13.42 Lecture Notes Spring 2004 5/3/04 Anti Rolling Tanks Anti-rolling tanks are commonly used to reduce ship rolling motions. In order to maximize the effectiveness of the anti rolling tank we must determine the optimal dimensions of the tank. We will consider a U-shaped tank with width, d, and length, L, where h(t) represents the fluctuation in height of the water column about its equilibrium position as a function of time. The motion of the water column and the motion of the vessel are coupled and can be readily determined using basic equation of motion for a dynamical system. First we can start with the simple case of a u-shaped water column, essentially in the absence of the vessel. The equation of motion for a simple u-shaped water column is () () () 02wwwmht bht cht++= , (1) h(t) L x4(t) zxwhere h(t) is the elevation differential of the water column, mw is the mass of the water per unit length into the page (wmdLρ=), and the restoring coefficient is simply wcgdρ= . Figure 2. Simple u-shaped water column. The natural frequency of the water motion in this column is 2/2wwwcgmLω== (2) When this u-channel is fitted into a ship with roll motion x4(t) a coupled set of free differential equations governing x4(t) and h(t) can be found. Ship motion: ()44 44 4 44 4 44 4 4(/2)Iaxbxcx gdB hfρ+++=− + (3) Water column motion: 4() () ()2wwwmh t b h t c h t gdB xρ++=− (4) The extra terms appearing on the RHS of equations (3) and (4) represent the cross coupling restoring terms. In equation (3), the cross coupling restoring coefficient comes from the motion of the water column. As the ship rolls, the water column motion yields a moment that acts opposite to the rolling direction. The magnitude of the motion can be through of as ()1/2wMmgB=− ⋅ where B/2 is the moment arm (half the beam of the vessel). The term f4 is simply the external moment acting on the vessel about the 1-axis. h(t) x4(t) zxThe RHS of equation (4) is determined similarly to that in equation (3). The liquid in the anti-rolling tank is forced by the ship motion and the resulting excitation moment is contrary to the motion of the vessel. As the ship rolls, the motion of the water on both sides of the tank contribute to the excitation moment ()()442/2sin2/2wMgd B x gd B xρρ=− ≈− (5) (recall, for small motions, 44sinxx≈). The coupled equations of motion can be written in matrix form: 44 444444 4444 1212100210002wwwIabxxxfccbhh hmcc+ ++ = (6) where 12c gdBρ= . In order to determine the optimal width of the anti-rolling tank for maximum roll reduction at resonance, we consider the following form of the motions and force: {}44ˆ() Reitxtxeω= (7) {}ˆ() Reitht heω= (8) {}44ˆ() Reitftfeω= (9) Considering the simple ship, in the absence of the roll tank, the resonant (natural) frequency (squared) is simply 24444 44cIaω=+. (10) Using equations (7) to (9), we can rewrite (6) in terms of the amplitudes of the motions and force:244 44 44 44 12442121()ˆˆ2ˆ102wwwIa cib cxfhcmcibωωωω−+ ++=−++ (11) Next, using equations (10) and (2), equation (11) becomes 44c−44c+44 124422121ˆˆ2ˆ10()2wwwib cxfhcmibωωω ω+=−−+. (12) Let ()222wwmωωλ−= be the “tuning” factor that can help us determine the optimal u-tank design. Now equation (12) becomes 444 124121ˆˆ2ˆ0wxib cfhcibωλω=+. (13) We can eliminate ˆh from the equation for 4ˆx 412ˆˆwxchibλω=−+ (14) to get the relationship between 4ˆxand 4ˆf: 1122444 4ˆˆwcxib fibωλω−=+ (15) 21144 12 44 44 122244 4()ˆˆˆwwwwib ib c ib ib b cxxfib ibωλ ω ωλ ωλω λω +− − −== ++ (16)()4422144 44 122422144 44 122ˆˆˆwwwwibxfib b b cbifbibb cλωωλ ωωλωλ ω+=−−−−+ + (17) It can be shown that 4ˆxhas a minimum magnitude at 0λ=. For the case 0λ= the amplitude of motion for the liquid in the u-tube is (from eq. (14)) 412 412ˆˆˆwwxcxchiib bωω=− = (18) and the motion of the vessel is ()4422144 122ˆˆwwbfxibb cωω=+. (19) The cross coupling restoring term c12 can decrease the roll motion. The anti-rolling tank acts like an additional damper at the resonance condition – note the phase of the water column relative to the ship roll motion. Since we consider only the real parts of the motion, ˆ() sinht h tω=and 44ˆ() cosxtx tω= are clearly 90º out of phase. The optimal tank length is determined by setting the natural frequency of the vessel, eq. (10), to the natural frequency of the water column, eq. (2): 224444 442/2wwwccgIa m Lωω====+ (20) 44 44442IaLgc+∴=
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