Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014


Dividing the moment by displacement will give the heeling arm (HZ):


    cos ∆


(8)


Displacement, trim and initial heel are determined from the draught readings. For each mass movement, the pendulum deflection is used to calculate the change of heel and adding this to the initial heel gives the actual heel. The average heel angle from all pendulums should be used. For each mass movement, the hull model can then be interrogated at the actual displacement, trim and heel to derive the value of KN.


Each righting arm can be calculated from KN using the standard equation:


    sin  cos (9)


where KGI and TCGI are the respective KG and TCG with the inclining masses in their initial positions.


The sign of the last term in Equation 9, TCGI cos is dependent on the sign convention being used. If, for example, distances to starboard and heel to starboard are both positive or both negative, then the sign of the term will be negative as shown. If distances and heel angle are of opposite signs, then the term will be positive.


Substituting HZ for GZ from Equation 6 into Equation 9:     sin  cos


(10)


This heeling arm equation is the basis for the proposed new method.


5.2 THE SOLUTION FOR TCGI


When = 0, sin= 0 and cos, so Equation 10 reduces to:


   (11)


Equation 11 can be re-arranged to give a solution for TCGI:   


upright hydrostatics. However, it (12)


KN0 is identical to TCB0 and could therefore be found from


is more


convenient to calculate KN0 with the other KN values which will be required. KN0 can be expected to be close to zero, but will only be exactly so if both hull and appendages are truly symmetric about the centreline. The actual value should generally be calculated.


HZ0 can be found from the trendline through the HZ points when plotted against heel angle and is the value of HZ when heel = 0, i.e. the intercept of the trendline.


A third-order polynomial trendline should be used because it can closely match non-linear data sets which


B-102 Figure 3: Illustration of Equation 13


5.4 INTERPRETATION OF TCGI In Equation 9, KGI and TCGI are


include a point of inflection - such as generally occur near equilibrium in GZ plots.


When there is known to be a discontinuity in GZ within the range of heels at the inclining experiment, the points should be divided into two sets, either


side of the


discontinuity, and only the set which spans upright used to determine HZ0. If the discontinuity is exactly at upright, both sets may be used and HZ0 taken to be the mean of the two intersections.


5.3 THE SOLUTION FOR KGI


Equation 10 can be re-arranged as:  sin     cos


and the solution for KG1 is therefore:


     cos sin


(14)


For each mass move, KGI sin(from Equation 13) is plotted against sinhe value of KGI is equal to the slope of the trendline and, as all points should lie on a straight line, a linear trendline is used to find an average result.


(13)


assumed to be


constants. Over the duration of the inclining experiment, the KG should remain constant, with care being taken not to shift the inclining masses vertically. The masses (or mass groups) should always be placed on the deck and not stacked on top of each other.


The TCG of the system will change as the inclining masses are shifted from side to side and it is important to distinguish between the constant TCGI and the instantaneous TCG after each mass shift. It will be shown that TCGI is the TCG of the complete system — ship plus inclining masses — with


©2014: The Royal Institution of Naval Architects


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