  We study the spectrum of non-homogeneous partially hinged plates having
structural engineering applications. A possible way to prevent instability
phenomena is to maximize the ratio between the frequencies of certain
oscillating modes with respect to the density function of the plate; we prove
existence of optimal densities and we investigate their analytic expression.
This analysis suggests where to locate reinforcing material within the plate;
some numerical experiments give further information and support the theoretical
results.
