## 6 Conclusions  
The main contribution of this article is the creation of a localisation and quantification method for damage identification, which is based on polynomial regression, hence creating an analytical comparative model. This approach provides relatively accurate information about the position of the damage wherever it is located within the scope of the examined model and about the extent of the damage. The focus of this publication is to demonstrate the theoretical principles on which these methods are based. On the other hand, experiments confirming localisation assumptions were already performed, and the research has yet to be extended by a detailed experiment using polynomial regression to localise and quantify the damage. Based on the presence of natural frequency drops compared to the reference system (\(\Delta f_{0j}\)), it is possible to claim that damage is present in the system. According to mode shapes, specifically their curvatures, and by assigning natural frequency drops to specific curvatures of individual mode shapes, it is possible to create a reference model to which the examined model is compared. The result of this comparison is a correlation function \(C_{L}\), which at a minimum corresponds to the identified location of the damage (\(x_{DI}\)). For the quantification of the extent of damage, linearisation of natural frequency decreases is considered, and the proportional damage \(\Delta d_{rN}=5\%\) was chosen as the nominal damage extent, thus creating a reference model for quantification. When comparing the examined model with the reference model, the slope of natural frequency decreases of the examined model is expressed as a function of proportional damage, which results in a correlation function \(C_{Q}\), the minimum of which implies the identified degree of the damage (\(\Delta d_{rI}\)), analogously to the location of damage in localisation. The localisation and quantification errors result from the polynomial regression of the curvatures of individual MSs relative to the position on the beam and the natural frequency decreases relative to the curvature, and thus the vector \(\Delta f_{0 Rj}\) contains a certain system error. Reduction in this error can be achieved by extending the model using an approximation based on the simulation of more damage locations. Additionally, an error is brought into the model by the non-zero width of the region simulating damage and by the linearisation of the natural frequency decreases for the purposes of the quantification method. Specifically, a significant improvement in the accuracy of the quantification method could be achieved by avoiding the linearisation of the natural frequency decreases in the reference model with respect to the extent of damage, which should be the aim of further research on this topic. The described principles could be applied to analyse more complex structures by applying a multi-coordinate approach and regression of them would be required. For example, in a structure consisting of multiple beams, each beam should be marked by an index and a regression function should be assigned to each beam. This method could be applied even to non-straight beams, where modal curvature, as defined in the text, could be replaced by change in curvature, and spatial transformations might be required as well. It is possible to perform regression even for different structural elements, such as plates, when working with multiple coordinates defining the position of the damage. Therefore, the method is not limited to 1D or 2D problems but can be applied to 3D problems, also. Conceptualization, M.M. ; Methodology, M.P. ; Software, M.P. ; Resources, M.M. and J.U. ; Writing--original draft, M.P. ; Writing--review & editing, M.P. ; Supervision, M.M. and J.U. All authors have read and agreed to the published version of the manuscript. The research in this paper was supported by the grant agency VEGA 1/0432/23 and VEGA 1/0497/23 of Ministry of Education, Science and Sport of the Slovak Republic.