Influence of partial immersion on steel plate dynamic properties

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 EFM 2018 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. 2 EPJ Web o Con e ences 213, 02047 (2019) h ps://doi.o g/10.1051/epjcon /201921302047 EFM 2018 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 3 EPJ Web o Con e ences 213, 02047 (2019) h ps://doi.o g/10.1051/epjcon /201921302047 EFM 2018 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.                                    w α M Q                                   i+1 =PT                                    w α M Q                                   i (6) The ans e ma ix is w i en in nex equa ion. PT=                                       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)                                       (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. 4 EPJ Web o Con e ences 213, 02047 (2019) h ps://doi.o g/10.1051/epjcon /201921302047 EFM 2018 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 5 EPJ Web o Con e ences 213, 02047 (2019) h ps://doi.o g/10.1051/epjcon /201921302047 EFM 2018 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 1. KRAMER, M. R., Z. LIU, Y. L. YOUNG. F ee i- b a ion o can ile e ed composi e pla es in ai and in wa e . Composi e S uc u es. 2013, 95, 254-263. DOI: 10.1016/j.comps uc .2012.07.017 . ISSN 02638223. 2. ERGIN, A., B. U ˘ GURLU. Linea ib a ion analysis o can ile e pla es pa ially subme ged in luid. Jou nal o Fluids and S uc u es. 2003, 17(7), 927-939. DOI: 10.1016/S0889-9746(03)00050-1. ISSN 08899746. 3. DE LA TORRE, O., X. ESCALER, E. EGUSQUIZA, M. FARHAT. Nume ical and expe imen al s udy o a nea by solid bounda y and pa ial subme gence e - ec s on hyd o oil added mass. 2014, 91, 1-9. DOI: 10.1016/j.comp luid.2013.12.003. ISSN 00457930. 4. DA LOZZO, E., F. AURICCHIO, G. CALVI, Added Mass Model o Ve ical Ci cula Cylinde Imme sed in Wa e , P oceedings o he 15 h Wo ld Con e ence on Ea hquake Enginee ing, 2012, Lisbon, Po ugal, pp. 1-10. 5. MURRAY, R. E., R. THRESHER, J. JONKMAN. Added mass e ec s on a ho izon al axis idal u bine us- ing FAST 8. Renewable Ene gy. 2018, 126, 987-1002. DOI: 10.1016/j. enene.2018.04.023. ISSN 09601481. 6. GREŠÁKOVÁ, K.. Expe imen al de e mina ion o he liquid in luence on an oscilla ing body. B no Uni e - si y o Technology, 2018 hdl.handle.ne /11012/82308. Diploma hesis. 7. ANSYS Inc., In oduc ion o acous ics (p oduc man- ual), ANSYS Acous ics Lec u es, 2016 8. AMS, A., W. KLEIN. VIB — Ve i ied Inclusions o C i ical Bending Vib a ions. Scien i ic compu a- ion wi h au oma ic esul e i ica ion. 1988, New Yo k: Sp inge -Ve lag, p. 91-98. Compu ing (Sp inge - Ve lag), 6. DOI: 10.1007/978-3-7091-6957-5. ISBN 0387820639. 6 EPJ Web o Con e ences 213, 02047 (2019) h ps://doi.o g/10.1051/epjcon /201921302047 EFM 2018