## 3 Principle of the Damage Localisation of the Structure
### Description of the Method  
The flow chart in Figure 3 roughly depicts the procedure of the localisation method. The method is based on a comparison of the examined mathematical model with a reference mathematical model. To create a reference mathematical model, it is necessary to know the MSs of the undamaged system \(v_{j}\) (\(j=3,4,\ldots,8\)) [27; 28; 29; 30]. For a clearer comparison of MSs, their normalisation is performed (Equation (5)) so that all values are in the interval <0;1> and normalised MSs \(v_{Nj}\) are attained, while the maximum amplitude is at the free end of the beam. This normalisation does not affect the results of the localisation. Subsequently, by polynomial regression of the 20 th degree of the \(v_{Nj}\)[31] and their consecutive derivation, their curvatures \(c_{Nj}\) (Equation (6)) are obtained [32]. Normalised MSs \(v_{Nj}\) and their curvatures \(c_{Nj}\) are depicted in Figure 4.  
\[v_{Nj}(x)=\frac{v_{j}(x)}{v_{j}(l)} \tag{5}\]  
\[c_{Nj}(x)=\frac{d^{2}v_{Nj}(x)}{dx^{2}}=\frac{M_{Nj}(x)}{EJ} \tag{6}\]  
Figure 4: Depiction of the normalised MSs and their curvatures with marked damage positions. The index \(k=1\), \(2\), \(\ldots\), \(10\) in the legend represents POS 1–10. Figure 3: Scheme of the localization method principle. Nominal damage to the beam \(\Delta d_{r}=5\%\) is separately simulated at ten different evenly distributed positions of the beam \(x_{Nk}\) (\(k=1,\ 2,\ \dots,\ 10\)), while \(x_{N1}=0.05\) m and \(x_{N10}=0.5\) m, so that this damage is placed in the largest possible range of curvatures of individual MS. For individual curvature of a specific MS, the RNFD is assigned (Figure 5) and each dependence of the RNFD on the curvature contains 11 elements (10--assigned RNFDs to specific curvatures, 1--assumed zero drop for zero curvature). The RNFD courses in the MS curvatures are obtained by polynomial regression of the second degree of these data. By merging these dependencies (\(c_{Nj}(x)\)--Figure 4 and \(\Delta f_{0 Rj}(c_{Nj})\)--Figure 5), it is possible to express the RNFDs as a vector that is a function of the position on the beam (Equation (7)). This dependence is shown graphically in Figure 6. By comparison of \(RNFDs\) of nominal damage at positions \(x_{Nk}\), and reference \(\Delta f_{0 Rj}(x)\), simulated by damage extent \(\Delta d_{r}=5\%\), it can be concluded that correlation of these quantities is relatively significant. \[\Delta f_{0 Rj}(x)=A_{j}\bigg{(}\sum_{i=1}^{n=20}a_{ij}x^{n-i}\bigg{)}^{2}+B_{ j}\bigg{|}\sum_{i=1}^{n=20}a_{ij}x^{n-i}\bigg{|}+C_{j} \tag{7}\]  
Equation (8) expresses the vector of ratios \(\Delta f_{0 Rj}^{*}\) of all RNFDs against the first RNFD. This vector can be used as a relevant mathematical model representing the reference system (index R) because this ratio is unique to the random position on the beam. \[\Delta f_{0 Rj}^{*}(x)=\frac{\Delta f_{0 Rj}(x)}{\Delta f_{0R1}(x)} \tag{8}\]  
Analogously, as a vector \(\Delta f_{0 Rj}^{*}\), the vector \(\Delta f_{0 Ej}^{*}\) is created, which is a mathematical model representing the examined system (index E). However, this vector already  
Figure 5: Relation between RNFDs and curvatures of MSs. Figure 6: RNFDs as a function of position on the beam. Vertical dashed lines represent positions \(x_{DE1}=0.5\) m and \(x_{DE2}=0.8\) m which are further used as an example in Sections 4 and 5 and red stars represent real RNFDs related to specific positions \(\Delta f_{0 Ej}(x_{k})\). corresponds to a specific damage position. By comparing the mentioned vectors using Equation (9), a correlation function is created, which acquires a minimum at the point of damage. \[v_{Nj}(x)=\frac{v_{j}(x)}{v_{j}(l)} \tag{5}\]0  
\[v_{Nj}(x)=\frac{v_{j}(x)}{v_{j}(l)} \tag{5}\]1  
\(x_{DI}\) represents the identified damage location (localisation target), which differs from the real damage location by the system error of the localisation method.