Estimation of pull-in instability voltage of Euler-Bernoulli micro beam by back propagation artificial neural network

Document Type : Reasearch Paper


Mechanical Engineering Group, Aligudarz Branch, Islamic Azad University, Aligudarz, Iran



The static pull-in instability of beam-type micro-electromechanical systems is theoretically investigated. Two engineering cases including cantilever and double cantilever micro-beam are considered. Considering the mid-plane stretching as the source of the nonlinearity in the beam behavior, a nonlinear size-dependent Euler-Bernoulli beam model is used based on a modified couple stress theory, capable of capturing the size effect. By selecting a range of geometric parameters such as beam lengths, width, thickness, gaps and size effect, we identify the static pull-in instability voltage. Back propagation artificial neural network with three functions have been used for modeling the static pull-in instability voltage of micro cantilever beam. The network has four inputs of length, width, gap and the ratio of height to scale parameter of beam as the independent process variables, and the output is static pull-in voltage of microbeam. Numerical data, employed for training the network and capabilities of the model in predicting the pull-in instability behavior has been verified. The output obtained from neural network model is compared with numerical results, and the amount of relative error has been calculated. Based on this verification error, it is shown that the back propagation neural network has the average error of 6.36% in predicting pull-in voltage of cantilever micro-beam.


Main Subjects

[1]  Khatami I., Pashai M. H., Tolou N., (2008), Comparative vibration analysis of a parametrically nonlinear excited oscillator using HPM and numerical method. Mathemat. Problems in Eng. 2008: 1-11.
[2]  Gasparini A. M., Saetta A. V., Vitaliani R. V., (1995), on the stability and instability regions of non-conservative continuous system under partially follower forces. Comput. Meth. Appl. Mech. Eng. 124: 63-78.
[3]  Osterberg P. M., Senturia S. D., (1997), M-TEST: A test chip for MEMS material property measurements using electrostatically actuated test structures. J. Microelectromech. Syst. 6: 107-118.
[4]  Osterberg P. M., Gupta R. K., Gilbert J. R., Senturia S. D., (1994), Quantitative models for the measurement of residual stress, poisson ratio and young.s modulus using electrostatic pull-in of beams and diaphragms. Proceedings of the Solid-State Sensor and Actuator Workshop. Hilton Head, SC.
[5]  Sadeghian H., Rezazadeh G., Osterberg P., (2007), Application of the generalized differential quadrature method to the study of pull-in phenomena of MEMS switches. IEEE/ASME J. Micro Electro Mech. Sys. 16: 1334-1340.
[6]  Salekdeh Y. A., Koochi A., Beni Y. T., Abadyan M., (2012), Modeling effect of three nano-scale physical phenomena on instability voltage of multi-layer MEMS/NEMS: Material size dependency, van der waals force and non-classic support conditions. Trends in Appl. Sci. Res. 7: 1-17.
[7]  Batra R. C., Porfiri M., Spinello D., (2007), Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater. Struct. 16: R23-R31.
[8]  Lin W. H., Zhao Y. P., (2008), Pull-in instability of microswitch actuators: Model review. Int. J. Nonlinear Sci. Numer.Simulation. 9: 175-184.
[9] Koiter W. T., (1964), Couple-stresses in the theory of elasticity: I and II. Proceed. Koninklijke Nederlandse Akademie van Wetenschappen Series B. 6717-6744.
[10] Mindlin R. D., Tiersten H. F., (1962), Effects of couple stresses in linear elasticity. Archive for Rational Mech. Analysis. 11: 415-448.
[11] Toupin R. A., (1962), Elastic materials with couple stresses. Archive for Rational Mech. Analysis. 11: 385–414.
[12] Anthoine A., (2000), Effect of couple-stresses on the elastic bending of beams. Int. J. Solids and Struc. 37: 1003-1018.
[13] Yang F., Chong A. C. M., Lam D. C. C., Tong P., (2002), Couple stress based strain gradient theory for elasticity. Int. J. Solids and Struc. 39: 2731-2743.
[14] Xia W., Wang L., Yin L., (2010), Nonlinear non-classical microscale beams: Static bending, post buckling and free vibration. Int. J. Eng. Sci. 48: 2044-2053.
[15] Asghari M., Rahaeifard M., Kahrobaiyan M. H., Ahmadian M. T., (2011), On the size-dependent behavior of functionally graded micro-beams. Mater. Design. 32: 1435-1443.
[16] Rong H., Huang Q. A., Nie M., Li W.,(2004), An analytical model for pull-in voltage of clamped multilayer beams. Sens. Actuators A. 116: 15-21.
[17] Yang F., Chong A. C. M., Lam D. C. C., Tong P., (2002), Couple stress based strain gradient theory for elasticity. Int. J. Solids and Struc. 39: 2731-2743.
[18] Shengli K., Shenjie Z., Zhifeng N., Kai W., (2011), The size-dependent natural frequency of Bernoulli. Euler microbeams. J. Eng. Sci. 46: 427-437.
[19] Ma H. M., Gao X. L., Reddy J. N., (2008), A microstructure dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. and Physics of Solids. 56: 3379-3391.
[20] Gupta R. K., (1997), Electrostatic pull-in test structure design for in-situ mechanical property measurements of microelectromechanical systems. Ph.D. Dissertation, Massachusetts Institute of Technology (MIT), Cambridge, MA.
[21] Zhao J., Zhou S., Wanga B., Wang X., (2012), Nonlinear microbeam model based on strain gradient theory. Appl. Mathemat. Modell. 36: 2674-2686.
[22] Freeman J. A., Skapura D. M., (1992), Neural networks: algorithms, applications, and programming techniques. Addision-Wesley.
[23] Gao D., Kinouchi Y., Ito K., Zhao Z., (2005), Neural networks for event extraction from time series: a back propagation algorithm approach. Future Gener. Comp. Sys. 21: 1096-1105.
[24] Rumelhart D. E., Hinton G. E., Williams R. J., (1986), Learning representations by back propagating error. Nature. 323: 533-536.
[25] Zhang H., Wei W., Mingchen Y., (2012), Boundedness and convergence of batch back-propagation algorithm with penalty for feedforward neural networks. Neurocomputing. 89: 141-146.
[26] Hongmei S., Gaofeng Z., (2011), Convergence analysis of a back-propagation algorithm with adaptive momentum. Neurocomputing. 74: 749-752.
[27] Demuth H., Beale M., (2001), Matlab Neural Networks Toolbox, User.s Guide, The Math Works, Inc.,