Radiation optimization of piezoelectric plates
Olivier DOARÉ, Emil GARNELL, Corinne ROUBY(1) IMSIA, ENSTA Paristech, France, {olivier.doare,emil.garnell,corinne.rouby} ensta-paristech.fr

Abstract
The goal of the present work is to control the modal dynamics of a piezo-electric loudspeaker in order to reachdirectivity of efficiency criteria. In this context, we study the dynamics and sound radiation of a piezoelectricplate biased by a static pressure on one side. This pressure inflates the plate and its forced dynamics around thisequilibrium position are then studied. With the help of von Karman plate theory, the static equilibrium positionof a plate of arbitrary shape is sought for before writing its linear dynamics around this equilibrium position.
This two-step problem is solved numerically using a finite elements software (Freefem++). This results in aseries of eigenmodes and eigenfrequencies for each value of the static pressure difference. Next the modal
forces due to the piezoelectric coupling are calculted. By considering a baffled plate, the acoustical radiation iscomputed using the Rayleigh integral. The numerical model is used to perform a parametric optimisation of thesystem.

Keywords: Piezoelectic plates, Loudspeaker, Optimisation, Genetic algorithm, Radiation

1INTRODUCTION

Numerous studies have focused on the dynamics of piezoelectric plates, with the aim of designing loudspeakers [4], reducing the acoustic transmission through panels [9], or controlling structural vibrations [2]. Often,a Kirchhoff-Love linear plate model taking into account piezoelectric coupling is considered [8]. An antisymmetric configuration of the piezoelectric layers is then considered in order to couple the electric field and thebending moments within the plate. Conversely, a symmetrical configuration produces only membrane forces
and therefore does not allow to generate a transverse displacement of the plate in the context of this linear
modeling. To couple the electric field to the transverse displacement in this context, the structure must have acurvature. It is then necessary to consider shells or prestressed plates. We propose here a plate configuration
prestressed by a pressure differential on which we can vary the degree of asymmetry of the piezoelectric layers.
Although this system can have many applications (sensors for non-destructive testing, microphone, energy harvesting...), we focus here on a speaker-type application and wish to optimize the radiation of the system. Oneof our objectives is to find a configuration with a response to an electrical signal dominated by the response ofa single mode of vibration, over the widest possible frequency range, by adjusting the shape of the electrodes
glued on the piezoelectric plate. There are several works in the literature seeking to achieve a modal forcing
by playing on the shape of the electrodes [5, 4, 10]. The approach developed in the present article will use thesame principles by integrating the prestresses, the degree of asymmetry and the acoustic radiation properties at
low frequencies into optimization.
First, we will derive a model of a piezoelectric plate prestressed by a static pressure differential and dynamically
excited by an electric voltage on the electrodes. This model will next be used to optimize various parametersin order to create a system radiating in a single structural mode. Finally, some optimization results will bepresented and analyzed.

2PIEZOELECTRIC PLATE MODEL

We are interested in the system represented on figure 1. The piezoelectric plate has a thickness h and an
arbitrary shape of characteristic dimension L. This plate consists of a linear and isotropic elastic material, char-6211

Figure 1. Schematic view of the problem : a plate defined by the border Σ where it is clamped. The interior
domain is denoted Ω. The plate consists of two piezoelectric layers separated by an electrode. Upper and lowerlayers are partially covered by an electrode, indicated by the orange color. The electrical potentials are V andcV on the upper and lower layers respectively.
acterized by its Young modulus E and Poisson’s ratio ν. The edge of the plate is denoted Σ. The displacement
is written ξ = uex + vey + wez . The equations describing the dynamics of this plate are obtained from the assumptions of Von Kármán [6] (thin plate, w at most of h and small derivatives in front of the unit, smalldeformations, Kirchhoff’s assumptions for the stresses in the median plane, and small in-plane displacements uand v compared to w).
Locally, the stresses in the plate are the sum of internal stresses due to the deformation and the stresses generated by the piezoelectric coupling:E(εxx + νεyy ) + e31 F E
1 − ν2E(εyy + νεxx ) + e31 F E
σyy =1 − ν2E
σxy =γxy1+ν

σxx =(1)

where E is the electric field along z. The latter equals 2V /h in the upper half and 2cV /h in the lower half.
It is assumed here that the piezoelectric coupling is only between the electric field along z and the uniaxial
stresses in the plane of the plate, and that this coupling is isotropic in the plane. In equations (1), e31 is theassociated piezoelectric coupling coefficient. The function F is a function that describes the fact that electrodes
partially corver the piezoelectric film. This factor equals 1 where the electrode is present, and 0 elsewhere.
The equations describing the nonlinear dynamics of the system taking into account the Von Kármán hypotheses
set out above are detailed in numerous works (for example [1]). The same development is carried out here
taking into account the piezoelectric coupling as introduced in equation (1). After introducing the membrane
forces Nx , Ny , Nxy , and the moments Mx , My , Mxy , the behavior equations link these six generalized forces todeformations through six equations written below in dimensionless form, using the following quantities for thescaling:sEh3
ρ(1 − ν 2 )Time: L2Lengths:
L
Pressure:(2)
Eh2
12(1 − ν 2 )L3
Membrane stresses:

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Eh
1 − ν2Moments:

Eh3
12L(1 − ν 2 )Voltage:

Ehe31 (1 − ν 2 )(3)The mechanical behaviour writes:Nx =∂u∂v 1+ν+∂x∂y 2∂w∂x

2+ν2∂w∂y2+ (1 + c)F v,

∂u 1 ∂w 2 ν ∂w 2
∂v+ν+Ny =++ (1 + c)F v,∂y∂x 2 ∂y2 ∂x1−ν ∂u ∂v ∂w ∂w
Nxy =++,2∂y ∂x ∂x ∂y
∂ 2w
∂ 2w
−
ν+ 3l(1 − c)F v,
∂ x2
∂ y2
∂ 2w
∂ 2w
My = − 2 − ν 2 + 3l(1 − c)F v,
∂y∂x
∂ 2w
Mxy = −(1 − ν),∂ x∂ yMx = −(4)
(5)
(6)
(7)
(8)(9)and equilibrium writes:∂ Nx ∂ Nxy+= 0,∂x∂y∂ Nxy ∂ Ny+= 0,∂x∂y∂ 2 Mxy∂ 2 Mx ∂ 2 My
∂ 2w
∂ 2w
∂ 2w
2p + ẅ +++2+12l
N+
N+
2N
= 0.
x
yxy
∂ x2
∂ y2
∂ x∂ y∂ x2∂ y2∂ x∂ y

(10)
(11)
(12)

These equations depend only on four dimensionless parameters: the Poisson’s ratio ν, the aspect ratio l = L/h,the dimensionless pressure differential p and the piezoelectric forcing symmetry parameter c, in addition toeventual parameters describing the shape of the plate. The clamped edge Σ, where the outgoing normal isdenoted n imposes:
u=v=w=03

grad w . n = 0andon Σ.

(13)Linear dynamics around an equilibrium state

We are interested in the response of the system subjected to a static pressure p0 when a voltage V (t) isimposed. The nonlinear equations presented in the previous section are first separated between a permanent partand low amplitude fluctuations. Order two quantities are then neglected and the weak form of the system iswritten to implement the finite element method with Freefem++ [7].
The nonlinear static problem of a membrane subject to p = p0 is first solved. Numerically, a gradient method
is implemented in Freefem++ to reach the nonlinear solution. This solution is represented in figure 2 for sometypical values of the parameters.
Next, the linear dynamics of the system around the nonlinear static equilibrium resulting from the static inflationpressure p is calculated. In practice, the finite element method is used to calculate the mass and stiffness
matrices M and K. It should be noted that the finite element stiffness matrix is not symmetric and the numericalmodal analysis of the problem then involves two families of modes: the right modes φ , resulting from the direct
eigenvalue problem, and the left modes ψ, given by the transposed eigenvalue problem. Thus the dynamics ofthe forced system in the modal basis involves an expansion along the right modes of the nine quantities of theproblem. In particular, the displacement along z writes
Nw(x, y,t) = ∑ qn (t)φwn (x, y),(14)n=16213

Figure 2. Static deformation of a plate submitted to a pressure differential p for l = 3.33 × 104 and ν = 0.25.
(a) Displacement w for p = 10−4 , (b) displacement w for p = 100, (c) displacement w(x, y = 1/2) for different
values of p in the range [10−4 , 103 ], (d) maximum value of w as function of p.
where the w subscript indicates that the eigenmode for w is considered. The nth component of the vector of themodal forces resulting from a projection on the left modes writes:
Fn =Z (1 + c)(ψNx n + ψNy n ) + 3l(1 − c)(ψMx n + ψMy n ) F vdΩ.

ΩIn this expression of the modal force the projections of F on two modal functions appear: one is related tothe membrane forces Nx + Ny , the other to the moments Mx + My . It should be noted the the projection on themoment is scaled by the aspect ratio l. For thin plates, l is very large (l = 3.33 × 104 in the case presented inthis article). Thus, for the membrane forces to contribute significantly to the modal force n, ψNx n + ψNy n must
be several orders of magnitude greater than ψMx n + ψMy n . This is what is observed in practice for large valuesof the pressure p. This is illustrated in figure 3 where φw1 , ψMx 1 + ψMy 1 and ψMx 1 + ψMy 1 are plotted for twoinflation pressures. We observe that when p increases, the membrane forces of this mode become very large
compared to the moments, so even if moments are in factor of a coefficient l = 3.33 × 104 , membrane forces
are dominant.
Let w = ℜ(ŵ(x, y)eiωt ) be the harmonic response of the plate when an input voltage of the form v = ℜ(v̂eiωt )6214

Figure 3. First eigenmode for two different pressures. The displacement of the first right mode is plotted on
the first line. The membrane stresses and moments of the first left mode are plotted on the second and thirdlines respectively. The mean value of these quatities and the eigenfrequencies are plotted as function of p on
the down-left and down-right figures respectively.
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is applied. The amplitude in the Fourier space ŵ then takes the form of a sum of modal contributions,Fn φwn (x, y).
2 − ω 2 − 2iηω ωωnnnŵ(x, y) = ∑(15)

In the above equation η is an arbitrary linear constant damping, which is added for consistency to take into
account different sources of damping in the mechanical system. Experiments on real systems would be necessary
to correctly model the damping. Assuming that radiated acoustic wavelengths are large compared to L, thepressure P (in pascals) radiated at a distance R0  L from the embedded plate in an infinite rigid plane takes
the simple form below,2πPFn χn= −eikr0 ∑ 2,22ρ f c0n ωn − ω − 2iηωn ωZwith χn =φwn ds,

(16)

Ω

where k is the acoustic wave number adimensioned by L and r0 = R0 /L. This low frequency approximationis the monopole radiation approximation, obtained in practice from the Rayleigh integral [3]. We thus see thatwith the approximations made above, the contribution to the radiation of a mode n of the structure is directly
proportional to the product Fn χn . Hence, the optimization of the following section is based on this particularquantity.

4

Optimisation of the electrodes’ geometry

The radiation associated with an input voltage v(t) of the system has been calculated in the previous section.
This modeling can then be used to optimize the geometry of the system for different purposes. We propose
here to work on a geometry of electrodes defined from concentric rings, parametrized by five rays and plottedin figure 4abc. This geometry defines the function F represented in figure 1 and appearing in the equation (15),which is 1 where the electrode is present and 0 elsewhere. The goal here is to maximize the radiation of theplate of one mode m, and minimize the other modes radiation. We thus seek to minimize the cost function:J=∑Nn=1,n6=m (χn Fn )2,(χm Fm )2(17)with the following constraints
0 < r1 < r2 < r3 < r4 < r5 <12and− 1 < c < 1.

(18)

This optimization was performed for different values of the parameters l, ν and p using a genetic algorithm
implemented under Matlab. In the following we will focus on the case m = 1.
In figure 4 are presented three cases of best local optima for three different inflation pressures. The frequency
response functions far from the plate are also plotted in figure 4. The shaded lines correspond to the frequency
response obtained if we make an error of +5% or −5% on the dimensions of the electrodes. The plate subjected
to a high inflation pressure seems both to give a more robust optimum and to increase the efficiency of theradiation.

5Conclusion

We presented a weakly nonlinear model of a plate of arbitrary shape forced by a static pressure differential
and by a dynamic loading by piezoelectric coupling. This model allows a quick calculation of the dynamics
and the acoustic radiation and could thus be used to optimise several parameters (electrode geometry r1 ...r5 andasymmetry coefficient of the piezoelectric forcing c) in order to maximize the radiation of one mode comparedto the others. This optimization problem has many local minima and although a long phase of exploration of the6216

(a)p= 0.01J = 0.00025195

c = 0.73753(c)p 0 = 100

J = 7.6807e-10

c = -0.609630(b)p0=1J = 0.00030626

c = 0.20239

Figure 4. Various electrode shapes (local minima of J) found by the genetic algorithm. Pink corresponds toF = 1, blue corresponds to F = 0.

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genetic algorithm with large populations is done for each case, we have no evidence that the global minimum.
However, the results show that we are able to isolate the radiation of a single mode.
We also note a very high sensitivity of the cost function J to the parameters of the problem, the minimum foundby the genetic algorithm, whether local or global, lies at the bottom of a very narrow well. This feature is not
mentioned in previous work on actuators or modal sensors based on a binary elecrode distribution (F = 0 or 1)[5], and can cause reliability problems in a practical application. The configuration proposed in this work canadvantageously overcome this problem, because two parameters can be adjusted after the design of the device,
the inflation pressure p0 and the asymmetry coefficient c of the piezoelectric forcing.
We are now working on other forms and other optimization objectives, as well as on the realization of optimized
prototypes.

References[1] M. Amabili. Nonlinear vibrations and stability of shells and plates. Cambridge University Press, 2008.
[2] I. Bruant, L. Gallimard, and S. Nikoukar. Optimal piezoelectric actuator and sensor location for active
vibration control, using genetic algorithm. Journal of Sound and Vibration, 329(10):1615–1635, 2010.
[3] A. Chaigne and J. Kergomard. Acoustique des Instruments de Musique. 2008.
[4] O. Doaré, G. Kergourlay, and C. Sambuc. Design of a circular clamped plate excited by a voice coil andpiezoelectric patches used as a loudspeaker. Journal of Vibration and Acoustics, 135(5):051025, 2013.
[5] A. Donoso and J. C. Bellido. Distributed piezoelectric modal sensors for circular plates. Journal of Soundand Vibration, 319(1-2):50–57, 2009.
[6] Y. C. Fung. Foundations of solid mechanics(Book on deformation and motion of elastic and plastic solids
including variational calculus and tensor analysis). Englewood Cliffs, N. J., Prentice-Hall, Inc. edition,
1965.
[7] F. Hecht and A. le Hyaric. http://www.freefem.org/ff++/index.htm.
[8] C. K. Lee. Theory of laminated piezoelectric plates for the design of distributed sensors actuators .1.
Governing equations and reciprocal relationships. Journal Of The Acoustical Society Of America, 87:1144–1158, 1990.
[9] N. Quaegebeur, P. Micheau, and A. Berry. Decentralized harmonic control of sound radiation and transmission by a plate using a virtual impedance approach. J. Acoust. Soc. Am., 125(5):2978–86, 2009.
[10] X. Zhang and Z. Kang. Topology optimization of piezoelectric layers in plates with active vibration control.
Journal of Intelligent Material Systems and Structures, 25(6):697–712, 2014.

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