Rolling Coin, Bicycles, Fish, Chaplygin Swimmer - Small Oscillations About Equilibrium

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 and their interesting properties and (2) the topic of small oscillations about an equilibrium point. Jump to small oscillations at .
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)..
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 info, visit Professor Andy Ruina's bicycle page: http://ruina.tam.cornell.edu/research/topics/bicycle_mechanics/overview.html.
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 just an analogy as a precise mechanical theory can be developed, and robots built based on the principles. See the eminent Professor Scott D. Kelly's work for more: https://mees.uncc.edu/directory/scott-david-kelly.
Rattlebacks, mysterious semi-ellipsoidal tops that have a preferred spin direction..
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..
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.
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, and the corresponding normal modes are given by the eigenvector..