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

D Hao; N Anh; Vilor Zakovorotnyy; N. Chiem

doi:doi:10.12737/6816

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

Published в Vestnik of Don State Technical University · Volume 14, Issue 4, 2014 · Pages 30–38 · Rubrics: Technical sciences

DOI 10.12737/6816

Received: 03.12.2014 Accepted: 30.12.2014 Published: 30.12.2014 Language of publication: ENG

Authors

D Hao · N Anh · Vilor Zakovorotnyy · N. Chiem

ABSTRACT

The subharmonic response of one third order of Duffing oscillator under harmonic and random excitations is in-vestigated for the first time by a technique combining the stochastic averaging method, the equivalent lineariza-tion method, and the technique of auxiliary function for Fokker-Planck equation. The averaged equations are line-ar 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.

KEYWORDS

Duffing oscillator subharmonic averaging method equivalent linearization auxiliary function harmonic excita-tion random excitation

Information Text References

Authors:

Type:

Article

Pages:

from 30 to 38

Status:

Published

Language:

Russian

Keywords:

Duffing oscillator, subharmonic, averaging method, equivalent linearization, auxiliary function, harmonic excita-tion, random excitation

Text

In this paper, we are concerned with the Duffing oscillator, which has been applied to model many mechanical systems and has attracted much attention as a typical nonlinear system. When the system is under only a harmonic excitation or random one, two popular tools used to study such a nonlinear system are the averaging method and equivalent linearization method, respectively. The former was originally given by Krylov and Bogolyubov [1] and then it was developed by Bogolyubov and Mitropolskiy [2-4] and was extended to systems under a random excitation with the works of Stratonovich [5], Khasminskii [6], and others, which were reviewed in survey paper by Mitropolskiy [3], Robert and Spanos [7] and Manohar [8]. The later, the stochastic equivalent linearization method, which has attracted many researchers due to its originality and capability for various applications in engineering, was first studies by Kazakov [9], who extended Krylov and Bogolyubov’s linearization technique [1] of deterministic problems to random problems. This method was also reviewed in some books by Roberts and Spanos [10], and Socha [11]

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 Me-chanics, 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 vibra-tion 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. DOIhttps://doi.org/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 sub-jected 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.

Доступ и загрузка

🗂 JATS XML ❝ To cite 📎 Citations

—

Views

—

Downloads

Citations

Индексируется в

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

Confirmation

Регистрация