Applied Mathematical Modelling

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1 Applied Mthemticl Modelling 37 (2013) Contents lists vilble t SciVerse ScienceDirect Applied Mthemticl Modelling journl homepge: Asymmetric free vibrtion of circulr plte in contct with incompressible fluid S. Triverdilo,, M. Shhmrdni, J. Mirzpour, R. Shbni b Deprtment of Civil Eng., Fculty of Eng., Urmi University, Urmi, Irn b Deprtment of Mechnicl Eng., Fculty of Eng., Urmi University, Urmi, Irn rticle info bstrct Article history: Received 1 Jnury 2011 Received in revised form 19 October 2011 Accepted 14 Februry 2012 Avilble online 22 Februry 2012 Keywords: Added mss Incompressible fluid Fourier Bessel series Vritionl formultion Vibrtion of circulr pltes in contct with fluid hs extensive pplictions in the industry. This pper derives dded mss nd frequencies for symmetric free vibrtion of coupled system including clmped circulr plte in contct with incompressible bounded fluid. Considering smll oscilltions induced by the plte vibrtion in the incompressible nd inviscid fluid, velocity potentil function is used to describe the fluid motion. Derivtion uses Kirchoff s thin plte theory. Two pproches re used to derive the free vibrtion frequency of the system. The solutions include n nlyticl solution employing Fourier Bessel series nd vritionl formultion pplied simultneously on the plte nd fluid. Strong correltion is found between free vibrtion frequencies of the two solutions. Finlly the effect of fluid depth on the dded mss nd free vibrtion frequencies of the coupled system is investigted. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Circulr pltes in contct with fluid hve extensive ppliction in engineering s micro pumps, coolnt in integrl-type nucler rector, circulr disk of butterfly vlves, etc. The dynmic chrcteristic of the plte in contct with fluid (wet plte) is different from tht of plte lone (dry plte). A portion of the fluid tht oscilltes with the plte by incresing the mss of the coupled system decreses the nturl free vibrtion frequency of the wet plte. This will ffect the performnce of the coupled system under dynmic loding. Theoreticl nd experimentl studies on the nturl frequencies of free-edge nnulr pltes resting on free surfce or completely submerged were crried out by Kwk nd Ambili [1]. Considering n unbounded fluid domin, they introduced nondimensionlized dded virtul mss incrementl (NAVMI) fctors in order to estimte the fluid effect on the individul nturl frequencies of the wet plte. Accounting for surfce wves Ambili nd Kwk [2] derived the response of circulr plte on free fluid surfce. They used perturbtion method to decompose the mixed boundry condition on the free surfce to two seprte problems involving very high nd very low frequency problems. Robinson nd Plmer [3] crried out modl nlysis of thin plte floting on the surfce of liquid nd oscillting t low frequencies nd derived governing eqution for the coupled system. Ginsberg nd Chu [4] developed vritionl pltform to derive the mode shpes of plte in contct with hevy fluid. Esmilzdeh et l. [5] derived the free vibrtion frequencies of structurl elements contining nd or submerged in fluid. They used potentil function to clculte hydrodynmic fluid pressure on the structure. The fluid depth hs importnt effect on the interction between fluid nd structure. Kwk nd Hn [6] investigted the effect of the fluid depth on the vibrtion of free edge circulr plte in contct with fluid. Considering the response of the system in high frequency Corresponding uthor. Tel.: ; fx: E-mil ddress: s.triverdilo@urmi.c.ir (S. Triverdilo) X/$ - see front mtter Ó 2012 Elsevier Inc. All rights reserved. doi: /j.pm

2 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) rnge, they ignored the effect of the surfce wves. Using Hnkel trnsformtion, they derived dul integrl eqution. Jeong nd Kim [7] considered the hydroelstic vibrtion of circulr plte submerged in bounded compressible fluid. Accounting for comptibility of deflection between plte nd fluid, they used Fourier Bessel series to solve the dynmic equilibrium eqution. Jeong [8] derived the dded mss nd free vibrtion frequencies for two identicl circulr pltes coupled by bounded fluid. Espinos nd Gllego-Jurez [9] studied the vibrtion of pltes submerged in wter pying ttention to the lower modes using nlyticl nd experimentl methods. Zhou nd Cheung [10] nlyzed the hydroelstic vibrtion of verticl rectngulr plte in contct with wter. Cho et l. [11] studied the modl chrcteristics of liquid storge tnk bffled with nnulr plte using coupled structurl coustic finite element method. Ostsevicius et l. [12] ccounting for viscous ir dmping studied the free nd forced vibrtion response of micro-bem. They evluted the response considering liner nd nonliner forms of Reynolds eqution. Hrrison et l. [13] investigted the response of plte oscillting ner rigid boundry ccounting for squeeze film-dmping effect. They showed experimentlly tht resonnce frequency increses s the distnce from the rigid boundry increses. This pper investigtes the free vibrtion of clmped circulr plte in contct with incompressible bounded fluid. Due to incompressibility of the plte, there is no possibility for xisymmetric vibrtion of the plte. Considering the comptibility of the deflections nd using the Fourier Bessel series, this pper derives dded mss nd frequencies for symmetric free vibrtion of the coupled system. The free vibrtion frequencies of the system re compred with those derived employing vritionl formultion. Finlly the effect of the fluid depth on the system s free vibrtion response is evluted. 2. Derivtion using Fourier Bessel series Fig. 1 depicts circulr plte plced over bounded incompressible nd inviscid fluid. Dynmic equilibrium eqution of the plte in contct with fluid is Dr 4 w þ q w ¼ P; where w is the plte deflection, D is bending stiffness of the plte, q the plte density nd P is the fluid pressure. In the cse of incompressible fluid contined in the rigid vessel, there will not be ny contribution from xisymmetric modes in the overll response. Considering the symmetric modes, the free vibrtion mode shpes of clmped circulr plte in the ir (dry plte) cn be written s [14] k ij r w ij ¼ I i ðk ij ÞJ i J i ðk ij ÞI i k ij r cosðihþ; i; j ¼ 1; 2;...; ð2þ where j is the number of dimetricl, i is the number of circulr, w ij is mode shpe ssocited with ith circulr node nd jth dimetricl, I i nd J i re Bessel functions of first nd second kind of order one, nd k ij is the frequency prmeter. Imposing boundry conditions of the clmped plte, the frequency prmeter could be obtined by solving the chrcteristic eqution J i 1 ðk ij ÞI i ðk ij Þ J i ðk ij ÞI i 1 ðk ij Þ¼0; i; j ¼ 1; 2;... ð3þ ð1þ Therefore the plte displcement could be rewritten s wðr; h; tþ ¼ X1 k ij r k ij r I i ðk ij ÞJ i J i ðk ij ÞI i cosðihþa ij ðtþ ¼ X1 w k ðrþ cosðihþa k ðtþ ¼ X1 w k ðr; hþa k ðtþ; i¼1 j¼1 ð4þ where A k nd w k re modl mplitude nd mode shpe corresponding to specific pir of i nd j. In other word, one to one correspondence between k nd pir of (i, j) is ssumed. Considering the free vibrtion of the plte, Eq. (1) tkes the following form r 4 w k k 4 k w k ¼ 0; k 4 k ¼ x2 k q D : ð5þ h z r Fig. 1. Clmped circulr plte in contct with fluid.

3 230 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) Tking into ccount the clmped boundry condition of the plte, the eigenvlue problem of Eq. (5) is self djoint [15], which mens tht the mode shpes of dry plte re orthogonl. Assuming n incompressible nd inviscid fluid, the fluid movement induced by vibrtion of plte could be described using velocity potentil function. This velocity potentil function u(r, z, h, t) should stisfy the Lplce eqution r 2 u ¼ 0 together with the following boundry conditions ou ¼ 0; or r¼ ou ¼ 0; oz z¼0 ou ¼ ow oz z¼h ot : Imposing boundry conditions (7,b), the solution for velocity potentil function becomes uðr; z; h; tþ ¼ X1 b ij r B ij ðtþj i cosðihþ cosh b ijz ¼ X1 u k ðrþ cosðihþ cosh b kz B k ðtþ ¼ X1 u k ðr; z; hþb k ðtþ; i¼1 j¼1 ð6þ ð7þ ð7bþ ð7cþ ð8þ where similr to the nottion used for the plte deflection, B ij (B k ) nd u k re modl mplitude nd mode shpe corresponding to specific pir of (i, j). The vlue of frequency prmeter b ij is derived from imposing the first boundry condition s J 0 i ðb ijþ ¼0: ð9þ Assuming no seprtion between the plte nd fluid, nd pplying the third boundry condition which describes comptibility of deflection between fluid nd the plte, we conclude tht B mn ðtþ b mn J b mn r m cosðmhþ sinh b mnh ¼ X1 k mn r k mn r _A mn ðtþ I m ðk mn ÞJ m J m ðk mn ÞI m cosðmhþ m¼1 n¼1 m¼1 n¼1 ð10þ Multiplying both sides by rj i (b ij r/)cos(ih) nd integrting on the plte wet surfce, we hve B k ¼ B ij ¼ X1 0 d im k c kl _ Al where k nd l re corresponding to (i, j) nd (m, n), respectively, nd d im is the Kronecker delt equl to one for i = m, nd is zero for i m. In Eq. (11) we hve k ¼ Z b ij sinhðb ij h=þ rj 2 i ðb ijr=þdr 0 ð12þ Z h i c kl ¼ rj i ðb ij r=þ I m ðk mn ÞJ m ðk mn r=þ J m ðk mn ÞI m ðk mn r=þ dr Noting tht J i (b ij r/) re not orthogonl to J m (k mn r/) nd I m (k mn r/), this eqution shows the coupling between different modes. Substituting this into Eq. (9) we hve uðr; z; h; tþ ¼ X1 d im k c kl u k cosðihþ cosh b kz _A l Assuming smll plte displcements, nd employing linerized Bernoulli s eqution, dynmic fluid pressure exerted on the circulr plte will be P ¼ q f d im k c kl u k cosðihþ cosh b kh A l ð14þ Now the equilibrium eqution could be expressed s D X1 r 4 ðw k cosðihþþa k þ q X1 w k cosðihþ A k þ q f ð11þ ð13þ d mi l c lk u l cosðihþ cosh b lh A k ¼ 0 ð15þ

4 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) Due to orthogonlity of the mode shpes of dry plte, we hve Z Z 2p rw l cosðmhþr 4 ðw k cosðihþþdhdr ¼ 0; k l 0 0 Z Z 2p 0 0 rw l cosðmhþw k cosðihþdhdr ¼ 0; k l ð16þ Tking into ccount the orthogonlity of mode shpes nd using N modes in the clcultion, the plte equilibrium eqution tkes the following form where X N d l1 d l1k A k þ XN ðe l1 d l1k þ g l1k Þ A k ¼ 0; l 1 ¼ 1;...; N ð17þ d l1 ¼ D e l1 ¼ q Z 0 Z 0 rw l1 r 4 w l1 dr rw 2 l1 dr X g l1k ¼ q N f d mi d mi1 c lk c ll1 l coshðb l h=þ ð18þ here it is ssumed tht index l 1 is correspondent to the mode designted by (i 1, j 1 ). Eq. (17) cn be expressed s da þðe þ gþ A ¼ 0 ð19þ where d ¼ digðd i Þ e ¼ digðe i Þ g ¼½g ij Š ð20þ To derive the free vibrtion frequencies, we ssume tht the response is hrmonic A ¼ Ae ixt Now Eq. (19) tkes the form of generlized eigenvlue problem ð21þ da ¼ x 2 ðe þ gþa Eigenvlues of this eqution gives squred free vibrtion frequency of the coupled system nd eigenvectors of it re mode shpes of the coupled system. Note tht due to introduction of fluid dded mss (the term g in the right hnd side) the mode shpes will be different from mode shpes of the dry plte. Knowing the eigenvectors of the coupled system the mode shpes of the wet plte could be clculted s bw k ¼ XN k1¼1 A k1 w k1 Now using Eq. (8) it is possible to evlute the fluid velocity field due to vibrtion of plte in the new mode shpe bw k. ð22þ ð23þ 3. Vritionl derivtion Applying vritionl principle simultneously on the plte nd fluid, it is possible to derive the governing eqution for coupled system. Simultneous ppliction of the vritionl principle on the plte nd fluid gretly simplifies the tretment of comptibility of deflection between plte nd fluid. Assuming tht the plte nd fluid re lwys in contct with ech other, the Lgrngin of the plte nd fluid reduces to [15] Z L ¼ r¼0 2 Z 2p 4q h¼0 2 ow ot 2 D 2 o 2 w or þ 1 ow 2 r or þ 1 o 2 w r 2 oh 2! 2 3 5r dr dh þ Z r¼0 Z 2p q f 1 ou u þ ow h¼0 2 oz ot u g 2 w2 r dr dh where the first nd second terms re the Lgrngin of the plte nd fluid, respectively. Decomposing plte deflection in terms of its dry mode shpes nd using fluid velocity potentil function of Eq. (8) nd equting equl to zero the vrition of Lgrngin (for derivtion see Appendix), we conclude tht ð24þ

5 232 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) ðd p þ d f ÞA þ e p A þ qf r t _ B ¼ 0 ð25þ sb þ r _ A ¼ 0 where the prmeters re defined in the ppendix. Eliminting B from Eq. (25) we hve ðd p þ d f ÞA þðe p þ q f r t s 1 rþ A ¼ 0 ð25bþ ð26þ Agin considering hrmonic free vibrtion response, we hve ðd p þ d f ÞA ¼ x 2 ðe p þ q f r t s 1 rþa ð27þ Compring with Eq. (22) it could be concluded tht in the vritionl derivtion the second term on the right hnd side replces the dded mss mtrix g of the Fourier Bessel derivtion. At the sme time the stiffness term includes n dditionl term (d f ) representing the fluid dded stiffness. Absence of this term in the formultion using Fourier Bessel derivtion is due to disregrding the term q f wg in the Bernoulli s hydrodynmic pressure (Eq. (14)). As will be shown in the subsequent section the dded stiffness hs no significnt effect on the free vibrtion frequencies. 4. Verifiction of results To verify the results, we use the work of Jeong nd Kim [7] on circulr plte submerged in bounded compressible fluid, where they compred their nlyticl results with those obtined from ANSYS [16]. In this section we only compre the results employing Fourier Bessel series with those of forementioned reference. In the next section the results employing the Fourier Bessel series compred to those of vritionl formultion. Fig. 2 depicts the submerged circulr plte in rigid continer s is considered by Jeong nd Kim. As could be seen the configurtion of the system is different from those considered in this study. In the configurtion considered in the reference pper, the dded mss hs two contributing components from fluid in the upper nd lower continment. To compre our results with those of Jeong nd Kim, we evlute seprtely the dded mss mtrix due to the upper (g 1 ) nd the lower (g 2 ) continments, nd then dd them to obtin the dded mss mtrix (g). Tble 1 gives the vlue of different prmeters used by Jeong nd Kim. The vlue of fluid bulk modulus is high representing nerly incompressible fluid. Tble 2 gives the vlue of the frequency obtined in this study using different number of circulr nd dimetricl compred to the corresponding vlues tken from the reference pper. Note tht only modes corresponding to symmetric response of the plte re tken from this reference. As could be seen, incresing the number of dimetricl considered in the nlysis, we rech t the sme vlues for frequency of different modes s reported by Jeong nd Kim. For exmple, incresing the number of the frequency of mode corresponding to one dimetricl nd one circulr node converges from Hz to Hz, which show exctly the sme vlue s reported in the reference work. The reference pper lso includes the results for fluid with lower bulk modulus representing more compressible fluids. Interesting point in the results is tht the chnge in the frequency of symmetric modes for more compressible fluids is smll. This mens tht compressibility hs no significnt effect on the symmetric modes nd explins why there is very good coincidence between the results of this pper (ssuming incompressible fluid) nd those of the reference pper (ccounting for compressibility). Fig. 3 depicts convergence nlysis for frequency of four modes with incresing number of dimetricl. Note tht lwys the convergence is from up nd the convergence rte is fster for modes with smller dimetricl. Also note tht convergence rte is independent of the number of circulr s it could be inferred from Tble 2. h 1 h 2 z r Fig. 2. Clmped submerged circulr plte s considered by Jeong nd Kim [7].

6 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) Tble 1 Vlue of different prmeters used in Ref. [7]. Plte rdius (mm) 120 Plte thickness (mm) 2 Plte Young modulus (GP) 69 Plte mss density (kg/m 3 ) 2700 Plte Poisson s rtio 0.3 h 1 (mm) 60 h 2 (mm) 20 Fluid density (kg/m 3 ) 1000 Fluid bulk modulus (GP) 22 Tble 2 Frequency of different modes clculted using different number circulr nd dimetricl compred to those of Ref. [7]. No. of circulr nd dimetricl f k No. of circulr nd dimetricl considered in the nlysis = 1,1 f k No. of circulr nd dimetricl considered in the nlysis = 1,2 f k No. of circulr nd dimetricl considered in the nlysis = 2,2 f k No. of circulr nd dimetricl considered in the nlysis = 2,3 f k No. of circulr nd dimetricl considered in the nlysis = 2,4 f k No. of circulr nd dimetricl considered in the nlysis = 2,5 f k Ref. [7] 1, , , , , , , , Fig. 3. Convergence nlysis for frequency of different modes with incresing number of dimetricl (the numbers in prenthesis denote the number of circulr nd dimetricl, respectively). 5. Simultion results In this section fter verifying the results of Fourier Bessel derivtion with those of vritionl formultion, the evolution of the free vibrtion frequencies s function of the fluid depth re evluted. The plte is ssumed to be mde of steel with rdius of 100 mm nd thickness of 2 mm. The fluid depth is vried from 5 mm to 100 mm to evlute its impct on the free vibrtion frequencies. The fluid nd plte densities re ssumed to be 1000 nd 7800 kg/m 3, respectively. Tble 3 compres the free vibrtion frequencies evluted using Fourier Bessel series nd vritionl formultion for fluid depth of 100 mm. As could be inferred from this tble ccounting for fluid dded stiffness hs no meningful impct

7 234 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) Tble 3 Comprison of the free vibrtion frequency (Hz) evluted using Fourier Bessel series nd vritionl formultion for dimetricl nd circulr of 1 4 (h = 100 mm). Mode number Circulr node, dimetricl node Fourier Bessel series Vritionl formultion Discrepncy (%) Chnge in the vibrtion frequency between dry nd Dry plte Dry plte Wet plte Dry plte Wet plte wet pltes Wet plte Ignoring fluid stiffness Considering fluid stiffness 1 1, , , , , , , , , , , , , , , , Fig. 4. Evolution of free vibrtion frequency with number of nodl dimeters nd circles in dry plte. on the vibrtion frequencies. On the other hnd, compring the frequencies derived using two pproches for dimetricl nd circulr of 1 to 4 revels tht the discrepncy between results of the two pproches is negligible. The discrepncy between results of the two formultions remins nerly the sme for dry nd wet pltes. Figs. 4 nd 5 depict the evolution of the free vibrtion frequency for modes corresponding to the different number of circulr nd dimetricl of dry nd wet pltes. This shows tht t ech circulr number the lrgest dded mss is for the cse of dimetricl node of one. At the sme time lrgest contribution comes from modes with smllest number of circulr nd then smllest number of dimetricl. For high frequency modes the chnge in the vibrtion frequency for the wet plte is miniml, which shows tht fluid-plte interction does not ffect these modes substntilly. In other words, coupling minly ffects the lower free vibrtion frequencies. This increses the difference between higher nd lower frequencies nd consequently reduces the contribution of the higher frequencies in the overll response. The lst column of Tble 1 evlutes the chnge in the vibrtion frequency for ech mode due to presence of the fluid. Plotting this chnge in frequency for different modes in Fig. 6 gives n impression tht fluid presence hs lrger impct on the lower frequencies. Incresing the vibrtion frequency, there is continuous decrese in the chnge of vibrtion frequency from dry to wet plte. This could be explined by tking into ccount the fluid velocity distribution of different modes cross continer height. Fig. 7 plots fluid velocity vector for different modes. This figure shows tht with the increse in the frequency lesser portion of the fluid in the continer re ffected by the plte vibrtion. In other word, while for first mode (m = 1, n = 1) nerly ll of the fluid in the continer is ffected by the plte induced vibrtion, for higher modes incresingly lrger portion of the fluid in the lower portion of the continer remins unffected by the plte vibrtion. This reduction

8 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) Fig. 5. Evolution of free vibrtion frequency with number of nodl dimeters nd circles in dry plte. Fig. 6. The chnge in the vibrtion frequencies of wet nd dry pltes for different modes. Fig. 7. Plot of fluid velocity vector for different modes: () m =1,n = 1, (b) m =1,n = 3, (c) m =3,n = 1, (d) m =3,n =3.

9 236 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) Tble 4 Effect of off-digonl terms in dded mss mtrix on the free vibrtion frequency of wet plte (h = 100 mm). Mode number Circulr node, dimetricl node Wet plte Discrepncy (%) Accounting for off-digonl terms Ignoring off-digonl terms 1 1, , , , , , , , , , , , , , , , Fig. 8. Compring free vibrtion modes shpes of dry nd wet pltes: () m =1,n = 1, (b) m =1,n =2. in the fluid velocity in the lower prt of the continer reduces the dded mss of the fluid nd consequently the chnge in the ssocited frequency becomes lesser. Tble 4 compres the free vibrtion frequency clculted using Fourier Bessel formultion ccounting for nd ignoring off-digonl terms in dded mss mtrix. Noting tht d nd e re digonl mtrices, the off-digonl terms in the eigenvlue problem of Eq. (22) re come from dded mss mtrix g. As it is evident from this tble the impct of off-digonl terms on

10 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) Fig. 9. Evolution of free vibrtion frequency of different modes with fluid depth: () m = 1, (b) m = 2 (c) m = 3, (d) m =4. the vibrtion frequency is smll, nd this difference reduces for higher frequency. This is prtly due to the fct tht higher frequencies re less ffected by fluid-plte interction. Incresing the number of modes considered in the nlysis the condition number of mtrices e+g nd d increses rpidly. The condition number is defined s the rtio of lrgest eigenvlue (squred circulr frequency) to smllest one nd controls the ccurcy of numericl solution when working with mtrix. Increse in the condition number results in the sensitivity of the numericl solution nd even smll errors in the numericl method (e.g due to roundoff error) could led to lrge errors in the clcultion of the eigenvectors. To void this error in the clcultion of eigenvectors, it is necessry to dopt preconditioner [17]. Preconditioner is usully used to reduce the condition number of ill-conditioned mtrixes. One of the common methods to reduce the condition number is digonl scling method. In this method digonl positive definite mtrix found so tht by pre nd post multiplying it in the interested mtrix the resulted mtrix becomes well-conditioned with lower condition number. Fig. 8 compres the two mode shpes corresponding to the circulr node of one for wet nd dry pltes, where the eigenvectors of the wet plte is clculted employing digonl scling s preconditioner [18]. As could be seen there is smll differences between mode shpes of the wet nd dry pltes. In other words, in greement with usul ssumption [9,19], it is possible to ssume tht the mode shpes of the dry plte remins pplicble for the wet plte s well. Shown in Fig. 9 re the evolutions of the free vibrtion frequencies of wet plte with fluid depth. Although in lower depths the viscosity hs predominnt effect on the fluid flow nd current potentil model will not be pplicble, however to derive the evolution of dded mss with fluid depth nlyses re done for fluid depth of mm. Closeness of the plte to the rigid boundry in the cse of smll h increses the dded mss of the fluid, which represent itself in the lower frequency of different modes for this vlues of h when compred to the lrger h vlues. Increse in the dded mss due to proximity to the rigid boundry is well known behvior (e.g. [13,20,21]). As could be inferred from these figures for ech frequency there is threshold fluid depth, where fter this threshold there is no chnge in the frequency of the wet plte with the increse in the fluid depth. This threshold frequency is smller for higher frequencies. For rnge of h/ between 0.2 nd 1, the frequencies of different modes remins unchnged with the vrition of the fluid depth. In other word, for usul rnge of the fluid depth the free vibrtion frequencies of the wet plte re not sensitive to the vrition in the fluid depth. 6. Conclusion The effect of fluid-structure interction on the free vibrtion frequencies of circulr plte in contct with incompressible bounded fluid is derived. Becuse of incompressibility of the fluid, free vibrtion modes of the wet plte should be symmetric. Two formultions re used to derive the free vibrtion frequencies of the coupled model. The first formultion uses Fourier Bessel series ccounting for comptibility of the deflection between plte nd the fluid in their interfces. The

11 238 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) second formultion employs vritionl principle pplied simultneously on the plte nd fluid. Strong correltion is found between the results of the two formultions. Finlly the evolution of the free vibrtion frequencies of the wet plte with incresing fluid depth is derived. Results show tht for usul rnge of the fluid depth the free vibrtion frequencies effectively remin unchnged with smll vrition in the fluid depth. Appendix Decomposing plte deflection nd fluid velocity function using Eqs. (4) nd (8), we conclude tht Z Z!! "! 2p q L ¼ _A k w k _A l w l D A k w k;rr þ 1 A k w k;r þ 1 A r¼0 h¼0 2 2 r r 2 k w k;hh!# X1 A l w l;rr þ 1 A l w l;r þ 1 Z Z "! 2p r r 2 um1 A l w l;hh r dr dh þ q f 1 B k u k;z r¼0 h¼0 2 þ X1 _A k w k! B l u l! g 2 Tking the vrition of the Lgrngin gives Z Z " 2p dl ¼ q X1 A k w k w l þ D X1 r¼0 h¼0 A k w k! þ 1 r 3 w k;r w l;hh7 þ 1 r 2 w k;hhw l;rr þ 1 r 3 w k;hh w l;r þ 1 r 4 w k;hh w l;hh A k B l u l! A l w l!#r dr dh ða1þ w k;rr w l;rr þ 1 r w k;rr w l;r þ 1 r w 2 k;rr w l;hh þ 1 r w k;r w l;rr þ 1 r w 2 k;r w l;r þ q f g X1 Z Z 2p þ q f 1 B k u k;z u l 1 B k u l;z u k þ X1 r¼0 h¼0 2 2 A k w k w l þ X1 Equting equl to zero the vrition of Lgrngin nd rewriting the resulting eqution we hve _B k u k w l!#da l r dr dh _A k w k u l!db l r dr dh ða2þ ðd p þ d f ÞA þ e p A þ qf r t B _ ¼ 0 sb þ r A _ ¼ 0 ða3þ where Z d p;kl ¼ r¼0 Z 2p h¼0 D w k;rr w l;rr þ 1 r w k;rr w l;r þ 1 r w 2 k;rr w l;hh þ 1 r w k;r w l;rr þ 1 r w 2 k;r w l;r r dr dh þ 1 r 3 w k;r w l;hh þ 1 r 2 w k;hh w l;rr þ 1 r 3 w k;hh w l;r þ 1 r 4 w k;hh w l;hh Z Z 2p d f ;kl ¼ q f g w k w l r dr dh r¼0 h¼0 Z Z 2p e 0 kl ¼ q w k w l r dr dh r¼0 h¼0 Z Z 2p r kl ¼ w l u k r dr dh r¼0 h¼0 Z Z 2p s kl ¼ r¼0 h¼0 1 2 ð u k u l;z þ u l u k;z Þr dr dh ða4þ References [1] M.K. Kwk, M. Ambili, Hydroelstic vibrtion of free-edge nnulr pltes, ASME Journl of Vibrtion nd Acoustics 121 (1999) [2] M. Ambili, M.K. Kwk, Vibrtion of circulr pltes on fluid surfce effect of surfce wves, J. Sound Vib. 226 (3) (1999) [3] N.J. Robinson, S.C. Plmer, A model of rectngulr plte floting on n incompressible liquid, J. Sound Vib. 142 (3) (1990) [4] J.H. Ginsberg, P. Chu, Asymmetric vibrtion of hevily fluid-loded circulr plte using vritionl principles, Journl of Acoustic Society of Americ 91 (2) (1992) [5] M. Esmilzdeh, A.A. Lkis, M. Thoms, L. Mrcouiller, Three-dimensionl modeling of curved structures contining nd/or submerged in fluid, Finite Element in Anlysis nd Design 44 (2008) [6] M.K. Kwk, S.-B. Hn, Effect of fluid depth on the hydroelstic vibrtion of free- edge circulr plte, J. Sound Vib. 230 (1) (2000) [7] K.-H. Jeong, K.-J. Kim, Hydroelstic vibrtion of circulr plte submerged in bounded compressible fluid, J. Sound Vib. 283 (2005) [8] K.-H. Jeong, Free vibrtion of two identicl circulr pltes coupled with bounded fluid, J. Sound Vib. 260 (2003) [9] F.M. Espinos, A.G. Gllego-Jurez, On the resonnce frequencies of wter-loded circulr plte, J. Sound Vib. 94 (1984) [10] D. Zhou, Y.K. Cheung, Vibrtion of verticl rectngulr plte in contct with wter on one side, Erthquke Eng. Struct. Dynm. 29 (2000)

12 S. Triverdilo et l. / Applied Mthemticl Modelling 37 (2013) [11] J.R. Cho, H.W. Lee, K.W. Kim, Free vibrtion nlysis of bffled liquid-storge tnks by the structurl coustic finite element formultion, J. Sound Vib. 258 (5) (2002) [12] V. Ostsevicius, R. Duksevicius, R. Gidys, A. Plevicius, Numericl nlysis of fluid structure interction effects on vibrtions of cntilever microstructure, J. Sound Vib. 308 (2007) [13] C. Hrrison, E. Tvernier, O. Vncuwenberghe, E. Donzier, K. Hsu, A.R.H. Goodwin, F. Mrty, B. Mercier, On the response of resonting plte in liquid ner solid wll, Sensors nd Actutors A 134 (2007) [14] L. Meirovitch, Principles nd Techniques of Vibrtions, Prentice-Hll Interntionl, [15] H. Isshiki, S. Ngt, Vritionl principles relted to motions of n elstic plte floting on wter surfce, in: Proceedings of the Eleventh Interntionl Offshore nd Polr Engineering Conference, Stvnger Norwy, 2001, pp [16] P. Kohnke, ANSYS theory, element nd commnd references, relese 5.4, SAS IP, [17] P. Wriggers, Nonliner Finite Element Methods, Springer, Berlin, [18] G.H. Golub, C.F. vn Lon, Mtrix Computtions, John Hopkins University Press, Bltimore, [19] M. Ambili, G. Dlpiz, C. Sntolini, Free-edge circulr pltes vibrting in wter, Modl Anlysis, The Interntionl Journl of Anlyticl nd Experimentl Modl Anlysis 10 (3) (1995) [20] A.I. Korotkin, Added msses of ship structure, Springer Science, [21] NCER, A review of dded mss nd fluid inertil forces, Nvl Civil Engineering Report, CR , 1982.