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Primary resonance of Duffing oscillator with fractional-order derivative. (English) Zbl 1252.35274

Summary: In this paper the primary resonance of Duffing oscillator with fractional-order derivative is researched by the averaging method. At first the approximately analytical solution and the amplitude-frequency equation are obtained. Additionally, the effect of the fractional-order derivative on the system dynamics is analyzed, and it is found that the fractional-order derivative could affect not only the viscous damping, but also the linear stiffness, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. This conclusion is remarkably different from the existing research results about nonlinear system with fractional-order derivative. Moreover, the comparisons of the 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 parameters of the fractional-order derivative, i.e. the fractional coefficient and the fractional order, on the amplitude-frequency curves are investigated, which are different from the traditional integer-order Duffing oscillator.

MSC:

35R11 Fractional partial differential equations
35A25 Other special methods applied to PDEs
34C29 Averaging method for ordinary differential equations

Software:

sysdfod; DFOC
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References:

[1] Podlubny, I., Fractional differential equations (1998), Academic: Academic London · Zbl 0922.45001
[2] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003
[3] Das, S., Functional fractional calculus for system identification and controls (2008), Springer-Verlag: Springer-Verlag Berlin · Zbl 1154.26007
[4] Caponetto, R.; Dongola, G.; Fortuna, L.; Petras, I., Fractional order systems: modeling and control applications (2010), World Scientific: World Scientific New Jersey
[5] Monje, C. A.; Chen, Y. Q.; Vinagre, B. M.; Xue, D. Y.; Feliu, V., Fractional-order systems and controls: fundamentals and applications (2010), Springer-Verlag: Springer-Verlag London
[6] Petras, I., Fractional-order nonlinear systems: modeling, analysis and simulation (2011), Higher Education Press: Higher Education Press Beijing · Zbl 1228.34002
[7] 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)
[8] Yang, S.; Shen, Y., Recent advances in dynamics and control of hysteretic nonlinear systems, Chaos Soliton Fract, 40, 1808-1822 (2009)
[9] Machado, J. A.T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun Nonlinear Sci Numer Simulat, 16, 1140-1153 (2011) · Zbl 1221.26002
[10] Machado, J. A.T.; Galhano, A. M.S., Fractional dynamics: a statistical perspective, ASME J Comput Nonlinear Dynam, 3, 2, 021201-1-021201-5 (2008)
[11] Li, G.; Zhu, Z.; Cheng, C., Dynamical stability of viscoelastic column with fractional derivative constitutive relation, Appl Math Mech, 22, 3, 294-303 (2001) · Zbl 1116.74364
[12] 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)
[13] Wang, Z.; Du, M., Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system, Shock Vib, 18, 257-268 (2011)
[14] 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) · Zbl 0901.73030
[15] 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) · Zbl 1273.70037
[16] Pinto, C. M.A.; Machado, J. A.T., Complex-order van der Pol oscillator, Nonlinear Dynam, 65, 3, 247-254 (2011)
[17] Atanackovic, T. M.; Stankovic, B., On a numerical scheme for solving differential equations of fractional order, Mech Res Commun, 35, 429-438 (2008) · Zbl 1258.65103
[18] Cao, J.; Ma, C.; Xie, H.; Jiang, Z., Nonlinear dynamics of Duffing system with fractional order damping, ASME J Comput Nonlinear Dynam, 5, 041012-1-041012-6 (2010)
[19] 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) · Zbl 1129.37015
[20] Wu, X.; Lu, H.; Shen, S., Synchronization of a new fractional-order hyperchaotic system, Phys Lett A, 373, 2329-2337 (2009) · Zbl 1231.34091
[21] Chen, J. H.; Chen, W. C., Chaotic dynamics of the fractionally damped van der Pol equation, Chaos Soliton Fract, 35, 188-198 (2008)
[22] Lu, J., Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys Lett A, 354, 305-311 (2006)
[23] Qi, H.; Xu, M., Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel, Mech Res Commun, 34, 210-212 (2007) · Zbl 1192.76008
[24] Wahi, P.; Chatterjee, A., Averaging oscillations with small fractional damping and delayed terms, Nonlinear Dynam, 38, 3-22 (2004) · Zbl 1142.34385
[25] 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) · Zbl 1173.93031
[26] Padovan, J.; Sawicki, J. T., Nonlinear vibrations of fractionally damped systems, Nonlinear Dynam, 16, 321-336 (1998) · Zbl 0929.70017
[27] Borowiec, M.; Litak, G.; Syta, A., Vibration of the Duffing oscillator: effect of fractional damping, Shock Vib, 14, 29-36 (2007)
[28] 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)
[29] Sanders, J. A.; Verhulst, F.; Murdock, J., Averaging methods in nonlinear dynamical systems (2007), Springer Science+Business Media: Springer Science+Business Media New York · Zbl 1128.34001
[30] Burd, Vladimir, Method of averaging for differential equations on an infinite interval: theory and applications (2007), Taylor & Francis Group · Zbl 1396.34002
[31] Nayfeh, A. H.; Mook, D. T., Nonlinear oscillations (1979), John Wiley: John Wiley New York
[32] Yang, S.; Nayfeh, A. H.; Mook, D. T., Combination resonances in the response of the duffing oscillator to a three-frequency excitation, Acta Mech, 131, 235-245 (1998) · Zbl 0938.70017
[33] Den Hartog, J. P., Mechanical vibrations (1956), McGraw-Hill: McGraw-Hill New York · Zbl 0071.39304
[34] Timoshenko, S.; Young, D. H.; Weaver, W., Vibration problems in engineering (1974), John Wiley: John Wiley New York
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