Title: A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points,
Abstract: We consider the bi-Laplacian eigenvalue problem for the modes of vibration of
a thin elastic plate with a discrete set of clamped points. A high-order
boundary integral equation method is developed for efficient numerical
determination of these modes in the presence of multiple localized defects for
a wide range of two-dimensional geometries. The defects result in
eigenfunctions with a weak singularity that is resolved by decomposing the
solution as a superposition of Green's functions plus a smooth regular part.
This method is applied to a variety of regular and irregular domains and two
key phenomena are observed. First, careful placement of clamping points can
entirely eliminate particular eigenvalues and suggests a strategy for
manipulating the vibrational characteristics of rigid bodies so that
undesirable frequencies are removed. Second, clamping of the plate can result
in partitioning of the domain so that vibrational modes are largely confined to
certain spatial regions. This numerical method gives a precision tool for
tuning the vibrational characteristics of thin elastic plates.