Road Vehicle Performance
EXAMPLE 2.7 BRAKE-FORCE PROPORTIONING
( )
/ r rl
f r max
f rl
l h f
BFR l h f
µ µ
+ +
= − + (2.30)
where
BFRf/r max = the brake force ratio, allocated by the vehicle’s braking system, that results in maximum (optimal) braking forces, and
Other terms are as defined previously.
It follows that the percentage of braking force that the braking system should allocate to the front axle (PBFf) for maximum braking is
/
100 100
f 1
f r max
PBF = − BFR
+ (2.31)
and the percentage of braking force that the braking system should allocate to the rear axle (PBFr) for maximum braking is
/
100
r 1
f r max
PBF = BFR
+ (2.32)
EXAMPLE 2.7 BRAKE-FORCE PROPORTIONING
A car has a wheelbase of 100 inches and a center of gravity that is 40 inches behind the front axle at a height of 24 inches. If the car is traveling at 80 mi/h on a road with poor pavement that is wet, determine the percentages of braking force that should be allocated to the front and rear brakes (by the vehicle’s braking system) to ensure that maximum braking forces are developed.
SOLUTION
The coefficient of rolling resistance is
2.9 Principles of Braking
31
impending slide). Applying Eq. 2.30 gives( )
Using Eq. 2.31, the percentage of the force allocated to the front brakes should be
/
and using Eq. 2.32 (or simply 100 − PBFf), the percentage of the force allocated to the rear brakes should be
It is clear from Eq. 2.30 that the design of a vehicle’s braking system is not an easy task because the optimal brake-force proportioning changes with both vehicle and road conditions. For example, the addition of vehicle cargo and/or passengers will change not only the weight of the vehicle (which affects frl in Eq. 2.30), but also the distribution of the weight, shifting the height of the center of gravity and its location along the vehicle’s longitudinal axis, and this will change the optimal brake force proportioning (BFRf/rmax). This is particularly problematic for trucks because of the large weight and center of gravity differences between loaded and unloaded conditions. Similarly, changes in road conditions produce different coefficients of adhesion, again changing optimal brake force proportioning. As a result of the uncertainties in weight and road conditions, vehicle designers often choose a compromise value of brake force proportioning that, on average, provides good braking but is rarely, if ever, optimal.
It is important to note that studies have indicated that if wheel lockup is to occur, it is preferable to have the front wheels lock first because having the rear wheels lock first can result in uncontrollable vehicle spin. Front-wheel lockup results in loss of steering control, but the vehicle will at least continue to brake in a straight line.
Technological advancements in braking systems since the late 1970s have resulted in vehicles that are increasingly capable of proportioning brake forces in a manner that is closer to optimal and avoids the dangerous rear-wheel–first lockup due to front-wheel underbraking.
Because true optimal brake force proportioning is seldom achieved in standard non-antilock braking systems, it is useful to define a braking-efficiency term that reflects the degree to which the braking system is operating below optimal. Simply stated, braking efficiency is defined as the ratio of the maximum rate of deceleration, expressed in g’s (gmax), achievable prior to any wheel lockup to the coefficient of road adhesion:
b gmax
Ș =
ȝ (2.33)
where
ηb = braking efficiency,
gmax = maximum deceleration in g units (with the absolute maximum = µ), and µ = coefficient of road adhesion.
2.9.3 13B13B13BAntilock Braking Systems
Many modern cars have braking systems designed to prevent the wheels from locking during braking applications (antilock braking systems). In theory, antilock braking systems serve two purposes. First, they prevent the coefficient of road adhesion from dropping to slide values (see Table 2.4). Second, they have the potential to raise the braking efficiency to 100%. In practice, designing an antilock braking system that avoids slide coefficients of adhesion and achieves 100% braking efficiency (ηb = 1.0) is a difficult task. This is because most antilock braking system technologies detect which wheels have locked and release them momentarily before reapplying the brake on the locking wheel. The wheel lock detection speed, speed of brake force reallocation, and braking system design (the amount of braking forces that can be accommodated by the vehicle’s front and rear brake discs and calipers) all impact the overall effectiveness of the antilock braking system. Early antilock braking systems often fell short of achieving 100% braking efficiency, and in many cases, an expert driver operating a non-antilock braking car could modulate the brakes to achieve shorter stopping distances than cars equipped with antilock brakes.
However, advances in antilock braking system technology continue to bring us closer to 100% braking efficiency.
2.9.4 14B14B14BTheoretical Stopping Distance
With a basic understanding of brake force proportioning and the resulting braking efficiency, attention can now be directed toward developing expressions for
2.9 Principles of Braking
33
minimum stopping distances. By inspection of Fig. 2.6, it can be seen that the relationship among stopping distance, braking force, vehicle mass, and vehicle speed isγb = mass factor accounting for moments of inertia during braking, which is given the value of 1.04 for automobiles [Wong 2008], and
Other terms are as defined in Fig. 2.6.
Integrating to determine stopping distance (S) gives
1
Substituting in the resistances (see Fig. 2.6), we obtain
1
W sin θg = grade resistance (positive for uphill slopes and negative for downhill slopes), and
Other terms are as defined previously.
To simplify notation, let resistance, frl, is constant and can be approximated by using the average of initial (V1) and final (V2) speeds in Eq. 2.5 [V = (V1 + V2) / 2]. With this assumption (which
introduces only a very small amount of error), and letting m = W/g and Fb = µW,
With braking efficiency considered, the actual braking force is
b b
F =η µW (2.41)
Therefore, by substitution into Eq. 2.40, the theoretical stopping distance is
12 aerodynamic resistance is ignored (due to its comparatively small contribution to braking), integration of Eq. 2.35 gives the theoretical stopping distance as