This paper deals with the period of free rolling of ship theoretically and experimentally, and the following results are obtained.
(1)
F (θ, θ), the resistance couple for the free rolling of a ship, which could be assumed as an arbitrary function of θ and θ, is expanded as follows.
F (θ, θ) = (
I0η) + (
I10θ±
I20θ
2+
I30θ
3) - (
I1θ·θ±
I2θ·θ
2),
where θ is a transverse inclination of the ship, and
I0,
I1 I2,
I10,
I20 and
I30 are the various coefficients which do not contain θ. It is well known that
I0, the coefficient of the first term, is an additional mass moment of inertia and the second terms are the frictional, eddy-making and wave-making resistances against the rotational motion of the ship. And it was found that the third terms indicate the mutual effect of the amount of the second order between the additional mass moment of inertia and the damping on each other, and it was proved experimentally that the effect of this terms on the damping is very small, but the effect on the additional mass moment of inertia comes to the great amount in the case of a ship with bilge keels, whereas this can be neglected as small in the case without bilge keels.
(2)
T, the half period of the free rolling, is approximately expressed as follows.
T_??_π√ (
I+
I0)
H/
WhG3_??_
T0√
H/
G3,
where
H=1+4θ/2θ
m (
f1θ
0+
f2θ
02),
G3=1+1/2
k1θ
02+1/3
k2θ
04,
W=weight of displacement,
h=metacentric height,
θ
0=initial amplitude,
θ
m=mean amplitude,
Δθ=decrease of amplitude within the range of one swing,
(
I+
I0) =apparent mass moment of inertia for small θ
0,
T0=free rolling period for small θ
0.
And
G3, a non-dimensional factor which is a function of the form of the statical stability curve, indicates the effect of the ship's form, and
H, a non-dimensional factor which is deduced from the terms, proportional to θ·θ and θ·θ
2, indicates the effect of the damping on the virtual mass moment of inertia, where
f1 and
f2 are constants to be evaluated ty the experiments. And the correctness of this formula is proved by the experiments.
(3) By the experiment results it, was proved that in the case of a ship with bilge keels the coefficients
f1 and
f2 have the values of the considerable amounts, but without bilge keels those values are so small that, putting
H_??_1. the period can be approximately calculated by the following simple formula.
T_??_
T0/√
G3.
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