PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Increased Radiation Efficiency Using Band Gap EffectJaesoon JUNG1; Cheol-Ho JEONG1; Jakob S. JENSEN212

Department of Electrical Engineering, Technical University of Denmark, Denmark

Department of Mechanical Engineering, Technical University of Denmark, Denmark

ABSTRACTPeriodic arrangements of elements, such as scatterers, local resonators, and inertial amplifications can induce
band gap phenomena, which result in vibration isolation in mechanical systems due to a significant reductionin the energy transmission from source to receiver. This effect has led to vibro-acoustic applications focusing
mainly on the reduction of structure-borne noise. Here, we propose to broaden the application range byutilizing the band gap effect to increase the efficiency of sound radiation from vibrating mechanical structures.
The increase in radiation efficiency is caused by the enhanced vibration localization near the excitation point.
In order to illustrate the phenomenon, a one-dimensional mass-spring model with periodically placed local
resonators is examined and with this theoretical model vibration localization and sound radiation efficiency
are analyzed.
Keywords: Band gap, Sound radiation, Radiation efficiency, Plate

1. INTRODUCTION
Passive control of vibro-acoustic response by utilizing novel materials has long been an interesting
subject of research. As a promising candidate, the metamaterials based on the band gap phenomenon,
which is realized by periodically arranging local resonators, has received great attention recently [13]. Liu et al. demonstrated that the band gap-based metamaterial can reduce the structure-borne noisetransmission significantly. This metamaterial was fabricated with a periodic resonator composed of alead ball and a silicone rubber coating [1]. After this pioneering work, the band gap has led to many
vibro-acoustic applications focusing mainly on the reduction of structure -borne noise [2,3]. For theseapplications, it is generally regarded that the reduction of vibration results in the reduction of noise.
However, the efficiency of sound radiation has received less attention , although it plays an importantrole in the interaction between vibration and sound. Furthermore, in a previous study, it was foundthat the efficiency of sound radiation may increase and may thus neutralize the benefit of band gap
for noise reduction [4].
Based on the increased radiation efficiency, in this work, we propose a way to broaden theapplication range by utilizing the band gap to enhance sound radiation. The key idea of this approachis confining vibrations at a localized zone using the band gap. Fig. 1 describes the problem in which
resonators bound the sound radiation zone with spatial periodicity of a. To illustrate this idea, atheoretical model based on a thin plate with periodically placed single degree of freedom resonators
is developed in Section 2. Using the theoretical model, characteristics of vibration and sound radiation
are examined in Section 3. To support the theoretical study, an experimental validation is carried out
in Section4. Finally, Section 5 concludes this work.

1

jjung@elektro.dtu.dk4227

Figure 1 Description of the problem

2.

THEORETICAL BACKGROUND

2.1 Structure modelling
We consider a periodic unit cell structure, which is composed of a thin plate and a mass-spring
resonator as described in Fig. 2(a). Assuming a harmonic motion, free vibration of the plate can be
described with thin plate theory as,Z 2 U w  D 4 w 0,(1)where ω is the angular frequency, w is the vertical displacement, ‫׏‬4 is the biharmonic operator, ρ isthe mass density, and D is the bending rigidity (D = Eh 3 /12(1-ν)) which is expressed with the elastic
modulus E, the thickness h, and the Poisson’s ratio ν. Furthermore, the equation of motion of themass-spring resonator can be expressed as,

Z 2 mr wr  kr wr  wp0,(2)

where mr and kr are the mass and stiffness of the resonator, wr and w p are dofs of the resonator and theplate at the connection point. Employing the finite element method, Eq. (1) can be discretized andcombined with Eq. (2) as a matrix equation as,

Z 2M  K w 0,

(3)

where M, K are the mass and stiffness matrices, and w is the displacement vector.
With the assumption that the unit cell structure is infinitely periodic, wave functions propagatingin the band gap structure follow the Bloch theorem [3], which is stated as,w raw r e jka ,(4)where r is a position vector inside a unit cell, a is the lattice vector (a = (ax ,a y)), and k is the bending
wave vector (k = (k x,k y)). The term e jka represents the amplitude and phase change, which isdetermined by the wave vector k and lattice vector the a.
Combining Eq. (3) and Eq. (4), a dispersion equation which defines the relation between wave
vector k and frequency ω can be obtained as [2,3],

Z 2M k  K k w

0,(5)where the tilde symbol represents the modification of matrices.
To obtain a dispersion relation for the band gap structure, Eq. (5) needs to be solved for every
possible combinations of wave vectors, but due to symmetry, we may confine the wave vectors to the4228

irreducible Brillouin zone (IBZ) which is described in Fig. 2(b). Usually the wave vectors are further
reduced to the boundary of the IBZ, (i.e., Γ→X, X→M, and M→ Γ) because the dispersion curve
obtained by solving Eq. (5) along the contour contains sufficient information to identify band gaps.

Figure 2 Unit cell structure (a) and wave vector space (b)2.2 Sound radiation modelling
The sound pressure p, which is radiated from a baffled and flat plate can be calculated based on
the Rayleigh integral as [5],jZU0 ve jkR
dS ,2S ³S R r, r0p(6)where ρ 0 is the mass density of air (1.2 kg/m 3 ), k is the wave number of acoustic wave (=ω/c0 , c 0 =343 m/s), R is the distance between the field point r 0 and the surface point r, v is the surface normal
velocity, and S is the surface area. Here, the plate is considered stiff enough to neglect the air loading.
Using Eq. (6) the radiation efficiency [5] which is dimensionless quantity measuring the efficiencyof sound radiation can be calculated as,V3,
U0c0 ³ vv*dS(7)­ pv* ½
Re
³S ®¯ 2 ¾¿ dS ,

(8)

S

3

where Π is the sound power and the asterisk denotes the complex conjugate.

3. SIMULATION RESULTS
3.1 Simulation conditions
Using the theoretical model, simulations are performed to examine the vibration and soundradiation features of the band gap structure. A thin acrylic plate with dimensions of 800 mm × 600mm × 3 mm is considered. The material properties of the acrylic are elastic modulus of 3.1 GPa,
Poisson’s ratio of 0.38, mass density of 1165 kg/m 3, and a loss factor of 0.2. To create a band gap,
768 resonators are periodically distributed on the plate with the periodicity of 25 mm. The resonatorparameters (i.e., mr = 5.2 g, k r = 18.47 kN/m) are determined to adjust band gap at the frequency rangeof interest (i.e., 300-500 Hz). Every edges of the plate are clamped and a surface force of 1 N/m2 isapplied at the center. Finally, a square sound radiation zone is created at the center of the plate byremoving resonators.
3.2 Band gap property
The propagation of bending waves is analyzed with the dispersion curve which describe the relation
between frequency and wave vector. Using Eq. (5), the dispersion curve is obtained as shown in Fig.

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3(a). In this figure, the band gap, which mismatches with wave vectors, is identified ranging from 300
Hz to 549 Hz. At the upper bound of the band gap, i.e., 559 Hz, the flexural wave dispersion coincides
with the acoustic wave dispersion since the periodic resonators slow down the speed of propagationof flexural waves. Around this frequency, sound radiates effectively due to the matching between
acoustic wavelength and flexural wavelength, which is known as the coincidence effect.
In the finite periodic structure, the band gap can be analyzed with eigen frequency distribution asshown in Fig. 3(b). In contrast to the distributed eigenfrequency spectrum of the bare structure, theband gap structures have separated eigenfrequency spectra. The fully filled band gap structure has noeigenfrequencies in the band gap range where no standing waves are formed. When the square zone
is created by removing resonators, several discretely located eigenfrequencies intrude the empty band.
These are localized modes. The small zone with 50 mm edge length has a single localized mode at
487 Hz whereas the larger zone with 100 mm edge length has double localized modes at 348 Hz and506 Hz. The formation of the localized modes is determined by the parameters of the zone for example,dimensions, material properties, and shape.

Figure 3 Dispersion curve (a) and eigenfrequency distributions (b)Fig 4(a) displays the mode shapes of the larger zone with 100 mm edge length. For an ordinary
plate structure, as the order of the mode increases mode shapes become distributed and complicated.
However, the localized modes are simple and advantageous for sound radiation due to the small
cancellation between acoustic waves in the near field. To verify this concept, modal radiation
efficiencies, which is calculated using the mode shape vectors and Eq. (7) is presented in Fig. 4(b).
The localized modes B and C show logarithmically increasing radiation efficiency at the frequency
range from 100 to 1000 Hz. This smooth behavior attribute to the small interference of acoustic waves
due to the simple mode shapes. On the other hand, the distributed mode shapes at A and D show
fluctuating radiation efficiencies caused by a large interference of acoustic waves. For frequencies
below 500 Hz, the localized mode B outperforms other modes with highly increased radiation
efficiency. Thus, the localized mode can be used to enhance sound radiation at the frequency of interestg g from 300 to 500 Hz.
rangingFigure 4 Mode shapes of the 100 mm zone (a) and modal radiation efficiency (b)4230

3.3 Sound radiation
To analyze the sound radiation of the band gap structure, sound pressure levels (SPLs) 1 m awayfrom the center of the plate are calculated using Eq. (6). Fig. 5 displays SPLs for four different cases.
For small zones (Full and 50 mm), the SPLs are significantly reduced but gradually increase within
the band gap. On the other hand, for the larger zone with 100 mm edge length, the SPL increases overthe bare structure within the band gap frequency. For this case, the SPL peak appears at 348 Hz whichcorresponds to the frequency of localized modes. The smooth SPL response is obtained due to thesmall number of peaks.

Figure 5 Sound pressure levels for different cases
At the lower bound of the band gap (300 Hz), dips appear for the cases of small zones (full and the50 mm). These dips can be interpreted with the cancellation of acoustic waves on the surface due tothe out of phase vibration induced by the resonator. Fig. 6(a) illustrates the cancellation by describing
the vibration velocity field of the full band gap structure at 300 Hz. These cancellation effects are
alleviated for the large zone (100 mm) where the out of phase vibration is distant from the center.
At the upper bound of the band gap (548 Hz), common peaks appear for band gap structures. Thesepeaks match with the coincidence between acoustic and flexural wave s as shown in Fig. 3(a). The
uniform vibration velocity field displayed in Fig. 4(b) supports the occurrence of coincidence effectat the upper bound of the band gap.
The radiation efficiencies of band gap structures are displayed in Fig. 7. The peaks and dips of theradiation efficiency at the upper and lower bounds of the band gap are observed. For the large zone,y increases byy 10 dB above 400 Hz in comparisonp
the radiation efficiency
with the bare structure.

Figure 6 Vibration velocity field at the dip frequency (a) and peak frequency (b)4231Figure 7 Radiation efficiency4. CONCLUSIONSIn this work, we have demonstrated the increased efficiency of sound radiation using a band gapstructure. A theoretical model is developed based on the thin plate theory and the Bloch theorem.
Mass-spring resonators are periodically arranged on the plate to create the band gap ranging from 300
to 549 Hz. The sound radiation from the vibrating band gap structure is modelled using the Rayleighintegral formulation. From simulations, we found that the band gap structures localize vibration at acertain zone and the localized vibrations amplify the sound pressures with small number of resonances.
These effects lead to the smoothed frequency response and the increased efficiency of sound radiation.

ACKNOWLEDGEMENTS
This work was supported by the Technical University of Denmark (Signature project).

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