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Accuracy Improvement of the Method of Multiple Scales for Nonlinear Vibration Analyses of Continuous Systems with Quadratic and Cubic Nonlinearities

機(jī)譯：具有二次和三次非線性的連續(xù)系統(tǒng)非線性振動(dòng)分析的多尺度方法的精度改進(jìn)

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摘要

This paper proposes an accuracy improvement of the method of multiple scales (MMSs) for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. As an example, we treat a shallow suspended cable subjected to a harmonic excitation, and investigate the primary resonance of the mth in-plane mode (Ω≈ω) in which Ω and ω_m are the driving and natural frequencies, respectively. The application of Galerkin's procedure to the equation of motion yields nonlinear ordinary differential equations with quadratic and cubic nonlinear terms. The steady-state responses are obtained by using the discretization approach of the MMS in which the definition of the detuning parameter, expressing the relationship between the natural frequency and the driving frequency, is changed in an attempt to improve the accuracy of the solutions. The validity of the solutions is discussed by comparing them with solutions of the direct approach of the MMS and the finite difference method.
機(jī)譯：本文提出了對(duì)具有二次和三次非線性的連續(xù)系統(tǒng)進(jìn)行非線性振動(dòng)分析的多尺度（MMS）方法的準(zhǔn)確性的改進(jìn)。例如，我們對(duì)經(jīng)受諧波激勵(lì)的淺懸索進(jìn)行處理，并研究第m面內(nèi)模式（Ω≈ω）的主諧振，其中Ω和ω_m分別是驅(qū)動(dòng)頻率和固有頻率。 Galerkin程序在運(yùn)動(dòng)方程中的應(yīng)用產(chǎn)生了具有二次和三次非線性項(xiàng)的非線性常微分方程。通過(guò)使用MMS的離散化方法獲得穩(wěn)態(tài)響應(yīng)，在該方法中，改變表示自然頻率和驅(qū)動(dòng)頻率之間關(guān)系的失諧參數(shù)的定義，以試圖提高解的精度。通過(guò)與MMS直接法和有限差分法的解進(jìn)行比較，討論了解的有效性。

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  • 來(lái)源
    《Mathematical Problems in Engineering》 |2010年第4期|p.389-400|共12頁(yè)
  • 作者

    Akira Abe;

  • 作者單位

    Department of Information Systems Engineering, Asahikawa National College of Technology,2-2-1-6 Syunkodai, Asahikawa, Hokkaido 071-8142, Japan;

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