Geometrically nonlinear, steady state vibration of viscoelastic beams. (March 2017) - Wielentejczyk, Przemysław; Lewandowski, Roman
Abstract: The problem of geometrically non-linear steady state vibrations of beams excited by harmonic forces is considered in this paper. The beams are made of a viscoelastic material defined by the classic 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 the solution to the problem. The von Karman theory is applied to describe the effects of geometric nonlinearities of beam deformations. The equations of motions are derived using the finite element methodology. A polynomial approximation of bending moments is used. The order of basis functions is set so as to obtain a 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 and the continuation method is applied for solving them. The tangent matrix of equations of amplitudes is determined in an explicit form. The stability of steady-state solution is also examined. The resonance curves for beams supported in a different way are shown and the results of calculation are briefly discussed. Highlights: Nonlinear steady-state vibration of beams made of viscoelastic Zener material is investigated. The HBM is used for the system of which the stress-strain relationship is given as a differential equation containing derivatives of both stress and strain. Some an unexpected behavior of the beams is found. The stability of the steady-state solution is also examined.