Ocean waves are not regular, as are assumed in the ordinary theory of rolling, and accordingy the max angle of roll deduced therefrom is not directly applicable in practical case. In this paper, the waves are not assumed to be regular, but only one wave, which passes the ship in one swing, is considered as sine form, and the limiting angle of swing possible in that ease is found. The angle of swing is generally given by
θ=
K-∞Σ
n=1θ
n cos
nkt.But a large swing can safely be represented by
θ=
K-θ
0cos
κt…(1)
This fact, valid both for apparent and absolute rollings, can be mathematically proved in both cases(Appendix)and also is observed from the record of actual rolling on sea as Fig.1
Assuming the form(1), only the swing, whose final amplitude is not less than the initial one, need to be taken into consideration for the present problem, then this condition gives the relation that σ must be less than unity. The final amplitude of swing θ
f is given by
θ
f = [2
eσ
8+
ae(1-σ
2)] sinσπ/2/
e√4σ
2(
e2σ
2-1)
2+
ae2e2(1-σ
2)
2tan
2σπ/2…(2)
The maximum value of θ
f for various σ is the desired limiting angle, which occurs, practically σ= 1/
e. Consequently θ
fmax occurs at σ=1 for
e>1, and at σ= 1/
e for
e>1, the results being as follows,
θ
fmax=1/√(
e2-1)
2 +(2
aee/π)
2 for
e<1 or
Tw<
TsT=
Tw.
θ
fmax=2+
ae(
e2-1)/
aee2(
e2-1)cosπ/2
e for
e>1 or
Tw>
TsT=
Tsθ
fmax=π/2
ae for
e=1 or
Tw=
TsT=
Tw=
Ts.…(3)
The effective extinction coef.
ae in these expression is to be determined by the following equations
ae=
a+√2
bγΘ
w/
e [(1-
e2)
2+(2
aee/π)
2] for
e<1.
ae=
a+2
bγΘ
w/
aee2(
e2-1)cosπ/2
e for
e>1.
ae=
a+
bγΘ
w/2
aeπ for
e=1.…(4)
For the absolute rolling, the limiting angles are given by multiplying
e2 to the results (3), and the
ae, by taking
e2γΘ
w instead of γΘH
w in (4).
These results are calculated and given, in Fig. 2.
The same method can also be applied for the swing on waves affectedby wind, in which the, new. variable θ=θ-φ is to be used instead of θ and the
fmax is given by(3), then the limitting angle is equal to θ
fmax plus φ.
These treatment is based on the assumption of isochronous stability curve, but the above results can also be applied to the non-isochronous case. The approximate θ'
fmax for non-isochronous ship can be got by taking the same area below the stability curve in both cases.
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