Modeling of rolling

MODELING OF ROLLING by

V. V. Semenov-Tjan-Shanskj Translated by

Michail Aleksandrov

The Department of Naval Architecture and Marine Engineering The University of Michigan

College of Engineering April 1970

The differential equation of the rolling of a ship in calm water can be written as

78+ 2/Vé -4- Dk8 = O,

where - moment of inertia of the ship and added masses 2N - coefficient of the moment of resistance forces Dh - stability coefficient

e and G - angle of the heel, angular velocity and acceleration.

Equation (1) describes the ship oscillations. With a known record of rolling 0(t) and an initial heel, 0, this equation helps to find the interconnection between its coef-ficients. We must establish the criterions of ship motion and dependence of the moment of resistance forces upon these criterions. The resistance is caused by such qualities of water as weight, viscosity and surface tension. _{For the}

con-sidered case, gravity plays the most substantial part due to the nature of the forces generating the rolling motion.

After normalization, equation (1) can be written as

Ö + 21á + K28 O,

where the resistance coefficient is: 2N

and the frequency of free oscillation is:

K

The resistance coefficient as can be seen from (2) , may be represented as a function of several parameters

where p - mass density of water p - viscosity

e - surface tension

In different experiments the interrelations between damping coefficients and the ship's characteristics were formed accor-ding to the wish and intuition of researcher. The theory of modeling helps to reduce the influence of a personal approach

in interpreting the experimental data. After selecting the most important constants which describe the phenomenon under

consideration this theory logically leads to the final result. The application of the modeling theory, Reference [1] , is

given in this paper.

Let us find the criterions affecting the ship motion using the analysis of their dimensions. To accomplish that we must assume that a power functional relation exists between

them which means that formula (5) can be rewritten as

q TT rz p 5 t

2t4K ,P

.0

The components here have the following dimensions:

= 1; [2b] = 1/sec; [k] = 1/sec; [p] = kg sec2/m

[q] = m/sec2; [p] = kg sec/rn2; [01 = kg/m

After substituting in (6) we have

i _{- (1)q} 1 )m (kg .

sec)n

(kg)t

(7)

sec sec m 2 m2 m

sec

Due to equality of dimensions in the right and left sides of (7) we have (kg) (rn)

(sec)

'

-in

p =

5 = f-

Substituting p, s and t in (6) we obtain: 'n pi

t-m-n

t'-m-'hi

28K ,O

_C.8(j

where c - non-dimensional coefficient. The non-dimensional parameters

2

can be taken as criterions for described motion. The number of such criterions can be found according to the known 'u-theorem.

The number (Q) of components in expression (5) is

Q6

the number Q of components with different dimensions is 5 (25,,,o,0q,,u, 6);

the number of components with independent dimensions is:

QN

the number of criterions according to Tr-theorem is:

fK

tK=0p0H2

₍

-)

The total number of criterions must be

=

- 3

,o&

In this case the criterion 00 was discarded as non-dimensional. Taking into account

p

and bearing in mind that from (4) we can obtain

(9)

the criterions from the right-hand side of equation (8) may be given as

K

1/2

li

Vr

if

r

We=

r=

Replacing (9) and (10) in (8) and using new notation m -

i = X;

=c.[(4)2.

The coefficient C, and parameters q, x and y must be obtained from experiment. In expression (11) the ratio reflects the load distribution. The product

Y

reflects the scale factor and the product

2 4

f-e

reflects only the physical properties of the fluid.

To calculate the parameters C, q, x and y, let us use the experimental data employing the least squares technique. First we have to rewrite (11) as follows

e- dX.e where

d

=

(h)2

6

p.6'

-

then we have

Introducing new notations

c;

d=7;

the previous expression can be written as

Here A, , n and can be obtained from experiments.

If the test was repeated n times, then for the ith test

4ß+

+r71 +

Using the condition that the sum

5=

must have a minimum value and therefore

55 êS

=o;

_-a-=0

?°

we have four equations to obtain b, q, x and y.

x7+,&, =

+

_{+ xL 7 + y}

+

+ x27±

=

+

±x!7+

It was accepted in the previous experiments on the basis that influence of surface tension is negligible and consiquently instead of (5) we have

2E

Applying the theory of modeling we can get (instead of 8) 28 _9/Ic-,L/'1

,,n-i

=c8o

i

=0;

Let us denote 1) I d E /2 '/ -'I

-

()

26 j

7=f;

_7C=;

_=;

_Q'=7,

and finally we have

nJ3+

+ x7= L

4tt

Ed7,

which is the solution of (17).

The appearance of component in (11) and (17) is indi-cative. It has the same role as in several previous

experimental investigations. The replacement of B by 2 is justified, because this ratio gives the relationship between restoring and inertial forces. The combination of numbers Fr, Re and We in formulas (11) and (12) can be used to solve the problem of the scale effect.

6

REFERENCES

[1] Aigenson, L. S., "Modeling," Soviet Science, 1952