Vibrating plate

The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.

There are several theories that have been developed to describe the motion of plates. The most commonly used are the Kirchhoff-Love theory and the Mindlin-Reissner theory. Solutions to the governing equations predicted by these theories can give us insight into the behavior of plate-like objects both under free and forced conditions. This includes the propagation of waves and the study of standing waves and vibration modes in plates.

The governing equations for the dynamics of a Kirchhoff-Love plate are

where ${\displaystyle u_{\alpha }}$ are the in-plane displacements of the mid-surface of the plate, ${\displaystyle w}$ is the transverse (out-of-plane) displacement of the mid-surface of the plate, ${\displaystyle q}$ is an applied transverse load, and the resultant forces and moments are defined as

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