In this study an efficient reduced-order model for a MEMS device is developed and investigations of the nonlinear static and the dynamic behavior are performed. The device is constituted of an imperfect microbeam under an axial load and an electric excitation. The imperfections, typically due to microfabrication processes, are simulated assuming a shallow arched initial shape. The axial load is deliberately added with an elevated value. The structure has a bistable static configuration of double potential well with possibility of escape. We derive a single-mode reduced-order model via the Ritz technique and the Padé approximation. This model, while simple, is able to combine both a sufficient accuracy, which enables to detect the main qualitative features of the device response up to elevated values of electrodynamic excitation, and a remarkable computational efficiency, which is essential for systematic global nonlinear dynamic simulations. We illustrate the nonlinear phenomena arising in the device, such as the coexistence of various competing in-well and cross-well attractors, which leads to a considerable versatility of behavior. We discuss their physical meaning and their practical relevance for the engineering design of the microstructure, since this is an uncommon and very attractive aspect in applications.

References

References
1.
Senturia
,
S. D.
,
2001
,
Microsystem Design
,
Kluwer Academic Publishers
,
Dordrecht
.
2.
Younis
,
M. I.
,
2011
,
MEMS Linear and Nonlinear Statics and Dynamics
,
Springer
.
3.
Bao
,
M.
,
2005
,
Analysis and Design Principles of MEMS Devices
,
Elsevier
,
Amsterdam
.
4.
Pelesko
,
J. A.
, and
Bernestein
,
D. H.
,
2003
,
Modeling MEMS and NEMS
,
Chapman & Hall/CRC
.
5.
Lifshitz
,
R.
, and
Roukes
,
M. L.
,
1999
, “
Thermoelastic Damping in Micro- and Nanomechanical Systems
,”
Phys. Rev. B
,
61
(
8
), pp.
5600
5609
.10.1103/PhysRevB.61.5600
6.
Rhoads
,
J. F.
,
Shaw
,
S. W.
, and
Turner
,
K. L.
,
2010
, “
Nonlinear Dynamics and Its Applications in Micro- and Nanoresonators
,”
J. Dyn. Syst., Meas., Control
,
132
(
3
), p.
034001
.10.1115/1.4001333
7.
Krylov
,
S.
,
Harari
,
I.
, and
Cohen
,
Y.
,
2005
, “
Stabilization of Electrostatically Actuated Microstructures Using Parametric Excitation
,”
J. Micromech. Microeng.
,
15
, pp.
1188
1204
.10.1088/0960-1317/15/6/009
8.
Das
,
K.
, and
Batra
,
R. C.
,
2009
, “
Symmetry Breaking, Snap-Through, and Pull-In Instabilities Under Dynamic Loading of Microelectromechanical Shallow Arch
,”
Smart Mater. Struct.
,
18
(
11
), p.
115008
.10.1088/0964-1726/18/11/115008
9.
Krylov
,
S.
,
Ilic
,
B. R.
,
Schreiber
,
D.
,
Seretensky
,
S.
, and
Craighead
,
H.
,
2008
, “
The Pull-In Behavior of Electrostatically Actuated Bistable Microbeams
,”
J. Micromech. Microeng.
,
18
(
5
), p.
55026
.10.1088/0960-1317/18/5/055026
10.
Younis
,
M. I.
,
Ouakad
,
H. M.
,
Alsaleem
,
F. M.
,
Miles
,
R.
, and
Cui
,
W.
,
2010
, “
Nonlinear Dynamics of MEMS Arches Under Harmonic Electrostatic Actuation
,”
J. Microelectromech. Syst.
,
19
(
3
), pp.
647
656
.10.1109/JMEMS.2010.2046624
11.
Hung
,
E. S.
, and
Senturia
,
S. D.
,
1999
, “
Generating Efficient Dynamical Models for Micro-Electro-Mechanical Systems From a Few Finite-Element Simulation Runs
,”
J. Microelectromech. Syst.
,
3
, pp.
280
289
.10.1109/84.788632
12.
Nayfeh
,
A. H.
,
Younis
,
M. I.
, and
Abdel-Rahman
,
E. M.
,
2005
, “
Reduced-Order Models for MEMS Applications
,”
Nonlinear Dyn.
,
41
, pp.
211
236
.10.1007/s11071-005-2809-9
13.
Younis
,
M. I.
,
Abdel-Rahman
,
E. M.
, and
Nayfeh
,
A. H.
,
2003
, “
A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS
,”
J. Microelectromech. Syst.
,
12
(
5
), pp.
672
680
.10.1109/JMEMS.2003.818069
14.
Ouakad
,
H. M.
, and
Younis
,
M. I.
,
2010
, “
The Dynamic Behavior of MEMS Arch Resonators Actuated Electrically
,”
Int. J. Non-Linear Mech.
,
45
, pp.
704
713
.10.1016/j.ijnonlinmec.2010.04.005
15.
Krylov
,
S.
, and
Dick
,
N.
,
2010
, “
Dynamic Stability of Electrostatically Actuated Initially Curved Shallow Micro Beams
,”
Continuum Mech. Thermodyn.
,
22
, pp.
445
468
.10.1007/s00161-010-0149-6
16.
Ouakad
,
H. M.
, and
Younis
,
M. I.
,
2010
, “
Nonlinear Dynamics of Electrically Actuated Carbon Nanotube Resonators
,”
J. Comput. Nonlin. Dyn.
,
5
, p.
011009
.10.1115/1.4000319
17.
Rhoads
,
J. F.
,
Shaw
,
S. W.
, and
Turner
,
K. L.
,
2006
, “
The Nonlinear Response of Resonant Microbeam Systems With Purely-Parametric Electrostatic Actuation
,”
J. Micromech. Microeng.
,
16
, pp.
890
899
.10.1088/0960-1317/16/5/003
18.
Gottlieb
,
O.
, and
Champneys
,
A. R.
,
2005
, “
Global Bifurcations of Nonlinear Thermoelastic Microbeams Subject to Electrodynamic Actuation
,”
IUTAM Symposium on Chaotic Dynamics and Control of System and Processes in Mechanics Solid Mechanics and its Applications
, edited by
G.
Rega
and
F.
Vestroni
,
Springer
,
Berlin
, Vol.
122
, pp.
47
57
.
19.
Lenci
,
S.
, and
Rega
,
G.
,
2006
, “
Control of Pull-In Dynamics in a Nonlinear Thermoelastic Electrically Actuated Microbeam
,”
J. Micromech. Microeng.
,
16
, pp.
390
401
.10.1088/0960-1317/16/2/025
20.
Ruzziconi
,
L.
,
Younis
,
M. I.
, and
Lenci
,
S.
,
2010
, “
Dynamical Integrity for Interpreting Experimental Data and Ensuring Safety in Electrostatic MEMS
,”
IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design
,
July
,
Aberdeen
.
21.
Towfighian
,
S.
,
Heppler
,
G. R.
, and
Abdel-Rahman
,
E. M.
,
2010
, “
A Low Voltage Controller for a Chaotic Microresonator
,”
ASME IDETC
, Paper No. 28990.
22.
Haghighi
,
H. S.
, and
Markazi
,
A. H. D.
,
2010
, “
Chaos Prediction and Control in MEMS Resonators
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
, pp.
3091
3099
.10.1016/j.cnsns.2009.10.002
23.
Dick
,
A. J.
,
Balachandran
,
B.
,
DeVoe
,
D. L.
, and
Mote
, Jr.,
C. D.
,
2006
, “
Parametric Identification of Piezoelectric Microscale Resonators
,”
J. Micromech. Microeng.
,
16
, pp.
1593
1601
.10.1088/0960-1317/16/8/021
24.
Wolf
,
K.
, and
Gottlieb
,
O.
,
2002
, “
Nonlinear Dynamics of a Noncontacting Atomic Force Microscope Cantilever Actuated by a Piezoelectric Layer
,”
J. Appl. Phys.
,
91
, pp.
4701
4709
.10.1063/1.1458056
25.
Ashhab
,
A.
,
Salapaka
,
M. V.
,
Dahleh
,
M.
, and
Mezić
,
I.
,
1999
, “
Melnikov-Based Dynamical Analysis of Microcantilevers in Scanning Probe Microscopy
,”
Nonlinear Dyn.
,
20
, pp.
197
220
.10.1023/A:1008342408448
26.
Arafat
,
H. N.
,
Nayfeh
,
A. H.
, and
Abdel-Rahman
,
E. M.
,
2008
, “
Modal Interactions in Contact-Mode Atomic Force Microscopes
,”
Nonlinear Dyn.
,
54
, pp.
151
166
.10.1007/s11071-008-9388-5
27.
Mestrom
,
R. M. C.
,
Fey
,
R. H. B.
,
Phan
,
K. L.
, and
Nijmeijer
,
H.
,
2010
, “
Simulations and Experiments of Hardening and Softening Resonances in a Clamped–Clamped Beam MEMS Resonator
,”
Sens. Actuators, A
,
162
, pp.
225
234
.10.1016/j.sna.2010.04.020
28.
Mestrom
,
R. M. C.
,
Fey
,
R. H. B.
,
Phan
,
K. L.
,
van Beek
,
J. T. M.
, and
Nijmeijer
,
H.
,
2008
, “
Modelling the Dynamics of a MEMS Resonator: Simulations and Experiments
,”
Sens. Actuators, A
,
142
(
1
), pp.
306
315
.10.1016/j.sna.2007.04.025
29.
Zhang
,
W. M.
,
Meng
,
G.
, and
Chen
,
D.
,
2007
, “
Stability, Nonlinearity and Reliability of Electrostatically Actuated MEMS Devices
,”
Sensors
,
7
, pp.
760
796
.10.3390/s7050760
30.
Zhang
,
W. M.
, and
Meng
,
G.
,
2007
, “
Nonlinear Dynamic Analysis of Electrostatically Actuated Resonant MEMS Sensors Under Parametric Excitation
,”
IEEE Sens. J.
,
7
, pp.
370
380
.10.1109/JSEN.2006.890158
31.
Towfighian
,
S.
,
Seleim
,
A.
,
Abdel-Rahman
,
E. M.
, and
Heppler
,
G. R.
,
2011
, “
A Large-Stroke Electrostatic Micro-Actuator
,”
J. Micromech. Microeng.
,
21
, p.
075023
.10.1088/0960-1317/21/7/075023
32.
Elata
,
D.
, and
Abu-Salih
,
S.
,
2005
, “
Analysis of a Novel Method for Measuring Residual Stress in Micro-Systems
,”
J. Micromech. Microeng.
,
15
, pp.
921
927
.10.1088/0960-1317/15/5/004
33.
Abu-Salih
,
S.
, and
Elata
,
D.
,
2006
, “
Experimental Validation of Electromechanical Buckling
,”
J. Microelectromech. Syst.
,
15
, pp.
1656
1662
.10.1109/JMEMS.2006.886015
34.
Ruzziconi
,
L.
,
Bataineh
,
A. M.
,
Younis
,
M. I.
, and
Lenci
,
S.
,
2012
, “
Nonlinear Dynamics of a MEMS Resonator: Theoretical and Experimental Investigation
,”
ICNPAA Congress on Mathematical Problems in Engineering
,
Aerospace and Sciences
,
Vienna
.
35.
Ruzziconi
,
L.
,
Lenci
,
S.
, and
Younis
,
M. I.
, “
An Imperfect Microbeam Under an Axial Load and Electric Excitation: Nonlinear Phenomena and Dynamical Integrity
” Int. J. Bifurcation and Chaos, in press.
36.
Alsaleem
,
F. M.
,
Younis
,
M. I.
, and
Ruzziconi
,
L.
,
2010
, “
An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically
,”
J. Microelectromech. Syst.
,
19
, pp.
794
806
.10.1109/JMEMS.2010.2047846
37.
Villaggio
,
P.
,
1997
,
Mathematical Models for Elastic Structures
,
Cambridge University Press
,
Pisa
.
38.
Rega
,
G.
, and
Troger
,
H.
,
2005
, “
Dimension Reduction of Dynamical Systems: Methods, Models, Applications
,”
Nonlinear Dyn.
,
41
, pp.
1
15
.10.1007/s11071-005-2790-3
39.
Steindl
,
A.
, and
Troger
,
H.
,
2001
, “
Methods for Dimension Reduction and Their Application in Nonlinear Dynamics
,”
Int. J. Solids Struct.
,
38
, pp.
2131
2147
.10.1016/S0020-7683(00)00157-8
40.
Reddy
,
J. N.
,
1984
,
Energy and Variational Methods in Applied Mechanics, with an Introduction to the Finite Element Method,
Wiley
,
Canada
.
41.
Nayfeh
,
A. H.
,
Kreider
,
W.
, and
Anderson
,
T. J.
,
1995
, “
Investigation of Natural Frequencies and Mode Shapes of Buckled Beams
,”
AIAA J.
,
33
(
6
), pp.
1121
1126
.10.2514/3.12669
42.
Brezinski
,
C.
, and
Redivo Zaglia
,
M.
,
1991
,
Extrapolation Methods. Theory and Practice
,
North-Holland
.
43.
Nusse
,
H. E.
, and
Yorke
,
J. A.
,
1998
,
Dynamics. Numerical Explorations
,
Springer-Verlag, NY
.
You do not currently have access to this content.