## 3 Decomposition of the Diametrical Mode Shapes for Disk-Blade-Disk Configuration  
A method to characterize the diametrical mode shapes of disk-blades-disk structures is proposed in this section. This method helps to have a better knowledge of which diametrical components are participating in a diametrical mode shape of the structure.  
As mentioned before, based on the ideas of the Fourier transform for periodical functions, the deformation at the periphery can be discomposed as a sum of single diametrical components, which characterize the diametrical modes of simple disk structures. Assuming that the diametrical mode shape is real (all points moving in phase or counterphase to each other), it is enough to discompose the periphery at the time when the deformation of all the points is at its maximum (\(t=0\) for Equation (3)).  
Mode shapes obtained experimentally or by means of numerical simulation are generally measured in a number of discrete points. Therefore, the discrete version of the Fourier transform (i.e., the FFT algorithm [45]) is used. To exemplify the method in order to discompose a diametrical mode shape, it will be applied to discompose a diametrical mode shape of a simple disk obtained with numerical simulation (Figure 5):  
* It is assumed that the response is measured in \(N\)' equidistant points.  
* In order to get a good resolution in the diametrical components information, this deformation is virtually elongated \(\lambda\) times (10 times in this example) in space. The first and the last point of the final sequence correspond to the displacement at angle \(0^{\circ}\) (Figure 5b).
* The FFT is applied to the resulting peripheral deformation. The transform is according to the following equation: \[X_{k}=\sum_{i=1}^{N}X_{i}e^{-\beta xk\frac{i-1}{N}}\text{ \ for }k=1,\ldots,N\] (5)
* In this equation \(X_{i}\) is the sequence of points of the elongated periphery that has \(N\) points. \(X_{k}\) is the sequence of complex values after the transformation (Figure 5c).
* The first N/2 values of the sequence \(X^{\prime}_{k}\) are plotted in Magnitude against the diametrical components. To do this, the following transformation with the index \(k\) has to be done. \[n=\frac{k-1}{\lambda}\] (6) \(n\) represents the diametrical components. In this case, \(\lambda\) = 10 is considered, because the peripheral deformation is elongated 10 times. The magnitude is represented in terms of 0-peak value and, therefore, to obtain \(X^{\prime}_{k}\) from the \(X_{k}\)-values: \[X_{k}^{\prime}=X_{k}\cdot 2/N\] (7)  
Figure 5: Procedure of mode shape decomposition: (**a**) Periphery of the disk, (**b**) Periphery elongated 10 times, (**c**) Magnitude after applying FFT algorithm, (**d**) Magnitude vs. Diametrical component.  
It is useful to notice the following consequences:  
* The resolution in the diametrical components is \(1/\lambda\), where \(\lambda\) is the number of times that the peripheral deformation is virtually elongated. To accurately see the diametrical components, \(\lambda\) should be higher or equal to \(2\) (\(\lambda\geq 2\)).
* The maximum diametrical component that can be detected on the mode shape depends on the number of measured points on the periphery (equidistant points). \(n_{max}=N^{\prime}/2\) where \(N^{\prime}\) is the number of measured points on the periphery.  
For the example case, it is clearly seen (Figure 5d) that the unique diametrical component in the analyzed mode shape is the component \(n=2\). This number indicates the number of nodal diameters in this diametrical mode, which is also appreciated in Figure 5a.