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This paper concerns the dynamic analysis of composite beams containing elastic and viscoelastic (VE) layers. A method for determination of the dynamic characteristics of multilayered beams with VE layers (i.e., natural frequencies, non-dimensional damping ratios and modes of vibration) is presented. The Euler-Bernoulli beam theory and the Timoshenko theory are used to describe the elastic and VE layers, respectively. To describe the mechanical properties of a VE layer, the four-parameter rheological model with fractional derivative is applied. A number of particular rheological models are particular cases of this general model. The virtual work principle and the finite element method together with the Laplace transform are used to derive the equation of motion in the frequency domain. The dynamic characteristics of a beam with VE layers are obtained as the solution to a properly defined nonlinear eigenvalue problem. The continuation method is adopted for solving the nonlinear eigenproblem. Several conclusions concerning the accuracy of the method and variability of results are presented on the basis of numerical studies. [ABSTRACT FROM AUTHOR]

• The subspace iteration method is used to solve nonlinear eigenvalue problem. • The continuation method is used to solve the nonlinear Hermitian eigenproblem. • The method presented allows determination of only a part of eigenvalues. • The method was tested for structures with viscoelastic elements. • The models of VE elements are described with the help of fractional derivatives. The paper presents an extension of the subspace iteration method for application in systems with viscoelastic damping elements, which are described by both classical and fractional models. The presented method enables determination of only a certain number of eigenvalues and associated eigenvectors. This is very useful, especially when the considered problem has many degrees of freedom, in which case it would be very time-consuming or even impossible to determine all the eigenvalues. At the same time, from the engineering point of view, it is not necessary to know all the eigenvalues. In this paper, the subspace iteration method is used for the first time to solve the nonlinear eigenproblems which appear in the dynamic analysis of systems with viscoelastic damping elements. The proposed solution consists of assuming the number of eigenvalues to be determined, taking the initial point of iteration and then solving the nonlinear reduced eigenproblem with Hermitian matrices in each iterative loop. The solution to the reduced eigenproblem is obtained using the continuation method, and it is an additional novelty in the paper. The correctness and effectiveness of the proposed approach are illustrated with numerical examples. [ABSTRACT FROM AUTHOR]