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