supp04 - 2.20 (13.021) Marine Hydrodynamics Supplemental...

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Unformatted text preview: 2.20 (13.021) Marine Hydrodynamics Supplemental Problems Mechanical Engineering 13.021 Supplemental Problems . . . . . . . . . . . . . . . . . . . . . A. ADDED MASS . . . . . . . . . . . . . . . . . . . . . Page 2 A. ADDED MASS Aa1. A sphere of volume in a fluid of density is located at a point (0, ,0) with respect to a certain coordinate system. In terms of this coordinate system , (a) identify whether each of the added mass coefficients are zero (0), or non-zero ( ) (do not work out any values). (b) If the sphere has generalized velocity of (0, ,0,0,0, ), the total kinetic energy of the surrounding fluid is . Aa2. A sphere is located at relative to a given coordinate system as shown. In a table for the added mass coefficients , , mark all the values: “ ” if it is positive, “ ” if it is negative, and “0” if it is zero. Aa3. A circular cylinder has radius =1cm and length =1m. Its added mass in water can be es- timated as = ; = ; and = . If the cylinder is translating through water with velocity =0.5m/s, assuming potential flow and ignoring the mass of the cylinder itself, the total amount of work required to bring the cylinder to rest is . Aa4. A certain body with added mass coefficients has constant velocities and all other , , . In terms of the added mass coefficients : (a) the forces and moments on the body (in the body coordinates) are = ; = ; = ; and = ; (b) the linear momentum of the sur- rounding fluid in the direction is ; and (c) the total kinetic energy in the fluid is . Aa5. A certain body has nonzero added mass coefficients only on the diagonal, i.e., = . For a body motion given by = and = , and all other , =0, the forces and mo- ments on the body in terms of are = , = , = , = , = , = . The total kinetic energy in the fluid at time =1 is . Aa6. A sphere of volume 1 m accelerates at = 2m/s while at the same time the surrounding fluid (density = 1 kg/m ) is accelerated at = 1 m/s . The horizontal force on the sphere is = . If remains the same, this force will vanish if = . Aa7. A 2D circular cylinder of radius moves in an unbounded fluid at . Assuming potential flow, the total amount of work done to bring this cylinder to rest is per unit width. Aa8. A two-dimensional square box of width 2 accelerates to the right at an acceleration of 1 while at the same time the surrounding fluid also accelerates to the right at 1 (both with respect to a fixed coordinate system). The total horizontal force on the square is N/m to the [right] [left]. Aa9. A sphere of radius and density is released in a current of velocity where . (a) In the absence of gravity, calculate the initial ( ) horizontal acceleration of the sphere. (b) If the current is absent but there is gravity, cal- culate the initial vertical acceleration of the sphere. (c) If both the current and gravity are present, calculate the angle (relative to the vertical) the sphere will tend to move initially....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.

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supp04 - 2.20 (13.021) Marine Hydrodynamics Supplemental...

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