## 2 Dynamic Modeling Process of the Fiber Woven Composite Shaft-Disk Rotor
### Domain Decomposition and Coordinate Mapping Technique for Perforated Webs  
In order to reduce weight, facilitate machining and handling, and enable the system's critical speed to avoid the operating speed, holes are often cut in the rotor web. These are commonly round holes, sector holes, etc. The web structure with holes, unlike a simple ring plate, is not easy to cut in the circumferential and radial directions for dividing regular sector cells when modeling. Therefore, a domain decomposition method for the web with arbitrary complex hole patterns is given below, and a coordinate mapping technique is introduced for subsequent dynamic modeling of the rotor. As an example, a ring plate with \(N\) round holes with inner/outer diameters of \(R_{i}/R_{o}\) can be uniformly decomposed into \(N\) sector plates with holes in the center, as shown in Figure 4. Since the modeling of the ring plate is usually performed in a rotating coordinate system, the sector plate is also divided into sub-domains in the \(r\)-\(\theta\) coordinate system to ensure that the segmentation lines are easy-to-describe straight lines in the \(r\)-\(\theta\) coordinate system. As in Figure 4, the \(i^{\text{th}}\) sector plate is decomposed into four irregular quadrilateral domains. It should be noted that the domain decomposition should avoid the tangent point of the hole-shaped boundary and the parallel line of the \(r/\theta\) coordinate axis; otherwise, it will affect the accuracy of the subsequent coordinate mapping, which will lead to the computational results of the model with the problems of non-convergence and large error. Figure 3: Mechanical properties of orthogonal plain-weave carbon fiber-reinforced resin matrix composites [7]. After the subdomain decomposition of the open ring plate, each subdomain can be directly regarded as a plate cell for solving the energy matrix of the cell based on the DQFEM. However, the unit is an irregular quadrilateral, which needs to be converted to a regular rectangular unit by means of the coordinate mapping technique. As shown in Figure 5, for any irregular quadrilateral cell in the \(r\)-\(\theta\) coordinate system, assuming that it has \(p\times q\) mapping nodes, the nodes here are only used to reflect the mapping relationship, which is different from the nodes used in the differential product in Section 2.3. By mapping Equation (1) [37], irregular quadrilaterals in the \(r\)-\(\theta\) coordinate system can be converted into square cells with \(-1\leq\xi\leq 1\) and \(-1\leq\eta\leq 1\) in the \(\xi\)-\(\eta\) coordinate system. \[ \begin{array}{l}r\ =\ \mathbf{R}_{qp}\Psi_{qp},\theta\ =\ \mathbf{\Theta}_{qp}\Psi_{qp},\\ \mathbf{R}_{qp}\ =\ \big{[}r_{11},r_{12},\ldots,r_{ijr},\ldots,r_{qp}\big{]},\\ \mathbf{\Theta}_{qp}\ =\ \big{[}\theta_{11},\theta_{12},\ldots,\theta_{ij},\ldots, \theta_{qp}\big{]},\\ \Psi_{qp}\ =\ \big{[}\theta_{11},\theta_{12},\ldots,\varphi_{ijr},\ldots, \varphi_{qp}\big{]}^{\mathsf{T}}\end{array}  \tag{1}\]  
Figure 4: Subdomain decomposition of a perforated ring plate. Figure 5: Coordinate mapping. (\(r_{ij}\), \(\theta_{ij}\)) denotes the coordinates of the mapping points distributed on the edges or inside of the quadrilateral cell. \(\varphi_{ij}\) denotes the mapping function corresponding to the _ij_ mapping point (\(r_{ij}\), \(\theta_{ij}\)), and the specific expression [15] is  
\[\varphi_{ij}(\zeta,\eta)\ =\ \left(\prod_{s\ =\ 1,s\
eq i}^{p}\frac{\zeta-\xi_{s}}{ \xi_{ij}-\xi_{s}}\right)\left(\prod_{l\ =\ 1,l\
eq j}^{q}\frac{\eta-\eta_{l}}{\eta_{ij}-\eta_{l}}\right) \tag{2}\]  
After the ring plate irregular unit has been coordinate-mapped, the two-dimensional differential product weight coefficient matrix for modeling using the DQFEM needs to be changed accordingly. The existing coordinates (\(r\), \(\theta\)) are functions with respect to (\(\xi\), \(\eta\)), and the displacement function f of the structure is a composite function of (\(\xi\), \(\eta\)).