AIAA 2002-0666

Characterization of Compliant-Backplate Helmholtz Resonators for an Electromechanical Acoustic Liner

Stephen Horowitz, Toshikazu Nishida, Louis Cattafesta, and Mark Sheplak

University of Florida Gainesville, FL

40th Aerospace Sciences Meeting & Exhibit

14-17 January 2002 / Reno, NV

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics, 1801 Alexander Bell Drive, Suite 500, Reston, Virginia 20191-4344.

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Abstract This paper presents the theoretical modeling and

experimental characterization of compliant-backplate Helmholtz resonators. This study is the first phase in the development of an actively tuned electromechanical acoustic liner. This liner is based upon a Helmholtz resonator that uses a compliant piezoelectric composite backplate. In this study, Helmholtz resonators with compliant aluminum diaphragms were modeled using lumped elements and characterized in a normal-incidence impedance tube. The pressure amplification factor, acoustic impedance, and backplate mode shape were measured. Overall, good qualitative agreement was found between the experimental results and the lumped element models.

1 Introduction Acoustic liners are designed to provide a desired

acoustic impedance boundary condition that suppresses the propagation of noise in an engine nacelle. The impedance of an acoustic liner is fixed by geometry and flight environment. The optimum impedance will change with engine operating condition, such as at take-off, cut-back, and landing.

Clearly, it is desirable to tune the liner acoustic impedance for optimum performance. Various active techniques have been investigated to modify the acoustic impedance, such as face plate heating and bias flow through the face plate [1]. The ultimate goal of this research is to develop an electromechanical acoustic liner that tunes the acoustic impedance to optimize attenuation characteristics. The primary element of this liner is a Helmholtz resonator with a compliant piezoelectric composite backplate coupled to a tunable electrical

£

Graduate Student, Student Member AIAA ¥

Associate Professor ¶

Assistant Professor, Senior Member AIAA §

Assistant Professor, Member AIAA Copyright 2002 by the University of Florida. Published by

the American Institute of Aeronautics and Astronautics, Inc. with permission.

filter network. If the filter network is shunted to the piezoelectric backplate, additional degrees of freedom can be added to the liner. This enables the electromechanical acoustic liner to match the absorption characteristics of multi-layer liners. In addition, the impedance of the electromechanical acoustic liner can be tuned in-situ.

In the absence of any electrical components, an electromechanical Helmholtz resonator mimics the behavior of a two degree-of-freedom (DOF) double-layer liner. The impedance of the compliant backplate serves the same function as the second layer in the cell, as shown in Figure 1.

(a) (b)

Figure 1: Single cell of an electromechanical acoustic liner (a) and a passive 2DOF acoustic liner (b) showing how the compliant backplate serves the function of the second layer.

In this paper, isotropic compliant-backplate Helmholtz resonators are analyzed and experimentally characterized in a normal-incidence impedance tube, in order to establish a baseline case before proceeding to a full piezoelectric-composite backplate. The paper is organized as follows. Section 2 describes the lumped element model, and Section 3 explains the experimental setup. Section 4 presents a comparison between the model and experiment, and Section 5 presents conclusions and discusses future work.

2 Lumped Element Model Lumped element modeling was used to analyze

and predict the behavior of the compliant backplate Helmholtz resonator [2-4]. The lumped element system was then represented using an equivalent circuit representation. Circuit analysis was performed on the equivalent circuit to yield input impedance and cavity pressure amplification.

Characterization of Compliant-Backplate Helmholtz Resonators for an Electromechanical Acoustic Liner

Stephen Horowitz,1£ Toshikazu Nishida,1¥ Louis Cattafesta,2¶ and Mark Sheplak2§

Interdisciplinary Microsystems Group

1Department of Electrical and Computer Engineering 2Department of Aerospace Engineering, Mechanics, and Engineering Science

University of Florida Gainesville, Florida 32611-6250

(352) 392-3983, (352) 392-7303 (FAX), ms@aero.ufl.edu

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2.1 Equivalent Circuit Representation At low frequencies, where the wavelength is

much larger than the largest physical dimension of the device, a distributed system can be lumped into idealized discrete circuit elements. Under this approximation, the linearized conservation of mass and Eulers equation are replaced by equivalent Kirchoff’s laws for volume velocity and pressure drop.

A conventional Helmholtz resonator can be lumped into three idealized elements. The neck of the resonator constitutes a pipe through which frictional losses are incurred. In addition, the air that moves through the neck possesses a finite mass and kinetic energy. Therefore, the neck possesses both dissipative and inertial components. The compressible air in the cavity stores potential energy and is modeled as a compliance. In a conventional Helmholtz resonator, it is implicitly assumed that the walls of the cavity are perfectly rigid. If one or more of the cavity walls displaces due to an applied pressure, the impedance of the wall(s) must also be accounted for.

P2

MaDRaN MaN

MaDrad

RaDrad

CaCP1

CaD

Q

Figure 2: Equivalent acoustic circuit representation of a Helmholtz resonator with a compliant backplate.

The equivalent circuit for this system is shown in Figure 2, where 1P and 2P represent the incident and cavity acoustic pressures, respectively. In the notation below, the first subscript denotes the domain (e.g., “a” for acoustic), and the second subscript describes the element (e.g., “D” for diaphragm). The acoustic compliance and mass of the compliant backplate diaphragm are aDC and aDM , respectively. The acoustic radiation impedance of the diaphragm is represented by a mass, aDradM , and a resistance,

aDradR . Although not shown, an acoustic resistance could also be included in series with aDC and aDM to model structural damping effects. aCC is the acoustic compliance of the cavity, while aNR and

aNM are the acoustic resistance and mass of the fluid in the neck, respectively. The neck resistance generally includes the nonlinear resistance associated with the orifice discharge.

The structure of the equivalent circuit is explained as follows. An incident pressure drives a volume flow rate Q through the orifice. This flow can either compress the fluid in the cavity or can displace the backplate. Physically, this is represented as a volume velocity divider.

The linear acoustic resistance in the neck of length L and radius 0a is given in Ingard [5] as

( )

22

220

116 1

28 1 4aN air

L kR c kx

ax x

′ = + + +

ρππ

,{1}

where airρ is the density and c is the speed of sound, k c= ω is the wavenumber, and

0

2

2airxa

=

µρ ω

{2}

and

( )0161

3a

L L s′ ′= + −π

, {3}

where 20 tubes a A′ = π is the open-area fraction, and

tubeA is the cross-sectional area of the normal impedance tube used in the experiments.

The acoustic mass of the neck is similarly defined as

( ) 2

0

4 103 3 10aN air

LM

x a

′= − +

ρπ

. {4}

As mentioned above, the acoustic radiation mass aDradM can be modeled, to first order, as a piston in

an infinite baffle if the circular diaphragm is mounted in a plate that is much larger in extent than the diaphragm radius [6]

2

83

airaDradM

aρπ

≅ . {5}

The radiation resistance can be approximated, for low values of ka as

23

2air

aDradRc

ρ ωπ

≅ . (6)

In this analysis, the backplate is modeled as a clamped circular plate. The deflection of a clamped circular plate of radius a and thickness h subjected to a uniform pressure P is given by [7]

224

( ) 164Pa r

w rD a

= − . {7}

The flexural rigidity D is defined as

( )

3

212 1Eh

Dν

=−

, {8}

4

where ν is Poisson’s ratio and E is the elastic modulus of the material.

The effective acoustic compliance of a clamped circular plate is derived by equating the total distributed potential energy stored in the deflection of the plate to a lumped spring and is found to be

( )6 2

3

1

16aD

aC

Eh

−=

π ν. {9}

Similarly the effective acoustic mass of the diaphragm is derived by equating the total distributed kinetic energy stored in the velocity of the plate to a lumped mass and is given by

2

95aD

hM

aρ

π= , {10}

where ρ is the density of the material.

2.2 Acoustic Input Impedance The input impedance of the compliant-backplate

Helmholtz resonator is given by

1 1

1 1

aD aDrad aDradaD aC

in aN aN

aD aDrad aDradaD aC

sM sM RsC sC

Z sM RsM sM R

sC sC

+ + +

= + ++ + + +

.{11}

A plot of the normalized specific acoustic input resistance ( )in tube airR A cρ and input reactance

( )in tube airX A cρ for a Helmholtz resonator having a neck with a length of 3.18 mm and a diameter of 4.72 mm, a cavity volume of 1950 mm3, and a compliant backplate consisting of an aluminum shim with 0.051 mm thickness is shown in Figure 3.

Figure 3: Normalized specific acoustic resistance and reactance of a compliant-backplate Helmholtz resonator.

2.3 Cavity Pressure Amplification The transfer function given by the ratio of cavity

pressure to incident pressure, 2 1P P , represents the pressure amplification of the resonator. This transfer function can be found using standard circuit analysis techniques and is given by

2

1

1 1

1 1

aD aDrad aDradaC aD

inaD aDrad aDrad

aD aC

sM sM RsC sCP

P Z M sM RsC sC

+ + + = + + + +

{12}

The anti-resonance, which occurs at the frequency at which the numerator equals zero, is dependent only upon the effective mass and compliance of the backplate. Physically, the anti-resonance of this transfer function is due to the mechanical resonance of the backplate, which prevents sound pressure from building up in the cavity.

For a Helmholtz resonator with the same dimensions as before, the pressure amplification spectrum is shown in Figure 4. The frequency response shows two resonant peaks separated by an anti-resonance.

Figure 4: Magnitude and phase of the theoretical pressure amplification transfer function of a compliant-backplate Helmholtz resonator.

3 Experimental Setup Characterization of the compliant backplate

Helmholtz resonator was conducted in the Interdisciplinary Microsystems Laboratory at the University of Florida. Four different compliant-backplate Helmholtz resonators were tested in a normal incidence impedance tube. The impedance tube consists of a 0.965 m long, 25.4 mm by 25.4 mm square duct that permits characterization in a known, plane wave acoustic field at frequencies up to 6.7

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kHz, the cut-on for the (1,0) mode in the impedance tube.

Input impedance and pressure amplification measurements were taken for the compliant-backplate Helmholtz resonator for a range of backplate thicknesses. In addition, scanning laser vibrometer measurements provided the low frequency mode shapes. For each set of measurements, the resonator was mounted flush to the end of the impedance tube, as shown in Figure 5. This setup permits simultaneous measurements of resonator acoustic impedance, pressure amplification and mechanical mode shape.

Cav. Mic

Incident Mic

Mic. 1Mic. 2

1 in

Compliant backplate

LV

Figure 5: Schematic of impedance tube terminated by compliant-backplate Helmholtz resonator.

Four microphones were used for simultaneous acoustic pressure measurements. Two Brüel and Kjær (B&K) type 4138 microphones were flush mounted in a rotating plug to the side of the impedance tube, as shown in Figure 5. The plug allowed for convenient microphone switching between impedance measurements to average out any amplitude or phase calibration differences between the microphones. The other two microphones were used to measure the pressure amplification transfer function. One microphone was flush mounted in the sidewall of the resonator cavity to measure the cavity pressure. The second microphone was flush mounted to the end face of the impedance tube to measure the acoustic pressure incident to the resonator. This microphone also served as a reference to ensure a constant SPL at the neck of the resonator.

The microphones were connected to a Brüel and Kjær PULSE Multi-Analyzer System Type 3560. The PULSE system served as the power supply and data acquisition unit for the microphones, and also generated the source waveform. The source waveform consisted of a periodic random signal that was band-limited between 1 kHz and 7.4 kHz. This signal was fed through a Techron 7540 power supply amplifier to a JBL Pro 2426H compression driver. The compression driver was connected to a tapered transition piece and mounted to the far end of the impedance tube. The transition piece served to couple the circular throat of the compression driver to the square duct of the impedance tube.

The Helmholtz resonators were constructed of modular aluminum plates. The modular design allows for parts to be interchanged for testing a variety of resonator geometries. The front plate consists of a 109 mm x 71.1 mm x 3.175 mm aluminum plate. It contains one 4.6 mm diameter, 3.175 mm deep hole that serves as the neck of the resonator. The cavity plate contains a 12.7 mm diameter, 15.2 mm deep hole that serves as the resonator cavity. The compliant backplates were constructed of thin aluminum shim stock, ranging in thickness from 0.127 mm down to 0.025 mm. To provide proper clamping of each compliant backplate, a 6.35 mm thick, 25.4 mm diameter ring containing a 12.7 mm diameter hole was mounted to the backside of each compliant sheet and tightened against the cavity plate.

To determine the normal incidence acoustic impedance, frequency response measurements were taken using the two microphones in the rotating plug. The frequency response was then used with the multi-point method [8] to determine the acoustic impedance. For the pressure amplification and impedance frequency response measurements were obtained at 3200 bins from 0 Hz to 6.4 kHz, yielding a binwidth of 2 Hz with 8000 averages. Data below 1 kHz was discarded, as there was no source excitation below 1 kHz.

To measure the mode shapes for each of the backplates, a Polytec Vibrascan Laser Vibrometer with an OFV 055 vibrometer scanning head was used. The scanning head was controlled by a Polytec OFV 3001S vibrometer controller. The backplate was deflected under a sinusoidal acoustic pressure of 125 dB (re 20 µPa) at 1 kHz. Due to driver efficiency issues, the compression driver was limited to frequencies greater than or equal to 1 kHz. The excitation frequency of 1 kHz was chosen to stay within this limitation while also staying well below the first resonance of each backplate, which was greater than or equal to 2246 Hz. By staying below this frequency, the assumption of quasi-static mode excitation was expected to hold well.

4 Experimental Results and Discussion Four backplates were characterized to obtain an

estimate of the acoustic behavior of the resonator for various configurations. Normal-incidence, specific acoustic impedance and pressure amplification transfer functions and quasi-static mode shapes were taken for each configuration. The impedance data is necessary to verify the model parameters and to aid in the design of the electromechanical acoustic liner. The pressure amplification spectrum is useful to provide an estimate of resonant frequencies, while

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the mode shape data verify the assumption of clamped plate behavior.

The mode shapes of each of the clamped circular backplate are shown in Figure 6. The measured mode shapes match the theoretical predictions in Eq. {7} for the static mode shape of a clamped circular plate, thus verifying proper clamping and plate behavior.

1 0.5 0 0.5 1 0

0.2

0.4

0.6

0.8

1

0.025mm data 0.051mm 0.076 mm0.127mm Predicted mode shape

r/a

w(r

)/w

(0)

'

Figure 6: Measured mode shapes for each backplate along with a theoretical mode shape for a clamped circular plate.

4.1 Resonator with 0.127 mm thick plate Experimental results are shown below for the

Helmholtz resonator possessing a 0.127 mm thick backplate. The magnitude and phase of the pressure amplification transfer function obtained for this backplate is shown in Figure 7 for an incident acoustic pressure of roughly 80 dB (re 20 µPa). The experimental data is overlaid with two theoretical curves, based on Eq {12}. Because of some uncertainty in the measurement of the backplate thickness, two theoretical curves are shown. The two curves reflect the 95% confidence integral estimates of the thickness. The results show a behavior similar to a conventional rigid walled resonator, with a single peak occurring in the frequency range tested. This peak occurred near 1996 Hz. Due to the small compliance of this backplate, the theoretical results were not sensitive to the uncertainty in the backplate thickness. The setup did not permit measurement of the second resonant peak and anti-resonance for this backplate, as it occurred beyond the cut-on for the (1,0) mode in the impedance tube.

Figure 7: Plot of the magnitude and phase of the pressure amplification obtained for the Helmholtz resonator with the 0.127 mm backplate.

The reflection coefficient was measured using the two-microphone method. The specific acoustic impedance was then found from the measured reflection coefficient. The real and imaginary components of the normalized specific acoustic impedance are shown in Figure 8. Two theoretical curves are also shown, each one corresponding to the maximum and minimum expected thickness as previously described.

In Figure 8, the reactance passes through zero at the resonance frequency shown in Figure 7. The specific acoustic resistance at this frequency was found to be around 0.7. Note the broadband increase in the normalized resistance between 2000 Hz and 6000 Hz. The disagreement between prediction and experiment may be due a nonlinear resistive effect. This error may also arise from a systematic uncertainty in the impedance measurement at near unity reflection coefficients.

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Figure 8: Plot of the normalized, specific acoustic resistance and reactance for the Helmholtz resonator with a 0.127 mm backplate.

4.2 Resonator with 0.076 mm thick plate

The same experiments were performed on the Helmholtz resonator possessing a 0.076 mm thick plate. The pressure amplification frequency response function is shown below in Figure 9. This backplate is sufficiently compliant to generate the two amplification peaks and the anti-resonance within the tested frequency range. The second resonant frequency is 4274 Hz and the anti-resonance is visible at 4104 Hz. In addition, the first peak has shifted down to about 1934 Hz.

Figure 9: Plot of the measured magnitude and phase of the pressure amplification obtained for the Helmholtz resonator with the 0.076 mm backplate.

The normalized specific acoustic resistance and reactance are shown in Figure 10. A sharp drop occurs in the reactance between 4068 Hz and 4254 Hz. A broadband increase in resistance is also observed, but an additional local peak and local minimum occurs due to the backplate resonance. From both the pressure amplification and impedance frequency response function, the theoretical curves over-predicted the actual resonant frequencies of the Helmholtz resonator by at least 11%. Similar issues exist with orifice impedance nonlinearities and measurement uncertainty.

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Figure 10: Plot of the specific acoustic resistance and reactance for the Helmholtz resonator with 0.076 mm backplate.

4.3 Resonator with 0.051 mm thick plate The pressure amplification frequency response

function of the Helmholtz resonator possessing a 0.051 mm thick backplate is shown in Figure 11. The first resonance has shifted down to 1912 Hz. Furthermore, the pressure amplification anti-resonance, and second resonance peak have shifted down to around 3912 Hz and 4112 Hz, respectively.

Figure 11: Plot of the magnitude and phase of the pressure amplification obtained for the Helmholtz resonator with the 0.051 mm backplate.

The normalized specific acoustic resistance and reactance are shown in Figure 12. For this backplate, the sharp drop in the reactance corresponds to the second resonance peak. The theoretical curves under-predicted the actual resonant frequencies of the Helmholtz resonator by at least 5.8%.

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Figure 12: Plot of the measured specific acoustic resistance and reactance for the Helmholtz resonator with the 0.051 mm thick backplate.

4.4 Resonator with 0.025 mm thick plate

The most compliant backplate tested had a thickness of 0.025 mm. The pressure amplification frequency response function is shown in Figure 13. The first resonance peak occurs around 1325 Hz, while the anti-resonance and second peak occur near 1860 Hz and 2730 Hz, respectively. The local extrema agree well with the normalized impedance data shown in Figure 14. The predicted resonance frequencies agree well with the data in this case. Furthermore, the predicted normalized impedance more closely agrees with the measurements than in previous cases. This is likely due to reduced nonlinear orifice effects.

Figure 13: Plot of the magnitude and phase of the pressure amplification spectrum obtained for the Helmholtz resonator with the 1 mil backplate.

5 Conclusions and Future Work A lumped element model of a compliant

backplate Helmholtz resonator has been developed and compared with experiment. Pressure amplification and impedance frequency response functions were obtained for four different backplate thicknesses. Overall, good qualitative agreement was found between the lumped element model and measurements. The primary disagreement between theory and experiment is hypothesized to be due to either nonlinear impedance effects or a systematic measurement error at near-unity reflection coefficients. Interestingly, the best agreement is found for the most compliant diaphragm and may possibly correspond to reduced nonlinear effects. Future work will test this hypothesis. In addition, a nonlinear orifice resistance will also be incorporated into the model.

Work is currently underway to extend this analysis and characterization to an electromechanical

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acoustic liner. The piezoelectric-composite backplate in this liner allows the compliance to be altered with electrical components to shift the resonant frequencies and tune the impedance.

Figure 14 : Plot of the measured specific acoustic resistance and reactance of the Helmholtz resonator with a 1mil thick backplate.

6 Acknowledgements Financial support for this project was provided

by NASA Langley Research Center (Grant #NAG-1-2261) and is monitored by Dr. Michael G. Jones.

7 References 1. Bielak, G.W., J.W. Premo, and A.S. Hersh,

Advanced Turbofan Duct Liner Concepts. 1999, NASA-LaRC.

2. Lindsay, J.F. and S. Katz, Dynamics of Physical Circuits and Systems. 1978, Champaign, Illinois: Matrix Publishers, Inc.

3. Rossi, M., Acoustics and Electroacoustics. 1988, Artech House: Norwood, MA. p. 245-308.

4. Hunt, F.V., Electroacoustics: The Analysis of Transduction, and Its Historical Background. 1982: Acoustical Society of America.

5. Ingard, K.U., "Notes on Sound Absorbers (N3)," 1999, Kittery Point, ME., p. 80.

6. Blackstock, D.T., Fundamentals of Physical Acoustics. 2000: John Wiley & Sons, Inc.

7. Timoshenko, S.P. and S. Woinowsky-Krieger, Theory of Plates and Shells. 2nd ed. ed. 1959, New York: McGraw-Hill.

8. .Jones, M.G. and T.L. Parrott, Evaluation of a Multi-Point Method for Determining Acoustic Impedance. Mechanical Systems and Signal Processing, 1989. 3(1): p. 15-35.