Page 370 - Automotive Engineering Powertrain Chassis System and Vehicle Body

Braking systems C HAPTER 12.1

which can be combined with equation 12.1.32 and the If, however, the ratio has been set so that the rear
static axle loads, equations 12.1.26 and 12.1.27, to give axle locks in preference to the front, then the brake
force generated at the rear axle when about to lock is
Pzh given by:
R ¼ F þ (12.1.37)
f
f
l
Pzh T r ¼ mR r
R r ¼ F r (12.1.38)
l Pzh (12.1.45)
¼ m F r
The above are in accord with those given in the EEC l
Directive and they show that a change in axle load in In this case, the brake force that is generated at the
favour of the front axle occurs during a braking ma- front axle is not necessarily the limiting value and its
noeuvre. In order for each axle to be simultaneously on magnitude is found from the brake ratio as:
the verge of locking, the brake force generated at each
axle must be in direct proportion to the vertical axle load. x
This means that to fully utilize the available tyre–ground T ¼ T r f
f

adhesion, the braking system must support an infinitely x r Pzh x (12.1.46)
variable brake ratio. ¼ m F r f
l x r
Consider first the case of a vehicle in which the brake
ratio is fixed. If the ratio has been set so that the front which leads to a total brake force of:
axle locks in preference to the rear, then the brake force
generated at the front axle when about to lock is given by:
T ¼ Pz ¼ T þ T r
f

T ¼ mR f Pzh x f Pzh
f
¼ m F r þ m F r
Pzh (12.1.39) l x r l
¼ m F þ
f
l Pzh 1
¼ m F r (12.1.47)
l x r
During the same braking event, the rear axle is also gen-
eratingabrakeforcethathasnotexceededits limitingvalue and this can be solved for the deceleration as a proportion
and this is found by considering the vehicle brake ratio: of g to be:
x f T f
R ¼ ¼ (12.1.40) z ¼ lmFr (12.1.48)
x r T r
Pðlx r þ mhÞ
from which Direct solution of equations 12.1.44 and 12.1.48 for z
is straightforward, however, greater insight to the me-
x r Pzh x r
T r ¼ T f ¼ m F þ (12.1.41) chanics of the braking process can be gained through the
f
x f l x f following graphical solution that deals with each axle in
turn.
leading to a total brake force of: The adhesion force acting between the front tyres
and ground depends upon the ratio of the tangential
T ¼ Pz ¼ T þ T r forces at the front and rear wheels due to the brake
f

Pzh Pzh x r torques. It is therefore linked to the fixed brake ratio
¼ m F þ l þ m F þ l x (12.1.42) and so the front adhesion force, T f , as a proportion of
f
f
f
the total is given by:
which reduces to:
T ¼ x T
f
f
Pzh 1 (12.1.49)
T ¼ Pz ¼ m F þ (12.1.43) ¼ x Pz
f
f
l x f
This equation can be rearranged to yield the maximum which when normalized to the vehicle weight, P,
value of deceleration as a proportion of g as: becomes:
mF f T f
z ¼ (12.1 44) ¼ x z (12.1.50)
f
Pðlx mhÞ P
f
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