# Damage Detection Using Modal Rotational Mode Shapes Obtained with a Uniform Rate CSLDV Measurement
## 4 Experimental Validation
### Results and Discussions of Three Damaged Cases  
#### 4.2.1 Case 1: A Single Crack  
For Case 1, the first four mode frequencies were 49.02 Hz, 237.70 Hz, 311.52 Hz, and 781.25 Hz, respectively. For each mode, a sinusoidal signal of the mode frequency was sent to the speaker to generate the acoustic excitation. The uniform-rate scanning CSLDV tests were assigned on the damaged plate to measure the corresponding mode shapes. The displacement mode shapes at these frequencies are shown in Figure 7, respectively. Obviously, the mode shapes look dense and smooth due to the high resolution and anti-noise ability of the uniform-rate scanning CSLDV. However, the discontinuous change of the displacement mode shape can hardly be seen for the bending modes (modes 1 and 3), while the torsional modes show a clear abrupt change at the crack location (modes 2 and 4).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Damage Case** & **Location** & **Size** & **Percent Length** \\ \hline Case 1 & Crack 1 & 13 mm \(\times\) 1 mm & 16.25\% \\ Case 2 & Crack 1 \& Crack 2 & 13 mm \(\times\) 1 mm \& 22 mm \(\times\) 1 mm \\ Case 3 & Crack 1 & 6 mm \(\times\) 1 mm & 7.5\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: The detailed description of three damage cases.  
Figure 7: The first four mode shapes of a single crack plate using uniform-rate scanning CSLDV. (**a**) The first bending mode at 49.02 Hz. (**b**) The first torsion mode at 237.70 Hz. (**c**) The second bending mode at 311.52 Hz. (**d**) The second torsion mode at 781.25 Hz.  
Without a priori knowledge of the undamaged plate, the undamaged mode shapes were extracted from the damaged ones using the polynomial function fitting method. The fit values of these modes against the polynomial orders are plotted in Figure 8. It can be seen that the fit value increased rapidly at lower orders and gradually converged at higher orders. Based on the convergence threshold of fit value, the minimum number of polynomial coefficients of the undamaged mode shapes were determined as (x = 9, y = 3) at 49.02 Hz, (x = 10, y = 10) at 237.70 Hz, (x = 9, y = 7) at 311.52 Hz, and (x = 13, y = 8) at 781.25 Hz, respectively.  
Based on the damaged and the reconstructed undamaged mode shapes, the modal rotational mode shapes of the x and y directions with the interval spacing 1.0 mm were derived and the corresponding normalized modal rotational damage indicators (RDI) of these modes were plotted in Figure 9. It is shown that the modal rotational damage indicators of the first mode are very messy in both the x and y directions and cannot identify any crack location. The reason is attributed to the fact that the first mode shape is not sensitive to the crack and the tiny changes of displacement mode shape can be easily interfered with by the noise. But for other three mode shapes, the Crack 1 location can be located by the modal rotational damage indicators in both directions. Moreover, the RDI of y direction is much clearer and more accurate. Not only the crack location, but also the crack span can be clearly identified. This is consistent with the conclusion of simulation analysis.  
Figure 8: The fit value of first four modes against polynomial function orders. (**a**) The first bending mode at 49.02 Hz. (**b**) The first torsion mode at 237.70 Hz. (**c**) The second bending mode at 311.52 Hz. (**d**) The second torsion mode at 781.25 Hz.