34,601 research outputs found
Nonlinear modes of clarinet-like musical instruments
The concept of nonlinear modes is applied in order to analyze the behavior of
a model of woodwind reed instruments. Using a modal expansion of the impedance
of the instrument, and by projecting the equation for the acoustic pressure on
the normal modes of the air column, a system of second order ordinary
differential equations is obtained. The equations are coupled through the
nonlinear relation describing the volume flow of air through the reed channel
in response to the pressure difference across the reed. The system is treated
using an amplitude-phase formulation for nonlinear modes, where the frequency
and damping functions, as well as the invariant manifolds in the phase space,
are unknowns to be determined. The formulation gives, without explicit
integration of the underlying ordinary differential equation, access to the
transient, the limit cycle, its period and stability. The process is
illustrated for a model reduced to three normal modes of the air column
Converting between quadrilateral and standard solution sets in normal surface theory
The enumeration of normal surfaces is a crucial but very slow operation in
algorithmic 3-manifold topology. At the heart of this operation is a polytope
vertex enumeration in a high-dimensional space (standard coordinates).
Tollefson's Q-theory speeds up this operation by using a much smaller space
(quadrilateral coordinates), at the cost of a reduced solution set that might
not always be sufficient for our needs. In this paper we present algorithms for
converting between solution sets in quadrilateral and standard coordinates. As
a consequence we obtain a new algorithm for enumerating all standard vertex
normal surfaces, yielding both the speed of quadrilateral coordinates and the
wider applicability of standard coordinates. Experimentation with the software
package Regina shows this new algorithm to be extremely fast in practice,
improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the
journal styl
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