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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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References

  1. SCHREYER, H. L. and MASUR, E. F. Buckling of shallow arches. Journal of the Engineering Mechanics Division, 92, 1–20 (1966)

    Google Scholar 

  2. PLAUT, R. H. Influence of load position on the stability of shallow arches. Zeitschrift für angewandte Mathematik und Physik, 30(3), 548–552 (1979)

    Article  Google Scholar 

  3. LEVITAS, J., SINGER, J., and WELLER, T. Global dynamic stability of a shallow arch by Poincare-like simple cell mapping. International Journal of Non-Linear Mechanics, 32(2), 411–424 (1997)

    Article  MATH  Google Scholar 

  4. BRESLAVSKY, I., AVRAMOV, K. V., MIKHLIN, Y., and KOCHUROV, R. Nonlinear modes of snap-through motions of a shallow arch. Journal of Sound and Vibration, 311(1), 297–313 (2008)

    Article  Google Scholar 

  5. CAI, J. G., FENG, J., CHEN, Y., and HUANG, L. F. In-plane elastic stability of fixed parabolic shallow arches. Science in China, 52(3), 596–602 (2009)

    Article  MATH  Google Scholar 

  6. PI, Y. L. and BRADFORD, M. A. Nonlinear dynamic buckling of shallow circular arches under a sudden uniform radial load. Journal of Sound and Vibration, 331(18), 4199–4217 (2012)

    Article  Google Scholar 

  7. CHEN, J. S. and LIN, J. S. Dynamic snap-through of a shallow arch under a moving point load. Journal of Vibration and Acoustics, 126(4), 514–519 (2004)

    Article  Google Scholar 

  8. CHEN, J. S. and LI, Y. T. Effects of elastic foundation on the snap-through buckling of a shallow arch under a moving point load. International Journal of Solids and Structures, 43(14), 4220–4237 (2006)

    Article  MATH  Google Scholar 

  9. CHEN, J. S. and LIN, J. S. Stability of a shallow arch with one end moving at constant speed. International Journal of Non-Linear Mechanics, 41(5), 706–715 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. ABDELGAWAD, A., ANWAR, A., and NASSAR, M. Snap-through buckling of a shallow arch resting on a two-parameter elastic foundation. Applied Mathematical Modelling, 37, 7953–7963 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. HA, J., GUTMAN, S., SHON, S., and LEE, S. Stability of shallow arches under constant load. International Journal of Non-Linear Mechanics, 58(1), 120–127 (2014)

    Article  Google Scholar 

  12. PLAUT, R. H. Snap-through of arches and buckled beams under unilateral displacement control. International Journal of Solids and Structures, 63, 109–113 (2015)

    Article  Google Scholar 

  13. BLAIR, K. B., FARRIS, T. N., and KROUSGRILL, C. M. Nonlinear dynamic response of shallow arches to harmonic forcing. Journal of Sound and Vibration, 194(3), 353–367 (1992)

    Article  Google Scholar 

  14. LAKRAD, F. and SCHIEHLEN, W. Effects of a low frequency parametric excitation. Chaos Solitons and Fractals, 22(5), 1149–1164 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. YI, Z. P., WANG, L. H., KANG, H. J., and TU, G. Y. Modal interaction activations and nonlinear dynamic response of shallow arch with both ends vertically elastically constrained for two-to-one internal resonance. Journal of Sound and Vibration, 333(21), 5511–5524 (2014)

    Article  Google Scholar 

  16. TIEN, W. M., NAMACHCHIVAYA, N. S., and BAJAJ, A. K. Non-linear dynamics of a shallow arch under periodic excitation I: 1:2 internal resonance. International Journal of Non-Linear Mechanics, 29(3), 349–366 (1994)

    Article  MATH  Google Scholar 

  17. TIEN, W. M., NAMACHCHIVAYA, N. S., and MALHOTRA, N. Non-linear dynamics of a shallow arch under periodic excitation II: 1:1 internal resonance. International Journal of Non-Linear Mechanics, 29(3), 367–386 (1994)

    Article  MATH  Google Scholar 

  18. MALHOTRA, N. and NAMACHCHIVAYA, N. S. Chaotic motion of shallow arch structures under 1:1 internal resonance. Journal of Engineering Mechanics, 123(6), 620–627 (1997)

    Article  Google Scholar 

  19. MALHOTRA, N. and NAMACHCHIVAYA, N. S. Chaotic dynamics of shallow arch structures under 1:2 resonance. Journal of Engineering Mechanics, 123(6), 612–619 (1997)

    Article  Google Scholar 

  20. DENG, L., WANG, F., and HE, W. Dynamic impact factors for simply-supported bridges due to vehicle braking. Advances in Structural Engineering, 18(6), 791–801 (2015)

    Article  Google Scholar 

  21. DENG, L., CAO, R., WANG, W., and YIN, X. F. A multi-point tire model for studying bridge-vehicle coupled vibration. International Journal of Structural Stability and Dynamics, 16(8), 1550047 (2016)

    Article  Google Scholar 

  22. BI, Q. and DAI, H. H. Analysis of non-linear dynamics and bifurcations of a shallow arch subjected to periodic excitation with internal resonance. Journal of Sound and Vibration, 233(4), 553–567 (2000)

    Article  Google Scholar 

  23. CHEN, J. S. and LIAO, C. Y. Experiment and analysis on the free dynamics of a shallow arch after an impact load at the end. Journal of Applied Mechanics, 72(1), 103–115 (2005)

    MATH  Google Scholar 

  24. YI, Z. P., WANG, L. H., and ZHAO, Y. Y. Nonlinear dynamic behaviors of viscoelastic shallow arches. Applied Mathematics and Mechanics (English Edition), 30(6), 771–777 (2009) https://doi.org/10.1007/s10483-009-0611-y

    Article  MATH  Google Scholar 

  25. YI, Z. P., WANG, L. H., KANG, H. J., and TU, G. Y. Modal interaction activations and nonlinear dynamic response of shallow arch with both ends vertically elastically constrained for two-to-one internal resonance. Journal of Sound and Vibration, 333(21), 5511–5524 (2014)

    Article  Google Scholar 

  26. DING, H., HUANG, L. L., MAO, X. Y., and CHEN, L. Q. Primary resonance of traveling vis-coelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38(1), 1–14 (2017) https://doi.org/10.1007/s10483-016-2152-6

    Article  MathSciNet  MATH  Google Scholar 

  27. REGA, G. Nonlinear vibrations of suspended cables — part I: modeling and analysis. Applied Mechanics Reviews, 57(6), 443–478 (2004)

    Article  Google Scholar 

  28. PAKDEMIRLI, M., NAYFEH, S. A., and NAYFEH, A. H. Analysis of one-to-one autoparametric resonances in cables — discretization vs. direct treatment. Nonlinear Dynamics, 8(1), 65–83 (1995)

    MathSciNet  Google Scholar 

  29. ZHAO, Y. Y., WANG, L. H., CHEN, D. L., and JIANG, L. Z. Non-linear dynamic analysis of the two-dimensional simplified model of an elastic cable. Journal of Sound and Vibration, 255(1), 43–59 (2002)

    Article  Google Scholar 

  30. SRINIL, N., REGA, G., and CHUCHEEPSAKUL, S. Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, part I: theoretical formulation and model validation. Nonlinear Dynamics, 48(3), 231–252 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. SRINIL, N. and REGA, G. Two-to-one resonant multi-modal dynamics of horizontal/inclined cables, part II: internal resonance activation, reduced-order models and nonlinear normal modes. Nonlinear Dynamics, 48(3), 253–274 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. PAULSEN, W. and MANNING, M. Finding vibrations of inclined cable structures by approximately solving governing equations for exterior matrix. Applied Mathematics and Mechanics (English Edition), 36(11), 1383–1402 (2015) https://doi.org/10.1007/s10483-015-1990-7

    Article  MathSciNet  MATH  Google Scholar 

  33. ZHAO, Y. and WANG, L. On the symmetric modal interaction of the suspended cable: three-to-one internal resonance. Journal of Sound and Vibration, 294(4/5), 1073–1093 (2006)

    Article  Google Scholar 

  34. GUO, T. D., KANG, H. J., WANG, L. H., and ZHAO, Y. Y. Cable’s mode interactions under vertical support motions: boundary resonant modulation. Nonlinear Dynamics, 84(3), 1259–1279 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. GUO, T. D., KANG, H. J., WANG, L. H., and ZHAO, Y. Y. A boundary modulation formulation for cable’s non-planar coupled dynamics under out-of-plane support motion. Archive of Applied Mechanics, 86(4), 729–741 (2016)

    Article  Google Scholar 

  36. GUO, T. D., KANG, H. J., WANG, L. H., and ZHAO, Y. Y. Cable dynamics under non-ideal support excitations: nonlinear dynamic interactions and asymptotic modeling. Journal of Sound and Vibration, 384, 253–272 (2016)

    Article  Google Scholar 

  37. FUJINO, Y., WARNITCHAI, P., and PACHECO, B. M. An experimental and analytical study of autoparametric resonance in a 3DOF model of cable-stayed-beam. Nonlinear Dynamics, 4(2), 111–138 (1993)

    Google Scholar 

  38. FUJINO, Y. and XIA, Y. Auto-parametric vibration of a cable-stayed-beam structure under random excitation. Journal of Engineering Mechanics, 132(3), 279–286 (2006)

    Article  Google Scholar 

  39. GATTULLI, V., MORANDINI, M., and PAOLONE, A. A parametric analytical model for nonlinear dynamics in cable-stayed beam. Earthquake Engineering and Structural Dynamics, 31(6), 1281–1300 (2002)

    Article  Google Scholar 

  40. LENCI, S. and RUZZICONI, L. Nonlinear phenomena in the single-mode dynamics of a cable-supported beam. International Journal of Bifurcation and Chaos, 19(3), 923–945 (2009)

    Article  MathSciNet  Google Scholar 

  41. KANG, H. J., GUO, T. D., ZHAO, Y. Y., FU, W. B., and WANG, L. H. Dynamic modeling and in-plane 1:1:1 internal resonance analysis of cable-stayed bridge. European Journal of Mechanics- A/Solids, 62, 94–109 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. CAO, D. Q., SONG, M. T., ZHU, W. D., TUCKER, R. W., and WANG, C. H. T. Modeling and analysis of the in-plane vibration of a complex cable-stayed bridge. Journal of Sound and Vibration, 331(26), 5685–5714 (2012)

    Article  Google Scholar 

  43. CAETANO, E., CUNHA, A., GATTULLI, V., and LEPIDI, M. Cable-deck dynamic interactions at the International Guadiana Bridge: on-site measurements and finite element modelling. Structural Control and Health Monitoring, 15(3), 237–264 (2010)

    Article  Google Scholar 

  44. CONG, Y. Y., KANG, H. J., and SU, X. Y. Cable-stayed shallow arch modeling and in-plane free vibration analysis of cable-stayed bridge with CFRP cables. Chinese Journal of Solid Mechanics, 39(3), 316–327 (2018)

    Google Scholar 

  45. NAYFEH, A. H. and MOOK, D. T. Nonlinear Oscillations, Wiley, New York (1979)

    MATH  Google Scholar 

  46. CONG, Y. Y., KANG, H. J., and GUO, T. D. Planar multimodal 1:2:2 internal resonance analysis of cable-stayed bridge. Mechanical Systems and Signal Processing, 120, 505–523 (2019)

    Article  Google Scholar 

  47. KANG, H. J., ZHAO, Y. Y., and ZHU, H. P. In-plane non-linear dynamics of the stay cables. Nonlinear Dynamics, 73(3), 1385–1398 (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Civil Engineering, Hunan University, Changsha, 410082, China

    Yunyue Cong, Houjun Kang & Tieding Guo

  2. Key Laboratory for Damage Diagnosis of Engineering Structures of Hunan Province, Hunan University, Changsha, 410082, China

    Houjun Kang

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  1. Yunyue Cong
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  2. Houjun Kang
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  3. Tieding Guo
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Correspondence to Houjun Kang.

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

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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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  • DOI: https://doi.org/10.1007/s10483-019-2497-8

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