In luence o pa ial imme sion on s eel pla e dynamic p ope -
ies
Jan K˚u eˇ
cka1,? and Vladimí Habán1,??
1Vik o Kaplan Depa men o Fluid Enginee ing, B no Uni e si y o Technology, Technická 2896/2, B no, 616 69, Czech Republic
Abs ac . Eigen equency o objec d ops when i is subme ged in he wa e compa ed o an ai exposi ion. In
his a icle he e ec o pa ial subme sion on dynamic p ope ies o s eel pla e is in es iga ed. Changes di e
by a ype o pa icula modeshape because o ela i e posi ion o nodes and an inodes o ee su ace. Dynamic
p ope ies a e in luenced mainly by posi ion o an inodes, in which maximal displacemen occu s. Compa ison
o expe imen , FEM simula ion in ANSYS and cus om code analy ical solu ion is p esen ed. Added e ec s o
luid on he s eel pla e a e discussed.
1 In oduc ion
Change o na u al equency occu es when objec is sub-
me ged in wa e . Imme sion in wa e causes he d op
o eigen equencies o pa icula modes o 50 - 70% o
alue in ai (some imes called d y equency and oppo-
si e we equency). The e is di e en change o a io be-
ween we and d y equency o di e en modes. Bend-
ing ( lex) modes a e gene aly mo e in luenced compa ed
o o sion ( wis ing) modes which expe ience smalle e-
quency d op. E ec s on dynamic p ope ies o composi e
pla es we e s udied in [1] and compa ison o ee ib a ion
FEM simula ion and analy ical me hod is gi en. F equen-
cies o di e en shapes and imme sion induced change
is p esen ed bu only o comple ly subme ged composi e
pla e.
Compa ison o compu a ional esul s wi h added mass
coe icien eplacing e ec s o su uning wa e is p esen ed
in [2]. Analogy o his app oach is used in his pape bu
implemen ed in a di e en way as an added densi y in a
ans e ma ix me hod. Added mass e ec s wi h a espec
o a subme sion o a NACA p o ile in a FEM simula ion
and expe imen wss a subjec o s udy [3]. This con ibu-
ion is in a way simila , FEM analysis and expe imen is
compa ed and an e ec o pa ial subme sion on a dynamic
p ope ies o s udied geome y is p esen ed.
S udy [4] ocused on a analy ical o semi-analy ical
me hod o calcula e equency o pa ially imme sed ci -
cula cylinde and discussed added mass e ec as a de ined
coe icien . The e is p esen ed o mula o simple calcu-
la ion o equency du ing pa ial subme sion using his
coe icien .
Mo e p ac ical use o added mass e ec and a e-
quency change o solid body is p esen ed in [5] ocused
?e-mail: 133644@ u b .cz
??e-mail: [email p o ec ed].cz
on a ma ine hyd okine ic u bine and i s dynamics in lu-
enced by a subme sion in a wa e .
This con ibu ion p o ides a compa ison o a e-
quency change o s eel pla e caused by pa ial imme sion
in wa e . Besides expe imen he used me hods a e ans e
ma ix me hod (TMM) and FEM analysis in ANSYS so -
wa e package. Which one o hese wo me hods p o ides
be e esul s compa ed o expe imen al measu emen is
examined.
2 Expe imen
The expe imen consis ed o measu ed s eel pla e o de-
ined size equiped wi h accele ome e s and ei he ully
subme ged (no subme ged) o pa ially while hanging on
a s eel cables. A e exci a ion he esponse o senso s was
eco ded and la e ans e ed o ampli ude- equency space
using Fou ie ans o m. The expe imen al da a we e ob-
ained as a pa o diploma hesis [6].
Dimensions o pla e a e 550 mm (L) 80 mm (b) and
6 mm ( ), alues we e chosen so he na u al equencies
would be low enough o senso s and equency change
o we ed pla e would be big enough o measu e and com-
pa e o o he me hods p esen ed in he a icle. Two senso s
we e sc ew moun ed in op co ne s. Geome ical p ope -
ies a e in Fig. 1. Dis inc ion o o sion and bending modes
is possible wi h his assembly, because senso s will ha e
di e en phase angle in case o a o sion mode and equal
o a bending eigen equency.
The measu emen p ocedu e s a ed wi h exci a ion o
s ill hanging s eel pla e wi h ei he a hi o a hamme ,
modal hamme o in case o almos (and ully) subme ged
pla e hamme s iking a ool ouching a s eel pla e. This
was ollowed by an a bi a y numbe o seconds o da a
ga he ing du ing shock damping be o e nex exci a ion.
© The Au ho s, published by EDP Sciences. This is an open access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License 4.0
(h p://c ea i ecommons.o g/licenses/by/4.0/).
EPJ Web o Con e ences 213, 02047 (2019) h ps://doi.o g/10.1051/epjcon /201921302047
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80
550
6
Fig. 1. Dimensions and moun ing posi ions o accele a ion sen-
so s
This p ocedu e was epea ed wi h a ious ools a o di -
e en le els o subme sion.
Due o equipmen limi a ions he case o almos com-
ple ly and ully subme ged posi ions o pla e could be
measu ed only wi h one a ailable wa e p oo senso . The
o de o modes and change o o de o i h mode om
o sion o bending (d y i h o sion mode becomes six h
we equency) happens be o e he s eel pla e is ully sub-
me ged. Also he measu emen was consis en h ough
epea ed e en s so his limi a ion was deemed negligible.
The esul s o expe imen al measu emen o d y and we
( ully subme ged pla e) a e in Tab. 1.
Iden i ica ion o o sion shapemode is a he simple
due o moun ing posi ion o senso s. In Fig. 1 a e moun -
ing posi ions in op co ne s o pla e and in Fig. 3 is ap-
pa en di e ence o o sion modes. Bending modes a e
ib a ing in a same phase, he phase angle di e ence is
null and o sion modes ha e phase di e ence equal o π
(Pi). This way he change o o de o modeshapes could
be de e mined.
In Fig. 2 is expe imen aly ob ained end o equency
change o second o sion mode and ou h bending mode
du ing subme sion ( hey a e i h and six h modes in o e -
all o de ). This g aph indica es ha he change happens
when he e is abou 100 mm o pla e s ill s icking up om
he wa e . Ano he poin o no e in his pic u e is a end o
o sion mode equency du ing subme sion. Wi h a knowl-
edge o a six h shape mode in Fig. 3 i is e iden ha his
Table 1. Expe imen al esul s (bold alues o we equency
indica es change be ween lex and wis mode)
Mode D y equency [Hz] We eq. [Hz]
1 106.1 74
2 293.5 206
3 430.6 355
4 571 417
5 878 703 (6)
6 948 721 (5)
0150 300 450 600
700
800
900
Subme sion le el [mm]
F equency [Hz]
To sion mode
Flex mode
Fig. 2. Second o sion and ou h bending mode du ing imme -
sion, expe imen al da a
end is d i en by loca ion o nodes and an i-nodes wi h e-
spec o wa e le el. When he an i-node (loca ion o maxi-
mum ampli ude o he s eel pla e mo emen ) is subme ged
apid equency d op ollows and ice e sa, desc ibed o -
sion mode has a h ee an i-nodes and wo nodes in lengh -
wise di ec ion (di ec ion o a subme sion) and he line con-
nec ing poin s has a h ee signi ican ecessi e pa s. This
Fig. 3. Shapes o i s six eigen modes, hi d and six h mode a e o sion ( wis ing) ones es a e bending ( lex) modes. Depic ed o de
is alid o sumbe ged s eel pla e.
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e ec is much less ob ious in a equency-subme sion de-
pendency o he compa ed lex mode because his mode
has a six an i-nodes and so he g adual subme sion e ec
is no isible in an ele en s ep expe imen p ocess.
3 Simula ion
Analogical se up was used o simula e same si ua ion us-
ing FEM sol e ANSYS wi h Acous ic ex ension which
makes a ailable acous ic capabli ies o use in g aphic in-
e ace. Modal analysis module was used o ind eigen
equencies o pa icula modes. Simula ion was spli in o
h ee pa s, simula ion o he we pla e wi h s uc u al and
luid pa , simula ion o he s uc u al pa only (since he
e ec o ai as a su ounding medium is insigni ican ) and
hi d se up was pa ame ical case o g adual subme sion
o s uc u al pa in wa e and ai .
Compa ed o expe imen he geome y was in a way
simpli ied. S uc u al pa was only box wi h no holes o
hanging opes, no sc ew holes o senso moun ings, no
cham e on edges and he model was missing all pa s o
senso s, cabling e c. Fluid pa was a cylinde su ounding
he men ioned s uc u al pa and dimensions we e chosen
as heigh 1.5 m and diame e 0.8 m. Choice o his size
is mo i a ed by ha ing domain big enough so i ac s like
Fig. 4. FEM analysis compu a ional domain (line wi h a ow
ends indica es he pla e subme sion mo emen du ing consecu-
i e simula ions. Posi ion o he pla e in he pic u e is 440 mm
subme sion.
Table 2. Simula ion pa ame e se ings
Pa ame e Symbol [Uni ] Value
densi y (wa e ) ρw[kg ·m−3] 1000
densi y (ai ) ρa[kg·m−3] 1.204
densi y (s eel) ρs[kg ·m−3] 7850
sound speed (wa e ) cw[m ·s−1] 1500
sound speed (ai ) ca[m ·s−1] 343
Young m. (s eel) E[Pa] 208 ·109
Poisson a io µ[1] 0.3
iscosi y (wa e ) νw[Pa ·s] 1 ·10−3
iscosi y (ai ) νa[Pa ·s] 1 ·10−5
bulk iscosi y (w.) νw(b)[Pa ·s] 1 ·103
bulk iscosi y (a.) νa(b)[Pa ·s] 1 ·102
in ini e space so only na u al equencies o s uc u al pa
a e gained. This cylinde was spli 850 mm in di ec ion
o axis in o wo pa s, he bigge cylind ic pa was used
as acous ic body o wa e and a smalle one se ed as an
ai medium (Fig. 4). S uc u al pa was placed using di-
mension pa ame e ha was changed du ing compu a ion
o g adualy subme ge he pla e in o a wa e wi hou a need
manualy edi his alue a e e e y simula ion s ep.
Pa ame e s used in simula ion a e w i en in Tab. 2.
Damping p ope ies we e used only o a luid pa only
and he damping was used o limi and iden i y speci ic
modes. Mesh size o a en housand quad a ic HEX20 el-
emen s was used and i was su icen coun o p o ide ac-
cu a e esul s and u he mesh e inemen did no change
equency o s abili y alues. Recommended numbe o
elemen s [7] (6 quad a ic elemen s in acous ic domain)
was aken in accoun . Bounda y condi ion o a igid wall
was on a h ee ou e aces o cylinde and FSI in e ace
( luid s uc u e in e ac ion) was on con ac su aces be-
ween s eel and wa e (ai ).
Model wi h s uc u al pa only was used o es ima e
Young modulus o he measu ed s eel pla e since he alue
o i s na u al equency was known om expe imen .
The e is numbe o means o iden i y whe he he
mode is bending o o sional wi hou a look on a de o med
body. Since he damping pa ame e s o wa e and ai we e
p esen he alue o mode s abili y numbe could be used
o ack pa icula mode du ing equency change caused
by subme sion in simula ion s eps. Ano he way is mon-
i o di ec ional de o ma ion in ce ain loca ion, using an
ei he co ne o a pla e esul s in a bigge de o ma ion in
case o wis ing modes compa ed o bending ones. These
p o ided a use ul app oach o ack modes in a da a so
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0150 300 450 600
700
800
900
Subme sion le el [mm]
F equency [Hz]
To sion mode
Flex mode
Fig. 5. Second o sion and ou h bending mode du ing imme -
sion, simula ion da a
o example he p e iously men ioned change o o de o
modes could be easily ob ained wi hou isual check on
each simula ion esul . This me hod was checked on a
smalle equency esolu ion and hen used in u he sim-
ula ions.
In Fig. 5 is isible same si ua ion wi h ype o mode o -
de change his ime wi h compu a ional da a. The change
be ween modeshapes happens a a big highe subme sion
le el compa ed o g aph om expe imen . Simula ion al-
lowed smalle s eps o subme sion so on he lex mode
end he e a e isible pa s when he i e nodes and six
an i-nodes we e being subme ged.
4 Cus om code
Analy ical solu ion using TMM ( ans e ma ix me hod)
was c ea ed o compa e wi h expe imen al and compu a-
ional da a. The solu ion is based on equa ion desc ibing
bending o a s eel pla e. G adual change o densi y was
used o include added mass e ec o pa ial imme sion. The
de ails o his app oach a e desc ibed in his sec ion. Sim-
ila app oach o na u al equencies o o o is sol ed in
[8].
Di e en ial equa ion o bending o a pla e can be w i -
en in ollowing o m.
∂2w
∂ 2+c2·
I
A
·
∂4w
∂x4=0 (1)
Nex he e is need o a s a e ec o and a ela ions
be ween i s elemen s.
uT= w α M Q!(2)
∂w
∂x=α(3)
M=−E·I·
∂α
∂x(4)
Q=∂M
∂x(5)
Wi h hese equa ions and ew modi ica ions o mula
o banding can be w i en using ans e ma ix.
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w
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=PT
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w
α
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Q
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(6)
The ans e ma ix is w i en in nex equa ion.
PT=
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Sp(βx)Cp(βx)
β
Sn(βx)
β2EI
Sn(βx)
β3EI
Cn(βx)βSp(βx)Cn(βx)
βEI
SN(βx)
β2EI
Sn(βx)β2EI Cn(βx)βEI S p(βx)Cp(βx)
β
Cp(βx)β3EI S n(βx)β2EI Cn(βx)βSp(βx)
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(7)
Elemen s Spand Cpa e Rayleigh-K ylo unc ions. In
case o Snand Cn he sign be ween hype bolic and simple
goniome ic unc ion is nega i e.
Sp(βx)=1
2(sinh βx+sin βx)(8)
Cp(βx)=1
2(cosh βx+cos βx)(9)
Pa ame e βcon ains equency elemen ω.
β4=ω2·A
c2·I(10)
Speed o sound is a unc ion o Young’s modulus and
ma e ial densi y.
c2=E
ρ(11)
Table 3. Simula ion pa ame e se ings
Pa ame e Symbol [Uni ] Value
densi y (s eel) ρs[kg ·m−3] 7850
densi y o subme ged
pa
ρ2[kg ·m−3] 15800
Young modulus E[Pa] 208 ·109
A ea A[m2] 4.8 ·10−4
Second a ea momen I[m4] 1.44 ·10−9
Solu ion o eigen alue was ob ained by inding an ex-
eme alue in an assumed equency span.
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Pa ame e s o compu a ion we e same as in FEM AN-
SYS simula ion bu no all o hem a e aplicable, geome y
is limi ed o s uc u al pa only and ma e ial p ope ies o
s eel a e used. Bu o subs i u e e ec o subme sion, pla e
is spli in wo hal s acco dingly o imme sion le el and he
subme ged pa has a di e en densi y. This alue was de-
e mined by eigen equency o ully subme ged pla e so
he densi y o whole pla e is he unkonwn alue. Du ing
ollowing compu a ions his alue was cons an and only
change was a leng h a io be ween wo pa s wi h di e en
densi ies (Tab. 3).
The eason o his app oach is ha when he s uc u e
ib a es he su ounding media ib a es as well. So he
inc ease o densi y subs i ues he e ec o wa e a ound
s eel pla e.
5 Resul s
Compa ison o h ee di e en me hods used o ind sec-
ond na u al equency is p esen ed in Fig. 6. The ends
a e same o all o hem, wi h e iden quick equency de-
c ease in he i s , middle and inal pa o subme sion ac-
co ding o he loca ion o an i-nodes. Bu he e is also
isible disag eemen , none o he da a se s i s wi h o he
p ecisely h ough he whole in e al. The analy ic and
0150 300 450 600
70
80
90
100
110
Subme sion le el [mm]
F equency [Hz]
TMM
FEM
exp.
Fig. 6. Fi s na u al equency du ing imme sion
0150 300 450 600
200
250
300
Subme sion le el [mm]
F equency [Hz]
TMM
FEM
exp.
Fig. 7. Second na u al equency du ing imme sion
0150 300 450 600
400
500
600
Subme sion le el [mm]
F equency [Hz]
TMM
FEM
exp.
Fig. 8. Thi d na u al equency du ing imme sion
FEM app oach a e in ag eemen in he las pa o subme -
sion bu no so well du ing he ini ial pa . I seems ha
bo h compu a ional app oaches ha e a di e en limi a ions
compa ed o each o he and also a e missing some physi-
cal phenomena ha is con ained in expe imen al measu e-
men .
In Fig. 7 and in Fig. 8 he second and hi d bending
modes (second and ou h mode o e all, hi d one is o -
sional) equencies o pla e in ela ion o imme sion a e
p esen ed. The end esembles si ua ion o a i s na u al
equency, bo h ull imme sion and non-subme ged con-
di ion esul s in accu a e and app oxima ely same alues
o equency. T ends show simila cou se bu mo e o less
di e in i s alues along he way.
Added mass e ec s in his con ibu ion a e educed o
an added densi y as is desc ibed in Cus om code sec ion.
Since he added densi y is cons an o whole imme sed
pa i would be possible o calcula e single alue o added
mass. This is no as use ul as in si ua ion o ha monic i-
b a ion o body in one di ec ion o which case he added
mass is easily ela ed o simple olume o liquid ha i-
b a es wi h s uc u e. In p esen ed cases he olume o
luid a ound he body ha ib a es is mo e complica ed
because i is de e imed by a mode shape o a pa icula
na u al equency.
6 Conclusion
In his pape he in luence o imme sion on dynamic p op-
e ies namely na u al equencies o s eel pla e a e p e-
sen ed. To compa e da a wi h expe imen ally ob ained al-
ues FEM simula ion and TMM code is used. The na u al
equencies o a ully subme ged s eel pla e and a s eel
pla e wi hou in luence o su ounding wa e a e p e y
much p ecise bu be ween hese poin s he se s di e .
Change o o de o i h and six h modeshape is cap-
u ed in expe imen and in FEM simula ion. The change
happens in a bi di e en le el o subme sion bu ha can
be in pa explained by expe imen esolu ion wi h ele en
s eps o imme sion.
Compa ison o i s h ee bending modes shows a di -
e ence be ween used me hods and expe imen . The ans-
e ma ix based me hod seems o be mo e p ecise hen
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FEM simula ion in a middle pa o subme sion in e al.
This is despi e o a simpli ied app oach o added mass e -
ec using cons an added densi y o subme ged pa o
s eel pla e. Plausible explana ion o disag eemen wi h ex-
pe imen a e physical in luences included in expe imen
bu ommi ed in FEM simula ion and analy ically based
TMM compu a ion. Despi e hese di e ences his me h-
ods o e use ul desc ip ion o beha io o equency o
he s eel pla e du ing g adual imme sion.
The p esen ed wo k was pe o med wi h suppo o TA ˇ
CR
p ojec TE02000232 Special Ro a y Machines Enginee ing Cen-
e.
Nomencla u e
A[m2] a ea
b[m], [mm] wid h o he pla e
c[m s−1] speed o sound
E[Pa] Young modulus
I[m4] second momen o a ea
L[m], [mm] leng h o he pla e
M[Nm] o que
P ans e ma ix
Q[N] Shea o ce
[m], [mm] hickness o he pla e
us a e ec o
w[m] de lec ion, de o ma ion
x[m] dimension coo dina e
α[ ad] angle o inclina ion
ν[Pa s] iscosi y
ω[ ad s−1] eigen alue, angula equency
Re e ences
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EPJ Web o Con e ences 213, 02047 (2019) h ps://doi.o g/10.1051/epjcon /201921302047
EFM 2018