Analysis of full-scale measurements of manoeuvrability by trial and error method

Report No. 332

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

L

October 1911

ANALYSIS OF FULL-SCALE MEASUREMENTS OF MANOEUVRABILITY BY TRIAL AND ERROR METHOD

by

Contents:

Summary.

List of Figures.

1,. Introduction.

Mathematical model and performance index. Some examples of identification.

1. Conclusion. References.

Summary.

It is inevitable lo include non-linear terms in the mathematical model of ship's manoeuvrabiiity, especially in the case of full ships.

For the identification of non-linear systems, usually, a trial and error method, which is governed by an optiniization technique is employed. The Simplex Method [i] (a non-linear optimization technique by Neider and Mead) is one of the most efficient methods at this moment. In this report,

an application of this was made for the identification of manoeuvrability of a 200.000 DWT tanker using full-scale measurements [2].

List of Figures, Identification procedure. Computation time. Norrbin's model 14/14. Norrbin's model 20/20. Longitudinal velocity 1/1.

6,

Longitudinal velocity 20/20.

Î. Van Leeuwen's model 114/lIt.

Van Leeuwen's model 20/20.

1. Introduction.

The Nomoto's first order model of ship manoeuvrability [3] has beer uèd

for a long time because of its simplicity in predicting a ship manoeuvre and easy analysis of full-scale measurements. But in the case of full ships, such as super tankers recently built, the range of ship manoeuvres which can

be described by a linear equation is restricted to a very small range. If we

warft to predict ship manoeuvres in a wide range, we have to use a non-linear equation. Consequently, we have problems what kind of mathematical model we should use and how we can analyse the result of full-scale measurement to i1entify the parameters in that model.

In the identification of non-linear models, the procedure of Fig. i is generally used. First of all, the form of model should be chósen, then a certain time

history of input is supplied to both, the system an the model, and after comparing the outpiats of system and model, the parameters of a model are determined so as to minimize the difference of the above mentioned signals by the logic of the identification method.

For the block of "Logic" in Fig. 1, an optimization technique is applied.

That is, after expressing the difference of both signals in a performance

index, which is a scalai' function of a set of parameters of the model, this performance index is minimized by an optimization technique.

When we identify the ship's manoeuvrabilitr by this method, it is possible

to make use of .any kind of result which is obtained by any kind of input, if it ha enough information to identify the system. So, we should consider the

problem what is the best time history of the input for this method. Often

the Zig-zag manoeuvre test is used to make the identification of a linear model easier.

The Simplex method which will be applied for the identification here is one of the most efficient optimization technique at this moment.

The intention of this report is to examine the possibility of the procedure of Fig. i in the analysis of full scale measurements. The details of an Algol program, which was made for this purpose, can be found in another

2. Mathematical iodel and performance index.

According to the identification method we are applying, it is necessary to specify the form of the mathematical model first.

Theoretically, the Simplex method can manage any complicated model (with many parameters).

In practice, however, the more parameters there are, the more computation time will be needed.

Consequently, the model should be as simple as possible to prevent a long computation time.

Norrbin [5] improved the inadequacy of Nomoto' first order model by introducing a simple non-linear term, e.g.

= KS - r - (i)

This seems to be sufficient as long as we are concerned with the time

history of the course of ships only. However, especiaily in the case of super tankers, it is well-known that the decrease of shipt s velocity caused by the

motion of a ship is not small

[6]

, and it is desirable to get some information about the longitudinal motion without losing simplicity.

Recently, Van Leeuwen proposed a rather simple model [î] including a longitu-dinal equation as follows:

(KS - r

-2

Tu

(u0 - u)

_{- Kr}

where:

u0 : approaching ship velocity

u : instantaneous ship velocity

r = (L/u)r : non-dimensional rate of turn with respect to the

instantaneous ship velocity.

He assumed that if the space history of ship motions (r* and ) are the same, then the force and the moment which act upon a ship are proportional to the square of the instantaneous velocity. He further assumed a proportional

relation-ship between r* and 6 when the motion of a relation-ship is caused by the rudder only. Eq. (2) is the result of above assumptions.

-2-In Eq. (3), he introduced

only the most significant contributing term to speed loss in order to keep it simple.

For our purpose, however, _{it is convenient to use a dimensional equation}

because the results of _{full scale measurements will be given not in a space} history but in _{a time history. So, Eq. (2) should be re-written as follows:}

T' = K6u2 - ru - ar3/u

₍₄₎

We can agree that _{moments caused by r* and r*3}

are ptportional to u, but the moment due to 6 should _{be proportional to the incoming velocity at the} rudder. This incoming velocity _{seems to be more dependent upon the number}

of propeller revolutions (n) than upon the ship's speed itself. So, it would

be better to modify Eq. () a little.

T = _{K6n2 - ru - ar3/u}

If we make n and _{u non-djiensjona1 by no and u0, we get:}

T = K6(n/n )2 _{- r(u/u0) -}

_ar3(%/u)

(5)

When we consider

n and u to be constant, T, K and a in Eq. (5) have the same

values of t ho s e in Norrbin'

s model. If the information about n is not available,

and if the output of the engine _{does not change, then it may be possibLe to} assume n/n0 i.

ConsideHng the steady state _{values of turning, we get from Eq.}

u/u0 i - Kü/u0.r2

putting this relation into Eq. (5) and taking = o, we get:

K6(n/n)2

_{= r - (K/u0 - a)r3 + 0(r5)}

This relatjön should _{be fitted to the result of the spiral test.} case of Norrbin's model, K6 _{= r - ar3).}

3

(3)

In this report, the Norrbin's model and the modified Van Leeuwen's model which is discussed above will be applied.

Next, we have to specify the performance index which measures the difference between the attual system and the mathematical model.

Mostly, the following function is used for this purpose: T

J=;

_{0JII's%m II}

2

dt

state vector of the system = state vector of the model

However, when the difference between )C

and%m is expected to be

small,

it is better to use absolute values instead of square of norm.

So,we use the function as follds:

T

=

J

{ _I

_-

+ 10 Jr5 -

_rml } dt (8)

The second order term is multiplied by 10 in order to make the contributions of both terms of the same order.

3. Some examples of identification.

In this section, the results of the identification of a 200.000 DWT tanker will be presented.

The Shipbuilding Laboratory in Deift has a set of complete time histories of Zig-Zag trials of that ship, which includes rudder angle, course, rate of

turn and velocity of the ship for /1, 114/114, 20/20 and 30/30 Zig-Zag trials

at every second. So, we can make use of them for our purpose assuming that these data have enough amount of information.

In the program of the Simplex method, we have to calculate the whole time history of the ship motion for each set of parameters to evaluate the

perfor-mance index by a Runge-Kutta procedure. It costs about four seconds by IBM-360/65 which includes 1400 steps of calculation for i and r. That means if we identify a complicated model, it will need a very long computation time. For example, a rough sketch of computatiön time according to the number of parameters of a model is shown in Fig. 2.

From that figure, it can be seen that we should restrict the number f

parameters to four at the most, and that it is very convenient if it is possible to divide a model into several parts (to divide a 6-parameters model into a 2- and 14-parameters model).

The number of parameters we should obtain are listed as follows:

where ôr : residual rudder angle.

Since we have data of r and u, we can consider Eq. (3) independent of Eq. (5). So, we can obtain parameters of both models within 15 to 20 minutes. When we have results of the spiral test, we can obtain K and a by Eq. (7).

0f course, it makes our identification process much eas±er.

Bech's reverse spiral test

[8]

will be very helpful especially in determining K of an unstable ship. The parameters obtained by Simplex Method are listed in Table 1.

5

No. of parameters Remark

Norrbin's Model y. eeuwen's Model 14 6 T, K, a, cSr T, K, a, r'

T, K

u

Table 1.

Time histories of a ship and an obtained model are compared in Figs. 3 to

8.

Though we can see in those figures that models are fitted to full scale measurements reasonably well, sets of parameters obtained varies rather much according to respective trials with the exception of the K/T-values.

If we really need such a change in the value of parameters according to the diffeient input, there is no merit to use a non-linear model.

Several reasons can be considered for these features:

(i) Imperfection of models.

It is said [9] that, in the analysis of Zig-Zag trials, the change of para-meters according to respective inputs is caused by the simpldfication of the

linear part as well as the non-linear effect.

If we use the second order model for the linear part, the unsatisfactory fitting at the initial stage may be improved. In that case, however, we have to

obtain two more parameters (T1, T2 and T3 instead of T). It will cost too much computation time by the Simplex Method and the result of the spiral test

will be needed.

(2) Lack of information in the input.

This seems to be the most significant reason.

It was a dominant feature that only the parameter K/T had converged very quickly in the procedure of the Simplex Method.

When we consider the step response of the linear part of the Norrbin's model (Fig. 9), K i

c3

r

-r -r

6 -- 1/T ('1/sec) K/T (1/see2)

aIT

(sec/deg2) r (deg) Tu (sec) Ku (m.sec/deg2) Norrbin 111/114 _{-0.0012 0.00019 0.069 0.91 -} -20/20 -0.0022 0.00018 0.0811 1.75 - -y. Leeuwen 114/111 -0.0032 0.00019

o.o6

1.014 600 _37.11 20/20 -0.000514 0.00019 0.0111

1.62

510 140.1

L

we can see that the initial slope of the response is given by K/T-value (so we can say that it is probably the most significant parameter) and that, after the K/T-value is decided, the saturated value of r is given by l/T. On the other hand, when we include the non-linear part, this saturated value is also governed by alT as well as l/T if the amplitude of r reaches to a certain value. Consequently, when the measured results have a feature of the uniform amplitude like the Zig-Zag trial has, there will be a mutual dependence

between the values of l/T and

_aIT.

To prevent these phenomena, it will be necessary to provide the input signal with some random characteristics.

(3) Initial condition of r.

Since the full scale measurements used here were disturbed by waves, it was difficult to specify the initial condition of r correctly. So, r(o)o is used for all identifications.

This initial condition affects rather much in obtaining parameters, when the measured rate of turn was used an initial condition, a completely different

set of parameters was obtained.

To eliminate this effect, it will be necessary to adopt some suitable filters or to use a much longer time history of measurement.

Comparing the Norrbin's and Van Leeuwen's model, Van Leeuwen's model is a little better in describing a saturating tendency of the aniplitude of . In the

prediction of a manoeuvre of super tankers, however, it is becoming important to predict the change of ship velocity as well as rate of turn. So, Van Leeuwen's model will be very important in this respect. In addition, it will be very

convenient if the modified Van Leeuwen's model of Eq. (5), which includes n explicitly, can cover a certain range of n.

li. Conclusion.

An application of the Simplex Method to the identification of the ship xnanoeuvrab±lity was made.

The results are not so muchprom.sing, because this method is too much time

consuming in the calculation, especially when the number of parameters to be obtained is increased. Almost all of the computation time of this method is consumed in a Runge-Kutta procedure to obtain a time history of ship motions. So, if we can use a hybrid computer, the computation time may be reduced to a large extent.

In addition to this, we have to consider a suitable time history of input for this method to complete the procedure.

-8-References:

[i] J,A. Neider and R. Mead:

"A Simplex Method for Function Minimization" Computer J. 7,

1965

C.C. Giansdorp and 14. Buitenhek:

"Manoeuvring Triais with a

200.000

tons Tanker" Report No. 2}48, Shipbuilding Laboratory,

University of Technology, Deift

₁₉₆₉

K. Nomoto:

"Analysis of Kempf's Standard Manoeuvre Test and Proposed Steering Quality Indices"

Fist Symposium on Ship Manoeuvrability,

₁₉₆₀

T. Koyaina:

"An Algol Program of the Simplex Method" Report No. 307, Shipbuilding Laboratory, University of Technology, Delft 1971

N. Norrbin:

"On the Design and Analysis of the Zig-Zag Test on Base of Quasi-Linear Frequency Response"

10th ITTC London,

₁₉₆₃

H. Eda and CL. Crane

"Steering Characteristics of Ships in Calm Water and Waves"

TSNAME Vol. 75, 1966

[î] G. van Leeuwen:

"A simplified Non-linear Model of a Manoeuvring Ship" Report No.

262,

Shipbuilding Laboratory,

University of Technoiogr, Délit 1970.

- 10

(w&S 8325) M.I. Bech:

"The Reverse Spiral Test as Applied to Large Ships" Shipping World and Shipbuilders, November 1968

S. Motora and M. Fujino:

"On the Modified Zig-Zag Manoeuvre to Obtain the Course-Keeping Qualities o Less Stable Ships"

Ship

>J Model

Logic

Fig.1

Identification procedure

o

4

6

No. of parameters

T

time

Fig. 9

Step response of

-H Ship

8.

:RH Logic

f--H Model

Fig. i Identification procedure

1pow UqJJON

fl/fl

iepow

uJJo4

£

iI,

Uap (I!'2

u

m.-5

Fig.8 Longituding vetodty 20120

- meod

- modeL

t5Ø

FIg. 5 Longitudinat veLocIty 14/14

__

A

-500

/

1000

V

wY

i

Fig. 8 y. Leouwen modeL 20/20

measured

--

modeL

Fig. 9' Step response of