Abstract
In this paper the primary resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied analytically and numerically. At first the approximately analytical solution is obtained by the averaging method, and it is found that the fractional-order derivative could affect the dynamical properties of VDP oscillator, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude–frequency equation for steady-state solution is established, and the corresponding stability condition is also presented based on Lyapunov theory. Then, the comparisons of several different amplitude–frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two fractional parameters, i.e., the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated for some typical excitation amplitudes, which are different from the traditional integer-order VDP oscillator.
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References
Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press, Beijing (2011)
Podlubny, I.: Fractional Differential Equations. Academic, London (1998)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008)
Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D.Y., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)
Caponetto, R., Dongola, G., Fortuna, L., Petras, I.: Fractional Order Systems: Modeling and Control Applications. World Scientific, New Jersey (2010)
Rossikhin, Y.A., Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63, 010801-1–010801-52 (2010)
Yang, S., Shen, Y.: Recent advances in dynamics and control of hysteretic nonlinear systems. Chaos Soliton Fract. 40, 1808–1822 (2009)
Machado, J.A.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16, 1140–1153 (2011)
Machado, J.A.T., Galhano, A.M.S.: Fractional dynamics: a statistical perspective. ASME J. Comput. Nonlinear Dyn. 3(2), 021201-1-1-5 (2008)
Li, G., Zhu, Z., Cheng, C.: Dynamical stability of viscoelastic column with fractional derivative constitutive relation. Appl. Math. Mech. 22(3), 294–303 (2001)
Wang, Z., Du, M.: Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system. Shock Vib. 18, 257–268 (2011)
Wang, Z., Hu, H.: Stability of a linear oscillator with damping force of fractional order derivative. Sci. Chin Phys. Mech. Astron. 53(2), 345–352 (2010)
Pinto, C.M.A., Machado, J.A.T.: Complex-order forced van der Pol oscillator. J. Vib. Control. 18(14), 2201–2209 (2012)
Pinto, C.M., Machado, J.T.: Complex-order van der Pol oscillator. Nonlinear Dyn. 65(3), 247–254 (2011)
Rossikhin, Y.A., Shitikova, M.V.: Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems. Acta Mech. 120, 109–125 (1997)
Tavazoei, M.S., Haeri, M., Attari, M., Bolouki, S., Siami, M.: More details on analysis of fractional-order van der Pol oscillator. J. Vib. Control. 15(6), 803–819 (2009)
Atanackovic, T.M., Stankovic, B.: On a numerical scheme for solving differential equations of fractional order. Mech. Res. Commun. 35, 429–438 (2008)
Cao, J., Ma, C., Xie, H., Jiang, Z.: Nonlinear dynamics of Duffing system with fractional order damping. ASME J .Comput. Nonlinear Dyn. 5: 041012-1-2-6 (2010)
Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M.: Chaotic dynamics of the fractionally damped Duffing equation. Chaos Soliton Fract. 32, 1459–1468 (2007)
Chen, J.H., Chen, W.C.: Chaotic dynamics of the fractionally damped van der Pol equation. Chaos Soliton Fract. 35, 188–198 (2008)
Lu, J.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A. 354, 305–311 (2006)
Chen, L., Zhu, W.: The first passage failure of SDOF strongly nonlinear stochastic system with fractional derivative damping. J. Vib. Control. 15(8), 1247–1266 (2009)
Qi, H., Xu, M.: Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel. Mech. Res. Commun. 34, 210–212 (2007)
Wahi, P., Chatterjee, A.: Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dyn. 38, 3–22 (2004)
Wu, X., Lu, H., Shen, S.: Synchronization of a new fractional-order hyperchaotic system. Phys. Lett. A 373, 2329–2337 (2009)
Padovan, J., Sawicki, J.T.: Nonlinear vibrations of fractionally damped systems. Nonlinear Dyn. 16, 321–336 (1998)
Huang, Z., Jin, X.: Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative. J. Sound Vib. 319, 1121–1135 (2009)
Borowiec, M., Litak, G., Syta, A.: Vibration of the Duffing oscillator: effect of fractional damping. Shock Vib. 14, 29–36 (2007)
Shen, Y.J., Yang, S.P., Xing, HJ.: Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative. Acta Phys. Sinica 61(11), 110505-1-6.(2012)
Shen, Y.J., Yang, S.P., Xing, H.J.: Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative (II). Acta Phys. Sinica 61(15), 150503-1-9 (2012)
Shen, Y.J., Yang, S.P., Xing, H.J., Gao, G.S.: Primary resonance of Duffing oscillator with fractional-order derivative. Commun. Nonlinear Sci. Numer. Simul. 17(7), 3092–3100 (2012)
Shen, Y.J., Yang, S.P., Xing, H.J., Ma, H.X.: Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives. Int. J. Non-Linear Mech. 47(9), 975–983 (2012)
Nayfeh, A.H., Mook, D.T.: Nonlinear oscillations. John Wiley, New York (1979)
Leung, A.Y.T., Yang, H.X., Guo, Z.J.: The residue harmonic balance for fractional order van der Pol like oscillators. J. Sound Vib. 331, 1115–1126 (2012)
Sardar, T., Ray, S.S., Bera, R.K., Biswas, B.B.: The analytical approximate solution of the multi-term fractionally damped van der Pol equation. Phys. Scr. 80, 025003-1-6 (2009)
Xie, F., Lin, X.Y.: Asymptotic solution of the van der Pol oscillator with small fractional damping. Phys. Scr. 136, 014033-1-4 (2009)
Acknowledgments
The authors are grateful to the support by the National Natural Science Foundation of China (No. 11072158 and 11372198), the Program for New Century Excellent Talents in University (NCET-11-0936), and the Cultivation Plan for Innovation Team and Leading Talent in Colleges and Universities of Hebei Province (LJRC018).
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Shen, YJ., Wei, P. & Yang, SP. Primary resonance of fractional-order van der Pol oscillator. Nonlinear Dyn 77, 1629–1642 (2014). https://doi.org/10.1007/s11071-014-1405-2
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DOI: https://doi.org/10.1007/s11071-014-1405-2