Category:Floating Elastic Plate

Introduction

The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of physical structures such as a floating break water, an ice floe or a VLFS). The equations of motion were formulated more than 100 years ago and a discussion of the problem appears in Stoker 1957. The problem can 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.

The solution methods are divided up into those for Two-Dimensional Floating Elastic Plate and those for a Three-Dimensional Floating Elastic Plate.

Linear Thin Elastic Plate Theory

We present here the theory of an elastic plate in a vacuum, concentrating of the two-dimensional problem.

There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler Beam theory (other beam theories include the Timoshenko Beam theory and Reddy-Bickford Beam theory where shear deformation of higher order is considered). For a Bernoulli-Euler Beam, the equation of motion is given by the following

[math]\displaystyle{ \partial_x^2\left(\beta(x)\partial_x^2 \zeta\right) + \gamma(x) \partial_t^2 \zeta = p }[/math]

where [math]\displaystyle{ \beta(x) }[/math] is the non dimensionalised flexural rigidity, and [math]\displaystyle{ \gamma }[/math] is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that [math]\displaystyle{ h }[/math] is the thickness of the plate, [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \zeta }[/math] is the plate vertical displacement) .

The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).

[math]\displaystyle{ \partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0 }[/math]

at the edges of the plate.

The problem is subject to the initial conditions

[math]\displaystyle{ \zeta(x,0)=f(x) \,\! }[/math]

[math]\displaystyle{ \partial_t \zeta(x,0)=g(x) }[/math]

Linear Elastic Plate on Water

We begin with the linear equations for a fluid. The kinematic condition is the same

[math]\displaystyle{ \frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; }[/math]

but the dynamic condition needs to be modified to include the effect of the the plate

[math]\displaystyle{ \rho g\zeta + \rho \frac{\partial\Phi}{\partial t} = D \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} , \ z=0; }[/math]

We also have Laplace's equation

[math]\displaystyle{ \Delta \Phi = 0,\,\,-h\lt z\lt 0 }[/math]

and the usual non-flow condition at the bottom surface

[math]\displaystyle{ \partial_z \Phi = 0,\,\,z=-h, }[/math]

where [math]\displaystyle{ \zeta }[/math] is the surface displacement, [math]\displaystyle{ \Phi }[/math] is the velocity potential, and [math]\displaystyle{ \rho }[/math] is the fluid density.

Frequency Domain Equations

If we make the assumption of Frequency Domain Problem that everything is proportional to

[math]\displaystyle{ \exp (i\omega t)\, }[/math]

the equations become

[math]\displaystyle{ i\omega\zeta = \partial_z\phi , \ z=0; }[/math] [math]\displaystyle{ \rho g\zeta + i\omega\rho \phi = D \partial_x^4 \eta -\omega^2 \rho_i h \zeta, \ z=0; }[/math] [math]\displaystyle{ \Delta \phi = 0,\,\,-h\lt z\lt 0 }[/math] [math]\displaystyle{ \partial_z \phi = 0,\,\,z=-h, }[/math]

where [math]\displaystyle{ \zeta }[/math] is the surface displacement and [math]\displaystyle{ \phi }[/math] is the velocity potential in the frequency domain.

These equations can be simplified by defining [math]\displaystyle{ \alpha = \omega^2/g }[/math], [math]\displaystyle{ \beta = D/\rho g }[/math] and [math]\displaystyle{ \gamma = \rho_i h/\rho }[/math] to obtain

[math]\displaystyle{ \Delta \phi = 0, \;\;\; -h \lt z \leq 0, }[/math] [math]\displaystyle{ \partial_z \phi = 0, \;\;\; z = - h, }[/math] [math]\displaystyle{ \beta \partial_x^4\partial_z \phi - \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\; z = 0, \;\;\; -L \leq x \leq L, }[/math]

We consider the problem of small-amplitude waves which are incident on finite floating elastic plate occupying water surface for [math]\displaystyle{ -L\lt x\lt L }[/math]. These equations are derived in Floating Elastic Plate The submergence of the plate is considered negligible. We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction. We also assume that the plate edges are free to move at each boundary, although other boundary conditions could easily be considered using the methods of solution presented here. We begin with the Frequency Domain Problem for a semi-infinite Floating Elastic Plates in the non-dimensional form of Tayler 1986 (Dispersion Relation for a Floating Elastic Plate). We also assume that the waves are normally incident (incidence at an angle will be discussed later).

[math]\displaystyle{ \Delta \phi = 0, \;\;\; -h \lt z \leq 0, }[/math] [math]\displaystyle{ \partial_z \phi = 0, \;\;\; z = - h, }[/math] [math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt -L,\,\,{\rm or}\,\,x\gt L }[/math] [math]\displaystyle{ \partial_x^2\left\{\beta(x) \partial_x^2\partial_z \phi\right\} - \left( \gamma(x)\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\; z = 0, \;\;\; -L \leq x \leq L, }[/math]

where [math]\displaystyle{ \alpha = \omega^2 }[/math], [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] are the stiffness and mass constant for the plate respectively. The free edge conditions at the edge of the plate imply

[math]\displaystyle{ \partial_x^3 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math] [math]\displaystyle{ \partial_x^2 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math]

Nonlinear Elastic Thin Plate on Water

The nonlinear plate equations can be found by include We begin with the nonlinear equations for a fluid. The kinematic condition is

[math]\displaystyle{ \frac{\partial\zeta}{\partial t}+\frac{\partial\Phi}{\partial x} \frac{\partial\zeta}{\partial x} =\frac{\partial\Phi}{\partial z}, \ z=\zeta(x,t) }[/math]

the dynamic condition is

[math]\displaystyle{ \partial_x^2 \left( \frac{ D \partial_x^2 \zeta}{(1+(\partial_x\zeta^2))^{3/2}} \right) + \rho_i h \frac{\partial^2 \zeta}{\partial t^2} = - \rho\frac{\partial\Phi}{\partial t}-\rho\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - \rho g \zeta, \ z=\zeta(x,t) }[/math]

There are different versions of the nonlinear boundary condition for a plate and this one is based on Parau and Dias 2002. We also have Laplace's equation

[math]\displaystyle{ \Delta \Phi = 0,\,\,-h\lt z\lt \zeta, }[/math]

and the usual non-flow condition at the bottom surface

[math]\displaystyle{ \partial_z \Phi = 0,\,\,z=-h. }[/math]