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



297 



Fig. 44 Calculated and experimental rigid-body midship bend- 

 ing moments for a destroyer model in regular waves at 6 fps. 

 Model length 5.71 ft, wave length 7.14 ft, wave height 1.43 in. 

 (from Dalzell, 1959) 



sinusoidal and thus do not reveal any impulsive forces. 

 Experimental data indicate that the motions (heave and 

 pitch) of the model seem to be little affected by whatever 

 impulsive forces cause the vibration. Hence, it was 

 decided to utilize the Korvin-Kroukovsky method to pre- 

 dict motions, then to evaluate the hj^drodynamic loads 

 at successive instants of time during the cycle assuming 

 the calculated motions and sinusoidal waves. This 

 evaluation was to be done taking into account the varia- 

 tion during the cycle of buoyancy and added hydrody- 

 namic mass brought about by the geometry of the model." 

 An inertial force exerted by a fluid on a body results 

 from the rate of change of the fluid momentum: 



Force = y (m w) 



(18) 



where m" is the hydrodynamic mass and iv the relative 

 velocity of the body with respect to the fluid.-* In a 

 seaplane landing (used by Szebehely to formulate ship 



slaniming analysis) the duration of the impact is short, 

 the velocity is assumed to remain constant, and the force 

 is defined as 



Force = w dm" /dt 



(19) 



In the linearized analysis of ship motions, as developeil by 

 Korvin-Kroukovsky and Jacobs (3-1957), a wall-sided 

 ship is assumed, the hydrodynamic mass is held constant, 

 and the force is expressed as 



Force = m" dw/dl 



(20) 



Considering a relatively long duration of a destroyer im- 

 pact and variable draft and wetted beam, Dalzell modi- 

 fied Korvin Kroukovsky and Jacobs' expressions to in- 

 clude variations of both the hydrodynamic mass and 

 velocity; i.e., expressed the force as 



Force = ih"w + m" 



{■1\) 



^' Symbol w is used here for the vertical velocity of an impacting 

 body. 



Quasi-static wave-caused bending moment resulting 

 from this analysis is shown in Fig. -14 (extended Korvin 

 method) for comparison with the linearized analysis of 

 Jacobs (1958) (constant coefficients), and the mean line 

 drawn through the oscillatory experimental data. 



In discussing results of a pioneering attempt at a 

 rational analysis, it is often advisable to consider sepa- 

 ratel.y the magnitudes and curve shapes (or trends). In 

 the present case an apparently improved method of 

 analysis resulted in large exaggeration of the bending 

 moment. On the other hand, the analysis demon- 

 strated a nonsinusoidal behavior of the bending moment 

 and thus confirmed earlier findings of Horn (1910) and 

 Hazen and Nims (3-1940). Demonstrated departures 

 from a sinusoidnl curve include a sharper peak at the 

 maximum amplitude and steeper flanks of the curve, 

 which are capable of exciting vibrations of a slender hull. 



Calculations of the vibratory responses were made 

 using calculated quasi-static bending-moment cvirve as an 

 e.xciting function. Basic expressif)ns for the vibratory 

 response were taken from Frankland (1942) and Timo- 

 shenko.^^ Natural frequency and damping were de- 

 termined experimentally. Only a single vibratory mode 

 was present since the model consisted of two rigid halves 

 jointed and kept aligned by the dynamometer flexure bar. 

 The integration f)f equations of motion was replaced by 

 the summation of finite differences, and calculations were 

 carried through seven wave-encounter cycles in order to 

 eliminate the transient response caused by uncertain 

 initial conditions. The cyclic bending-moment varia- 

 tions were found to repeat themselves beginning with 

 the fourth cycle. 



Model speed of 6.0 fps was chosen for the analysis be- 

 cause at this speed the period of wave encounter was an 

 exact multiple of the model's natural frequency and the 

 calculations were thereby simplified. Unfortunately, 

 at the nearest test speed of 6.1 fps test results exhibited a 

 rather exaggerated amount of vibration, which did 



2' "Vibration Problems in Engineering," D Van Nostrand Co., 

 New York, N. Y., third edition, 1955. 



