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Nonlinear normal modes of a shallow arch with elastic constraints for two-to-one internal resonances

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Abstract

The nonlinear normal modes (NNMs) of an elastically constrained (EC) shallow arch in the case of two-to-one internal resonance are constructed, and the effects of the vertical and rotational elastic boundary constraints are studied. The multiple scales method is directly applied to obtain the second-order uniform-expansion solution and the modulation equations from the dimensionless integral–partial–differential equation of motion. The elastic constraints have a corresponding relationship with the coefficients of modulation equations and influence the natural frequencies and modes, as demonstrated by solving the algebraic eigenvalue equation. The stability of the uncoupled single-mode and coupled-mode motions for the nonlinear system is investigated. Then the shape functions, activation conditions and space–time evolutions accounting for the two-to-one internally resonant NNMs for vertical and rotational elastic constraints are examined. The results show that the vertical and rotational elastic constraints play a fundamental role in the nonlinear dynamic phenomena of the EC shallow arch.

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Acknowledgments

The authors gratefully acknowledge the support given to this research by the National Natural Science Foundation of China (Nos. 11002030, 11032004 and 51178059). Valuable comments and useful suggestions by two anonymous reviewers are also gratefully acknowledged.

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Authors and Affiliations

  1. School of Civil Engineering and Architecture, Changsha University of Science and Technology, Changsha, 410114, People’s Republic of China

    Zhuangpeng Yi

  2. Department of Civil and Environmental Engineering, Rice University, Houston, TX, 77005, USA

    Ilinca Stanciulescu

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  1. Zhuangpeng Yi
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Correspondence to Zhuangpeng Yi.

Appendix

Appendix

$$\begin{aligned} g_1(x)= & {} b\pi \phi ''_m \int _0^1 \phi '_m \sin {2\pi x} \text {d}x \nonumber \\&+\,b\pi ^2\cos 2\pi x\int _0^1(\phi '_m)^2 \text {d}x, \end{aligned}$$
(39)
$$\begin{aligned} g_2(x)= & {} b\pi \phi ''_n \int _0^1 \phi '_n \sin {2\pi x} \text {d}x \nonumber \\&+\,b\pi ^2\cos 2\pi x\int _0^1(\phi '_n)^2 \text {d}x, \end{aligned}$$
(40)
$$\begin{aligned} g_3(x)= & {} b\pi \phi ''_m \int _0^1 \phi '_n \sin 2\pi x \text {d}x \nonumber \\&+\,b\pi \phi ''_n \int _0^1 \phi '_m \sin 2\pi x \text {d}x \nonumber \\&+\,2b\pi ^2 \cos 2\pi x \int _0^1 \phi '_m \phi '_n \text {d}x. \end{aligned}$$
(41)
$$\begin{aligned} \chi _1(x)= & {} b\pi (\varGamma ''_1+2\varGamma ''_2)\int _0^1 \phi '_m \sin 2\pi x\text {d}x \nonumber \\&+\,b\pi \phi ''_m \int _0^1(\varGamma '_1+2\varGamma '_2)\sin 2\pi x\text {d}x \nonumber \\&+\,2b\pi ^2 \cos 2\pi x \int _0^1 \phi '_m (\varGamma '_1+2\varGamma '_2) \text {d}x \nonumber \\&+\,\frac{3}{2}\phi ''_m \int _0^1 (\phi '_m )^2\text {d}x, \end{aligned}$$
(42)
$$\begin{aligned} \chi _2(x)= & {} b\pi (\varGamma ''_5+\varGamma ''_6)\int _0^1 \phi '_n \sin 2\pi x\text {d}x \nonumber \\&+\,b\pi \phi ''_n \int _0^1 (\varGamma '_5+\varGamma '_6) \sin 2\pi x\text {d}x \nonumber \\&+\,\phi ''_m \int _0^1 (\phi '_n)^2 \text {d}x + 2\phi ''_n \int _0^1 \phi '_m \phi '_n \text {d}x \nonumber \\&+\,2b\pi \varGamma ''_4 \int _0^1 \phi '_m \sin 2\pi x\text {d}x \nonumber \\&+\,2b\pi \phi ''_m \int _0^1 \varGamma '_4 \sin 2\pi x\text {d}x \nonumber \\&+\,2b\pi ^2 \cos 2\pi x\int _0^1( 2 \phi '_m \varGamma '_4 \nonumber \\&+\,\phi '_n \varGamma '_5 + \phi '_n \varGamma '_6) \text {d}x, \end{aligned}$$
(43)
$$\begin{aligned} \chi _3(x)= & {} b\pi (\varGamma ''_3+2\varGamma ''_4)\int _0^1 \phi '_n \sin 2\pi x\text {d}x \nonumber \\&+\,b\pi \phi ''_n \int _0^1(\varGamma '_3+2\varGamma '_4)\sin 2\pi x\text {d}x \nonumber \\&+\,2b\pi ^2 \cos 2\pi x \int _0^1 \phi '_n (\varGamma '_3+2\varGamma '_4) \text {d}x \nonumber \\&+\,\frac{3}{2}\phi ''_n \int _0^1 (\phi '_n )^2\text {d}x, \end{aligned}$$
(44)
$$\begin{aligned} \chi _4(x)= & {} b\pi (\varGamma ''_5+\varGamma ''_6)\int _0^1 \phi '_m \sin 2\pi x\text {d}x \nonumber \\&+\,b\pi \phi ''_m \int _0^1 (\varGamma '_5+\varGamma '_6) \sin 2\pi x\text {d}x \nonumber \\&+\,\phi ''_n \int _0^1 (\phi '_m)^2 \text {d}x + 2\phi ''_m \int _0^1 \phi '_m \phi '_n \text {d}x \nonumber \\&+\,2b\pi \varGamma ''_2 \int _0^1 \phi '_n \sin 2\pi x\text {d}x \nonumber \\&+\,2b\pi \phi ''_n \int _0^1 \varGamma '_2 \sin 2\pi x\text {d}x \nonumber \\&+\,2b\pi ^2 \cos 2\pi x\int _0^1( 2 \phi '_n \varGamma '_2 \nonumber \\&+\,\phi '_m \varGamma '_5 + \phi '_m \varGamma '_6) \text {d}x, \end{aligned}$$
(45)

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Yi, Z., Stanciulescu, I. Nonlinear normal modes of a shallow arch with elastic constraints for two-to-one internal resonances. Nonlinear Dyn 83, 1577–1600 (2016). https://doi.org/10.1007/s11071-015-2432-3

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