## 3 Dynamic Response of Systems with Viscoelastic Elements  
The analysis will focus on a structure with viscoelastic damping elements, described using rheological models that incorporate fractional derivatives. The general equation of motion can be expressed as follows:  
\[\mathbf{M}\ddot{\mathbf{q}}(t)+\mathbf{C}\ddot{\mathbf{q}}(t)+\mathbf{K} \mathbf{q}(t)+\int_{0}^{t}\overset{\sim}{\mathbf{G}}(t-\tau)\dot{\mathbf{q}}( \tau)d\tau=\mathbf{P}(t) \tag{4}\]where \(\mathbf{M}\) and \(\mathbf{K}\) are the mass and stiffness matrices, respectively, \(\mathbf{C}\) is the damping matrix associated with the structure, and \(\overset{\sim}{\mathbf{G}}(t-\tau)\) is a matrix of damping kernel functions whose form depends on the adopted model. \(\mathbf{P}(t)\) represents the vector of excitation forces and \(\mathbf{q}(t)\) is the vector of displacements. In structural dynamics for matrix \(\overset{\sim}{\mathbf{G}}(t-\tau)\), rheological models are commonly used, with fractional models being employed in this article. As part of this analysis, the Scott-Blair viscoelastic element (Figure 1) will be introduced. Its behavior is described by the following equation:  
\[u(t)=cD_{t}^{\alpha}x(t) \tag{5}\]  
where \(c\) and \(\alpha\) are the parameters of the described element \((0<\alpha\leq 1)\), the symbol \(D_{t}^{\alpha}(\bullet)\) denotes the fractional derivative of order \(\alpha\) with respect to time, and \(x(t)\) is the difference in displacements of the damper nodes. \(u(t)\) denotes the force in the viscoelastic element. A more extensive discussion of fractional calculus can be found in [48]. The following Riemann-Liouville definition of the fractional derivative was applied in the article:  
\[D_{t}^{\alpha}x(t)=\frac{d^{\alpha}x(t)}{dt^{\alpha}}=\frac{1}{\Gamma(1- \alpha)}\frac{d}{dt}\int\limits_{0}^{t}\frac{x(s)}{(t-s)^{\alpha}}ds \tag{6}\]  
where \(\Gamma\) denotes the gamma special function. As demonstrated in [49; 50; 51], the application of this definition is justified in the context of viscoelastic materials. In this article, Kelvin or Maxwell models will be considered, both classical and fractional (Figure 2). Classical models are described in [52] by [PERSON] et al., in [53] by [PERSON], and in [54] by [PERSON] et al., and a description of fractional models can be found in [50] by [PERSON] et al., in [51] by [PERSON] and [PERSON], and in [55] by [PERSON] and [PERSON]. The force in the damping element, in the case of the Kelvin model, is described by the following equation:  
\[u(t)=k_{0}x(t)+c_{0}D_{t}^{\alpha}x(t) \tag{7}\]  
and for the Maxwell model, it is described as follows:  
\[v_{1}u(t)+D_{t}^{\alpha}u(t)=k_{1}D_{t}^{\alpha}x(t) \tag{8}\]  
where \(c_{0}\) and \(c_{1}\) are the damping coefficients of the Kelvin and Maxwell model, respectively, and \(k_{0}\) and \(k_{1}\) are the stiffness parameters of the Kelvin and Maxwell model, respectively. Additionally, \(v_{1}=k_{1}/c_{1}\), and by achieving \(\alpha=1\), one can obtain the classical models. Figure 1: Scott–[PERSON] viscoelastic element. Figure 2: Models of viscoelastic elements: (**a**) classical Kelvin model; (**b**) fractional Kelvin model, (**c**) classical Maxwell model; (**d**) fractional Maxwell model. After applying the Laplace transformation, Equation (4) can be written as follows:  
\[\Big{(}s^{2}\mathbf{M}+s\mathbf{C}+\mathbf{K}+\mathbf{G}(s)\Big{)}\widetilde{ \mathbf{q}}(s)=\widetilde{\mathbf{P}}(s) \tag{9}\]  
where \(\widetilde{\mathbf{q}}(s)\) and \(\widetilde{\mathbf{P}}(s)\) represent the Laplace transform of \(\mathbf{q}(t)\) and \(\mathbf{P}(t)\), respectively, and \(\mathbf{G}(s)\) is the matrix of damping kernel functions in the Laplace domain. Its form depends on the adopted rheological model of the viscoelastic element, the considered structure, and the placement of the viscoelastic element. It can be expressed as the sum of matrices for individual viscoelastic elements, denoted as \(\mathbf{G}(s)=\sum_{k=1}^{r}\mathbf{G}_{k}(s)\) (where \(r\) denotes the number of viscoelastic elements).