A proposal of a method to specify the permissible region of instability in the steering characteristics of ships

Report No. 299

February 1971

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

A PROPOSAL OF A METHOD TO SPECIFY THE PERMISSIBLE REGION OF INSTABILITY IN THE STEERING CHARACTERISTICS OF SHIPS

by

Contents.

Summary.

List of figures. Introduction.

The simple human operator model proposed. Requirements for limit cycle.

if. Numerical examples.

5.

Conclusion. References.

Summary.

The way to specify the permissible region of instability of ships is proposed from the viewpoint that ships should be controllable for any human operator without difficulty.

In the case of the steering of the ship (course keeping), where the motions of ships are small, we can approach that problem with a linear

equation. The importance of the time constant T, as well as the propor-tionality coefficient K, was stressed. Assuming a very simple human operator model, which can be followed by any human operator, the permissible region was specified as a combination of T and K.

List of figures.

Fig. 1. The bang-bang control (i).

Fig. 2. The bang-bang cóntrol (2). Fig. 3. The bang-bang cOntrol (3).

Fig. 14. _{Controlability of the unstable ship.}

Fig. 5. Control law. Fig

6.

Limit cycle.

Fig.

_7.

An example of permissible region.

Fig.

8.

The time history of point A. Fig.

_9.

The time history of point B. Fig.1O. The time history of point C. Fig.11. The time history of point D. Fig.12. The time history of point E.

1. Introduction.

It has been considered that ships should be stable on their course to be steered satisfactory. But from recent experience, we have found that

it is not so difficult to steer the large tankers straight on course as we had feared, even if most of them are unstable. On the other hand, it is very difficult to steer the small unstable ship which has the same width of

hysteresis loop in the result of the Dieudonn Spiral test.

From this ezperience, we should deduce that the diffiulty in steering the unstable ship depends largely on the time constant of the ship.

For the same statical characteristics, the unstable ship with the small value of absolute time constant changes its course too quickly to be

followed by the human operator, but in case of the ship with large absolute time constant, the human operator has enough time to check the ship's

course.

It has been known that the unstable ship can be steered satisfactory

when it is steered automatically with the sufficient amount of rate-cöntrol( i).

For example, let us consider the Nomoto's model,

T + r = KiS (i)

T > O, K < O stable ship T < O, K > O : unstable ship

when the rudder angle is changed by the control law of

6kr

(2)

Eq. (i) becomes

T? + (1-kX)r = O

T.

1-MC r + r O

and the new apparent time constant T/(1-kK) can be made positive by k which is larger than i/K when the ship is unstable.

(3)

ç)

t.

When concerned with the steering of ships by human operators, we must consider it as a man-machine system which combines a human operator

with a ship. The attempts to model the human operator(to find the control law of the human operator as E. (2) ) have been carried out and

further experiments are continuing now, but we have not been able to

find

satisfactory results from them. Certainly, it would be reasonable to attempt to find the characteristics of human operator first, and after that to consider the whole man-machine system, but we can not wait for that. We can -measure the value of T and K now, but we do not know whether these values are satisfactory or not. We an estimate the value of T and K of the ship which we are going to design, but we do not know in what range these values should be situated.

The analysis of the ship manoeuvres has considerably progressed but the criterion to specify ship manoeuvrability is still lac}Ung. We have to know the region in which T and K should be selected, even if it is a very

rough estimation.

There may be another approach to determine it, for example:4 assuming that with a very simple control law which can be followed easily by any húman operators, the ship may be steered straight on course within a tolerable range of errors, then we may specify the steering characteris-tics of the ship satisfactory. By this approach we may be able to propose the permissible range of instability.

2. The simple human operator model proposed.

The simplest control law would be a bang-bang control which is related to the course error.

= 'S sign (tji)

I

tj)>O o

p< o

But it can be proved by the limit cycle analysis using the describing function that the unstable ship can not be steered by this control. Some examples are shown in Figures 1 - 3.

To steer the unstable ship we have to pay attention to the rate of turn as in Eq. (i).

When we solve the eqta.ton of motion (Eq. (i) ) with the fixed rudder angle and the initial condition r(o) = ró, we get

t t

r(t) KçS0(1 - e - ) + r0e

t (5)

= + (r0 KESc,) e

-Therefore, if the ship is unstable (T <'o) and when the rudder angle is kept constant, r(t) increases infinitely in the direction which is deter-mined by the sign of r8 - KS0. But it also means that the turning direction of the unstable ship with the initial tate of turn rd can be changed by keeping the rudder angle which is larger than r0/K, even if only the linear system is considered.

It can be understood easily when we see the result of the spiral test

(Fig. )4). This curve shows the equilibrium points of ? = o of the equation

Tf' + f(r) KS

= (KcS

-

f(r) ) o

Therefore,the yaw acceleration is positive in the left part of this curve,

and negative in the right. So, the point A which is specified by the set of values (_cSc,, r) will move upwaì'ds until it gets to the point P if the rudder angle is not changed.

But if the rudder angle is changed to (Point B) crossing the equili-brium curve, the rate of turn of the ship will begin to decrease immediately and changes the ship's course in the opposite direction.

From the consideration above, it can be seen that we can make a ship operate in the limit cycle of amplitude r5 by the control law illustrated in Fig. 5.

An example of this limit cycle is shown in Fig. 6 (unstable ship).

The reason that we must introduce a hysteresis loop in the conrol lw is the sensory threshold of a human operator to detect the rate of turn of the ship. The human operator cannot detect a very small rate of turn. So, when the thip changes the turning direction, he can detect it after the

rate of turn is increased to a certain level. If we do not introduce a

hysteresis loop, a human operator will feel difficulty to follow this control law.

Although the course of a ship can not be guaranteed by this control law, it would not be difficult to correct it when the rate of turn is limited within a certain range.

In the following discussion, we will use this control law as a simple and feasible control, and consider the conditions which the limit cycle should satisfy.

3. Requirements for limit cycle.

When a ship is steered with the control law of Fig. 5, the period of the limit cycle varies according to the characteristics of ships. The time consumed to move form the point B to point C and D to A is

-r5

K60+r5 -T log

assuming the time time t _{= tCD = O, we get the period of limit cycle}

Kó0+r5

tCD = -2T log

Now, we should restrict this period within _{a certain range,}

tmin<tß

_CD

<t

_max

because if the period is too short, the human operator has to change the rudder angle so frequently and he will tire soon, and if the period is too long, the amplitude of course deviation becomes _{too large. This deviation}

can be estimated, assuming the change of rate of turn is sinusoidal, as

r5tCD

2iî

1dr

r T dr

= circular frequency of the limit cycle.

Certainly, the condition which is given by Eq. (8) _{can specify only a part} of the necessary conditions which ships should satisfy, but we are able to get soem information from it which combines the static _{and dynamic} charac-teristics of ships.

(6)

(9)

(1)

14 Numerical Examples.

We have to know the values &, r, tj

and tmax to specify the per-missible region of T and K, but we are still lacking the information

about these values.

So, let us try to guess them only to know the tendency of this criterion. For example, the following values may be said not to be unreasonable.

= 50

r = 0,05°/sec

tmin = 30 sec

tmx

= 125 sec

The last value, tmax = 125 sec, means i 1.00 according to the Eq. (9).

The permissible region of instability which is obtained with -these values is shown in Fig.

_7.

In Figures

8 -

12, examples of the time histories of r and i are shown for five sets of (T, K), which are marked on Fig. 7.

In the case of points A and B, the motions of the ships are moderate. But when a ship has the characteristics of point C, the helmsman has to work too busily, and at points D and E, the amplitude of _4' becomes too large, 1.7° and 1.LL° respectively.

Concerning the estimated values we used here, perhaps we can not change and (which determines tmax) largely, but nobody knows the correct value of r. If it is possible to choose r sma1ler, we can specify tmax larger and we will get a larger permissible region than Fig.

_7.

5. Conclusion.

A method to obtain the permissible range of T and K is proposed.

This report proposes only the method by which we can _{arrange our way of} thinking about the instability of ships, and does not have an intention to propose a numerical result. If this method would be accepted, the next step we should take is to get the reliable value of r8. The other values may be estimated from the practical experience of crews.

Certainly, the value of r8 will be changed by the installaUon of navi-gational aids such as "rate gyro", but it can be considered as the reward for the investment.

The values of T and K in this method should be completely linear para-meters. So, to make a practical reference to this methòd, it is necessary to get the linear characteristics of ships instead of the linearized ones, For this purpose, Bech's reverse spiral test will make a contribution to get the linear K as the slope at the origin of the r - curve.

To get the linear T from the full scale measurement, it will be necessary to have a test with small course deviations, such as the modified zig-zag trial proposed by Motora et. al. (2).

Certainly, this method can be employed only for the steering requirement. For another purposes such as course changing, some different requirements should be added.

References:

Motora, S. and Koyaina, T.:

"On the improvement of the manoeuvrability of ships by automatic control".

J.S.N.A. Japan, Vol. 116, 1961t.

Motora, S. and Fujino, M.:

"On the modified zig-zag manoeuvrer to obtain the course-keeping qualities of less stable ship".

12th ITTC Report,

_1969.

q',o

5O O

-5.0

6 50

5.0

o

- 5.0

§

5.0

P

t I I

50

100 150

Fig.3

The bangbang control (3)

T=lOOs, K=O.1/s

200

250

t.

sec.

5.0

A

-60

//

/

/

/'

/

/

/

Fig.4

_{Controllability of}

the unstable ship

/

/

/

/

/

/

/

K6= f(r)

czO

/

V

L

o

u

rs

-60

Fig. 5

Control

law

o

Fig. 6

Limit cycle

o

o

200

400

600

T sec

/

O

D

o

OB

E

C

Fig. 7

An example of permissible region

K/sec

deg. degdeg.

lèc.

1.0 0.05

5.0

0

00

-1.0 -0.05 -5.0

150

\ 200

,1"

₀

/

/

sec.

s

.

q

_r

deg dei. deg

4ec

1.0

0.05 5.0

o o

1 .0 0.05 5.0

Fig. 9

Time history of point B

\

200

t

(j)

r

_ô

deg

deq.-deg

-sec

1.0

0.05

5.0

00

LO

0.05

5.0

r

ÁtvA

₅₀

JAIk

₅₀

IA

s

.

(1)rô

deg deg_.deg

-fec

1.0 0.05 5.0 o o

-1.0 -0.05 -5.0

100

/

\

_N

_/

ah

't,

Fig. 11

Time history of point D

q

r

ô

deg

deg_.deg

1ec

1.0 0.05

5.0

00 0

-1.0 -0.05 -5.0

6

Fig. 12

Time history of point E

'1'

N

_\

\