# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## 3 Reduced-Order Modeling
### Parametric Projection of Disc Mistuning  
Unlike the projection of blade mistuning, the coordinates between adjacent sectors of the disc are directly coupled, and the disc mistuning matrix cannot be expressed in block-diagonal form. Therefore, the mistuning projection method in the previous section cannot be directly employed. Nevertheless, the idea of representing the overall modes by normal modes of a relatively simple component and projecting the mistuning matrix onto these normal modes can be utilized. Of course, some modifications to the formulations are needed to account for the coupling issue of the disc model. Firstly, the disc part mode \(\mathbf{\Phi}_{t}^{d}\  
where \(\hat{\mathbf{\Phi}}_{t}^{d}\) denotes the expanded modes with respect to \(\mathbf{\Phi}_{t}^{d}\). Detailed deductions on this proposition are presented, for brevity, in Appendix B. Similar to the expression of \(\mathbf{\Phi}_{t}^{d}\) in Equation (17), the expanded modes \(\hat{\mathbf{\Phi}}_{t}^{d}\) with respect to \(\hat{\mathbf{\Phi}}_{t}^{d}\) can also be expressed by cyclically symmetric modes. \[\hat{\mathbf{\Phi}}_{t}^{d}=\left(\mathbf{E}\otimes\mathbf{I}_{N_{d}}\right) \underset{h=1,2,\cdots,N}{\text{Bdiag}}\left(\hat{\mathbf{\Phi}}_{h}^{d}\right), \tag{20}\]  
where \(\hat{\mathbf{\Phi}}_{h}^{d}\) denotes the expanded cyclic modes of \(\overline{\mathbf{\Phi}}_{h}^{d}\), and  
\[\overline{\mathbf{\Phi}}_{h}^{d}=\left\{ \begin{array}{l}\overline{\mathbf{\Phi}}_{h, R}^{d}\\ \overline{\mathbf{\Phi}}_{h,I}^{d}\end{array} \right\},\quad\hat{\mathbf{\Phi}}_{h}^{d}= \left\{ \begin{array}{l}\overline{\mathbf{\Phi}}_{h,R}^{d}\\ \overline{\mathbf{\Phi}}_{h,I}^{d}\\ \overline{\mathbf{\Phi}}_{h,L}^{d}\end{array} \right\}, \tag{21}\]  
where \(\overline{\mathbf{\Phi}}_{h,L}^{d}\) is a matrix dependent on \(\overline{\mathbf{\Phi}}_{h,R}^{d}\); for \(h\)ND modes,  
\[\overline{\mathbf{\Phi}}_{h,L}^{d}=e^{j(h-1)\theta_{0}}\mathbf{R}_{d}(\theta_{0}) \overline{\mathbf{\Phi}}_{h,R}^{d}, \tag{22}\]  
where \(\mathbf{R}_{d}(\theta_{0})\) denotes a rotation transformation matrix, and \(\mathbf{R}_{d}(\theta_{0})=\mathbf{I}_{cd}\otimes\mathbf{R}_{e}(\theta_{0})\), where \(\mathbf{I}_{cd}\) is an identity matrix of dimensions \(N_{cd}\), and \(N_{cd}\) is the number of nodes on cyclic symmetry interface. Then, similar to that shown in Equation (14), the expanded cyclic modes \(\hat{\mathbf{\Phi}}_{h}^{d}\) are represented by the normal modes of the disc component. \[\hat{\mathbf{\Phi}}_{h}^{d}=\mathbf{\Phi}_{fre}^{d}\overline{\mathbf{p}}_{h}^{d}, \tag{23}\]  
where \(\mathbf{\Phi}_{fre}^{d}\) denotes the normal mode matrix of a tuned sector model of the disc with the left and right interfaces, as well as the adjacent interface between the disc and shaft, being set to be free, and \(\overline{\mathbf{p}}_{h}^{d}\) denotes the participation factor. Herein, the free constraint is employed on the disc-shaft adjacent interface instead of a fixed one, because the latter one may not accurately describe the 1 ND coupling modes between shaft and disc, leading to non-negligible errors in forced responses. The employment of free modes avoids such issues and presents satisfactory results in both frequencies and forced responses.