Document Type : Article

**Authors**

Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, P.O. Box 15719-14911, Iran.

**Abstract**

This paper investigates the optimal configuration for a partially two-layered circular capacitive microplate subjected to AC-DC electrostatic actuation. To this end, the static deflection due to DC electrostatic actuation, natural frequency of vibration about static position and primary resonance response due to AC electrostatic actuation are studied. Primarily, the nonlinear equations of motion are derived through classical laminated plate theory (CLPT). Then, the static position and natural frequency of vibration around static position are obtained using Galerkin approach. The linear mode shapes of non-uniform microplate i.e. a microplate coated as partial by a second layer are used as comparison functions. The forced vibration equations around static position are separated using Galerkin method, and solved by the multiple scale perturbation theory. Firstly, the impact of changes in the second layer radius on the variations of static and dynamic response of the system is studied while its thickness remains constant. Then, the effect of changes in the second layer thickness is studied while its radius remains constant. Finally, the impact of simultaneous change in the radius and thickness of the second layer is studied while its volume remains constant.

**Keywords**

**Main Subjects**

References:

1. Younis, M.I. MEMS, Linear and Nonlinear Statics and Dynamics, Springer, New York (2011).

2. Poloei, E., Zamanian, M., and Hosseini, S.A.A. "Nonlinear vibration analysis of an electrostatically excited micro cantilever beam coated by viscoelastic layer with the aim of finding the modified configuration", Structural Engineering and Mechanics, 61(2), pp. 193- 207 (2017).

3. Vogl, G.W. and Nayfeh, A.H. "A reduced-order model for electrically actuated clamped circular plates", Journal of Micromechanics and Microengineering, 15(4), p. 684 (2005).

4. Vogl, G.W. and Nayfeh, A.H. "Primary resonance excitation of electrically actuated clamped circular plates", Nonlinear Dynamics, 47(1), pp. 181-192 (2007).

5. Liao, L.D., Chao, P.C., Huang, C.W., et al. "DC dynamic and static pull-in predictions and analysis for electrostatically actuated clamped circular microplates based on a continuous model", Journal of Micromechanics and Microengineering, 20(2), p. 025013 (2009).

6. Bertarelli, E., Colnago, A., Ardito, R., et al. "Modelling and characterization of circular microplate electrostatic actuators for micropump applications", In Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems, 16th International Conference on, pp. 1-7, IEEE (2015).

7. Wang, K.F., Wang, B., and Zhang, C. "Surface energy and thermal stress effect on nonlinear vibration of electrostatically actuated circular micro-/nanoplates based on modified couple stress theory", Acta Mechanica, 228(1), pp. 129-140 (2017).

8. Shooshtari, A. and Dalir, M.A. "Nonlinear free vibration analysis of clamped circular fiber metal laminated plates", Scientia Iranica, Transactions B, Mechanical Engineering, 22(3), p. 813 (2015).

9. Khorshidi Paji, M., Dardel, M., Pashaei, M.H., et al. "The effect of radial force on pull-in instability and frequency of rigid core circular and annular plates subjected to electrostatic field", Scientia Iranica, 25(4), pp. 2111-2129 (2018).

10. Karimzadeh, A., Ahmadian, M.T., and Rahaeifard, M. "Effect of size dependency on in-plane vibration of circular micro-rings", Scientia Iranica, 24(4), pp. 1996-2008 (2017).

11. Dowlati, S., Rezazadeh, G., Afrang, S., et al. "An accurate study on capacitive microphone with circular diaphragm using a higher order elasticity theory", Latin American Journal of Solids and Structures, 13(4), pp. 590-609 (2016).

12. Shahriari, B., Zargar, O., Baghani, M, et al. "Free vibration analysis of rotating functionally graded annular disc of variable thickness using generalized differential quadrature method", Scientia Iranica, 25(2), pp. 728-740 (2018).

13. Jallouli, A., Kacem, N., Bourbon, G., et al. "Pullin instability tuning in imperfect nonlinear circular microplates under electrostatic actuation", Physics Letters A, 380(46), pp. 3886-3890 (2016).

14. Saghir, S., Bellaredj, M.L., Ramini, A., et al. "Initially curved microplates under electrostatic actuation: theory and experiment", Journal of Micromechanics and Microengineering, 26(9), p. 095004 (2016).

15. Batra, R.C., Porfiri, M., and Spinello, D. "Vibrations and pull-in instabilities of microelectromechanical von Karman elliptic plates incorporating the Casimir force", Journal of Sound and Vibration, 315(4), pp. 939-960 (2008).

16. Batra, R.C., Porfiri, M., and Spinello, D. "Reducedorder models for microelectromechanical rectangular and circular plates incorporating the Casimir force", International Journal of Solids and Structures, 45(11),pp. 3558-3583 (2008).

17. Wang, Y.G., Lin, W.H., Li, X.M, et al. "Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force", Applied Mathematical Modelling, 35(5), pp. 2348-2357 (2011).

18. Yang, W.D., Kang, W.B., andWang, X. "Thermal and surface effects on the pull-in characteristics of circular nanoplate NEMS actuator based on nonlocal elasticity theory", Applied Mathematical Modelling, 43, pp. 321-336 (2017).

19. Zhang, Y. "Large de ection of clamped circular plate and accuracy of its approximate analytical solutions", Science China Physics, Mechanics & Astronomy, 59(2), p. 624602 (2016).

20. Medina, L., Gilat, R., and Krylov, S. "Bistable behavior of electrostatically actuated initially curved micro plate", Sensors and Actuators A: Physical, 248, pp. 193-198 (2016).

21. Medina, L., Gilat, R., and Krylov, S. "Modeling strategies of electrostatically actuated initially curved bistable micro plates", International Journal of Solids and Structures, 118, pp. 1-13 (2017).

22. Caruntu, D.I. and Oyervides, R. Frequency response reduced order model of primary resonance of electrostatically actuated MEMS circular plate resonators, Communications in Nonlinear Science and Numerical Simulation, 43, pp. 261-270 (2017).

23. Caruntu, D.I. and Oyervides, R. "Voltage response of primary resonance of electrostatically actuated MEMS clamped circular plate resonators", Journal of Computational and nonlinear dynamics, 11(4), p. 041021(2016).

24. Vahdat, A.S., Rezazadeh, G., and Afrang, S. "Improving response of a MEMS capacitive microphone filtering shock noise", Microelectronics Journal, 42(5), pp. 614-621 (2011) .

25. Sharafkhani, N., Rezazadeh, G., and Shabani, R. "Study of mechanical behavior of circular FGM microplates under nonlinear electrostatic and mechanical shock loadings", Acta Mechanica, 223(3), pp. 579-591 (2012).

26. Shabani, R., Sharafkhani, N., Tariverdilo, S., et al. "Dynamic analysis of an electrostatically actuated circular micro-plate interacting with compressible fluid", Acta Mechanica, 224(9), pp. 20-25 (2013).

27. Li, Z., Zhao, L., Ye, Z., Wang, H., et al. "Resonant frequency analysis on an electrostatically actuated microplate under uniform hydrostatic pressure", Journal of Physics D: Applied Physics, 46(19), p. 195108 (2013).

28. Nabian, A., Rezazadeh, G., Haddad-derafshi, M., et al. "Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure", Microsystem Technologies, 14(2), pp. 235-240 (2008).

29. Tian-Jie, C. "Model to analyze micro circular plate subjected to electrostatic force", Sensors & Transducers, 153(6), p. 129 (2013).

30. Mohammadi, V., Ansari, R., Shojaei, M, et al."Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates", Nonlinear Dynamics, 73(3), pp. 1515-1526 (2013).

31. Ansari, R., Gholami, R., Shojaei, M.F, et al. "Surface stress effect on the pull-in instability of circular nanoplates", Acta Astronautica, 102, pp. 140-150 (2014).

32. Rashvand, K., Rezazadeh, G., Mobki, H., et al. "On the size-dependent behavior of a capacitive circular micro-plate considering the variable length-scale parameter", International Journal of Mechanical Sciences,77, pp. 333-342 (2013).

33. Salehipour, H., Nahvi, H., and Shahidi, A.R. "Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity", Physica E: Low-dimensional Systems and Nanostructures, 66, pp. 350-358 (2015).

34. Liu, C.C. "The stability and non-linear vibration analysis of a circular clamped microplate under electrostatic actuation", Smart Science, 5(3), pp. 132-139 (2017).

35. Deshpande, M. and Saggere, L. "An analytical model and working equations for static de ections of a circular multi-layered diaphragm-type piezoelectric actuator", Sensors and Actuators A: Physical, 136(2), pp. 673-689 (2007).

2. Poloei, E., Zamanian, M., and Hosseini, S.A.A. "Nonlinear vibration analysis of an electrostatically excited micro cantilever beam coated by viscoelastic layer with the aim of finding the modified configuration", Structural Engineering and Mechanics, 61(2), pp. 193- 207 (2017).

3. Vogl, G.W. and Nayfeh, A.H. "A reduced-order model for electrically actuated clamped circular plates", Journal of Micromechanics and Microengineering, 15(4), p. 684 (2005).

4. Vogl, G.W. and Nayfeh, A.H. "Primary resonance excitation of electrically actuated clamped circular plates", Nonlinear Dynamics, 47(1), pp. 181-192 (2007).

5. Liao, L.D., Chao, P.C., Huang, C.W., et al. "DC dynamic and static pull-in predictions and analysis for electrostatically actuated clamped circular microplates based on a continuous model", Journal of Micromechanics and Microengineering, 20(2), p. 025013 (2009).

6. Bertarelli, E., Colnago, A., Ardito, R., et al. "Modelling and characterization of circular microplate electrostatic actuators for micropump applications", In Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems, 16th International Conference on, pp. 1-7, IEEE (2015).

7. Wang, K.F., Wang, B., and Zhang, C. "Surface energy and thermal stress effect on nonlinear vibration of electrostatically actuated circular micro-/nanoplates based on modified couple stress theory", Acta Mechanica, 228(1), pp. 129-140 (2017).

8. Shooshtari, A. and Dalir, M.A. "Nonlinear free vibration analysis of clamped circular fiber metal laminated plates", Scientia Iranica, Transactions B, Mechanical Engineering, 22(3), p. 813 (2015).

9. Khorshidi Paji, M., Dardel, M., Pashaei, M.H., et al. "The effect of radial force on pull-in instability and frequency of rigid core circular and annular plates subjected to electrostatic field", Scientia Iranica, 25(4), pp. 2111-2129 (2018).

10. Karimzadeh, A., Ahmadian, M.T., and Rahaeifard, M. "Effect of size dependency on in-plane vibration of circular micro-rings", Scientia Iranica, 24(4), pp. 1996-2008 (2017).

11. Dowlati, S., Rezazadeh, G., Afrang, S., et al. "An accurate study on capacitive microphone with circular diaphragm using a higher order elasticity theory", Latin American Journal of Solids and Structures, 13(4), pp. 590-609 (2016).

12. Shahriari, B., Zargar, O., Baghani, M, et al. "Free vibration analysis of rotating functionally graded annular disc of variable thickness using generalized differential quadrature method", Scientia Iranica, 25(2), pp. 728-740 (2018).

13. Jallouli, A., Kacem, N., Bourbon, G., et al. "Pullin instability tuning in imperfect nonlinear circular microplates under electrostatic actuation", Physics Letters A, 380(46), pp. 3886-3890 (2016).

14. Saghir, S., Bellaredj, M.L., Ramini, A., et al. "Initially curved microplates under electrostatic actuation: theory and experiment", Journal of Micromechanics and Microengineering, 26(9), p. 095004 (2016).

15. Batra, R.C., Porfiri, M., and Spinello, D. "Vibrations and pull-in instabilities of microelectromechanical von Karman elliptic plates incorporating the Casimir force", Journal of Sound and Vibration, 315(4), pp. 939-960 (2008).

16. Batra, R.C., Porfiri, M., and Spinello, D. "Reducedorder models for microelectromechanical rectangular and circular plates incorporating the Casimir force", International Journal of Solids and Structures, 45(11),pp. 3558-3583 (2008).

17. Wang, Y.G., Lin, W.H., Li, X.M, et al. "Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force", Applied Mathematical Modelling, 35(5), pp. 2348-2357 (2011).

18. Yang, W.D., Kang, W.B., andWang, X. "Thermal and surface effects on the pull-in characteristics of circular nanoplate NEMS actuator based on nonlocal elasticity theory", Applied Mathematical Modelling, 43, pp. 321-336 (2017).

19. Zhang, Y. "Large de ection of clamped circular plate and accuracy of its approximate analytical solutions", Science China Physics, Mechanics & Astronomy, 59(2), p. 624602 (2016).

20. Medina, L., Gilat, R., and Krylov, S. "Bistable behavior of electrostatically actuated initially curved micro plate", Sensors and Actuators A: Physical, 248, pp. 193-198 (2016).

21. Medina, L., Gilat, R., and Krylov, S. "Modeling strategies of electrostatically actuated initially curved bistable micro plates", International Journal of Solids and Structures, 118, pp. 1-13 (2017).

22. Caruntu, D.I. and Oyervides, R. Frequency response reduced order model of primary resonance of electrostatically actuated MEMS circular plate resonators, Communications in Nonlinear Science and Numerical Simulation, 43, pp. 261-270 (2017).

23. Caruntu, D.I. and Oyervides, R. "Voltage response of primary resonance of electrostatically actuated MEMS clamped circular plate resonators", Journal of Computational and nonlinear dynamics, 11(4), p. 041021(2016).

24. Vahdat, A.S., Rezazadeh, G., and Afrang, S. "Improving response of a MEMS capacitive microphone filtering shock noise", Microelectronics Journal, 42(5), pp. 614-621 (2011) .

25. Sharafkhani, N., Rezazadeh, G., and Shabani, R. "Study of mechanical behavior of circular FGM microplates under nonlinear electrostatic and mechanical shock loadings", Acta Mechanica, 223(3), pp. 579-591 (2012).

26. Shabani, R., Sharafkhani, N., Tariverdilo, S., et al. "Dynamic analysis of an electrostatically actuated circular micro-plate interacting with compressible fluid", Acta Mechanica, 224(9), pp. 20-25 (2013).

27. Li, Z., Zhao, L., Ye, Z., Wang, H., et al. "Resonant frequency analysis on an electrostatically actuated microplate under uniform hydrostatic pressure", Journal of Physics D: Applied Physics, 46(19), p. 195108 (2013).

28. Nabian, A., Rezazadeh, G., Haddad-derafshi, M., et al. "Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure", Microsystem Technologies, 14(2), pp. 235-240 (2008).

29. Tian-Jie, C. "Model to analyze micro circular plate subjected to electrostatic force", Sensors & Transducers, 153(6), p. 129 (2013).

30. Mohammadi, V., Ansari, R., Shojaei, M, et al."Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates", Nonlinear Dynamics, 73(3), pp. 1515-1526 (2013).

31. Ansari, R., Gholami, R., Shojaei, M.F, et al. "Surface stress effect on the pull-in instability of circular nanoplates", Acta Astronautica, 102, pp. 140-150 (2014).

32. Rashvand, K., Rezazadeh, G., Mobki, H., et al. "On the size-dependent behavior of a capacitive circular micro-plate considering the variable length-scale parameter", International Journal of Mechanical Sciences,77, pp. 333-342 (2013).

33. Salehipour, H., Nahvi, H., and Shahidi, A.R. "Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity", Physica E: Low-dimensional Systems and Nanostructures, 66, pp. 350-358 (2015).

34. Liu, C.C. "The stability and non-linear vibration analysis of a circular clamped microplate under electrostatic actuation", Smart Science, 5(3), pp. 132-139 (2017).

35. Deshpande, M. and Saggere, L. "An analytical model and working equations for static de ections of a circular multi-layered diaphragm-type piezoelectric actuator", Sensors and Actuators A: Physical, 136(2), pp. 673-689 (2007).

Transactions on Mechanical Engineering (B)

March and April 2020Pages 715-729