1886.] 



Straining of Ships caused by Rolling. 



25 



position, iv is the weight of the unit of length and its contents, and 



% -h 2 is the moment of inertia of the unit of length about the axis of 

 9 



rotation. 



In order to form an equation of energy and work, we require to 

 assume an axis of rotation for the ship ; and the assumption here 

 made is, that the axis of rotation is a principal longitudinal axis 

 through the centre of gravity Gr of the whole ship and her contents. 

 A ship's axis of rotation is not, in reality, fixed ; but that may for 

 the present be disregarded. The important point in connexion with 

 it is that, whatever position the instantaneous axis may occupy at 

 any given moment, it is the axis about which each unit of length of 

 the ship is then rotating, with the same angular velocity. This con- 

 dition follows from the rigidity of the ship, or rather from the 

 structure being so nearly rigid that any motion of one part relatively 

 to another, about the axis of rotation, is so v small that it may be 

 neglected. 



When the unit of length shown in section in fig. 1 is inclined to an 

 angle 6 from the upright, the principal forces which act upon it are — 

 first, the weight w of every part of the ship and her contents that is 

 contained in this length, acting vertically downwards through its 

 centre of gravity g ; and, secondly, the weight of the volume of dis- 

 placement d for the unit of length under consideration, acting ver- 

 tically upwards in a line, dm, through its centre. These forces are 

 equivalent to the couple d X gz, and a vertical force at g equal to 

 w—d. 



Let Gr be the point in which the axis of rotation through the 

 centre of gravity of the ship intersects the section in fig. 1. Then 

 the moment which resists the inclination of the section at any angle & 

 will be the resultant of the two coiiples dxgz and — (w—d)Ga. Let 

 w — d=8. The work done in inclining the unit of length in fig. 1 to 



© 



the angle of inclination 9 will be d\ gzdO—S\ GadO. If a curve be 



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constructed with the length of the ship for an abscissa line, and the 

 f© f© 



values of d\ gzdO— 8\ GadO for ordinates — these values being set off 

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at points in the length to which the sections for which they are cal- 

 culated correspond — it will represent the longitudinal distribution of 

 the work done in opposition to the action of the righting moments. 

 The base line in fig. 2 represents the length of the ship. Suppose 

 the first ordinate to be at the plane of division for which the section 

 of the ship is as shown at fig. 1, we then require to determine the 



value of d\ ' gzdO— h\ " GadO, at this section, which may be readily 

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done. 



