Sec. 62.5 



ESTIMATE OF ADDED LIQUID MASS 



433 



If a rigid boundary is introduced under a 

 decelerating ship, as when it runs suddenly into 

 shallow water, the kinetic energy in the unsteady 

 flow around the ship is increased, probably 

 because the particles are no longer free to follow 

 a minimum-energy pattern. Hence, the added 

 mass of the entrained water begins to increase 

 appreciably by the time the bed clearance under 

 the ship has diminished to less than its mean 

 draft. This is in accordance with the results 

 derived analytically by H. Lamb, L. M. Milne- 

 Thomson, and others, which indicate that the 

 added mass of entrained liquid increases as a 

 solid boundary of infinite extent is approached. 



If the decelerating body suddenly approaches 

 a limiting vertical boundary of limited extent 

 but of large proportions compared to itself, such 

 as a small tug which surges up to a large ship at 

 too high a speed, or an exercise torpedo which 

 runs into the side of a hull, the analytic study 

 indicates that the added mass of the water 

 around the smaller decelerating body should also 

 increase. However, it is reported that in one of 

 the few known cases where this theorem has 

 been applied in practice, the observed data 

 indicated that the added mass of the smaller 

 body was reduced, so that it became easier to 

 stop within a given distance or time interval. 

 The smaller body had the general form of an 

 ellipsoid of revolution and it was approaching 

 the larger body by a sidling motion. It is entirely 

 possible that other factors were present in the 

 latter case and were not taken into account. 



62.5 Estimating the Added-Mass Coefficients 

 of Vibrating Ships in Confined Waters. The 

 effect of shallow water upon a vibrating ship is 

 discussed in Sec. 35.13 of Volume I; Fig. 35. G 

 illustrates schematically the flow around the 

 sections of a ship form in vertical vibration. Sec. 

 35.14 explains that ship vibration, particularly 

 in the vertical direction, is greatly magnified in 

 shallow water. This is also discussed by F. M. 

 Lewis in a paper "Ship Vibration" [Proc. World 

 Eng'g. Cong., Tokyo, 1929, Vol. XXIX, Part 1, 

 publ. in 1931, pp. 203-204]. T. W. Bunyan states 

 that the various critical vibration frequencies 

 and amplitudes, in a transverse direction, are 

 also affected by restricted waters such as the 

 Suez Canal [IME, Apr 1955, Vol. LXVII, p. 100]. 



F. M. Lewis, in the reference cited, states that 

 from physical reasoning and analytic study the 

 inertial effect of the added mass of entrained water 

 is greater in shallow water than in deep water. 



Despite verification of the foregoing by the ex- 

 periments of J. J. Koch and C. W. Prohaska, to 

 be mentioned presently, and the full-scale tests 

 by R. T. McGoldrick on the Great Lakes ore 

 carrier E. J. Kulas [TMB Rep. 762, Jun 1951, 

 esp. Table 1 on p. 4 and p. 11], F. H. Todd and 

 W. J. Marwood report that, for one ship case 

 at least, the opposite result was found [NECI, 

 1947-1948, Vol. 64, p. D127]. 



Assuming an increase in added-liquid mass in 

 shallow water, the direct result of the increase in 

 total mass is to decrease the natural frequency 

 of the ship, so that resonant vibration in a 

 frequency range below that of the exciting forces 

 at the operating speed in deep water might occur 

 within that lowered range in shallow water. For 

 example, the blade frequency for the single-screw 

 drive of the ABC transom-stern ship of Part 4 

 at the designed speed in deep water, is, from 

 Fig. 78.Nb, Z(rpm) = 4(109.7) = 438.8 cycles 

 per min. The sixth-moded vertical vibration of 

 the hull, which for certain reasons might be 

 objectionable, is assumed to occur at 380 cycles 

 per min. At the sustained speed of 18.7 kt, this 

 is still well below the blade frequency at that 

 reduced speed, likewise in deep water. However, 

 when running in the river below Port Correo the 

 propeller rpm might be reduced to say 81, and the 

 blade frequency to 324 cpm. The shallow-water 

 effect on the added-liquid mass might be great 

 enough to lower the sixth-moded resonant fre- 

 quency from 380 to 324 cpm. Not only would 

 this be exactly in the running range but there 

 would be an enormous magnification effect with 

 the nominal bed clearance of only 4 ft under the 

 ship. K. Wendel mentions a case similar to this 

 on page 71 of TMB Translation 260, July 1956. 

 The problem of the naval architect is to deter- 

 mine, and if possible to predict in advance, the 

 magnitude and effect of the changes such as 

 this in added mass, frequency, and ampUtude. 



The specific information known to be available 

 concerning the effect of shallow water on the 

 added mass of the water around a ship, when 

 vibrating vertically, is limited to the experi- 

 mental data of: 



(1) Koch, J. J., "Eine experimentelle Method 

 zur Bestimmung der reduzierten Masse des 

 mitschwingenden Wassers bei Schiffsschwing- 

 ungen (Experimental Method for Determining 

 the Virtual Mass for Oscillations of Ships)," 

 Ing.-Arch., 1933, Vol. IV, Part 2, pp. 103-109; 



