The vibrations of thin plates

Abstract

We describe the equations of motion of an incompressible elastic body $Ω$ 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 $C^{k,α}$ 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 $Ω$, 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 $Ω$, 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.