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Unformatted text preview: Dynamics and Stability application to submerged bodies, vortex streets and vortexbody systems Eva Kanso University of Southern California CDS 140B – Introduction to Dynamics February 5 and 7, 2008 Fish swim by coupling of their shape changes with the surrounding fluid Flow field around a Carangiform fish based on PIV data (Muller et al. 1997) Trout swimming in an experimentally gen erated vortex street (Liao et al. 2003) Fish swimming in school (Stakiotakis et al. 1999) 2 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. 3 Outline Today ◦ Basic Concepts in Newtonian, Lagrangian and Hamiltonian Mechanics ◦ Equilibria and Stability ◦ Stability of Kirchhoff’s equation for a Rigid Body in Potential Flow Thursday ◦ Locomotion of an Articulated Body in Potential Flow ◦ Infinite Vortex Street ◦ Interaction of a Solid Body with Point Vortices 4 Basic Concepts 5 Newtonian, Lagrangian and Hamiltonian Mechanics Physical space Newtonian Mechanics r 1 r 2 r 3 Configuration space Lagrangian Mechanics q Phase space (q,p) Hamiltonian Mechanics 6 Newtonian Mechanics – single particle e1 e2 e3 r Balance of Linear Momentum. The motion of the particle is governed by Newton’s second law: force = mass × acceleration, F ( r , ˙ r ,t ) = ˙ p where the dot denotes the derivative with respect to time t and p = m ˙ r is the linear momentum of the particle. Balance of Angular Momentum. Define the angular momentum π of a particle and the moment M acting upon it as: π = r × p , and M = r × F , M = ˙ π Energy. The kinetic energy T is defined as T = 1 2 m ˙ r · ˙ r . For m constant, d T d t = F · ˙ r 7 Newtonian Mechanics – single particle Conservative Force. A force is said to be conservative if it depends only on the position r and is such that the work it does is independent of the path taken. For a closed path , the work done vanishes. I F · d r = 0 ⇐⇒ ∇ × F = It is a deep property of flat space 1 R 3 that ∇× F = implies we may write the force as F =∇ V ( r ) for some potential function V ( r ) (also referred to as potential energy )....
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 Fall '10
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 Angular Momentum, Kinetic Energy, Hamiltonian mechanics, Lagrangian mechanics, Noether's theorem

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