Eva_LectWk5

Eva_LectWk5 - Dynamics and Stability application to...

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Unformatted text preview: Dynamics and Stability application to submerged bodies, vortex streets and vortex-body 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 :12141227. 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 , 255289. 3 Outline Today Basic Concepts in Newtonian, Lagrangian and Hamiltonian Mechanics Equilibria and Stability Stability of Kirchhoffs 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 Newtons 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 of a particle and the moment M acting upon it as: = r p , and M = r F , M = 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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Eva_LectWk5 - Dynamics and Stability application to...

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