4-1. Tyre Dynamics

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reinforcing materials such as fabric and wire. Tire companies were first started in the early 20th century. Today over 1 billion tires are produced annually.
Tyre Dynamics

Modelling of Automotive Systems

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Wheels have been used for at least 5700 years. Near the northern side of the Caucasus several graves were found, in which since 3700 BC people had been buried on wagons or carts.

Modelling of Automotive Systems

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Tires or tyres are pneumatic enclosures used to protect and enhance the effect of wheels. The earliest tires were bands of iron made by by skilled craftsman, known as a wheelwright. The first practical pneumatic tire was made by the Scot John Boyd Dunlop for his son's bicycle, in an effort to prevent the headaches his son had while riding on rough roads. The pneumatic tire also has effect of vastly reducing rolling resistance compared to a solid tire on rough ground. It is able to "absorb" bumps in the road as it rolls over them without experiencing a reaction force opposite to the direction of travel, as is the case with a solid (or foam-filled) tire. Pneumatic tires are made of a flexible elastomer material such as rubber with reinforcing materials such as fabric and wire. Tire companies were first started in the early 20th century. Today over 1 billion tires are produced annually. Modelling of Automotive Systems

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Veff

Modelling of Automotive Systems

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Effective tyre radius reff − radius which relatesωw to the linear longitudial velocityof the wheelVeff Veff = reff ωw

(it is faster than Vx when in traction)

Vx = reff ω eff

φ=

a reff

reff =

a

φ φ

ωw = ,

t t − time taken by an element of tyre to move through half the contach patch

Modelling of Automotive Systems

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Veff = reff ω w = rstat = rw −

a , t

Fz kt

kt − virtical tyre stiffness rstat = rw cos φ ,

a = rw sin φ

Therefore reff =

a

φ

=

rw sin φ

φ

r and φ = cos −1  stat  rw

 , 

i.e.   −1  rstat     sin  cos    rw    reff = rw r  cos −1  stat   rw 

Modelling of Automotive Systems

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Since reff =

a

φ

=

rw sin φ

φ

,

reff < rw

On the other hand, since a rstat

= tan φ ,

then a = rstat tan φ , or rref φ = rstat tan φ reff =

tan φ

φ

rstat ,

therefore reff > rstat

(tan φ > φ )

rstat < reff < rw Modelling of Automotive Systems

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Longitudinal tyre force is a function of slip ratio It is linear near slip ratio = 0 Fxf = Cσf σ xf Fxr = Cσr σ xr

Modelling of Automotive Systems

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Longitudinal tyre force Longitudin al tyre force Fxf and Fxr depends on a) slip ratio b) normal load c) friction coefficien t of tyre - road interface Longitudin al Slip : s = reff ω w − Vx reff − effective radius Slip ratio are defined as

σx = σx =

s reff ω w

=

reff ω w − Vx reff ω w

s reff ω w − Vx = Vx Vx

for tyre on wheel in tractio n mode for tyre on wheel in braking mode or on driven whe el

For small linear region, Fxf = C σf σ xf Fxr = C σr σ xr where Cσf , Cσr are called tyre stiffiness Modelling of Automotive Systems

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Longitudinal tyre force Brush model Net velocity at treads :

s = reff ω w − Vx for driving wheel, the tread element is bending s > 0 in direction of backwards The bending defection generate longitudinal force - proportional to s - proportional to time duration for which the tread element stays in the contact patch, or inversely proportional to the velocity

Veff

Therefore,

Fxf = Cσf

s reff ω w

For driven wheel s = reff ω w − Vx < 0 for driven wheel, the tread element is bending in direction of backwards. Tyre is subject to backwards ground force (resistance). Modelling of Automotive Systems

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Rolling resistance • Normal load causes tyre deformation ain contact patch • Damping in the tyre material dissapates energy • Viscousity causes forward half force stronger

Modelling of Automotive Systems

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Rolling resistance Rxf + Rxr = − f ( Fzf + Fzr ) f = 0.01 − 0.04 Rx ⋅ rstat = Fz ⋅ ∆x

Modelling of Automotive Systems

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Normal tyre forces Fzf =

1 (− Faero ha − m&x&h sin θ + mglr cos θ ) l f + lr

Fzr =

1 (Faero ha + m&x&h sin θ + mgl f cos θ ) l f + lr

Modelling of Automotive Systems

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