Decomposition Analysis of an Automotive Powertrain Design ...

S = 1 - (π / 30) r e N d /V . (18)
The force that can be transmitted at the wheel without the wheel slipping (adhesion) is limited by
the product of the normal force and the coefficient of adhesion. The wheel slips when the axle
driving torque exceeds the adhesion torque according to
T d / r e > µ o F nd (19)
and upon slip, the tractive force becomes,
F d = F nd µ (20)
where F nd is the normal force at the driving wheel. The adhesion coefficient, µ, depends on slip
and tire material as shown in Figure 6. The traction coefficient is either experimentally determined
(SAE Standard J1987) or represented analytically. Starkey et al. (1988) suggest the relation
µ = 1.07 µ o (1 - e -17.73 S/(S-1) ) - 0.285 S / (S-1). (21)
In either case, for partitioning considerations, the functional dependence remains unchanged. The
entire behavior is explicitly represented by
where
F d = F nd µ(S) + (1 - |sign (S)| ) (T d - (π / 30)I d dN dr /dt ) / r e (22)
1 for S > 0
sign(S) = { -1 for S < 0
0 for S = 0.
Clearly, the driving force F d is discontinuous. When adhesion is lost, the tractive force
becomes zero, T d accelerates the wheel, slip increases, µ increases with S, and maximum traction
is regained.
The driveline is represented as a stiff shaft and speed reducer with the inertias, I ds , I fd , and
the final drive ratio, ξ fd , given in Appendix A. The relationship between drive shaft speed and
wheel speed is