## 3 Existence of Synchronous Motion  
In what follows, it is assumed that  
\[m_{R1}=m_{R2}=m_{R3}=m_{R},\ r_{R1}=r_{R2}=r_{R3}=r_{R} \tag{10}\]  
Additionally, the average phase among the exciter and three CRs is assumed to be \(\varphi\), and their phase differences are \(2\alpha_{1}\), \(2\alpha_{2}\) and \(2\alpha_{3}\), i.e.,  
\[\varphi=(\varphi_{1}+\varphi_{R1}+\varphi_{R2}+\varphi_{R3})/4,\ \varphi_{1}-\varphi_{R1}=2\alpha_{1}\]  
\[\varphi_{R1}-\varphi_{R2}=2\alpha_{2},\ \varphi_{R2}-\varphi_{R3}=2\alpha_{3} \tag{11}\]  
Then, Equation (11) is transformed to  
\[\varphi_{1}=\varphi+\frac{3}{2}\alpha_{1}+\alpha_{2}+\frac{1}{2}\alpha_{3}= \varphi+\varsigma_{1}\]  
\[\varphi_{R1}=\varphi-\frac{1}{2}\alpha_{1}+\alpha_{2}+\frac{1}{2}\alpha_{3}= \varphi+\varsigma_{R1}\]  
\[\varphi_{R2}=\varphi-\frac{1}{2}\alpha_{1}-\alpha_{2}+\frac{1}{2}\alpha_{3}= \varphi+\varsigma_{R2}\]  
\[\varphi_{R3}=\varphi-\frac{1}{2}\alpha_{1}-\alpha_{2}-\frac{3}{2}\alpha_{3}= \varphi+\varsigma_{R3} \tag{12}\]  
Since a vibrating system's motion is periodic, the variation in average mechanical angular velocity \(\dot{\varphi}\) is periodic as well. If the least common multiple period is assumed to be \(T_{\rm LCOMP}\), then the following should be satisfied:  
\[\omega_{\rm m}=\frac{1}{T_{\rm LCOMP}}\int_{t^{\prime}}^{t^{\prime}+T_{\rm L COMP }}\dot{\varphi}(t)\mathrm{d}t=\mathrm{constant} \tag{13}\]  
According to the average method of modified small parameters [22; 23; 24; 25; 26; 27], it is assumed that the instantaneous change coefficient of \(\dot{\varphi}\) around \(\omega_{\rm m}\) and that of \(\dot{\alpha}_{i}\) to be \(\varepsilon_{0}\) and \(\varepsilon_{i}\) (\(\varepsilon_{0}\) and \(\varepsilon_{i}\) are functions of time \(t\), \(i=1,\ 2,\ 3\)). This yields  
\[\dot{\varphi}=(1+\varepsilon_{0})\omega_{\rm m},\ \dot{\alpha}_{i}=\varepsilon_{i} \omega_{\rm m},\ i=1,\ 2,\ 3 \tag{14}\]  
Introducing Equation (14) into the derivative of Equation (12), we get  
\[\dot{\varphi}_{1}=(1+\varepsilon_{0}+\frac{3}{2}\varepsilon_{1}+\varepsilon_ {2}+\frac{1}{2}\varepsilon_{3})\omega_{\rm m}=(1+\upsilon_{1})\omega_{\rm m}\]  
\[\dot{\varphi}_{R1}=(1+\varepsilon_{0}-\frac{1}{2}\varepsilon_{1}+\varepsilon_ {2}+\frac{1}{2}\varepsilon_{3})\omega_{\rm m}=(1+\upsilon_{R1})\omega_{\rm m}\]  
\[\dot{\varphi}_{R2}=(1+\varepsilon_{0}-\frac{1}{2}\varepsilon_{1}-\varepsilon_ {2}+\frac{1}{2}\varepsilon_{3})\omega_{\rm m}=(1+\upsilon_{R2})\omega_{\rm m}\]  
\[\dot{\varphi}_{R3}=(1+\varepsilon_{0}-\frac{1}{2}\varepsilon_{1}-\varepsilon_ {2}-\frac{3}{2}\varepsilon_{3})\omega_{\rm m}=(1+\upsilon_{R3})\omega_{\rm m}\]  
\[\ddot{\varphi}_{1}=\dot{\upsilon}_{1}\omega_{\rm m},\ \ddot{\varphi}_{R1}=\dot{ \upsilon}_{R1}\omega_{\rm m},\ \ddot{\varphi}_{R2}=\dot{\upsilon}_{R2}\omega_{\rm m},\ \ddot{\varphi}_{R3}=\dot{\upsilon}_{R3}\omega_{\rm m} \tag{15}\]  
At the steady-state, the \(\ddot{\varphi}_{1}\) and \(\ddot{\varphi}_{Ri}\) can be neglected in the first three formulae of Equation (9), and the responses are given as:  
\[x=-\frac{r_{m}r}{\mu_{x}}\cos(\varphi+\varsigma_{1}+\gamma_{x})-\sum_{i=1}^{3} \frac{\eta_{m}\eta_{r}r_{m}r}{\mu_{x}}\cos(\varphi+\varsigma_{Ri}+\gamma_{x})\]\[y=\frac{r_{m}r}{\mu_{y}}\sin(\varphi+\varsigma_{1}+\gamma_{y})+\sum_{i=1}^{3}\frac {\eta_{m}\eta_{r}r_{m}r}{\mu_{y}}\sin(\varphi+\varsigma_{Ri}+\gamma_{y})\]  
\[\psi=-\sum_{i=1}^{3}\frac{\eta_{m}\eta_{r}r_{m}r_{r}}{\mu_{\psi}l_{e}}\sin( \varphi+\varsigma_{Ri}+\beta_{i}+\gamma_{\psi}) \tag{16}\]  
The electromagnetic torque of the induction motor can be written as the following expression [25; 26] in the situation of operating in the vicinity of \(\omega_{\rm m}\):  
\[T_{\rm e1}=T_{\rm e01}-k_{\rm e01}v_{1} \tag{17}\]