Vibration of plates

This permits a two-dimensional plate theory to give an excellent approximation to the actual three-dimensional motion of a plate-like object.

[1] There are several theories that have been developed to describe the motion of plates.

[5] 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.

The topic of plate vibrations is treated in books by Leissa,[6][7] Gontkevich,[8] Rao,[9] Soedel,[10] Yu,[11] Gorman[12][13] and Rao.

[14] The governing equations for the dynamics of a Kirchhoff-Love plate are where

is the transverse (out-of-plane) displacement of the mid-surface of the plate,

(upwards), and the resultant forces and moments are defined as Note that the thickness of the plate is

and that the resultants are defined as weighted averages of the in-plane stresses

For an isotropic and homogeneous plate, the stress-strain relations are where

The strain-displacement relations for Kirchhoff-Love plates are Therefore, the resultant moments corresponding to these stresses are If we ignore the in-plane displacements

, the governing equations reduce to The above equation can also be written in an alternative notation: In solid mechanics, a plate is often modeled as a two-dimensional elastic body whose potential energy depends on how it is bent from a planar configuration, rather than how it is stretched (which is the instead the case for a membrane such as a drumhead).

However, the resulting partial differential equation for the vertical displacement w of a plate from its equilibrium position is fourth order, involving the square of the Laplacian of w, rather than second order, and its qualitative behavior is fundamentally different from that of the circular membrane drum.

For free vibrations, the external force q is zero, and the governing equation of an isotropic plate reduces to or This relation can be derived in an alternative manner by considering the curvature of the plate.

For small deformations, the mean curvature is expressed in terms of w, the vertical displacement of the plate from kinetic equilibrium, as Δw, the Laplacian of w, and the Gaussian curvature is the Monge–Ampère operator wxxwyy−w2xy.

The total potential energy of a plate Ω therefore has the form apart from an overall inessential normalization constant.

Here μ is a constant depending on the properties of the material.

The kinetic energy is given by an integral of the form Hamilton's principle asserts that w is a stationary point with respect to variations of the total energy T+U.

The resulting partial differential equation is For freely vibrating circular plates,

, and the Laplacian in cylindrical coordinates has the form Therefore, the governing equation for free vibrations of a circular plate of thickness

is Expanded out, To solve this equation we use the idea of separation of variables and assume a solution of the form Plugging this assumed solution into the governing equation gives us where

The general solution of this eigenvalue problem that is appropriate for plates has the form where

is the order 0 modified Bessel function of the first kind.

(and there are an infinite number of roots) and from that find the modal frequencies

the first term inside the sum in the above equation gives the mode shape.

from the initial conditions by taking advantage of the orthogonality of Fourier components.

We seek to find the free vibration modes of the plate.

Assume a displacement field of the form Then, and Plugging these into the governing equation gives where

is a constant because the left hand side is independent of

From the right hand side, we then have From the left hand side, where Since the above equation is a biharmonic eigenvalue problem, we look for Fourier expansion solutions of the form We can check and see that this solution satisfies the boundary conditions for a freely vibrating rectangular plate with simply supported edges: Plugging the solution into the biharmonic equation gives us Comparison with the previous expression for

we use initial conditions and the orthogonality of Fourier components.