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Dr. Shane Ross, Virginia Tech. Lecture 26 of a course on analytical dynamics (Newton-Euler, Lagrangian dynamics, and 3D rigid body dynamics). We discuss (1) various examples of nonholonomic systems a…
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Rolling Coin, Bicycles, Fish, Chaplygin Swimmer - Small Oscillations About Equilibrium
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- 1 Dr. Shane Ross, Virginia Tech. Lecture 26 of a course on analytical dynamics (Newton-Euler, Lagrangian dynamics, and 3D rigid body dynamics). We discuss (1) various examples of nonholonomic systems a…
- 2 Rolling coin example. How fast does it have to go before it falls over? We find the critical speed for a U.S. quarter (about 0.2 m/s).
- 3 Bicycle stability. What makes a bicycle stable? It's hard to give a simple physical explanation, but one can find a minimum critical speed necessary for the bicycle to be self-stabilizing. For more …
- 4 Analogy between nonholonomic rolling systems and aquatic animals. It's as if the swimmers push off the water, just as a roller skater pushes off the ground at the point of contact. This is more than…
- 5 Rattlebacks, mysterious semi-ellipsoidal tops that have a preferred spin direction.
- 6 Small oscillations about equilibrium in an N degree of freedom system. Normal modes are the modes of purely oscillatory motion. It is naturally formulated in a Lagrangian setting.
- 7 We work out the details in 1 degree of freedom, based on a Taylor series expansion of the 1-dimensional potential energy function about the equilibrium point
- 8 N degrees of freedom, Taylor series expansion of the N-dimensional potential energy function about the equilibrium point. Solving for the natural frequencies is analogous to an eigenvalue program, a…
