## 4 Vibration-Based Road Friction Monitoring
### Low Frequency Vibrations
It was showed that the tyre first natural frequency is only function of constant tyre parameters and that the friction potential is strongly related to the strength of this resonance frequency. In particular, authors defend that the tyre longitudinal stiffness \(C_{\lambda}\), and more specifically the gradient of the longitudinal force with respect to the slip velocity (defined as a factor \(\alpha\) in this work), affects greatly the prominence of the frequency magnitude portrait. In order to corroborate experimentally the transfer function model, [PERSON] et al. performed several tests using a passenger car on dry asphalt, iced road and submerged road surfaces (Submerged asphalt with rubber partitions for hydroplaning test). The field trials were conducted at \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{3}{c}{**Longitudinal Stiffness Variation \(\Delta C_{\lambda}\) (\%)**} \\ \hline
**Test** & **Type 1** & **Type 2** \\ \hline Cold to steady-state temperature & \(-17\)\% & \(-21\)\% \\ \(-10\)\% pressure & \(17\)\% & \(15\)\% \\ \(-20\)\% pressure & \(29\)\% & \(28\)\% \\ Reduced tread (2.5 mm) & \(34\)\% & \(91\)\% \\ +200 kg (normal load ) & \(13\)\% & \(7\)\% \\ +400 kg (normal load ) & \(60\)\% & \(42\)\% \\ Wet road & \(4\)\% & \(-2\)\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Longitudinal stiffness changes for different tyre temperature, inflation pressure, tread depth, normal load, and road surface wetness [129]. a constant speed 16.6 m/s (60 km/h). The algorithm detected changes in the longitudinal stiffness \(C_{\lambda}\) when travelling from asphalt to ice road surface and vice versa, as well as when travelling from dry to submerged asphalt (hydroplaning). [PERSON] et al. (2015) [4], presented a method that detected the road friction potential based on the resonance frequency of the driveline of an in-wheel motor drive system, Figure 22. The hypothesis was based on a linear function between the maximum road friction coefficient \(\mu_{\text{max}}\) and the longitudinal stiffness \(C_{\lambda}\). Despite this not being explicitly mentioned, such function was presented previously by [PERSON] et al. [43]. [PERSON] et al. derived a formula showing that the natural frequency varied, depending on the longitudinal stiffness \(C_{\lambda}\), in the range 15-22 Hz. The velocity of the vehicle did not change the natural frequency value, but the shape of the frequency response function. [PERSON] et al. tested their method using a vehicle at constant speed and managed to detect changes in the resonance frequency for two different road surfaces: cement road and glossy ice road surface. Additional results on other intermediate rigid surfaces (e.g., wet asphalt) would have been of interest in order to assess whether the linear dependence between the tyre longitudinal stiffness and the road friction potential still holds. [PERSON] and [PERSON] followed the same reasoning and presented a method for detecting the road friction coefficient based on the wheel vibrations [50]. Once again, it was proposed by the authors that the tyre slip slope presents a clear correlation with the maximum road grip coefficient. Authors presented experimental data from previous works obtained on dry, wet, snow and icy road surfaces and remarked that normalised slip slope values (expressed as a factor of slip slope values obtained in dry concrete, and designated as LKX) could be grouped into 0.13-0.20 for wet ice, approximately 0.33 for snow and 0.69 for dry rough ice. [PERSON] and [PERSON] employed the MF-Swift tyre model to enhance the frequency response models presented in previous works and concluded that the first natural frequency of the tyre's vibrations and the damping ratio depend on longitudinal slip stiffness. Specifically, the influence of the tyre longitudinal stiffness on the damping ratio is greater compared to the influence of the natural frequency.