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Forced response of quadratic nonlinear oscillator: comparison of various approaches

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Abstract

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of different analytical approximate approaches. The forced vibration of snap-through mechanism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincaré method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are compared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method predicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the LindstedtPoincaré method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Wenan Jiang & Liqun Chen

  2. College of Sciences, Shanghai University, Shanghai, 200444, China

    Guoce Zhang & Liqun Chen

  3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China

    Liqun Chen

Authors
  1. Wenan Jiang
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  2. Guoce Zhang
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  3. Liqun Chen
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Corresponding author

Correspondence to Liqun Chen.

Additional information

Project supported by the State Key Program of National Natural Science Foundation of China (No. 11232009) and the National Natural Science Foundation of China (No. 11572182)

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Jiang, W., Zhang, G. & Chen, L. Forced response of quadratic nonlinear oscillator: comparison of various approaches. Appl. Math. Mech.-Engl. Ed. 36, 1403–1416 (2015). https://doi.org/10.1007/s10483-015-1991-7

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  • DOI: https://doi.org/10.1007/s10483-015-1991-7

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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