Piezoelectric heterogeneous bimorphs have extensive applications in the MEMS area. In order to formulate their displacement distribution more conveniently, a concise analytical solution is described herein. This solution integrates all of the important influence factors, such as geometry and material properties, into a single expression. Secondary effects such as shear, residual stress induced curvature, and electrostiction, are not considered, but they are found to be minimally important based on the accuracy of the predicted results. The expression is verified using the finite element method, and shown to be straightforward compared to the more complicated couple-field FEM analysis for a specific range of materials and thicknesses. The method is subsequently shown to be capable of quickly estimating the displacement in a bimorph beam, making it a useful tool for designing piezoelectric structures.

Issue Section:
Technical Briefs
Keywords:
piezoelectric materials, micromechanical devices, beams (structures), lead compounds, finite element analysis, elastic deformation
Topics:
Deflection, Design, Displacement, Finite element model, Geometry, Steel, Stress, Metals, Elastic moduli, Electric fields, Finite element methods, Piezoelectric materials, Finite element analysis, Materials properties, Shear (Mechanics)
1.
Itoh
,
T.
, and
Suga
,
T.
,
1993
, “
Development of a Force Sensor for Atomic Force Microscopy Using Piezoelectric Thin Films
,”
Nanotechnology
,
4
, pp.
218
–
224
.
2.
Luginbuhl
,
Ph.
,
Racine
,
G. A.
,
Lerch
,
Ph.
,
Romanowicz
,
B.
,
Brooks
,
K. G.
,
de Rooij
,
N. F.
,
Renaud
,
Ph.
, and
Setter
,
N.
,
1996
, “
Piezoelectric Cantilever Beams Actuated by PZT sol-gel Thin Film
,”
Sens. Actuators, A
,
54
, pp.
530
–
535
.
3.
Smits
,
Jan G.
, and
Choi
,
Wai-shing
,
1991
, “
The Constituent Equations of Piezoelectric Heterogeneous Bimorphs
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
383
(
),
May, pp.
256
–
270
.
4.
Smits
,
Jan G.
, and
Dalke
,
Susan I.
, and
Cooney
,
Thomas K.
,
1991
, “
The Constituent Equations of Piezoelectric Bimorphs
,”
Sens. Actuators, A
,
28
, pp.
41
–
61
.
5.
Yocum
,
Matthew
, and
Abramovich
,
Haim
,
2002
, “
Static Behavior of Piezoelectric Actuated Beams
,”
Comput. Struct.
,
80
, pp.
1797
–
1808
.
6.
Saravanos
,
Dimitris A.
, and
Heyliger
,
Paul R.
,
1999
, “
Mechanics and Computational Models for Laminated Piezoelectric Beams, Plates, and Shells
,”
Appl. Mech. Rev.
,
52
(
10
), pp.
305
–
319
.
7.
Hwang
,
Woo-Seok
, and
Park
,
Hyun Chul
,
1993
, “
Finite Element Modeling of Piezoelectric Sensors and Actuators
,”
AIAA J.
,
31
(
5
), May, pp.
930
–
937
.
8.
Cappelleri
,
D. J.
,
Frecker
,
M. I.
,
Simpson
,
T. W.
, and
Snyder
,
A.
,
2002
, “
Design of a PZT Bimorph Actuator using a Metamodel-based Approach
,”
ASME J. Mech. Des.
,
124
, pp.
354
–
357
.
9.
Devoe
,
D. L.
, and
Pisano
,
A. P.
,
1997
, “
Modeling and Optimal Design of Piezoelectric Cantilever Microactuators
,”
J. Microelectromech. Syst.
,
6
(
3
), pp.
266
–
270
.
10.
Allik
,
H.
, and
Hughes
,
J. R.
,
1970
, “
Finite Element Method for Piezoelectric Vibration
,”
Int. J. Numer. Methods Eng.
,
2
, pp.
151
–
157
.
11.
Borrelli
,
A.
, and
Patria
,
M. C.
,
1999
, “
Saint-Veanat’s Principle for a Piezoelectric Body
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
,
59
(
3
), pp.
1098
1098
.
12.
Benjeddou
,
A.
,
Trindade
,
M. A.
, and
Ohayon
,
R.
,
1997
, “
A Unified Beam Finite Element Model for Extension and Shear Piezoelectric Actuation Mechanisms
,”
J. Intell. Mater. Syst. Struct.
,
8
, pp.
1012
–
1025
.
13.
Krommer
,
M.
, and
Irschik
,
H.
,
2002
, “
An Electromechanically Coupled Theory for Piezoelastic Beams Taking into Account the Charge Equations of Electrostatics
,”
Acta Mech.
,
154
, pp.
141
–
158
.
14.
Krommer
,
M.
, and
Irschik
,
H.
,
1999
, “
On the Influence of the Electric Field on Free Transverse Vibration of Smart Beams
,”
Smart Mater. Struct.
,
8
, pp.
401
–
410
.
15.
Krommer
,
M.
,
2001
, “
On the Correction of the Bernoulli-Euler Beam Theory for Smart Piezoelectric Beams
,”
Smart Mater. Struct.
,
10
, pp.
668
–
680
.
16.
Mouturet
,
V.
, and
Nogarede
,
B.
,
2002
, “
Optimal Dimensioning of a Piezoelectric Bimorph Actuator
,”
European Physical Journal of Applied Physics
,
17
, pp.
107
–
118
.
17.
Muralt
,
P.
,
2000
, “
Ferroelectric Thin Films for Micro-Sensors and Actuators: a Review
,”
J. Micromech. Microeng.
,
10
, pp.
136
–
146
.
18.
Smits
,
J. G.
, and
Choi
,
W.-S.
,
1991
, “
The Constituent Equations of Piezoelectric Heterogeneous Bimorphs
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
38
(
3
), pp.
256
–
270
.
19.
Smits, J. G., and Dalke, S. I., 1989, “The Constituent Equations of Piezoelectric Bimorphs,” IEEE-Ultrasonic Symposium, pp. 781–784.
20.
Elka, E., Elata, D., and Abramovich, 2003, “The Electromechanical Response of Multi-Layered Piezoelectric Structures,” TME Report No. 478, Faculty of Mechanical Engineering, Technion, I.I.T., 32000 Halfa, Israel, July.
Copyright © 2004
 by ASME
You do not currently have access to this content.