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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Low-Frequency Free Vibration of Rods with Finite Strain

Journal of Applied Nonlinear Dynamics 3(1) (2014) 85--93 | DOI:10.5890/JAND.2014.03.007

A.M. Baghestani, S.J. Fariborz$^{1}$ , S.M. Mousavi$^{2}$

$^{1}$ Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Avenue, Tehran, Iran

$^{2}$ Department of Civil and Structural Engineering, Aalto University, PO box 12100, FI-00076, Finland

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The governing differential equation for the free vibration of a rod undergoing finite strain is obtained by means of Hamilton’s principle. The equation contains quadratic as well as cubic nonlinear terms. For the low-frequency analysis of rods, the two harmonics solution is considered for the equation. The Galerkin method is employed to convert the partial differential equation to a system of two nonlinear ordinary differential equations. These equations are solved utilizing generalized differential quadrature(GDQ) and continuation methods to obtain the backbone curves and also mode shapes of vibration for rods with two different kinds of boundary conditions.


  1. [1]  Bojadziev, G.N. and Lardner, R.W. (1973), Monofrequent oscillations in mechanical systems governed by second order hyperbolic differential equations with small non-linearities, International Journal of Non-Linear Mechanics , 8, 289-302.
  2. [2]  Cveticanin, L. and Uzelac, Z. (1999), Nonlinear longitudinal vibrations of a rod, Journal of Vibration and Control, 5, 827-849.
  3. [3]  Cveticanin, L. (2009), Application of homotopy-perturbation to non-linear partial differential equations, Chaos, Solitions and Fractals, 40, 221-228.
  4. [4]  Andrianov, I.V. and Danishevs'kyy, V.V. (2002), Asymptotic approach for non-linear periodical vibrations of continuous structures, Journal of Sound and Vibration, 249(3), 465-481.
  5. [5]  Parashar, S.K. and Wagner, U.V. (2004), Nonlinear longitudinal vibrations of transversally polarized piezoceramics: experiments and modeling, Nonlinear Dynamics, 37, 51-73.
  6. [6]  Murnaghan, F.D. (1951), Finite Deformation of an Elastic Solid, Dover, New York.
  7. [7]  Vanhille, C. and Campos-Pozuelo, C. (2007), Numerical analysis of strongly nonlinear extensional vibrations in elastic rods, IEEE Trans. Ultrasonic, 54, 96-106.
  8. [8]  Luo, A.C.J. (2010), On a nonlinear theory of thin rods, Communications in Nonlinear Science and Numerical Simulation, 15, 4181-4197.
  9. [9]  Abedinnasab, M.H. and Hussein, M.I. (2013),Wave dispersion under finite deformation, Wave Motion, 50(3), 374-388.
  10. [10]  Malvern, L.E. (1969), Introduction to the Mechanics of a Continuous Medium, Prentice Hall, Englewood Cliffs, New Jersey.
  11. [11]  Campos-Pozuelo, C., Vanhille, C., and Gallego-Juarez, A. (2006), Comparative study of the nonlinear behavior of fatigued and intact samples of metallic alloys, IEEE Transactions On Ultrasonics , 53, 175-184.
  12. [12]  Zong, Z. and Zhang, Y. (2009), Advanced Differential Quadrature Methods, CRC Press.
  13. [13]  Tornabene, F. and Viola, E. (2008), 2-D Solution for free vibrations of parabolic shells using generalized differential quadrature method, European Journal of Mechanics-A/Solids, 27, 1001-1025.
  14. [14]  Allgower, E.L. and Georg, K. (1990), Numerical Continuation Methods: An Introduction, Springer-Verlag.