Advertisement

Journal of Mechanical Science and Technology

, Volume 29, Issue 3, pp 1207–1215 | Cite as

Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams

  • Farzad EbrahimiEmail author
  • Majid Ghadiri
  • Erfan Salari
  • Seied Amir Hosein Hoseini
  • Gholam Reza Shaghaghi
Article

Abstract

In this study, the applicability of differential transformation method (DTM) in investigations on vibrational characteristics of functionally graded (FG) size-dependent nanobeams is examined. The material properties of FG nanobeam vary over the thickness based on the power law. The nonlocal Eringen theory, which takes into account the effect of small size, enables the present model to be effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton’s principle. The obtained results exactly match the results of the presented Navier-based analytical solution as well as those available in literature. The DTM is also demonstrated to have high precision and computational efficiency in the vibration analysis of FG nanobeams. The detailed mathematical derivations are presented and numerical investigations performed with emphasis placed on investigating the effects of several parameters, such as small scale effects, volume fraction index, mode number, and thickness ratio on the normalized natural frequencies of the FG nanobeams. The study also shows explicitly that vibrations of FG nanobeams are significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

Keywords

Differential transformation method Functionally graded material Nanobeam Nonlocal elasticity Vibration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Ebrahimi and A. Rastgoo, Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers, Smart Materials and Structures, 17 (1) (2008) 015044.CrossRefGoogle Scholar
  2. [2]
    F. Ebrahimi, A. Rastgoo and A. A. Atai, A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate, European Journal of Mechanics-A/Solids, 28 (5) (2009) 962–973.CrossRefzbMATHGoogle Scholar
  3. [3]
    S. Iijima, Helical microtubules of graphitic carbon, Nature, 354 (6348) (1991) 56–58.CrossRefGoogle Scholar
  4. [4]
    R. H. Baughman, A. A. Zakhidov and W. A. de Heer, Carbon nanotubes—the route toward applications, Science, 297 (5582) (2002) 787–792.CrossRefGoogle Scholar
  5. [5]
    X. Wang and H. Cai, Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes, Acta materialia, 54 (8) (2006) 2067–2074.CrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Maranganti and P. Sharma, Length scales at which classical elasticity breaks down for various materials, Physical review letters, 98 (19) (2007) 195504.CrossRefGoogle Scholar
  7. [7]
    A. C. Eringen and D. G. B. Edelen, On nonlocal elasticity, International Journal of Engineering Science, 10 (3) (1972) 233–248.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    J. Choi, M. Cho and W. Kim, Surface effects on the dynamic behavior of nanosized thin film resonator, Applied Physics Letters, 97 (17) (2010) 171901.CrossRefGoogle Scholar
  9. [9]
    M. Cho, J. Choi and W. Kim, Continuum-based bridging model of nanoscale thin film considering surface effects, Japanese Journal of Applied Physics, 48 (2R) (2009) 020219.CrossRefGoogle Scholar
  10. [10]
    W. Kim, S. Y. Rhee and M. Cho, Molecular dynamics-based continuum models for the linear elasticity of nanofilms and nanowires with anisotropic surface effects, Journal of Mechanics of Materials and Structures, 7 (7) (2013) 613–639.CrossRefGoogle Scholar
  11. [11]
    J. H. Ko, J. Jeong, J. Choi and M. Cho, Quality factor in clamping loss of nanocantilever resonators, Applied Physics Letters, 98 (17) (2011) 171909.CrossRefGoogle Scholar
  12. [12]
    Q. Wang and V. K. Varadan, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart Materials and Structures, 15 (2) (2006) 659.CrossRefGoogle Scholar
  13. [13]
    H. Niknam and M. M. Aghdam, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation, Composite Structures, 119 (2015) 452–462.CrossRefGoogle Scholar
  14. [14]
    W. Ann and A. Mehta, The use of functionally graded poly-SiGe layers for MEMS applications, Materials Science Forum, 492 (2005).Google Scholar
  15. [15]
    M. Asghari, M. T. Ahmadian, M. H. Kahrobaiyan and M. Rahaeifard, On the size-dependent behavior of functionally graded micro-beams, Materials & Design, 31 (5) (2010) 2324–2329.CrossRefGoogle Scholar
  16. [16]
    A. E. Alshorbagy, M. A. Eltaher and F. F. Mahmoud, Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, 35 (1) (2011) 412–425.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    L.-L. Ke and Y.-S. Wang, Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures, 93 (2) (2011) 342–350.CrossRefMathSciNetGoogle Scholar
  18. [18]
    L.-L. Ke, Y.-S. Wang, J. Yang and S. Kitipornchai, Nonlinear free vibration of size-dependent functionally graded microbeams, International Journal of Engineering Science, 50 (1) (2012) 256–267.CrossRefMathSciNetGoogle Scholar
  19. [19]
    M. A. Eltaher, A. E. Samir and F. F. Mahmoud, Free vibration analysis of functionally graded size-dependent nanobeams, Applied Mathematics and Computation, 218 (14) (2012) 7406–7420.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    M. A. Eltaher, A. E. Alshorbagy and F. F. Mahmoud, Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams, Composite Structures, 99 (2013) 193–201.CrossRefGoogle Scholar
  21. [21]
    M. A. Eltaher, A. E. Samir and F. F. Mahmoud, Static and stability analysis of nonlocal functionally graded nanobeams, Composite Structures, 96 (2013) 82–88.CrossRefGoogle Scholar
  22. [22]
    M. Simsek and H. H. Yurtcu, Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory, Composite Structures, 97 (2013) 378–386.CrossRefGoogle Scholar
  23. [23]
    T. R. Tauchert, Energy principles in structural mechanics, NY: McGraw-Hill (1974).Google Scholar
  24. [24]
    I. H. A.-H. Hassan, On solving some eigenvalue problems by using a differential transformation, Applied Mathematics and Computation, 127 (1) (2002) 1–22.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    Z. G. Wang, Axial vibration analysis of stepped bar by differential transformation method, Applied Mechanics and Materials, 419 (2013) 273–279.CrossRefGoogle Scholar
  26. [26]
    M. Malik and H. H. Dang, Vibration analysis of continuous systems by differential transformation, Applied Mathematics and Computation, 96 (1) (1998) 17–26.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Farzad Ebrahimi
    • 1
    Email author
  • Majid Ghadiri
    • 1
  • Erfan Salari
    • 1
  • Seied Amir Hosein Hoseini
    • 2
  • Gholam Reza Shaghaghi
    • 1
  1. 1.Department of Mechanical Engineering, Faculty of EngineeringImam Khomeini International UniversityQazvinIran
  2. 2.Department of Mechanical EngineeringUniversity of ZanjanZanjanIran

Personalised recommendations