The nonlinear modal coupling between the vibration modes of an arch-shaped microstructure is an interesting phenomenon, which may have desirable features for numerous applications, such as vibration-based energy harvesters. This work presents an investigation into the potential nonlinear internal resonances of a microelectromechanical systems (MEMS) arch when excited by static (DC) and dynamic (AC) electric forces. The influences of initial rise and midplane stretching are considered. The cases of one-to-one and three-to-one internal resonances are studied using the method of multiple scales and the direct attack of the partial differential equation of motion. It is shown that for certain initial rises, it is possible to activate a three-to-one internal resonance between the first and third symmetric modes. Also, using an antisymmetric half-electrode actuation, a one-to-one internal resonance between the first symmetric and the second antisymmetric modes is demonstrated. These results can shed light on such interactions that are commonly found on micro and nanostructures, such as carbon nanotubes.

References

References
1.
Malihi
,
S.
,
Beni
,
Y. T.
, and
Golestanian
,
H.
,
2017
, “
Dynamic Pull-In Stability of Torsional Nano/Micromirrors With Size-Dependency, Squeeze Film Damping and van der Waals Effect
,”
Opt. Int. J. Light Electron Opt.
,
128
, pp.
156
171
.
2.
Shojaeian
,
M.
,
Beni
,
Y. T.
, and
Ataei
,
H.
,
2016
, “
Electromechanical Buckling of Functionally Graded Electrostatic Nanobridges Using Strain Gradient Theory
,”
Acta Astronaut.
,
118
, pp.
62
71
.
3.
Marotti de Sciarra
,
F.
, and
Barretta
,
R.
,
2014
, “
A Gradient Model for Timoshenko Nanobeams
,”
Physica E
,
62
, pp.
1
9
.
4.
Canadija
,
M.
,
Barretta
,
R.
, and
Marotti de Sciarra
,
F.
,
2016
, “
A Gradient Elasticity Model of Bernoulli–Euler Nanobeams in Non-Isothermal Environments
,”
Eur. J. Mech. A/Solids
,
55
, pp.
243
255
.
5.
Barretta
,
R.
,
Čanadija
,
M.
, and
Marotti de Sciarra
,
F.
,
2016
, “
A Higher-Order Eringen Model for Bernoulli–Euler Nanobeams
,”
Arch. Appl. Mech.
,
86
(
3
), pp.
483
495
.
6.
Akgöz
,
B.
, and
Civalek
,
Ö.
,
2014
, “
Mechanical Analysis of Isolated Microtubules Based on a Higher-Order Shear Deformation Beam Theory
,”
Compos. Struct.
,
118
, pp.
9
18
.
7.
Ebrahimi
,
F.
, and
Barati
,
M. R.
,
2016
, “
Size-Dependent Thermal Stability Analysis of Graded Piezomagnetic Nanoplates on Elastic Medium Subjected to Various Thermal Environments
,”
Appl. Phys. A
,
122
(
10
), p.
910
.
8.
Shaat
,
M.
,
Akbarzadeh Khorshidi
,
M.
,
Abdelkefi
,
A.
, and
Shariati
,
M.
,
2016
, “
Modeling and Vibration Characteristics of Cracked Nano-Beams Made of Nanocrystalline Materials
,”
Int. J. Mech. Sci.
,
115–116
, pp.
574
585
.
9.
Ouakad
,
H. M.
, and
Younis
,
M. I.
,
2010
, “
Nonlinear the Dynamic Behavior of MEMS Arch Resonators Actuated Electrically
,”
Int. J. Non-Linear Mech.
,
45
(
7
), pp.
704
713
.
10.
Tsai
,
N.-C.
,
Sue
,
C.-Y.
, and
Lin
,
C.-C.
,
2008
, “
Design and Dynamics of an Innovative Micro Gyroscope Against Coupling Effects
,”
Microsyst. Technol.
,
14
(
3
), pp.
295
306
.
11.
Sanchez
,
R.
, and
Renard
,
P.
,
2006
, “
Design of a Micro-Satellite for Precise Formation Flying Demonstration
,”
Acta Astronaut.
,
59
(
8–11
), pp.
862
872
.
12.
Comi
,
C.
,
Corigliano
,
A.
,
Ghisi
,
A.
, and
Zerbini
,
S.
,
2013
, “
A Resonant Micro Accelerometer Based on Electrostatic Stiffness Variation
,”
Meccanica
,
48
(
8
), pp.
1893
1900
.
13.
Yoo
,
S. K.
,
Lee
,
J. H.
,
Yun
,
S.-S.
,
Gu
,
M. B.
, and
Lee
,
J. H.
,
2007
, “
Fabrication of a Bio-MEMS Based Cell-Chip for Toxicity Monitoring
,”
Biosens. Bioelectron.
,
22
(
8
), pp.
1586
1592
.
14.
Choi
,
N.-J.
,
Lee
,
Y.-S.
,
Kwak
,
J.-H.
,
Park
,
J.-S.
,
Park
,
K.-B.
,
Shin
,
K.-S.
,
Park
,
H.-D.
,
Kim
,
J.-C.
,
Huh
,
J.-S.
, and
Lee
,
D.-D.
,
2005
, “
Chemical Warfare Agent Sensor Using MEMS Structure and Thick Film Fabrication Method
,”
Sens. Actuators B
,
108
(
1–2
), pp.
177
183
.
15.
Hafiz
,
M.
,
Kosuru
,
L.
,
Ramini
,
A.
,
Chappanda
,
K.
, and
Younis
,
M.
,
2016
, “
In-Plane MEMS Shallow Arch Beam for Mechanical Memory
,”
Micromachines
,
7
(
10
), p.
191
.
16.
Hafiz
,
M. A. A.
,
Kosuru
,
L.
, and
Younis
,
M. I.
,
2016
, “
Microelectromechanical Reprogrammable Logic Device
,”
Nat. Commun.
,
7
, p.
11137
.
17.
Leung
,
A. Y. T.
, and
Fung
,
T. C.
,
1990
, “
Non-Linear Steady State Vibration and Dynamic Snap Through of Shallow Arch Beams
,”
Earthquake Eng. Struct. Dyn.
,
19
(
3
), pp.
409
430
.
18.
Chen
,
L.-Q.
, and
Jiang
,
W.-A.
,
2015
, “
Internal Resonance Energy Harvesting
,”
ASME J. Appl. Mech.
,
82
(
3
), p.
031004
.
19.
Das
,
K.
, and
Batra
,
R. C.
,
2009
, “
Symmetry Breaking, Snap-Through and Pull-In Instabilities Under Dynamic Loading of Microelectromechanical Shallow Arches
,”
Smart Mater. Struct.
,
18
(
11
), p.
115008
.
20.
Ouakad
,
H. M.
,
2013
, “
An Electrostatically Actuated MEMS Arch Band-Pass Filter
,”
Shock Vib.
,
20
(
4
), pp.
809
819
.
21.
Alkharabsheh
,
S. A.
, and
Younis
,
M. I.
,
2013
, “
Statics and Dynamics of MEMS Arches Under Axial Forces
,”
ASME J. Vib. Acoust.
,
135
(
2
), p.
021007
.
22.
Nayfeh
,
A. H.
, and
Emam
,
S. A.
,
2008
, “
Exact Solution and Stability of Postbuckling Configurations of Beams
,”
Nonlinear Dyn.
,
54
(
4
), pp.
395
408
.
23.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
1989
, “
Modal Interactions in Dynamical and Structural Systems
,”
ASME Appl. Mech. Rev.
,
42
(
11
), pp.
175
201
.
24.
Balachandran
,
B.
, and
Nayfeh
,
A. H.
,
1990
, “
Nonlinear Oscillations of a Harmonically Excited Composite Structure
,”
Compos. Struct.
,
16
(
4
), pp.
323
339
.
25.
Alkharabsheh
,
S. A.
, and
Younis
,
M. I.
,
2013
, “
Dynamics of MEMS Arches of Flexible Supports
,”
J. Microelectromech. Syst.
,
22
(
1
), pp.
216
224
.
26.
Ouakad
,
H. M.
, and
Younis
,
M. I.
,
2014
, “
On Using the Dynamic Snap-Through Motion of MEMS Initially Curved Microbeams for Filtering Applications
,”
J. Sound Vib.
,
333
(
2
), pp.
555
568
.
27.
Ghayesh
,
M. H.
,
Farokhi
,
H.
, and
Alici
,
G.
,
2015
, “
Internal Energy Transfer in Dynamical Behavior of Slightly Curved Shear Deformable Microplates
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
4
), p.
041002
.
28.
Medina
,
L.
,
Gilat
,
R.
, and
Krylov
,
S.
,
2017
, “
Latching in Bistable Electrostatically Actuated Curved Micro Beams
,”
Int. J. Eng. Sci.
,
110
, pp.
15
34
.
29.
Antonio
,
D.
,
Zanette
,
D. H.
, and
López
,
D.
,
2012
, “Frequency Stabilization in Nonlinear Micromechanical Oscillators.
Nat. Commun.
,
3
, p.
806
.
30.
Srinil
,
N.
, and
Rega
,
G.
,
2007
, “
Two-to-One Resonant Multi-Modal Dynamics of Horizontal/Inclined Cables—Part II: Internal Resonance Activation, Reduced Order Models and Nonlinear Normal Modes
,”
Nonlinear Dyn.
,
48
(
3
), pp.
253
274
.
31.
Chen
,
L. Q.
,
Zhang
,
G.-C.
, and
Ding
,
H.
,
2015
, “
Internal Resonance in Forced Vibration of Coupled Cantilevers Subjected to Magnetic Interaction
,”
J. Sound Vib.
,
354
, pp.
196
218
.
32.
Westra
,
H. J. R.
,
van der Zant
,
H. S. J.
, and
Venstra
,
W. J.
,
2012
, “
Modal Interactions of Flexural and Torsional Vibrations in a Microcantilever
,”
Ultramicroscopy
,
120
, pp.
41
47
.
33.
Chin
,
C.
, and
Nayfeh
,
A.
,
1999
, “
Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams
,”
Nonlinear Dyn.
,
20
(
2
), pp.
131
158
.
34.
Nayfeh
,
A. H.
,
Chin
,
C.
, and
Nayfeh
,
S. A.
,
1996
, “
On Nonlinear Normal Modes of Systems With Internal Resonance
,”
ASME J. Vib. Acoust.
,
118
(
3
), pp.
340
345
.
35.
Malhotral
,
N.
, and
Namachchivaya
,
N.
,
1997
, “
Chaotic Motion of Shallow Arch Structures Under 1:1 Internal Resonance
,”
J. Eng. Mech.
,
123
(
6
), pp.
620
627
.
36.
Tien
,
W.
,
W.-M.
,
Sri Namachchivaya
,
N.
, and
Malhotra
,
N.
,
1994
, “
Non-Linear Dynamics of a Shallow Arch Under Periodic Excitation-II—1: 1 Internal Resonance
,”
Int. J. Non-Linear Mech.
,
29
(
3
), pp.
367
386
.
37.
Nayfeh
,
A.
,
Lacarbonara
,
W.
, and
Chin
,
C.-M.
,
1999
, “
Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances
,”
Nonlinear Dyn.
,
18
(
3
), pp.
253
273
.
38.
El-Bassiouny
,
A. F.
,
Kamel
,
M. M.
, and
Abdel-Khalik
,
A.
,
2003
, “
Two-to-One Internal Resonances in Nonlinear Two Degree of Freedom System With Parametric and External Excitations
,”
Math. Comput. Simul.
,
63
(
1
), pp.
45
56
.
39.
El-Bassiouny
,
A. F.
,
2005
, “
Three-to-One Internal Resonance in the Nonlinear Oscillation of Shallow Arch
,”
Phys. Scr.
,
72
(
6
), pp.
439
450
.
40.
Bi
,
Q.
, and
Dai
,
H. H.
,
2000
, “
Analysis of Non-Linear Dynamics and Bifurcations of a Shallow Arch Subjected to Periodic Excitation With Internal Resonance
,”
J. Sound Vib.
,
233
(
4
), pp.
557
571
.
41.
Nayfeh
,
A. H.
, and
Pai
,
P. F.
,
2004
,
Linear and Nonlinear Structural Mechanics
,
Wiley
,
New York
.
42.
Batra, R, C.,
Porfiri
,
M.
, and
Spinello
,
D.
,
2006
, “
Capacitance Estimate for Electrostatically Actuated Narrow Microbeams
,”
Micro Nano Lett.
,
1
(
2
), pp.
71
73
.
43.
Batra
,
R. C.
,
Porfiri
,
M.
, and
Spinello
,
D.
,
2006
, “
Electromechanical Model of Electrically Actuated Narrow Microbeams
,”
J. Microelectromech. Syst.
,
15
(
5
), pp.
1175
1189
.
44.
Nayfeh
,
A. H.
,
2000
,
Nonlinear Interactions
,
Wiley InterScience
,
New York
.
45.
Tondl
,
A.
,
Ruigrock
,
T.
,
Verhulst
,
F.
, and
Nabergoj
,
R.
,
2000
,
Autoparametric Resonance in Mechanical Systems
,
Cambridge University Press
,
Cambridge, UK
.
46.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1995
,
Nonlinear Oscillations
,
Wiley InterScience
,
New York
.
47.
Younis
,
M. I.
, and
Nayfeh
,
A. H.
,
2003
, “
A Study of the Nonlinear Response of a Resonant Microbeam to an Electric Actuation
,”
Nonlinear Dyn.
,
31
(
1
), pp.
91
117
.
48.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
,
1995
,
Applied Nonlinear Dynamics
,
Wiley
,
New York
.
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