# Study on Influence of Multi-Parameter Variation of Bladed Disk System on Vibration Characteristics
## 2 Modal Calculation of Bladed Disk System System Based on Group Theory
\begin{array}{ccc}{}^{(d)}\Lambda_{mm}&\mu_{mk}&\mu _{mt}+\mu_{mt^{\prime}}\cos\pi\alpha\\ \mu_{mk}&{}^{(b)}\mathbf{I}_{kk}&\mu_{kt}+\mu_{kt^{\prime}}\cos\pi\alpha\\ \mu_{mt}+\mu_{mt^{\prime}}\cos\pi\alpha&\mu_{kt}+\mu_{kt^{\prime}}\cos\pi&{}^{ (d)}\overline{m}_{tt}+{}^{(d)}\overline{m}_{t^{\prime}t^{\prime}}+\Big{(}{}^{ (d)}\overline{m}_{t^{\prime}t}+{}^{(d)}\overline{m}_{tt^{\prime}}\Big{)}\cos \pi\alpha\end{array} \right] \tag{8}\]  
\[[M_{r}^{\text{Im}}]=\left[ \begin{array}{ccc}0&0&\mu_{mt^{\prime}}\\ 0&0&\mu_{kt^{\prime}}\\ -\mu_{t^{\prime}m}&-\mu_{t^{\prime}k}&{}^{(d)}\overline{m}_{tt^{\prime}}-{}^{ (d)}\overline{m}_{t^{\prime}t}\end{array} \right]\sin\pi\alpha \tag{9}\]  
\[[M_{r}^{\text{Im}}]=\left[ \begin{array}{ccc}{}^{(d)}\Lambda_{mm}&0&0\\ 0&{}^{(b)}\mathbf{I}_{kk}&0\\ 0&0&{}^{(d)}\overline{k}_{tt}+{}^{(d)}\overline{k}_{t^{\prime}t^{\prime}}+ \Big{(}{}^{(d)}\overline{k}_{t^{\prime}t}+{}^{(d)}\overline{k}_{t^{\prime}t} \Big{)}\cos\pi\alpha\end{array} \right] \tag{10}\]  
\[[K_{r}^{\text{Im}}]=\left[ \begin{array}{ccc}0&0&0\\ 0&0&0\\ 0&0&{}^{(d)}\overline{k}_{tt^{\prime}}-{}^{(d)}\overline{k}_{t^{\prime}t} \end{array} \right]\sin\pi\alpha \tag{11}\]  
Here, \(m=k=6\), that is, the first six eigenvalues and the eigenvector of one are taken, to reduce the calculation scale. There are:  
\[\mu_{mt^{\prime}}=\mu^{\text{T}}{}_{t^{\prime}m},\ \ \ \mu_{kt^{\prime}}=\mu^{\text{T}}{}_{t^{ \prime}k},\ \ \ {}^{(d)}\overline{m}_{tt^{\prime}}={}^{(d)}\overline{m}_{t^{\prime}t}^{\text{T}} \ \ \ {}^{(d)}\overline{k}_{t^{\prime}t}={}^{(d)}\overline{m}_{tt^{\prime}}^{\text{T}} \tag{12}\]  
By the property of the block circular matrix, let \(\{\bar{\xi}_{r}\}=\{\overline{\xi}_{r}\}e^{ir\pi}\), and plug in:  
\[([K_{r}]-\lambda_{r}[M_{r}])\{\overline{\xi}_{r}\}=0 \tag{13}\]  
\[\{\overline{\xi}_{r}\}=\left\{\overline{\xi}_{r}^{\text{Re}}\right\}+i\left\{ \overline{\xi}_{r}^{\text{Im}}\right\} \tag{14}\]  
By expanding the equation, the real characteristic equation of \(2n\) order is obtained:  
\[\left[ \begin{array}{ccc}K_{r}^{\text{Re}}&-K_{r}^{\text{Im}}\\ K_{r}^{\text{Im}}&K_{r}^{\text{Re}}\end{array} \right]\left\{ \begin{array}{ c}\overline{\xi}_{r}^{\text{Re}}\\ \overline{\xi}_{r}^{\text{Im}}\end{array} \right\}=\lambda_{r}\left[ \begin{array} []{ccc}M_{r}^{\text{Re}}&-M_{r}^{\text{Im}}\\ M_{r}^{\text{Im}}&M_{r}^{\text{Re}}\end{array} \right]\left\{ \begin{array}{ c}\overline{\xi}_{r}^{\text{Re}}\\ \overline{\xi}_{r}^{\text{Im}}\end{array} \right\} \tag{15}\]  
\[r=1,2,\cdots N_{f};\,N_{f}=\left\{ \begin{array}{c}N/2-1,\ N\,\text{is\,even \,numbers}\\ (N-1)/2,\ N\,\text{is\,odd\,numbers}\end{array} \right. \tag{16}\]  
when \(r=0\) and \(N/2(N\) is even), the equation degenerates into a real feature problem, i.e.,  
\[ \begin{array}{c}[K_{0}]\{\overline{\xi}_{0}\}=\lambda_{0}[M_{0}]\{\overline{ \xi}_{0}\}\\ [K_{N/2}]\{\overline{\xi}_{N/2}\}=\lambda_{N/2}[M_{N/2}]\{\overline{\xi}_{N/2}\} \end{array}  \tag{17}\]  
The basic physical parameters of vibration mainly include vibration frequency and mode. The natural frequency of each component of the rotor of a turbine includes the natural frequency of rotating state and the natural frequency of non-rotating state. The latter depends on the material characteristics, collection characteristics and boundary conditions of the components, and is not dependent on external factors. The component structure material is determined, and the natural frequency of its non-rotating state is also determined accordingly. Since all rotor components are continuous elastomers, there are multiple natural vibration frequencies. For each \(r\), the \(2n\) eigenvalues of the real eigenvalue problem are \(n\) double roots, and the eigenvectors associated with each double roots \(\lambda_{r}\) are orthogonal to each other's two-dimensional eigensubspace, that is, there are two mutually orthogonal modes. \[\left\{\overline{\xi}_{r}^{(1)}\right\}=\left\{ \begin{array}{c} \mathrm{Re}\!\left\{\overline{\xi}_{r_{j}}\right\}\\ \mathrm{Im}\!\left\{\overline{\xi}_{r_{j}}\right\}\end{array}