## 5 Experimental Test
### Mode Shape Decomposition of the Diametrical Modes for the Standing Disk-Blades-Disk Structure  
In order to use mode shape decomposition for the diametrical modes of disk-blades-disk assemblies, presented in Section 3, the several first modes of the structure are determined experimentally for the non-rotating case. For this purpose, the following procedure is done: The structure is excited with the upper PZT using a chirp signal of 4 s. The response is measured with the other PZT (used as sensor) and with the Laser Doppler Vibrometer (LDV), which is directly pointing to some point of the periphery on the upper surface of the disk-blades-disk. For one position of the Laser, the excitation is applied five times to check the repeatability of the experiments and the coherence between signals. Once the experiment is performed for one position of the Laser, the structure is turned to an angle of 2\(\pi\)/35 rad. The aforementioned procedure is repeated for this position, and so on. Therefore, after one complete turn of the structure, the deformation on the periphery is obtained in 35 equidistant points (Figure 9). Figure 10 shows the excitation signal, the response measured with the PZT, and the response measured with the LDV when the LDV is directed to point 1 (Figure 9). Figure 8: Experimental setup. After acquiring the time signals for the 35 points, the Frequency Response Functions (FRF) were obtained. In the calculated FRFs, the signal of the LDV is used as a response and the signal of the PZT (exciter) is used as a reference signal. The FRFs using the PZT (sensor) as a response and PZT (exciter) as a reference are also calculated to check the repeatability of the experiments. Figure 11 shows the FRF of the LDV and the FRF of the PZT for two different positions of the LDV. The modes shown in this figure are diametrical modes, which will be analyzed in the standing case and in the rotating case (following section). As can be seen, when changing the measured point, the response of the LDV obviously changes (Figure 11a), while the response of the PZT (Figure 11b) does not (since the only change between both tests is the focusing position of the LDV). Therefore, in Figure 11b, both lines appear almost overlaid. This validates the repeatability of the experiments and the use of both PZTs for the rotating case. Figure 11: Analyzed diametrical modes. Experimental response of the structure measuring different positions of the shaded-disk with the LDV (**a**) and the PZT (**b**). Standing structure (not rotating). Figure 9: Modal analysis of the bladed disk. Positions measured with the LDV. With the FRFs of the 35 points on the periphery in the experimental case, the Operating Deflection Shape (ODS) in the axial direction is obtained. As an example, the deformation of the second mode (326 Hz, Figure 11) is shown in Figure 12a. Figure 12b shows the mode shape obtained with numerical simulation, which is based on a modal analysis of the bladder-disk using a FEM (Finite Element Method) Model with commercial software ANSYS [37]. A tetrahedral mesh, using SOLID186 [37] elements, was used for the calculation. A mesh sensitivity analysis was conducted, changing the number of elements of the mesh such that it does not affect the results. The optimal mesh had 20940 elements. The results obtained show good accuracy of the experimentation in natural frequency values. For the simulation, the response on the periphery is obtained in 225 points due to mesh discretization. In order to compare both experimental and simulation results, the amplitude is normalized in both cases. The comparison between experimental and simulation results for the periphery of the analyzed modes (Figure 11) is shown in Figure 13. In the same figure, the decomposition of the mode shape, according to the method presented in Section 3, is also shown for both experimentation and simulation procedures. It is seen that there is a good correlation between the experimental and numerical results.