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Unformatted text preview: Dynamics and Stability application to submerged bodies and vortexbody systems Eva Kanso University of Southern California CDS 140B – Introduction to Dynamics February 5 and 7, 2008 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 BodyVortex System, to appear in J. Fluid Mech. 11. Kanso E., J.E. Marsden, C.W. Rowley and J. MelliHuber [2005], Locomotion of articulated bodies in a perfect fluid, Int. J. Nonlin. Science , 15 , 255–289. 2 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 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 = , ∇ × ψ  ∞ = 0 4 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 = , ∇ × ψ  ∞ = 0 5 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 ◦ 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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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
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