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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]