Vien and Lee 



frequency. Thus, introduction of supplemental damping into the equa- 

 tion of motion is needed to achieve a reasonable prediction of cata- 

 maran motions. The supplemental damping is introduced into the 

 equations of motion in a form linearly proportional to the oscillation 

 velocity. This damping is found to depend on the ratio of the ship speed 

 to the celerity of motion-generated waves. This fact implies that the 

 interaction between the wave systems, created by oscillation and for- 

 ward speed, is important and should be included in the evaluation of 

 damping. 



Prediction of statistical averages of motion amplitudes for 

 catamarans in irregular seas is made by using the frequency respon- 

 se-amplitude operator in conjunction with the Pierson and Moskowitz 

 (1964) sea spectrum. The probable frequency of a water contact with 

 the cross-deck structure is computed for given conditions such as 

 height of the cross-deck structure from the waterline, significant 

 wave height, and forward speed. The formula used is based on a 

 truncated Rayleigh's probability distribution for slamming and is 

 similar to the formula developed by Ochi (1964) for bow slamming 

 of monohull ships. 



The expressions are developed for various loadings contri- 

 buted by inertial and hydrodynamic effects, such as bending and 

 torsion moments and shear forces on both the cross-deck structure 

 and the supporting strut (shear and bending only). 



Presentation of the work on motion of catamarans is given 

 in the following order. In section 2, the subjects covered are : 

 formulation of equations of motion, derivation of the hydrodynamic 

 coefficients and derivation of an expression to estimate the number 

 of slammings of the cross-deck structure per given period in regu- 

 lar and irregular waves. In Section 3, the derivation of expressions 

 for various hydrodynamic loads on catamarans is given. In Section 

 4, a presentation of comparisons of theoretical and experimental 

 results is made, and concluding remarks are given. 



1.2- Motion . 



Equations of Motion 



The assumptions or conditions made in this paper for stu- 

 dying motions of catamarans are as follows. A catamaran which is 

 made of two symmetrical hulls is cruising with a constant speed, 

 while it is experiencing an undulatory motion due to sea waves. The 

 sea waves are assumed to be made of a linear sum of unidirectional 



468 



