NORMAL-MODES FOR LARGE-AMPLITUDE VIBRATION OF A CANTILEVER BEAM

We consider the non-linear normal modes for finite amplitude vibrations of a cantilever beam by using an extension of the invariant manifold techniques recently developed by the authors. Detailed results from two examples are presented. The first considers non-linearities which arise from finite deformations, including inertial non-linearities. In this case it is shown that if one simply accounts for the kinematic effects of foreshortening, the linear modes of the cantilever beam work remarkably well for describing the actual non-linear mode shapes of the beam up to quite large amplitudes. A second example demonstrates the manner in which the method handles non-linearities in the boundary conditions, in this case, a non-linear spring attached at the free end of the beam.