## 4 Conclusions  
In this paper, an adapted two-step approach to simulate the nonlinear vibrations of a hyperelastic solid undergoing large deformations in contact with rigid plane is presented. The methodology allows not only the separation between the dynamics operating at  
Figure 19: Time evolution of the vibrations displacement of the linearized solution with the new boundary condition (**right**) (**—**) and the reference solution (**left**) for frictionless contact (**—**), \(\mu=0.1\) (**—**), \(\mu=0.3\) (**—**), \(\mu=0.5\) (**—**) and \(\mu=1.0\) (\(y=\text{width}/2\), \(r=R_{e}\)). Figure 18: Time evolution of the vibrations displacement of the linearized solution (**—**) with the new boundary condition (**—**) and the reference solution (**—**) for various values of \(\mu\) (\(y=\text{width}/2\), \(r=R_{e}\)). different scales but also shows numerical robustness and consistency. The Coulomb friction law is used in the contact interface. This numerical approach is applied to a finite element model corresponding to a grooved cylinder. The linearization process seems to work very well for frictionless contact; however, significant differences can be observed for the case with frictional contact, especially for higher values of \(\mu\). The change of the boundary condition in the contact active zone seems to improve the linearized solution. A much more precise condition in the contact active zone would probably depend on the friction (slip or stick) regime. As previously mentioned, the example proposed and discussed in this paper corresponds to a grooved cylinder undergoing large deformation in contact with rigid plane. One of the most crucial next steps is to be able to provide an illustration of the feasibility of the proposed approach on a tire structure. Numerous future research can be carried out to increase the understanding of non-linear phenomena on tire structures. Although it is not possible to give an exhaustive list of topics of interest, some studies that seem to have a certain priority and interest may be considered:  
* First of all, it can be noted that damping has been neglected in modeling; however, experiments show that vibration energy is not transported to the opposite of contact patch due to strong damping of tire structure for such relatively high frequencies. The vibration energy is thus concentrated close to the entrance and the exit of contact patch. Moreover the vibrations evolution in the circumferential direction shows the opposite observation; therefore, the addition of a suitable damping model seems important and is one of the priority issues to be explored. The inclusion of the Kelvin-Voigt model, often used to treat the viscosity of tire rubber, seems to be an interesting option [28]. * In a similar way, the modeling of friction at the contact interface should be completed by the consideration of the coefficient of rolling friction [29] in order to reproduce more realistic physical phenomena involved in tire/pavement interaction vibration and noise. Indeed, the consideration of only the coefficient of sliding friction force as proposed in this study may be not sufficient to accurately reproduce real phenomena of tire vibration response. * It should be interesting to apply the proposed two-step approach on a real industrial tire structure and to compare the results with full experimental tests. One of the most important challenges may be to identify the different sources of nonlinear vibrations in order to identify and quantify the first-order physical phenomena involved. In this context, more realistic modeling including both tire rubber viscosity and Hertzian contact for polymorph materials or the model of contact 3D multi-layered structures with orthotropic material should be developed to reproduce more specifically the design of a real wheel tire. * Extension of the prediction of non-linear vibrations in a real environment and in real conditions would also be interesting with the consideration of a more realistic pavement with imperfections.