View source for Linear Wave-Body Interaction

{{Ocean Wave Interaction with Ships and Offshore Structures
 | chapter title = Linear Wave-Body Interaction
 | next chapter = [[Added-Mass, Damping Coefficients And Exciting Forces]]
 | previous chapter =  [[Ship Kelvin Wake]]
}}

{{incomplete pages}}

[[Image:Rigid_body.jpg|thumb|right|600px|Rigid body motions]]

We consider a [[Linear Plane Progressive Regular Waves|Linear Plane Progressive Regular Wave]] interacting with a floating body in two dimensions (the main concepts survive almost with no change in the more practical three-dimensional problem).

== Introduction ==

We derive here the equations of motion for a body in [[Linear Plane Progressive Regular Waves]] in the frequency domain in 
two dimensions. We begin with the equations for a fixed body and then consider the equations for a rigid body.

== Frequency Domain Equations ==

{{standard linear problem notation}}
[[Variable Bottom Topography]]
can also easily be included.

{{standard linear wave scattering equations}}

The simplest case is for a fixed body 
where the operator is <math>L=0</math> and we will consider this boundary condition
first before moving on to floating bodies.

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Equations for a Fixed Body in Frequency Domain ==

The equations for a fixed body are

{{standard linear wave scattering equations without body condition}}
{{frequency domain equations for a rigid body}}
plus the radiation conditions.  

We decompose the potential as
<center>
<math>
\phi = \phi^{\mathrm{I}} + \phi^{\mathrm{D}}
</math>
</center>
where <math>\phi^{\mathrm{I}}</math> is the incident potential and <math>\phi^{\mathrm{D}}</math>
is the diffracted potential.  The boundary condition for the diffracted potential is
<center><math>
\Delta\phi^{\mathrm{D}}=0, \, -h<z<0,\,\,\,\mathbf{x} \in \Omega
</math></center>
<center><math>
\partial_n\phi^{\mathrm{D}} = 0, \, z=-h,
</math></center>
<center><math>
 \partial_n \phi^{\mathrm{D}}  = \alpha \phi,\,z=0,\,\,\mathbf{x} \in F,
</math></center>
plus 
<center><math>
 \partial_n \phi^{\mathrm{D}}  = - \partial_n \phi^{\mathrm{I}},\,\, \mathbf{x} \in \partial\Omega,
</math></center>

Code to calculate the solution (using a slighly modified method) can be found 
[[Boundary Element Method for the Radiation Potential in Finite Depth]]

== Equations for the Radiation Potential in Frequency Domain ==

We decompose the body motion into the rigid body modes of motion. Associated with
each of these modes is a potential which must be solved for 
The equations for a floating body are

{{standard linear wave scattering equations without body condition}}
{{frequency domain equations for the radiation modes}}

{{sommerfeld radiation condition two dimensions for radiation}}

We denote the solution for each of the radiation potentials by
<math>\phi_m^{mathbf{R}}</math> and the total potential is written as
<center><math>
\phi = \phi_
</math>
</center

== Calculation of the RAO ==

The equations of motion for <math> \xi_K(t)\,</math> follow from Newton's law applied to each mode in two dimensions. The same principles apply with very minor changes in three dimensions

We begin by considering the equation in surge. 
<center><math> \mathbf{M} \frac{\mathrm{d}^2\xi_1}{\mathrm{d}t^2} = F_{1\omega} ( \xi_1, \dot\xi_1, \ddot\xi_1, t) </math></center>
where <math> \frac{\mathrm{d}\xi_1}{\mathrm{d}t} = \dot\xi_1 \, </math> and <math> F_{1\omega} \, </math> is the force on the body due to the fluid pressures, by virtue of linearity, <math> F_{1\omega} \,</math> will be assumed to be a linear functional of <math> \xi_1, \dot\xi_1, \ddot\xi_1 \, </math>. [[Memory effects]] exist when surface waves are generated on the free surface, so <math> F_{1\omega} \,</math> depends in principle on the entire history of the vessel displacement. We adopt here the [[Frequency Domain Problem|frequency domain]] formulation where the vessel motion has been going on over an infinite time interval, <math> (-\infty, t)\,</math> with <math> e^{i\omega t}\,</math> dependence.

We will therefore set:
<center><math> \xi_K(t) = \mathrm{Re} \left\{ \Pi_K e^{i\omega t} \right\}, \qquad K=1,3,4 </math></center>
In this case we can linearize the water induced force on the body as follows:
<center><math> F_{1\omega}(t) = X_1(t) - A_{11} \ddot \xi_1 - A_{13} \ddot \xi_3 - A_{14} \ddot \xi_4 </math></center>

<center><math> - B_{11} \dot \xi_1 - B_{13} \dot \xi_3 - B_{14} \dot \xi_4 </math></center>

<center><math> - C_{11} \xi_1 - C_{13} \xi_3 - C_{14} \xi_4 \,</math></center>

<center><math> = X_1(t) - \sum_j \left[ A_{1j} \ddot \xi_j + B_{1j} \dot \xi_j + C_{1j} \xi_j \right] </math></center>

The same expansion applies for other modes, namely Heave (<math> K = 3 \, </math>) and Roll (<math> K=4 \, </math>). In sum:

<center><math> F_{K\omega} (t) = X_K - \sum_j \left[ A_{Kj} \ddot \xi_j + B_{Kj} \dot \xi_j + C_{Kj} \xi_j \right], \qquad K = 1,3,4 </math></center>

* The added-mass matrix <math> A_{Kj} \,</math> represents the added inertia due to the acceleration of the body in water with acceleration <math>\ddot\xi_j\,</math>.

* The damping matrix <math> B_{Kj}\,</math> governs the energy dissipation into the fluid domain in the form of surface waves.

* The hydrostatic restoring matrix <math> C_{Kj} \, </math> represents the system stifness due to the hydrostatic restoring forces and moments.

For harmonic motions, the matrices <math> A_{Kj} \, </math> and <math> B_{Kj} \, </math> are functions of <math> \omega\,</math>, so we write <math> A_{Kj} (\omega), \ B_{Kj} (\omega)\, </math>. This functional form will be discussed below. The hydrostatic matrix <math> C_{Kj} \, </math> is independent of <math>\omega\,</math> and many of its elements are identically equal to zero. Collecting terms in the left-hand side and denoting by <math> M_{Kj}\,</math> the body inertia matrix:

=== Surge ===

<center><math> \sum_j \left[ -\omega^2 \left( M_{1j} + A_{1j} \right) + i\omega B_{1j} + C_{1j} \right] \Pi_j = \mathbf{X}_1 (\omega), \quad j=1,3,4 </math></center>

=== Heave ===

<center><math> \sum_j \left[ -\omega^2 \left( M_{3j} + A_{3j} \right) + i\omega B_{3j} + C_{3j} \right] \Pi_j = \mathbf{X}_3 (\omega), \quad j=1,3,4 </math></center>

=== Roll ===

<center><math> \sum_j \left[ -\omega^2 \left( I_G + A_{4j} \right) + i\omega B_{4j} + C_{4j} \right] \Pi_j = \mathbf{X}_4 (\omega)</math></center>

The extension of these equations to six degrees of freedom is straightforward. However before discussing the general case we will study specific properties of the 2D Problem for the sake of clarity.

== Symmetric body ==

Consider a body symmetric about the <math> X = 0\,</math> axis.

[[Image:Symmetric.jpg|thumb|right|600px|Symmetric body]]

For a body symmetric port/starboard:

* Verify that Heave is decoupled from Surge and Roll. In other words the Surge and Roll motions do not influence Heave and vice versa:

<center><math> \left[ -\omega^2 \left( M + A_{33} \right) + i\omega B_{33} + C_{33} \right] \Pi_3 = A </math></center>

* The only nonzero hydrostatic coefficients are <math> C_{33} \, </math> and <math> C_{44} \, </math>. Verify that this is the case even for non-symmetric sections.

* Surge and Roll are coupled for symmetric and non-symmetric bodies. The coupled equation of motion becomes:

<u>Surge-Roll</u>

<center><math> \sum_{j=1,4} \left[ -\omega^2 \left( M_{ij} + A_{ij} \right) + i\omega B_{ij} + C_{ij} \right] \Pi_j = \mathbf{X}_i, \quad i,j = 1,4 </math></center>

* When Newton's law is expressed about the center of gravity:

<center><math> M_{14} = M_{41} = 0, \ M_{11} = M, \ M_{44} = I_G </math></center>

where <math> I_G\,</math> is the body moment of inertia about the center of gravity. If the equations are to be expressed about the origin of the coordinate system, then the formulation must start with respect to <math> G\,</math> and expressions derived w.r.t. <math> O \,</math>, via a coordinate transformation.

* The exciting forces <math> \mathbf{X}_1, \mathbf{X}_3 \,</math> are defined in an obvious manner along the X- and Z-axis. The Roll moment <math> \mathbf{X}_4 \, </math> is defined initially about <math> G\,</math>.

Need to derive definitions for the coefficients that enter the Heave & Surge-Roll equations of motion:

<center><math> M = \rho \forall, \qquad \forall = \ </math> volume of water displaced by body (archimedian principle of buoyancy) </center>

<center><math> C_{33} = \rho g A_\omega = \rho g B, </math></center>

<center><math> A_\omega = </math> body waterplane area  = <math>B</math>  (Beam in two dimensions)</center>

<center><math> \left( C_{44} \right)_G = \rho g \frac{B^3}{12} = \ </math> Roll restoring moment due to a small angular displacement </center>
<center> about the center of gravity. Verify for all wall and non-wall sided sections </center>  

<center><math> C_{44} \equiv \left( C_{44} \right)+G \ne \left( C_{44} \right)_O; \quad </math> Derive an expression for <math> \ \left( C_{44} \right)_O \ </math> in terms of <math> \ \left( C_{44} \right)_G. </math></center>

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/666E84F4-5679-47FD-BD7B-9D39877DE5A1/0/lecture9.pdf here]