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.
осциллятор Дюффинга, субгармоника, метод усреднения, эквивалентная линеаризация, вспомогательная функция, гармонические возбуждения, случайные возбуждения
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]
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