Understanding and Prediction of Ship Motions 



second derivative of displacement, the equation should be of second order. In 

 other words, it should contain a term mxg, where m is the mass of the ship. The 

 differential equation to be chosen is now not unique, without further considera- 

 tions, whereas Eq. (2) above was uniquely determined by the value of A3 3(0)). 

 Now, any equation of the form 



a X3 + b X3 + ex = f(t) (3) 



will suffice, provided only that 



A33 = ioib + (c - co^a) . 



The only a priori restriction on the values of a and c is that the linear combina- 

 tion (c - co^a) should have the proper value. 



It is here that a physical idea is introduced. We know that if a ship is given 

 a steady displacement in heave from its equilibrium position, there will be a 

 steady restoring force approximately proportional to the amplitude of the dis- 

 placement. We let the quantity ex denote this steady force component, and we 

 note that c is independent of frequency, co. Then the parameter a can be uniquely 

 determined from A3 3. 



Quite often the quantity ex is referred to as a buoyancy force, but it must 

 be recognized that this is not entirely true. It is easily shown experimentally 

 that c varies with speed, and it does in fact include hydrodynamic as well as 

 hydrostatic effects. 



The quantity a will usually (but not always ! ) be found to be larger than the 

 ship mass, m. It is then common to define an "added mass" equal to (a - m); 

 this is the apparent increase in inertia which the ship experiences because it is 

 accelerating the surrounding water. Of course, it is not a quantity which is 

 characteristic of the ship, for in fact it depends on frequency. 



Finally, the quantity b, which is viniquely determined from the value of A33, 

 can be considered as a damping coefficient. This is easily seen from Eq. (3). 

 At least in the case of heave motion, most of the damping will appear physically 

 in the form of radiated waves, and this quantity can be more reliably calculated 

 than any of the other parameters considered here. 



The major advantage of this approach is that to some extent the dynamics 

 of the ship itself can be separated from the hydrodynamic problem. This ap- 

 pears most clearly when the above ideas are extended to include all six degrees 

 of freedom. In the rotational modes, in particular, the moments of inertia can 

 be varied easily without changing the hull shape or the hydrodynamic forces or 

 moments. If the coefficients A.^^ are all known and if they have been broken 

 down into hydrodynamic and ship inertial components, the changes in Aj^ (and 

 thus in the motions) due to variations in mass distribution can be calculated. 



Furthermore, ship motions are often most critical near resonance, for then 

 they are largest in amplitude. Near resonance, the amplitude is very largely 

 controlled by the amount of damping, and it is the damping which is most readily 

 calculated in the above framework. 



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