PARAMETERS INFLUENCING TYRE MODELLING FOR SUSPENSION DESIGN

Saurabh Ranjan Sharma, AUE, For Student Formula Competetion 2015

ABSTRACT

The purpose of this document is to describe important tyre characteristics and their effects on vehicle

performance. The said vehicle to be designed and manufactured for SUPRA SAE India, Students Formula

2015. The characteristics considered are those that the judges have deemed important for discussion on

tyres at the competition and have actually proved to be crucial for precise design of an automobile

suspension.

INTRODUCTION

Importance of a tyre in a racing competition cannot be overemphasized given that it is the only means

of contact between the road and the vehicle. The tyres are required to produce the forces necessary to

control the vehicle. It will be safe to say that they are at the heart of vehicle handling and performance.

Insight into the various tyre parameters will help us students in many ways. Knowledge of these

characteristics and their effects on racecar performance can give the engineer insight into performance

optimization. A firm grasp on what influences a tyre’s behavior and how it affects the overall vehicle

dynamics will better prepare the students to score higher during design judging competition. However,

tyre itself is very complex. What we can do is study the specific tyre characteristics in relation with the

racecar dynamics for which purpose we are designing the prototype in the first place.

The Parameters required for the design are:

1. Co-efficient of friction

2. Vertical stiffness

3. Slip angle

4. Slip ratio

5. Camber angle

6. Cornering stiffness

7. Camber stiffness

8. Self-aligning torque

9. Pneumatic trail

10. Effective rolling radius

11. Current condition of the tyre in use from the previous vehicle.

SAE CO-ORDINATE SYSTEM

TIRE MODEL ADOPTED

Tyres are complex composites containing many layers of different materials, anisotropic. Hence its

material properties and structure cannot be used to derive its behavior. Simpler empirical models of the

tyre has been adopted to derive its behavior. Though no model truly addresses the complexity of a real

world tyre, realistic results can be obtained when empirical stiffness values are used. The simplest of the

models is the elastic foundation model (Ref: 1).

Each element in an elastic foundation model is considered to be an independent spring. This

model allows for discontinuity in the displacement distribution and slope of center line.

CO-EFFICIENT OF FRICTION

- Defined as the unit-less ratio of friction force to normal force, not proportional to surface area

of contact.

In case of tyres, this is wrong. Because of viscoelastic behavior of rubber, deformation occurs in tyre,

both elastic and plastic in a non-linear fashion due to natural property of polymer chains.

As the tyre is loaded, total frictional force increases with surface area but co-efficient of friction

is reduced. Hence, coefficient of friction of a tyre can be greater than 1. It can depend upon many

factors: atmospheric dust, humidity, temperature, vibration and extent of contamination.

As the temperature rises due to some reason (continuous skidding/high speeds), the coefficient of

friction attains highest value in the optimal temperature range then starts decreasing. When slip angle

becomes large, the rear of the patch begins to slide and attains low coefficient of friction.

Availability of coefficient of friction facilitates Modelling of Braking system and is required for the

further calculations.

VERTICAL TYRE STIFNESS

The tyre acts as a spring, a part of the suspension system when in contact with the road and hence plays

an important part in the overall suspension design.

Thus as any conventional spring, the tyres have a stiffness/rate.

F = Kt*x

Kt = F/x N/m

This rate is related to the overall ride rate of a vehicle by:

1/Kr = 1/Kw + 1/Kt

SLIP ANGLE (α)

- Defined as the angle between the wheel’s direction of travel and wheel heading.

The slip angle produces a component of lateral force (FY). This lateral force acts from behind the center

line of the wheel in a direction that it attempts to realign the tyre. Evident from elastic foundation

model, there is a friction limited value of the lateral force due to the slip angle.

c.d ≤ (µ.FV) / l c: foundation stiffness

d: tyre center line displacement

l: tyre footprint length

Maximum non-slide force, FY = 0.5*c*l2*α

0.5*c*d*l = 0.5*c*(µ.FV) / l*c*l = 0.5* µ.FV

At high slip angles, rear of the footprint slides on the road surface: Lesser capacity for lateral force,

reduction of self-aligning torque.

When not completely sliding, lateral force is independent of coefficient of friction, depends on

foundation stiffness.

FY = Fα = Cα*α = CS*FV*α [Cα = CS*FV] CS: Cornering stiffness coefficient

Cα: Cornering stiffness

Since central/drag force ratio reduces with slip angle, higher cornering stiffness is desirable

𝐹𝑠

𝐹𝑑=

𝐶𝛼∙𝛼∙cos 𝛼

µ𝑟∙𝐹𝑣+𝐶𝛼∙𝛼∙sin 𝛼

Ref. Fig:

SLIP RATIO (% SLIP)

- Slip in longitudinal plane

SR = (Ω-Ω)/ Ω = Ω*Re/V*cos α – 1 Ω: Angular velocity of driven wheel

Ω: Angular velocity of free rolling wheel

Re: Effective rolling radius

V: Velocity

CAMBER ANGLE

FV = F¥ = C¥*¥ = Cc*FV*¥ ¥: Camber angle

C¥: Camber stiffness

Cc: Camber stiffness coefficient

Camber stiffness is rate of change of camber force with camber angle (∆F¥/∆¥)

TOTAL LATERAL FORCE

FY = Fα + F¥ = Cα*α + C¥*¥

FY = FV*(CSα + CC¥)

SELF ALIGNING TORQUE

- Resultant of lateral force and pneumatic trail

Trail may also be induced mechanically from suspension geometry with presence of castor and kingpin

offset.

MZ = FY*t = FY.l/6 = (c*l2*tan α)/12

MZ = µ2*FV2/ (4*c*l tan α) – (µ3* FV

3)/(6*c2*l3*tan2α)

As slip angle increases, lateral force increases, self-aligning torque decreases. This means that as the

slip angle decreases and lateral force begins to reach its limit, the driver loses the feedback through the

tyres. However this loss of feedback alerts him of the slide beforehand.

Self-aligning torque also affected by tyre pressure.

PNEUMATIC TRAIL

- Moment arm through which lateral force acts.

As per foundation stiffness model, the lateral force acts from behind the wheel’s center line.

Now, considering the contact patch as near triangular distribution,

t = MZ/FY=l/6

EFFECTIVE ROLLING RADIUS

Front and rear tyre loaded radius is required to calculate the CG location of non-suspended mass.

REFERENCES

1. Dixon, J.C. (1996). Tires, Suspension and Handling.

2. Smith, Nicholas D. (2004). Understanding parameters influencing Tire Modelling.

3. Milliken & Milliken (1995). Racecar vehicle dynamics.