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Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables’ external excitations

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Abstract

The nonlinear dynamic behaviors of a double cable-stayed shallow arch model are investigated under the one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. The in-plane governing equations of the system are obtained when the harmonic excitation is applied to cables. The excitation mechanism due to the angle-variation of cable tension during motion is newly introduced. Galerkin’s method and the multi-scale method are used to obtain ordinary differential equations (ODEs) of the system and their modulation equations, respectively. Frequency- and force-response curves are used to explore dynamic behaviors of the system when harmonic excitations are symmetrically and asymmetrically applied to cables. More importantly, comparisons of frequency-response curves of the system obtained by two types of trial functions, namely, a common sine function and an exact piecewise function, of the shallow arch in Galerkin’s integration are conducted. The analysis shows that the two results have a slight difference; however, they both have sufficient accuracy to solve the proposed dynamic system.

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Author information

Correspondence to Houjun Kang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11572117, 11502076, and 11872176)

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Cite this article

Cong, Y., Kang, H. & Guo, T. Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables’ external excitations. Appl. Math. Mech.-Engl. Ed. 40, 977–1000 (2019). https://doi.org/10.1007/s10483-019-2497-8

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Key words

  • nonlinear dynamics
  • cable-stayed system
  • internal resonance
  • primary resonance
  • multi-scale method

Chinese Library Classification

  • O322

2010 Mathematics Subject Classification

  • 65P40