## 1 Introduction
Advantages: they consider the range in which the parameter changes without knowing its exact distribution, allow for the relatively easy determination of the lower and upper bounds of the response function, and are relatively easy to code; disadvantage: the obtained results may be overestimated. In this article, the dynamic response of structures with viscoelastic elements is analyzed. It is assumed that the behavior of viscoelastic material is described using rheological models, both classical and fractional. The method presented considers uncertainties in the design parameters of the structure as well as damper parameters. A method for determining the frequency response function is presented, which is one of the most important tools for assessing the dynamic response of structures. However, there are very few publications addressing this issue for structures with viscoelastic elements. Interval analysis is employed in this article, addressing a drawback of this approach, namely significant overestimation. To avoid this, the element-by-element technique is applied. Previous works, ref. [42] by [PERSON] et al., ref. [43] by [PERSON] and [PERSON], and [44] by [PERSON] et al., have shown that this technique reduces overestimation in both statics and dynamics problems. Its application to systems with viscoelastic elements is presented here for the first time. The formulation of equations of motion using the element-by-element technique is general and applicable to various rheological models. Laplace transformation is used to express the equations of motion in the frequency domain, representing them as a system of interval equations. Their solution involves individual elements of the FRF matrix, solved using [PERSON]'s fixed-point theorem. Three methods are compared in computations: the proposed method, the vertex method, and the direct arithmetic of interval numbers. Direct analysis leads to significant overestimations, while the proposed method aligns more closely with the vertex method and reduces computational costs significantly. The examples presented demonstrate that the proposed method is a valuable tool for assessing the lower and upper bounds of the response of structures with viscoelastic dampers. Additionally, the obtained results in one example were compared with those obtained using Taylor series expansion. A novelty of this work is the application of the fixed-point iteration method to find the lower and upper bounds of the frequency response function for systems with viscoelastic damping elements, also described using fractional derivatives. The presented method can be applied to account for the influence of uncertainties in the parameters of viscoelastic damping devices on the response of the device or the entire structure, which is an advantage compared to traditional methods that do not consider parameter uncertainties. The article is organized as follows: Section 2 provides a brief introduction to interval analysis; Section 3 presents the formulation of equations of motion for systems with viscoelastic elements, considering uncertain design parameters; Section 4 demonstrates the application of [PERSON]'s fixed-point theorem to solve systems of interval equations for determining the frequency response function; Section 5 presents the method of overestimation reduction, involving the formulation of equations of motion using the element-by-element technique with the use of Lagrange multipliers; Section 6 presents examples; and Section 7 concludes the article.