33 Vehicle Mechanics 2.6.1 c onsTanT F TR , l eVel r oad In the first case, we will assume a level road condition where the propulsion unit for an EV exerts a constant tractive force. The level road condition implies that β ( s ) = 0. We will assume that the EV is initially at rest, which implies v (0) = 0. The free body diagram at t = 0 is shown in Figure 2.13a. Assume F TR (0) = F TR > C o mg , i.e., the initial tractive force is capable of overcoming the initial rolling resistance. Therefore, dv (0) Σ = F m (0) ( a m 0) = dt dv (0) ⇒ − F C TR 0 mg = m . dt dv (0) Since F C TR (0) > 0 mg , > 0 and the velocity v increases.
Once the vehicle starts to move, the dt forces acting on it changes. At t > 0, dv dv Σ = F m ⇒ − F F TR AD − = F m roll dt dt ρ TR sg 2 dv ⇒ − F v n [ ( ) t C ] D F A v ( ) t v − + sgn[ ( ) t m ] ( g C ( ) C v 2 0 1 t m ) = 2 dt dv Assuming v t ( ) > 0 for t > 0 and solving for , dt dv F ρ = − TR gC 0 1 − + C A D F gC v 2 dt m 2 m Let us define the following constants for a constant F TR acceleration F K TR 1 0 = − gC > 0 m K 2 1 = + C A D F gC > 0 2 m ρ dv which gives = − K K 2 1 2 v . dt F road F AD m F F TR roll F TR m F roll mg (a) (b) FIGURE 2.13 Forces acting on the vehicle on a level road: (a) free body diagram at t = 0 and (b) forces on the vehicle at t > 0.