4.2 Restoring forces for surface vessels

For surface vessels, the restoring forces will depend on the craft’s metacentric height, the location of the CG and the CB as well as the shape and size of the water plane. Let $A_{wp}$ denote the water plane area and:

$$G{M}_T=\text{transverse metacentric height (m)}$$ $$G{M}_L=\text{longitudinal metacentric height (m)}$$

The metacentric height $G{M}_i$ where $i=(T,L)$ is the distance between the metacenter $M_i$ and CG

Definition metacenter

The theoretical point $M_i$ at which an imaginary vertical line trough the CB intersects another imaginary vertical line trough a new $C{B}_1$ created when the body is displaced, or tilted, in the water.

Hydrostatics of floating vessels

For a floating vessel at rest, buoyancy and weight are in balance such that:

$$mg=\rho g\nabla$$

The hydrostatic force in heave is recognized as the difference of the gravitational and buoyancy forces:

$$Z=mg-\rho g(\nabla +\delta (z)=-\rho g\delta \nabla (z)$$

where z is the displacement in heave and $\delta \nabla (z)$ is the change in the displaced water.

Change in the displaced water can be written as:

$$\delta \nabla (z)={\int }_0^{z}A(\zeta )d\zeta$$

$A(\zeta )$ is the waterplane area of the vessel as a function of the heave position.

Perturbed buoyancy force vector due to variations in displaced volume

$$\delta f_b^n=\left[\begin{array}{c}0\\ 0\\ -\rho g{\int }_0^{z}A(\zeta )d\zeta \end{array}\right]$$

From this it follows that the total restoring force vector is:

$$f_r^b=R^T({\Theta }_{nb})(f_g^n+f_b^n+\delta f_b^n)=-\rho g\left[\begin{array}{c}-\sin \theta \\ \cos \theta \sin \varphi \\ \cos \theta \cos \varphi \end{array}\right]{\int }_0^{z}A(\zeta )d\zeta$$

where we have exploited that $f_g^n=-f_b^n$

Restoring moment

The moment arm in roll and pitch are $G{M}_T\sin \varphi$ and $G{M}_L\sin \theta$, respectively.

The restoring moment in body frame is equal to:

$$m_r^b=r_{GM}^b\times f_b^b=-\rho g\nabla \left[\begin{array}{c}G{M}_T\sin \varphi \cos \theta \cos \varphi \\ G{M}_L\sin \theta \cos \theta \cos \varphi \\ -G{M}_L\cos \theta +G{M}_T\sin \varphi \sin \theta \end{array}\right]$$

6-DOF generalized gravity and buoyance forces

Linear (small angle) theory for boxed-shaped vessels

Assumes that $\varphi$, $\theta$, $z$ are small such that:

$g(\eta )\approx G\eta$

We have computed the G matrix in CF, that is the center of flotation:

$$G^{CF}=diag(0,0,\rho g{A}_{wp},\rho g\nabla G{M}_T,\rho g\nabla G{M}_L,0)$$

We need to transform this expression to the CO.

If we transform $G^{CF}$ to the CO, two additional coupling terms $G_{35}=G_{53}$ appears, this is shown below:

These coupling terms depend on the location of CF with respect to CO:

$$r_{bf}^b={\left[\begin{array}{ccc}LCF& 0& 0\end{array}\right]}^T$$

Computation of GM values

Usually computed by software.

The GM values can be computed for given moment of areas, CG and CB using these formulas.

Computation of metacenter height for surface vessels

The metacenter height M can be computed by using basic hydrostatics

$$G{M}_T=B{M}_T-BG$$ $$G{M}_L=B{M}_L-BG$$

For small roll and pitch angles the transverse and longitudinal radius of curvature can be approximated by:

$$B{M}_T=\frac{I_T}{\nabla }$$ $$B{M}_L=\frac{I_L}{\nabla }$$

Moments of area about the waterplane:

$$I_L=\int {\int }_{Awp}{x}^{2}dA$$ $$I_T=\int {\int }_{Awp}{y}^{2}dA$$

For conventional ships an upper bound on these integrals can be found by considering a rectangular waterplane area $A_{wp}=BL$ where B and L are the beam and length of the hull upper bounded by:

$$I_L<\frac{1}{12}{L}^{3}B$$ $$I_T<\frac{1}{12}{B}^{3}L$$

Metacenter M, center of gravity G and center of buoyancy B for a submerged and a floating vessel. The reference is the keel line K.

Metacenter stability

A floating vessel is said to be:

Transverse metacentrically stable if $G{M}_T\ge G{M}_{T,min}>0$ Transverse metacentrically stable if $G{M}_L\ge G{M}_{L,min}>0$

The longitudinal stability requirement is easy to satisfy for ships since the pitching motion is quite limited. This corresponds to a large $G_{ML}$ value. The lateral requirement, however, is an important design criterion used to prescribe sufficient stability in roll to avoid that the vessel does not roll around. The vessel must also have damage stability (stability margins) in case of accidents.

Typically, in roll $G_{MT,min}>0.5m$ while in pitch $G{M}_{L,min}$ is much larger (more than 100.0 m).