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# How to calculate value of Rolling radius ?

Rolling radius is the effective radius of the tire when the tire is rotating and moving forward on

the ground . The value of rolling radius can be calculated either using the wheel forward & wheel angular velocity or by measuring the distance traveled as a function of wheel revolution.

Here , is the mathematical way of finding the rolling radius of tire:-

In this figure we have,

• Rg - Geometric radius of tire

• Rh - loaded height of tire

• 2*a - length of contact patch

• 2*θ - Tire print angle

• ω - Angular velocity

• V - wheel forward velocity

The tire vertical deflection is:-

Rh=Rg*cosθ

a =RgSinθ (1)

Rg-Rh = Rg (1-cosθ) (2)

If the motion of the tire is compared to the rolling of a rigid disk with radius Rw, then the tire must move a distance,

a = Rw * θ (3)

From equation (1) & (3) we get ,

Rw = Rg* Sinθ/θ

Expanding Sinθ/θ ,using taylor series ,

Sinθ = θ - (1/6)*θ^3

Taking only 2 terms from taylor series,

Rw=Rg*( 1 - (1/6)*θ^2 ) (4)

For Cosθ , taylor series can be written as,

Cosθ = 1 - (θ^2)/2

θ^2 = 2*(1- Cosθ )

From equation (2) we can write,

θ^2 = 2*(1- Rh/Rg) (5)

Putting (5) in equation (4) we get,

Rw = Rg*[ 1 - 1/3*(1-Rh/Rg) ]

## Rw = (2/3)*Rg + (1/3)*Rh

(Above equation is the value of rolling radius).

Rh is a function of tire load Fz,

Rh = Rg - (Fz/kz)

Where, kz is the vertical stiffness.

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