Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives

International Journal of Non-Linear Mechanics

Volumn 47, Issue 9, 2012, Pages 975-983

  Shen, Yongjun ^a   Yang, Shaopu ^a   Xing, Haijun ^a   Ma, Huaixiang ^a  

^a SHIJIAZHUANG TIEDAO UNIVERSITY   (China)

Author keywords

Amplitude frequency curves; Averaging method; Duffing oscillator; Fractional order derivative; Primary resonance; Vibration control

Indexed keywords

AMPLITUDE-FREQUENCY; AVERAGING METHOD; DUFFING OSCILLATOR; FRACTIONAL-ORDER DERIVATIVES; PRIMARY RESONANCE;

CIRCUIT RESONANCE; OSCILLATORS (MECHANICAL); VIBRATION CONTROL;

STIFFNESS;

EID: 84864407721     PISSN: 00207462     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijnonlinmec.2012.06.012     Document Type: Article

References (40)

1. I. Podlubny Fractional Differential Equations 1998 Academic London
2. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of Fractional Differential Equations 2006 Elsevier Amsterdam
3. S. Das Functional Fractional Calculus for System Identification and Controls 2008 Springer-Verlag Berlin
4. R. Caponetto, G. Dongola, L. Fortuna, and I. Petras Fractional Order Systems: Modeling and Control Applications 2010 World Scientific New Jersey
5. C.A. Monje, Y.Q. Chen, B.M. Vinagre, D.Y. Xue, and V. Feliu Fractional-order Systems and Controls: Fundamentals and Applications 2010 Springer-Verlag London
6. I. Petras. Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation 2011 Higher Education Press Beijing
7. Y.A. Rossikhin, and M.V. Shitikova Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results Applied Mechanics Reviews 63 2010 010801-1 52
8. S. Yang, and Y. Shen Recent advances in dynamics and control of hysteretic nonlinear systems Chaos, Solitons and Fractals 40 4 2009 1808 1822
9. J.A.T. Machado, V. Kiryakova, and F. Mainardi Recent history of fractional calculus Communications in Nonlinear Science and Numerical Simulation 16 3 2011 1140 1153
10. J.A.T. Machado, and A.M.S. Galhano Fractional dynamics: a statistical perspective ASME Journal of Computational and Nonlinear Dynamics 3 2 2008 021201-1 021201-5
11. G. Li, Z. Zhu, and C. Cheng Dynamical stability of viscoelastic column with fractional derivative constitutive relation Applied Mathematics and Mechanics 22 3 2001 294 303
12. Z. Wang, and H. Hu Stability of a linear oscillator with damping force of fractional order derivative Science China: Physics, Mechanics & Astronomy 53 2 2010 345 352
13. Z. Wang, and M. Du Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system Shock and Vibration 18 1-2 2011 257 268
14. M.S. Tavazoei, M. Haeri, M. Attari, S. Bolouki, and M. Siami More details on analysis of fractional-order van der Pol oscillator Journal of Vibration and Control 15 6 2009 803 819
15. C.M.A. Pinto, and J.A.T. Machado Complex-order van der Pol oscillator Nonlinear Dynamics 65 3 2011 247 254
16. C.M.A. Pinto, and J.A.T. Machado Complex-order forced van der Pol oscillator Journal of Vibration and Control 2011 10.1177/1077546311429150
17. Y.A. Rossikhin, and M.V. Shitikova Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems Acta Mechanica 120 1997 109 125
18. W. Deng Numerical algorithm for the time fractional Fokker-Planck equation Journal of Computational Physics 227 2 2007 1510 1522
19. T.M. Atanackovic, and B. Stankovic On a numerical scheme for solving differential equations of fractional order Mechanics Research Communications 35 2008 429 438
20. J. Cao, C. Ma, H. Xie, and Z. Jiang Nonlinear dynamics of Duffing system with fractional order damping ASME Journal of Computational and Nonlinear Dynamics 5 2010 041012-1 6
21. A. Palfalvi Efficient solution of a vibration equation involving fractional derivatives International Journal of Non-Linear Mechanics 45 2010 169 175
22. L.J. Sheu, H.K. Chen, J.H. Chen, and L.M. Tam Chaotic dynamics of the fractionally damped Duffing equation Chaos, Solitons and Fractals 32 2007 1459 1468
23. X. Wu, H. Lu, and S. Shen Synchronization of a new fractional-order hyperchaotic system Physics Letters A 373 2009 2329 2337
24. J.H. Chen, and W.C. Chen Chaotic dynamics of the fractionally damped van der Pol equation Chaos, Solitons and Fractals 35 2008 188 198
25. J. Lu Chaotic dynamics of the fractional-order Lü system and its synchronization Physics Letters A 354 2006 305 311
26. H. Qi, and M. Xu Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel Mechanics Research Communications 34 2007 210 212
27. P. Wahi, and A. Chatterjee Averaging oscillations with small fractional damping and delayed terms Nonlinear Dynamics 38 2004 3 22
28. L. Chen, and W. Zhu The first passage failure of SDOF strongly nonlinear stochastic system with fractional derivative damping Journal of Vibration and Control 15 8 2009 1247 1266
29. L. Chen, and W. Zhu Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations International Journal of Non-Linear Mechanics 46 2011 1324 1329
30. J. Padovan, and J.T. Sawicki Nonlinear vibrations of fractionally damped systems Nonlinear Dynamics 16 1998 321 336
31. M. Borowiec, G. Litak, and A. Syta Vibration of the Duffing oscillator: effect of fractional damping Shock and Vibration 14 2007 29 36
32. Z. Huang, and X. Jin Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative Journal of Sound and Vibration 319 2009 1121 1135
33. Y. Shen, S. Yang, and H. Xing Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative Acta Physica Sinina 61 11 2012 110505-1 6
34. Y. Shen, S. Yang, H. Xing, and G. Gao Primary resonance of Duffing oscillator with fractional-order derivative Communications in Nonlinear Science and Numerical Simulation 12 7 2012 3092 3100
35. J.A. Sanders, F. Verhulst, and J. Murdock Averaging Methods in Nonlinear Dynamical Systems 2nd ed. 2007 Springer ScienceBusiness Media New York
36. Vladimir Burd Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications 2007 Taylor & Francis Group
37. A.H. Nayfeh, and D.T. Mook Nonlinear Oscillations 1979 John Wiley New York
38. S. Yang, A.H. Nayfeh, and D.T. Mook Combination resonances in the response of the Duffing oscillator to a three-frequency excitation Acta Mechanica 131 1998 235 245
39. J.P. Den Hartog Mechanical Vibrations 4th ed. 1956 McGraw-Hill New York
40. S. Timoshenko, D.H. Young, and W. Weaver Vibration Problems in Engineering 5th ed. 1974 John Wiley New York