## 1 Introduction  
An important issue while solving engineering-related problems is taking into account the uncertainty of design parameters. Many parameters may have physical and geometric uncertainties as a result of modeling inaccuracies, measurement errors, manufacturing errors, etc. Parameter uncertainties can be dealt with using three methods: probabilistic methods, fuzzy theory, and interval values. In probabilistic methods, the parameters are described as random variables, assuming that the probability distribution is known. Some examples of probabilistic methods are the Monte Carlo method [1], perturbation techniques [2], and spectral methods [3]. However, probabilistic methods are often labor intensive. The fuzzy theory method is used to analyze uncertainties especially when sufficiently reliable stochastic data are not available [4]. In the interval values method, uncertainties are modeled as interval values, whose parameters are uncertain but bounded. In 1979, [PERSON] [5] introduced the concept of interval analysis. Since then, this method has been used to address numerous static and dynamic problems. In [6], interval analysis was used to solve the eigenproblem of systems with uncertain parameters. The interval finite element method was used in [7, 8], where the eigenvalue problem of systems with parameters described by interval values was addressed. The mass and stiffness matrices were expressed using the matrix perturbation theory. In [9], this method was applied in damped systems. In [10], it was demonstrated that the range of the structure response obtained using the perturbation theory, together with the interval analysis, includes the range obtained by the probabilistic method. In [11], the modal interval analysis method was used to analyze natural frequencies, eigenvectors, and frequency response functions. The solution for systems with uncertain-but-bounded parameters can also be determined using the vertex method [12, 13]. This method assumes that the solution is sought only for the limit values of the parameters and consists in solving the problem for each combination of them. Thus, it is necessary to select the lower and upper limits of the structure response among the solutions obtained. This method yields an exact or close-to-exact solution but is labor intensive as it requires \(2^{r}\) combinations of parameters (where \(r\) is the number of uncertain parameters). Hence, numerous other methods for calculating the lower and upper bounds of the response function have recently been proposed. In [14], to determine natural frequencies, a combination of the interval finite element method with the element-by-element approach was presented. The advantage of this approach is that overestimation of the obtained solution is prevented. A similar approach was used in [15] to determine the lower and upper limits of frequency response functions. In this approach, equations describing the dynamic response of the structure are written as a system of interval linear equations, which is solved iteratively using [PERSON]'s fixed-point theorem. In [16], the frequency response function was analyzed using the Laplace transform, but this method is suitable only for small uncertainties. In [17], a method for deriving approximate explicit expressions of frequency response functions was presented. In [18], the time response of a structure with uncertainties that are described as random and/or interval quantities was presented. In this study, three models of uncertainty were proposed: the first applied the generalized polynomial chaos theory to uncertainties described as random quantities, the second used the Legendre metamodel to uncertainties described as interval quantities, whereas the third was a hybrid one that takes into account the possibility of both random and interval uncertainties. In [19], the dynamic response of the structure, which is subjected to uncertain excitations, was analyzed.