Applied Mathematics and Mechanics

, Volume 40, Issue 1, pp 85–96 | Cite as

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

  • Dongxing CaoEmail author
  • Yanhui Gao


The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded (AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions. Furthermore, assuming polynomial distributions of Young’s modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.

Key words

axially functionally graded (AFG) beam non-uniform natural frequency asymptotic development method 

Chinese Library Classification


2010 Mathematics Subject Classification



  1. [1]
    NGUYEN, D. K. Large displacement response of tapered cantilever beams made of axially functionally graded material. Composites Part B: Engineering, 55(9), 298–305 (2013)CrossRefGoogle Scholar
  2. [2]
    NIE, G. J., ZHONG, Z., and CHEN, S. Analytical solution for a functionally graded beam with arbitrary graded material properties. Composites Part B: Engineering, 44(1), 274–282 (2013)CrossRefGoogle Scholar
  3. [3]
    ZHANG, J. H. and ZHANG, W. Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a honeycomb sandwich plate. Acta Mechanica, 223(5), 1047–1066 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    NIU, Y., HAO, Y., YAO, M., ZHANG, W., and YANG, S. Nonlinear dynamics of imperfect FGM conical panel. Shock and Vibration, 2018, 4187386 (2018)Google Scholar
  5. [5]
    SHENG, G. G. and WANG, X. Nonlinear vibration control of functionally graded laminated cylindrical shells. Composites Part B: Engineering, 52(9), 1–10 (2013)CrossRefGoogle Scholar
  6. [6]
    HAO, Y. X., LI, Z. N., ZHANG, W., LI, S. B., and YAO, M. H. Vibration of functionally graded sandwich doubly curved shells using improved shear deformation theory. Science China Technological Sciences, 61(6), 791–808 (2018)CrossRefGoogle Scholar
  7. [7]
    LEE, J. W. and LEE, J. Y. Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression. International Journal of Mechanical Sciences, 122, 1–17 (2017)CrossRefGoogle Scholar
  8. [8]
    LI, X. F., WANG, B. L., and HAN, J. C. A higher-order theory for static and dynamic analyses of functionally graded beams. Archive of Applied Mechanics, 80(10), 1197–1212 (2010)CrossRefzbMATHGoogle Scholar
  9. [9]
    AVCAR, M. and ALWAN, A. S. Free vibration of functionally graded Rayleigh beam. International Journal of Engineering and Applied Sciences, 9(2), 127–127 (2017)CrossRefGoogle Scholar
  10. [10]
    SINA, S. A., NAVAZI, H. M., and HADDADPOUR, H. An analytical method for free vibration analysis of functionally graded beams. Materials and Design, 30(3), 741–747 (2009)CrossRefGoogle Scholar
  11. [11]
    YANG, X., WANG, S., ZHANG, W., QIN, Z., and YANG, T. Dynamic analysis of a rotating tapered cantilever Timoshenko beam based on the power series method. Applied Mathematics and Mechanics (English Edition), 38(10), 1425–1438 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    LEUNG, A. Y. T., ZHOU, W. E., LIM, C. W., YUEN, R. K. K., and LEE, U. Dynamic stiffness for piecewise non-uniform Timoshenko column by power series—part I: conservative axial force. International Journal for Numerical Methods in Engineering, 51(5), 505–529 (2001)CrossRefzbMATHGoogle Scholar
  13. [13]
    HEIN, H. and FEKLISTOVA, L. Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets. Engineering Structures, 33(12), 3696–3701 (2011)CrossRefGoogle Scholar
  14. [14]
    WANG, X. and WANG, Y. Free vibration analysis of multiple-stepped beams by the differential quadrature element method. Applied Mathematics and Computation, 219(11), 5802–5810 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    ZHAO, Y., HUANG, Y., and GUO, M. A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory. Composite Structures, 168, 277–284 (2017)CrossRefGoogle Scholar
  16. [16]
    CHEN, L. Q. and CHEN, H. Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model. Journal of Engineering Mathematics, 67(3), 205–218 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    YAN, Q. Y., DING, H., and CHEN, L. Q. Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations. Applied Mathematics and Mechanics (English Edition), 36(8), 971–984 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    DING, H., TANG, Y. Q., and CHEN, L. Q. Frequencies of transverse vibration of an axially moving viscoelastic beam. Journal of Vibration and Control, 23(20), 1–11 (2015)MathSciNetGoogle Scholar
  19. [19]
    DING, H., HUANG, L. L., MAO, X. Y., and CHEN, L. Q. Primary resonance of traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38(1), 1–14 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    CHEN, R. M. Some nonlinear dispersive waves arising in compressible hyperelastic plates. International Journal of Engineering Science, 44(18), 1188–1204 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    ANDRIANOV, I. V. and DANISHEVS’KYY, V. V. Asymptotic approach for non-linear periodical vibrations of continuous structures. Journal of Sound and Vibration, 249(3), 465–481 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    NAYFEH, A. H. and MOOK, D. T. Nonlinear Oscillations, John Wiley & Sons, New York, 54–56 (1979)zbMATHGoogle Scholar
  23. [23]
    LENCI, S., CLEMENTI, F., and MAZZILLI, C. E. N. Simple formulas for the natural frequencies of non-uniform cables and beams. International Journal of Mechanical Sciences, 77(4), 155–163 (2013)CrossRefGoogle Scholar
  24. [24]
    TARNOPOLSKAYA, T., HOOG, F. D., FLETCHER, N. H., and THWAITES, S. Asymptotic analysis of the free in-plane vibrations of beams with arbitrarily varying curvature and crosssection. Journal of Sound and Vibration, 196(5), 659–680 (1996)CrossRefGoogle Scholar
  25. [25]
    KUKLA, S. and RYCHLEWSKA, J. An approach for free vibration analysis of axially graded beams. Journal of Theoretical and Applied Mechanics, 54(3), 859–870 (2016)CrossRefGoogle Scholar
  26. [26]
    HUANG, Y. and LI, X. F. A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. Journal of Sound and Vibration, 329(11), 2291–2303 (2010)CrossRefGoogle Scholar
  27. [27]
    HUANG, Y., YANG, L. E., and LUO, Q. Z. Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section. Composites Part B: Engineering, 45(1), 1493–1498 (2013)CrossRefGoogle Scholar
  28. [28]
    HUANG, Y. and RONG, H. W. Free vibration of axially inhomogeneous beams that are made of functionally graded materials. International Journal of Acoustics and Vibration, 22(1), 68–73 (2017)MathSciNetCrossRefGoogle Scholar
  29. [29]
    XIE, X., ZHENG, H., and ZOU, X. An integrated spectral collocation approach for the static and free vibration analyses of axially functionally graded nonuniform beams. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231(13), 2459–2471 (2016)Google Scholar
  30. [30]
    AKGÖZ, B. and CIVALEK, Ö. Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory. Composite Structures, 98, 314–322 (2013)CrossRefGoogle Scholar
  31. [31]
    FANG, J. S. and ZHOU, D. Free vibration analysis of rotating axially functionally graded tapered Timoshenko beams. International Journal of Structural Stability and Dynamics, 16(5), 197–202 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    SHAHBA, A., ATTARNEJAD, R., and HAJILAR, S. Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams. Shock and Vibration, 18(5), 683–696 (2011)CrossRefGoogle Scholar
  33. [33]
    SHAHBA, A., ATTARNEJAD, R., MARVI, M. T., and HAJILAR, S. Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Composites Part B: Engineering, 42(4), 801–808 (2011)CrossRefGoogle Scholar
  34. [34]
    SHAHBA, A. and RAJASEKARAN, S. Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling, 36(7), 3094–3111 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    RAO, S. S. Vibration of Continuonus Systems, John Wiley & Sons, Canada, 317–326 (2007)Google Scholar
  36. [36]
    NAYFEH, A. H. Introduction to Perturbation Techniques, John Wiley & Sons, New York, 18–21 (1981)zbMATHGoogle Scholar
  37. [37]
    KRYZHEVICH, S. G. and VOLPERT, V. A. Different types of solvability conditions for differential operators. Electronic Journal of Differential Equations, 2006(100), 1–24 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mechanical EngineeringBeijing University of TechnologyBeijingChina
  2. 2.Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical StructuresBeijingChina

Personalised recommendations