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Journal of Engineering Mathematics

, Volume 110, Issue 1, pp 97–121 | Cite as

Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method

  • Davit Ghazaryan
  • Vyacheslav N. Burlayenko
  • Armine Avetisyan
  • Atul Bhaskar
Article
  • 196 Downloads

Abstract

Free vibrations of non-uniform cross-section and axially functionally graded Euler–Bernoulli beams with various boundary conditions were studied using the differential transform method. The method was applied to a variety of beam configurations that are either axially non-homogeneous or geometrically non-uniform along the beam length or both. The governing equation of an Euler–Bernoulli beam with variable coefficients was reduced to a set of simpler algebraic recurrent equations by means of the differential transformations. Then, transverse natural frequencies were determined by requiring the non-trivial solution of the eigenvalue problem stated for a transformed function of the transverse displacement with appropriately transformed its high derivatives and boundary conditions. To show the generality and effectiveness of this approach, natural frequencies of various beams with variable profiles of cross-section and functionally graded non-homogeneity were calculated and compared with analytical and numerical results available in the literature. The benefit of the differential transform method to solve eigenvalue problems for beams with arbitrary axial geometrical non-uniformities and axial material gradient profiles is clearly demonstrated.

Keywords

Differential transform method Free vibrations Functionally graded material Non-uniform cross-sectional beam 

Notes

Acknowledgements

This research has been carried out under the financial support of the Erasmus Mundus post-doctoral exchange program ACTIVE, Grant Agreement No. 2013-2523/001-001 at the University of Southampton.

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Davit Ghazaryan
    • 1
    • 3
  • Vyacheslav N. Burlayenko
    • 2
    • 3
  • Armine Avetisyan
    • 1
  • Atul Bhaskar
    • 3
  1. 1.Department of Information Technology and AutomationNational Polytechnic University of ArmeniaYerevanRepublic of Armenia
  2. 2.Department of Applied MathematicsNational Technical University ‘KhPI’KharkivUkraine
  3. 3.Faculty of Engineering and the EnvironmentUniversity of SouthamptonSouthamptonUK

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