  We consider the nonlinear nonlocal beam evolution equation introduced by
Woinowsky- Krieger. We study the existence and behavior of periodic solutions:
these are called nonlinear modes. Some solutions only have two active modes and
we investigate whether there is an energy transfer between them. The answer
depends on the geometry of the energy function which, in turn, depends on the
amount of compression compared to the spatial frequencies of the involved
modes. Our results are complemented with numerical experiments, overall, they
give a complete picture of the instabilities that may occur in the beam. We
expect these results to hold also in more complicated dynamical system
