Publications

The common thread of the research work presented in this thesis is the implementation of analytical and numerical methods to describe the dynamic behavior of nonlinear mechanical systems, which have several characteristic time scales and may be stochastic. I have developed these activities within the framework of two research areas in vibration and acoustics.

The first area concerns the passive attenuation of vibrations using nonlinear absorbers known as NESs (Nonlinear Energy Sinks). In this context, I am particularly interested in the control of self-sustained oscillations. The latter generally occur in dynamical systems whose equilibrium solution becomes unstable in favor of an oscillating solution, usually periodic. As the amplitude of these oscillations can be very large, it is desirable to be able to attenuate them. NESs are also oscillators with the particularity of possessing an essentially nonlinear stiffness, which gives them the ability to resonate at any frequency. When an NES is coupled to a self-sustaining oscillator, the resulting dynamic system has a number of solutions, some of which correspond to low-amplitude oscillations. The aim is to understand how to favor these solutions over others where the NES is almost inoperative.

The second area concerns the study of transient phenomena in single-reed musical instruments such as clarinets and saxophones. This work is situated within the general context of the study of strategies followed by musicians during transient phases: note attacks, transitions between two notes, extinction transients, etc. In particular, it aims to correlate gestures (or time evolutions of control parameters) with sound results. In this context, my work focuses on the study of attack transients. From the point of view of dynamical systems, this leads me to study the influence of the time variation of bifurcation parameters on the emergence of a periodic solution, but also on the basins of attraction when multistability occurs.

Although far apart in terms of application, it turns out that the systems studied in these two research areas (the self-oscillating mechanical system coupled to a NES on the one hand, and the instrument during attack transients on the other) have a common feature. Both are modeled by systems of differential equations, which reveal a small parameter highlighting their fast-slow nature. The time evolution of state variables in such fast-slow systems is characterized by a succession of fast and slow epochs. This common nature also enables these systems to be studied within a common mathematical framework. Within the mechanics and acoustics community, the originality of my approach lies above all in the appropriation of theoretical results known to mathematicians on singularly perturbed ordinary differential equations with the aim of understanding the complex behaviors of the concrete mechanical systems under consideration.

For each of the above-mentioned areas of research, my main contributions to the state of the art are presented, together with the prospects I envisage for the short- and medium- terms.

The research presented in this thesis is a contribution to the study of attack transients in clarinet-like instruments. The main objective is to understand the behavior of the instrument when the mouth pressure of the musician is increased slowly through time at a constant rate. Although previous research proves that oscillations can appear at a value of the mouth pressure corresponding to 1/3 of the pressure needed to close the reed (the static oscillation threshold), numerical simulations and in vitro experiments show that for gradual increases of the mouth pressure, the audible sound generally appears when mouth pressure reaches a much higher value, called the dynamic oscillation threshold. This phenomenon is referred to as bifurcation delay in this work. A major part of this work follows an analytical approach, using the foundations of dynamic bifurcation theory to study the bifurcation delay in a simple and well known clarinet model (the "Raman" model). The properties of the dynamic oscillation threshold are related to indicators of the time variation of the mouth pressure such as its initial value and its slope. One of the remarkable features of the bifurcation delay is its strong dependence on noise, including that arising from round-off errors of the computer. The properties of the dynamic threshold are different according to whether the noise can be ignored or not. Additionally, an artificial mouth is used on a clarinet-like instrument to show that the bifurcation delay is not only a numerical phenomenon. Experimental observations performed on a clarinet-like instrument blown by an artificial mouth prove that bifurcation delay exists not only on numerical simulations, but also on real-life systems. These observations show that the properties of the bifurcation delay observed in low-precision simulations are similar to experimental ones.