# A Reduced-Order Model for the Vibration Analysis of Mistuned Blade-Disc-Shaft Assembly
## 3 Reduced-Order Modeling  
### Model Order Reduction via Cyclic Symmetry Analysis  
Model order reduction techniques are often employed, combined with finite element models, to significantly reduce the computational requirements during analysis. It is much more urgent for mistuned bladed discs, as mistuning is highly random and thousands of mistuning patterns are often needed to study the statistical vibration characteristics. In this paper, the ideas presented in References [28; 29; 30] are employed and further developed for blade-disc-shaft assemblies. From a general perspective of view, these methods belong to the mode superposition method, whereas the modes used are those of a tuned bladed disc rather than a certain mistuned case. According to these approaches, the normal modes of the tuned blade-disc-shaft assemblies are used as the basis for projecting the mistuned physical model into a modal domain, which can be represented as  
\[\mathbf{u}=\mathbf{\Phi}_{l}\mathbf{q}, \tag{3}\]  
where \(\mathbf{\Phi}_{l}\) denotes the mode matrix of the tuned assembly in static condition, where the matrices due to rotation are not included during eigenvalue analysis, and \(\mathbf{q}\) represents the modal coordinate vector. The effectiveness of this coordinate transformation for mistuned blade-disc-shaft assemblies is validated in the next section. In the vibration analysis of bladed discs, the cyclic symmetry properties are often employed to reduce the computation requirements [31; 32]. A tuned blade-disc-shaft assembly is also cyclically symmetric if the shaft is axially symmetric. In the analysis of a rotating blade-disc-shaft assembly, the rotordynamic effects should not be neglected. However, few relative investigations considered these effects. As the rotordynamic matrices are all speed-dependent, the Coriolis matrix is skew-symmetric, and the centrifugal stiffening matrix depends on the stress distributions due to centrifugal forces, whether these matrices can be included in cyclic symmetry analysis should be rigorously verified. It can be proven that one matrix could be incorporated in cyclic symmetry analysis if such a matrix of the sector model has the following relationship with that of its adjacent sector. \[\widehat{\mathbf{M}}_{s}=[\mathbf{R}_{s}(\theta_{0})]^{\mathrm{T}}\widehat{ \mathbf{M}}_{s+1}\mathbf{R}_{s}(\theta_{0}), \tag{4}\]where \(\widehat{\mathbf{M}}_{s}\) and \(\widehat{\mathbf{M}}_{s+1}\) denote the matrices of \(\mathrm{sth}\) and \((s+1)\)th sectors, respectively, and \(\mathbf{R}_{s}(\theta_{0})\) represents a rotation transformation matrix in term of the sector angle. \[\mathbf{R}_{s}(\theta_{0})=\mathbf{I}_{N_{n}}\otimes\mathbf{R}_{e}(\theta_{0}), \tag{5}\]  
where \(\mathbf{I}_{N_{n}}\) denotes an identity matrix of dimensions \(N_{n}\), where \(N_{n}\) is the number of nodes in a sector model, the symbol \(\otimes\) denotes the Kronecker product, and \(\mathbf{R}_{e}(\theta_{0})\) is an elementary matrix. Taking the rotation axial along the \(z\)-axis as an example, \(\mathbf{R}_{e}(\theta_{0})\) can be expressed as  
\[\mathbf{R}_{e}(\theta_{0})=\left[ \begin{array}{ccc}\cos\theta_{0}&\sin \theta_{0}&0\\ -\sin\theta_{0}&\cos\theta_{0}&0\\ 0&0&1\end{array} \right]. \tag{6}\]  
Detailed proof of this proposition, for brevity, is presented in Appendix A. According to the explanation in Appendix A, the Coriolis, centrifugal stiffening, and spin softening matrices can all be incorporated in cyclic symmetry analysis. In this circumstance, the mode matrix \(\mathbf{\Phi}_{t}\) represented in terms of cyclically symmetric modes can be used as the projecting basis during model order reduction.