424 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 623 



general nature of the flow pattern for vertical and 

 lateral unsteady motions, corresponding to 2-noded 

 vibration in vertical and horizontal planes. The 

 problem is how to estimate or to derive numerical 

 values for the added mass of entrained liquid, or 

 for the added mass moment of inertia, around an 

 actual surface-ship hull in unsteady motion in 

 any or all of its degrees of freedom. 



The problem of most frequent application, and 

 at least of major interest, is that of finding the 

 added- or virtual-mass coefficients for a ship in 

 vibration. This involves a separate — and addi- 

 tional — group of degrees of freedom. Many years 

 ago H. W. NichoUs found that, for a rectangular- 

 section model vibrating vertically, the added- 

 mass coefficient was approximated by 



Cj^M = 0.37 I + 0.20 



(62.i) 



where B was the beam and H the draft of the 

 model. For the model which he used. Cam worked 

 out as 0.78. For a triangular-section model he 

 found it to be 0.70 [TMB Rep. 395, Feb 1935, 

 p. 9]. 



Much later it was stated by F. H. Todd 

 [SNAME, 1947, Vol. 55, p. 160] that, for vertical 

 vibration only: 



"In calculations on the natural frequency of ship hulls 

 (in 2-noded vibration), the total virtual mass of the hull 

 and water together varies from 2 to 4 times the ship 

 displacement. The variation is linear with beam-draft ratio, 

 and is given approximately by the line 



Virtual inertia factor = 



+ 1.2 



The virtual inertia factor is defined as the ratio of (dis- 

 placement + entrained water) to displacement." 



Here the virtual inertia factor corresponds to the 

 virtual-mass coefficient Cvm ■ The added-mass 

 coefficient Cam is Cvm — 1.0 or 



Ca 



-1-0.2 



(62. ii) 



where 0.2 is the viscous-resistance or damping 

 factor. 



For the ABC ship of Part 4, the added-mass 

 coefficient approximated by this method would be 



Cam = !(§) + 0-2 = 0.936 + 0.2 = 1.136 



For the Gopher Mariner at the heavy displace- 

 ment [SNAME, 1955, pp. 436-494, esp. pp. 451, 



473], at a mean draft of 24.42 ft, the corresponding 

 value by this method is 



^AM — 



+ 0.2 = 1.037 -f- 0.2 = 1.237 



However, the data from Table 1 of TMB 

 Report 1022, of May 1956, based on the 1929 

 method of F. M. Lewis and the reduction factor 

 of J. L. Taylor for 2-noded bending of an ellipsoid 

 [SNAME, 1955, p. 471], give a Cam of only 

 0.978 for 2-noded vertical vibration. 



Supplementing the foregoing, an excellent 

 summary of the adaptation of knowledge con- 

 cerning the added mass of the entrained water to 

 ship-vibration problems was given by F. H. 

 Todd in 1947 [SBMEB, May, Jun, Jul, 1947, 

 Vol. 54, pp. 307-312, 358-362, 400-403, respec- 

 tively; ASNE, Feb 1948, pp. 86-110, esp. pp. 

 101-104]. 



The use of a straight-line function embodying 

 the B/H ratio as a means of determining the 

 added-liquid mass for vertical vibration, de.'jpite 

 its unexpected applicability to many types of 

 ship, can at best be considered only a first approxi- 

 mation, to be used in an early stage of the design 

 when the hull form is not yet known. Moreover, 

 it takes no account of the fore-and-aft distribu- 

 tion of the added mass of entrained hquid, other 

 than to assume that it is the same as on all other 

 ships for which reliable data are available. 



A somewhat different method, for use after 

 the ship is structurally complete but before it is 

 placed in commission, is perhaps more applicable 

 to ship appendages than to hulls proper. It 

 involves a comparison of the resonant frequency 

 of the appendage structure in air with the reson- 

 ant frequency of the same structure in water. 

 The structure can be set in motion by a vibration 

 generator, first when the ship is on the building 

 ways or in the building dock, with the appendage 

 surrounded by air. It is again vibrated when the 

 vessel is in the water, with the vibration generator 

 (above the water) imparting its periodic forces 

 through some kind of flexible connection with 

 the appendage. 



Then 



(.An A, J' 



Uin WateJ 



(62.iii) 



(.A.A,>)^ 



