Oscillating microplates attached to microbeams is the main part of many microresonators. There are several body and contact forces affecting a vibrating microbeam. Among them are some forces appearing to be significant in micro and nano size scales. Accepting an analytical approach, we present the mathematical modeling of a microresonator a nonlinear model for MEMS are presented, which accounts for the initial deflection due to polarization voltage, mid-plane stretching, and axial loads as well as the nonlinear displacement coupling of electric force. The equations are nondimensionalized and the design parameters are developed. However, the main purpose of this investigation is to present an applied model to simulate the squeeze-film phenomena. We separate the two characteristics of the squeeze-film phenomena and model the damping and stiffness effects individually. Motion of the microplate and flow of the gas underneath is similar to the function of a decoupler plate in hydraulic engine mounts (Golnaraghi and Nakhaie Jazar 2001, Nakhaie Jazar and Golnaraghi 2002). More specifically, the squeeze-film damping is qualitatively similar to the function of decoupler plate in hydraulic engine mounts, which is an amplitude dependent damping (Christopherson and Nakhaie Jazar 2005). It means squeeze-film damping effect is a positive phenomenon to isolate the vibration of microplate from the substrate. Following Golnaraghi and Nakhaie Jazar (2001), and utilizing the aforementioned similarity, we model the squeeze-film damping force, fsd, by a cubic function, where the coefficient Cs is assumed constant and must be evaluated experimentally. In the simplest case, we present the following fifth degree function to simulate the spring force, fss, of the squeeze film phenomenon, simply because at low amplitudes, w ≈ 0, the fluid layer is not strongly squeezed and there is no considerable resistance. On the other hand, at high amplitudes, w ≈ d, there is not much fluid to react as a spring. In addition, speed is proportionally related to the squeezeness of trapped fluid. The coefficient ks assumed constant and must be evaluated experimentally. The coefficients cs and ks are dependent on geometry as well as dynamic properties of the fluid, but assumed to be independent of kinematics of the microbeam such as displacement and velocity. Therefore, this investigation presents two mathematical functions to describe stiffness and damping characteristics of squeeze-film phenomena in a reduced-order model of microresonators.

1.
Abdel-Rahman
E. A.
,
Nayfeh
A. H.
, and
Younis
M. I.
, (
2004
), “
Finite-Amplitude Motions of Beam Resonators and Their Stability
”,
Journal of Computational and Theoretical Nanoscience
,
1
(
4)
,
385
391
.
2.
Abdel-Rahman
E. M.
,
Younis
M. I.
, and
Nayfeh
A. H.
, (
2002
), “
Characterization of the Mechanicsl Behavior of an Electrically Actuated Microbeam
”,
Journal of Micromrchanical and Microengineering
,
12
,
759
766
.
3.
Andrews M., Harris I, and Turner G., (1993), A comparison of squeeze-film theory with measurements on a microstructure Sensors Actuators A 36 79–87
4.
Andrews M.K. and Harris P.D., (1995), Damping and gas viscosity measurements using a microstructure Sensors Actuators A 49 103–8
5.
Bao
M.
,
Yang
H.
,
Sun
Y.
, and
French
P. J.
, (
2003
),
Modified Reynolds equation and analytical analysis of squeeze-film air damping of perforated structures
J. Micromech. Microeng.
13
795
800
6.
Blech
J. J.
, (
1983
),
On isothermal squeeze films
J. Lubric. Technol.
105
615
20
7.
Burgdorfer
A.
, (
1959
),
The influence of the molecular mean free path on the performance of hydrodynamic gas lubricated bearing
,
Journal of Basic Engineering
,
81
,
94
99
.
8.
Chen J. and Kang S.M., (2000), “An algorithm for automatic model reduction of nonlinear MEMS devices,” in Proc. IEEE Int. Symp. Circuits and Syst., May 28-31, pp. 445–448.
9.
Chen
J.
,
Kang
S.
,
Zou
J.
,
Liu
C.
, and
Schutt-Aine´
J. E.
, (
2004
),
Reduced-Order Modeling of Weakly Nonlinear MEMS Devices With Taylor-Series Expansion and Arnoldi Approach
,
J. of Microelectromechanical Systems
,
13
(
3)
,
441
451
.
10.
Chen
J.
,
Kang
S.
,
Zou
J.
,
Liu
C.
, and
Schutt-Aine´
J. E.
, (
2004
),
Reduced-Order Modeling of Weakly Nonlinear MEMS Devices With Taylor-Series Expansion and Arnoldi Approach
,
J. of Microelectromechanical Systems
,
13
(
3)
,
441
451
.
11.
Chen Y., and White J., (2000) “A quadratic method for nonlinear model order reduction,” in Proc. Int. Symp. on Modeling and Simulation of Microsystem Conf., Mar.
12.
Christopherson
J.
, and
Nakhaie
Jazar G.
, (
2005
), “
Optimization of Classical Hydraulic Engine Mounts Based on RMS Method
”,
Journal of the Shock and Vibration
,
12
(
12)
,
119
147
.
13.
Darling R.B., Hivick C., and Xu J., (1998), Compact analytical modeling of squeeze film damping with arbitrary venting conditions using a green’s function approach Sensors Actuators A 70 32–41
14.
E.
,
Mehri
B.
, and
Nakhaie
Jazar G.
, (
1997
), “
Existence of Periodic Solution for Equation of Motion of Simple Beams with Harmonically Variable Length
”,
Journal of Vibration and Acoustics
,
119
,
485
488
.
15.
Fedder G.K., (1994), “Simulation of Micromechanical Systems,” Ph.D. dissertation, University of California at Berkeley.
16.
Golnaraghi
M. F.
, and
Nakhaie
Jazar G.
, (
2001
), “
Development and Analysis of a Simplified Nonlinear Model of a Hydraulic Engine Mount
”,
Journal of Vibration and Control
,
7
(
4)
,
495
526
.
17.
Griffin W. S., Richardson H. H., and Yamanami S., (1966), A study of fluid squeeze-film damping ASME J. Basic Eng. 451–6
18.
Gupta
A.
,
Denton
J. P.
,
McNally
H.
,
Bashar
R.
(
2003
) “
NovelFabrication Method for Surface Micromachined Thin Single-Crystal Silicon Cantilever Beams
.”
Journal of Microelectromechanical Systems.
12
(
2)
,
185
185
.
19.
Gupta R. K., and Senturia S. D., (1997), “Pull-in time dynamics as a measure of absolute pressure,” in Proc. The Tenth Annual International Workshop on Micro Electro Mechanical Systems, New York, NY, pp. 290–294.
20.
Harmany
Z.
, (
2003
),
Effects of vacuum pressure on the response characteristics on MEMS cantilever structures
,
NSF EE REU PENN STATE Annual Research Journal
Vol.
1
,
54
64
.
21.
Houlihan
R.
, and
Kraft
M.
, (
2005
),
Modeling squeeze film effects in a MEMS accelerometer with a levitated proof mass
,
J. Micromech. Microeng
,
15
,
893
902
.
22.
Hung
E. S.
, and
Senturia
S. D.
, (
1999
),
Generating efficient dynamical models for microelectromechanical systems from a few finite element simulation runs
,
J. Microelectromech. Syst.
,
8
,
280
289
.
23.
Khaled
A. R. A.
,
Vafai
K.
,
Yang
M.
,
Zhang
X.
, and
Ozkan
C. S.
, (
2003
), “
Analysis, control and augmentation of microcantilever deflections in bio-sensing systems
”,
Sensors and Actuators B
,
7092
,
1
13
.
24.
Kwok
P. Y.
,
Weinberg
M. S.
,
Breuer
K. S.
, (
2005
),
Fluid Effects in Vibrating Micromachined Structures
,
Journal of Microelectromechanical Systems
,
14
(
4)
,
770
781
.
25.
Langlois
W E
, (
1962
),
Isothermal squeeze films
,
Q. Appl. Math.
20
131
50
26.
Lyshevski, S. E., (2001), “Nano- and Microelectromechanical Systems, Fundamentals of Nano- and Microengineering”, CRC Press, Boca Raton, Florida.
27.
Madou, M. J., (2002), Fundamentals of Microfabrication: The Science of Miniaturization, Second Edition. Foreword pp 1–2, 536–542. CRC Press: Boca Raton, FL.
28.
Mahmoudian N., Rastgaar Aagaah M., Nakhaie Jazar G., and Mahinfalah M., (2004), “Dynamics of a Micro Electro Mechanical System (MEMS)”, International Conference on MEMS, NANO, and Smart Systems, Banff, Alberta - Canada, August 25–27.
29.
Malatkar P., (2003), “Nonlinear vibrations of cantilever beams and plates”, Ph.D. Thesis in Mechanical Engineering, Virginia Polytechnic Institute and State University.
30.
Meirovitch, L., (1997), “Principles and technologies of vibrations”, Prentice Hall.
31.
Mukherjee
T.
,
Fedder
G. K.
, and
Blanton
R. D.
, (
1999
), “
Hierarchical design and test of integrated microsystems
,”
IEEE Design Test
, vol.
16
, pp.
18
27
, Oct.–Dec..
32.
Najar
F.
,
Choura
S.
,
El-Borgi
S.
,
Abdel-Rahman
E M
and
Nayfeh
A H
, (
2005
), “
Modeling and design of variable-geometry electrostatic microactuators
”,
J. Micromech. Microeng
,
15
(
3)
419
429
.
33.
Nakhaie
Jazar G.
, and
Golnaraghi
M. F.
, (
2002
), “
Nonlinear Modeling, Experimental Verification, and Theoretical Analysis of a Hydraulic Engine Mount
”,
Journal of Vibration and Control
,
8
(
1)
, pp.
87
116
.
34.
Nayfeh
A. H.
and
Younis
M. I.
, (
2004
), “
Modeling and Simulations of Thermoelastic Damping in Microplates
”,
Journal of Micromechanics and Microengineering
,
14
,
1711
1717
.
35.
Nayfeh A. H., and Mook D. T., (1979), Nonlinear Oscillations, John Wiley & Sons, New York.
36.
Nayfeh A. H., Younis M. I., Abdel-Rahman E. A., (2005), “Reduced-Order Modeling of MEMS”, Third MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 14-17.
37.
Nayfeh
A. H.
, and
Younis
M. I.
, (
2004
), “
A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping
”,
Journal of Micromechanics and Microengineering
,
14
,
170
181
.
38.
Pan
F.
,
Kubby
J.
,
Peeters
E.
,
Tan
A.
, and
Mukherjee
S.
, (
1998
), “
Squeeze film damping effect on the dynamic response of a MEMS torsion mirror
,”
J. Micromech. Microeng.
, vol.
8
, no.
3i
, pp.
200
208
.
39.
Rewienski M.J., (2003), “A Trajectory Piecewise-Linear Approach to Model Order Reduction of Nonlinear Dynamical Systems”, Ph.D., thesis, Department of Electrical Engineering, Massachusetts Institute of Technology.
40.
Shi
F.
,
Ramesh
P.
, and
Mukherjee
S.
, (
1996
), “
Dynamic analysis of micro-electro-mechanical systems
,”
Int. J. Numer. Meth. Eng.
, vol.
39
, no.
24
, pp.
4119
4139
.
41.
Shi
F.
,
Ramesh
P.
, and
Mukherjee
S.
, (
1996
), “
Dynamic analysis of micro-electro-mechanical systems
,”
Int. J. Numer. Meth. Eng.
, vol.
39
, no.
24
, pp.
4119
4139
.
42.
Starr J. B., (1990), Squeeze-film damping in solid-state accelerometers, in Proc. Tech. Dig. IEEE Solid-State Sensors and Actuators Workshop, Hilton Head Island, SC, pp. 44–47.
43.
Sudipto
K
, and
Aluru
N. R.
, (
2004
), “
Full-Lagrangian schemes for dynamic analysis of electrostatic MEMS
”,
Journal of Microelectromechanical Systems
,
13
(
5)
,
737
758
.
44.
Sun Y, Chan W K and Liu N, (2002), A slip model with molecular dynamics J. Micromech. Microeng. 12 316–22
45.
Veijola T and Mattila T, (2001), Compact squeezed-film damping model for perforated surface Proc. IEEE 11th Int. Conf. on Solid-State Sensors, Actuators and Microsystems pp 1506–9
46.
Veijola T, Kuisma H and Lahdenpera J, (1998), The influence of gas-surface interaction on gas-film damping in a silicon accelerometer Sensors Actuators A 66 83–92
47.
Vogl
G. W.
, and
Nayfeh
A. H.
, (
2005
), “
A Reduced-Order Model for Electrically Actuated Clamped Circular Plates
,”
Journal of Micromechanics and Microengineering
,
15
,
684
690
.
48.
White A., (2002), Review of Some Current Research in Microelectromechanical Systems (MEMS) with Defence Applications, DSTO Aeronautical and Maritime Research Laboratory, Fishermans Bend Vic, Australia, pp 10.
49.
Yang
J. L.
,
Ono
T.
, and
Esashi
M.
, (
2002
), “
Energy dissipation in submicrometer thick single-crystal 116 cantilevers
”,
Journal of Microelectromechanical Systems
,
11
(
6)
,
775
783
.
50.
Yang Y.J., and Senturia S. D., (1997), Effect of air damping on the dynamics of nonuniform deformations of microstructures, in Proc. IEEE International Conference on Solid-State Sensors and Actuators, New York, NY, 1093–1096.
51.
Yang Y. J., (1998), Squeeze-film damping for MEMS structures, MS Thesis, Electrical Engineering, Massachusetts Institute of Technology.
52.
Yang Y. J., Marc-Alexis Gretillat, Stephen D. Senturia. (1997), “Effect of Air Damping on the Dynamics of Nonuniform Deformations of Microstructures, International Conference on Solid-State Sensors and Actuators. Chicago, June 16-19, pp 1094–1096.
53.
Yang Y-J and Senturia S D, (1996), Numerical simulation of compressible squeezed-film damping Proc. Solid-State Sensor and Actuator Workshop pp 76–9
54.
Younis M. I., and Nayfeh A. H., (2005), “Modeling Squeeze-Film Damping of Electrostatically Actuated Microplates Undergoing Large Deflections,” ASME 20th Biennial Conference on Mechanical Vibration and Noise, 5th International Conference on Multibody Systems, Nonlinear Dynamics and Control, DETC2005-84421, Long Beach, CA, September 24-28.
55.
Younis M. I., (2001), “Investigation of the Mechanical Behavior of Microbeam-Based MEMS Devices”, MS. thesis in Mechanical Engineering, Virginia Polytechnic Institute and State University, December.
56.
Younis M. I., and Nayfeh A. H., (2005), “Dynamic Analysis of MEMS Resonators under Primary-Resonance Excitation,” ASME 20th Biennial Conference on Mechanical Vibration and Noise, DETC2005-84146, Long Beach, CA, September 24-28.
57.
Younis
M. I.
, and
Nayfeh
A. H.
, (
2003
), “
A study of the nonlinear response of a resonant microbeam to electric actuation
”,
Journal of Nonlinear Dynamics
,
31
,
91
117
.
58.
Younis, M., I., (2004), “Modeling and Simulation of Micrielectromecanical System in Multi-Physics Fields”, Ph.D., thesis, Mechanicsl Engineering, Virginia Polytechnic Institute and State University.
59.
Younis
M., I.
,
Abdel-Rahman
E., M.
,
Nayfeh
A.
, (
2003
), “
A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS
”,
Journal of Microelectromechanical Systems
,
12
(
5)
,
672
680
.
60.
Zhang
C.
,
Xu
G.
, and
Jiang
Q.
, (
2004
), “
Characterization of the squeeze film damping effect on the quality factor of a microbeam resonator
”,
Journal of Micromechanics and Microengineering
,
14
,
1302
1306
.
61.
Zhao
Z.
,
Dankowicz
H.
,
Reddy
C. K.
, and
Nayfeh
A. H.
, (
2004
), “
Modelling and Simulation Methodology for Impact Microactuators
,”
Journal of Micromechanics and Microengineering
,
14
,
775
784
.
This content is only available via PDF.