## 4 Model Simulations with FEM-3D
### Simulation Results with Motion  
To determine the influence of the motion in the proposed models, in this section, it is presented the FEM-3D simulation of the models including motion in the moving parts. Thus, Table 4 shows the velocities and its corresponding slips (see Equation (2)) selected to study the effect of the motion in the analyzed models. More specifically, we analyze the thrust force, the levitation force, and the electric currents. It is important to notice that the friction losses are not taken into account in the present work. Figure 30 shows the evolution of the thrust force of each model for all values shown in Table 4. To understand this figure, it is necessary to consider the following regions inside:  
* Region I (low velocities zone): \(0.6<s<1\leftrightarrow 0\frac{m}{s}<v_{moving\_part}<8\frac{m}{s}\). With these velocities, the behavior of the thrust force had the same evolution as in standstill conditions, where this force obtained its highest value in model 4 (see Equation (20)). \[\overset{\rightarrow}{F}_{y\_model\_4}>\overset{\rightarrow}{F}_{y\_model\_3}> \overset{\rightarrow}{F}_{y\_model\_2}>\overset{\rightarrow}{F}_{y\_model\_1}\] (20)  
Between this range of slips, the developed thrust forces for each model increased their values until the slip reached the value of 0.6. At this point, the thrust forces from model 2, model 3, and model 4 reached their maximum value around 226.2, 262.447, and 284.778 N, respectively. * Region II (Medium velocities zone): \(0.5<s<0.6\leftrightarrow 8\frac{m}{s}<v_{moving\_part}<10\frac{m}{s}\). This region was characterized by a non-uniformity evolution of the thrust force. Model 2, model 3, and model \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Dynamic Conditions for the Motion Analysis**} \\ \hline Slip & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ Velocity (m/s) & 20 & 18 & 16 & 14 & 12 & 10 & 8 & 6 & 4 & 2 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Slips values and linear velocities simulated with FEM-3D. Figure 30: Thrust force versus slips. 4 presented their inflection points between these velocities and after that continued with an evolution similar to region I (see Equation (21)). \[\overset{\rightarrow}{F}_{y\_model\_4}>\overset{\rightarrow}{F}_{y\_model\_3}> \overset{\rightarrow}{F}_{y\_model\_2}\] (21) However, the thrust force developed in model 1 continued to increase from 8 to 10 m/s, where the thrust force was nearly 294 N (with a slip around 0.5). This value was the highest thrust force obtained in the present work. \(0<s<0.5\leftrightarrow 10\frac{m}{s}<v_{moving\_part}<20\frac{m}{s}\). In this region, the evolution of the thrust force decreased in all models. In general, model 1 presented a higher thrust force than others. The behavior of these forces in model 2, model 3, and model 4 were very similar. The thrust forces decreased as the velocity of the secondary part reached the synchronous velocity (20 m/s and s = 0), when model 2 developed the highest thrust force (around 41 N) at slip equal to zero. This phenomenon appeared due to the end-effect and it was not present in asynchronous rotatory machines because the torque obtained at synchronous velocity was equal to zero. Next, Figure 31 shows the effect of the motion for the levitation force. This figure gives us the following information:  
* Model 1, model 2, and model 3 presented two regions of work depending on the selected slip. The change-point between the levitation zone and the attraction zone varied in the function of the models. So, this change of zone occurred when the values of slips were 0.55 in model 1, 0.6 in model 2, and 0.72 in model 3. However, model 4 did not operate under levitation condition. For all slip values, model 4 developed an attraction force that reached the maximum value near to 568 N with a velocity equal to 12 m/s (slip equal to 0.4). * Two regions are shown in Figure 31: Region I: \(0.5<s<1\leftrightarrow 0\frac{m}{s}<v_{moving\_part}<10\frac{m}{s}\).