## 5 Experimental Test
### Mode Shape Decomposition of the Diametrical Modes for the Standing Disk-Blades-Disk Structure
This transformation gives the frequency content of the response in terms of the diametrical components since it is precisely what it is obtained isolating \(n\) from Equation (13). To normalize the amplitude of the vibration of the rotating sensor (PZT) and of the stationary sensor (LDV), the following procedure is used. For the bladed-disk with no rotation, the same harmonic excitation as in the rotating case (in some cases reducing 1 or 2 Hz the excited frequency due to  
Figure 14: Excitation and response of the bladed-disk when exciting the third mode of Figure 11 PZT exciter, PZT sensor and LDV. (**a**) Time signals (**b**) Autospectrum. the slight increase in the natural frequency values [43; 49], when the structure rotates) is applied to the disk-blades-disk structure. The LDV, in this case, focuses on the point with maximal peripheral deformation, which is determined by the modal analysis performed in the previous section. This test gives the values \(A_{PZT,\,\Omega=0\,\mathrm{Hz},f=\textit{excited}}\) and \(A_{LDV,\,\Omega=0\,\mathrm{Hz},f=\textit{excited}}\) for each of the analyzed modes, which are the amplitude of the response of the bladder-disk at the natural frequency considered (_fexcited_). Therefore, for each of these \(i\)-modes:  
\[A_{norm,PZT}=\frac{A_{PZT,\,\Omega=\,5\,\mathrm{Hz}}}{A_{PZT,\,\Omega=\,0 \,\mathrm{Hz},f=\textit{excited}}},A_{norm,LDV}=\frac{A_{LDV,\,\Omega=\,5\, \mathrm{Hz}}}{A_{LDV,\,\Omega=\,0\,\mathrm{Hz},f=\textit{excited}}} \tag{16}\]  
Note that \(A_{PZT,\,\Omega=5\,\mathrm{Hz},A_{LDV,\,\Omega=\,5\,\mathrm{Hz}}}\) represent all the values of the spectra shown in Figure 14 and \(A_{PZT,\,\,\Omega=0\,\mathrm{Hz},f=\textit{excited}}\), \(A_{LDV,\,\Omega=0\,\mathrm{Hz},f=\textit{excited}}\) are just single numbers. The normalization of the amplitude compares the amplitudes obtained with the PZT (Rotating Frame) and the LDV (Stationary Frame), and sees if they are in accordance with Equation (14). As an example, the normalization in frequency and amplitude of the mode analyzed in Figure 14 is shown in Figure 15.  
As deduced in Section 4, the experimental frequencies obtained from the stationary frame are directly related to the mode shape decomposition method proposed in this paper. Therefore, the decomposition in diametrical components of the peripheral deformation obtained with experimental and numerical modal analysis (Section 5.2) is necessary to understand what is obtained from the stationary frame. For this purpose, the mode shape decompositions of the analyzed modes (Figure 13) are simply separated into a negative and positive part, according to Equation (14). Therefore, if A(n) is considered as the function obtained with the mode shape decomposition (functions shown in Figure 13), the following function \(A^{\prime}(n)\) can be calculated, for each of the analyzed \(i\)-modes, according to:  
Figure 11:0  
\(A^{\prime}(n)\), which is defined for positive and negative \(n\), can be directly compared with the experimental results obtained with the LDV, for the rotating disk-blades-disk, after having normalized the Frequency and Amplitude (Equations (15) and (16)). This is performed for all the diametrical modes analyzed in this paper obtaining Figure 11:1  
Figure 11:2 Normalized response of the PZT and LDV for the third mode in Figure 11:3  
Figure 11:4 Experimental results for the frequencies detected with the stationary sensor (LDV) and its relationship to the diametrical components obtained with the mode shape decomposition. As seen in Figure 16, the main components obtained with the mode shape decomposition are closely related to the natural frequencies of the rotating structure that are observed from the stationary frame. This means that the frequencies observed from the stationary frame can be predicted with the method proposed for the mode shape decomposition. Furthermore, the amplitudes for the experimental results are slightly lower than the ones predicted with the mode shape decomposition.