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http://dx.doi.org/10.12989/sss.2021.28.5.699

Jumps phenomenon elimination of a Duffing oscillator using pole placement control method  

Mahmoudi, Reza (Faculty of Civil Engineering, University of Tabriz)
Ghaffarzadeh, Hosein (Faculty of Civil Engineering, University of Tabriz)
Yang, T.Y. (Department of Civil Engineering, University of British Columbia)
Publication Information
Smart Structures and Systems / v.28, no.5, 2021 , pp. 699-709 More about this Journal
Abstract
This paper presents a numerical and analytical study in the time-frequency domain to control the bifurcation and instability in a forced Duffing oscillator by a linear state feedback control. The proposed method evolves minimizing computational expenses of analytical approaches by an approximate method to suppress the responses of the dynamical system based on pole placement theory. The instability frequency range of Duffing oscillator is identified by approximate analytical methods. Bifurcation and jump points of Duffing oscillator are identified in the frequency domain by perturbation and harmonic balance methods for average and strong nonlinearity of the system, respectively. The Caughey method is used to linearize Duffing oscillator to solve system in the state space form. A linear state feedback controller with pole placement is applied to system in the time domain. The observed controlling force is added to approximate solution equation in frequency domain which vanished bifurcation length. The results reveal that the proposed method can be beneficial in reducing dynamic responses and eliminating jump points of system with high accuracy.
Keywords
bifurcation and jump; gain coefficient; harmonic balance; linearization; perturbation; pole placement;
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1 Friswell, M.I. and Penny, J.E.T. (1994), "The accuracy of jump frequencies in series solutions of the response of a Duffing oscillator", J. Sound Vib., 169(2), 261-269. https://doi 10.1006/jsvi.1994.1018   DOI
2 Murata, A., Kume, Y. and Hashimoto, F. (1987), "Application of catastrophe theory to the forced vibration of a diaphragm air spring", J. Sound Vib., 112(1), 31-44. https://10.1016/S0022-460X(87)80091-3   DOI
3 Brennan, M.J., Kovacic, I., Carrella, A. and Waters, T.P. (2008), "On the jump-up and jump-down frequencies of the Duffing oscillator", J. Sound Vib., 318, 1250-1261. https://doi.org/10.1016/j.jsv.2008.04.032   DOI
4 Krack, M. and Gross, J. (2019), Harmonic balance for nonlinear vibration problems, Springer International Publishing. https://doi.org/10.1007/978-3-030-14023-6   DOI
5 Lei, Y., Xu, W., Shen, J. and Fang, T. (2006), "Global synchronization of two parametrically excited Systems using active control", Chaos Solit. Fract., 28, 428-436. https://doi.org/10.1016/j.chaos.2005.05.043   DOI
6 Loria, A., Panteley, E. and Nijmeijer, H. (1998), "Control of the chaotic Duffing equation with uncertainty in all parameters", IEEE Transact. Circuits Syst. I: Fundamental Theory and Applications, 45(12), 1252-1255. http://doi.org/10.1109/81.736558   DOI
7 Seemann, W. and Gausmann, R. (2001), "A note on the strong nonlinear behavior of piezoceramics excited with a weak electric field", SPIE Smart Struct. Mater., 4333, 131-140. https://doi.org/10.1117/12.432749   DOI
8 Park, J.H. (2005a), "Chaos synchronization of a chaotic system via nonlinear control", Chaos Solit. Fract., 25, 579-584. https://doi.org/10.1016/j.chaos.2004.11.038   DOI
9 Park, J.H. (2006), "Synchronization of Genesio chaotic system via back-stepping approach", Chaos Solit. Fract., 27, 1369-1375. https://doi.org/10.1016/j.chaos.2005.05.001   DOI
10 Peleg, K. (1979), "Frequency response of non-linear single degree-of-freedom systems", Int. J. Mech. Sci., 21, 75-84. https://doi.org/10.1016/S0022-460X(85)80150-4   DOI
11 Storer, D.M. and Tomlinson, G.R. (1993), "Recent developments in the measurement and interpretation of higherorder transfer functions from nonlinear structures", Mech. Syst. Signal Process., 7(2), 173-189. https://doi.org/10.1006/mssp.1993.1006   DOI
12 Ucar, A., Lonngren, K.E. and Bai, E.W. (2006), "Synchronization of the unified chaotic systems via active control", Chaos Solit. Fract., 27, 1292-1297. https://doi.org/10.1016/j.chaos.2005.04.104   DOI
13 Verhulst, F. (1996), Nonlinear Differential Equations and Dynamical Systems, Springer, New York, USA.
14 Wagg, D. and Neild, S. (2010), Nonlinear Vibration with Control, Springer, New York, USA.
15 Newland, D.E. (1993), Introduction to Random Vibrations, Spectral and Wavelet Analysis, New York: Longman.
16 Mahmoud, G.M., Aly, S.A. and Farghaly, A.A. (2007), "On chaos synchronization of a complex two coupled dynamos system", Chaos Solit. Fract., 33, 178-187. https://doi.org/10.1016/j.chaos.2006.01.036   DOI
17 Mahmoudi, R., Ghaffarzadeh, H., Ahani, E. and Katebi, J. (2019), "Sliding mode control of linear structures and a Duffing system using active tendons", Proceedings of the Institution of Civil Engineers - Eng. Computat. Mech., 172(3), 106-117. https://doi.org/10.1680/jencm.18.00033   DOI
18 Nayfeh, A.H., Mook, D.T. and Holmes, P. (1980), Nonlinear Oscillations, Wiley, New York, USA.
19 Ott, E., Grebogi, C. and Yorke, J.A. (1990), "Controlling chaos", Phys. Rev. Lett., 64(11), 1196-1199. https://doi.org/10.1103/PhysRevLett.64.1196   DOI
20 Wang, Y., Guan, Z.H. and Wang, H.O. (2003), "Feedback and adaptive control for the synchronization of Chen system via a single variable", Phys. Lett. A, 312, 34-40. https://doi.org/10.1016/S0375-9601(03)00573-5   DOI
21 Worden, K. (1996), "On jump frequencies in the response of a Duffing oscillator", J. Sound Vib., 198(4), 522-525. https://doi.org/10.1006/jsvi.1996.0586   DOI
22 Worden, K. and Tomlinson, G.R. (2001), Nonlinearity in Structural Dynamics, Detection, Identification and Modelling, University of Sheffield, UK.
23 Wu, J., Chen, W., Yang, F., Li, J. and Zhu, Q. (2015), "Global adaptive neural control for strict-feedback time- delay systems with predefined output accuracy", Inform. Sci., 301, 27-43. https://doi.org/10.1016/j.ins.2014.12.039   DOI
24 Beltran-Carbajal, F. and Silva-Navarro, G. (2014), "Active vibration control in Duffing mechanical systems using dynamic vibration absorbers", J. Sound Vib., 333, 3019-3030. https://doi.org/10.1016/j.jsv.2014.03.002   DOI
25 Contreras-Lopez, J., Ornelas-Tellez, F. and Espinosa-Juarez, E. (2019), "Nonlinear optimal control for reducing vibrations in civil structures using smart devices", Smart Struct. Syst, Int. J., 23(3), 307-318. http://doi.org/10.12989/sss.2019.23.3.307   DOI
26 Baghaei, K., Ghaffarzadeh, H., Hadigheh, A. and Dias-da-Costa, D. (2019), "Chattering-free sliding mode control with a fuzzy model for structural applications", Struct. Eng. Mech., Int. J., 69(3), 307-315. http://doi.org/10.12989/sem.2019.69.3.307   DOI
27 Chen, X. and Liu, C. (2010), "Passive control on a unified chaotic system", Nonlinear Anal.: Real World Applicat., 11, 683-687. https://doi.org/10.1016/j.nonrwa.2009.01.014   DOI
28 Chen, S. and Lu, J. (2002), "Synchronization of an uncertain chaotic system via adaptive control", Chaos Solit. Fract., 14, 643-647. https://doi.org/10.1016/S0960-0779(02)00006-1   DOI
29 Dinca, F. and Teodosiu, C. (1973), "Nonlinear and Random Vibrations", SIAM Review, 17(3), 578.   DOI
30 Luo, X.S., Zhang, B. and Qin, Y.H. (2010), "Controlling chaos in space-clamped FitzHugh-Nagumo neuron by adaptive passive method", Nonlinear Anal.: Real World Applicat., 11, 1752-1759. https://doi.org/10.1016/j.nonrwa.2009.03.029   DOI
31 Ghandchi-Tehrani, M., Wilmshurst, L.I. and Elliott, S.J. (2015), "Bifurcation control of a Duffing oscillator using pole placement", J. Vib. Control, 21(14), 2838-2851. https://doi.org/10.1177/1077546313517586   DOI
32 Agiza, H.N. and Yassen, M.T. (2001), "Synchronization of Rossler and Chen chaotic dynamical systems using active control", 278(4), 191-197. https://doi.org/10.1016/S0375-9601(00)00777-5   DOI
33 Park, J.H. (2005b), "On synchronization of unified chaotic systems via nonlinear control", Chaos Solit. Fract., 25, 699-704. https://doi.org/10.1016/j.chaos.2004.11.031   DOI
34 Kovacic, I. and Brennan, M.J. (2011), The Duffing equation: nonlinear oscillators and their behaviour, John Wiley, London, UK.
35 Chang, R.J. (2017), "Extension of nonlinear stochastic solution to include sinusoidal excitation illustrated by Duffing oscillator", J. Computat. Nonlinear Dyn., 12(5), 051030. https://doi.org/10.1115/1.4037105   DOI
36 Efimov, D. and Perruquetti, W. (2016), "On condition of oscillations and multi-homogeneity", Math. Control Signals Syst., 28(1), 1-37. https://doi.org/10.1007/s00498-015-0157-y   DOI
37 Elabbasy, E.M., Agiza, H.N. and El-Dessoky, M.M. (2004), "Synchronization of modified Chen system", Int. J. Bifurc. Chaos, 14, 3969-3979. https://doi10.1142/S0218127404011740   DOI
38 Glendinning, P. (1994), Stability, Instability and Chaos: an Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, Cambridge, UK.
39 Jezequel, L. and Lamarque, C.H. (1991), "Analysis of nonlinear dynamic systems by the normal form theory", J. Sound Vib., 149(3), 429-459. https://doi.org/10.1016/0022-460X(91)90446-Q   DOI