# Page:EB1911 - Volume 24.djvu/999

ROLLING or smpsl
937
SHIPBUILDING

respectively. Froude gave his reasons for expecting the resistance to vary partly as the first and partly as the second power of the angular velocity. The latter part he considered would be due to surface friction and the head resistance of keels and deadwood, and the

FIG. 29.*CUl'V€S of extinction. A, light, and B, deep draught, no bilge keels; C, light, and D, deep draught, with bilge keels. former to the resistance caused by the creation of a small wave at each roll, which, by travelling away from the ship, would cause dissipation of energy. F roude's views have been confirmed by the accuracy with which'the expression -§ LZ=a.9-l-b.9' may be made to fit the curve of extinction of practically any ship by the judicious selection of the coefficients a and b. M. Bertin has, however, d .

preferred an expression equivalent to -(~£=b.6”, while other French investigators have preferred an expression equivalent to d9

- 11.6. - .

On substituting the value of a in equation (7) it becomes:il 2

0=AeT sm -\$4-B, . . . (8)

a simplified form of the equation for resisted rolling when the coefficient b is neglected.

For the “ Revenge " the following equations represent the curves of extinction given in fig. 29:

For deep draught:

without bilge keels - Q = -01230 + ~OO25@2 For light draught:

with, -§§ =-065 0+-017 0*

without bilge keels-gg =-015 o+-002892 with, ,, , T5-§ =-084 9+-019 G2

-%~a2;-{-Kg-I-Wm 0-®1s1n gli! =O,

which becomes on substitution (K being expressed in terms of a)(Z20 20. J0 1r2 1r2 . 1r

1112+ T dt+T”6 ' T2 ~ @1 Sm TK

The general solution is-al

T . 2

@=Ae sm (Q /1-j'a+v +A.e».S1n (.%-B., <9> where T 2 T2 TT

1 f 2 i@ ~ -1 2a 1

Axz- (I -T12) -|- "Z T12 and [31 -tan mm and A and B are arbitrary.

The first term represents a free oscillation of the ship, which in time dies out, leaving a forced oscillation in the period of the waves. From observations on rolling, however, it is found that, owing to the departure from exact uniformity in the waves'encountered;a ship seldom, if ever, completely forsakes her own natural period of rolls; for each slight alteration in the wave period T1 introduces afresh terms involving the free oscillations of the ship. In the synchronizing conditions where T =T1, the forced oscillation is represented by ' Ir, rt

0 = -E91 COS T,

the amplitude being limited entirely by the resistance; the phase is; before that of the wave slope. The vessel is then upright in mid-height, and inclined to its maximum angle on the crest and in the hollow of the wave. The maximum amplitude 6 is given by E-®, =a.@. Since the right-hand term represents the decrement of roll due to resistance, the left-hand side must represent the increment of roll due to the wave in this synchronizing steady motion. If this latter relation be assumed to hold when the resistance to motion is represented by the more general decremental equation, then the maximum amplitude (9 is given by § @, =a.®-l-b.®'.

In 1894 and 1895 M. Bertin, at the Institution of Naval Architects, extended this relation to cases in which T1 is not equal to T, obtaining at the same time not simply the angles of steady rolling for these cases, but the maximum angles passed through on the way to the steady condition; to these maximum angles he gave the name of “ apogee " rolls.

In 1896, at the Institution of Naval Architects, Mr R. E. Froude investigated the probable maximum amplitude of roll under the influence of a non-synchronous and non-harmonic swell; He imagined three identical ships, A, B and C, the first rolling in still water, and the two others placed in the same swell assumed recurrent in period 2T1, but not necessarily harmonic. Assuming resistance to vary as g, then denoting the vessels by suffixes, the effective wave slope by 01, and constants by K, K' and K”, de Ke Kf.

Yrs+ at + Ar°»

(1203 IZHB,

»-(E-f-Kg?-+K 0B»K”o, ,

lizgc 0c

Eg-i- +K'0C - K”61.

1 See papers on this subject read before the Institution of Naval Architects in 1900 by Professor Bryan and in 1905 and 1909 by Mr

A. W. Johns. “ See Trans. Inst. Naval Arch., 1875.