Analytical and numerical study of the influence of different support types in the nonlinear vibrations of beams

This paper aims at studying the influence of the different support types on the free nonlinear vibration frequency of a beam, both by analytical and numerical approaches and its dependence on the vibration amplitude. First, an analytical study of two cases of simply supported beams is carried out by using the nonlinear normal modes (NNM) and the multiple scale methods to obtain the analytical expressions of the nonlinear frequencies as a function of the vibration amplitude. Such results have been compared and discussed with other authors’ results. In addition, a nonlinear finite element model of these two cases showed that the analytical and numerical results are in good agreement. Additionally, several numerical studies have been carried out for different support types. A total of seven different boundary conditions have been numerically analysed and the corresponding frequency-amplitude relations have been obtained and compared. In addition, the effect of the axial inertial forces has been enhanced by adding concentrated masses at one end of the beam. It was found for instance that the hinged-hinged beam shows a hardening behaviour, which depends on the beam slenderness, or that the hinged-simply supported beam results in the largest softening behaviour. The fundamental causes of nonlinearity are, on one hand the coupling of the midline deformation and bending and, on the other hand, the coupling of axial inertial forces and bending. The significance of the first of these effects depends on the beam material and geometrical properties.