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| − | = Introduction =
| + | This page has moved to [[:Category:Floating Elastic Plate|Floating Elastic Plate]]. |
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| − | The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
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| − | physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated
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| − | more than 100 years ago and a discussion of the problem appears in [[Stoker_1957a|Stoker 1957]]. The problem can
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| − | be divided into the two and three dimensional formulations which are closely related.
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| − | = Two Dimensional Problem =
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| − | = Equations of Motion =
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| − | For a thin plate floating on the surface of the water, the motion of the plate is derived from the Classical Thin Plate Theory which was developed by Kirchhoff. The equation of motion for a thin plate (also known as Kirchhoff's equation) is given
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| − | by the following
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| − | <math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p</math>
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| − | where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the plate,
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| − | <math>h</math> is the thickness of the plate (assumed constant), <math> p</math> is the pressure
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| − | and <math>\eta</math> is the plate displacement.
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| − | The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
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| − | pressure at the surface), i.e.
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| − | <math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>
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| − | where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
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| − | is the velocity potential. The velocity potential is governed by Laplace's equation through out
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| − | the fluid domain subject to the free surface condition and the condition of no flow through the
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| − | bottom surface. If we denote the region of the fluid surface covered in the plate (or possible
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| − | multiple plates) by <math>P</math> and the free surface by <math>F</math> the equations of motion for the
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| − | [[Frequency Domain Problem]] with frequency <math>\omega</math> for water of
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| − | [[Finite Depth]] are the following. At the surface
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| − | we have the dynamic condition
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| − | <math>D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
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| − | i\omega \rho \phi, \, z=0, \, x\in P</math>
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| − | <math>0=
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| − | \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>
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| − | and the kinematic condition
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| − | <math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>
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| − | the equation within the fluid is [[Laplace's Equation]]
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| − | <math>\nabla^2\phi =0 </math>
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| − | and we have the no-flow condition through the bottom boundary
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| − | <math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>
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| − | (so we have a fluid of constant depth with the bottom surface at <math>z=-h</math> and the
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| − | free surface or plate covered surface are at <math>z=0</math>).
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| − | <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
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| − | are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
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| − | the thickness and flexural rigidity of the plate.
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| − | Finally we need to include
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| − | some boundary conditions at the edge of the plate. The most common boundary conditions
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| − | in pratical applications are that the edges are free, this means that we have the additional
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| − | conditions that
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| − | <math>\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0</math>
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| − | at the edges of the plate.
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| − | = Solution Method =
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| − | There are many different methods to solve the corresponding equations ranging from highly analytic such
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| − | as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
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| − | applicable and have advantages in different situations.
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