11.6.2 Millot’s model for free-reed instability

An enhanced model for free-reed behaviour was proposed by Millot and Baumann [1]. It adds one extra ingredient to Fletcher’s model (discussed in detail in section 11.6.1), as sketched in Fig. 1. A tube of length $L_2$ and cross-sectional area $S_2$ is placed in between the volume $V$ and the reed. The pressure in the main volume, $p_1(t)$, might now be different from the pressure at the inner face of the reed, $p_2(t)$. The other variables remain the same as in the discussion of section 11.6.1: flow rate $u(t)$ past the reed, reed tip displacement $x(t)$, reed properties $\omega_r$, $Q_r$ and $m$, and the reed area function $F(x)$.

Figure 1. Millot’s model, with the reed connected to a chamber of volume $V$ by a tube of length $L_2$ and cross-sectional area $S_2$.

Most of the equations are virtually unchanged. The Bernoulli expression relating to the flow past the reed involves the pressure $p_2$:

$$p_2 \approx \dfrac{\rho_0 u^2}{2 C^2 F(x)^2} . \tag{1}$$

The same is true for the equation describing the reed dynamics:

$$\ddot{x}+\dfrac{\omega_r}{Q_r}\dot{x}+\omega_r^2 x = K_p p_2 . \tag{2}$$

The equation relating the rate of change of pressure inside the chamber to the net volume flow rate of air into it obviously involves the pressure $p_1$:

$$\dot{p_1}=\dfrac{\rho_o c^2}{V}[U_0 -u – K_x \dot{x}] . \tag{3}$$

Finally, we use the “Helmholtz resonator” approximation to treat the “plug” of air in the new tube as if it was a rigid mass: Newton’s law then states

$$S_2(p_1-p_2)=\rho_0 S_2 L_2\dfrac{d}{dt}\left(\dfrac{u+K_x \dot{x}}{S_2} \right) . \tag{4}$$

This gives an equation linking $p_1$ and $p_2$. We can re-write it slightly by making use of the formula for the Helmholtz resonance frequency of the chamber and tube, from section 4.2.1:

$$\omega_h^2=\dfrac{c^2 S_2}{VL_2} . \tag{5}$$

Equation (4) then becomes

$$p_1-p_2=\dfrac{\rho_0 c^2}{\omega_h^2 V} \dfrac{d}{dt}\left(u+K_x \dot{x} \right) . \tag{6}$$

We then follow the same procedure as in section 11.6.1, setting $p_1=\bar{p_1}+ p’_1 e^{i \omega t}$ and so on. The only real change is the new equation (6), which is linear and so is easy to deal with. After a little algebra, we can reach a result that is almost identical to equation (16) from section 11.6.1:

$$\left[ -\omega^2 + i \omega \dfrac{\omega_r}{Q_r} + \omega_r^2 \right]x’ = -K_p \dfrac{i \omega D + E}{i \omega A’ + B} x’$$

$$=-K_p \dfrac{(i \omega D + E)(-i \omega A’ + B)}{\omega^2 A’^2 + B^2} x’ . \tag{7}$$

The only difference is that the constant $A$ has been replaced by

$$A’=\dfrac{A}{1-\omega^2/\omega_h^2}=\dfrac{V}{\rho_0 c^2 (1-\omega^2/\omega_h^2)} . \tag{8}$$

The new condition for instability is thus

$$\dfrac{\omega_r}{Q_r} < K_p\dfrac{EA’-DB}{\omega_r^2 A’^2 + B^2} \tag{9}$$

where $A’$ is evaluated at $\omega=\omega_r$, using the same approximation as before that the reed generally vibrates close to its natural frequency. The only terms in this equation which are not necessarily positive are $A’$ and $E=U_0 \dfrac{\bar{F}’}{\bar{F}}$. But notice that these terms appear as the product $A’E$, so if the signs of both are changed, the stability threshold will be exactly the same. Instability requires this product to be positive, so either an opening reed with $\omega_r < \omega_h$ or a closing reed with $\omega_r > \omega_h$.

We can relate this pattern to Fletcher’s impedance formulation of the stability condition, described at the end of section 11.6.1. The input impedance that would be measured at the reed, for the Millot model, can easily be evaluated from the equations above: the result is

$$Z_M(\omega) = \dfrac{\rho_0 c^2}{i \omega V}\left[1-\omega^2/\omega_h^2\right] .\tag{10}$$

We saw in section 11.6.1 that for a closing reed, with $\bar{F}’ < 0$, the imaginary part of the impedance must be positive for instability, whereas for an opening reed it must be negative. Equation (10) then shows that if $\omega_h < \omega$, a closing reed could be unstable, whereas if $\omega_h > \omega$ then the opening reed could be unstable.

[1] L. Millot and Cl. Baumann, “A proposal for a minimal model for free reeds”, Acta Acustica united with Acustica 93, 122—144 (2007).