# Static and Dynamic Analysis of Non-Pneumatic Tires Based on Experimental and Numerical Methods
## 4 Dynamic Study: Steady State Analysis
#### Steady State Analysis  
The rolling simulation consisted of two main steps: in the first step, the FE model was built rotating the 60\({}^{\circ}\) sector and the footprint analysis was reproduced imposing the same vertical displacement of the static tests (20 mm). The static footprint is an essential requirement for the second step, in which the steady state analysis was performed. The first simulation step is fundamental to check if the radial stiffness of the FE model  
Figure 13: FE model of the 60\({}^{\circ}\) sector. built for the dynamic simulation matches the one of the FE model built for the static characterization, verifying that it behaves the same way. This check was carried out calculating the percentage error between the values of Forces obtained for homologous values of Displacements for the two different models, finding that the maximum value of percentage error was 6%. The computational cost of the footprint analysis for this model was higher than the one previously described due to the modeling of the tread along the entire circumference of the tire and the modeling of the spokes with 3D elements that led to a higher number of elements and nodes. In fact, the previous FE model was constituted by 139,543 nodes and 129,042 elements, whilst the model created for the steady state analysis was made up of 274,984 nodes and 805,836 elements. Once the footprint analysis was realized, the steady state analysis was performed. At first, it was mandatory to define the longitudinal velocity of the center rim and the angular velocity of the tire, the value of which is found by means of an iterative process. In the free rolling motion, the following equations are verified:  
\[\mathrm{V}=\Omega*R_{e},\;F_{x}=0,\;M_{y}=0, \tag{2}\]  
where V is the speed of the center of the rim, \(\Omega\) is the angular velocity of the tire, \(R_{e}\) is the effective rolling radius of the tire, \(F_{x}\) is the longitudinal force, and \(M_{y}\) is the torque. When the simulation is set, both \(\Omega\) and \(R_{e}\) are unknown and the only input is the longitudinal velocity V, set equal to 10 km/h. The iteration started with an approximated value of \(R_{e}\) obtained using the following equation deduced from geometric consideration [29]:  
\[R_{e}=R-\frac{\delta}{3} \tag{3}\]  
where \(R\) is the nominal radius and \(\delta\) is the vertical deflection imposed in the footprint analysis. This equation gives a first attempt at effective rolling radius \(R_{e}\) equal to 323.4 mm and from the first equation of (2) can be derived an angular velocity \(\Omega\) of 8.6 rad/s. With these inputs, the simulation was set imposing at first, an angular velocity of 7.7 rad/s, and subsequently, it was varied to a value of 9.7 rad/s. Covering this range of values of angular velocity, the one corresponding to a value of longitudinal force \(F_{x}\) equal to zero will be the angular velocity of free rolling condition. In Figure 14 is represented the longitudinal force \(F_{x}\) vs. the angular velocity \(\Omega\) diagram, from which can be observed that \(F_{x}\) is zero for an angular velocity of 8.425 rad/s. For the first Equation (2), the effective rolling radius \(R_{e}\) is  
\[R_{e}=\frac{V}{\Omega}=\frac{2777.8\;\mathrm{mm/s}}{8.425\;\mathrm{rad/s}}=329.7\;\mathrm{mm}, \tag{4}\]  
where the velocity \(V\) of 10 km/h was transformed into mm/s. Figure 14: Longitudinal force vs. angular velocity for the calculation of the effective rolling radius. The effective rolling radius is necessary also for the slip ratio definition:  
\[S=-\frac{(V-\Omega*R_{e})}{V}*100 \tag{5}\]  
with which it is possible to define the longitudinal force vs. slip diagram, shown in Figure 15 for three different values of vertical deflection. The diagrams are not symmetric with respect to the y-axis, representing a different behavior in braking and driving condition.