116 



THEORY OF SEAKEEPING 



« 2 



Ui 



a 



2 



-a 

 -o 

 < 



l_^ Poin+s on Grims Low 



*• Frequency Approximation 



Fig. 6 Variation of added mass coefficient with frequency 

 (from Golovato, 1957) 



cepted only afs a rough confirmation of the order of mag- 

 nitude of the added masses. The roughly indicated 10 

 per cent reduction in added masses is clearly caused by 

 the surface effects, since Prohaska's models spanned the 

 width of the test tank and three-dimensional effects were 

 absent. 



Dimpker's (19154) and Holstein's (1936) tests were 

 made at frequencies in the range estimated for ship mo- 

 tions in waves. These tests will be discussed in greater 

 detail later in connection with damping (in Section 3.2) 

 since damping forces as well as added masses were deter- 

 mined. Holstein's measurements of added masses ap- 

 pear to be too erratic to be useful. Wendel (1950), esti- 

 mated the experimental errors in these measurements 

 and demonstrated that added masses are considerably 

 smaller than indicated by the submerged-prism theory. 

 Dimpker, like Holstein, made tests at a series of fre- 

 cjuencies governed by the stiffness of the retaining 

 springs. He did not publish information on frequencies 

 directly, but on spring stiffnesses. The data given in 



the published paper do not appear to be sufficient to 

 calculate the frequencies. 



To summarize the results on the oscillation of partially 

 submerged prisms: None of the tests made so far gives 

 sufficient information on the frefjuency of oscillations to 

 permit evaluation of the added mass versus frequency 

 relationship. The tests have generally indicated that 

 experimentally' measured added-mass coefficients of 

 bodies on the water surface are smaller than those for 

 deeply submerged prisms. Evidently, additional ex- 

 perimental research is needed. All of the tests de- 

 scribed in the foregoing were made in small tanks and it 

 can lie questioned whether the test data were not af- 

 fected by wave reflections from tank ends. While wave- 

 absorbing beaches have been used in towing tanks for 

 many years, it was not realized until recently how diffi- 

 cult it is to prevent wave reflections. 



In the work just described a strictly pragmatical ap- 

 proach was taken. Reference should he made to Wein- 

 blum (1952) and Keulegan and Carpenter (1956) for 

 the less evident aspects of the inertial force and added- 

 mass concepts. In particular, for bodies at the water 

 surface the hydrodynamic force is connected with wave 

 formation. It depends therefore not only on instantane- 

 ous conditions but on the past history of motions as well. 

 Added mass becomes a definite concept only when corre- 

 lated with a definite tj'pe of motion. The added masses 

 in harmonic oscillation are not necessarily identical with 

 the added masses in, for instance, uniform acceleration 

 of a body. In the tests of Dimpker, Holstein, and Pro- 

 haska, the added masses were derived from the natural 

 period of decaying oscillations. It can be questioned 

 whether added masses so obtained are identical with 

 those occurring in sustained harmonic oscillations. 



3 Ship forms oscillating on the water surface. 

 Golovato (1957a, 6) reported on experiments with a 

 harmonically heaving ship model restrained from pitch- 

 ing." The model had lines composed of parabolic arcs, 

 following Weinblum (1953), and had a prismatic coef- 

 ficient of 0.655. The inertial and damping forces in 

 heaving were calculated from the amplitude and phase 

 lag of the motion records as compared with the records 

 of the harmonic exciting force. Fig. 6 shows the coef- 

 ficient of accession to inertia k^ plotted versus nondimen- 

 sional frequency w{B/g)^^''. A horizontal arrow at 

 about A'j = 0.93 shows the value calculated by using 

 F. M. Lewis' (1929) data; i.e., neglecting surface wave 

 effects. The curve shown by heavy dots is Grim's 

 (1953 a,b) asymptotic evaluation of the added mass for 

 low frequencies. The experimental data at low fre- 

 quencies are somewhat uncertain because of model inter- 

 ference with waves reflected from the sides of the towing 

 tank. At higher frequencies the coefficient fcj is shown 

 to be independent of the Froude number. 



Fig. 6 covers a wide range of frequencies and the pic- 

 ture may be misleading unless the range important in 



" The author understands that similar experiments also were 

 made with pitching oscillations, but the results have not yet been 

 published. 



