## 2 Materials and Methods
### Theory and Modeling
#### 2.2.2 Coupling Model
With the increase of cross-sectional diameter (\(\Phi\)3.5, \(\Phi\)4), outer ring diameter (\(\Phi\)44, \(\Phi\)47, \(\Phi\)54) and number (2, 6) of rubber O-rings, the supporting stiffness and damping coefficients increase correspondingly, and show obvious frequency-dependent characteristics. Figure 6: Supporting element of the rubber O-ring. (**a**) Flexible support; (**b**) equivalent linear spring-damping model. If node \(i\) and node \(j\) are coupled through the rubber O-rings, based on the theory of the two-degree-of-freedom spring-mass-damping system [30], the global coupling matrices of the stiffness and the damping of the winding rotor can be written as  
\[\mathbf{K}_{\text{Rubber}}=\left[ \begin{array}{cccc}0&&&0\\ &(\mathbf{K}_{\text{rubber}})_{ii}&&-(\mathbf{K}_{\text{rubber}})_{ij}\\ &&0&\\ &-(\mathbf{K}_{\text{rubber}})_{ji}&&(\mathbf{K}_{\text{rubber}})_{ji}\\ 0&&&0\\ \end{array} \right]_{208\times 208} \tag{10}\]  
\[\mathbf{C}_{\text{Rubber}}=\left[ \begin{array}{cccc}0&&&0\\ &(\mathbf{C}_{\text{rubber}})_{ii}&&-(\mathbf{C}_{\text{rubber}})_{ij}\\ &&0&\\ &-(\mathbf{C}_{\text{rubber}})_{ji}&&(\mathbf{C}_{\text{rubber}})_{jj}\\ 0&&&\mathbf{0}\\ \end{array} \right]_{208\times 208} \tag{11}\]  
(2) Rolling Bearing  
The rolling bearing is a rotating part which is affixed to the rotating shaft and the bushing, as shown in Figure 8a. Its main function is to transmit rotation and support the shaft. Usually, to simplify computing, the bearing is equivalent to the spring-damping unit shown in Figure 8b. The damping coefficient of multiple rubber rings installed in parallel is far higher than the bearing's. Therefore, this paper ignores the impact of bearing damping on the winding rotor when building the model (\(c_{\text{bx}}=c_{\text{bz}}=0\)), and with the assumption that the bearing stiffness is symmetrical (\(k_{\text{bx}}=k_{\text{by}}\)), the equivalent bearing stiffness coefficient is \(k_{s}=6\times 10^{7}\) N/m [31]. The global coupling stiffness matrix of the rolling bearing is similar with the rubber O-ring. Figure 7: The curves of supporting stiffness and supporting damping of rubber O-ring. (**a**) Equivalent supporting stiffness curve; (**b**) equivalent supporting damping curve. When the bearing is coupled with other parts, just the coupling effect of equivalent radial stiffness is considered an equivalent axial stiffness of the bearing and is ignored in this paper. The bearing also restricts the displacement, but not including the rotation. Therefore, the global coupling matrix form of equivalent stiffness and damping of the bearing is the same as the rubber O-ring. (3) Interference Connection  
When the shaft is coupled with the sleeve, it is assumed that nodes \(i\) and \(j\) are coupled by the interference connection, which restricts the total of eight coordinates of two coupled nodes just in their stiffness. Because the shaft is fixed rigidly to the sleeve, there is no relative motion between them during rotation, hence the damping between them is not considered. The equivalent stiffness of the interference connection was obtained by the static analysis, and the coefficient is \(k_{s}=1.036\times 10^{8}\) N/m. The global coupling stiffness matrix of the interference connection is similar to that of the rubber O-ring.