## 1 Introduction
The authors also applied the element-by-element technique, which significantly reduced overestimation resulting from the dependencies between interval parameters. However, this method is only suitable for handling small uncertainties. The procedure for deriving approximate explicit values of FRFs for systems with uncertainties was presented in [15] by [PERSON] et al. The effectiveness of the Rational Series Expansion in estimating the lower and upper bounds of FRF values was demonstrated. The proposed method also effectively mitigates overestimations. In the work [16] written by [PERSON] et al., an approach based on two-dimensional Chebyshev polynomials was introduced for estimating the dynamic response of structures with interval uncertainties. The authors utilized the decomposition of the dynamic response function into univariate and bivariate functions and fitted them using Chebyshev polynomials. This approach facilitates the transformation of the dynamic analysis of nonlinear systems with uncertain parameters into solving simpler one-dimensional and two-dimensional equations. In studies concerning uncertain design parameters, obtaining the response of structures, particularly in dynamic problems, is greatly facilitated by expanding the sought value into a Taylor series around the central value. This approach utilizes the sensitivities of the sought function with respect to changes in uncertain design parameters. In the work [17], [PERSON] et al. applied this approach to estimate the dynamic response of damped systems. In reference [18] written by [PERSON] et al., it was shown that using a second-order Taylor series enables the accurate estimation of structural responses even with large uncertainties, although the focus was on undamped systems. In reference [19], [PERSON] and [PERSON] proposed the identification of critical combinations of uncertain parameters by maximizing the objective function through second-order Taylor series expansion. The structure under consideration included dampers, with uncertain parameters extending to damper characteristics as well. Additionally, in the works [20] by [PERSON] and [PERSON] and [21] by [PERSON], structures featuring viscoelastic elements were analyzed. These studies presented algorithms that notably reduced the number of combinations needed to determine the lower and upper bounds of the sought dynamic response of the structure. One method that enables the incorporation of uncertainties in design parameters is stochastic analysis ([PERSON] et al. in [22; 23]). This approach, particularly when parameters are described as interval numbers, was applied by [PERSON] and [PERSON] in [24] using affine arithmetic. This application helped reduce the overestimation of the determined structural response. Instead, [PERSON] et al., in [25], introduced a hybrid approach where the boundaries of probabilistic characteristics are determined through interval arithmetic. Additionally, an approach utilizing Chebyshev polynomials to solve problems with multiple uncertain variables was introduced. The authors also address scenarios where the analyzed systems are described using mixed random and interval parameters. In reference [26] by [PERSON], a method was proposed to determine the lower and upper bounds of the mean value and standard deviation assuming the presence of mixed uncertain parameters. This method is based on expanding a series of random and interval quantities with respect to uncertain parameters. A comparison with the Monte Carlo method was also conducted, demonstrating a significant reduction in computational costs. In reference [27], [PERSON] et al. considered the time response of structures and proposed a hybrid model combining generalized polynomial chaos theory and the Legendre metamodel method. On the other hand, [PERSON] et al., in refence [28], presented a method based on genetic algorithms for computing optimization problems with hybrid uncertain parameters.