Fletcher gave a rather general analysis of the linearised instability threshold of a nonlinear valve subjected to a mean flow, with acoustical feedback on one or both of its faces [1]. In a later paper [2], he then applied this approach to the specific case of a free reed attached to a rigid chamber of volume $V$, as sketched in Fig. 1. The predictions were compared with the results of careful experiments, giving generally quite good agreement — some examples were shown in section 11.6.
Fletcher’s theoretical model is related closely to the models we have used earlier for reed woodwind instruments and for brass instruments. It is based on three equations. We first need to define some notation. There is a steady inflow of air to the chamber, with volume flow rate $U_0$. The pressure inside the chamber is $p_0+p(t)$, where $p_0$ is the steady atmospheric pressure. It is assumed to be uniform throughout the volume, in a similar way to our discussion of the Helmholtz resonator back in section 4.2.1. The volume flow rate of air past the reed is $u(t)$, and it is assumed that the pressure outside is simply atmospheric.
Exactly as we did before, we model the reed as a damped harmonic oscillator representing the fundamental bending mode. It is assumed to have a natural frequency $\omega_r$ and a Q-factor $Q_r$, and the tip displacement of the reed is denoted $x(t)$. The reed has length $L$ and width $w$, and it has effective modal mass $m$ per unit area. The area through which the flow $u(t)$ is squeezed for a given position of the vibrating reed is denoted $F(x)$ — we will say something about how this area function can be calculated a little later.
The first of our model equations is the Bernoulli expression relating to the flow past the reed:
$$p \approx \dfrac{\rho_0 u^2}{2 C^2 F(x)^2} . \tag{1}$$
This is identical to the expression we have used before, except for the inclusion of a vena contracta coefficient $C$, for which Tarnolpolsky et al. suggest the value 0.61 for the jet geometry of interest here. The second equation describes the reed dynamics:
$$\ddot{x}+\dfrac{\omega_r}{Q_r}\dot{x}+\omega_r^2 x = K_p p \tag{2}$$
where dots denote time derivatives and where $K_p =1.5 w L/m$: the value 1.5 here is suggested by Tarnopolsky et al. based on the mode shape of the lowest cantilever mode shape of the reed. The third equation relates the rate of change of pressure inside the chamber to the net volume flow rate of air into it:
$$\dot{p}=\dfrac{\rho_0 c^2}{V}[U_0 – u – K_x \dot{x}] \tag{3}$$
where $\rho_0$ is the density of air and $c$ is the speed of sound. This is the same equation we used for the Helmholtz resonator in section 4.2.1, except for the final term in the square brackets. This describes the component of volume flow generated directly by the motion of the reed. It involves a constant $K_x=0.4wL$, where the suggested numerical value 0.4 is again based on the cantilever mode shape.
For the purposes of a linearised stability analysis, we now assume that
$$u=\bar{u}+u’ e^{i \omega t} \mathrm{,~~} p=\bar{p}+p’ e^{i \omega t} \mathrm{,~~} x=\bar{x}+x’ e^{i \omega t} \tag{4}$$
where for example $\bar{p}$ is the mean value and $p’$ is the (complex) amplitude of a small harmonic fluctuation at frequency $\omega$. We substitute these expressions into the three equations, linearise with respect to all the primed quantities where necessary, and then equate separately the steady terms and terms involving $e^{i \omega t}$. Equations (2) and (3) are both linear, so this involves very little effort. The results from equation (2) are
$$\omega_r^2 \bar{x}=K_p \bar{p} \tag{5}$$
and
$$\left[ -\omega^2 + i \omega \dfrac{\omega_r}{Q_r} + \omega_r^2 \right]x’ = K_p p’ . \tag{6}$$
Corresponding results from equation (3) are
$$\bar{u}=U_0 \tag{7}$$
and
$$i \omega p’ = – \dfrac{\rho_0 c^2}{V} [u’+i \omega K_x x’] . \tag{8}$$
For equation (1) we need to work a little harder. First, we take the first two terms of a Taylor expansion of $F(x)$:
$$F(x) \approx F(\bar{x}) + x’ e^{i \omega t} \dfrac{dF}{dx}(\bar{x})=\bar{F}+\bar{F}’ x’ e^{i \omega t}\tag{9}$$
where for brevity we define $\bar{F}=F(\bar{x})$ and $\bar{F}’=dF(\bar{x})/dx$.
So now equation (1) gives
$$\bar{p}+p’e^{i \omega t}\approx \dfrac{\rho_0}{2C^2}\left( \bar{u}+u’e^{i \omega t} \right)^2\left(\bar{F}+\bar{F}’ x’ e^{i \omega t} \right)^{-2}$$
$$\approx \dfrac{\rho_0}{2C^2 \bar{F}^2}\left( U_0^2 + 2 U_0 u’ e^{i \omega t} \right) \left(1-2\dfrac{\bar{F}’}{\bar{F}} x’ e^{i \omega t} \right)$$
$$\approx \dfrac{\rho_0}{2C^2 \bar{F}^2}\left( U_0^2 +2 U_0 u’ e^{i \omega t} – 2U_0^2 \dfrac{\bar{F}’}{\bar{F}} x’ e^{i \omega t} \right)\tag{10}$$
and so
$$\bar{p} =\dfrac{\rho_0 U_0^2}{2C^2 \bar{F}^2}\tag{11}$$
and
$$p’=\dfrac{\rho_0 U_0}{C^2 \bar{F}^2}\left(u’-U_0 \dfrac{\bar{F}’}{\bar{F}}x’ \right). \tag{12}$$
We are interested in thresholds for pressure, so $\bar{p}$ is the variable of interest. Equation (5) gives the static displacement of the reed tip in response to that mean pressure, then equation (11) determines the input flow rate $U_0$ necessary to create the desired pressure.
We now eliminate $u’$ between equations (8) and (12) to obtain
$$-\dfrac{i \omega V}{\rho_0 c^2}p’ – i \omega K_x x’ = U_0 \dfrac{\bar{F}’}{\bar{F}} x’ + \dfrac{U_0}{2 \bar{p}}p’ \tag{13}$$
so that
$$(i \omega A +B)p’=-(i \omega D + E)x’ \tag{14}$$
with
$$A=\dfrac{V}{\rho_0 c^2} \mathrm{,~~} B=\dfrac{C^2 \bar{F}^2}{\rho_0 U_0} \mathrm{,~~} D=K_x \mathrm{,~~} E=U_0 \dfrac{\bar{F}’}{\bar{F}} . \tag{15}$$
Now we can substitute in equation (6):
$$\left[ -\omega^2 + i \omega \dfrac{\omega_r}{Q_r} + \omega_r^2 \right]x’ = -K_p \dfrac{i \omega D + E}{i \omega A + B} x’$$
$$=-K_p \dfrac{(i \omega D + E)(-i \omega A + B)}{\omega^2 A^2 + B^2} x’ . \tag{16}$$
In order for an instability to occur, the imaginary part of the right-hand side expression must at least compensate for the damping term on the left-hand side, so the condition for instability is
$$\dfrac{\omega_r}{Q_r} < K_p\dfrac{EA-DB}{\omega_r^2 A^2 + B^2} \tag{17}$$
where we have substituted $\omega_r$ for $\omega$ in the denominator because we always expect our free reed to vibrate close to its natural frequency. The constants $A$, $B$ and $D$ are always positive, but $E$ depends on the sign of $\bar{F}’$. So we certainly need $\bar{F}’>0$ to have any chance of satisfying the instability condition — in other words, it must behave as an “opening reed” in the immediate vicinity of the equilibrium displacement $\bar{x}$.
To calculate a reasonable approximation to the area function $F(x)$, we can follow a similar approach to previous authors [2,3]. For a given displaced position of the reed, we can think of connecting the edge of the reed, all the way round, to the nearest point on the slot in the reed plate, with a kind of “curtain”. The area of that curtain is what we want: it is the minimum area that can be used to block any flow past the reed.
This calculation could be done using the cantilever mode shape, but for the work reported here a simpler approach has been used, in which the reed is envisaged as being rigid and flat, hinged to the base plate along one edge. Three cases have to be considered: the reed lying entirely outside the slot, the reed tip lying within the thickness of the slot in the base plate, and the reed tip having emerged on the other side. For each case it is straightforward to compute the area function, armed with the reed dimensions together with the thickness of the base plate, the stand-off distance of the “hinge” above the plate and the clearance around the edges. Some examples for different geometrical configurations were shown in section 11.6.
We should mention a small complicating factor. Fletcher suggests that another term could be included in equation (1), to take some account of the fact that the air flow past the reed is not really a steady flow. He argues [2] that a simple application of the non-steady version of Bernoulli’s equation would lead to
$$p \approx \dfrac{\rho_0 u^2}{2 C^2 F(x)^2} + \dfrac{d}{dt} \left[ \dfrac{\rho_0 u \delta}{C F(x)} \right] \tag{18}$$
in place of equation (1), where $\delta$ is an estimate of the length over which the jet past the reed remains laminar: we might expect it to be of the order of a few millimetres. Following through the same linearisation procedure, one extra complex factor appears. The simplest way to summarise the result is to say that everything follows identically if we replace $C^2$ by
$$\dfrac{C^2}{1+i \omega \delta C \bar{F}/U_0}. \tag{19}$$
This complicates the algebra somewhat, but when the effect of making this change is tested numerically for the geometry of any of the reeds we will be concerned with, the effect turns out to be very slight. Our earlier neglect of the extra term seems to be justified.
The stability analysis described above is specific to the situation sketched in Fig. 1, but we could note that Fletcher’s original version [1] was couched in more general terms. This involves keeping equations (6) and (12), but replacing equation (8) with the more general version
$$p’=-Z(\omega) \left(u’+i \omega K_x x’ \right) \tag{20}$$
where $Z(\omega)$ is the impedance presented to one face of the reed. The negative sign arises because the impedance we would measure would involve positive volume flow into the acoustic system, whereas we have defined our flow rate positive into the reed. We can recover the previous case by recalling that the impedance of a volume $V$ is
$$Z_{volume}=\dfrac{\rho_0 c^2}{i \omega V}. \tag{21}$$
In this more general framework, the condition for instability becomes
$$\dfrac{\omega_r^2}{Q_r} < \mathrm{Im}\left[\dfrac{K_p(U_0 \bar{F}’/\bar{F} + i \omega K_x)}{-Y(\omega)-C^2 \bar{F}^2/\rho_0 U_0}\right] \tag{22}$$
where $\mathrm{Im}$ denotes the imaginary part, and $Y=1/Z$ is the admittance of the acoustic system. Again, the expression could all be evaluated at $\omega=\omega_r$ for an explicit approximation. Using the notation from equation (16), for instability we require
$$\dfrac{\omega_r^2}{Q_r} <-K_p \mathrm{Im} \left[ \dfrac{i \omega D + E}{Y +B}\right]=-K_p\mathrm{Im} \left[\dfrac{(i \omega D + E)(B+Y^*)}{|Y|^2+B^2}\right]$$
$$=-K_p\left[\dfrac{(\omega D (B+\mathrm{Re} Y) – E \mathrm{Im} Y}{|Y|^2+B^2}\right] . \tag{23}$$
Certainly for this condition to be satisfied, the final expression must be positive. The constants $B$ and $D$ are positive, so the first term in the numerator always acts against the requirement. Because $Y$ is a driving-point admittance, $\mathrm{Re} Y > 0$ and so lossiness in the acoustical system always tends to discourage instability. For instability we always require $E\mathrm{Im} Y > 0$: so a closing reed with $E<0$ needs $\mathrm{Im} Y<0$ while an opening reed with $E>0$ needs $\mathrm{Im} Y>0$. In terms of impedance, these sign conditions are reversed: $Y=1/Z=Z^*/|Z|^2$ so $\mathrm{Im} Y$ has the opposite sign to $\mathrm{Im} Z$.
[1] N. H. Fletcher: “Autonomous vibration of simple pressure-controlled valves in gas flows”, Journal of the Acoustical Society of America 93, 2172—2180 (1993).
[2] A. Z. Tarnopolsky, N. H. Fletcher and J. C. S. Lai, “Oscillating reed valves — an experimental study”, Journal of the Acoustical Society of America 108, 400—406 (2000).
[3] L. Millot and Cl. Baumann, “A proposal for a minimal model for free reeds”, Acta Acustica united with Acustica 93, 122—144 (2007).