  We describe the equations of motion of an incompressible elastic body
$\Omega$ in 3-space acted on by an external pressure force, and the Newton
iteration scheme that proves the well-posedness of the resulting initial value
problem for its equations of motion on $C^{k,\alpha}$ spaces. We use the first
iterate of this Newton scheme as an approximation to the actual vibration
motion of the body, and given a (finite) triangulation $K$ of it, produce an
algorithm that computes it, employing the direct sum of the space of PL vector
fields associated to the oriented edges and faces of the first barycentric
subdivision $K'$ of $K$ (the metric duals of the Whitney forms of $K'$ in
degree one, and the metric duals of the local Hodge $*$ of the Whitney forms in
degree two, respectively) as the discretizing space. These vector fields, which
capture the algebraic topology properties of $\Omega$, encode them into the
solution of the weak version of the linearized equations of motion about a
stationary point, the essential component in the finding of the first iterate
in the alluded Newton scheme. This allows for the selection of appropriate
choices of $K$, relative to the geometry of $\Omega$, for which the algorithm
produces solutions that accurately describe the vibration of thin plates in a
computationally efficient manner. We use these to study the resonance modes of
the vibration of these plates, and carry out several relevant simulations, the
results of which are all consistent with known vibration patterns of thin
plates derived experimentally.
