## 3 Numerical Results
### Quasi-Static Rolling
#### 3.2.1 Preamble-Frictionless Contact  
The analysis of the quasi-static rolling is initiated by considering a frictionless contact. Although this first case studied, based on frictionless contact, does not make sense for the reality of tire-road interaction, as it does not correspond to the real phenomena of tire contact, this preliminary study is carried out only to illustrate the feasibility and validity of the proposed two-steps approach on the grooved cylinder under study. In addition, this first application with frictionless contact will serve as a reference for a qualitative comparison with the following case, which considers the impact of friction. So the main objective of this section is to study the kinematics and the contact pressure of the grooved cylinder in order to validate both the mechanical modeling and the associated numerical methodology. Figure 4 shows the time evolution of the quasi-static displacement components. The displacement field is, as expected, periodic. The longitudinal displacement \(U_{\mathrm{x}}\) slows down at the entry of the contact patch (\(t\approx 0.05\) s) and speeds up at the exit of the contact patch (\(t\approx 0.055\) s) to reach the rim longitudinal position. The longitudinal deformation decreases as we move away from the contact zone. The same kinematical behavior is \begin{table}
\begin{tabular}{c c} \hline \hline External radius & \(R_{e}=0.1\) m \\ Internal radius & \(R_{l}=0.02\) m \\ Width & \(w=0.1\) m \\ Young Modulus & \(E=2\times 10^{8}\) Pa \\ Poisson coefficient & \(\
u=0.3\) \\ Density & \(\rho=900\) kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Prototype mechanical properties. observed for tires in [23]. In fact, the radial distance between the tread pattern and the rotation axis is minimal at the contact patch and reach its maximum at the opposite of the contact patch. The vertical displacement \(U_{x}\) shows that the non-penetration condition is well respected (\(0.05\) s \(\leq t\leq 0.055\) s) on the contact patch. The lateral displacement amplitude \(U_{y}\) decreased as we move away from the contact patch due to the Poisson effect. The lateral deformation is obviously greater in the contact active zone and is null at the opposite of the contact patch (\(t\approx 0.07\) s). Furthermore, the lateral displacement has a symmetric distribution; the deformation at the entry of the contact zone is the same at the exit of the contact patch and the maximum is reached at the contact patch center (\(t\approx 0.052\) s). The displacement of the node occupying the rim zone shows that the Dirichlet boundary conditions are well respected. Figure 5 shows that the lateral position does not impact the longitudinal and vertical displacement. The lateral displacement amplitude decreases toward 0 as we move to the contact patch center. This result was expected since the plane (\(y=0\)) is a symmetry plane of the structure. Equation (35) shows that the quasi-steady acceleration appears as the excitation force of the vibrations response. The acceleration evolution in the circumferential direction and in the frequency domain is displayed in order to have an idea on how the vibrations will be excited. Figure 6 shows the time evolution of the acceleration in the rotating frame. The angle \(\theta=0\) deg denotes the contact patch center and \(\theta<0\) deg designates the contact patch exit. From a time perspective, the acceleration's maximum is reached when the block edge enters (\(t\approx 0.052\) s) and leaves (\(t\approx 0.0563\) s) the contact patch. From a circumferential point of view, the maximum of the vertical and lateral acceleration is located at the contact patch boundaries and have a similar distribution at the entry (\(\theta\approx 20\) deg) and at the exit (\(\theta\approx-17\) deg) of the contact active zone. The vertical acceleration is null in the  
Figure 4: Time evolution of the quasi-static displacement for \(r=R_{e}\) (—), \(r=(R_{e}+R_{i})/2\) (—) and \(r=R_{i}\) (—) with \(y=\text{width}/2\).