## 2 Dynamic Modeling Process of the Fiber Woven Composite Shaft-Disk Rotor
### Energy Derivation of System Structure Units
#### 2.3.1 Potential and Kinetic Energy of the Rotating Shaft Units  
Based on the Timoshenko beam theory, the shaft in the rotor can be equated to a six-degree-of-freedom beam, and the elastic deformation displacement at a point (\(x_{\rm S},y_{\rm S},z_{\rm S}\)) on the beam is:  
\[\overline{w}_{\rm S}\ =\ w_{\rm S}(x_{\rm S})-y_{\rm S}\varphi_{\rm S,z}(x_{\rm S})+ z_{\rm S}\varphi_{\  
\[ \begin{bmatrix}\sigma_{\text{D},r}\\ \sigma_{\text{D},\theta}\\ \tau_{\text{D},r\theta}\\ \tau_{\text{D},rx}\\ \tau_{\text{D},\theta x}\end{bmatrix} = \begin{bmatrix}Q_{11}&Q_{12}&0&0&0\\ Q_{21}&Q_{22}&0&0&0\\ 0&0&Q_{66}&0&0\\ 0&0&0&Q_{44}&0\\ 0&0&0&0&Q_{55}\end{bmatrix}\begin{bmatrix}\varepsilon_{\text{D},r}\\ \varepsilon_{\text{D},\theta}\\ \tau_{\text{D},\theta}\\ \tau_{\text{D},rx}\\ \tau_{\text{D},\theta x}\end{bmatrix} = \begin{bmatrix}\frac{E_{11}}{1-\mu_{12} \mu_{21}}&\frac{\mu_{21}E_{11}}{1-\mu_{12}\mu_{21}}&0&0&0\\ \frac{\mu_{12}E_{22}}{1-\mu_{12}\mu_{21}}&\frac{E_{22}}{1-\mu_{12}\mu_{21}}&0 &0&0\\ 0&0&G_{12}&0&0\\ 0&0&0&G_{23}&0\\ 0&0&0&0&G_{13}\end{bmatrix}\begin{bmatrix}\varepsilon_{\text{D},r}\\ \varepsilon_{\text{D},\theta}\\ \tau_{\text{D},\theta r}\\ \tau_{\text{D},rx}\\ \tau_{\text{D},\theta x}\end{bmatrix}  \tag{17}\]  
Since the first-order shear deformation theory is applied to the ring plate, only \(E_{11}\), \(E_{22}\), \(\mu_{12}\), and \(\mu_{21}\) are introduced for orthotropic anisotropic materials, and they satisfy \(E_{11}\mu_{21}=E_{22}\mu_{12}\), i.e., \(Q_{12}=Q_{21}\). In summary, the potential energy of the ring plate can be found to be:  
\[ \begin{split} V_{\text{D}}&=\ \frac{1}{2}\int\int\int \left(\begin{array}{c}\sigma_{\text{D},r}\varepsilon_{\text{D},r}+\sigma_{ \text{D},\theta}\varepsilon_{\text{D},\theta}+\tau_{\text{D},\theta}\tau_{ \text{D},\theta}\\ +\kappa_{\text{D}}\tau_{\text{D},r}\gamma_{\text{D},r}+\kappa_{\text{D}}\tau_{ \text{D},\theta r}\gamma_{\text{D},\theta r}\\ \end{array}\right)r_{\text{D}}d\tau_{\text{D}}d\theta_{\text{D}}d\chi_{\text{D }}\\ &=\frac{1}{2}\int\int\int\left(\begin{array}{c}Q_{11}r_{\text{D}} \bigg{(}\frac{\partial u_{\text{D}}}{\partial r_{\text{D}}}+x_{\text{D}}\frac {\partial\varphi_{\text{D},\theta}}{\partial r_{\text{D}}}\bigg{)}^{2}+\frac{ \partial_{22}}{\tau_{\text{D}}}\bigg{(}\frac{\partial\varphi_{\text{D}}}{ \partial\theta_{\text{D}}}+u_{\text{D}}-x_{\text{D}}\bigg{(}\frac{\partial \varphi_{\text{D},r}}{\partial\theta_{\text{D}}}-\varphi_{\text{D},\theta} \bigg{)}\bigg{)}^{2}\\ +2Q_{12}\bigg{(}\frac{\partial u_{\text{D}}}{\partial r_{\text{D}}}+x_{\text{D }}\frac{\partial\varphi_{\text{D},\theta}}{\partial r_{\text{D}}}\bigg{)} \bigg{(}\frac{\partial\varphi_{\text{D}}}{\partial\theta_{\text{D}}}+x_{\text{ D}}-x_{\text{D}}\bigg{(}\frac{\partial\varphi_{\text{D},r}}{\partial\theta_{ \text{D}}}-\varphi_{\text{D},\theta}\bigg{)}\bigg{)}\\ +Q_{66}r_{\text{D}}\bigg{(}\frac{1}{r_{\text{D}}}\frac{\partial u_{\text{D}}}{ \partial\theta_{\text{D}}}+\frac{\partial\varphi_{\text{D}}}{\partial r_{ \text{D}}}-\frac{\varphi_{\text{D}}}{r_{\text{D}}}+x_{\text{D}}\bigg{(}\frac{ \partial\varphi_{\text{D},\theta}}{\partial\theta_{\text{D}}}-r_{\text{D}} \frac{\partial\varphi_{\text{D},r}}{\partial r_{\text{D}}}+\varphi_{\text{D}, r}\bigg{)}\bigg{)}^{2}\\ +\kappa_{\text{D}}Q_{44}r_{\text{D}}\bigg{(}\frac{\partial w_{\text{D}}}{ \partial r_{\text{D}}}+\varphi_{\text{D},\theta}\bigg{)}^{2}+x_{\text{D}}Q_{55}r _{\text{D}}\bigg{(}\frac{1}{r_{\text{D}}}\frac{\partial w_{\text{D}}}{ \partial\theta_{\text{D}}}-\varphi_{\text{D},r}\bigg{)}^{2}\end{array}\right) \end{split} \right)d\tau_{\text{D}}d\theta_{\text{D}}d\chi_{\text{D}}\\ &\text{where the shear correction factor}\ \kappa_{\text{D}}=5/6.\end{split} \tag{18}\]  
For a multi-component coupled geared rotor, a multi-body dynamics approach is used to calculate the kinetic energy of the disk in the system. In the absolute coordinate system \(\text{C}_{0}\), the position vector of a point (\(x_{\text{D}}\), \(r_{\text{D}}\), \(\theta_{\text{D}}\)) on the disk is shown in Equation (19).