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Old 12-12-2012, 08:13 PM   #11
catweasel67
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The maximum braking

The maximum braking allows to achieve the maximum deceleration, that is:


The deceleration is a function of the wheelbase p, of the vertical h and horizontal position of the center of gravity and of the coefficients of friction, not of the vehicle mass.

The ratio between the front wheel braking force and the total available braking force depends only from the geometrical properties and the coefficients of friction:



Figure 6. Deceleration lines and braking ratio
wheelbase p=1.4 m; height of the center of gravity h=0.7 m; horizontal position of the c.o.g. b=0.7 m

The above figure shows that the deceleration increases with the coefficients of friction. Due to the load transfer the front wheel braking force is bigger than the rear wheel braking force.
The braking ratio between the front and the rear wheel are expressed by the red lines.
The horizontal axis represents a rear wheel only braking condition; the vertical axis a front wheel only braking condition.
If the coefficients of friction are low, the importance of the rear braking force is not negligible, as in high friction condition.
The maximum deceleration before the forward tilting is 1 g.
As an example consider braking a vehicle with a deceleration equal to 0.5g; it is possible to reach the desiderate decelerationusing different ratio braking. Braking using the front wheel only, requires a front coefficient of friction equal to 0.68 (point A).
If the used braking ratio is 80% on the front wheel, 20% on the rear wheel, the same 0.5g deceleration requires a front coefficient of friction equal to 0.55 and equal to 0.4 on the rear wheel (point B).
Which is then the optimum braking to achieve the 0.5g deceleration?
If the maximum coefficients of friction are the same for both the wheels, Figure 7 shows that the maximum deceleration is achieved when both the tires are used in the same way.
As an example consider a coefficient of friction equal to 0.8 for both the tires; the maximum deceleration (0.8g) is achieved with a braking ratio equal to 90:10.
Using the front brake only, the maximum deceleration is equal to 0.67g; using only the rear brake is equal to 0.29g.
In slipping condition the coefficient of friction is equal to 0.4, the maximum deceleration is equal to 0.4g and the optimum braking ratio is 30:70.
Se il fondo stradale è più scivoloso e i coefficienti di aderenza di entrambe le ruote risultano pari a 0.4 la frenata ottimale si ha con una diversa ripartizione (30/70) e fornisce una decelerazione pari a 0.4 g.

The 45° line represents the condition mf= mr and is the optimum braking; this line intersects different braking ratio lines as function of the desiderated deceleration.


Figure 7. Braking action of dry(0.8) and wet(0.4) surfaces

Figure 8 shows that the optimum braking line is tangent to the 50:50 braking ratio; it do not intersect the ration curves with the rear wheel braking force bigger than the front wheel braking force.
The above consideration is valid even if the static load is bigger on the rear wheel.
The optimum braking line is always tangent to the ratio curve having the same values of the static loads ratio.

As an example, if the static loads ratio is 45:55 (45% on the front wheel, 55% on the rear wheel), the optimum ratio curve is the 45:55 and is tangent in the origin to the optimum braking line.

Figura 8. Braking action of dry(0.8) and wet(0.4) surfaces
wheelbase p=1.4 m; height of the center of gravity h=0.7 m; horizontal position of the c.o.g. b=0.7 m


or. put another way.

use both brakes to maximise braking.
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