  The paper deals with a nonlinear evolution equation describing the dynamics
of a non homogeneous multiply hinged beam, subject to a nonlocal restoring
force of displacement type. First, a spectral analysis for the associated
weighted stationary problem is performed, providing a complete system of
eigenfunctions. Then, a linear stability analysis for bi-modal solutions of the
evolution problem is carried out, with the final goal of suggesting optimal
choices of the density and of the position of the internal hinged points in
order to improve the stability of the beam. The analysis exploits both
analytical and numerical methods; the main conclusion of the investigation is
that non homogeneous density functions improve the stability of the structure.
