Linear Wave-Body Interaction

Linear wave-body interactions

• Consider a 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

[math]\displaystyle{ \zeta(t): \quad \mbox{ambient wave elevation. Regular or random with definitions to be given below}. \, }[/math] [math]\displaystyle{ \xi_1(t): \quad \mbox{Body surge displacement} \, }[/math] [math]\displaystyle{ \xi_3(t): \quad \mbox{Body heave displacement} \, }[/math] [math]\displaystyle{ \xi_4(t): \quad \mbox{Body roll displacement} \, }[/math]

Linear theory

• Assume: [math]\displaystyle{ \left| \frac{\partial\zeta}{\partial x} \right| = O(\varepsilon) \ll 1 \, }[/math]

Small wave steepness. Very good assumption for gravity waves in most cases, except when waves are near breaking conditions.

• Assume [math]\displaystyle{ \left| \frac{\xi_1}{A} \right| = O(\varepsilon) \ll 1 \, }[/math]

[math]\displaystyle{ \left| \frac{\xi_3}{A} \right| = O(\varepsilon) \ll 1 \, }[/math] [math]\displaystyle{ \left| \xi_4 \right| = O(\varepsilon) \ll 1 \, }[/math]

These assumptions are valid in most cases and most bodies of practical interest, unless the vessel response at resonance is highly tuned or lightly damped. This is often the case for roll when a small amplitude wave interacts with a vessel weakly damped in roll.

• The vessel dynamic responses in waves may be modelled according to linear system theory:

By virtue of linearity, a random seastate may be represented as the linear super position of plane progressive waves;

[math]\displaystyle{ \zeta(x,t) = \sum_j A_j \cos ( K_j x - \omega_j t + \epsilon_j ) \, }[/math]

where in deep water: [math]\displaystyle{ K_j = \frac{\omega_j^2}{g} \, }[/math].

According to the theory of St. Denis and Pierson, the phases [math]\displaystyle{ \epsilon\, }[/math], are random and uniformly distributed between [math]\displaystyle{ ( - \pi, \pi ] \, }[/math]. For now we assume them known constants:

At [math]\displaystyle{ X=0\, }[/math]:

[math]\displaystyle{ \zeta(t) = \sum_j A_j \cos ( \omega_j t - \epsilon_j ) \, }[/math] [math]\displaystyle{ = \mathfrak{Re} \sum_j A_j e^{i\omega_j t - i \epsilon_j} }[/math]

And the corresponding vessel responses follow from linearity in the form:

[math]\displaystyle{ \xi_K (t) = \mathfrak{Re} \sum_j \Pi_K (\omega_j) e^{i\omega_j t - i\epsilon_j}, \qquad K = 1, 3, 4 }[/math]

Where [math]\displaystyle{ \Pi_K (\omega) \, }[/math] is the complex Rao for mode [math]\displaystyle{ K\, }[/math]. It is the object of linear seakeeping theory. In the frequency domain to derive equations for [math]\displaystyle{ \Pi\omega)\, }[/math]. The treatment in the stochastic case is then a simple exercize in linear systems.

• The equations of motion for [math]\displaystyle{ \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

Surge:

[math]\displaystyle{ \mathbf{M} \frac{d^2\xi_1}{dt^2} = F_{1\omega} ( \xi_1, \dot\xi_1, \ddot\xi_1, t) }[/math]

where [math]\displaystyle{ \frac{d\xi_1}{dt} = \dot\xi_1 \, }[/math] and [math]\displaystyle{ F_{1\omega} \, }[/math] is the force on the body due to the fluid pressures, by virtue of linearity, [math]\displaystyle{ F_{1\omega} \, }[/math] will be assumed to be a linear functional of [math]\displaystyle{ \xi_1, \dot\xi_1, \ddot\xi_1 \, }[/math].

• Memory effects exist when surface waves are generated on the free surface, so [math]\displaystyle{ F_{1\omega} \, }[/math] depends in principle on the entire history of the vessel displacement.

• We will adopt here the frequency domain formulation where the vessel motion has been going on over an infinite time interval, [math]\displaystyle{ (-\infty, t)\, }[/math] with [math]\displaystyle{ e^{i\omega t}\, }[/math] dependence.

We will therefore set:

[math]\displaystyle{ \xi_K(t) = \mathfrak{Re} \left\{ \Pi_K e^{i\omega t} \right\}, \qquad K=1,3,4 }[/math]

In this case we can linearize the water induced force on the body as follows:

Surge

[math]\displaystyle{ F_{1\omega}(t) = X_1(t) - A_{11} \ddot \xi_1 - A_{13} \ddot \xi_3 - A_{14} \ddot \xi_4 }[/math] [math]\displaystyle{ - B_{11} \dot \xi_1 - B_{13} \dot \xi_3 - B_{14} \dot \xi_4 }[/math] [math]\displaystyle{ - C_{11} \xi_1 - C_{13} \xi_3 - C_{14} \xi_4 }[/math] [math]\displaystyle{ = X_1(t) - \sum_j [ A_{1j} \ddot \xi_j + B_{1j} \dot \xi_j + C_{1j} \xi_j ] }[/math]

The same expansion applies for other modes, namely Heave ([math]\displaystyle{ K = 3 \, }[/math]) and Roll ([math]\displaystyle{ K=4 \, }[/math]). In sum:

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

• The added-mass matrix [math]\displaystyle{ A_{Kj} \, }[/math] represents the added inertia due to the acceleration of the body in water with acceleration [math]\displaystyle{ \ddot\xi_j\, }[/math].

• The damping matrix [math]\displaystyle{ B_Kj\, }[/math] governs the energy dissipation into the fluid domain in the form of surface waves.

• The hydrostatic restoring matrix [math]\displaystyle{ C_{Kj} \, }[/math] represents the system stifness due to the hydrostatic restoring forces and moments.

For harmonic motions, the matrices [math]\displaystyle{ A_{Kj} \, }[/math] and [math]\displaystyle{ B_{Kj} \, }[/math] are functions of [math]\displaystyle{ \omega\, }[/math], or

[math]\displaystyle{ A_{Kj} (\omega), \ B_{Kj} (\omega)\, }[/math]

This functional form will be discussed below. The hydrostatic matrix [math]\displaystyle{ C_{Kj} \, }[/math] is independent of [math]\displaystyle{ \omega\, }[/math] and many of its elements are identically equal to zero.

Collecting terms in the left-hand side and denoting by [math]\displaystyle{ M_{Kj}\, }[/math] the body inertia matrix:

Surge: