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http://dx.doi.org/10.12989/sss.2014.14.5.743

Period doubling of the nonlinear dynamical system of an electrostatically actuated micro-cantilever  

Chen, Y.M. (Department of Mechanics, Sun Yat-sen University)
Liu, J.K. (Department of Mechanics, Sun Yat-sen University)
Publication Information
Smart Structures and Systems / v.14, no.5, 2014 , pp. 743-763 More about this Journal
Abstract
The paper presents an investigation of the nonlinear dynamical system of an electrostatically actuated micro-cantilever by the incremental harmonic balance (IHB) method. An efficient approach is proposed to tackle the difficulty in expanding the nonlinear terms into truncated Fourier series. With the help of this approach, periodic and multi-periodic solutions are obtained by the IHB method. Numerical examples show that the IHB solutions, provided as many as harmonics are taken into account, are in excellent agreement with numerical results. In addition, an iterative algorithm is suggested to accurately determine period doubling bifurcation points. The route to chaos via period doublings starting from the period-1 or period-3 solution are analyzed according to the Floquet and the Feigenbaum theories.
Keywords
micro-cantilever; incremental harmonic balance method; Floquet theory; period doubling; chao;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 Borzi, B., Vona, M., Masi, A. et al. (2013), "Seismic demand estimation of RC frame buildings based on simplified and nonlinear dynamic analyses", Eartq. Struct., 4(2).
2 Anishchenko, V.S., Astakhov, V.V., Neiman, A.B. et al. (2002), Nonlinear dynamics of chaotic and stochastic systems, Springer-Verlag Berlin Heidelberg.
3 Ashhab, M., Salapaka, M.V., Dahleh, M. and Mezic, I. (1999), "Dynamic analysis and control of microcantilevers", Automatica, 35, 1663-1670.   DOI   ScienceOn
4 Bayat, M. and Pakar, I. (2012), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech., 43(3), 337-347.   DOI   ScienceOn
5 Chan, E.K. and Dutton, R.W. (2000), "Electrostatic micromechanical actuator with extended range of travel", J. Microelectromech. S., 9(3), 321-328.   DOI   ScienceOn
6 Chen, Y.M., Meng, G. and Liu, J.K. (2010), "An iterative method for nonlinear dynamical system of an electrostatically actuated micro-cantilver", Phys. Lett. A, 374, 3455-3459.   DOI   ScienceOn
7 De, S.K. and Aluru, N.R. (2006), "Complex nonlinear oscillations in electrostatically actuated microstructures", J. Microelectromech. S., 15(2), 355-369.   DOI   ScienceOn
8 Feigenbaum, M. (1978), "Qualitative universality for a chaos of nonlinear transformations", T. Stat. Phys., 19, 5-32.
9 Ferri, A.A. (1986), "On the equivalence of the incremental harmonic balance method and the harmonic balance-Newton Raphson method", J. Appl. Mech, - ASME, 53, 455-457.   DOI
10 Hu, S.Q. and Raman, A. (2006), "Chaos in atomic force microscopy", Phys. Rev. Lett., 96, 036107.   DOI   ScienceOn
11 Fu, Y.M. and Zhang, J. (2009), "Nonlinear static and dynamic response of an electrically actuated viscoelastic microbeam", Acta Mech. Sinca, 25(2), 211-218.   DOI   ScienceOn
12 Hassani, F.A., Payam, A.F. and Fathipour, M. (2010), "Design of a smart MEMS accelerometer using nonlinear control principles", Smart Struct. Syst., 6(1), 1-16.   DOI   ScienceOn
13 Hornstein, S. and Gottlieb, O. (2008), "Nonlinear dynamics, stability and control of the scan process in noncontacting atomic force microscopy", Nonlinear Dynam., 54, 93-122.   DOI
14 Kacem, N., Arcamone, J., Perez-Murano, F. and Hentz, S. (2010), "Dynamic range enhancement of nonlinear nanomechanical resonant cantilever for highly sensitive NEMS gas/mass sensor applications", J. Micromech. Microeng., 20(4), 1-9, 045023.
15 Lau, S.L. and Cheung, Y.K. (1981), "Amplitude incremental variational principle for nonlinear vibration of elastic systems", J. Appl. Mech. - ASME, 48(4), 959-964.   DOI
16 Lyshevski, S.E. (1997), "Nonlinear microelectromechanic systems (MEMS) analysis and design via the Lyapunov stability theory", Proceeding of the 40th IEEE Conference on Decision and Control Orlando, FL USA.
17 Leung, A.Y.T. and Fung, T.C. (1990), "Construction of chaotic regions", J. Sound Vib., 131(3), 445-455.
18 Liu, S., Davidson, A. and Liu, Q. (2003), "Simulating nonlinear dynamics and chaos in a MEMS cantilever using Poincare mapping", Proceedings of the IEEE, Transducers'03, the 12th International Conference on Solid State Sensors Actuators and Microsystems, Boston, 8-12 June.
19 Liu, S., Davidson, A. and Liu, Q. (2004), "Simulation studies on nonlinear dynamics and chaos in a MEMS cantilever control system", J. Micromech. Microeng., 14(7), 1064-1073.   DOI   ScienceOn
20 Mahmoodi, S.N. and Jalili, N. (2009), "Piezoelectrically actuated microcantilevers: An experimental nonlinear vibration analysis", Sens. Actuat. A, 150(1), 131-136.   DOI   ScienceOn
21 Najar, F., Nayfeh, A.H., Abdel-Rahman, E.M., Choura, S. and El-Borgi, S. (2010b), "Dynamics and global stability of beam-based electrostatic microactuators", J. Vib. Control, 16(5), 721-748.   DOI   ScienceOn
22 Manna, M.C., Bhattacharyya, R. and Sheikh, A.H. (2010), "Nonlinear dynamic response and its control of rubber components with piezoelectric patches/layers using finite element method", Smart Struct. Syst., 6(8), 89-903.
23 Meng, G., Zhang, W.M., Huang, H., Li, H.G. and Chen, D. (2009), "Micro-rotor dynamics for micro-electro-mechanical systems (MEMS)", Chaos Soliton. Fract., 40(2), 538-562.   DOI   ScienceOn
24 Najar, F., Nayfeh, A.H., Abdel-Rahman, E.M., Choura, S. and El-Borgi, S. (2010a), "Nonlinear analysis of MEMS electrostatic microactuators: primary and secondary resonances of the first mode", J. Vib. Control, 16(9), 13-21.
25 Price, R.H., Wood, J.E. and Jacobsen, S.C. (1989), "Modeling considerations for electrostatic forces in electrostatic microactuators", Sensor. Actuat. A, 20(1-2), 107-114.   DOI   ScienceOn
26 Nayfeh, A.H. and Younis, M.I. (2005), "Dynamics of MEMS resonators under superharmonic and subharmonic excitations," J. Micromech. Microeng., 15(10), 1840-1847.   DOI   ScienceOn
27 Nayfeh, A.H., Younis, M.I. and Abdel-Rahman, E.M. (2007), "Dynamic pull-in phenomenon in MEMS resonators", Nonlinear Dynam., 48(1-2), 153-163.   DOI
28 Passiana, A., Muralidharana, G., Mehtaa, A., Simpson, H., Ferrell, T.L. and Thundat, T. (2003), "Manipulation of microcantilever oscillations", Ultramicroscopy, 97(1-4), 391-399.   DOI   ScienceOn
29 Raghothama, A. and Narayanan, S. (1999), "Non-linear dynamics of a two-dimensional airfoil by incremental harmonic balance method", J. Sound Vib., 226(3), 493-517.   DOI   ScienceOn
30 Senturia, S.D. (1998), "Simulation and design of microsystems: a 10-year preserve", Sens. Actuat. A, 67, 1-7.   DOI   ScienceOn
31 Shen, J.H., Lin, K.C., Chen, S.H. and Sze, K.Y. (2008), "Bifurcation and route-to-chaos analyses for Mathieu-Duffing oscillator by the incremental harmonic balance method", Nonlinear Dynam., 52(4), 403-414.   DOI
32 Xu, L., Lu, M.W. and Cao, Q. (2003) "Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise nonlinearities by incremental harmonic balance method", J. Sound Vib., 264(4), 873-882.   DOI   ScienceOn
33 Towfighian, S., Hppler, G.R. and Abdel-Rhman, E.M. (2011), "Analysis of a chaotic electrostatic micro-oscillator", J. Comput. Nonlinear Dyn., 6(1), 1-10, 011001   DOI
34 Zhang, W.M., Meng, G. and Chen, D. (2007), "Stability, nonlinearity and reliability of electrostatically actuated MEMS devices", Sensors, 7, 760-796.   DOI
35 Urabe. M. (1965) "Galerkin's procedure for nonlinear periodic systems", Arch. Ration. Mech. An., 20(2), 120-152.
36 Waris, M.B. and Ishihara, T. (2012), "Dynamic response analysis of floating offshore wind turbine with different types of heave plates and mooring systems by using a fully nonlinear model", Coupled Syst. Mech., 1(3), 247-268.   DOI   ScienceOn
37 Zhang, W.M. and Meng, G. (2005), "Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS", Sens. Actuat. A, 119(2), 291-299.   DOI   ScienceOn