## 2 Dynamic Model of the Vibration System  
Figure 1 shows the experimental vibration machine and its dynamic model. Comparing Figure 1a,b, the dynamic model of the vibration system consists of a rigid body(Num 1), three vibration motors including DC motor and ER(Code M1-3), springs(Num 2), and a fixed base(Num 3). The springs connect the base with the rigid body. Three DC motors drive three ERs in the same rotational direction, respectively. In Figure 1b, \(o_{i}\) are the rotational centers of ERs, \(r\) is the eccentric radius of three ERs, \(\varphi_{i}\) denote ER rotates about its spin axis, \(o\) is the mass center of the vibration system, \(l_{i}\) is the distance between the  
rotational center \(o_{i}\) and the center of mass \(o\), \(\beta_{i}\) is the angle between \(oo_{i}\) and \(x\)-axis, \(m\) is the mass of the rigid body, \(m_{i}\) are the masses of ERs, \(k_{x},k_{y},k_{\psi}\) are stiffness of springs, \(f_{x},f_{y},f_{\psi}\) are damping of springs, and \(i=1,2,3\). The mass center \(o\) translations are \(x\) and \(y\), and angular rotation is \(\psi\).  
To analyze the dynamics of the vibration system driven by three ERs using control synchronization, the coupling characteristic of three ERs in the state of vibration synchronization should be investigated firstly. At present, the average method of the modified small parameter is one of the most systematic methods to study vibration synchronization [10; 11; 12]. Considering the paper length, we only give the key equations using this method, and the detailed deduced process can be seen in reference [33].  
The electromechanical mathematic model of the vibration system is as follows [33]:  
\[M\ddot{x}+f_{x}\dot{x}+k_{x}x=\sum_{i=1}^{3}v_{i}m_{i}r\Big{(}\dot{\varphi}_{i }^{2}\cos\varphi_{i}+\ddot{\varphi}_{i}\sin\varphi_{i}\Big{)},\]  
\[M\ddot{y}+f_{y}\dot{y}+k_{y}y=\sum_{i=1}^{3}m_{i}r\Big{(}\dot{\varphi}_{i}^{2} \sin\varphi_{i}-\ddot{\varphi}_{i}\cos\varphi_{i}\Big{)},\]  
\[J_{\psi}\ddot{\psi}+f_{\psi}\dot{\psi}+k_{\psi}\psi=\sum_{i=1}^{3}v_{i}m_{i}rl_ {i}\Big{[}\ddot{\varphi}_{i}^{2}\sin(\varphi_{i}-v_{i}\beta_{i})-\ddot{\varphi }_{i}\cos(\varphi_{i}-v_{i}\beta_{i})\Big{]},\]  
\[J_{i}\ddot{\varphi}_{i}+f_{i}\dot{\varphi}_{i}=T_{ei}-m_{i}r\big{[}\ddot{y}\cos \varphi_{i}-v_{i}\ddot{x}\sin\varphi_{i}+l_{i}\ddot{\psi}\cos(\varphi_{i}-v_ {i}\beta_{i})+v_{i}l_{i}\dot{\psi}^{2}\sin(\varphi_{i}-v_{i}\beta_{i})\big{]},\ i=1,2,3, \tag{1}\]  
where \(T_{Li}=m_{i}r\big{[}\ddot{y}\cos\varphi_{i}-v_{i}\ddot{x}\sin\varphi_{i}+l_{i} \ddot{\psi}\cos(\varphi_{i}-v_{i}\beta_{i})+v_{i}l_{i}\dot{\psi}^{2}\sin( \varphi_{i}-v_{i}\beta_{i})\big{]}\), \(T_{ei}\) are the electromagnetic torques of three motors, \(J_{i}\approx m_{i}r^{2}\) are the moment of inertia of ERs, \(J_{\psi}\) is the equivalent moment of the vibration system, \(M=m+m_{1}+m_{2}+m_{3}\), \(f_{1}\) and \(f_{2}\) are the damping coefficient of the axis of the motor, \(v_{i}=-1\) denotes the motor rotates clockwise, and \((\dot{\bullet})\) denotes \(\mathrm{d}^{2}\big{(}\bullet)/\mathrm{d}t^{2}\).  
The steady-state system responses are obtained as follows [33]:  
\[x=\sum_{i=1}^{3}\frac{v_{i}rm_{i}\dot{\varphi}_{i}^{2}\cos(\varphi_{i}-\gamma _{xi})}{k_{x}-M\dot{\varphi}_{i}^{2}},\]  
\[y=\sum_{i=1}^{3}\frac{rm_{i}\dot{\varphi}_{i}^{2}\sin(\varphi_{i}-\gamma_{yi}) }{k_{y}-M\dot{\varphi}_{i}^{2}},\]  
\[\psi=\sum_{i=1}^{3}\frac{v_{i}rm_{i}\dot{\varphi}_{i}^{2}l_{i}\sin(\varphi_{i} -v_{i}\beta_{i}-\gamma_{yi})}{k_{\psi}-I_{\psi}\dot{\varphi}_{i}^{2}}, \tag{2}\]  
Figure 1: The vibration machine is excited by three eccentric rotors (ERs): (**a**) the vibration machine; (**b**) the dynamic model.