MIT2_019S11_SK2 - 2.019 Design of Ocean Systems Lecture 6...

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2.019 Design of Ocean Systems Lecture 6 Seakeeping (II) February 21, 2011
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Wave Radiation Problem ζ 3 ( t )= ¯ ζ 3 cos( ω t ) ˙ ζ 3 ( t ω ¯ ζ 3 sin( ω t ) ¨ ζ 3 ( t ω 2 ¯ ζ 3 cos( ω t ) x z ω , λ ,V p g 2 a ~n Total: P ( t ρ ∂φ ρ gz Hydrodynamic: P d ( t ρ = P ¯ d cos( ω t ψ ) t t Hydrodynamic Force : ZZ ¯ F 3 ( t S B P d n z d S = F 3 cos( ω t ψ ) ¯ ¯ = F 3 cos ψ cos( ω t )+ F 3 sin ψ sin( ω t ) ¯ ¯ F 3 cos ψ F 3 sin ψ = ζ 3 ω 2 ζ ¨ 3 ( t ) ζ 3 ω ζ ˙ 3 ( t ) ¯ ¯ = A 33 ζ ¨ 3 ( t ) B 33 ζ ˙ 3 ( t ) A 33 : Added mass; B 33 : Wave damping
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Physical Meaning of Wave Damping ζ 3 ( t )= ¯ ω , λ ,V p g z ζ ¨ 3 ( t ω 2 ζ ¯ 3 cos( ω t ) x 2 a Energy flux out EV g Control Volume Energy flux out Averaged power into the fluid by the body: ζ 3 cos( ω t ) ˙ ζ 3 ( t ω ¯ ζ 3 sin( ω t ) g Z T E ¯ in = 1 { F 3 ( t ) } ζ ˙ 3 ( t )d t T 0 Z T n o = 1 A 33 ζ ¨ 3 ( t ) ζ ˙ 3 ( t )+ B 33 ζ ˙ 3 ( t ) ζ ˙ 3 ( t ) d t = B 33 ( ζ ¯ 3 ω ) 2 / 2 T 0 Averaged energy flux out of the control volume: E ¯ flux =2 V g E 2 V g a 2 d E ¯ Conservation of energy: E ¯ in E ¯ =0 d t B 33 ( a/ ζ ¯ 3 ) 2 > 0 B 33 =0 if a =0 corresponding to ω = , 0
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Mathematical Formulation of Heave Radiation Problem z ζ 3 ( t )= cos ( ω t ) y Φ tt + g Φ z =0 η ( t Φ t /g x ~n Radiation condition: Generated waves must propagate away from the body Hydrodynamic Pressure: Radiation Force: Radiation Moment: Deep water condition: 2 Φ ( x, y, z, t )=0 Φ z →−∞ 0a s P d ( x, t ρ Φ t R F ~ R ( t P d d s S B R M ~ R ( t S B P d ( ~x × )d s
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Frequency-Domain Formulation of Heave Radiation Problem ¯ ζ 3 =1 y ω 2 φ 3 + g φ 3 z =0 η ¯ = i ωφ 3 /g x Radiation condition ~n Deep water condition: Let: ζ 3 ( t )=cos ω t = <{ e i ω t } Φ ( ~x, t )= <{ φ 3 ( ~x )e i ω t } η ( x, y, t <{ η ¯( x, y )e i ω t } P d ( <{ p d ( )e i ω t } F ~ R ( f ~ e i ω t M ~ R ( t <{ ~ i ω } t m e t <{ } p d = i ρωφ 3 ( ) R f ~ =
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MIT2_019S11_SK2 - 2.019 Design of Ocean Systems Lecture 6...

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