Dynamic analysis of an underwater suspension system in ocean currents with method of averaging

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1 Indin Journ of Geo Mrine Sciences Vo. (),November 7, pp. -7 Dynmic nysis of n underwter suspension system in ocen currents with method of verging Xing Li,, Jifeng ui,, Min Zho,,* & Tong Ge, Stte Key Lbortory of Ocen Engineering, Shnghi Jio Tong University, Shnghi,, hin obortive Innovtion enter for Advnced Ship nd Deep-Se Exportion (ISSE), Shnghi, hin *[E-mi: min.zho@sjtu.edu.cn] Received 7 October 5 ; revised 3 October Underwter suspension system is simpified into singe degree of freedom mode with restoring force nd qudrtic dmping. Noniner mthemtic mode of the suspension unit osciting in sti wter nd in ocen currents is estbished. And its nytic soutions re obtined with method of verging. Resuts show tht nytic soution with the method of verging hve sm differences with numeric soution bsed on the fourth-order Runge-Kutt nd cn effectivey describe the underwter oscition, nd soution with weker noniner term hs the better pproximtion. With rge mss, short tether ength nd rge current veocity, the nytic soution obtined with method of verging is coser to the numeric soution. [Key words: Underwter suspension system; Dynmic mode; Trnsverse oscition; Method of verging; Ocen currents] Introduction Underwter tethered systems re vehices, toos or other pckges ttched to tether (or cbe) nd suspended from surfce vesse, which hve been widey used in the mrine environments for exportion, inspection nd engineering opertions. They re vit toos tht provide sfe nd effective ccess to deep wter. Due to different ims nd tsks, the tether types re vrious, miny contining neutr buoyncy tether nd rmoured umbiic cbe. Neutr buoyncy tether is mosty used on submersibes ike ROV, towfish nd ine rry sonr, nd the dynmic behvior of towed object hs ttrcted much ttention from reserchers,3,,5,. Armoured umbiic cbe hs submerged weight in se wter nd owns greter structur strength thn neutr buoyncy tether, but more motion restrictions to the suspended object, too. The rmoured umbiic cbe is commony ppied in underwter suspension systems in ocen engineering, such s trencher, hevy duty ROV, TMS of ROV system, TLP founion, hevy underwter opertion toos nd so on. When the underwter suspension system is owered down into the wter, vertic nd trnsverse oscition behviors hppen due to extern excittions, which infuence the sfety nd stbiity of the system. Vertic oscition behvior subject to surfce excittion hs redy been widey studied 7, becuse sck cbe nd rge snp ods cn occur during vertic oscition behvior in rough ses, which my cuse structur dmge to the tether nd endnger the recovery of the underwter suspension object. But ess ttention re pid to the trnsverse oscition behvior, which is so quite importnt in underwter octing, time nd position contro of hnging work nd controing the operting rnge of suspension systems. There re two kinds of modeing techniques vibe to predict the response of tethered systems: the continuous nytic methods nd the discrete numeric modes 7. Discrete numeric modes re vid for some noniner properties ike qudrtic drg nd sptiy vrying properties of cbes. The noniner couping motion principe between the tether nd the vehice cn be incuded in these modes. The most prevent numeric pproches used nowdys in determining the hydrodynmic performnce of n underwter tethered system re

2 INDIAN J. MAR. SI., VOL., NO., NOVEMBER 7 the umped mss method 9, the finite difference method, nd the finite eement method,3. Numeric methods re effective toos but they so hve sever imittions. For exmpe, they need much ccution time nd cnnot provide quick estimtes of dynmic chrcteristics of tethered systems. But nytic methods re not constrined by these imittions. They cn ccurtey nd concisey summrizes the retionship between vribes, nd vues of independent vribes cn be esiy obtined through inverse ccution corresponding to the dependent vribes. Drisco et. once deveoped continuous onedimension nytic mode which coud simute verticy tethered system subject to surfce excittion 7. The mpitude of ship motion tht triggers snp od coud be predicted by soving the trnsfer function of ship-cge motion. Min purpose of this pper is to estbish noniner one-dimension nytic mode tht represents the trnsverse oscition behvior of tethered suspension system in ocen currents, nd find the nytic soution with method of verging, which wi be usefu to predict the oscition chrcteristics. Mteris nd Methods In the re ocen environment, the underwter suspension system usuy works with the surfce ship foting nd the tether ength vrying. Its motion chrcteristics hve gret compexity. Therefore, the underwter suspension system is simpified s penduum to get the eqution of vibrtion dmping, nd some ssumptions re mde s foowing: The underwter suspension object oscites in sm mpitude, nmey the mximum penduum nge is ess thn 5, due to excittion forces induced by currents or ship motion through tethers; Without considering the tether's estic extension, effects of grvity nd fow resistnce of the tether re negected, becuse they re too much smer compred to the forces directy cted on the submerged object which hs too much more weight thn the tether; Regrdess of motion of the mother ship, the trnsport inerti force nd oriois inerti force re zero, nmey F Ie = nd F I =. The chnge of position where the tether is connected to the surfce ship is ignored. Fig. shows typic underwter suspension system where the ship nd underse unit re connected by ong estic tether. The system is simpified into singe degree of freedom mode with restoring force nd qudrtic dmping. In order to nyze the motion of the underwter suspension object, the coordinte frme xoy is first estbished s shown in Fig., nd the origin O is set in the center of the bottom of ship. The coordinte xis x is horizont, nd coordinte xis y is vertic nd directed from top to bottom. The ngur dispcement between tether nd coordinte xis y is defined s φ, which is positive when the suspension object moves to the right side of xis y. current o y Fig. Definition of coordinte system nd symbos The underwter suspension object is cted upon by the foowing forces 5 : () Grvity G mg, buoyncy F Vg mf g ; () Added mss force due to cceertion du m, m is the dded mss, u is the motion veocity of the suspension object; (3) Tether tension F t ; () onsidering the infuence of ocen current, the drg force on the underwter suspension object cn be written s F k u u cos ( u u cos ) D c c ds ds d ws uc cos uc cos where d is drg coefficient reted with Reynods number, w is se wter density, S is projected re of the submerged object, s is the object s dispcement, u c is ocen current veocity, nd is the sm defection nge. Due to D'Aembert principe, the bsic eqution of retive motion dynmics is estbished, nd projected to the tngent xis t of the oscition trjectory, then we cn obtin d s d s m ( G F)sin m () ds ds d ws uc cos uc cos F t F G x F D

3 LI et.: DYNAMI ANALYSIS OF AN UNDERWATER SUSPENSION SYSTEM IN OEAN URRENTS When the suspension object mkes micrompitude vibrtion, defection nge is sm ( 5 ), nd sin, cos, s, then Eq. () cn be written s d ( m m ) ( G F) () d d d ws uc uc dws Setting dmping prmeter nd ( m m ) ngur frequency ( G F) ( m m ), Eq. () cn be rewritten in the foowing form d d uc d uc (3) Eq. (3) is the differenti eqution of the underwter suspension object mking micrompitude vibrtion in ocen current environment, describing the rues of oscition with squre dmping. The eqution hs the term of ocen currents, which is the key point different from convention simpe penduum equtions. Due to the effect of the dmping term, the mpitude decys constnty. The hydrodynmic dded forces nd moments come bout from the cceertion of the fuid prtices when they encounter the submerged object. Since there is no retive cceertion in the norm direction between submerged object nd its surrounding body of fuid, the norm dded mss is ignored. If the forces re projected to the norm xis n of the oscition trjectory, we cn obtin the tether tension d Ft m ( G F)cos d d d ws uc uc sin () Soution procedures with method of verging In the bove section, the differenti eqution governing the oscition of the suspension object in ocen currents with singe degree of freedom is obtined. As the vibrtion eqution hs the term of bsoute vue, method of verging is proper wy to sove nd get the nytic soution. As we know, if moving in ocen currents, the underwter suspension object wi not oscite round the y xis nd wi hve bnce nge, which cn be got from sttic equiibrium. If etting im ( t) nd im ( t) in Eq. (3), we t t cn obtin the bnce nge uc dwsuc im ( t), which is consistent t ( G F) with soution by sttic equiibrium. onsider the noniner oscition eqution of the underwter suspension object f (, ) (5) The soution cn be expressed in the foowing form provided tht nd θ re considered s functions of t rther thn constnts ccording to method of verging. In order to describe the bnce nge, constnt A is introduced. So the soution gives A ( t)cos[ t ( t)] () ( t) sin[ t ( t)] (7) where uc dwsuc A im ( t) t ( G F) (8) nd the foowing equtions describing the sow vritions of t () nd () t f ( cos, sin ) sind (9) f ( cos, sin ) cos d () where t. Eq. (9) nd Eq. () re ced the verging equtions. ompring Eq. (3) nd (5), we know d uc d uc f () uc uc sin sin For the underwter suspension object, () In the ide condition of sti wter with current veocity being zero, Eq. () cn be written s d d f sin sin () Substituting f into Eq. (9) nd Eq. (), we obtin (3) 3 () thus t (5) 3 ()

4 INDIAN J. MAR. SI., VOL., NO., NOVEMBER 7 3 so we hve t cos ( t ) (7) 3 where nd re undetermined prmeters. In gener, giving initi conditions () nd (), substituting them into Eq. (7), nd yied or 9, 3 (8) 9 rccos 3 9, 3 rccos 9 3 (9) Substituting Eq.(8) into Eq. (7) mkes the divergence of in the initi time, not stisfying the physic scene, thus bndoning Eq.(8). () onsidering the ocen current condition, uc uc m m the vue of infuences the ( G F) bsoute vue symbo of Eq. (). As the mpitude mximum is genery 5, if the ocen current veocity is rge nd tether ength is short, u c cn be rger thn, nd Eq. () cn be rewritten s d uc d uc uc f sin () Substituting f into (9) nd (), we hve uc () () Thus we obtin u c t / e (3) () hence, u c u t c e cos( t ) (5) where nd re undetermined prmeters. Giving initi conditions () nd (), nd substituting them into Eq. (5),, cn be got s 3 c c c, 3 3 uc u u u rccos u c or u u u u 3 c c c rccos 3 3 c u c, () (7) The bove prmeter group stisfies the physic scene, but hs opposite sign in () nd (7), in () differs by phse of from (7). If considering vrying between / /, the prmeters in () cn be bndoned. Eq. (7) nd Eq. (5) re the pproximte nytic soutions, nd, dopt the prmeters in (9) nd (7) respectivey. It cn be seen from Eq. (3) nd Eq. () tht initiy the rte t which the mpitude of the response decreses is proportion to the squre of the mpitude of the initi disturbnce in sti wter nd to the mpitude of the initi disturbnce in ocen currents. Thus when the mpitude of the initi disturbnce is rge, the initi decy is sower in sti wter thn in ocen currents. If the initi disturbnce is sm, the opposite is expected to be true. According to Eq. (7) nd Eq. (5), the mpitude in sti wter decys gebricy, whie the mpitude in ocen currents decys exponentiy with time. Given initi prmeters nd ocen current conditions, when t, mpitude tends to, without retionship with initi. keeps constnt, resuting tht the soution with the method of verging for the first order pproximtion hs no modifiction on the oscition frequency. Resuts nd Discussions Different prmeters re given to the nytic expressions to nyze the effects of the different types of dmping, nd vie the nytic soution for cses with nd without ocen currents. For simpicity,

5 ( o ) ( o ) ( o ) ( o ) LI et.: DYNAMI ANALYSIS OF AN UNDERWATER SUSPENSION SYSTEM IN OEAN URRENTS.5, m.5m, m.5m d f conditions give () 5 /8, (). f. The initi Oscition in sti wter Resuts of nytic soution with the method of verging nd numeric soution bsed on the fourth-order Runge-Kutt re compred nd presented s beow (soid nd dotted ines, respectivey). is n effective expicit or impicit itertive method for soving noniner ordinry differenti eqution, nd it hs high fideity for noniner soution. The sm difference between nytic nd numeric resuts shows the effectiveness of the method of verging. The noniner term is determined by both dmping prmeter nd retive ngur veocity. Athough the dmping prmeter is rger thn one, with the sm term of retive ngur veocity, the nytic soution sti hs retive stisfied pproximtion. Oscition in ocen currents The retionships between defection nge nd time with different mss re potted in Fig. 3, nd resuts obtined from nytic soution re compred with numeric resuts. As shown in the grph, the ccuted vues of defection nge with method of verging hve simir trend to those of numeric resuts, but the mpitude of numeric resuts re smer thn nytic ones. With the increse of time, defection nge first oscites, nd then grduy dmps into stedy nge. The stedy nge with both verging nd is most the sme. As the mss increses, the difference between defection nges obtined with method of verging nd numeric method becomes smer due to the smer dmping prmeter ε. The osciting period T keeps the sme, but the defection nge dmps fster to stedy sttus with the smer mss () Oscition during the first 3 seconds - 8 () m=5 t, ε=8., ω= (b) Oscition during the 3 seconds Fig. omprison between nytic soution nd numeric soution bsed on the fourth-order Runge-Kutt in short time nd ong time (m= t, = m, u c = m/s, ε=8., ω=.98) (b) m= t, ε=.35, ω=.8

6 ( o ) ( o ) ( o ) ( o ) INDIAN J. MAR. SI., VOL., NO., NOVEMBER (c) m= t, ε=.53, ω=.8 Fig. 3 Defection nge vs time with different mss (=5 m, u c =.5 m/s). Fig. shows the retionships between defection nge nd time with different tether ength. It cn be seen tht the mpitude of numeric resuts re smer thn nytic ones. As the tether ength increses, the dmping prmeter ε grows rger nd the difference of defection nge obtined with method of verging becomes bigger from those with numeric method. Since the osciting period gives T ( m m ) / ( G F), the osciting period T increses with onger tether ength, which is obvious in Fig.. Besides, the defection nge dmps to stedy sttus with simir time of s for both methods. In short, tether ength infuences both the dmping prmeter ε nd period T () =5 m, ε=.53, ω=.8 - (b) = m, ε=.35, ω= (c) = m, ε=8., ω=. Fig. Defection nge vs time with different tether ength (m=t, u c =.5 m/s). onsidering the infuence of ocen current, the resuts of different current veocities obtined from nytic soution re compred with numeric resuts nd potted in Fig. 5. The comprison shows tht the defection nge dmps to stedy sttus in shorter time with stronger current veocity, though the osciting period keeps the sme. According to the eqution in section, ocen current doesn t infuence the dmping prmeter ε. So the defection nges obtined with method of verging nd hve simir ccurcy in environment of different ocen current veocities.

7 ( o ) ( o ) ( o ) LI et.: DYNAMI ANALYSIS OF AN UNDERWATER SUSPENSION SYSTEM IN OEAN URRENTS () u c =.5 m/s, ε=.53, ω= (b) u c = m/s, ε=.53, ω=.8-8 (c) u c =.5 m/s, ε=.53, ω=.8 Fig. 5 Defection nge vs time with different ocen current veocity (m=t, =5m). oncusions The mpitude ttenution is sensitive to vrition of the dmping prmeter ε, which is ffected by some physic properties, ike the mss, tether ength nd current veocity. With rge mss, short tether ength nd rge current veocity, the nytic soution obtined with method of verging is coser to the numeric soution. The noniner term is determined by both dmping prmeter nd retive ngur veocity. The nytic soution with weker noniner term hs the better pproximtion. Method of verging hs ony first-order pproximte precision of soving noniner eqution. But for this mode, method of verging cn be simpe nd effective to sove the probem, nd resut in stisfctory ccurcy. If more ccurte nytic soutions re required, other nytic methods need to be tried. The oscition mode buit in this pper is retivey ide, nd some conditions ike the esticity, grvity nd drg force of tether re ignored, which exist in prctic probems nd cn be studied further. Acknowedgements The support of High Technoogy Ship Reserch nd Progrm of Ministry of Industry nd Informtion Technoogy of the Peope s Repubic of hin (Grnt No. 539 []), nd the Speciized Reserch Fund for the Doctor Progrm of Higher Eduction of hin (Grnt No. 73) for this work is grtefuy cknowedged. References Lueck, R.G., Drisco, F.R. nd Nhon, M., A wveet for predicting the time-domin response of verticy tethered systems, Ocen Eng.,, 7():- 53. Fng, M.., hen, J. H., Luo, J. H. nd Hou,.S., On the behvior of n underwter remotey operted vehice in uniform current, Mrine Techno., 8, 5():-9. 3 Zhu, K.Q., Zheng, D.., i, Y., Yu,.L., Wng, R., Liu, Y.L., nd Zhng, F., Noniner hydrodynmic response of mrine cbe-body system under rndom dynmic excittions, J. Hydrodyn., Ser. B, 9, (): Shin, H.K., Ryue, J.S., Ahn, H.T., Seo, H.S. nd Kwon, O.., Study on dynmic behvior nysis of towed ine rry sensor, Int. J. Nv. Architect. Ocen Eng.,, (): -. 5 Yu, S.., Yuh, J. nd Kim, J., Armess underwter mnipution using sm depoybe gent vehice connected by smrt cbe, Ocen Eng., 3, 7:9-59. Prk, J. nd Kim, N., Dynmics modeing of semisubmersibe utonomous underwter vehice with towfish towed by cbe, Int. J. Nv. Architect. Ocen Eng., 5, 7(): Drisco, F.R., Lueck, R.G. nd Nhon, M., The motion of deep-se remotey operted vehice system, Prt : nytic mode, Ocen Eng.,, 7(): 57-7.

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