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1/3 SUBHARMONIC RESPONSE OF DUFFING OSCILLATOR UNDER PERIODIC AND RANDOM EXCITATIONS

https://doi.org/10.12737/6816

Abstract

The subharmonic response of one third order of Duffing oscillator under harmonic and random excitations is investigated for the first time by a technique combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function for Fokker-Planck equation. The averaged equations are linear zed so that the stationary density function of the approximate response can be found exactly by the technique of auxiliary function. The one third order subharmonic response obtained by the present technique is validated by numerical simulation. The significant contribution of this work is that it may lead to a new trend in investigating subharmonic oscillators in random nonlinear systems.

About the Authors

N. D. Anh
Institute of Mechanics, Hanoi, Vietnam
Viet Nam


Vilor Lavrentyevich Zakovorotny
Don State Technical University, Russia
Russian Federation


D. N. Hao
University of Information Technology, HochiMinh City, Vietnam
Viet Nam


N. Х. Chiem
University of Le Quy Don, Hanoi, Vietnam
Viet Nam


References

1. Krylov, N. M. Bogoliubov, N. N. Introduction to nonlinear mechanics. (trans: Solomon Lefschetz of excerpts from two Russian monographs). Princeton University Press, Michigan, 1947. ─ 472 p.

2. Bogoliubov, N. N., Mitropolsky, Y. A. Asymptotic methods in the theory of nonlinear oscillations. Moscow: Nauka, 1963 (in Russian). ─ 572 p.

3. Mitropolsky, Y. A. Averaging method in non-linear mechanics. International Journal of Nonlinear Mechanics, Pergamon Press Ltd., 1967, vol. 2, pp. 69-96.

4. Mitropolsky, Y. A., Dao N. V., Anh, N. D. Nonlinear oscillations in systems of arbitrary order. Kiev: Naukova-Dumka, 1992 (in Russian). ─ 344 p.

5. Stratonovich, R. L. Topics in the Theory of Random Noise. Vol. II, New York: Gordon and Breach, 1967. ─ 472 p.

6. Khasminskiy, R. Z. A limit theorem for the solutions of differential equations with random right-hand sides. Theory of Probability and Its Applications, 1966, vol. 11, pp. 390-405.

7. Roberts, J. B., Spanos, P. D. Stochastic averaging: An approximate method of solving random vi-bration problems. International Journal of Nonlinear Mechanics, 1986, no. 21(2), pp. 111-134.

8. Manohar, C. S. Methods of nonlinear random vibration analysis. Sãdhanã, 1995, vol. 20, pp. 345-371.

9. Kazakov, I. E. An approximate method for the statistical investigation for nonlinear systems. Proc. of Zhukovsky Air Force Engineering Academy, 1954, vol. 394, pp. 1–52 (in Russian).

10. Roberts, J. B., Spanos, P. D. Random Vibration and Statistical Linearization. Dover Publications Inc., Mineola, New York, 1999. ─ 176 p.

11. Socha, L. Linearization Methods for Stochastic Dynamic System, Lecture Notes in Physics. Springer, Berlin, 2008. ─ 391 p.

12. Elishakoff, I., Andrimasy, L., Dolley, M. Application and extension of the stochastic linearization by Anh and Di Paola. Acta Mechanica, 2009, vol. 204, iss. 1-2, pp. 89-98.

13. Anh, N. D., Hieu, N. N., Linh, N. N. A dual criterion of equivalent linearization method for nonlinear systems subjected to random excitation. Acta Mechanica, 2012, vol. 223, iss. 3, pp. 645-654.

14. Anh, N. D., Zakovorotny, V. L, Hieu, N. N., Diep, D. V. A dual criterion of stochastic linearization method for multi-degree-of-freedom systems subjected to random excitation. Acta Mechanica, 2012, vol. 223, iss. 12, pp. 2667-2684.

15. Nayfeh, A. H., Mook, D. T. Nonlinear oscillations. Wiley-Interscience, 1995. ─ 275 p.

16. Mitropolsky, I. A., Dao, N. V. Applied asymptotic methods in nonlinear oscillations. Springer-Science +Business Media, B.V. DOI 10.1007/978-94-015-8847-8. 1997. ─ 341 p.

17. Kelly, S. G. Mechanical vibrations: Theory and applications. Cengage Learning, 2012. ─ 475 p.

18. Davies, H. G., Rajan, S. Random superharmonic and subharmonic response: Multiple time scaling of a duffing oscillator. Journal of Sound and Vibration, 1988, vol. 126, iss. 2, pp. 195-208.

19. Dimentberg, M. F., Iourtchenko, D. V., Ewijk, O. V. Subharmonic response of a quasi-isochronous vibroimpact system to a randomly disordered periodic excitation. Nonlinear Dynamics, 1998, vol. 17, pp. 173-186.

20. Haiwu, R., Xiangdong, W., Wei, X., Tong, F. Subharmonic response of a single-degree-of-freedom nonlinear vibroimpact system to a randomly disordered periodic excitation. Journal of Sound and Vibration, 2009, vol. 327, pp. 173-182.

21. Li, F. M., Yao, G. 1/3 Subharmonic resonance of a nonlinear composite laminated cylindrical shell in subsonic air flow. Composite Structures, 2013, vol. 100, pp. 249-256.

22. Huang, Z. L., Zhu, W. Q., Suzuki, Y. Stochastic averaging of strongly non-linear oscillators under combined harmonic and white noise excitations. Journal of Sound and Vibration, 2000, vol. 238, pp. 233-256.

23. Haiwu, R., Wei, X., Guang, M., Tong, F. Response of a Duffing oscillator to combined deterministic harmonic and random excitation. Journal of Sound and Vibration, 2001, vol. 242, iss. 2, pp. 362-368.

24. Anh, N. D., Hieu, N. N. The Duffing oscillator under combined periodic and random excitations. Probabilistic Engineering Mechanics, 2012, vol. 30, pp. 27-36.

25. Narayanan, S., Kumar, P. Numerical solutions of Fokker-Planck equation of nonlinear systems subjected to random and harmonic excitations. Probabilistic Engineering Mechanics, 2012, vol. 27, pp. 35-46.

26. Anh, N. D. Random oscillations in non-autonomous mechanical systems with random parametric excitation. Ukranian Mathematical Journal, 1985, vol. 37, pp. 412-416.

27. Anh, N. D. Two methods of integration of the Kolmogorov-Fokker-Planck equations (English). Ukr. Math. J., 1986, vol. 38, pp. 331-334; trans. from Ukr. Mat. Zh. 1986, vol. 38, iss. 3, pp. 381-385.

28. Lutes, L., Sarkani, S. Stochastic Analysis of Structural Dynamics. Upper Saddle River, New Jersey: Prentice Hall, 1997. ─ 276 p.


Review

For citations:


Anh N.D., Zakovorotny V.L., Hao D.N., Chiem N.Х. 1/3 SUBHARMONIC RESPONSE OF DUFFING OSCILLATOR UNDER PERIODIC AND RANDOM EXCITATIONS. Vestnik of Don State Technical University. 2014;14(4):39-49. https://doi.org/10.12737/6816

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