## 6 Examples
### Frame with Kelvin Models  
The structure under consideration is a frame with dampers described by the fractional Kelvin model, as shown in Figure 14. The parameters of the frame are as follows: the mass of the floor is \(m=1000\) kg, and the stiffness of the storey is \(k=10\),000 N/m. Structural damping is assumed to be proportional and described by the relationship \(\mathbf{C}=0.001\ \mathbf{K}+1.0\ \mathbf{M}\). Additionally, all dampers are assumed to be the same and described by the following parameters: \(k_{0}=1000\) N/m, \(c_{0}=500\) Ns\({}^{\alpha}\)/m, and \(\alpha=0.8\). The natural frequencies and the nondimensional damping ratios for the analyzed frame are presented in Table 5. The parameters of the dampers \(c_{0}\) and \(k_{0}\) are uncertain, with 5% uncertainty assumed for each parameter independently. The parameters of the dampers can be described as follows:  
\[c_{0i}^{I}=c_{0i}^{C}\left(1+\delta_{i}^{I}\right),\ k_{0i}^{I}=k_{0i}^{C} \left(1+\delta_{i}^{I}\right) \tag{42}\]  
where \(\delta_{i}^{I}=\left[-0.05,\ 0.05\right]\). The matrix \(\mathbf{D}_{1}^{I}\) appearing in Equation (36) has the following form:  
\[\bar{\mathbf{D}}_{1}^{I}=\left(\mathrm{i}\lambda\right)^{a}\bar{\mathbf{C}}_{ 0}^{C}\bar{\mathbf{A}}_{1}^{I}+\bar{\mathbf{K}}_{0}^{C}\bar{\mathbf{A}}_{2}^ {I} \tag{43}\]  
where the matrices \(\bar{\mathbf{C}}_{0}^{C}\) and \(\bar{\mathbf{K}}_{0}^{C}\) contain the central values of the damping coefficients and stiffness coefficients of the dampers, respectively, while \(\bar{\mathbf{A}}_{1}^{I}\) and \(\bar{\mathbf{A}}_{2}^{I}\) contain the uncertainties of these parameters. All matrices in Equation (43) are constructed using the element-by-element technique, and transformation (34) has been applied in the calculations. The real and imaginary parts of selected elements of the FRF matrix are shown in Figures 15-18. Figure 16: Graphs of \(H_{22}\): (**a**) real part; (**b**) imaginary part. Figure 14: Four-story building with three dampers. Figure 15: Graphs of \(H_{11}\): (**a**) real part; (**b**) imaginary part. Based on Figures 15-18, it is evident that the presented method yields results close to the vertex method. Moreover, compared to the DIM, it significantly reduces overestimation resulting from the direct analysis in interval arithmetic, especially at lower frequencies. This confirms the advantages of the presented method. Additionally, considering the computational cost of the presented method and the vertex method, the presented method also holds an advantage. In the case of the vertex method and six varying parameters, the number of cases to be considered for each analyzed frequency is \(2^{6}=64\), whereas with the presented method, the solution is typically obtained after a few iterations and is relatively straightforward to implement. In Tables 6 and 7, the relative error for the presented method and the direct method is shown for selected values of frequency and two uncertainty values: 5% and 1%. The absolute errors of the magnitude for \(H_{11}\) were calculated. The vertex method was adopted as a comparative method. Based on the results presented in Tables 6 and 7, it can be observed that the error values are always smaller in the case of the presented method. The largest differences are for lower frequencies, which confirms the observations from the \(H_{11}\) plots, where the largest overestimations for the direct method are indeed in this area. The obtained results confirm the superiority of the presented approach over the direct interval method. Figure 17: Graphs of \(H_{33}\): (**a**) real part; (**b**) imaginary part. Figure 18: Graphs of \(H_{44}\): (**a**) real part; (**b**) imaginary part. In Table 8, the computation time for \(H_{11}\) is presented for various uncertainty values. Based on the example provided, it can be observed that the discussed iterative method is more efficient, the smaller the uncertainties are. This is associated with a lower number of iterations needed to find a solution.