Stability analysis

The condition of initial stability for any floating structure is expressed using the metacentric heights (see stability).

\[\begin{split}\left\{ \begin{array}{lcl} \overline{GM_T} &\gt 0\\ \overline{GM_L} &\gt 0 \end{array} \right.\end{split}\]

where

  • \(\overline{GM_T}\) is the transverse metacentric height,
  • \(\overline{GM_L}\) is the longitudinal metacentric height,

these height can be computed as

\[\overline{GM} = \overline{KB} + \overline{BM} - \overline{KG}\]

where

  • \(K\) is the lowest point on the vertical going through the gravity center of the box, which means \(\overline{KG} = H/2\)
  • \(B\) is the buoyancy center of the box.
  • \(\overline{BM}\) is the transverse/longitudinal metacentric radius

For a parallelepipedic box, these radii are given simply as

\[\begin{split}\left\{ \begin{array}{lcl} \overline{BM_T} &=& \dfrac{L^2}{12T}\\ \overline{BM_L} &=& \dfrac{B^2}{12T} \end{array} \right.\end{split}\]

where \(T\) is the draught.

For a parallelepipedic box of uniform density \(c = \dfrac{\rho_{parallelepiped}}{\rho_{water}}\), this draught is

\[T = H \times c\]

The buoyancy center of the box is located at the center of the immersed volume

\[\overline{KG} = \dfrac{H \times c}{2}\]

The conditions of initial stability thus yield

\[\begin{split}\left\{ \begin{array}{lcl} \overline{GM_T} = \dfrac{H}{2}(c-1) + \dfrac{B^2}{12H\times c} &\gt 0\\ \overline{GM_L} = \dfrac{H}{2}(c-1) + \dfrac{L^2}{12H\times c} &\gt 0 \end{array} \right.\end{split}\]

For the previous conditions (\((L,B,H) = (8,4,2)m\) and \(c = 0.5\)), these conditions are true for any density.

Unstable box

For a box of dimensions \((L,B,H) = (5,5,5)m\), the conditions are true only for the density values given in the figure Fig. 28.

stability conditions of a box

Fig. 28 Stability conditions of a box, depending on the density \(c\), courtesy [Gilloteaux]

For a density \(c = 0.5\), and an initial roll angle \(\phi = 2^{\circ}\), the metacentric heights computed on the box mesh are \(GM_T = GM_L = -0.41667\). Negative values induce unstable initial behavior of the box.

Linear approximation

In linear approximation, the roll and pitch restoring coefficients are computed based on the metacentric heights:

\[\begin{split}\left\{ \begin{array}{lcl} K_{44} &=& \rho g V \overline{GM_T}\\ K_{55} &=& \rho g V \overline{GM_L} \end{array} \right.\end{split}\]

where \(V\) is the displacement volume.

The roll solution in linear approximation is given in the figure Fig. 29

roll solution in linear approximation

Fig. 29 Roll solution in linear approximation

Nonlinear approximation

In nonlinear approximation, the hydrostatic force and torque are computed on the mesh, following the box in its motions. This means that the metacentric heights can be computed also for each position. A \(2^{\circ}\) initial roll angle will induce a roll and a pitch motion to reach a stable position, see below and figure Fig. 31

stabilization of the box

Fig. 30 Stabilization of the box

The damping coefficients are taken at \(1E5\) for the rotation degrees of freedom, in order to reduce the computation time.

roll (orange), pitch (blue) and yaw (green) solutions in nonlinear approximation

Fig. 31 Roll (orange), pitch (blue) and yaw (green) solutions in nonlinear approximation

The metacentric heights are shown in figure Fig. 32 : they both start at the negative value, given above, and finish at a positive value, indicating that the box reached a stable position.

transversal (red) and longitudinal (violet) metacentric heights in nonlinear approximation

Fig. 32 Transversal (red) and longitudinal (violet) metacentric heights in nonlinear approximation

Box with a growing density

The same box, with a varying density is considered, along with the nonlinear hydrostatic approximation :

\[c(t) = 0.1 + 0.8 \dfrac{t}{200}\]

The following Fig. 33, Fig. 34 and Fig. 35 illustrates the behavior of the box. We can find the two density values \(c_1 = 0.211\) and \(c_2 = 0.789\) for which the metacentric heights become negative. The box turns over slightly after theses two density values, with a delay due to the inertia and damping forces. The first turn over ends up on the orientation previously observed (roll at 45 degrees and pitch around 33 degrees). For the second turn over, the box recovers its initial orientation (zero roll and pitch) but with a 15 degrees yaw angle.

stabilization of the box

Fig. 33 Stabilization of the box, with varying density

heave (blue), roll (orange), pitch (green) and yaw (red) solutions with varying density

Fig. 34 heave (blue), roll (orange), pitch (green) and yaw (red) solutions with varying density

box density (violet), transversal (pink) and longitudinal (brown) metacentric heights with varying density

Fig. 35 box density (violet), transversal (pink) and longitudinal (brown) metacentric heights with varying density

References

[Gilloteaux]Gilloteaux, J. C. (2007). Mouvements de grande amplitude d’un corps flottant en fluide parfait. Application à la récupération de l’énergie des vagues (Doctoral dissertation).