Vehicle Dynamics and safety

5 Tyre characteristics and modelling

5.1 Introduction

The handling performance and directional response of a vehicle are greatly

influenced by the mechanical force and moment generating characteristics

of the tyres. In road vehicle dynamics the manner in which a vehicle accel-

erates, brakes and corners is controlled by the forces generated over four

relatively small tyre contact patches. If the tread pattern and the road

texture are also considered it is clear that the area of frictional contact

is reduced even more significantly. Figure 5.1 shows the deflection of a

vehicle’s tyres under hard cornering and helps to illustrate the significant

requirements on the tyre to produce forces that control the relatively large

mass of the vehicle.

It is not intended here to discuss the construction of the tyre carcass, mater-

ials or tread pattern. This is addressed by more general texts on vehicle

dynamics (Gillespie, 1992) or more focused books on the subject of tyres

(French, 1989; Moore, 1975). Rather this chapter will start by describing

the mechanisms required to generate the vertical tyre forces that support

the vehicle, the longitudinal forces required for driving and braking and the

lateral forces needed for cornering. The distribution of pressure and stress

will also generate local moments acting at the tyre contact patch. A good

Fig. 5.1 An example of tyre deflection under hard cornering (courtesy of Auto

Motor und Sport)

Multibody Systems Approach to Vehicle Dynamics

254 X SAE

Left

Y Y

SAE SAE

F F Direction of travel

Y Y Right X SAE

Y Y

SAE SAE

F F

Y Y

Fig. 5.7 Generation of tyre lateral forces due to plysteer

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.

5.3 The tyre contact patch

5.3.1 Friction

The classical laws of friction as often taught to undergraduates can be sum-

marized 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 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) coefficent 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

Tyre characteristics and modelling 255

Direction of sliding Tyre material

Adhesive forces due to molecular bonding

Road surface

Fig. 5.8 Frictional force component due to adhesion

F

F Loading Unloading δ

Fig. 5.9 Hysteresis in rubber

conditions such as high tyre pressures. The concept of a coefficient of fric-

tion associated with static and sliding conditions will, however, prove use-

ful 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 these 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 displace-

ment 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 Dynamics

256 Direction of sliding Loading

Unloading

Road surface

Fig. 5.10 Loading and unloading of tyre rubber in the contact patch

be zero it can be seen from Figure 5.10 that an element of rubber in the

contact patch will be subject to continuous compressive loading and

unloading.

In the idealized situation of no friction as the tyre slides over the irregular

road surface compressive forces normal to the surface are generated as the

rubber is loaded and unloaded. Due to the hysteresis in the rubber the sum

of the loaded forces is greater than the sum of the unloaded forces resulting

in, for example here, a resultant braking force opposing the direction of

sliding.

5.3.2 Pressure distribution in the tyre contact patch

In order to understand the manner by which forces and moments are gen-

erated in the contact patch of a rolling tyre an initial appreciation of the

stresses acting on an element of tread rubber in the contact patch is

required. Each element will be subject to a normal pressure p and a shear

$

stress acting in the road surface. In theory the element will not slip on the

$

road if p where is the coefficient of friction between the tread rub-

ber and the road surface.

The pressure distribution depends on tyre load and whether the tyre is sta-

tionary, rolling, driven or braked. The pressure distribution is not uniform

and will vary both along and across the contact patch. In order to under-

stand the mechanics involved with the generation of forces and moments in

the contact patch some simplification of the pressure distribution will be

adopted here starting with Figure 5.11 where typical pressure distributions

in the tyre contact patch for a stationary tyre and the effects of inflation

pressure are considered.

Generally the pressure rises steeply at the front and rear of the contact

patch to a value that is approximately equal to the tyre inflation pressure.

Overinflation causes an area of higher pressure in the centre of the contact

patch while underinflation leads to an area of reduced pressure in the

centre of the patch.

When the tyre is rolling 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.

Multibody Systems Approach to Vehicle Dynamics

260 m t c

k z

z z

P {X }

SAE 1

{Z }

SAE 1

Fig. 5.14 Vertical tyre force model based on a linear spring damper

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 for-

mulated to simulate the behaviour of the aircraft on the runway, particu-

larly 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 for-

ward velocity V of the wheel relative to the road surface is given by

R

V (5.15)

e Tyre characteristics and modelling 261

R

V = e

O R u Tread

R

t R

t

material

V = V =

u R u

R

l e Compression

B D P C A {X }

Rear Front SAE 1

R

t

V = e

Tangential velocity R

t

V =

of tread relative to O 1

R

t

V = e

Direction of slip relative

to the road surface

Longitudinal

shear stress

Fig. 5.15 Generation of slip in a free rolling tyre

The tread material approaching the front of the contact patch will have a

t

tangential velocity V relative to the wheel centre O given by

R

t (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 tread material just before

it enters the contact patch.

As the tread material enters the contact patch at point A the rearward tangen-

tial velocity relative to the wheel centre is just slightly greater than the forward

velocity of the vehicle. This results in initial rearward slip of tread material

relative to the road surface between point A and C. At point C it is assumed

Multibody Systems Approach to Vehicle Dynamics

262 Undeformed

tyre Deformed

tyre Rear Front