46 Electric and Hybrid Vehicles 2.8 PROPULSION SYSTEM DESIGN The steady-state maximum velocity, maximum gradability and the velocity equations can be used in the design stage to specify the power requirement of a particular vehicle. Let us consider the tractive power requirement for initial acceleration, which plays a significant role in determining the rated power of the propulsion unit. The initial acceleration is specified as 0 to v f in t f s. The design problem is to solve for F TR starting with a set of variables including vehicle mass, rolling resistance, aerodynamic drag coefficient, percent grade, wheel radius, etc., some of which are known, while others have to be assumed.
The acceleration of the vehicle in terms of these variables is given by = = dv F F − a TR RL (2.32) dt m The tractive force output of the electric motor for an EV or the combination of electric motor and internal combustion engine for a HEV will be a function of the vehicle velocity. Furthermore, the road load characteristics are also function of velocity resulting in a transcendental equation to be solved to determine the desired tractive power from the propulsion unit. The other design requirements also play a significant role in determining the tractive power.
The problem is best handled by a computer simula- tion where the various equations of Sections 2.5 and 2.6 can be used iteratively to calculate the tractive force and power requirements from the propulsion unit for the given set of specifications. The insight gained from the scenarios discussed in this chapter will be used later to specify the power ratings of the electric motor and internal combustion engines for electric and hybrid vehicles in the next chapter. PROBLEMS 2.1 A length of road is straight and it has a profile in the x–y plane described by f ( ) x x = ∗ 200ln 7.06 10 − 4 ( ) + 1,416 . where 0 3 ≤ ≤ x mile = 15,840 ft. ; f ( x ) and x are in feet. a.
Plot the road profile in the x–y plane for 0 1 ≤ ≤ x 5,840 ft. b. Derive an expression for β ( x ). Calculate β (500 ft.). c. Derive an expression for percent grade ( x ). Calculate percent grade (500 ft.). d. Derive an expression for tangential road length s ( x ), such that s (0) = 0. Calculate s (500 ft.) e. Can you find an expression for x ( s ), i.e., can you express x as a function of s ? Show some steps in your attempt. 2.2 A straight roadway has a profile in the x – y plane given by f ( ) x x f f = ≤ 4.1 ) for 0 x f ≤ = 2 mile 10,560 ft. f ( x f ) and x f are in ft. a. Derive an expression for β ( x f ). Calculate β (1 mile). b.
Calculate the tangential road length, s from 0 to 2 miles.