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Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

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Abstract

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integro-differential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with Galerkin’s method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the anti-symmetric modes being excited. Bifurcations including the saddle-node, Hopf, period-doubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.

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Authors and Affiliations

  1. Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou, 510275, China

    Junda Li & Jianliang Huang

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  1. Junda Li
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  2. Jianliang Huang
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Correspondence to Jianliang Huang.

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Citation: LI, J. D. and HUANG, J. L. Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base Harmonic excitations. Applied Mathematics and Mechanics (English Edition), 41(12), 1881–1896 (2020) https://doi.org/10.1007/s10483-020-2694-6

Project supported by the National Natural Science Foundation of China (Nos. 11972381 and 11572354) and the Fundamental Research Funds for the Central Universities (No. 181gzd08)

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Li, J., Huang, J. Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations. Appl. Math. Mech.-Engl. Ed. 41, 1881–1896 (2020). https://doi.org/10.1007/s10483-020-2694-6

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  • DOI: https://doi.org/10.1007/s10483-020-2694-6

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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