Some corner effects on the loss of selfadjointness and the non-excitation of vibration for thin plates and shells
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Many time-dependent partial differential equations modelling mechanical vibrations have rigid body motions or non-trivial steady states as solutions which cannot be regarded as vibrations. For an energy-conserving second-order distributed parameter vibrating system such as a vibrating membrane or an elastodynamic solid, the initial states with non-zero strain energy will indeed excite vibrations. However, for elastic vibrations modelled by higher-order partial differential equations such as the thin Kirchhoff plate and the shallow circular cylindrical shell, the presence of corners will contribute extra static strain-energy terms to the original energy bilinear form. We are able to find some states containing such positive strain energy which does not excite vibrations. The collection of all such states forms a subspace of dimension l - 3, where l is the number of corners, provided that l > 3 and that not all of the corner points are collinear on the plane. As a consequence, the (spatial parts of the) operators also lose their selfadjointness. Such corner effects can clearly be seen from several concrete examples on rectangular domains.
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