a Kirchhoff-Love plate are where, for a plate with density , and {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equations governing the dynamics of Kirchhoff-Love plates |- | The total kinetic energy (more precisely, action of kinetic energy) of the plate is given by Therefore, the variation in kinetic energy is We use the following notation in the rest of this section. Then For a Kirchhof-Love plate Hence, Define, for constant through the thickness of the plate, Then Integrating by parts, The variations and are zero at and . Hence, after switching the sequence of integration, we have Integration by parts over the mid-surface gives Again, since the variations are zero at the beginning and the end of the time interval under consideration, we have For the dynamic case, the variation in the internal energy is given by Integration by parts and invoking zero variation at the boundary of the mid-surface gives If there is an external distributed force acting normal to the surface of the plate, the virtual external work done is From the principle of virtual work, or more precisely, Hamilton's principle for a deformable body, we have . Hence the governing balance equations for the plate are |} Solutions of these equations for some special cases can be found in the article on vibrations of plates. The figures below show some vibrational modes of a circular plate. Isotropic plates The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected. In that case we are left with one equation of the following form (in rectangular Cartesian coordinates): where is the bending stiffness of the plate. For a uniform plate of thickness , In direct notation For free vibrations, the governing equation becomes {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of dynamic governing equations for isotropic Kirchhoff-Love plates |- | For an isotropic and homogeneous plate, the stress-strain relations are where are the in-plane strains. The strain-displacement relations for Kirchhoff-Love plates are Therefore, the resultant moments corresponding to these stresses are The governing equation for an isotropic and homogeneous plate of uniform thickness in the absence of in-plane displacements is Differentiation of the expressions for the moment resultants gives us Plugging into the governing equations leads to Since the order of differentiation is irrelevant we have . Hence If the flexural stiffness of the plate is defined as we have