March 29, 2013 -
12:00 pm to 1:00 pm
Location: Cupples I, Room 6 |
Host: Prof. Renato Feres
A Senior Honors Thesis Presentation.
Abstract: The great variety of mechanical systems can be sorted into two broad classes. The first class is comprised of the so called holonomic mechanical systems; the second class consists of every other mechanical system, and they are called nonholonomic mechanical systems. The differing behaviors between the two systems have at their mathematical heart the condition of integrability and involutivity. Holonomic systems are represented by a configuration manifold and a tangent sub-bundle which is involutive. On the other hand the tangent sub-bundle of a nonholonomic system, naturally, fails this condition. One can introduce random motion into a mechanical system by constructing a diffusion process on its configuration manifold. This has been fairly extensively researched for holonomic systems. However, comparatively little work of the same variety has been done on nonholonomic systems. Introducing random motion into a mechanical system even so simple as the bi-planar bicycle produces some very interesting behavior. One might expect that, due to the simple and symmetric nature of the bi-planar bicycle, if one were to look only at the motion of the center of mass, one would observe standard Brownian motion on the plane. However, the diffusion actually demonstrated is not rotationally symmetric. In fact, the overall shape of the diffusion is directly dependent on two parameters, those being the distance between the two wheels, denoted as , and the radius of the wheels themselves, denoted r. More precisely, the shape of the diffusion is dependent on the ratio r/ and r. As the ratio r/ tends to infinity the diffusion reduces to pure Brownian motion on the plane, and as the ratio approaches infinity the diffusion reduces to 1 dimensional Brownian motion. Another interesting phenomenon which arose in the bi-planar bicycle after randomness was added was diffusion in the transversal direction. In the purely smooth case such motion would be impossible. This may turn out to be a general feature of nonholonomic mechanical systems.
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