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• Harmonic vibrations of VE laminate plates with exponential complex-conjugate solution form and harmonic balance method. • The use of refined zig-zag theory for the description of layered plate kinematics. • Application of fractional Zener material model with separation of volumetric and deviatoric strain. The paper is devoted to the theoretical formulation of the harmonic vibrations problem for laminate plates in the von Kàrmàn geometrically non-linear regime. The plate layers are modelled as viscoelastic using the fractional Zener material with the linear elastic material being a special case. The model of the material is formulated with the separation of the volumetric and deviatoric strain. The laminate plate kinematics is described using the refined zig-zag theory. The vibrations of the plate are formulated using the alternative exponential form with complex-conjugate amplitudes. The harmonic balance method and the time-averaging are applied to derive the amplitude equation of the problem in hand. The efficiency and correctness of the formulation is verified in the continuation paper Part 2 – numerical analysis. [ABSTRACT FROM AUTHOR]

• FE solution of harmonic vibrations for VE layered plates with exponential complex-conjugate form and refined zig-zag theory. • Effective application of the continuation method to solve the discretised amplitude equation. • Numerous analyses of the influence of various plate and formulation parameters on the resonance behaviour of laminate plates. The paper is devoted to the numerical analysis of the harmonic vibrations of laminate plates with viscoelastic layers in the von Kàrmàn geometrically non-linear regime. The amplitude equation derived in the prequel paper Part 1 – theoretical background is discretized using the 8-noded bi-quadratic plate finite elements. The response curves are obtained using the continuation method to solve the discretised amplitude equation treated as the equation with the frequency of vibrations as a parameter. The proposed method of solution is verified by comparing the results with those from available literature. Furthermore, several numerical analyses are performed, including testing of FE mesh density, plate aspect ratio, various VE materials, support conditions and layers layout as well as the possibility of simplification of the complex-valued zig-zag function used in the description of plate kinematics. Several results of practical importance are found and discussed in the conclusions. [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]

• 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]