Simplified methods for calculation of damping using records of free nonlinear rolling with large amplitude decay

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SIMPLIFIED METHODS FOR CALCULATION OF DAMPING COEFFICIENTS USING RECORDS OF FREE NONLINEAR ROLLING WITH LARGE

AMPLITUDE DECAY

by

R. V. Borisov

Translated by

Michail Aleksandrov

Department of Naval Architecture and Marine Engineering The University of Michigan

College of Engineering

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Large amplitude decay is the constant phenomenon of the

free nonlinear rolling of the ships with bilge keels, in the case of deck submergence in model rolling tests, etc. Several diff-erent methods can be used to interpret the records of model rolling tests. Some of them are inadequately precise for the

rolling with large amplitude decay, some are quite accurate but very complicated. The methods described here enable one to make simple and accurate calculations of damping coefficients with large and small rolling amplitudes decay.

Solving equations for free nonlinear motion, with a harmonic approximation of heel angle and stepwise representation of maxi-mum amplitudes, (Figure l.a) S. N. Blagovescheuskyv obtained

formulas for damping coefficients

3 B

2 - (I) (1)

where and are the amplitudes of decayed rolling, taken from record.

D - displacement (model)

n0 - frequency of rolling for given amplitude

Pd1 and 2di - dynamic stability levers taken in accordance

)

with and

Q.1.

b)

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3. I. Voitkunsky, Reference [2], introduced the linear representation for maximum amplitudes (Figure l.b)

4n(t)=

[4-(-)t]

The damping coefficient according to this is

JZD((df

td.,)

W

fl?(&S )[1_1(1--fii)]

1 1'f.

G. I. Faddeev, Reference [3], on the basis of the expon-ential law (Figure l.c)

is obtained

W=-;

n:(e+e? )I f-f [f-f F(p)} where 11 2-n 4_, _p I

'2+7p2+9,

If p o (small degree of decay) correction factor F(p) - o. Function F(p) is given in Figure 2.

a'Om(t)

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F(p) 0.5 0.3 0,2 a, 0 Fig. 2

Formula (3) gives better, more accurate results than (1)

AG

and (2), and for amplitude decay factor ₌ _0.1 _{0.6 it can}

Qi+l

be simplified. Using series representation for in and = x we get 3 4

i

x

x x

p=__(x+++T

+.);

E(p)o2704±07,3±o7,z+2 3 3 3 2'

4 ± 4,

2G(f---)(f-X+I).

Substituting this in (3) we have

.D14-ç

) [4'47x4 825x'-.- f,825x2-f 2 ₁

4

n' 8'(/) J

(f x+x2)(42z1x'i. (Z71OsfVx2+2)J

After simplifications

3 .D(t,, t

_{) [}

x2 1

W4

Z(/)

[/+r+--+-+iJ

where A = 0.006x2 - 0.005x3 - 0.555x - term of negligible

magnitude. Note that expression in brackets is series represen-tation of eX, so finally we have

3

_D(t'4f)

eX

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Nondimensional damping coefficient

If _{A IA}

L2z

Pt7

For small decay

esf

349

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Calculations made according to formula (7) showed only 1.5% dif f-erence with the results obtained by (3) for = 0.5. For the linear resistance law

-/(

-iili9

. Substituting w from (7)

aria &

a- '

we have C L

2'=----e '/f--

,-I 26

24O

4& (9)

W

_"(4-4)

e2'

+4A It rt(A+4A)9.' f--i

x

Let 2,d. - 2.d. = 9..(® )AO, where = Q1 (1 - - average angle

1

i+l

1 then Dt'4) 8 2

4cj x

ti=e jr,

where n0 can be calculated using test data for a given amplitude. The magnitude of n0 is very near to the frequency given by Pavlenno formula, Reference [51. For the angle of heel which corresponds to the maximum on the stability diagram, we can write

we can find

349

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Calculations can be made in table form (table 1) where U)

and are plotted in Figure 4 or 5 against

i917

and

9C

The magnitude O° must be corrected according to Figure 3.

These graphs are valid for large (

'

-and small decay of amplitudes.

In the case of the S-type stability diagram W

and 2

L obtained from Figures 4 and 5 must be multiplied by

6 4° 2° 0 Fig. 3 Table 1 5

cq

LSO _2/u I 5 6 7 ,V

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BIBLIOGPAPHY

Blagoveschensky, S.N. "Ship Motion," 1953

Voitkunsky, J.I., "Calculation of Resistance Coefficients and Added Mass Coefficients Using Results of Model

Rolling Tests," Proceedings LSI, vol XIII, 1954

Faddeev, G. I., "Interpretation of Records of Free Rolling With Large Decay," LSI Proceedings, vol. XXXVI, 1963

Glotov, V. K. "Rational Methods for the Calculation of Rolling Parameters," LSI Proceedings, vol. XXXI 1960

Pavlenko, G.E., "Theory of Ship Rolling with Reference to Ship Safety at Sea," Pros. of AS of USSE p 12, 1947