2015
1
4
4
72
Local and Global Approaches to Fracture Mechanics Using Isogeometric Analysis Method
2
2
The present research investigates the implementations of different computational geometry technologies in isogeometric analysis framework for computational fracture mechanics. NURBS and Tsplines are two different computational geometry technologies which are studied in this work. Among the features of Bspline basis functions, the possibility of enhancing a Bspline basis with discontinuities by means of knot insertion makes isogeometric analysis method a suitable candidate for modeling discrete cracks. Also, the repetition of two different control points between two patches can create a discontinuity in and demonstrates a singularity in the stiffness matrix. In the case of a predefined interface, nonuniform rational Bsplines are used to obtain an efficient discretization. Tsplines constitute a type of computational geometry technology with the possibility of local refinement and with no topologically rectangular arrangement of control points. Therefore, Tsplines can decrease superfluous control points which do not have any major effects on the geometry. Various numerical simulations demonstrate the suitability of the isogeometric approach in fracture mechanics.
1

168
180


Abdolghafoor
Khademalrasoul
Ph.D. Student of Civil Engineering, Shahrood University of technology, Iran.
Ph.D. Student of Civil Engineering, Shahrood
Iran
ag.khadem@yahoo.com


Reza
Naderi
Department of Civil Engineering. Shahroud University of technology.
Department of Civil Engineering. Shahroud
Iran
rz_naderi@yahoo.com
Fracture mechanics
Isogeometric analysis method
Knot insertion
NURBS
Tspline
[[1] Bhardwaj G., Singh I.V., Mishra B.K., Bui T.Q., “Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions”, Composite Structures, Vol. 126, pp. 347359, 2015. ##[2] Singh I.V., Bhardwaj G., Mishra B.K., “A new criterion for modeling multiple discontinuities passing through an element using XIGA”, Journal of Mechanical Science and Technology, Vol. 29, No. 3, pp. 11311143, 2015. ##[3] Rabczuk T., Belytschko T., Cracking particles: a simplified meshfree method for arbitrary evolving cracks, International Journal for Numerical Methods in Engineering, Vol. 61 No. 13, pp. 23162343, 2004. ##[4] Naderi R., Khademalrasoul A., “Fully Automatic Crack Propagation Modeling in Interaction with Void and Inclusion without Remeshing” Modares Mechanical Engineering, Vol. 15 No. 7, pp. 261273, 2015. (In Persian) ##[5] Zhuang Z., Liu Z., Cheng B., Liao J., “Chapter 2  Fundamental Linear Elastic Fracture Mechanics.In”: Zhuang Z, Liu Z, Cheng B, Liao J, editors. Extended Finite Element Method. Oxford: Academic Press, pp. 1331, 2014. ##[6] Daxini S.D., Prajapati J.M., “A Review on Recent Contribution of Meshfree Methods to Structure and Fracture Mechanics Applications”, The Scientific World Journal, 2014. ##[7] Chen T., Xiao Z.G., Zhao X.L., Gu X.L., “A boundary element analysis of fatigue crack growth for welded connections under bending”, Engineering Fracture Mechanics, Vol. 98, pp. 4451, 2013. ##[8] Hughes T.J.R., Cottrell J.A., Bazilevs Y., “Isogeometric Analysis Toward integration of CAD and FEM”, 2009. ##[9] Cottrell J.A., Hughes T.J.R., Bazilevs Y., “Isogeometric Analysis: Toward Integration of CAD and FEA”, Wiley, 2009. ##[10] Hughes T.J.R., Cottrell J.A., Bazilevs Y., “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 39–41, pp. 41354195, 2005. ##[11] Bazilevs Y., Calo V.M., Cottrell J.A., Evans J.A., Hughes T.J.R., Lipton S., et al., “Isogeometric analysis using Tsplines”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, No. 5–8, pp. 229263, 2010. ##[12] Ghorashi S.S., Valizadeh N., Mohammadi S., “Extended isogeometric analysis for simulation of stationary and propagating cracks”, International Journal for Numerical Methods in Engineering, Vol. 89, No. 9, pp. 10691101, 2012. ##[13] De Luycker E., Benson D.J., Belytschko T., Bazilevs Y., Hsu M.C., “XFEM in isogeometric analysis for linear fracture mechanics”, International Journal for Numerical Methods in Engineering, Vol. 87, No. 6, pp. 541565, 2011. ##[14] Evans J.A., Bazilevs Y., Babuška I., Hughes T.J.R., “nWidths, sup–infs, and optimality ratios for the kversion of the isogeometric finite element method”, Computer Methods in Applied Mechanics and Engineering, Vol. 198 No. 21–26, pp. 17261741, 2009. ##[15] Cottrell J.A., Hughes T.J.R., Reali A., “Studies of refinement and continuity in isogeometric structural analysis”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 41–44, pp. 41604183, 2007. ##[16] Cottrell J.A., Reali A., Bazilevs Y., Hughes T.J.R., “Isogeometric analysis of structural vibrations, Computer Methods in Applied”, Mechanics and Engineering, Vol. 195, No. 41–43, pp. 52575296, 2006. ##[17] Akkerman I., Bazilevs Y., Kees C.E., Farthing M.W., “Isogeometric analysis of freesurface flow”, Journal of Computational Physics, Vol. 230, No. 11, pp. 41374152, 2011. ##[18] Bazilevs Y., Akkerman I., “Large eddy simulation of turbulent Taylor–Couette flow using isogeometric analysis and the residualbased variational multiscale method”, Journal of Computational Physics, Vol. 229, No. 9, pp. 34023414, 2010. ##[19] Bazilevs Y., Calo V.M., Cottrell J.A., Hughes T.J.R., Reali A., Scovazzi G., “Variational multiscale residualbased turbulence modeling for large eddy simulation of incompressible flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, No. 1–4, pp. 173201, 2007. ##[20] Bazilevs Y., Hsu M.C., Scott M.A., “Isogeometric fluid–structure interaction analysis with emphasis on nonmatching discretizations, and with application to wind turbines”, Computer Methods in Applied Mechanics and Engineering, Vol. 249–252, pp. 2841, 2012. ##[21] Li K., Qian X., “Isogeometric analysis and shape optimization via boundary integral”, ComputerAided Design, Vol. 43, No. 11, pp. 14271437, 2011. ##[22] Hassani B., Tavakkoli S.M., Moghadam N.Z., “Application of isogeometric analysis in structural shape optimization”, Scientia Iranica, Vol. 18, No. 4, pp. 846852, 2011. ##[23] Rots J., “Smeared and discrete representations of localized fracture”, International Journal of Fracture, Vol. 51, No. 1, pp. 4559, 1991. ##[24] Schellekens J.C.J., De Borst R., “On the numerical integration of interface elements”, International Journal for Numerical Methods in Engineering, Vol. 36, No. 1, pp. 4366, 1993. ##[25] Simo J.C., Oliver J., Armero F., “An analysis of strong discontinuities induced by strainsoftening in rateindependent inelastic solids”, Computational Mechanics, Vol. 12, No. 5, pp. 277296, 1993. ##[26] Oliver J., “Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 2: numerical simulation”, International Journal for Numerical Methods in Engineering, Vol. 39, No. 21, pp. 36013623, 1996. ##[27] Belytschko T., Black T., “Elastic crack growth in finite elements with minimal remeshing”, International Journal for Numerical Methods in Engineering, Vol. 45, No. 5, pp. 601620, 1999. ##[28] Babuška I., Zhang Z., “The partition of unity method for the elastically supported beam”, Computer Methods in Applied Mechanics and Engineering, Vol. 152, No. 1–2, pp. 118, 1998. ##[29] Babuska I., Melenk, J, “The Partition of unity method”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 727–758, 1997. ##[30] Bhardwaj G., Singh I.V., Mishra B.K., “Stochastic fatigue crack growth simulation of interfacial crack in bilayered FGMs using XIGA”, Computer Methods in Applied Mechanics and Engineering, Vol. 284, pp. 186229, 2015. ##[31] Bhardwaj G., Singh I.V., Mishra B.K., Kumar V., “Numerical Simulations of Cracked Plate using XIGA under Different Loads and Boundary Conditions”, Mechanics of Advanced Materials and Structures, 2015. ##[32] Scott M.A., Li X., Sederberg T.W., Hughes T.J.R., “Local refinement of analysissuitable Tsplines”, Computer Methods in Applied Mechanics and Engineering, Vol. 213–216, pp. 206222, 2012. ##[33] Piegl L.A., Tiller W., The Nurbs Book, SpringerVerlag GmbH, 1997. ##[34] Rogers D.F., “An introduction to NURBS: with historical perspective”, Morgan Kaufmann Publishers, 2001. ##[35] De Boor C., “On calculating with Bsplines”, Journal of Approximation Theory, Vol. 6, No.1, pp. 5062, 1972. ##[36] Cox M.G., “The Numerical Evaluation of BSplines”, IMA Journal of Applied Mathematics, Vol. 10, No. 2, pp. 134149, 1972. ##[37] Sederberg T.W., Zheng J., Bakenov A., Nasri A., Tsplines and TNURCCs, ACM SIGGRAPH 2003 Papers, San Diego, California, 882295: ACM, pp. 477484, 2003. ##[38] Buffa A., Cho D., Sangalli G., “Linear independence of the Tspline blending functions associated with some particular Tmeshes”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, No. 23–24, pp. 14371445, 2010. ##[39] Buffa A., Cho D., Kumar M., “Characterization of Tsplines with reduced continuity order on Tmeshes”, Computer Methods in Applied Mechanics and Engineering, Vol. 201–204, pp. 112126, 2012. ##[40] Scott M.A., Simpson R.N., Evans J.A., Lipton S., Bordas S.P.A., Hughes T.J.R., et al., “Isogeometric boundary element analysis using unstructured Tsplines”, Computer Methods in Applied Mechanics and Engineering, Vol. 254, pp. 197221, 2013.##]
Vibration analysis of a rotating closed section composite Timoshenko beam by using differential transform method
2
2
This study introduces the Differential Transform Method (DTM) in the analysis of the free vibration response of a rotating closed section composite, Timoshenko beam, which features material coupling between flapwise bending and torsional vibrations due to ply orientation. The governing differential equations of motion are derived using Hamilton’s principle and solved by applying DTM. The natural frequencies are calculated and the effects of the bendingtorsion coupling, the slenderness ratio and several other parameters on the natural frequencies are investigated using the computer package, Mathematica. Wherever possible, comparisons are made with the studies in open literature.
1

181
186


Saeed
Talebi
Department of Mechanical
Engineering, University of
Isfahan
Department of Mechanical
Engineering, University
Iran
saeed.talebi68@gmail.com


Hamed
Uosofvand
department of mechanical engineering, university of kashan, Kashan, Iran
department of mechanical engineering, university
Iran
mr.uosofvand@gmail.com


Alireza
Ariaei
Department of Mechanical Engineering, Faculty of Engineering, University of Isfahan, isfahan, iran
Department of Mechanical Engineering, Faculty
Iran
ariaei@eng.ui.ac.ir
Rotating beam
Composite
Natural frequency
Mode shape
DTM
[[1] Thakkar D., Ganguli R., “Helicopter vibration reduction in forward ﬂight with induced shear based piezoceramic actuation”, Smart Mater. Struct., 30 (3), pp. 599608, 2004. ##[2] Kumar S., Roy N., Ganguli R., “Monitoring low cycle fatigue damage in turbine blades using vibration characteristics”, Mech. Syst. Signal Process, 21 (1), pp.480–501, 2007. ##[3] Zhou J.K., “Differential Transformation and its Application for Electrical Circuits”, Wuhan, Huazhong University Press, Wuhan, China, 1986. ##[4] Ozdemir O.O., Kaya M.O., “Flap wise bending vibration analysis of a rotating tapered cantilever BernoulliEuler beam by differential transform method”, Journal of Sound and Vibration, 289, pp.413–420, 2006. ##[5] Mei C., “Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam”, Computers and Structures, 86, pp.1280–1284, 2008. ##[6] Kaya M. O., Ozdemir Ozgumus O., “Flexural–Torsional Coupled Vibration Analysis of Axially Loaded Closed Section Composite Timoshenko Beam by Using DTM”, Journal of Sound and Vibration, (35), pp. 495506, 2007. ##[7] Li J., Shen R., Hua H., Jin X., “Bendingtorsional coupled vibration of axially loaded composite Timoshenko thinwalled beam with closed crosssection”, Composite Structures 64, pp. 2335, 2004.##]
Dynamical Behavior of a Rigid Body with One Fixed Point (Gyroscope). Basic Concepts and Results. Open Problems: a Review
2
2
The study of the dynamic behavior of a rigid body with one fixed point (gyroscope) has a long history. A number of famous mathematicians and mechanical engineers have devoted enormous time and effort to clarify the role of dynamic effects on its movement (behavior) – stable, periodic, quasiperiodic or chaotic. The main objectives of this review are: 1) to outline the characteristic features of the theory of dynamical systems and 2) to reveal the specific properties of the motion of a rigid body with one fixed point (gyroscope).This article consists of six sections. The first section addresses the main concepts of the theory of dynamical systems. Section two presents the main theoretical results (obtained so far) concerning the dynamic behavior of a solid with one fixed point (gyroscope). Section three examines the problem of gyroscopic stabilization. Section four deals with the nonlinear (chaotic) dynamics of the gyroscope. Section five is a brief analysis of the gyroscope applications in engineering. The final section provides conclusions and generalizations on why the theory of dynamical systems should be used in the study of the movement of gyroscopic systems.
1

187
206


Svetoslav
Nikolov
Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. BonchevStr., Bl. 4, Bulgaria
Institute of Mechanics, Bulgarian Academy
Iran
s.nikolov@imbm.bas.bg


Nataliya
Nedkova
University of Transport, G. Milev Str., 158, 1574 Sofia, Bulgaria
University of Transport, G. Milev Str., 158,
Iran
nataliya_nedkova@abv.bg
Gyroscopic systems
theory of dynamical systems
dynamical behavior
[[1] Afraimovich, V., Gonchenko, S., Lerman, L., Shilnikov, A. and Turaev, D., “Scientific Heritage of L.P. Shilnikov”, Regular and Chaotic Dynamics, Vol. 19, No. 4, pp. 435460, 2014. ##[2] Alligood, K., Sauer, T. and Yorke, J., Chaos. An Introduction to Dynamical Systems, Springer, NewYork, 1996. ##[3] Anchev, A., “On the Stability of Permanent Rotations of a Heavy Gyrostat”, J. of Appl. Math. and Mech., Vol. 26, No. 1, pp. 2228, 1962. ##[4] Andronov, A., Witt, A. and Chaikin, S., Theory of Oscillations, AddisonWesley, Reading, MA, 1966. ##[5] Аndronov, А. And Pontryagin, L., “Systemes grossieres”, DAN USSR, Vol. 14, pp. 247251, 1937. ##[6] Аrnold, V., Ordinary Differential Equations. Fourth Ed., Igevsk, 2000 (in Russian). ##[7] Arnold, V., Afraimovich, V., Iliaschenko, Yu. and Shilnikov, L., Bifurcation Theory. Nauka, Moscow, 1986 (in Russian). ##[8] Arrowsmith, D. and Place, C., Dynamical Systems: differential equations, maps and chaotic behaviour, Chapman & Hall, London, 1992. ##[9] Aslanov, V. and Doroshin, A., “Chaotic Dynamics of an unbalanced Gyrostat”, J. of Applied Mathematics and Mechanics, Vol. 74, pp. 524535, 2010. ##[10] Bachvarov, S., Mechanics. Part I, Stand. Print, Sofia, 2001 (in Bulgarian). ##[11] Banhi, V. and Savin, A., “Molecular Gyroscopes and Biological Affects of Weak Extremely Lowfrequency Magnetic Fields”, Physical Review E, Vol. 65, pp. 051912, 2002. ##[12] Bardin, B., “On the Orbital Stability of Pendulum Like Motions of Rigid Body in the BobylevSteklov Case”, Regular and Chaotic Dynamics, Vol. 15, No. 6, pp. 704716, 2010. ##[13] Barreira, L. and Valls, C., Dynamical Systems: An Introduction, Springer, London, 2013. ##[14] Basak, I., “Explicit Solution of the ZhukovskiVolterra Gyrostat”, Regular and Chaotic Dynamics, Vol. 14, No. 2, pp. 223236, 2009. ##[15] Bautin, N. and Leontovich, E., Methods and Approaches for Qualitative Investigation of Two Dimensional Dynamical Systems, Nauka, Moscow, 1989 (in Russian). ##[16] Bloch, A., Nonholonomic Mechanics and Control. Springer, New York, 2003. ##[17] Bolotin, S., “The Hill Determinant of a Periodic Trajectories”, Mathematika, Mechanika, Vol. 3, pp. 3034, 1988. ##[18] Borisov, A. and Mamaev, I., Dynamics of Rigid Body, MoscowIjevsk, 2001. ##[19] Borisov, A., Kilin, A. and Mamaev, I., “Absolute and Relative Choreographies in Rigid Body Dynamics”, Regular and Chaotic Dynamics, Vol. 13, No. 3, pp. 204220, 2008. ##[20] Borisov, A., Kilin, A. and Mamaev, I., “Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point”, Regular and Chaotic Dynamics, Vol. 13, No. 3, pp. 221233, 2008. ##[21] Brizard, A., An Introduction to Lagrangian Mechanics, World Scientific, Singapore, 2008. ##[22] Bulatovic, R., “The Stability of Linear Potential Gyroscopic Systems in Cases when the Potential Energy has a Maximum”, Prikl. Mat. Mekh., Vol. 61, No. 3, pp. 385389, 1997. ##[23] Burra, L., Chaotic Dynamics in Nonlinear Theory, Springer, New York, 2014. ##[24] Butenin, N., Neimark, Yu. and Fufaev, N., Introduction in Theory of Nonlinear Oscillations, Nauka, Мoscow 1976 (in Russian). ##[25] Carlson, J. and Doyle, J., “Highly Optimized Tolerance: robustness and design in complex systems”, Phys. Rev. Lett., Vol. 84, pp. 25292532, 2000. ##[26] Chang, D. and Marsden, J., Gyroscopic forces and collision avoidance with convex obstacles. In: New Trends in Nonlinear Dynamics and Control and Their Application, Springer, Berlin, Vol. 295, pp. 145159, 2003. ##[27] Chen, H., “Chaos and Chaos Synchronization of a Symmetric Gyro with Linear Plus Cubic Damping”, J. of Sound and Vibrations, Vol. 255, No. 4, pp. 719740, 2002. ##[28] Chen, H. and Ge, Zh., “Bifurcations and Chaos of a Twodegree of Freedom Dissipative Gyroscope”, Chaos, Solitons and Fractals, Vol. 24, pp. 125136, 2005. ##[29] Chetayev, N., The Stability of Motion, Pergamon Press, New York 1961. ##[30] Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, McGrawHill Inc, London, 1987. ##[31] Coutinho, M.: Guide to Dynamics Simulations of Rigid Bodies and Particle Systems. Springer, London (2013) ##[32] Deriglazov, A., Classical Mechanics: Hamiltonian and Lagrangian formalism, Springer, New York, 2010. ##[33] Desloge, E., Classical Mechanics, Vol. I, John Wiley & Sons, New York, 1982. ##[34] Elipe, A., Arribas, M. and Riaguas, A., “Complete Analysis of Bifurcations in the Axial Gyrostat Problem”, J. Phys. A: Math. Gen.,Vol. 30, pp. 587601, 1997. ##[35] Elmandouh, A., “New Integrable Problems in the Dynamics of Particle and Rigid Body”, Acta Mech., Vol. 226, No. 11, pp. 37493762, 2015. ##[36] Elmandouh, A., “New Integrable Problems in Rigid Body Dynamics with Quartic Integrals”, Acta Mech., Vol. 226, No. 8, pp. 24612472, 2015. ##[37] Eueliri, L., “Theoria Motus Corporum Solidorum seu Rigidorum”. Griefswald, A. F. Rose, 1785; or Eueleri, L., Opera Omnia Ser. 2 Teubner, 3, 1948 and 4, 1950. ##[38] Fan, Y. and Chay T., “Crisis and topological entropy”, Physical Review E, Vol. 51, pp. 10121019, 1995. ##[39] Farivar, F., Shoorehdeli, M., Nekoui, M. and Teshnehlab, M., “Chaos Control and Generalized Projective Synchronization of Heavy Symmetric Chaotic Gyroscope Systems via Gaussian Radial Basis Adaptive Variable Structure Control”, Chaos, Solitons and Fractals, Vol. 45, pp. 8097, 2012. ##[40] Ge, Z., Chen, H. and Chen, H., “The Regular and Chaotic Motions of Symmetric Heavy Gyroscope with Harmonic Excitation”, J. of Sound and Vibration, Vol. 198, No. 2, pp. 131147, 1996. ##[41] Gluhovsky, A., “Nonlinear Systems that are Superpositions of Gyrostats”, Sov. Phys. Dokl., Vol. 27, pp. 823825, 1982. ##[42] Gluhovsky, A., “The structure of Energy Conserving Loworder Models”, Physics of Fluids, Vol. 11, No. 2, pp. 334343, 1999. ##[43] Gonchenko, S. and Ovsyannikov, I., “On Bifurcations of Threedimensional Diffeomorphisms with a Nontransversal Heteroclinic Cycle Containing Saddlefoci”, Nonlinear Dynamics, Vol. 6, No. 1, pp. 6177, 2010 (in Russian). ##[44] Gradwell, G., Khonsari, M. and Ram, Y., “Stability Boundaries of a Conservative Gyroscopic System”, J. of Applied Mechanics, Vol. 70, pp. 561567, 2003. ##[45] Gray, G., Kammer, D., Dobson, I. and Miller, A., “Heteroclinic Bifurcations in Rigid Bodies Containing Internally Moving Parts and a Viscous Damper”, ASME Applied Mechanics, Vol. 66, 720728, 1999. ##[46] Guckenheimer, J. and Holmes, Ph., Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields, Springer, New York, 1992. ##[47] Han, Zh. and Wang, S., “Multiple Solutions for Nonlinear Systems with Gyroscopic Terms”, Nonlinear Analysis, Vol. 75, pp. 57565764, 2012. ##[48] Hartman, Ph., Ordinary Differential Equations, John Willey & Sons, London, 1964. ##[49] Hauger, W., “Stability of a Gyroscopic Nonconservative System”, J. of Applied Mechanics, Vol. 42, pp. 739740, 1975. ##[50] Hill G., “On the Part of the Motion of the Lunar Perigee which is a Function of the Mean Motion of the Sun and Moon”, Acta Math., Vol. 8, pp. 136, 1886. ##[51] Idowu, B., Vincent, U. and Njah, A., “Synchronization of Chaos in Nonidentical Parametrically Excited Systems”, Chaos, Solitons and Fractals, Vol. 39, pp. 23222331, 2009. ##[52] Kirillov, O., “Gyroscopic Stabilization in the Presence of Nonconservative Forces”, Dokl. Acad. Nauk, Vol. 416, pp. 451456, 2007. ##[53] Kliem, W. and Pommer, Ch., “Indefinite Damping in Mechanical Systems and Gyroscopic Stabilization”, Z. Angew. Math. Phys. (ZAMP), Vol. 60, pp. 785795, 2009. ##[54] Kowalevski, S., “Sur le Probleme de la Rotation d’un corps Solide Antor d’un Point Fixe”, Acta Math., Vol. 12, pp. 177232, 1889. ##[55] Kozlov, V., “Gyroscopic Stabilization and Parametric Resonance”, J. of Applied Maths and Mechs., Vol. 65, No. 5, pp.715721, 2001. ##[56] Krechetnikov, R. and Marsden J., “On Destabilizing Effects of Two Fundamental Nonconservative Forces”, Physica D, Vol. 214, pp. 2532, 2006. ##[57] Krechetnikov, R. and Marsden J., “Dissipation Induced Instabilities in Finite Dimensions”, Reviews of Modern Physics, Vol. 79, pp. 519553, 2007. ##[58] Krechetnikov, R. and Marsden J.” Dissipation Induced Instability Phenomena in Infinite Dimensional Systems”, Arch. Rational Mech. Anal., Vol. 194, pp. 611668, 2009. ##[59] Kuznetsov, Yu., Elements of Applied Bifurcation Theory, 2 ed., Springer, New York, 1998. ##[60] Lakhadanov, V., “On Stabilization of Potential Systems”, PMM, Vol. 39, No. 1, pp. 4550, 1975. ##[61] Lancaster, P., “Stability of Linear Gyroscopic Systems: A review”, Linear Algebra and its Applications, Vol. 439, pp. 686706, 2013. ##[62] Landa, P., Nonlinear Oscillations and Waves in Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 1996. ##[63] Leimanis, E., The General Problem of Motion of Coupled Rigid Bodies about a Fixed Point, Springer, Berlin, 1965. ##[64] Leipnik, R. and Newton, T., “Double Strange Attractor in Rigid Body Motion with Linear Feedback Control”, Phys. Lett., Vol. 86, No. 2, pp. 6367, 1981. ##[65] Leonhardt, U. and Piwnicki, P., “Ultrahigh Sensitivity of Slowlight Gyroscope”, Physical Review A, Vol. 62, pp. 055801, 2000. ##[66] Leontovich, Е. and Мayer, A., “Оn Trajectories which Definite the Qualitative Separation of Sphere of Trajectories”, DAN USSR, Vol. 14, pp. 251254, 1937. ##[67] Liu, Y. and Rimrott, F., “Global Motion of a Dissipative Asymmetric Gyrostat”, Archive of Applied Mechanics Vol. 62, pp. 329337, 1992. ##[68] Luo, J., Nie, Y., Zhang, Y. and Zhou, Z., “Null Result for Violation of the Equivalence Principle with Freefall Rotating Gyroscopes”, Physical Review D, Vol. 65, pp. 042005, 2002. ##[69] Markeev, A., “The Dynamics of a Rigid Body Colliding with a Rigid Surface”, Regular and Chaotic Dynamics, Vol. 13, No. 2, pp. 96129, 2008. ##[70] Marsden, J., Lectures on Mechanics, CambridgeUniversity Press, London Math. Society Lecture Notes Series 174., 1992. ##[71] Merkin D., Gyroscopic Systems, Nauka, Moscow, 1924 (in Russian). ##[72] Metelicin, I., Theory of Gyroscope. Theory of stability, Nauka, Moscow, 1977 (in Russian). ##[73] Morozov, V. and Kalenova, V., “Stability of Nonautonomous Mechanical Systems with Gyroscopic and Dissipative Forces”, XII Russian Control Conference, Moscow 1619 June 2014, pp. 18881894, 2014. ##[74] Neimark, Yu., Method of Points Map in Theory of Nonlinear Oscillations, Nauka, Moscow, 1972 (in Russian). ##[75] Neimark, Yu. and Landa, P., Stochastic and Chaotic Oscillations, Kluwer Acad. Publishers, Singapore, 1992. ##[76] Nikolov, S., and Nedkova, N., “Gyrostat Model Regular and Chaotic Behaviour”, J. of Theoretical and Applied Mechanics, Vol. 45, No 4, pp. 1530, 2015. ##[77] Panchev, S.: Theory of Chaos. Sofia, Acad. Publ.“prof. Marin Drinov”, Sofia, 2001 (in Bulgarian). ##[78] Pakniyat, A., Salarieh, H. and Alasty, A., “Stability Analysis of a New Class of MEMS Gyroscopes with Parametric Resonance”, Acta Mech. Vol. 223, No. 6, pp. 11691185, 2012. ##[79] Pali, J. and Smale, S., “Structure Stability Theorems”, Mathematics, Vol. 13, No. 2, pp. 145155, 1969. ##[80] Poston, T. and Stewart, I., Catastrophe Theory and its Applications, PITMAN, London, 1978. ##[81] Rumiantsev, V., “On the Stability of Motion of Gyrostat”, J. Appl. Math. Mech., Vol. 25, pp. 919, 1961. ##[82] Ruelle, D. and Takens, F., “On the Nature of Turbulence”, Commun. in Math. Phys., Vol. 20, pp. 167192, 1971. ##[83] Seyranian, A., Stoustrup, J. and Kliem, W., “On Gyroscopic Stabilization”, Zangew Math Phys (ZAMP), Vol. 46, pp. 255267, 1995. ##[84] Sheu, L., Chen, H., Chen, J., Tam, L., Chen, W. and et al., Chaos in the NewtonLeipnik System with Fractional Order”, Chaos, Solitons and Fractals, Vol. 36, pp. 98103, 2008. ##[85] Shilnikov, L., “On a New Type of Bifurcation of Multidimensional Dynamical Systems”, DAN USSR, Vol. 189, No 1, pp. 5962, 1969. ##[86] Shilnikov, L., On a New Type Bifurcation of Multidimensional Dynamical Systems”, Sov. Math., Vol. 10, pp. 13681371, 1969. ##[87] Shilnikov, L., “A Contribution to the Problem of the Structure of an Extended Neighborhood of a Rough Equilibrium State of Saddlefocus Type”, Math.USSR Sbornik, Vol. 81(123), pp. 92103, 1970. ##[88] Shilnikov, L., Shilnikov, A., Turaev, D. and Chua, L., Methods of Qualitative Theory in Nonlinear Dynamics. Part II, World Scientific, Singapore, 2001. ##[89] Sonechkin, D., Stochasticity in the Model of General Circulation Atmosphere, Gidrometeoizdat, Leningrad, 1984 in Russian. ##[90] Stringari, S., “Superfluid Gyroscope with Cold Atomic Gases”, Physical Review Letters, Vol. 86, No. 21, pp. 47254728, 2001. ##[91] Thomson, W. and Tait, P., Treatise on Natural Philosophy. vol. I, part I, Cambridge University Press, 1879. ##[92] Tong, Chr. and Gluhovsky, A., “Gyrostatic Extensions of the HowardKrishnamutri Model of the Thermal Convection with Shear”, Nonlin. Processes Geophys., Vol. 15, pp. 7179, 2008. ##[93] Tong, Chr., “Lord Kelvin’s Gyrostat and its Analogs in Physics, Including the Lorenz Model”, Am. J. Phys. Vol. 77, No.6, pp. 526537, 2009. ##[94] Tsai, N., Wu, B., “Nonlinear Dynamics and Control for Singleaxis Gyroscope Systems”, Nonlinear Dyn., Vol. 51, pp. 355364, 2008. ##[95] Uitni, H., “Peculiarities of Map in Euclidian Space”, Мathematics, Vol. 13, No. 2, pp. 105123, 1969. ##[96] Wang, Ch. and Yau, H., “Nonlinear Dynamic Analysis and Sliding Mode Control for a Gyroscope System”, Nonlinear Dyn., Vol. 66, pp. 5365, 2011. ##[97] Will, C., “Covariant Calculation of General Relativistic Effects in an Orbiting Gyroscope Experiment”, Physical Review D, Vol. 67, pp. 062003, 2003. ##[98] Woodman, O.: An Introduction to Internal Navigation. University of Cambridge Press (2007) ##[99] Zhuravlev, V., Grounding in Theoretical Mechanics, Fizmatlit, Moscow, 2001 (in Russian).##]
Buckling Analysis of Cantilever Nanoactuators Immersed in an Electrolyte: A Close Form Solution Using DuanRach Modified Adomian Decomposition Method
2
2
A new modified Adomian Decomposition Method (ADM) was utilized to obtain an analytical solution for the buckling of the nanocantilever actuators immersed in liquid electrolytes. The nanoactuators in electrolytes are subject to different nonlinear forces including ionic concentration, van der Waals, external voltage and electrochemical forces. The Duan–Rach modified Adomian decomposition method was used to obtain a full explicate solution for the buckling of nanoactuators free of any undetermined coefficients. The results were compared with those of Wazwas ADM and of a finite element method available in the literature and excellent agreement was found between them.
1

207
219


Mohammad
Ghalambaz
Department of Mechanical Engineering, Dezful Branch, Islamic Azad University, Dezful, Iran.
Department of Mechanical Engineering, Dezful
Iran
m.ghalambaz@gmail.com


Mehdi
Ghalambaz
Department of Mechanical Engineering, Dezful Branch, Islamic Azad University, Dezful, Iran.
Department of Mechanical Engineering, Dezful
Iran
ghalambaz.mehdi@gmail.com


Mohammad
Edalatifar
Department of Electrical Engineering, Dezful Branch, Islamic Azad University, Dezful, Iran.
Department of Electrical Engineering, Dezful
Iran
m.edalatifar@gmail.com
Nanoactuator
Duan and Rach ADM
Analytic Solution
Electrolyte
[[1] Martin, O. Gouttenoire, V. Villard, P. Arcamone, J. Petitjean, M. Billiot, G., G., Philippe, J., Puget, P., Andreucci, P., Ricoul, F. and Dupré, C., “Modeling and design of a fully integrated gas analyzer using a μGC and NEMS sensors”, Sensors and Actuators B: Chemical, Vol. 194, pp.2208, 2014. ##[2] Jóźwiak, G. Kopiec, D. Zawierucha, P. Gotszalk, T. Janus, P. Grabiec, P., and , and Rangelow, I. W., “The spring constant calibration of the piezoresistive cantilever based biosensor”, Sensors and Actuators B: Chemical, Vol. 170, pp. 201206, 2012. ##[3] Ekinci, K. L. and Roukes, M. L.,” Nanoelectromechanical systems”., Review of Scientiﬁc Instruments, Vol. 76, No.6, 061101, 2005. ##[4] Guthy, C., Belov, M. Janzen, A. Quitoriano, N. J. Singh, A. Guthy, C., Belov, M., Janzen, A., Quitoriano, N.J., Singh, A., Wright, V.A., Finley, E., Kamins, T.I. and Evoy, S., “Largescale arrays of nanomechanical sensors for biomolecular fingerprinting. Sensors and Actuators B” Chemical, Vol. 187, pp. 111117, 2013. ##[5] Choi, W.Y., Osabe, T., and Liu, T. J. K., “Nanoelectromechanical nonvolatile memory (NEMory) cell design and scaling, Electron Devices” IEEE Transactions on, Vol. 55, No.2, pp. 34823488, 2008. ##[6] Dumas, N., Trigona, C. Pons, P. Latorre, L., and Nouet, P, “Design of smart drivers for electrostatic MEMS switches”, Sensors and Actuators A: Physical, Vol. 167, pp. 422432, 201. ##[7] Boyd, J. G., and Kim, D.,”Nanoscale electrostatic actuators in liquid electrolytes”, Journal of colloid and interface science, Vol. 301, No. 2, pp. 542548, 2006. ##[8] Noghrehabadi, A., Eslami, M., and Ghalambaz, M., “Influence of size effect and elastic boundary condition on the pullin instability of nanoscale cantilever beams immersed in liquid electrolytes”, International Journal of NonLinear Mechanics, Vol. 52, pp. 7384, 2013. ##[9] Boyd, J. G. and Lee, J., “Deflection and pullin instability of nanoscale beams in liquid electrolytes”, Journal of colloid and interface science, Vol. 356, pp. 38794, 2011. ##[10] Wazwaz, A. M., “The numerical solution of sixthorder boundary value problems by the modiﬁed decomposition method”, Applied Mathematics and Computation, Vol.118, pp. 311325, 2001. ##[11] Alam, M. K., Rahim, M. T., Avital, E. J., Islam, S., Siddiqui, A. M., & Williams, J. J. R., “Solution of the steady thin film flow of nonNewtonian fluid on vertical cylinder using Adomian Decomposition Method”, Journal of the Franklin Institute, Vol. 350, No. 4, pp. 818839, 2013. ##[12] Koochi, A., Kazemi, A. S., Tadi Beni, Y., Yekrangi, A., and Abadyan, M., “Theoretical study of the effect of Casimir attraction on the pullin behavior of beamtype NEMS using modified Adomian method. Physica”, E: Lowdimensional Systems and Nanostructures, Vol. 43, No. 2, pp. 625632, 2010. ##[13] Koochi, A. L. I., HosseiniToudeshky, H. Ovesy, H. R., and Abadyan, M.,” Modeling the Influence of Surface Effect on Instability of NanoCantilever in Presence of Van Der Waals Force”, International Journal of Structural Stability and Dynamics, Vol. 13, pp. 1250072, 2013. ##[14] Soroush, R. Koochi, A. L. I. Kazemi, A. S., and Abadyan, M., “Modeling the Effect of Van Der Waals Attraction on the Instability of Electrostatic Cantilever and DoublySupported NanoBeams Using Modified Adomian Method”, International Journal of Structural Stability and Dynamics, Vol. 12, 1250036, 2012. ##[15] Kuang, J. H., and Chen, C. J., “Adomian decomposition method used for solving nonlinear pullin behavior in electrostatic microactuators”, Mathematical and Computer Modelling, Vol. 41, pp. 14791491, 2005. ##[16] Koochi, A. Kazemi, A. S. Noghrehabadi, A. Yekrangi, A., and Abadyan, M., “New approach to model the buckling and stable length of multi walled carbon nanotube probes near graphite sheets”, Materials & Design, Vol. 32, 29492955, 2011. ##[17] Noghrehabadi, A., Ghalambaz, M., and Ghanbarzadeh, A.,” A new approach to the electrostatic pullin instability of nanocantilever actuators using the ADM–Padé technique”, Computers & Mathematics with Applications, Vol. 64, pp. 28062815, 2012. ##[18] Duan, J. S. and Rach, R., “A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations”, Applied Mathematics and Computation, Vol. 218, pp. 409041118, 2011. ##[19] Duan, J. S., Rach, R., Wazwaz, A. M., Chaolu, T., and Wang, Z.,” A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions”, Applied Mathematical Modelling, Vol. 37, pp. 86878708, 2013. ##[20] Adomian, G. and Rach, R., “Inversion of nonlinear stochastic operators”, J Math Anal Appl, Vol. 91, pp. 39–46, 1983. ##[21] Duan, J. S., Rach, R., and Wang, Z.,” On the effective region of convergence of the decomposition series solution”, Journal of Algorithms & Computational Technology, Vol. 7, pp. 227248, 2013. ##[22] Yazdanpanahi, E., Noghrehabadi, A., and Ghalambaz, M.,” Pullin instability of electrostatic doubly clamped nano actuators: Introduction of a balanced liquid layer (BLL)”, International Journal of NonLinear Mechanics, Vol. 58, pp. 128138, 2014.##]
Analytical Solution of Linear, Quadratic and Cubic Model of PTT Fluid
2
2
An attempt is made for the first time to solve the quadratic and cubic model of magneto hydrodynamic Poiseuille flow of PhanTheinTanner (PTT). A series solution of magneto hydrodynamic (MHD) flow is developed by using homotopy perturbation method (HPM). The results are presented graphically and the effects of nondimensional parameters on the flow field are analyzed. The results reveal many interesting behaviors that warrant further study on the equations related to nonNewtonian fluid phenomena.
1

220
228


Naeem
Faraz
Shanghai University, Shanghai China
Shanghai University, Shanghai China
Iran
nfaraz_math@yahoo.com


Hou
Lei
Shanghai University, Shanghai China
Shanghai University, Shanghai China
Iran
houlei@staff.shu.edu.cn


Yasir
Khan
Hafr al Batin Saudia Arabia
Hafr al Batin Saudia Arabia
Iran
yasirmath@yahoo.com
PhanTheinTanner (PTT) model
homotopy perturbation method
Nonlinear
[[1] LM. Maria, M. Hana, N. Sarka, Global existence and uniqueness result for the diffusive peterlin viscoelastic model, Nonlin. Anal. Meth. Appl.120 (2015) 154170.##[2] Z. Ting, Global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data, Nonlin. Anal. Meth. Appl. 100 (2014) 5977. ##[3] G. Matthias, G. Dario, N. Manuel, Lptheory for a generalized nonlinear viscoelastic fluid model of differential type in various domains, Nonlin. Anal. Meth. Appl. 75 (2012) 50155026. ##[4] F. Ettwein, M. Ruzicka, B. Weber, Existence of steady solutions for micropolar electrorheological fluid flows, Nonlin. Anal. Meth. Appl. 125 (2015) 129. ##[5] F. J. SuarezGrau, Asymptotic behavior of a nonNewtonian flow in a thin domain with Navier law on a rough boundary, Nonlin. Anal. Meth. Appl. 117 (2015) 99123. ##[6] Y. Ye, Global existence and blowup of solutions for higherorder viscoelastic wave equation with a nonlinear source term, Nonlin. Anal. Meth. Appl.112 (2015) 12946. ##[7] R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of polymeric liquids, 1Fluid Mechanics second edition, John Wiley & Sons, Inc. 1987. ##[8] N. PhanThien and R. I. Tanner, A new constitutive equation derived from network theory, J. NonNewtonian Fluid Mech. 2(1977) 353–365. ##[9] L. Quinzani, R. Armstrong, R. Brown, Use of coupled birefringence and LDV studies of flow through a planar contraction to test constitutive equations for concentrated polymer solutions. J. Rheolm. 39 (1955) 1201–1228. ##[10]A. Baloch, P. Townsend, M. Webster, On vortex development in viscoelastic expansion and contraction flows. J Non Newton Fluid Mech. 65 (1996) 133–149. ##[11]J. Tichy, B. BouSaid B, (2008) The PhanThien and Tannermodel applied to thin film spherical coordinates: applications for lubrication of hip joint replacement. J Biomech Eng, 130 (2008) 021012. ##[12]A. M. Siddiqui, Q. A. Azim, A. Ashraf et al, Exact Solution for Peristaltic Flow of PTT Fluid in an Inclined Planar Channel and Axisymmetric Tube, Int.J. Nonlin. Sci. Num. Sim. 10 (2009) 7591 ##[13]L. Ferras, J. Nobrega, F. Pinho, Analytical solutions for channel flows of PhanThienTanner and Giesekus fluids under slip. J. Non Newton Fluid Mech. 171 (2012) 97–105 ##[14]P. J. Oliveira and F. T. Pinho, Analytical solution for fullydeveloped channel and pipe flow of PhanThien, Tanner fluids, J. Fluid Mech. 387 (1999) 271–280. ##[15]F. T. Pinho and P. J. Oliveira, Analysis of forced convection in pipes and channels with simplified PhanThien Tanner Fluid, Int. J. Heat Mass Transfer. 43(2000) 2273–2287. ##[16]Hou Lei, V. Nassehi, Evaluation of stress effecting flow in rubber mixing, Nonlin. Anal. Meth. Appl. 47 (2001) 18091820. ##[17]Hou Lei, Member, IAENG, D.Z. Lin, B.Wang, H.L. Li, L. Qiu,Computational Modelling on the Contact Interface with Boundarylayer Approach , Pro. Worl. Cong. Eng., I (2011) July 6 – 8, London, U.K. ##[18]Hou Lei, H. L. Li, H. Wang , L. Qiu, Stochastic Analysis in the ViscoElastic Impact Condition, Conference on Chemical Engineering and Advanced Materials (CEAM) VIRTUAL FORUM Naples 2009 ##[19]Hou Lei, J. Zhao and L. Qiu, The nonNewtonian fluid in the collision, Appl. Mech. Mat. 538 (2014) 7275. ##[20]Z. Shaoling, Hou Lei, Decoupled algorithm for solving PhanThienTanner viscoelastic fluid by finite element method, Comp. Math. App. 69 (2015) 423437. ##[21]Hou Lei, Li, Hanling, Zhang Jiajian; et al Boundarylayer eigen solutions for multifield coupled equations in the contact interface, App. Math. Mech., 31 (2010) 719732. ##[22]N. Faraz, Study of the effects of the Reynolds number on circular porous slider via variational iteration algorithmII, Comp. Math. App. 61 (2011) 19911994. ##[23]N. Faraz, Y. Khan, D. S. Shankar, Decompositiontransform method for Fractional Differential Equations, Int. J. Nonl. Sci. Num. Sim. 11 (2010) 305310. ##[24]Y. Khan, N. Faraz, S. Kumar, et al, A Coupling Method of Homotopy Perturbation and Laplace Transformation for Fractional Models, Uni. Pol. Buch. Sci. Bull.Ser. AApp. Math. Phy. 74 (2012) 5768. ##[25]N. Faraz, Hou Lei, Y. Khan, Homotopy Perturbation Method for Thin Film Flow of a Maxwell Fluid over a Shrinking/Stretching Sheet with Variable Fluid Properties International Conference On Mechanics And Control Engineering, MCE (2015) 5257. ##[26]N. Faraz, Y. Khan, Study of the Rate Type Fluid with Temperature Dependent Viscosity, Zeitschrift Fur Naturforschung Section AA Journal of Physical Sciences. 67 (2011) 460468. ##[27]Y. Khan, Q. Wu, N. Faraz; et al, Heat Transfer Analysis on the Magnetohydrodynamic Flow of a NonNewtonian Fluid in the Presence of Thermal Radiation: An Analytic Solution, Zeitschrift Fur Naturforschung Section AA Journal Of Physical Sciences. 67 (2012) 147152. ##[28]Y. Khan, N. Faraz, Y. Ahmet; et al. A Series Solution of the Long Porous Slider, Tribology Transactions. 54 (2011)187191. ##[29]F. Talay Akyildiz, K. Vajravelu, Magnetohydrodynamic flow of a viscoelastic fluid, Physics Letters A. 372 (2008) 33803384. ##]
Pullin behavior of a biomass sensor based on an electrostatically actuated cantilevered CNT with consideration of rippling effect
2
2
This paper examines the pullin behavior of a biomass sensor with a cantilevered CNT actuated electrostatically by considering rippling deformation. Although this phenomenon can remarkably change the behavior of CNT, its effect on the performance of a CNTbased mass sensor has not been investigated thus far. This investigation is based on modified EulerBernoulli beam theory and rippling effect is entered into the equations related to the cantilevered CNTbases sensor. The impact of other properties like different masses, mechanical damping and intermolecular force is studied in this paper, as well. The results reveal that rippling deformation decreases the pullin voltage and tip deflection of CNT but enhances the pullin time. Results related to the impact of other mentioned properties are presented, too. The results are compared with other pullin sensor equations in the literature and “molecular dynamics”, in both of which an excellent agreement is seen, to verify the soundness of this study.
1

229
239


Nazanin
Farjam
Shahid Chamran University of Ahvaz
Shahid Chamran University of Ahvaz
Iran
nazanin.farjam@gmail.com
Biomass sensor
Rippling Deformation
CNT
Pullin instability
[[1] Keivan Kiani, Hamed Ghaffari, Bahman Mehri, Application of elastically supported singlewalled carbon nanotubes for sensing arbitrarily attached nanoobjects, Current Applied Physics 13 (2013) 107120. ##[2] Ho Jung Hwang, Jeong Won Kang, Carbonnanotubebased nanoelectromechanical switch, Physica E: Lowdimensional Systems and Nanostructures 27 (2005) 163175. ##[3] K. Azizi, S. Majid Hashemianzadeh, Sh. Bahramifar, Density functional theory study of carbon monoxide adsorption on the inside and outside of the armchair singlewalled carbon nanotubes, Current Applied Physics 11 (2011) 776782. ##[4] Muhammad A. Hawwa, Hussain M. AlQahtani, Nonlinear oscillations of a doublewalled carbon nanotube, Computational Materials Science 48 (2010) 140143. ##[5] S.K. Georgantzinos, N.K. Anifantis, Carbon nanotubebased resonant nanomechanical sensors: a computational investigation of their behavior, Physica E: Lowdimensional Systems and Nanostructures 42 (2010) 17951801. ##[6] M.Z. Atashbar, B. Bejcek, S. Singamaneni, Carbon nanotube based biosensors. Vienna, Austria, in: IEEE Sensor Conference (October 24th27th, 2004), pp.10481105. ##[7] R.F. Gibson, E.O. Ayorinde, Y.F. Wen, Vibrations of carbon nanotubes and their composites: a review, Composites Science and Technology 67 (2007) 128. ##[8] T. Murmu, S. Pradhan, Thermomechanical vibration of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Materials Science 46 (2009) 854e859. ##[9] S. Gupta, R. Batra, Continuum structures equivalent in normal mode vibrations to single walled carbon nanotubes, Computational Materials Science 43 (2008) 715723. ##[10]K. Tonisch, V. Cimalla, F. Will, F. Weise, M. Stubenrauch, A. Albrecht, M. Hoffmann, O. Ambacher, Nanowirebased electromechanical biomimetic sensor, Physica E: Lowdimensional Systems and Nanostructures 37 (2007) 208211. ##[11]J. Zhu, Pullin instability of two opposing microcantilever arrays with different bending rigidities, International Journal of Mechanical Sciences 50 (2008) 5568. ##[12]M. Rasekh, S.E. Khadem, Pullin analysis of an electrostatically actuated nanocantilever beam with nonlinearity in curvature and inertia, International Journal of Mechanical Sciences 53 (2011) 108115. ##[13]Dequesnes, M., Tang, Z., Aluru, N.R., Static and Dynamic Analysis of Carbon NanotubeBased Switches,Journal of Engineering Materials and Technology, 126(3) (2004) 230237. ##[14]I. Mehdipour, A. ErfaniMoghadam, C. Mehdipour Application of an electrostatically actuated cantilevered carbon nanotube with an attached mass as a biomass sensor, Current Applied Physics 13 (7), 14631469. ##[15]Wang, X.Y., Wang, X., Numerical simulation for bending modulus of carbon nanotubes and some explanations for experiment, Composites: Part B, 35 (2004) 79–86. ##[16]Payam Soltani, D. D. Ganji, I. Mehdipour1 and A. Farshidianfar, Nonlinear vibration and rippling instability for embedded carbon nanotubes, Journal of Mechanical Science and Technology 26 (4) (2012) 985992. ##[17]Koochi, A., Kazemi, A.S., Noghrehabadi, A., Yekrangi, A., Abadyan, M., New approach to model the buckling and stable length of multi walled carbon nanotube probes near graphite sheets, Materials and Design, 32 (2011) 29492955. ##[18]Hayt, W.H., Buck, J.A., Engineering electromagnetic, 6th ed. New York: McGrawHill, (2001). ##[19]Abbasnejad, B., Rezazadeh, G., Shabani, R., Stability analysis of a capacitive fgm microbeam using modified couple stress theory, Acta Mechanica Solida Sinica, 26(4) (2013) 427440.##]