## 4 Interval Equations System Solving
\begin{bmatrix}\mathbf{K}^{I}+\mathbf{i}\lambda\mathbf{C}^{I}-\lambda^{2} \mathbf{M}^{I}+\mathbf{G}^{I}(\lambda)&\mathbf{X}^{I}\\ \mathbf{X}&0\end{bmatrix}\begin{bmatrix}\widetilde{\mathbf{q}}(\lambda)^{I} \\ \beta^{I}\end{bmatrix} = \begin{bmatrix}\widetilde{\mathbf{P}}(\lambda)^{I}\\ 0\end{bmatrix}  \tag{29}\]  
where interval vectors and matrices are constructed using the element-by-element technique. This technique will be demonstrated based on the construction of the component matrix \(\mathbf{G}^{I}(\lambda)\). Let us consider a three-story frame, as shown in Figure 3, with two dampers described by the Kelvin model. The schematic illustrates the forces \(u_{i}\), representing the interactions of individual dampers with each floor. Figure 3: A frame with two Kelvin dampers. The matrix containing uncertain damping parameters \(c_{0k}^{I}\) will be considered. According to [51] by [PERSON] and [PERSON] and [52] by [PERSON] et al., for one damper \(k\), the matrix can be expressed as follows:  
\[\mathbf{C}_{0k}^{I}= \begin{bmatrix}c_{0k}^{I}&-c_{0k}^{I}\\ -c_{0k}^{I}&c_{0k}^{I}\end{bmatrix} = \begin{bmatrix}c_{0k}^{C}&-c_{0k}^{C}\\ -c_{0k}^{C}&c_{0k}^{C}\end{bmatrix} \left( \begin{bmatrix}1&0\\ 0&1\end{bmatrix} + \begin{bmatrix}\delta_{k}^{I}&0\\ 0&\delta_{k}^{I}\end{bmatrix} \right)=\mathbf{C}_{0k}^{C}\left(\mathbf{I}+ \delta_{k}^{I}\right) \tag{30}\]  
where \(c_{0k}^{I}\) is the uncertain damping parameter of damper \(k\), \(c_{0k}^{C}\) is the central value of parameter \(c_{0k}^{I}\), and \(\delta_{k}^{I}\) denotes the uncertainty of parameter \(c_{0k}^{I}\). If we consider two dampers, it is possible to express the matrix \(\mathbf{C}_{0}^{I}\) in the following form:  
\[\mathbf{C}_{0}^{I}= \begin{bmatrix}c_{01}^{I}&-c_{01}^{I}&0&0\\ -c_{01}^{I}&c_{01}^{I}&0&0\\ 0&0&c_{02}^{I}&-c_{02}^{I}\\ 0&0&-c_{02}^{I}&c_{02}^{I}\end{bmatrix} = \tag{31}\] \[= \begin{bmatrix}c_{01}^{C}&-c_{01}^{C}&0&0\\ -c_{01}^{C}&c_{01}^{C}&0&0\\ 0&0&c_{02}^{C}&-c_{02}^{C}\\ 0&0&-c_{02}^{C}&c_{02}^{C}\end{bmatrix} \left( \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} + \begin{bmatrix}\delta_{1}^{I}&0&0&0\\ 0&\delta_{1}^{I}&0&0\\ 0&0&\delta_{2}^{I}&0\\ 0&0&0&\delta_{2}^{I}\end{bmatrix} \right)=\mathbf{C}_{0}^{C}\left(\mathbf{I}+ \boldsymbol{\Delta}^{I}\right)\]  
It is worth noting that Formula (31) aims to illustrate only the idea of the element-by-element technique using the example of coefficients for two dampers, and it is not the final form of the matrix appearing in the equation of motion. From the above expression, it is clear that one uncertain parameter associated with the damper appears four times, meaning the uncertainties \(\delta_{k}^{I}\) overlap and cause overestimation. According to the above notation, the remaining matrices can be expressed as follows:  
\[\mathbf{K}^{I}=\mathbf{K}^{C}\left(\mathbf{I}+\boldsymbol{\Delta}_{1}^{I} \right),\ \mathbf{C}^{I}=\mathbf{C}^{C}\left(\mathbf{I}+\boldsymbol{\Delta}_{2}^{I} \right),\ \mathbf{M}^{I}=\mathbf{M}^{C}\left(\mathbf{I}+\boldsymbol{\Delta}_{3}^{I}\right),\ \mathbf{G}^{I}(\lambda)=\mathbf{G}^{C}(\lambda)\left(\mathbf{I}+\boldsymbol{ \Delta}_{4}^{I}\right) \tag{32}\]  
and the equation of motion takes the following form:  
\[\begin{split}&\left[ \begin{bmatrix}\mathbf{K}^{C}+\mathrm{i} \lambda\mathbf{C}^{C}-\lambda^{2}\mathbf{M}^{C}+\mathbf{G}^{C}(\lambda)& \mathbf{X}^{T}\\ \mathbf{X}&\mathbf{0}\end{bmatrix} + \begin{bmatrix}\mathbf{K}^{C}&\mathbf{0} \\ \mathbf{0}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\boldsymbol{\Delta}_{1}^{I}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} +\mathrm{i}\lambda \begin{bmatrix}\mathbf{C} ^{C}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\boldsymbol{\Delta}_{2}^{I}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} \\ -\lambda^{2} \begin{bmatrix}\mathbf{M}^{C}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\boldsymbol{\Delta}_{3}^{I}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} +