6,754 research outputs found
Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: pro or contra?
The phenomenon of a topological monodromy in integrable Hamiltonian and
nonholonomic systems is discussed. An efficient method for computing and
visualizing the monodromy is developed. The comparative analysis of the
topological monodromy is given for the rolling ellipsoid of revolution problem
in two cases, namely, on a smooth and on a rough plane. The first of these
systems is Hamiltonian, the second is nonholonomic. We show that, from the
viewpoint of monodromy, there is no difference between the two systems, and
thus disprove the conjecture by Cushman and Duistermaat stating that the
topological monodromy gives a topological obstruction for Hamiltonization of
the rolling ellipsoid of revolution on a rough plane.Comment: 31 pages, 11 figure
Global Formulation and Control of a Class of Nonholonomic Systems
This thesis study motion of a class of non-holonomic systems using geometric mechanics, that provide us an efficient way to formulate and analyze the dynamics and their temporal evolution on the configuration manifold. The kinematics equations of the system, viewed as a rigid body, are constrained by the requirement that the system maintain contact with the surface. They describe the constrained translation of the point of contact on the surface. In this thesis, we have considered three different examples with nonholonomic constraint i-e knife edge or pizza cutter, a circular disk rolling without slipping, and rolling sphere. For each example, the kinematics equations of the system are defined without the use of local coordinates, such that the model is globally defined on the manifold without singularities or ambiguities. Simulation results are included that show effectiveness of the proposed control laws
The energy–momentum method for the stability of non-holonomic systems
In this paper we analyze the stability of relative equilibria of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit
both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for
holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the the rolling disk, the roller racer, and the rattleback top
Poisson structures for reduced non-holonomic systems
Borisov, Mamaev and Kilin have recently found certain Poisson structures with
respect to which the reduced and rescaled systems of certain non-holonomic
problems, involving rolling bodies without slipping, become Hamiltonian, the
Hamiltonian function being the reduced energy. We study further the algebraic
origin of these Poisson structures, showing that they are of rank two and
therefore the mentioned rescaling is not necessary. We show that they are
determined, up to a non-vanishing factor function, by the existence of a system
of first-order differential equations providing two integrals of motion. We
generalize the form of that Poisson structures and extend their domain of
definition. We apply the theory to the rolling disk, the Routh's sphere, the
ball rolling on a surface of revolution, and its special case of a ball rolling
inside a cylinder.Comment: 22 page
Rocking and rolling: a can that appears to rock might actually roll
A beer bottle or soda can on a table, when slightly tipped and released,
falls to an upright position and then rocks up to a somewhat opposite tilt.
Superficially this rocking motion involves a collision when the flat circular
base of the container slaps the table before rocking up to the opposite tilt. A
keen eye notices that the after-slap rising tilt is not generally just
diametrically opposite the initial tilt but is veered to one side or the other.
Cushman and Duistermaat (2006) recently noticed such veering when a flat disk
with rolling boundary conditions is dropped nearly flat. Here, we generalize
these rolling disk results to arbitrary axi-symmetric bodies and to
frictionless sliding. More specifically, we study motions that almost but do
not quite involve a face-down collision of the round container's bottom with
the table-top. These motions involve a sudden rapid motion of the contact point
around the circular base. Surprisingly, like for the rolling disk, the net
angle of motion of this contact point is nearly independent of initial
conditions. This angle of turn depends simply on the geometry and mass
distribution but not on the moment of inertia about the symmetry axis. We
derive simple asymptotic formulas for this "angle of turn" of the contact point
and check the result with numerics and with simple experiments. For tall
containers (height much bigger than radius) the angle of turn is just over
and the sudden rolling motion superficially appears as a nearly symmetric
collision leading to leaning on an almost diametrically opposite point on the
bottom rim.Comment: 10 pages, 5 figure
Nonholonomic Dynamics
Nonholonomic systems are, roughly speaking, mechanical
systems with constraints on their velocity
that are not derivable from position constraints.
They arise, for instance, in mechanical systems
that have rolling contact (for example, the rolling
of wheels without slipping) or certain kinds of sliding
contact (such as the sliding of skates). They are
a remarkable generalization of classical Lagrangian
and Hamiltonian systems in which one allows position
constraints only.
There are some fascinating differences between
nonholonomic systems and classical Hamiltonian
or Lagrangian systems. Among other things: nonholonomic
systems are nonvariational—they arise
from the Lagrange-d’Alembert principle and not
from Hamilton’s principle; while energy is preserved
for nonholonomic systems, momentum is
not always preserved for systems with symmetry
(i.e., there is nontrivial dynamics associated with
the nonholonomic generalization of Noether’s
theorem); nonholonomic systems are almost Poisson
but not Poisson (i.e., there is a bracket that together
with the energy on the phase space defines
the motion, but the bracket generally does not satisfy
the Jacobi identity); and finally, unlike the
Hamiltonian setting, volume may not be preserved
in the phase space, leading to interesting asymptotic
stability in some cases, despite energy conservation.
The purpose of this article is to engage
the reader’s interest by highlighting some of these
differences along with some current research in the
area. There has been some confusion in the literature
for quite some time over issues such as the
variational character of nonholonomic systems, so
it is appropriate that we begin with a brief review
of the history of the subject
Dynamics of a Rolling Disk in the Presence of Dry Friction
In this paper we are interested in the dynamics and numerical treatment of a rolling disk on a flat support. The objective of the paper is to develop a numerical model which is able to simulate the dynamics of a rolling disk taking into account various kinds a friction models (resistance against sliding, pivoting and rolling). A mechanical model of a rolling disk is presented in the framework of Non-smooth Dynamics and Convex Analysis. In an analytical study, approximations are derived for the energy decay of the system during the final stage of the motion for various kinds of frictional dissipation models. Finally, the numerical and analytical results are discussed and compared with experimental results available in literatur
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