## 2 Materials and Methods
### Theory and Modeling
#### 2.2.2 Coupling Model  
In order to illustrate the coupling relationship of between node \(i\) and node \(j\), a two-degrees-of-freedom spring-mass-damping system is used, and the draw free-body diagrams of the system as shown in Figure 5.  
The coordinates describing the positions of the masses, \(x_{i}(t)\) and \(x_{j}(t)\), are measured from their respective static equilibrium positions. \(F_{i}(t)\) and \(F_{j}(t)\) denote the forces acting on the node masses \(m_{i}\) and \(m_{j}\), respectively. The equations of motion of the masses \(m_{i}\) and \(m_{j}\) can be derived and written as  
\[\left[ \begin{array}{cc}m_{i}&0\\ 0&m_{j}\end{array} \right]\left\{ \begin{array}{c}\ddot{x}_{i}\\ \ddot{x}_{j}\end{array} \right\}+\left[ \begin{array}{cc}c_{i}+c_{s}&-c_{s} \\ -c_{s}&c_{s}+c_{j}\end{array} \right]\left\{ \begin{array}{c}\dot{x}_{i}\\ \dot{x}_{j}\end{array} \right\}+\left[ \begin{array}{cc}\dot{x}_{i}+k_{s}&-k_{ s}\\ -k_{s}&k_{s}+k_{j}\end{array} \right]\left\{ \begin{array}{c}x_{i}\\ x_{j}\end{array} \right\}=\left\{ \begin{array}{c}F_{i}(t)\\ F_{j}(t)\end{array} \right\} \tag{8}\]  
where \(k_{s}\) and \(c_{s}\) are the equivalent supporting stiffness and damping between two coupling nodes. \(k_{i}\) and \(c_{i}\) are the stiffness and damping of node \(i\), and \(k_{j}\) and \(c_{j}\) are the stiffness and damping of node \(j\). (1) Rubber O-Ring  
With the development of the textile industry, the winding speed is getting higher and higher, which leads to the rotational speed of the winding rotor to exceed the second- or third-order critical speed. To avoid heavy vibration when the winding rotor passes through the critical speed points, the structure of flexible support is adopted. The flexible coupling support with rubber O-rings is shown in Figure 6a. Figure 5: Spring-mass-damping system. The rubber O-ring is equivalent to a spring-damping element. In practice, the radial compression of the rubber O-ring installed in the winding rotor is less than 0.1 mm, and it is much smaller than the cross-sectional diameter of the rubber O-ring by 3.5-4 mm. Therefore, the supporting stiffness and damping of the rubber O-ring can be approximated as the linear elastic range, hence the spring-damping element may be further equivalent to the linear spring-damping element of the single-degree-of-freedom system in the X direction, as shown in Figure 6b. The rubber O-ring is installed between the grooved bush and the supporting arm. The groove width is often larger than the cross-sectional diameter of the rubber O-ring. In addition, the groove is filled with grease. Therefore, a single rubber ring can only bear radial load, and multiple rubber rings installed in parallel produce a certain bending effect through their radial stiffness. Naturally, the rubber O-ring acts as a supporting element and does not limit the rotation. When node \(i\) and node \(j\) are coupled through rubber O-rings, it is assumed that the rubber O-rings only restrict the displacement, but not the rotation. Therefore, stiffness and damping coupled nodes can be expressed as  
\[ \begin{split}\mathbf{K}_{\text{rubber}}&=\left[ \begin{array}{cc}\mathbf{I}_{2}&0\\ 0&0\end{array}\right]_{4\times 4}k_{\text{r}ij}\\ \mathbf{C}_{\text{rubber}}&=\left[\begin{array}{cc}\mathbf{I}_{2}&0\\ 0&0\end{array}\right]_{4\times 4}c_{\text{r}ij}\end{split}  \tag{9}\]  
where \(\mathbf{I}_{2}\) is an identity matrix of order 2. \(k_{\text{r}}\) and \(c_{\text{r}}\) are the equivalent stiffness and damping coefficients of the rubber O-rings, respectively. The coefficients of the equivalent stiffness and damping coefficients of the rubber O-ring can be obtained through the forced non-resonance method [14], and the frequency-supporting stiffness curve and the frequency-supporting damping curve are presented in Figure 7.  
Three types of rubber O-rings are used in the winding rotor shown in Figure 1, and they are always used in groups of 2, 4 and 6 rubber O-rings.