# Damage Identification Based on Adding Mass for Liquid-Solid Coupling Structures
## 4 Damage Identification Experiment of a Submerged Cantilever Steel Beam
### Damage Identification
#### 4.4.3 Damage Identification  
When the additional mass is determined, it is necessary to analyze how frequencies change with the mass position to verify the reliability and validity of information obtained by the adding mass method. Based on the effective information, the error norm of the theoretical and experimental frequencies can be expressed as the objective function, Equation (9). The damage identification can be carried out by modifying the damage factors of substructures. The Patternsearch program in MATLAB is used to find the damage factor by minimizing the value of the objective Equation (9). In the optimization, the FE model of the undamaged structure is developed using the damage factor \(\mu\), and the according to frequencies \(\omega_{ji}^{F}\) of damaged structure is calculated. \[T(\mu)=\sum_{j}\sum_{i}\big{(}\frac{\omega_{ji}^{F}(\mu)-\omega_{ji}^{\ \ \ 2}}{ \omega_{ji}}\big{)}^{\ \ \ 2}, \tag{9}\]  
where, \(\omega_{ji}^{F}\) and \(\omega_{ji}\) are, respectively, the \(j\)-th (\(j\) is from 1 to \(k\), \(k\) is the number of frequencies measured in the experiment) frequency computed from FE analysis and measured experimentally when the additional mass is added to the \(i\)-th (\(i\) is form 1 to \(n\), \(n\) is the number of substructures) substructure. (1) Single Damage  
Usually, sensors can not be put into water, so the acceleration sensor is attached to the top of the cantilever beam out of the water, as shown in Figure 6. The 0.3 kg iron block is attached to the beam using a high-strength magnet. Mass is gradually moved from the first substructure at the bottom of the beam. When the iron block moves to a new position, the beam is stimulated, and the corresponding excitation and time-domain acceleration responses are acquired at the same time. According to the acceleration data, the first four frequencies of the structure are identified with the mass attached at different positions. Figure 11 shows the curves relating the first four frequencies of the undamaged and damaged cantilever beam structures with the additional mass at the immersion depth of 0.4 m.  
Figure 11: Experimental and computed values of the first four natural frequencies of the cantilever beam with a single damage. It can be seen from Figure 11 that the frequencies are different when the additional mass is added at different positions. This indicates that a large amount of modal information can be obtained by adding masses at different positions. Besides, the maximum frequency error between measured values and computed values of the first four frequencies of undamaged structures is less than 5.48%, demonstrating the rationality of the model. For damaged structures, the experimental frequencies are also consistent with theoretical frequencies. According to the obtained data from the experiment and the FE model when considering the attached fluid mass, damage can be identified by Equation (9). The results are shown in Figure 12. The figure indicates that the substructure 9 is damaged, besides the damage factor is about 0.5. The other substructures are not damaged, which is consistent with the theoretical result. Damage identification results of the structure are generally the same for different immersion depths. This indicates that the adding mass method is reliable for damage identification at various immersion depths. (2) Multiple Damages  
The additional mass and dynamic simulation of multiple damages are carried out in the same way as the single damage scenario, and the curves relating the first four natural frequencies with the mass position can be obtained. The immersion depth of 0.4 m is taken as an example, as shown in Figure 13. Damage identification is performed according to Equation (9), and the identification results are shown in Figure 14.  
It can be seen from Figure 13 that in the case of multiple damages, the four natural frequencies of the damaged structure are lower than those of the undamaged structure, so the data are reliable.