Strip Theory Of Ship Motions. Heave & Pitch

Ship motions with forward speed

• Ship advances in the positive x-direction with constant speed [math]\displaystyle{ U\, }[/math]

• Regular waves with absolute frequency [math]\displaystyle{ \omega_0\, }[/math] and direction [math]\displaystyle{ \beta\, }[/math] are incident upon the ship

• The ship undergoes oscillatory motions in all six degrees of freedom [math]\displaystyle{ \xi_j(t), \quad j=1,\cdots,6\, }[/math].

• The ship seakeeping problem will be treated in the frequency domain under the assumption of linearity.

• Ship-hull boundary condition

The relevant rigid body velocities for a ship translating with a constant forward velocity and heaving and pitching in head waves are:

[math]\displaystyle{ \vec{U}=U\vec{L}; \quad \mbox{Forward speed} \, }[/math] [math]\displaystyle{ \vec{\dot{\xi}}_3(t)=\dot{\xi}_3(t)\vec{k}; \quad \mbox{Heave velocity (positive up)} \, }[/math] [math]\displaystyle{ \vec{\dot{\xi}}_5(t)=\dot{\xi}_5(t)\vec{j}; \quad \mbox{Pitch angular velocity (positive,bow down)} \, }[/math]

Let [math]\displaystyle{ \vec{n}(t)\, }[/math] the time-dependent unit normal vector to the instantaneous position of the ship hull. Let [math]\displaystyle{ \vec{n_0}\, }[/math] be its value when the ship is at rest.

• The body boundary condition will be derived for small heave & pitch motions and will be stated on the mean position of the ship at rest.

The nonlinear boundary condition on the exact position of the ship hull is:

[math]\displaystyle{ \frac{\partial\Phi}{\partial n}=\overrightarrow{V} \cdot \vec{n} \, }[/math]

where,

[math]\displaystyle{ \Phi=\Phi_3 + \Phi_5 \, }[/math] [math]\displaystyle{ \overrightarrow{V} \cong U \vec{l} + \vec{K} \left( \dot{\xi}_3 - X \dot{\xi}_5 \right) \, }[/math]

• The unsteady velocity potential [math]\displaystyle{ \Phi\, }[/math] has been written as the linear superposition of the heave and pitch components. The total rigid-body velocity is the sum of vertical velocity due to the ship heave and pitch.

• The normal vector[math]\displaystyle{ \vec{n}(t)\, }[/math] is defined by

[math]\displaystyle{ \vec{n}(t) = \vec{n}_0 + \vec{\xi}_5 \times \vec{n}_0 \, }[/math]

where [math]\displaystyle{ \vec{\xi}_5(t)\, }[/math] is the ship pitch rotation angle, and:

[math]\displaystyle{ \vec{n}_0 \equiv (n_1,n_2,n_3) \, }[/math]

Keeping only the unsteady components:

[math]\displaystyle{ \vec{V} \cdot \vec{n} = U \xi_5 n_3 + \left( \dot{\xi}_3 - X \dot{\xi_5} \right) n_3 = \frac{\partial\Phi}{\partial n} }[/math]

• Note that the steady component [math]\displaystyle{ U n_1 \, }[/math] has already been accounted for in the statement of the steady flow. Let

[math]\displaystyle{ \Phi=\mathbb{R}\mathbf{e}\left(\phi e^{i\omega t} \right), \quad \phi=\Pi_3 X_3 + \Pi_5 X_5 }[/math]

where [math]\displaystyle{ \phi_3\, }[/math] and [math]\displaystyle{ \phi_5\, }[/math] are the unit-amplitude heave & pitch complex velocity potentials in time-harmonic flow. Also:

[math]\displaystyle{ \xi_3(t) = \mathbb{R}\mathbf{e} \left\{ \Pi_3(\omega) e^{i\omega t} \right\}, \quad \xi_5(t) = \mathbb{R}\mathbf{e} \left\{ \Pi_5(\omega) e^{i\omega t} \right\} }[/math]

It follows that:

[math]\displaystyle{ \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S}_B \, }[/math] [math]\displaystyle{ \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S}_B \, }[/math]

• Note the forward-speed effect in the pitch boundary condition and no such effect in heave.

Ship-hull boundary conditions

• Diffraction problem: [math]\displaystyle{ \phi=\phi_7 \, }[/math]

[math]\displaystyle{ \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \frac{\partial}{\partial n} \equiv \vec{n} \cdot \nabla \, }[/math]

where:

[math]\displaystyle{ \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta - iKY\sin\beta} \, }[/math]

• Radiation problem: [math]\displaystyle{ \phi = \phi_3 + \phi_5 \, }[/math]

We will consider the special but important case of heave & pitch which are coupled and important modes to study in the ship seakeeping problem.

• Heave and pitch are the only modes of interest in head waves ([math]\displaystyle{ \beta = 180^\circ \, }[/math]) when all other modes of motion (Roll-Sway-Yaw) are identically zero for a ship symmetric port-starboard. Surge is nonzero but generally small for slender ships and in ambient waves of small steepness.

• Bernoulli equation

The linear hydrodynamic disturbance pressure due to unsteady flow disturbances is given relative to the ship frame:

[math]\displaystyle{ P = - \rho \left(\frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi \, }[/math]

where [math]\displaystyle{ \Phi\, }[/math] is the respective real potential.

Radiation problem

[math]\displaystyle{ \Phi = \mathbb{R}\mathbf{e} \left\{ \phi e^{i\omega t} \right\} \, }[/math] [math]\displaystyle{ P = \mathbb{R}\mathbf{e} \left\{ \mathbb{P} e^{i\omega t} \right\} \, }[/math] [math]\displaystyle{ \mathbb{P} = - \rho \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \Pi_3 \phi_3 + \Pi_5 \phi_5 \right) }[/math]

where the complex velocity potentials satisfy the 2D boundary value problems derived earlier for slender ships.

• The hydrodynamic pressure will be integrated over the ship hull to obtain the added-mass and damping coefficients next:

[math]\displaystyle{ F_i(t) = \iint_{\bar{S}_B} P n_i dS, \quad i = 3,5 \, }[/math]

The expressions derived below extend almost trivially to all other modes of motion. In general,

[math]\displaystyle{ n_i \equiv \vec{n} = \left(n_1,n_2,n_3\right);\quad i=1,2,3 }[/math] [math]\displaystyle{ n_i \equiv \vec{X} \times \vec{n} = \left(n_4,n_5,n_6 \right); \quad i=4,5,6 }[/math]

So for heave:

[math]\displaystyle{ n_i = n_3\, }[/math]

And pitch:

[math]\displaystyle{ n_i = n_5 = - X n_3 + X n_1 \simeq - X n_3 \left(n_1 \ll n_3 \right) \, }[/math]

Expressing [math]\displaystyle{ F_i \, }[/math] in terms of the heave & pitch added mass & damping coefficients when the ship is forced to oscillate in calm water, we obtain when accounting for cross-coupling effects:

[math]\displaystyle{ F_i^H (t) = - \sum_{j=3,5} \left[ A_{ij} \frac{d^2\xi_j}{dt^2} + B_{ij} \frac{d\xi_j}{dt} \right] }[/math]

Hydrostatic restoring effects are understood to be added to [math]\displaystyle{ F_i^H\, }[/math].

Introducing complex notation: