# Damage Detection Using Modal Rotational Mode Shapes Obtained with a Uniform Rate CSLDV Measurement
## 4 Experimental Validation
### Results and Discussions of Three Damaged Cases
#### 4.2.2 Case 2: Multi Cracks  
For Case 2, the corresponding first four mode frequencies are 49.61 Hz, 234.38 Hz, 299.61 Hz, and 717.58 Hz. The displacement mode shapes at these frequencies are plotted in Figure 10, respectively.  
Figure 9: The RDIs colored plot of first four modes of a single crack plate. (**a**) RDIs of the first mode in the x direction. (**b**) RDIs of the first mode in the y direction. (**c**) RDIs of the second mode in the x direction. (**d**) RDIs of the second mode in the y direction. (**e**) RDIs of the third mode in the x direction. (**f**) RDIs of the third mode in the y direction. (**g**) RDIs of the fourth mode in the x direction. (**h**) RDIs of the fourth mode in the y direction.  
Results are the same as Case 1 and only the torsional modes can clearly reveal the discontinuity of the displacement mode shapes at the crack locations. Using the same procedure, mode shapes of the undamaged state can be reconstructed with determined function orders based on the fit value convergence threshold.  
The normalized modal rotational damage indicators are plotted in Figure 11. For the first mode, the RDIs failed to detect and localize the damage as this mode is the bending mode and the modal rotational mode shapes in the x direction are close to zero. Therefore, RDIs of this mode are actually noises. For the third mode, the RDIs can only reflect the damage caused by Crack 2. This is because Crack 2 is much larger than Crack 1 and located at the large amplitude of vibration mode. However, modal rotational damage indicators of the torsional modes in the y direction are much better to identify the damage locations clearly, except the RDIs of the torsional modes in the x direction. This is because changes of the modal rotational mode shapes are mainly along the y direction and therefore the modal rotational mode shapes in the x direction can be easily contaminated by noise. From the RDIs of the second mode in the y direction, two cracks can be clearly located and the indicators at Crack 2 position are more significant than those at Crack 1 position, as shown in Figure 11d. It indicates that the damage severity can also be identified. However, the results derived from the fourth mode seem hardly to identify the location of Crack 1. It is because the damage indicators are normalized by the maximum value and illustrated by color plots, so the color of the relatively small Crack 1 is much lighter than that of the major crack. If the area is further zoomed, the surface plot of the rotational damage indicator of the second torsional mode in the y direction is replotted in Figure 12. It clearly shows that the damage indicator of Crack 1 is less than 20% of the indicator of Crack 2. When the left bottom quarter of the figure is enlarged separately, the indicator of Crack 1 is quite obvious. Therefore, the local scanning and damage detection can effectively identify the relatively small crack of the multiple damage case.  
Figure 10: The first four measured mode shapes of the multi cracks plate using uniform-rate scanning CSLDV. (**a**) The first bending mode. (**b**) The first torsion mode. (**c**) The second bending mode. (**d**) The second torsion mode.  
Figure 11: The RDIs colored plot of multi crack plate in the x and y directions. (**a**) RDIs of the first mode in the x direction. (**b**) RDIs of the first mode in the y direction. (**c**) RDIs of the second mode in the x direction. (**d**) RDIs of the second mode in the y direction. (**e**) RDIs of the third mode in the x direction. (**f**) RDIs of the third mode in the y direction. (**g**) RDIs of the fourth mode in the x direction. (**h**) RDIs of the fourth mode in the y direction.