F. C. Moon (ed.), Applications of Nonlinear and Chaotic Dynamics in Mechanics, Kluwer Academic Publisher, Dordrecht, 1998, pp

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1 F. C. Moon (ed.), Applications of Nonlinear and Chaotic Dynamics in Mechanics, Kluwer Academic Publisher, Dordrecht, 1998, pp DELAY, NONLINEAR OSCILLATIONS AND SHIMMYING WHEELS G. STÉPÁN Department of Applied Mechanics Technical University of Budapest, H-1521 Budapest, Hungary 1. Introduction The lateral vibration of towed wheels the so-called shimmy is a well-known classical example of self-excited nonlinear oscillations. The developments of more and more sophisticated wheel suspension systems and tires in vehicles, as well as the appearance of semi-trailers, caravans, high speed motorcycles and other large towed vehicle systems require a continuous analysis of this phenomenon. The paper calls the attention for the complexity of the modeling issues and for that of the nonlinear dynamics of the phenomenon strictly from the view-point of the wheel only. This means that the complex dynamics of the vehicle itself is simplified and only those key elements of the mechanical models are considered and analyzed which are related to the wheel ground connection. The resulting finite and infinite dimensional dynamical systems exhibit a great variety of nonlinear oscillations such as stable and unstable periodic and quasi-periodic oscillations, chaotic and transient chaotic behavior. The infinite dimensional phase space is due to the presence of a special kind of memory, or time delay in the system which is related to the deformed tire contact surface keeping a record of the past motion of the wheel. The common meaning of the word shimmy is a dance back to the early thirties of this century. This shows that the phenomenon named by the same word, shimmy, has been studied for several decades already (the earliest scientific study the author could find is that of Schlippe et al. (1941). However, the phenomenon has not been fully explored. This is due to two important reasons. One problem is caused by the fact that the vehicle itself is a complex dynamical system having important nonlinearities and many degrees of freedom (DOF) serving several lowfrequency linear vibration modes which may all be important components of the several types of dynamical behavior at different running speeds and conditions. This part of the problem is somewhat resolved by the appearance of multi-body dynamics and good commercial computer codes. The other part of the problem is related to the wheel-ground contact. One of the most challenging tasks of continuum mechanics is the contact problems which need long computation time even

2 in stationary cases and when the most advanced finite element programs are used (see Kalker (1990) for railway wheels and Böhm et al. (1989) for tires). The dynamic contact problems need special codes, tremendous computational effort and still, there are no analytical examples and results to check the calculations which are available for a couple of parameter settings only. In this study, the simplest possible mechanical structures are considered, with the lowest number of mechanical DOF which still exhibit shimmy motion. On the other hand, several wheel models are considered, and these models are analyzed from nonlinear vibration view-point. The models have 4 important elements: the vehicle, the king pin, the caster, and the wheel. In accordance with the above goals, the vehicle is always modeled as a rigid body running straight with a constant speed v. Clearly, the towed wheel may have a straight stationary running if all the mechanical parts are rigid. Shimmy may occur if at least one of the king pin, the caster, and the wheel is elastic. Two groups of the basic models can be classified this way. If the wheel is rigid then the king pin (and/or the caster) is considered to be elastic. These non-holonomic models may describe realistic systems at low speed, like the trolleys at supermarkets or airports (see Plaut, 1996). Also, when long semi-trailers are modeled, an elastic king-pin may have an important effect on the dynamics. The often chaotic and/or transient chaotic dance of these wheels can be experienced on these simple trolleys where the king pin is loose (see Stépán, 1991). The control of this motion has been studied recently via feedback linearization by Goodwine et al. (1997). If the wheel is elastic then shimmy may occur even with rigid caster and king pin. These models are realistic for pneumatic tires in the case of short caster length. However, the description of the contact of the elastic wheel and rigid ground is much more complicated than the case of the elastic king pin. For engineering applications in the middle range of the towing speed, the quasi-stationary idea of the lateral creep force and moment helps to simplify this problem and to achieve reasonable agreement with experiments (see Pacejka, 1988). These models use a holonomic mechanical model. As successful studies, the paper of Scheidl et al. (1985) can be mentioned here for tractor-semi-trailer systems, the analysis of [Sharp] for motorcycles, and that of Fratile et al. (1995) for caravans. Although, the creep force idea can strongly be criticized from nonlinear dynamics point of view, the conclusions of these studies are really useful and important: they give advice how to pack in a caravan not to increase its moment of inertia about its vertical axis, etc. However, the most complex phenomena which can be studied by these models are the different structures of stable and unstable limit cycles only. In the case of the elastic wheel models of airplane front landing gears, Schlippe (1941) realized in the late thirties already the presence of a past effect in the system. The wheel contact region behaves like a memory in the system, it records the past position and orientation of the shimmying wheel. First, this memory effect was modeled by using the assumption that the contact line is straight. This resulted a delay differential equation with discrete delays. Since the theory of delay differential equations was missing from the mathematical literature at that

3 Figure 1: Model of a towed wheel. time, the mathematical analysis suffered of several simplifications. In the final section of this paper, those models are established and analyzed which describe this memory effect more precisely, using the contact line deformation as a distributed parameter in the system. The resulting coupled partial and integral differential equation is transformed into a general functional differential equation which describes the memory effect. Stability analysis and nonlinear oscillations are presented based on the existing mathematical theory of these infinite dimensional systems. 2. Mechanical model The mechanical model shown in Figure 1 has 4 parts. The rigid vehicle runs with a constant velocity v in the ξ direction providing a non-stationary geometric constraint for the structure. Note that the system is not conservative due to this constraint. The king pin at the point A has its axis in the vertical direction z, and it is attached to the vehicle via a spring of overall stiffness s. The qualitative effect of the viscous damping of this support will also be mentioned, but it is neglected in the analytical calculations. The rigid caster of mass m c has a length l. This length and the towing speed v will serve as the two bifurcation parameters. The homogeneous wheel of mass m w has a radius R. Its central part is assumed to be rigid. The tire is elastic, modeled as a thin stretched string (see Pacejka, 1988).

4 This string is characterized by the relaxation length σ which depends on the tension in the string and the specific lateral stiffness c of the tire. The circumferential elasticity of the tire is neglected. The half-contact-length is denoted by a, and the wheel center has a constant distance b = R 2 a 2 from the ground. If the above geometrical constraints are considered only, the following general coordinates are chosen: p is the lateral displacement of the king pin, ψ is the angle of rotation of the caster, ϕ is that of the wheel, and q(x) is the lateral displacement of the tire contact points. The scalar coordinates p, ψ, ϕ as well as the function q(x) depend on the time t, of course. The lateral shape q of the deformed tire is described in the moving coordinate system (x, y, z) attached to the caster. The wheel can roll, slide, or creep. It rolls if all the wheel-ground contact points P have zero velocity, it slides if all of them has non-zero velocity. We speak about creep if some parts of the contact line q stick to the ground, some others slip. The static and dynamic coefficients µ s, µ d of friction are used to describe constraints and friction forces between the wheel and the ground. 3. Rigid tire If the tire has no lateral elasticity, that is c, the contact region is reduced to a single contact point P O, q(0, t) 0, a = 0, σ = 0. Then the system is non-holonomic with the kinematical constraint v P = 0. This results a system of two first order scalar ordinary differential equations for the general coordinates p, ψ, ϕ in accordance with the formula ( ) ( ) vp x v cos ψ + ṗ sin ψ R ϕ v P = = v sin ψ + q cos ψ l ψ. (1) v P y Thus, the system can be described by a single quasi-velocity which can be the angular velocity Ω = ψ of the caster. The Appell-Gibbs equation coupled to the above definition of the quasi-velocity and the kinematical constraints (1) provides the four dimensional system of first order scalar equations ψ = Ω, ( Ω = v l 1 cos 2 ψ mw 2m c ( ) tan2 ψ ) tan 2 ψ Ω + ( s lm c p mw ) tan ψ 2m c cos ψ Ω2 ( cos ψ + m w R 2 4m c l cos ψ + 6 tan 2 ψ cos ψ 2 ), ṗ = v tan ψ + Ωl cos ψ, v + Ωl sin ψ ϕ = R cos ψ. (2) Since ϕ is a cyclic coordinate, it does not appear in the first 3 equations, and the trajectories of the rolling system can uniquely be represented in the three dimensional subspace ψ, ψ, p of the caster angle and angular velocity and the king pin lateral displacement.

5 Figure 2: Stability charts and bifurcation diagrams. The linear stability analysis of the straight rolling (the trivial solution of (2)) gives a critical value of the caster length: 3mw l > l cr = R. (3) 2m c Figure 2 shows the corresponding stability chart. The stability limit at zero viscous damping is represented by the line at κ = 0. In this case, the speed of towing does not appear in the stability condition. However, it does appear when κ > 0. The corresponding stability limit is the result of a lengthy calculation when the viscous damping is included in the mathematical model. Thus, short caster length and increasing speed destabilizes the towed wheel. At the critical value of the caster length, the Hopf bifurcation calculation (see details in Stépán, 1991) based on the third degree truncation of (2) presents a subcritical case, that is there exists an unstable limit cycle around the asymptotically stable zero solution for some values of l l cr > 0. This is also presented by the bifurcation diagram of Figure 2, where A is the amplitude of the oscillation. The numerical simulation of (2) does not show any attractor outside this limit cycle. In fact, the trajectories run to some singular surfaces at ψ = ±π/2. Clearly, the real structure will not roll anymore when the vibrations are so violent. The wheel is going to slide when the static friction is not able to provide the necessary constraining force for rolling. Consequently, the above equations (2) of rolling describe the system only for that region of the phase space where the friction force F at the contact point P satisfies the condition F µ s N = µ s (m c /2 + m w )g. (4) This condition can be represented in the phase space as codimension 1 surfaces.

6 Within these surfaces the trajectories are described by (2). When the trajectories hit these surfaces, they switch to the dynamics of the sliding wheel. The corresponding equations of motion are the Lagrangian equations of the holonomic system where the kinematical constraint v P = 0 is released. The 3 dimensional system of the second order equations in the 6 dimensional phase space of the general coordinates and velocities assumes the form: ( 1 3 m c + ( ) ) 1 + R2 4l 2 mw l 2 ( 1 2 m ) c + m w l cos ψ 0 ψ ( 1 2 m ) c + m w l cos ψ mc + m w 0 p = m wr 2 ϕ v P x l v 2 P x +v2 P y ( 1 2 m ) c + m w l ψ 2 sin ψ sp v P x sin ψ+v P y cos ψ µ ) d(m c /2 + m w g (5) v P y v 2 P x +v2 P y R v 2 P x +v2 P y where the coordinates of the contact point velocity v P x,p y has to be substituted from (1). The numerical simulation of the equations (2) and (5) and the precise check of the changes between the two dynamics are difficult. The general methods of Glocker et al. (1995) are needed if the system is more complicated and consists of several bodies which may all slide or stick. The above case is somewhat simpler, although orbitally stable periodic motion as well as chaotic or transient chaotic motion can also be experienced outside the unstable limit cycle for different values of the parameters (see the bifurcation diagrams in Figure 2). Figure 3 explains a case when transient chaos occurs (for simulation results and parameter values see Stépán, 1991). The plane ψ, ψ represents the 4 dimensional phase space of rolling. The thick trajectories start running outwards around the unstable limit cycle presented by a thin ellipse in the figure. When the trajectory hits the condition of rolling calculated from (4) and approximated by two planes here at ± ψ sl, the wheel starts sliding. The speed v P of the contact point represents the next 2 dimension in the phase space of Figure 3. When the thick trajectory changes to a thin one and leaves the space of rolling, it follows the dynamics of (5) in the 6 dimensional phase space of sliding. This dynamics is strongly dissipative due to the sliding friction. The trajectories enter the space of rolling several times. When they enter it within the conditions of rolling, and outside the unstable limit cycle, they start spiraling outwards again, and the wheel shows a chaotic dance. This may disappear after a stochastically varying time of transient period when the trajectory arrives back to the space of rolling within the direct domain of attraction of the zero solution, and the straight running of the wheel is approached asymptotically. The above structure of the trajectories can also be followed by an approximate 1 dimensional Poincare section of the trajectories considered along the ψ axis. This

7 Figure 3: Phase space structure and discrete map. cartoon of the transient chaotic wheel motion is presented on the discrete map of Figure 3. Note, that the above described transient chaotic motion occurs for certain parameter values only, when the static friction is about 2 times greater than the dynamic one. In other cases chaotic motion or just simpler periodic motions may coexist with the asymptotically stable straight running and they are separated by the unstable periodic motion. 4. Elastic tire and creep force If the tire is elastic, there is a contact region of length 2a between the wheel and the ground (see Figure 4). In the meantime, the king pin is assumed to be rigid, that is s. It can be shown that there is always a region at the rear end E of the contact line where the contact points slide, although the contact points at the other part of the wheel may still stick to the ground. This phenomenon is called creep. In case of a constant (!) drift angle, the calculation of the resultant force and couple of the distributed lateral (partly dynamic and partly static) friction forces is presented by Pacejka (1988). Since the drift angle ψ is fixed in this calculation, v L = v, the shape of the contact line can analytically be determined, and it depends on the lateral deformation q 0 = q(a,.) of the leading edge L only. Thus, the lateral creep force F y and moment M Oz can be presented in the form { 22.2 ca 2 σ F y (q 0 ) = q 0, q 0 < q 0sl = σ ca µ 2 d N, (6) µ d N sign q 0, q 0 q 0sl M Oz (q 0 ) = { 0.16µd Na sin ( 69.8 ca2 1 σ µ d N q 0), q0 < q 0sl, (7) 0, q 0 q 0sl

8 Figure 4: Creep force and moment. where µ d N = µ d (m c /2 + m w )g is the overall friction force when the wheel fully slides, that is when q 0 > q 0sl. Now, this stationary creep force and moment is used instead of the real kinematic constraint along the contact line to describe the dynamics of the towed wheel. In the three dimensional phase space ψ, ψ, q 0 of the caster angle, angular velocity and the lateral displacement of the leading edge L, the equations of motion assume the simple form I A ψ = lfy (q 0 ) M Oz (q 0 ), (8) q 0 = v sin ψ + (l a) ψ v σ q 0 cos ψ, (9) where I A is the mass moment of inertia of the wheel and the caster with respect to the z axis at the king pin A. The first equation (8) describes a simple holonomic system, while the second equation (9) still preserves a kinematical constraint with respect to the leading edge L and also a no kink at L condition introduced by Pacejka (1988). Although the above equations are the results of a complicated simplification of the wheel-ground contact, the linear stability analysis of the zero solution of (8,9) with (6,7) gives a similar condition to that of the case of the rigid tire in (3): l > l cr = a + σ. (10) Moreover, the Hopf bifurcation analysis of the 3rd degree truncation of (8,9) with (6,7) presents a subcritical case again (for details see Stèpàn, 1992), thus both the stability chart and the bifurcation diagram of Figure 2 are valid for this model, too. However, there is no chaotic attractor or transient chaos outside the unstable limit cycle. An orbitally asymptotically stable limit cycle exists, however, when

9 the viscous damping at the tire-ground contact is also included in the model (see the κ > 0 case in the bifurcation diagram of Figure 2). In this case, the speed v of towing shows up in the linear stability condition, just as in the case of the rigid tire. As mentioned in the introduction, the inclusion of a stationary creep force in the dynamic equations simplifies the mathematical model tremendously (compare equations (2,5) to (8,9)), the nonlinearities are partly put into the nonlinear creep force and moment characteristics, but in the meantime, most parts of the complex dynamics are lost. Even some linear stability effects are missing in the above models. For example, experiments show that the stability of the straight rolling is regained sometimes with increasing speed, then lost again, etc. These issues are discussed in the subsequent section. 5. The memory effect of the elastic tire In this section, the condition of rolling is enforced for the whole contact line between the wheel and the ground. Although it is true for large values of static coefficient µ s of friction only, and never true in the vicinity of the rear edge E, this model still serves important and new nonlinear oscillation phenomena. The main advantage of this model is, that it takes into account the actual and dynamically varying shape q(x, t) of the contact region. To simplify the calculations, we restrict ourselves to the case of rigid carcass, and the elasticity of the tread elements are modeled only. This means that the relaxation length is negligible, that is σ = 0, and the lateral displacement q(a, t) of the leading edge L is also zero as follows from (9). The equations of motion with respect to the scalar ψ and the function q(x,.) assume the form a I A ψ(t) = c (l x)q(x, t)dx (11) a q(x, t) = v sin ψ(t) + (l x) ψ(t) + q (x, t) ( v cos ψ(t) q(x, t) ψ(t) ), (12) x [ a, a], t [t 0, ), and q(a, t) = 0. (13) where prime denotes the derivative with respect to the moving coordinate x attached to the caster. In these equations, the partial differential equation (PDE) (12) with the boundary condition (13) describes the kinematical constraint v P = 0 for any point P along the contact line in accordance with Figure 1. This system of integral and partial differential equations describes the dynamics in an infinite dimensional phase space. As mentioned in the Introduction, the contact line keeps a history of the wheel past positions and orientations while the contact points stick to the ground, and these past values determine the actual lateral force system which appear in the right hand side of the integral-differential equation (11). This can be interpreted mathematically in the following way. The PDE part (12) of the equations has a travelling wave-like solution. This can be constructed by giving the absolute coordinates (ξ, η) of the contact point

10 P as ξ(x, t) = vt (l x) cos ψ(t) q(x, t) sin ψ(t), η(x, t) = (l x) sin ψ(t) + q(x, t) cos ψ(t). Actually, the time derivatives of these equations also lead to the above PDE (12). This time, however, we are going to change the space variable x to the time delay τ which is the time needed for the leading edge L at x = a to travel backwards (relative to the caster) to the actual point P. This means that ξ(x, t) = ξ(a, t τ), η(x, t) = η(a, t τ). From the above expressions, simple algebra gives l x = vτ cos ψ(t) + (l a) cos(ψ(t) ψ(t τ)), (14) q(x, t) = vτ sin ψ(t) + (l a) sin(ψ(t) ψ(t τ)), (15) dx dτ = v cos ψ(t) + (l a) ψ(t τ) sin(ψ(t) ψ(t τ)). (16) Note that a linear approximation provides explicit expression for the connection of the delay and space coordinate in the physically obvious form τ a x v Introduce the dimensionless bifurcation parameters L and V for the caster length and the speed of towing by: L = l a,. T = 2a v, V = 1 αt, where α 2 = 2ac(l 2 + a 2 /3)/I A is the square of the natural angular frequency of the system when its speed is zero. With the help of formula (15), q(x, t) can be eliminated from (11), and the variable x of integration is changed to τ via formulae (14,16). The use of the dimensionless time t := t/t and the new dimensionless variable ϑ = τ/t of integration results in the 3rd degree truncated form of a nonlinear delay-differential equation (DDE) V 2 7 ψ(t) + ψ(t) 6 ψ3 (t) = L 1 L 2 + 1/3 ( 0 (L 1 2ϑ)ψ(t + ϑ)dϑ + f(ψ t ) + g(ψ t, ψ ) t ). (17) 1 where the odd functions f and g are calculated as functions of ψ t defined by

11 ω/α L 2 S S 1 P S V Figure 5: Stability chart and vibration frequencies. ψ t (ϑ) = ψ(t + ϑ), ϑ [ 1, 0] : f(ψ t ) = 5 2 ψ2 (t) 0 (L 1 2ϑ)ψ(t + ϑ)dϑ 1 +2ψ(t) 0 1 (L 1 ϑ)ψ2 (t + ϑ)dϑ (L ϑ)ψ3 (t + ϑ)dϑ, g(ψ t, ψ t ) = 1 2 ψ2 (t) 0 1 (L 1 2ϑ)2 ψ(t + ϑ)dϑ ψ(t) 0 1 (L 1 ϑ)(l 1 2ϑ)ψ(t + ϑ) ψ(t + ϑ)dϑ (L 1) 0 1 (L 1 2ϑ)ψ2 (t + ϑ) ψ(t + ϑ)dϑ. Its only unknown is the scalar ψ, but due to the presence of the integration of its past values on [t 1, t], the phase space is still infinite dimensional: it is the space

12 3 L 2 1 U κ = 0 κ = κ = κ = U V Figure 6: Stability charts with viscous damping. of the continuous functions ψ t on [ 1, 0]. For more details of the theory see e.g. Kuang (1993). The asymptotic stability of the zero solution of (17) is checked by means of the characteristic function associated with the linear part of (17) when the ψ(t) = Kexp (λt) trial solution is substituted there: D(λ) = V 2 λ L 1 L 2 + 1/3 (L 1 e λ λ 1 + e λ λ ) e λ λ 2 This characteristic function has infinitely many characteristic roots in accordance with the infinite dimensional nature of the problem. The necessary and sufficient condition of asymptotic stability in the linear systems is the same as in the case of ODEs, that is Reλ j < 0, j = 1, 2,.... The method presented in Theorem 2.19 in Stépán (1989) gives the stability chart of Figure 5 where a complex structure of stability domains appear. The stability limit at L = 1 l = a also appears just as in the case of the creep force model in (10) with the actual assumption σ = 0, but there are also many alternating borders of stability with Hopf bifurcations along them. The corresponding vibration frequencies are also presented above the stability chart. This model presents several (infinitely many) bifurcation points as the running speed v is decreased. However, if viscous damping is added to the model at the tire, the number of these bifurcation points become finite, at great values of the damping only a couple of them survives. This is shown in Figure 6 where the κ > 0 stability limits are the result of a long calculation not discussed here. There is also another important nonlinear oscillation phenomenon which appears in this delay model of the tire. At the parameter values V=0.1319, L= (18)

13 of the point P in the stability chart of Figure 5, quasiperiodic oscillations appear with 2 vibration frequencies: one is somewhat more than the natural frequency α of the structure at v = 0, the other one is at about the half of it. The codimension 1 and 2 Hopf bifurcation analysis requires a long and tedious algebraic calculation using the nonlinear functions f, g in (17), and it serves the structure of stable and unstable limit cycles and tori embedded in a 2 or 4 dimensional center manifold of the infinite dimensional phase space. Actually, the calculations are very similar to those of Campbell (1995) and Stépán (1995) for models in population dynamics and robotics. At this point we refer only to a simple and useful application of Theorem 3.24 in Stépàn (1989) or in Stépán (1997). The zero solution of the delay-differential equation 0 ẍ(t) + c 0 x(t) = c 1 x(t + θ)dθ (19) is asymptotically stable if and only if 1 0 < c 1 < c 0 and c 0 4j 2 π 2 j = 1, 2,.... (20) The linear part of the delay model (17) is the same as (19) when the caster length L. In this case c 0 = c 1 = 1/V 2. Clearly, the first stability condition of (20) is just violated: there exist a λ = 0 characteristic root in this case. The meaning of this root is obvious from the physical model, since the position of the wheel in η direction is neutral in the case of infinitely long caster. The second conditions of (20) give those critical speeds where self-excited vibrations are likely to occur at finite caster length which are still greater than a: V = 1 2π ( 0.159), 1 4π, 1 6π,.... This means that the stability limits of Figure 5 tend to these values as L, and the corresponding unstable regions shrink and then disappear, since the shimmying wheel is always stable for very long caster apart of some critical speeds. At these critical speeds one condition of the Hopf Bifurcation Theorem is not satisfied, namely the critical characteristic roots do not cross the imaginary axis with nonzero derivative with respect to the bifurcation parameter. 5. Conclusions Three basic models of the shimmying wheels have been considered and analyzed here from nonlinear oscillations point of view. The model of rigid tires may exhibit chaotic and transient chaotic motion. The creep force models describe the partially sliding and rolling elastic tire with the help of a stationary tire model fitted in the dynamic model. This model gives some simple and useful results for engineering design, but does not present most of the complex behavior of the shimmying wheels. The more precise modeling of the dynamic contact problem of the tire and the ground may include the memory effect of the tire. This effect appeared

14 in the early models of the shimmying wheel (see Schilppe, 1941). These models must be reconsidered again, and the recent developments of mathematics both in bifurcation theory and in functional analysis of infinite dimensional systems provide much better understanding of the complex motion of the elastic wheels of motorcycles, cars, trucks, airplanes. Acknowledgments This research was partially supported by the Danish Council for Scientific and Industrial Research Grant No M, the Hungarian Scientific Research Foundation OTKA Grant No , the US-Hungarian Science and Technology Program Grant No. 336, and the Greek-Hungarian Technology Program. The author thanks Professor Hans Pacejka and Professor Hans True for the useful discussions. References Böhm, F., Kollatz, M., (1989) Some Theoretical Models for Computation of Tire Nonuniformities, Fortschrittberichte VDI, , VDI Verlag, Berlin. Campbell, S. A., Bélair, J., Ohira, T., and Milton, J. (1995) Complex dynamics and multistability in a damped oscillator with delayed negative feedback, Journal of Dynamics and Differential Equations 7, Fratila, D., Darling, J. (1995) Improvements to car/caravan handling and high speed stability through computer simulation, in Proceedings of the ASME International Mechanical Engineering Congress (San Francisco, 1995) 95-WA/MET-9 pp Glocker, Ch., Pfeiffer, F. (1995) Multi impacts with friction in rigid multibody systems, Nonlinear Dynamics 7 (1995) Goodwine, B., Stépán, G. (1997) Controlling unstable rolling phenomena, Journal of Vibration and Control, accepted for publication. Guckenheimer, J. and Holmes, P. (1986) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag, New York. Kuang, Y. (1993) Delay Differential Equations, Academic Press, Boston. Kalker, J. J. (1990) Wheel-rail rolling contact, Wear 144, Pacejka, H. (1988) Modelling of the pneumatic tyre and its impact on vehicle dynamic behaviour, Research report i72b, TU Delft. Plaut, R. H. (1996) Rocking instability of a pulled suitcase with two wheels, Acta Mechanica 117, Scheidl, R., Stribersky, A., Troger, H., Zeman, K. (1985) Nonlinear stability behaviour of a tracktor-semitrailer in downhill motion, in Proceedings of the 9th IAVSD Symposium (Linköping, 1985) pp Schlippe, B. v., Dietrich, R. (1941) Shimmying of a pneumatic wheel, Lilienthal-Gesellschaft für Luftfahrtforschung, Bericht 140, translated for the AAF in 1947 by Meyer & Company, pp Sharp, R. S., Jones, C. J. (1980) A comparison of tyre represetations in a simple wheel shimmy problem, Vehicle System Dynamics 9, Stépán, G. (1989) Retarded Dynamical Systems, Longman, London. Stépán, G. (1991) Chaotic motion of wheels, Vehicle System Dynamics 20, Stépán, G. (1992) Nonlinear modelling of shimmying wheels, in P. Christiansen (ed.), Future Directions of Nonlinear Dynamics in Physical and Biological Systems, NATO ASI series, Plenum, New York, pp Stépán, G., Haller, G. (1995) Quasiperiodic oscillations in robot dynamics, Nonlinear

15 Dynamics 8, Stépán, G. (1997) Delay-differential equation models for machine tool chatter, F. C. Moon (ed.), Nonlinear Dynamics of Material Precessing and Manufacturing, Wiley, N.Y.