1
In luence o Roll Mo ion on Added Resis ance in Wa es in Full and
Ballas Load Condi ions
Sao i Yoko a, Masa u Tsujimo o and Ma iko Ku oda 1
1 Na ional Ma i ime Resea ch Ins i u e, Na ional Ins i u e o Ma i ime, Po , and A ia ion Technology, Tokyo, Japan
Abs ac . As mo emen s o imp o e he sa e y o ma ine anspo a ion and educe G eenhouse Gas (GHG)
emissions in he shipping indus y ha e inc eased, in e es in ship pe o mance in ac ual seas has also inc eased.
To co ec ly e alua e ship pe o mance in ac ual seas, accu a e es ima ion o he ex e nal o ce ac ing on a ship
is impo an . So a , many o mulas o es ima ing he added esis ance in heading wa es ha e been p oposed,
and he accu acy has been imp o ed s eadily. The e a e also se e al me hods o es ima ing he added esis ance
in oblique wa es, bu he e is no es ima ion o mula ha akes in o accoun he oll mo ion. The au ho s
p oposed a p ac ical co ec ion me hod o imp o e he accu acy o he added esis ance in oblique wa es by
conside ing he oll mo ion. In his me hod, he equency esponse o he added esis ance in wa es can be
exp essed, and i was con i med ha he p oposed equa ion conside ing he oll mo ion was mo e consis en
wi h he expe imen al alue. The au ho s conduc ed a ank es o in es iga e he e ec o di e en load
condi ions on he added esis ance in wa es conside ing he oll mo ion. In his pape , ank es s a e conduc ed
o in es iga e he e ec o he oll mo ion on he added esis ance in wa es o a ull ship in a di e en load
condi ion. Calcula ed esul s and ank es s esul s in egula wa es a e compa ed. In he ull load condi ion,
he peak equency o he oll ampli ude is in long wa es and he e ec on he added esis ance in wa es is
small. In he ballas load condi ion, he peak equency o oll ampli ude is in sho e wa es han ha o he ull
load condi ion, and he e ec on he added esis ance in wa es is shown. When he expe imen al esul s o he
oll ampli ude a e used in he es ima ion o he added esis ance in wa es, i is con i med ha he es ima ion
accu acy is imp o ed.
Keywo ds: added esis ance in wa es, oll mo ion, model es , ank es , oblique wa es, KVLCC2.
1. In oduc ion
As mo emen s o imp o e he sa e y o ma ine anspo a ion and educe G eenhouse Gas (GHG) emissions
in he shipping indus y ha e inc eased, in e es in ship pe o mance in ac ual seas has also inc eased. In o de o
co ec ly e alua e ship pe o mance in ac ual seas, accu a e es ima ion o he ex e nal o ce ac ing on a ship is
impo an . Resis ance due o wa es, namely, added esis ance in wa es, is one o he main componen s o ex e nal
o ce in ac ual seas, and hus mus be es ima ed accu a ely.
In es ima ions o he added esis ance in egula wa es, he wa e e lec ion componen o he added esis ance
is gene ally appended o he adia ion and di ac ion componen s. The adia ion and di ac ion componen s o
he added esis ance a e exp essed by he o mula de i ed by Ma uo[1]. A semi-empi ical o mula o he wa e
e lec ion componen has been p oposed by Fujii and Takahashi[2], and a p ac ical co ec ion me hod was
p esen ed by Tsujimo o e al.[3] The added esis ance in wa es es ima ed by his me hod has been e i ied many
imes, and he accu acy o es ima ions has also been s udied by compa ison wi h onboa d moni o ing da a. The
es ima ion accu acy o his me hod has been demons a ed, especially o head wa es, and he me hod has now
become a gene al es ima ion me hod.
I is necessa y o es ima e he added esis ance in wa es wi h high accu acy no only o head wa es, bu also
o oblique wa es. S udy o he added esis ance in oblique wa es was began wi h s udy o compu a ional
e alua ion by Hosoda’s[4] , and he esul s o calcula ions by he es ima ion o mula and es esul s we e compa ed
by Ma uo and Iwase[5] and Fujii and Takahashi[2].
Se e al ecen epo s ha e examined he ela ionship be ween he added esis ance in wa es and he oll
mo ion o a ship. Yoshida e al.[6] conduc ed expe imen al esea ch on he added esis ance in oblique wa es o a
la ge blun ship and no ed he ela ionship be ween he oll mo ion and he added esis ance. They con i med ha
he added esis ance becomes la ge a he equency whe e he oll ampli ude has i s peak alue. In addi ion, a es
by Valan o e al.[7] con i med he e ec o he oll mo ion on he added esis ance in beam and qua e ing wa es.
16 h In e na ional Symposium on P ac ical Design o Ships and O he Floa ing S uc u es PRADS 2025
Ann A bo , MI, USA, Oc obe 19 h – 23 d 2025
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These esul s sugges he possibili y ha he con ibu ion o he oll mo ion o he added esis ance in wa es is no
negligible, depending on he ype o ship and i s load condi ion. The e o e, he au ho s p oposed a p ac ical
co ec ion me hod o imp o e he accu acy o he added esis ance in oblique wa es by conside ing he oll
mo ion[8]. I has been con i med ha his p oposed me hod imp o es he accu acy o es ima ion o he added
esis ance in oblique wa es o ine ships.
This pape desc ibes he e ec o he oll mo ion o he added esis ance in wa es and he compa ison o he
es esul s and es ima ion esul s in he ull and ballas load condi ions.
2. Es ima ion Me hod and Tes Resul
2.1. Calcula ion me hod conside ing oll mo ion
In he con en ional calcula ion me hod, he added esis ance in egula wa es RAW comp ises he added
esis ance mainly due o ship mo ion RAWM and i s co ec ion RAWR, which is so called added esis ance due o wa e
e lec ion. RAWM is calcula ed by Ma uo’s heo em[1]. RAWR is in oduced as a co ec ion e m o RAWM om he
iewpoin o accu acy[3] and is calcula ed acco ding o Eq. (1). RAWR comp ises he blun ness coe icien , B , he
coe icien o d a and equency,
α
d, and he e ec o ad ance speed, 1+
α
U. The coe icien o ad ance speed,
CU, can be de e mined by ank es s o he empi ical o mula. In his pape , CU in head wa es is de e mined by he
ank es s in egula head wa es and CU in oblique wa es is de e mined by he empi ical o mula.
2
1(1 )
2
AWR a max d U
R g BB
ρζ α α
= +
(1)
()
UU
CF
αα
=
(2)
whe e
ρ
is he luid densi y, g is he g a i a ional accele a ion,
ζ
a is he ampli ude o egula wa es, Bmax is he
ship b ead h,
α
is he angle be ween he ship heading and egula wa es (0 deg. is de ined as head wa es) and F
is he F oude numbe .
Conside ing ha he added esis ance in wa es is a ec ed by he oll mo ion, he added esis ance due o oll
mo ion, RAWRoll, is exp essed in sepa a ely as Eq. (3).
AW AWM AWR AW Roll
RR R R=++
(3)
Fo es ima ion o RAWRoll , he concep o Ge i sma & Beukelman’s me hod[9] is expanded, wi h u he de ails
p o ided in ou 2021 publica ion[8]. RAWRoll is exp edssed in Eq. (4).
22
44
4/
e xx a
AW Roll
pp
k Bk
RdL
ωφ
=
(4)
whe e k is he ci cula wa e numbe ,
ω
e is he encoun e ci cula equency o wa es,
φ
a is he ampli ude o he
oll angle, Lpp is he leng h be ween pe pendicula s, B44 is he oll damping coe icien and kxx is he adius o
la e al ine ia.
2.2. Tes esul s
In o de o examine he ela ionship be ween RAWRoll and he oll mo ion, he expe imen al esul s and he
es ima ed esul s a e compa ed.
In his pape , he objec ship is selec ed as KVLCC2 and he model is shown in Fig. 1. The p incipal dimensions
o he ship a e shown in Table 1. This ship is no equipped wi h bilge keels. The expe imen s we e ca ied ou a
he Ac ual Sea Model Basin[10]. The leng h o he model is abou 4.4 m, and he es speed is F =0.142.
3
Fig. 1 Model o KVLCC2.
Table 1 P incipal dimensions o KVLCC2.
I em Uni
Value
KVLCC2
Condi ion
-
Full
Ballas
Leng h be ween pe pendicula s (Lpp)
m
320.0
Ship b ead h (Bmax)
m
58.0
Midship d a (dm)
m
20.8
10.0
A d a (da)
m
20.8
11.4
Fo e d a (d )
m
20.8
8.6
T ans e se me acen ic heigh (GM)
m
5.71
17.8
Non-dimensional la e al adius o
gy a ion (kxx/Bmax)
- 0.40 0.35
Non-dimensional longi udinal adius
o gy a ion (kyy/Lpp)
- 0.25 0.26
Blun ness coe icien in head wa es
(B )
- 0.41 0.24
Coe icien o ad ance speed in head
wa es (CU)
- 14.9 12.2
The expe imen al and es ima ed esul s o he equency esponse o he oll ampli ude and he added esis ance
in oblique wa es a e shown in Fig. 2 and Fig.3, espec i ely in he ull and ballas load condi ions. In hese igu es,
λ
is he wa e leng h, k is he ci cula wa e numbe and KAW is he non-dimensional coe icien o he added
esis ance in wa es shown in Eq. (5). In o de o ensu e he ep oducibili y, expe imen s a e aken wice a each
wa e leng h.
22
4
AW
AW
a max pp
R
KgB L
ρζ
=
(5)
The oll ampli ude in he expe imen s is selec ed o he wa e heigh equi alen o 3 m in ull scale, which is
he same as in he calcula ion.
In (a-2), (b-2) and (c-3) in Fig.2 and Fig. 3, he symbols ep esen he expe imen al esul s and he solid line
ep esen s he calcula ion esul s. The oll mo ion is calcula ed using he ex inc ion pa ame e s
φ
m and ∆
φ
m ob ained
by he ee oll es s. The quad a ic coe icien s a and b in Eq. (8) a e ob ained using hese pa ame e s[11]. Table
2 shows he quad a ic coe icien s o a and b a he speed o each es o he added esis ance in wa es.
1mii
φ φφ
+
∆=−
(6)
1
2
ii
m
φφ
φ
+
+
=
(7)
2
m mm
ab
φ φφ
∆= +
(8)
whe e i deno es he numbe o a peak o ough in he ee oll es s and he uni o
φ
i is deg ee.
The ela ionship be ween he oll damping coe icien B44 and he quad a ic coe icien s is shown in Eq. (9).
44
2 180
m
g GM
B ab
φ
ρφ
πω π
∇
= +
(9)
whe e
ωφ
is he oll na u al equency and
∇
is he displacemen olume o he ship.
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Table 2 Quad a ic coe icien s in oll mo ion o KVLCC2
Ship (load condi ion)
T
φ
(F =0) [s]
a (F =0.142) [-]
b (F =0.142) [1/deg.]
KVLCC2 (Full)
19.5
0.020
0.005
KVLCC2 (Ballas )
11.0
0.098
0.005
The es ima ion o he oll mo ion in ela ion o B44 has been in es iga ed using ull-scale ship da a[12].
Compa ison on he s anda d de ia ion o he oll mo ion be ween es ima ed esul s and onboa d moni o ing da a
show in good ag eemen . In his case, he oll damping coe icien s a e de e mined by he ank es esul s and he
pa ame e s ela ed o he cen e o g a i y a e de e mined by he designed load condi ion.
Focusing on he esul o ull load condi ion shown in Fig. 2, he peak o he equency esponse unc ion o he
oll ampli ude is in long wa es om he es ima ed esul s. The e o e, he oll e ec o he added esis ance in
wa es appea s in long wa es. Al hough he es ima ion esul shows in good ag eemen wi h he es esul , he
es ima ed esul in (c-2) o Fig. 2 is excessi e esul compa ed o he es esul . The equency esponse unc ions
o oll phase shown in (a-3), (b-3) and (c-3) o Fig. 2 a e equi alen o he expe imen al esul and he es ima ion
in all wa e di ec ions.
(a-1) Added esis ance in bow
wa es
(a-2) Roll ampli ude in bow wa es
(a-3) Roll phase in bow wa es
(b-1) Added esis ance in beam
wa es
(b-2) Roll ampli ude in beam
wa es
(b-3) Roll phase in beam wa es
(c-1) Added esis ance in
qua e ing wa es
(c-2) Roll ampli ude in qua e ing
wa es
(c-3) Roll phase in qua e ing
wa es
Fig. 2 F equency esponse unc ion o added esis ance (le ), oll ampli ude (middle) and oll phase ( igh ) in
ull load condi ion.
Nex , (a-2) and (b-2) in Fig.3 show ha he peak o he oll ampli ude in ballas load condi ion appea s in sho e
wa es han ha o ull load condi ion. The es ima ion esul s o he peak equency is in good ag eemen wi h he
es esul s. Howe e , he es ima ion esul s a e o e es ima ed in e ms o he oll ampli ude compa ed o he es
esul s. Thus he added esis ance in wa es is o e es ima ed. (c-2) in Fig. 3 shows ha es ima ion esul o he oll
ampli ude is in good ag eemen wi h he es esul . (c-1) in Fig. 3 shows ha he oll e ec on he added esis ance
in wa es shows small in qua e ing wa es in bo h he es ima ion and es esul s. The equency esponse unc ions
o oll phase shown in (a-3), (b-3) and (c-3) o Fig. 3 a e equi alen o he expe imen al esul and he es ima ion
in all wa e di ec ions.
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F om hese esul s shown in Fig. 2 and Fig. 3, i is ound ha he es ima ion o he oll ampli ude is o e es ima ed.
The e o e, he case whe e he expe imen al esul s a e used o he oll ampli ude is examined in es ima ion o
added esis ance in wa es. Fig. 4 shows he esul o es ima ing he added esis ance in wa es by in e pola ing he
expe imen al esul s o he oll ampli ude in ballas load condi ion. The lines in (a-2’) and (b-2’) o Fig. 4 a e
in e pola ed expe imen al esul s. The lines in (a-1’) and (b-1’) o Fig. 4 a e he added esis ance in wa es
calcula ed om he in e pola ion esul s. As a esul o calcula ions using he in e pola ed alue, he added
esis ance in wa es shows p ac ical esul s wi hou excess alue. Howe e , compa ed o he es esul s o he
added esis ance in wa es, calcula ed esul s a e s ill some o e es ima ed.
(a-1) Added esis ance in bow
wa es
(a-2) Roll ampli ude in bow
wa es
(a-3) Roll phase in bow wa es
(b-1) Added esis ance in beam
wa es
(b-2) Roll ampli ude in beam
wa es
(b-3) Roll phase in beam wa es
(c-1) Added esis ance in
qua e ing wa es
(c-2) Roll ampli ude in qua e ing
wa es
(c-3) Roll phase in qua e ing
wa es
Fig. 3 F equency esponse unc ion o added esis ance (le ), oll ampli ude (middle) and oll phase ( igh ) in
ballas load condi ion.
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(a-1’) Added esis ance in bow wa es
(a-2’) Roll ampli ude in bow wa es
(b-1’) Added esis ance in beam wa es
(b-2’) Roll ampli ude in beam wa es
Fig. 4 F equency esponse unc ion and es in e pola ion o added esis ance (le ) and oll ampli ude ( igh )
in ballas load condi ion.
F om he p e ious esea ch[8], he oll e ec on added esis ance in wa es is signi ican in ine ships. The e o e
he esul s o his s udy a e cha ac e ized as he damping componen o he oll mo ion. In he case o a la ge ull
ship like he a ge ship, he oll ampli ude is o e es ima ed since he oll ampli ude is smalle han he ine ship.
The e o e, i is conside ed ha he in luence o he pa ame e s in Eq. (4) equi es o examine.
3. Conclusions
In his s udy, he es ima ion o he added esis ance in wa es conside ing he oll mo ion is discussed by using
a ship model o KVLCC2. The esul s a e as ollows.
In he ull load condi ion, he peak equency o he oll ampli ude o he es ima ion shows good ag eemen wi h
he expe imen al esul , bu he oll ampli ude o e es ima es in he es ima ion. The peak equency o he oll
ampli ude is in long wa es and he e ec on he added esis ance in wa es is small.
In he ballas load condi ion, he peak equency o he oll ampli ude appea s in sho e wa es han ha o he
ull load condi ion and he e ec on he added esis ance in wa es is la ge han in he ull load condi ion. The
accu acy o he es ima ion o he oll ampli ude a ec s ha o he added esis ance in wa es. When he
expe imen al esul s a e used in he es ima ion o he added esis ance in wa es, i is ound ha he accu acy is
imp o ed.
Acknowledgmen s
This esea ch was suppo ed by JSPS KAKENHI G an Numbe JP24K07913. The au ho s would like o exp ess
ou hea el g a i ude o M . Daisuke Wako, M . Ryohei Fukasawa, Ms. Akiko Saku ada, M . Ta suya Hamada
and Ms. Azumi Kaneko o Na ional Ma i ime Resea ch Ins i u e o conduc ing he expe imen s oge he in his
esea ch.
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