Sec. 62.3 



ESTIMATE OF ADDED LTOITID MASS 



431 



represents the length of the ship; see Fig. 37 on 

 page 472 of SNAME, 1955. 



It now becomes necessary to return to a con- 

 sideration of the second method for determining 

 the added mass of the entrained liquid aroimd the 

 circular rod of diagram 2 in Fig. 62.E, when 

 flexing in 2-noded vertical vibration. F. M. Lewis 

 adopted the schematic method shown in that 

 diagram because he felt that the possible error 

 involved in assuming pure shear deflection, rather 

 than bending deflection, was less than the error 

 involved in shifting from the ellipsoid of revolution 

 (or its lower half) to the actual underwater hull 

 of a ship, at least for the proportions correspond- 

 ing to those of a ship. However, in 1930 J. Lock- 

 wood Taylor published the values of a reduction 

 factor for an elHpsoid of revolution, vibrating 

 vertically in an ideal liquid with 2 nodes and 3 

 loops, by considering pure bending deflection 

 rather than pure shear deflection. This means 

 that, as indicated in diagram 5 of Fig. 62. G, the 

 transverse planes separating the several segments 

 remain normal to the bent axis of the ellipsoid. 

 The segments are thus thin on the concave side 

 and thick on the convex side, as they are in a 

 simple bent beam. Taylor found reduction-factor 

 values as much as 8 per cent below those of Lewis. 



A graph of J. L. Taylor's factor for 2-noded 

 pure bending is published by C. W. Prohaska 

 [ATMA, 1947, Fig. 25, p. 197]; also by R. T. 

 McGoldrick and V. L. Russo [SNAME, 1955, 

 Fig. 35, p. 471]; it appears in TMB Report 739, 

 October 1953, Fig. 2 on page 14. It is presented 

 here, along with those of Lewis, in Fig. 62.1, 

 supplemented by Taylor's values for 3-noded 

 bending vibration of an ellipsoid, given on page 

 170 of his 1930 INA paper, reference (9) of Sec. 

 62.8. Prohaska's diagram indicates graphically 

 that the reduction factors are for an ellipsoid 

 flexing in 2-noded vibration; those of TMB 

 Report 739 and of the 1955 SNAME reference 

 do not. 



Since the ship may be assumed to bend more 

 nearly hke a simple beam when vibrating ver- 

 tically, at least in the lower modes of vibration, 

 with 2, 3, and possibly 4 nodes, and since the 

 data of J. L. Taylor for an ellipsoid vibrating in 

 this manner are available, it is present practice 

 (1956) to use Taylor's (smaller) reduction factor 

 for 2-noded vibration rather than that of Lewis. 

 Many years ago E. B. Moullin pointed out that 

 Lewis' added-mass values had to be reduced by 

 10 per cent [INA, 1930, p. 179]. It is not clear 



whether he had in mind what has later come to be 

 the difference between Lewis' reduction factor 

 for pure shear and J. L. Taylor's reduction 

 factor for pure bending, or some other effect. In 

 any case, a recent re-analysis of model and ship 

 vibration data indicates that the method of 

 F. M. Lewis, combined with the reduction factor 

 of J. L. Taylor for 3-diml flow, give values of the 

 added weight of the entrained water around a 

 vibrating ship which are still too high, at least 

 for 2- and 3-noded vertical vibration [McGoldrick, 

 R. T., and Russo, V. L., SNAME, 1955, p. 490]. 

 Further study and analysis are required before 

 additional refinements in these prediction methods 

 can be attempted. 



A comparison of the procedures followed by 

 F. M. Lewis, C. W. Prohaska, K. Wendel, and 

 others in predicting the added mass of the 

 entrained liquid about the underwater hull of a 

 ship, vibrating vertically in its fundamental 

 2-noded or 3-noded frequency, indicates a number 

 of discrepancies with the assumptions of Sec. 

 62.1 which are not discussed there: 



(1) Although the limiting conditions set up for 

 the conformal transformation appear to take 

 account of the free-surface effects, as do the 

 electric analogies set up by J. J. Koch, it is by no 

 means clear that these cover adequately the 

 situation for ship-shaped sections, even in an 

 ideal liquid having mass but no viscosity. For 

 example, most of the ship sections developed by 

 Lewis and mentioned in the reference have 

 vertical sides at the waterline although most of 

 those developed by Prohaska do not. There is a 

 question whether full compensation has been 

 made in the analytic process for the flare or 

 tumble home to be found on actual ships in this 

 region. K. Wendel comments that in Koch's 

 experiment "••• it is possible to satisfy the 

 boundary condition only approximately; •••" 

 [TMB Transl. 260, p. 34]. 



(2) The analytic method described takes it for 

 granted that the ship, built to the shape shown 

 by the lines, has boundaries that remain rigid 

 locally, even though the ship flexes as a whole. 

 It is perfectly possible, and .indeed quite probable, 

 that some flat or nearly flat areas of hull plating, 

 lying generally normal to the direction of vibra- 

 tory motion, "give" or yield or deflect under the 

 external accelerative pressures and forces by 

 amounts of the same order of magnitude as the 

 hull deformations. Carried to the limit, one would 



