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


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.


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



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.


  1. 1.
    Miyamoto Y, Kaysser W, Rabin B, Kawasaki A, Ford R (1999) Functionally graded materials: design, processing and applications. Springer, New YorkCrossRefGoogle Scholar
  2. 2.
    Sadowski T (2009) Non-symmetric thermal shock in ceramic matrix composite (CMC) materials. In: de Borst R, Sadowski T (eds) Solid mechanics and its applications. Lecture notes on composite materials—current topics and achievements, vol 154. Springer, Netherlands, pp 99–148Google Scholar
  3. 3.
    Burlayenko VN, Altenbach H, Sadowski T, Dimitrova SD (2016) Computational simulations of thermal shock cracking by the virtual crack closure technique in a functionally graded plate. Comput Mater Sci 116(15):11–21CrossRefGoogle Scholar
  4. 4.
    Burlayenko VN (2016) Modelling thermal shock in functionally graded plates with finite element method. Adv Mater Sci Eng 2016:7514638Google Scholar
  5. 5.
    Şimşek M (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl Eng Des 240(4):697–705CrossRefGoogle Scholar
  6. 6.
    Abrate S (1995) Vibration of non-uniform rods and beams. J Sound Vib 185(4):703–716MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Attarnejad R, Semnani SJ, Shahba A (2010) Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams. Finite Elem Anal Des 46(10):916–929MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guo SQ, Yang SP (2014) Transverse vibrations of arbitrary non-uniform beams. Appl Math Mech 35(5):607–620MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garijo D (2015) Free vibration analysis of nonuniform Euler Bernoulli beams by means of Bernstein pseudospectral collocation. Eng Comput 31:813–823CrossRefGoogle Scholar
  10. 10.
    Elishakoff I, Candan S (2001) Apparently first closed-form solution for vibrating: inhomogeneous beams. Int J Solids Struct 38(19):3411–3441MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Candan S, Elishakoff I (2001) Apparently first closed-form solution for frequencies of deterministically and/or stochastically inhomogeneous simply supported beams. J Appl Mech 68:176–185CrossRefzbMATHGoogle Scholar
  12. 12.
    Li QS (2000) A new exact approach for determining natural frequencies and mode shapes of non-uniform shear beams with arbitrary distribution of mass or stiffness. Int J Solids Struct 37(37):5123–5141CrossRefzbMATHGoogle Scholar
  13. 13.
    Ait Atmane H, Tounsi A, Meftah SA, Belhadj HA (2010) Free vibration behavior of exponential functionally graded beams with varying cross-section. J Vib Control 17(2):311–318MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li XF, Kang YA, Wu JX (2013) Exact frequency equations of free vibration of exponentially functionally graded beams. Appl Acoust 74(3):413–420CrossRefGoogle Scholar
  15. 15.
    Tang AY, Wu JX, Li XF, Lee KY (2014) Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams. Int J Mech Sci 89:1–11CrossRefGoogle Scholar
  16. 16.
    Alshorbagy AE, Eltaher MA, Mahmoud FF (2011) Free vibration characteristics of a functionally graded beam by finite element method. Appl Math Model 35(1):412–425MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shahba A, Attarnejad R, Marvi MT, Hajilar S (2011) Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Compos B Eng 42(4):801–808CrossRefGoogle Scholar
  18. 18.
    Burlayenko VN, Altenbach H, Sadowski T, Dimitrova SD, Bhaskar A (2017) Modelling functionally graded materials in heat transfer and thermal stress analysis by means of graded finite elements. Appl Math Model 45:422–438MathSciNetCrossRefGoogle Scholar
  19. 19.
    Huang Y, Li X-F (2010) A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J Sound Vib 329(11):2291–2303CrossRefGoogle Scholar
  20. 20.
    Hein H, Feklistova L (2011) Free vibrations of non-uniform and axially functionally graded beams using haar wavelets. Eng Struct 33(12):3696–3701CrossRefGoogle Scholar
  21. 21.
    Bambill DV, Felix DH, Rossi RE (2010) Vibration analysis of rotating Timoshenko beams by means of the differential quadrature method. Struct Eng Mech 34(2):231–245CrossRefGoogle Scholar
  22. 22.
    Rajasekaran S (2013) Differential transformation and differential quadrature methods for centrifugally stiffened axially functionally graded tapered beams. Int J Mech Sci 74:15–31CrossRefzbMATHGoogle Scholar
  23. 23.
    Pukhov GE (1978) Computational structure for solving differential equations by Taylor transformations. Cybernet Syst Anal 14(3):383–390CrossRefGoogle Scholar
  24. 24.
    Pukhov GE (1978) Taylor transformations and their applications in electrical and electronics. Naukova Dumka, Kiev (in Russian)zbMATHGoogle Scholar
  25. 25.
    Pukhov GE (1982) Differential transforms and circuit-theory. Int J Circuit Theory Appl 10(3):265–276CrossRefzbMATHGoogle Scholar
  26. 26.
    Pukhov GE (1982) Differential analysis of circuits. Naukova Dumka, Kiev (in Russian).
  27. 27.
    Pukhov GE (1980) Differential transformations of functions and equations. Naukova Dumka, Kiev (in Russian).
  28. 28.
    Pukhov GE (1986) Differential transformations and mathematical modeling of physical processes. Naukova Dumka, Kiev (in Russian).
  29. 29.
    Pukhov GE (1988) Approximate methods of mathematical modelling based on the use of differential T-transformations. Naukova Dumka, Kiev (in Russian).
  30. 30.
    Pukhov GE (1990) Differential spectrums and models. Naukova Dumka, Kiev (in Russian).
  31. 31.
    Bervillier C (2012) Status of the differential transformation method. Appl Math Comput 218(20):10158–10170MathSciNetzbMATHGoogle Scholar
  32. 32.
    Simonyan SH, Avetisyan AG (2010) Applied theory of differential transforms. Chartaraget, YerevanzbMATHGoogle Scholar
  33. 33.
    Avetisyan AG, Simonyan SH, Ghazaryan DA (2009) Solution of linear time optimal control problems in domain of differential transformations. Bull Tomsk Politech Univ 315(5):5–13Google Scholar
  34. 34.
    Avetisyan AG, Avinyan VR, Ghazaryan DA (2013) A method for solving silvester type parametric matrix equation. Proc NAS RA SEUA 66(4):376–383Google Scholar
  35. 35.
    Ozgumus OO, Kaya MO (2006) Flapwise bending vibration analysis of double tapered rotating eulerbernoulli beam by using the differential transform method. Meccanica 41(6):661–670CrossRefzbMATHGoogle Scholar
  36. 36.
    Mei C (2008) Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam. Comput Struct 86(11–12):1280–1284CrossRefGoogle Scholar
  37. 37.
    Abdelghany SM, Ewis KM, Mahmoud AA, Nassar MM (2015) Vibration of a circular beam with variable cross sections using differential transformation method. Beni-Suef Univ J Basic Appl Sci 4:185–191CrossRefGoogle Scholar
  38. 38.
    Wattanasakulpong N, Charoensuk J (2015) Vibration characteristics of stepped beams made of FGM using differential transformation method. Meccanica 50:1089–1101MathSciNetCrossRefGoogle Scholar
  39. 39.
    Suddounga K, Charoensuka J, Wattanasakulpong N (2014) Vibration response of stepped FGM beams with elastically end constraints using differential transformation method. Appl Acoust 77:20–28CrossRefGoogle Scholar
  40. 40.
    Shahba A, Rajasekaran S (2012) Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials. Appl Math Model 36(7):3094–3111MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rajasekaran S, Tochaei EN (2014) Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order. Meccanica 49:995–1009MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ebrahimi F, Mokhtari M (2015) Vibration analysis of spinning exponentially functionally graded Timoshenko beams based on differential transform method. Proc Inst Mech Eng Part G 229(14):2559–2571CrossRefGoogle Scholar
  43. 43.
    Weaver W Jr, Timoshenko SP, Young DH (1990) Vibration problems in engineering, 5th edn. Wiley, New YorkGoogle Scholar
  44. 44.
    Antia HM (2002) Numerical methods for scientists and engineer, 2nd edn. Birkhäuser Verlag, BaselzbMATHGoogle Scholar

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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