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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. The plate is assumed to be isotropic while the water motion is irrotational and inviscid.
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| − | = Two Dimensional Problem =
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| − | == Equations of Motion ==
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| − | When considering a two dimensional problem, the <math>y</math> variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler beam which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the Timoshenko beam theory and Reddy-Bickford beam theory where shear deformation of higher order is considered.
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| − | For a Bernoulli-Euler beam on the surface of the water, the equation of motion 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 beam,
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| − | <math>h</math> is the thickness of the beam (assumed constant), <math> p</math> is the pressure
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| − | and <math>\eta</math> is the beam vertical displacement.
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| − | The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).
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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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| − | 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 beam (or possible
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| − | multiple beams) 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 governed by [[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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| − | == 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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| − | = Three Dimensional Problem
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