Hybrid vehicles - Propulsion Systems for Hybrid Vehicles - page 38

the slip versus speed becomes very non-linear. Noting that engine speed is most
often given in rpm, the conversion to rad/s is included in (1.18) for consistency with
mks units used in all of the road load expressions:

2 p
60

n e r w
z 0 ð 1 s Þð m = s Þ

V ¼ ð 1 : 18 Þ

The next section develops the relationships between wheel slip and its impact on
engine speed for the given value of vehicle speed used in the road load expressions.

1.5.2 Friction and wheel slip
The maximum tractive force that can be applied to the driven wheels is limited by
the coefficient of friction at the tyre patch and road surface. Assuming that the
vehicle is front wheel drive, the limiting tractive effort becomes

F trac lim ¼ m ð s Þ F f norm ð N Þ ð 1 : 19 Þ

where the normal force on the driven axle is given by the mass split for the
front axle from (1.3) plus one half the unsprung mass, m us , listed in Table 1.13.
Equation (1.20) sums up the normal force for a front wheel drive vehicle on level
terrain: roman N at the end

g cos

B
L ð

m us
2

F f norm ¼ m a þ N p m p Þþ ð a Þð N Þ ð 1 : 20 Þ

The speed dependant coefficient of friction can be approximated for a given
value of peak static friction as

mð s Þ¼m pk ½ a ð 1 e bs

Þ cs ð 1 : 21 Þ

A typical wheel slip curve is shown in Figure 1.26. The constants in (1.21) are
empirically determined by curve fitting experimental wheel slip data. For the wheel
used in this example, and for m pk = 0.85, the coefficients are determined to be
a = 1.1; b = 0.20; and c = 0.0035.
The wheel slip is then determined by equating the road surface condition
dependant traction limit, (1.26), with the vehicle propulsion system tractive effort
given as (1.11). Figure 1.26 is a plot of (1.21) for the values given.
Note from Figure 1.26 that wheel traction reaches its peak value at a slip of
20% on dry concrete or asphalt. The trace in Figure 1.26 commences at the wheel
freewheeling point (0,0) and ends at the friction value for locked wheel skid (100%
slip). With a single occupant, (1.19) predicts the traction limit versus wheel slip
shown in Figure 1.27 using data from Table 1.13 on level terrain.
To get an appreciation for the traction limit defined in Figure 1.27 it can be
seen that for the vehicle mass with one occupant and a wheel slip value of 20% the
maximum acceleration achievable is

F trac lim
ð m a þ m us þ N p m p Þ g ¼ 0 : 522

a ¼ ð g Þ ð 1 : 22 Þ