The effect of radial force on pull-in instability and frequency of rigid core circular and annular plates subjected to electrostatic field

Document Type : Article


1 -

2 Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Postal Code: 47148-71167, Shariati Street, Babol, Mazandaran, Iran


In this work static pull-in instability and frequency analysis of circular and annular plates in electrical field was studied. The plate is modeled based on classical plate theory with nonlinear Von Kármán strain-displacement field. The governing equation of motion and boundary conditions were obtained using Hamilton principle. For this purpose potential and kinetic energies and the work done by radial and electrostatic force are obtained. Governing partial differential equations were reduced to ordinary differential equations by Galerkin's method. Then, static pull-in instabilities of clamped circular plate and annular plate with clamped-clamped and clamped-simply boundary conditions were analyzed by arc-length continuation method. The effect of rigid core, radial load, geometric nonlinearity, inner radius and boundary conditions on pull-in instability and frequency of the plate has been studied.


Main Subjects

1. Nathanson, H.C., Newell, W.E., Wickstrom, R.A.,
and Davis, J.R. The resonant gate transistor", IEEE
Transactions on Electron Devices, 14, pp. 117-133
2. Taylor, G. The coalescence of closely spaced drops
when they are at di erent electric potentials", Proceedings
of the Royal Society of London, A: Mathematical,
Physical and Engineering Sciences, 306, pp. 423-434
3. Chao, P.C., Chiu, C., and Liu, T.-H. DC dynamic
pull-in predictions for a generalized clamped-clamped
micro-beam based on a continuous model and bifurcation
analysis", Journal of Micromechanics and
Microengineering, 18, p. 115008 (2008).
4. Krylov, S. Lyapunov exponents as a criterion for the
dynamic pull-in instability of electrostatically actuated
microstructures", International Journal of Non-Linear
Mechanics, 42, pp. 626-642 (2007).
5. Hung, E.S. and Senturia, S.D. Extending the travel
range of analog-tuned electrostatic actuators", Journal
of Microelectromechanical Systems, 8, pp. 497-505
6. Bochobza-Degani, O. and Nemirovsky, Y. Modeling
the pull-in parameters of electrostatic actuators with
a novel lumped two degrees of freedom pull-in model",
Sensors and Actuators A: Physical, 97, pp. 569-578
7. Bochobza-Degani, O. and Nemirovsky, Y. Experimental
veri cation of a design methodology for torsion
actuators based on a rapid pull-in solver", Microelectromechanical
Systems, Journal of, 13, pp. 121-130
8. Madinei, H., Rezazadeh, G., and Azizi, S. Stability
and bifurcation analysis of an asymmetrically electrostatically
actuated microbeam", Journal of Computational
and Nonlinear Dynamics, 10, p. 021002 (1-8)
9. Hsu, M. De
ection analysis of electrostatic microactuators
using the di erential quadrature method",
Tamkang Journal of Science and Engineering, 9, pp.
97-106 (2006).
10. Batra, R.C., Por ri, M., and Spinello, D. Electromechanical
model of electrically actuated narrow
microbeams", Journal of Microelectromechanical Systems,
15, pp. 1175-1189 (2006).
11. Huang, Y.-T., Chen, H.-L., and Hsu, W. An analytical
model for calculating the pull-in voltage of
micro cantilever beams subjected to tilted and curled
e ects", Microelectronic Engineering, 125, pp. 73-77
12. Beni, Y.T. and M. Heidari, Numerical study on
pull-in instability analysis of geometrically nonlinear
Euler-Bernoulli microbeam based on modi ed couple
stress theory", International Journal of Engineering
and Applied Sciences, 4, pp. 41-53 (2012).
13. Zhang, Y. and Zhao, Y.-P. Numerical and analytical
study on the pull-in instability of micro-structure
under electrostatic loading", Sensors and Actuators A:
Physical, 127, pp. 366-380 (2006).
14. Chowdhury, S., Ahmadi, M., and Miller, W. A
comparison of pull-in voltage calculation methods for
MEMS-based electrostatic actuator design", 1st International
Conference on Sensing Technology, November,
Palmerston North, New Zealand, pp. 21-23 (2005)
15. Mojahedi, M. and Rahaeifard, M. Static de
and pull-in instability of the electrostatically actuated
bilayer microcantilever beams", International Journal
of Applied Mechanics, 7, pp. 1-15 (2015).
16. Baghania, M., Asgarshamsib, A., and Goharkhaha,
M. Analytical solution for large amplitude vibrations
of microbeams actuated by an electro-static force",
Scientia Iranica, Transactions B, Mechanical Engineering,
20, pp. 1499-1507 (2013).
17. Rahaeifard, M. and Ahmadian, M. On pull-in instabilities
of microcantilevers", International Journal of
Engineering Science, 87, pp. 23-31 (2015).
18. Gholami, R., Ansari, R., and Rouhi, H. Studying
the e ects of small scale and Casimir force on the
non-linear pull-in instability and vibrations of FGM
2126 M. Khorshidi Paji et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2111{2129
microswitches under electrostatic actuation", International
Journal of Non-Linear Mechanics, 77, pp. 193-
207 (2015).
19. Xiao, Y., Wang, B., and Zhou, S. Pull-in voltage analysis
of electrostatically actuated MEMS with piezoelectric
layers: A size-dependent model", Mechanics
Research Communications, 66, pp. 7-14 (2015).
20. Srinivas, D. Electromechanical dynamics of simplysupported
micro-plates", International Journal of
Computational Engineering Research, 2, pp. 1388-1395
21. Mukherjee, S., Bao, Z., Roman, M., and Aubry, N.
Nonlinear mechanics of MEMS plates with a total
Lagrangian approach", Computers & structures, 83,
pp. 758-768 (2005).
22. Zhao, X., Abdel-Rahman, E.M., and Nayfeh, A.H.
A reduced-order model for electrically actuated microplates",
Journal of Micromechanics and Microengineering,
14, pp. 900-906 (2004).
23. Zand, M.M., Rashidian, B., and Ahmadian, M. Contact
time study of electrostatically actuated microsystems",
Scientia Iranica, Transactions B, Mechanical
Engineering, 17, pp. 346-357 (2010).
24. Wang, K., Kitamura, T., and Wang, B. Nonlinear
pull-in instability and free vibration of micro/
nanoscale plates with surface energy-A modi ed
couple stress theory model", International Journal of
Mechanical Sciences, 99, pp. 288-296 (2015).
25. Saif, M., Alaca, B.E., and Sehitoglu, H. Analytical
modeling of electrostatic membrane actuator for micro
pumps", Journal of Microelectromechanical Systems,
8, pp. 335-345 (1999).
26. Wang, Y.G., Lin, W.H., Li, X.M., and Feng, Z.J.
Bending and vibration of an electrostatically actuated
circular microplate in presence of Casimir force",
Applied Mathematical Modelling, 35, pp. 2348-2357
27. Soleymani, P., Sadeghian, H., Tahmasebi, A., and
Rezazadeh, Gh. Pull-in instability investigation of circular
micro pump subjected to nonlinear electrostatic
force", Sensors & Transducers, 69, pp. 622-628 (2006).
28. Nayfeh, A.H., Younis, M.I., and Abdel-Rahman, E.M.
Reduced-order models for MEMS applications", Nonlinear
Dynamics, 41, pp. 211-236 (2005).
29. Reddy, J.N., Theory and Analysis of Elastic Plates and
Shells, CRC Press (2006).
30. Magrab, E.B., Vibrations of Elastic Systems: With
Applications to MEMS and NEMS, Springer Science
& Business Media, 184 (2012).
31. Timoshenko, S.P. and Gere, J.M., Theory of Elastic
Stability, Courier Corporation (2009).
32. Pelesko, J.A. and Bernstein, D.H., Modeling Mems and
Nems, CRC press (2002).
33. Rao, S.S., Vibration of Continuous Systems, John
Wiley & Sons (2007).
34. Seydel, R., Practical Bifurcation and Stability Analysis,
Springer Science & Business Media, 5 (2009).
35. Gal^aan-Vioque, J., Krauskopf, B., and Osinga, H.M.,
Numerical Continuation Methods for Dynamical Systems:
Path Following and Boundary Value Problems,
Springer (2007).
36. Nayfeh, A.H. and Balachandran, B., Applied Nonlinear
Dynamics: Analytical, Computational and Experimental
Methods, John Wiley & Sons (2008).
37. Parseh, M., Dardel, M., and Ghasemi, M.H. Investigating
the robustness of nonlinear energy sink in steady
state dynamics of linear beams with di erent boundary
conditions", Communications in Nonlinear Science
and Numerical Simulation, 29, pp. 50-71 (2015).
38. Vogl, G.W. Nonlinear dynamics of circular plates
under electrical loadings for capacitive micromachined
ultrasonic transducers (CMUT)", PhD Dissertation
Blacksburg, VirginiaTech (2006).