Principles of naval engineering - Historic Naval Ships Association

PRINCIPLES OF NAVAL ENGINEERINGFigure 3-22. — Loss of righting arm when center of gravity ismoved off the center line.147.38loss of righting arm involved in an athwartshipmovement of G is equal at any angle of heel toG1G2 X cos 6 . This variable distance (G1G2 xcos 6 ) is called the ship's inclining arm; whenthis value is multiplied by the displacement,W, the product is the ship's inclining moment.The expression G^T = G1G2 x cos(9may bestated as: The inclining arm is equal to theathwartship shift in the center of gravity timesthe cosine of the angle o f heel. The cosine ofthe angle of heel is a ratio which can be foundby consulting a table of cosines. If the incliningarm is computed for 10 ,20°, 30°, etc. bymultiplying G1G2 by the cosine of the properangle, a curve of inclining arms can be obtainedby plotting values of G1G2 x cos^ verticallyagainst angles of heel horizontally, which resultsin a cosine curve. Note that the cosinecurve (fig. 3-23) is just the opposite of the sinecurve (fig. 3-20) but is otherwise identical inshape.Just as the sine curve was superimposedon the GZ curve, so may the cosine curve besuperimposed on the stability curve to show theeffect on stability of moving a weight athwartship.The cosine curve has been placed on theoriginal stability curve, corrected for the actualheight of the center of gravity. The dotted area(fig. 3-23) is that portion of the curvewhich waslost due to the weight shift, and the lined areais the remaining or residual portion of the curve.The residual maximum righting arm js ABwhich develops at an angle of about 37 . Thenew range of stability is from ?0° to 50°.The new curve of static stability can beplotted on the base by transferring down theintercepted distances between the cosine curveand the old GZ curve. For example, in figure3-24 the old righting arm at 37° was AD, theloss of righting arm (inclining arm) at thisangle is AC, leaving a residual GZ of CD. Thisvalue has been plotted up from the base as ABto provide one point on the final curve. Theresidual stability may be analyzed on the newcurve as follows:48