Non-linear vibration of beams with internal resonance by the hierarchical finite-element method

The hierarchical finite-element (HFEM) and the harmonic balance methods (HBM) are used to investigate the geometrically non-linear free and steady-state forced vibrations of uniform, slender beams. The beam analogue of von Karman's non-linear strain-displacement relationships are employed and the middle plane in-plane displacements are included in the model. The equations of motion are developed by applying the principle of virtual work and are solved by a continuation method, 1:3 and 1:5 internal resonances are discovered and their consequences are discussed. The convergence properties of the HFEM are analyzed and it is demonstrated that the HFEM model requires far fewer degrees of freedom than the h-version of the FEM models presented in the Literature. (C) 1999 Academic Press.