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is measured in feet or metres and is the controlling factor of the boat’s initial stability. The distance BM is called the metacentric radius and the distance GZ the righting lever.


The naval architect now has to use three formulae, all of which are fun- damental to the study of ship stability.


The first which is proven using advanced mathematics states that:


BM = I/ m (1)


and the second, which is the fundamental formula of the Inclining Experiment, states that:


GM = wd/Δ.tan θ m (2)


The third states that: GM = KB + BM – KG m (3)


where BM = the metacentric radius


GM = the metacentric height


KG = height of the centre of gravity above the keel


I = second moment of the waterplane area


d = distance the inclining weights are moved


w = inkling weight θ = angle of heel


Δ = vessel’s displacement weight = volume of displacement


m m


m m4 m kg


degrees kg m3


Tan θ is a mathematical function and is obtained from either a booklet of mathematical tables or by pressing the appropriate button on a hand calculator. is pronounced Velta.


Of the items in the above list, all, except the second moment of area of the waterplane (I), can be measured directly. Textbooks on naval architecture give the method of computing I using the so-called Simpson’s Rules although, these days, it is calculated using a standard computer program. For the marine surveyor at the bottom of a drydock such luxuries are not available, and he has to rely on a surprisingly accurate but very simple empirical formula which states that:


I = CW


where CW


TM LWL .BWL2 /12CB TM m4 (4)


= the coefficient of waterplane area = the mean draught


- m TM = (0.41 + 0.66)/2


we calculate as follows: from formula 4 I = 0.8485 x 14.86 x 2.0452


12 x 0.7922 x 0.535


And from formula 1: BM = 10.37/12.88


= 0.805 m


Taking KG as 0.75 x 1.14, KB as 0.53 x 0.535 and using formula 3, we obtain:


GM = 0.53 x 0.535 + 0.805 – 0.75 x 1.14 = 0.23 m


That is satisfactory and shows that the boat has a reasonably good initial stability.


As a matter of interest, naval architects use, as a preliminary design figure, an assumed value of GM to be 20% of the waterline breadth. The figure of 0.23 m for the above narrowboat is 11.25% of her waterline breadth.


The Report • March 2019 • Issue 87 | 37 = 0.535 m


Now, using the above formulae and the data in Table 1 and taking the mean draught as:


K Figure 3 Vessel heeled due to Movement of Weight w. The other symbols can be identified from Table 1.


For the narrowboat whose details are given in Table 1, the value of CW


which is the area of the waterplane


divided by the waterline length and breadth was 0.8485 and may be assumed for most modern boats to be 0.85.


B


B1 S


W


w θ G


d


M Z


L


= 10.37


m4


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