SHIP MOTIONS 



159 



tical value to date. However, theoretical research into 

 rolling since Froude seems to have been rendered inef- 

 fective by adherence to his methods. This does not 

 apply to the more advanced theory of i-oU stabilization. 



]iy intuitive reasoning W. Froude reduced the three 

 conventional coupled equations to the form for a simple 

 harmonic oscillator, ecjuation 2-(25), in which he ini- 

 tially neglected the damping (coefficient B = 0). He 

 assumed that a ship with a small lieam and draft in com- 

 parison to the wave length moves in the same manner as 

 the water which it displaces; i.e., that its center of 

 buoyancy has the same orbital motion as the orbital 

 motion of water particles at its depth. He observed that 

 the resultant of the gravity and acceleration forces in 

 orbital motion is always normal to the water surface at 

 any point on the wave. The exciting moment in rolling 

 is then simply e.xpressed as the product of this resultant 

 force, the metacentric height and th(> roll angle measured 

 between the ship mast and the local normal to the water 

 surface. However, Froude neglected the contriiiutions 

 of acceleration forces to the magnitude of the resultant 

 and used the constant gravity force in computing the 

 righting moment. His final result is approximately 

 eciuixalent to roll-side sway coupling, wliilc paitially ac- 

 counting for the hea\ing motion. 



In the form used by W. Froude (18()1) the resultant 

 eciuation of a simple harmonic oscillator without damp- 

 ing takes the form'' 



rf-0 47r- 



dt- 



T„- 



{<f> - ") 



(7) 



where </> is the angle lietween the shi]) mast and the ver- 

 tical and a is the local inclination of the water surface. 

 Assuming that a ship is upright and at rest before waves 

 take effect, the subsequent nKjtions are given by the 

 equation 



TT H 1 / . ., irt 7',, . 2wt\ ,, 

 d> = I sni- sm — (S 



where Tn is the natural period of undampetl ship rolling 

 and Tc is the wave-encounter period. The motions of a 

 ship are sh<nvn, therefore, to consist of two sujxapo.sed 

 systems of oscillations, one at a ship's natural period 7'ij, 

 the other at the wave encounter period 1%. In general, 

 the resultant motion will show a series of beats of in- 

 creasing and decreasing amplitude of indi\'idual oscilla- 

 tions. As the ratio Ta/Te approaches unity, however, 

 the oscillations rapidly grow with each consecutive cycle, 

 reaching catastrophic proportions. Introduction of 

 damping checks the unlimited growth of oscillati(Mis, but 

 damping of ships in roll is generally small at zero speed 

 and certain characteristics of an undamped motion per- 

 sist to a large extent. The most important of these is the 

 existence of a high and sharp peak on the curve of the 

 magnification factor (</>„,ax/amax) at svnchronism (7',, — 



Most of the usable information on ship rolling comes 

 from the foregoing simple relationships. It consists of 

 i-ecognition of the importance of avoiding .synchronism 

 with the waves, Ta/Tc = f , and of having a ship's natural 

 period Tn sufficiently large, i.e., having a metacentric 

 height sufficiently small, to avoid synchronism with waves 

 frequently met in Nature. The.se basic conclusions re- 

 main valid even when physical conditions of ship opera- 

 tion deviate widely from the idealized conditions as- 

 sumed by Froude. Ship speed anil wave direction weie 

 later included as factors by others, but only in the sense 

 of affecting the period of wave encounter T,.* Charts 

 were constructed (Niedermair, 1936; Maiming, 1042) 

 clearly showing the fa\'orable and unfavorable combina- 

 tions of ship speed and heatling. A large amount of ob- 

 servations at sea (for instance Hebecker, 1940, Meckel, 

 1941) clearly demonstrated the advantage of large nat- 

 ural roll periods for normal types of surface ships as 

 recommended fii'st liy W. Froude. 



Under the a.ssumptions made by Froude, a ship mo\es 

 with the surrounding water and, therefore, there is no 

 flow of water relative to the ship. The water flow in 

 waves is not interfered with and the pressure acting on 

 the ship is the .same as that which would exist in the water 

 if the ship were not there. This assumption which later 

 became known as the "Froude-Kriloff hypothesis," is the 

 direct conse(iuence of physical conditions for a small shi]) 

 on long r(>gular waves coming exactly from a beam di- 

 rection. 



In the foregoing treatment by Froude, water disturb- 

 ances caused Ity a rolling ship and all ship side-sway mo- 

 tions with respect to water (i.e., apart from the horizontal 

 component of orbital wave motion) were neglected. 

 Rankine (I8(54a) showed that there is a certain effect on 

 the water flow in the case of deep draft or short waves. 

 Rankine (18(i4r) also commented on the effects of the 

 vertical acceleratif>n neglected by Froude. Under the 

 physical conditions assumed by Froude, these considera- 

 tions were only second-order corrections and did not 

 modify the simple basic conclusions. Apparently the 

 matter did not receive further attention. 



2.21 Deviations from Froude's assumptions. The 

 fact that physical conditions change drastically for a long 

 ship in oblique waves has largely been overlooked. Under 

 such conditions a section of a ship's length, which is in a 

 certain relation to a wave, cannot freely participate in 

 the wave motion, being restrained by other parts of the 

 ship which are located differently in respect to the wave 

 form. The flow of water around each section is un- 

 avoidable and it brings about changes in pressure dis- 

 tribution and in the forces acting on a ship. The.se forces 

 have not yet been taken into account in the analysis of 

 ship rolling. \ certain amount of theoretical supporting 

 material for th(^ calculation of forces cau.sed by lateral 

 water flows can be found in tlie work of Ursell (1946, f948a 



* The effects of the mass and restoring force are here represented 

 in terms of the natural period T^. This corresponds to e(|Uution 

 2-(.32) with K = and w„ = 27r/r„. 



* Neither changes of hydrodynaniie forces nor couplings with 

 other modes of motion were taken into accoimt. These inevitalily 

 develop with forward speed and with obliquity of wave crests. 



