APPENDIX A - OSCILLATION OF A SHIP WITH 

 SIX DEGREES OF FREEDOM 



Consider a ship floating on a uniform stream and making oscillations 



of small amplitude about a fixed point. Take the x-axis in the direction 



of the uniform stream and the z-axis vertically upwards. The fluid motion 



around the ship is expressed by the velocity potential of the form 



= Ux + U<J) + <}> 



(301) 



where U is the velocity of the uniform flow and (J)., gives the periodical 



disturbance. Since the amplitude of oscillation is small, the steady 



potential, Utj)., is determined from the boundary condition when the hull is 



fixed at an average position and identical with the velocity potential for 



the uniform motion of the ship. Now we consider the coordinate system 



x_, y fi , z , which is fixed to the ship with the origin at the center of 



gravity. The coordinates of the center of gravity at the average position 



are written as x , y , z , and the displacement of the center of gravity 

 (j (j (j 



as 5-j » 5o» Kr,- Designating the angle of rotation about each axis by 

 1 , 6„, 6„, and omitting higher order terms, one can write 



x o = x " x g " h + 



> 3 <y-y G > - e 2 (z-z G ) 



o 



y - y G - C 2 + e i( z_z G ) - e 3 (x-x G ) 



z n = z - z r ~ ^i + 6 9 (x-x p ) - 8 n (y-y_) 



(302) 



If the hull surface is expressed by the equation 



F(x ,y ,z ) = 



(303) 



the boundary condition on the hull surface is given by the equation 



107 



