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 [1] and Acoustics/Active Control
2. Passive systems: achieve sound attenuation by using [2]

   2.1. reactive devices such as Helmholtz resonators and expansion chambers.

   2.2. resistive materials such as acoustic linings and porous membranes

3. Hybrid systems: use both active and passive elements to achieve sound reduction [3]
4. 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 [4]. 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.

Figure 1. Helmholtz resonator and vibration absorber

Parameters definition

Parameter	definition	Parameter	definition
$M_{a}$	Acoustic mass of the resonator	$C_{a}$	Acoustic compliance
$\rho$	Density of the fluid	P	pressure at the neck entrance
ω	Excitation frequency	S	Cross section area of the neck
a	Radius of the neck	L	Actual neck length
y	Displacement in the direction pointing inward the neck	V	Cavity volume of the Helmholtz resonator
F	Force applied in the y direction at the resonator neck entrance	Leff	Effective neck length (the mass inside the neck and the mass near the neck edges)

Theoretical analysis of Helmholtz resonators

For a neck which is flanged at both ends, L_eff 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

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

or considering $P=\rho R\ T$ and $c={\sqrt {\gamma RT}}$, 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

$R_{r}={\frac {\rho ck^{2}\ S^{2}}{2\pi }}$

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

$R_{v}=2R_{s}S{\frac {\ (L+a)}{\rho ca}}$

where R_s for a sufficiently large neck diameter is

$R_{s}=0.00083{\sqrt {\frac {w}{2\pi }}}$, where ω 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:

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

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.

Figure 2. open duct system with a side branch Helmholtz resonator with electrical circuit analogy near junction point A

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 .

Figure 3. Effect of Resonator Volume on sound attenuation

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.

Figure 4. Effect of Resonator Damping on sound attenuation

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.

Figure 5. Variable volume Helmholtz resonator

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.

Figure 5. Equivalent circuit for duct system in figure 2

Figure 6. Simplified equivalent circuit

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

$Z_{1}={\frac {\rho c}{S_{p}}}({\frac {1}{4}})(ka)^{2}+j0.6ka$

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

${\frac {Z_{2}}{\frac {\rho c}{S_{p}}}}={\frac {{\frac {Z_{1}}{\frac {\rho c}{S_{p}}}}+jtan(kL_{1})}{1+j{\frac {Z_{1}}{\frac {\rho c}{S_{p}}}}tan(kL_{1})}}$

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

${\frac {Z_{4}}{\frac {\rho c}{S_{p}}}}={\frac {{\frac {Z_{3}}{\frac {\rho c}{S_{p}}}}+jtan(kL_{2})}{1+j{\frac {Z_{3}}{\frac {\rho c}{S_{p}}}}tan(kL_{2})}}$

finally the impedance at the speaker is given by

${\frac {Z_{sys}}{\frac {\rho c}{S_{enc}}}}={\frac {{\frac {Z_{4}}{\frac {\rho c}{S_{enc}}}}+jtan(kL_{enc})}{1+j{\frac {Z_{4}}{\frac {\rho c}{S_{enc}}}}tan(kL_{enc})}}$

where S_enc is the cross section area of the speaker enclosure, and L_enc 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.

${\begin{bmatrix}P_{d}(l)\\U_{d}(l)\end{bmatrix}}$=${\begin{bmatrix}cos(kL)&-jsin(kL){\frac {\rho c}{S_{p}}}\\-j{\frac {sin(kL)}{\frac {\rho c}{S_{p}}}}&cos(kL)\end{bmatrix}}{\begin{bmatrix}P_{d}(0)\\U_{d}(0)\end{bmatrix}}$

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.

Appendix A: Matlab Code for straight pipe with Helmholtz resonator

%This Matlab code is used for calculating pressure at the place of
%microphone for the pipe without Helmholtz resonator
clear all
clc
V0=1;
for f=10:1:200;
   omega=2*pi*f;
   freq(f-9)=f;
   c=343;                                     % speed of sound
   rho=1.21;                                  % density of the medium (air)
   ap=0.0254;                                 % radius of the pipe
   Sp=pi*(ap^2);                              % cross sectional area of the pipe
   k=omega/c;                                 % wave number
   Lenc=0.10;                                 %distance separating the enclosure aperture and the speaker face
   L1=0.34+(0.6*ap);                          % length of the pipe separating the termination from the resonator
   L2=0.62+(0.85*ap);                         % length of the pipe between the resonator and speaker enclosure
   Lx=0.254;                                  % length of the pipe from HR to microphone
   Lm=Lx+L2;                                  %distance from speaker to microphon
   Ld=L1+L2;
   Rm=1;                                      % loudspeaker coil resistance
   Bl=7.5;
   Cm=1/2000;                                 %Complience of the speaker
   OmegaN=345;                                %Natural frequency of the speaker(55HZ)
   a=0.0425;                                  %effective radius of the diaphragm
   Senc=pi*(a^2);                             % cross sectional area of the speaker enclosure  (Transformer ratio)
   Mm=0.01;                                   %Air load mass on both side of the driver
   Gamma=1.4;                                 %Specific heat ratio for air
   P0=10^5;
   b=0.38;
   Pref=20e-6;                                %Reference pressure for sound pressure level
       
   %calculate the system impedance
   Z1=(rho*c*Sp)*(((1/4)*((k*ap)^2)+i*(0.6*k*ap))/(Sp^2));      %open ended unflanged pipe
   % impedance at point 4( after speaker)
   Z4=(rho*c/Sp)*(((Z1/(rho*c/Sp))+i*tan(k*Ld))/(1+i*(Z1/(rho*c/Sp))*tan(k*Ld))); 
   Zsys=(rho*c/Senc)*(((Z4/(rho*c/Senc))+i*tan(k*(Lenc)))/(1+i*(Z4/(rho*c/Senc))*tan(k*(Lenc)))); 
   %calculating the impedance of the loud speaker 
   Induct=(Mm/(Senc^2));                                                   
   Resist=((Bl^2)/((Senc^2)*(Rm)));
   Drivimp=Resist+(i*omega*Induct)+(1/(i*omega*Cm));           % impedance moadel for speaker
   Impedance=Zsys+ Drivimp;
   Voltage=V0*Bl/((Rm)*Senc);
   Velocity=Voltage/Impedance;
   Pressure=Velocity*Zsys;
   Pressure=Pressure*sqrt(2);
   Velocity=Velocity*sqrt(2);
   TR1=[Pressure;Velocity];
   TR15=[cos(k*(Lenc)) -i*(rho*c/Senc)*sin(k*(Lenc)); -i*(sin(k*(Lenc)))/(rho*c/Senc) cos(k*(Lenc))];
   TR3=[cos(k*Lm) -i*(rho*c/Sp)*sin(k*Lm); -i*(sin(k*Lm))/(rho*c/Sp) cos(k*Lm)];
   Tf=TR3*TR15*TR1;
   Pmike=Tf(1)/(sqrt(2));                                % pressure at microphone
   Vmike=Tf(2);                                          % velocity at microphone
   MagP(f-9)=20*log10(Pmike/Pref);
end;
nondim=freq/(OmegaN/2/pi);
plot(nondim,MagP,'k-.');
title('frequency response of system with Helmholtz Rsonator');
xlabel('Normalized Excitation Frequency (Hz)');
ylabel('Sound pressure level (dB)');