# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## Appendix A
\begin{array}{ccccccccc}\widehat{\mathbf{Y}}^{(1 )}_{1}&\widehat{\mathbf{Y}}^{(1)}_{2}&0&0&\cdots&0&0&\widehat{\mathbf{Y}}^{(N )}_{3}\\ \widehat{\mathbf{Y}}^{(1)}_{3}&\widehat{\mathbf{Y}}^{(2)}_{1}&\widehat{ \mathbf{Y}}^{(2)}_{2}&0&\cdots&0&0&0\\ 0&\widehat{\mathbf{Y}}^{(2)}_{3}&\widehat{\mathbf{Y}}^{(3)}_{1}&\widehat{ \mathbf{Y}}^{(3)}_{2}&\cdots&0&0&0\\ 0&0&\widehat{\mathbf{Y}}^{(3)}_{3}&\widehat{\mathbf{Y}}^{(4)}_{1}&\cdots&0&0&0 \\ 0&0&0&\widehat{\mathbf{Y}}^{(4)}_{3}&\cdots&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&0&\cdots&\widehat{\mathbf{Y}}^{(N-2)}_{3}&\widehat{\mathbf{Y}}^{(N-1)}_ {1}&\widehat{\mathbf{Y}}^{(N-1)}_{2}\\ \widehat{\mathbf{Y}}^{(N)}_{2}&0&0&0&\cdots&0&\widehat{\mathbf{Y}}^{(N-1)}_ {3}&\widehat{\mathbf{Y}}^{(N)}_{1}\end{array} \right] \tag{14}\]  
where \(\widehat{\mathbf{Y}}^{(s)}_{1}\), \(\widehat{\mathbf{Y}}^{(s)}_{2}\), and \(\widehat{\mathbf{Y}}^{(s)}_{3}\) for \(s=1,2,\cdots,N\) denote \(2\times 2\) block matrices, and  
\[\widehat{\mathbf{Y}}^{(s)}_{1}=\left[ \begin{array}{cc}\widehat{\mathbf{Y}}^{( s-1)}_{LL}+\widehat{\mathbf{Y}}^{(s)}_{RR}&\widehat{\mathbf{Y}}^{(s)}_{RI}\\ \widehat{\mathbf{Y}}^{(s)}_{IR}&\widehat{\mathbf{Y}}^{(s)}_{II}\end{array}  \right],\;\;\widehat{\mathbf{Y}}^{(s)}_{2}=\left[ \begin{array}{cc}\widehat{ \mathbf{Y}}^{(s)}_{RL}&0\\ \widehat{\mathbf{Y}}^{(s)}_{IL}&0\end{array} \right],\;\;\widehat{\mathbf{Y}}^{(s )}_{3}=\left[ \begin{array}{cc}\widehat{\mathbf{Y}}^{(s)}_{LR}&\widehat{ \mathbf{Y}}^{(s)}_{LI}\\ 0&0\end{array} \right]. \tag{15}\]The cyclically symmetric coordinate system is equivalent with several independent local Cartesian coordinate systems attached to each sector. Thus, the coordinate transformation between the global Cartesian coordinate system and the cyclically symmetric coordinate system can be expressed as  
\[\widehat{\mathbf{u}}=\left[ \begin{array}{c}\widehat{\mathbf{u}}^{(1)}\\ \widehat{\mathbf{u}}^{(2)}\\ \widehat{\mathbf{u}}^{(3)}\\ \widehat{\mathbf{u}}^{(4)}\\ \widehat{\mathbf{u}}\\ \vdots\\ \widehat{\mathbf{u}}^{(N)}\end{array} \right]=\left[ \begin{array}{ccccc} \mathbf{I}_{\mathrm{S}}&0&0&0&\cdots&0\\ 0&\mathbf{R}(\theta_{0})&0&0&\cdots&0\\ 0&0&\mathbf{R}(2\theta_{0})&0&\cdots&0\\ 0&0&0&\mathbf{R}(3\theta_{0})&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0&\cdots&\mathbf{R}[(N-1)\theta_{0}]\end{array} \right]\left[ \begin{array}{c}\widetilde{\mathbf{u}}^{(1)}\\ \widetilde{\mathbf{u}}^{(2)}\\ \vdots\\ \widetilde{\mathbf{u}}^{(n)}\\ \vdots\\ \overline{\mathbf{u}}^{(N)}\end{array} \right]=\mathbf{R}_{c}\widetilde{ \mathbf{u}}, \tag{10}\]  
where \(\mathbf{R}(s\theta_{0})\) for \(s=1,2,\cdots,N-1\) represents the rotation transformation matrix in terms of the angle \(s\theta_{0}\), and \(\mathbf{I}_{\mathrm{S}}\) is an identity matrix. It can be easily understood that the rotation transformation matrix \(\mathbf{R}(s\theta_{0})\) is equal to the products of \(\mathbf{R}(\theta_{0})\)\(s\) times, i.e., \(\mathbf{R}(s\theta_{0})=\left[\mathbf{R}(\theta_{0})\right]^{s}\). For brevity in expression, \(\mathbf{R}(\theta_{0})\) is simply represented by \(\mathbf{R}_{0}\), and \((\mathbf{R}_{0})^{N}=\mathbf{I}_{\mathrm{S}}\). By applying this coordinate transformation to \(\widehat{\mathbf{Y}}\), such a matrix in a cyclically symmetric coordinate system becomes \(\widetilde{\mathbf{Y}}=\mathbf{R}_{c}^{\mathrm{T}}\widehat{\mathbf{Y}} \mathbf{R}_{c}\), and  
\[\widetilde{\mathbf{Y}}=\left[\begin{array}{ccccc}\widehat{\mathbf{Y}}_{1}^{ (1)}&\widehat{\mathbf{Y}}_{1}^{(1)}\mathbf{R}_{0}&0&\cdots&0&\widehat{\mathbf{ Y}}_{1}^{(N)}(\mathbf{R  
Thus, the matrix \(\widetilde{\mathbf{Y}}\) can be expressed as a block-circulated form as follows:  
\[\widetilde{\mathbf{Y}}=\left[