### The effective rolling radius

The effective rolling radius is not the same as the loaded tyre radius Rl, with the latter being defined as the vertical distance between the wheel centre and the horizontal surface. A free rolling tyre rotates around a point near the contact patch. For a rigid wheel on a flat horizontal surface, this point coincides with the single contact point between tyre and road, and the forward speed Vx equals angular speed time (loaded = unloaded) radius.

For a pneumatic tyre, the distance between points at the circumference of the tyre and the wheel centre varies from a value close to the unloaded radius just before entering the contact area to the same value as the loaded radius just at the projection point of the wheel centre on the contact area. At that point, the peripheral velocity of the tread (relative to the wheel centre) coincides with the horizontal velocity V of the wheel centre.

Moving out of the contact area, the tread regains its original length and the peripheral velocity returns to Ω.R with R the unloaded radius. As a consequence, the spin speed of the wheel with a pneumatic tyre under conditions of free rolling is less than that of a rigid wheel and:

It means that the centre of rotation of the wheel usually lies somewhere below the surface. The effective rolling tyre under free rolling also behaves different with varying tyre load compared to the loaded tyre radius. A loaded radius behaves almost linear in the tyre load Fz, i.e. the tyre behaves as a linear spring in vertical direction. The effective rolling radius varies significantly with tyre load. This can be described based on empirical fit as follows [1]:

with tyre deflection ρ, tyre deflection ρ0 for nominal tyre load Fz0, and fitparameters B, D, E which may vary according to:

3 < B < 12 :            B stretches the effective tyre characteristic curve along the Fz -axis (ordinate). B large means a large slope at Fz= 0.
0.2 < D < 0.4 :        shift from asymptote at high wheel loads
0.03 < E < 0.25 :     with low values of E for stiff tyres

Effective and loaded tyre radius under conditions of free rolling

An example of the variation of Rl and Re is shown in figure above for B = 10, D = 0.25 and E = 0.05. The tyre stiffness is taken as 2·105 N/m. The unloaded radius R is taken as 0.32 m and we choose Fz0 = 4000 N. We have also varied the parameters to illustrate the range of possible effective rolling radius characteristics.

The effective rolling radius turns out to increase with increasing speed and increasing inflation pressure. The variation with speed is strongly dependent on the tyre carcass structure.

A radial-ply tyre rolling radius appears to be almost constant for varying speed in contrast with the diagonal-ply (bias-ply) tyre. This phenomenon has to do with the radial response of the tyre to higher circumferential speeds.