Vibrating plate
Encyclopedia
In solid mechanics
, a branch of mathematics
and physics
, a plate is 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). A vibrating plate can be modeled in a manner analogous to a vibrating drum
. However, the resulting partial differential equation
for the vertical displacement u of the plate from its equilibrium position
is fourth order, involving the square of the Laplacian of u, rather than second order,
and its qualitative behavior is fundamentally different from that of the drum.
The potential energy density of a plate depends how the plate is deformed, and so on the mean curvature
and Gaussian curvature
of the plate. For small deformations, the mean curvature is expressed in terms of u, the vertical displacement of the plate from kinetic equilibrium, as Δu, the Laplacian of u, and the Gaussian curvature is the Monge–Ampère operator uxxuyy−u. 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 u is a stationary point with respect to variations
of the total energy T+U. The resulting partial differential equation is
Solid mechanics
Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a plate is 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). A vibrating plate can be modeled in a manner analogous to a vibrating drum
Vibrations of a circular drum
The vibrations of an idealized circular drum, essentially an elastic membrane of uniform thickness attached to a rigid circular frame, are solutions of the wave equation with zero boundary conditions....
. However, the resulting partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
for the vertical displacement u of the plate from its equilibrium position
is fourth order, involving the square of the Laplacian of u, rather than second order,
and its qualitative behavior is fundamentally different from that of the drum.
The potential energy density of a plate depends how the plate is deformed, and so on the mean curvature
Mean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....
and Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
of the plate. For small deformations, the mean curvature is expressed in terms of u, the vertical displacement of the plate from kinetic equilibrium, as Δu, the Laplacian of u, and the Gaussian curvature is the Monge–Ampère operator uxxuyy−u. 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
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
asserts that u is a stationary point with respect to variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
of the total energy T+U. The resulting partial differential equation is