Electrostriction is a recent actuation mechanism which is being explored for a variety of new micro- and millimeter scale devices along with macroscale applications such as artificial muscles. The general characteristics of these materials and the nature of actuation lend itself to possible production of very rich nonlinear dynamic behavior. In this work, principal parametric resonance of the second mode in in-plane vibrations of appropriately designed electrostrictive plates is investigated. The plates are made of an electrostrictive polymer whose mechanical response can be approximated by Mooney Rivlin model, and the induced strain is assumed to have quadratic dependence on the applied electric field. A finite element model (FEM) formulation is used to develop mode shapes of the linearized structure whose lowest two natural frequencies are designed to be close to be in 1:2 ratio. Using these two structural modes and the complete Lagrangian, a nonlinear two-mode model of the electrostrictive plate structure is developed. Application of a harmonic electric field results in in-plane parametric oscillations. The nonlinear response of the structure is studied using averaging on the two-mode model. The structure exhibits 1:2 internal resonance and large amplitude vibrations through the route of parametric excitation. The principal parametric resonance of the second mode is investigated in detail, and the time response of the averaged system is also computed at few frequencies to demonstrate stability of branches. Some results for the case of principal parametric resonance of the first mode are also presented.

References

References
1.
Petersen
,
K. E.
,
1982
, “
Silicon as a Mechanical Material
,”
Proc. IEEE
,
70
(
5
), pp.
420
457
.
2.
Craighead
,
H. G.
,
2000
, “
Nanoelectromechanical Systems
,”
Science
,
290
(
5496
), pp.
1532
1535
.
3.
Pelrine
,
R.
,
Kornbluh
,
R.
,
Pei
,
Q.
, and
Joseph
,
J.
,
2000
, “
High-Speed Electrically Actuated Elastomers With Strain Greater Than 100 Percent
,”
Science
,
287
(
5454
), pp.
836
839
.
4.
Petralia
,
M. T.
, and
Wood
,
R. J.
,
2010
, “
Fabrication and Analysis of Dielectric-Elastomer Minimum-Energy Structures for Highly-Deformable Soft Robotic Systems
,”
IEEE/RSJ Int. Conference on Intelligent Robots and Systems
(
IROS
), Taipei, Taiwan, Oct. 18–22.
5.
Xia
,
F.
,
Tadigadapa
,
S.
, and
Zhang
,
Q. M.
,
2005
, “
Electroactive Polymer Based Microfluidic Pump
,”
Sens. Actuators A
,
125
(
2
), pp.
346
352
.
6.
Tripathi
,
A.
, and
Bajaj
,
A. K.
,
2013
, “
Computational Synthesis for Nonlinear Dynamics Based Design of Planar Resonant Structures
,”
ASME J. Vib. Acoust.
,
135
(
5
), p.
051031
.
7.
Tripathi
,
A.
, and
Bajaj
,
A. K.
,
2014
, “
Design for 1:2 Internal Resonances in in-Plane Vibrations of Plates With Hyperelastic Materials
,”
ASME J. Vib. Acoust.
,
136
(
6
), p.
061005
.
8.
Tripathi
,
A.
,
2015
, “
On Computational Synthesis and Dynamic Analysis of Nonlinear Resonant Systems With Internal Resonances
,”
Ph.D. thesis
, Purdue University, West Lafayette, IN.https://docs.lib.purdue.edu/dissertations/AAI10075590/
9.
Lallart
,
M.
,
Richard
,
C.
,
Sukwisut
,
P.
,
Petit
,
L.
,
Guyomar
,
D.
, and
Muensit
,
N.
,
2012
, “
Electrostrictive Bending Actuators: Modeling and Experimental Investigation
,”
Sens. Actuators A
,
179
, pp.
169
177
.
10.
Liu
,
Y.
,
Ren
,
K. L.
,
Hofmann
,
H. F.
, and
Zhang
,
Q.
,
2005
, “
Investigation of Electrostrictive Polymers for Energy Harvesting
,”
IEEE Trans. Ultrason., Ferroelectr. Freq. Control
,
52
(
12
), pp.
2411
2417
.
11.
Svanberg
,
K.
,
1987
, “
The Method of Moving Asymptotes—A New Method for Structural Optimization
,”
Int. J. Numer. Methods Eng.
,
24
(
2
), pp.
359
373
.
12.
Richards
,
A. W.
, and
Odegard
,
G. W.
,
2010
, “
Constitutive Modeling of Electrostrictive Polymers Using a Hyperelasticity-Based Approach
,”
ASME J. Appl. Mech.
,
77
(
1
), pp.
1
15
.
13.
Nayfeh
,
A. H.
,
2000
,
Nonlinear Interactions: Analytical, Computational, and Experimental Methods
,
Wiley-Interscience
,
New York
.
14.
Bajaj
,
A. K.
,
Chang
,
S. I.
, and
Johnson
,
J. M.
,
1994
, “
Amplitude Modulated Dynamics of a Resonantly Excited Autoparametric Two Degree-of-Freedom System
,”
Nonlinear Dyn.
,
5
(
4
), pp.
433
457
.
15.
Tripathi
,
A.
, and
Bajaj
,
A. K.
,
2016
, “
Topology Optimization and Internal Resonances in Transverse Vibrations of Hyperelastic Plates
,”
Int. J. Solids Struct.
,
81
, pp.
311
328
.
16.
Cook
,
A. H.
,
Malkus
,
D. S.
,
Plesha
,
M. E.
, and
Witt
,
R. J.
,
2004
,
Concepts and Applications of Finite Element Analysis
,
Wiley-Interscience
,
New York
.
17.
Bendsoe
,
M. P.
, and
Sigmund
,
O.
,
1999
, “
Material Interpolation Schemes in Topology Optimization
,”
Arch. Appl. Mech.
,
69
(
9–10
), pp.
635
654
.
18.
George
,
E. D.
, Jr.
,
Haduch
,
G. A.
, and
Jordan
,
S.
,
1988
, “
The Integration of Analysis and Testing for the Simulation of the Response of Hyperelastic Materials
,”
Finite Element Anal. Des.
,
4
(
1
), pp.
19
42
.
19.
He
,
W.
,
Bindel
,
D.
, and
Govindjee
,
S.
,
2012
, “
Topology Optimization in Micromechanical Resonator Design
,”
Optim. Eng.
,
13
(2), pp. 271–292.
20.
Chadwick
,
P.
,
1999
,
Continuum Mechanics, Concise Theory and Problems
,
Dover Publications
,
New York
.
21.
Balachandran
,
B.
, and
Nayfeh
,
A. H.
,
1990
, “
Nonlinear Motions of Beam-Mass Structure
,”
Nonlinear Dyn.
,
1
(
1
), pp.
39
61
.
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