29 Vehicle Mechanics resistance and the vertical load on the wheel is known as the coefficient of rolling resistance C 0 . The rolling resistance force on a roadway of slope β is sgn[ v m ] c g C os β ( ) + ≠ xT 0 1 C v 2 xT if v xT 0 F roll = ( ) F F TR − = gxT x if v F T T 0 and R g − ≤ F C xT 0 mg cos β (2.3) sgn[ F F TR − = gxT x ]( C m 0 0 g v cos ) β β if 0 a F F − > C m T T nd R gxT g cos Typically 0.004 C 0 0.02 (unitless) and C 1 ≪ C 0 (s 2 /m 2 ). C 0 mg is the maximum rolling resistance at standstill. The sgn [ ] v xT is the signum function given as f 0 v xT ≥ sgn [ ] 1 i v xT = .
− 1 if 0 v xT The aerodynamic drag force is the viscous resistance of the air working against the motion of the vehicle. The force is given by F v AD = + sgn[ xT ]{0.5 ρ C A ( ) v v 2 D F xT 0 } (2.4) where ρ is the air density in kg/m 3 , C D is the aerodynamic drag coefficient (dimensionless and typically is 0.2 C D 0.4), A F is the equivalent frontal area of the vehicle and v 0 is the head-wind velocity. 2.4 DYNAMICS OF VEHICLE MOTION The tractive force is the force supplied by the electric motor in an EV and by the combination of electric motor and internal combustion engine in a HEV to overcome the road load.
The dynamic equation of motion in the tangential direction is given by dv k m xT m = − F F TR RL (2.5) dt where k m is the rotational inertia coefficient to compensate for the apparent increase in the vehicle’s mass due to the on-board rotating mass. Typical value of k m is between 1.08 and 1.1, and it is dimen- dv sionless. Additional explanation on rotational inertia appears in Section 2.7.4. XT is the accelera- dt tion of the vehicle. The dynamic equations can be represented in the state space format for simulation of an EV or a HEV system. The motion described by Equation 2.5 is the fundamental relationship required for dynamic simulation of the vehicle system.
v xT is one of the state variables of the vehicle dynamical system. The second equation needed for modeling and simulation is the velocity equation where either s or x F can be used as the state variable. The slope of the roadway β will be an input to the simulation model, which may be given in terms of the tangential roadway distance s as β = β ( ) s or in terms of the horizontal distance as β = β ( ) x F . If β is given in terms of s , then the second state variable equation is ds = v xT . (2.6) dt