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The problem of non-linear, steady state vibration of beams, harmonically excited by harmonic forces is investigated in the paper. The viscoelastic material of the beams is described using the Zener rheological model with fractional derivatives. The constitutive equation, which contains derivatives of both stress and strain, significantly complicates the solution to the problem. The von Karman theory is applied to take into account geometric nonlinearities. Amplitude equations are obtained using the finite element method together with the harmonic balance method, and solved using the continuation method. The tangent matrix of the amplitude equations is determined in an explicit form. The stability of the steady-state solution is also examined. A parametric study is carried out to determine the influence of viscoelastic properties of the material on the beam's responses. [ABSTRACT FROM AUTHOR]

Abstract: Frame structures with viscoelastic dampers mounted on them are considered in this paper. Viscoelastic (VE) dampers are modelled using two, three-parameter, fractional rheological models. The structures are treated as elastic linear systems. The equation of motion of the whole system (structure with dampers) is written in terms of state-space variables. The resulting matrix equation of motion is the fractional differential equation. The proposed state space formulation is new and does not require matrices with huge dimensions. The paper is devoted to determine the dynamic properties of the considered structures. The nonlinear eigenvalue problem is formulated from which the dynamic parameters of the system can be determined. The continuation method is used to solve the nonlinear eigenvalue problem. Moreover, results of typical calculations are presented. [ABSTRACT FROM AUTHOR]