## 6 Conclusions
### Nomenclature  
\begin{tabular}{l l} \(f_{1}\) & damping coefficient of an induction motor's rotor \\ \(f_{x^{\prime}}\), \(f_{y^{\prime}}\), \(f_{\varphi}\) & damping constants of the vibrating system in \(x\)-, \(y\)- and \(\psi\)- directions, respectively, \(f_{\varphi}=(f_{x}x_{y}^{2}+f_{y}x_{x}^{2})/2\) \\ \(f_{Ri}\) & friction coefficient between CR \(i\) and the inner wall of the corresponding cavity, \(i=1,\ 2,\ 3\) \\ \(\mathbf{F}_{i}\) & damping matrix of spring \(i\), \(i=1,\ 2,\ 3,\ 4,\ \mathbf{F}_{1}=\mathbf{F}_{3}=\mathrm{diag}(f_{x}/2,\ 0)\), \(\mathbf{F}_{2}=\mathbf{F}_{4}=\mathrm{diag}(0,\ f_{y}/2)\) \\ \(g\) & acceleration due to the gravity, and gravity is in \(y\)- direction \\ \(I_{Ri}\) & moment of inertia of CR \(i\) about its spin axis \(o_{Ri}\), \(i=1,\ 2,\ 3\) \\ \(J\) & moment of inertia of the vibrating system, \(J=J_{m}+m_{1}r^{2}+3m_{R}(r_{R}^{2}+l_{0}^{2})\) \\ \(I_{1}\) & moment of inertia of an exciter, \(J_{1}=m_{1}r^{2}\) \\ \(J_{m}\) & moment of inertia of the rigid frame \\ \(R_{Ri}\) & moment of inertia of CR \(i\), \(J_{Ri}=I_{Ri}+m_{R}r_{R}^{2}\), \(i=1,2,3\) \\ \(k_{\phi 1}\) & stiffness coefficient of angular velocity of an induction motor operating at angular velocity \(\omega_{\mathrm{m}}\) \\ \(k_{x}\), \(k_{y}\), \(k_{\phi}\) & spring constants of the vibrating system in \(x\)-, \(y\)- and \(\psi\)-directions, respectively, \(k_{\phi}=(k_{x}l_{y}^{2}+k_{y}l_{x}^{2})/2\) \\ \(\mathbf{K}_{i}\) & stiffness matrix of spring \(i\), \(i=1,\ 2,\ 3,\ 4,\ \mathbf{K}_{1}=\mathbf{K}_{3}=\mathrm{diag}(k_{x}/2,\ 0)\), \(\mathbf{K}_{2}=\mathbf{K}_{4}=\mathrm{diag}(0,\ k_{y}/2)\) \\ \(l_{0}\) & length from the rotary center of an exciter to the centroid of an arbitrary cavity \\ \(l_{\mathrm{e}}\) & equivalent rotating radius of the vibrating system about the centroid of the rigid frame, \(J=Ml_{\mathrm{e}}^{2}\) \\ \(L_{m}\) & mutual inductance between the stator and rotor windings \\ \(L_{r}\) & inductance of the rotor windings \\ \(L_{s}\) & inductance of the stator windings \\ \(m\) & mass of the rigid frame \\ \(m_{1}\) & mass of an exciter \\ \(m_{R}\) & mass of the standard CR \\ \(m_{R1}\) & mass of CR \(i\), \(i=1,\ 2,\ 3\), \(m_{R1}=m_{R2}=m_{R3}=m_{R}\) \\ \(M\) & total mass of the vibrating system, \(M=m+m_{1}+3m_{R}\) \\ \(N_{i}\) & reaction force acting on CR \(i\) (\(i=1,\ 2,\ 3\)) by the inner wall of the corresponding cavity \\ \(r\) & eccentric radius of an exciter \\ \(r_{0}\) & radius of the standard CR \\ \(r_{0i}\) & radius of CR \(i\), \(i=1,\ 2,\ 3\), \(r_{01}=r_{02}=r_{03}=r_{0}\) \\ \(r_{I}\) & \(r_{I}=l_{0}/l_{\mathrm{e}}\) \\ \(r_{m}\) & mass ratio of an exciter to the vibrating system, \(r_{m}=m_{1}/M\) \\ \(r_{R}\) & rotational radius of the standard CR rotating about the centroid of the standard cavity along the inner wall, \\ \(r_{R}=R-r_{0}\) & rotational radius of CR \(i\) (\(i=1,\ 2,\ 3\)) rotating about the centroid of the corresponding cavity along the inner wall, \\ \(r_{R1}=r_{R2}=r_{R3}=r_{R}\) \\ \(R\) & radius of the standard cavity \\ \(R_{i}\) & radius of cavity \(i\), \(i=1,\ 2,\ 3\), \(R_{1}=R_{2}=R_{3}=R\) \\ \(R_{r}\) & resistance of the rotor windings \\ \(R_{s}\) & resistance of the stator windings \\ \(\mathbf{R}\) & matrix of rotation, \(\mathbf{R}=\left[ \begin{array}{cc}\cos\psi&\sin\psi\\ -\sin\psi&\cos\psi\end{array}