34 Electric and Hybrid Vehicles 2.6.1.1 Velocity Profile The velocity profile for the constant F TR , level road case can be obtained from solving for v from the dv equation above, which gives (see Figure 2.14) dt K v t ( ) = 1 tanh 1 2 K 2 ( ) K K t (2.11) The terminal velocity can be obtained by taking the limit of the velocity profile as time approaches infinity. The terminal velocity is K V T = = lim ( v t ) . 1 ⇒ = K K 1 2 K V 2 T t →∞ K 2 2.6.1.2 Distance Traversed The distance traversed by the vehicle can be obtained from the following relation: ds ( ) t = = v t ( ) V K T T tanh( 2 V t ) dt The distance as a function of time is obtained by integrating the above equation:
1 s ( ) t = ln [ ] cosh( ) K V 2 T t (2.12) K 2 The starting acceleration is often specified as 0 to v f m/s in t f s, where v f is the desired velocity at the end of the specified time t f s. The time to reach the desired velocity and distance traversed during that time is given by 1 t = cosh − 1 ( ) K s f e 2 f and (2.13) K V 2 T 1 s f T = ln [ ] cosh( ) K V 2 t f , respectively. (2.14) K 2 The desired time can also be expressed as 1 K tanh 1 t = − 2 f f v (2.15) K K 1 2 K 1 where v f is the velocity after time t f . v(t) t FIGURE 2.14 Velocity profile for a constant F TR on a level road. K 1 K 2