Mooring and Positioning of Vehicles in a Seaway 



The equations of motion in regular waves, for six degrees of 

 freedom, are formulated according to linear theory by the balance 

 of inertial, damping, restoring, exciting, and coupling forces and 

 moments. Both hydrodynamic and hydrostatic effects due to the body- 

 fluid interaction are included in the analysis, together with the 

 influences of the naooring system. The longitudinal motions (heave, 

 pitch, and surge) are coupled to each other, and similarly, the 

 lateral motions (sway, yaw, and roll) are also coupled. There is 

 no coupling between the two planes of motions, in accordance with 

 linear theory. 



The hydrodynamic forces and nnoments such as damping, 

 exciting effects due to waves, etc. , are determined by application 

 of the methods of slender-body theory. Essentially, this theory 

 nnakes the assumption that, for an elongated body where a transverse 

 dimension is small compared to its length, the flow at any cross 

 section is independent of the flow at any other section; therefore, 

 the flow problem is reduced to a two-dimensional problem in the 

 transverse plane. The forces at each section are found by this 

 method, and the total force is found by integrating over the length 

 of the body. A description of the application of slender-body theory 

 to calculate the forces acting on submerged bodies and surface ships 

 in waves is presented in [ 2] , where simplified interpretations of 

 force evaluation in terms of fluid momentum are also given. The 

 hydrostatic and mooring forces and moments are combined with the 

 hydrodynamic terms, resulting in linear combinations of terms that 

 are proportional to acceleration, velocity and displacement in the 

 various degrees of freedom. All of these expressions, when related 

 to the appropriate ship inertial reactions by Newton's law, lead to the 

 set of six linear coupled differential equations of motion. 



Solutions of the equations are found for regular sinusoidal seas 

 with varying wave length and heading relative to the barge. The 

 response amplitude operators are found from these solutions together 

 with the phases of the motions relative to the systemi of regular waves. 

 Assuming a knowledge of the oncoming irregular sea conditions (e.g. 

 in terms of sea state, as specified by an associated surface-elevation 

 energy spectrom from information in [ 3] ) , the set of energy spectra 

 for the ship motions are determined. Information on average values 

 and probabilities of relatively high values of the amplitudes of oscil- 

 lations in the ship-motion time histories for the different degrees of 

 freedom are found fronm the ship-motion energy spectra in accordance 

 with the methods of [ 1] . Cross- spectra are also used to determine 

 the energy spectra and hence the various average values and the pro- 

 babilities for the remiaining quantities of interest, such as load- 

 displacement time histories and other quantities which are linear 

 combinations of the ship motions and their time rates of change (the 

 presence of lowering lines for placing loads on the ocean floor is 

 considered in this analysis). These energy spectra may also be 

 obtained from the solutions of the differential equations by linear 

 superposition, and explicit use of cross-spectra here is necessary 



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