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Planar frame structures made of a viscoelastic material are considered in the paper. The technically very important structures made of a homogenous material are contemplated. A family of rheological models (classic and fractional) are used to describe the mechanical properties of the viscoelastic material. In particular, the dynamic characteristics of the structures are of interest. A numerically very efficient method is proposed to determine such characteristics. The method requires the solution to the linear eigenvalue problem for corresponding elastic structures and the solution to a nonlinear, algebraic equation. The presented method is much more efficient than other methods where, very often, the continuation method is used to solve the nonlinear eigenvalue problem. The influence of temperature changes on dynamic characteristics is analyzed using the frequency–temperature principle. The results of several parametric analyses are presented and discussed. For the first time, the influence of temperature on the dynamic characteristics of beams has been studied in detail. [ABSTRACT FROM AUTHOR]

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]

• 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]

• 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]