Iteratioal Applied Mechaics Vol 46 o March (Russia Origial Vol 46 o October ) OLIEAR OSCILLATIOS OF A FLOATIG ELASTIC PLATE ÀÅ Buatov ad ÀÀ Buatov The multi-scales method is used to derive third-order equatios of gravitatioal bedig oscillatios of a thi elastic plate floatig o the surface of a homogeeous perfect icompressible fluid of fiite depth The equatios icorporate the compressive force ad oliear acceleratio of vertical displacemets of the plate Based o these equatios the deflectio of the plate ad the velocity potetial of the fluid iduced by a travelig periodic wave of fiite amplitude are expaded ito asymptotic series to terms of the third order of smalless The depedece of the oscillatio characteristics o the elastic modulus ad thicess of the plate compressive force the iitial legth ad steepess of the wave is aalyzed Keywords: thi elastic plate fluid of fiite depth gravitatioal bedig oscillatios iitial wave legth wave steepess Itroductio The oscillatios of a floatig elastic plate were studied i a liear formulatio i [ 7 8 4] disregardig the compressive force ad i [ 6 4] taig the compressive force ito accout The oliear oscillatios of a floatig absolutely flexible plate were studied i [9] The fiite-amplitude bedig oscillatios of a floatig elastic plate were aalyzed i [ 4] disregardig the fact that the acceleratio of vertical displacemet is oliear We will study the oscillatios of a logitudially compressed elastic plate floatig o the surface of a homogeeous perfect icompressible fluid of costat depth i which a periodic wave of fiite amplitude propagates We will use the method of multiple-scale asymptotic expasios [5] ad tae ito accout the fact that the acceleratio of the vertical displacemets of the plate is oliear Problem Formulatio Goverig Equatios Cosider a thi elastic plate floatig o the surface of a homogeeous perfect icompressible fluid of costat depth The plate ad fluid are ubouded i the horizotal directios Let the oscillatios of the plate be oliear ad the motio of the fluid be potetial With dimesioless variables x x z z t g t * ( / g ) * where is the wave umber the problem is to solve the Laplace equatio xx zz ( x H z ) () for the velocity potetial ( x z t) with the followig boudary coditios: D 4 4 Q 4 p x x z x t p t x z () at the plate fluid iterface ( z ) ad Marie Hydrophysical Istitute atioal Academy of Scieces of Uraie Kapitasaya St Sevastopol Uraie 99 e-mail: ewislad@listru Traslated from Priladaya Mehaia Vol 46 o pp 6 7 October Origial article submitted Jue 5 9 6-795//46-9 Spriger Sciece+Busiess Media Ic 9
z () o the bottom ( z H) The iitial coditios (t =)are f( x) t (4) D I () (4) the followig otatio is used: D g D Eh Q Q ( ) g h Å h ad are the ormal elastic modulus thicess desity ad Poisso s ratio of the plate; Q is the logitudial compressive force per uit width of the plate; ( xtis ) the deflectio of the plate or the rise of the plate fluid iterface; is the desity of the fluid; g is the acceleratio of gravity The velocity potetial ad the deflectio of the plate are related by the iematic coditio t x x z (5) The term with the factor i the dyamic coditio () is the iertia of the vertical displacemets of the plate The first term i bracets of this expressio characterizes the oliearity of the vertical acceleratio of the plate which was eglected i [ 4] Equatios for oliear Approximatios To solve problem () (5) we will use the multiple-scales method [5] Let us itroduce two ew variables T t T t slowly varyig from t Ò where is a small yet fiite ad let the followig expasios hold: f f O( ) O( ) f f f f O( ) () Substitutig from () ito () ad () we obtai the followig equalities up to terms of the third order of smalless: z z z x z Let us cosider dyamic () iematic (5) ad iitial (4) coditios ad represet the velocity potetial of the plate fluid iterface z i the form ( xt ) ( xt) z( xt) zz( xt) () Substitutig f f ( x t ) ad z ( xt ) ito () ad (5) respectively expressig the partial derivative with respect to time by the chai rule as t T T T cosiderig the depedece of o x ad t i () ad collectig the coefficiets of lie powers of ad equatig them to zero we obtai the equatios x z ( x H z ) () 4
4 4 D Q x4 x F * zt T (z ) (4) T z L ( ) (5) z ( z H) (6) f ( x) T G t ( ) (7) to fid oliear approximatios I (4) (7): F F * F F F F L G ( ) T z T T z x z T z L T z T z x x z T F T z T T x x z z T z T z z x x z T z T z T z z T T x x 4 4 T z T z z T T z T 5 4 T z T z z 5 T T z zt L 6 z x x x x T T z x 6 x z x z F z xz x F 7 xz x x z x xz x z 4
7 G G xz x xz T T T ote that the terms F ad F o the right-had sides of the dyamic coditios (4) for the secod ( ) ad third ( ) approximatios are due to the oliearity of the acceleratio of the vertical displacemets of the plate Deflectio of the Plate ad Velocity Potetial of the Fluid Equatios () (7) represet the geeral case of usteady oscillatios of fiite amplitude Let us solve these equatios for travelig periodic waves specifyig the first approximatio ( ) of the deflectio of the plate (the rise of the plate fluid iterface) i the form of a wave cos x T ( T T ) () travelig i the egative directio of the x-axis The the iematic coditio (5) yields z si z () To satisfy the boudary coditio (6) o the bottom we represet i the form Substitutig () ito () we obtai b b +cosh( zh)si () +sih H Hece the followig equalities hold: b si b + sih H + cosh( zh) (4) The dispersio equatio below follows from the dyamic coditio (4) with () (4) for oscillatios i liear approximatio: 4 ( Q D )( tahh) tah H (5) The expressio for ( Ò Ò ) i () follows from the subsequet approximatios Substitutig ad ito the right-had side of Eqs (4) ad (5) for the secod approximatio solvig the problem for ad assumig that there is o pricipal harmoic we obtai a cos b si (6) a 4 tah H b + cosh ( z H) ( tahh cothh ) tahh 4 + cosh HtahH ( 5tahH cothh ) ( 4Q 6D ) 4 ( 4Q 6D ) tahh ( tah H) 4 It appears that the fuctio does ot deped o Ò because ( Ò ) The solutios for the first () (4) (5) ad the secod (6) approximatios determie the right-had sides of dyamic (4) ad iematic (5) statemets of the problem for the third approximatio ( ) Elimiatig the terms resposible for secularity we obtai formulas for ad : a cos b si a b + cosh ( z H) + cosh H where ( l l ) tah H l l l ( 9Q 8D ) 4 4
5 l l l l a( cothh 6cothH) cothhcoth H 8 l a 5 coth H cothh ( cothh 5cothH) cothh 8 5 l a cothhcothh cothhcothh cothh 8 ( 9Q 8D ) tahh ( tah H) 4 T l l4( H) coth l a( cothh cothh) cothhcoth H 8 5 l4 l4 l4 l4 a cothh cothh ( cothh cothh) cothh 8 5 l4 a cothhcothh cothhcothh cothh 4 Thus we have the followig third-order expressios for the deflectio ad velocity potetial : a cos b si xt ( ) a Passig to the dimesioal quatities ( * g / x x z / t t/ g a where a is the amplitude of the iitial harmoic) we get acos a a cos a a cos a g / b si a g b si a g b si x ( ) t a g (7) (hereafter the subscript beeath x z t ad the asteris o are omitted) 4 Aalysis of the Results Solutio (7) is valid beyod small eighborhoods of the resoat values of the wave umber that are the positive real roots ad of the equatios ad respectively I the deep-water approximatio ( Í ) these equatios become D ( 76) Q 4 D ( ) Q 4 4 Rejectig the terms represetig the iertia of the plate ( ) we arrive at If there is o compressio ( Q ) the Q Q 4D 4D / Q 9Q 56D 78D / 4
m m 4 4 4 h m 4 h m a b Fig H m H m 4 7 4 75 5 6 7 75 8 85 65 6 6 9 Q m 6 9 Q m 4 5 55 4 45 5 55 6 a Fig b / 4 4D / 4 9D I the shallow-water approximatio ( H ) the resoat values ad are the roots of the equatios solvig which yields 4HD 5D H Q 4 9HD D H Q (8) 4 5 8 H H H Q 6 ( ) 5D / 5 9H 9 H( H Q ) 5D / eglectig the iertia of the plate ( i (8)) leads to Q /( 5D ) Q /( D ) As the umerical result demostrate if the depth of the fluid remais costat the roots ad of the equatios ad icrease with decreasig elastic modulus of the plate ad icreasig compressive force A icrease i the depth of the fluid also leads to a icrease i ad However the upper limit of the depth at which this effect is sigificat decreases with decreasig elastic modulus but icreases with decreasig compressive force If Å ad Q remai costat a decrease i the thicess of the plate leads to a icrease i ad Figure illustrates the depedece of (dashed lies) ad (solid lies) o the thicess of a plate with / 87 4 (ice) for H = m Curves ad correspod to 9 ad 5 8 M/m forq i Fig a ad toq adq D for Å 9 /m i Fig b Figures a ad b shows the isolies of ad o the plae H Q for h = 5 m ad Q D m If the plate is absolutely flexible ((Å Q = ) the solutio (7) for a fluid of fiite depth is valid for ay values of the wave umber [9] 44
/ a / a x m 4 6 8 x m Fig Fig 4 The distributio (calculated by formulas (7)) of the vertical displacemet of the plate fluid iterface (plate deflectio) i the directio of wave propagatio shows that the structure of perturbatios depeds ot oly o the flexural rigidity of the plate ad the compressive force but also o the fluid depth ad the wave legth ( / ) ad steepess ( a ) of the iitial pricipal harmoic The effect of these parameters ca be judged from the chage i the cotributio of the higher harmoics to oscillatios ad hece the deflectio wave profile This is illustrated by Figs ad 4 plotted for Í m t 7 sec 5 m ad Í m t sec m respectively for the same ad / as i Fig The solid dashed ad dash-ad-dot lies correspod to Q = D 95 D for Å 9 /m h m à m The curves of ( x) demostrate the effect of the compressive force o the phase velocity of the flexural wave of the plate It decreases with icreasig compressive force Q for ay ad fixed elastic modulus E The phase lag (the wave propagates i the egative directio of the x-axis) is due also to the icrease i the elastic modulus at costatq ad If adq is fixed the phase velocity icreases with E Whe there is o compressio (Q = ) a icrease i E leads to a icrease i the phase velocity of the flexural wave for all The oliearity of the acceleratio of vertical displacemets causes the phase of oscillatios to shift i the directio of wave motio ad their amplitude to wealy icrease if the plate is absolutely flexible (E = ) If the plate is elastic the effect of the oliearity of the acceleratio o the phase is strog whe It is maifested as a phase lag o matter whether the compressive force is preset or abset This effect weaes with icreasig wavelegth of the iitial pricipal harmoic Coclusios Usig the multiple-scales method we have derived the equatios for three oliear approximatios of the solutio for the oscillatios of a floatig elastic plate with the compressive force ad the oliear acceleratio of vertical displacemets tae ito accout For a travelig periodic wave of fiite amplitude we have obtaied asymptotic expasios of the third order describig the bedig of the plate ad the velocity potetial of the fluid These expasios are valid beyod small eighborhoods of two resoat values of the wave umber The depedece of the bedig characteristics ad resoat wave umbers of a ice plate o its thicess ad elastic modulus the compressive force ad the fluid depth has bee aalyzed as a example It has bee show that these parameters have a effect o the phase velocity of the deflectio wave ad the cotributio of higher harmoics to oscillatios ad hece ad o the deflectio wave profile REFERECES A E Buatov Effect of logitudial compressio o trasiet vibratios of a elastic plate floatig o the surface of a liquid It Appl Mech 7 o 75 79 (98) A E Buatov ad L V Cheresov Trasiet vibratios of a elastic plate floatig o a liquid surface It Appl Mech 6 o 8 878 88 (97) O M Gladu ad V S Fedoseo oliear steady-state oscillatios of a elastic plate floatig o the surface of a fluid of fiite depth Izv A SSSR Meh Zhid Gaza o 46 54 (989) 4 R V Gol dshtei ad A V Marcheo Log waves i the ice fluid system subject to ice pressure i: Electrical ad Mechaical Properties of Ice [i Russia] Gidrometeoizdat Leigrad (989) pp 88 5 5 A H ayfeh Perturbatio Methods Wiley ew Yor (97) 45
6 V I Pozhuev ad P Polyaova Effect of a movig load o a ice cover: ostatioary problem Stroit Meh Rasch Sooruzh o 6 46 5 (99) 7 V A Tacheo ad V V Yaovlev Usteady bedig-gravitatioal waves i fluid-plate system It Appl Mech o 66 7 (984) 8 D E Kheisi Oscillatrios of a ifiite plate floatig o the surface of a perfect fluid: ostatioary problem Izv A SSSR Meh Mashiostr o 6 67 (96) 9 A E Buatov ad A A Buatov Propagatio of surface wave of fiite amplitude i a basi with floatig broe ice It J Offsh Polar Eg 9 o 6 66 (999) A E Buatov ad V V Zharov Formatio of the ice cover s flexural oscillatios by actio of surface ad iteral ship waves Part Surface waves It J Offsh Polar Eg o 7 o (997) D G Daffy The respose of floatig ice to a movig vibratig load Cold Reg Sci Tech 5 64 (99) A D Kerr The critical velocities of a movig o a floatig ice plate that is subjected to i-plae forces Cold Reg Sci Tech o 6 67 76 (98) R M S M Schules R J Hosig ad A D Seyd Waves due to a steadily movig source o a floatig ice plate Part J Fluid Mech 8 97 8 (987) 4 V A Squire R J Hosig A D Kerr ad P J Laghore Movig Loads o Ice Plates Kluwer Dordrecht (996) 46