Nonlinear Free Vibrations of Planar Elastic Beams: A Unified Treatment of Geometrical and Mechanical Effects

The exact equations of motion of a planar, initially straight, beam are determined within the large displacement framework, by considering geometrical nonlinearities and linear elastic behaviour of the material. With the aim of investigating the behaviour also for low slenderness, shear deformations and rotational inertia are taken into account, together with axial inertia. An axial linear spring is added to one end of the beam, permitting us to investigate the effect of varying boundary conditions, from the hinged-supported (stiffness=0) to the hinged-hinged (stiffness=∞) limit cases. The Poincaré-Lindstedt method is applied to obtain an approximate analytical solution. The nonlinear frequency correction ω2, responsible for the hardening vs softening nonlinear behaviour, is determined. Preliminary results on its dependence on the system parameters are illustrated.