The single track model

The single track model Dr. M. Gerdts Uniersität Bayreuth, SS 2003 Contents 1 Single track model 1 1.1 Geometry.................................... 1 1.2 Computation of slip angles........................... 2 1.3 Longitudinal tyre forces............................ 2 1.4 Lateral tyre forces................................ 4 1.5 Air resistance.................................. 4 1.6 Equations of motion I.............................. 4 1.7 Equations of motion II............................. 5 1.8 Constraints................................... 5 1.9 Parameters................................... 5 i

1 1 Single track model 1.1 Geometry l l h e SP α F Lx δ α ψ F Ly F s F u α h h F uh F sh Notation:, h elocity front/rear wheel δ steering angle elocity α, α h slip angle front/rear wheel β side slip angle ψ yaw angle F s, F sh lateral tyre forces F u, F uh longitudinal tyre forces l, l h distance from center of graity to front/rear wheel distance from center of graity to drag mount point e SP F Lx, F Ly air resistance in longitudinal and lateral direction m mass of car The steering angle δ is related to the steering wheel angle δ w by δ w = i L δ.

2 1 SINGLE TRACK MODEL 1.2 Computation of slip angles The slip angles are gien by ( ) l ψ sin β α = δ arctan, cos β ( ) lh ψ + sin β α h = arctan. cos β Explanation: Since the car body does not expand or shrink, the elocity components in the longitudinal direction of the car body hae to be equal: cos β = h cos α h = cos(δ α ). In the lateral direction the difference between the elocities is gien by the yaw angle elocity: h sin α h = l h ψ + sin β, sin(δ α ) = l ψ sin β. Combining these four equations yield the aboe formulas for the respectie slip angles. 1.3 Longitudinal tyre forces The car has rear wheel drie. The drier controls the braking force F B 0, the gear i {1, 2, 3, 4, 5} and the accelerator pedal position φ. The latter will result in the torque M wheel (φ, i) = i g (i) i t M mot (φ, i), where M mot (φ, i) = f 1 (φ) f 2 (w mot (i)) + (1 f 1 (φ))f 3 (w mot (i)) denotes the motor torque and w mot (i) = i g(i) i t R 1 1 S denotes the rotary frequency of the motor depending on the gear i and the longitudinal slip S. For conenience, the slip is neglected, e.g. S = 0. If the slip is not neglected, it is defined by 1, if R ϕ, R ϕ S = R ϕ 1, otherwise

1.3 Longitudinal tyre forces 3 ϕ denotes the rotary frequency of the wheel and is gien by the differential equation I R ϕ = F uh R. The functions f 1, f 2 and f 3 are gien by f 1 (φ) = 1 exp( 3φ), f 2 (w mot ) = 37.8 + 1.54 w mot 0.0019 wmot, 2 f 3 (w mot ) = 34.9 0.04775 w mot. braking force: The braking force is distributed on the front and rear wheels by the formulas F B = 2 3 F B, F Bh = 1 3 F B such that F B + F Bh = F B holds. rolling resistance: The rolling resistance force at the front and rear wheel, respectiely, is gien by F R = f R () F z, F Rh = f R () F zh where f R () = f R0 + f R1 100 + f R4 is the friction coefficient and F z = m l h g l + l h, F zh = m l g l + l h ( ) 4 ( in [km/h]), 100 denote the static tyre loads at the front and rear wheel, respectiely. longitudinal force front wheel: longitudinal force rear wheel: F u = F B F R. F uh = M wheel(φ, i) R F Bh F Rh

4 1 SINGLE TRACK MODEL 1.4 Lateral tyre forces The lateral tyre forces are functions of the respectie slip angles (and the tyre loads, which are constant in our model). A simple model is the AT-model (arcustangens-model): F s = c AT 1 arctan(c AT 2 α ), F sh = c AT 1 arctan(c AT 2 α h ). A famous model is the magic formula of Pacejka: F s = D sin (C arctan (B α E (B α arctan(b α )))), F sh = D h sin (C h arctan (B h α h E h (B h α h arctan(b h α h )))). The slope of F s at α = 0 is gien by B C D and similar for F sh. 1.5 Air resistance F Lx = 1 2 c w ρ A 2, F Ly = 1 2 c y ρ A 2 R Notation: c w ρ A c y R air drag coefficient air density effectie flow surface elocity lateral air drag coefficient lateral air elocity 1.6 Equations of motion I ẋ = cos(ψ β), ẏ = sin(ψ β), = 1 m [(F uh F Lx ) cos β + F u cos(δ + β) (F sh F Ly ) sin β F s sin(δ + β)], β = w z 1 m [(F uh F Lx ) sin β + F u sin(δ + β) + (F sh F Ly ) cos β + F s cos(δ + β)], ψ = w z, ẇ z = 1 I zz [F s l cos δ F sh l h F Ly e SP + F u l sin δ]

1.7 Equations of motion II 5 1.7 Equations of motion II ẍ = 1 m [(F uh F Lx + F u cos δ F s sin δ) cos ψ (F sh F Ly + F u sin δ + F s cos δ) sin ψ] = 1 m [(F uh F Lx ) cos ψ + F u cos(δ + ψ) F s sin(δ + ψ) (F sh F Ly ) sin ψ] ÿ = 1 m [(F uh F Lx + F u cos δ F s sin δ) sin ψ + (F sh F Ly + F u sin δ + F s cos δ) cos ψ] = 1 m [(F uh F Lx ) sin ψ + F u sin(δ + ψ) + F s cos(δ + ψ) + (F sh F Ly ) cos ψ] ψ = 1 I zz [F s l cos δ F sh l h F Ly e SP + F u l sin δ] side slip angle: absolute elocity: (ẏ ) β = ψ arctan ẋ = ẋ 2 + ẏ 2 1.8 Constraints The steering angle is restricted by δ 0.5263 [rad]. The steering angle elocity is restricted by δ 0.5 [rad/s]. The braking force F B is restricted by 0 F B 15000 [N]. The accelerator pedal position φ is restricted by 0 φ 1. 1.9 Parameters Car:

6 1 SINGLE TRACK MODEL m 1239 [kg] car mass g 9.81 [m/s 2 ] acceleration due to graity l 1.19016 [m] distance from center of graity to front wheel l h 1.37484 [m] distance from center of graity to rear wheel e SP 0.5 [m] distance from center of graity to drag mount point R 0.302 [m] wheel radius I zz 1752 [kgm 2 ] moment of inertia i L 21.1 steering wheel transmission I R 1.5 moment of inertia of wheel Drag: c w 0.3 air drag coefficient ρ 1.249512 [N/m 2 ] air density A 1.4378946874 [m 2 ] effectie flow surface c y 0.3 lateral air drag coefficient Gear shift: i g (1) 3.91 first gear i g (2) 2.002 second gear i g (3) 1.33 third gear i g (4) 1.0 fourth gear i g (5) 0.805 fifth gear i t 3.91 motor torque transmission Tyre: AT-model c AT 1 3317.99 tyre coefficient AT-model c AT 1h 2872.29 tyre coefficient AT-model c AT 2 30.0 tyre coefficient AT-model c AT 2h 30.0 tyre coefficient AT-model Tyre: Pacejka-model

1.9 Parameters 7 B 10.96 tyre coefficient Pacejka-model (stiffness factor) C 1.3 tyre coefficient Pacejka-model (shape factor) D 4560.40 tyre coefficient Pacejka-model (peak alue) E 0.5 tyre coefficient Pacejka-model (curature factor) B h 12.67 tyre coefficient Pacejka-model (stiffness factor) C h 1.3 tyre coefficient Pacejka-model (shape factor) D h 3947.81 tyre coefficient Pacejka-model (peak alue) E h 0.5 tyre coefficient Pacejka-model (curature factor) Rolling resistance: f R0 0.009 coefficient f R1 0.002 coefficient f R4 0.0003 coefficient