Reduced-Order Model for the Nonlinear Dynamics of Cables
Publication: Journal of Engineering Mechanics
Volume 148, Issue 9
Abstract
Formulation of governing equations for an elastic cable have a long and dated history. A unified framework to detect the dynamics of such systems is detailed, justified, and assessed numerically. Modal analyses are performed in the Frenet basis, which describes the motion in the local frame and accounts accurately for the system’s physics. In this article, a methodology to produce arbitrary reduced-order models for cable nonlinear dynamics is provided. The results obtained from the latter via direct time integration are challenged numerically via a comparison with results of the nonlinear finite-element method.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
The authors confirm that the data supporting the findings of this study are available within the article.
Acknowledgments
The authors gratefully thank the Ministère de la Transition Écologique for financial support.
References
Antman, S. 1995. Nonlinear problems of elasticity. New York: Springer.
Arena, A., A. Pacitti, and W. Lacarbonara. 2016. “Nonlinear response of elastic cables with flexural-torsional stiffness.” Int. J. Solids Struct. 87 (Jun): 267–277. https://doi.org/10.1016/j.ijsolstr.2015.09.019.
Benedettini, F., and G. Rega. 1987. “Non-linear dynamics of an elastic cable under planar excitation.” Int. J. Non Linear Mech. 22 (6): 497–509. https://doi.org/10.1016/0020-7462(87)90039-4.
Benedettini, F., and G. Rega. 1997. “Experimental investigation of the nonlinear response of a hanging cable. Part II: Global analysis.” Nonlinear Dyn. 14 (2): 119–138. https://doi.org/10.1023/A:1008259120942.
Benedettini, F., G. Rega, and R. Alaggio. 1995. “Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions.” J. Sound Vibr. 182 (5): 775–798. https://doi.org/10.1006/jsvi.1995.0232.
Benedettini, F., G. Rega, and F. Vestroni. 1986. “Modal coupling in the free nonplanar finite motion of an elastic cable.” Meccanica 21 (1): 38–46. https://doi.org/10.1007/BF01556315.
Berlioz, A., and C.-H. Lamarque. 2005. “A non-linear model for the dynamics of an inclined cable.” J. Sound Vib. 279 (3–5): 619–639. https://doi.org/10.1016/j.jsv.2003.11.069.
Bertrand, C., V. Acary, C.-H. Lamarque, and A. Ture Savadkoohi. 2020a. “A robust and efficient numerical finite element method for cables.” Int. J. Numer. Methods Eng. 121 (18): 4157–4186. https://doi.org/10.1002/nme.6435.
Bertrand, C., C. Plut, A. T. Savadkoohi, and C.-H. Lamarque. 2020b. “On the modal response of mobile cables.” Eng. Struct. 210 (May): 110231. https://doi.org/10.1016/j.engstruct.2020.110231.
Crisfield, M. 1997. Nonlinear finite element analysis of solid and structures. New York: Wiley.
Davis, M. 2010. Lecture notes: Finite difference methods, course in mathematics and finance. London: Imperial College.
Ferretti, M., D. Zulli, and A. Luongo. 2019. “A continuum approach to the nonlinear in-plane galloping of shallow flexible cables.” In Advances in mathematical physics, 2019. London: Hindawi.
Floquet, G. 1883. “Sur les équations différentielles linéaires à coefficients périodiques.” Ann. Sci. Ec. Norm. Super. 12: 47–88. https://doi.org/10.24033/asens.220.
Galileo, G. 1638. “Discorsi e dimostrazioni matematiche intorno a due nuove scienze.” Edizione Naz. sotto gli auspicii di sua maesta il re d’Italia 7: 186.
Gatulli, V., M. Lepidi, F. Potenza, and U. di Sabatino. 2019. “Modal interactions in the nonlinear dynamics of a beam–cable–beam.” Nonlinear Dyn. 96 (4): 2547–2566. https://doi.org/10.1007/s11071-019-04940-8.
Goodey, W. 1961. “On the natural modes and frequencies of a suspended chain.” Q. J. Mech. Appl. Math. 14 (1): 118–127. https://doi.org/10.1093/qjmam/14.1.118.
Hagedorn, P., and B. Schäfer. 1980. “On non-linear free vibrations of an elastic cable.” Int. J. Non Linear Mech. 15 (4–5): 333–340. https://doi.org/10.1016/0020-7462(80)90018-9.
Huygens, C. 1646. “Correspondance no 21, letter to mersenne, proposition 8.” Dutch Acad. Sci. 36.
Irvine, H., and T. Caughey. 1974. “The linear theory of free vibrations of a suspended cable.” Proc. R. Soc. A 341 (1626): 299–315. https://doi.org/10.1098/rspa.1974.0189.
Irvine, H. M. 1992. Cable structures. New York: Dover.
Kirchoff, G. 1876. Vorlesungen über Mathematische Physik: Mechanik. Leipzig, Germany: Druck une Verlag von B.G.
Kloppel, K., and K. Lie. 1942. “Die lotrechten Eingenschwingungen des Hangebrucken.” Bauingenieur 23: 277.
Lacarbonara, W. 2013. Nonlinear structural mechanics, the elastic cable: From formulation to computation. New York: Springer.
Lacarbonara, W., and A. Pacitti. 2008. “Nonlinear modeling of cables with flexural stiffness.” Math. Probl. Eng. 2008 (2008): 370767. https://doi.org/10.1155/2008/370767.
Lee, C., and N. Perkins. 1992. “Nonlinear oscillations of suspended cables containing a two-to-one internal resonance.” Nonlinear Dyn. 3 (6): 465–490. https://doi.org/10.1007/BF00045648.
Lee, C., and N. Perkins. 1995. “Three-dimensional oscillations of suspended cables involving simultaneous internal resonance.” Nonlinear Dyn. 8 (1): 45–63. https://doi.org/10.1007/BF00045006.
Leibniz, G. 1691a. “Solutions to the problem of the catenary, or funicular curve, proposed by M. Jacques Bernoulli.” Acta Eruditorum.
Leibniz, G. 1691b. “The string whose curve is described by bending under its own weight, and the remarkable resources that can be discovered from it by however many proportional means and logarithms.” Acta Eruditorum.
Luongo, A., G. Rega, and F. Vestroni. 1982. “Monofrequent oscillations of a non-linear model of a suspended cable.” J. Sound Vibr. 82 (2): 247–259. https://doi.org/10.1016/0022-460X(82)90533-8.
Luongo, A., G. Rega, and F. Vestroni. 1984. “Planar non-linear free vibrations of an elastic cable.” Int. J. Non Linear Mech. 19 (1): 39–52. https://doi.org/10.1016/0020-7462(84)90017-9.
Marigo, J. J. 2014. “Mécanique des Milieux Continus I.” Accessed December 1, 2018. https://cel.archives-ouvertes.fr/cel-01023392.
Pai, P., and A. Nayfeh. 2012. “Fully nonlinear model of cables.” AIAA J. 30 (12): 2993–2996. https://doi.org/10.2514/3.48990.
Pérès, J. 1953. Mécanique générale. Paris: Masson & Cie.
Perkins, N. 1992. “Modal interactions in the non-linear response of elastic cables under parametric/external excitation.” Int. J. Non Linear Mech. 27 (2): 233–250. https://doi.org/10.1016/0020-7462(92)90083-J.
Pugsley, A. 1949. “On the natural frequencies of suspension chains.” Q. J. Mech. Appl. Math. 2 (4): 412–418. https://doi.org/10.1093/qjmam/2.4.412.
Rannie, W., and T. von Karman. 1941. “The failure of the Tacoma narrows bridge.” In Federal works agency applications, VI. Washington, DC: Federal Works Agency.
Rega, G. 1996. “Non-linearity, bifurcation and chaos in the finite dynamics of different cable models.” Chaos Solitons Fractals 7 (10): 1507–1536. https://doi.org/10.1016/S0960-0779(96)00092-6.
Rega, G. 2004. “Nonlinear vibrations of suspended cables—Part I: Modeling and analysis.” Appl. Mech. Rev. 57 (6): 443–478. https://doi.org/10.1115/1.1777224.
Rega, G., R. Alaggio, and F. Benedettini. 1997. “Experimental investigation of the nonlinear response of a hanging cable. Part I: Local analysis.” Nonlinear Dyn. 14 (2): 89–117. https://doi.org/10.1023/A:1008246504104.
Rega, G., and F. Benedettini. 1989. “Planar non-linear oscillations of elastic cables under subharmonic resonance conditions.” J. Sound Vibr. 132 (3): 367–381. https://doi.org/10.1016/0022-460X(89)90631-7.
Rega, G., N. Srinil, and R. Alaggio. 2008. “Experimental and numerical studies of inclined cables: Free and parametrically-forced vibrations.” J. Theor. Appl. Mech. 46 (3): 621–640.
Rega, G., F. Vestroni, and F. Benedettini. 1982. “Parametric analysis of large amplitude free vibrations of a sus pended cable.” Int. J. Solids Struct. 20 (2): 95–105. https://doi.org/10.1016/0020-7683(84)90001-5.
Rohrs, J. 1851. “On the oscillations o a suspension cable.” Trans. Cambridge Philos. Soc. 9: 379–389.
Routh, E. 1868. Advanced rigid dynamics. New York: Macmillan.
Saxon, D., and A. Cahn. 1953. “Modes of vibration of a suspended chain.” Q. J. Mech. Appl. Math. 6 (3): 273–285. https://doi.org/10.1093/qjmam/6.3.273.
Simpson, A. 1972. “On the oscillatory motions of translating elastic cables.” J. Sound Vibr. 20 (2): 177–189. https://doi.org/10.1016/0022-460X(72)90420-8.
Sofi, A. 2013. “Nonlinear in-plane vibrations of inclined cables carrying moving oscillators.” J. Sound Vibr. 332 (7): 1712–1724. https://doi.org/10.1016/j.jsv.2012.11.012.
Sofi, A., and G. Muscolino. 2007. “Dynamic analysis of suspended cables carrying moving oscillators.” Int. J. Solids Struct. 44 (21): 6725–6743. https://doi.org/10.1016/j.ijsolstr.2007.03.004.
Soler, J. 1970. “Dynamic response of single cables with initial sag.” J. Franklin Inst. 290 (4): 377–387. https://doi.org/10.1016/0016-0032(70)90192-4.
Sundararajan, P., and S. Noah. 1997. “Dynamics of forced nonlinear systems using shooting/arc-length continuation methods—Application to rotor systems.” ASME J. Vib. Acoust. 119 (1): 9–20. https://doi.org/10.1115/1.2889694.
Triantafyllou, M. 1984. “The dynamics of taut inclined cables.” Q. J. Mech. Appl. Math. 37 (3): 421–440. https://doi.org/10.1093/qjmam/37.3.421.
Visweswara Rao, G., and R. Iyengar. 1991. “Internal resonance and non-linear response of a cable under periodic excitation.” J. Sound Vibr. 149 (1): 25–41. https://doi.org/10.1016/0022-460X(91)90909-4.
Wang, L., and G. Rega. 2010. “Modelling and transient planar dynamics of suspended cables with moving mass.” Int. J. Solids Struct. 47 (20): 2733–2744. https://doi.org/10.1016/j.ijsolstr.2010.06.002.
Warminski, J., D. Zulli, G. Rega, and J. Latalski. 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.
Wu, Q., K. Takahashi, and S. Nakamura. 2005. “Formulae for frequencies and modes of in-plane vibrations of small-sag inclined cables.” J. Sound Vibr. 279 (3–5): 1155–1169. https://doi.org/10.1016/j.jsv.2004.01.004.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 148 • Issue 9 • September 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Oct 20, 2021
Accepted: Mar 31, 2022
Published online: Jul 9, 2022
Published in print: Sep 1, 2022
Discussion open until: Dec 9, 2022
ASCE Technical Topics:
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.