Sec. 62.S 



ESTIMATE OF ADDED LIQUID MASS 



43'J 



Schematic Flexure f^ttern of hr 

 in Vertical Vibration 



/ 



For Outer Portion 

 JL,qu,d = (°0625)TT^(a^-b^)^ 

 per Unit Lenqth 



Fig. 62.N Method of Estimatino the Added 



Liquid Mass fob a Large, Thin, Cantilever 



Structure in Lateral Vibration 



Cantilever structures, comprising a category 

 which embodies most of the appendages listed, 

 sway with a motion somewhat resembling that of 

 a tree in a gusty wind, illustrated schematically 

 in Fig. 62.N. The root of the cantilever is rela- 

 tively rigid, so that most of the motion occurs 

 near the tip, as with the Washington skeg men- 

 tioned previously. In the absence of a specific 

 analytic solution covering this case, a reasonable 

 approximation is achieved by assuming that the 

 cantilever has a semielliptic section and that it is 

 hinged to the hull at its midlength, at a point 

 corresponding to the root attachment of the 

 cantilever. This produces a motion, approximately 

 normal to the plane of the thin appendage, which 

 is greater than that of the cantilever structure 

 near the root but less at the tip. Experience 

 indicates that the kinetic energies and added 

 masses in the two cases are of the same order of 

 magnitude. 



From the second diagram at the top of Fig. 

 62.A a 2-diml elUptic-section cylinder in unsteady 

 oscillatory motion about an axis at its center has 

 an added mass moment of inertia J l of 

 (0.125)7rp(a^ — h^Y per unit length, where a is 

 half the semimajor axis and h is half the semiminor 

 one. 



As an example of this method, take the case of 

 the thin, vertical centerline skeg mider the transom 

 stern of the ABC ship of Part 4, for which sections 

 are indicated in Fig. 66.P and a profile in Fig. 

 66.Q. This skeg is of variable depth, below the 



hull, but it may be assumed from Fig. 66.Q to 

 have an average depth, normal to the centerline 

 buttock in the vicinity, of about 16.5 ft. This is 

 the semimajor axis of an equivalent semielliptic 

 section. The semiminor axis is estimated from 

 Fig. 66. P as 2.5 ft. Then, for standard salt water 

 at 59 deg F, 15 deg C, and for half of the elliptic 

 section, the moment of inertia of the added- 

 liquid mass, about the point of attachment of 

 the skeg to the hull, is 



Jl = (0.0625)7rp(a' - hy 



= 0.0625(3. 1416)I.9905[(16.5)' - (2.5)']' 

 = 27,654 slug-ft' per ft length, 



the latter reckoned generally parallel to" the 

 centerline buttock in the vicinity. 



62.8 Partial Bibliography on Added-Mass and 

 Damping Effects. There are hsted here a number 

 of references in the technical literature relating to 

 the added or entrained masses around bodies, 

 ships, and typical forms of interest to the marine 

 architect. This list includes the references men- 

 tioned throughout the present chapter. 



An excellent historical summary of the develop- 

 ment of means for evaluating and taking account 

 of the added mass of entrained water around a 

 vibrating ship is given by R. T. McGoldrick in 

 the introduction and text of TMB Report 395, 

 issued in February 1935. The bibliography on 

 page 30 lists most of the early papers of import- 

 ance, beginning with that of Otto Schlick in 

 INA, 1884. 



A bibhography on vibration, containing 73 

 references, was collected by the SNAME Hull 

 Structure Committee and pubhshed as part of the 

 work on Project S-7 in SNAME Bulletin, Janu- 

 ary 1952, pages 14-15. A supplementary list of 

 30 references was published in SNAME Bulletin, 

 October 1952, page 27. 



References pertaiiiing to added-hquid mass 

 effects, published subsequent to 1924, include: 



(1) NichoUs, H. W., "Vibration of Ships," INA, 1924 



pp. 141-163 



(2) Taylor, J. L., "Ship Vibration Periods," NECI, 



1927-1928, Vol. 44, pp. 143-176 



(3) Cole, A. P., "The Natural Periods of Vibration of 



Ships," lESS, 1928-1929, Vol. LXXII, pp. 43-86 



(4) Kempf, G., and Helm, K., "Auslaufmeeaungen am 



Sehiff und am Modell Dampfer Hamburg (Retarda- 

 tion Measurements on the Ship and on the Model 

 of the Steamer Hamburg)," WRH, 7 Sep 1928, 

 pp. 336-340. These authors found a virtual-mass 

 coefficient for retardation and straight-ahead 



