Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014
the inclining masses in their initial position. The initial angle of heel, which is unlikely to be exactly upright, is found from the draught readings.
When upright, the ship may have a non-zero TCB0 due to asymmetry of the hull and appendages. There could also be an inherent off-centre location of the transverse centre of gravity, TCGI, due to the ship loading. Either, or both, would require a mass shift of moment w d0 to bring the ship to upright equilibrium.
At upright equilibrium TCB0, TCG0 and KN0 are identical. The summation of transverse moments gives:
∆ ∆
This can be expanded and rearranged to:
∆
At upright, when cos, Equation 8 reduces to:
∆
Substituting into Equation 16 gives:
(17) (18) which is identical to Equation 12, the solution for TCGI.
This confirms that the constant term TCGI is the initial TCG of the system (ship plus inclining masses) with no heeling moment, i.e. with the inclining masses on board and in their initial positions, prior to the first mass movement.
5.5 CONSIDERATION OF LIQUID MOVEMENT
In interpreting TCGI it was assumed that the TCG of the ship alone is a constant; it is common practice to assume that the centre of gravity of the ship, together with any liquids aboard, remains fixed throughout the inclining experiment. An adjustment
is then made to the as- inclined fluid KG to obtain the solid KG.
Experience has shown that, even with small angles of heel, the transverse moment of inertia of the liquid surfaces may change
significantly and it is
necessarily valid to assume that the total free-surface moment (FSM) is constant throughout the experiment.
As it is now a simple matter to determine values of free- trimming KN at any angle — and this method depends upon that capability — so also moments of transference of liquids in tanks at any combination of heel and trim can be calculated. The difference between initial and heeled KNs is the arm of transference.
If specific software with this capability is not available, then each tank can be treated as a ship hull. KN values run in fresh water ( = 1.0), with displacement set to the
©2014: The Royal Institution of Naval Architects
By using regular shapes, the characteristics could be calculated exactly to ensure accuracy. Both classic and new methods were used to analyse KG and, in every case, the classic method underestimated KG by the amounts shown in Table 1. Less than 1× 10-8 m error, the limit of accuracy of the calculation, was found when using the new method.
The difference in KG between the classic method and the new method has also been investigated by re-analysing the inclining experiments for a number of ship types.
B-103 not
6. VALIDATION BY CALCULATION The accuracy of the proposed new method has been tested by mathematical analysis of regular prisms. A range of TCGs was used to induce initial heel. The actual shapes used were:
a box with 8 m beam and 4 m draught
a wedge with 45o deadrise and 4 m draught; and a wedge with 30o deadrise and 1.52 m draught.
Three initial angles of heel were assumed: 0o, 0.8o and 2.0o. Table1: Calculated Differences in KG (m)
(16) (15)
5.6 BENEFITS OF THE NEW METHOD The proposed new method makes no reference to the metacentre and is free of the associated errors. It can be used on any hull
form and, within the constraints of liquids on board, to any angle of inclination.
Curves of limiting KG are, to a large extent, derived from GZ values for the intact and damaged ship. GZs, in turn, are based on KN data. Therefore, the use of KN data to evaluate
the inclining experiment ensures a correlation between actual KG and limiting KG. good
volume of liquid, will deliver arms of transference in global coordinates. The products of these and the mass of liquid will be the moments of transference.
For each mass shift, the corresponding moment of transference should be included to give the total heeling moment. Note that, if moments of transference are in ship coordinates, then they should be added to the term w d but,
if in global coordinates, then they should be added to the term w d cos
Care must still be exercised to ensure that liquids do not flow out of any tank due to excessive heel.
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