## 3 Numerical Results
### Vibration Response
#### 3.3.2 Coefficient of Friction Impact  
The aim of this section is to study the coefficient of friction impact on the linearization process and on the vibrations response. A comparison between the displacement components of the linearized solution with the reference solution for various values of \(\mu\) is displayed. A new boundary condition is proposed to improve the linearization method. Finally, the coefficient of friction impact on the vibrations amplitude is investigated.  
Unlike the frictionless case, a greater difference between the vibrations is observed in Figure 17 for all values of \(\mu\). The differences are especially observed in the contact active zone for the longitudinal and lateral vibrations displacement. Increasing the value of \(\mu\) makes the vibrations constant in the contact patch for the reference solution. Indeed, the quasi-static rolling results show that increasing \(\mu\) favors the stick regime. Hence, the longitudinal and lateral vibrations displacements remain constant in the contact patch. The constant value is not equal to 0 since the inertia changes the contact stiffness [22]. The boundary condition (31) used to treat the contact constraint is no longer valid for  
Figure 16: Time evolution of the vibration velocity in the circumferential direction (\(r=R_{e}\), \(y=\mathrm{width}/2\)) using the linearization method.  
Figure 15: Time evolution of the vibration displacement of the linearized solution for \(r=R_{e}\) (—), \(r=(R_{e}+R_{i})/2\) (—) and \(r=R_{i}\) (—) \(-y=\mathrm{width}/2\) (**top**) and for \(y=\mathrm{width}/2\) (—), \(y=\mathrm{width}/4\) (—) and \(y=0\) (—) \(-r=R_{e}\) (**bottom**).  
friction. A simple change of this condition is to impose an homogeneous displacement in the contact active zone as follows  
\[\mathbf{w}=\mathbf{0}\quad\text{ on }\ \Gamma_{c}^{t} \tag{39}\]  
The results of the boundary conditions change are given in Figure 18 that illustrates the time evolution of the vibrations field for various values of \(\mu\). A clear improvement of the linearized solution with respect to the reference solution is observed whether inside or outside the contact patch and especially for high values of \(\mu\). The biggest difference is observed for \(\mu=0.1\). Indeed, for this value of \(\mu\), the nodes in the contact patch operate in the slip regime, which is clearly incompatible with the new boundary condition but still better than the previous one especially outside the contact zone. A more precise boundary condition to treat the frictional contact should depend on the contact regime. Now, we compare between the vibrations for different values of \(\mu\) for both solutions.  
Figure 19 shows the vibrations sensitivity with respect to the coefficient of friction. Both solutions give more or less the same observations expect for \(\mu=0.1\) due to the incompatibility between the contact boundary condition and the friction slip regime. Larger vibrations are observed for the frictionless case for both solutions. Increasing \(\mu\) results in damping the vibrations amplitude until reaching an invariant vibration response for \(\mu>0.3\). This translates the saturation effect observed for the quasi-static rolling. The same saturation phenomenon is observed in [27] for advancing pin-loaded contacts with friction where the equations of motion are solved using the Green's functions.  
Figure 17: Time evolution of the vibrations displacement of the linearized solution (—) and the reference solution (—) for various values of \(\mu\) (\(y=\text{width}/2\), \(r=R_{e}\)).