# Pipeline Damage Identification Based on Additional Virtual Masses
## 2 Damage Identification Method Based on Additional Virtual Masses
### Damage Identification Based on Sensitivity  
Assume that the global structure consists of \(n\) components, each defined as a substructure. Let the structural damage be modeled in terms of stiffness reduction ratios of the substructures. If the damage factor of the \(i\)th substructure is defined as \(\mathbf{\mu}_{i}\), then \(\mathbf{\mu}=\{\mathbf{\mu_{1}},\mathbf{\mu_{2}},\ldots,\mathbf{\mu_{n}}\}^{\mathsf{T}}\). \(\mathbf{K}_{i}\) is the original undamaged stiffness matrix of the \(i\)th substructure expressed in the global DOFs. Hence, the global stiffness matrix \(\mathbf{K}(\mathbf{\mu})\) of the damaged structure is assembled as  
\[\mathbf{K}(\mathbf{\mu})=\sum_{i=1}^{n}\mathbf{\mu}_{i}\mathbf{K}_{i}. \tag{5}\]  
By adding a virtual mass to the real structure, the dynamic characteristics of the resulting virtual structure \(\mathbf{G}_{i}(\mathbf{\mu},\mathbf{m})\) are changed compared with the original real structure \(\mathbf{G}_{i}(\mathbf{\mu},\mathbf{0})\). In this manner, the virtual structure can be designed and purposely used for damage identification. Here, the virtual mass is added to increase the relative sensitivity of natural frequencies with respect to damage parameter \(\mathbf{\mu}_{i}\). Assuming the damage vector \(\mathbf{\mu}\), the characteristics of the \(k\)th natural frequency \(\widetilde{\mathbf{\omega}}_{ki}(\mathbf{\mu},\mathbf{m})\) of the corresponding virtual structure (symbolically denoted by \(\mathbf{G}_{i}(\mathbf{\mu},\mathbf{m})\)) depend on three parameters: order \(\mathbf{k}\) of natural frequency, virtual mass \(\mathbf{m}\), and its placement \(\mathbf{i}\). The virtual mass does not affect stiffness, and as such, the stiffness matrix of such a virtual structure is still \(\mathbf{K}(\mathbf{\mu})\). Let \(\mathbf{R}_{ki,l}(\mathbf{\mu},\mathbf{m})\) denote the absolute sensitivity of \(\widetilde{\mathbf{\omega}}_{ki}(\mathbf{\mu},\mathbf{m})\) with respect to the extent of damage \(\mathbf{\mu}_{l}\) of the \(l\)th substructure  
\[\mathbf{R}_{ki,l}(\mathbf{\mu},\mathbf{m})=\frac{\partial\widetilde{\mathbf{\omega}}_{ki}(\bm {\mu},\mathbf{m})}{\partial\mathbf{\mu}_{l}}=\frac{\widetilde{\mathbf{\varphi}}_{ki}^{ \mathsf{T}}(\mathbf{\mu},\mathbf{m})\mathbf{K}_{l}\widetilde{\mathbf{\varphi}}_{ki}(\mathbf{\mu}, \mathbf{m})}{2\widetilde{\mathbf{\omega}}_{ki}(\mathbf{\mu},\mathbf{m})} \tag{6}\]  
where \(\widetilde{\mathbf{\varphi}}_{ki}(\mathbf{\mu},\mathbf{m})\) is the \(k\)th mass-normalized global mode shape of the virtual structure \(\mathbf{G}_{i}(\mathbf{\mu},\mathbf{m})\). According to the definition of sensitivity, if \(\mathbf{\mu_{0}}\) is the initial value of the damage factor, then a first-order approximation occurs between changes in the damage factor \(\Delta\mathbf{\mu}=\mathbf{\mu}-\mathbf{\mu_{0}}\) and structure frequency \(\mathbf{\Delta}\widetilde{\mathbf{\omega}}_{ki}=\widetilde{\mathbf{\omega}}_{ki}(\mathbf{\mu}, \mathbf{m}_{ki})-\widetilde{\mathbf{\omega}}_{ki}(\mathbf{\mu_{0}},\mathbf{m}_{ki})\)  
\[\mathbf{\Delta}\widetilde{\mathbf{\omega}}_{ki}=\sum_{l=1}^{n}\mathbf{R}_{ki,l}(\mathbf{\mu_{0} },\mathbf{m})\mathbf{\Delta}\mathbf{\mu}_{l} \tag{7}\]  
We collect all frequencies \(\widetilde{\mathbf{\omega}}_{ki}(\mathbf{\mu},\mathbf{m}_{ki})\) in all structures \(\mathbf{G}_{i}(\mathbf{\mu},\mathbf{m}_{ki})\), where \(\mathbf{k}=\mathbf{i_{1}},\mathbf{i_{2}},\ldots,\mathbf{i_{j_{i}}}\). \(\mathbf{i}=\mathbf{1},\mathbf{2},\ldots\mathbf{n}\). Then, formula (7) is arranged as in the following sequence:  
\[\mathbf{\Delta}\widetilde{\mathbf{\omega}}=\mathbf{R}\mathbf{\Delta}\mathbf{\mu}, \tag{8}\]where  
\[\left\{\begin{array}{c}