Comparing Nonlinear Free Vibrations of Timoshenko Beams with Mechanical or Geometric Curvature Definition

The nonlinear free oscillations of a planar, initially straight Timoshenko beam are investigated by means of the asymptotic development method. Attention is focus on the difference in considering the “mechanical” vs the “geometric” curvature of the axis of the beam, which are different for extensible beams. A comparison of the results obtained by the two models is proposed, and it is shown when they are equivalent and when they give different nonlinear behaviours. A parametric analysis showing the effects of the slenderness (arbitrary, not necessarily large) and of the stiffness of the right-end axial spring is performed.