# Stereo-DIC Measurements of a Vibrating Bladed Disk: In-Depth Analysis of Full-Field Deformed Shapes
## 1 Introduction
Additionally, since SLDV are 1D sensors, three separate devices would be needed to obtain 3D information, requiring a proper calibration and control strategy to ensure that all the sensors point at the same location over time. This results in complex and expensive hardware setups. For these reasons, a _vision-based_ system can be a valid alternative solution to capture the modal shapes of industrial bladed disks, performing a _full-field_ dynamic acquisition of the displacements. The Digital Image Correlation (DIC), augmented by the stereoscopic feature, stereo-DIC or 3D-DIC [20], is quite effective for this application mainly for two reasons: the high complexity of the target geometry, as mentioned before, and the large or almost complete accessibility of the blade surfaces, at least for the unshrouded blisks, such as that tested in this work. To obtain a three-dimensional displacement map, stereo image pairs of the vibrating target are simultaneously acquired with two synchronized cameras [21; 22]. An interesting comparison between stereo-DIC and 3D scanner LDV was provided by [PERSON] et al. [23]. The three-dimensional LDV is obtained with three scanning laser heads to obtain the velocity vector, instead of just a component. The obtained results were quite accurate both in terms of frequencies and modal shapes, although only a simple plate was considered as test article. However, significant differences can be observed in terms of acquisition and analysis times as well as costs. Acquisitions with high frames per second (fps) are usually required in 3D-DIC to identify rigid and small structures, resulting in high-frequency modes and fast harmonic responses. Therefore, two high-speed cameras are required for stereo vision, and they need to be synchronized in time, even within nanoseconds [22]. [PERSON] and [PERSON] [24] used two cameras with a maximum frame rate of 5000 fps, which obviously implies a Nyquist frequency of 2500 Hz, and a harmonic at 1230 Hz was indeed properly detected. [PERSON] and [PERSON] [25] just used a single high-speed camera and a setup involving four planar mirrors to compensate the single point of vision and still obtain a three-dimensional reconstruction of the vibration response. Their camera was set at 500 fps, having a target maximum natural frequency of just 55 Hz. [PERSON] et al. [26] again used a four-mirror setup for achieving a pseudo-stereo system, using a very high-speed camera up to 12,500 fps. They measured frequency response functions (FRF) up to 1000 Hz, and the modal shape reconstruction was quite successful for a mode at frequency 600 Hz, thus much lower than the Nyquist frequency. On the contrary, the subsampling strategy, or alternatively mentioned as downsampling, can be performed to overcome the Nyquist rate limitation [27; 28; 29]. If the signal to be reconstructed is just a sinusoidal harmonic, a sampling time can be chosen as slightly different (usually higher) than an integer multiple of the period. In principle, according to this approach, any vibration frequency could be measured, but two issues should be considered. The harmonic content of the signal must be purely sinusoidal, and this can be obtained as the result of a single harmonic response, provided that the exciting load is again a pure sine function and any nonlinearity is avoided, especially in terms of boundary constraints. The other requirement is that the exposure time of the camera, at maximum shutter speed, is short enough with respect to the vibration period. In this paper, the 3D-DIC was applied to a bladed disk and the downsampling strategy was adopted to find the deformed shape of this structure, up to the high-frequency range in the order of several kHz. Two low frame-rate cameras, with a fast shutter time, have been assembled in a stereo set-up, and a digital projector has been used to project a structured light pattern, thus strengthening the stereo matching and enhancing the 3D-DIC reconstruction.