# Damage Identification Based on Adding Mass for Liquid-Solid Coupling Structures
## 2 Damage Identification Based on the Adding Mass Method  
For submerged structures, it is often a challenge to locate the measuring points due to the influence of liquid. Moreover, structural modes are not sensitive to local damage, which becomes even more complicated for coupling analyses. The adding mass method can overcome these difficulties to some extent. More positions additional masses are added at, more modal information can be obtained, which can lead to more accurate damage identification. Therefore, the adding mass method is employed in this work to identify the damage of submerged structures. A cantilever beam is taken as an example (see Figure 1), the structure is divided into \(n\) substructures, which means that the additional mass can take one of the \(n\) positions. The global structure is denoted as \(S_{i}\), which refers to the structure with the mass added to the \(i\)-th substructure. Dynamic experiments are conducted on the structure \(S_{i}\). It is assumed that the first \(k\) natural frequencies of the structure can be obtained noted as \(\omega_{i}=\{\omega_{1i},\omega_{2i},\ldots,\omega_{ki}\}\), where \(\omega_{ji}\) represents the \(j\)-th natural frequency of the structure with the additional mass added to the \(i\)-th substructure. From the first to the \(n\)-th substructure, the total natural frequency matrix can be noted as \(\overline{\omega}=\{\omega_{1},\omega_{2},\ldots,\omega_{i},\ldots,\omega_{n}\}\).  
The additional mass can change the dynamic characteristics of the structure, and structural modes can also vary depending on the location of the additional mass. Generally, larger additional masses can lead to greater changes in structural responses. This results in higher effects on the structural dynamic characteristics and local sensitivity. In these circumstances, the same order mode is irrelevant, and the accuracy of damage identification is improved. However, if the additional mass is too large, it may cause difficulty for experiment operation or even structural collapse. If the additional mass is too small, the sensitivity is low, and the effect of additional mass may not be observed. Therefore, the appropriate additional mass is selected according to the relative sensitivity of the natural frequency \(R_{ji,l}(\mu,m)\) (see Equation (1)), which is calculated by the FE model. The optimal additional mass is determined using the undamaged structure with the damage factor \(\mu=1\). Finally, the mass \(m\) is modified to achieve the optimal value of \(R_{ji,l}(\mu,m)\).  
\[R_{ji,l}(\mu,m)=\frac{\partial\omega_{ji}(\mu,m)}{\omega_{ji}(\mu,m)\partial \mu_{l}}=\frac{\Psi_{ji}^{\mathrm{T}}(\mu,m)K_{l}\Psi_{ji}(\mu,m)}{2{\omega_{ ji}}^{2}(\mu,m)}, \tag{1}\]  
where \(R_{ji,l}\) is the relative sensitivity of the \(j\)-th order natural frequency of the \(l\)-th substructure, with the additional mass at the \(i\)-th position; \(\omega_{ji}\) and \(\Psi_{ji}\) are the \(j\)-th order natural frequency and mode shape of  
Figure 1: Schematic diagram of additional masses.  
the damaged structure \(S_{i}\); \(K_{l}\) is the undamaged stiffness matrix of the \(l\)-th substructure; and \(\mu_{l}\) is the ratio of stiffness after and before the damage of the \(l\)-th substructure.  
In summary, the steps of the additional mass method include: (1) successively attaching the selected appropriate mass to each substructure of the structure to build new structures, (2) obtaining the natural frequencies information of each new structure by dynamic experiments, (3) using the obtained frequencies to identify structural damage.