## 2 Background
### Road-Rubber Friction
#### 2.2.1 Tyre Properties as a Function of Temperature and Stress Frequency  
As mentioned previously, the friction generated in the road-rubber interface is the result of a set of complex interactions between the tyre and the road, which can be summarised by two stress mechanisms: road roughness effect and molecular adhesion [2]. Such effects arise from the viscoelastic properties of the tyre, which can be approximated by a spring \(K\) connected in parallel to a damper of damping coefficient \(\eta\)[2], Figure 3a. Therefore, the force generated by the tyre can be expressed as a function of the deformation \(y\), the rate of deformation \(\dot{y}\), and the tyre properties.  
\[F_{tyre}=Ky+\eta\dot{y} \tag{8}\]  
Type properties \(K,\eta\) are function of temperature, see Figure 3b. The tyre's modulus of elasticity, represented by the spring \(K\), is maximum below the glass transition temperature \(T_{g}\), and reduces for temperatures greater than that. The tyre's damping mechanism, also known as hysteresis or energy  
Figure 3: (**a**) The tyre as a viscoelastic material can be approximated as a spring \(K\) connected in parallel to a damper \(\eta\); (**b**) The influence of temperature on the tyre as a viscoelastic material. The modulus of elasticity is maximum for temperatures lower to the glass transition temperature. Hysteresis (energy loss) is maximum at the glass transition temperature [2].  
loss, represented by \(\eta\) is maximum at the glass transition temperature. For temperatures different to that, the energy loss is smaller. As a matter of fact, winter types are designed to have a lower glass transition temperature compared to summer tyres. Consequently, when only the tyre force between the tyre and the road is considered, winter tyres generate larger forces at low temperatures compared to summer tyres and summer tyres generate larger forces at increased temperatures [2].  
Tyre properties are also a function of stress frequency, the frequency at which the load is applied, see Figure 4. At low excitation frequencies/velocity \(\dot{y}\), the damper \(\eta\) does not contribute much, therefore the influence of the spring \(K\) is dominant. The opposite happens at higher frequencies, where the damper \(\eta\) becomes more important. Regarding the energy loss, this becomes maximum at an intermediate range of frequencies and falls for frequencies different to that. At low excitation frequencies, the energy loss is small because the damping mechanism is not important. At higher excitation frequencies there is not enough time for the tyre's rubber molecules to return to equilibrium and thus rubber remains constantly in tension. Therefore, the tyre's modulus of elasticity is low in the low excitation frequency range and high in the high-frequency excitation range. A careful observation reveals that maximum energy loss occurs when the modulus of elasticity changes.  
Hence, to understand how a tyre behaves, the tyre type needs to be considered and the temperature and stress frequency monitored [55]. When temperature increases the material becomes softer (lower modulus of elasticity), while when stress frequency increases the material becomes more rigid (higher modulus of elasticity). Other parameters, such as the tyre inflation pressure, also influence the rigidity of the tyre [56], and therefore are expected to be monitored.