# 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  
The group theory is one of the main methods for vibration analysis of circularly symmetric structures. If N blades are identical, that is, the same geometrical, material and natural vibration frequency, then the bladed disk system is an N-order circularly symmetric structure. From the perspective of group theory, the N-order cyclic symmetric structure is about the symmetry of C\({}_{\text{N}}\) groups, so the N-order cyclic symmetric structure is also known as the C\({}_{\text{N}}\) structure. The vibration of the bladed disk system is analyzed by using group theory and modal synthesis. By using the property of block circulation matrix \([T]\), the motion equation of the whole system is reduced to a complex characteristic equation of a lower order and transformed into a real characteristic equation. Then the software of real symmetric matrix eigenvalue is used to solve the problem. It can be inferred from the properties of the cyclic symmetry structure that the node displacements in adjacent sectors of a cyclic symmetry structure have the same transfer matrix, and they are satisfied by:  
\[\{\delta_{k+1}\}=[T]\{\delta_{k}\} \tag{1}\]  
where, \(T\) is the transfer matrix and \(k=1\),..., \(N\) is the number of blades, and \(N\) is the total number of blades. The displacement \(\left\{{}^{(d)}\delta_{t}\right\}\) and \(\left\{{}^{(b)}\delta_{t^{\prime}}\right\}\) are not independent. If displacement is considered as a complex number, the relation between them is:  
\[\left\{{}^{(b)}\delta_{t^{\prime}}\right\}=\left\{{}^{(b)}\delta_{t}\right\}e ^{j\pi\alpha} \tag{2}\]  
\[\{p_{t^{\prime}}\}=\{p_{t}\}e^{j\pi\alpha} \tag{3}\]  
where, \(i=\sqrt{-1}\) is the unit of pure virtual quantity; \(r=0,1,2,\cdots,N/2\) is the number of the pitch diameter, \(\alpha=2\pi/N\) is the rotation period, and \(N\) is the total number of blades, that is, the order of cyclic symmetry. The total transformation can be derived from above as follows:  
\[\{q\}=\left\{ \begin{array}{c}\bar{\chi}_{m}\\ \eta_{k}\\ P_{t}\\ P_{t^{\prime}}\end{array} \right\}=\left[ \begin{array}{ccc}I_{mm}&0&0\\ 0&I_{kk}&0\\ 0&0&I_{tt}\\ 0&0&I_{tt}e^{j\pi\alpha}\end{array} \right]\left\{ \begin{array}{c}\xi_{m}\\ \eta_{k}\\ P_{t}\end{array} \right\}=[H_{r}]\{\xi_{r}\} \tag{4}\]  
The mass matrix and stiffness matrix of the bladed disk system in \(\xi_{r}\) coordinates are obtained by the functional invariant principle:  
\[ \begin{array}{rl}[M_{r}]&=[H_{r}]^{*}[\mu_{r}][H_{r}]\\ &=\left[\begin{array}{ccc}I_{mm}&\mu_{mk}&\mu_{mt}+e^{i\pi\alpha}\mu_{mt^{ \prime}}\\ \mu_{mk}&I_{kk}&\mu_{tt}+e^{i\pi\alpha}\mu_{kt^{\prime}}\\ \mu_{mt}+e^{i\pi\alpha}\mu_{mt^{\prime}}&\mu_{kt}+e^{i\pi\alpha}\mu_{kt^{ \prime}}&{}^{(d)}\overline{m}_{tt}+{}^{(d)}\overline{m}_{tt^{\prime}t^{\prime} }+e^{-i\pi\alpha}{}^{(d)}\overline{m}_{t^{\prime}t}+e^{i\pi\alpha}{}^{(d)} \overline{m}_{tt^{\prime}}\end{array}\right]\end{array}  \tag{5}\]  
\[ \begin{array}{rl}[K_{r}]&=[H_{r}]^{*}[k_{r}][H_{r}]\\ &=\left[\begin{array}{ccc}{}^{(d)}\Lambda_{mm}&0&0\\ 0&{}^{(b)}\Lambda_{kk}&0\\ 0&0&{}^{(d)}\overline{k}_{tt}+{}^{(d)}\overline{k}_{t^{\prime}t^{\prime}}+e^{- i\pi\alpha}{}^{(d)}\overline{k}_{t^{\prime}t}+e^{i\pi\alpha}{}^{(d)}\overline{k}_{tt^{ \prime}}\end{array}\right]\end{array}  \tag{6}\]  
The matrices in the Equations (5) and (6) are the Hermite matrices. The equation of the complex motion of the disk system is obtained by:  
\[[M_{r}]\Big{\{}\bar{\xi}\Big{\}}+[K_{r}]\{\bar{\xi}\}=0 \tag{7}\]  
In order to extract characteristic pairs of the Hermite matrix bundle \([M_{r}]\), \([K_{r}]\), the real and imaginary parts of them were written as \(\big{[}M_{r}^{Re}\big{]}\), \(\big{[}K_{r}^{Re}\big{]}\) and \(\big{[}M_{r}^{Im}\big{]}\), \(\big{[}K_{r}^{Im}\big{]}\), i.e.,  
\[[M_{r}^{\text{Re}}]=\left[