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

Abstract

In this paper we exploit the method of numerical continuation combined with orthogonal collocation to determine the steady-state oscillations of a model of flute-like instruments that is formulated as a nonlinear neutral delay differential equation. The delay term in the model causes additional complications in the analysis of the behaviour of the model, in contrast to models of other wind instruments which are formulated as ordinary differential equations. Fortunately, numerical continuation provides bifurcation diagrams that show branches of stable and unstable static and periodic solutions of the model and their connections at bifurcations, thus enabling an in depth analysis of the global dynamics. Furthermore, it allows us to predict the thresholds of the different registers of the flute-like instrument and thus to explain the classical phenomenon of register change and the associated hysteresis.

Regime Change Thresholds in Recorder-Like Instruments: Influence of the Mouth Pressure Dynamics

“…Thereby,ab ifurcation diagram provides am ore global knowledge on the different possible solutions of the system, and an easier interpretation of different phenomena observed experimentally or in time-domain simulations, as illustrated for example in [40,41,9]. This will be illustrated by Figure 12 provided in section 3, which represents such adiagram of the state-of-the-art model of flutelikei nstruments, in terms of oscillation frequencyo ft he periodic solutions with respect to the blowing pressure.…”

“…The aim is not here to propose am ethod to analyse the physical model of flute-likei nstruments, which allows to explain and predict the regime change, as Coltman [1,8], Sawada and Sakaba [2], Auvray et al [3] and Terrien et al [9]. Conversely,ap henomenological modelling of the influence of the blowing pressure dynamics on the regime change threshold is proposed.…”

“…Therefore, successive time-domain simulations of P m steps (see Figure 18)with different amplitudes are carried out, to determine the function f (ΔP )t hrough Equation (9).Such simulations have been achievedfor the two fingerings F 4 and G 4 ,inboth cases for transitions from the first to the second register.T he results are represented in Figure 19 with respect to ΔP . In the twocases, it seems that the results followasquare root function: the linear correlation coefficients between ΔP and (1/t dyn ) 2 are of 0.96 for the F 4 fingering and 0.97 for the G 4 fingering.…”

“…However, their results have not been validated through comparison with experimental data. More recent studies have demonstrated that the resolution of the complete non linear model through numerical methods dedicated to computation of periodic solutions allows to explain and predict, under quasi-static hypothesis, the hysteresis associated to register change [9]. Finally,i ts eems that few studies have focused, in terms of regime change thresholds, on other control parameters (i.e.…”

“…From am ore theoretical point of view, as the cause of register change in recorder-likei nstruments has recently been identified as abifurcation of the system [4,9], corresponding to aloss of stability of aperiodic solution branch, it suggests to consider the results of dynamic bifurcations theory [17]. This theory takes into account time evolution of the bifurcation parameters.…”