Some notes on the application of K-T analysis to marginally stable ships

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Sc.ieepsbouwkun&

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jeNOTES ON THE APPLICATION OF K-T ANALYSIS TO MARGINALLY STABLE

SHIPS. by K. Nomoto

1. INTERPRETATION OF Z- TEST RESULTJ FOR MARGINALLY

STABLE

SHIP-IN TERMS OF THE FIRST-ORDER SYST&M APPROXIMATION.

Figs. 1 - 4 illustrate a typical example of K-T analysis to Z-test results for marginally stable ships. Tne principal dimer.sions

and other particulars of this ship are listed in Table 1. TABLE 1

( Ad?

'

LxBxD

220 x 31.09 x 16.07 ( m ) dm

11.65

67, 200T

L/B

7.08

CB

_0.82

B/d

2.67

i/fd

-

0.116

Ar/Ld

1/77

14.6 kt.

A few words should be given here on the

5

degree zig-zag test

(Fig. 4). At opening the manoeuvre, the 6hip had a negative initial angular velocity as is liable to be for marginally stable or unstable ships, so that after 98 sec. the null rate of turn was

realized under the effect of the first rudder action. So far as employed the assumption of the first-order system approximation, we can simply shift the time origin to this instant.

Using the usual procedure of defining K and T value out of Z-teats, we obtained the negative K and T as described in the fi:pare.

It may sound rather strange, but this does sometimes really occur

for marginally stable or slightly unstable ships. _{If the ship was} much less stable, 5 degree Z-manoeuvre could not be realized, however.

In synthesizing the ship motion yOe() using the K and T thus derived, we should make use of Taylor expansion of

£07,

otherwise*

the numerical error of computation would spoil the result hopelessly.

In this application the ship response to rudder movement is so sluggish (very large positive or negative T), that we can appro-ximate the rudder angle time history in rectiangular fashion, as shown in the sketch below.

5

tecti4mrdor

cif/Prix/met/

p 4.

/?gyfor this rectiangular

Set)

is

composed of the skeet& excited by a series of stepwise

.Stells.

The step response of p is simply;

t

V.;

i<

_C/r)

ztArt

The step response of

'P is

1

51;

di

=

KT;,

I (z - /) -

17/77

Ari,

I-

_z

_tr/

)3

t)

/

r

f 1

tri

Obviously from this reduction, motion of a ship

with

negative K and T under zig-zag steering is a diverging oscillation. Yet for a slight instability so far, the motion can be apparently an ordinary one and bear the usual K-T analysis.

Glancing over Figs. 1 through 4, it may

be said that

the

first-order system theory can provide a fair interpretation even for

mar-ginally stable ships.

2. COHERENCY BET/'EN THE INDICES K AND T DERIVED FROM Z-TESTS AND

r' - 6C CHARACTERISTICS OBTAINED THROUGH THE SPIRAL TESTS.

Fig. 5 shows the values of 1/KI and 1/T' derived from Z-tests for marginally stable or slightly unstable tankers, against r'm that is nondimensional average angular rate of turn ciuring the

manoeuvre. As is seen in the figure, 1/K' and 1/T. decrease their values with decreasing r'm, which implies smaller rudder

angle

employed.

This marked tendency results from the considerable decrease in yaw-damping

with

decreasing r'm, in other words, non-linear yaw-damping. Here should be called upon the physical interpretation of K and T,

that is;

K turning force provided by the rudder /

=

yaw-damping /i)

inertia of the ship yaw-damping/

It is worth noting that 1/K'

zinc! 1/T' of each indivicaual ship

become null at largely the same r'm. This implies that the

yaw-damping becomes null At this critical r'm. For the ship motion whose average r' is less than this r'm, the ship will be unstable on the average and vice versa. Here again the concept

of

"linear on the average" should be called up.

Now we can relate the indices with r' - 4 characteristics obtained through theepiral tests.

and

Accord i_n:: to thu "linear on

the averaize"

concept, the average

K' derved

from

a

Z-test

is to correspond with

the slope of the

straight line

A3 in

the sketch below, where A and 3 are the

points

on the r' - S curve, whose r' values ,are equal to the values

of r' during the test. The larz;er

the

rudder anzle employed, the larger the r' peak values

and the

smaller the K' value.

As a particular case, suppose a Z-monoeuvre in which the r' peak values correspond to the points Ao and B. The derived K' would then be infinitive and 1/K' be null (1/T' too). This

case should correspond to the critical r'm aforementioned indeed.

Through

this reanoning we can estinate the loop height of r'- S

curve from a series of Z-tents with a few different rudder angles.

Considering that r'm for a Z-test is nearly equal to 0.7 x r' peak value, the critical r'm should be multiplied with 1/0.7 to yield a half height of r' - O loop.

The results for the present examples are seen on the tahle in Fig.5 and one can say it is fairly coherent results if taking account of the extremely simplified assumi.tion involved.

CONCLUSION

In principle it is obvious that the first-order system approximation

for

ship

motion in steering nhouid not be applied to unstable ships where considerable nonlinearity intervenes. Norrbin has then suggested some modification for the

method taking

account of a simple nonlinear term in Alimgn Rapport Zr. 12, 1965 and really it should be a promising

approach.

On the other hand, however, introducing the concept of "linear on the average" the first-order system theory is found to be able to well interprete the Z-test results even for marginally stable or slightly unstable ships in conunction with r'- p characteristics obtained from the spiral tests.

Considering the extreme sim?licity and extensive mathematical. convenience of the theory, it should still be one of the good sources

for manoeuvrability measure of ships including marginally stable ones.

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