Frequency Response Analysis of a Capacitive Micro-beam Resonator Considering Residual and Axial Stresses and Temperature Changes Effects

Document Type: Research Paper


Department of Mechanical Engineering, Khoy Branch, Islamic Azad University


This paper presents a study on the frequency response of a capacitive micro-beam resonator under various applied stresses. The governing equation whose solution holds the answer to all our questions about the mechanical behavior is the nonlinear electrostatic equation. Due to the nonlinearity and complexity of the derived equation analytical solution are not generally available; therefore, the obtained differential equation has been solved by using a step by step linearization scheme and a Galerkin based reduced order model. The obtained static pull-in voltages have been validated by previous reports and a good agreement has been achieved. The dynamic behavior of the beam under residual, axial and thermal stresses has been investigated. It has been shown that applying the positive residual stress and negative temperature changes shifts right the frequency response and decrease the vibration amplitude and vice versa. Also, it has been shown that applying the bias DC voltage beside the exciting AC voltage decreases the stiffness of the system and so, shifts left the frequency response and increases the vibration amplitude. 


[1] Basso M., Giarre L., Dahleh M., Mezic I., 1998,Numerical analysis of complex dynamics in atomic force microscopes, Proceedings of the IEEE International Conference on Control Applications 2:1026–1030.

[2] Fritz J., Baller M.K., Lang H.P., Rothuizen H., Vettiger P., Meyer E., Gntherodt H.J., Gerber C., Gimzewski J.K., 2001,Translating bio-molecular recognition into nanomechanics , Science 288:316–318.
[3] Sidles JA., 1991, Noninductive detection of single proton-magnetic resonance, Applied Physics Letters 58:2854-2856.

[4] Nabian A., Rezazadeh G., Haddad-derafshi M., Tahmasebi A., 2008, Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure, Microsyst Technol 14:235–240.

[5] Fathalilou M., Motallebi A., Rezazadeh G., Yagubizade H., Shirazi K., Alizadeh Y., 2009, Mechanical Behavior of an Electrostatically-Actuated Microbeam under Mechanical Shock, Journal of Solid Mechanics 1: 45-57.

[6] Senturia SD., 2001, Microsystem Design, Norwell, MA: Kluwer.
[7] Zhang Y., Zhao Y., 2006, Numerical and analytical study on the pull-in instability of micro- structure under electrostatic loading, Sensors and Actuators A: Physical 127: 366-367.

[8] Rezazadeh G., Sadeghian H., Abbaspour E., 2008, A comprehensive model to study nonlinear behaviour of multilayered micro beam switches, Microsyst Technol 14: 143.

[9] Sadeghian H., Rezazadeh G., 2007, Application of the Generalized Differential Quadrature Method to the Study of Pull-In Phenomena of MEMS Switches, Journal of Microelectromechanical Systems 16(6):1334-1340.

[10] Osterberg P.M., Senturia S.D., 1997, M-TEST a test chip for MEMS material property measurement using electrostatically actuated test structures, Journal of Microelectromechanical Systems 6(2):107-118.

[11] Senturia SD., Aluru N, White J., 1997, Simulating the behavior of MEMS devices, IEEE Computing in Science and Engineering 4(1): 30–43.
[12] Abdel-Rahman E.M., Younis M.I., Nayfeh A.H., 2002, Characterization of the mechanical behavior of an electrically actuated microbeam , Journal of Micromechanics and Microengineering 12:759–766.

[13] Rezazadeh G., Tahmasebi A., Zubtsov M., 2006, Application of Piezoelectric Layers in Electrostatic MEM Actuators: Controlling of Pull-in Voltage, Microsyst Technol 12 : 1163-1170.

[14] Mukherjee T., Fedder G.K., White J., 2000, Emerging simulation approaches for micromachined devices, IEEE Transactions on Computer- Aided Design of Integrated Circuits and Systems 19: 1572–1589.

[15] Senturia SD., 1998, CAD challenges for microsensors, microactuators, and Microsystems, Proceedings of the IEEE 86: 1611–1626.

[16] Talebian S., Rezazadeh G., Fathalilou M., Yagubizade H., 2010, Effect of Temperature on Pull-in Voltage and Natural Frequency of an Electrostatically Actuated Microplate, Mechatronics 20(6):666-673.