SOME CORNER EFFECTS ON THE LOSS OF SELFADJOINTNESS AND THE NON-EXCITATION OF VIBRATION FOR THIN PLATES AND SHELLS

Many time-dependent partial differential equations modelling mechanica l vibrations have rigid body motions or non-trivial steady states as s olutions which cannot be regarded as vibrations. For an energy-conserv ing second-order distributed parameter vibrating system such as a vibr ating 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 su ch as the thin Kirchhoff plate and the shallow circular cylindrical sh ell, the presence of corners will contribute extra static strain-energ y terms to the original energy bilinear form. We are able to find some states containing such positive strain energy which does not excite v ibrations. The collection of all such states forms a subspace of dimen sion l-3, where l is the number of corners, provided that l > 3 and th at not all of the corner points are collinear on the plane. As a conse quence, the (spatial parts of the) operators also lose their selfadjoi ntness. Such corner effects can clearly be seen from several concrete examples on rectangular domains.