19th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007
Methodology for electro mechanical simulation of piezoelectric ceramics PACS: 43.38.Fx 1
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Morcillo López, Miguel Ángel ; Cesteros Morante, Beatriz ; Cordero Izquierdo, Roberto ; 1 2 Hidalgo Otamendi, Antonio ; Fernández, Jose Francisco 1 Fundación CIDAUT; Parque tecnológico de Boecillo, P.209, 47151 Boecillo. Valladolid. Spain;
[email protected];
[email protected];
[email protected];
[email protected] 2 Instituto de Cerámica y Vidrio, C/Kelsen, 5. Madrid. E-28049. Spain;
[email protected] ABSTRACT In this paper a design methodology of components with piezoelectric elements, as active or movement generators, is presented. In recent years the development of smart structures that use piezoelectric sensors and actuators has increased to obtain the desired mechanical behaviour. Simulation is a widely used tool in the designing of products that allows a better understanding of the model behaviour, and a reduction of the number of prototypes. This leads to a reduction in costs and development time, and increases competitiveness. The Acoustics and Vibrations Department of the CIDAUT Foundation along with the Electroceramics Department of ICV have developed a methodology for the simulation of piezoelectric elements that takes into account their electrical and mechanical properties using a general finite element software. Numerical results have been correlated with experimental results. INTRODUCTION Simulation tools allow us to predict structural behaviour from a design (shape, dimensions, material properties…). A design can be discretized into elements where different behaviour laws can be applied according to the physical phenomena to study. Besides the prediction of structural behaviour, simulation tools are used to optimize a design, with the objective of minimizing/maximizing a response such as velocity vibration in a point, reducing mass… Simulation codes used in mechanical behaviour are more and more versatile, and allow taking into account electrical, acoustical, thermal behaviour and different couplings. Traditional codes used to solve coupled electric problems work in 2D domains, whereas in mechanical simulation powerful tool are used to calculate complex 3D structures with millions of d.o.f.s. Next how to study piezoelectric behaviour coupled to structural vibrations will be shown from the thermal behaviour model using the commercial software MSC.Nastran. A cymbal structure made of a piezoelectric disc with a preformed brass sheet on each side will be modelled using this method. This approximation allows studying complex structures which require a high computational power. THEORY FOR PIEZO ELECTRIC CERAMIC SIMULATION When using the finite element method to solve a mechanical problem, first it is necessary to mesh the structure in elements where basic behaviour laws can be applied and certain hypothesis are assumed. The following equations describe piezoelectric behaviour:
ε = s Eσ − d t E
Eq. 1
D = dσ + p S E
Where σ are mechanical stresses, ε are strains, E represents the electric field, D is the electric E S displacement, s is the mechanical compliance matrix for a constant field E, p is the permittivity matrix for a fixed state of deformation, and d is the piezoelectric matrix. The electric field E is related to the electrical potential φ by: Eq. 2
E = − gradφ And the mechanical strain ε to the mechanical displacement by:
Eq. 3
ε = Bu
Developing these equations and applying Maxwell and classical mechanic laws R.Lerch [1] obtains:
Mu&& + Duu u& + K uu u + K uφ φ = FB + FS + F p
Eq. 4
K ut φ u + K φφ φ = QS + QP
Where u is the displacement vector, φ is the electric potential, M is the mass matrix, D is the damping matrix, Kuu is the mechanical stiffness matrix, KuΦ is the piezoelectric coupling matrix, KΦΦ is the dielectric matrix. FB are mechanical volume forces, FS are mechanical surface forces, FP are mechanical point forces, QS are surface electrical charges and QP point charges. If we consider the electric potential φ known, the system of equations is reduced to the first equation (in Eq. 4) where the variable to solve is displacement u. MSC/NASTRAN, one of the most widely used finite element solver, offers no specific element to model piezoelectric structures directly. Rather, the analogy between piezoelectric strain and thermally induced strain is used. Piezoelectric coefficients characterizing the actuator are input as thermal expansion coefficients, and the voltage as a temperature increment. This formulation may lead to the resolution of the problem using the matrix formula implemented in MSC.Nastran for MAT9 anisotropic material:
σ 11 G11E E σ 22 G21 σ G E 33 = 31 τ 12 G41E E τ 23 G51 τ G E 13 61
G12E
G13E
G14E
G15E
E 22 E 32 E 42 E 52 E 62
E 23 E 33 E 43 E 53 E 613
E 24 E 34 E 44 E 54 E 64
E 25 E 35 E 45 E 55 E 65
G G G G G
G
G
G
G
G
G
G
G
G
G
G G G G G
G16E ε 11 A1 G26E ε 22 A2 G36E ε 33 A3 − (φ − φ ref G46E γ 12 A4 G56E γ 23 A5 G66E γ 13 A6
)
Eq. 5
Next the methodology used to implement piezoelectric material properties in this formulation to solve the coupled mechanical electric problem is discussed.
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19 INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID
MATERIAL CHARACTERIZATION The following table shows the piezoelectric material properties obtained from tests and bibliography: Table 1. Piezoelectric material properties
Elastic constants
Dielectric constants
Piezoelectric constants
PZT 5A S11E (m2/N) S12E (m2/N) S13E (m2/N) S33E (m2/N) S44E (m2/N) S66E (m2/N) Density (kg/m3) ε11S/ε0S ε33S/ε0S d31(m/V) d33(m/V) d24(m/V) d15(m/V)
1.64E-11 -5.74E-12 -7.22E-12 1.88E-11 4.75E-11 4.43E-11 7750.00 1035.972 938.708 -1.71E-10 3.74E-10 5.84E-10 5.84E-10
APPLICATION EXAMPLE As an application example a finite element model of a cymbal is shown. The objective is to optimize the brass structure to obtain maximum response amplification. The cymbal consists of a piezoelectric disc 1mm thick and 12mm in diameter. On each side of the disc a 0.3mm thick stamped brass sheet is bonded with a 0.02mm thick epoxy lamina. The following figures show the test and finite element models.
Figure 1. Cymbal test model
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19 INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID
8.7m m 4.7m m
Figure 2. Finite element model The finite element model consists of 16,368 solid elements and 20,611 nodes. The following hypotheses have been posed: • Linear behaviour • Small displacements and strains • Piezoelectric material with anisotropic behaviour A frequency response analysis is made for a constant electric potential 10V in a frequency range up to 2000Hz. Figure 3 shows the response of a point in the upper sheet of the cymbal structure:
Figure 3. Simulated frequency response The calculation of a model like this takes only a few minutes. In order to validate this simulation methodology, the vibration of the metallic structure has been measured when an electric potential is applied to the piezoelectric disk. Figure 4 compares experimental and simulated results:
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19 INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID
Figure 4. Experimental (red) and simulated (blue) responses The resonance frequency for the simulated model is slightly lower than the measured one. The damping in the simulated model needs to be updated. CONCLUSIONS This paper shows how to simulate coupled electrical-mechanical behaviour with commercial structural software. Although lacking final tuning of the model, a methodology has been presented to obtain validated and robust results. This analysis tool is quite powerful, and allows the simulation of complex structures with low calculation times. References [1] Reinhard Lerch. Simulation of Piezoelectric devices by two and three dimensional finite elements. IEEE Transactions on Ultrasonic, ferroelectric and frequency controls Vol 37 N2 May 1990 [2] P. Ochoa, M. Villegas, J.L Pons, P. Ledinger, J.F, Fernández. Tunability of cymbals as Piezocoposite transducers. Journal of Electroceramics. February 26, 2005
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