Eva_Lect2Wk5 - Dynamics and Stability application to...

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Dynamics and Stability application to submerged bodies and vortex-body systems Eva Kanso University of Southern California CDS 140B – Introduction to Dynamics February 5 and 7, 2008
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References 1. Marsden and Ratiu, Introduction to Mechanics and Symmetry 2. Marsden, Lectures on Mechanics 3. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos 4. Arnold, Mathematical Methods of Classical Mechanics 5. Lamb, Hydrodynamics 6. Saffman, Vortex Dynamics 7. Newton, The N vortex problem 8. Lenoard, N. E. [1997], Stability of a Bottom Heavy Vehicle 9. Shashikanth, B.N., J.E. Marsden, J.W. Burdick, and S.D. Kelly [2002], The Hamil- tonian structure of a 2D rigid circular cylinder interacting dynamically with N Point vortices, Phys. of Fluids , 14 :1214–1227. 10. Kanso, E., and B. Oskouei [2007], Stability of a Coupled Body-Vortex System, to appear in J. Fluid Mech. 11. Kanso E., J.E. Marsden, C.W. Rowley and J. Melli-Huber [2005], Locomotion of articulated bodies in a perfect fluid, Int. J. Nonlin. Science , 15 , 255–289. 2
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Outline Tuesday Basic Concepts in Newtonian, Lagrangian and Hamiltonian Mechanics Equilibria and Stability Stability of Kirchhoff’s equation for a Rigid Body in Potential Flow Today Locomotion of an Articulated Body in Potential Flow Interaction of a Solid Body with Point Vortices 3
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Motion in inviscid, incompressible fluid Inviscid fluid: u ∂t + u · ∇ u = - 1 ρ p. Incompressibility: div u = 0 Fluid velocity: u = φ + ∇ × ψ Potential function: Δ φ = 0, B . C . φ · n | B = normal velocity of body , φ | = 0 Stream potential vector: Δ ψ = - ω , ω = ∇ × u B . C . ∇ × ψ · n | B = 0 , ∇ × ψ | = 0 4
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Motion in inviscid, incompressible fluid Inviscid fluid: u ∂t + u · ∇ u = - 1 ρ p. Incompressibility: div u = 0 Fluid velocity: u = φ + ∇ × ψ Potential function: Δ φ = 0, B . C . φ · n | B = normal velocity of body , φ | = 0 Stream potential vector: Δ ψ = - ω , ω = ∇ × u B . C . ∇ × ψ · n | B = 0 , ∇ × ψ | = 0 5
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Motion in potential flow The solids and fluid are studied as one dynamical system in terms of the solid variables only. The fluid effect is encoded in the added masses. B 3 B 1 B 2 β 1 β 2 β 3 ( x 1 ,y 1 ) ( x 2 ,y 2 ) ( x 3 ,y 3 ) e 1 e 2 ( β i , x i , y i ) : orientation and position of body B i ( Ω i , V i ) : angular and translational velocities of B i . Here, Ω i = Ω i b 3 , V i = V x i b i 1 + V y i b i 2 Main idea: write, following Kirchhoff, φ = 3 i =1 ϕ i x V x i + ϕ i y V y i + ϕ i β Ω i 6
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Incompressible, irrotational fluid: ∇ · u = 0 and u = φ = Δ φ = 0 Boundary conditions: φ · n | B = normal velocity of body φ | = 0 Following Kirchhoff, φ = 3 i =1 ϕ i x v x i + ϕ i y v y i + ϕ i β Ω i Δ ϕ i x = 0 B . C .
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