## 6 Examples
### Frame with Maxwell Model  
The frame shown in Figure 14 is considered, but with dampers described using the Maxwell model. The parameters of the frame remain the same as in example 6.3, and the structural damping is assumed in a similar manner. The dampers are characterized by the following parameters: \(k_{1}=1000\) N/m, \(c_{1}=500\) Ns\({}^{\alpha}\)/m, and \(\alpha=0.8\). Table 10 presents the natural frequencies and nondimensional damping ratios for the analyzed frame. As previously mentioned, the focus is on the influence of uncertainties in the parameters of viscoelastic elements. Therefore, it is assumed that the parameters and the dampers \(c_{1}\) and \(k_{1}\) are uncertain, with uncertainty of 5%. It is further assumed that these parameters can change independently and are described as follows:  
\[c_{1i}^{I}=c_{1i}^{C}(1+\delta_{i}^{I}),\ \ k_{1i}^{I}=k_{1i}^{C}(1+\delta_{i}^{I}) \tag{44}\]  
where \(\delta_{i}^{I}=[-0.05,\ 0.05]\). This time, unlike the Kelvin model, to express the relationship between uncertainty and the central value of the parameter in Equation (36) as a multiplication, it is necessary to use the formulation of equations of motion with internal variables. A comprehensive explanation of constructing such equations of motion for a frame with viscoelastic dampers can be found in [52] by [PERSON] et al. This formulation expands the dimension of the problem by the number of internal variables, yet it enables the application of the presented method to this damper model as well. The matrix \(\bar{\mathbf{D}}_{1}^{I}\) has the following form:  
\[\mathbf{D}_{1}^{I}=(\mathrm{i}\lambda)^{a}\bar{\mathbf{C}}_{1}^{C}\bar{ \mathbf{A}}_{1}^{I}+\bar{\mathbf{K}}_{1}^{C}\bar{\mathbf{A}}_{2}^{I} \tag{45}\]  
the matrices \(\bar{\mathbf{C}}_{1}^{C}\) and \(\bar{\mathbf{K}}_{1}^{C}\) have the following structures:  
\[\bar{\mathbf{C}}_{1}^{C}=\begin{bmatrix}\mathbf{C}_{1,ss}^{C}&\mathbf{C}_{1, dd}^{C}&\mathbf{0}\\ \mathbf{C}_{1,dd}^{C}&\mathbf{C}_{1,dd}^{C}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix},\ \bar{\mathbf{K}}_{1}^{C}=\begin{bmatrix}\mathbf{K}_{1,ss}^{C}&\mathbf{K}_{1, sd}^{C}&\mathbf{0}\\ \mathbf{K}_{1,dd}^{C}&\mathbf{K}_{1,dd}^{C}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix} \tag{46}\]  
Additional matrices \(\mathbf{C}_{1,sd}^{C}\), \(\mathbf{C}_{1,ds}^{C}\), \(\mathbf{C}_{1,dd}^{C}\), \(\mathbf{K}_{1,sd}^{C}\), \(\mathbf{K}_{1,ds}^{C}\), and \(\mathbf{K}_{1,dd}^{C}\) are associated with additional degrees of freedom related to internal variables, while matrices \(\mathbf{C}_{1,ss}^{C}\) and \(\mathbf{K}_{1,ss}^{C}\) are associated with degrees of freedom of the analyzed structure ([PERSON] et al. [41]). The matrices are constructed using the element-by-element technique, and transformation (34) is applied. The results, in the form of selected elements of the FRF matrix, are shown in Figures 24-27.  
Figure 23: Graphs of \(H_{11}\) for parameter variability of 2%: (**a**) real part; (**b**) imaginary part.  
Figure 26: Graphs of \(H_{33}\): (**a**) real part; (**b**) imaginary part.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode** & **Natural Frequency \(\omega\) [rad/s]** & **Nondimensional Damping Ratio \(\gamma\) [-]** \\ \hline
1 & 1.1121 & 0.4638 \\
2 & 3.2319 & 0.1692 \\
3 & 4.9942 & 0.1160 \\
4 & 6.1657 & 0.0980 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Dynamic characteristics of the frame from example 6.4.  
Figure 24: Graphs of \(H_{11}\): (**a**) real part; (**b**) imaginary part.  
Figure 25: Graphs of \(H_{22}\): (**a**) real part; (**b**) imaginary part.  
In this case, it can also be observed that the applied method closely resembles the results obtained from the vertex method and significantly reduces overestimation resulting from calculations involving interval numbers.  
In addition, it is worth noting that all calculations were conducted using MATLAB with the IntLab toolbox (Rump [60]), which provided essential support for interval arithmetic computations.