## 2 Materials and Methods
### Theory and Modeling
#### 2.2.1 Finite Element Theory of Rotor  
Because the winding rotor is the cantilever beam structure, the motion and deformation mainly come from the radial direction during the high-speed rotating of the winding rotor. Therefore, this paper did not consider the deformation and rotation in the \(Z\)-axis (length) direction. The cylindrical shaft beam element is shown in Figure 4, \(P\) and \(Q\) are two nodes of the beam element, and the local element spatial coordinates \(o-xyz\) are established with the node \(P\), \(z\) is tangent along the axis direction of the element, \(x\) and \(y\) are defined in the transverse direction of the element. \(l\) is the length of beam element and \(s\) is the distance between node \(P\) and arbitrary point on the centerline. Figure 3: Finite element model of the winding rotor. According to the structural features and the working characteristics of the winding rotor, axial motion and torsion were neglected, the translation of the cross-section centerline was given by the two distances (\(u,v\)), and rotation was given by the two angles (\(\theta_{x},\ \theta_{y}\)) due to bending and shear deformation. The generalized coordinates of the beam element can be expressed as following. \[\mathbf{e}=\left\{u_{Px}\,v_{Py}\,\theta_{Px}\,\theta_{Py}\,u_{Qx}\,v_{Qy}\, \theta_{Qx}\,\theta_{Qy}\right\}^{\text{T}} \tag{1}\]  
where \(u_{Px}\) and \(u_{Qx}\) are the translation of the two nodes \(P\) and \(Q\) along the \(x\)-axis direction, respectively; \(v_{Px}\) and \(v_{Qx}\) are the translation of the two nodes \(P\) and \(Q\) along the \(y\)-axis direction, respectively; \(\theta_{Px}\) and \(\theta_{Qx}\) are the angular deformations rotating around the \(x\) axis; and \(\theta_{Py}\) and \(\theta_{Qy}\) are the angular deformations rotating around the \(y\) axis. The generalized displacement of arbitrary points on the centerline is a position and time-dependent function, and the displacement can be expressed by displacement interpolation functions. For the shaft beam element, linear-displacement interpolation functions \(\mathbf{\Psi}_{r}\) and angular-displacement interpolation functions \(\mathbf{\Phi}_{r}\) can be written as [29]  
\[\mathbf{\Psi}_{r}=\left\{ \begin{array}{c}\Psi_{1}(s)\\ \Psi_{2}(s)\\ \Psi_{3}(s)\\ \Psi_{4}(s)\end{array} \right\}=\left\{ \begin{array}{c}\frac{1}{1+\chi} \big{(}2a^{3}-3a^{2}+1+\chi(1+a)\big{)}\\ \frac{1}{1+\chi}\big{(}a^{3}-2a^{2}+a+\frac{1}{2}\chi a(1+a)\big{)}\\ \frac{1}{1+\chi}\big{(}-2a^{3}+3a^{2}+\chi a\big{)}\\ \frac{1}{1+\chi}\big{(}a^{3}-a^{2}+\frac{1}{2}\chi a^{2}-\frac{1}{2}\chi a \big{)}\end{array} \right\}^{\text{T}} \tag{2}\]  
\[\mathbf{\Phi}_{r}=\left\{ \begin{array}{c}\Phi_{1}(s)\\ \Phi_{2}(s)\\ \Phi_{3}(s)\\ \Phi_{4}(s)\end{array} \right\}=\left\{ \begin{array}{c}\frac{1}{(1+\chi)l} \big{(}6a(a-1)\big{)}\\ \frac{1}{1+\chi}\big{(}3a^{2}-4a+1+\chi(1-a)\big{)}\\ \frac{1}{(1+\chi)l}\big{(}6a(a-1)\big{)}\\ \frac{1}{1+\chi}\big{(}3a^{2}-2a+\chi a\big{)}\end{array} \right\} \tag{3}\]  
where \(a=s/l\); \(\chi\) is the transverse shear effect and \(\chi=12 EI/K^{\prime}A_{b}Gl^{2}\); \(E\) is the elastic modulus of beam element; \(I\) is the second moment of area and \(I=\pi(D^{4}-d^{4})/64\), \(D\) and \(d\) are the outer diameter and inner diameter of the beam element; \(A_{b}\) is the cross-section area of beam element; \(K^{\prime}\) is the transverse shear form factor; and \(G\) is the shear modulus. The generalized displacement of a typical point internal to the beam element is approximated by the relation [15]  
\[\left\{ \begin{array}{c}u_{x}(s,t)\\ v_{y}(s,t)\\ \theta_{x}(s,t)\\ \theta_{y}(s,t)\end{array} \right\}=\left\{ \begin{array}{cccccccc}\Psi_{1}&0 &0&\Psi_{2}&\Psi_{3}&0&0&\Psi_{4}\\ 0&\Psi_{1}&-\Psi_{2}&0&0&\Psi_{3}&-\Psi_{4}&0\\ 0&-\Phi_{1}&\Phi_{2}&0&0&-\Phi_{3}&\Phi_{4}&0\\ \[PERSON]_{1}&0&0&\[PERSON]_{2}&\Phi_{3}&0&0&\Phi_{4}\end{array} \right\}^{\text{T}} \mathbf{e}(t) \tag{4}\]  
Figure 4: Cylindrical shaft beam element.