Chapter 3-STABILITY AND BUOYANCY 



Since moving a weight which is already aboard 

 will cause no change in displacement, there can 

 be no change in M, the metacenter. If M re- 

 mains fixed, then the upward movement of the 

 center of gravity results in a loss of metacentric 

 height: 



GjM 



GM - GG 



1 



where 



G,M = new metacentric height (after weight 

 movement), in feet 



GM = old metacentric height (before weight 

 movement), in feet 



GGj^ = rise in center of gravity, in feet 



If the ammunition on the main deck is moved 

 down to the 6th deck, the positions of G and Gj 

 will be reversed. The shift in G can be found 

 from the same formula as before, the only dif- 

 ference being that GGj^ now becomes a gain in 

 metacentric height instead of a loss (fig. 3-18). 



If a weight is moved vertically downward, the 

 ship's center of gravity, G, will move straight 

 down on the centerline and the correction is addi- 

 tive. In this case the sine curve is plotted below 

 the abscissa. The final stability curve is that 

 portion of the curve above the sine correction 

 curve. 



A vertical shift in theship'scenter of gravity 

 changes every righting arm throughout the entire 

 range of stability. If the ship is at any angle of 

 heel, such as 6 in figure 3-18, the righting arm 

 is GZ with the center of gravity at G. But if the 

 center of gravity shifts to G^ as the result of a 

 vertical weight shift upward, the righting arm 

 becomes Gj^Zj^, which is smaller than GZ by the 

 amount of GR. In the right triangle GRGj, the 

 angle of heel is at Gj; hence the loss of the 

 righting arm may be found from 



GG^ X sin e vertically against angles of heel 

 horizontally, which results in a sine curve. When 

 plotted, the curve is as illustrated in figure 3- 19. 



The sine curve may be superimposed on the 

 original stability curve to show the effect on 

 stability characteristics of moving the weight up 

 in a ship. Inasmuch as displacement is un- 

 changed, the righting arms of the old curve need 

 be corrected for the change of G only, and no 

 other variation occurs. Consequently, if GGj^ x 

 sin e is deducted from each GZ on old stability 

 curve, the result will be a correct righting arm 

 curve for the ship after the weight movement. 



In figure 3-20 a sine curve has been super- 

 imposed on an original stability curve. The dotted 

 area is that portion of the curve which was lost 

 due to moving the weight up, whereas the lined 

 area is the remaining or residual portion of the 

 curve. The residual maximum righting arm is 

 AB and occurs at an angle of about 37°. The new 

 range of stability is from 0°to 53°. 



The reduced stability of the new curve be- 

 comes more evident if the intercepted distances 

 between the old GZ curve and the sine curve are 

 transferred down to the base, thus forming a 

 new curve of static stability (fig, 3-21). Where 

 the old righting arm at 30° was AB, the new one 

 has a value of CB, which is plotted up from the 

 base to locate point D (CB = AD) and thus a point 

 is established at 30° on the new curve. A series 

 of points thus obtained by transferring inter- 

 cepted distances down to the base line delineates 

 the new curve, which maybe analyzed as follows: 



GM is now the quantity represented by EF. 



Maximum righting arm is now the quantity 

 represented by HI. 



Angle at which maximum righting arm oc- 

 curs is 37 °. 



Range of stability is from 0° to 53°. 



Total dynamic stability is represented by the 

 shaded area. 



GR = GG^ X sin $ 



This equation may be stated in words as: The 

 loss of righting arm equals the rise in the center 

 of gravity times the sine of the angle of heel . The 

 sine of the angle of heel is a ratio which can be 

 found by consulting a table of sines. 



If the loss of GZ is found for 10°, 20°, 30°, 

 and so forth by multiplying GG^ by the sine of 

 the proper angle, a curve of loss of righting 

 arms can be obtained by plotting values of 



HORIZONTAL WEIGHT SHIFT 



When the ship is upright, G lies in the fore 

 and aft centerline, and all weights on board are 

 balanced. Moving any weight horizontally will 

 result in a shift in G in an athwartship direc- 

 tion, parallel to the weight movement. B and G 

 are no longer in the same vertical line and an 

 upsetting moment exists at 0° inclination, which 

 will cause the ship to heel until B moves under 

 the new position of G. In calm water the ship will 



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