Principles of naval engineering - Historic Naval Ships Association

Chapter 3-STABILITY AND BUOYANCYSince moving a weight which is already aboardwill cause no change in displacement, there canbe no change in M, the metacenter. If M remainsfixed, then the upward movement of thecenter of gravity results in a loss of metacentricheight:whereGjM GM - GG 1G,M = new metacentric height (after weightmovement), in feetGM = old metacentric height (before weightmovement), in feetGGj^ = rise in center of gravity,in feetIf the ammunition on the main deck is moveddown to the 6th deck, the positions of G and Gjwill be reversed. The shift in G can be foundfrom thesame formula as before, the only differencebeing that GGj^ now becomes a gain inmetacentric height instead of a loss (fig. 3-18).If a weight is moved vertically downward, theship's center of gravity, G, will move straightdown on the centerline and the correction is additive.In this case the sine curve is plotted belowthe abscissa. The final stability curve is thatportion of the curve above the sine correctioncurve.A vertical shift in theship'scenter of gravitychanges every righting arm throughout the entirerange of stability. If the ship is at any angle ofheel, such as 6 in figure 3-18, the righting armis GZ with the center of gravity at G. But if thecenter of gravity shifts to G^ as the result of avertical weight shift upward, the righting armbecomes Gj^Zj^, which is smaller than GZ by theamount of GR. In the right triangle GRGj, theangle of heel is at Gj; hence the loss of therighting arm may be found fromGR = GG^ X sin $This equation may be stated in words as: Theloss of righting arm equals the rise in the centerof gravity times the sine of the angle of heel . Thesine of the angle of heel is a ratio which can befound 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 ofthe proper angle, a curve of loss of rightingarms can be obtained by plotting values ofGG^ X sin e vertically against angles of heelhorizontally, which results in a sine curve. Whenplotted, the curve is as illustrated in figure 3- 19.The sine curve may be superimposed on theoriginal stability curve to show the effect onstability characteristics of moving the weight upin a ship. Inasmuch as displacement is unchanged,the righting arms of the old curve needbe corrected for the change of G only, and noother variation occurs. Consequently, if GGj^ xsin e is deducted from each GZ on old stabilitycurve, the result will be a correct righting armcurve for the ship after the weight movement.In figure 3-20 a sine curve has been superimposedon an original stability curve. The dottedarea is that portion of the curve which was lostdue to moving the weight up, whereas the linedarea is the remaining or residual portion of thecurve. The residual maximum righting arm isAB and occurs at an angle of about 37°. The newrange of stability is from 0°to 53°.The reduced stability of the new curve becomesmore evident if the intercepted distancesbetween the old GZ curve and the sine curve aretransferred down to the base, thus forming anew curve of static stability (fig, 3-21). Wherethe old righting arm at 30° was AB, the new onehas a value of CB, which is plotted up from thebase to locate point D (CB = AD) and thus a pointis established at 30° on the new curve. A seriesof points thus obtained by transferring intercepteddistances down to the base line delineatesthe new curve, which maybe analyzed as follows:GM is now the quantity represented by EF.Maximum righting arm is now the quantityrepresented by HI.Angle at which maximum righting arm occursis 37 °.Range of stability is from 0° to 53°.Total dynamic stability is represented by theshaded area.HORIZONTAL WEIGHT SHIFTWhen the ship is upright, G lies in the foreand aft centerline, and all weights on board arebalanced. Moving any weight horizontally willresult in a shift in G in an athwartship direction,parallel to the weight movement. B and Gare no longer in the same vertical line and anupsetting moment exists at 0° inclination, whichwill cause the ship to heel until B moves underthe new position of G. In calm water the ship will45