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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.

## 2 comentarios

Now see you calculate rolling-radius. I read it as rolling-circumference, so the distance the tire makes in one cicle.

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I also once determined calculating rolling radius like this. But later realised that the treathlength on the ground represents a certain angle. So calculation is a bit different.

For small angles marginal, but for agricultural tyres larger angles, so can not be neclected anymore.

Also have idea, where the missing part of treath is going to.

When the treath goes of the ground it moves a little backward to the underground, also yust before comming to the ground. Example : rollingradius 3.07 times unloaded diameter, then 3.14159-3.07= 0.07159 times diameter say 100cm is 7.159cm lost treath/4= 1.8cm rounded. Just off the ground 1.8cm running backward, then after the next 1.8cm running forward on the same place.

Yust before going to…

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