Genetic Algorithm and ANN for Estimation of SPIV of Micro Beams
الموضوعات :
1 - Department of Mechanical Engineering,
Aligudarz Branch, Islamic Azad University, Aligudarz, Iran
الکلمات المفتاحية: Euler-Bernoulli, Modified couple stress theory, Nonlinear micro-beam, genetic algorithms, Artificial Neural Networks, Static pull-in instability,
ملخص المقالة :
In this paper, the static pull-in instability (SPIV) of beam-type micro-electromechanical systems is theoretically investigated. Herein, modified strain gradient theory in conjunction with Euler–Bernoulli beam theory have been used for mathematical modeling of the size dependent instability of the micro beams. 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. Two common beam-type systems including double-clamped and clamped-free cantilever have been investigated. 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 (ANN) with three functions have been used for modelling the static pull-in instability voltage of micro beam. Effect of the size dependency on the pull-in performance has been discussed for both micro-structures. 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. The number of nodes in the hidden layer, learning ratio and momentum term are optimized using genetic algorithms (GAs). Numerical data, employed for training the network and capabilities of the model in predicting the pull-in instability behaviour 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. Resultant low relative error value of the ANN model indicates the usability of the BPN in this area.
[1] Tadi Beni, Y., Karimipour, I., and Abadyan, M., “Modeling the Instability of Electrostatic Nano-Bridges and Nano-Cantilevers Using Modified Strain Gradient Theory”, Applied Mathematical Modelling, Vol. 39, No.9, 2015, pp. 2633-2648.
[2] Gasparini, A. M., Saetta, A. V., and Vitaliani, R. V., “On the Stability and Instability Regions of Non-Conservative Continuous System under Partially Follower Forces”, Computer Methods in Applied Mechanics and Engineering, Vol. 124, No. (1-2), 1995, pp. 63-78.
[3] Osterberg, P. M., Senturia, S. D., “M-TEST: a Test Chip for MEMS Material Property Measurements Using Electrostatically Actuated Test Structures”, Journal of Microelectromechanical Systems, Vol. 6, No. 2, 1997, pp. 107-118.
[4] Osterberg, P. M., Gupta, R. K., Gilbert, J. R., and Senturia, S. D., “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, 1994.
[5] Sadeghian, H., Rezazadeh, G., Osterberg, P., “Application of the Generalized Differential Quadrature Method to the Study of Pull-in Phenomena of MEMS Switches”, Journal of Microelectromechanical Systems, Vol. 16, No. 6, 2007, pp. 1334-1340.
[6] Salekdeh, Y. A., Koochi, A., Beni, Y. T., and Abadyan, M., “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 Applied Sciences Research, Vol. 7, No. 1, 2012, pp. 1-17.
[7] Batra, R. C., Porfiri, M., Spinello, D., “Review of Modeling Electrostatically Actuated Microelectromechanical Systems”, Smart Materials and Structures, Vol. 16, No. 6, 2007, R23-R31.
[8] Lin, W. H., Zhao, Y. P., “Pull-in Instability of Micro-Switch Actuators: Model Review”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 9, No. 2, 2008, pp.175-184.
[9] Koiter, W. T., “Couple-Stresses in the Theory of Elasticity: I and II.”, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series B, 1964, pp. 6717-6744.
[10] Mindlin, R. D., Tiersten, H. F., “Effects of Couple-Stresses in Linear Elasticity”, Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, 1962, pp. 415-448.
[11] Toupin, R. A., “Elastic Materials with Couple-Stresses, Archive for Rational Mechanics and Analysis”, Vol. 11, No. 1, 1962, pp. 385-414.
[12] Anthoine, A., “Effect of Couple-Stresses on the Elastic Bending of Beams”, International Journal of Solids and Structures, Vol. 37, No. 7, 2000, pp. 1003-1018.
[13] Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P., “Couple Stress Based Strain Gradient Theory for Elasticity”, International Journal of Solids and Structures, Vol. 39, No. 10, 2002, pp. 2731-2743.
[14] Xia, W., Wang, L., and Yin, L., “Nonlinear Non-Classical Microscale Beams: Static Bending, Post Buckling and Free Vibration”, International Journal of Engineering Science, Vol. 48, No. 12, 2010, pp. 2044-2053.
[15] Asghari, M., Ahmadian, M. T., Kahrobaiyan, M. H., and Rahaeifard, M., “On the Size-Dependent Behavior of Functionally Graded Micro-Beams”, Materials and Design, Vol. 31, No. 5, 2010, pp. 2324-2329.
[16] Rong, H., Huang, Q. A., Nie, M., and Li, W., “An Analytical Model for Pull-in Voltage of Clamped–Clamped Multilayer Beams”, Sensors and Actuators A: Physical, Vol. 116, No. 1, 2004, pp. 15-21.
[17] Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P., “Couple Stress Based Strain Gradient Theory for Elasticity”, International Journal of Solids and Structures, Vol. 39, No. 10, 2002, pp. 2731-2743.
[18] Shengli, K., Shenjie, Z., Zhifeng, N., and Kai, W., “The Size-Dependent Natural Frequency of Bernoulli–Euler Micro-Beams”, Journal of Engineering Science, Vol. 46, No. 5, 2008, pp. 427-437.
[19] Ma, H. M., Gao, X. L., and Reddy, J. N., “A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory”, Journal of the Mechanics and Physics of Solids, Vol. 56, No. 12, 2008, pp. 3379-3391.
[20] Tadi Beni, Y., Koochi, A., and Abadyan, M., “Theoretical Study of the Effect of Casimir Force, Elastic Boundary Conditions and Size Dependency on the Pull-In Instability of Beam-Type NEMS”, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 4, 2011, pp.979-988.
[21] Zhao, J., Zhou, S., Wanga, B., and Wang, X., “Nonlinear Microbeam Model Based on Strain Gradient Theory”, Applied Mathematical Modelling, Vol. 36, No. 6, 2012, pp. 2674-2686.
[22] Freeman, J. A., Skapura, D. M., “Neural Networks: Algorithms, Applications, and Programming Techniques”, Addision-Wesley, 1992.
[23] Gao, D., Kinouchi, Y., Ito, K., and Zhao, Z., “Neural Networks for Event Extraction from Time Series: A Back Propagation Algorithm Approach, Future Generation Computer Systems”, Vol. 21, No. 7, 2005, pp. 1096-1105.
[24] Rumelhart, D. E., Hinton, G. E., and Williams, R. J., “Learning Representations by Back Propagating Error”, Nature, Vol. 323, 1986, pp. 533-536.
[25] Zhang, H., Wei, W., and Mingchen, Y., “Boundedness and Convergence of Batch Back-Propagation Algorithm with Penalty for Feedforward Neural Networks”, Neurocomputing, Vol. 89, 2012, pp. 141-146.
[26] Holland, J. H., “Adaption in Natural and Artificial Systems”, Ann Arbor: University of Michigan Press, 1975.
[27] He, Y., Guo, D., and Chu, F., “Using Genetic Algorithms and Finite Element Methods to Detect Shaft Crack for Rotor-Bearing System”, Mathematics and Computers in Simulation, Vol. 57, No. 1-2, 2001, pp. 95-108.
[28] Wong, M. L. D., Nandi, A. K., “Automatic Digital Modulation Recognition Using Artificial Neural Network and Genetic Algorithm”, Signal Processing, Vol. 84, No. 2, 2004, pp. 351-365.
[29] Tang, K. S., Man, K. F., Kwong, S., and He, O., “Genetic Algorithms and Their Applications”, IEEE Signal Processing Magazine, Vol. 13, No. 6, 1996, pp. 22-37.
[30] Demuth, H., Beale M., Matlab Neural Networks Toolbox, User’s Guide, The Math Works, Inc., http://www.mathworks.com, 2001.