325_add_Added Masses of Ship Structures.pdf - 316 7 Elastic One-Dimensional Oscillations of an Elongated Body in Fluid surface of the cylinder depend on

325_add_Added Masses of Ship Structures.pdf - 316 7 Elastic...

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316 7 Elastic One-Dimensional Oscillations of an Elongated Body in Fluid surface of the cylinder depend on x and θ as follows: v(r = a,θ,x) = v 0 cos kx cos θ. (7.4) Solutions of the Laplace equation for velocity potential 2 ϕ ∂x 2 + 2 ϕ ∂r 2 + 1 r ∂ϕ ∂r + 1 r 2 2 ϕ ∂θ 2 = 0 , taking into account ( 7.4 ), can be represented in the form ϕ(x,r,θ) = R(r) cos kx cos θ ; this leads to the following ODE for the function R : R + 1 r R k 2 + 1 R 2 R = 0 . (7.5) Solution of Eq. ( 7.5 ), vanishing for r → ∞ , is given by the Bessel function of sec- ond kind of first order R(r) = CK 1 (kr) where an arbitrary constant C is determined from the boundary condition ( 7.4 ): ∂ϕ ∂r r = a = v 0 cos ks cos θ = CkK 1 (ka) cos kx cos θ ; C = v 0 kK 1 (ka) . Therefore, the expression for velocity potential near vibrating cylinder looks as follows: ϕ(x,r,θ) = v 0 kK 1 (ka) K 1 (kr) cos kx cos θ. (7.6) Computing the kinetic energy of fluid using ( 7.6 ) we find T = − 1 2 ρ 2 π 0 ϕ ∂ϕ ∂r a dθ = − 1 2 ρ K 1 (ak) akK 1 (ak) a 2 πv 2 0 cos 2 kx.
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  • Fall '09
  • Kinetic Energy, Boundary value problem, Normal mode, reduction coefficients

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