Christophe Vergez

A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions

Combining the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time-domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called asymptotic numerical method (ANM) continuation technique. Several classical examples are presented to illustrate the use of this numerical approach.

A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities

Abstract In this paper, we extend the method proposed by Cochelin and Vergez [A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324 (2009) 243–262] to the case of non-polynomial nonlinearities. This extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical systems. The principle remains to transform the original ODE system into an extended polynomial quadratic system for an easy application of the harmonic balance method (HBM). The transformation of non-polynomial terms is based on the differentiation of state variables with respect to the time variable, shifting the nonlinear non-polynomial nonlinearity to a time-independent initial condition equation, not concerned with the HBM. The continuation of the resulting algebraic system is here performed by the asymptotic numerical method (high order Taylor series representation of the solution branch) using a further differentiation of the non-polynomial algebraic equation with respect to the path parameter. A one dof vibro-impact system is used to illustrate how an exponential nonlinearity is handled, showing that the method works at very high order, 1000 in that case. Various kinds of nonlinear functions are also treated, and finally the nonlinear free pendulum is addressed, showing that very accurate periodic solutions can be computed with the proposed method.

An Instrumented Saxophone Mouthpiece and its Use to Understand How an Experienced Musician Plays

An instrumented saxophone mouthpiece has been developed to measure, during the player’s performance, the evolution of important variables : the mouth pressure, the mouthpiece pressure and the force applied on the reed by the lower lip. Moreover, according to the pressure signals in the mouth and in the mouthpiece, the instantaneous ratio of the vocal tract input impedance and of the saxophone input impedance is estimated at frequencies multiple of the playing frequency (using the concept of Gabor mask). On the selected sound examples, analyses reveal many aspects of the strategies of the player. First of all, the role of the vocal tract in the characteristics of the sound production is sometimes prominent. Secondly, the sound production on the desired note (and register) as well as pitch correction seem to be the result of complementary adjustments of the mouth pressure and of the lip pressure on the reed. This is not in agreement with musicians feeling, since they often claim to let their force on the reed unchanged during the note and from note to note.

Quasistatic nonlinear characteristics of double-reed instruments

This article proposes a characterization of the double reed in quasistatic regimes. The nonlinear relation between the pressure drop, deltap, in the double reed and the volume flow crossing it, q, is measured for slow variations of these variables. The volume flow is determined from the pressure drop in a diaphragm replacing the instruments bore. Measurements are compared to other experimental results on reed instrument exciters and to physical models, revealing that clarinet, oboe, and bassoon quasistatic behavior relies on similar working principles. Differences in the experimental results are interpreted in terms of pressure recovery due to the conical diffuser role of the downstream part of double-reed mouthpieces (the staple).

Inversion of a physical model of a trumpet

We deal with the inversion of a physical model of a trumpet, i.e. how should the model be controlled in order to obtain a given sound. It is shown that this problem is ill-posed since an infinity of inputs can produce the same sound. Then, a criterion based on the slowness of the musicians gestures compared to the sound signal evolution is used: this leads to a physically pertinent solution. Finally, we present some simulation results.

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.

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.

Nonlinear Dynamics in Physical Models: Simple Feedback-Loop Systems and Properties

This work establishes a new approach to the functioning of musical instruments that is oriented toward sound synthesis by computer, and its application to musical creation and composition. In this work, we only consider musical instruments that produce sustained sound. Our approach, which relies on the theory of nonlinear dynamical systems, provides theoretical results regarding instruments, their models, a class of equations with delay, and sound synthesis itself. Experimental and practical results open new sonic possibilities in terms of sound material and the control of sound synthesis, both of which are important to performers and composers of contemporary music. Out of these results, new possibilities also arise for the control of chaotic sounds and the control of the proportion of nonperiodic components introduced in sound by chaotic behavior. Therefore, new artificial instruments can be designed that fulfill the fundamental properties of a musical instrument: richness of the sonic space, expressivity, flexibility, predictability, and ease of control of sonic results. A model of brass instruments simulated on a workstation and played in real time which exemplifies these remarkable properties is presented in a second article in this issue, entitled “Nonlinear Dynamics in Physical Models: From Basic Models to True Musical-Instrument Models.” The notion of a “model” is essential for a better comprehension and use of the properties of sound analysis and synthesis methods. Many physical models of musical instruments have been proposed and studied by various authors (for an overview, see Smith’s [1996] work, for instance). The approach described in this article is based on some advantages and difficulties specific to physical models. From a fundamental point of view, it appears that the complexity of physical models comes in part from their nonlinear nature. Therefore, their study should rely on the increasingly rich theory of nonlinear dynamical systems. However, developers should not merely build models and deliver them to musicians; rather, they should help musicians understand the models by conceiving abstractions of the models and offering useful explanations. Above all, it is necessary for the user to understand the structure of the space of instrumental sounds. In particular, this comprehension is indispensable for elaborating the control of synthesis models that are at the same time efficient and musically pertinent. To fulfill the requirements mentioned above, we have developed models as archetypes, i.e., models that retain the essence of the behavior of a class of instruments while disregarding all details that are not useful for understanding what is typical of that class. The existence of delayed-feedback loops in the equations of the models is a characteristic

Blowing machine for wind musical instrument : toward a real-time control of the blowing pressure

Blowing machines were developed since the early stage of research in musical acoustics because they are necessary for the study of musical instruments. The system under study is based on a previously developed blowing machine dedicated to the study of the physics of wind instruments (trumpet, reed instruments, recorder) for which the blowing pressure was tuned by hand through a manometer. In this paper, the blowing machine is connected to pressured air through a servo-valve driven from a computer within a closed loop according to a (possibly) time-varying target pressure. The system being non linear, a robust controller has been implemented (PID + gain scheduling + feedforward) with step response performances reaching up to 5 ms to 10 ms.

Response of an artificially blown clarinet to different blowing pressure profiles

Using an artificial mouth with an accurate pressure control, the onset of the pressure oscillations inside the mouthpiece of a simplified clarinet is studied experimentally. Two time profiles are used for the blowing pressure: in a first set of experiments the pressure is increased at constant rates, then decreased at the same rate. In a second set of experiments the pressure rises at a constant rate and is then kept constant for an arbitrary period of time. In both cases the experiments are repeated for different increase rates. Numerical simulations using a simplified clarinet model blown with a constantly increasing mouth pressure are compared to the oscillating pressure obtained inside the mouthpiece. Both show that the beginning of the oscillations appears at a higher pressure values than the theoretical static threshold pressure, a manifestation of bifurcation delay. Experiments performed using an interrupted increase in mouth pressure show that the beginning of the oscillation occurs close to the stop in the increase of the pressure. Experimental results also highlight that the speed of the onset transient of the sound is roughly the same, independently of the duration of the increase phase of the blowing pressure.