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• Efficient solution procedure for non-linear eigen-value problem based on homotopy and quasi-Newton methods. • Numerical verification of the proposed formulation and testing of various solution variants. • Extensive analyses of results sensitivity to the model parameters. In this paper the numerical procedure to solve the non-linear eigen-value problem describing the free vibrations of the laminate plates with viscoelastic (VE) layers is presented. The derivation of that main equation based on the refined zig-zag theory and using fractional Zener material model to describe VE material is presented in the companion paper Lewandowski R, Litewka P. Wielentejczyk P. Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 1. Theoretical background. The solution obtained using the continuation method provides the required dynamic characteristics of the plates – the frequency and the damping ratio. In the iterative part of each continuation step either the full Newton or the quasi-Newton iterative procedure are used. Several numerical examples are solved to verify accuracy, efficiency and versatility of the proposed formulation. Numerous parametric studies are carried out to determine the influence of various solution variants and combinations of material parameters onto the results describing the damping properties of laminate plates. [ABSTRACT FROM AUTHOR]

• Pioneer application of refined zig-zag theory in dynamic analysis of laminate plates. • Separate fractional Zener material description for volumetric and deviatoric strains. • Derivation of efficient finite element formulation leading to the non-linear eigen-value problem. The refined zig-zag theory is used to solve the dynamic problems for composite sandwich plates made from the elastic and the viscoelastic (VE) layers. The linear constitutive relations are used to model physical properties of the VE layers separately for volumetric and deviatoric strains. The fractional Zener model is used in the formulation of VE material. The Laplace transformation and the finite element method are used to derive the discretized nonlinear eigenvalue problem. The required dynamic characteristics of the plates – the frequency and the damping ratio, fundamental in design of structural elements in many fields of engineering, can be obtained using the procedure described in the companion paper Litewka P. Lewandowski R, Wielentejczyk P. Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 2. Numerical analysis. [ABSTRACT FROM AUTHOR]

• The refined zig-zag theory is used to solve the dynamic problem for composite beams with viscoelastic layers. • The fractional Zener model is used in analysis. • The dynamic characteristics of multi-layered viscoelastic beams are of interest. • The hierarchical FEM and the Laplace transform are used to derive the nonlinear eigenvalue problem. • The continuation method is adopted to solve the nonlinear eigenvalue problem. In the present paper the displacement based refined zig-zag theory is used, for the first time, to solve dynamic problems of composite and sandwich beams made of both elastic and VE layers. The linear constitutive relations are used to model physical properties of the VE layers. The rheological models can be in the classical or fractional versions, but the main focus is on the fractional Zener model. The dynamic characteristics of considered beams are evaluated. The hierarchical finite element and the Laplace transformation are used to derive the nonlinear eigenvalue problem and its solutions yield the required dynamic characteristics of beams. The continuation method is adopted to solve this problem numerically. Accuracy, effectiveness and efficiency of the proposed method are verified in several examples. The results of the presented calculations are summarized with some conclusions. [ABSTRACT FROM AUTHOR]