Vehicle Dynamics and safety

Generation of tyre lateral forces due to plysteer

It is evident that the opposite occurs to conicity in that switching a tyre from the left to the right of the vehicle does not reverse the lateral force direction. Thus for a vehicle fitted with tyres all exhibiting the same plysteer there will be a tendency for the vehicle to drift off a straight course without some steering correction. A correction will modify the course of the vehicle but will cause the rear wheels to 'track' to the side of the front wheels so that the vehicle progresses with a crab like motion, albeit imperceptible to the driver.

The tyre contact patch

Friction

The classical laws of friction as often taught to undergraduates can be summarized as:

1. Friction is a property of two contacting surfaces. It does not make sense to discuss friction as if it were a material property.
2. Frictional force is linearly proportional to normal force and can be defined using a coefficient of friction (frictional)

1. The frictional force between two surfaces is directly proportional to the normal force pressing the surfaces together.

2. The coefficient of friction is a constant that depends on the nature of the surfaces in contact (e.g. roughness, material properties) and is independent of the normal (force/normal force).

3. The coefficient of friction is independent of contact area between the two surfaces.

4. The static coefficient of friction (stiction) is greater than the kinetic (sliding) coefficient of friction.

5. The coefficient of friction is independent of sliding speed.

A detailed treatment of this subject with regard to tyres is given by Moore (1975), where it is shown that the above laws are flawed, or limited in certain conditions such as high tyre pressures. The concept of a coefficient of friction associated with static and sliding conditions will, however, prove useful for describing the tyre models used later in this chapter.

For tyres, the friction generated between the tread rubber and the road surface is generated through two mechanisms:

Being adhesion and hysteresis. The adhesive component, shown in Figure 5.8, results from molecular bonds generated between the exposed surface atoms of rubber and road material in the contact area. This is the larger component of friction on dry roads but is greatly reduced when the road surface is contaminated with water or ice. Hence the use of 'slick' tyres, with no tread and increased surface contact area, for racing on dry roads.

In order to understand the hysteresis mechanism consider a block of rubber subjected to an increasing and then a decreasing load as shown in Figure 5.9. As the rubber is loaded and unloaded it can be seen that for a given displacement the force F is greater during the loading phase than the unloading phase.

If we continue to consider the situation where a non-rotating tyre is sliding over a non-smooth surface with a coefficient of friction assumed to Multibody Systems Approach to Vehicle Dynamics256 Direction of sliding Loading Unloading Road surface.

5.3.2 Pressure distribution in the tyre contact patch

It will be shown later that pressure distribution in the contact patch is not symmetric and is greater towards the front of the contact patch.

A linear model of tyre vertical force may need to be extended to a non-linear model for applications involving very heavy vehicles or studies where the tyre encounters obstacles in the road or terrain of a similar size to the contact patch or smaller. This could also be applicable for parallel work in the aircraft industry where established tyre models have been formulated to simulate the behaviour of the aircraft on the runway, particularly on landing, and potential problems with wheel shimmy (Smiley, 1957; Smiley and Horne, 1960). Where a non-linear model of vertical tyre force is required, the most straightforward approach would be to represent the stiffness-based component of the force by a cubic spline.

Interpolation of measured static force–displacement data.

5.4.3 Longitudinal force in a free rolling tyre (rolling resistance)

Under normal driving conditions a tyre is continually subject to a wide range of tractive driving and braking forces. This section discusses the formulation of driving and braking forces under pure slip conditions, i.e. straight-line motion only. The more complex situation of combined slip, for example simultaneous braking and cornering, is addressed later in this chapter.

As a starting point it can be shown that slip will always be present in the tyre contact patch even in the absence of tractive driving and braking forces. Consider first the free rolling tyre shown in Figure 5.15 and the mechanism that leads to the generation of longitudinal slip. The model used in Figure 5.15 has simplifications but will help to develop an initial understanding. As the tyre rolls forward the radius reduces as tread material approaches point A at the front of the tyre contact patch.

At this point we can say that the forward velocity V of the wheel relative to the road surface is given by:

RV = e^{O R u TreadRt Rtmaterial}V = V = u R uRl e CompressionB D P C A {X }Rear Front SAE 1RtV = eTangential velocity RtV = of tread relative to O 1RtV = eDirection of slip relative to the road surface Longitudinal shear stress

The tread material approaching the front of the contact patch will have a tangential velocity V relative to the wheel centre O given by:

Rt (5.16)V u

As the tread material gets close to the start of the contact patch, the tyre radius decreases causing the tangential velocity of the tread material to decrease, causing circumferential compression of the tread material just before it enters the contact patch.

As the tread material enters the contact patch at point A, the rearward tangential velocity relative to the wheel centre is just slightly greater than the forward velocity.

obser-vations (Moore, 1975) indicate that the tangential speed does not reduce tothis level. Between point D and B the radius recovers to a value greater thanthe effective rolling radius causing the direction of slip to reverse again to arearward direction.

It is clear that the direction of slip changes several times as tread movesthrough the contact patch resulting in the distribution of longitudinal shearstress of the type shown at the bottom of Figure 5.15. The shear stress isplotted to be consistent with the SAE reference frame and is not symmetricwith the net effect being to produce an overall force, the rolling resistance,acting in the negative X direction.

SAE

It should be noted that the two-dimensional model presented is notfully representative as components of lateral slip are also introduced ina free rolling tyre due to deformation of the side walls as shown inFigure 5.16.

As the tyre carcass deforms in the vicinity of the contact patch the defor-mation of the side walls creates

additional inwards movement of the tread material (Moore, 1975). This causes the contact patch to assume an hour-glass shape creating an effect referred to as 'squirm' (Gillespie, 1992) as the tread material moves through the contact patch.

Before moving on to consider the driven or braked tyre we will now consider the rolling resistance forces generated in a free rolling tyre. Rolling resistance results from energy losses in the tread rubber and side walls. Energy loss in the tread rubber is produced by