120 



THEORY OF SEAKEEPING 



cussed in the published paper. " The oscillatory motions 

 of the model were recoi'ded on a rotating drum. The 

 damping was defined by the logarithmic decrement 5 as 

 a function of the frequency ca of the oscillation and the 

 mean submergence (/. 



For a 60-deg prism the mean submergence (over the 

 oscillating cycle) is eciual to the beam at mean waterline. 

 Dimpker showed that data for tests at varying fre- 

 quencies and submersions collapsed into a single curve 

 when plotted as 5/d- versus w"d. These amplitude and 

 frequency parameters were initially defined in a non- 

 dimensional form. After the constant quantities, such 

 as the mass involved, the acceleration of gravity g and 

 the water density p were omitted, the parameters took 

 the form indicated in the foregoing. The resultant curve 

 is reproduced on Fig. 10. Data for the cylinder are given 

 in Fig. 11. In this case the immersed shape varies with 

 the draft d and it is not possible to make a generalized 

 plot. It is interesting to note that the maximum damp- 

 ing occurs at an immersion of about 2.5 cm; i.e., half- 

 radius. The decrease of damping with further immersion 

 is in agreement with the general tendency shown by the 

 theories of Holstein and Havelock (increase of / in equa- 

 tion 22). 



Dimpker (1934) evaluated the virtual masses for a 

 wedge and a cylinder on the basis of changes in the 

 natural period resulting from changes of immersion and 

 frequency. The frequencies were given in terms of 

 spring constants, and the significance of results cannot be 

 seen readily since insufficient data were given for re- 

 calculation. 



Unfortunately, the e.xperiments of Holstein and Dimp- 

 ker appear to be the only published data in direct veri- 

 fication of the theory in regard to sectional damping 

 coefficients. Verification of other theoretical methods 

 is indirect. This consists of calculating ship motions 

 using the coefficients evaluated on the basis of the previ- 

 ously mentioned methods and of accepting the success- 

 ful motion prediction as justification for the method of 

 calculation. However, it is far from being a reliable 

 verification in view of the number of steps involved and 

 the complexity of the calculations. Grim (1953) justi- 

 fied his theoretical damping curves by an analysis of 

 coupled pitching and heaving oscillations of several .ship 

 models. The oscillations were induced by means of ro- 

 tating unbalanced masses; i.e., with a known value of 

 the exciting function. Korvin-Kroukovsky and Jacobs 

 (1957) successfully used Grim's damping coefficients in 

 the analysis of several ship models which had been tested 

 in towing tanks. Korvin-Kroukovsky and Lewis (1955) 

 and Korvin-Kroukovsky (1955c) previously made simi- 

 lar use of Havelock's coefficients. 



Direct measurements of damping on a ship model were 

 made by Golovato (1957a and 5) and Gerritsma (1957c and 

 d, 1958, I960)." The ship model used by Golovato was a 



l|. 



OL 



0.5 



1.0 



1.5 

 oj -J B,'g 



J.O 



2-5 



'" Dimpker's (1934) paper is a part of a Gottingen dissertation 

 prepared under the guidance of M. Schuler and L. Prandtl. 



" Added mass data from these tests were discussed in Section 

 3.15-3. 



Fig. 12 Comparison of measured damping in heave with one 



calculated by strip theory using Holstein-Havelock method 



(from Gerritsma, 1957c) 



mathematically defined form, symmetrical fore and aft. 

 ^'arious hydrodynamic forces were measured experimen- 

 tally in a simple heaving motion (pitching restrained) 

 induced by a mechanical oscillator. The results for 

 damping are shown in Fig. 14 which also includes damp- 

 ing as compvited by Ha^'elock's and Grim's methods. 

 The general trend of the damping-force \'ariations as 

 given by all three methods is identical, but both calcu- 

 lated methods gi\'e higher damping than the experimen- 

 tal values. (In connection with ship motion analysis, it 

 is suggested that the reader concentrate his attention on 

 the abscissa range 0.9 to 1.5.) 



Fig. 7, taken from Gerritsma (1957rf), shows a com- 

 parison of damping-force coefficients measured on a 

 Series 60, 0.()0 block coefficient model with those com- 

 puted by Korvin-Kroukovsky and Jacobs (1957) by 

 means of the strip theory, using Grim's (1953) material. 

 Figs. 12 and 13 show a similar comparison with the damp- 

 ing computed by Korvin-Kroukovsky (1955c) using the 

 Holstein-Havelock method. 



In comparing the heave damping data of Golovato and 

 Gerritsma, it is obser\-ed that the measured damping 

 of the Series 60 model is much higher than that of the 

 idealized model : the maximum nondimensional value in 

 Fig. 12 is about 3.7 as against 2 in Fig. 14. A part of 

 this drastic increase can be explained by the presence of 

 inclined ship sides in the afterbody of the Series 60 

 model, while Golovato's idealized model was wall-sided. 



The theoretically evaluated damping is also higher for 

 the Series 60 model, but the increase is not as drastic as 

 for the measured damping. This difi'erence may result 

 from the fact that the inclination of ship sides at the 



