Engineering Acoustics/Noise control with self-tuning Helmholtz resonators

Introduction
Many engineering systems create unwanted acoustic noise. Noise may be reduced using engineering noise control methods. One noise control method popular in mufflers is the Helmholtz resonator, see here. It is comprised of a cavity connected to the system of interest through one or several short narrow tubes. The classical examples are in automobile exhaust systems. By adding a tuned Helmholtz resonator, sound is reflected back to the source.

Helmholtz resonators have been exploited to enhance or attenuate sound fields at least since ancient Greek times where they were used in ancient amphitheaters to reduce reverberation. Since this time, Helmholtz resonators have found widespread use in reverberant spaces such as churches and as mufflers in ducts and pipes. The Helmholtz resonator effect underlies the phenomena of sunroof buffeting seen here. One advantage of the Helmholtz resonator is its simplicity. However, the frequency range over which Helmholtz resonators are effective is relatively narrow. Consequently these devices need to be precisely tuned to the noise source to achieve significant attenuation.

Noise and vibration control
There are four general categories for noise and vibration control:


 * 1)  Active systems: load or unload the unwanted noise by using actuators such as loudspeakers  and  Acoustics/Active Control
 * 2)  Passive systems: achieve sound attenuation by using
 * 2.1. reactive devices such as Helmholtz resonators and expansion chambers.
 * 2.2. resistive materials such as acoustic linings and porous membranes
 * 1)  Hybrid systems: use both active and passive elements to achieve sound reduction
 * 2)  Adaptive-passive systems: use passive devices whose parameters can be varied in order to achieve optimal noise attenuation over a band of operating frequencies.

Lumped element model of the Helmholtz resonator
The Helmholtz resonator is an acoustic filter element. If dimensions of the Helmholtz resonator are smaller than the acoustic wavelength, then dynamic behavior of the Helmholtz resonator can be modelled as a lumped system see. It is effectively a mass on a spring and can be treated so mathematically. The large volume of air is the spring and the air in the neck is the oscillating mass. Damping appears in the form of radiation losses at the neck ends, and viscous losses due to friction of the oscillating air in the neck. Figure 1 shows this analogy between Helmholtz resonator and a vibration absorber.



Theoretical analysis of Helmholtz resonators
For a neck which is flanged at both ends, Leff is approximately:

$$ L_\text{eff} = \ L + \ 1.7 \ a $$

The acoustic mass of a Helmholtz resonator is given by

$$ M_a = L_\text{eff}\ \rho\ S $$

The stiffness of the resonator is defined as the reciprocal of the compliance, and it is defined as

$$ k_r = \frac{dF}{dy} $$

where

$$ F = P\ S$$

For adiabatic system with air as an ideal gas, the thermodynamic process equation for the resonator is

$$PV^{\gamma} = \text{constant} \,$$ Differentiating this equation gives

$$V^\gamma dP + P \gamma V^{\gamma -1} dV = 0 \,$$

The change in the cavity volume is

$$ dV = -Sdy\ $$

substituting these into differential equation, it can be re-casted as

$$V^\gamma \frac{dF}{S} - P \gamma V^{\gamma -1} S dy = 0 \Rightarrow \frac{dF}{dy} = \frac{P \gamma S^2}{V} = k_r$$

or considering $$ P = \rho R\ T$$ and $$c=\sqrt{\gamma R T}$$, resonator stiffness is then:

$$ k_r =\frac{\rho c^2\ S^2}{V} $$

where c is the speed of sound, and $$\rho$$ is the density of the medium.

Two source of damping in the Helmholtz resonator can be considered: sound radiation from the neck and viscous losses in the neck, which in many cases can be neglected compared to radiation losses.

1. Sound radiation from the neck: Sound radiation resistance is a function of the outside neck geometry. For a flanged pipe, the radiation resistance is approximately

$$ R_r =\frac{\rho c k^2\ S^2}{2\pi} $$

where k is the wave number, $$ k =\frac{w}{c} $$

2. Viscous losses in the neck: The mechanical resistance due to viscous losses can be considered as

$$ R_v =2R_s S\frac{ \ (L+a)}{\rho c a} $$ where Rs for a sufficiently large neck diameter is

$$ R_s =0.00083\sqrt{\frac{w}{2\pi}} $$, where &omega; is the excitation frequency.

The mechanical impedance of the mechanical system is defined as the ratio of the driving force and the velocity of the system at the driving point.The mechanical impedance of a driven mass-spring–damper system is

$$\hat{Z}_m=\frac{\hat{F}}{\hat{u}}= R_m+j(\omega m - \frac{k_r}{w})$$

according to the analogy between Helmholtz resonator and mass-spring–damper system (vibration absorber), the mechanical impedance of a Helmholtz resonator is obtained by replacing mass and damping from Helmholtz resonator system in above equation:

$$\hat{Z}_{mres}= (R_v+\frac{\rho c k^2\ S^2}{2\pi})+j(\omega \rho L_{eff} S - \frac{\rho c^2\ S^2}{wV})$$

The natural frequency of a Helmholtz resonator,w0, is the frequency for which the reactance is zero:

$$ w_0 =c\sqrt{\frac{S}{L_\text{eff}V}} $$,

and the acoustic impedance of the Helmholtz resonator is

$$\hat{Z}_{res}= \frac{\hat{Z}_{mres}}{\ S^2}$$

Resonance occurs when the natural frequency of the resonator is equal to the excitation frequency. Helmholtz resonators are typically used to attenuate sound pressure when the system is originally at resonance.A simple open-ended duct system with a side branch Helmholtz resonator and the analogous electrical circuit of the system is shown below. For an undamped resonator, the impedance at resonance is zero, and therefore according to electrical analogy in Fig.2 the Helmholtz resonator becomes a short circuit. There is no current flowing in the elements in the right. On the other hand, the undamped Helmholtz resonator at resonance causes all reflection of acoustic waves back to the source, while in damped resonator some current will flow through the branch to the right of the Helmholtz resonator and reduce the magnitude of attenuation.



1- Effect of Resonator Volume on sound attenuation

Figure 3 shows the frequency response of the above duct system without Helmholtz resonator, and with two different volume Helmholtz resonators with the same natural frequency. The excitation frequency axis is normalized with respect to the fundamental frequency of the straight pipe system, which was also chosen as the natural frequency of the resonator. The maximum attenuation of sound pressure for duct systems with side branch Helmholtz resonators occurs when the natural frequency of the resonator is equal to the excitation frequency. By comparing two curves with different colors, blue and gray, it can be seen that to increase the effective bandwidth of attenuation of a Helmholtz resonator, the device should be made as large as possible. It should be mention that in order to minimize the effects of standing waves within the device, the dimensions do not exceed a quarter wavelength of the resonator natural frequency.



2- Effect of Resonator Damping on sound attenuation

The effect of Helmholtz resonator damping(Resulting from radiation resistance and viscous losses in the neck) on the frequency response of the duct system is shown in Figure 5. The lightly damped Helmholtz resonator is not robust with respect to changes in the excitation frequency, since the sound pressure in the duct system can be amplified if the noise frequency shifts to the vicinity of either of the two system resonances. To increase the robustness of Helmholtz resonator with respect to changes in the excitation frequency, damping is useful to add to the resonators to decrease the magnitude of the resonant peaks. Such increase in robustness decreases performance, since the maximum attenuation is significantly less for heavily damped Helmholtz resonators. The motivation for creating a tunable Helmholtz resonator stems from this trade off between robustness and performance. A tunable Helmholtz resonator, capable of adjusting its natural frequency to match the excitation frequency, would be able to guarantee the high performance of a lightly damped Helmholtz resonator and track changes in frequency.



Adaptive Helmholtz resonator
The tunable Helmholtz resonator is a variable volume resonator, which allows the natural frequency to be adjusted.As shown in Figure 5, a variable volume Helmholtz resonator can be achieved by rotating an internal radial wall inside the resonator cavity with respect to an internal fixed wall. The movable wall is fixed to the bottom end plate which is attached to a DC motor to provides the motion to change the volume.



To determine the sound pressure and volume velocity at any position along the duct such as the microphone position in Figure 2, we should first determine the pressure and velocity at the speaker.





The acoustic impedance for the system termination, which is an unflanged open pipe is approximately

$$ Z_1 =\frac{\rho c}{S_p}(\frac{1}{4})(ka)^2+j 0.6 ka$$

where Sp is the cross sectional area of the pipe, and ap is the radius of the pipe. The impedance at point 2 is

$$ \frac{Z_2}{\frac{\rho c} {S_p}}=\frac{\frac{Z_1}{\frac{\rho c} {S_p}}+j tan(kL_1)}{1+j\frac{Z_1}{\frac{\rho c} {S_p}}tan(kL_1)}$$

where L1 is the length of the pipe separating termination form the resonator. The resonator acoustic impedance is the same as what is shown above. The acoustic impedance at point 3 is given by

$$ Z_3 =\frac{Z_2 Z_{res}}{Z_2+ Z_{res}}$$

The impedance at point 4 can be determined by

$$ \frac{Z_4}{\frac{\rho c} {S_p}}=\frac{\frac{Z_3}{\frac{\rho c} {S_p}}+j tan(kL_2)}{1+j\frac{Z_3}{\frac{\rho c} {S_p}}tan(kL_2)}$$

finally the impedance at the speaker is given by

$$ \frac{Z_{sys}}{\frac{\rho c} {S_{enc}}}=\frac{\frac{Z_4}{\frac{\rho c} {S_{enc}}}+j tan(kL_{enc})}{1+j\frac{Z_4}{\frac{\rho c} {S_{enc}}}tan(kL_{enc})}$$

where Senc is the cross section area of the speaker enclosure, and Lenc is the length of the enclosure aperture from the speaker.

From figure 6 with the system impedance (Z_{sys}), the pressure and velocity at the speaker can be determined. considering transfer matrices, the pressure and velocity at any location in the duct system may be computed from the pressure and velocity at the speaker.The first transfer matrix may be used to relate the pressure and velocity at the point downstream in a straight pipe to the pressure and velocity at the origin of the pipe.

$$\begin{bmatrix} P_d(l) \\ U_d(l) \end{bmatrix}$$=$$\begin{bmatrix} cos (k L) & -j sin (k L) \frac{\rho c}{S_p}  \\ -j \frac{sin (k L)}{\frac{\rho c}{S_p}} & cos (k L) \end{bmatrix} \begin{bmatrix} P_d(0) \\ U_d(0) \end{bmatrix}$$

The second matrix relates the pressure and velocity immediately downstream of the sidebranch, to the pressure and velocity immediately before the side branch.

$$\begin{bmatrix} P_d(3) \\ U_d(3) \end{bmatrix}$$=$$\begin{bmatrix} 1 & 0  \\  -\frac {1}{Z_{res}} & 1 \end{bmatrix} \begin{bmatrix} P_d(2) \\ U_d(2) \end{bmatrix}$$

Correct combination of theses transfer matrices may be used to determine the pressure occurring in the system at the location of the microphone in figure 2.