Etapa Jurisdiccional

For a wheel to roll in a longitudinal (straight line) direction, it requires either a torque or an external force driving the motion. This motion cannot exist without a frictional component at the tyre-road interface. Resulting from friction, longi- tudinal and transverse traction forces are formed. These forces are related to the corresponding normal force of the wheel.

Normal force

The vertical reaction vector exerted by a tyre load on the road's surface is known as the normal force Fz. The magnitude and position of the normal force mainly

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depends on the total weight of the vehicle, the wheel's location with reference to the vehicle's centre of gravity, the slope of the road and the longitudinal acceleration of the vehicle. As the tyre rotates it is subjected to deformation in the contact patch. The tyre material is deected vertically as it enters the contact patch and bounces back as it exits the contact patch. The energy spent deforming the tyre is not completely recovered as it returns to its original shape, resulting in internal damping (Rajamani, 2011). The loss of energy results in an asymmetric vertical load and is represented by a force on the tyre known as the rolling resistance Fr

(Figure 2.6). Haney (2003) states that the asymmetry becomes more exaggerated with speed.

Figure 2.6: Tyre normal force.

Rolling resistance is in the opposite direction to the vehicle's motion an can be calculated with Equation 2.1. The variable ∆x is not easily measurable and therefore scholars such as Rajamani (2011) and Moore (1975) indicate that Fr is simply taken

as a proportionate of Fz with a constant rolling resistance coecient f. According

to Rajamani (2011) f has a range of 0.01 to 0.04 and has a typical value of 0.015 for passenger vehicles.

Fr =

Fz(∆x)

Rs (2.1)

2. LITERATURE REVIEW Longitudinal traction force

As the tyre rolls and deects in the contact patch, a reduction of the undeected radius Ro results in the formation of traction forces Fx in the longitudinal direction

between the tread rubber and the road. Moore (1975) explains the shape of the traction force as follows: Due to the change in radius, an eective radius Re is

selected to calculate the forward velocity V . If ω denotes the angular velocity of the wheel V is calculated with Equation 2.2. It is apparent that Rs< Re< Ro.

V = ω × Re (2.2)

Figure 2.7: Tyre longitudinal traction force.

The tangential velocity of point g on the undeected tread surface is given by ω×Ro,

relative to the wheel centre o. As g approaches the entry point a of the contact patch, its velocity decreases causing circumferential compression in the tread band, hence the velocity of point a relative to o exceeds the underlying road velocity, ω × Re.

This causes rearward longitudinal slip or attempted slip in the region a to b (Figure 2.7). Slip is thus dened as the dierence in the axle velocity V and the equivalent rotational velocity ω × R. From b to centre of contact c and to point d (the mirror image of b), the eective rolling radius is marginally less than Re. This results in a

limited amount of forward slip. Within the last quarter of the contact patch from d to 22

e, the velocity of the tread elements increases resulting in rearward slip or attempted slip.

This slip direction reverses a number of times and is represented by a sinusoidal-like wave form with a positive and a negative peak. Tielking and Roberts (1987) indicated that the slip direction changed again at the exit point of the contact patch, resulting in a second positive peak value. The absolute dierence of the peak values is equal to the rolling resistance Fr. De Beer et al. (2005) indicate that the longitudinal

stress is approximately 12% of the associated vertical stress. Transverse traction force

Deection and bending of the tyre side walls during rolling cause inward lateral movement of the tread rubber. The tread rubber assumes an hourglass shape over the distance of the contact patch. The lateral movement results in an outward shear stress distribution over the length and across the width of the contact patch (Figure 2.8). The magnitude of the transverse shear depends on tyre construction, tread rubber, vertical load and ination pressures. Under inated tyres and heavy load

Figure 2.8: Tyre transverse traction force.

conditions are the major contributors to the transverse traction force Fy (De Beer

et al., 2005). Scholars such as Tielking and Roberts (1987) and De Beer et al. (2005) indicate that the transverse traction stresses are greater than longitudinal traction stresses. De Beer et al. (2005) estimated the transverse traction stress at 17% of the associated vertical stress which is 5% more than the longitudinal traction stress.

2. LITERATURE REVIEW Braking and acceleration

With the onset of braking the rolling resistance increases Frto a substantially greater

breaking force Fb (Figure 2.10). The severity of braking is measured by the brake

slip ratio sb and is dened in Equation 2.3, where wro is the angular velocity of the

wheel in free rolling and wbr the angular velocity of the wheel in braking (Moore,

1975).

sb =

wro− wbr

wro

|_v=constant (2.3) A braking torque Tb applied to the wheel reduces the angular velocity below the

free rolling value, resulting in a positive brake slip ratio. Brake slip ratio is zero for free rolling and unity for locked-wheel braking (Moore, 1975). The variation of a braking force coecient Fb/Fz with slip ratio is illustrated in Figure 2.9 and

peaking at approximately 0.2 to 0.3. According to Moore (1975) braking within the vicinity of zero to point A is desirable since directional stability is preserved by rolling rather than locked-wheel sliding. Once point A is reached, sb rapidly reaches

unity, resulting in tyre slide or locked-wheel braking. Miller et al. (2001) indicated that general wheel slip is between 0% and 3% under normal driving conditions.

Figure 2.9: Relationship between Fb/Fzand sb; Fd/Fz and sd (Moore, 1975).

During braking the axle to ground height Rs reduces in comparison to free rolling

conditions. The distribution of vertical pressure shifts more towards the front half of the contact patch, increasing the asymmetry oset ∆x. The braking force Fb

has an approximated triangular distribution over the contact patch and the total longitudinal traction force consists of Fb superimposed on Fr (Figure 2.10).

Acceleration is in many aspects similar to braking though in the opposite direction. Instead of wheel-lock once point A (Figure 2.9) is reached, the drive slip ratio sd

rapidly reaches unity resulting in wheel spin. The drive slip ratio is dened in Equation 2.4, where wro is the angular velocity of the wheel in free rolling and wdr

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the angular velocity of the wheel subject to a driving torque (Moore, 1975). sd=

wdr− wro

wdr

|_v=constant (2.4) The driving torque Td results in a longitudinal traction force Fd with a similar

distribution along the contact patch as Fb, but the motion is in the direction of

travel as illustrated in Figure 2.10. The total longitudinal traction force consists of Fd superimposed on Fr.

Figure 2.10: Distribution of Fz and Fx for braking and acceleration.