XXIV ICTAM, 21-26 August 2016, Montreal, Canada

ANALYSIS OF VEHICLE HANDLING FOR FRONT DRIVE AND REAR WHEEL DRIVE VEHICLES Sergei S. Avedisov ,∗ Chaozhe R. He, Wubing B. Qin, and Gábor Orosz

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, 48105, USA

Summary Steady state handling characteristics of rear wheel drive (RWD) and front wheel drive (FWD) vehicles are compared. Each configuration is modeled by a bicycle model equipped with a brush model, and the nonlinear dynamics are derived through the Appelian framework. The steady state solutions of the nonlinear equations are investigated via numerical continuation for understeer and oversteer scenarios. The results are summarized using bifurcation diagrams for several different angles.

INTRODUCTION

As shown in [2, 3], numerical bifurcation analysis can be used to characterize vehicle handling. One may obtain the global picture of the various steady states while varying parameters including states with high lateral acceleration that cannot be obtained by traditional analysis [1]. In prior works, due to geometrical simplifications, no distinction was made between rear wheel drive (RWD) and front wheel drive (FWD) vehicles. In this paper we use the Appelian framework to derive the nonlinear dynamics of bicycle models for automotive steering considering rear wheel drive (RWD) and front wheel drive (FWD) configurations. We show that these two configurations yield different dynamical equations, and analyze the steady state motions for understeer and oversteer scenarios and different steering angles. This analysis can lay the foundation for the design of future vehicle control systems tailored to the drive type of the vehicle.

DERIVATION OF

F (α) 0.7 M(α) 0.04 y F v Figure 1(a) shows the bicy- (a) FF F N (b)aN (c) x F vQ 0.6 Q 0.02 cle model for automotive steer- γ 0.5 yR ψ l MF 0 FR αF ing. The bicycle model is com- F 0.4 d ψ −0.02 xR 0.3 y MR P G posed of the body of length `, αR −0.04 vP 0.2 v mass m and mass moment of iner- a R −0.06 a R 0.1 tia J with two massless . 0 −0.08 c  x 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 The is located at α [rad] α [rad] distance d in front of the rear . Figure 1: (a): Bicycle model with the steering angle γ, generalized coordinates x,y and ψ, and the The are modeled using the geometrical parameters ` and d. (b): Lateral tire force F (scaled by the normal force N) as a function brush tire model with a parabolic of the α. (c): Aligning moment M (scaled by aN) as a function of the slip angle α. pressure distribution.

The motion of the bicycle is described by the position of the center of mass (x,y) and the heading angle ψ. Using the constraint that the rear wheel (front wheel) is rolling with a constant velocity v0 in the RWD (FWD) model and assuming quasi-steady lateral tire deformations, we can obtain the equations for the lateral velocity σ1 = −sin(φ)x˙ + cos(φ)y˙ and yaw rate σ2 = φ˙ though the Appellian framweork. The handling dynamics of the RWD bicycle model are given by m 0 σ˙  −mv σ   F + F cos(γ)  1 = 0 2 + R F , (1) 0 Jc σ˙2 0 −d FR + (` − d)FF cos(γ) + MR + MF where the forces FR = F(αR) and FF = F(αF) are shown in Figure 1 (b) as the function of the slip angle α. Similarly, the moments MR = M(αR) and MF = M(αF) are depicted in Figure 1 (c). The slip angles for the RWD dynamics are given by

−σ1 + (d − a)σ2 v0 sin(γ) − (σ1 + (` − d)σ2)cos(γ) − a(σ2 + γ˙) tan(αR) = , tan(αF) = . (2) v0 v0 cos(γ) + (σ1 + (` − d)σ2)sin(γ) Similarly, the handling dynamics of the FWD bicycle model are given by

 ( )  " m 2 #  −m tan γ (σ + σ (` − d) − v sin(γ))γ − m v0σ2 + m(` − d)tan(γ)σ 2 2 m(` − d)tan (γ) σ˙1 cos2(γ) 1 2 0 ˙ cos(γ) 2 cos (γ) =   2 2 2 σ˙ tan(γ) m(` − d)tan (γ) Jc + m(` − d) tan (γ) 2 −m(` − d) 2 (σ1 + σ2(` − d) − v0 sin(γ))γ˙ − m(` − d)tan(γ)σ1σ2 cos (γ) (3) " 1 # FR + ( ) FF + cos γ . −d F + (`−d) F + M + M R cos(γ) F R F The forces and moments are still given by Figure 1 (b) and (c), respectively, while the slip angles become

(−σ1 + (d − a)σ2)cos(γ) v0 sin(γ) − σ1 − (` − d)σ2 − a(σ2 + γ˙)cos(γ) tan(αR) = , tan(αF) = . (4) v0 − σ1 sin(γ) − (` − d)σ2 sin(γ) v0 cos(γ)

π 2 In traditional steady state handling analysis the simplifications a ≈ 0, γ, αR, αF  2 , F(α) ≈ 2ka α, M(α) ≈ 0 are used yielding the steady states

2k a2`γ ` v γ mg d ` − d  σ ∗ = R + K 0 , σ ∗ = , K = − , (5) 1 2 2 us 2 ` v0 us 2 2 2kRa `d − mv (` − d) v0 g + Kus 2` kFa kRa 0 v0 g for the lateral velocity and yaw rate both for RWD and FWD. Here Kus represents the understeer coefficient, and for Kus > 0(Kus < 0) the vehicle understeers (oversteers) according to the SAE convention.

COMPARISON OF STEADY STATE CORNERING FOR FWD AND RWD CONFIGURATIONS

Kus < 0, γ = 1◦ Kus > 0, γ = 5◦ In this section we compare the RWD FWD RWD FWD 8 8 8 8 σ1∗ σ1∗ σ1∗ σ1∗ m m m m steady state solutions of (1,2) and [ s ] 6 [ s ] 6 [ s ] 6 [ s ] 6 (3,4) for SAE understeer (Kus > 0) 4 4 4 4 and SAE oversteer (K < 0) sce- 2 2 2 2 us 0 0 0 0 narios for different steering angles. −2 −2 −2 −2 Figure 2 shows the stable and un- −4 −4 −4 −4 −6 −6 −6 −6

−8 −8 −8 −8 stable steady state solutions as sol- 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 id green and dashed red curves, re- v0 [m/s] v0 [m/s] v0 [m/s] v0 [m/s] spectively, as well as the traditional Figure 2: Bifurcation diagrams for the RWD/FWD vehicle models for various Kus and γ values. (SAE) steady state cornering solu- The left two figures denote the SAE oversteer configuration (Kus < 0), while the right two figures tions (5), as gray curves while vary- denote an SAE understeer configuration (Kus > 0). The solid gray line represents the SAE cornering solution. Stable solutions of (1,2) and (3,4) are marked by solid green curves, while unstable ing the speed v0 for different values of the steering angle γ. The equi- solutions are marked by dashed red curves. The blue crosses denote fold and pitchfork bifurcations, while the black pluses denote Hopf bifurcations. In the oversteer configuration the vertical black librium value of σ1 is denoted by ∗ dashed line represents the value of v0 when the vehicle loses stability in the simplified case (5). σ1 . ◦ For Kus < 0, γ = 1 the low-speed stable solution loses stability via fold bifurcation for both FWD and RWD configurations below the critical speed predicted by the SAE solution. At high speed the RWD dynamics yield a stable solution corresponding to negative σ1 born via Hopf bifurcation, whereas the FWD dynamics yield a stable solution corresponding to positive σ1 born via fold bifurcation. ◦ For Kus > 0, γ = 5 the low-speed stable solution also loses stability via fold bifurcation for both RWD and FWD configurations, where the SAE solution does not show a critical speed at all. Stable solutions at high speed arise via Hopf bifurcations for both RWD and FWD configurations. In the RWD case these stable solutions correspond to negative σ1 and in the FWD case they correspond to positive σ1. In general we observe that the stable solutions arising at high speeds are characterized by negative σ1 in RWD dynamics and by positive σ1 in FWD dynamics. Hence the RWD vehicle tends to oversteer at high speeds, whereas the FWD vehicle tends to understeer at high speeds. This confirms with common notions about the handling characteristics of FWD and RWD in extreme scenarios [4]. This demonstrates the predictive power of the models developed in this paper.

References

[1] Gillespie T. D.: Fundamentals of Vehicle Dynamics. Society of Automotive Engineers. 1992. [2] Ono E., Hosoe S., Tuan H. D., Doi S.: Bifurcation in vehicle dynamics and robust front wheel steering control. IEEE Transactions on Control System Technology: 6(3), 412-420, 1998. [3] Rossa F. D., Mastinu G., Piccardi C.: Bifurcation analysis of an automobile model negotiating a curve. Vehicle Systems Dynamics: International Journal of Vehicle Mechanics and Mobility: 50(10), 1539-1562, 2012. [4] Heisler H.: Advanced Vehicle Technology. Society of Automotive Engineers International. 2002.