No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
Dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation
I. Podlubny, Fractional Differential Equations ( Academic, London, 1998).
R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, Fractional Order Systems: Modeling and Control Applications ( World Scientific, New Jersey, 2010).
S. Das, Functional Fractional Calculus for System Identification and Controls ( Springer-Verlag, Berlin, 2008).
I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation ( Higher Education Press, Beijing, 2011).
Y. G. Yang, W. Xu, X. D. Gu et al., “ Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise,” Chaos, Solitons Fractals 77, 190–204 (2015).
L. C. Chen, T. L. Zhao, W. Li, and J. Zhao, “ Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order PIλDμ feedback controller,” Nonlinear Dyn. 83(1–2), 529–539 (2016).
X. H. Li, J. Y. Hou, and J. F. Chen, “ An analytical method for Mathieu oscillator based on method of variation of parameter,” Commun. Nonlinear Sci. Numer. Simul. 37, 326–353 (2016).
A. H. Nayfeh, Nonlinear Oscillations ( Wiley, New York, 1973).
J. A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, 2nd ed. ( Springer-Verlag, New York, 2007).
A. Y. T. Leung and S. K. Chui, “ Nonlinear vibration of coupled Duffing oscillators by an improved incremental harmonic balance method,” J. Sound Vib. 181(4), 619–633 (1995).
P. J. Ju and X. Xue, “ Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators,” Appl. Math. Modell. 38(24), 6144–6152 (2014).
M. Heydari, G. B. Loghmani, and S. M. Hosseini, “ An improved piecewise variational iteration method for solving strongly nonlinear oscillators,” Comput. Appl. Math. 34(1), 215–249 (2015).
Y. J. Shen, S. P. Yang, and X. D. Liu, “ Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method,” Int. J. Mech. Sci. 48(11), 1256–1263 (2006).
S. L. Lau and W. S. Zhang, “ Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method,” ASME J. Appl. Mech. 59, 153–160 (1992).
H. Li, J. Cao, and C. Li, “ High-order approximation to Caputo derivatives and Caputo-type advection–diffusion equations (III),” J. Comput. Appl. Math. 299, 159 (2016).
T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres, “ Fractional variational calculus with classical and combined Caputo derivatives,” Nonlinear Anal. 75(3), 1507–1515 (2012).
Y. J. Shen, S. P. Yang, H. J. Xing et al., “ Primary resonance of Duffing oscillator with fractional-order derivative,” Commun. Nonlinear Sci. Numer. Simul. 17(7), 3092–3100 (2012).
Article metrics loading...
In this paper, the computation schemes for periodic solutions of the forced fractional-order Mathieu-Duffing equation are derived based on incremental harmonic balance (IHB) method. The general forms of periodic solutions are founded by the IHB method, which could be useful to obtain the periodic solutions with higher precision. The comparisons of the approximate analytical solutions by the IHB method and numerical integration are fulfilled, and the results certify the correctness and higher precision of the solutions by the IHB method. The dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation is investigated by the IHB method. Then, the effects of the excitation frequency, fractional order, fractional coefficient, and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed. At last, the detailed results are summarized and the conclusions are made, which present some useful information to analyze and/or control the dynamical response of this kind of system.
Full text loading...
Most read this month