Analytical solution for nonlinear dynamic response of the viscoelastic microbeam under electrical actuation based upon micropolar theory of elasticity

Authors

1 M.Sc., Mech. Eng., Advanced Light Weight Composites Research Center, University of Sistan and Baluchestan, Zaheden, Iran.

2 Assist. Prof., Mech. Eng., Advanced Light Weight Composites Research Center, University of Sistan and Baluchestan, Zaheden, Iran.

Abstract

In this paper, the dynamic response of electro actuated viscoelastic microbeam is investigated and micropolar theory of elasticity has been used to consider the effects of size in microstructure. Euler-Bernoulli beam theory and Hamilton’s principle with considering viscoelastic integral constitutive equations, the midplane stretching effect, the axial residual stress and electrostatic force has been used to obtain the equation of motion and the boundary condition of fixed-fixed viscoelastic microbeam. Therefore, the nonlinear integro-differential equation in Volterra integral equation form is obtained. Galerkin method will be used, in order to solve the nonlinear partial integro-differential governing equation and then it converted to the ordinary integro-differential equation. By using the fourth order Runge - Kutta method, we can obtain the response (transverse displacement) of the electro actuated viscoelastic microbeam. In the following, the effect of initial gap value and material length scale parameter on the viscoelastic microbeam behavior are investigated. In the end, the viscoelastic microbeam is simulated in the FE software and the problem is analyzed in quasi-static form. In order to validate, the simulation result is compared with the result obtained from the quasi-static solution of the viscoelastic microbeam.

Keywords

Main Subjects

References

[1] Cleland AN, Roukes ML (1996) Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals. Appl Phys 69(18): 2653-2655.
[2] Wineman AS, Rajagopal KR (2000) Mechanical response of polymers: An introduction. Cambridge University Press, Cambridge.
[3] Altenbach H, Eremeyev V (2011) Computational modelling and advanced simulations. Springer, Berlin.
[4] Bethe K, Baumgarten D, Frank J (1990) Creep of sensors elastic elements: metals versus non-metals. Sens Actuators A 23(3): 844-849.
[5] Teh K, Lin L (1999) Time-dependent buckling phenomena of polysilicon micro beams. Microelectron J 30(11): 1169–1172.
[6] Tuck K, Jungen A, Geisberger A, Ellis M, Skidmore G (2005) A study of creep in polysilicon MEMS devices. J Eng Mater Technol 127(1): 90-96.
[7] Lee H, Zhang P, Bravman J (2005) Stress relaxation in free-standing aluminum beams. Thin Solid Films 476(1): 118-124.
[8] McLean M, Brown W, Vinci R (2010) Temperature-dependent viscoelasticity in thin Au films and consequences for MEMS devices. J Microelectromech Syst 19(6): 1299-1308.
[9] Pamidighantam S, Puers R, Baert K, Tilmans H (2002) Pull-in voltage analysis of electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions. J Micromech Microeng 12(4): 458-466.
[10] Yang F, Chong A, Lam D, Tong P (2002) Couple stress based strain gradient theory of elasticity. Int J Solids Struct 39(10): 2731-2743.
[11] Rahaeifard M, Kahrobaiyan M, Asghari M, Ahmadian M (2011) Static pull-in analysis of microcantilevers based on the modified couple stress theory. Sens Actuators A 171(2): 370-374.
[12] Yin L, Qian Q, Wang L (2011) Size effect on the static behavior of electrostatically actuated microbeams. Acta Mech Sin 27(3): 445-451.
[13] Kong S (2013) Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory. Appl Math Model 37(12): 7481-7488.
[14] Baghani M (2012) Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. Int J Eng Sci 45: 99-105.
[15] Zamanzadeh M, Rezazadeh G, Jafarsadeghi-Poornaki I, Shabani R (2013) Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes. Appl Math Model 37(10): 6964-6978.
[16] Shaat M, Mohamed SA (2014) Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories. Int J MechSci 18: 208-217.
[17] Khanchegardan A, Amiri A, Rezazadeh G (2015) Thermo-diffusive coupling effect on the damping ratio based on modified couple stress theory in micro-beam resonators. Modares Mechanical Engineering 15(9): 116-124. (in Persian)
[18] Fu Y, Zhang J, Bi R (2009) Analysis of the nonlinear dynamic stability for an electrically actuated viscoelastic microbeam. Microsyst Technol 15(9): 763-769.
[19] Fu Y, Zhang J (2009) Nonlinear static and dynamic responses of an electrically actuated viscoelastic microbeam. Acta Mech Sin 25(2): 211-218.
[20] Zhang J, Fu Y (2012) Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory. Meccanica 47(7): 1649-1658.
[21] Attia M (2017) Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges. Meccanica 52(10): 2391-2420.
 [22] Christensen R (1971) Theory of viscoelasticity. Academic Press, New York.
[23] Sadd M (2009) Elasticity: Theory, applications, and numerics. 2th edn. Academic Press, Massachusetts.
[24] Rajagopal K, Wineman A (2008) A quasi-correspondence principle for Quasi-Linear viscoelastic solids. Mech Time-Dep Mater 12(1): 1-14.
[25] Osterberg P, Senturia S (1997) M-TEST: A test chip for MEMS material property measurement using electrostatically actuated test structures. J Microelectromech Syst 6(2): 107-118.
[26] Emam SA, Nayfeh AH (2009) Postbuckling and free vibrations of composite beams. Compos Struct 88(4): 636-642.
[27] Park SK, Gao XL (2006) Bernoulli-Euler beam model based on a modified couple stress theory. J Micromech Microeng 16(11): 2355-2359.
[28] Rao S (2007) Vibration of continuous systems. John Wiley & Sons, New Jersey.
[29] Batra RC, Porfiri M, Spinello D (2006) Electromechanical model of electrically actuated narrow microbeams. J Microelectromech Syst 15(5): 1175-1189.
[30] Salehi Kolahi MR, Moeinkhah H (2018) Non-linear vibration of curved microbeam under electrostatic actuation by using reduced order model and finite element simulation. Modares Mechanical Engineering 17(12): 514-522. (in Persian)
[31] Tajaddodianfar F, Yazdi MH, Pishkenari HN (2014) Dynamics of bistable initially curved shallow microbeams: Effects of the electrostatic fringing fields. Proc AIM IEEE 1279-1283.
[32] Ajri M, Seyyed Fakhrabadia MM, Rastgooa A (2018) Analytical solution for nonlinear dynamic behavior of viscoelastic nano-plates modeled by consistent couple stress theory. Lat Am J Solids Stru 15(9): 175-198.
[33] Mokhtari Amir Majdi MA, Tahani M (2018)  Sizedependent analysis of micro-bridge gyroscopes under the combined effects of instantaneous DC voltage and harmonic base excitations. Modares Mechanical Engineering 18(01): 231-238. (in Persian)
[34] Andakhshideh A, Maleki S, Marashi (2018)  Investigation of nonlinear pull-in phenomena         in functionally graded micro-beams under electrostatic excitation. Journal of Solid and Fluid Mechanics 8(03): 137-151. (in Persian)
Journal of Solid and Fluid Mechanics
Volume 9, Issue 3
October 2019
Pages 125-138

Files

Share

How to cite

Statistics