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 permits a two-dimensional plate theory to give an excellent approximation to the actual three-dimensional motion of a plate-like object. 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 Uflyand-Mindlin. The latter theory is discussed in detail by Elishakoff. 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 topic of plate vibrations is treated in books by Leissa, Gontkevich, Rao, Soedel, Yu, Gorman and Rao.
Kirchhoff-Love plates
The governing equations for the dynamics of a Kirchhoff-Love plate are
N α β , β = J 1 u ¨ α M α β , α β + q ( x , t ) = J 1 w ¨ − J 3 w ¨ , α α {\displaystyle {\begin{aligned}N_{\alpha \beta ,\beta }&=J_{1}~{\ddot {u}}_{\alpha }\\M_{\alpha \beta ,\alpha \beta }+q(x,t)&=J_{1}~{\ddot {w}}-J_{3}~{\ddot {w}}_{,\alpha \alpha }\end{aligned}}}
where u α {\displaystyle u_{\alpha }} are the in-plane displacements of the mid-surface of the plate, w {\displaystyle w} is the transverse (out-of-plane) displacement of the mid-surface of the plate, q {\displaystyle q} is an applied transverse load pointing to x 3 {\displaystyle x_{3}} (upwards), and the resultant forces and moments are defined as
N α β := ∫ − h h σ α β d x 3 and M α β := ∫ − h h x 3 σ α β d x 3 . {\displaystyle N_{\alpha \beta }:=\int _{-h}^{h}\sigma _{\alpha \beta }~dx_{3}\quad {\text{and}}\quad M_{\alpha \beta }:=\int _{-h}^{h}x_{3}~\sigma _{\alpha \beta }~dx_{3}\,.}
Note that the thickness of the plate is 2 h {\displaystyle 2h} and that the resultants are defined as weighted averages of the in-plane stresses σ α β {\displaystyle \sigma _{\alpha \beta }} . The derivatives in the governing equations are defined as
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