APPENDIX A 

 HYDROSTATIC RESTORING FORCES AND SPRING CONSTANTS 



Hydrostatic restoring forces and spring constants are computed for 

 the two-dimensional analysis under the following assumptions: 



(a) The body rotates about the origin of the coordinate system 

 and all forces and moments are computed about that point. 



(b) The body has vertical sides in the region of its waterplane. 



(c) All motions are small. 



1 . Sway Motion. 



In the horizontal plane the body is in neutral equilibrium. There- 

 fore, there are no hydrostatic restoring forces and 



"^11 " "^12 " "^13 " °' '■^■■^^ 



2. Heave Motion. 



Vertical displacement of the body results in a change in the buoy- 

 ant volume of the body and consequently a change in the buoyant force on 

 the body. Since this force must be perpendicular to the waterline, 

 there is no change in the horizontal force as a result of vertical dis- 

 placement and 



KH^^ = 0. (A-2) 



If one considers a small vertical displacement, 6y, there is a resulting 

 change in volume: 



6V = - 6yA (for 6y + upwards) . 



Here, A is the waterplane area. The vertical force then is: 

 w ^ 



F = m^^&y = - pgA^6y, 



^22 " P8\ " PS[x,3 - x^]. (A-3) 



In this equation x^ and x^^ denote the sides of the body as shown in the 

 Figure in this appendix. Since the vertical force may be regarded as act- 

 ing at the centroid of the waterplane area, x^^, the moment may be expressed. 



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