents the free lateral vibrations of the tre brush elements. 2.3 Equation of motion of the caster-wheel sstem The equation of motion of the caster-wheel rigid-bod sstem can be derived with the help of Newton s Law: a β J A ψ(t) = k (l )q(, t)d k (l + R cos(α + χ))w(χ, t) Rdχ, (8) a where J A is the mass moment of inertia of the caster-wheel sstem with respect to the ais at the king pin A. The equation of motion (8) is coupled to (2) and (7). The boundar conditions (BCs) of the PDEs are: q(a, t) =w(β,t), w(,t)=q( a, t), w (,t)= Rq ( a, t). (9) The first two BCs ensure the continuit of the lateral deformation at the leading edge L and at the trailing edge R, respectivel. The third one (no kink at R) refers to d dt w(,t)= d dtq( a, t), which describes the initial lateral speed of the deformation waves that propagate along the circumference of the tre outside the contact patch. 3

4 3 TIME DELAYED TYRE MODEL 3.1 Travelling wave in the contact patch The memor effect of the tre/ground contact patch is known since von Schlippe alread identified this propert of tres in [6]. In case of rolling, the tre particles stick to the ground and keep their positions during the contact. Consequentl, their positions at the time instant t were determined when the reached the ground at the leading edge at the earlier time instant t τ 1 (). The time dela τ 1 () characteries the time needed for a tre particle to travel backwards in the contact patch from the leading edge L to its actual position (see Figure 1). Thus, the travelling wave solution of the PDE (2) can be determined in the form: q(, t) =(l a + vτ 1 ())ψ(t) (l a)ψ(t τ 1 ()) + q(a, t τ 1 ()) (1) for [ a, a]. In this linear approimation, the time dela has a phsicall obvious form: τ 1 () = a v. (11) 3.2 Travelling wave solution outside the contact patch Since the translational speed of tre particles along the circumference of tre is approimatel constant when the vibrations are small, a traveling wave solution can also be composed for (7). The application of Duhamel s integral formula leads to w(χ, t) =w(,t τ 2 (χ)) cos(ω c τ 2 (χ)) + 1 d ω c dt w(,t τ 2(χ)) sin(ω c τ 2 (χ)) + 1 τ2(χ) l + R sin α + v )) ω c R ϑ ψ(t τ2 (χ)+ϑ)sin(ω c (τ 2 (χ) ϑ))dϑ + 1 ω c τ2(χ) where the time dela is 2v ψ(t τ 2 (χ)+ϑ)cos (α + v ) R ϑ sin(ω c (τ 2 (χ) ϑ))dϑ, (12) τ 2 (χ) = Rχ v. (13) 3.3 Overall description of the lateral deformations Using the second and the third BCs of (9), the travelling wave solution (12) can be composed as a function of the trailing edge deformation of the contact patch: w(χ, t) =q( a, t τ 2 (χ)) cos(ω c τ 2 (χ)) + 1 d ω c dt q( a, t τ 2(χ)) sin(ω c τ 2 (χ)) + 1 τ2(χ) l + R sin α + v )) ω c R ϑ ψ(t τ2 (χ)+ϑ)sin(ω c (τ 2 (χ) ϑ))dϑ + 1 ω c τ2 (χ) 2v ψ(t τ 2 (χ)+ϑ)cos (α + v ) R ϑ sin(ω c (τ 2 (χ) ϑ))dϑ. Using the travelling wave solution (1), the trailing edge deformation can be described: q( a, t τ 2 (χ)) = (l + a)ψ(t τ 2 (χ)) (l a)ψ(t τ 2 (χ) T 1 ) + q(a, t τ 2 (χ) T 1 ), (14) (15) where T 1 = τ 1 ( a) 2a v (16) 4

5 is the time interval that is needed for a tre particle to travel in the contact patch from the leading edge L to the trailing edge R. The material time derivative at the leading edge can be calculated via (2): where d dt q( a, t τ 2(χ)) = vψ(t τ 2 (χ)) + (l + a) ψ(t τ 2 (χ)) (17) After the substitution of (15) and (17) into (14), the first BC of (9) leads to q(a, t) =((l + a)ψ(t T 2 ) (l a)ψ(t T 1 T 2 )) cos(ω c T 2 ) + q(a, t T 1 T 2 )cos(ω c T 2 )+ 1 (vψ(t T 2 )+(l + a) ω ψ(t ) T 2 ) sin(ω c T 2 ) c + 1 l + R sin α + v )) ω c R ϑ ψ(t T2 + ϑ)sin(ω c (T 2 ϑ))dϑ + 1 ω c 2v ψ(t T 2 + ϑ)cos (α + v ) R ϑ sin(ω c (T 2 ϑ))dϑ, T 2 = τ 2 (β) Rβ 2R(π α) (19) v v refers to the time needed for a tre particle to travel along the circumference of the tre from the trailing edge R to the leading edge L. Using (1) and (14) with (15) in (8), one can eliminate the deformation functions q(, t) and w(χ, t). As a consequence, the corresponding PDEs (2) and (7) can be omitted, and the governing equations of the sstem reduce to: ( J A ψ(t)+2ak l 2 + a2 3 ω c ω c ω c ( l + R cos ) ψ(t) =kv ( α + v R τ 2 T1 (l a + vτ 1 )((l a)ψ(t τ 1 ) q(a, t τ 1 )) dτ 1 )) ((l + a)ψ(t τ 2 ) (l a)ψ(t T 1 τ 2 )) cos(ω c τ 2 )dτ 2 l + R cos α + v )) R τ 2 q(a, t T 1 τ 2 )cos(ω c τ 2 )dτ 2 l + R cos α + v )) ( R τ 2 vψ(t τ 2 )+(l + a) ψ(t ) τ 2 ) sin(ω c τ 2 )dτ 2 l + R cos α + v )) τ 2 R τ 2 l + R cos α + v R τ 2 )) τ 2 l + R sin α + v )) R ϑ ψ(t τ2 + ϑ)sin(ω c (τ 2 ϑ))dϑ dτ 2 2v ψ(t τ 2 + ϑ)cos (α + v ) R ϑ sin(ω c (τ 2 ϑ))dϑ dτ 2, which is coupled to (18). It can be seen that the substitution of the leading edge lateral deformation q(a, t) leads to a neutral tpe distributed dela differential equation. 4 STABILITY ANALYSIS One can substitute the eponential trial solutions ψ(t) = P e λt and q(a, t) = Qe λt into the linear governing equations (18) and (2) and can calculate the characteristic function D(λ) of the sstem. Due to its compleit, we do not present the formula of the characteristic function here. Self-ecited vibrations of the caster-wheel sstem can be identified where the rectilinear motion becomes unstable. According to the D-subdivision method, the stabilit boundaries can be determined if we substitute λ =iωinto the characteristic function with the real positive angular frequenc ω and we separate its real and imaginar parts to be eros. Unfortunatel, the boundaries cannot be calculated in closed form but b means of numerical methods, one can construct stabilit charts. For eample, using the multi-dimensional bisection method [17], stabilit boundaries can be determined in the multidimensional parameter space of the model. Figure 2 shows some of these boundaries in the v l plane for the parameters a =.4 m, R =.2m, k =6kN/m 2, ρa =.4kg/m and J A =.16 kg/m 2, which are based on the laborator (18) (2) 5

6 4th International Tre Colloquium l [m] v [m/s] 2 Figure 2: Stabilit boundaries in the v l parameter plane v [m/s] 4 15 f [H] 2 2 Figure 3: The variation of the characteristic roots versus the towing speed eperiments in [14]. The boundaries characterie the domains, where different kinds of self-ecited vibrations occur with different frequencies. As it can be observed, lot of boundaries appear in the investigated parameter domain. Let us consider the caster length l =.5 m, and let us show some properties of the emerging selfecited vibrations against the longitudinal speed v. The characteristic roots having positive real parts are shown in Figure 3 for the speed range m/s. As it can be observed, the self-ecited vibrations emerge in the whole speed range. The different projections of this chart are plotted in Figure 4. Both ver low (1 H) and higher (8 H) vibration frequencies can be identified in the figure. Of course, our model does not consider the damping, which can strongl influence these result. The full stabilit analsis in the presence of damping will be the task of future research. 5 TYRE DEFORMATIONS To check the theoretical results and to obtain information about the tre deformations, numerical simulation was carried out. The original IDE-PDEs sstem (8), (2) and (7) were discretied b means of simple finite difference method. The number of spatial mesh points in the contact patch was chosen to 2 while outside the contact patch it was 3. The initial conditions for the numerical simulations were set to ero initial caster angle and ero lateral deformations, but non-ero initial angular speed of the 6

7 4th International Tre Colloquium m/s 18. m/s 11. m/s 6 f [H] 17. m/s v [m/s] m/s m/s 11. m/s 17. m/s v [m/s] Figure 4: The real parts of the characteristic roots and corresponding vibration frequencies against the towing speed v =11. m/s v =17. m/s v =18. m/s v =18.4 m/s Figure 5: Tre lateral deformation at different towing speeds caster was considered. Figure 5 shows the representative tre deformation at different towing speeds, which speeds are also marked in Figure 4. Obviousl, as the wheel speed increases the number of the waves decreases outside the contact patch, compare the deformations at the speeds 11. and 17. m/s. It is also worth to observe that small variation in the towing speed can strongl influence the vibration frequencies and the corresponding tre deformations, see the cases that relate to the towing speeds 18. and 18.4 m/s. 6 CONCLUSION A mechanical model was constructed to investigate the self-ecited vibration of towed tres. The delaed tre/ground contact model was implemented. Moreover, the lateral tre deformation was also considered outside the contact patch b means of the so-called brush model that also takes into account the mass of the tre particles. The governing equations of the sstem were determined as an IDE-PDE sstem and were transformed into the form of dela differential equations of the neutral tpe. It was shown that the interaction of the contact patch and the non-contacting tre particles lead to tre vibrations with various frequencies in a wide towing speed range. Although, damping was not considered in our stud, some of the detected 7

8 vibrations can be the origin of noise and heat generation in practice even in the presence of damping. Acknowledgement This research was partl supported b the János Bolai Research Scholarship of the Hungarian Academ of Sciences and b the Hungarian National Science Foundation under grant no. OTKA PD REFERENCES [1] V. Cossalter Motorccle Dnamics, Lulu Editor, 26. [2] R.S. Sharp, S. Evangelou, and D.J.N. Limebeer, Advances in the Modelling of Motorccle Dnamics, Multibod Sstem Dnamics 12 (24), pp [3] I.J.M. Besselink, Shimm of Aircraft Main Landing Gears, Technical Universit of Delft, The Netherlands, 2. [4] N. Terkovics, S. Neild, M. Lowenberg, and B. Krauskpof, Bifurcation Analsis of a Coupled Nose Landing Gear-Fuselage Sstem, in Proceedings of AIAA 212 paper No. AIAA , Minneapolis, Minnesote, USA, 212, pp [5] C. Howcroft, B. Krauskopf, M. Lowenberg, and S. Neild, Effects of Freepla on Aircraft Main Landing Gear Stabilit, in Proceedings of AIAA 212 paper No. AIAA , Minneapolis, Minnesote, USA, 212, pp [6] B. Schlippe and R. Dietrich, Das Flattern Eines Bepneuten Rades (Shimming of a pneumatic wheel), in Bericht 14 der Lilienthal-Gesellschaft für Luftfahrtforschung English translation is available in NACA Technical Memorandum 1365, pages , , 1954, 1941, pp , [7] L. Segel, Force and moment response of pneumatic tires to lateral motion inputs, Journal of Engineering for Industr, Transactions of the ASME 88B (1966), pp [8] H.B. Pacejka, The Wheel Shimm Phenomenon, Technical Universit of Delft, The Netherlands, [9] G. Stépán, Dela, nonlinear oscillations and shimming wheels, inproceedings of Smposium CHAOS 97 Kluwer Ac. Publ., Dordrecht, Ithaca, N.Y., 1998, pp [1] J. Cesbron, F. Anfosso-Lédée, D. Duhamel, H.P. Yin, and D.L. Houédec, Eperimental stud of tre/road contact forces in rolling conditions for noise prediction, Journal of Sound and Vibration 32 (29), pp [11] P.B.U. Andersson and W. Kropp, Rapid tre/road separation: An eperimental stud of adherence forces and noise generation, Wear 266 (29), pp [12] D. O Bo and A. Dowling, Tre/road interaction noise Numerical noise prediction of a patterned tre on a rough road surface, Journal of Sound and Vibration 323 (29), pp [13] R.J. Pinnington, Tre-road contact using a particle-envelope surface model, Journal of Sound and Vibration 332 (213), pp [14] D. Takács and G. Stépán, Micro-shimm of towed structures in eperimentall uncharted unstable parameter domain, Vehicle Sstem Dnamics 5 (212), pp [15] D. Takács and G. Stépán, Contact patch memor of tres leading to lateral vibrations of fourwheeled vehicles, Philosophical Transactions of the Roal Societ A: Mathematical, Phsical and Engineering Sciences 371 (213). [16] D. Takács, G. Oros, and G. Stépán, Dela effects in shimm dnamics of wheels with stretched string-like tres, European Journal of Mechanics A/Solids 28 (29), pp [17] D. Bachrath and G. Stépán, Bisection method in higher dimensions and the efficienc number, Periodica Poltechnica 56 (212), pp