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Vibration of plates
Encyclopedia
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Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
s. 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 suggests that a two-dimensional plate theory
Plate theory
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions . The typical thickness to width ratio of a plate...
will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.
There are several theories that have been developed to describe the motion of plates. The most commonly used are the Kirchhoff-Love theory
Kirchhoff–Love plate theory
The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions...
and the
Mindlin-Reissner theory
Mindlin–Reissner plate theory
The Mindlin-Reissner theory of plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1951 by Raymond Mindlin . A similar, but not identical, theory had been proposed earlier by Eric Reissner in...
. 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.
Kirchhoff-Love plates
The governing equations for the dynamics of a Kirchhoff-Love plate are
where
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Note that the thickness of the plate is
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where the Latin indices go from 1 to 3 while the Greek indices go from 1 to 2. Summation over repeated indices is implied. The
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For a uniformly thick plate of thickness
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Isotropic plates
For an isotropic and homogeneous plate, the stress-strain relations are
where
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for Kirchhoff-Love plates are
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Therefore, the resultant moments corresponding to these stresses are
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If we ignore the in-plane displacements
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Free vibrations
For free vibrations, the governing equation of an isotropic plate is
Circular plates
For freely vibrating circular plates,
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Therefore, the governing equation for free vibrations of a circular plate of thickness
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Expanded out,
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To solve this equation we use the idea of separation of variables
Separation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
and assume a solution of the form
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Plugging this assumed solution into the governing equation gives us
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where
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The left hand side equation can be written as
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where
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appropriate for plates has the form
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where
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Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of the first kind and
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From these boundary conditions we find that
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We can solve this equation for
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For a given frequency
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of
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Rectangular plates
Consider a rectangular plate which has dimensions
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Assume a displacement field of the form
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Then,
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and
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Plugging these into the governing equation gives
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where
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From the left hand side,
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where
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Since the above equation is a biharmonic eigenvalue problem, we look for Fourier expansion
solutions of the form
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We can check and see that this solution satisfies the boundary conditions for a freely vibrating
rectangular plate with simply supported edges:
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Plugging the solution into the biharmonic equation gives us
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Comparison with the previous expression for
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number of solutions with
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Therefore the general solution for the plate equation is
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To find the values of
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we get,
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See also
- BendingBendingIn engineering mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically...
- Bending of platesBending of platesBending of plates or plate bending refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the...
- Infinitesimal strain theory
- Kirchhoff–Love plate theoryKirchhoff–Love plate theoryThe Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions...
- Linear elasticityLinear elasticityLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
- Mindlin–Reissner plate theoryMindlin–Reissner plate theoryThe Mindlin-Reissner theory of plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1951 by Raymond Mindlin . A similar, but not identical, theory had been proposed earlier by Eric Reissner in...
- Plate theoryPlate theoryIn continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions . The typical thickness to width ratio of a plate...
- Stress (mechanics)
- Structural acousticsStructural acousticsStructural acoustics is the study of the mechanical waves in structures and how they interact with and radiate into adjacent fluids. The field of structural acoustics is often referred to as vibroacoustics in Europe and Asia. People that work in the field of structural acoustics are known as...