VERSITYCSouthampton(4Symmetricdynamicbehaviour in regular wavesEquationsofmotion&ResponsesExamplesfrom:HydroelasticityofShips,R.E.D.Bishop&W.G.Price,CUP1979Hydrodynamics and HydroelasticityProf.P.Temarel,WUT,July 2015NIVERSITYOSummary-Last lectureSouthamptonFluidactions fora Strip (2D)theoryformulation using:.Relativemotion(displacement)andtotalderivativeconcepts:Use ofvelocity potentials to obtain added mass and fluid dampingvaluesforeachstrip (or section)inconjunctionwithConformalmappingforaccuraterepresentationof shipsections(Lewisormulti-parameterconformalmapping)Summary-ThislectureGeneralised equationsofmotionforthe (entire)shipin regularwavesusing2Dhydroelasticity-unifiedtheoryincludingboth rigidbodymotionsanddistortionsSolution ofgeneralised equations of motion; obtaining principal·coordinatesExamples ofsymmetric responses:e.g.motions, distortions, bendingmoments,shearforces
1 (4) Symmetric dynamic behaviour in regular waves Equations of motion & Responses Examples from: Hydroelasticity of Ships, R.E.D. Bishop & W.G. Price, CUP 1979 Hydrodynamics and Hydroelasticity Prof. P. Temarel, WUT, July 2015 2 Summary – Last lecture • Fluid actions for a Strip (2D) theory formulation using: Relative motion (displacement) and total derivative concepts • Use of velocity potentials to obtain added mass and fluid damping values for each strip (or section) in conjunction with • Conformal mapping for accurate representation of ship sections (Lewis or multi-parameter conformal mapping) Summary – This lecture • Generalised equations of motion for the (entire) ship in regular waves using 2D hydroelasticity – unified theory including both rigid body motions and distortions • Solution of generalised equations of motion; obtaining principal coordinates • Examples of symmetric responses: e.g. motions, distortions, bending moments, shear forces
JIVERSITYOReminder: DisplacementSouthampton:Derivatives of displacement:()Withrespecttotimeand()withrespecttocoordinatealongshipRememberingthatTotalDerivative&absoluteverticaldisplacementareCD.a-uow(x,t- Z pr(0)wr(x)Dtataxr=0ThenoODZ pr(t)wr(x)-UE pr(t)wr(x)w(x,t)=Dtr=0r=0andD28088Z pr(0)wr(x)-2U 2 pr()w()+U2 w(x,t)=Z pr(t)w(x)Dt2r=0r=0r=03JIVERSITYOGeneralised Fluid actions (1):SouthamptonRadiation&RestoringtermsRememberingthattheexternalforcewasexpressedas11Fs(t)=ws(x)F(x,t)dx= -Ws(x)H(x,t)dx+Ws(x)Z(x,t)dx=-H,(t)+三s(t)0LetuslookatthefirsttermD2 w(x)+[N(x)-Um(x)Dw(x,t)Hs(t)=+pgB(x)w(x.t)/dxws(x/m(x)Df2DtSubstitutingforthetotalderivatiesofw(x,t).T[Pr(t)wr(x)-2U pr(t)w(x)+U2pr(t)w(x)dxws(xm(x) >r=01[ws(x)[N(x)-Um(x)] [pr(t)w(x)-U pr(t)w(x)]ldr=0ows(x)pg B(x)ZPr(t)wr(x) dxr=02
2 3 Reminder: Displacement • Derivatives of displacement: (. ) With respect to time and (‘) with respect to coordinate along ship ( ) ( ) 0 2 ( ) ( ) 0 ( ) ( ) 2 0 ( , ) 2 Dt 2 D and ( ) ( ) 0 ( ) ( ) 0 ( , ) Dt D Then Remembering that Total Derivative& absolute vertical displacement are x wr t pr r pr t wr x U r pr t wr x U r w x t x wr t pr r pr t wr x U r w x t x U Dt t D ( ) ( ) 0 w(x,t) p t w x r r r 4 Generalised Fluid actions (1): Radiation & Restoring terms w ( x ) g B( x ) p (t )w ( x ) dx w ( x ) N( x ) Um ( x ) p (t )w ( x ) U p (t )w ( x ) dx w ( x ) m( x ) p (t )w ( x ) U p (t )w ( x ) U p (t )w ( x ) dx w( x,t ) g B( x )w( x,t ) dx Dt D w( x,t ) N( x ) Um ( x ) Dt D w( x,t ) H t w ( x ) m( x ) F t w ( x )F( x,t )dx w ( x )H( x,t )dx w ( x )Z( x,t )dx H t t r r r s L r r r r r s L r r r r r r r s L s L s s s s L s L s L s 0 0 0 0 2 0 0 2 2 0 0 0 0 2 Substituting for the total derivaties of ( ) Let uslook at the first term ( ) ( ) ( ) Remembering that the externalforce was expressed as
IVERSITYOGeneralised Fluid actions (1):SouthamptonRadiation&RestoringtermsIn this expression we have terms involving p(t), p(t),as well as p(t)The latter appears in all three terms, which we will associate withadded mass, fluid damping and hydrostatic restoring coefficientsof the ship in water.Hence, we need to do something about p(t) appearing inthe first two.In regular waves pr(t)= prexp(-ioet); Thus p,(t)=-e pr(t)or pr(t)=_ ProGeneralised Fluid actions ():JIVERSITYOSouthamptonRadiation&RestoringtermsUsing thisrelationshipand interchangingsummation and integration,wegetu2[m(x)wj(x)ws(x)]m(x)wr(x)ws(x)-L2QeHs(t)=pr(t)[UTr=00[N(x)w-(x)ws(x)[m(x)w(x)ws(x)QeOL+ Z pr(0J (N(x)wr(x)ws(x)-2Um(x)wi(x)ws(x)-Um(x)wr(x)ws(x)dxr=00L8+ Z pr() pg B(x)wr(x) ws(x)dxr=00orLHs(t)=Z [ArsPr(t)+Brspr(t)+CrsPr(t)]for s=0, 1,2,3..r=03
3 5 Generalised Fluid actions (1): Radiation & Restoring terms 2 2 or ( ) In regular waves ( ) exp( ); Thus ( ) ( ) Hence, we need to do something about ( ) appearing inthe first two. of the ship in water. added mass,fluid damping and hydrostatic restoring coefficients The latter appears in all three terms, which we will associate with In this expression we have termsinvolving ( ), ( ), as well as ( ). e r r r r e r e r p p t p t p i t p t p t p t p t p t p t 6 Generalised Fluid actions (1): Radiation & Restoring terms [ ( ) ( ) ( )] for s 0,1, 2, 3,. 0 ( ) or ( ) ( ) ( ) 0 ( ) 0 ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) 0 ( ) ( ) ( ) 2 2 ( ) ( ) ( ) 2 ( ) ( ) ( ) 2 2 ( ) ( ) ( ) 0 ( ) 0 ( ) Using this relationship and interchanging summation and integration, we get t Crs pr t Brs pr t Ars pr r t Hs x dx ws x wr g B x L t pr r x dx ws x wr x U m x ws x wr x U m x ws x wr N x L t pr r dx x ws x wr m x e U x ws x wr N x e U x ws x wr m x e U x ws x wr m x L t pr r t Hs
UNIVERSITYOFGeneralisedAdded MassMatrixSouthamptonAnelementoftheaddedmassorinertiamatrixfortheshipis:LU21Ars = [ m(x)wr(x)w,(x)dx-[m(x)w(x)w,(x)dxo00U2!ULJ N(x)w,(x)ws(x)dx - m'(x)w,(x)ws(x)dx-2/QeoQe0Using integration by parts u dv= uvl - [v duwithu=mws (u'=m'ws +mw',)and v=w[ m(x)w(x)w,(x)dx=m(x)w;(x)wg(x) [60LLm(x)w(x)ws(x)dx-[m(x)w,(x)ws(x)dx00UNIVERSITYOFGeneralised Added MassMatrixSouthamptonThusU2LArs =J m(x)wr(x)ws(x)dx++ (w ()dt0U2UIm(x)w(x)ws(t) 16-j N(x)wi(x)ws(x)dx-o00This is as per equation (7.16)of the book (*), omitting the underlined termswhich use a different expression (Theory B, Salvesen et al 1970) that hasmoretermsinvolvingforwardspeed(*) HydroelasticityofShips, Bishopand Price,19794
4 7 Generalised Added Mass Matrix m x w x w x dx m x w x w x dx m x w x w x dx m x w x w x m w m w m w w m x w x w x dx U N x w x w x dx U m x w x w x dx U A m x w x w x dx r s L r s L L r s r s L s s s r r s L e r s L e r s L e r s L rs ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) with u (u ) and v Using integration by parts u dv u v v du, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) An element of the added mass or inertia matrix for the ship is: 0 0 0 0 0 2 2 0 2 0 2 2 0 8 Generalised Added Mass Matrix (*) Hydroelasticity of Ships, Bishop and Price,1979 more termsinvolving forward speed which use a different expression (Theory B,Salvesen et al1970) that has This is as per equation (7.16) of the book (*), omitting the underlined terms ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ' ( ) Thus 2 0 2 0 2 0 2 2 0 L r s e r s L e r s L e r s L rs m x w x w x U N x w x w x dx U m x w x w x dx U A m x w x w x dx
VERSITYGen. Added Mass Matrix-discussion SouthamptonelementArs:added inertia at sthmodeduetomotion (ordistortion)at.rthmodedepends both on sectional added mass &fluid damping values; hencedependsonshapeand size of (mean orclam water)underwatershape:dependson mass distribution andhull flexibility,for distortion modes.depends on encounter frequency @e.fora shipwithpointedends;lasttermdisappears:speed dependence exists for both diagonal and off-diagonal termsnote, however, for heave as w'=o no speed dependence;forpitchasw,o,thereisspeeddependence:Note division between Ars and Crs may seem arbitrary, but is based onwheresometermsbelongmorenaturally9IVERSITYOGeneralised Fluid Restoring matrix SouthamptonAn element of the fluid restoring (or stiffness) matrix isLpg B(x)wr(x) ws(x)dxCrs =0elementCrs:fluidstiffnessatsthmodeduetomotion (ordistortion)at rth mode;or CrsPris changein sth generalisedforce duetobuoyancyforcechanges due to distortion Prwrasbuoyancychanges areregarded hydrostatic,justifiesinclusion instiffnessorrestoringmatrix:Note,however,itdoesnotimplyhydrostaticloading5
5 9 Gen. Added Mass Matrix-discussion • element Ars: added inertia at sth mode due to motion (or distortion) at rth mode • depends both on sectional added mass & fluid damping values; hence depends on shape and size of (mean or clam water) underwater shape • depends on mass distribution and hull flexibility, for distortion modes • depends on encounter frequency ωe • for a ship with pointed ends; last term disappears • speed dependence exists for both diagonal and off-diagonal terms note, however, for heave as w’0=0 no speed dependence; for pitch as w’1≠0, there is speed dependence • Note division between Ars and Crs may seem arbitrary, but is based on where some terms belong more naturally 10 Generalised Fluid Restoring matrix C g B x w x w x dx r s L rs ( ) ( ) ( ) An element of the fluid restoring (or stiffness)matrix is 0 • element Crs: fluid stiffness at sth mode due to motion (or distortion) at rth mode; • or Crspr is change in sth generalised force due to buoyancy force changes due to distortion prwr • as buoyancy changes are regarded hydrostatic, justifies inclusion in stiffness or restoring matrix • Note, however, it does not imply hydrostatic loading
VERSITYGeneralised Fluid Damping matrix SouthamptonAn element of the fluid damping matrix1Brs =J (N(x)wr(x)ws(x)-2U m(x)wi(x)ws(x)-Um(x)w (x)ws(x)dx0Using integration by parts Judv= u v- fv du.withu=w,ws (u'=w,ws+wsw,)andv=mJ m(x)w(n)ws(x)dx=m(x)wr(x)ws() /6T.1[m(x)w,(x)w,(x)dx-[m(x)w,(x)w,(x)dx00VERSITYOGeneralised Fluid Damping matrix SouthamptonThusLBrs = JN(x)wr(x)ws(x)dx0L+U [m(x)[wr(x)ws(x)-w(x)ws(x)]dx0-U m()wr(t)ws() /This is as per equation(7.17)of the book (*),omitting the underlined termswhich usea different expression (TheoryB,Salvesen etal 1970)that hasmore terms involving forward speed(*) Hydroelasticity of Ships, Bishopand Price,19796
6 11 Generalised Fluid Damping matrix L r s r s L L r s r s L r s r s s s r s r s r s L rs m x w x w x dx m x w x w x dx m x w x w x dx m x w x w x w w w w w w m B N x w x w x U m x w x w x U m x w x w x dx 0 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) with u (u ) and v Using integration by parts u dv u v v du, ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) An element of the fluid damping matrix 12 Generalised Fluid Damping matrix (*) Hydroelasticity of Ships, Bishop and Price,1979 more termsinvolving forward speed which use a different expression (Theory B,Salvesen et al1970) that has This is as per equation (7.17) of the book (*), omitting the underlined terms 0 ( ) ( ) ( ) ( ) ( )] 0 ( )[ ( ) ( ) 0 ( ) ( ) ( ) Thus L x ws x wr U m x x dx ws x wr L x ws x wr U m x L x dx ws x wr N x Brs
Generalised Fluid Damping matrix Southampton.element Brs:fluid damping at sth modeduetomotion (or distortion)at rth mode;:or Brs P, is the sth generalised fluid damping force, due to motion atrth mode; i.e. damping of p, due to motion at prviceversaforBsrPsdepends bothon sectional addedmass &fluid dampingvalues; hencedepends on shape and sizeof (mean or clam water)underwatershape.depends on mass distribution andhull flexibility,for distortion modes.dependsonencounterfrequencyefor a ship with pointed ends speed dependence exists for off-diagonalterms only,sinceBrs+Bsr=o,Timman-Newman relationship.intuitively one can argue the damping is provided bythe diagonal termswiththeoff-diagonaltermsprovidingthecoupling13IVERSITYOGeneralised structural dampingSouthamptonAssumed diagonal duetolack ofdata,namelybrr=2vrorarrwhere w, dry hull natural frequency (rad/s), arr generalised mass anddampingfactor Vr=2or,with 8,thelogarithmic decrementthat canbemeasured.bestmeasurementsfromfree oscillation decaytests,e.g.slamoranchordrop in calm water,both involve influence of water (e.g.in slam addedmass and fluid damping, latter small; in anchor drop added masseffects)Empirical formulaeavailable,based on tests; widelyused Kumai (1958)82=3.5L and8,=8,(0,/02)0.75 forr>2AsKumai (1958)formulaearebasedon small cargoships,usuallyincrementedbasedonexperienceTypicalvaluesV2=0.006 (destroyer), 0.004 (tanker ballast),0.002 (tanker loaded)V3=0.009 (destroyer), 0.007 (tanker ballast), 0.005 (tanker loaded)7
7 13 Generalised Fluid Damping matrix • element Brs: fluid damping at sth mode due to motion (or distortion) at rth mode; • or is the sth generalised fluid damping force, due to motion at rth mode; i.e. damping of ps due to motion at pr • vice versa for • depends both on sectional added mass & fluid damping values; hence depends on shape and size of (mean or clam water) underwater shape • depends on mass distribution and hull flexibility, for distortion modes • depends on encounter frequency ωe • for a ship with pointed ends speed dependence exists for off-diagonal terms only, since Brs+Bsr=0, Timman-Newman relationship • intuitively one can argue the damping is provided by the diagonal terms with the off-diagonal terms providing the coupling rs r B p sr s B p 14 Generalised structural damping • Assumed diagonal due to lack of data, namely brr=2νrωrarr where ωr dry hull natural frequency (rad/s), arr generalised mass and damping factor νr=2πδr, with δr the logarithmic decrement that can be measured. • best measurements from free oscillation decay tests, e.g. slam or anchor drop in calm water, both involve influence of water (e.g. in slam added mass and fluid damping, latter small; in anchor drop added mass effects) • Empirical formulae available, based on tests; widely used Kumai (1958) δ2=3.5L and δr= δ2(ωr /ω2)0.75 for r>2 • As Kumai (1958) formulae are based on small cargo ships, usually incremented based on experience • Typical values ν2=0.006 (destroyer), 0.004 (tanker ballast), 0.002 (tanker loaded) ν3=0.009 (destroyer), 0.007 (tanker ballast), 0.005 (tanker loaded)
NIVERSITYOFGeneralised fluid & structural dampingSouthampton0@VL/g=4WeVL/g=9.60eVL/g=17rb'rrB'rB'B0o8.981.850.21106.131.690.2320.632.119.240.2515.81.760.16310.149.71.740.1629.9non-dimensionalisation b'rr=(brr VL/g)/arr,B'rr=(Brr VL/g)/arr.Tanker in ballast; mode r=2 wet resonancefrequency3.6 rad/s or21.4/VL/g.Structural dampingpredominantathigherfrequencies15GeneralisedFluidactions (2):NIVERSITYOSouthamptonWaveexcitation.Regularwaves;.both incidentwave (Froude-Krylov)and diffraction componentsFinally thegeneralised waveexcitation is:L三s(t)=J ws(x)Z(x,1)dx=0L[ (-2m(x)-io[N(x)-Um(x)]+ pg B(x))5(x,t)ws dx0based on what was derived before, and (x,t) being the waveelevationat centre line.This is as per equation (7.19)of the book (*),omitting theunderlined terms which use a different expression (Theory B, Salvesen et al 1970)that hasmoreterms involvingforward speed16(*)Hydroelasticity of Ships, Bishop & Price, 19798
8 15 Generalised fluid & structural damping r b’rr ωe√L/g=4 ωe√L/g=9.6 ωe√L/g=17 B’rr B’rr B’rr 0 0 8.98 1.85 0.21 1 0 6.13 1.69 0.23 2 0.63 9.24 2.11 0.25 3 15.8 10.1 1.76 0.16 4 29.9 9.7 1.74 0.16 • non-dimensionalisation b’rr=(brr √L/g)/arr, B’rr=(Brr √L/g)/arr • Tanker in ballast; mode r=2 wet resonance frequency 3.6 rad/s or 21.4/ √L/g • Structural damping predominant at higher frequencies 16 Generalised Fluid actions (2): Wave excitation • Regular waves; • both incident wave (Froude-Krylov) and diffraction components (*) Hydroelasticity of Ships, Bishop & Price,1979 that has more termsinvolving forward speed underlined terms which use a different expression (Theory B,Salvesen et al1970) at centre line.Thisis as per equation (7.19) of the book (*), omitting the based on what was derived before, and (x,t) being the wave elevation ( ) ( ) ( ) ( )} ( , ) 2 { 0 ( ) ( , ) 0 ( ) Finally the generalised wave excitation is: dx ws m x i N x Um x g B x x t L x Z x t dx ws L t s
NIVERSITYOGeneralisedwaveexcitationSouthampton.in,firsttwoterms (dependenton sectional addedmass andfluiddamping)denotethediffractioncontribution-usuallysmallcomparedtolastterm.in ,last term dueto incident wave pressure,including Smith effect-usuallydominateswaveexcitationDependence on forward speed, wave frequency and heading.Lineardependenceonregularwaveamplitude:dependson mass distribution andhull flexibility,for distortion modesImportanttounderstand howtheintegrand in,varieswithmodalindexandwavefrequencyNIVERSITYOFGeneralised waveexcitationSouthamptonVariation of integrand alongloadedtanker,9m/shead regular wavesr=o.heaver=1, pitch00-100.220.01A-880.250.13LMaximummagnitudesaroundL/α=1Variationalongshipsreflectstherelevantmodeshape9
9 17 Generalised wave excitation • in Ξs first two terms (dependent on sectional added mass and fluid damping) denote the diffraction contribution – usually small compared to last term • in Ξs last term due to incident wave pressure, including Smith effect – usually dominates wave excitation • Dependence on forward speed, wave frequency and heading • Linear dependence on regular wave amplitude • depends on mass distribution and hull flexibility, for distortion modes • Important to understand how the integrand in Ξs varies with modal index and wave frequency 18 Generalised wave excitation • Variation of integrand along loaded tanker; 9m/s head regular waves r=0, heave r=1, pitch • Maximum magnitudes around L/λ=1 • Variation along ships reflects the relevant mode shape
FONVGeneralised Equations of motion Southampton.Generalised fluid actionsJ ws(x)H(x.1)dx+J ws(x)Z(x.1)dx=-Hs()+三s(0)Fs()= ws(x)F(x1)dx=-)000orinmatrixformF(t) = -A p(t) - Bp(t) - Cp(t)+三(t)andsinceap(t)+ b p(t) +c p(t) =F(t).then(a + A) p(t)+(b + B)p(t)+ (c + C) p(t) = 三(t)For 三(t)=三exp(-iet) i.e.regular wavethen p(t)=pexp(-ioet)whichcanbe solvedfrom[-wé(a+A)-iwe(b+B)+(c+C)]p=三19ERSITYOGeneralised Equations of motion Southampton[-(a+ A)-i0(b+ B)+(c+C)]p= =whereothe encounterfrequencyn total number of modes (rigid+ distortion)A: Generalised Added mass or inertia matrix (n x n) ; @e dependentB : Generalised Hydrodynamic Damping matrix (n x n); @dependentC : Generalised Restoring matrix (n x n)a : Generalised mass matrix (n x n)b:Generalisedstructural dampingmatrix (nxn)c:Generalisedstiffnessmatrix (nxn)三:Generalised waveexcitationvector (n xi);includes incident waveanddiffraction; oand o dependent20p: principal coordinate amplitude vector (n x 1)10
10 19 Generalised Equations of motion • Generalised fluid actions • or in matrix form • and since • then ( ) ( ) ( ) 0 0 0 F t w ( x )F( x,t )dx w ( x )H( x,t )dx w ( x )Z( x,t )dx H t t s s s L s L s L s F(t) A p(t) Bp(t) Cp(t) Ξ(t) a p(t) b p(t) cp(t) F(t) (a A)p(t) (b B) p(t) (c C)p(t) Ξ(t) [ ( ) ( ) ( )] which can be solved from then (t) exp( ω t) For (t) exp( ω t) i.e.regular wave e e a A b B c C p Ξ p p Ξ Ξ e 2 e ω iω -i -i 20 Generalised Equations of motion where e the encounter frequency n total number of modes (rigid+ distortion) A : Generalised Added mass or inertia matrix (n x n) ; e dependent B : Generalised Hydrodynamic Damping matrix (n x n); e dependent C : Generalised Restoring matrix (n x n) a : Generalised mass matrix (n x n) b : Generalised structural damping matrix (n x n) c : Generalised stiffness matrix (n x n) : Generalised wave excitation vector (n x1); includes incident wave and diffraction; e and dependent p : principal coordinate amplitude vector (n x 1) [ ω ( ) iω ( ) ( )] e 2 e a A b B c C p Ξ