Abstract
Modeling transverse vibration of nonlinear strings is investigated via numerical solutions of partial-differential equations and an integro-partial-differential equation. By averaging the tension along the deflected string, the classic nonlinear model of a transversely vibrating string, Kirchhoff’s equation, is derived from another nonlinear model, a partial-differential equation. The partial-differential equation is obtained via neglecting longitudinal terms in a governing equation for coupled planar vibration. The finite difference schemes are developed to solve numerically those equations. An index is proposed to compare the transverse responses calculated from the two models with the transverse component calculated from the coupled equation. A steel string and a rubber string are treated as examples to demonstrate the differences between the two models of transverse vibration and their deviation from the full model of coupled vibration. The numerical results indicate that the differences increase with the amplitude of vibration. Both models yield satisfactory results of almost the same precision for vibration of small amplitudes. For large amplitudes, the Kirchhoff equation gives better results.
Similar content being viewed by others
References
Arosio A. and Panizzi S. (1996). On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348: 305–330
Bilbao S. and Smith J.O. (2005). Energy conserving finite difference schemes for nonlinear strings. Acustica 91: 299–311
Chen L.Q. (2005). Analysis and control of transverse vibrations of axially moving strings. Appl. Mech. Rev. 58: 91–116
Chen L.Q. (2005). Principal parametric resonance of axially accelerating viscoelastic strings constituted by the Boltzmann superposition principle. Proc. R. Soc. Lond. A 461: 2701–2720
Countryman M. and Kannan R. (1992). Forced oscillations of elastic strings. Q. Appl. Math. L(1): 57–71
Christie I. (1984). A Galerkin method for a nonlinear integro-differential wave system. Comput. Methods Appl. Mech. Eng. 44: 229–237
Carrier G.F. (1945). On the nonlinear vibration problem of the elastic string. Q. J. Appl. Math. 3: 157–165
Carrier G.F. (1949). A note on the vibrating string. Q. J. Appl. Math. 7: 97–101
Kirchhoff G. (1867). Vorlesungen über Mathematische Physik: Mechanik. Druck und Verlag von B. G. Teubner, Leipzig
Kobayashi T. (2004). Boundary position feedback control of Kirchhoff’s non-linear strings. Math. Methods Appl. Sci. 27: 79–89
Irvine H.M. and Caughey T.K. (1974). The linear theory of free vibrations of a suspended cable. Proc. R. Soc. Lond. A 341: 299–315
Larkin N.A. (2002). Global regular solutions for the nonhomogeneous Carrier equation. Math. Prob. Eng. 8: 15–31
Leissa A.W. and Saad A.M. (1994). Large amplitude vibrations of strings. J. Appl. Mech. 61: 296–301
Liu I.S. and Rincon M.A. (2003). Effect of moving boundaries on the vibrating elastic string. Appl. Numer. Math. 47: 159–172
Molteno T.C. and Tufillaro N.B. (2004). An experimental investigation into the dynamics of a string. Am. J. Phys. 72: 1157–1167
Mote D.C. (1966). On the nonlinear oscillation of an axially moving string. J. Appl. Mech. 33: 463–464
Murthy G.S.S. and Ramakrishna B.S. (1965). Nonlinear character of resonance in stretched strings. J. Acoust. Soc. Am. 38: 461–471
Narasimha R. (1968). Nonlinear vibration of an elastic strings. J. Sound Vib. 8: 134–155
Nayfeh A.H. and Mook D.T. (1979). Nonlinear Oscillations. Wiley, New York
Oplinger D.W. (1960). Frequency response of a nonlinear stretched string. J. Acoust. Soc. Am. 32: 1529–1538
Peradze J. (2005). A numerical algorithm for the nonlinear Kirchhoff string equation. Numer. Math. 102: 311–342
Shahruz S.M. (1999). Boundary control of Kirchhoff’s non-linear string. Int. J. Control 72: 560–563
Thomas J.W. (1995). Numerical Partial Differential Equations: Finite Difference Methods. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, LQ., Ding, H. Two nonlinear models of a transversely vibrating string. Arch Appl Mech 78, 321–328 (2008). https://doi.org/10.1007/s00419-007-0164-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-007-0164-7