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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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| − | The equation for a elastic plate which is governed by Kirkoffs 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 Bernouilli 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} - \omega^2 \rho_i h \eta =
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| − | \rho g \frac{\partial \phi}{\partial z} + 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>). 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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