СУБГАРМОНИЧЕСКИЙ ОТКЛИК ТРЕТЬЕГО ПОРЯДКА ДЛЯ ОСЦИЛЛЯТОРА ДУФФИНГА, ВОЗМУЩЕННОГО ГАРМОНИЧЕСКИМ И СЛУЧАЙНЫМ ВОЗДЕЙСТВИЕМ
Аннотация и ключевые слова
Аннотация (русский):
В статье впервые исследуется субгармонический отклик третьего порядка осциллятора Дуффинга на основе метода стохастического усреднения и одновременно стохастической линеаризации. При этом используется разрабатываемый авторами метод вспомогательных функций для уравнения Фоккера – Планка. Усредненные уравнения линеаризованы так, что плотностная стационарная функция приближенного отклика может быть получена точно с помощью метода вспомогательной функции. Полученные на основе разработанного метода решения сравниваются с численными решениями. Значение этой работы заключается в том, что предложенный метод может привести к новой тенденции в исследовании субгармонических осцилляторов в слу-чайных нелинейных систем.

Ключевые слова:
осциллятор Дюффинга, субгармоника, метод усреднения, эквивалентная линеаризация, вспомогательная функция, гармонические возбуждения, случайные возбуждения
Текст

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]

Список литературы

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. Боголюбов, Н. Н., Митропольский, Ю. A. Asimptoticheskie metodyi v teorii nelineynyih kolebaniy. [Asymptotic methods in the theory of nonlinear oscillations.] Москва: Наука, 1963 (in Russian). ─ 572 с.

3. Mitropolsky, Y. A. Averaging method in non-linear mechanics. International Journal of Nonlinear Me-chanics, Pergamon Press Ltd., 1967, no. 2, pp. 69-96.

4. Митропольский, Ю. A., Нгуен Ван Дао, Нгуен Донг Ань. Nelineynyie kolebaniya v sistemah proizvolnogo poryadka. [Nonlinear oscillations in systems of arbitrary order]. Киев : Наукова думка, 1992. ─ 344 с. (in Russian).

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, vol. 21, iss. 2, pp. 111-134.

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

9. Казаков, И. Е. Priblizhennyiy metod statisticheskogo issledovaniya nelineynyih sistem. [An approxi-mate method for the statistical investigation for nonlinear systems.] Москва: Изд. ВВИА им. Н. Е. Жуковского, 1954, т. 394, с. 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.

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