Kaplan 



■^ff- = -^ = - [ z + i cos a'Csin \ • ^ - cos >/ • e )] (41) 



where <^, 9, and \\t are the rotational barge motions, roll, pitch 

 and yaw, respectively, and the superscript y denotes the boom 

 azimuth angle. These quantites are derived on the same basis as 

 those for center-lowered loads, it being assumed that the boom 

 pivots about the ship CG. The instantaneous magnitudes of these 

 quantities thus appear as linear combinations of the instantaneous 

 ship-motion solutions. 



Each motion of the barge in response to a regular sinusoidal 

 wave having a given frequency and propagating in a given direction 

 will also be sinusoidal, of the same frequency, but will, in general, 

 possess a different phase. In addition, the amplitude of each motion 

 will, in general, differ from that of the wave, the ratio of the former 

 to the latter being a function of the wave frequency and the heading 

 of the wave relative to the heading of the barge, and this amplitude- 

 ratio function is known as the response amplitude operator for the 

 particular motion of interest. In order to arrive at an effective 

 characterization of the barge motions in a random sea, in which 

 case these motions themselves have a random nature, the function 

 known as the spectral energy density, or the energy spectrum, of 

 each motion must be found. This spectrum is a measure of the vari- 

 ation of the squares of the amplitudes of the sinusoidal components 

 of the motion, as a function of frequency and wave direction. The 

 total area under the spectral- energy density curve contains much of 

 the statistical information on average amplitudes, near-maximum 

 amplitudes, etc. , for the particular motion considered. For an 

 arbitrary motion, represented by the i-subscript, the energy 

 spectrum of that motion, due to the effects of irregular waves, is 

 given by 



$^*''^ (cu) = |Ti^(co)|^ A'(a)) (42) 



for a particular fixed barge heading in a unidirectional irregular 

 sea, where A (co) is the wave spectrum and | Tj- | is the response 

 amplitude operator for that heading. 



For computational purposes in the present study, the Neumann 

 Pierson spectral- energy description of the seaway has been adopted, 

 and calculations made for these particular sea states, corresponding 

 to three particular wind speeds. The following table illustrates the 

 conditions. 



1036 



