Regime change and oscillation thresholds in recorder-like instruments

Abstract

Based on results from the literature, a description of sound generation in a recorder is developed. Linear and non-linear analysis are performed to study the dependence of the frequency on the jet velocity. The linear analysis predicts that the frequency is a function of the jet velocity. The non-linear resolution provides information about limit cycle oscillation and hysteretic regime change thresholds. A comparison of the frequency between linear theory and experiments on a modified recorder shows good agreement except at very low jet velocities. Although the predicted threshold for the onset of the first regime shows an important deviation from experiments, the hysteresis of threshold to higher regimes is accurately estimated. Furthermore, a qualitative analysis of the influence of different parameters in the model on the sound generation and regime changes is presented.

Calculation of the Steady-State Oscillations of a Flute Model Using the Orthogonal Collocation Method

“…As underlined in the previous section and in [3], even the simplest model of flute-like instrument includes an additional delay compared to models of other wind instruments. As they do not deal with delayed differential equations, the different software mentioned above are not helpful for the present study.…”

“…In the same way, if the strong dependance of the frequency on the jet velocity U j , highlighted in figure 10, is a well-known behaviour of both models and real instruments (see for example [1,3,22,43,44,45]), a linear analysis of the model only gives a rough estimation of the frequency evolution, and does not distinguinsh between stable and unstable parts of the branch. As highlighted in figure 10, the bifurcation diagram not only predicts precisely the frequency evolution along the branch, but also the stabilisation of the frequency slightly above the resonance frequency (observed experimentally for example in [3]). Through the computation of stability properties of the branch of periodic solutions, it finally gives information about the minimum and maximum frequencies that can be observed for a given periodic regime.…”

“…Especially, it highlights that the methods used here predict precisely the saturation of the oscillation amplitude (figure 9), a commonly observed behaviour in experiments and simulations [1,3,5,43], which is not explained or predicted by the often-used linear analysis of the model [3]. In the same way, if the strong dependance of the frequency on the jet velocity U j , highlighted in figure 10, is a well-known behaviour of both models and real instruments (see for example [1,3,22,43,44,45]), a linear analysis of the model only gives a rough estimation of the frequency evolution, and does not distinguinsh between stable and unstable parts of the branch. As highlighted in figure 10, the bifurcation diagram not only predicts precisely the frequency evolution along the branch, but also the stabilisation of the frequency slightly above the resonance frequency (observed experimentally for example in [3]).…”

“…Such a formulation highlights the presence of a delayed derivative termż(t −τ ), and thus the neutral nature of the system, already underlined in [3,35].…”

“…Timedomain simulations (see for example [1,2,3]) allow the computation of both transients and steady-state periodic and non-periodic oscillations, and are thus informative about the diversity of dynamical regimes. Nevertheless, these methods are very sensitive to initial conditions, and do not necessary provide information about coexistence of multiple solutions.…”