Kaplan 



Figures 9 and 10 show, for each of the six barge motions, 

 the variation of total spectral energy with barge heading, for each 

 of the three sea states considered. The ordinate plotted for each of 

 the curves is the r.m.s, value, aj , for the time history of the barge 

 motion represented, and it will be convenient to refer to the variance 

 (Tj of the function as the total energy. The r.m.s. value of any 

 time-history function is therefore the square root of its total energy 

 (assuming, here, as always, the mean value of any time -history 

 function to be zero. 



Representative examples of calculated spectral energy density 

 functions for the case of the center-lowered load are presented for 

 two sea states and two barge headings relative to the predominant 

 wind direction in Fig. 11, The spectral energy density functions 

 shown for the load were calculated from those of the fundamental set 

 of cross-spectral energy density functions, i.e. those of the six 

 barge motions. Since the time histories of the load and amplitude 

 operators for the former may be obtained by forming appropriate 

 linear combinations of the complex response operators, Tj^, for 

 the barge motions, and calculating their squared absolute values. 



The r.m.s. values (as defined here) were obtained for all 

 quantities of interest for the load lowering operation such as dis- 

 placements, accelerations, tensions, etc, as well as the forces in 

 the mooring cables, for each sea state and barge heading relative to 

 the waves. Similarly variations of these quantities as a function of 

 the boom azimuth angle were found, from which an optimum boom 

 angle (which minimizes the r.m.s. values of any one of the time 

 histories of interest) may be determined. As an example of results 

 obtained for a 20C ton load lowered in a State 5 sea with a crosswind 

 barge heading and with the optimum boom azimuth angle (here 180°, 

 i.e. boom over the stern), the r,m.s. value of the added-dynamic 

 line tension given by Fig. 12 is (2. 38)(200)/32. 2 = 14.8 tons. From 

 data on the normal probability curve for this r.m. s, value, it can 

 be shown that the downward force of impact on the bottom would 

 exceed 25 tons approximately 2.3% of the time, if the instant of im- 

 pact were allowed to occur at random. For a center-lowered load 

 under the same conditions, the r.m.s. value of its acceleration is 

 1 . 17(200)/32. 2 = 7.27 tons, and the downward impact force on the 

 bottom would exceed 14.3 tons approximately 2.3% of the time. 



The r.m.s. value of the fluctuating component of the force in 

 each of the four mooring cables is shown in Fig. 13 as a function of 

 barge heading for each sea state. The four are seen to have nearly 

 the same r.m.s. value for any particular barge heading in a Sea 

 State 3, with the actual values varying between 300 and 600 lb. For 

 a Sea State 4 the range is from 1100 to 1750 lb, with the differences 

 between r.m.s. values for the four fluctuating cable forces being as 

 much as 150 lb. For Sea State 5, the range is from 1800 to 2700 lbs 

 with differences in r.m.s. values between cables of 250 lb. In all 



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