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Nonlinear Dynamics

, Volume 97, Issue 4, pp 2201–2218 | Cite as

Nonlinear planar modeling of massive taut strings travelled by a force-driven point-mass

  • M. Ferretti
  • S. N. Gavrilov
  • V. A. Eremeyev
  • A. LuongoEmail author
Original paper

Abstract

The planar response of horizontal massive taut strings, travelled by a heavy point-mass, either driven by an assigned force, or moving with an assigned law, is studied. A kinematically exact model is derived for the free boundary problem via a variational approach, accounting for the singularity in the slope of the deflected string. Reactive forces exchanged between the point-mass and the string are taken into account via Lagrange multipliers. The exact model is consistently simplified via asymptotic analysis, which leads to condense the horizontal displacement as a passive variable. The dynamic increment of tension, with respect the static one, is neglected in the governing equations, but evaluated a posteriori, as a higher-order quantity in a perturbation perspective. The equations are solved and rearranged in the form of an integral equation coupled with an integro-differential equation, thus extending a procedure already introduced in the literature. Numerical results, showing the importance of the horizontal reactive force on the quality of motion, are discussed, generalizing those relevant to massless strings.

Keywords

Traveling mass Taut string Free boundary problem Nonlinear contact force Integral equation 

Notes

Acknowledgements

V.A.E. acknowledges the support of the Government of the Russian Federation (contract No. 14.Y26.31.0031).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest to declare.

References

  1. 1.
    Frỳba, L.: Vibration of Solids and Structures Under Moving Loads, vol. 1. Springer, Berlin (2013)zbMATHGoogle Scholar
  2. 2.
    Bajer, C.I., Dyniewicz, B.: Numerical Analysis of Vibrations of Structures Under Moving Inertial Load, vol. 65. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  3. 3.
    Smith, C.E.: Motions of a stretched string carrying a moving mass particle. J. Appl. Mech. 31(1), 29–37 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Derendyayev, N.V., Soldatov, I.N.: The motion of a point mass along a vibrating string. J. Appl. Math. Mech. 61(4), 681–684 (1997)CrossRefGoogle Scholar
  5. 5.
    Gavrilov, S.N.: Nonlinear investigation of the possibility to exceed the critical speed by a load on a string. Acta Mech. 154(1–4), 47–60 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gavrilov, S.N.: The effective mass of a point mass moving along a string on a Winkler foundation. J. Appl. Math. Mech. 70(4), 582–589 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bersani, A.M., Della Corte, A., Piccardo, G., Rizzi, N.L.: An explicit solution for the dynamics of a taut string of finite length carrying a traveling mass: the subsonic case. Z. für Angew. Math. Phys. 67(4), 108 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, L., Rega, G.: Modelling and transient planar dynamics of suspended cables with moving mass. Int. J. Solids Struct. 47(20), 2733–2744 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Al-Qassab, M., Nair, S., O’leary, J.: Dynamics of an elastic cable carrying a moving mass particle. Nonlinear Dyn. 33(1), 11–32 (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Rao, G.V.: Linear dynamics of an elastic beam under moving loads. J. Vib. Acoust. 122(3), 281–289 (2000)CrossRefGoogle Scholar
  11. 11.
    Lee, H.P.: Transverse vibration of a Timoshenko beam acted on by an accelerating mass. Appl. Acoust. 47(4), 319–330 (1996)CrossRefGoogle Scholar
  12. 12.
    Stokes, G.G.: Discussion of a differential equation relating to the breaking of railway bridges. Printed at the Pitt Press by John W, Parker (1849)Google Scholar
  13. 13.
    He, W.: Vertical dynamics of a single-span beam subjected to moving mass-suspended payload system with variable speeds. J. Sound Vib. 418, 36–54 (2018)CrossRefGoogle Scholar
  14. 14.
    Bajer, X.I., Dyniewicz, B., Shillor, M.A.: Gao beam subjected to a moving inertial point load. Math. Mech. Solids 23(3), 461–472 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shadnam, M.R., Mofid, M., Akin, J.E.: On the dynamic response of rectangular plate, with moving mass. Thin-walled Struct. 39(9), 797–806 (2001)CrossRefGoogle Scholar
  16. 16.
    Nikkhoo, A., Hassanabadi, M.E., Azam, S.E., Amiri, J.V.: Vibration of a thin rectangular plate subjected to series of moving inertial loads. Mech. Res. Commun. 55, 105–113 (2014)CrossRefGoogle Scholar
  17. 17.
    Luongo, A., Piccardo, G.: Dynamics of taut strings traveled by train of forces. Contin. Mech. Thermodyn. 28(1–2), 603–616 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ferretti, M., Piccardo, G., Luongo, A.: Weakly nonlinear dynamics of taut strings traveled by a single moving force. Meccanica 52(13), 3087–3099 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ferretti, M., Piccardo, G.: Dynamic modeling of taut strings carrying a traveling mass. Contin. Mech. Thermodyn. 25(2–4), 469–488 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yang, B., Tan, C.A., Bergman, L.A.: On the problem of a distributed parameter system carrying a moving oscillator. In: Tzou, H.S., Bergman, L.A. (eds.) Dynamics and Control of Distributed Systems, pp. 69–94. Cambridge University Press (1998) Google Scholar
  21. 21.
    Cazzani, A., Wagner, N., Ruge, P., Stochino, F.: Continuous transition between traveling mass and traveling oscillator using mixed variables. Int. J. Non-Linear Mech. 80, 82–95 (2016)CrossRefGoogle Scholar
  22. 22.
    Dyniewicz, B., Bajer, C.I.: Paradox of a particle’s trajectory moving on a string. Arch. Appl. Mech. 79(3), 213–223 (2009)CrossRefzbMATHGoogle Scholar
  23. 23.
    Dyniewicz, B., Bajer, C.I.: New feature of the solution of a Timoshenko beam carrying the moving mass particle. Arch. Mech. 62(5), 327–341 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gavrilov, S.N., Eremeyev, V.A., Piccardo, G., Luongo, A.: A revisitation of the paradox of discontinuous trajectory for a mass particle moving on a taut string. Nonlinear Dyn. 86(4), 2245–2260 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Luongo, A., Zulli, D.: Mathematical Models of Beams and Cables. Wiley, Hoboken (2013)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.International Research Center on Mathematics and Mechanics of Complex SystemsUniversity of L’AquilaL’AquilaItaly
  2. 2.Department of Civil, Construction-Architectural and Environmental EngineeringUniversity of L’AquilaL’AquilaItaly
  3. 3.Institute for Problems in Mechanical Engineering RASSt. PetersburgRussia
  4. 4.Peter the Great St. Petersburg Polytechnic University (SPbPU)St. PetersburgRussia
  5. 5.Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland
  6. 6.Research Institute for MechanicsNational Research Lobachevsky State University of Nizhni NovgorodNizhni NovgorodRussia

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