Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Parameter analyses of suspended cables subjected to simultaneous combination, super and sub-harmonic excitations

  • Zhao, Yaobing (College of Civil Engineering, Huaqiao University) ;
  • Zheng, Panpan (College of Civil Engineering, Huaqiao University)
  • Received : 2019.12.12
  • Accepted : 2020.11.02
  • Published : 2021.07.25
⟨ Previous Next ⟩

Abstract

The present study is dealing with a geometrically nonlinear model of the suspended cable subjected to multi-frequency excitations. In particular, two of the super, sub-harmonic, and combination resonances are excited simultaneously here. The nonlinear integro-differential equations of the suspended cable are introduced, together with the Galerkin method, to obtain a reduced-order model, whose responses are investigated by solving the reduced ordinary differential equations. Then, the obtained single-mode discretization equations are solved using the method of multiple scales in the frequency regions of three simultaneous resonant cases, with the stability characteristics determined. Effects of parameters on resonance characteristics are carried out by investigating several numerical examples. Numerical results demonstrate that the two-frequency excitation has significant influences on the dynamical behaviors of the nonlinear system. Each harmonic excitation component's contribution to the overall resonant responses is mainly dependent on its excitation amplitude. The stable steady-state solutions are confirmed by using numerical integration, and the number of steady-state solutions varies from one to seven as to different parameters of the system in simultaneous resonances. Besides, it is of great significance to include the effects of excitation phase differences on nonlinear vibration characteristics.

Keywords

Acknowledgement

The research described in this paper was financially supported by the National Natural Science Foundation of China (11602089), and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-YX505).

References

  1. Arafat, H.N. and Nayfeh. A.H. (2003), "Non-linear responses of suspended cables to primary resonance excitations", J. Sound Vib., 266(2), 325-354. https://doi.org/10.1016/S0022-460X(02)01393-7.
  2. Bayat, M., Paker, I. and Bayat, M. (2017), "Nonlinear vibration of multi-body systems with linear and nonlinear springs", Steel Compos. Struct., 25(4), 497-503. https://doi.org/10.12989/scs.2017.25.4.497.
  3. Bemmo, D.T., Siewe, M.S. and Tchawoua, C. (2011), "Nonlinear oscillations of the FitzHugh-Nagumo equations under combined external and two-frequency parametric excitations", Phys. Lett. A., 375(19), 1944-1953. https://doi.org/10.1016/j.physleta.2011.02.072.
  4. Bitar, D., Kacem, N., Bouhaddi, N. and Collet, M. (2015), "Collective dynamics of periodic nonlinear oscillators under simultaneous parametric and external excitations", Nonlinear Dynam., 82(1-2), 749-766. https://doi.org/10.1007/s11071-015-2194-y.
  5. Bauomy, H.S. and El-Sayed, A.T. (2018), "Vibration performance of a vertical conveyor system under two simultaneous resonances", Arch. Appl. Mech., 88(8), 1349-1368. https://doi.org/10.1007/s00419-018-1375-9.
  6. Breslavsky, I.D. and Amabili, M. (2018), "Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation", Nonlinear Dynam., 93(1), 53-62. https://doi.org/10.1007/s11071-017-3983-2.
  7. Benedettini, F., Rega, G. and Alaggio, R. (1995), "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. https://doi.org/10.1006/jsvi.1995.0232.
  8. Cong, Y. and Kang, H. (2019), "Planar nonlinear dynamic behavior of a cable-stayed bridge under excitation of tower motion", Eur. J. Mech. A-Solid., 76, 91-107. https://doi.org/10.1016/j.euromechsol.2019.03.010.
  9. Cong, Y., Kang, H. and Yan, G. (2021) "Investigation of dynamic behavior of a cable-stayed cantilever beam under two-frequency excitations", Int. J. Nonlinear Mech., 129, 103670. https://doi.org/10.1016/j.ijnonlinmec.2021.103670.
  10. Chen, H., Hou, L., Chen, Y. and Yang, R. (2017), "Dynamic characteristics of flexible rotor with squeeze film damper excited by two frequencies", Nonlinear Dynam., 87(4), https://doi.org/10.1007/s11071-016-3204-4.
  11. Cheng, G. and Zu, J.W. (2004), "Dynamics of a dry friction oscillator under two-frequency excitations", J. Sound Vib., 275(35), 591-603. https://doi.org/10.1016/j.jsv.2003.06.027.
  12. Deng, H.Q., Li, T.J., Xue, B.J. and Wang, Z.W. (2015), "Analysis of thermally induced vibration of cable-beam structures", Struct. Eng. Mech., 53(3), 443-453. https://doi.org/10.12989/sem.2015.53.3.443.
  13. Ding, H. and Zu, J.W. (2013), "Periodic and chaotic responses of an axially Accelerating viscoelastic beam under two-frequency excitations", Int. J. Appl. Mech., 5(2), 1350019. https://doi.org/10.1142/S1758825113500191.
  14. Ding, H. (2016), "Steady-state responses of a belt-drive dynamical system under dual excitations", Acta Mech. Sinica., 32(1), 156-169. https://doi.org/10.1007/s10409-015-0510-x.
  15. Fang, F., Xia, G. and Wang, J. (2018), "Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations", Acta Mech. Sinica., 34(3), 561-577. https://doi.org/10.1007/s10409-017-0743-y.
  16. Gattulli, V., Lepidi, M., Potenza, F. and Sabatino, U.D. (2019), "Modal interactions in the nonlinear dynamics of a beam-cablebeam", Nonlinear Dynam., 96(4), 2547-2566. https://doi.org/10.1007/s11071-019-04940-8.
  17. Guo, T., Kang, H., Wang, L., Liu, Q. and Zhao, Y. (2018), "Modal resonant dynamics of cables with a flexible support: a modulated diffraction problem", Mech. Syst. Signal Pr., 106, 229-248. https://doi.org/10.1016/j.ymssp.2017.12.023.
  18. Irvine, H. (1981), "Cable Structures", MIT Press.
  19. Ilyas, S., Alfosail, F.K. and Younis, M.I. (2019a), "On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators", J. Comput. Nonlinear Dynam., 14, 041006-1-8. https://doi.org/10.1115/1.4042694.
  20. Ilyas, S., Alfosail, F.K., Bellaredj, M.L.F. and Younis, M.I. (2019b), "On the response of MEMS resonators under generic electrostatic loadings: experiments and applications", Nonlinear Dynam., 95(3), 2263-2274. https://doi.org/10.1007/s11071-018-4690-3 .
  21. Kacem, N., Baguet, S., Duraffourg, L., Jourdan, G., Dufour, R. and Hentz, S. (2015), "Overcoming limitations of nanomechanical resonators with simultaneous resonances" Appl. Phys. Lett., 107, 073105. http://doi.org/10.1063/1.4928711.
  22. Kalita, B. and Dwivedy, S.K. (2019), "Nonlinear dynamics of a parametrically excited pneumatic artificial muscle (PAM) actuator with simultaneous resonance condition", Mech. Mach. Theory., 135, 281-297. https://doi.org/10.1016/j.mechmachtheory.2019.01.031.
  23. Kim, K.W., Park, N.C., Song, W.K., Kim, M.K. and Parnichkun, M. (2019), "Sensitivity Analysis of Cable Parameters on Tension of a Suspended Cable under Combination Resonances", Int. J. Struct. Stab. Dynam., 19(3), 1950025. https://doi.org/10.1142/S0219455419500251.
  24. Kang, H., Guo, T., Zhu, W., Su, J. and Zhao, B. (2019), "Dynamical modeling and non-planar coupled behavior of inclined CFRP cables under simultaneous internal and external resonances", Appl. Math. Mech.-Engl., 40(5), 649-678. https://doi.org/10.1007/s10483-019-2472-6.
  25. Kang, H., Cong, Y. and Yan, G. (2020), "Theoretical analysis of dynamic behaviors of cable-stayed bridges excited by two harmonic forces", Nonlinear Dynam. 95(3), 2263-2274. https://doi.org/10.1007/s11071-020-05763-8
  26. Luongo, A. and Zulli, D. (2013), "Mathematical Models of Beams and Cables". New York, Wiley.
  27. Li, Z. J., Li, P., He, Z. and Cao, P. (2013), "Static and free vibration analysis of shallow sagging inclined cables", Struct. Eng. Mech., 45(2), 145-157. https://doi.org/10.12989/sem.2013.45.2.145.
  28. Lacarbonara, W. (2013), "Nonlinear Structural Mechanics: Theory", Dynamical Phenomena and Modeling, 1st edn. Springer.
  29. Li, H. Y., Li, J., Lang, T. Y. and Zhu, X. (2017), "Dynamics of an axially moving unidirectional plate partially immersed in fluid under two frequency parametric excitation", Int. J. Nonlinear Mech., 99, 31-39. https://doi.org/10.1016/j.ijnonlinmec.2017.10.019.
  30. Mao, X.Y., Ding, H., Lim, C.W. and Chen, L.Q. (2016), "Superharmonic resonance and multi-frequency responses of a supercritical translating beam", J. Sound Vib., 385, 267-283. https://doi.org/10.1016/j.jsv.2016.08.032.
  31. Michon, G., Manin, L., Remond, D., Dufour, R. and Parker, R.G. (2008), "Parametric instability of an axially moving belt subjected to multifrequency excitations: experiments and analytical validation", J. Appl. Mech., 75(4), 041004-1-8. https://doi.org/10.1115/1.2910891.
  32. Nguyen, H. (2013), "Simultaneous resonances involving two mode shapes of parametrically-excited rectangular plates", J. Sound Vib., 332(20), 5103-5114. https://doi.org/10.1016/j.jsv.2013.04.010.
  33. Nayfeh, A.H. (1985), "The response of non-linear single-degreeof-freedom systems to multifrequency excitations", J. Sound Vib., 102(3), https://doi.org/10.1016/S0022-460X(85)80150-4.
  34. Nayfeh, A.H. and Jebril, A.E.S. (1987), "The response of twodegree-of-freedom systems with quadratic and cubic nonlinearities to multi-frequency parametric excitations", J. Sound Vib., 115(1), 83-101. https://doi.org/10.1016/0022-460X(87)90493-7.
  35. Nayfeh, A.H. and Mook, D.T. (1979), "Nonlinear Oscillations", John Wiley Sons, Inc.
  36. Nayfeh A.H. and Pai, F.P. (2004), "Linear and Nonlinear structural Mechanics", A John Wiely Sons, Inc. Hoboken, New Jersy.
  37. Plaut, R.H., Haquang, N. and Mook, D.T. (1986), "Simultaneous resonances in non-linear structural vibrations under twofrequency excitation", J. Sound Vib., 106(3), 361- https://doi.org/10.1016/0022-460X(86)90184-7.
  38. Plaut, R.H., Gentry, J.J. and Mook, D.T. (1990), "Resonances in non-linear structural vibrations involving two external periodic excitations", J. Sound Vib., 140(3), 371-379. https://doi.org/10.1016/0022-460X(90)90756-P.
  39. Plaut, R.H. and Hsieh, J.C. (1985), "Oscillations and instability of a shallow-arch under two-frequency excitation", J. Sound Vib., 102(2), 189-201. https://doi.org/10.1016/s0022-460x(85)80052-3.
  40. Rezaiee-Pajand, M., Masoodi, A.R. and Rajabzadeh-Safaei, N. (2019), "Nonlinear vibration analysis of carbon nanotube reinforced composite plane structures", Steel Compos. Struct., 30(6), 493-516. https://doi.org/10.12989/scs.2019.30.6.493.
  41. Rega, G. (2004a), "Nonlinear vibrations of suspended cables, Part I: Modeling and analysis", Appl. Mech. Rev., 57(6), 443-478. https://doi.org/10.1115/1.1777224.
  42. Rega, G. (2004b), "Nonlinear vibrations of suspended cables, Part II: Deterministic phenomena", Appl. Mech. Rev., 57(6), 479-514. https://doi.org/10.1115/1.1777225.
  43. Rezaei, M., Talebitooti, R. and Rahmanian, S. (2019), "Efficient energy harvesting from nonlinear vibrations of PZT beam under simultaneous resonances", Energy, 182, 369-380. https://doi.org/10.1016/j.energy.2019.05.212.
  44. Rega, G., Lacarbonara, W., Nayfeh, A.H. and Chin, C.M. (1999), "Multiple resonances in suspended cables: direct versus reduced-order models", Int. J. Nonlinear Mech., 34(5), 901-924. https://doi.org/10.1016/S0020-7462(98)00065-1.
  45. Sun, C., Zhao, Y., Peng, J., Kang, H. and Zhao, Y. (2018), "Multiple internal resonances and modal interaction processes of a cable-stayed bridge physical model subjected to an invariant single-excitation", Eng. Struct., 172(1), 938-955. https://doi.org/10.1016/j.engstruct.2018.06.088.
  46. Sahoo, B. (2019), "Nonlinear dynamics of a viscoelastic traveling beam with time-dependent axial velocity and variable axial tension", Nonlinear Dynam., 97(1), 269-296. https://doi.org/10.1007/s11071-019-04969-9.
  47. Sahoo, B. (2020), "Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under twofrequency parametric excitations and internal resonance". Nonlinear Dynam, 99, 945-979. https://doi.org/10.1007/s11071-019-05264-3.
  48. Sahoo, B. (2021), "Nonlinear vibration analysis of a hinged-clamped beam moving with pulsating speed and subjected to internal resonance", Int. J. Struct. Stab. Dynam., 21(8), 2150117. https://doi.org/10.1142/S0219455421501170.
  49. Sahoo, B., Panda, L.N. and Pohit, G. (2015a), "Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation", Int. J Nonlinear Mech., 78, 35-44. https://doi.org/10.1016/j.ijnonlinmec.2015.09.017.
  50. Sahoo, B., Panda, L. and Pohit, G. (2015b), "Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam", Nonlinear Dynam., 82(4), 1721-1742. https://doi.org/10.1007/s11071-015-2272-1.
  51. Sahoo, B, Panda, L.N. and Pohit, G. (2017), "Stability, bifurcation and chaos of a traveling viscoelastic beam tuned to 3:1 internal resonance and subjected to parametric Excitation", Int. J. Bifur. Chaos, 27(2), 1750017. https://doi.org/10.1142/S0218127417500171.
  52. Warminski, J., Zulli, D., Rega, G. and Latalski, J. (2016), "Revisited modelling and multimodal nonlinear oscillations of a sagged cable under support motion", Meccanica., 51(11), 2541-2575. https://doi.org/10.1007/s11012-016-0450-y.
  53. Yi, Z., Yuan, M., Tu, G. and Zeng, Y. (2019), Modeling of the multi-cable supported arch and a novel technique to investigate the natural vibratory characteristics", Appl. Math. Model., 75, 640-662. https://doi.org/10.1016/j.apm.2019.05.055.
  54. Yang, S.P., Nayfeh, A.H. and Mook, D.T. (1998), "Combination resonances in the response of the Duffing oscillator to a threefrequency excitation", Acta Mech., 131(3), 235-245. https://doi.org/10.1007/bf01177227.
  55. Zhang, W.F., Liu, Y.C., Ji, J. and Teng, Z.C. (2014), "Analysis of dynamic behavior for truss cable structures", Steel Compos. Struct., 16(2), 117-133. https://doi.org/10.12989/scs.2014.16.2.117.
  56. Zhao, Y., Guo, Z., Huang, C., Chen, L. and Li, S. (2018a), "Analytical solutions for planar simultaneous resonances of suspended cables involving two external periodic excitations", Acta Mech., 229(11), 4393-4411. https://doi.org/10.1007/s00707-018-2224-1.
  57. Zhao, Y., Huang, C., Chen, L. and Peng, J. (2018b), "Nonlinear vibration behaviors of suspended cables under two-frequency excitation with temperature effects", J. Sound Vib., 416, 279-294. https://doi.org/10.1016/j.jsv.2017.11.035.
  58. Zhao, Y., Lin, H., Chen, L. and Wang, C. (2019), "Simultaneous resonances of suspended cables subjected to primary and superharmonic excitations in thermal environments", Int. J. Struct. Stab. Dyam., 19(12), 1950155 https://doi.org/10.1142/S0219455419501554.
  59. Zhao, Y., Zheng, P., Lin, H. and Huang, C. (2021), "Temperature effects on dynamic properties of suspended cables subjected to dual harmonic excitations", J. Shock Vib., 8883036. https://doi.org/10.1155/2021/8883036.