Piezoelectrically excited, resonant, elastic beams find wide use as piezoelectric fans, optical choppers, MEMS sensors, and piezoelectric motors. The devices consist of either one piezoelectric ceramic patch (piezopatch) bonded on one side (asymmetric configuration), or of two oppositely poled patches placed symmetrically on either side of a thin, flexible elastic beam (symmetric configuration). Field equations of the coupled structure governing the coupled longitudinal and bending motions of the resonator are derived using linear constitutive equations, slender beam approximations, and Hamilton’s principle. Analytical solutions are found to the coupled eigenvalue problem. Eigenvalues and eigenfunctions for the short-circuited and open-circuited configurations are predicted analytically and are found to be in excellent agreement with results from three-dimensional finite element simulations. Electromechanical coupling factors (EMCF) are computed using the analytical and finite element model and optimal resonator geometries are identified for maximal EMCF. The EMCF predictions are also compared with experiments for an asymmetrically configured resonator. The analytical solution provides a convenient tool for the optimal design of such devices.

Keywords:
vibrations, piezoelectric materials, eigenvalues and eigenfunctions, finite element analysis, dynamic response, crystal resonators
Topics:
Dynamic response, Finite element analysis, Optimization, Resonance, Eigenvalues, Boundary-value problems, Eigenfunctions, Finite element model, Equations of motion, Piezoelectric materials
1.
Toda
,
M.
,
1979
, “
Theory of air flow generation by a resonant type PVF2 bimorph cantilever vibrator
,”
Ferroelectrics
,
22
, pp.
911
–
918
.
2.
Yoo
,
J. H.
,
Hong
,
J. I.
, and
Cao
,
W.
,
2000
, “
Piezoelectric ceramic bimorph coupled to thin metal plate as cooling fan for electronic cooling
,”
Sens. Actuators, A
,
79
, pp.
8
–
12
.
3.
Campbell Jr., J. S., Glezer, W. Z., and Hartley, J. G., 1998, “Thermal management of a laptop with synthetic air microjets,” IEEE Intersociety Conference on Thermal Phenomena, pp. 43–50.
4.
Crawley
,
E. F.
, and
de Luis
,
J.
,
1987
, “
Use of piezoelectric actuators as elements of intelligent structures
,”
AIAA J.
,
25
, pp.
1373
–
1385
.
5.
Crawley
,
E. F.
, and
Anderson
,
E. H.
,
1990
, “
Detailed model of piezoceramic actuation of beams
,”
J. Intell. Mater. Syst. Struct.
,
1
, pp.
5
–
25
.
6.
Main
,
J. A.
,
Garcia
,
E.
, and
Howard
,
D.
,
1994
, “
Optimal placement and sizing of paired piezoactuators in beams and plates
,”
Smart Mater. Struct.
,
3
, pp.
373
–
381
.
7.
Abramovich
,
H.
,
1998
, “
Deflection control of laminated composite beams with piezoceramic layers-closed form solutions
,”
Compos. Struct.
,
43
, pp.
217
–
231
.
8.
Gibbs
,
G. P.
, and
Fuller
,
C. R.
,
1992
, “
Excitation of thin beams using asymetric piezoelectric actuators
,”
J. Acoust. Soc. Am.
,
6
, pp.
3221
–
3227
.
9.
Pan
,
X.
, and
Hansen
,
C. H.
,
1994
, “
Piezoelectric crystal vs point force excitation of beams and plates
,”
J. Intell. Mater. Syst. Struct.
,
5
, pp.
363
–
370
.
10.
Brennan
,
M. J.
,
Elliot
,
S. J.
, and
Pinnington
,
R. J.
,
1995
, “
The dynamic coupling between piezoceramic actuators and a beam
,”
J. Acoust. Soc. Am.
,
102
, pp.
1931
–
1942
.
11.
Wolf, K.-D., 2000, “Electromechanical energy conversion in asymmetric piezoelectric bending actuators,” Ph.D. thesis, Technische Universita¨t Darmstadt, Germany.
12.
Pan
,
J.
,
Hansen
,
C. H.
, and
Snyder
,
S. D.
,
1992
, “
A study of the response of a simply supported beam to excitation by a piezoelectric actuator
,”
J. Intell. Mater. Syst. Struct.
,
3
, pp.
3
–
16
.
13.
Rivory
,
J. F.
,
Hansen
,
C. H.
, and
Pan
,
J.
,
1994
, “
Further studies of the dynamic response of a simply supported beam excited by a pair of out-of-phase piezoelectric actuators
,”
J. Intell. Mater. Syst. Struct.
,
5
, pp.
654
–
663
.
14.
Kim
,
S. J.
, and
Jones
,
J. D.
,
1996
, “
Influence of piezo-actuator thickness on the active vibration control of a cantilever beam
,”
J. Intell. Mater. Syst. Struct.
,
6
, pp.
610
–
623
.
15.
Rontong
,
J. A.
,
Wright
,
J. R.
,
Wynne
,
R. J.
, and
Tomlinson
,
G. R.
,
1997
, “
Modeling of a hybrid constrained layer/piecewise approach to active damping
,”
ASME J. Vibr. Acoust.
,
119
, pp.
120
–
130
.
16.
Lobontiu
,
N.
,
Goldfarb
,
M.
, and
Garcia
,
E.
,
1999
, “
Achieving maximum tip deflection during resonant excitation of piezoelectrically actuated beams
,”
J. Intell. Mater. Syst. Struct.
,
10
, pp.
900
–
913
.
17.
Barboni
,
R.
,
Mannini
,
A.
,
Fantini
,
E.
, and
Gaudenzi
,
P.
,
2000
, “
Optimal placement of PZT actuators for the control of beam dynamics
,”
Smart Mater. Struct.
,
9
, pp.
110
–
120
.
18.
Wang
,
Q.
, and
Wang
,
C. M.
,
2000
, “
Optimal placement and size of piezoelectric patches on beams from the controllability perspective
,”
Smart Mater. Struct.
,
9
, pp.
558
–
567
.
19.
Bu¨rmann
,
P.
,
Raman
,
A.
, and
Garimella
,
S. V.
,
2002
, “
Dynamics and topology optimization of piezoelectric fans
,”
IEEE Trans. Compon., Packag. Manuf. Technol., Part A
,
25
, pp.
592
–
600
.
20.
Tiersten, H. F., 1969, Linear Piezoelectric Plate Vibrations, Plenum, New York.
21.
Waterloo Maple, Inc., 2002, Academic (ACD) Maple 8 student edition for Windows, Waterloo, Ontario, Canada.
22.
Berlincourt
,
D. A.
,
Corran
,
D. R.
, and
Jaffe
,
H.
,
1964
, “
Piezoelectric and piezomagnetic materials and their function in transducers
,”
Phys. Acoust.
,
1A
, pp.
169
–
270
.
23.
Hunt, F. V., 1954, Electroacoustics, Wiley, New York.
24.
ANSYS Inc., 2001, ANSYS 6.0 Documentation.
25.
IRE Standards on Piezoelectric Crystals, 1961, Proc. IRE, 49, pp. 1162–1169.
Copyright © 2005
 by ASME
You do not currently have access to this content.