Nonlinear Dynamics

, Volume 86, Issue 4, pp 2245–2260 | Cite as

A revisitation of the paradox of discontinuous trajectory for a mass particle moving on a taut string

  • S. N. Gavrilov
  • V. A. Eremeyev
  • G. PiccardoEmail author
  • A. Luongo
Original Paper


A nonlinear extended regularized model of taut string carrying a moving point mass is proposed with the intent to contribute to the solution of the paradox of particle’s discontinuous trajectory. Introducing a coupling between transversal and longitudinal string displacements, an additional equation expressing the so-called wave pressure force arises whose closed-form expression can be obtained imposing that the load-structure combined system is dissipation free. In this context, paradoxical situations in the classic model lead to the emergence of wave resistance forces in the new proposed model that can significantly influence the motion of the inertia particle. Comparisons between classic and improved solutions are presented highlighting the possibility that mass particle can reach the remote support, or can return to the initial support, or do not come to any of the string support according to the values of four dimensionless parameters governing the extended problem.


Traveling mass Taut string Wave pressure force Discontinuity trajectory paradox Nonlinear coupling 


  1. 1.
    Luongo, A., Piccardo, G.: Dynamics of taut strings traveled by train of forces. Contin. Mech. Thermodyn. 28(1–2), 603–616 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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, New York (1998)CrossRefGoogle Scholar
  3. 3.
    Pesterev, A.V., Bergman, L.A., Tan, C.A., Tsao, T.-C., Yang, B.: On asymptotics of the solution of the moving oscillator problem. J. Sound Vib. 260, 519–536 (2003)CrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Gavrilov, S.N., Indeitsev, D.A.: The evolution of a trapped mode of oscillations in a string on an elastic foundation—moving inertial inclusion system. PMM J. Appl. Math. Mech. 66(5), 825–833 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ouyang, H.: Moving-load dynamic problems: a tutorial (with a brief overview). Mech. Syst. Signal Process. 25, 2039–2060 (2011)CrossRefGoogle Scholar
  7. 7.
    Bajer, C.I., Dyniewicz, B.: Numerical Analysis of Vibrations of Structures Under Moving Inertial Loads. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  8. 8.
    Pesterev, A.V., Bergman, L.A.: An improved series expansion of the solution to the moving oscillator problem. J. Vib. Acoust. ASME 122, 54–61 (2000)CrossRefGoogle Scholar
  9. 9.
    Rao, G.V.: Linear dynamics of an elastic beam under moving loads. J. Vib. Acoust. ASME 122, 281–289 (2000)CrossRefGoogle Scholar
  10. 10.
    Al-Qassab, M., Nair, S., O’Leary, J.: Dynamics of an elastic cable carrying a moving mass particle. Nonlinear Dyn. 33, 11–32 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Wang, L., Rega, G.: Modelling and transient planar dynamics of suspended cables with moving mass. Int. J. Solids Struct. 47, 2733–2744 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Smith, C.E.: Motions of a stretched string carrying a moving mass particle. J. Appl. Mech. 31(1), 29–37 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bajer, C.I., Dyniewicz, B.: Virtual functions of the space-time finite element method in moving mass problems. Comput. Struct. 87, 444–455 (2009)CrossRefGoogle Scholar
  14. 14.
    Gavrilov, S.: Nonlinear investigation of the possibility to exceed the critical speed by a load on a string. Acta Mech. 154, 47–60 (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ferretti, M., Piccardo, G.: Dynamic modeling of taut strings carrying a traveling mass. Contin. Mech. Thermodyn. 25(2–4), 469–488 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Gavrilov, S.N.: Transition through the critical velocity for a moving load in an elastic waveguide. Tech. Phys. 45(4), 515–518 (2000)CrossRefGoogle Scholar
  18. 18.
    Stronge, W.J.: An accelerating force on a string. J. Acoust. Soc. Am. 50, 1382–1383 (1971)CrossRefGoogle Scholar
  19. 19.
    Kaplunov, Y.D., Muravskii, G.B.: Vibrations of an infinite string on a deformable foundation under action of a uniformly accelerating moving load. Passage through critical velocity. Izv. Akad. Nauk USSR Mek. Tverd. Tela 1, 155–160 (1986). (in Russian)Google Scholar
  20. 20.
    Gavrilov, S.: Non-stationary problems in dynamics of a string on an elastic foundation subjected to a moving load. J. Sound Vib. 222(3), 345–361 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gao, Q., Zhang, J., Zhang, H.W., Zhong, W.X.: The analytical solutions for the wave propagation in a stretched string with a moving mass. Wave Motion 59, 1–28 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mote, C.D.: On the nonlinear oscillation of an axially moving string. J. Appl. Mech. Trans. ASME 33, 463–464 (1966)CrossRefGoogle Scholar
  23. 23.
    Dyniewicz, B., Bajer, C.I.: Paradox of a particles trajectory moving on a string. Arch. Appl. Mech. 79(3), 213–223 (2009)CrossRefzbMATHGoogle Scholar
  24. 24.
    Luongo, A., Ferretti, M., D’Annibale, F.: Paradoxes in dynamic stability of mechanical systems: investigating the causes and detecting the nonlinear behaviors. SpringerPlus 5(1), 1–22 (2016)CrossRefGoogle Scholar
  25. 25.
    Luongo, A., D’Annibale, F.: A paradigmatic minimal system to explain the Ziegler paradox. Contin. Mech. Thermodyn. 27(1–2), 211–222 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Stokes, G.G.: Discussion of a differential equation relating to the breaking of railway bridges. Math. Phys. Pap. 2, 178–220 (1883)Google Scholar
  27. 27.
    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
  28. 28.
    Nicolai, E.L.: On pressure of vibrations. Annals of St. Petersburg Polytechnic Institute. Sect. Tech. Nat. Sci. Math. 18(1), 49–60 (1912). (In Russian)Google Scholar
  29. 29.
    Havelock, T.H.: Some dynamical illustrations of the pressure of radiation and of adiabatic invariance. Philos. Mag. Ser. 47(280), 754–771 (1924)CrossRefGoogle Scholar
  30. 30.
    Nicolai, E.L.: On a dynamical illustration of the pressure of radiation. Philos. Mag. Ser. 49(289), 171–177 (1925)CrossRefzbMATHGoogle Scholar
  31. 31.
    Vesnitski, A.I., Kaplan, L.E., Utkin, G.A.: The laws of variation of energy and momentum for one-dimensional systems with moving mountings and loads. PMM J. Appl. Math. Mech. 47(5), 692–695 (1983)CrossRefzbMATHGoogle Scholar
  32. 32.
    Andrianov, V.L.: The resistance to the motion of loads along elastic directions caused by the radiation of waves in them. PMM J. Appl. Math. Mech. 57(2), 383–387 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Denisov, G.G.: On the wave pressure on an obstacle in the case of transversal oscillations of the string. Izv. RAN. Mek. Tverd. Tela 5, 187–192 (2001). (In Russian)Google Scholar
  34. 34.
    Gavrilov, S.N.: The effective mass of a point mass moving along a string on a Winkler foundation. PMM J. Appl. Math. Mech. 70(4), 582–589 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Denisov, G.G., Novilov, V.V., Smirnova, M.L.: The momentum of waves and their effect on the motion of lumped objects along one-dimensional elastic systems. PMM J. Appl. Math. Mech. 76(2), 225–234 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  37. 37.
    Ogden, R.W.: Non-linear Elastic Deformations. Dover, Mineola (1997)Google Scholar
  38. 38.
    Lurie, A.I.: Nonlinear Theory of Elasticity. North-Holland, Amsterdam (1990)zbMATHGoogle Scholar
  39. 39.
    Biot, M.A.: Mechanics of incremental deformations. In: Theory of Elasticity and Viscoelasticity of Initially Stressed Solids and Fluids, Including Thermodynamics Foundation and Applications to Finite Strain. Wiley, New York/London (1965)Google Scholar
  40. 40.
    Fu, Y.B., Ogden, R.W.: Nonlinear stability analysis of pre-stressed elastic bodies. Contin. Mech. Thermodyn. 11, 141–172 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Altenbach, H., Eremeyev, V.A.: Vibration analysis of non-linear 6-parameter prestressed shells. Meccanica 49(8), 1751–1761 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Truesdell, C.A.: The Elements of Continuum Mechanics. Springer, Berlin (1966)zbMATHGoogle Scholar
  43. 43.
    Truesdell, C.: Rational Thermodynamics, 2nd edn. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  44. 44.
    Rayleigh, Lord: On the pressure of vibrations. Philos. Mag. S6(3), 338–346 (1902)CrossRefzbMATHGoogle Scholar
  45. 45.
    Cazzani, A., Malagú, M., Turco, E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1–2), 139–156 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Greco, L., Cuomo, M.: An isogeometric implicit G1 mixed finite element for Kirchhoff space rods. Comput. Methods Appl. Mech. Eng. 298, 325–349 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Gosselin, F.P., Païdoussis, M.P.: Dynamical stability analysis of a hose to the sky. J. Fluids Struct. 44, 226–234 (2014)CrossRefGoogle Scholar
  48. 48.
    Rizzi, N.L., Varano, V., Gabriele, S.: Initial postbuckling behavior of thin-walled frames under mode interaction. Thin Walled Struct. 68, 124–134 (2013)CrossRefGoogle Scholar
  49. 49.
    Bersani, A.M., Giorgio, I., Tomassetti, G.: Buckling of an elastic hemispherical shell with an obstacle. Contin. Mech. Thermodyn. 25(2), 443–467 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Metrikine, A.V.: Parametric instability of a moving particle on a periodically supported infinitely long string. J. Appl. Mech. ASME 75(1), 11006 (2008)CrossRefGoogle Scholar
  51. 51.
    Andreaus, U., Chiaia, B., Placidi, L.: Soft-impact dynamics of deformable bodies. Contin. Mech. Thermodyn. 25(2–4), 375–398 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Andreaus, U., Baragatti, P., Placidi, L.: Experimental and analytical investigations of the forced response of a cantilever beam contacting a deformable and dissipative obstacle under harmonic excitation. Int. J. Non-linear Mech. 80, 96–106 (2016)CrossRefGoogle Scholar
  53. 53.
    Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Aminpour, H., Rizzi, N.: A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis. Math. Mech. Solids 21(2), 168–181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Pagnini, L.C.: Model reliability and propagation of frequency and damping uncertainties in the dynamic along-wind response of structures. J. Wind Eng. Indus. Aerodyn. 59(2–3), 211–231 (1996)CrossRefGoogle Scholar
  56. 56.
    Pagnini, L., Repetto, M.P.: The role of parameter uncertainties in the damage prediction of the alongwind-induced fatigue. J. Wind Eng. Ind. Aerodyn. 104–106, 227–238 (2012)CrossRefGoogle Scholar
  57. 57.
    Andreaus, U., dell’Isola, F., Porfiri, M.: Piezoelectric passive distributed controllers for beam flexural vibrations. J. Vib. Control 10(5), 625–659 (2004)CrossRefzbMATHGoogle Scholar
  58. 58.
    Porfiri, M., Dell’Isola, F., Frattale Mascioli, F.M.: Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers. Int. J. Circuit Theory Appl. 32(4), 167–198 (2004)CrossRefzbMATHGoogle Scholar
  59. 59.
    Shen, H., Qiu, J., Ji, H., Zhu, K., Balsi, M., Giorgio, I., dell’Isola, F.: A low-power circuit for piezoelectric vibration control by synchronized switching on voltage sources. Sens. Actuators A Phys. 161(1), 245–255 (2010)CrossRefGoogle Scholar
  60. 60.
    Giorgio, I., Galantucci, L., Corte, Della A., Del Vescovo, D.: Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications. Int. J. Appl. Electromagn. Mech. 47(4), 1051–1084 (2015)Google Scholar
  61. 61.
    D’Annibale, F., Rosi, G., Luongo, A.: Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces. Meccanica 50(3), 825–839 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  63. 63.
    Gurtin, M.E.: Configurational Forces As Basic Concepts of Continuum Physics. Springer, Berlin (2000)zbMATHGoogle Scholar
  64. 64.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, 10th edn. National Bureau of Standards, Washington (1972)zbMATHGoogle Scholar
  65. 65.
    Frýba, Ladislav: Vibration of Solids and Structures Under Moving Loads. Thomas Telford, London (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering RASSt. PetersburgRussia
  2. 2.Peter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia
  3. 3.Faculty of Mechanical Engineering and AeronauticsRzeszów University of TechnologyRzeszówPoland
  4. 4.DICCA - University of GenoaGenoaItaly
  5. 5.M&MoCSUniversity of L’AquilaL’AquilaItaly

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