Motion and Resistance of a Low-Waterplane Catamaran 



to) 



where A- k , B^ and F, are frequency and speed dependent. 



For slender catamarans we can assume that no coupling 

 exists between the motions on the horizontal and vertical planes so 

 that 



A ik ■ B i k ■ ° 



for the following combinations of i and k 

 i = 2, 4, 6 for k =3, 5 



or 



= 3, 5 for k = 2, 4, 6 



As has been seen previously, the hydrodynamic coefficients 

 appearing in the equations of motion can be obtained, if the solution 

 of the velocity potentials (^ (i = 2, . . . , 6) are known. In the solu- 

 tion of Cp-j , the flow around each transverse section is assumed 

 to be two-dimensional, and, thus, the variable x enters as a para- 

 meter in the expressions for 0j_ . A correct mathematical deve- 

 lopment to lead to such an approximation from the slender-body 

 theory is given by Ogilvie and Tuck (1969), and a comprehensive 

 review of this approximation is given in Newman (1970). 



Approximation of three-dimensional flow by two-dimensio- 

 nal flow as described previously is often called the strip approxima- 

 tion. The strip approximation of the hydrodynamic coefficients has 

 been quite successful in the prediction of motions of monohull ships. 

 Regardless of the possibility that the two-body interference of the 

 flow between the hulls may weaken the two-dimensional approxima- 

 tion, the strip approximation is applied in this work to check both 

 the reliability of the approximation and the areas which might be 

 improved should the approximation prove unreliable. 



The assumption of linear excitation-response relationship 

 and the strip approximation lead finally to determination of the hydro- 

 dynamic coefficients shown in Table 1. The solution of two-dimen- 

 sional added inertia and damping is obtained by using the method of 

 source distribution. The description of the method for heave added 

 mass and damping is given in Lee, Jones, and Bedel (1971). 



For illustration, the derivation of A 53 , B 5 _ and B 26 

 will be shown here. From Equation (74) and the strip approximation, 

 we have 



533 



