Motion and Resistance of a Low-Waterplane Catamaran 



regular waves of different frequencies. The response of a catamaran 

 to these waves is assumed linear in amplitudes and frequencies. The 

 amplitudes of the waves and the motions are assumed to be small, 

 and, consequently, the fluid disturbance generated by the motions of 

 waves and ship is also assumed small. The depth of the ocean is 

 assumed infinitely deep and the effects of wind and current on the mo- 

 tion are not considered. 



Within a linear approximation of the motion and with the 

 conditions prescribed in this work it is convenient to choose Oxyz, 

 a coordinate system representing the mean position of the catamaran 

 as the reference frame for which the equations of motion are to be 

 formulated. When the catamaran has only steady translation, the 

 Oxz plane coincides with the longitudinal plane of symmetry of the 

 catamaran, the Oxy plane coincides with the calm water surface, the 

 Oz axis is directed upward, the Ox axis is directed toward the bow, 

 and the Oy axis is directed toward the port side. Since the wave 

 excitation is assumed to be of harmonic nature in time, the equations 

 can be formulated in a frequency domain. 



With the conditions stated in the foregoing paragraph, the 

 linearly coupled motion of a catamaran in six degrees of freedom 

 can be written with the motion-generated displacements from the 

 mean position denoted by £• ; where the values of i represent 1 for 

 surge, Z for sway, 3 for heave, 4 for roll, 5 for pitch, and 6 for 

 yaw, as 



6 



1^ i(\A 4. a W-ur t x r- t - Tr( e ) o" J wt 



k 



r, {( M i k + A ik >H + B i k * k + c i k * k = F r e " JW (i) 



for i = 1.2. . . . , 6. The equation shown above is a degenerate case of 

 the equations of motion of floating bodies in waves, formulated in the 

 time domain which has the form of integro -differential equations as 

 shown by Cummins (1962) and Ogilvie (1964). In Equation (l) M^ is 

 the mass or moment of inertia of the catamaran, A^ , the added iner- 

 tia, Bj_k , the damping, C^k , the restoring constants, Fj( e ', the wave 

 excitation in the form of complex amplitude, and j is the imaginary 

 unit associated only with a harmonic -time function. 



The expression "added mass (or inertia)" which will be 

 frequently referred to in this paper is used for mathematical conve- 

 nience. Thus, it does not have the same meaning as the classical 

 added mass which is an intrinsic property of the geometry of the body 

 only and is independent of motion, frequency, and forward speed. The 

 mathematical relation between the added mass of the classical defini- 

 tion given in Lamb's Hydrodynamics and the one referred to in this 



469 



