References
1. Batra, R.C., Por ri, M. and Spinello, D. \Vibrations of
narrow microbeams predeformed by an electric eld",
Journal of Sound and Vibration, 309(3-5), pp. 600-612
(2008).
2. Taylor, G.I. \The coalescence of closely spaced drops
when they are at di erent electric potentials", Proceedings
of the Royal Society A, 306(1478), pp. 423-434
(1968).
3. Nathanson, H.C., Newell, W.E., Wickstrom, R.A.
and Davis, J.R. \The resonant gate transistor", IEEE
Transactions on Electron Devices, 14(3), pp. 117-133
(1967).
4. Osterberg, P.M. \Electrostatically actuated microelectromechanical
test structures for material property
measurement", PhD Dissertation, Massachusetts Institute
of Technology, (1995).
5. Rochus, V., Rixen, D.J. and Golinval, J.C. \Electrostatic
coupling of MEMS structures: transient
simulations and dynamic pull-in", Nonlinear Analysis:
Theory, Methods & Applications, 63(5-7), pp. 1619-
1633 (2005).
6. Krylov, S. \Lyapunov exponents as a criterion for the
dynamic pull-in instability of electrostatically actuated
microstructures", International Journal of Nonlinear
Mechanics, 42(4), pp. 626-642 (2007).
7. Abdel-Rahman, E.M., Younis, M.I. and Nayfeh, A.H.
\Characterization of the mechanical behavior of an
electrically actuated microbeam", Journal of Micromechanics
and Microengineering, 12(6), pp. 759-766
(2002).
8. De, S.K. and Aluru, N.R. \Complex nonlinear oscillations
in electrostatically actuated microstructures",
Journal of Microelectromechanical Systems, 15(2), pp.
355-369 (2006).
9. Moghimi Zand, M. and Ahmadian, M.T. \Characterization
of coupled-domain multi-layer microplates
in pull-in phenomenon, vibrations and dynamics",
International Journal of Mechanical Sciences, 49(11),
pp. 1226-1237 (2007).
10. Moghimi Zand, M. and Ahmadian, M.T. \Vibrational
analysis of electrostatically actuated microstructures
considering nonlinear e ects", Communications in
Nonlinear Science and Numerical Simulation, 14(4)
pp. 1664-1678 (2009).
11. Koiter, W.T. \Couple-stresses in the theory of elasticity
I and II", Proceedings Koninklijke Nederlandse
Akademie van Wetenschappen, Series B; 67 pp. 17-44
(1964).
12. Mindlin, R.D. and Tiersten, H.F. \E ects of couplestresses
in linear elasticity", Archive for Rational
Mechanics and Analysis, 11(1), pp. 415-448 (1962).
13. Asghari, M., Kahrobaiyan, M.H., Rahaeifard, M. and
Ahmadian, M.T. \Investigation of the size e ects in
Timoshenko beams based on the couple stress theory",
Archive of Applied Mechanics, 81(7), pp. 863-874
(2011).
14. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P.
\Couple stress based strain gradient theory for elasticity",
International Journal of Solids and Structures,
39(10), pp. 2731-2743 (2002).
15. Park, S.K. and Gao, X.L. \Bernoulli-Euler beam
model based on a modi ed couple stress theory", J.
Micromech. Microeng, 16(11), pp. 2355-2359 (2006).
16. Kong, S., Zhou, S. Nie, Z. and Wang, K. \The sizedependent
natural frequency of Bernoulli-Euler microbeams",
Int. J. Eng. Sci., 46(5), pp. 427-437 (2008).
17. Ma, H.M., Gao, X.L. and Reddy, J.N. \A microstructure
dependent Timoshenko beam model based on a
modi ed couple stress theory", J. Mech. Phys. Solids,
56(12), pp. 3379-3391 (2008).
18. Xia, W., Wang, L. and Yin, L. \Nonlinear nonclassical
microscale beams: static bending, postbuckling and
free vibration", Int. J. Eng. Sci., 48(12), pp. 2044-
2053 (2010).
19. Asghari, M., Kahrobaiyan, M.H. and Ahmadian, M.T.
\A nonlinear Timoshenko beam formulation based on
the modi ed couple stress theory", Int. J. Eng. Sci.,
48(12), pp. 1749-1761 (2010).
20. Mindlin, R.D. \Micro-structure in linear elasticity",
Archive for Rational Mechanics and Analysis, 16(1),
pp. 51-78 (1964).
21. Mindlin, R.D. \Second gradient of strain and surfacetension
in linear elasticity", International Journal of
Solids and Structures, 1(4), pp. 417-438 (1965).
22. Mindlin, R.D. and Eshel, N.N. \On rst straingradient
theories in linear elasticity", International
Journal of Solids and Structures 4(1), pp. 109-124
(1968).
23. Wang, B., Zhao, J. and Zhou, S. \A micro scale
Timoshenko beam model based on strain gradient
elasticity theory", European Journal of Mechanics -
A/Solids, 29(4), pp. 591-599 (2010).
24. Sedighi, H.M. \Size-dependent dynamic pull-in instability
of vibrating electrically actuated microbeams
based on the strain gradient elasticity theory", Acta
Astronautica, 95, pp. 111-123 (2014).
25. Kuang, J.H. and Chen, C.J. \Dynamic characteristics
of shaped micro-actuators solved using the di erential
quadrature method", Journal of Micromechanics and
Microengineering, 14(4), pp. 647-655 (2004).
26. Konig, E.R. and Wachutka, G. \Multi-parameter homotopy
for the numerical analysis of MEMS", Sensors
and Actuators A: Physical, 110(1), pp. 39-51 (2004).
27. Nayfeh, A.H. and Younis, M.I. \A new approach to
the modeling and simulation of
exible microstructures
under the e ect of squeeze lm damping", Journal of
Micromechanics and Microengineering, 14(2), pp. 170-
181 (2004).
28. Younis, M.I. \Modeling and simulation of microelectromechanical
systems in multi-physics elds", PhD
Dissertation, Virginia Polytechnic Institute and state
University (2004).
184 R. Derakhshan et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 175{185
29. Liao, S.J. \An approximate solution technique which
does not depend upon small parameters: a special example",
International Journal of Nonlinear Mechanics,
30(3), pp. 371-380 (1995).
30. Moghimi Zand, M., Ahmadian, M.T. and Rashidian,
B. \Semi-analytic solutions to nonlinear vibrations of
microbeams under suddenly applied voltages", Journal
of Sound and Vibration, 325(1-2), pp. 382-396 (2009).
31. Bao, M. and Yang, H. \Squeeze lm air damping in
MEMS", Sensors and Actuators A: Physical, 136(1),
pp. 3-27 (2007).
32. Rajabi, F. and Ramezani, S. \A nonlinear microbeam
model based on strain gradient elasticity theory", Acta
Mechanica Solida Sinica, 26(1), pp. 21-34 (2013).
33. He, X.J., Wu, Q., Wang, Y., Song, M.X. and Yin,
J.H. \Numerical simulation and analysis of electrically
actuated microbeam-based MEMS capacitive switch",
Microsystem Technologies, 15, pp. 301-307 (2009).
34. Batra, R.C., Por ri, M. and Spinello, D. \Capacitance
estimate for electrostatically actuated narrow
microbeams", IET Micro & Nano Letters, 1(2), pp.
71-73 (2006).
35. Batra, R.C., Por ri, M. and Spinello, D. \Electromechanical
model of electrically actuated narrow
microbeams", Journal of Microelectromechanical Systems,
15(5), pp. 1175-1189 (2006).
36. Liao, S.J., Beyond Perturbation: Introduction to Homotopy
Analysis Method, In Chapman & Hall/CRC,
336 Pages, Boca Raton, London, New York, Washington
DC (2004).
37. Liao, S.J. and Cheung, A.T. \Application of homotopy
analysis method in nonlinear oscillations", ASME
Journal of Applied Mechanics, 65(4), pp. 914-922
(1998).
38. Liao, S.J. \Analytic approximate technique for free
oscillations of positively damped systems with algebraically
decaying amplitude", International Journal
of Nonlinear Mechanics, 38(8), pp. 1173-1183 (2003).
39. Li, S. and Liao, S.J. \Analytic approach to solve
multiple solutions of a strongly nonlinear problem",
Applied Mathematics and Computation, 169(2), pp.
854-865 (2005).
40. Hayat, T. and Sajid, M. \On analytic solution for
thin lm
ow of a fourth grade
uid down a vertical
cylinder", Physics Letter A 361, pp. 316-322 (2007).
41. Abbasbandy, S. \The application of the homotopy
analysis method to nonlinear equations arising in heat
transfer", Physics Letter A 360(1), pp. 109-113 (2006).
42. Derakhshan, R. \E ect of curved micro-beam on natural
frequency and pull-in voltage considering strain
gradient theory", M.Sc. Thesis, Sharif University of
Technology (2013).
43. Roozbahani, M., Heydarzadeh, N. and Moghimi Zand,
M. \Analytical solutions to nonlinear oscillations of a
microbeam using higher order beam theory", Scientia
Iranica B, 23(5) pp. 2179-2193 (2016).
44. Younis, M. and Nayfeh, A. \A study of the nonlinear
response of a resonant microbeam to an electric actuation",
Nonlinear Dynamics, 31(1), pp. 91-117 (2003).
45. Mojahedi, M., Moghimi Zand, M. and Ahmadian, M.T
\Static pull-in analysis of electrostatically actuated
microbeams using homotopy perturbation method",
Applied Mathematical Modelling34(4), pp. 1032-1041
(2010).
46. Osterberg, P.M. \Electrostatically actuated micromechanical
test structure for material property measurement",
PhD Dissertation, Massachusetts Institute of
Technology; (1995).
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