Ogilvie 



how they resolve the problem left over from the work of Peters and Stoker: How 

 do we explain away the predicted infinite amplitudes of motion at resonance? 



In Newman's formulation, the lowest order forces have the following orders 

 of magnitude: 



excitation by incident wave, /3S; 



hydrostatic restoring force, /0y; 



ship inertial reactions, /3y; 



added mass and damping forces, ji'^y . 



Away from resonance, the excitation must be equal to the sum of hydrostatic 

 plus ship inertia forces. Under these circumstances, it is clear that we can set: 



7 = S + smaller terms. 



At resonance, the forces of order (iy total zero, and so the excitation force must 

 eqvial the added mass and damping forces, which are the lowest order non- 

 vanishing forces. Therefore, at resonance 



7 = — + smaller terms. 



Since y must still be a small parameter, we must require that S « /3, if the 

 perturbation analysis is to remain valid. Such a requirement appears reason- 

 able. 



As a practical approach, if we were to try to use Newman's formulas for 

 the forces, we could now follow the procedure which is usual in perturbation 

 analyses, viz., absorb the small parameters into the force and motion variables. 

 We would then calculate the forces, including the higher order added mass and 

 damping forces, and from these calculate the motions. Away from resonance, 

 the higher order contributions should be negligible (if the conditions of the the- 

 ory are really satisfied), and the results should reduce to those of Peters and 

 Stoker. At and near resonance, the higher order forces should dominate the 

 lower order forces and control the predicted responses. 



This approach is logical, at least insofar as a ship may really be consid- 

 ered as thin, but the results are not very usefvil because of their complexity. 

 The damping coefficients are the only elements of the problem which fall out in 

 a fairly simple fashion, and it has already been seen that at least some of these 

 can be calculated in a much simpler manner, from radiation considerations. In 

 order to evaluate the potential usefulness of the thin ship idealization in predict- 

 ing ship motions, some calculations of damping coefficients have been made for 

 Series 60 models and compared with experiments by Gerritsma, Kerwin, and 

 Newman (1962). Figure 7 shows some typical results from their paper. The 

 heave damping coefficient (b3 3) is plotted against frequency for a sequence of 

 values of Froude number. The ship concerned is the Cg = 0.60 form of the 

 Series 60. The agreement is at least qualitatively good. 



56 



