## 3 Numerical Results
### Vibration Response
#### 3.3.1 Preample-Frictionless Contact  
As for the quasi-static rolling, a frictionless contact is considered first. As a reminder, this first preliminary study based on the prediction of the nonlinear vibration response with frictionless contact only aims to illustrate the feasibility of the proposed two-steps approach. A comparison between the linearization method solution and the reference solution, which represents the difference between the dynamic and the quasi-static response is presented. Then, the vibrations velocity spectrum is analyzed. Finally, the vibrations spatial evolution is visualized to assess the global vibration behavior of the grooved cylinder.  
Figure 13 illustrates the comparison between the linearized solution and the reference solution. Both solutions are, as expected, periodic. Both solutions have the same vibration frequency with a small difference with respect to the phase. The vertical vibrations are null in the contact patch for both methods; therefore, treating the normal contact constraint as a Dirichlet-like boundary condition seems to be the right choice. The longitudinal vibrations maximum amplitude is located in the contact patch and have the same order of magnitude as the vertical vibrations. The lateral vibrations are located in the contact zone and decrease towards 0 as we move away from the contact zone. The magnitude of the lateral vibrations are much lower than the vertical and longitudinal components.  
Figure 12: Time evolution of the total contact force (\(y=\text{width}/2\) and \(r=R_{e}\)) for frictionless contact (—), \(\mu=0.1\) (—), \(\mu=0.3\) (—), \(\mu=0.5\) (—) and \(\mu=1.0\) (—).  
Figure 14 shows the vibrations velocity spectrum. Both solutions give more or less the same vertical and longitudinal velocities. The main difference between the methods are observed in the lateral direction. A set harmonics around the multiples of the blocks frequency with the same modulation effect described in the quasi-static rolling results is observed for both solutions. The lateral vibrations velocity spectrum has a frequency comb distribution. The harmonics around the multiples of the rotation frequency have vanished for the longitudinal and vertical vibrations. Hence, the linearization method allows us to separate the dynamic behaviors operating at different scales. The vertical and longitudinal vibration velocity magnitude is more or less the same and greater than the magnitude of the lateral velocity. The same observation was found for the quasi-steady acceleration.  
Figure 15 illustrates the vibrations variation in the radial and lateral directions. The vibrations amplitude decreases as we move away from the tread pattern in the radial direction. The lateral vibrations amplitude decreases as we move towards the contact center in the lateral direction whereas the vertical and longitudinal vibrations remain constant. The same observation was found for the quasi-static displacement in Figures 4 and 5.  
Figure 14: Frequency spectrum of the vibration velocity for (\(y=\text{width}/2\), \(r=R_{e}\)) of the linearized solution (—) and the reference solution (—).  
Figure 13: Time evolution of the vibrations displacement for (\(y=\text{width}/2\), \(r=R_{e}\)) of the linearized solution (—) and the reference solution (—) (full simulation (**top**) and zoom around one period (**bottom**)).  
Figure 16 shows the vibrations velocity variation in the circumferential direction. Unlike the quasi-steady acceleration, the mechanical behavior at the entry of the contact patch is no longer similar to the behavior at the exit of the contact patch. The vertical vibrations are, as expected, null in the contact patch and at the opposite of the contact patch (\(\theta=180\) deg). The lateral vibrations maximum is situated at the contact patch entry and decreases towards 0 as we move away from the contact active zone. The longitudinal vibrations velocity has a more or less symmetric distribution with respect to the plane (\(z=R_{i}\)).