Crack Influences on the Static and Dynamic Characteristic of a Micro-Beam Subjected to Electro Statically Loading
Subject Areas : EngineeringA.R Shahani 1 , G Rezazadeh 2 , A Rahmani 3
1 - Department of Applied Mechanics, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
2 - Mechanical Engineering Departments, Urmia University, Urmia, Iran
3 - Department of Applied Mechanics, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran
Keywords: MEMS, Stability Analysis, Cracked micro-beam,
Abstract :
In the present work the pull-in voltage of a micro cracked cantilever beam subjected to nonlinear electrostatic pressure was studied. Two mathematical models were employed for modeling the problem: a lumped mass model and a classical beam model. The effect of crack in the lumped mass model is the reduction of the effective stiffness of the beam and in the beam model; the crack is modeled as a massless rotational spring the compliance of which is related to the crack depth. Using these two models the pull-in voltage is extracted in the static and dynamic cases. Stability analysis is also accomplished. It has been observed that the pull-in voltage decreases as the crack depth increases and also when the crack approaches the clamped support of the beam. The finding of this research can further be used as a non-destructive test procedure for detecting cracks in micro-beams.
[1] Senturia S. D., 2001, Microsystem Design, Boston, Kluwer.
[2] Younis M. I., Nayfeh A. H., 2003, A study of the nonlinear response of a resonant micro-beam to an electric actuation, Journal of Nonlinear Dynamics 31: 91-117.
[3] Elata D., Bamberger H., 2006, On the dynamic pull-In of electrostatic actuators with multiple degrees of freedom and multiple voltage sources, Journal of Microelectromechanical Systems 15: 131-140.
[4] 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: 366-380.
[5] Rezazadeh G., Fathaliou M., Sadeghi M., 2011, Pull-in voltage of electro-statically actuated micro-beams in terms of lumped model pull-in voltage using novel design corrective coefficients, Sensing and Imaging 12: 117-131.
[6] Zhang W. M., Yan H., Peng Z. K., Meng G., 2014, Electrostatic pull-in instability in MEMS/NEMS: a review, Sensors and Actuators A: Physical 214: 187-214.
[7] Choi B., Lovell E. G., 1997, Improved analysis of micro-beams under mechanical and electrostatic loads, Journal of Micromechanics and Micro engineering 7: 24-29.
[8] Chowdhury S., Ahmadi M., Miller W. C., 2005, A closed-form model for the pull-in voltage of electrostatically actuated cantilever beams, Journal of Micromechanics and Micro engineering 15: 756-763.
[9] Chao P. C. P., Chiu C. W., Liu T. H., 2008, DC dynamic pull-in predictions for a generalized clamped–clamped micro-beam based on a continuous model and bifurcation analysis, Journal of Micromechanics and Micro engineering 18: 1-14.
[10] Krylov S., 2007, Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures, International Journal of Non-Linear Mechanics 42: 626-642.
[11] Varvani-Farahani A., 2005, Silicon MEMS components: a fatigue life assessment approach, Microsystem Technologies 11: 129-134.
[12] Hill M J., Rowcliffe D. J., 1947, Deformation of silicon at low temperatures, Journal of Materials Science 9: 1569-1576.
[13] Muhlstein C. L, Brown S. B., Ritchie R. O., 2001, High-cycle fatigue and durability of polycrystalline silicon films in ambient air, Sensors and Actuators A: Physical 94: 177-188.
[14] Ando T., Shikida M., Sato K., 2001, Tensile-mode fatigue testing of silicon films as structural materials for MEMS, Sensors and Actuators A: Physical 93: 70-75.
[15] Motallebi A., Fathalilou M., Rezazadeh G., 2012, Effect of the open crack on the pull-in instability of an electrostatically actuated micro-beam, Acta Mechanica Solida Sinica 25: 627-637.
[16] Sourki R., Hoseini S. A. H., 2016, Free vibration analysis of size-dependent cracked micro-beam based on the modified couple stress theory, Applied Physics A 413: 1-11.
[17] Tadi Beni Y., Jafaria A., Razavi H., 2015, Size effect on free transverse vibration of cracked nano-beams using couple stress theory, International Journal of Engineering 28: 296-304.
[18] Loya J. A., Aranda-Ruiz J., Fern J., 2014, Torsion of cracked Nano-rods using a nonlocal elasticity model, Journal of Physics D : Applied Physics 47: 115304-115315.
[19] Barr A. D. S., Christides S., 1984, One-dimensional theory of crack Euler-Bernoulli beams, International Journal of Mechanical Sciences 26: 639-639.
[20] Rizos P. F., Aspragathos N., Dimarogonas A. D., 1990, Identification of crack location and magnitude in cantilever beam from the vibration modes, Journal of Sound Vibration 138: 381-388.
[21] Chondros T. G., Dimarogonas A. D., Yao J., 1998, A continuous cracked beam vibration theory, Journal of Sound Vibration 215: 17-34.
[22] Behzad M., Meghdari A., Ebrahimi A 2008 A linear theory for bending stress-strain analysis of a beam with an edge crack, Engineering Fracture Mechanics 75: 4695-4705.
[23] Li Q. S., 2002, Free vibration analysis of non-uniform beams with an arbitrary number of cracks and concentrated mass, Journal of Sound Vibration 252: 509-525.
[24] Binici B., 2005, Vibration of beams with multiple open cracks subjected to axial force, Journal of Sound Vibration 287: 277-295.
[25] Zhang G. P., Wang Z. G., 2008, Fatigue of small-scale metal materials: from micro to nano-scale structural integrity and microstructural worthiness, Springer Netherlands 152: 275-326.
[26] Sadeghian H., Goosen H., Bossche A., Thijsse B., Van Keulen F., 2011, On the size-dependent elasticity of silicon nano-cantilevers: impact of defects, Journal of Physics D : Applied Physics 44: 072001.
[27] Shengli K., Shenjie Z., Zhifeng N., Kai W., 2009, Static and dynamic analysis of micro beams based on strain gradient elasticity theory, International Journal of Engineering Science 47: 487-498.
[28] 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, Applied Mathematical Modeling 37: 6964-6978.
[29] Akbaş S. D., 2016, Free vibration of edge cracked functionally graded micro-scale beams based on the modified couple stress theory, International Journal of Structural Stability and Dynamics 16: 1750033.
[30] Torabi K., Dastgerdi J., 2012, An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nano-beams using a nonlocal elasticity model, Thin Solid Films 520: 6595-6602.
[31] Gounaris G. D., Papadopoulos C. A., Dimarogonas A. D., 1996, Crack identification in beams by coupled response measurement, Computational and Structural 58: 409-423.
[32] Shifrin E. I., Ruotolo R., 1999, Natural frequencies of a beam with an arbitrary number of cracks, Journal of Sound Vibration 223: 409-423.
[33] Dimarogonas A. D., Paipettis S. A., 1983, Analytical Methods in Rotor Dynamics, Elsevier Applied Science, London.
[34] Dimarogonas A. D., 1976, Vibration Engineering, Paul, West Publishers.
[35] Rezazadeh G., Tahmasebi A., Zubostow M., 2006, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage, Microsystem Technologies 12: 1163-1170.
[36] Dominicus J., Ijntema Harrie A. C., 1992, Static and dynamic aspects of an air-gap capacitor, Sensors and Actuators A: Physical 35: 121-128.
[37] Lee K. B., 2011, Principles of Micro-Electromechanical Systems, John Wiley & Sons, New Jersey.
[38] Younis M. I., 2011, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York.
[39] Rochus V., Rixen D. J. and Golinval J.C., 2005, Electro-static coupling of MEMS structures: transient simulations and dynamic pull-in, Nonlinear Analysis, Theory, Methods & Applications 63: 1619-1633.
[40] Vytautas O., Rolanas D., 2010, Microsystems Dynamics, Springer, New York.
[41] Osterbeg P. M., 1995, Electro-Statically Actuated Micro-Electromechanical Test Structures for Material Property Measurements, PhD Dissertation Massachusetts Institute of Technology.
[42] Joglekar M. M., Pawaskar D. N., 2011, Closed-form empirical relations to predict the static pull-in parameters of electrostatically actuated micro-cantilevers having linear with variation, Microsystem Technologies 17: 35-45.
[43] Abbasnejhad B., Rezazadeh G., Shabani R., 2011, Stability analysis of a capacitive FGM micro-beam using modified couple stress theory, Acta Mechanica Solida Sinica 26: 427-440.