Lightweight Electric/Hybrid Vehicle Design

Design for optimum body-structural and running-gear performance efficiency 229
third and fourth gears respectively. Actual plots would of course depend on the torque/speed
characteristics of electric drives.
8.8.3 ROTARY INERTIA, ROLLING AND GRADIENT RESISTANCES
While translatory-mass inertia calculation is relatively easy, that of rotary mass is more difficult.
It is necessary to sum the rotating inertia of the separate transmission components by relating
relative torque to the driving axle as T = ΣIaR i 2 – for moment of inertia I and angular acceleration
a. Since wheel circumferential speed is equal to vehicle translatory velocity, an ‘equivalent mass’
can be considered as concentrated at the rolling radius – having the same effect on the inertia of
translatory motion as the summation of individual rotary inertias.
Thus effective inertia mass M = M r i e 2αd = ΣIR2 where equivalent mass M = ΣIR e 2 /r2 , the effective
mass being M = M + M = MY where Y is termed the rotary mass factor, a valuable tool of the
i e
method. To find its value, the rotary masses are divided into parts rotating with wheels – and those
rotating with power unit. The latter gain in importance in the lower ranges since M is proportional
e
to R2 . Average values quoted in the literature for a fully laden commercial vehicle are fourth-gear,
1.09, then 1.2, 1.6 through to 2.5 in first. The value of Y is obtained from
Y = 1 + M e /M = 1 + [(ΣI w /Mr 2 ) + (ΣIR 2 /Tr 2 )]
where ΣI w is wheel inertia and I e is that for ‘engine speed’ parts and typically Y = 1 + (0.4 +
0.0025R 2 ).
Rolling resistance R r can be expressed as a dimensionless coefficient f (= 0.013 for radial tyres)
while transmission resistance is obtained from a summation of the power consumed at various
stages along the powertrain. Overall efficiency E depends on the transmission system and is
conventionally assumed as E = 0.90 in direct drive and 0.85 in lower gears. Air resistance is
obtained from
R a = C d AρV 2 /2g
for vehicle speed V, gravity acceleration g, projected frontal area A and air density ρ (C d rising to
0.8 for a bluff box van). Gradient resistance is the component of gravitational force parallel to
vehicle attitude, (b). Since W d μ = fW + W tan θ max , maximum gradient = Wm − f.
8.8.4 ACCELERATIVE PERFORMANCE
Selection of gear ratio will depend upon the chosen operation for the vehicle. The ratio for maximum
acceleration can be estimated from
G max = (Rr/T e )±[(Rr 2 /T e ) + (Wr 2 /g + I w /r)/I e ] 1/2
where motion resistance R = R r + R a and I e and I w are engine and wheel inertias. Engine speed
relates to vehicle speed as
V = 2pN e r . 3600/12 . 60E s G . 5280 = N e r/168GE s
and tyre slip efficiency E s can be assumed equal to 0.965 for this calculation.
Thus N e and T e for each gear reduction may be tabulated against road speed by means of the
engine torque/speed curve. Motion resistance forces are subtracted from P to give free tractive