## 5 Experimental Test
### Mode Shape Decomposition of the Diametrical Modes for the Standing Disk-Blades-Disk Structure
The peripheral deformation is quite similar for the analyzed modes, and the main diametrical components detected for each mode shape are the same, when analyzing the disk-blades-disk assembly experimentally and numerically. The MPC (Modal Phase Collinearity) [2] close to one (experimental results) shows that the mode shapes are real mode shapes (all points moving in phase or counterphase to each other). For the simulation, this is automatically achieved, since no damping effects are considered (undamped system) and, therefore, the modes are real. The discrepancy in natural frequencies values (between experimental and numerical results) is less than 7% for the analyzed modes. Figure 12: Response of the periphery for the same diametrical mode (326 Hz and second mode). (**a**) Experimental, (**b**) Numerical simulation. Tests with the Disk-Blades-Disk Structure Rotating and Detection of Natural Frequencies with LDV from the Stationary Frame  
To determine which peaks, corresponding to the first natural frequencies of the rotating structure (diametrical mode shapes), are obtained with a stationary sensor (LDV), the following procedure is used. The rotating speed of the structure is fixed as 5 Hz. In this condition, the disk-blades-disk structure is excited with a harmonic excitation (pure sine), created with the exciter PZT. Each of the analyzed diametrical modes in this paper (Figure 13) is excited. The response is simultaneously measured with the PZT attached to the rotating structure (sensor) and with the LDV on the stationary frame. Figure 14 shows the excitation and response from both rotating and stationary frames for the third mode analyzed (Figure 11). From the time signals, it is observed that, when the excitation starts (Figure 14a), the dynamic response of the rotating structure is clearly affected, as seen on the rotating frame sensor. This response is clearly dominated by the excitation signal. From the stationary frame (LDV), a change in signal behavior is also appreciated during the time of the excitation. The response observed with the LDV during that time is the superposition of the response due to the PZT-excitation and the response due to the low frequency motion of the rotating structure. This motion, which is mainly due to imperfections on the surface and misalignment of the rotating structure, produces low frequency components around the rotating speed of the disk and first harmonics when the structure rotates [19, 48]. Nevertheless, for the frequency range of the analyzed diametrical modes, only residual harmonics can be appreciated. These harmonics are not important in comparison with the level of peaks caused by the excitation of the natural frequencies (Figure 14b (LDV response)). Figure 13: Diametrical modes analyzed with experimentation and simulation. Peripheral deformation and mode shape decomposition. Analyzing the response from the rotating frame in the frequency domain (Figure 14b (PZT)), it can be seen that only the excited frequency, which is a natural frequency of the structure, appears. Furthermore, from the stationary frame, the frequencies detected are symmetrically shifted with respect to the excited frequency. It is to be mentioned that due to the rotation, the exact value of some natural frequencies experiences a slight increase [43; 49], with respect to the stationary case. This increase is less than 2 Hz (0.5%) with respect to the non-rotating case value, for all the analyzed diametrical modes (Figure 11). To relate the frequency modulations and the diametrical components detected with the mode shape decomposition, the following transformation in the frequency axis is performed for each of the \(i\)-analyzed diametrical modes:  
\[f_{norm}=\frac{f-f_{excited}}{\Omega} \tag{15}\]  
\(f\) represents all the x-axis values (Figure 14), \(f_{excited}\) is the frequency excited with the patch (the natural frequency from the rotating frame) and \(\Omega\) the rotating speed of the disk-blades-disk structure (which is always 5 Hz in these paper).