PRINCIPLES OF NAVAL ENGINEERING 



147.38 



Figure 3-22. — Loss of righting arm when center of gravity is 

 moved off the center line. 



loss of righting arm involved in an athwartship 

 movement of G is equal at any angle of heel to 

 G1G2 X cos 6 . This variable distance (G1G2 x 

 cos 6 ) is called the ship's inclining arm; when 

 this value is multiplied by the displacement, 

 W, the product is the ship's inclining moment. 



The expression G^T = G1G2 x cos(9may be 

 stated as: The inclining arm is equal to the 

 athwartship shift in the center of gravity times 

 the cosine of the angle o f heel. The cosine of 

 the angle of heel is a ratio which can be found 

 by consulting a table of cosines. If the inclining 

 arm is computed for 10 , 20°, 30°, etc. by 

 multiplying G1G2 by the cosine of the proper 

 angle, a curve of inclining arms can be obtained 

 by plotting values of G1G2 x cos^ vertically 

 against angles of heel horizontally, which re- 

 sults in a cosine curve. Note that the cosine 

 curve (fig. 3-23) is just the opposite of the sine 

 curve (fig. 3-20) but is otherwise identical in 

 shape. 



Just as the sine curve was superimposed 

 on the GZ curve, so may the cosine curve be 



superimposed on the stability curve to show the 

 effect on stability of moving a weight athwart- 

 ship. The cosine curve has been placed on the 

 original stability curve, corrected for the actual 

 height of the center of gravity. The dotted area 

 (fig. 3-23) is that portion of the curvewhich was 

 lost due to the weight shift, and the lined area 

 is the remaining or residual portion of the curve. 

 The residual maximum righting arm js AB 

 which develops at an angle of about 37 . The 

 new range of stability is from ?0° to 50°. 



The new curve of static stability can be 

 plotted on the base by transferring down the 

 intercepted distances between the cosine curve 

 and the old GZ curve. For example, in figure 

 3-24 the old righting arm at 37° was AD, the 

 loss of righting arm (inclining arm) at this 

 angle is AC, leaving a residual GZ of CD. This 

 value has been plotted up from the base as AB 

 to provide one point on the final curve. The 

 residual stability may be analyzed on the new 

 curve as follows: 



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