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Research Papers

Resonance Oscillation of Third-Order Forced van der Pol System With Fractional-Order Derivative

[+] Author and Article Information
Nguyen Van Khang

Department of Applied Mechanics,
Hanoi University of Science and Technology,
Hanoi 100000, Vietnam
e-mail: khang.nguyenvan2@hust.edu.vn

Bui Thi Thuy

Department of Mechanics,
Hanoi University of Mining and Geology,
Hanoi 100000, Vietnam
e-mail: thuybt167ncs@gmail.com

Truong Quoc Chien

Department of Applied Mechanics,
Hanoi University of Science and Technology,
Hanoi 100000, Vietnam
e-mail: chienams@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 13, 2015; final manuscript received April 27, 2016; published online May 24, 2016. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 11(4), 041030 (May 24, 2016) (5 pages) Paper No: CND-15-1371; doi: 10.1115/1.4033555 History: Received November 13, 2015; Revised April 27, 2016

This study aims to investigate the harmonic resonance of third-order forced van der Pol oscillator with fractional-order derivative using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the harmonic solution is then obtained by means of Lyapunov theory. A comparison between the traditional integer-order of forced van der Pol oscillator and the considered fractional-order one follows the numerical simulation. Finally, the numerical results are analyzed to show the influences of the parameters in the fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability.

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References

Miller, K. S. , and Ross, B. , 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Oldham, K. B. , and Spanier, J. , 1974, The Fractional Calculus, Academic Press, Boston, MA.
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, London.
Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. , 1993, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, The Netherlands.
Baleanu, D. , Machado, J. A. T. , and Luo, A. C. J. , 2012, Fractional Dynamics and Control, Springer, New York.
Baleanu, D. , Diethelm, K. , Scalas, E. , and Trujillo, J. J. , 2012, Fractional Calculus Models and Numerical Methods, World Scientific, Singapore.
Bagley, R. L. , and Torvik, P. J. , 1983, “ A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol., 27(3), pp. 201–210. [CrossRef]
Bagley, R. L. , and Torvik, P. J. , 1985, “ Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures,” AIAA J., 23(6), pp. 918–925. [CrossRef]
Rossikin, Y. A. , and Shitikova, M. V. , 1997, “ Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids,” ASME Appl. Mech. Rev., 50(1), pp. 15–57. [CrossRef]
Zhang, W. , and Shimizu, N. , 1998, “ Numerical Algorithm for Dynamic Problems Involving Fractional Operator,” Int. J. JSME Ser. C, 41(3), pp. 364–370. [CrossRef]
Shimizu, N. , and Zhang, W. , 1999, “ Fractional Calculus Approach to Dynamic Problems of Viscoelastic Materials,” Int. J. JSME Ser. C, 42(4), pp. 825–837. [CrossRef]
Fukunaga, M. , Shimizu, N. , and Nasuno, H. , 2009, “ A Nonlinear Fractional Derivative Model of Impulse Motion for Viscoelastic Materials,” Phys. Scr., 136, p. 01410.
Fukunaga, M. , and Shimizu, N. , 2011, “ Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law,” ASME J. Comput. Nonlinear Dyn., 6(2), p. 021005. [CrossRef]
Fukunaga, M. , and Shimizu, N. , 2013, “ Comparison of Fractional Derivative Models for Finite Deformation With Experiments of Impulse Response,” J. Vib. Control, 20(7), pp. 1033–1041. [CrossRef]
Khang, N. V. , and Chien, T. Q. , 2016, “ Subharmonic Resonance of Duffing Oscillator With Fractional-Order Derivative,” ASME J. Comput. Nonlinear Dyn., 11(5), p. 051018. [CrossRef]
Wahi, P. , and Chatterjee, A. , 2004, “ Averaging Oscillations With Small Fractional Damping and Delayed Terms,” Nonlinear Dyn., 38, pp. 3–22. [CrossRef]
Nishimoto, K. , 1989, “ Nishimoto's Fractional Calculus of Elementary Functions,” International Conference of Fractional Calculus and Its Applications, Nihon University, Tokyo, Japan, pp. 112–122.
Tseng, C.-C. , Pei, S.-C. , and Hsia, S.-C. , 2000, “ Computation of Fractional Derivatives Using Fourier Transform and Digital FIR Differentiator,” Signal Process., 80(1), pp. 151–159. [CrossRef]
Munkhammar, J. D. , 2004, “ Rieman–Liouville Fractional Derivatives and Taylor–Rieman Series,” Project Report, Uppsala University, Uppsala, Sweden, Report No. 2004:7.
Attari, M. , Haeri, M. , and Tavazoei, M. S. , 2010, “ Analysis of a Fractional Order van der Pol-Like Oscilattor Via Describing Function Method,” Nonlinear Dyn., 61, pp. 265–274. [CrossRef]
Barbosa, R. S. , Tenreiro Machado, J. A. , Vinagre, B. M. , and Calderón, A. J. , 2007, “ Analysis of the van der Pol Oscillator Containing Derivatives of Fractional Order,” J. Vib. Control, 13(9–10), pp. 1291–1301. [CrossRef]
Chen, J.-H. , and Chen, W.-C. , 2008, “ Chaotic Dynamics of the Fractionally Damped van der Pol Equation,” Chaos Solitons Fractals, 35(1), pp. 188–198. [CrossRef]
Ge, Z.-M. , and Hsu, M.-Y. , 2007, “ Chaos in a Generalized van der Pol System and in Its Fractional Order System,” Chaos Solitons Fractals, 33(5), pp. 1711–1745. [CrossRef]
Ge, Z.-M. , and Hsu, M.-Y. , 2008, “ Chaos Excited Chaos Synchronizations of Integral and Fractional Order Generalized van der Pol Systems,” Chaos Solitons Fractals, 36(3), pp. 592–604. [CrossRef]
Ge, Z.-M. , and Zhang, A.-R. , 2007, “ Chaos in a Modified van der Pol System and in Its Fractional Order Systems,” Chaos Solitons Fractals, 32(5), pp. 1791–1822. [CrossRef]
Tavazoei, M. S. , Heari, M. , Attari, M. , Bolouki, S. , and Siami, M. , 2009, “ More Details on Analysis of Fractional-Order van der Pol Oscillator,” J. Vib. Control, 15(6), pp. 803–819. [CrossRef]
Carla, M. A. P. , and Tenreira Machado, J. A. , 2011, “ Complex-Order van der Pol Oscillator,” Nonlinear Dyn., 65(3), pp. 247–254. [CrossRef]
Shen, Y. , Wei, P. , Sui, C. , and Yang, S. , 2014, “ Subharmonic Resonance of van der Pol Oscillator With Fractional—Order Derivative,” Math. Probl. Eng., 2014, p. 738087.
Wei, P. , Shen, Y.-J. , and Yang, S.-P. , 2014, “ Super-Harmonic Resonance of Fractional-Order van der Pol Oscillator (in Chinese),” Acta Phys. Sin., 63(1), p. 010503.
Mitropolskii, I. A. , and Dao, N. V. , 1997, Applied Asymptotic Methods in Nonlinear Oscillations, Kluwer Academic Publisher, Dordrecht, The Netherlands.
Dao, N. V. , 1998, Stability of Dynamic Systems With Examples and Solved Problems, VNU Publishing House, Hanoi, Vietnam.
Dao, N. V. , 1979, Nonlinear Oscillations of Higher Order Systems, NCSR Vietnam, Hanoi, Vietnam.
Dao, N. V. , 1979, “ Nonlinear Oscillation of Third Order Systems—Part 1: Autonomous Systems,” J. Tech. Phys., 20(4), pp. 511–519.
Dao, N. V. , 1980, “ Nonlinear Oscillation of Third Order Systems—Part 2: Non-Autonomous Systems,” J. Tech. Phys., 21(1), pp. 125–134.
Dao, N. V. , 1980, “ Nonlinear Oscillation of Third Order Systems—Part 3: Parametric Systems,” J. Tech. Phys., 21(2), pp. 253–265.
Golmankhaneh, K. A. , Arefi, R. , and Baleanu, D. , 2015, “ Synchronization in a Nonidential Fractional Order of a Proposed Modified System,” J. Vib. Control, 21(6), pp. 1154–1161. [CrossRef]
Sanders, J. A. , and Verhulst, F. , 1985, Averaging Methods in Nonlinear Dynamical Systems, Springer, New York.

Figures

Grahic Jump Location
Fig. 1

The amplitude–frequency curve, where δp = 0

Grahic Jump Location
Fig. 2

The amplitude–frequency curve, where δp = 1 and p = 0.25

Grahic Jump Location
Fig. 3

The amplitude–frequency curve, where δp = 1 and p = 0.5

Grahic Jump Location
Fig. 4

The amplitude–frequency curve, where δp = 1 and p = 0.75

Grahic Jump Location
Fig. 5

The amplitude–frequency curve, where δp = 1 and p = 0.5 (the circlets denote the solution by numerical integration)

Grahic Jump Location
Fig. 6

The amplitude–frequency curves corresponding to δp = 1 and different values of p

Grahic Jump Location
Fig. 7

The amplitude–frequency curves corresponding to p = 0.5 and different values of δp

Grahic Jump Location
Fig. 8

The amplitude–frequency curves corresponding to different values of E

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