# Structural Damage Localization by the Principal Eigenvector of Modal Flexibility Change
## 3 Numerical Example
From Figures 11 and 12 both damage localization methods can achieve satisfactory results by using error-free data. Figure 11: **(a)** Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 31; **(b)** Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements 5 and 31 (\(\times 10^{-4}\)). Figure 10: **(a)** Damage index (PE curvature) map using the first four exact modes when the defects occurred in elements 5 and 17; **(b)** Damage index (ULS variation curvature) map using the first four exact modes when the defects occurred in elements 5 and 17 (\(\times 10^{-3}\)). Figure 12: (**a**) Damage index (PE curvature) map using the first four exact modes when the defects occurred in elements 5 and 31; (**b**) Damage index (ULS variation curvature) map using the first four exact modes when the defects occurred in elements 5 and 31 (\(\times 10^{-3}\)). Figure 5:0 (**a**) Damage index (PE) map using the first four modes with 3% noise when elements 5 and 31 are damaged; (**b**) Damage index (ULS variation) map using the first four modes with 3% noise when elements 5 and 31 are damaged (\(\times 10^{-4}\)). From Figure 13, one can see that the indexes of PE and ULS variation can still identify the damaged elements 5 and 31 even if 3% noise is considered. According to Figure 14, it is apparent that the curvature indexes are seriously affected by the measurement error, and the damaged segments cannot be determined by these curvature maps with noise. Figures15-18 presented the damage localization results for damage case 5 using the exact modal data and the data with 3% noise, respectively. From Figures 15a and 16a, one can see that the damaged elements 7, 19 and 33 were clearly identified by PE and PE curvature. From Figure 15b, the ULS variation method fails to detect the damaged element 19. It can be concluded that the PE algorithm can precisely identify multiple defect locations, whereas the ULS method occasionally missed multiple damage locations. Figure 5:1 (**a**) Damage index (PE curvature) map using the first four modes with 3% noise when elements 5 and 31 are damaged; (**b**) Damage index (ULS variation curvature) map using the first four modes with 3% noise when elements 5 and 31 are damaged (\(\times 10^{-3}\)). Figure 5:2 _Cont._  
Figure 5:3 (**a**) Damage index (PE curvature) map using the first four modes without noise when elements 7, 19 and 33 are damaged; (**b**) Damage index (ULS variation curvature) map using the first four modes without noise when elements 7, 19 and 33 are damaged (\(\times 10^{-3}\)). Figure 5:4 (**a**) Damage index (PE) map using the first four exact modes when the defects occurred in elements 7, 19 and 33; (**b**) Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements 7, 19 and 33 (\(\times 10^{-4}\)). Figure 5:5 **(a)** Damage index (PE) map using the first four modes with 3% noise when elements 7, 19 and 33 are damaged; **(b)** Damage index (ULS variation) map using the first four modes with 3% noise when elements 7, 19 and 33 are damaged (\(\times 10^{-4}\)). Figure 5:6 **(a)** Damage index (PE curvature) map using the first four modes with 3% noise when elements 7, 19 and 33 are damaged; **(b)** Damage index (ULS variation curvature) map using the first four modes with 3% noise when elements 7, 19 and 33 are damaged.