# A Comparison between Two Reduction Strategies for Shrouded Bladed Disks
## 2 Background
### Cyclic Symmetry  
Bladed disks belong to the particular category of the so-called cyclically symmetry structures [20], which are defined as a special class of periodic structures, determined by a finite number \(N\) of identical substructures (one blade and the corresponding disk sector), each one constituted by \(J\) degrees of freedom and forming on the whole a closed ring.  
The cyclically symmetric structures are characterized by the fact that the deflection of one substructure is strictly linked to the deflection of the previous one, having the same amplitude and a constant phase difference \(\varphi\) from it. In other words, the same instantaneous deflected shape reappears after successive intervals of time \(t\), rotated round an additional substructure each time, thus generating what are generally called 'rotating mode shapes'.  
Figure 2 describes the simplified geometry of the fundamental section of a \(N\)-sectors bladed disk, introducing the distinction between the interface complex coordinates \(\mathbf{q_{H}}\) and \(\mathbf{q_{L}}\) and all the remaining inner dofs \(\mathbf{q_{I}}\). More in detail, \(\mathbf{q_{H}}\) defines the nodes belonging to the High sector interface, whereas \(\mathbf{q_{L}}\) identifies the coordinates at the Low interface.  
The dofs vector of the fundamental sector is  
\[\mathbf{q}=\left\{\begin{array}{c}\mathbf{q_{H}}\\ \mathbf{q_{I}}\\ \mathbf{q_{L}}\end{array}\right\} \tag{1}\]  
and the cyclic symmetry conditions can be applied to it by means of a set of complex constraint equations,  
\[\mathbf{q_{H}}=\mathbf{q_{L}}e^{i\varphi}, \tag{2}\]  
which link the interface coordinates \(\mathbf{q_{H}}\) and \(\mathbf{q_{L}}\), where the phase angle \(\varphi\) is defined as:  
\[\varphi=h\frac{2\pi}{N}. \tag{3}\]  
The 'harmonic index' \(h\) in Equation (3) can assume any integer value between \(0\) and \(N-1\), corresponding to an integer number of waves in the circumferential direction. It is worth mentioning  
Figure 2: Fundamental sector of a simplified bladed disk.  
that mode shapes characterized by \(h=0\) and \(h=N/2\) correspond to standing waves, while mode shapes with \(0<h<N/2\) rotate clockwise and those with \(h>N/2\) rotate counter-clockwise.  
Once the full set of stiffness and mass matrices of the fundamental sector are obtained, the matrices of the full bladed disk can be finally assembled thanks to the circularity property of the matrices of cyclically symmetric structures. In fact, if \(\mathbf{K_{BD}}\) is the block-diagonal matrix, whose \(h^{\text{th}}\) block is the stiffness matrix corresponding to the \(h^{\text{th}}\) harmonic index, the stiffness matrix of the whole bladed disk can be obtained [20] as  
\[\mathbf{K}=[(\mathbf{E_{N}^{H}}\otimes\mathbf{I_{P}})^{-1}\cdot\mathbf{K_{BD}}\cdot(\mathbf{E_{N}} \otimes\mathbf{I_{P}})^{-1}]\cdot\mathbf{N}, \tag{4}\]  
where:  
* \(\mathbf{E_{N}}\) is an \(\mathbf{N}\times\mathbf{N}\) Fourier matrix (\(\mathbf{N}\) is the number of blocks, coincident with the number of nodal diameters \(h\) in this case), and \(\mathbf{E_{N}^{H}}\) is its conjugate transpose.
* \(\mathbf{I_{P}}\) is the identity matrix of dimension \(\mathbf{P}\), \(\mathbf{P}\) being the dimension of each block of matrix \(\mathbf{K_{BD}}\).
* \(\otimes\) is the Kronecker product.