SHIP MOTIONS 



161 



nous"; i.e., the natural period To is eoiistant and inde- 

 pendent ol' the angle of roll. 



Subsequent theoretical studies concentrated on the 

 dynamics of noni.sochronous motion; i.e., the motion re- 

 sulting from a righting arm differing from the one shown 

 by equation (9). Two relatively recent examples of such 

 work will be cited.* Vedeler (1953) assumed the equa- 

 tion of motion for rolling in smooth water to be 



<!> + hicj> + b2<p- + co'- (0 - A-0^) =0 (10) 



In this case nonlinearity in damping as well as in right- 

 ing arm is included. When a simple harmonic exciting 

 function is put on the right-hand side and the equation is 

 solved, the "magnification factor" (the ratio of the am- 

 plitude of rolling to the wave slope) is found to be (after 

 sufficient time to damp out transients) that shown in Fig. 

 5. In this figure n designates the "tuning factor" n = 

 To/Z. 



The dotted cin-\-e, peaked at n = 1, indicates the usual 

 form of the magnification-factor curve for a simple har- 

 monic oscillator. The dotted curves swinging upward and 

 to the left result from nonlinear equation (10). To the 

 left of « = 0.80 the curves indicate an unstable condition 

 and only the part to the right, shown by a solid line, is 

 considered to apply in reality. This part is character- 

 ized by the discontinuous jumps from one branch of 

 theoretical curve to the other along vertical lines B-C and 

 B'-C. 



The other example is from Bavunann (19.55). The 

 equation of rolling in regular side waves is written as 



J4> + B{T,) + A[l + iS{t)]h{4>') ^0, (11) 



where 



J = moment of inertia (including added water 

 masses) 



4> = angle of ship with respect to vertical 



<j>' = angle of ship with respect to the normal to 

 wave surface (i.e., to "apparent vertical") 



B = damping coefficient as a function of period 

 of ship oscillation of a gi\'en anqilitude 



A = weight of a ship 



/3 = vertical acceleration in waves in terms of 

 gravity acceleration 

 h (4>') = righting arm as a function of inclination </>' 



T, = ship period as a function of amplitude of os- 

 cillation 



Froude investigated the motion only for small am- 

 plitudes of rolling for which equation (9) can be assumed 

 valid. Baumann investigated ecjuation (11) for all 

 angles up to the angle </>„ at which the righting arm 

 vanishes. Baumann evaluated the erjuation numerically 

 and presented the results in numerous graphs for dif- 

 ferent forms of righting-arm curves. An example is re- 

 produced in Fig. 6. The abscissa is the ratio of the 

 square of the period of rolling in waves T^ to that in 

 smooth water at small angles To. The ordinate is the 

 ratio of the amplitude of the angle 0, designated as f, 



' An additionat paper by Robb (1958) has just appeared. 



:5-0.5 

 1^ 



0.15 



0.10 



0.05 



0.5 



1.0 



rv^/Vco) 



1.5 



2.0 



Fig. 6. Calculated steady nonlinear rolling of a ship in regular 

 waves (from Baumann, 1955). 4, is rolling amplitude 0„ angle 

 of list at which righting arm vanishes , Tn is period of rolling in 

 waves and To (O) is natural period of rolling at very small 

 amplitudes 



to the angle (/>„ for the vanishing righting arm. The in- 

 sert sketch shows the curve of righting arm forwhich com- 

 putations were made. The curves are labelled according 

 to the wave height-to-length ratio, and the data repre- 

 sent the steady ship rolling after transients are damped 

 out. 



The appearance of the curves in Figs. 5 and 6 is strik- 

 ing and has been used by some to explain lurching. An 

 extremely important feature is, however, overlooked; 

 namely, these figures describe a fully established motion, 

 which results from a sustained action of regular waves 

 after all transients arc damped out. However, waves in 

 nature are never regular, and the sustained excitation 

 needed to develop the indicated behavior does not occur. 

 Even the so-called "regular swells" show on the records 

 a large degree of irregularity. It should be emphasized 

 that the behavior indicated by Figs. 5 and 6 will not 

 manifest itself after a few oscillations, but requires a 

 continuous su.stained excitation. In realistic sea con- 

 ditions, high wa\'es usually are encountered in small 

 groups, and two or three wa\'es in such a group may ex- 

 hibit an apparent regularity. This is not sufficient to 

 bring about the odd beha\'if)r described in the figures. 

 It is, therefore, the author's opinion that this type of re- 

 search in rolling has taken an artificial direction. It can- 

 not be too strongly emphasized that investigations of ship 

 motions in regular waves are meaningful only when con- 

 sidered as the material to be further operated upon by 

 statistical methods in order to represent the ship behavior 

 in the natural irregular sea. 



W. Froude (18G1-1875) did not discuss explicitly the 

 behavior of .ships in irregular .seas but he did emphasize 

 transient behavior in rolling. In this respect his con- 

 clusions appear to be more realistic than the conclusions 

 of later writers based on the implied I'egularity. The 

 author suspects that if transient instead of su.stained be- 

 havior were considered in the foregoing examples, the ac- 

 tion of a nonlinear system in the first few oscillations 

 would not differ materially from a linear one, with suit- 

 ably chosen constant coefficients. 



