Naval Architecture by Cecil Hobart Peabody

Author:Cecil Hobart Peabody
Language: eng
Format: epub, pdf
Publisher: J. Wiley & Sons; [etc., etc.]
Published: 1911-03-25T05:00:00+00:00

Fig. 141.

where r^—ais the metacentric height GM. Consequently the righting moment of any angle is

Equating this righting moment to the moment required to produce the angular acceleration as expressed by equation (i),

~gdt' whence

IT' ;^<?=-6<?. (3)

in which the constant h depends on the acceleration due to gravity, and on the properties of the ship r^ — a and p^. This equation is readily integrated as follows:

D(r^-a)e^-p; ^ .,,

r ddd^e r

/My •'• [dtj

= -b^e^+c (4)

At the end of a roll the ship comes to rest at the maximum inclination tf^, and at that instant the angular velocity is zero, so that

and consequently

dO ,

-j^=Wej-e\ (5)

The time of a single roll is considered to be the interval between

the maximum incUnation on one side, and the maximum inclination

on the other side; so that the time of rolling from the erect position

(when 0=0) to the maximum inclination (when 6=dJ) can be deter-

• mined by integrating between the limits o and 0^. Thus

'^». do

I If . e V-^in

2 frL <?^Jo ft 2

• •

m r-~ I — , ■• • • • • • • |OJ

^/<K/fI ARCHITECTURE.

Another expression is commonly used in which p^ is replaced by its value from the equation

which gives

/

I-P.'D,

""^gDir.^ay

(7)

in which D is the displacement in tons and I is the moment of inertia of the weight of the ship in terms of tons and feet. These equations may be compared with the equation for oscillation of a simple pendulum:

/

=->lj

(8)

Hoseley's Graphical Method. —Starting from isochronous rolling with an involute for the metacentric curve, the time of roUing of a ship with any form of metacentric curve can be determined by aid of a graphical method due to Moseley. It is not necessary, however, to draw the metacentric curve in order to apply the method; it is sufficient to have the curve of statical stability.

In Fig. 142 let OM represent the curve of righting moments, or

the curve of statical stability for the given ship; and let OW be the curve of dynamical stability corresponding. At a given angle of inclination 5, the righting moment is dm, and may be represented by w. Equating the righting moment to the moment required to produce the angular acceleration of rolling,

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