TITLE:
The Lagrangian Method for a Basic Bicycle
AUTHORS:
F. Talamucci
KEYWORDS:
Nonholomic Constraints; Lagrangian Equations; Pseudovelocities; Nonlinear Order Differential Equations; Stability
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.2 No.4,
March
21,
2014
ABSTRACT:
The ground plan in order to disentangle the hard problem of modelling the
motion of a bicycle is to start from a very simple model and to outline the
proper mathematical scheme: for this reason the first step we perform lies in a
planar rigid body (simulating the bicylcle frame) pivoting on a horizontal
segment whose extremities, subjected to nonslip conditions, oversimplify the
wheels. Even in this former case, which is the topic of lots of papers in
literature, we find it worthwhile to pay close attention to the formulation of
the mathematical model and to focus on writing the proper equations of motion
and on the possible existence of conserved quantities. In addition to the first
case, being essentially an inverted pendulum on a skate, we discuss a second
model, where rude handlebars are added and two rigid bodies are joined. The
geometrical method of Appell is used to formulate the dynamics and to deal with
the nonholonomic constraints in a correct way. At the same time the equations
are explained in the context of the cardinal equations, whose use is habitual
for this kind of problems. The paper aims to a threefold purpose: to formulate
the mathematical scheme in the most suitable way (by means of the
pseudovelocities), to achieve results about stability, to examine the
legitimacy of certain assumptions and the compatibility of some conserved
quantities claimed in part of the literature.