This paper investigates the internal energy transfer and modal interactions in the dynamical behavior of slightly curved microplates. Employing the third-order shear deformation theory, the microplate model is developed taking into account geometric nonlinearities as well as the modified couple stress theory; the initial curvature is modeled by an initial imperfection in the out-of-plane direction. The in-plane displacements and inertia are retained, and the coupled out-of-plane, rotational, and in-plane motion characteristics are analyzed. Specifically, continuous models are developed for kinetic and potential energies as well as damping and external works; these are balanced and reduced via Lagrange's equations along with an assumed-mode technique. The reduced-order model is then solved numerically by means of a continuation technique; stability analysis is performed by means of the Floquet theory. The possibility of the occurrence of modal interactions and internal energy transfers is verified via a linear analysis on different natural frequencies of the system. The nonlinear resonant response of the system is obtained for the cases with internal energy transfer, and energy transfer mechanisms are analyzed; as we shall see, the presence of an initial curvature affects the system dynamics substantially. The importance of taking into account small-size effects is also shown by discovering this fact that both the linear and nonlinear internal energy transfer mechanisms are shifted substantially if this effect is ignored.

References

References
1.
Rembe
,
C.
, and
Muller
,
R. S.
,
2002
, “
Measurement System for Full Three-Dimensional Motion Characterization of MEMS
,”
J. Microelectromech. Syst.
,
11
(
5
), pp.
479
488
.
2.
Ghayesh
,
M. H.
,
Farokhi
,
H.
, and
Amabili
,
M.
,
2013
, “
Nonlinear Behaviour of Electrically Actuated MEMS Resonators
,”
Int. J. Eng. Sci.
,
71
, pp.
137
155
.
3.
Younis
,
M. I.
,
Abdel-Rahman
,
E. M.
, and
Nayfeh
,
A.
,
2003
, “
A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS
,”
J. Microelectromech. Syst.
,
12
(
5
), pp.
672
680
.
4.
De
,
S. K.
, and
Aluru
,
N. R.
,
2004
, “
Full-Lagrangian Schemes for Dynamic Analysis of Electrostatic MEMS
,”
J. Microelectromech. Syst.
,
13
(
5
), pp.
737
758
.
5.
Liu
,
S.
,
Davidson
,
A.
, and
Lin
,
Q.
,
2004
, “
Simulation Studies on Nonlinear Dynamics and Chaos in a MEMS Cantilever Control System
,”
J. Micromech. Microeng.
,
14
(
7
), pp.
1064
1073
.
6.
Sulfridge
,
M.
,
Saif
,
T.
,
Miller
,
N.
, and
Meinhart
,
M.
,
2004
, “
Nonlinear Dynamic Study of a Bistable MEMS: Model and Experiment
,”
J. Microelectromech. Syst.
,
13
(
5
), pp.
725
731
.
7.
Zhang
,
W. M.
, and
Meng
,
G.
,
2007
, “
Nonlinear Dynamic Analysis of Electrostatically Actuated Resonant MEMS Sensors Under Parametric Excitation
,”
IEEE Sens. J.
,
7
(
3
), pp.
370
380
.
8.
Duwel
,
A.
,
Candler
,
R. N.
,
Kenny
,
T. W.
, and
Varghese
,
M.
,
2006
, “
Engineering MEMS Resonators With Low Thermoelastic Damping
,”
J. Microelectromech. Syst.
,
15
(
6
), pp.
1437
1445
.
9.
Mestrom
,
R. M. C.
,
Fey
,
R. H. B.
,
van Beek
,
J. T. M.
,
Phan
,
K. L.
, and
Nijmeijer
,
H.
,
2008
, “
Modelling the Dynamics of a MEMS Resonator: Simulations and Experiments
,”
Sens. Actuators, A
,
142
(
1
), pp.
306
315
.
10.
Bergers
,
L. I. J. C.
,
Hoefnagels
,
J. P. M.
,
Delhey
,
N. K. R.
, and
Geers
,
M. G. D.
,
2011
, “
Measuring Time-Dependent Deformations in Metallic MEMS
,”
Microelectron. Reliab.
,
51
(
6
), pp.
1054
1059
.
11.
Haghighi
,
H. S.
, and
Markazi
,
A. H. D.
,
2010
, “
Chaos Prediction and Control in MEMS Resonators
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
10
), pp.
3091
3099
.
12.
Farokhi
,
H.
, and
Ghayesh
,
M.
, “
Size-Dependent Behaviour of Electrically Actuated Microcantilever-Based MEMS
,”
Int. J. Mech. Mater. Des.
(in press).
13.
Gholipour
,
A.
,
Farokhi
,
H.
, and
Ghayesh
,
M.
,
2014
, “
In-Plane and Out-Of-Plane Nonlinear Size-Dependent Dynamics of Microplates
,”
Nonlinear Dyn.
,
79
(
3
), pp.
1771
1785
.
14.
Ghayesh
,
M. H.
, and
Farokhi
,
H.
,
2015
, “
Nonlinear Dynamics of Microplates
,”
Int. J. Eng. Sci.
,
86
, pp.
60
73
.
15.
Farokhi
,
H.
, and
Ghayesh
,
M. H.
,
2015
, “
Nonlinear Dynamical Behaviour of Geometrically Imperfect Microplates Based on Modified Couple Stress Theory
,”
Int. J. Mech. Sci.
,
90
, pp.
133
144
.
16.
Ghayesh
,
M. H.
,
Farokhi
,
H.
, and
Alici
,
G.
,
2015
, “
Subcritical Parametric Dynamics of Microbeams
,”
Int. J. Eng. Sci.
,
95
, pp.
36
48
.
17.
Das
,
K.
, and
Batra
,
R. C.
,
2009
, “
Pull-in and Snap-Through Instabilities in Transient Deformations of Microelectromechanical Systems
,”
J. Micromech. Microeng.
,
19
(
3
), p.
035008
.
18.
Slava
,
K.
,
Bojan
,
R. I.
,
David
,
S.
,
Shimon
,
S.
, and
Harold
,
C.
,
2008
, “
The Pull-In Behavior of Electrostatically Actuated Bistable Microstructures
,”
J. Micromech. Microeng.
,
18
(
5
), p.
055026
.
19.
Lam
,
D. C. C.
,
Yang
,
F.
,
Chong
,
A. C. M.
,
Wang
,
J.
, and
Tong
,
P.
,
2003
, “
Experiments and Theory in Strain Gradient Elasticity
,”
J. Mech. Phys. Solids
,
51
(
8
), pp.
1477
1508
.
20.
McFarland
,
A. W.
, and
Colton
,
J. S.
,
2005
, “
Role of Material Microstructure in Plate Stiffness With Relevance to Microcantilever Sensors
,”
J. Micromech. Microeng.
,
15
(
5
), p.
1060
.
21.
Fleck
,
N. A.
,
Muller
,
G. M.
,
Ashby
,
M. F.
, and
Hutchinson
,
J. W.
,
1994
, “
Strain Gradient Plasticity: Theory and Experiment
,”
Acta Metall. Mater.
,
42
(
2
), pp.
475
487
.
22.
Zhao
,
J.
,
Zhou
,
S.
,
Wang
,
B.
, and
Wang
,
X.
,
2012
, “
Nonlinear Microbeam Model Based on Strain Gradient Theory
,”
Appl. Math. Modell.
,
36
(
6
), pp.
2674
2686
.
23.
Akgöz
,
B.
, and
Civalek
,
Ö.
,
2013
, “
A Size-Dependent Shear Deformation Beam Model Based on the Strain Gradient Elasticity Theory
,”
Int. J. Eng. Sci.
,
70
, pp.
1
14
.
24.
Salamat-Talab
,
M.
,
Shahabi
,
F.
, and
Assadi
,
A.
,
2012
, “
Size Dependent Analysis of Functionally Graded Microbeams Using Strain Gradient Elasticity Incorporated With Surface Energy
,”
Appl. Math. Modell.
,
37
(
1–2
), pp.
507
526
.
25.
Mohammadi
,
H.
, and
Mahzoon
,
M.
,
2013
, “
Thermal Effects on Postbuckling of Nonlinear Microbeams Based on the Modified Strain Gradient Theory
,”
Compos. Struct.
,
106
, pp.
764
776
.
26.
Farokhi
,
H.
,
Ghayesh
,
M. H.
, and
Amabili
,
M.
,
2013
, “
Nonlinear Dynamics of a Geometrically Imperfect Microbeam Based on the Modified Couple Stress Theory
,”
Int. J. Eng. Sci.
,
68
, pp.
11
23
.
27.
Ghayesh
,
M. H.
,
Farokhi
,
H.
, and
Amabili
,
M.
,
2014
, “
In-Plane and Out-of-Plane Motion Characteristics of Microbeams With Modal Interactions
,”
Compos. Part B: Eng.
,
60
, pp.
423
439
.
28.
Ghayesh
,
M. H.
,
Amabili
,
M.
, and
Farokhi
,
H.
,
2013
, “
Three-Dimensional Nonlinear Size-Dependent Behaviour of Timoshenko Microbeams
,”
Int. J. Eng. Sci.
,
71
, pp.
1
14
.
29.
Hashemi
,
S. H.
, and
Samaei
,
A. T.
,
2011
, “
Buckling Analysis of Micro/Nanoscale Plates Via Nonlocal Elasticity Theory
,”
Physica E
,
43
(
7
), pp.
1400
1404
.
30.
Sharma
,
J. N.
, and
Sharma
,
R.
,
2011
, “
Damping in Micro-Scale Generalized Thermoelastic Circular Plate Resonators
,”
Ultrasonics
,
51
(
3
), pp.
352
358
.
31.
He
,
L.
,
Lou
,
J.
,
Zhang
,
E.
,
Wang
,
Y.
, and
Bai
,
Y.
,
2015
, “
A Size-Dependent Four Variable Refined Plate Model for Functionally Graded Microplates Based on Modified Couple Stress Theory
,”
Compos. Struct.
,
130
, pp.
107
115
.
32.
Wang
,
B.
,
Zhou
,
S.
,
Zhao
,
J.
, and
Chen
,
X.
,
2011
, “
A Size-Dependent Kirchhoff Micro-Plate Model Based on Strain Gradient Elasticity Theory
,”
Eur. J. Mech. A. Solids
,
30
(
4
), pp.
517
524
.
33.
Ramezani
,
S.
,
2012
, “
A Shear Deformation Micro-Plate Model Based on the Most General Form of Strain Gradient Elasticity
,”
Int. J. Mech. Sci.
,
57
(
1
), pp.
34
42
.
34.
Tahani
,
M.
,
Askari
,
A. R.
,
Mohandes
,
Y.
, and
Hassani
,
B.
,
2015
, “
Size-Dependent Free Vibration Analysis of Electrostatically Pre-Deformed Rectangular Micro-Plates Based on the Modified Couple Stress Theory
,”
Int. J. Mech. Sci.
,
94–95
, pp.
185
198
.
35.
Roque
,
C. M. C.
,
Ferreira
,
A. J. M.
, and
Reddy
,
J. N.
,
2013
, “
Analysis of Mindlin Micro Plates With a Modified Couple Stress Theory and a Meshless Method
,”
Appl. Math. Modell.
,
37
(
7
), pp.
4626
4633
.
36.
Asghari
,
M.
,
2012
, “
Geometrically Nonlinear Micro-Plate Formulation Based on the Modified Couple Stress Theory
,”
Int. J. Eng. Sci.
,
51
, pp.
292
309
.
37.
Thai
,
H.-T.
, and
Choi
,
D.-H.
,
2013
, “
Size-Dependent Functionally Graded Kirchhoff and Mindlin Plate Models Based on a Modified Couple Stress Theory
,”
Compos. Struct.
,
95
, pp.
142
153
.
38.
Ansari
,
R.
,
Gholami
,
R.
,
Faghih Shojaei
,
M.
,
Mohammadi
,
V.
, and
Darabi
,
M. A.
,
2015
, “
Size-Dependent Nonlinear Bending and Postbuckling of Functionally Graded Mindlin Rectangular Microplates Considering the Physical Neutral Plane Position
,”
Compos. Struct.
,
127
, pp.
87
98
.
39.
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
.
40.
Ouakad
,
H. M.
, and
Younis
,
M. I.
,
2010
, “
The Dynamic Behavior of MEMS Arch Resonators Actuated Electrically
,”
Int. J. Non Linear Mech.
,
45
(
7
), pp.
704
713
.
41.
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
.
42.
Ouakad
,
H. M.
,
2013
, “
An Electrostatically Actuated MEMS Arch Band-Pass Filter
,”
Shock Vib.
,
20
(
4
), pp.
809
819
.
43.
Mohammad
,
T.
, and
Ouakad
,
H.
,
2014
, “
Static, Eigenvalue Problem and Bifurcation Analysis of MEMS Arches Actuated by Electrostatic Fringing-Fields
,”
Microsyst. Technol.
44.
Medina
,
L.
,
Gilat
,
R.
,
Ilic
,
B.
, and
Krylov
,
S.
,
2014
, “
Experimental Investigation of the Snap-Through Buckling of Electrostatically Actuated Initially Curved Pre-Stressed Micro Beams
,”
Sens. Actuators, A
,
220
, pp.
323
332
.
45.
Medina
,
L.
,
Gilat
,
R.
, and
Krylov
,
S.
,
2014
, “
Symmetry Breaking in an Initially Curved Pre-Stressed Micro Beam Loaded by a Distributed Electrostatic Force
,”
Int. J. Solids Struct.
,
51
(
11–12
), pp.
2047
2061
.
46.
Medina
,
L.
,
Gilat
,
R.
, and
Krylov
,
S.
,
2012
, “
Symmetry Breaking in an Initially Curved Micro Beam Loaded by a Distributed Electrostatic Force
,”
Int. J. Solids Struct.
,
49
(
13
), pp.
1864
1876
.
47.
Yang
,
F.
,
Chong
,
A. C. M.
,
Lam
,
D. C. C.
, and
Tong
,
P.
,
2002
, “
Couple Stress Based Strain Gradient Theory for Elasticity
,”
Int. J. Solids Struct.
,
39
(
10
), pp.
2731
2743
.
48.
Park
,
S. K.
, and
Gao
,
X. L.
,
2006
, “
Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory
,”
J. Micromech. Microeng.
,
16
(
11
), p.
2355
.
49.
Ma
,
H. M.
,
Gao
,
X. L.
, and
Reddy
,
J. N.
,
2008
, “
A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory
,”
J. Mech. Phys. Solids
,
56
(
12
), pp.
3379
3391
.
50.
Gao
,
X. L.
,
Huang
,
J. X.
, and
Reddy
,
J. N.
,
2013
, “
A Non-Classical Third-Order Shear Deformation Plate Model Based on a Modified Couple Stress Theory
,”
Acta Mech.
,
224
(
11
), pp.
2699
2718
.
51.
Amabili
,
M.
,
2004
, “
Nonlinear Vibrations of Rectangular Plates With Different Boundary Conditions: Theory and Experiments
,”
Comput. Struct.
,
82
(
31–32
), pp.
2587
2605
.
You do not currently have access to this content.