Simulation of full-scale results of manoeuvring trials of a 200.000 tons tanker with a simple mathematical model

flaot: No. 301

riaroh.

19fl

r

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

Simulation of Full-Scale Resulta of

Manoeuv'ing r1a1n of a 200.000 -tone

i?anker witb a Simple Mathematical Model.

y

Lirb of Symbol..

a

t integer with the valu.

i, in eq (r)

pine sign indicating coures stability

minus sign indicating course instability

a

tontant faotor in sq (2)

AS/A0

i blade ursa ratio

b

s non linear damping

$ constant. associated with thrust coefficient polynomial

Day,

i constants a.sooiat.d with torque coefficient polynomial

D

i diameter of the prop.11er

t advance ratio of th. propeller

K

t proportionality constant

KL

i proportionality constant

s torque coefficient

Kp

s thrust ooffioient

- x

longitudinal ma.. (inclusiva added mase)

n

s reveut tens pez' .eooM

s maximum revolutions er seoond

s pitch ratio

s torque delivered to propeller

s maximum torque d.livered to propeller

R

s rseistancss

r

irateofturn

T

s thrust

s

T1, T2, T

s time constants

u

$ forward speed

i steam inl.t eatio

¿

_{t rudder angle}

¿SR

s rudder ingl. due to disturbances

s oours. angle

f

s density of water

Liet of Figures1

Fig. 1: ReBulte of the Die*onn4 epiral test in fully laden condition.

Pig. 2: Resulta of a 7/7 zigzagtrial in fully laden condition. Pig. 3: Results of a 14/14 zigzagtriai in fully laden condition. Pig. 4: Resulte of a 20/20 zigzagtrial in fUlly laden condition. Pig. 5: Results of a 30/30 zigzagtrial in fully laden esn&ttion.

Pig. 6 Resulta of the Diedonn spiral teat

in ballasted. condition.

Pig.. 7* Resulte of a 7/7 zigzagtria]. in baflasted. oondition. Pig. 8: Resulta of a 14/14 zigza.gtrial in ballasted condition.

Fig. 9: Results of a

20/20 zigzagtrial in bai.lasted condition.

Pig.l0s Results of a 30/30 zigzagtrial in ballasted condition.

Pig.Us

Result of a 20 degrees turning circle in kUasteti condition.

Pig..12: Bode diagrams for fully laden

condition.

Contenta.

Summary

IntroduotiGn.

The mathematical model and the determination of the coefficiente. Reaulte of the eimulation of full-iode manoeuvrai.

3.1 In fully laden condition.

.2 In ballasted oondition.

3.3. Diecuielon of the reault. of the iimulation.

Resulte of the computation of trequenoy-reapons trials. Conclusione.

Summary

A simple mathematical model, the we1l.kiown second order differential equation with a non-linear term and a simple forward speed equation, hae been adopted for the simulation of full scale triale with a 2OOOOO tone

tanker. The cosfliolente in this mathniatioal model er. adjustid in en imperiosi way; they are constant for one power setting for the whole rang. of rudder angle,

The result. of the simulation have been presented compared to the valu.. measured during the full scal. trial.,

With the determined coefficients, fr.quenoy..respone oharaoter.tatioe bave

been aaloulated.

i. Introduction.

The purpose of this report is to provide a, mathematical model which is

simple, possesses all signifiant rnanoeuvring qualities, and describes

fu].l-aoale manosuvres accurately enough for practical goals.

Th. results of the wanoeuvring trials of a t-o1aas Thitob Shell TanPer (i) have been usict for the aimulation.

This mathematical model can be used. for either course keeping

atidiea,

i.e. adjusting autopilot characteristics to the ehip's dynanic behaviour due. to small disturbances, or Indicating the turning capacity. To this

end a speed equation can not be dispensed. with, because of the large

epeed op ocouriUrig at the greater rudderanglee usually observed with

mammoth tank.ra. In the mathematical model as presented in the next section, rate of turn - forward speed interference has been omitted due to the fact that the speed measurements made during the full-scale trials seem to be not reliable enough to base mora sophisticated mathematical modele upon.

In the course of time, various attempts have been made to construct

emperla..]. mathematical models to describe turning-,

coureekeepirig and

course changing characteristics. A linear approach has been presented by Nomoto (2) ad a simple non4in.ar model has been given by Nrbin (3)

and Beo?i (4). Recently van Leauwen (5) published a simple non..linear mathematical model which includes the forward speed-rate of turn inter..

ferenoe.

This model oan be used for all msnoeuvring aspects. To

determine

this

interference it le necessary to have reliable speed measurements which were not available in this case. Therefore Be'a mathematical model expanded with a speed equatò has been used for the simulation.

2. The ulathematical model and. The determination of the coefficiente.

2.1 The mathematical model.

The mathematical model used in the simulation is given by the following three euatione.

3.

The drif t equation, whioh is very important l'or the hydrodynamics of the

ship is ignored by applying the assumption that dritt velocity and rate of turn have a time independent relation during manouvringp

!

+ (T1 +

+ sr + br3 - K(T36+j...5r)

Ça-x)+T(l-a)-R.-KLr2

(i)

'. r.

The first equation is in faot Nomoto's

s.00nd

order equation where a non-'

linear damping term has been added as auggest.d by Beob.

The righthand side .f the second equation i.

mainly du.

to the longitudinal

component of the centrifugal foro.. mi effect of the rudder oheracteri.-tie. has be*n inored upon the speed drop. Although this is not quite true,

..pedi..11y in the first transient otag. of motion, rudder influence seams

to be small compared

to the influence of

the centrifugal foro..

The left hamt cid, of th. .qation has been oenetruoted with the well-kiown

.leiunt., ship'. assi inolusive the added mass, the thrust of the propeller,

the thrust deduction factor and the ahip's resistance.

2.2 The d.t.mination of the coefficients.

The lst'.hand oid. ooeffioisnt. of the second equation in

Cl)

can be salsul4t.d, assuming that the (complicated) natur, of the propeller..

hull interference will not change du. to different flou conditions d*ring mane.uvring. Added mass is assumed to be 4 percent of the

ship's mass.

It is to be noted that this simulation ii done for one partioular setting of

the

steam throttle, i.ø. 100% stesa inlet.

Goodwin et. al. (6) published an equation for the behaviour of a

turbine,

+ 1 a (2)

Por "a", the value 1 has been recommended. In this os.. the steam in..

ist ratio is always and are steady state values of

this. quantities on a straight course with maximum power. Reference ii made to table i for the values of

the

eignifint quantities.

t.

The tortue of the propei1r hae been given assuming that the given propeller produces the same thrust and absorbes the came

torq1ue of a similar N6MB 13-.esriea propellor. In (7) the following

formulae have been given

T -

/)

X

J

_a

pnD4

(3)

. K,4'.

Drfe/4b)X (P)Y J

J

b

Combining (2) with

(3b)

_{for a given speed}

resulte in a rpm-value

which, when inserted in (3a),

gives a. thrust.

Typical values of speed and rpm with totally open throttle are given

in tab.l 2.

Th. resistance dt the ship has been computed assuming that resistance can be approximated by a ptrabola with speed as the independent

variabl, which gives reasonable reu1ta eapeodallr at the lower Proud. numbers,

All remaining coefficients have been found by a trial- and

error-method.

The coefficients a, b, K could be determined by the results of the Diendonn spiral test.

The time constants have been adjusted with all zigzatrials, ,aeewning

a fixed, ratio between the time constants, in order bo get one set of coefficients for ali. zigzagtrials.

The ooeficient KL bag been determined by the results of turning circles during yarda-trials.

The coefficients ae tabulated in tabel ₃ for the fully laden and baflasted condition respectively.

3. teeu1ts of the simulation of full scale manoeuvres.

3.1 In fully laden condition.

In fig. i the results of the Di.donn _{eptral. test are given.} It is obvious that the width of the byetsresialoop is also present in the speed rudderangle graph.

The simulation of the sigzagbrials has been done with an IBM 360/65 digital computer using a standard Runge-Kutta routin, for the

integration of the &ifferetia1 eqeatione.. Weather influences havi

been

accounted for by choosing an appropriate value for ¿p

The result., given in figure. 2 through 5 how a satisfactory

agree-ment between full saale data end simulation,

3.2 In ballacted condition.

In fig. 6 the resulta of the Di.1donn spiral test are given. The resulte of the zigzagtrialu can

be

seen in

figuree 7 through 10.

The saine simulation procedure baa been followed here as in the fully laden condition.

The resulte show a catiafaotory

agreement. in the baflacted. oase,

two turning circle. were available, The resulte are given in fig. 11 and when compared to full scale valu.s it can b. seen thkt in particular the overshoot of the sate of turn is not predicted.

This ii due to the fact that 'the rate of turn - ship's speed inter..

ferino. hs

been

nece.eari].y neglected

33 Discussion of th. results of the simulation.

Th. results show that the

mathematloal model chosen can simulate the

ship's characteristics to such an extent

that for

engineering pur..

poses the recuits aan be used. Nevertheless, an important featur, of

the rate of turn time h.tstor during an turning circle is not

adequatly simulated. To this end a mathematical model should be

constructed with r*, the dimensionless ratio of tate of turñànd ship's

speed, as a dependent vsxtble. The determination of the coeffioienta

in euch a model can only be

done when reliable speed measurements during manosuvring trial. are available. The determination of the

coeffiotente by a trial and error method. is of course a weak point

in the simulation process. More sophisticated methode should be used to determine the coefficients (8, On the other hand, auch

methods are generally highly time-consuniing with respect to a digital computer.

6.

In meet cassa one restricta

oneself to en optimal fitting of only

one rur or to an optimal fitting of the main characteristics je. overehootangis and periodtim. of a zigmag together with.the steady

stat. characteristics of a spiral test. This means, in fact, that different asta of coefficients result from different run..

The advantage of this trial- and errorinethod is that one gets one

set of ooeffioienta whioh describes the ship's behaviour accurately

enough.

It is, however, very likely that this final set of coefficients is far from optimal in the sense of the lest squares approzimation.

Oth.r methods of th. determination of coefficiente by using phase

p3.n pl.t seem not to give more accurate results, because to get

a phase plane plot the angula.r acceleration of the ship is needed.

This quantity, however, must be computed by analogous or digital

differentiation of the small and therefore not too accurata rate of.

turn signal. It ii generally impossible to measur. the angular

acoelaration itself.

4.

Rqu]4s

of, the computation of frequency - reeponu trials.

The adjuetiig' cf the autopilot oharaoterietios is very often done

with the aid of the Bode diagrams. The amplitud.- and phase

characteristics are very importent to determine the overall responøe of the system i.e. ship and autopilot. In the figuras 12 and 13 the frequency response graphe are presented for the fully laden and ballasted condition.

5. Conclusions

The following conclusions can be drawn:

A relatively- simple non-linear ma.thsmatisal model

with constant

coefficients can provide reasonable results for both course keep±ng

and turning qualities,

The resulta of th. speed

equation ¡suit be

considered to be an indication of the speed doop rather tian an

accurate prediction.

More accurate speed information during manoeuvring i. greatly needed in order to construct more reliable and precia. equations.

7.

s

References:

(i) Glsnadorp, C.C. and M. Buitethek

"Mnoeuvring Trials with a 200.000 tons Tanker".

Report 248, Shipbuilding Laboratory of the De]! t Univarsity of Technology. August 1969.

Nomoto, K.

*tjy5j5

of Kempt'a Standard Manoeuvre Teat arid Proposed Steering Quality Indices".

DPMS report 1461, First Symposium on Ship Manoeuvrabilit3r, 1960

Norbin, N.H.

"Zigzag Test Teohnique and Analysis with prelimenary Statietiosi

Results".

Report 12 Swedish Stat. Shipbuilding Experimental Tank, june 1965. Beak, M. and. L. Wagner Smitt

"Analogue Simulation of Ship Manoeuvres".

I'A Report Hy 14. September 1969. Lyngby, Denmark. Van Leeuwen, G.

"A .btplified non-linear model of a manoeuvring h.tp'.

Report 262. Shipbuilding Laboratory of the Delft University of Technology, February 1970

Goodwin et.al.

"The Practical Application of Computera in Marine Engineering". Transactions I.M.E. Vol. 80 No, ₇ july 1967

Van Lnnieren, W.P.A., J.D. van Manen, M.W.CE Oosterveld. "The Wageningen B-aorew series"

Sohip en Werf. 6 maart 1970/ 20 mart _1970.

Koywna, T.

"Identifioation problem of Ship Manoeuvrebility".

Report of the Shipbuilding Laboratory- of the D.lft University of Technology. (to be printed.)

Tabla 1.

Ship part ioulera

Ship 1.ngbh Breadth

Draft (fuily laden condition)

Forward draft (ballasted condition) Aft.rw. draft (balla.t.d condition) Mean draft (ballaited oondition)

Displao.ment (fully laden condition)

Di.plaoement (ballasted condition)

Propeller Number of blades Torque maximum Rudder area Design rpm

3l000

in

47.16 in

18.90 n

7.30

in

u oo in

9.15

in

238,000m3

i06,000m3

i

4

0.700

0,5143

210,000 kgin

75.3

in3

85

Pable 2.

R.p.m. - apead relationehip with totally open steam throttle

during mano.uvring.

Ftl1y laden oondition

i

Ballasted condition

Spe.d (kn)

rpm

Speed (kn)

rpm

80.4

16.8

85.0

14

79.8

15.8

84,4

13

79.1

14.8

83.7

12

78.5

15.8

_83.1

11

_77.9

12.8

82.5

10

_77,3

11.8

_81.9

9

76.7

10.8

81.4

8

76.2

9.8

_80.9

7

75.7

8.8

80.4

6

75.3

7,8

79,9

5

74.9

6.8

_79.5

4

74.5

5.8

790

4.8

78.7

Tab.]. ₃

Coefficients of the matbematioal model.

u11y laden condition Ballasted condition

T1 250 80 10 3 P3 20 6 a -1 1 b 80 000

1.20O

K

-00434

-0.0471

In -

24953

11436

KL 3600000 _1750000

SPIRAL TEST FULL CONDITION

SPIRAL TEST FULL CONDITION

Fig. i : Results of the Dìeudonn spiral test In fully laden

condition O.10

jo

-32 -29 S.B. -24 -20 -16 -12 -B -4

-.1

/'o

4 8 12 16 20 24 PORT 28 32 o o

.0

- Oh o 16 o

0fb

o o Ukn. 8 o 00 0 o

t.

2 -32 -29 -24 s.a -20 -16 -12 -8 -4 0 4 B 12 16

w

20 21. PoRT 28 32

-I"

L b 31 2 10 o -10

20

30

-¿0

-50

ZIG-ZAG TRIAL 7/7 FULL

Fig. 2 : Results of a

7/7

zìgzagtrìal in fully laden conditIon

ZIG-ZAG TRIAL 14/16 FULL

Fig. 3 Results of a

lf

114 _{zigzagtrìal in fully laden condition}

uu

II

I

.4k

t

20»

Ai

o i

ii

.i'.

01 5',

ii

7,,

°'

I _'u '0 1 '0 1510 16u0 17''

::1:

W

r i

:- Ti

1o1!Lo!

!

TT

i .

r

1'

f

/

L

7\:

I

L

i

f

':

6oIÁ1II!Ii\iso

\

6O 50 ¿0 30 20 lo

-lo

20 -30 - ¿o 50 ZIG-ZAG TRIAL 20/20:F(JLL

Fig. 14 : Results of' a 20(20 _{zigzagtrìai in fully laden condition}

10 80 11)0

ZIG-ZAG TRIAL30/30 FULL

32

-

28

-s.a 24

-

20

-

16

-o 12 r 8 V/sec 03 02 -01 C 12 16 60 20 21. PORT 28 32 32 24 s.a 20 16 12 o ₁₂ o o C e 12 21. PORT 28 32

SPIRAL TEST BALAST CONcIrION

Fig. 6 : Results of the D±eudonn spirai teat in baliasted

condition o 16 u o 14 20 16

50

V

LO 6° 30 20 10

-i a

-20

-30 -¿O

-50

Z 6-ZAG TRIAL /7 BALLASr.

Results of a

_7/7

zigzagtrìal in ballasted' condition

ZIG-ZAG TRIAL 14/14 BALLAST

ZIG-ZAG TRIAL 20/20. BALLAST.

9 : Results of a 20/20 zigzagtrial In bal].asted condition

ZIG-ZAG TRIAL 30/30 BALLAST.

Fig. 11: Result of a 20 degrees turning circle in ballasted condition 340 320 300 280 260 240 ?20 TURNING BALLAST S.B. 2d CIRCLE

./

/

.7.

/

JO io 160 140 120 lOO 80 60 £0 20 o

/

/

/.

.7

/

/

/

1

/

/ 20 i 44 210 3(0 £00 500 6C0 700 t sac. BC 0 900 £0

6=3

4

o

5

tf

5

014

0.2

0.2

-0.4

Lgw

-0.6

-0.8

-1.0

-1.2

-1.4

-1.6

-1.8.

Fig. 12: Bode diagrams for fully laden condition

s;

-2.0

-1.5

;

-io

-0.

20

10

-10

1gw

-20

-30

-40

-50

-60

-70

-80

-90

f

/2

6=10

g11

Lg

tul

T2

o

-0.8

-1.0

04

0.2

1gw

-O.2

-0.4

-0.6

-1.4

-1.8

Fig. 13: Bode diagrams for bailasted condition

-2.

'6=5

-1.

j

20

10

1gw

-20

-30

-40

-50

-60

-70

-80.

-90