# Stereo-DIC Measurements of a Vibrating Bladed Disk: In-Depth Analysis of Full-Field Deformed Shapes
## 5 Investigation of the Blade Deformed Shapes  
The described testing procedure allows for the visualization of the harmonic response of the blisk under a sinusoidal excitation at any frequency. It is worth noting that, for lightly damped systems such as the blisks, the deformed shapes are expected to be characterized by all the points moving in phase or out of phase. This means that the phase difference between two points of the blisk should be approximately 0\({}^{\circ}\) or 180\({}^{\circ}\). Nevertheless, in the case of blisks it is really common to have a certain mistuning level, which causes any theoretical tuned mode to split into two separate modes, having really close frequencies and asymmetric response amplitudes. These two modes, which are really close in frequency, cause the impossibility to separate their shapes during a harmonic response analysis. In this scenario, indeed, the measured deformed shape is the combination of the two mistuned modes. In addition to this, since at the investigated peak frequencies the two excited modes are close to the resonance, the excitation force and the responses are in general neither in phase nor out of phase, thus the phases are different for the two modes since their natural frequencies are slightly different. As a consequence, the measured deformed shape may present points with phase differences which are neither 0\({}^{\circ}\) nor 180\({}^{\circ}\). Both the amplitude asymmetry and the phase shifting can be investigated with the proposed setup. Since Equation (1) is used to set the cameras' frame rate (\(n_{\rm P}=1\)), the measured response describes one full period along the acquisition time. Thus, it is possible to compute the Discrete Fourier Transform (DFT) of each point of the displacement map and then evaluate the amplitude and the phase of the first harmonic for each measurement  
Figure 7: Displacement maps for the 6357 Hz excitation frequency, along the directions of the cylindrical reference system of the blisk. point. This elaboration was performed, as an example, on the analyzed blisk for the tangential displacement of the tests at 2568 Hz and 6357 Hz frequencies, and the results are reported in Figure 8.  
In a theoretical scenario of perfectly symmetric and isolated modes, all the blades would show the same displacement amplitude for those modes with a Number of Nodal Diameters (NND) equal to zero, which are the modes experiencing the highest excitation when the load is applied at the hub of the blisk. On the contrary, mistuned blisks will show asymmetries in the amplitude maps and any phase value in the range 0-180\({}^{\circ}\), as discussed before. To this extent, Figure 8a confirms that each blade has a different amplitude at 2568 Hz, thus denoting a certain mistuning level. On the other hand, Figure 8b shows that all the blades vibrate with the same phase of about \(-55^{\circ}\), while the disk is vibrating with a phase of about \(125^{\circ}\), i.e., perfectly out of phase with respect to the blade. This distribution of the phases is common for the zero NND modes. Since all the points move in phase or out of phase, it is possible to state that only the shape of one mode was excited in this test. It is interesting to point out that the zero NND modes, in perfectly tuned blisks, are not  
Figure 8: DFT elaborations of the test results: (**a**) amplitude at 2568 Hz, (**b**) phase at 2568 Hz, (**c**) amplitude at 6357 Hz, (**d**) phase at 6357 Hz. couples of degenerate modes. Thus, the presence of any mistuning can only determine amplitude asymmetry. On the other hand, the test at 6357 Hz demonstrated a different behavior. Figure 8c again shows that the deformed shape was localized in few blades which had higher amplitudes, while Figure 8d shows that all the possible phase values between \(-180^{\circ}\) and \(180^{\circ}\) were measured. In order to improve the readability, Figure 8b,d were enhanced by adding the white solid lines representing the blade contours.