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Bifurcations and chaos of an inclined cable. (English) Zbl 1176.74089
Summary: The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kovačič and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.

74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
74H65 Chaotic behavior of solutions to dynamical problems in solid mechanics
Full Text: DOI
[1] Irvine, H.M., Caughey, T.K.: The linear theory of free vibrations of a suspended cable. Proc. R. Soc. Lond. Ser. A 341, 299–315 (1974) · doi:10.1098/rspa.1974.0189
[2] Irvine, H.M.: Cable Structures. MIT Press, Cambridge (1981)
[3] Carrier, G.F.: On the non-linear vibration problem of the elastic string. Q. Appl. Math. 3, 157–165 (1945) · Zbl 0063.00715
[4] Carrier, G.F.: A note on vibration string. Q. Appl. Math. 7, 97–101 (1949) · Zbl 0033.03003
[5] Meirovitch, L.: Elements of Vibration Analysis. McGraw-Hill, New York (1975) · Zbl 0359.70039
[6] West, H.H., Geschwindner, L.F., Suhoski, J.E.: Natural vibrations of suspension cables. J. Struct. Div. ST11, 2277–2291 (1975)
[7] Henghold, W.M., Russell, J.J.: Equilibrium and natural frequencies of cable structures (A nonlinear finite element approach). Comput. Struct. 6, 267–271 (1976) · Zbl 0332.73081 · doi:10.1016/0045-7949(76)90001-8
[8] Perkins, N.C.: Modal interactions in the nonlinear response of elastic cables under parametric/excitation. Int. J. Non-Linear Mech. 27(2), 233–250 (1992) · Zbl 0794.73033 · doi:10.1016/0020-7462(92)90083-J
[9] Takahashi, K., Konishi, Y.: Non-linear vibrations of cables in three dimensions, Part II: out-of-plane vibrations under in-plane sinusoidally time-varying load. J. Sound Vib. 118(1), 85–97 (1987) · doi:10.1016/0022-460X(87)90256-2
[10] Rega, G., Lacarbonara, W., Nayfeh, A.H., Chin, C.M.: Multiple resonances in suspended cables: direct versus reduced-order models. Int. J. Non-Linear Mech. 34(5), 901–924 (1999) · Zbl 1068.74562 · doi:10.1016/S0020-7462(98)00065-1
[11] Benedettini, F., Rega, G., Alaggio, R.: Nonlinear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions. J. Sound Vib. 182(5), 775–798 (1995) · doi:10.1006/jsvi.1995.0232
[12] Rega, G., Lacarbonara, W., Nayfeh, A.H., Chin, C.-M.: Multimodal resonances in suspended cables via a direct perturbation approach. In: Proceedings of ASME DETC97, vol. DETC97/VIB-4101, Sacramento, CA, 14–17 September 1997
[13] Pilipchuk, V.N., Ibrahim, R.A.: Non-linear modal interactions in shallow suspended cables. J. Sound Vib. 227(1), 1–28 (1999) · doi:10.1006/jsvi.1999.2326
[14] Zheng, G., Ko, J.M., Ni, Y.O.: Super-harmonic and internal resonances of a suspended cable with nearly commensurable natural frequencies. Nonlinear Dyn. 30, 55–70 (2002) · Zbl 1049.74620 · doi:10.1023/A:1020395922392
[15] Nayfeh, A.H., Arafat, H., Chin, C.M., Lacarbonara, W.: Multimode interactions in suspended cables. J. Vib. Control 8, 337–387 (2002) · Zbl 1107.74314 · doi:10.1177/107754602023687
[16] Zhao, Y.Y., Wang, L.H., Chen, D.L., Jiang, L.Z.: Non-linear dynamics analysis of the two-dimensional simplified model of an elastic cable. J. Sound Vib. 255(1), 43–59 (2002) · doi:10.1006/jsvi.2001.4151
[17] Malhotra, N., Sri Namachchivaya, N., McDonald, R.J.: Multipulse orbits in the motion of flexible spinning discs. J. Nonlinear Sci. 12(1), 1–26 (2002) · Zbl 1132.74303 · doi:10.1007/s00332-001-0367-y
[18] Haller, G., Wiggins, S: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrödinger equation. Physica D 85, 311–347 (1995) · Zbl 0890.58048 · doi:10.1016/0167-2789(95)00120-S
[19] Yeo, M.H., Lee, W.K.: Evidence of global bifurcations of an imperfect circular plate. J. Sound Vib. 293(1), 138–155 (2006) · doi:10.1016/j.jsv.2005.09.035
[20] Kovačič, G., Wiggins, S.: Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation. Physica D 57, 185–225 (1992) · Zbl 0755.35118 · doi:10.1016/0167-2789(92)90092-2
[21] Reilly, O.M., Holmes, P.J.: Non-linear, non-planar and non-periodic vibrations of a string. J. Sound Vib. 153(3), 413–435 (1992) · Zbl 0924.73127 · doi:10.1016/0022-460X(92)90374-7
[22] Reilly, O.M.: Global bifurcations in the forced vibration of a damped string. Int. J. Non-Linear Mech. 28(3), 337–351 (1993) · Zbl 0776.73038 · doi:10.1016/0020-7462(93)90040-R
[23] Zhang, W., Tang, Y.: Global dynamics of the cable under combined parametrical and external excitations. Int. J. Non-Linear Mech. 37(4), 505–526 (2002) · Zbl 1341.74066 · doi:10.1016/S0020-7462(01)00026-9
[24] Tien, W., Sri Namachchivaya, N., Bajaj, A.K.: Non-linear dynamics of a shallow arch under periodic excitation-I. 1:2 internal resonances. Int. J. Non-Linear Mech. 29(3), 349–366 (1994) · Zbl 0808.73043 · doi:10.1016/0020-7462(94)90007-8
[25] Tien, W., Sri Namachchivaya, N., Malhotra, N.: Non-linear dynamics of a shallow arch under periodic excitation-II. 1:1 internal resonance. Int. J. Non-Linear Mech. 29(3), 367–386 (1994) · Zbl 0808.73044 · doi:10.1016/0020-7462(94)90008-6
[26] Sanders, J.A., Verhulst, F.: Averaging Methods in Non-linear Dynamical System. Applied Mathematical Sciences, vol. 59. Springer, New York (1985) · Zbl 0586.34040
[27] Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field. Springer, New York (1983) · Zbl 0515.34001
[28] Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Ind. Univ. Math. J. 21, 193–225 (1971) · Zbl 0246.58015 · doi:10.1512/iumj.1971.21.21017
[29] Nusse, H.E., York, J.A.: Dynamics: Numerical Explorations. Springer, New York (1997)
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