LOADS ACTING ON A SHIP AND THE ELASTIC RESPONSE OF A SHIP 



295 



(ockl number of ufidal points) vauisli amidships. How- 

 ever, tlu'y will he sigiiiheant at a certain point forward 

 of amidships where the total vibration- caused bending 

 moment should be at its maximum. Ochi's experiments, 

 Fig. 42, show indeed the maximum to occur about 7 

 per cent forwai'd of amidships. 



5.53 Added mass and damping. I'he \'il)ration of 

 ships is governed by the same laws of dynamics as os- 

 cillations in waves and the reader is referred to C-hapter 2 

 for much of the discussion. The \-ibration analysis is 

 based on the strip method of evaluating a ship's mass 

 and structural-propert J' distributions. The effective mass 

 is a mass of a ship's section of unit length plus a certain 

 imaginary added mass, the acceleration of which gives 

 the same force as that caused by water pi'ossures. The 

 evaluation of the added mass is usually based on theoreti- 

 cal work of F. M. Lewis (3-192!)), .J. Lockwood Taylor 

 (3-1930/)), and Prohaska (3-1947). Xo c'omplete and 

 reliable experimental verification of this material appears 

 to be available. Alotlel experiments may indeed be 

 ciuestionable because of certain water-separation efTects 

 during a part of an oscillation cycle (Keuligan and Car- 

 penter, 3-195(3; T. B. Abell, 3-191()). Theory indicates 

 that free-water surface effects (wave making) should not 

 be signilicaut at vibration freciuencies and the data, 

 given by the in\-estigators mentioned, should, therefore, 

 be directly apijlicable. 



The largest uncertainty is in the effect of three- 

 dimensionality on the added masses obtained by two- 

 dimensional strip theory. F. M. Lewis (3-1929) and 

 J. Lockwood Taylor (3-19306) made analyses of this 

 effect at various vibration modes, and Lewis evaluated 

 the residts for the first and second modes. The im- 

 portance of higher modes in slam-caused vilirations makes 

 it desirable (if not mandatory) to extend these calcula- 

 tions to higher modes. In this connection it is necessary 

 to mention the paper by Macagno and Landweber 

 (3-1958) in which it was demonstrated that results of 

 the analysis strongly depend on the assumed nature of 

 a body's deflection; i.e., on niovements of a body surface 

 element in shear and trending deflections, including ro- 

 tation of sections. 



The evaluation of the damping in vibration appears to 

 be more uncertain than e\'aliiation of added masses. 

 The following ([notation from Ochi (1958t') may serve as 

 an introduction to this subject: "We have little data 

 which is sufficient to estimate the damping coefficient in 

 ship vibration, especially to estimate the external (water) 

 and the internal (structural) damping coefficient. More- 

 over, there are some difi'erences between numerical 

 values given in the following papers; 



"[J. Lockwood Taylor (3-19306)] gives the following 

 simple values for small amplitude of two node vibraticjn 

 in full size ship: 



b/pA = 0.025 (for 80 rpm) 

 = 0.032 (for 99 rpm) 

 = 0.0()(i (for 148 rpm) 



"where 



6 = damping coellicicnt 



pA = mass of ship ])er unit length 



"McCioldrick (1954) mentions from the analysis of many 

 data on full-scale experimental works that the damping 

 in ship vibration appears to increase with fretiuency and 

 the value of b/pAu (where oj is a frec|uency in radians) is 

 constant. He gives 0.034 as the mean \-alue of h/pAo: 

 for all modes of ship vibration. Kinnai (1958) recently 

 discussed damping factors in the higher modes of ship 

 vibration taking into account the effects of shear de- 

 flection, rotary inertia, internal damping and also made 

 some experiments. On the other hand, Sezewa (193()) 

 made a theoretical consideration on four damping factors 

 in ship vibration, namely, ( 1 ) water friction, (2) genera- 

 tion of pressure wave, (3) generation of surface wave, 

 (4) structural damping force. He concluded that a 

 generation of sm'face waves as well ;is a structural damp- 

 ing force are the main sources of damping, especially the 

 former is pronounced in light draft condition." 



As the result of his investigation, (Jchi found that the 

 internal damping of his brass towing-tank model formeil 

 80 per cent of the total in the first mode and essentially 

 100 per cent in higher modes. He also found theoreti- 

 cally that the damping is proportional to the fourth 

 power of the ratio a„/L, where a,, is the characteristic 

 number of the /;th mode and L the length of the bar. 



The reader's attention is called to three forms of ex- 

 pressing the damping of a vibrating system: 



(o) The term "damping coefficient" has been used for 

 the coefficient h of the velocity-dejjendent term (y or i) 

 of differential ecjuations of o.scillatory motion [for in- 

 stance eciuation (13)]. 



{b) The factor e"**' occurs in solutions of free x'ibrating 

 systems and is a measure of the rate of amplitude decay. 

 Kumai (1958) refers to the ([uantity q as the "damping 

 factor." 



((■) The "logarithmic decrement," 6, which is con- 

 nected with the damping factor by the relationship 



:ini CO 



(15) 



where w is the natural frequency. 



Ochi's solution of eciuation (13) (for ])ure flexural de- 

 flection) resulted in the evaluation of the damping factor 



b, + ^ha„/Ly 

 2m 



(Iti) 



where he used ^ = 4.07 X 10^ for a steel structure. 



Kumai (1958) computed Table 11 which shows con- 

 tributions of various factors to the logarithmic decre- 

 ment based on calculations for a 32,000-ton tanker. 

 The last column gives the empiricall.y obtained logarith- 

 mic decrement for vertical vibration of ships from 2()0 

 to GfiO ft long. This relationship for the two-noded vibra- 

 tion is 



5,. = C/L (IG) 



where C is a coefficient with the value between 3 and 4, 

 and L is a ship's length. For higher modes the logarith- 



