Proceedings of the XII International Symposium on Dynamic Problems of Mechanics (DINAME 7), P.S.Varoto and M.A.Trindade (editors), ABCM, Ilhabela, SP, Brazil, February 6 - March, 7 Wheel Dynamics Georg Rill 1 1 Uniersity of Applied Sciences Regensburg, Galgenberstr. 3, 9353 Regensburg, Germany Abstract: The dynamics of the wheel rotation is mainly influenced by the driing or the braking torque and the longitudinal tire force. Today different tire models are aailable. Structural tire models are ery complex. Here, the dynamics of the wheel rotation is dominated by the complex tire model. Within handling models, the steady state tire forces and torques are generated as functions of the longitudinal and lateral slip. Depending on the slip definition the dynamics of a wheel depends now on the ehicle elocity or the angular elocity of the wheel. Here, the wheel dynamics will become more and more stiff if the ehicle slows down or the wheel is close to a locking situation. This causes problems in drie away and braking to stand still maneuers. In this paper a simple quarter car model is used to study the wheel dynamics. Different model approaches and the performance of explicit and implicit ode solers are inestigated. Keywords: Vehicle Dynamics, Wheel, Tire, ode-soler INTRODUCTION Today, numerical simulation of system dynamics is a standard in the design of cars and trucks. Usually the ehicles are modeled by Multibody Systems (Rauh, 3 as well as Weinfurter et al., 4). A typical model for a passenger car consists of seeral subsystems, Fig. 1. Complex ehicle models are used to inestigate the ride comfort and the handling Drie train Drier Chassis Engine Suspension Wheel/Tire Steering System Figure 1 Vehicle modeled by subsystems performance as well as to generate load data for life time prediction analysis. Additionally to these off-line simulation tasks real-time applications, typically used for the design of ehicle control systems and the test of electronic control units, become more and more popular. In order to achiee real time performance either the model complexity has to be reduced (Pankiewicz, 3) or sophisticated modeling techniques and appropriate ode-solers (Rill, 6c) hae to be used. Among many other problems (Rill and Chucholowski, 5 as well as Rill 6a) the wheel dynamics becomes critical in particular in drie away and braking to stand still maneuers. The dynamics of the wheel rotation is mainly influenced by the driing or the braking torque and the longitudinal tire force. Usually the driing torque, if present, is transmitted ia the driing shaft to the wheel. The torsional stiffness of the driing shaft proides a simple torque-based interface between the wheel rotation and a separate drie train model. This interface is moderately stiff and can therefore be mastered by standard ode-solers. The dynamics of a braked wheel is quite delicate, because a wheel may lock in an instant. Here, either a soft braking torque model or an implicit ode soler is needed. SIMPLE VEHICLE MODEL Equations of Motion In order to focus on the dynamics of the wheel motion a quite simple ehicle model will be used, Fig.. The linear momentum for the chassis and the angular momentum for the wheel result in m = F x, (1)
Wheel Dynamics Ω F x m Θ r T Figure Simple chassis/wheel model Θ Ω = T r F x, () where m is the corresponding chassis mass, Θ and r denote the inertia and the radius of the wheel. Within this simple model the wheel suspension and the tire deflection was not taken into account. Finally, T describes the driing or braking torque. Longitudinal Tire Force Today, different tire models are aailable (Lugner and Plöchl, 6). Structural tire models are ery complex. They are computer time consuming and they need a lot a data. Usually, they are used for stochastic ehicle ibrations occurring during rough road rides and causing strength-releant component loads. Here, the dynamics of the wheel rotation is dominated by the complex tire model. Handling models are characterized by a useful compromise between user-friendliness, model-complexity and efficiency in computation time on the one hand, and precision in representation on the other hand (Hirschberg et al., ). Here, the steady state tire forces and torques are generated as functions of the longitudinal and lateral slip. The longitudinal force F x can be described as a function of the longitudinal slip s x, Fig. 3. The longitudinal 4 Longitudinal force F x [N] 3 1-1 - -3-4 -4-3 - -1 1 3 4 Longitudinal slip s x [] Figure 3 Longitudinal force characteristics of a typical passenger car tire slip is defined by s x = r Ω, (3) r Ω where r seres in this simple approach also for the dynamic rolling radius. Linearized Equations of Motion In normal driing situations the longitudinal slip is ery small and the wheel is close to the rolling condition Ω r. (4) Hence, disturbances of a normal driing situation with = and Ω = /r may be described by = + (5) and Ω = + Ω. (6) r For small disturbances and r Ω the longitudinal slip can be simplified to s x = + r Ω + r Ω r Ω. (7)
G. Rill But, for small slip alues the steady state force characteristics can be approximated by a linear function where df describes the initial inclination of the longitudinal tire characteristics F x (s x ). and F x df s x, (8) Assuming a constant driing elocity = const the equations of motion can be written as = r Ω } {{ } ẋ = 1 m Ω = T Θ r Θ This set of linear differential equations can now be arranged in matrix form df 1 df 1 m m df r df r r Ω Θ Θ df (r Ω ) (9) df (r Ω ). (1) } {{ } A }{{} x + T Θ, (11) where the disturbances in the ehicle elocity and the circumferential elocity of the wheel r Ω were used as state ariables. Eigenalues The solution of the homogenous state equation ẋ = Ax is gien by x(t) = x e λt, where the eigen-alues λ and eigenectors x are defined by (A λi)x =. (1) Non-triial solutions x are possible if det A λi = or ( df )( 1 m λ df r ) ( )( Θ λ df 1 df r ) = (13) m Θ will hold. The resulting quadratic equation has the solution λ 1 = and λ = df ( r Θ + 1 m ). (14) The second eigenalue and hence the dynamic of the system depends on the driing elocity. For small driing elocities the system becomes ery stiff. Hence, numerical integration algorithms based on explicit formulas must reduce the step size with a dropping driing elocity in order to maintain the stability. NUMERICAL INTEGRATION MatLab-Function The equations of motion (1) and () and a simple nonlinear longitudinal force characteristics are coded in the following MatLab function function xp=f(t,x) simple ehicle model including wheel dynamics data global mass Theta r df Fxmax Torque states =x(1); ehicle elocity o=x(); wheel ang. el. longitudinal slip (nonlinear) sx = (r*o-)/abs(r*o);
Wheel Dynamics longitudinal force (simple nonlinear function) fx = df*sx; if fx > +Fxmax, fx=+fxmax; end if fx < -Fxmax, fx=-fxmax; end equations of motion p = fx/mass; linear momentum op = (Torque-r*fx)/Theta; angular momentum state deriaties xp = [ p; op ]; as a system of first order differential equations. Vehicle Data Typical data for a passenger car are proided in Tab. 1. Chassis mass (quarter car) m = 4 kg Inertia of wheel Θ = 1. kgm Wheel radius r =.3 m Initial incl. long. force char. df = 1 N/ Maximum long. force Fx max = 3 N Driing torque T = 1 Nm Table 1 Typical data for passenger car Explicit Euler Integration The explicit Euler integration x(t + h) = x(t) + h f (t,x) (15) is numerically stable as long as the integration step size h and the eigenalues λ of the dynamic system will satisfy Applied to Eq. (14) one gets or h df 1 + hλ < 1. (16) ( r h df Θ + 1 m ( r ) (17) Θ + 1 m ). (18) The step size of a stable explicit Euler solution depends on the ehicle data and the driing speed. Hence, in real-time applications where a constant step size must be used driing manoeuer with cannot be handled by an explicit Euler algorithm. Here, for an integration step size of h =.5 ms the explicit Euler algorithm is stable as long as will hold. > 1.9375 m/s (19) The Integration was started at t = with (t = ) = = ms and Ω(t = ) = /r. This describes a ehicle which is rolling backwards. The constant driing torque of T = 1 Nm will then accelerate the ehicle forward. The results are plotted in Fig. 4. As predicted the explicit Euler algorithm becomes numerically unstable if the absolute alue of the elocity drops below a critical alue. Here, the numerical instability produces non-physical oscillations in the circumferential wheel elocity at r Ω 1.8 m/s. On the basis of the linearized equations the critical elocity was predicted in Eq. 19 with a slightly larger alue.
G. Rill -1.6-1.7-1.8-1.9 -. -.1 r Ω -. -.3 -.4 -.5..4.6.8 1. time [s] Figure 4 Results of the explicit Euler algorithm Comparison to Implicit Formula A comparison of the MatLab solers ode3 (explicit Runge-Kutta (,3) pair of Bogacki and Shampine) with ode3s (a modified implicit Rosenbrock formula of order ) is shown in Fig. 5. For both formulas the defaults error alues of 1 Step Size [s] 1-1 1 - implicit (ode3s) 1-3 explicit (ode3) 1-4 1-5 -.5 -. -1.5-1. -.5 Vehicle elocity [m/s] Figure 5 Step sizes for explicit and implicit soler RelTol = 1 3 and AbsTol = 1 6 were applied. In order to maintain stability the explicit Runge-Kutta algorithm has to reduce the step size nearly proportional to the absolut alue of the ehicle elocity. Here, the implicit algorithm is approximately 5 times faster than the explicit one. Een at small absolute elocities it runs with considerable large step sizes. Modified Slip For a locked wheel or at stand still when Ω = will hold, the longitudinal slip in Eq. (3) is no longer defined. A small modification in the slip definition s x = r Ω, () r Ω + num where a small but finite elocity num > was added to denominator aoids this problem. Now, the explicit Euler algorithm is stable if + num > h ( r df Θ + 1 ). (1) m
Wheel Dynamics will hold. Hence, by setting num = m/s a stable transition from negatie to positie elocities is possible here, Fig. 6. The simple explicit Euler algorithm can master drie away and stand still maneuers. But, the modified slip definition 1..5 -.5-1. -1.5 -. -.5 r Ω.5 1. 1.5..5 3. time [s] 3.5 4. Longitudinal slip [-].1.8.6.4 physical. modified 1 3 4 Figure 6 Stable solution with explicit Euler algorithm for num = m/s slows down the wheel dynamics especially at low wheel angular elocities and may hence produce results with poor accuracy. Figure 6 shows that the longitudinal slip based on physics will tend to infinity at rω whereas the modified slip stay constant thus generating a constant longitudinal force which correspond to the constant driing torque. Howeer, in normal driing situations, where r Ω num holds, the difference between the physically based slip and the modified slip will hardly be noticeable. In general, an implicit algorithm should be used to integrate the wheel rotation. Then, een the dynamics of a braked wheel can be mastered without problems (Rill, 6a). DYNAMIC TIRE FORCES Standard Approach According to an der Jagt () the tire forces show a dynamic behaior which can be approximated by a first order differential equation T x Ḟx D = F x Fx D, () where the time constant T x is deried from the relaxation length, T x = Using Eqs. (3), (8) and (3) the differential equation () reads as which, finally results in r Ω. (3) r Ω r Ω ḞD x = df Fx D (4) r Ω Ḟ D x = df For Ω = this equation is still defined; it simplifies to (r Ω ) r Ω Fx D. (5) Ḟ D x = df. (6) Hence, the longitudinal force F D x is increased or decreased according to the magnitude and sign of ehicle elocity. Wheel/Tire Dynamics The dynamic tire force F D x instead of the steady state alue F x must be used in the equations of motion (1) and () now. According to Eqs. (4) to (8) the equations of motion can be linearized. One gets m = F D x, Θ Ω = T r F D x, Ḟ D x = df (r Ω ) F D x, (7)
G. Rill where the term r Ω in the denominator of Eq. (3) was replaced by its linearized alue. Arranged in matrix form they read as 1 m r Ω = r Θ r Ω + T, (8) Θ Ḟx D df df Fx D l }{{} x l }{{ x }}{{} ẋ A x Eigenalues The eigenalues of the corresponding system matrix A are now gien by λ 1 = and λ /3 = ( ) ( ) df 1 ± i m + r. (9) Θ For = the eigenalues λ /3 are purely imaginary. Hence, at stand still the wheel and the longitudinal tire force will perform undamped motions. Figure 7 shows the eigenalues calculated with the ehicle data gien in Tab. 1 and a relaxation length of =.7 m which is a typical alue for a normal passenger car tire. The stability region of the explicit Euler algorithm is a circle with radius R = 1/h touching the origin from the left. Again, to achiee stable solutions with the explicit Euler algorithm a ery small step size must be used. Stability region of explicit Euler algorithm =1 1/h Im(λ) = λ 1 4 4 - -15-1 -5 λ 1 Re(λ) =1 λ 3 = Figure 7 Eigenalues of ehicle with dynamic tire force and stability region of the explicit Euler algorithm Simulation Results For a step size of h =.5s the solution becomes unstable if the absolute circumferential elocity drops below 1m/s, Fig. 8. Within the Euler solution the dynamic tire force Fx D was limited to a maximum alue of Fx max = 3 N. Howeer, the dynamics of the ehicle with dynamic tire forces can be handled quite well with any kind of implicit algorithm or higher order explicit formula, Fig. 8. This simple dynamic tire model coers the whole elocity range. An enhanced dynamic tire model can be found in Rill, 6b. Here, braking to stand still and drie away manoeures can be simulated in good accuracy because here, no changes in the wheel/tire dynamics at low elocities by modifications in the slip definition are necessary. At stand still the dynamic tire model automatically changes to a stick slip model.
Wheel Dynamics F x [N] 1-1 - Euler explicit: h=.5 r Ω 1 3 4 3 1-1 - -3 1 3 4 Fx [N] ode3 and ode3s 1 r Ω -1-1 3 4 35 34 33 3 31 3 1 3 4 Step sizes [s] 1 1-1 1-1 -3 1-4 1-5 implicit: ode3s explicit: ode3 -.5 - -1.5-1 -.5.5 Figure 8 Simulation results and step sizes of explicit and implicit algorithms CONCLUSION In this paper a simple quarter car model was used to analyze the wheel/tire dynamics. Using steady state tire models the wheel/tire dynamics strongly depends on the elocity of the ehicle. A slip modification which slows down the wheel/tire dynamics at low elocities is needed to oercome the singularity at stand still. Dynamic tire models operate in the whole elocity range. It was shown further that the explicit Euler algorithm is not suitable for soling the wheel/tire dynamics especially at low elocities. Here, implicit algorithms perform best. In real time applications where a constant step size is needed the implicit Euler algorithm has proofed its efficiency, Rill, 6b. If the oerall ehicle model cannot be integrated by an implicit soler a co-simulation of the oerall ehicle with the subsystem wheel/tire is recommended. REFERENCES Hirschberg, W, Rill, G., Weinfurter, H., User-Appropriate Tyre-Modeling for Vehicle Dynamics in Standard and Limit Situations, Vehicle System Dynamics, Vol. 38, No., pp. 13-15. Lisse: Swets & Zeitlinger, an der Jagt, P., The Road to Virtual Vehicle Prototyping; new CAE-models for accelerated ehicle dynamics deelopment, PhD-Thesis, Tech. Uni. Eindhoen, Lugner, P.; Pacejka, H. and Plöchl, M., Recent adances in tyre models and testing procedures, Vehicle System Dynamics, Vol. 43, No. 67, pp. 413-436, 5 Pankiewicz, E. and Rulka, W., From Off-Line to Real Time Simulations by Model Reduction and Modular Vehicle Modeling, in: Proceedings of the 19th Biennial Conference on Mechanical Vibration and Noise Chicago, Illinois, 3 Rauh, J., Virtual Deelopment of Ride and Handling Characteristics for Adanced Passenger Cars, Vehicle System Dynamics, Vol. 4, Nos. 1-3, pp. 135-155, 3 Rill, G., A Modified Implicit Euler Algorithm for Soling Vehicle Dynamics Equation, Multibody System Dynamics, Vol. 15, No. 1, pp 1-4, 6 Rill, G., First Order Tire Dynamics, Proceedings of the III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, 6 Rill, G., Vehicle Modeling by Subsystems To appear in: Journal of Brazilian Society of Mechanical Sciences and Engineering - RBCM, 6 Rill, G. and Chucholowski, C., Modeling Concepts for Modern Steering Systems, ECCOMAS Multibody Dynamics, Madrid, Spain, 5 Weinfurter, H., Hirschberg, W., Hipp, E., Entwicklung einer Störgrößenkompensation für Nutzfahrzeuge mittels Steerby-Wire durch Simulation, in: Berechnung und Simulation im Fahrzeugbau, VDI-Bericht 1846, S. 93-941, VDI Verlag Düsseldorf, 4 RESPONSIBILITY NOTICE The author is the only responsible for the printed material included in this paper.