296 



THEORY OF SEAKEEPING 



Table 1 1 Numerical Results of Components of Logarithmic Decrement in Higher 

 Symmetric Modes of Vertical Vibration of a Ship. From Kumoi (1958) 



No. of 

 nodes 



2 



4 



6 



8 



A'. 

 cpm 



62.04 

 173.8U 

 289.30 

 401.00 



0,01813 

 0.(12410 

 0.02L'(;0 

 0.02180 



0.00313 

 0.02110 

 0.04070 

 . 00000 



5r 



0.00032 



0,00010 



(10000 

 , 00004 



0,0216 

 0,0453 

 (l,0(i34 

 0.0818 



Oemp 



0.0216 

 0.0467 

 0.0687 

 0.0890 



Contributed by: | — normal viscosity, j; — tangential viscositj', f — external damping. 



inic (decrement 5„ is related to the one at two-noded fre- 

 quency, 



5„ = d,iw„/o,,y^' (17) 



where oj,, is the natural frequency of the «-noded \'ibra- 

 tion and coj that of the fundamental two-noded mode. 



5.54 Similarity conditions in ship vibrations. The 



rapid growth of the damping coefhcient with vibration 

 frecjuency indicates that towing-tank models cannot 

 directly represent the condition.s of a full-size ship. 

 Slam-caused vibrations in models are attenuated rapidly 

 because of the heavy damping associated with high ^'ibra- 

 tion frequency. In ships, on the other hand, the damp- 

 ing associated with a low fretiuency is small and vibra- 

 tions persist a long time. This observation does not 

 make model tests less valuable but it shows that inter- 

 pretation of model tests can only be made with proper 

 regard to the vibration-res]:)onse theory. 



In metal models used by Ochi and Sato, (Sections 4.2 

 and 4.3), the structural characteristics affecting bending 

 deflections were similar to those of full-size ships. How- 

 ever, the reduced number of longitudinals and reduced 

 support of the skin may ha\-e resulted in different shear 

 characteristics. Careful consideration of these appears 

 to be needed in view of the large effect of tangential vis- 

 cosity on damping which is shown by Table 11. 



Jasper (1958 appendix) investigated the scaling law for 

 geometrically and structurally similar ships f)perating in 

 similar seas. The results are summarized in the follow- 

 ing cjuotation : "Thus it has been shown that the wave- 

 induced stresses in similar ships, operating at the same 

 speed-length ratio in similar seas, vary as the length of 

 the ship for a suddenly applied step load, and as the 

 square root of the length for an instantaneously applied 

 impulsi\'e load. The ordinary, slow varying, wave- 

 induced stresses may be expected to vary as the length 

 of the ship." This summary indicates that ciuasi-static 

 wave-caused bending stress and vibration-caused bend- 

 ing stress should be considered as two distinct compo- 

 nents to be added to obtain the total stress. A percen- 

 tage increase of the wave-caused bending stress by slam- 

 ming appears to be an untenable concept, except when 

 comparing ships of similar size and type. 



5.55 Dalzell's analysis of destroyer model vibration. 

 Only the dynamics of the \-ibratory ship resjjonse to a 

 slam was discussed in the foregoing sections. The prin- 

 ciples on the basis of which the impact force in a slam can 

 be estimated were outlined in Chapter 2 but it appears 



that no actual evaluation of the force in a typical slam 

 has been made. Ochi used for his vibration analysis 

 the forces measured on a model in a towing tank. Fur- 

 thermore, the available methods, based on Wagner's 

 work, apply only to a sharp slam (i.e., bottom impact), 

 similar in nature to the impact experienced in a seaplane 

 landing. No methods of analysis were heretofore pub- 

 lished for a relatively slow developing bow immersion, 

 such as was observed by Warnsinck and St. Denis 

 (3-1957). Dalzell's (1959) work appears to be the first 

 attempt at an analysis of hydrodynamic forces involved 

 in such a case, as well as of the resultant vibratory re- 

 sponse of a towing-tank model. 



Tlie scope and objectives of this work best can be 

 stated by quoting from Dalzell's introduction: "A num- 

 ber of investigations of the midship bending moments 

 experienced by jointed wooden models in regular head 

 waves at the Davidson Laboratory (Lewis, 1954; Lewis 

 and Dalzell, 3-1958; Dalzell, 1959) have indicated 

 the presence of nonsinusoidal forces at some time during 

 each cycle, usually when wave length and model speed 

 produce large motion amplitudes. The evidence takes 

 the form of records showing vibration of the jointed 

 model. Tests in irregular head waves show that the 

 vibration occurs under conditions thought to be similar 

 to those under which full-size ship slamming takes place. 

 The purpose of this investigation was to determine 

 whether the impulsive forces on the model could be cal- 

 culated, utilizing existing theoretical methods and data." 



"It was felt worthwhile to work with a regular wave 

 case for simplicity, and the case chosen for study was that 

 of a 5.71-ft destroyer model" at 6.0 ft/sec in regular 

 waves 7.14 ft long. Experimental records for this case 

 had been obtained and indicated both large motions and 

 large model vibrations. (A complete description of the 

 model and experiments maj^ be found in Lewis and Dal- 

 zell, 1958). Further, the ship motion theory developed 

 by Korvin-Ivroukovsky had been used to predict motions 

 for this case with reasonably good agreement with ex- 

 periment (see Korvin-Kroukovsky and Jacobs, 3-1957). 

 The theory can also be used to predict bending moment 

 response in regular waves, and this had been done for the 

 case luider study by Jacobs (1958). Since these methods 

 utilize coupled etiuations of motion with constant co- 

 efficients for a rigid body, the calculated motion and 

 force responses to an assumed sinusoidal wave are also- 



' The model represents a destro3-er 383 ft in length. 



