304 



THEORY OF SEAKEEPING 



conditions of submergence, velocity, and acceleration at 

 each ship section while computing these latter on the 

 basis of the linear theory of motions. Only a pioneer- 

 ing attempt in this direction was made by Dalzell (1959) 

 and further work of this type is suggested. 



8.5 Ship Motions in Seven Degrees of Freedom. '^^ 

 Measiu'ements of stresses on ships at sea (E. V. Lewis, 

 1957c), shown in Figs. 46 and 47, indicate frequent 

 occurrences of high bending stresses in other than head 

 waves. The necessary material for rational analysis is 

 completely lacking in this case. First of all, a reliable 

 knowledge of the directional sea spectrum is necessary 

 and is not yet available. Next, the calculations of ship 

 motions in se\'en degrees of freedom'^ must be formu- 

 lated. The formulation of the ecjuations of motions 

 has been accomplished by Kriloff (3-1898) and is also 

 found in aeronautical literature. A large amount of 

 theoretical and experimental research is needed, how- 

 ever, for evaluation of the necessarj^ coefficients of 

 differential equations of motion. This subject was dis- 

 cussed in greater detail in Chapter 3. However, the 

 subject is again emphasized here because the data of 

 E. V. Lewis' (1957c) report indicate the significance of 

 oblique sea conditions in causing large ship bending 

 moments. 



8.6 — Slamming: 8.61 — Bottom impact. The problem 

 of ship slamming has l>een treated quite inadequately in 

 the past. No clear distinction has been made between 



(a) the high pressure significant for the bottom plating, 



(b) the total bottom impact force causing ship vibrations 

 and affecting ship bending moments, and (c) the force 

 resulting from the submersion of a flared bow of ships 

 designed for high speed. The loads, listed under (6) 

 and (c), are applied to a ship's structure with such a 

 rapidity that ship vibrations are excited. The relation- 

 ship between the load and the stress cannot be deter- 

 mined in this case by the rules of statics and it is neces- 

 sarj^ to investigate the elastic response of a ship's struc- 

 ture. It appears that little attention has been given 

 so far to the study of these elastic effects. 



Theoretical evaluation of bottom pressures in slam- 

 ming has been developed only in the past few years by 

 Szebehely and his associates at the David Taylor jNIodel 

 Basin. This activity was based on theories of H. Wag- 

 ner which had been found valid in seaplane engineering 

 for surfaces of large deadrise (say over 10 deg). In the 

 case of ships, the theory apparently gave good results 

 in the estimate of the total impact force when the edges 

 of the wetted area reached the turn of the bilge. It 

 indicated, however, peak pressures sometimes exceeding 

 1300 psi, which apparently are many times higher than 

 those measured at sea or estimated from ship-damage 

 observations. 



Attention should be called to the fact that these high 

 pressures are indicated both by Wagner's expanding- 

 plate and Wagner's spray-root theories. The fii'st of 

 these, however, is intended to represent the total force 



^^ Surge, sidesway, heave, rolling, yawing, pitching and rudder 

 motion. 



acting on a plate, but does not represent correctly the 

 flow condition at the edge of the wetted area. The 

 spray-root theory represents these conditions correctly, 

 provided the deadrise angle is not too small. It is a 

 potential-flow theor}' and it remains valid as long as the 

 fluid velocities change not too rapidly with distance; 

 i.e., the velocity gradient is small. If the velocity 

 gradient is large, the \'iscosity becomes significant and the 

 potential flow breaks down. These conditions can be 

 expected to occur at a spray root of a V-section with a 

 \-ery small deadrise angle. Flat bottoms or bottoms 

 with excessivel.y small deachise should be avoided in the 

 fore parts of ships. As long as they are used, however, 

 research on the impact of such surfaces is recommended. 

 It should represent an extension of Wagner's work but 

 should not be an indiscriminate application of it. The 

 physical conditions existing at a low deadrise should be 

 taken into account. 



The application of Wagner's expanding-plate theory 

 to the calculation of the total impact force involves the 

 concept of an added mass. Very high forces at low 

 deadrises result from the consideration of the added 

 masses of water as though they were real masses of a 

 fixed magnitude. In this approach a force of infinite 

 magnitude is predicted in the case of a flat-bottom 

 impact. After reaching this point, the writers on the 

 subject usually invoke the elasticity of water or of a 

 body's structure in order to explain this physically im- 

 possible result. However, a quantitative analysis of 

 the effect of these elasticities is lacking. Two alternate 

 research programs can be suggested here: (a) The de- 

 \'elopment of an impact theory in which the h.ydrody- 

 namic mass is deri\'ed on the basis of true physical con- 

 ditions^" and is not assumed in advance, (b) The 

 quantitative evaluation of the effects of water's and 

 ship structure's elasticity. 



The research directed to obtaining reliable experi- 

 mental data on impact pressures is recommended. As 

 this was suggested in Chapter 2, the mean pressure over a 

 certain bottom area should be measured rather than a 

 peak pressure at an isolated gage. The size of the area 

 should correspond to a typical area of a ship's bottom 

 plating between supports. 



8.62 Slamming — submersion of a flared bow. In 

 the foregoing paragraphs attention was concentrated on 

 the aspects of slamming which affect the bottom plating 

 and which define the total forces resulting from bottom 

 impacts of cargo ships. It is suggested that such cal- 

 culations as were made by Bledsoe (3-1956) for cargo- 

 type ships be extended to cover the full submersion of a 

 flared destroyer bow. In this case the impact lasts an 

 appreciable amount of time" and the impact force must 

 he presented as a function of time. Typical combina- 

 tions of the .ship and wave motions should be assumed. 

 These can be obtained either fiom model tests or from 



"> Not forgetting the extremely rapid changes of the flow pattern 

 with time, and the effects of viscositj- when the velocity gradient 

 l)ecomes excessive. 



''' Of the order of 1 sec. 



