Nonlinear responses of a cable-beam coupled system under parametric and external excitations. (English) Zbl 1293.74212
Summary: This paper analytically investigates the nonlinear responses of a cable-beam coupled system under the combined effects of internal and external resonance. The cable is considered a geometric nonlinearity, and the beam is considered as Euler-Bernoulli model, but it is coupled by fixing it at one end. The coupled nonlinear differential equations are formulated by using the Hamilton principle. The spatial problem is solved by using Galerkin’s method to simplify the governing equations to a set of ordinary differential equations. Applying the multiple time scales method to the ordinary differential equations, the first approximate solutions and solvability condition are derived. The effects of the cable sag to span ratio, mass ratio, and stiffness ratio on the nonlinear responses are investigated. The results show good agreement between the analytical and numerical solutions especially near the external resonance frequency.
MSC:
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74H10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics |
Keywords:
cable-beam coupled system; amplitude-frequency internal resonance; external resonance; nonlinear responseReferences:
| [1] | Yu W.Q., Chen F.Q.: Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables. Arch. Appl. Mech. 81, 1231-1252 (2011) · Zbl 1271.74229 · doi:10.1007/s00419-011-0520-5 |
| [2] | Nayfeh A.H., Arafat H.N., Chin C.M., Lacarbonara W.: Multi-mode interactions in suspended cables. J. Vib. Control 8, 337-387 (2002) · Zbl 1107.74314 · doi:10.1177/107754602023687 |
| [3] | Younesian D., Esmailzadeh E.: Non-linear vibration of variable speed rotating viscoelastic beams. Nonlinear Dyn. 60, 193-205 (2010) · Zbl 1189.74049 · doi:10.1007/s11071-009-9589-6 |
| [4] | Ghayesh M.H.: Subharmonic dynamics of an axially accelerating beam. Arch. Appl. Mech. 82, 1169-1181 (2012) · Zbl 1293.74175 · doi:10.1007/s00419-012-0609-5 |
| [5] | Wang L., Ni Q., Huang Y.Y.: Bifurcations and chaos in a forced cantilever system with impacts. J. Sound Vib. 296(4-5), 1068-1078 (2006) · doi:10.1016/j.jsv.2006.03.016 |
| [6] | Mamandi A., Kargarnovin M.H., Farsi S.: An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions. Int. J. Mech. Sci. 52, 1694-1708 (2010) · doi:10.1016/j.ijmecsci.2010.09.003 |
| [7] | Mamandi A., Kargarnovin M.H.: Dynamic analysis of an inclined Timoshenko beam traveled by successive moving masses/forces with inclusion of geometric nonlinearities. Acta Mech. 218, 9-29 (2011) · Zbl 1300.74027 · doi:10.1007/s00707-010-0400-z |
| [8] | Holubová G., Matas A.: Initial-boundary value problem for the nonlinear string-beam system. J. Math. Anal. Appl. 288, 784-802 (2003) · Zbl 1037.35087 · doi:10.1016/j.jmaa.2003.09.028 |
| [9] | Yau J.D., Yang Y.B.: Vibration reduction for cable-stayed bridges traveled by high-speed trains. Finite Elem. Anal. Des. 40, 341-359 (2004) · doi:10.1016/S0168-874X(03)00051-9 |
| [10] | Xia Y., Fujino Y.: Auto-parametric vibration of a cable-stayed-beam structure under random excitation. ASCE J. Eng. Mech. 132, 279-286 (2006) · doi:10.1061/(ASCE)0733-9399(2006)132:3(279) |
| [11] | Zhang W., Cao D.X.: Studies on bifurcation and chaos of a string-beam coupled system with two degrees-of-freedom. Nonlinear Dyn. 45, 131-147 (2006) · Zbl 1138.74345 · doi:10.1007/s11071-006-2423-5 |
| [12] | Cao D.X., Zhang W.: Global bifurcations and chaotic dynamics for a string-beam coupled system. Chaos Solitons Fractals 37, 858-875 (2008) · Zbl 1163.74025 · doi:10.1016/j.chaos.2006.09.072 |
| [13] | Gattulli V., Lepidi M.: Nonlinear interactions in the planar dynamics of cable-stayed beam. Int. J. Solids Struct. 40, 4729-4748 (2003) · Zbl 1041.74514 · doi:10.1016/S0020-7683(03)00266-X |
| [14] | Gattulli V., Lepidi M., Macdonald J.H.G., Taylor C.A.: One-to-two global-local interaction in a cable-stayed beam observed through analytical, finite element and experimental models. Int. J. Nonlinear Mech. 40, 571-588 (2005) · Zbl 1349.74216 · doi:10.1016/j.ijnonlinmec.2004.08.005 |
| [15] | Zu J.W., Cheng G.: Dynamic analysis of an optical fiber coupler in telecommunications. J. Sound Vib. 268, 15-31 (2003) · doi:10.1016/S0022-460X(02)01575-4 |
| [16] | Wu Q., Takahashi K., Okabayashi T., Nakamura S.: Response characteristics of local vibrations in stay cables on an existing cable-stayed bridge. J. Sound Vib. 261, 403-420 (2003) · doi:10.1016/S0022-460X(02)01088-X |
| [17] | Brownjohn J.M.W., Cheong B., Lee J.: Dynamic performance of a curved cable-stayed bridge. Eng. Struct. 21, 1015-1027 (1999) · doi:10.1016/S0141-0296(98)00046-7 |
| [18] | Caetano E., Cunha A., Taylor C.A.: Investigation of dynamic cable—deck interaction in a physical model of a cable-stayed bridge. Part II: seismic response. Earthq. Eng. Struct. 29, 499-521 (2000) · doi:10.1002/(SICI)1096-9845(200004)29:4<499::AID-EQE919>3.0.CO;2-A |
| [19] | Au F.T.K., Cheng Y.S., Cheung Y.K., Zheng D.Y.: On the determination of natural frequencies and mode shapes of cable-stayed bridges. Appl. Math. Model. 25, 1099-1115 (2001) · Zbl 1169.74303 · doi:10.1016/S0307-904X(01)00035-X |
| [20] | Nayfeh A.H., Mook D.T.: Nonlinear oscillations. Wiley, New York (1979) · Zbl 0418.70001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
