13.42 LECTURE 17:FORWARD SPEED EFFECTS AND MODEL TESTINGSPRING 2003cA. H. TECHET & M.S. TRIANTAFYLLOU1. Model Testing of Seakeepingzxη(xm,t)bowUsTowing tank experiments can be done to determine the characteristics of a model vessel in waves.(1) Place a fixed model at position along the tank.(2) Generate a regular incident wave train with frequency, ω, and amplitude, a.(3) Measure the surface elevation at xmfar from the model.(4) Measure the heave force F3(t) and pitch moment F5(t), wave amplitude at the model(5) Compare the surface elevation at xmwith the force signal to determine the phase offset between thetwo signals, ψ = ωt1.13.42 Spring 2003.12 SPRING 2003cA. H. TECHET & M.S. TRIANTAFYLLOU(6) Take x=0 at the midship point. So that the surface elevation at the ship is:eη(0, t) = a eiωtAnd the elevation at xmiseη(xm, t) = a ei(ωt−kxm)= eη(0, t) e−ikxmIf xm> 0 then the surface elevation at xmlags the surface elevation at x = 0, by a phase ofψ1= kxm; else if xm< 0 then the surface elevation at x = 0 lags the surface elevation at xm, bya phase of ψ1= kxm.(7) The phase lag between the forcing signal and the amplitude of the waves at the body can be foundsimply by combining the phase shift between the elevation at xmand the force with the expectedphase shift in surface elevations due to location difference between x = xmand x = 0.ψ0= ψ + ψ1= ω t1− k xmF3(t) = RenbF3ei(ω t−ψ0)obFI3+bFD3=bF3/a eiψ0Where the LHS of this equation is the excitation force.In order to find the added mass and damping coefficients we can force the body in calm water and measurethe forces:FORWARD SPEED EFFECTS & MODEL TESTING 3x(t) = bx cos ωt = Rebx eiωtF3(t) = RenbF3ei(ωt−ψ)bF3e−iψ= bx(m + A33) ω2− iωB33− C33bxSeparating the real and imaginary part of the above equations:bF3cos ψ = bx (m + A33) ω2− C33bxbF3sin ψ = bx B33ω(1.1)we can use this data to get the added mass and damping coefficients.4 SPRING 2003cA. H. TECHET & M.S. TRIANTAFYLLOU2. Speed Effectsβω, aλzxTake a ship moving with speed U cos β against waves with wavelength, amplitude and frequency, λ, a, ω.These waves have a phase speed (the speed at which each crest moves)Cp= ω/k,and the ship has a forward speed relative to the phase speed:Urel= U cos β + Cp.Since the ship is moving, an observer on this ship’s bow would observe waves coming in at a frequency,ωe, the encounter frequency, leading to an apparent wave phase speed, Cpe:ωek= U cos β + Cpωe= U k cos β + ωUsing the deep water dispersion relationship: ω|ω| = kg we can rewrite the encounter frequency as afunction of the incident wave train frequency and forward ship speed:ωe= ω +ωgU cos βFORWARD SPEED EFFECTS & MODEL TESTING 5Typically an observer on the ship will report back with his or her observed frequency, ωe, and wish totranslate that information to arrive back at the wave frequencyω =g2U cos β(−1 ±s1 + 4ωeU cos βg),provided4ωeU cos βg> −1.In a head seas configuration, the incident wave angle, β, is between ±π/2 so that cos β > 0, and ωe> 0always for all ω. In following (stern) seas the incident wave angle, β, is between3π2andπ2so that cos β < 0.When |U cos β| > Cpthe ship is overtaking the waves and ωe< 0, and when |U cos β| = Cpthen theencounter frequency is zero.It is possible to meet some ambiguity when observing incident wave trains. For example if the magnitudeof the encounter frequency and angle are measured, it may be possible for the sign of ωeto be uncertain.This could result from a misdiagnosis of the incident wave angle, β versus β + π.The idea of encounter frequency ωehas implications on the sea spectrum analysis of ship motion in waves.Since the spectrum is representative of the available energy in the waves we can write a relationship betweenthe spectrum on the waves with frequency, ω, and the complementary spectrum with encounter frequency,ωe, such that energy is preserved:S(ωe) |dωe| + S(ω) |dω|S(ωe) =S(ω)|dωe/dω|For deep water,dωedω= 1 +2ωgU cos βSuch that the spectrum in terms of the encounter frequency is6 SPRING 2003cA. H. TECHET & M.S. TRIANTAFYLLOUS(ωe) =S(ω)|1 +2ωgU cos β|From the equation above we can see that S(ωe) goes to infinity as the denominator goes to zero at2ωgU cos β = −1Given the forward velocity, U , and the incident wave angle, β, againωe= ω +ω|ω|gU cos β• If β is in−π2,π2then ωe> 0 for ω > 0 and ωe< 0 for ω < 0.• If β is inπ2,3π2then ωe< 0 for ω > 0 and ω >gU1cos β.FORWARD SPEED EFFECTS & MODEL TESTING 78 SPRING 2003cA. H. TECHET & M.S. TRIANTAFYLLOU3. Speed Effects on ForcesWith respect to the ship’s reference frame for a ship in head seas:φ = −U x + φs(x, y, z) + φT(x, y, z, t),where the first term is solely the potential due to the forward motion of the ship (or in the reference frameof the ship, the apparent speed of the oncoming water), the second term is the contribution from steadydiffraction and the final term is unsteady term.The body boundary condition for a vessel in both heave and pitch can be written in familiar terms:∇φ · bn =~VB· bnFORWARD SPEED EFFECTS & MODEL TESTING 9where bn is the unit vector normal to the body surface and is only used to get the correct velocity direction.For small angular motions, ~α, we can write the normal vector asbn = bn0+ ~α × bn0∇φ = −Ubi + ∇φs+ ∇φT~VB· bn = ∇φ · (bn0+ ~α × bn0)−Ubi + ∇φs· bn0= 0In the perpendicular direction to the surface normal bn, ∇φsis negligible and we arrive at the condition:∇φT· bn0− Ubi · (~α × bn0) =~VB· bn0In heave and pitching motionsbi · (~ω × bn0) = x5nz− x6nyVertical velocity is ˙x3+ Uθ and we can get∂φT∂n=~VB· bn0+ U {x5nz− x6ny}10 SPRING 2003cA. H. TECHET & M.S. TRIANTAFYLLOUφ5= φ05+Uiωφ03~F = −Z ZSBp bn dS 'Z ZSBρ∂φ∂tT − U∂φT∂xbn0dS +~Fh(From Salvensen, Tuck, Faltinsen Trans. SNAME 1970, Vol. 78.)3.1. Heave, U = 0:(m + A33) ˙x3+ B33¨x3+ C33x3= F3(t)FORWARD SPEED EFFECTS & MODEL TESTING 113.2. Heave, U 6= 0: The ship is moving at velocity, U , such that the wave encounter frequency is ωeandthe ship is also heaving at ωe:x3(t) = bx3eiωetThe force is a combination of incident and diffractive forces and is a function of both ω and ωe:bF3=bF3I+bF3d=bF3(ω, ωe)The incident force component is primarily dependent on the basic frequency, ω, however the diffraction forceis