44 Electric and Hybrid Vehicles The wheel dynamics can be achieved from the balance of forces and torques as − F F − = dv Rw roll m w (2.27) dt ω T F − + b F r J = d wh TR gyT roll wh w (2.28) dt where T TR is the driving or tractive torque, m w is the mass of the wheel and r wh is the wheel radius. T TR is zero for the trailing wheel. The tractive force is the force supplied by the electric motor in an EV and by the combination of electric motor and IC engine in a HEV to overcome the road load.
Substituting Equations 2.27 and 2.28 in Equation 2.26, we can write T ω TR − − b − − = + dv J d F gyT F F AD gxT R F m r w ( ) m + w wh (2.29) r wh r wh dt r wh dt It can be easily concluded that the road load force of Equation 2.29 has the same components as that of Equation 2.1 and that the term T TR / r wh plays the role of the traction force. To accelerate the vehicle, the traction force has to overcome rolling resistance, aerodynamic load, weight component and the equivalent force of the rear portion of the car.
The right-hand side of Equation 2.29 can be written in simplified form as dv m m + + J ω ( ) w d wh = dv w k m m (2.30) dt r wh dt dt dv The term k m m is an equivalent force of inertia accounting for the translating mass of the quarter dt car plus the inertia of the rotating masses. Finally, the general form of the dynamic equation of motion in the tangential direction assumes the same form as Equation 2.5: dv k m m T = − F F R RL R − F r (2.31) dt where k m is the rotational inertia coefficient. In Equation 2.30, the tangential speed of the wheel can also differ from the spin velocity of the wheel, according to the value of the slip.
2.7.5 T racTion l imiT and c onTrol The maximum rolling resistance force that can be sustained at the tire–road interface depends on the amount of vehicle slip and the vertical load on the wheel. A slip-dependent friction coefficient µ ( s ) and the vertical load define the rolling resistance on the wheel as F s roll (s) ( = µ ) F gyT where F gyT is the normal force on the wheel. Accordingly, wheel equations can be rewritten as dv − F F Rw − = roll ( ) s m w dt d ω T F TR − + gyT r b F oll w ( ) s r h w = J wh dt