## 3 Numerical Results
### Quasi-Static Rolling
#### 3.2.1 Preamble-Frictionless Contact
Figure 5: Time evolution of the quasi-static displacement for \(y=\text{width}/2\) (—), \(y=\text{width}/4\) (—) and \(y=0\) (—) with \(r=R_{e}\). contact zone since the vertical displacement is constant in the contact patch. The lateral acceleration decreases towards 0 as we move away from the contact patch. The longitudinal acceleration has a shear distribution in the contact patch. An acceleration at the entry of the contact patch and a deceleration at the exit patch are observed. The balance between the acceleration and the deceleration varies with respect to the contact patch form. Figure 7 shows the associated quasi-steady acceleration spectrum plotted with respect to the order. The order is defined as the frequency over the rotation speed and it is used to analyze the spectrum distribution independently of the rotation frequency. The fundamental frequency is equal to the rotation frequency (\(\text{Order}=1\)). Two families of harmonics appear since various non-linearities are taken into account in the model. The first family of harmonics operates around the multiples of the rotation frequency \(n\omega,\ n\in\mathbb{N}\). The second family of harmonics operates around 10 k\(\omega,\ k\in\mathbb{N}\), with a modulation effect. It concerns the contact forces frequency since the structure is composed of 10 blocks. This modulation phenomena is due to the rotation effect [24,25]. In fact, the kinematical quantities are calculated with respect to the fixed frame while the contact forces are connected to the rotating frame. This phenomena is not observed in the lateral direction since it represents the rotation axis direction. The vertical and longitudinal accelerations have the same order of magnitude, which is greater than the one of the lateral acceleration. Finally, the acceleration amplitude decreases as we move away from the contact patch. Figure 8 shows the contact pressure distribution with respect to the position of the groove (leading and trailing edge). When a block enters the contact zone, the contact  
Figure 6: Time evolution of the quasi-steady acceleration in the circumferential direction (\(y=\text{width}/2\) and \(r=R_{e}\)). pressure is concentrated at the entry. As the block advances, the contact pressure moves to the block center and becomes similar to the distribution of the cylindrical contact pressure described in Hertz theory [26]. As the block exits the contact patch, the pressure becomes concentrated in the trailing edge with the same distribution as the one observed for the leading edge. A small variation of the total contact force is observed. Since the deflection \(d\) is constant and the contact patch form is changing, the load supported by the structure varies. The last figure shows the spectrum of the total force variation \(\delta F_{z}\), which represents the difference between the total contact force and its mean value. It shows that the variation spectrum is composed of harmonics located around the multiples of the blocks frequency. In the following, the Coulomb friction law is now considered and its impact on the quasi-static rolling is investigated.