Skip to main content
Log in

Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

The free vibration of axially functionally graded (FG) non-uniform beams with different boundary conditions is studied using Differential Transformation (DT) based Dynamic Stiffness approach. This method is capable of modeling any beam (Timoshenko or Euler, centrifugally stiffened or not) whose cross sectional area, moment of Inertia and material properties vary along the beam. The effectiveness of the method is confirmed by comparing the present results with existing closed form solutions and numerical results. In FG beams, flexural rigidity and mass density may take majority of functions including polynomials, trigonometric and exponential functions (converted to polynomial expressions). DT based Dynamic stiffness approach is proved to be a versatile and simple approach compared to many other methods already proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Abbreviations

A 0 :

Area of the section at the root

A(x):

Cross sectional area at any section

b(x):

Breadth of the cross section at any section

d(x)=E(x)I(x):

Flexural rigidity at any section

E(x):

Modulus of elasticity of the axially graded material at any section

e :

Taper ratio

G(x):

Modulus of rigidity at any section

h(x):

Depth of the cross section at any section

I(x):

Moment of Inertia at any section

I 0 :

Moment of Inertia of the section at the root

K(x)=κA(x)G(x):

Shear rigidity at any section

K :

Structural stiffness matrix

KG :

Geometric stiffness matrix

L :

Length of the beam

mr=1:

Depth taper only

mr=2:

Both width and depth taper

M T :

Tip mass at the free end

M(x):

Moment at any section

M :

Mass matrix

nt :

Number of terms

nr :

Material non-homogeneity factor

P(x):

Centrifugal force or compressive load

p(x):

Lateral load

\(p = \frac{\mathit{PL}^{2}}{\mathit{EI}_{0}}\) :

Buckling load parameter

R :

Hub radius

s :

Summation index

T :

Typical material property

T a ,T z :

Typical material property for Alumina and Zirconia respectively

V(x):

Shear force

w :

Lateral deflection of the centre line of the beam

x, y and z :

Cartesian coordinate axes

y(x):

Function

β :

Tip mass parameter

δ=R/L :

Non-dimensional parameter for hub radius \(\eta = \sqrt{\varOmega^{2} + \omega^{2}}\)

κ :

Shear correction factor

λ :

Rotating speed parameter where \(\lambda = \varOmega\sqrt{\frac{\rho_{0}A_{0}L^{4}}{E_{0}I_{0}}}\)

μ :

Natural frequency parameter where \(\mu = \omega\sqrt{\frac{\rho_{0}A_{0}L^{4}}{E_{0}I_{0}}}\)

Ω :

Angular rotation speed in radians/sec

φ :

Shape function for θ

ρ(x):

Mass density of axially graded material at any section

ψ :

Shape function for bending rotation

θ :

Bending rotation

ω :

Natural frequency

\(\xi = \frac{x}{L} \) :

Non-dimensional variable

ς :

Damping factor

\(\nabla = \frac{\mathrm{d}}{\mathrm{d}x}\) :

Operator

References

  1. Koizumi M (1993) The concept of FGM. Ceramic Trans Funct Grad Mater 34:3–10

    Google Scholar 

  2. Koizumi M (1997) FGM activities in Japan. Composites, Part B, Eng 28:1–4

    Article  Google Scholar 

  3. Elishakoff I, Perez A (2005) Design of a polynomially inhomogeneous bar with a tip mass for specified mode shape and natural frequency. J Sound Vib 287(4–5):1004–1012

    Article  ADS  Google Scholar 

  4. Elishakoff I, Pentaras D (2006) Apparently the first closed-form solution of inhomogeneous elastically restrained vibrating beams. J Sound Vib 298(1–2):439–445

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Banerjee JR (1997) Dynamic stiffness formulation for structural elements: a general approach. Comput Struct 63:101–103

    Article  MATH  Google Scholar 

  6. Banerjee JR (2000) Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method. J Sound Vib 233(5):857–875

    Article  ADS  MATH  Google Scholar 

  7. Wang G, Wereley NM (2004) Free vibration analysis of rotating blades with uniform tapers. AIAA J 42(12):429–437

    Article  Google Scholar 

  8. Banerjee JR, Su H, Jackson DR (2006) Free vibration of rotating tapered beams using the dynamic stiffness method. J Sound Vib 298(4–5):1034–1054

    Article  ADS  Google Scholar 

  9. Vinod KG, Gopalakrishnan S, Ganguli R (2007) Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements. Int J Solids Struct 44:5875–5893

    Article  MATH  Google Scholar 

  10. Doyle JF (1977) Wave propagation in structures, 2nd edn. Springer, Berlin (Chap 5)

    Google Scholar 

  11. Wright AD, Smith VR, Thresher TW, Wang JLC (1982) Vibration modes of centrifugally stiffened beams. ASME J Appl Mech 49(2):197–202

    Article  MATH  Google Scholar 

  12. Huang Y, Li XF (2010) A new approach for free vibration of axially functionally graded beams with non-uniform cross section. J Sound Vib 329(11):2291–2303

    Article  ADS  Google Scholar 

  13. Mabie HH, Rogers CB (1972) Transverse vibration of double-tapered cantilever beams. J Acoust Soc Am 51:1771–1774

    Article  ADS  Google Scholar 

  14. Downs B (1977) Transverse vibration of cantilever beam having unequal breadth and depth tapers. ASME J Appl Mech 44:737–742

    Article  Google Scholar 

  15. Gupta RS, Rao SS (1978) Finite element eigen value analysis of tapered and twisted Timoshenko beams. J Sound Vib 56(2):187–200

    Article  ADS  MATH  Google Scholar 

  16. Dawe DJ (1978) A finite element for the vibration analysis of Timoshenko beams. J Sound Vib 60(1):11–20

    Article  ADS  MATH  Google Scholar 

  17. To CWS (1981) A linearly tapered beam finite element incorporating shear deformation and rotary inertia for vibration analysis. J Sound Vib 78(4):475–484

    Article  ADS  Google Scholar 

  18. Lees AW, Thomas DL (1982) Unified Timoshenko beam finite element. J Sound Vib 80(3):355–366

    Article  ADS  MATH  Google Scholar 

  19. Lee SY, Lin SM (1994) Bending vibrations of rotating non-uniform Timoshenko beams with an elastically restrained root. ASME J Appl Mech 61:949–955

    Article  MATH  Google Scholar 

  20. Du H, Lim MK, Liew KK (1994) A power series solution for vibration of a rotating Timoshenko beam. J Sound Vib 175(4):505–523

    Article  ADS  MATH  Google Scholar 

  21. Nagaraj VT (1996) Approximate formula for the frequencies of a rotating Timoshenko beam. J Aircr 33:637–639

    Article  Google Scholar 

  22. Lin SC, Hsiao KM (2001) Vibration analysis of a rotating Timoshenko beam. J Sound Vib 240(2):303–322

    Article  ADS  MATH  Google Scholar 

  23. Choi DT, Chou YT (2001) Vibration analysis of elastically supported turbo machinery blades by the modified differential quadrature methods. J Sound Vib 240(5):937–953

    Article  ADS  Google Scholar 

  24. Irie T, Yamada G, Takahashi I (1979) Determination of the steady state response of a Timoshenko beam of varying section by the use of the spline interpolation technique. J Sound Vib 63(2):287–295

    Article  ADS  MATH  Google Scholar 

  25. Irie T, Yamada G, Takahashi I (1980) Vibration and stability of a non-uniform Timoshenko beam subjected to follower force. J Sound Vib 70(4):503–512

    Article  ADS  MATH  Google Scholar 

  26. Lee SY, Lin SM (1992) Exact vibration solutions for non-uniform Timoshenko beams with attachments. AIAA J 30(12):2930–2934

    Article  ADS  MATH  Google Scholar 

  27. Shahba A, Attarnejad R, Tavanaie Marvi M, Hajilar S (2011) Free vibration and stability of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Composites, Part B, Eng 42(4):801–808

    Article  Google Scholar 

  28. Attarnejad R, Semnani SJ, Shahba A (2010) Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams. Finite Elem Anal Des 46:916–929

    Article  MathSciNet  Google Scholar 

  29. Attarnejad R, Shahba A (2011) Basic displacement functions for centrifugally stiffened tapered beam. Int J Numer Methods Biomed Eng 27:1385–1397

    MATH  Google Scholar 

  30. Attarnejad R, Shahba A (2011) Basic displacement functions in analysis of centrifugally stiffened tapered beams. Arab J Sci Eng 36:841–853

    Article  Google Scholar 

  31. Zhou JK (1986) Differential transformation and its application for electrical circuits. Huazhong University Press, China

    Google Scholar 

  32. Chen CK, Ju SP (2004) Application of differential transformation to transient advective-dispersive transport equation. J Appl Math Comput 155(1):25–38

    Article  MathSciNet  MATH  Google Scholar 

  33. Arikoglu A, Ozkol I (2004) Solution of boundary value problems for integro-differential equations by using differential transformation method. J Appl Math Comput 168(2):1145–1158

    MathSciNet  Google Scholar 

  34. Bert CW, Zeng H (2004) Analysis of axial vibration of compound bars by differential transformation method. J Sound Vib 275(3–5):641–647

    Article  ADS  Google Scholar 

  35. Kaya MO (2006) Free vibration analysis of a rotating Timoshenko beam by differential transformation method. Aircr Eng Aerosp Technol 78:194–203

    Article  Google Scholar 

  36. Banerjee JR, Sobey AJ (2002) Energy expressions for rotating tapered Timoshenko beam. J Sound Vib 254(4):818–822

    Article  ADS  Google Scholar 

  37. Wilson EL (2002) Three dimensional static and dynamic analysis of structures. Computers and Structures, Berkeley

    Google Scholar 

  38. Wakashima K, Hirano T, Nino M (1990) Space applications of advanced structural materials. ESA SP 303:97

    Google Scholar 

  39. Nakamura T, Wang T, Sampath S (2000) Determination of properties of graded materials by inverse analysis and instrumented indentation. Acta Mater 48:4293–4306

    Article  Google Scholar 

  40. Ozgumus OO, Kaya MO (2010) Vibration analysis of a rotating tapered Timoshenko beam using DTM. Meccanica 45:33–42

    Article  MATH  Google Scholar 

  41. Attarnejad R, Shahba A (2011) Dynamic displacement functions in free vibration analysis of centrifugally stiffened tapered beams. Meccanica 46(6):1267–1281

    Article  MathSciNet  Google Scholar 

  42. Tang B (2008) Combined dynamic stiffness matrix and precise time integration method for transient forced vibration response analysis of beams—short communication. J Sound Vib 309:868–876

    Article  ADS  Google Scholar 

  43. Zhong WX, Williams FW (1994) A precise time step integration method. Proc IME C J Mech Eng Sci 208:427–430

    Article  Google Scholar 

  44. Wang CM, Wang CY, Reddy JN (2005) Exact solutions for buckling of structural members. CRC Press, Boca Raton

    Google Scholar 

  45. Hodges DH, Rutkowski MJ (1981) From vibration analysis of rotating beam by a variable order finite element method. AIAA J 19:1459–1466

    Article  ADS  MATH  Google Scholar 

  46. Zarrinzadeh H, Attarnejad R, Shahba A (2012) Free vibration of rotating axially functionally graded tapered beams. Proc IME G J Aero Eng 226(4):363–379

    Article  Google Scholar 

  47. Ozgumus OO, Kaya MO (2006) Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method. Meccanica 41:661–670

    Article  MATH  Google Scholar 

  48. Elishakoff I (2001) Inverse buckling problem for inhomogeneous columns. Int J Solids Struct 38(3):457–464

    Article  MathSciNet  MATH  Google Scholar 

  49. Huang Y, Li XF (2011) Buckling analysis of nonuniform and axially graded columns with varying flexural rigidity. ASCE J Eng Mech 137(1):73–81

    Article  Google Scholar 

  50. Rajasekaran S (2012) Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods. Appl Math Model. doi:10.1016/j.apm.2012.09.024

    Google Scholar 

  51. Shahba A, Rajasekaran S (2012) Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials. Appl Math Model 36:3094–3111

    Article  MathSciNet  MATH  Google Scholar 

  52. Bervillier (2012) State of the differential transformation method. J Appl Math Comput 218:10158–10170

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author thanks the management and Principal Dr. R. Rudramoorthy of PSG College of Technology for providing necessary facilities to complete the research work reported in this paper. The author also thanks the anonymous reviewers for their advice and suggestions towards enhancing the quality of the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. Rajasekaran.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rajasekaran, S. Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach. Meccanica 48, 1053–1070 (2013). https://doi.org/10.1007/s11012-012-9651-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-012-9651-1

Keywords

Navigation