23,821 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
Oscillation threshold of a clarinet model: a numerical continuation approach
This paper focuses on the oscillation threshold of single reed instruments.
Several characteristics such as blowing pressure at threshold, regime
selection, and playing frequency are known to change radically when taking into
account the reed dynamics and the flow induced by the reed motion. Previous
works have shown interesting tendencies, using analytical expressions with
simplified models. In the present study, a more elaborated physical model is
considered. The influence of several parameters, depending on the reed
properties, the design of the instrument or the control operated by the player,
are studied. Previous results on the influence of the reed resonance frequency
are confirmed. New results concerning the simultaneous influence of two model
parameters on oscillation threshold, regime selection and playing frequency are
presented and discussed. The authors use a numerical continuation approach.
Numerical continuation consists in following a given solution of a set of
equations when a parameter varies. Considering the instrument as a dynamical
system, the oscillation threshold problem is formulated as a path following of
Hopf bifurcations, generalizing the usual approach of the characteristic
equation, as used in previous works. The proposed numerical approach proves to
be useful for the study of musical instruments. It is complementary to
analytical analysis and direct time-domain or frequency-domain simulations
since it allows to derive information that is hardly reachable through
simulation, without the approximations needed for analytical approach
Harmonic analysis of oscillators through standard numerical continuation tools
In this paper, we describe a numerical continuation method that enables
harmonic analysis of nonlinear periodic oscillators. This method is formulated
as a boundary value problem that can be readily implemented by resorting to a
standard continuation package - without modification - such as AUTO, which we
used. Our technique works for any kind of oscillator, including electronic,
mechanical and biochemical systems. We provide two case studies. The first
study concerns itself with the autonomous electronic oscillator known as the
Colpitts oscillator, and the second one with a nonlinear damped oscillator, a
non-autonomous mechanical oscillator. As shown in the case studies, the
proposed technique can aid both the analysis and the design of the oscillators,
by following curves for which a certain constraint, related to harmonic
analysis, is fulfilled.Comment: 20 pages, 4 figure
Systems control theory applied to natural and synthetic musical sounds
Systems control theory is a far developped field which helps to study stability, estimation and control of dynamical systems. The physical behaviour of musical instruments, once described by dynamical systems, can then be controlled and numerically simulated for many purposes.
The aim of this paper is twofold: first, to provide the theoretical background on linear system theory, both in continuous and discrete time, mainly in the case of a finite number of degrees of freedom ; second, to give illustrative examples on wind instruments, such as the vocal tract represented as a waveguide, and a sliding flute
Proving Abstractions of Dynamical Systems through Numerical Simulations
A key question that arises in rigorous analysis of cyberphysical systems
under attack involves establishing whether or not the attacked system deviates
significantly from the ideal allowed behavior. This is the problem of deciding
whether or not the ideal system is an abstraction of the attacked system. A
quantitative variation of this question can capture how much the attacked
system deviates from the ideal. Thus, algorithms for deciding abstraction
relations can help measure the effect of attacks on cyberphysical systems and
to develop attack detection strategies. In this paper, we present a decision
procedure for proving that one nonlinear dynamical system is a quantitative
abstraction of another. Directly computing the reach sets of these nonlinear
systems are undecidable in general and reach set over-approximations do not
give a direct way for proving abstraction. Our procedure uses (possibly
inaccurate) numerical simulations and a model annotation to compute tight
approximations of the observable behaviors of the system and then uses these
approximations to decide on abstraction. We show that the procedure is sound
and that it is guaranteed to terminate under reasonable robustness assumptions
- âŚ