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A Calculation ofShip Turning Motion Taking Coupling
Effect Due to Heel into Consideration
by Masayoshi HIRANO, Member
Juiashi TAKASHINA
59 ?
Ij ;I r?.
IJ dl4 ,..J Reprinted from TRANSACTiONS OFTHE WEST-JAPAN SOCIETY OF NAVAL ARCHITECTS
71
A Calculation of Ship Turning Motion Taking Coupling
Effect Due to Heel into Consideration
by Masayoshi HIRANO*. Member Junshi TAKASHINA
Summary
This paper presents a thet-hod to calculate ship turning motion taking hu cou--
-pling effect due to leel into consideration for a ship which perförms large heel in
her turning motion. The turning motion is calculated being based upon the coupled equations of surge. sway. yaw and heel. For two kinds of roil.on,roLl-off ships
computations are made, and the computed results are compared with the results of model experiment which was carried out using 5.0 m long and seIfpropelled models. Both th computed and the z;eren:ai asuits ara in a sfa:tar7
agrec-ment. and furthermore the computed results explain very vell the speed effect on the turning ability, namely the reductioq of the turning radius accordiig to the increase of the ship speed. The major conclusions obtained in che study uf this paper are that thé tùrning motion of a ship with large heel, such as a
roll-on/roll-o ship. should be treated together with the- motion of heel
simultaneous-ly. and that the calculatiôn method proposéd here is effective and useful for the
analysis-of the turning-motion of such a sip.
1. Introduction
When a ship enters itÒ turning motion in calm water, she generallyperforms motions in
four degrees of freedom. They are three components of the motion in horizontal plane, namely
surge. sway and yaw. and heel in addition. In thé studies ai ship turning motion, most of works have been made on the assumption that the hotizontal motions could be separated from other types of motion such as heel". Namely neglecting the coupling effectdue to heel the urning
motion has been calculated being based upon the coupled equations of the horizontal motions.
The motion of heel has been treated a1one3'4. although the effect dúe to the turning motion has
beén considered in the equation of heel. The assumption and the normal practice described
above, which have been employed in the analysis of the ship turning- motion. can be understood
to be quite reasonablé and acceptable for most of hips in general. because the horizontal mo-tions are predominant and the motion of heel is usually small in the turning motion of most of ships. But some ships. such as a roll-on/roll-off ship and a high speed container carrier,perform considerable heel in their turning motion. [n relation to this fact there arises a question about the adequacy of the assumption and the noral practice. described before, to these kinds of ships. The study in this paper started from this question.
Thé authors recantly had a chance co investigate the maneuverability of roll-on.roll-,,ff
ships. In connection to chis investigation, a model experiment of che turningmotion tvns carried
out using two kinds of 3.0 m long and self-propelled roll-on roil-off ship models. In this
experi-ment an interesting fact was found that the turniig radius decreased with large heel accordinz to increase of the ship speed within the approach speed range
of F (Froude nuber) :
72
while it is generally understood that the turning radius hardly changes with change of the ship speed in such a speed runge as that under the approach speed of E.: 0.3. Similar result to that of roll-on roll-iff ships was obtained experimentally by Mori et al.» on a high speed container
carrier vith quadruple screw propellers. Both roll-on ¡roll-oif ships and a container carrier,
de-scribed above, have similar characteristics of hull geometry, in which large beam to draft rati and large KG high position of the center of gravity) may be one of major causes of the large heel in the turing motion. The speed effect on the türning ability, namely the redúctioñ of the turning radius nccotding to the increase of the shi speed. could be attributed to the large heel
caused by centrifugal force, as pointed out by Mori et al.». This fact clearly suggests that the
turning motion of a ship wich large heel, such as a roll-on/roll-off ship, should be treated
to-gether with motion of heel simultaneously.
In this paper ari attempt is made to calculate the ship curing motion on the basis of the
cou-pled equations of the horizontal motions and the motion of heel. The computed results are com-pared wich the experimental results, and the validity of the method proposed here is examin-ed. [n the turning motion of some naval ships with such very high speed as that above F.,: 0.3. an another type of speed effect on the turning ability. flamely the increase of the turning radius according to the increase of the ship speed, can be seen due to the phenomenon oi wave-making at bow and due to considerable changes in trim and sinkage'. But in this paper this kind of
probLem is riot included limiting the application range of the calculation to convencional merchant
ships with approach speed under F.: 0,3 such as a roll-on/roll-off ship arid a container carrier.
Nomenclature
A,, =rudder area
a,, =rario of hydrodvriamic force, induced upon ship hull by rudder action, to rudder force d =drajt of ship (molded)
d maxjum draft at heeled condition (see Fig. 1(b))
= rudder normal force
GZ()=restoring moment lever of heel
h,, = vertical distance between calm water surface and point upon which lateral force
acts (see Fig. 1(b))
I1, r,,. ¡=moment of inertia of ship with respect to z, y and :axes respectively J.Jadded mothent of inertia of ship with respect to z and z-axes respectively
J,' =dimensionless added moment of inertia of ship with respect to z-axis ( Jj1/2pL'd) J,.. = advance coefficient (= u(1w)/nD,)
K = cocal heel moment
L. =length of sh:ïp betveen perpendiculars) m =mass of ship
mT.m,=added mass in z and y-axes direction respectively
nrÇ, =dimensionless added mass in .r and y-axes direction respectively (= ni,.n., 1, 2aDd) .V =total yaw moment
.V(ç) heel damping moment = turning radius
r = turning race
r' =dimensionless turning rate (= rL. V= L. &) = natural period oi roll
A Calculation of pningv1_otioa Taking Heel. Effect into Considération 73 u =ship speed in X àxià direction
V = ship speed (=(uau2)1I2)
V =effective rudder inflow speed
u =ship speed in y-ans direction
u' =dimensionless ship speed in y-axis direction(= u/Y) W =displacement of ship
X total force in x-axis direction
.tR =x-coordinâte of point opon which rudder force YRUDOER acts
Y = total force in y-axià direction
:q =:-coordinate of point upon which lateral förce acts
:
:-coordinate of point upon which rudder fòrce YRUDDER actsaR =effective rudder inflow angle
J rudder angle
=dimensionless linear damping coefficient of heel
2 =aspect ratio of rudder
=heel angle
2. Mathematical Model
2. 1. Equation of Motion
A set of coordinate axes with origin fixed at the center of gravity of the ship, as shown in Figs. 1(á) and 1(b). is used to describe the ship turning moon. Longitudinal and transverse
horizontal axes are represented by the x and y-axes respectively. By-reference to this coordinate
syste& the equations of the turning motion, in general, can be t+ritten in the following fornió)
considering the coupling effects between the horizontal motions and the motion of heel.
x,X
Fig. 1(à) Coordinate System (1)
Surge-: m(svr) = X
Sway: m(t+ur) = Y
Yaw : E!:+(!,,!::)suh1 l
±(1,,!)r
sin(c) =
Eieel : ¡gg
(1,,-1)' sin(.Q) = K
Assuming f,, = l then equations (i) become
---..--.
---.,
z
Figé 1(b) Coordinate System ()
Surge: 'n('zur)= X
Sway : m(±ur) = Y
Yaw : =
()
Heel :
=K
where X, Y. .V and K denote
total hvdrodvnamjc forces and moments generated by ship motions. propeller and rudder. The hydrodynarnjc terms can be expressed in the form
=
Y = YHU,.,.+ RUDDER
N =
K =Kfcu,.±KRDD
where the term with subscript HULL represents hydrodynamjc forces and moments produced by motions of ship hull (without propeller and rudder) and acting upon it", and the term with sub-script RUDDER represents rudder forces and moments including hydrodynamic forces and mo-Inents induced upon ship hull by rudder,actjon'.
2. 2. Longitudinal Force Acting upon Ship Hull and Propeller Thrust
According co the mathematical model
proposed by the Group-4MG, 'che longitudinal force arid the propeiler thrust XPRoPEER cari be writtri
= ri
PROPELl,E = (1).. T(J,)
where .V(tfl represents ship resistance as a funtio,j of u. and T(J,) use of propeller characteristic in open vater as a function of J,.
2. 3. Lateral Force and Yaw Moment Acting upon Ship Hull
Introducing dimensionless forms such as
y''lULl.
-
'L'
vKULt.-
_________1,2ßLdV:
can be obtained by making
(5)
theh the lateral force and the yaw moment.
excluding the added inertia terms. are considered to be functions of u',r' and u, in generai, taking the coupling effect due to heel
into consideration, But there is rio well-established mathematical models for the expression
of
and N,, in forms with inclusion of the heel effect. Hence the following approach is
attempted in this paper. Let us express and .V,., in the following form
= ,n u'm u' r'--Y(v', r')±Y,(u', r', ç')
= J'-_V,,j' r')--.Vf,,(Ju' r', ç,)
Y(u', r') and .V0(u', r') in equations (7) represent rieuvering motion without the heel efféct. Many expression of Y(ü', r') and .V0(u'. r'). Referring
Inoue". the following model is employed here.
YÇ0(u', r') = Y u' Y r'+. u'' u' - Y,, u: r'-- Y,,,, u' r'-- Y,, r';,"'
.V(u'r') = .V
'.V:r'--.V, ,,':' -.v:.,,.
r',";
Now Yj(r/, r', ) and ,V,(u'. r', ) in equations (7) represent che added terms due. co inclusion of the heel effect. Referring to the force measurement test result" of a container carrier model, an assumption is 'nade that the iinear terms vith respect to and the variation of the
CT)
the hydrodynamjc force and moment of
ma-mathematical models are available for the to the extensive studies made by Prof. S.
(3.)
A Calculation of Ship Turning Motion Taking HeeL Effect into Consideration 75 so-called linear terms, such as Y',, Y',, 4V and AV. due to heel are important in che expression of YÇ(u'. r', o) and V(u', r',e). Y(v'. r', e) and .VÇ(u', r'. ) then can be expressed in the form
YÇ,1(u', r', e) = Y
--Y(c = ç,)Y(ç = 0)1r'+[Y( = )Y(9 =0)],"
* Y ç±Y, u'!ç--Y',,, r'i
:V(u',r', e) = V [.V(9 = e) .V(ç = (i)] v'± (Nge = e),V, = (1)] r' (t))
* 9+N' ±N
r'!2. 4. Heel Moment Acting upon Ship Hull
The heel moment can be written
K5 'rz
-V()W.GZ(')..-. Y(:(
(10)where che coupling effects due to the horizontal motions are described in the form of :ff.
5. Rudder Force and Moment
The rudder. forces and moments including the hydrodyriarnic forces and moments induced
3hp rudder a::on, aa.e .YRUDDER, RUDOEa, .VgUØDft ad o *rittan in
the following form according to the MMG rnaciematical model".
= sin '3
= (1--a7)F cos 3
= (x.aq xq,)F cos 3
(ia.5)x F.
CoS 3Keoo = :R+aq :RR)FV cos 3
(ia,,)z F. cos J
where the hydrodynamic force induced .pon ship hull by rudder action is described in the form
of a cos 3. Thé rudder nornal force F' can he expressed in the form"
F =--,
-1Vsina
(12)where the study made by Dr. Y. Yoshimura et al.° is referred in che calculation of the effective
rudder inflow speed V.g. Numerical Calculations
3. 1. Ship Model
Two kinds of roll-on/roll-off ships, ship A (with single screw propeller and singie rudder) and Ship B (with twin screw propellers and twin rudders). are used in this study. The principal particulars of hull, propeller and rudder of the self-propelled modéls of both ships, with which the model experiment of the turning motion was carried out. are shown in Table t.
3. 2. Hydrodynarnic Forces and Moments
Force measurement test was not carried out for both Ship A and Ship B. Therefore the
hydrodynamic force and moment derivatives etc. in he equations of the turning motion were
obtained by estimate calculation. Results are summarized in Table 2. The methods of estima-tion of the derivatives etc. employed in this Paper are as follows.
The added inertia terms, namely m, nr. and J,. are determined by the estimate charts
given by Proj. S. Niotora.
The derivatives in equations (8) are estimated being based upon the results oi studies made by Prof. S. Inoue'.
The derivatives in equations (9) are estimated using th iorce measürement test result'. including the heel effect, of a container carrier model, which has length to beam raso of
76
& 7. beam to draft ratio of 2.97 and block coecient of 0.57. The same quantity as that obtained of the container carrier is used for Y'. and N. In Figs. 2(a)-2(d). the results of the force measurement test ot the container carrier mentioned above are shown in the form of the ratio of the so-called linear term at = o to that at = 0. namely Y(p =),!
YCç = 'O etc.. taking heel angle o in abscissa. The derivatives Y,.,. Y,1, .V,, and are determined using the slope at the craight tines drawn n Figs. 2(a-2(d). (4) The heel damping moment .V) can 5e expressed
=
2x(í-J)
(13)The result obtained by free roil test at running condition is utilized for the the timear
damping coethcient . and shown in Table 2.
iS) The following reintion can be written for che :-coordinate of che point. :. upon which
che lateral torce acts.
-
-
.-
Vr(14)
Table i Principal Particulars
SHIP A B HuU L 5. 000 (ro) 5.000 (ro) B 0.761 (ro) 0.734 (ro) d 0.21: (ro) 0.160 (ro) C5 0. 560 0. 696
LB
6.57 6.81 Bid 3.59 4.59 GM 0.040 (ro) 0.352 (ro) KG 0.327 (ro) 0.343 (rn) Rudder AR! Ld 1:56.5 X 11/80.0 X 2
1.35 1.38 Propellero,
PD:. 0.ó7 (ai).053 0.300 2 5 i 4Table 2 Hydrodynarnic Force Derivatives
SHIP A B A B mr rn m, m 3. 035 0.697 0. 026 r 0.370 -0.075 -0. 030 -0.065-0.021 Jzg/!Z 0.777 0. 600 0.760 0.600 -0. 333 -0. 044 -0.231-0.027 3.0 0.3 0.0 0.0
X,
Y 10 -0. 276 0.061 0.') j -0. 299 0.056 0. 0 0. 0 -0. 0076 0. 0 0.0 -0.0076 Y -0.505 0.045 -0.3370. 028 -0. 137 0.070 -0. 105 0.059 -0. 135 0. 119 0.30 0.30 -0.430 -0.270hd
-3.030 -0.061 j 0.53 -.3.030 -0.025 0.21 0.25A Calculation of Ship Turning Mótion Taking Heel Effect into Consideration
Nv()
1.5 1.5 1.0 O LO 0.5:-0 5 Ò 15 (deç.) 0 5 1Q 15 (eg.)Fig. 2(a) Linear Derivative Y as Function Fig. 2(b) Linear Derivative Y as Function
of Heel Angle o Heel Angle
Yr( 0) 1.5
15
-0 5 10 15 (ceg.)
Fig. 2Cc) Linear Derivative Y as Function of Heel Angle
,J.5
O S -- 0 15 !(ceç)
Fig. 2(d) Linear berivative .V as Function of Heel Angle
From equation (lfl. h7 cari be obtained using measured quantities of u, r and ø at steady turning. The results obtained in this manner, which are mean value about the turning mo-tions of several kinds of rudder angle. are shown in Table 2 in the forms of h/4
and h/
d for both ships. On the other hand, some estimate chartsLl) áre available for. estimation of :. Referring to these charts, the result of Ship A obtained by equation(14) and shownin Table 2 is considere4 to be quite adequate. But h of Ship B seems abnormally large.
namely it has the value thore than 1.0 even in the form of k&/d1. The large beam to draft ratio (=4.59) of Ship B rnight be one of major causes of that, but further discussion about that is not attempted here. The results shown in Table 2 are used in the numerical calcuitions for both 'ships.
3. 3 '4urnerical Resùlts and Discussion
Ac rst che computed results of Ship A, are shown in Figs. 3-7. The results of the model
e'cpetmenc vhicn vas arried out by the authors at the square basin oc Shto Researc'i Institute
of Japan (S. R. L) using the before-mentioned self-propeiled model, are also shown in these
ñgures.
The dimensionless turning raté at steady turning. r, is shown in Fig. 3 taking rudder angle in abscissa for three cases of the approach speed of F"0.2L. 0.26 and 0.30. It can be seen from Fig 3 that che computed results agree satisfactorily wich the. experimental results, and
explain very well the speed effect n the turning ability described in the Introduction. The compited result béing based upon the coupled equations of only the horizontal motions with exclusion of the heel effect. namely by the rst three of equatiOns (1). is also shown in Fig. 3
78
-20
Fig. 3 Maximum Heel Angle (Ship A) -40 -30 -20
/S'
-05Fig. 3 Turning Rate at Steady Turning
(Shio A' 20 ..'l-' I- -I 230 'u S.S G(ceç.) -40 .30 .20 .ï0 0 IO - 20 30 40 .I0
--.-20 dldeq.1 -"O 30-Z0 -IO 0 20 30 40Fig. 4 Speed Reduction at Steady Turning
(Ship A) -IO Ceeq.) .4Q -30 -20 .10 3 _I0 20 30 40 -IO .20
-Fig. 6 Heel Angle at Steady Turning
(Ship A)
r
-20 LFig. 7 Tithe Histories of Turning Rate arid
Heel Angle (Ship. A)
1.0 -0.5 .___.4 , _-_-;. ,
r,-,-'.'? /
4-,,
S (e eq.) -10 0 2ô 30 40 r(eeg.!sec.) (eeq.) Coi. -.. 0. 20 r -35'(Port) Fn0,25the computed r shows lower value than that 'I0
of the experimental result, in ocher words
larger türniag radius would be computed with- 0
lo io us.c
our the heel effect. The speed effect on the
turning ability can be inrerreted as
According to the increase of the ship speed, heel angle increases due to the
centrifugal force.
Consequently the rnomeñt derivative due
to u', .V, increases and the moment derivative due to r', N, decreases as clearly can be seeñ from Figs. 2(b) and 2(d).
Finally turning radius decreases by larger and smaller
The fact that considerable heel can be seen even at small rudder angle, shown in Figs. S
and 6. could suggest a possibility of che decrease of checourse keeping ability according to the increase of che ship speed. This problem is not treated here, but will be examined in another report.
The ratio of the speed reduction at steady turning. V3IV. is shown in, Fig. 4. Both the computed and the experimental results can be seen to be in fairly good agreement. The speed
i!2 22 :' -2
-' 3
.40 .30 -20 -lO O/
/ /;
//
r. 1.0 -as .10Fig. S Turning Rate at Steady Turning
(Ship B) 5(c.q.) 0 20 30 '.0 & ieeuj .40 .10 40 .IO O - 10 20 20 '0 IIi..
.10 F '-...
A Calculation of Ship Turning Motion Taking ifeeh Effect into. Consideration 79
on the turning ability shown in Fig. 3: .: :. .
-TheO Inäximum heel angle e is. shoWn in Fig. 5,. -Quite satisfactòry eemeat between the computed -and the experimeñtal results can be seen for all the cases of the ship speed.
The heel angle at steady turning 9, is shown in Fig 6 and quite satisfactory agreement between the computed and the experimental results can be seen for all the cases of the ship speed just like as the maximum heel angle described above. An interesting result, that s, has
the maximum value at some amount of rudder angle between 10 and 20 for three cases of the approach speed, can be seen in Fig. 6. while increases with the increase of rudder angle as shown in Fig. 5. The rudder angle which gives the maximum value to , depeñds upon the
ratio of magnitude between the ship speed and the turning rate at steady turniñg, as pointed
out and discussed by. Prof. S. moue2t'.
Time histories of. the turning rate and the heel angle from rudder execution to steady turn-ing are shown in Fig. 7. as one of the example, for the case of the approach speed
of F,0..26
and the rudder angle of J=-35 (the port turning). lt can be seen from Fig. 7 that both the computed and the experimental results are in satisfactory agreement.Nect tne computea results ot anip B are aiscussea Toe Lompucea reuts vi toe fleiia'.uii-less turning räte at steady turning, the ratio of the speed reduction at steady turning, the max-imum heel angle. the heel angle at steady turning, and time histories of the turning rate and
V., V.
dideg.)
.40 .30 .20 -10 0 10 20 30 40
Fig. Speed Reduction at Steady Turning
(Ship B) - -20 -IO O lO 20 30 0 40L Çs big.) I -Í ÇOlt i'... (d.ç.) 20 1a22 ' -n cl:l O22 -Í -.a4 O.?R _..l 20
2Sl' O
10 IO Sidig.).20 FIg. ti Heel Angie at Steady Turning
r eg.1seJ
(eeç.)
-20 abscissa together with the experimental
re-suits at S. R. I. for two casesof the approach
.10 --- speed of F,,_O.22 and O.2& It can be seen
from these figures that the computed results
20 t(sei generally give good agreement with the
exper-imental results although some disepancY is
seen in the ratio of- the speed reduction in
20 Fig. 9 The same discussion as that of Ship
Fig. 12 Time Histories of Turning Rate and
Heel Angle (Ship 3) A ca
be made of Ship B.
In spite of the fact
that fairly siple
mathematical model, shown in equations (9). is employed to describe the coupling effect due to
heel, the computed results of both Ship A and Ship B show satisfactory agreement with the
ecperitnental results It is realized that the turning motion of a ship with large heel such as a roll-oû/rOii-od ship.
iuh be
_ d together with the motion of heel simultaneously,and that the calculation metio4 proposed here is effective and useful for the analysis of the turning
motion of such a ship..
-- . FnsO.26
-4. CoucluSiOn.S
It is the purpose -of this paper to calculate the turning motion taking the coupling effect due
to heel into consideration for a ship hich performs large heel in her tirning motioñ. Computa-tions were made for two kinds of roll-on/roll-off ships. The computed results were compared with the experimental
results. and the validity of te
calculation method proposed here wasexarnned. F-urtherOre the speed effect on the turning ability etc were discussed. The results obtained in the study of this paper are suthmarzed as follows.
The turning motion of a ship with large heel. such as a roll-on ¡roll-off ship. should be treat-ed together with the motion of heel simultaneousW.
Ia spite of the fact that fairly simple mathematical model is employed to describe the heel
effect, the computed results give satisfactorY agreement with the experimental results. The calculation method proposed here is effective and useful for the analysis of the
turning motion of a ship with lar-ge heel!
In the turning motiôn of a ship with large heel, the speed effect -on the turning ability. that the turning radius decreases ac-cording to the increase of the ship speed. is seen under the approach speed of F,: 0. 3 -and this fact has already been found experimentally by
Mori et al. The above_mentioned speed effect is assured from both aspects of the
calcula-tion and the model experiment in this paper. kflowledgemefltS
The authors wish to express their sincere gratitude to Prof. S. taoue. Kyushu University. for
his invaluable guidance and encouragement, especially for his precious advice on
the hydrody namic force with the heel effect. given to the authors in the course of this work. Thank are also due to Dr. Y Takaishi. S. R. I.. for his kind arrangement of the model
experiment of S. R. I., and to Dr. A. Ogawa. S. R. I.. fôr his valuable discussion on equations of
motion. lo
the heel angle are shown in Figs. 8. 9. 10, U
References
A Calculation of Ship Turning Motion Taking Heel Effect into Consideration
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1970.
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S. moue; Measurement of Point in Vertical Direction upon Which Lateral Force Acts at Steady Turning (in Japanese). Technical Report. SP. 25. Technical Committee of the West-Ja-pan Society of Naval Architects. 1965.
