# Bending - Dynamic Bending of Plates - Dynamics of Thin Kirchhoff Plates

Dynamics of Thin Kirchhoff Plates

The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. The equations that govern the dynamic bending of Kirchhoff plates are

$M_{alphabeta,alphabeta} - q(x,t) = J_1~ddot{w}^0 - J_3~ddot{w}^0_{,alphaalpha}$

where, for a plate with density ,

$J_1 := int_{-h}^h rho~dx_3 ~;~~ J_3 := int_{-h}^h x_3^2~rho~dx_3$

and

$ddot{w}^0 = frac{partial^2 w^0}{partial t^2} ~;~~ ddot{w}^0_{,alphabeta} = frac{partial^2 ddot{w}^0}{partial x_alpha partial x_beta}$

The figures below show some vibrational modes of a circular plate.

• mode k = 0, p = 1

• mode k = 0, p = 2

• mode k = 1, p = 2

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