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



293 



natural t'rc(|uencies and the t'recjvieiicy of the excitini^ 

 force. 



The term "transient \'il)rations" will he used to define 

 the viliratory response of a structure to a force Pit,.v) 

 variable in time and in the point of application and 

 usually of a short duration. In the present exposition 

 the attention is concentrated on vibrations caused by a 

 slamming impact. In nearly flat-bottomed carf>;o ships 

 it may ha\-e a duration of a fraction of a second, in a 

 destroyer it may last 1.5 sec. Two separate phases are 

 involved here: (a) The development of the momentum 

 and energy content in a ship's hull during force action. 

 (6) The decay of free oscillations after the exciting force 

 ceases to act. The secimd pha.se is more conspicuous 

 to an observer aboard a ship and is important in defining 

 the \ibration-caused bending stress, which is superposed 

 on se\'eral cycles of the wave-caused ([uasi-static stress. 

 An investigation of the first phase is necessary, however, 

 in order t(j define the dynamic condition of the hull at 

 the end on the impact, after which a slow decay of the 

 vibration begins. 



Vil)rations in many modes are excited by a slam. Un- 

 like the response to a specified frec[uency of a forced 

 oscillation, the chstribution of the momenta and energy 

 among many vibration modes cannot be determined 

 a priori. The determination of this distribution is the 

 fundamental part of the prolilem discussed here. An- 

 other important part of the problem is the evaluation of 

 the damping connected with each viliration mode. 

 The damping affects the ability of a structure to absorb 

 the impact momentum and it controls directly the rate of 

 the vibration decay in each mode. Superposition of 

 vibrations of all modes, wdth tlifferent momentum con- 

 tents, different initial time lags and different rates of 

 decay, determines the entire behavior of the structure 

 in time. The cjuestion of damping will be further con- 

 sidered in the next section. 



Three methods of attacking the problem, outlineil in 

 the foregoing, appear to be available; namely, digital 

 calculations, electric analog, and theoretical analysis. 



The first method was outlined by McGoldrick et al 

 (1953), with reference to Jasper (1948) and Mathewson 

 (1949, 1950), and by Polacheck (1957). The ditt'erential 

 eciuations of the vibration theory had been converted into 

 finite-difference form and calculations were performed 

 by means of high-speed electronic computers. In this 

 approach it is neither necessary nor possible to distin- 

 guish between the behavior of different vibration modes. 

 The calculations give directly the total behavior of a 

 structiu'e, as it could have been observed in an experiment 

 without furnishing explanation as to reasons for this 

 behavior. This appears to be the only method which 

 currently can be applied to actual ships in which section 

 properties vary along the length. Two important ap- 

 proximations have to be made, hfiwever, with respect to 

 the damping. A certain mean value of the damping 

 coefficient has to be assumed, disregarding its dependence 

 on the vibration frecjuency, and this coefficient has to be 

 assumed as proportional to the masses of ship sections. 



The applicability of the electrical analog method to 

 ship-\'ibration problem was discussed by McGoldrick, 

 et al, with reference to Kron (1944) and Kapilolf (DTMB 

 Rep. 742). It appears that attention was concentrated 

 on determination of normal modes and natural fre- 

 quencies rather than on transient responses. 



Theoretical analyses of forced vibrations and transient 

 phenomena caused by slamming have lieen found to be 

 difficult, anil so far were only apjjlied to l)ars of uniform 

 section. The following is quoted from McG(jldrick et al 

 (1953): 



"The group working at the University of Michigan 

 soh'ed by means of operational calculus the partial differ- 

 ential C(|uation for the uniform bar subject to bending de- 

 flection only, having a uniformly distributed viscous 

 damping and acted upon by a transverse load whi(-h 

 was an arbitiary function of time and position along the 

 bar. This ixniuired the solution of the differential e()ua- 

 tion 



d.r 



dt- 



EI^+ >nj^f+b^ = Pi.v,t) 



dt 



"where b is the damping force per unit length of bar per 

 unit velocity, and P{.r, t) is the external force per unit 

 length varying both with x and t.-^ 



"A solution of this e(|uation was found 1)3' the opera- 

 tional method employing the Laplace transformation. 

 The derivation is given in the second progress report of 

 the University of Michigan on its contract with the 

 Office of Naval Research. -« 



"The solution shows that, whatever form the function 

 /-"(.r, I) takes, the response of the bar is expressible in a 

 series of normal modes; in other words, the system be- 

 haves in general like the systems whose small oscilla- 

 tions were studied by Rayleigh . . . Moreover, such a 

 system does not partake of wave motion in tlie ordinary 

 sense in that there is no fixed rate of propagation of a 

 flexural wave. If the bar is struck at one end, a finite 

 time will be reciuired before a finite motion takes place 

 at the other end, but the process is the result of com- 

 pounding motions in normal modes in each of which the 

 system deflects simultaneously at all points rather than 

 the result of a flexural wave traveling back and forth. 



"It also follows from the solution of the uniform bar 

 problem that in each normal mode the .system behaves 

 as a sj^stem of one degree of freedom would beha^-e and 

 as though tills mode only were present. The amplitude 

 produced in each mode by a given simple harmonic 

 driving force depends on the magnitude of the force, the 

 influence function, the effective mass, stitt'ness, and 

 damping constant of the S3^stem in that mode and on the 

 ratio of the frequency of the force to the natural fre- 

 quency of the mode. The ordinary resonance curve for a 

 system of one degree of freedom is applicable to each 

 normal mode indi\idually . . ." 



Uchi (195(ia, 1958(/, 1958() also obtained the .solution of 



26 1'reviouslv defined symbols are omitted from the quotation. 

 '"' ( )rmoiulroyd et al (1948). 



