## 3 Decomposition of the Diametrical Mode Shapes for Disk-Blade-Disk Configuration
### Application of the Decomposition in a Real Prototype Disk-Blades-Disk Structure  
This section shows the application of the method in a real prototype disk-blades-disk structure (Figure 6a). The structure is a pump-turbine impeller with seven curved blades. The diameter is approximately 3 m. This structure has been analyzed in previous researches [6; 30]. Only a diametrical mode of the structure is shown here as an example. The peripheral deformation of this mode is obtained by means of numerical simulation and through experimental modal analysis (Figure 6b). The amplitude of this deformation is normalized to 1 in both cases. The decomposition of the mode shape applying the methodology explained before is shown in Figure 6c. Figure 6: Diametrical mode of a real impeller (non-rotating structure): (**a**) Impeller analyzed, (**b**) Peripheral deformation of the diametrical mode with main component \(n=3\), (**c**) Amplitude of the components obtained with mode shape decomposition. It can be seen that, apart from the main component \(n=3\) (therefore, in the aforementioned references the mode is described as 3 ND), other components such as \(n=4\), 10, 11 are also found in the decomposition. Generally, these structures exhibits multiple components instead of a single diametrical component that simple disk-like structures have (Figure 5) [43]. Therefore, it is shown that the description of these mode shapes with a unique diametrical mode component [6; 26; 27; 28; 29; 30; 31; 32; 33] is incomplete. The next section shows the consequences of the natural frequencies detected with a stationary sensor, when the disk-blade-disk structure is in rotation. Diametrical Mode Shapes of a Rotating Disk-Blades-Disk Structure Observed from a Stationary Point of View and Its Relation to Mode Shape Decomposition  
This section shows the analytical relation between the components obtained with mode shape decomposition and the natural frequencies observed with a sensor on the stationary frame. Figure 7 helps us to understand the underlying problem. A sensor placed on the stationary reference frame (for example, a Laser Doppler Vibrometer) measures the transverse vibration of the rotating structure. As mentioned before (Section 2), for a structure rotating at low rotating speeds (neglecting inertia, shear and gyroscopic effects), it can be assumed that the mode shape in the standing case is the same as in the rotating case (considering the mode shape from the rotating frame), and also that the resonant frequency is approximately the same as in the standing case [43]. If the deformation of the \(i\)-diametrical mode of the bladed-disk is analyzed from the rotating frame [19; 43]:  
\[w_{p,i-rot}(\theta,t)=A_{i-rot}(\theta)\cdot\cos(\omega_{i-rot}t) \tag{8}\]  
In this section, subscripts \"-rot\" and \"-stat\" will be used to emphasize the difference between both reference systems. Using Equation (4), Equation (8) can be rewritten as:  
\[w_{p,i-rot}(\theta,t)=\sum_{n=0}^{\infty}A_{n,i-rot}\cdot\cos(n\theta+\varphi_ {n})\cdot\cos(\omega_{i-rot}t) \tag{9}\]  
If the resonance is analyzed from an observer on the stationary frame, the angular coordinate \(\theta\) (on the rotating frame) is changed for angular coordinate \(\beta\) (on the stationary frame) [19] with the following change of reference frame (Figure 7):  
\[\beta=\theta+\Omega\cdot t \tag{10}\]  
Figure 7: Rotating and stationary frame. This transformation is substituted in Equation (9) obtaining the peripheral deformation observed from the stationary frame:  
\[w_{p,i-stat}(\beta,t)=\sum_{n=0}^{\infty}A_{n,i-rot}\cdot\cos(n(\beta-\Omega \cdot t)+\varphi_{n})\cdot\cos(\omega_{i-rot}t) \tag{11}\]  
For simplicity, it will be assumed that the sensor on the stationary frame is located at \(\beta=0\). This assumption does not affect the frequency content of the function above. In this case, Equation (11) can be rewritten as:  
\[