## 5 Simulation Verification
Despite the non-identical friction coefficients, the rotational velocities of the synchronization for the three CRs (oscillating regularly around 991.7 r/min) are equal because of the regulating effect from the dynamic coupling torques. However, the difference in the friction coefficients makes the limit behavior of the coupled CRs transform from in-phase synchronization to partial synchronization (comparing Figure 12c,d with Figure 11c,d), i.e., the stable value of \(2\alpha_{2}\) changes from to \(344.5^{\circ}\), \(2\pi_{3}\) from \(0^{\circ}\) to \(16.2^{\circ}\). Note that there are small fluctuations in the stable phase differences among three CRs. The reason behind this phenomenon is that, as mentioned in Figures 4 and 5, the mismatch in the friction coefficients can cause the difference in the synchronization abilities between an exciter and a CR, i.e., in the second simulation, the intensity level of being synchronized by the exciter through the rigid frame in CR 2 is higher than that in CR 1 or CR 3. In Figure 12e,f (Figure 13e,f), it is observed that apart from identical vibrations in \(x\)- and \(y\)-directions, the vibration in \(\psi\)-direction is non-zero and obvious, which indicates a mixed form of motion consisting of a circular motion and a minor swing motion at the steady-state. Note that the swing movement fails to effectively transmit energy. Figure 11: Simulation results when \(f_{\rm{R1}}=0.08\), \(f_{\rm{R2}}=0.08\), \(f_{\rm{R3}}=0.08\): (**a**) rotational velocities of three CRs and an induction motor; (**b**) phase difference between an exciter and CR 1; (**c**) phase difference between CRs 1 and 2; (**d**) phase difference between CRs 2 and 3; (**e**) displacements in \(x\)- and \(y\)-directions; (**f**) angular displacement in \(\psi\)-direction; (**g**) reaction force acting on CR \(i\) (\(i=1,2,3\)) by the inner wall. Figure 12: Simulation results when \(f_{R1}=0.08\), \(f_{R2}=0.04\), \(f_{R3}=0.08\): (**a**) rotational velocities of three CRs and an induction motor; (**b**) phase difference between an exciter and CR 1; (**c**) phase difference between CRs 1 and 2; (**d**) phase difference between CRs 2 and 3; (**e**) displacements in \(x\)- and \(y\)-directions; (**f**) angular displacement in \(\psi\)-direction; (**g**) reaction force acting on CR \(i\) (\(i=1,2,3\)) by the inner wall. Figure 13: Simulation results when \(f_{R1}=0.08\), \(f_{R2}=0.06\), \(f_{R3}=0.04\): (**a**) rotational velocities of three CRs and an induction motor; (**b**) phase difference between an exciter and CR 1; (**c**) phase difference between CRs 1 and 2; (**d**) phase difference between CRs 2 and 3; (**e**) displacements in \(x\)- and \(y\)-directions; (**f**) angular displacement in \(\psi\)-direction; (**g**) reaction force acting on CR \(i\) (\(i=1,2,3\)) by the inner wall. Figure 14: Simulation results when \(f_{R1}=0.04\), \(f_{R2}=0.04\), \(f_{R3}=0.04\): (**a**) rotational velocities of three CRs and an induction motor; (**b**) phase difference between an exciter and CR 1; (**c**) phase difference between CRs 1 and 2; (**d**) phase difference between CRs 2 and 3; (**e**) displacements in \(x\)- and \(y\)-directions; (**f**) angular displacement in \(\psi\)-direction; (**g**) reaction force acting on CR \(i\) (\(i=1,2,3\)) by the inner wall. Figure 13 presents the results of computer simulation when \(f_{R1}=0.08\), \(f_{R2}=0.06\), \(f_{R3}=0.04\). When the vibrating system moves into the synchronized state at \(t=7\) s, the relevant steady-state values are as follows: the synchronous rotation velocity nears 991.9 r/min (or fluctuating around it); the phase differences are \(2\alpha_{1}=199^{\circ}\sim 212^{\circ}\) (average value \(205.5^{\circ}\)), \(2\alpha_{2}=351^{\circ}\sim 352^{\circ}\) (average value \(351.5^{\circ}\)), and \(2\alpha_{3}=352^{\circ}\sim 353^{\circ}\) (average value \(352.5^{\circ}\)). It should be stressed that the fluctuations of the stable values of \(2\alpha_{2}\) and \(2\alpha_{3}\) in Figure 13c,d are smaller than those from Figure 12c,d.