# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## Appendix A  
In this section, detailed proof of the proposed proposition in Section 3.1 is presented. Regarding the issue of including the rotodynamic matrices in cyclic symmetry analysis, the determination factors are whether the matrices can be transferred into block-circulated form in cyclically symmetric coordinates. Thus, the core content of the proposition is that one matrix can be transferred into block-circulated form in a cyclically symmetric coordinate system if it satisfies the relationship shown in Equation (4). As an example, a block-circulated matrix is shown herein as follows:  
\[\mathbf{Z}=\left[ \begin{array}{cccccccc}\mathbf{Z}_{1}&\mathbf{Z}_{2}&0&0& \cdots&0&\mathbf{Z}_{3}\\ \mathbf{Z}_{3}&\mathbf{Z}_{1}&\mathbf{Z}_{2}&0&\cdots&0&0\\ 0&\mathbf{Z}_{3}&\mathbf{Z}_{1}&\mathbf{Z}_{2}&\cdots&0&0\\ 0&0&\mathbf{Z}_{3}&\mathbf{Z}_{1}&\cdots&0&0\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ 0&0&\cdots&0&\mathbf{Z}_{3}&\mathbf{Z}_{1}&\mathbf{Z}_{2}\\ \mathbf{Z}_{2}&0&0&\cdots&0&\mathbf{Z}_{3}&\mathbf{Z}_{1}\end{array} \right] \tag{1}\]  
where \(\mathbf{Z}_{1},\mathbf{Z}_{2}\), and \(\mathbf{Z}_{3}\) are the sub-blocks of \(\mathbf{Z}\). Without loss of generality, an \(N\)-sector finite element model is used as an example for detailed deduction. For brevity, the DOFs on the right and left cyclic boundaries are represented by \"\(R\)\" and \"\(L\)\" and the others as \"\(I\)\", and the left part of the \(s\)th sector \(\widetilde{\mathbf{u}}_{L}^{(s)}\) is equal to the right part of the \((s+1)\)th sector \(\widetilde{\mathbf{u}}_{R}^{(s+1)}\). Then, the coordinates of this model in a global Cartesian coordinate system can be represented as \[\widehat{\mathbf{u}}=\left[ \begin{array}{c}\widehat{\mathbf{u}}^{(1)}\\ \widehat{\mathbf{u}}^{(2)}\\ \vdots\\ \widehat{\mathbf{u}}^{(s)}\\ \vdots\\ \widehat{\mathbf{u}}^{(N)}\end{array} \right],\quad\widehat{\mathbf{u}}^{(s)}= \left\{ \begin{array}{c}\widehat{\mathbf{u}}^{(s)}_{R}\\ \widehat{\mathbf{u}}^{(s)}_{I}\end{array} \right\},\;\;s=1,2,3,\cdots,N. \tag{12}\]  
Then, a certain structural matrix \(\widehat{\mathbf{Y}}\) of this model in a global Cartesian coordinate system has the following form:  
\[\widehat{\mathbf{Y}}=\left[ \begin{array}{ccccccccc}\widehat{\mathbf{Y}}^{(N )}_{IL}+\mathbf{Y}^{(1)}_{RR}&\widehat{\mathbf{Y}}^{(1)}_{RI}&\widehat{ \mathbf{Y}}^{(1)}_{RL}&0&0&0&0&\cdots&\widehat{\mathbf{Y}}^{(N)}_{LR}&\widehat {\mathbf{Y}}^{(N)}_{LI}\\ \widehat{\mathbf{Y}}^{(1)}_{IR}&\widehat{\mathbf{Y}}^{(1)}_{IR}&\widehat{ \mathbf{Y}}^{(1)}_{IR}&\widehat{\mathbf{Y}}^{(1)}_{RL}&0&0&0&\cdots&0&0\\ \widehat{\mathbf{Y}}^{(1)}_{LR}&\widehat{\mathbf{Y}}^{(1)}_{IR}&\widehat{ \mathbf{Y}}^{(1)}_{IR}+\widehat{\mathbf{Y}}^{(2)}_{RR}&\widehat{\mathbf{Y}}^ {(2)}_{RI}&\widehat{\mathbf{Y}}^{(2)}_{RL}&0&0&\cdots&0&0\\ 0&0&\widehat{\mathbf{Y}}^{(2)}_{LR}&\widehat{\mathbf{Y}}^{(2)}_{IR}&\widehat{ \mathbf{Y}}^{(2)}_{LI}&\widehat{\mathbf{Y}}^{(3)}_{RL}&\widehat{\mathbf{Y}}^ {(3)}_{RL}&\widehat{\mathbf{Y}}^{(3)}_{RL}&\cdots&0&0\\ 0&0&0&\widehat{\mathbf{Y}}^{(3)}_{IR}&\widehat{\mathbf{Y}}^{(3)}_{II}&\widehat {\mathbf{Y}}^{(3)}_{IL}&\widehat{\mathbf{Y}}^{(3)}_{IR}&\cdots&0&0\\ 0&0&0&0&\widehat{\mathbf{Y}}^{(3)}_{LR}&\widehat{\mathbf{Y}}^{(3)}_{LI}& \widehat{\mathbf{Y}}^{(3)}_{IL}&\widehat{\mathbf{Y}}^{(4)}_{RR}&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ \widehat{\mathbf{Y}}^{(N)}_{RL}&0&0&0&0&0&\cdots&\widehat{\mathbf{Y}}^{(N-1)}_ {LL}+\widehat{\mathbf{Y}}^{(N)}_{RR}&\widehat{\mathbf{Y}}^{(N)}_{RI}\\ \widehat{\mathbf{Y}}^{(N)}_{IL}&0&0&0&0&0&\cdots&\widehat{\mathbf{Y}}^{(N)}_{ IR}&\widehat{\mathbf{Y}}^{(N)}_{III}\end{array} \right]. \tag{13}\]  
For clarity, the above matrix is further expressed as block form as  
\[\widehat{\mathbf{Y}}=\left[