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

Yunyue CONG, Houjun KANG, Tieding GUO

doi:10.1007/s10483-019-2497-8

2019 , Vol. 40 >Issue 7: 977 - 1000

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

Articles

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

Expand

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

Received date: 2018-10-12

Revised date: 2019-01-08

Online published: 2019-07-01

Supported by

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

Fold

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.

Key words： singularity; velocity potential; the transient problem; the harmonic problem; internal resonance; multi-scale method; nonlinear dynamics; primary resonance; cable-stayed system

Cite this article

Yunyue CONG, Houjun KANG, Tieding GUO . Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables' external excitations[J]. Applied Mathematics and Mechanics, 2019 , 40(7) : 977 -1000 . DOI: 10.1007/s10483-019-2497-8

References

[1] SCHREYER, H. L. and MASUR, E. F. Buckling of shallow arches. Journal of the Engineering Mechanics Division, 92, 1-20(1966)
[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)
[3] LEVITAS, J., SINGER, J., and WELLER, T. Global dynamic stability of a shallow arch by Poincaré-like simple cell mapping. International Journal of Non-Linear Mechanics, 32(2), 411-424(1997)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[14] LAKRAD, F. and SCHIEHLEN, W. Effects of a low frequency parametric excitation. Chaos Solitons and Fractals, 22(5), 1149-1164(2004)
[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)
[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)
[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)
[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)
[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)
[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)
[21] DENG, L., CAO, R., WANG, W., and YIN, X. F. A multi-point tire model for studying bridgevehicle coupled vibration. International Journal of Structural Stability and Dynamics, 16(8), 1550047(2016)
[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)
[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)
[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
[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)
[26] DING, H., HUANG, L. L., MAO, X. Y., and CHEN, L. Q. Primary resonance of traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38(1), 1-14(2017) https://doi.org/10.1007/s10483-016-2152-6
[27] REGA, G. Nonlinear vibrations of suspended cables-part I:modeling and analysis. Applied Mechanics Reviews, 57(6), 443-478(2004)
[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)
[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)
[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)
[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)
[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
[33] ZHAO, Y. and WANG, L. On the symmetric modal interaction of the suspended cable:three-toone internal resonance. Journal of Sound and Vibration, 294(4/5), 1073-1093(2006)
[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)
[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)
[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)
[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)
[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)
[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)
[40] LENCI, S. and RUZZICONI, L. Nonlinear phenomena in the single-mode dynamics of a cablesupported beam. International Journal of Bifurcation and Chaos, 19(3), 923-945(2009)
[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 MechanicsA/Solids, 62, 94-109(2016)
[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)
[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)
[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)
[45] NAYFEH, A. H. and MOOK, D. T. Nonlinear Oscillations, Wiley, New York (1979)
[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)
[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)

Options

Outlines

模态框（Modal）标题

Abstract

Cite this article

References