Tyre Rolling Resistance Modelling

The Question

A long-standing rule of thumb sets bicycle tyre pressure at roughly one bar for every ten kilograms of rider-plus-bike weight, yet field practice increasingly favours markedly lower pressures. The engineering literature is oddly unhelpful here: although the mechanisms of rolling resistance are well understood — hysteretic losses in the rubber combined with mechanical suspension losses over road texture — many authors hold that rolling resistance cannot be modelled from first principles, and that empirical correlations are the best available tool. That conclusion is what this work set out to test.

First-Principles Model

Three established components combine into a tractable analytical model. Maxwell's viscoelastic constitutive model describes the hysteretic energy loss in the tyre rubber under cyclic deformation; Persson's rough-surface contact theory provides the contact mechanics between tyre and road texture; and steady-state quarter-car dynamics couples these to the macroscopic tyre–road interaction, including suspension compliance. The result is a closed-form expression for the rolling resistance coefficient as a function of tyre width, inflation pressure, vertical load, road roughness and rolling speed — derived without empirical fitting parameters and depending only on measurable material and geometric properties. The model addresses rolling resistance alone; grip and cornering stiffness are separate phenomena, and the single-corner quarter-car representation omits the chassis coupling that full bicycle dynamics would add to the load spectrum.

Main Result

At constant tyre stiffness, the effect of tyre width below 25 mm falls below the just-noticeable difference for amateur performance. Heavier riders pay a small rolling-resistance premium on smooth surfaces, but on rough terrain the relationship reverses: the larger contact patch distributes the roughness excitation more favourably. For road quality typical of Belgian roads, the model justifies inflation pressures considerably lower than the traditional one-bar-per-ten-kilograms rule.

Practical Interpretation

Because the optimum pressure depends strongly on surface roughness, there is a concrete performance case for a pressure-switching system that responds to road quality in real time — effectively managing the tyre–road contact dynamically — to exploit this variation across mixed-surface stages. More broadly, the exercise is a reminder that "first-principles modelling is not feasible" is often a statement about effort rather than possibility: with the right combination of viscoelasticity, contact theory and vehicle dynamics, a phenomenon widely treated as empirical yields a clean closed-form description.