# 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

Rw - effective/Rolling radius

2*a - length of contact patch

2*θ - Tire print angle

ω - Angular velocity

V - wheel forward velocity

Note :- Rolling radius is somewhere between loaded height and the geometric radius.

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.