## 4 Quantification of the Extent of Damage  
Once the damage position is identified, it is possible to assign characteristics to this position that lead to the development of the quantification method. The quantification method follows the scheme in Figure 9. In this case, the output is the identified extent of the damage \(\Delta d_{rI}\). Figure 8: Identified damage (**left**) and its relative error (**right**) according to the localisation method. Figure 7: Correlation function for localisation. For the needs of the quantification methodology, it is necessary to work with the nominal level of damage, according to which slopes of linearised RNFDs are assigned. The theory of quantification assumes that RNFDs have a linear, or character similar to linear, with relation to damage extent, which implies that the nominal value should be expressed from an interval of the most linear character possible, and at the same time, RNFD courses should be linearised in the best possible correlation with real course declines. Figure 10 shows that below 5% damage, differences between \(\Delta f_{0 Ej}\) (real RNFDs) and \(\Delta f_{0 Uj}\) (linearised RNFDs--Equation (10)) are relatively small, allowing the mentioned linearisation. Therefore, the nominal proportional rate of RNFDs is to be derived from the damage represented by \(\Delta d_{rN}=5\%\). The resulting linearisation error needs to be further considered in the application of the method. As in the case of localisation, where the output is the identified position of the damage (\(x_{DI}\)), when assessing the degree of a damage output is identified as the damage extent (\(\Delta d_{rl}\)). Similarly, it is necessary to compare the examined model against the reference model. The reference model is represented by RNFDs \(\Delta f_{0 Rj}\), while these decreases are not only a function of the location of the damage on the beam, but also the extent of the damage. After locating the damage and considering the nominal damage \(\Delta d_{rN}=5\%\), the vector of RNFDs \(\Delta f_{0 Rj}(x_{DI},\Delta d_{rN})\) is known. To compare the reference and examined models, it is  
Figure 10: Comparison of real and linearised RNFDs for damage extent \(\langle 0;10\rangle\%\). Colours assigned to specific RNFDs, resp. index j, are also valid for the subsequent figures. Figure 9: Scheme of the quantification method principle. appropriate to compare the slopes of these linearised models. The nominal slopes of the reference model are \(k_{Rj}\) (Equation (11)) and the RNFDs defined by the mentioned slopes are \(\Delta f_{0 Lkj}\) (Equation (12)). \[k_{Rj}=k_{j}(x_{D1},\Delta d_{rN})=\frac{\Delta f_{0 Rj}(x_{D1},\Delta d_{rN})}{ \Delta d_{rN}} \tag{11}\]  
\[\Delta f_{0 Lkj}=\frac{\Delta f_{0 Rj}}{\Delta d_{rN}}\Delta d_{r}=k_{Rj}\Delta d _{r} \tag{12}\]  
Equation (13) applies to the slope of the examined model \(k_{Ej}\), which implies that the slopes of the examined model are functions of \(\Delta d_{r}\), while \(\Delta d_{r}\in\langle 0;100\rangle\):  
\[k_{Ej}(\Delta d_{r})=\frac{\Delta f_{0 Ej}(x_{DE},\Delta d_{rE})}{\Delta d_{r}}, \tag{13}\]  
where \(x_{DE}\) is the real damage location of the examined model and \(\Delta d_{rE}\) is the real relative extent of the beam damage. For quantification, the damage rate of the examined system \(\Delta d_{rE}\) is considered as an unknown constant value.