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Plates
The equations of motion of a stiff plate are the (2+1)D generalization of those of a beam. We assume the plate to lie, when at rest, in the plane, and to be of thickness ; the deflection of the plate from its equilibrium state is assumed to be perpendicular to the plane. The plate material has density , as well as Young's modulus and Poisson's ratio , all of which are assumed, for the sake of generality, to be smooth positive functions of and . In particular, must be less than onehalf. The classical development depends on neglecting rotational inertia effects and makes various assumptions analogous to the ``plane sections remain plane and perpendicular to the neutral axis'' hypothesis that was used as the basis for the EulerBernoulli beam model [77]. The resulting equation of motion [6,113] can be written as
where
and used
. If the material parameters and the thickness are constant, then

(5.31) 
which is easily seen to be a direct generalization of (5.2). As such, we expect to find the same anomalous behavior of the resulting propagation velocities, which can become infinitely large in the highfrequency limit. Numerical integration of these equations via a waveguide mesh proceeds along exactly the same lines as in the case of the EulerBernoulli beam; in particular, we find a restriction on the space step/time step ratio similar to those that resulted in §5.1.2.
Because the development is so similar to the (1+1)D case, we will proceed directly to the more refined model of plate motion, which is a direct generalization of the Timoshenko theory for beams. First proposed by Mindlin, the model [77,120], can be written as system of eight PDEs [173]:

(5.32a) 


(5.33a) 

Here, we have written
where is the transverse displacement of the plate, and
is the pair of angles giving the orientation of the sides of a deformed differential element of the plate with respect to the perpendicular. (In the classical theory, for which crosssections of the plate are assumed to remain parallel to the plate normal, we have
.) In addition, we have the shear forces
and moments
, which are the (2+1)D generalizations of and . The system (5.31)(5.32) as a whole is known as Mindlin's system, although it is more commonly written s a system of three secondorder equations in the variables , and [77]. We have written Mindlin's system so that it is easy to see the decomposition into two separate subsystems, one in
and the other in
, with the coupling occurring via constantproportional terms in
,
, and . In particular, subsystem (5.31) is similar to the lossless parallelplate system (see §4.4), except for the coupling terms.
It is easy to see that this system is not, as written, symmetric hyperbolic. It is easy to symmetrize it by taking sums and differences of (5.32c) and (5.32d), in which case we get, in terms of the variable
,
where the stands for zero entries, and

(5.36) 

(5.37) 
The system defined by (5.33) is lossless, due to the antisymmetry of
. Also, note that
is positive definite (recall that is positive, and less than onehalf), but not diagonal^{}; this did not come up in any of the systems we have looked at previously, and will have interesting consequences in the circuit representations in the next section.
Subsections
Next: Maximum Group Velocity
Up: Applications in Vibrational Mechanics
Previous: Longitudinal and Torsional Waves
Stefan Bilbao
20020122