2005reading6 - 2.016 Hydrodynamics Reading #6 2.016...

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2.016 Hydrodynamics Reading #6 2.016 Hydrodynamics Prof. A.H. Techet Added Mass For the case of unsteady motion of bodies underwater or unsteady flow around objects, we must consider the additional effect (force) resulting from the fluid acting on the structure when formulating the system equation of motion. This added effect is added mass. Most floating structures can be modeled, for small motions and linear behavior, by a system equation with the basic form similar to a typical mass-spring-dashpot system described by the following equation: () (6.1) mx ±± + bx ± + kx = f t where m is the system mass, b is the linear damping coefficient, k is the spring coefficient, f(t) is the force acting on the mass, and x is the displacement of the mass. The natural frequency ω of the system is simply k . (6.2) = m In a physical sense, this added mass is the weight added to a system due to the fact that an accelerating or decelerating body (ie. unsteady motion: dU dt 0 ) must move some volume of surrounding fluid with it as it moves. The added mass force opposes the motion and can be factored into the system equation as follows: m x (6.3) mx + bx ± + kx = f t a where m a is the added mass. Reordering the terms the system equation becomes: ( mm ) + b x ± + k = f t + x (6.4) a From here we can treat this again as a simple spring-mass-dashpot system with a new mass m m =+ m such that the natural frequency of the system is now a k = k = (6.5) m m + m a It is important in ocean engineering to consider floating vessels or platforms motions in more than one direction. Added mass forces can arise in one direction due to motion in a different direction, and thus we can end up with a 6 x 6 matrix of added mass coefficients. version 3.0 updated 8/30/2005 -1- © 2005 A. Techet
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2.016 Hydrodynamics Reading #6 Looking simply at a body in two-dimensions we can have linear motion in two directions and rotational motion in one direction. (Think of these coordinates as if you were looking down on a ship.) Two dimensional motion with axis (x,y) fixed on the body. 1: Surge, 2: Sway, 6: Yaw The unsteady forces on the body in the three directions are: −= m 1 1 du 1 + m du 2 + m du 6 (6.6) F 1 2 dt 1 6 dt dt F 2 = m 2 du 1 + m du 2 + m du 6 (6.7) dt 2 dt 2 dt F 6 = m 6 du 1 + m du 2 + m du 6 (6.8) dt 6 dt 6 dt Where F 1 , F 2 , and F 6 , are the surge (x-) force, sway (y-) force and yaw moments respectively. It is common practice in Ocean Engineering and Naval Architecture to write the moments for roll, pitch, and yaw as F 4 , F 5 , and F 6 and the angular motions in these directions as X 4 , X 5 , and X 6. This set of equations, (6.6)-(6.8), can be written in matrix form, F = [] ± , M u du 1 dt m 11 m 12 m 16 ⎤⎜ F = m 21 m 22 m 26 du 2 (6.9) dt m 61 m 62 m 66 ⎥⎜ du 6 dt version 3.0 updated 8/30/2005 -2- © 2005 A. Techet
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2.016 Hydrodynamics Reading #6 Considering all six degrees of freedom the Force Matrix is: ± u 1 m 11 m 12 m 13 m 14 m 15 m 16 ⎤⎛ ⎜⎟ ± u 2 m m 21 m 22 m 23 m 24 m 25 m 26 36 ± m 31 m 32 m 33 m 34 m 35 u 3 F = (6.10) ± u 4 m
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.016 taught by Professor Alexandratechet during the Fall '05 term at MIT.

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2005reading6 - 2.016 Hydrodynamics Reading #6 2.016...

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