4.3 Load conditions and natural periods

The load condition will determine the heave, roll and pitch periods of a marine craft. The load condition varies over time (due to loading, offloading, fuel burning, water tanks, etc.)

$$(M_{RB}+A(\omega ))\ddot{\xi }+B(\omega )+B_{\nu }(\omega ))\dot{\xi }+C\xi ={\tau }_{ext}$$

This is a linear mass-damper-spring system where the seakeeping coordinates $\xi ={\left[\begin{array}{cccccc}{\zeta }_1& {\zeta }_2& {\zeta }_3& {\zeta }_4& {\zeta }_5& {\zeta }_6\end{array}\right]}^T$ are zero-mean wave-induced perturbations in {s} from an equilibrium state defined by a ship moving at constant heading $\psi$ and speed U. In calm water $\dot{\xi }=0$ and {s} coincides with {b}. The output from the hydrodynamic codes are frequency-dependent added mass A($\omega$) and potential damping B($\omega$). Viscous damping $B_{\nu }(\omega )$ is usually added using semi-empirical methods while the restoring matrix C = G.

In a linear system, the natural periods will be independent on the coordinate origin if they are computed using the 6- DOF coupled equations of motion. This is because the eigenvalues of a linear system do not change when applying a similarity transformation.

1-DOF decoupled analysis (natural periods)

The decoupled natural periods should be computed in the CF using the decoupled equations of motion. If not, the results can be very wrong since the eigenvalues of the decoupled equations depend on the coordinate origin as opposed to the 6-DOF coupled system.

Implicit equations for frequency: Must be solved by iteration since added mass is a function of frequency.

MATLAB implementation

Computation of natural periods in 6-DOF coupled system

MATLAB implementation

Natural periods as a function of load condition

The roll and pitch periods will depend strongly on the load condition while the heave is less affected.

Free surface effects

Many ships are equipped liquid tanks like ballast and anti-roll tanks. A partially filled tank is known as a slack tank and in these tanks the liquid can move and endanger the ship stability. The reduction of metacentric height $G{M}_T$ caused by the liquids in slack tanks is known as the free-surface effect. The mass of the liquid or the location of the tanks play no role, only the moment of inertia of the surface affects stability. The effective metacentric height corrected for slack tanks filled with sea water is:

$$G{M}_{T,eff}=G{M}_T-FSC$$ $$FSC=\sum _{r=1}^N\frac{\rho }{m}{i}_r$$

FSC = free-surface correction $i_r$ is the moment of inertia of the water surface

Payload effects

The metacentric height $G{M}_T$ is reduced on board a ship if a payload with mass $m_p$ is lifted up and suspended at the end of a rope of length h.

The effective metacentric height is:

$$G{M}_{T,eff}=G{M}_T-h\frac{m_p}{m}$$

where m is the mass of the vessel.

The destabilizing effect appears immediately after raising the load sufficiently to let it move freely.