Vibrations of thin plates with small clamped patches

Eigenvalues of fourth-order elliptic operators feature prominently in stability analysis of elastic structures. This paper considers out-of-plane modes of vibration of a thin elastic plate perforated by a collection of small clamped patches. As the radius of each patch shrinks to zero, a point constraint eigenvalue problem is derived in which each patch is replaced by a homogeneous Dirichlet condition at its centre. The limiting problem is consequently not the eigenvalue problem with no patches, but a new type of spectral problem. The discrepancy between the eigenvalues of the patch-free and point constraint problems is calculated. The dependence of the point constraint eigenvalues on the location(s) of clamping is studied numerically using techniques from numerical algebraic geometry. The vibrational frequencies are found to depend very sensitively on the number and centre(s) of the clamped patches. For a range of number of punctures, we find spatial clamping patterns that correspond to local maxima of the base vibrational frequency of the plate.