GEOMETRICALLY NONLINEAR, STEADY STATE VIBRATION OF VISCOELASTIC BEAMS BY FEM

Abstract eng:
Analysis of nonlinear vibrations of viscoelastic systems was devoted in papers [1-3]. In these studies, the Kelvin model was applied to describe the viscoelastic material. Damping properties were written down by ordinary or fractional derivatives. In the present paper, the problem of geometrically non-linear steady state vibrations of beams excited by harmonic forces is considered. Beams are made of a viscoelastic material defined by the classical Zener rheological model; the simplest model that takes into account all the basic properties of real viscoelastic materials. The constitutive stress-strain relationship for this type of material is given as a differential equation containing derivatives of both stress and strain. This significantly complicates a solution of the problem. Till now the problem of nonlinear vibrations of beams made of a viscoelastic material defined by the Zener model has not been analysed. To describe effects of geometric nonlinearities of beam deformations the von Karman theory is applied. The equations of motions are derived using the finite element methodology. A polynomial approximation of the bending moments is used. The order of basis functions is set so as to get the coherent approximation of moments and displacements. In the steady state solution of equations of motion only one harmonic is taken into account. The matrix equations of amplitudes are derived using the harmonic balance method. To solve them the Newton-Raphson method is applied. The tangent matrix of equations of amplitudes is determined. The resonance curves for beams supported in a different way are shown and results of calculation are briefly discussed.

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