## 5 Results and Discussion
### Vibration Synchronization and Control Synchronization  
#### 5.1.1 Numerical Results  
Combining the self-adjusting function of the vibration system, the phase difference is the most appropriate parameter for studying synchronization because it represents the different synchronous states. According to references [33; 35], we know that the equivalent radius ratio is the key parameter affecting the phase difference. Considering the actual structure of the vibration machine, we choose the equivalent radius ratio \(r_{\psi}<5\) to investigate the vibration synchronization. Since the motor does the periodic motion, the phase differences of ERs must be periodic change. Therefore, we choose \((-2\pi,2\pi)\) as the change range of the phase difference. Applying the method provided in Figure 3 in reference [33], we obtain some results as shown in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Motor Parameter** & **Unit** & **Value** \\ \hline Rated velocity: \(n\) & rpm & 3000 \\ Armature resistance: \(R\) & \(\Omega\) & 5.5 \\ Torque constant: \(K_{t}\) & Nm/A & 25.4 \\ Magnet constant: \(K_{e}\) & Vs/rad & 1.7 \\ Voltage: \(u\) & V & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the vibration motor. Figure 6: The experimental method: (**a**) the experimental flow; (**b**) the experimental equipment. In general, we make two kinds of comparison including the different fixed distances \(l_{i}\) of ER and the different fixed angles \(\beta_{i}\) of ER. Comparing these six group curves, the point \(r_{\psi}=\sqrt{2}\) is the cut-off point of the change of the phase difference. When \(r_{\psi}<\sqrt{2}\), each phase difference has two stable solutions, which explains that the vibration system driven by three ERs has two synchronous states. By contrast, each phase difference only has one solution except Figure 7a. Figure 7a has two group stable solutions because ERs are symmetrically mounted, which is a limiting case. It can be seen from Figure 7 that the absolute values of these phase differences are close to \(120^{\circ}\), which explains that the exciting forces of ERs cancel each other out. The phenomenon conforms to the principle of minimum potential energy, which explains why the vibration machine using vibration synchronization does not achieve the increase of the exciting resultant force and driving power. Because the vibration synchronization restricts the application of the three ERs structure, we propose control synchronization to solve this limitation of the vibration system. Therefore, control synchronization is to study how to control the absolution of the phase difference from \(120^{\circ}\) to \(0^{\circ}\). Through the above theoretical analysis, we know the change rule of the phase difference as shown in Figure 8. At each moment, three ERs have to rotate close the phase difference of \(\pm 120^{\circ}\) to keep the center of mass at rest. Although the resultant forces are offset, a torque is created, which causes the body to swing around the center of mass according to Equation (1). To provide the basis for the master-slave control scheme, we assume two-thirds of ERs have achieved control synchronization to study the change of the phase difference. Figure 9a adopts motor 1 and motor 3 as the control object using control synchronization. Then, since two ERs rotate in the same motion, they can be equivalent to ER 4. At this moment, the three-ERs system becomes a two-ERs system. Similarly, Figure 9b adopts motor 1 and motor 2 as the control object using control synchronization. Comparing with Figure 9a,b is more symmetrical. When two-thirds of ERs apply control synchronization, the three-ERs system turns into the two-ERs system. Taking Figure 9b as the study example, we give the changes of the phase difference between ER 3 and ER 4 as shown in Figure 10. It can be seen that \(r_{\psi}=\sqrt{2}\) is also the key point. The absolution of each phase difference is close to \(180^{\circ}\) when \(r_{\psi}<\sqrt{2}\) and that is close to \(0^{\circ}\) when \(r_{\psi}>\sqrt{2}\).