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)