Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure

Abstract

Reed instruments are modeled as self-sustained oscillators driven by the pressure inside the mouth of the musician. A set of nonlinear equations connects the control parameters (mouth pressure, lip force) to the system output, hereby considered as the mouthpiece pressure. Clarinets can then be studied as dynamical systems; their steady behavior being dictated uniquely by the values of the control parameters. Considering the resonator as a lossless straight cylinder is a dramatic yet common simplification that allows for simulations using nonlinear iterative maps.

This paper investigates analytically the effect of a linearly increasing blowing pressure on the behavior of this simplified clarinet model. When the control parameter varies, results from the so-called dynamic bifurcation theory are required to properly analyze the system. This study highlights the phenomenon of bifurcation delay and defines a new quantity, the dynamic oscillation threshold. A theoretical estimation of the dynamic oscillation threshold is proposed and compared with numerical simulations.

Appendices

Appendix A: Table of notation

1.1 A.1 Physical variables

Symbol	Explanation	Unit
Z c	characteristic impedance	Pa s m−3
K s	static stiffness of the reed per unit area	Pa m−1
P M	static closing pressure of the reed	Pa
H	opening height of the reed channel at rest	m
U	flow created by the pressure imbalance between the mouth and the mouthpiece	m3 s−1
U r	flow created by the motion of the reed	m3 s−1
U in	flow at the entrance of the resonator	m3 s−1
U A	flow amplitude parameter	m3 s−1
P m	musician mouth pressure	Pa
P	pressure inside the mouthpiece	Pa
ΔP	pressure difference P m −P	Pa
y	displacement of the tip of the reed	m
τ	round trip travel time of a wave along the resonator	s

1.2 A.2 Dimensionless variables

Symbol	Associated physical variable
γ	musician mouth pressure
ζ	flow amplitude parameter
u	flow at the entrance of the resonator
p	pressure inside the mouthpiece
r	reflexion function of the resonator
p +	outgoing wave
p −	incoming wave
p +∗	nonoscillating static regime of p + (fixed points of the function G)
ϕ	invariant curve
w	difference between p + and ϕ
ϵ	increase rate of the parameter γ
γ st	static oscillation threshold
γ dt	dynamic oscillation threshold
$\gamma_{\mathrm{dt}}^{\mathrm{th}}$	theoretical estimation of the dynamic oscillation threshold
$\gamma_{\mathrm{dt}}^{\mathrm{num}}$	value of γ when the system begins to oscillate (calculated numerically)

1.3 A.3 Nonlinear characteristic of the embouchure

Function	Associated representation	Definition
F	{u;p}	u=F(p)
G	{p +;p −}	p +=G(−p −)

Appendix B: Invariant curve

The invariant curve ϕ(γ,ϵ) is invariant under the mapping (23a)–(23b), it therefore satisfies the following equation:

$$ \phi(\gamma,\epsilon)=G \bigl(\phi(\gamma-\epsilon,\epsilon),\gamma \bigr). $$

(40)

First of all, the invariant curve is expanded into a power series of ϵ and only the first-order is retained:

$$ \phi(\gamma,\epsilon) \approx \phi_0(\gamma)+\epsilon \phi_1(\gamma). $$

(41)

Secondly, the function G is linearized around the curve p ^+∗(γ) of the fixed points:

(42)

where

$$ \partial_x G (x,y )=\frac{\partial G(x,y)}{\partial x}. $$

(43)

Then, we make a Taylor expansion of ϕ(γ−ϵ,ϵ):

(44)

Finally, neglecting the second-order terms in ϵ, Eq. (40) becomes:

(45)

To obtain the approximate analytical expression of the invariant cure ϕ, Eq. (45) is successively solved for the functions ϕ ₀(γ) and ϕ ₁(γ).

As expected, to order 0 we find

$$ \phi_0(\gamma)= p^{+*}(\gamma). $$

(46)

To order 1, we have to solve:

(47)

and, therefore,

$$ \phi_1(\gamma)=\frac{\partial p^{+*}(\gamma)}{\partial \gamma} \frac{\partial_x G (p^{+*}(\gamma),\gamma )}{\partial_x G (p^{+*}(\gamma),\gamma )-1}. $$

(48)

Finally, the expression of the invariant curve is

(49)