Frequency domain analysis of geometrically nonlinear vibrations of 3D beams

Abstract

A beam, p-version, hierarchical finite element is employed to study the dynamic behaviour of beams undergoing longitudinal, torsional, and bending free vibrations in any plane. Clamped-clamped, isotropic and elastic beams of circular cross section are analysed. The geometrical non-linearity is taken into account by considering a simplified version of Greens strain tensor. The mass and stiffness matrices are derived by the principle of the virtual work, and the harmonic balance method is employed to map the equations of motion to the frequency domain. The ensuing algebraic non-linear system is solved by a continuation method. Assuming a Fourier series where the constant term and the first three harmonics are considered, one concludes that internal resonances appear in bending and torsion.