Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-020-2694-6
Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations∗ Junda LI, Jianliang HUANG† Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, China (Received Feb. 4, 2020 / Revised Oct. 19, 2020)
Abstract The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential 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 symmetrybreaking 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. Key words nonlinear vibration, buckled beam, incremental harmonic balance method, bifurcation, subharmonic resonance Chinese Library Classification O322 2010 Mathematics Subject Classification
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74K10
Introduction
As basic structural elements, buckled beams, whose dynamic equations consist of both quadratic and cubic nonlinearities, have been investigated for several decades. They can be found in airplanes, rockets, missiles, buildings in cold regions, foundations of heavy-duty machines, and micro-electromechanical systems (MEMS)[1] . Different methods have been applied ∗ 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) (2020) https://doi.org/10.1007/s10483-020-2694-6 † Corresponding author, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11972381 and 11572354) and the Fundamental Research Funds for the Central Universities (No. 18lgzd08) c
Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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Junda LI and Jianliang HUANG
in the nonlinear analysis of beams. Tseng and Dugundji[2] used the harmonic balance method to obtain the per
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