Transmission loss of membrane-type acoustic metamaterials with...

Please cite this article as: Naify, Christina J., Chia-Ming Chang, Geoffrey McKnight, and Steven Nutt.

"Transmission loss of membrane-type acoustic metamaterials with coaxial ring masses." Journal of

Applied Physics 110, no. 12 (2011): 124903. http://dx.doi.org/10.1063/1.3665213 1

Transmission loss of membrane-type acoustic metamaterials

with coaxial ring masses Christina J. Naify1*, Chia-Ming Chang2, Geoffrey McKnight2

, Steven Nutt1

1. Department of Materials Science, 3651 Watt Way, VHE 602, University of Southern

California, Los Angeles, California 90089, USA

2. HRL Laboratories, 3011 Malibu Canyon Rd, Malibu, California 90265-4797, USA

Abstract: Transmission loss (TL) of membrane-type locally resonant acoustic metamaterials

(LRAMs) with added ring masses was analyzed using both finite element analysis and

experimental techniques. LRAM structures traditionally display high TL (over 40 dB) over a

narrow frequency band. TL peak frequency in membrane-type LRAMs is tuned by variations in

mass and membrane properties, with the objective of increasing the breadth of the TL peak. The

addition of a ring mass to the structure either increased the bandwidth of the TL peak or introduced

multiple peaks, depending on the number of rings, the distribution of mass between the center and

ring masses, and radii of the rings. The addition of ring mass increased the TL frequency

bandwidth by up to 360% (at a sound level 30 dB) compared to a structure of equivalent overall

weight with only a central mass. Finite element analysis was used to predict TL behavior of several

ring configurations, and TL for these configurations was measured to validate the model

predictions. Finally, finite element analysis was used to predict the mode shapes of the structure

under single-frequency excitation to understand the mechanisms responsible for the TL peaks.

Key words: Field emitters, cell membranes, acoustic resonance, finite element methods, materials

properties

*Corresponding author: [email protected]

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1. Introduction

Locally resonant acoustic metamaterials (LRAMs) have been developed recently which display

useful sound attenuation properties over narrow frequency bands [1-7]. LRAMs are heterogeneous

materials capable of blocking sound transmission at frequencies several orders of magnitude less

than the Bragg frequency. One of the main advantages of LRAMs is the ability to prevent sound

propagation at low frequencies (below 1 kHz) without the addition of significant mass or bulk.

Three-dimensional LRAM structures typically consist of a hard core material surrounded by a

soft rubber shell and supported by a rigid frame. In contrast, the development of two-dimensional

membrane-type LRAM (mass-weighted membranes) has led to structures that exceed the acoustic

mass law by 4× [8, 9]. Although a single cell of LRAM typically attenuates sound over only a

narrow frequency range, arrays of cells have been shown to provide multi–transmission loss (TL)

peak profiles [10, 11].

In addition to attempts to establish governing principals, the focus of LRAM studies has been

largely concentrated on establishment of computational techniques to predict results and to better

understand the physical mechanisms of the structures [12-14]. Since the initial establishment of

LRAMs and exploration of the governing principals, studies have expanded the possibilities of the

structures by using different geometries to increase the number of TL peaks available from a single

cell.

Although analysis of the structures has been shown to be increasingly accurate, limitations of

the effectiveness of the structures persist. The TL profile of a single cell of LRAM traditionally

consists of a single peak with limited frequency bandwidth. In an attempt to address some of these

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limitations, Hirsekorn et al. [15] presented a simulation of LRAM with elliptical masses (instead

of circular), which exploited the asymmetry to split the TL peak into degenerate, separate peaks.

Additionally, Gu et al. [16] showed that orientation of elliptical inclusions affected the resonance

frequency. Acoustic absorption of LRAMs has also been explored, with results showing absorption

peaks for which the frequencies are tuned by variation of mass magnitude [17]. As an additional

geometric variation, Larabi et al. [18] demonstrated that, by using alternating multi-coaxial

cylinders of soft rubber and hard material (steel), multiple TL peaks could be achieved, for which

the peak frequencies depended on the physical parameters of the layers.

Analysis of geometric variations in LRAMs has demonstrated effectiveness in producing

multiple peaks from a single cell. These studies, however, were presented without experimental

validation. Additionally, the performance of membrane-type LRAMs with variations in geometry

has not yet been explored. In this study, ring masses were added to traditional membrane-type

LRAMs. Finite element analysis was used as a predictive tool to determine ideal geometric

parameters, including the number of rings, the distribution of mass between the center and ring

masses, and the ring radii. Once desired geometries were determined analytically, experimental

validation was performed to measure the TL of samples with equivalent geometries. Variations of

the ring mass geometries in the structure addressed two key practical limitations regarding

LRAMs—the single-peak TL profile and the narrow frequency bandwidth of the peak.

The organization of this paper includes details of sample construction, finite element analysis,

and experimental evaluation in Sec. II. Section III presents the results obtained both analytically

and experimentally, while discussion of these results is reserved for Sec. IV. Conclusions and

implications of the results are laid out in Sec. V.

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2. Methods

2.1 Sample construction

The basic structure for membrane-type LRAMs consists of a membrane (polyetherimide) stretched

in tension, a support structure, and a centrally located mass. Tension was introduced into the

membrane during the structure fabrication process. A heat-activated adhesive was used to bond the

membrane to the support structure, and the difference in thermal expansion coefficient between the

membrane and support generated membrane tension. Mass was added to the center of the structure,

and the out-of-plane displacement of the membrane was measured using a position-sensitive

detector. Details of the calculation relating the membrane tension to out-of-plane displacement of

the membrane can be found in Refs. 9 and 19. The support structure used was a 1-mm-thick glass-

fiber/epoxy composite ring designed to fit snugly in the testing apparatus. Properties of the

membrane material used for both experimental measurement and finite element analysis (FEA)

simulation are shown in Table I. The central mass had a diameter of 3 mm and thickness of 1 mm,

while properties of the ring masses (18-8 stainless steel) used are shown in Table II.

Table 1: Properties of membrane material.

Property

Polyetherimide (PEI)

Diameter (mm) 24

Thickness (mm) 0.076

Tension (Pa) 5.7 x 106

Modulus (Pa) 3.6 x 109

Density (kg/m3) 1200

Poisson’s ratio 0.36

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Table 2: Properties of ring mass configurations 1A-1E (addition of multiple rings).

Configuration

Central mass (g)

Ring 1 mass (g)

Ring 2 mass (g)

Ring 3 mass (g)

Ring 1 radius: inner, outer (mm)



1A 0.16 -- -- -- -- -- --

1B 0.16 -- -- -- -- --

1C 0.16 0.16 -- -- 4, 5 -- --

1D 0.16 0.16 0.16 4, 5 7, 8 --

1E 0.16 0.16 0.16 0.16 4, 5 7, 8 10, 11

In this study, a geometric variation was introduced to membrane-type LRAMs in the form of a

ring mass bonded at a given radius on the membrane. Figure 1 shows a schematic of the

membrane-type LRAMs with a) one attached ring mass (radius 4 mm) and b) two attached ring

masses (radii 4 mm and 7 mm). Both the central mass and ring masses were bonded to the

membrane surface using an adhesive tape and aligned to be concentric with the membrane center.

Figure 1: Schematic of single cell with a) one ring mass and b) two ring masses. (A)

Support structure, (B) membrane, (C) central mass, (D) ring mass 1, (E) ring mass 2.

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2.2 Finite element analysis

To optimize the design of the ring geometries, finite element analysis (COMSOL,

multiphysics) was used to calculate the predicted TL of the structures. An axisymmetic model was

used for the analysis to facilitate calculation intensity. The initial pressure magnitude incident on

the structure was set at 1 Pa, and all physical properties (structure size, material properties) were

equivalent to those used experimentally (see Table I). In the analysis, a rigid boundary condition

was imposed at the edge of the support ring, consistent with the clamped boundary condition used

experimentally. The effects of damping were not accounted for in the analysis. Figure 2(a) shows a

schematic of the axisymmetric FEA model.

Figure 2: (Color online) FEA schematic of a) axisymmetric layout, including central

mass, one ring mass, support ring, and membrane as well as boundary conditions and

direction of incident sound and b) ring height adjustment with increasing radius.

The FEA analysis used to calculate TL treated the impedance tube setup as an acoustic-

structural interaction. The impedance tube walls were rigid, and the central mass, ring mass, and

membrane structures were modeled with freetriangular mesh elements. Pressure amplitudes at each

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end of the tube (upstream (pinc) and downstream (pt)) were calculated, and TL was then determined

using Eq. (1).

(1)

Ring masses were bonded to the membrane at various radii to investigate the effect of radius

variation. To maintain desired mass magnitudes of the rings relative to the center mass magnitude,

the heights of the rings were adjusted with variation in radius. The height of the ith ring (ti,ring) was

determined using the desired mass magnitude, m, desired radius, ri,inner, and material density ρ

according to Eq. (2).

(2)

A schematic of the variation of ring height with increasing radius is shown in Fig. 2(b). The width

of all of the rings used (router-rinner) was kept constant at 1 mm.

Tables II–IV show the properties (mass and diameter) of the rings simulated in each

configuration. The various configurations can be classified in three groups—addition of multiple

rings, variation in mass distribution, and variation in ring radius. Table II shows configurations

1A–1E, in which multiple rings of equal mass magnitude (up to three) were added to the structure.

Configuration 1A featured a central mass of 0.16 g, while configuration 1B included only a ring

mass (4 mm radius, 0.16 g). The mass distribution for configuration 1C consisted of a center mass

(0.16 g) and a single ring mass (4 mm radius, 0.16 g). Configuration 1D consisted of a center mass

(0.16 g) and two ring masses (4 mm radius, 0.16 g and 7 mm radius, 0.16 g). Configuration 1E

included the addition of a third ring with 10 mm radius (0.16 g) along with the rings from

configuration 1D.

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Table 3: Properties of ring mass configurations 2A and 2B (variation in mass

distribution).

Configuration

Central mass (g)

Ring 1 mass (g)


2A 0.24 0.24 4, 5

2B 0.08 0.40 4, 5

Table 4: Properties of ring mass configurations 3A–3C (variation in ring radius).

Configuration

Central mass (g)

Ring 1 mass (g)


3A 0.08 0.40 6, 7

3B 0.08 0.40 8, 9

3C 0.08 0.40 10, 11

Configurations 2A and 2B (Table III) each included one central mass and one ring mass with

radius of 4 mm, as well as a total mass (central plus ring) of 0.48 g. The difference between these

two configurations was the distribution of the masses. In configuration 2B, the ring mass

magnitude was 5× the magnitude of the central mass, while the mass in configuration 2A was

divided equally between the central mass and the ring (0.24 g each).

The mass distribution between center mass and ring mass in configurations 3A–3C (Table IV)

was constant (0.08 and 0.40 g, respectively), while the single ring radius was increased from 6 mm

to 10 mm.

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2.3 Experimental validation

Experimental measurements of the TL of the structures was performed using an impedance

tube (Brüel and Kjær model 4206, ASTM E2611-09, standard test method for measurement of

normal incidence sound transmission of acoustical materials based on the transfer matrix method

[20]). The inner diameter of the tube was 29 mm, and the structures were excited using a

broadband sound source over a frequency range of 100–6400 Hz. Samples were attached to tight-

fitting rings inserted in the tube to achieve a consistent, clamped boundary condition for each test.

TL was calculated using software which employed the transfer matrix method (PULSE software,

B&K). The impedance tube setup used to measure TL featured four microphones with an anechoic

termination at the end of the tube opposite the speaker.

3. Results

To provide a baseline comparison for the ring-mass structures, a membrane-type LRAM

structure is analyzed in which all of the mass is concentrated at the center of the structure (no

attached ring mass). Figure 3 shows the FEA-predicted TL for such a structure with center mass of

0.48 g. The TL profile for this configuration exhibits a single low-frequency minimum (288 Hz), a

single high-frequency minimum (3.458 kHz), and a single TL peak (436 Hz). The TL peak

corresponds to anti-resonance behavior, where minimal sound is transmitted across the structure,

while the TL minima correspond to resonance frequencies, where ∼100% of the sound is

transmitted.

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Figure 3: FEA-predicted TL of a structure with 0.48 g mass in center (no attached ring

mass).

As presented in previous studies [9], the frequency of the first resonance was dominated by

center mass magnitude along with membrane tension, while the second resonance frequency was

dominated by membrane mass and thickness. The frequency of the TL peak was tuned by varying

both membrane and mass properties. Increasing the magnitude of the mass resulted in a decrease in

TL peak frequency, while decreasing the membrane radius resulted in an increase in TL peak and

resonance frequencies.

Figure 4(a) shows FEA results of predicted TL for two additional structures. The first had a

central mass of 0.16 g (configuration 1A), while the second had no central mass, but instead only a

ring mass of radius 4 mm and mass 0.16 g (configuration 1B). The combination of these two mass

distributions formed the distribution in configuration 1C. The TL profiles of configurations 1A and

1B each show a single TL peak and two resonance dips. The frequencies of the peak and resonance

dips for configurations 1A and 1B are shown in Table V along with those of configuration 1C.

Additionally, acoustic mass law-predicted TL behavior is shown in Fig. 4(a) for a single partition

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structure with surface density equivalent to that of configuration 1A, 1B (0.16 g added to the

membrane), and configuration 1C (0.32 g added to the membrane). The acoustic mass law TL was

calculated using Eq. (3) [21]

(3)

where ω = 2πf, f is the frequency, ρs is the surface density of the samples (density per unit

thickness), and ρo and c are the density and speed of sound in the surrounding fluid (air),

respectively. The TL peak for configuration 1A increased 56 dB over the mass law prediction,

while the TL peak for configuration 1B increased 46 dB. The TL peak magnitudes for

configuration 1C were 36 dB and 49 dB above the mass law prediction.

Figure 4: (Color online) a) FEA results showing TL of configurations 1A (central mass

only), 1B (ring mass only), and 1C (central mass and one ring mass) as well as acoustic

mass law-predicted TL. b) FEA showing TL of configurations 1C (central mass and one

ring mass), 1D (central mass and two ring masses), and 1E (central mass and three ring

masses).

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Table 5: Resonance and peak TL frequencies for configurations 1A, 1B, and 1C.

Configuration

1st resonance

(Hz)

1st peak

(Hz)

2nd resonance

(Hz)

2nd peak

(Hz)

3rd resonance

(Hz)

1A 512 794 4073 -- --

1B 724 1905 6025 -- --

1C 478 912 1380 1995 6025

The addition of a single ring mass in configuration 1C resulted in a double-TL peak profile (at

912 Hz and 1995 Hz) as well as an additional resonance dip. Subsequent addition of rings to the

structure in configuration 1C (with two TL peaks) resulted in an additional TL peak with the

addition of each ring. For example, Fig. 4(b) shows predicted TL for configuration 1D, which

exhibited three TL peaks and three resonance dips, while configuration 1E showed four TL peaks

and four resonance dips.

Figure 5 shows the predicted TL for configurations 2A and 2B, which were constructed with

equivalent total mass and ring radii, but different mass distributions. Configuration 2A, where

center mass and ring magnitude were equivalent, resulted in a two-peak profile (724 Hz and 1.659

kHz). Configuration 2B, however, featured a non-uniform mass distribution, where the magnitude

of the ring mass was 5× the magnitude of the center mass. This non-uniform distribution yielded

one single, broad peak (circle-marker in Fig. 5). Although the total mass magnitude of

configuration 2B was equivalent to the total mass of the sample in Fig. 3 (featuring a single central

mass), the resulting TL profile covered a frequency range of roughly 5.5 kHz, as opposed to 3 kHz.

The TL peak for configuration 2B occurred at 1.096 kHz.

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Figure 5: (Color online) FEA-predicted TL of configurations 2A (uniform mass

distribution) and 2B (non-uniform mass distribution).

Figure 6 shows the effect of ring diameter on TL using configurations 3(A-C). Configuration

3A displayed a wide-band, two-peak profile with a frequency span of almost 10 kHz. As the radius

of the ring was further increased, the wide-band behavior diminished, and despite the non-uniform

mass distribution, a resonance dip appeared between the two TL peaks. Demonstrating this, the TL

behavior of configuration 3C, in which the ring mass radius was 10 mm, had a frequency

bandwidth of only about 2.5 kHz.

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Figure 6: (Color online) FEA-predicted TL of configurations 3A (ring mass radius 6

mm), 3B (ring mass radius 8 mm), and 3C (ring mass radius 10 mm).

To provide validation for the FEA model, the TL of select configurations was measured and

compared. Figure 7 shows the measured TL and corresponding predicted TL profiles for

configurations 1C, 1D, 2B, and 3C. These configurations were chosen to validate each of the three

types of variations examined analytically (multi-ring, mass distribution, and ring radius). The

materials used in the experimental samples were identical to those used in the FEA model. The

measured TL values provided validation for the model predictions for each configuration, and the

resonance dips and TL peak frequencies occurred within 15% of the FEA-predicted values.

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Figure 7: (Color online) FEA-predicted (dashed) and experimental (triangle) TL

profiles for a) configuration 1C (uniform mass distribution), b) configuration 1D

(multiple rings, central mass, and two ring masses), c) configuration 2B (non-uniform

mass distribution), and d) configuration 3C (ring radius variation, radius of 10 mm).

4. Discussion

The circular membrane with a single central mass exhibited both high-order resonance and

anti-resonance behavior, as expected, although the frequencies of these anti-resonances (TL peaks)

occurred well above the membrane resonance frequency. Variations in membrane properties, such

as membrane tension, thickness, or radius, shifted the anti-resonance peak frequencies, much like

they did the low frequency TL peaks [9]. However, the shifts were relatively small, and it was not

possible to reduce the anti-resonance peaks to below the membrane resonance simply by varying

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membrane properties. Additionally, variations in the membrane properties to tune high-frequency

anti-resonances could have the adverse effect of de-tuning lower frequency TL peaks. Because of

the constraints involved in tuning anti-resonance TL peaks by varying membrane properties,

implementation of ring masses afforded an appealing means of generating multi-peak TL profiles

at low frequencies. Note, however, that despite the creation of multi-peak TL profiles, some

drawbacks are associated with the use of ring masses. In particular, the addition of the ring mass

(configuration 1C) resulted in an increase in the overall mass of the structure compared to a

structure with no ring mass (configuration 1A). Nevertheless, despite an overall increase in mass,

TL peak increase over the mass law-predicted TL for a structure with 0.32 g central mass [9, 21]

was 36 dB at the low frequency peak.

FEA predictions of TL for a membrane-type LRAM were presented and compared to

experimentally obtained values. A single cell of LRAM with mass concentrated at the membrane

center produced a one-peak TL profile with narrow bandwidth (3.5 kHz, Fig. 3). The addition of

an annular ring mass with mass equivalent to the center mass resulted in multiple TL peaks, with a

distinct peak present for each ring added. Variations of the ring geometry (mass and radius) as well

as the distribution of mass between the center mass and ring resulted in distinctly different TL

profiles.

Analysis of configurations 1A (center mass only) and 1B (ring mass only) was included to

provide comparison with configuration 1C (center mass and ring mass). Configuration 1A and 1B

both displayed a single TL peak with frequency values similar to those of each of the two peaks in

configuration 1C (Fig. 4(a)). The difference in TL peak frequencies between configurations 1A

and 1B and configuration 1C was due to the combined effects of an increased mass magnitude and

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a decrease in membrane radius between the masses and the support ring. The increase in overall

applied mass magnitude (from 0.16 g in configurations 1A and 1B to 0.32 g in configuration 1C)

caused the frequency of the first resonance to decrease. The high-frequency resonance (6.025

kHz), which was dependent on membrane properties, was equal in magnitude for configurations

1B and 1C because of the equal distance between the ring mass and the support ring (8 mm).

In configurations 1C-E, up to three rings were added to the structure. The radii of the rings

were evenly distributed across the radius of the membrane. The resulting TL profile for a three-

ring (plus central mass) structure exhibited four distinct TL peaks (Fig. 4(b)). Although the overall

mass of the structure increased with the addition of each ring, the number of TL peaks also

increased. The first resonance frequencies of configurations 1C-E were 478 Hz, 549 Hz, and 630

Hz, respectively. The variation in frequency was due to the dependence of the first resonance

frequency on center mass magnitude. As more mass was concentrated away from the center of the

structure with increasing ring additions, the first resonance frequency also increased.

The effect of mass distribution uniformity was examined in configurations 2A and 2B. A

uniform mass distribution between the center mass and the single ring (configuration 2A, Fig. 5)

resulted in a two-peak TL profile. The first TL peak corresponded to the anti-resonance dominated

by the center mass, while the second TL peak corresponded to the anti-resonance dominated by the

ring mass. The mass was redistributed in configuration 2B so that the mass of the ring was 5× that

of the center mass, and, as a result, the two TL peaks moved closer together. The ring mass TL

peak decreased and the central mass TL peak increased, eventually resulting in a single, wide-band

peak. Although the single TL peak in configuration 2B showed a wider frequency bandwidth than

the sample in Fig. 3 (single central mass), the TL peak also occurred at higher frequency (1148 Hz

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versus 436 Hz). Thus, a trade-off arises between creating a wide-band peak and achieving a large

TL magnitude (over 40 dB) at low frequencies. A wide-band TL peak at low frequencies can be

obtained either by increasing the mass magnitude of the central and ring masses or by varying the

membrane properties. For example, a decrease in membrane tension resulted in a decrease in TL

peak frequency [9]. Other geometric variations, such as decrease in membrane thickness and

increase in membrane radius, can also be employed to achieve TL peaks at low frequencies.

Structure geometries and mass distributions can be altered to achieve a desired TL outcome (multi-

peak, single peak), and these geometries can be designed and optimized using FEA.

Improvements in sound insulation resulting from the addition of ring masses can be quantified

by examining the frequency bandwidth of the TL peak. The bandwidth of the TL peak (resonance-

to-resonance) for the structure with a 0.48 g center mass (Fig. 3) was 3.2 kHz. Adding a ring mass

and using the mass distribution of configuration 2B (Fig. 5) increased the frequency bandwidth to

5.5 kHz, an increase of 72%, with no change in the structure mass. The width of the TL peak at a

sound level of 30 dB for the 0.48 center mass structure was ∼ 300 Hz, while the width of the TL

peak for configuration 2B was 1.4 kHz, an increase of 366%.

Increasing the ring mass radius to 6 mm (as in configuration 3A, Fig. 6) resulted in an increase

in the frequency band of the TL peak (8.5 kHz). As the radius was further increased to 8 mm and

10 mm, a resonance dip was introduced, and a narrowing of the TL peak frequency band occurred.

As the ring diameter approached the radius of the cell, the effect of the ring was diminished. The

increase in ring diameter resulted in a structure whose behavior approached that of a cell with a

central mass and a smaller overall membrane radius. When a single mass was concentrated at the

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membrane center, the structure displayed a large TL peak, although the width of the peak was

narrow (100 Hz).

To better understand the behavior of the structures under excitation, FEA was used to predict

resonant and anti-resonant mode shapes at discrete frequencies (Fig. 8(b)). The frequencies

selected for the mode shape calculations were those of each TL peak and resonant frequency for

configuration 2A (see Fig. 5). General mode shapes for configuration 2A are shown in Fig. 8(b)

along with a schematic of the setup and qualitative phase information about the motion of the

central mass m1 relative to the ring mass m2. Displacements were also measured using a laser

vibrometer. The vibrometer measured out-of-plane displacements at 20 points swept radially

across the structure, which was excited at discrete frequencies using the impedance tube speaker

(experimental setup is detailed in Naify et al. [9]). Mode shapes measured experimentally are

shown in Fig. 8(d).

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Figure 7: (Color online) a) Schematic showing masses, membrane, and support ring. b)

Displacement profile (FEA) of the motion of the structure with configuration 2A

(uniform mass distribution) for the three resonance and two peak frequencies. Light

gray – component displacement up, dark gray – component displacement down, black –

no displacement. Arrows indicate points of inflection of the membrane across the

neutral axis (corresponding to no displacement). c) Measured out-of-plane

displacement profile of structure with configuration 2A.

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At the first resonance dip of the structure (416 Hz), m1 and m2 vibrated in-phase, and there was

no inflection point in the membrane. This behavior was consistent with the first resonance of a

structure in which all of the mass is concentrated at the membrane center (no ring mass). The mode

shape of the first TL peak (870 Hz) was also consistent with single-mass (no ring mass) behavior

in that both m1 and m2 were in-phase and the membrane showed an inflection point across the

neutral axis. Explanation of the relationship between this mode shape and the high-magnitude TL

at the given frequency has been documented in previous studies [9].

The dynamic response of the structure at the second resonance dip, which occurred between

the two TL peaks (1150 Hz), showed m1 and m2 were mutually out-of-phase, and, although an

inflection point occurred between the two masses, there was no inflection between the masses and

the support ring. At the second TL peak (1450 Hz), m1 and m2 were mutually out-of-phase,

although the inflection point between the masses and the support ring also appeared at a radius

similar to the inflection point at the first TL peak frequency. Finally, at the third resonance dip

(6020 Hz), both the central mass (m1) and the ring mass m2 were stationary, while the membrane

showed a displacement maximum. This behavior was consistent with the single-mass (no ring

mass) behavior shown previously (Fig. 3). Out-of-plane displacement measured experimentally

matched FEA-predicted mode shapes in the general shape of the structure under single-frequency

excitation.

Experimental measurements of TL provided validation of the accuracy of the predicted TL for

each of the compared configurations. Minor discrepancies between the experimental and FEA TL

values arose because of sample construction variables and simplifications or assumptions made in

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the model (e.g., no damping). Some of the imperfections in sample construction leading to error

included positioning both the central mass and positioning ring masses to be concentric with the

support ring. The fabrication technique for the ring masses resulted in the inner and outer

diameters of the ring not being perfectly concentric.

The FEA over-predicted the magnitude of the TL peaks, and this was attributed primarily to

the exclusion of damping from the analysis. Additionally, a perfectly clamped boundary condition

was used in the model, as well as a calculated tension value, both of which could cause deviations

from measured TL profiles. Although the FEA model used yielded reasonably accurate predictions

of the measured TL profile for the single-celled axisymmetric samples, more samples with more

complex geometry, such as a multi-celled array or square cells (where axisymmetric models would

not be used) will present greater computational challenges. Although increased complexity is

likely to result in larger discrepancies between predicted and measured values, this analytical

approach should be useful for predicting overall trends in performance for membrane-type LRAMs

with attached ring masses.

5. Conclusions

Membrane-type LRAMs with ring masses were fabricated and evaluated to determine the

effects of ring geometry on the TL performance of the structures. FEA was performed on the

structure to determine favorable geometries with the goals of achieving broadband sound

insulation and/or a multi-peak TL profile from a single cell. Once desired geometries were

achieved, experimental validation was undertaken to confirm the accuracy of the predictive

capabilities of the model.

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Multiple ring configurations were analyzed with three major types of variation: the number of

rings, the mass distribution, and the ring radii. Increasing the number of rings attached to the

sample resulted in a multi-peak TL profile, in which the number of additional peaks corresponded

to the number of attached rings. Variations in the uniformity of the mass distributed between the

central mass and ring mass resulted in either a multi-peak profile (uniform mass) or a broadband

TL peak (non-uniform mass). Increasing the radius of the ring to approach the membrane radius

had a counterproductive effect on the TL performance, decreasing the TL bandwidth.

The addition of ring masses to a membrane-type LRAM proved effective in both increasing the

frequency bandwidth of the TL peak and attaining multiple TL peaks from a single cell. In

contrast, previous reports had shown that acoustic metamaterials attenuated sound waves over only

a narrow frequency range. The results presented here demonstrate that the effectiveness of this

type of membrane-type structure can be extended to a broad frequency range (from 3.2 kHz to 5.5

kHz) using only a single cell and, thus, the use of multiple layers of LRAMs (the approach

currently used for broadband insulation) may not be necessary. The use of a single layer of LRAM,

in turn, would offer both weight and space-savings in the design of the structure, critical

parameters for aerospace and automotive applications. The addition of multiple ring masses to a

cell is an approach well-suited to applications in which multiple tones require attenuation. TL

profiles for various sound insulation needs are thus achievable by geometric variation of the

membrane-type LRAM structure without addition of mass.

Acknowledgements: The authors would like to thank HRL Laboratories for support of this work.

The authors would also like to thank Matt Sneddon for technical support and Ryan van

Schilfgaarde for assistance with sample preparation.

http://dx.doi.org/10.1063/1.3665213




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