## 3 Existence of Synchronous Motion
### Analysis of the Coupling Dynamic Characteristics of an Exciter and Three CRs  
Equations (18)-(21) can also be written as:  
\[T_{\rm e01}-f_{1}\omega_{\rm m}=T_{\rm u}(\chi_{\rm f1}+\chi_{\rm a1}) \tag{49}\]  
\[-f_{\rm R1}\eta_{m}\eta_{r}^{2}T_{\rm u}(2+\eta_{m}W_{\rm e01}) =T_{\rm u}(\chi_{\rm f2}+\chi_{\rm a2}) \tag{50}\] \[-f_{\rm R2}\eta_{m}\eta_{r}^{2}T_{\rm u}(2+\eta_{m}W_{\rm e01}) =T_{\rm u}(\chi_{\rm f3}+\chi_{\rm a3})\] (51) \[-f_{\rm R3}\eta_{m}\eta_{r}^{2}T_{\rm u}(2+\eta_{m}W_{\rm e01}) =T_{\rm u}(\chi_{\rm f4}+\chi_{\rm a4}) \tag{52}\]  
On the right sides of Equations (49)-(52), \(T_{\rm u}\chi_{\rm fi}\) and \(T_{\rm u}\chi_{\rm ai}\) (\(i=1,\ 2,\ 3,\ 4\)) are the load torques that a vibrating system acts upon four motors. In the expression of \(T_{\rm u}\chi_{\rm fi}\), the components independent of the phase differences represent the load torques originating from the vibration excited by its own exciter or CR (referred to as, simply, \(T_{Ai}\), \(i=1,\ 2,\ 3,\ 4\)); the components dependent on \(f_{Ri}\) cosine functions of phase differences describe the coupled load torques originating from the vibrations excited by others besides its own exciter or CR (referred to as, simply, \(T_{Bi}\), \(i=1,\ 2,\ 3,\ 4\)); the components dependent on \(f_{Ri}\) and sine functions of phase differences exist only in \(\chi_{\rm f2}\), \(\chi_{\rm f3}\) and \(\chi_{\rm f4}\), which depict the exclusive coupled load torques in CR \(i\) originating from the vibrations excited by the other CRs and an exciter (referred to as, simply, \(T_{Ci}\), \(i=1,\ 2,\ 3\)).  
In the expression of \(T_{\rm u}\chi_{\rm ai}\), the components dependent on sine functions of phase differences are to act as the coupled load torques of a phase leading exciter or CR and the coupled driving torques of a phase lagging exciter or CR, which are defined as the dynamic coupling torques among an exciter and three CRs (referred to as, simply, \(T_{\rm DCi}\), \(i=1,\ 2,\ 3,\ 4\)). In time of synchronous and steady states, the total dynamic coupling torques (referred to as, simply, \(T_{\rm DC}\), i.e., \(\sum\limits_{i=1}^{4}T_{\rm DCi}=T_{\rm DC}\)) are equal to zero. In other words, the total dynamic coupling torques do not generate or consume energy but transmit energy between an exciter and three CRs through the vibrating system. The other components dependent on \(f_{Ri}\) and cosine functions of phase differences express the exclusive coupled load torques in CR \(i\) originating from the vibrations excited by the other CRs and an exciter (referred to as, simply, \(T_{Di}\), \(i=1,\ 2,\ 3\)). Since an exciter and three CRs associate with each other by means of sine and cosine functions of phase differences, the phase differences are referred to as the coupled phases of an exciter and three CRs.  
The left sides of Equations (50)-(52) serve as a kind of friction torque from three CRs, which will dissipate vibration energy of the rigid platform and kinetic energy of three CRs. Due to the coupled torques \(T_{Bi}\), \(T_{Ci}\), \(T_{DCi}\) and \(T_{Di}\), the energy can be transferred from an exciter to three CRs through the moving rigid platform and compensates for three CRs' energy consumption.