Prediction of Acoustic Emission of a Rigid Electrodes DEAP Loudspeaker
Emiliano RUSTIGHI(1) , William KAAL(2) , Sven HEROLD(2) , Łukasz JANKOWSKI(3)(1) University of Southampton, United Kingdom, er@isvr.soton.ac.uk(2) Fraunhofer-Institut für Betriebsfestigkeit und Systemzuverlässigkeit LBF, Germany, william.kaal@lbf.fraunhofer.de andsven.herold@lbf.fraunhofer.de(3) Institute of Fundamental Technological Research (IPPT PAN), Poland, ljank@ippt.pan.pl

Abstract
Dielectric Electro-Active Polymers (DEAPs) are lightweight materials whose dimensions change signiﬁcantly
when subjected to electric stimulation. One form of DEAP construction consists of a thin layer of dielectric
sandwiched between two perforated rigid electrodes. They can be used as an actuator or a sensor and have thepotential to be an effective replacement for many conventional transducers. This paper refers to their use as loudspeakers. To date, ﬂat DEAP loudspeakers have been prototyped and tested but no numerical prediction of theiracoustic performance has been presented. In this paper an elemental model is presented. The electro-dynamic
behaviour of the electrodes and dielectric layers is taken into account. The acoustic impedance is calculated assuming bafﬂed conditions. The impedances of the individual layers are stacked together and preliminary results
are shown.
Keywords: Loudspeaker, DEAP, Sound power

1

INTRODUCTION

Dielectric Electric Active Polymers (DEAP) actuators are a new type of actuators characterised by large strokes,
low weight and low cost. They have been used in different applications among which robotics, haptics, energy
harvesting, vibration control, sensing and acoustics [1]. In their classical conﬁguration, an elastomer is sandwiched between two compliant electrodes and their working principle is based on compliant electrodes: whenvoltage is supplied to the electrodes, an electromechanical pressure squeezes the composite structure [2]. Because of the electromechanical pressure, both axial and transversal deformations are experienced. To obtaina useful stroke and force range for actuation purposes, several conﬁgurations are used, which may involve
mechanical bias (e.g., an external pressure, deformable frames or internal springs) or stacking and folding approaches.
A novel design for such actuators has been recently proposed which utilises rigid-perforated metal electrodes [3].
This design shows a more homogeneous strain distribution, highly conductive electrodes and a high breakdown
voltage across a large range of deformations. It proves to be especially advantageous for applications where
only moderate strains are needed, but good dynamics and load-bearing capabilities are required. In contrast
to conventional designs, such actuators only contract in one direction, whereas all other directions remain undeformed. In this way, it is possible to build actuators made of few layers only, which are of particular interest
to realise extremely ﬂat loudspeakers.
Dielectric elastomer loudspeakers with compliant electrodes have been proposed in different conﬁgurations [1,4]. In a similar way to electrostatic loudspeakers, a thin dielectric ﬁlm radiates sound in response to an oscillating voltage. Their operation voltage is of the order of 1 kV and characterised by a bias and a modulation
component. While the bias voltage applies an electromechanical pre-straining which is beneﬁcial to the overall performances, the modulation component of the voltage is responsible for the sound generation. Compared
to traditional loudspeakers they have a smaller mechanical impedance and are able to achieve larger displacements. Hence, a high sound pressure ﬁeld can be generated with loudspeakers of moderate size. Because of
their deformation mechanisms, they are in some ways analogous to monopole sources and they can potentially7345

show better directivity performance than traditional loudspeakers. However, since the displacement is driven by
Maxwell pressure law [2], they are inherently nonlinear.
A prototype of a ﬂat DEAP loudspeaker with perforated rigid electrodes has been previously built and tested [5].
In relation to compliant electrode loudspeakers, such design does not need mechanical bias and the increased
conductivity of the rigid electrodes allows for the use of less demanding ampliﬁers and to cover the full audible
frequency range. Such novel loudspeakers are lighter, cheaper and thinner than conventional loudspeakers. In
fact, differently from conventional coil loudspeakers, no resonating sound box is needed and an extremely ﬂat
conﬁguration can be obtained. In addition, since they do not need a mechanical bias, the sound emitting ﬁlm
can be manufactured in any desired shape. To date no model is available to predict the acoustic performance ofa rigid electrodes DEAP loudspeaker. In this paper an elemental model is presented, where the electrodes aremodelled as bending plates, the dielectric as a Winkler bedding and the acoustic impedance calculated assuming
bafﬂed conditions.

2

MECHANICAL MODEL OF THE ELECTRODES

The electrodes are assumed as extensionally rigid but compliant in bending. Hence, the electrodes are modeled as uniform thin circular plates of radius a with free boundary conditions and made of homogeneous andisotropic material. The appropriate boundary condition is then applied once they are combined with the dielectric layers. The analytical expressions of the natural frequencies fmn and mode shapes φmn given in [6] havebeen used to calculate the mobility matrix of each electrode by modal summation.
In order to do so the plate is discretised in element of equal area, ΔS, following the model suggested by Arenas[7]. In order to discretise the disc in elements of the same area, the radius is ﬁrstly divided into N equally
spaced elements. Each concentric rings then in divided in a series of areas multiple of the previous. In thispaper, the central area has been divided arbitrarily in four elements so to get 4N 2 elements in total. Hence thearea of each element will be
π  a 2
(1)
ΔS =
4 N
Figure 1 shows the elemental discretisation used in this paper (N = 8) and an example of calculated point
mobility at the centre of one of the elements. The properties of the disc are given in Table 1.
The perforation of the electrodes affects both density and bending stiffness of the electrodes. It has been
assumed that the perforation has no effect on the effective Poisson ratio of the electrodes. The perforationconsidered in this paper is a regular triangular pattern as shown in ﬁgure 2. The ligament efﬁciency is thenon-dimensional parameter used to characterise the perforation density
le =

d
p

(2)

where p is the pitch of the pattern and d = p − 2R where R is the radius of the perforation holes. Suchparameter conveniently span between 1 (no perforation) and 0 (maximum perforation) the full range of possible
perforation’s densities.
The effective density of the perforated plate ρe is considered to be reduced by the perforation using a mixing
law as:(3)ρe = ρ(1 − φA )where φA is the open area ratio, which, for circular perforations, is related to the ligament efﬁciency by φA =π√
(1 − le )2 . The effect the perforation has on the bending stiffness has been taken into account using the
2 3
simpliﬁed formula proposed by Soler and Hill [8]:De = D(1 − φA )λe

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(4)

30

0.06

|Receptance| (m/N)

20

Excitation
Response

10
0

0.04

-10
-20
-30

0.02

-40
-60 1
10
100

102

10 3

10 4

(m)

-50
0

‫ ס‬Receptance (deg)

80
60

-0.02

40
20
0

-0.04

-20
-40
-60

-0.06
-0.06

-80
-100
1
10

2

10

3

10

Frequency (Hz)

4

10

(a)

-0.04

-0.02

0
(m)

0.02

0.04

0.06

(b)

Figure 1. Point mobility (a) for a freely suspended disc computed at the centre of the marked area (b).
where D is the bending stiffness of the non-perforated plate and the parameter λe is obtained as 
3−φ d 1+φ d
13 + 3φ
d 2
1+
+
λe =
1−
8
4 p
2 p
p
with

3

h
2p − 1φ= h.
2p + 1(5)(6)

ELECTRO-MECHANICAL MODEL OF THE SOFT DIELECTRIC LAYER

3.1 Effective mechanical stiffness
The dielectric layers have been simply modelled as a Winkler bedding, i.e. as a set of parallel independent
impedances each of them representing the material underneath the discretised area ΔS. No viscoelastic behaviouris assumed here. The equilibrium of one element can be obtained using the Neo-Hookean hyperelastic material
expression for natural rubber in compression [9]:
1 Yd
+ pM = f (t)
(7)
λ− 2
λ
3
where λ is the stretch ratio and Yd is the effective dielectric Young’s modulus. Because of the perforation in
the electrodes the dielectric material (natural rubber) will tend to bulge into the holes. The macroscopic resultof this is a reduction of the effective Young’s modulus of a factor ψA , i.e. Yd = ψAY , with Y the nominal
dielectric Young’s modulus. Such factor has been estimated numerically by a FEM analysis using an equivalent
axial-symmetric model of an individual cell. The results obtained have been summarised in Figure 3 where also
the inﬂuence of the thickness of the dielectric layer is included. In the previous equation, f (t) is an appliedexternal force which is zero in this application and pM is the Maxwell pressure, which is the same as that of a7347

parallel plate rigid capacitor since there is no change in the area during deformation [1]:ε0 ε V (t) 2
pM =
2
z

(8)

where ε is the relative dielectric constant of the polymer, ε0 is the dielectric constant of free space, V is theapplied voltage and z is the ﬁlm thickness. When no external force is applied, the effect of the applied voltage
is to squeeze the dielectric layer. It is then possible to compute the equilibrium thickness z̄ to an applied
constant voltage as:
3εε0 z0 2V(9)z̄ = 3 z30 −2Yd
If a small external pressure is applied to the element we can linearise around the equilibrium position to obtain
the effective mechanical stiffness as a function of V .
1
݄ௗ = 0.11 mm
݄ௗ = 0.22 mm
݄ௗ = 0.33 mm

0.8

߰஺

0.6

ܴ
݀
‫݌‬
Figure 2. Perforation geometry:radius of the perforation R, pitchof the pattern p and ligament d.

0.4
0.2
0

1

0.8

0.6

0.4

0.2

0

݈௘

Figure 3. Stiffness correction factor ψA due to the perforation of
the electrodes compressing the dielectric layer.

3.2 Voltage driven response
When the loudspeaker is driven with a simple harmonic signal of frequency ω and amplitude VAC it is also
necessary to offset this signal by a constant VDC in order to avoid the reversing of the actuation. Since thevoltage is squared in equation (8), the alternate voltage has the effect of changing the stiffness but at the sametime creates parametric stiffness terms. The two parametric stiffnesses have been here neglected since muchsmaller than the non-parametric terms. In this case the loudspeaker driven excitation fV is composed of two
frequencies at the fundamental frequency ω and at twice the excitation frequency 2ω:

εε0 
(10)
fV = −(1 − ψA )ΔS 2 V12 eiωt +V22 ei2ωt
2z̄
2 /2.
where V1 = 2VDCVAC and V2 = VAC

4

CALCULATION OF ACOUSTIC EMISSIONS

4.1 Mobility assembling approach
In order to assemble the impedance matrices of each layer a procedure similar to the frequency based substructuring [10, 11] has been adopted. The methodology had however to be adapted to this particular case. In

7348

fact, since a free-free structure includes rigid body motions, its impedance or mobility matrix is singular andrank deﬁcient, and hence the inverse of the stiffness does not exist. For this reasons care must be taken into
assembling the substructures.
The ﬁrst step is to assemble the ﬁrst electrode to a grounded dielectric, as illustrated in Figure 4 . The assembled mobility matrix YGE is obtained applying compatibility and equilibrium conditions, and it expressedas(11)
YGE = YE (I + ZG YE )−1
where I is the identity matrix, ZG is the grounded dielectric impedance matrix and YE is the electrode mobility
matrix, which are given respectively by:ZG =−ikru,el
I
ω 

1
iω
2 (1 + iη ) − ω 2
Mmn ωmn
mn

YE = Bdiag

(12)


BT

(13)

where Mmn , ωmn , ηmn and B are the modal masses, natural frequencies, modal loss factors and modal matrix ofthe electrode.
Once the mobility of the ﬁrst dielectric electrode is obtained it is possible to assemble the next dielectric layer
where the internal dofs can be neglected in order to keep low the size of the ﬁnal mobility matrix. Figure 5
gives an illustration of the process. The ﬁnal assembled mobility matrix is given by
Y = ZFF − ZFI (I + YGE ZII )−1 YGE ZIF

−1

(14)

where the impedances matrices Zi j are sub-matrices of the full impedance matrix of the dielectric layer as
indicated in the following equation
fFR
ZFF ZFI vFR=(15)
ZIF ZII
fIR
vIR
Then again Equation 11 and Equation 14 can be used again for adding as many layers as requested.
(܎ீ , ‫) ீܞ‬+(܎ா , ‫ܞ‬ா )(܎ீா , ‫ீܞ‬ா ) (܎ோூ , ‫ܞ‬ோூ )(܎ீா , ‫ீܞ‬ா )+=Figure 4. Assembling the electrode toa grounded layer.

Figure 5.
actuator.

(܎ோி , ‫ܞ‬ோி )(܎ோி , ‫ܞ‬ோி )=Assembling a dielectric layer to a grounded

4.2 Emitted Sound Power
By deﬁnition the radiated sound power W is
W=



ΔS
Re vH ZA v
2

(16)

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where H is the complex conjugate transpose or Hermitian operator and ZA is the speciﬁc acoustic impedance
matrix. Assuming that each element can be approximated to a circular pistonic radiator the speciﬁc acoustic
impedance matrix can be expressed as in [12]:⎡⎤S1 (2kai )J1 (kai ) 2 sin(kd21 )cos(kd21 )ρck2 ΔSi)ρc 1 − J1 (2ka
+
i
+
i
2
·
·
·
ka
ka
2π
ka
kd
kd
i
i
i
21
21
⎢
⎥
2 sin(kd )⎢ 2⎥cos(kd21 )J1 (2kai )S1 (2kai )21(17)
ZA = ⎢ ρck ΔS 2 J1 (kai )
⎥
+
i
+
i
ρc
1
−
kai
kd21
kd21
kai
kai
⎣ 2π
⎦
..
..
.
.
where ρ is the air density, c is the sound speed in air, k = ωc is the wave number of sound, J1 is the ﬁrst-order
Bessel function, S1 is the Struve function and i is the imaginary unit, di j is the distance between the centersof two circular elements, respectively. Also the the radiation efﬁciency, deﬁned as ratio of the radiated soundpower over the mean radiation, and the kinetic energy of the top electrode can be easily computed.

5

SOME COMPUTATIONAL RESULTS

5.1 A loudspeaker prototype
A loudspeaker prototype was previously realised using largely available materials (see Figure 6). The electrodes
were made from 0.5 mm thick perforated mild steel sheets with 0.8 mm diameter holes and 1.5 mm pitch. The
R
dielectric material used was 0.11 mm thick latex exercise bands (Tan Extra-Thin Thera Band).
The electrode
diameters was 120 mm. The electrodes have two wings for powering which have been curved. This design
aims to lower their bending stiffness so that restrictions to vibration of the electrode is kept to a minimum.
The properties of such prototype are reported in Table 1 and will be used for the following calculations. A
measurement of the radiated sound power, as shown in Figure 7(a), was performed in a resonant box applying
a pseudo-random noise with a peak-to-peak voltage of about 80 V and an offset voltage of 800 V. Two, fourand ten layers loudspeakers were considered. It is noteworthy that the high response below 300 Hz was due
to background noise. It is also surprising that the high response at 500 Hz is not affected by the number oflayers.
Table 1. Calculations parameters.
a
VDC
ρe
he
p
R
le
Ee
νe

Description
Radius of the loudspeaker
Supply voltage offset
Density of electrodes
Thickness of electrodes
Pitch of perforation
Radius of perforation holes
Ligament efﬁciency
Young’s mod. of electrodes
Poisson ratio of electrodes

Value
0.06 m1000 V
5800 kgm−3
0.5 mm
1.5 mm
0.4 mm0.47
210e9 Pa
0.33

ηe
VAC
ε
hd
Ed
νd
ηd
ΨA

Description
Loss factor of electrodes
Total number of elements
Alternate supply voltage
Relative dielectric constantThickness of dielectric
Young’s mod. of dielectric
Poisson ratio of dielectric
Loss factor of dielectric
Perforation factor

Value
0.001
256
200 V
0.001
0.11 mm
5 MPa
0.45
0.05
0.66395.2 Preliminary results
The sound radiation power has been computed for the loudspeaker shown in Figure 6. Figure 7(b) shows thepredicted sound radiation power for 2, 4 and 10 layers. It has been assumed that only an alternate voltageof 80 V contributes to the loudspeaker excitation. The response at low frequency in the experimental results
is mainly due to background noise. The model shows instead a slope of 40dB per decade. This reﬂects amonopole radiation mode driven by a constant force over the frequency. In fact, both the monopole radiation

7350

(a)

(b)

Figure 6. Design of a DEAP loudspeaker (a) and its realisation (b).
50

50

40
Sound Power (dB re 10-12W)

Sound Power (dB re 10-12W)

40

2 layers
4 layers
10 layers

30

20

10

30

20

10

0

0

-10

-10

-20
101

102

10-1

103
Frequency (Hz)

10 4

100

10 1

݇ܽ

(a)

2 layers
4 layers
10 layers

-20
10 1

10 2

10 -1

103
Frequency (Hz)

104

100

101

݇ܽ

(b)

Figure 7. Predicted sound radiation powers for DEAP loudspeakers of 2, 4 and 10 layers: electrodes assumed
as rigid bodies (a) or ﬂexible plates (b). (k is the acoustic wavenumber)and the mobility of the structure grow at 20dB per decade. From these preliminary results it can be seen themodel is able to roughly predict the radiated sound power levels at higher frequencies. However, the resonantpeaks are not accurate and it seems that the model is far more stiff than the actual prototype. This is probably
caused by the imperfect planarity of the electrodes which reduces the contact area between dielectrics andelectrodes. Also the effect of increasing the number of layers is more apparent in the simulated results and at7351

very high frequencies the experimental sound power decreases more for a larger number of layers. This is dueto the ampliﬁer limited power in driving higher capacitive loads.

6

CONCLUSIONSIn this paper a new approach to model the acoustic emissions of a stacked DEAP loudspeaker with perforated
electrodes has been proposed. Preliminary numerical results have been compared with experimental data showing that the model is able to reproduce the physics of the real system. However, material properties needs tobe tuned in order to improve the match at resonances.

ACKNOWLEDGEMENTS
This project has been supported by the Royal Society International Exchanges scheme.

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