**The stability of a freely floating ship**

The PRS Research & Development Division developed theory of calculations of the *GZ*-curve for a freely floating vessel, available in the report.

Based on this theory, the computer software WinSEA used in PRS for stability calculations has been modified by Dr. Andrzej Laskowski, the author of the program. The user can choose three modes of calculating the *GZ*-curve: 1) “engineering”, related to the axis *O**x'* or *O**x''*, 2) “physical”, related to the axis *O**y* or *O**y'*, and 3) “natural”, related to the *z'*-axis, identical with the curve of minimum stability. There is also a zero option of “maximum stability”, for a ship with constant trim, normally not used.

The *O**x'yz'* system refers to a trimmed unit; it is defined by the PS and waterplane at an initial position. The *O**x''y'z'* system is rotated with respect to the *O**x'yz'* system around the axis *Oz'* by the angle of azimuth y. Both systems are fixed to the unit.

Calculations carried out for conventional ships show that the choice of the reference axis is meaningless. It has, however, some meaning for semi-submersible and jack-up platforms, due to a small ratio *L*/*B*, which entails a small difference between longitudinal and transverse stability. In the case of platforms with *an initial heel* it matters to which side the platform is inclined. If it is inclined in the direction of the initial heel, the *GZ*-curve exists at the entire range. If it is inclined in the opposite direction, in certain neighborhood of zero the *GZ*-curve is indefinite due to lack of longitudinal balance. This is due to the small ratio *L*/*B*, and more precisely, due to a small ratio of the principal moments of inertia of the waterplane in an upright position *J _{y}*/

In the range, where the *GZ*-curve is indefinite the unit is *unstable* with respect to the angle of twist, i.e., the ship can automatically rotate by 180° around the axis *z'*, assuming a heel towards the initial heel, where the ship is stable (it has a positive longitudinal metacentric height) with respect twist. This instability relative to the angle of twist is unfavorable to the safety of the ship.

Sample *GZ*-curves for three ships and two jack-ups are shown below.

Fig. 1. Body of fishing cutter

Fig. 2. *GZ*-curves for the cutter

Curve *c* in Fig. 2 concerns the constant trim, which occurs at the position of equilibrium.

Fig. 3. Barge

Fig. 4. *GZ*-curves for the barge

For units without an initial heel, the *GZ*-curves are anti-symmetric.

Fig. 5. Barge in damaged condition

Fig. 6. *GZ*-curve for damaged barge for heels to starboard (in the direction of initial heel)

Fig. 7. *GZ*-curve for damaged barge for inclinations on both sides

For units with an initial heel, the *GZ*-curves on both sides are different. For the side opposite to the initial heel they are larger, wherein in a certain neighbourhood of zero the *GZ*-curve of minimum stability does not exist, due to the lack of longitudinal balance. In the case of conventional ships, the interval of discontinuity exists also, but is imperceptible, of the order of angular minutes.

Fig. 8. Jack-up platform I in damaged condition, *L*/*B* = 1,36, *J _{y}*/

Fig. 9. *GZ*-curves for platform I inclined to starboard (in the direction of initial heel)

Curve *c* in Fig. 9 concerns the constant trim, as in the position of equilibrium.

Fig. 10. *GZ*-curve of minimum stability for platform I inclined to portside

For inclinations to portside, at the range a*’* Î á0, -2,5°ñ the *GZ*-curve does not exist. Outside this range the absolute values of the righting arms are larger, which is better seen in Fig. 12. On the plot summary (Fig. 11, Fig. 12) the discontinuity interval for the *GZ*-curve of minimum stability (curve *z**'*) is not well visible.

Fig. 11. GZ-curves for platform I inclined to portside

It is noteworthy that the GZ-curve for the reference axes *x'* and *y* almost coincide with each other. This happens in each case, which is well seen on the attached graphs. This comes from the fact the axes of rotation ** e** for these curves are very close to each other. Both curves, however, do not differ much from the curve of minimum stability (curve

Fig. 12. *GZ*-curves for platform I for inclinations on both sides

Normally, for a given heel angle the longitudinal balance of the ship occurs at four twist angles (four azimuths), corresponding to transverse inclinations to portside and starboard as well as to longitudinal inclinations (trims) by bow and aft. For the first two twist angles minima of the dynamic arm occur, and for the other – maxima.

In the range, where the *GZ*-curve does not exist there are only two zeros, corresponding to the minimum and maximum of the dynamic arm *l _{d}* (Fig. 13). The minimum corresponds to inclination towards the initial heel. The vicinity of the maximum corresponds to instable inclinations with respect to twist angle. It means that the ship will automatically rotate around the axis z' to the azimuth where the minimum of the dynamic arm

Fig. 13. Run of stability characteristics of rig II in function of twist angle Y for a' = 2°

Fig. 14. Jack-up platform II in damaged condition, *L*/*B _{max}* = 0,805,

Fig. 15. *GZ*-curves for platform II inclined to starboard

Fig. 16. *GZ*-curves for platform II inclined to portside

Fig. 17. *GZ*-curves for platform I for inclinations on both sides

Since the ratio *L*/*B _{max}* for platform II is smaller than 1, the discontinuity interval for the curve of minimum stability (curve

As before, in the range where the *GZ*-curve does not exist, there are only two zeros corresponding to the minimum and maximum of the dynamic arm *l _{d}* (Fig. 18).

Fig. 18. Run of curves *l*, *l _{d}* i

Fig. 19 shows the run of stability characteristics for the heel angle a*'* = 6° in function of azimuth for the reference axis *Ox''*. Identical characteristics are obtained for the reference axis *Oy*'. They are different from those for the reference axis *Oz'* (Fig. 18). Nonetheless, both indicate on common features – for the given heel angle the minimum and maximum dynamic arms occur at the same twist angles (azimuths).

Fig. 19. Run of righting arm *l* º *GZ* and dynamic *l _{d}* for rig II in function of azimuth Y,

for a heel angle less than the critical

From Fig. 19 it follows In addition that for the heel angle a*'* = 6° the platform cannot be longitudinally balance, when the azimuth is from the range y Î á86°, 106°ñ.

A few further conclusions can be drawn from the above calculations:

- at the initial range of stability all modes of calculations (including the fixed trim) yield practically the same results
- the choice of the reference axis is practically meaningless. It affects only the way the ship is balanced, without changing the direction of the righting moment, nor the area under the
*GZ*-curve, which is the lowest possible. This is a feature of a freely floating unit - for a ship with an initial heel, in a certain neighborhood of zero for inclinations on the opposite side, dependent on the ratio
*L*/*B*, the GZ-curve cannot be obtained, due to lack of longitudinal balance - for a freely floating ship only one GZ-curve is meaningful, related to transverse inclinations about the principal axis of inertia for an upright waterplane. For other azimuths, the
*GZ*-curves have intervals in which they do not exist.

For ships there is no revolution – any method of calculating the *GZ*-curve with free trim yields virtually the same curve. However, there is a revolutionary conclusion for platforms – there is only one *GZ*-curve related to the reference axis *Oz'*. In other words, for rigs *GZ*-curves for various azimuths do not exist, nonetheless the regulations require such calculations.