EFFECT OF ADDED MASS ON SUBMERGED VIBRATED PLATES

Transcription

1 EFFECT OF ADDED MASS ON SUBMERGED VIBRATED PLATES V.H. Vu 1, M. Thomas 1, A.A. Laks and L. Marcouller 3 1 Department o mechancal engneerng, École de technologe supéreure, Montréal, QC, Canada marc.thomas@etsmtl.ca Department o mechancal engneerng, École Polytechnque, Montréal, QC, Canada 3 Hydro-Québec s research nsttute, Varennes, QC, Canada ABSTRACT Vbraton o sold-lud nteracton systems requres a determnstc knowledge o braton analyss and structures couplng. It s edent that a structure bratng n a lud has to support an nteracte nertal eect caused by the lud whch changes the dynamc behaor o the sold tsel. Ths paper presents an expermental study on the eect o added mass on submerged plate structures whch are subjected to braton. Derent plates bratng n water at derent depths wth arous geometry ratos are studed usng a number o boundary condtons. Dynamc testng s conducted n tme domans to denty the modal parameters o the structures. The modal added mass alue and added mass coecent are then dered rom the change o modal requences rom ar to water. Ths s ollowed by a comparson o expermental results wth seeral numercal analyses ound n the lterature. 1. INTRODUCTION In sold-lud nteracton study, plate structures hae oten been consdered or two major reasons. Frstly, plates are wdely used n common structures or shps, heat exchangers, aerospace and cl engneerng. Secondly, the plate has an adantage n terms o structural conenence n modelng, analyss and testng. From a theoretcal pont o ew, there are many publshed artcles descrbng deelopments or dynamc analyss o plate structures. Lamb [1] conducted what s consdered the rst study on requency analyss o a plate-lud contact system usng the Raylegh method. Fu and Prce [] studed braton responses o cantleered ertcal and horzontal plates partally or totally mmersed n lud. They assumed that the plates brated n a sem-nnte lud medum. A nte element method and a sngularty dstrbuton panel approach were used to analyze the dynamc responses o plates n ar and also to determne the hydrodynamc coecents or each element n contact wth lud. Haddara and Cao [3] nestgated dynamc responses o plates n ar and submerged n lud under arous boundary condtons. They presented an approxmate soluton or the equaton o moton o a plate coupled wth lud and proded an analytcal added-mass actor dependng on the heght o the ree surace and the depth o lud under the plate. An approxmate expresson or the ealuaton o the modal added mass was dered. Ther results were n agreement wth Fu and Prce; when the plate depth attans 5% o ts length, rst requency bendng starts to change. Lang [4] took an alternate approach, usng the Raylegh Rtz method to dere the added mass actor or each mode o cantleer plate braton wth a correcton coecent or the aspect rato. These added mass actors were then used to calculate ree braton o the plate. Use o the nte element method or structural analyss s well known, howeer the problem o sold-lud nteracton s so complcated that, to date, no coupled element has been deeloped to model ths stuaton n commercal sotware. Recently, a research group at Polytechnque de Montreal deeloped a hybrd nte element model usng the Sanders thn plate method to analyze the braton o a arety o classes o plates n lud. The nteracte eects are dstnctly dered and nclude

2 nertal eects resultng n a pressure expresson whose product wth the structural shape uncton produces a rtual added mass schematc [5, 6]. In the eld o expermentaton, whch can be consdered as practcal research to aldate theoretcal models and applcatons, seeral separate tests on concrete structures are ound n the lterature. Powel and Robert [7] were the rst nestgators to experment wth crcular bound xed plates n contact wth lud. Ther obseratons reached slghtly hgher alues than those presented by Lamb [1]. Lndholm et al. [8] carred out an extense expermental study o the response o cantleer plates n ar and n contact wth lud. The plates wth derent aspect ratos and thcknesses were horzontally and ertcally placed or nclned. The results were compared wth theoretcal predctons based on smple beam theory, thn plate theory and chordwse hydrodynamc strp theory. An emprcal correcton actor was ntroduced to achee good theoretcal and expermental correlaton. It was concluded that the added mass actor changed wth submerged depth o the plate, but sgncant change occurred only when the submerged depth was less than about one hal the span length o the plate. Haddara and Cao [3] also reported a aldaton o analytcal analyss usng experments on a CFCF and a SFSF plate submerged n water. Natural requences and dampng rates were dented usng a Fourer transorm technque. Seeral mportant results can be cted as ollows: ) the change o the rst bendng requency mode was about 31% to 3%; ) the added mass actor seemed to ncrease steadly untl the depth o submergence reached about 10% o the length o the plate, then t leeled o to a steady alue; ) added mass alue was nersely proportonal to the natural requency, except or the torson modes whch had same added mass actor and ) there was a sharp ncrease n dampng when the plates were submerged n water, or all modes. The torson modes exhbted the smallest ncrease. It can be seen that a notable dsagreement exsts between expermental and theoretcal results. Snce the dentcaton method used n all o the aboe experments s the smple one-peak pckng method based on Fourer transormaton, the result s a hgh arance o dented requences and some low excted modes can be mssed. Ths paper s a ollow up to a preous artcle [9] and uses an nnoate dentcaton method on numerous plates to reew the case o braton o plates n water and accurately denty the modal parameters o the system and dere the modal added mass actor or each mode. Ths research can also be seen as a aldaton o the hybrd nte element model deeloped earler [5].. REVIEW OF ADDED MASS EFFECT Fg. 1 Sold-lud nteracton

3 Consder a structure (ST) and lud (FL) wth an nteracton surace (FS). At an arbtrary pont, a normal n and tangent t can be dented as shown n Fgure 1. In a acuum, the structure has ts own orthogonal modes X (r) whch correspond to ω natural requency and m generalzed mass. Dsplacement o the structure can be wrtten usng modal bases and a ector o modal partcpant actors (t) : a ( r t) = a ( t) Χ ( r) x, (1) s When the structure brates n a statc lud, the lnear nluence can be consdered as a orce F g on the structure. So, I we project on the th mode, one can obtan: s s s M & x ( t) + Cx& ( t) + Kx ( t) = F () [ m ] a&& + ω [ c ] a& + ω [ m ] a = [ F ] g (3) Ths orce F g s orgnally rom the nertal eect o lud around the structure, so the pressure on the structure s: The generalzed orce Then F F g g p ( r t) = a& ( t) p ( r) j j j g, (4) on the whole nteracton surace can be obtaned: FS ( r t) Χ ( r) ndfs = a& ( t) p j ( r) Χ ( r) ndfs = a& ma = p, (5) j [ m ] a& + [ c ] a& + ω [ m ] a = a& [ m ] FS ω (6) Or ([ m ] + [ m ]) a + ω [ c ] a + ω [ m ] a = 0 a a & & (7) It s edent that the eect o the lud has moded the mass matrx n the dynamc equaton. The matrx s called the matrx o added mass: m a ( r) Χ ( r) Wth the presence o added mass, one can mmedately arm that: The total mass o the system ncreases, The natural requences o the system decrease. m = p ndfs (8) a FS j The eect o added mass can now be calculated. We can obsere rom Equaton (8) that ths eect depends on the densty o the lud, the geometry o the structure, ts mode shapes and ts braton ampltude. Ths eect s not nluenced by lud low elocty or by the lud s scosty. In the case o stll water, only the eect o added mass s consdered. The matrx o added mass s always symmetrc and dentely poste.

4 3. STRUCTURES TESTING To prode a general case, two types o plates wth derent dmensons were studed to measure ther braton responses (Fgures and 3). The tests were conducted both n a acuum and n a water tank at derent depths n order to consder the eect o added mass. All three plates are steel, each weghng 787 kg/cm3, the elastc modulus s e11 Pa and the Posson rato s 0.9. Plate thckness s constant at 1.905mm. Table 1 shows the conguraton and dsposton o eght accelerometers on each plate. a3 a a B/ B B/ 60 A Plate A(mm) B(mm) a1(mm) a(mm) a3(mm) Fg. Plate CFFF test conguraton a1 a a a1 4 6 B A 100 Fg. 3 Plate CFCF test conguraton Plate A (mm) B (mm) a1 (mm) a (mm) Snce we are not concerned about the exctaton, an mpact hammer was used to excte the braton o the plate wthout any measurements o orce.

5 4. IDENTIFICATION METHOD The modal parameters o the plate were dented a a tme doman method usng a multarable autoregresse model smultaneously on all sensor response data. Least squares were used to estmate the model parameters and were mplemented wth QR actorzaton whch ges an unbased and ast computaton. The method s descrbed more detal n a companon paper, see [9]. 5. RESULTS AND DISCUSSION 5.1 Agreement between analyzed and dented alues To ew the accuracy o the dentcaton method the reader can reer to the companon paper whch has a crtcal assessment o the method. For these concrete plate structures one can compare the results o the rst e dented requences wth those dented usng nte element analyss (FEA) o plates n a acuum (Table 1; (1) s the dented alue and () s the FEA alue). Table 1: Idented and analyzed requences Plate (Hz) (Hz) (Hz) (Hz) Mode 5 (Hz) (1) () (1) () (1) () (1) () (1) () Agreement between repeated tests Consder plate number 4 results or three derent testng tmes. The rst test was done wth a consderable exctaton orce, whereas the two other tests use a reasonable orce alue. All tests were excted wth hammer mpacts at the same pont. On Fgure 4 one can see a dscrepancy between derent testng tmes. Durng the rst test wth the largest exctaton orce, all dented requences were lower than those o the two other tests whereas the dampng rates and mode shapes stablze at the hgher alues. Whle the dscrepances o requences and mode shapes are neglgble, a notceable araton s ound on dampng rates wth the amount up to 50%.

6 Frequency Dampng tme 1 tme tme 3 Test tme Mode 5 Dampng rate (%) tme 1 tme tme 3 Test tme Mode 5 MAC MAC tme 1 tme tme 3 Test tme Mode 5 Fg. 4 Three testng tmes (CFCF A500B01) 5.3 Eect o requency change and added mass Each plate s submerged n the water at derent depths and the depth/length rato ares rom 0 (n a acuum) to 0.6 (totally submerged). The dagrams below show the araton o seeral rst requences wth the poston o the plate n each case. 1,0 10,0 8,0 6,0 4,0,0 0,0 0,0 0,1 0, 0,3 0,4 0,5 0,6 50,0 40,0 30,0 0,0 10,0 0,0 0,0 0,1 0, 0,3 0,4 0,5 0,6 80,0 70,0 60,0 50,0 40,0 30,0 0,0 10,0 0,0 0,0 0,1 0, 0,3 0,4 0,5 0,6 D/L rato 160,0 140,0 10,0 100,0 80,0 60,0 40,0 0,0 0,0 0,0 0,1 0, 0,3 0,4 0,5 0,6 D/L rato Fg. 5 Frequency change o plate 1 (CFFF A377B01)

7 Fg. 6 Frequency change o plate (CFFF A01B01) D/L rato Fg. 7 Frequency change o plate 3 (CFFF A106B01)

8 D/L rato D/L rato Fg. 8 Frequency change o plate 4 (CFCF A500B01) D/L rato Fg. 9 Frequency change o plate 5 (CFCF A01B01) It can be seen rom these dagrams that eery requency mode decreases when the plate s submerged n water. Ths reducton s releant een at low depths o 0.1 tmes the plate s length. When the plate s submerged, the requences change slghtly as the depth ares and all cases are consdered to cease at a depth/length rato o 0.5. The requency change eect s most sgncant or the rst bendng mode. For a square CFFF plate ths largest alue o requency change s up

9 to 6%.The requency change alue s 69% or a rectangular CFCF plate, greater than the alue Haddara et Cao proded earler. It seems that the lower the requency alue, the greater the requency change except at torson modes where the change n requency alue s nearly dentcal. Ths result agrees well wth the conclusons o Haddara et Cao. To calculate the added mass actor, we consder that the structure s rgdty s retaned constant when t s submerged n lud. The change o requences thus exhbts only an aect o added mass on the structure. Thereore one can hae: n, M = K = ωn, M = ωn, C = ξ ωn, M ; C C + Ca = ξ ωn, M ( M M ) K = ω + ; = (9) Where K, M, C, ω, ξ and K, M, C, ω, ξ are the orthogonal modal rgdty, n, modal mass, modal dampng, natural requency and dampng rate o a mode n a acuum and lud respectely. denotes the modal added mass o ths mode. M a n, a From equaton (9), one can dere the added mass and added dampng actor M M a ω = ω n, n, C ξ ω a n, 1; = 1 C ξ ω n, (10) From ths deraton the ollowng are dagrams o added mass ersus the depth/length rato (D/L) o the plate n water Fg. 10 Added mass on plate 1 (CFFF A377B01)

10 Fg. 11 Added mass on plate (CFFF A01B01) Fg. 1 Added mass on plate 3 (CFFF A106B01)

11 Fg. 13 Added mass on plate 4 (CFCF A500B01) Fg. 14 Added mass on plate 5 (CFCF A01B01)

12 It s edent rom Fgures 10 to 14 that the added mass actor s derent or each mode and s most sgncant or the lowest requency mode exhbtng a mass alue more than ten tmes greater than the mass o ths mode n ar. Thereore ths rst mode o braton s the most mportant one n submerged plate braton. The assumpton o added lud olume n the work o Snha [10] s thus only ald or the rst mode o braton, where the lud added mass conssts o a cylnder around the plate. The torson modes stuated wthn range o the rst 5 modes seem to hae nearly dentcal added mass actor alues whch can be at least 45%. Ths s the same as the concluson reached by Haddara et Cao [3]. Lookng at the asymptote o the added mass, these Fgures show that the alue o the added mass actor seems to be asymptotc at a D/L rato o about 50%. But or rectangular plates (plate numbers 1, 3 and 4), conergence s earler than or square plates (plate numbers and 5). One can see that the acceptable D/L rato where the added mass actor ceases to change or rectangular plates s about 0.3 to 0.4 and or square plates s between 0.4 and 0.5. Obserng the two CFFF plates numbered 1 and 3, the aspect rato A/B o one plate s the nerse o the other (A/B=1.87 or plate 1, 0.53 or plate 3). The eect o change o added mass ollows the same rule and depends on the longer dmenson o the plate, whether ths sde s clamped or ree. 5.4 Eect o dampng change Smlar to requences, dampng rates are dented at derent depths and shown n the dagrams or plate number 4. Dampng rate (%) Dampng rate (%) Dampng rate (%) Dampng rate (%) Fg. 15 Dampng change eect on plate 4 Contrary to the requences, there s ether a decrease (mode 1, mode 4) or ncrease (mode, mode 3) o dampng when the plate s put nto water. Snce dampng s dcult to denty and can be a source o dscrepancy as dscussed earler, the dampng rato o a mode at derent

13 depths would be consdered constant the lud low s not turbulent at the exctaton tme. Ths means that turbulence causes addtonal dampng eects. It can also be seen that the rst mode exhbts sgncant added dampng alue at low depth postons, whch makes the total dampng rate abruptly ncrease. 5.5 Mode shapes The dented mode shapes are estmated usng the Modal Assurance Crteron (MAC, see [9]) whch descrbes the correlaton between the dented ector and the analyzed ector o each egen-mode. Fgure 16 shows the MAC o the rst 4 modes o plate 1. MAC MAC MAC MAC Fg. 16 Mode shapes o plate 1 (CFFFA377B01) From these Fgures one can see that the mode shapes are accurately dented when MAC alues are close to 1. The mode shapes o the rst ten modes are also unchanged when the structure moes rom a acuum nto lud. Only mode 3 s abnormal at seeral plate postons. Ths can be explaned by poor modal exctaton (low modal partcpaton actor).

14 6. CONCLUSION A relately complete experment on braton o a plate structure submerged n water s presented n ths paper, whch complements lud-structure nteracton lterature and aldates certan analytcally deeloped models. The eect o added mass on submerged plates and other types o structures s accurately dered usng a ast tme doman method. The ollowng conclusons can be drawn rom our obseratons and calculated results: ) The added mass actor s dstnctly obsered when the plate s submerged n lud. It changes sharply when the depth o the plate s less than one tenth o ts length. As the plate contnues lower nto the lud, the added mass actor changes slghtly and conerges at a specc depth, rangng rom 30% to 40% or rectangular plates or 40% to 50% or square plates. ) The alue o added mass s most sgncant or the rst bendng mode, whch can be more than ten tmes the modal mass n ar. Ths alue s smaller or hgher requency bendng modes but s nearly constant or all torson modes. In the rst e modes, the added mass actor attans at least 45% o the modal mass o the plate tsel. ) The dampng rato o a mode s not constant. It can be hgher when a notceable exctaton s appled. In almost all cases, dampng ncreases when the plate s submerged, howeer another mode exsts wth a dampng rate that s smaller than n ar, although ts alue seems to tend towards conergence when the plate submerges deeper. The braton o the plate can cause turbulence n the cnty o the plate and the result s an abrupt ncrease n the total dampng rate. ) Mode shapes are accurately dented, een when the plates are submerged n lud. In the range o the rst ten modes, there s a change or permutaton o eery mode shape. ) Further studes are recommended to more accurately denty the crtcal D/L rato where the requences start to change. The alue o conerged added mass or each mode should also be analytcally dered. Durng these urther tests and analyses, the eect o added or reduced dampng could be careully measured usng a reasonable exctaton orce. A smple model or the predcton o added dampng due to turbulence could be also deeloped. 7. REFERENCES 1. Lamb, H., On the bratons o an elastc plate n contact wth water, Proceedng o Royal Socety, A98, pp , Fu, Y., and PRICE, W. G., Interactons between a partally or totally mmersed bratng cantleer plate and the surroundng lud, Journal o Sound and Vbraton, 118 (3), pp , Haddara M. R., Cao, S., A study o the dynamc response o submerged rectangular lat plates, Marne Structures, 9, pp , Lang C.-C; Lao C.-C.; Ta Y.-S.; La W.-H, The ree braton analyss o submerged cantleer plates, Ocean Engneerng, Volume 8, Number 9, September 001, pp Y. Kerboua, A.A. Laks, M. Thomas and L. Marcouller, Hybrd method or braton analyss o rectangular plates. Nuclear Engneerng and Desgn. Volume 37, Issue 8, Aprl 007, Pages

15 6. Y.Kerboua, A.A.Laks, M. Thomas, L.Marcouller 005. école polytechnque de Montréal, EPM-RT Comportement dynamque des plaques rectangulares submerges. 7. Powell J. H, and Roberts, J. H. T., On the requency o braton o crcular daphragms, Proceedngs o the Physcal Socety, London, 35, pp , Lndholm, U.S., Kana, D. D., Chu, W. H. and Abramson, H.N., Elastc braton characterstcs o cantleer plates n water, Journal o Shp Research, pp11-, V.H. Vu, M. Thomas, A.A. Laks and L. Marcouller. Mult-autoregresse model or structural output-only modal analyss. Proceedngs o the 5th Semnar on machnery braton, Canadan Machnery Vbraton Assocaton, St John, Canada, K.Snha, J., S. Sandeep, and A.R. Rao, Added mass and dampng o submerged perorated plates. Journal o Sound and Vbraton, 00, Volume 60, Issue 3, p V.Hung. Vu s a PhD student at École de technologe supéreure, Montréal. Hs research project s about operatonal modal analyss o submerged structures n turbulent low. M. Thomas s a proessor at Department o Mechancal Engneerng, École de technologe supéreure, Montréal. He s an expert n structures mantenance, braton and control. Mr Thomas s the ormer Québec chapter presdent o Canadan machnery braton assocaton (CMVA). A.A. Laks s a proessor n the Department o Mechancal Engneerng, École Polytechnque de Montréal. Hs research nterests are random braton and waelet, plate and shell structures, nte element method, etc. L. Marcouller s a researcher at Hydro Quebec s research nsttute. He s an expert on braton analyss and smulaton o structures and machnes.