Vehicle Planar Motion Stability Study for Tyres Working in Extremely Nonlinear Region

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1 1 Vol. 23, No. 2, 2010 DOI: /CJME ***, available online at Vehicle Planar Motion Stability Study for Tyres Working in Extremely Nonlinear Region LIU Li 1, SHI Shuming 1, *, SHEN Shuiwen 2, and CHU Jiangwei 1 1 Transportation College, Jilin University, Changchun , China 2 Ricardo UK Ltd, Cambridge CB4 1YG, UK Received April 21, 2009; revised November 30, 2009; accepted January 11, 2010; published electronically January 12, 2010 Abstract: Many researches on vehicle planar motion stability focus on two degrees of freedom(2dof) vehicle model, and only the lateral velocity (or side slip angle) and yaw rate are considered as the state variables. The stability analysis methods, such as phase plane analysis, equilibriums analysis and bifurcation analysis, are all used to draw many classical conclusions. It is concluded from these researches that unbounded growth of the vehicle motion during unstable operation is untrue in reality thus one limitation of the 2DOF model. The fundamental assumption of the 2DOF model is that the longitudinal velocity is treated as a constant, but this is intrinsically incorrect. When tyres work in extremely nonlinear region, the coupling between the vehicle longitudinal and lateral motion becomes significant. For the purpose of solving the above problem, the effect of vehicle longitudinal velocity on the stability of the vehicle planar motion when tyres work in extremely nonlinear region is investigated. To this end, a 3DOF model which introducing the vehicular longitudinal dynamics is proposed and the 3D phase space portrait method is employed for visualization of vehicle dynamics. Through the comparisons of the 2DOF and 3DOF models, it is discovered that the vehicle longitudinal velocity greatly affects the vehicle planar motion, and the vehicle dynamics represented in phase space portrait are fundamentally different from that of the 2DOF model. The vehicle planar motion with different front wheel steering angles is further represented by the corresponding vehicle route, yaw rate and yaw angle. These research results enhance the understanding of the stability of the vehicle system particularly during nonlinear region, and provide the insight into analyzing the attractive region and designing the vehicle stability controller, which will be the topics of future works. Key words: vehicle dynamics, steering stability, nonlinear dynamics, phase space 1 Introduction It is crystal clear that the lost of the vehicle stability during icy and wet road conditions is caused by the nonlinearity of tyres. Yet, the development of the vehicle motion after losing its stability is largely unknown. Preventing tyres from nonlinear region by whatever the means plays a major part for a stably vehicle handling, but to regain the maneuverability is not by all means less critical. The later becomes viable only provided with necessary planar monition characteristics, which is the topic of this paper. Up to date, many researchers focus on two degrees of freedom(2dof) vehicle planar motions. The works of INAGAKI, et al [1] and ONO, et al [2] led to a conclusion that the vehicle had three equilibriums, one being stable focus while the other two unstable saddles, and that the stable region was rather narrow in the so-called β-ω phase plane spanned by the body side slip angle β and yaw rate ω. The * Corresponding author. shishuming@jlu.edu.cn This project is supported by National Natural Science Foundation of China (Grant No ) similar was also concluded by YOUNG, et al [3] and VINCENT [4], but the conclusions were only validate locally. SHEN, et al [5], adopted a geometrical method to explore the system from more global point of view for the first time. The complicated dynamics with bifurcation and limit-cycles were discovered. SHI, et al [6], discovered the stable region of vehicle cornering system spanned by the body slip angle and the yaw rate through the analysis of Lyapunov potential energy function surface. Unbounded growth of the vehicle motion during unstable operation is untrue in reality thus one limitation of the 2DOF model. The impact from the vehicle longitudinal motion is insignificant thus the vehicle longitudinal velocity is treated as a constant. This is the fundamental assumption of the 2DOF model, but this is intrinsically incorrect if tyres work in extremely nonlinear region. The coupling between the vehicle longitudinal and lateral motion becomes significant [7]. Consequently, the longitudinal velocity can not be treated as a constant. Therefore, a 3DOF (longitudinal velocity, lateral velocity and yaw rate) model is introduced and analyzed numerically in this paper. The comparison between 2DOF and 3DOF model in terms of the vehicle system dynamics

2 LIU Li, et al: Vehicle planar motion stability study for tyres working in extremely nonlinear region 2 is summarized and illustrated. 2 3DOF Vehicle Steering Model The bicycle model of vehicle steering system is shown in Fig. 1. The equations of 3DOF vehicle model are shown as follows [8] : mv (& x vyω) = Fx, mv (& y + vxω) = Fy, I & zω = Mz, where m Vehicle mass, m=1 500 kg; I z Yaw moment of inertia, I z = kg m 2 ; v x Longitudinal velocity; v y Lateral velocity; ω Yaw rate; F x Longitudinal force; F y Lateral force; M z Yaw moment. And Eq. (1) can be written as follows: mv ( & x vyω) = Flf cosδf Fsf sinδf + Flr cosδr ρ 2 Fsr sin δr Cair AL vx, 2 mv (& y + vxω) = Fsf cosδf + Flf sinδf + Fsr cosδr + Flr sin δr, I & zω = ( Flflf sinδf + Fsf lf cos δf ) ( Fl lr r sinδr + Fl sr r cos δr ), where δ f Front wheel steering angle; δ r Rear wheel steering angle; l f Distance from front axle to the mass center, l f =1.2 m; l r Distance from rear axle to the mass center, l r =1.3 m; F lf Longitudinal tyre force of front wheel; F sf Lateral tyre force of front wheel; F lr Longitudinal tyre force of rear wheel; F sr Lateral tyre force of rear wheel; C air Air resistance efficient; A L Vehicle frontal area; ρ Air density. (1) (2) In the classical 2DOF model, both the longitudinal tyre forces and air resistant force are neglected. The longitudinal velocity is treated as a constant. Consequently, the 2DOF vehicle steering model is a simplified version of Eq. (1) and Eq. (2) as follows: Fsf cosδf + Fsr cosδr v& y = vxω, m Fl sf f cosδ f Fl sr r cosδ r & ω =. I z For the vehicle planar motion, if the system is not constrained, all the state variables of the system are free to develop with time. This allows the investigation of the dynamical coupling between the lateral and longitudinal directions. However, as a consequence of the longitudinal velocity being constrained as a constant in 2DOF model, the dynamics represented by 2DOF could only be partially true. It is therefore reasonable to introduce the 3DOF model. In this paper, the focus is given to the interaction between longitudinal and lateral motions. Although the longitudinal tyre forces and air resistant force are neglected, the longitudinal velocity is now not a constant. The proposed 3DOF steering model is shown as follows: ( F sinδ + F sinδ ) sf f sr r v& x = vyω, m Fsf cosδf + Fsr cosδr v& y = vxω, m Fl sf f cosδf Fl sr r cosδr & ω =. I z The lateral tyre force depending on the side slip angle is represented by the magic formula model: (3) (4) F = Dsin( Carctan( Bα E( Bα arctan Bα), (5) where B, C, D, E are coefficients, F is the lateral tyre force and α is the side slip angle. The coefficients are listed in Table [4], while Fig. 2 graphically shows the relationship between the lateral tyre forces and side slip angles for the respective front and rear tyres. Table 1. Bicycle model tyre parameters y In F sr F lr δ r vy y CoG ω β ψ v F sf vx F lf δ f x CoG Tyre Coefficient B C D E Front tyre Rear tyre The side slip angles of both front and rear tyres are given as below: l r Fig. 1. Vehicle bicycle model l f x In vy + ω glf αf = δf arctan, vx (6)

3 3 vy ω glr αr = δr arctan. vx (7) Fig. 3. Local view of v y -ω phase plane trajectories Fig. 2. Lateral tyre force versus side slip angle for front and rear tyres in bicycle model. 3 Comparison of Phase Portrait Analysis between the 2DOF and 3DOF Models Vehicle motion can be visualized in a phase space spanned by longitudinal velocity, lateral velocity and yaw rate (v x, v y, and ω). However, conventional study of 2DOF model is more common to employ a phase plane spanned only by the two of just-mentioned state variables. Fig. 3 and Fig. 4 show the typical v y -ω portraits of the 2DOF model. Fig. 3 is the representation of the vehicle motion in a local region of v y and ω. The regions for v y and ω are ( 10, 10) m/s and ( 1, 1) rad/s, respectively. Simulation lasts 20 s. Note that, the red o indicates the end points of the simulation. Fig. 5 and Fig. 6 are other sets of vehicle trajectories in theβ-ω phase plane portrait resulting from 2DOF model. Similar to Fig. 3, Fig. 5 is the representation of the vehicle motion in a local region of β and ω. The results given in Fig. 3 and Fig. 5 are also presented by many researches [1, 4], which are used to distinguish the stable and unstable region for 2DOF model. Nevertheless, these two figuers only reflect the local portrait of the state variables. It is still largely unkown the development of the state variables if the vehicle loses its stablility. Fig. 4 and Fig. 6 show the more global view of vehicle motions in v y -ω and β-ω phase plane. It could be concluded from Fig. 4 and Fig. 6 that once the vehicle loses its stablility, the body side slip angle increases to π/2 rapidly, and that both the yaw rate and lateral velocity grow exponentially. Because the longitudinal velocity is treated as a constant, the energy of lateral and yaw motion becomes unbounded, which is contradictory to the vehicle motion in reality. Therefore, the global phase portrait derived from the 2DOF vehicle model reveals the limitation of the 2DOF model. Fig. 4. Global view of v y - ω phase plane trajectories Fig. 5. Local view of β-ω phase plane trajectories Fig. 6. Global view of β-ω phase plane trajectories

4 LIU Li, et al: Vehicle planar motion stability study for tyres working in extremely nonlinear region 4 Fig. 7 and Fig. 8 show the global view in v x -v y -ω phase space of the 3DOF with v y and ω being respectively region of ( 10, 10) m/s and ( 1, 1) rad/s. The initial vlaues of v x is 20m/s, and the intervals of v y and ω are 4 m/s and 0.4 rad/s, respectively. Fig. 8 is the projection of the portrait onto the v y -ω plane. grow larger in 3D phase space when initial longitudinal velocity is increased. Howerer, It is also indicated from Fig. 10 and Fig. 11 that the attractive region of equilibrium becomes narrow with the increase of initial longitudinal velocity. This trendency of losing stability could also be concluded from the study based on 2DOF model [1]. Fig. 7. v x -v y -ω phase space trajectories Fig. 9. v x -v y -ω phase space trajectories Fig. 8. v x -v y -ω phase space trajectories onto v y -ω plane Fig. 10. Local phase space trajectories onto v y -ω plane with initial longitudinal velocity being 20 m/s As previously mentioned, the state variable grows exponentially once it come into the unstable region for the 2DOF model. However, this would not happen to the 3DOF model. Since vehicle longitudinal velocity is not treated as a constant any more, part of the lateral and yaw motion converts into longitudinal motion and the trajectories which form closed and bounded orbits present multiplicity. 4 Phase Space Portraits of 3DOF 4.1 Phase space portraits with the variation of initial longitudinal velocity Fig. 9 shows the vehicle trajectories with v y and ω taking their initial value from the interval of ( 10,10) m/s and ( 1,1) rad/s and the initial longitudinal velocity being respectively 20 m/s and 40 m/s. Fig. 10 and Fig. 11 show the projections of the portrait onto v y -ω plane when v x is 20m/s and 40m/s, respectively. It could be concluded from Fig. 9 that the orbits formed by the vehicle trajectories Fig. 11. Local phase space trajectories onto v y -ω plane with initial longitudinal velocity being 40 m/s 4.2 Phase space portraits with the variation of front wheel steering angle Fig. 12 and Fig. 13 show the local phase space

5 5 trajectories onto v y -ω plane when the front wheel steering angle δ f is respectively 0.01 rad and 0.05 rad. It is shown in these two figures that the equilibrium positions vary with the front wheel steering angle, and there is no fixed-point equilibrium solution once the front wheel steering angle reaches and beyond 0.05 rad. Fig. 14. v x -v y -ω phase space trajectories with δ f being 0.01 rad and rad Fig. 12. Local phase space trajectories onto v y -ω plane with δ f being 0.01 rad Fig. 15. v x -v y -ω phase space trajectories with δ f being 0.05rad Fig. 13. Local phase space trajectories onto v y -ω plane with δ f being 0.05 rad For the sake of comparing the phase space the portraits with different front wheel steering angles, the constant initial values of the vehicle longitudinal, lateral speed and vehicle yaw rate are chosen, i.e., v x =20 m/s, v y =0 m/s and ω=0 rad/s. The phase portrait, vehicle yaw rate and yaw angle with different front wheel steering angle inputs (either constant or sine wave) are presented in the following sections Constant front wheel steering angle inputs Fig. 14 and Fig. 15 show the phase space trajectories with the front wheel steering angle being 0.01 rad, rad and 0.05 rad. Fig. 16 shows the projection of trajectories onto v y -ω plane with the front wheel steering angle δ f being 0.01 rad, rad and 0.05 rad, respectively. And Fig. 17 shows an enlargement of Fig. 16. Fig. 16. v x -v y -ω phase space trajectories onto v y -ω plane with δ f being 0.01rad, rad and 0.05rad Fig. 17. Enlargement of Fig. 16

6 LIU Li, et al: Vehicle planar motion stability study for tyres working in extremely nonlinear region Sine wave front wheel steering angle inputs Fig. 18 represents the results of the sine wave front wheel steering angle inputs. Frequencies of the sine waves are all 0.4 Hz, and the amplitudes are rad, 0.03 rad and 0.05 rad, respectively. Fig. 19 and Fig. 20 show the phase space trajectories with the amplitudes of the front wheel steering angle being rad, 0.03 rad and 0.05 rad. Fig. 21 shows the projection of trajectories onto v y -ω plane while Fig. 22 is the enlargement of Fig. 21. It is shown from these numerical experiments that when either the constant or sine wave but the amplitude of the front wheel steering angle is small, the v x -v y -ω phase space trajectories indicate the system is stable. Otherwise, the system loses its stability. Fig. 21. v x -v y -ω phase space trajectories onto v y -ω plane with the amplitude of δ f being rad, 0.03 rad and 0.05 rad Fig. 18. Front wheel steering angle input Fig. 22. Local enlargement of Fig Analysis of the Results Fig. 19. v x -v y -ω phase space trajectories with the amplitude of δ f being rad and 0.03 rad 5.1 Constant front wheel steering angle inputs Fig. 23, Fig. 24 and Fig. 25 show the state variables (v x, v y, ω) with the front wheel steering angle being 0.01 rad, rad and 0.05 rad, respectively. Fig. 26 and Fig. 27 show the vehicle motion in x-y plane. Fig. 20. v x -v y -ω phase space trajectories with the amplitude of δ f being 0.05 rad Fig. 23. State variables (v x, v y and ω) with δ f being 0.01 rad

7 7 Fig. 27. Vehicle route with δ f being 0.01 rad, rad and 0.05 rad Fig. 24. State variables (v x, v y and ω) with δ f being rad Fig. 25. State variables (v x, v y and ω) with δ f being 0.05 rad The results indicate that when the front wheel steering angle is small (0.01 rad and rad), the system is stable. Consequently, the vehicle motion forms a circle orbit in x-y plane, implying the vehicle yaw rate will be a constant. Thus, the yaw angle grows linearly with time. The vehicle has full capability to follow the steering commands. On the other hand, once the front wheel steering angle reaches 0.05rad, all the state variables v x, v y and ω start oscillating. It is interesting to see that the longitudinal velocity becomes negative, which means the vehicle is turning backward. This is further confirmed by the vehicle motion represented in x-y plane. The vehicle struggles to follow a circle orbit command initially, which is the reason why the radius of trajectories circles is continuously reduced. However, the vehicle loses its control and moves toward the negative x-direction eventually. All these are the evidences that the vehicle is unstable in these conditions. Fig. 28 and Fig. 29 show the lateral tyre forces given in the lateral tyre force vs. tyre side slip angle diagrams. It is shown in these figures that when the front wheel steering angle is small, both the side slip angle and lateral tyre force are in the linear region. However, the tyre clearly works in nonlinear region when the front wheel steering angle reaches 0.05 rad. Thus, working in nonlinear region of tyre(s) is the ultimate reason of the vehicle losing its stability. Fig. 26. Vehicle yaw angle with δ f being 0.01 rad, rad and 0.05 rad Fig. 28. Front tyre side slip angle and lateral force with δ f being 0.01 rad, rad and 0.05 rad

8 LIU Li, et al: Vehicle planar motion stability study for tyres working in extremely nonlinear region 8 Fig. 29. Rear tyre side slip angle and lateral force with δ f being 0.01rad, rad and 0.05rad 5.2 Sine wave front wheel steering angle input Fig. 30, Fig. 31 and Fig. 32 represent the development of the state variables v x, v y and ω with time when the sine wave steering input has the amplitudes of rad, 0.03 rad and 0.05 rad, repectively. For the sake of analysis, only the time history of 50 s is given. Fig. 33 comparies of vehicle motion in x-y plane while Fig. 34 is an enlargement of Fig. 33 in a region shown by a squre box. Fig. 32. State variables (v x, v y and ω) with the amplitude of δ f being 0.05 rad Fig. 33. Vehicle route with the amplitude of δ f being rad, 0.03 rad and 0.05 rad Fig. 30. State variables (v x, v y and ω) with the amplitude of δ f being rad Fig. 34. Local enlargement of Fig. 33 Fig. 31. State variables (v x, v y and ω) with the amplitude of δ f being 0.03 rad The results indicate that when the amplitude is small (e.g., rad and 0.03 rad), the vehicle is stable and can follow the so-called S route steering commands. However, once again when the amplitude reaches 0.05 rad, the interesting vehicle dynamics emerges. The yaw rate remains positive and the longitudinal velocity appears negative, implying the vehicle spins around incessantly and even turns backward occasionally. The vehicle has lost its stability. Fig. 35 and Fig. 36 show the lateral tyre forces given in the lateral tyre force vs. tyre side slip angle diagrams. Clearly, when the amplitude of the front wheel steering

9 9 angle is small (e.g., rad and 0.03 rad), the tyres work in the linear region. On the other hands, the tyres work in severe nonlinear region when the amplitude reaches 0.05 rad, which again is the cause of vehicle instability. Fig. 35. Front tyre side slip angle and lateral force with the amplitude of δ f being 0.015rad, 0.03 rad and 0.05rad Fig. 36. Rear tyre side slip angle and lateral force with the amplitude of δ f being 0.015rad, 0.03 rad and 0.05rad 6 Conclusions (1) 3D phase portrait is used to analyze the stability of vehicle planar motion represented by the 3DOF model. Comparing with that of the 2DOF model, it is clearly concluded that the vehicle longitudinal velocity greatly affects the vehicle planar motion dynamics when tyre works in extremely nonlinear region. (2) Vehicle system dynamics is fundamentally different from that represented by the 2DOF model. The state variables grow exponentially once they come into the unstable region in 2DOF model. However, this would not happen to 3DOF model. The trajectories form closed and bounded orbits. (3) The consequences of the vehicle motion with different front wheel steering angles are interpreted by investigating the corresponding vehicle route, yaw rate and yaw angle. These analyses enhance the understanding of the cause of the vehicle instability. (4) The phase portrait after vehicle losing its stability represented by 3DOF provides the insight into the vehicle motion and guidance for future vehicle stability control design. References [1] INAGAKI S, KSHIRO I, YAMAMOTO M. Analysis of vehicle stability in critical cornering using phase plane method[g]. SAE Paper, No , [2] ONO E, HOSOE S, TUAN H D, et al. Bifurcation in vehicle dynamics and robust front wheel steering control[j]. IEEE Transactions on Control System Technology, 1998, 6(3): [3] KO Young Eun, LEE Jang Moo. Estimation of the stability region of a vehicle in plane motion using a topological approach[j]. Int. J. of Vehicle Design, 2002, 30(3): [4] VINCENT Nguyen. Vehicle handling, stability, and bifurcation analysis for nonlinear vehicle models[d]. USA, Washington D C: University of Maryland, [5] SHEN Shuiwen, WANG Jun, SHI Peng. Nonlinear dynamics and stability analysis of vehicle plane motions[j]. Vehicle System Dynamics, 2007, 45(1): [6] SHI Shuming, MAO Zhenyong, XIANG Hui, et al. Research on nonlinear analysis methods of vehicle cornering stability[j]. Chinese Journal of Mechanical Engineering, 2007, 43(10): (in Chinese) [7] LIU Li, CHU Jiangwei, SHI Shuming, et al. Analysis on the influence of the vehicle longitudinal acceleration to the handling stability[j]. Journal of Vibration and Shock, 2009, 28(6): (in Chinese) [8] KIENCKE U, NIELSEN L. Automotive control systems[m]. New York: Springer, Biographical notes LIU Li, born in 1982, is currently a PhD candidate in Transportation College, Jilin University, China. Her research interests include vehicle operation simulation and control. liuli_jlu@yahoo.com.cn. SHI Shuming, born in 1965, is currently a professor in Jilin University, China. He received his PhD degree from Jilin University, China, in His research interests include vehicle operation simulation and control. shishuming@jlu.edu.cn SHEN Shuiwen, born in 1967, is currently a senior control engineer in Ricado UK Ltd, UK. He received his PhD degree from Jilin University, China, in His research interests include HEV and vehicle system dynamics and control. shuiwen_shen@hotmail.com CHU Jiangwei, born in 1962, is currently a professor in Jilin University, China. He received his PhD degree from Northeast Forestry University, China, in His research interests include Automobile remanufacture theory and technology. cjw_62@163.com