Skip to main content
SpringerLink
Log in

Large deflection of flexible tapered functionally graded beam

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

In this paper the semi-analytical analyses of the flexible cantilever tapered functionally graded beam under combined inclined end loading and intermediate loading are studied. In order to derive the fully non-linear equations governing the non-linear deformation, a curvilinear coordinate system is introduced. A general non-linear second order differential equation that governs the shape of a deflected beam is derived based on the geometric nonlinearities, infinitesimal local displacements and local rotation concepts with remarkable physical properties of functionally graded materials. The solutions obtained from semi-analytical methods are numerically compared with the existing elliptic integral solution for the case of a flexible uniform cantilever functionally graded beam. The effects of taper ratio, inclined end load angle and material property gradient on large deflection of the beam are evaluated. The Adomian decomposition method will be useful toward the design of tapered functionally graded compliant mechanisms driven by smart actuators.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ananthasuresh, G.K., Kota, S.: Designing compliant mechanisms. Mechanical Engineering 117(11), 93–96 (1995)

    Google Scholar 

  2. Xiao, S.F., Du, Q., Chen, B., et al.: Modal test and analysis of cantilever beam with tip mass. Acta Mechanica Sinica 18(4), 407–413 (2002)

    Article  Google Scholar 

  3. Cai, G.P., Hong, J.Z., Yang, S.X.: Model study and active control of a rotating flexible cantilever beam. International Journal of Mechanical Sciences 46(6), 871–889 (2004)

    Article  MATH  Google Scholar 

  4. Zhang, X.W., Yang, J.L.: Inverse problem of elastica of a variable-arc-length beam subjected to a concentrated load. Acta Mechanica Sinica 21(5), 444–450 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Liu, J.Y., Hong, J.Z., Cui, L.: An exact nonlinear hybridcoordinate formulation for flexible multibody systems. Acta Mechanica Sinica 23(6), 699–706 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Shin, K., Brennan, M.J.: Two simple methods to suppress the residual vibrations of a translating or rotating flexible cantilever beam. Journal of Sound and Vibration 312(1-2), 140–150 (2008)

    Article  Google Scholar 

  7. Huang, Y.A., Deng, Z.C., Xiong, Y.L.: High-order model and slide mode control for rotating flexible. Smart Structure Mechanism and Machine Theory 43(8), 1038–1054 (2008)

    MATH  Google Scholar 

  8. Piovan, M.T., Sampaio, R.: Vibrations of axially moving flexible beams made of functionally graded materials. Thin-Walled Structures 46(2), 112–121 (2008)

    Article  Google Scholar 

  9. Sankar, B.V.: An elasticity solution for functionally graded beams. Composites Science and Technology 61(5), 689–696 (2001)

    Article  Google Scholar 

  10. Rahimi, G.H., Davoodinik, A.R.: Thermal behavior analysis of the functionally graded Timoshenko’s beam. IUST International Journal of Engineering Science 19(5-1), 105–113 (2008)

    Google Scholar 

  11. Hagood, N.W., Chung, W.H.: Modeling of piezoelectric actuator dynamics for active structural control. Journal of Intelligent Material Systems and Structures 1(7), 327–354 (1990)

    Article  Google Scholar 

  12. Crawley, E.F.: Intelligent structures for aerospace: a technology overview and assessment. AIAA J. 32(8), 1689–1699 (1994)

    Article  Google Scholar 

  13. Gaudenzi, P., Barboni, R.: Static adjustment of beam deflections by means of induced strain actuators. Smart Mater. Struct. 8(2), 278–283 (1999)

    Article  Google Scholar 

  14. Bisshop, K.E., Drucker, D.C.: Large deflection cantilever beams. J. Quart. Appl. Math. 3(3), 272–275 (1945)

    Google Scholar 

  15. Frisch-Fay, R.: Flexible Bars. Washington, Butterworths (1962)

    MATH  Google Scholar 

  16. Howell, L.L, Midha, A.: Parametric deflection approximations for end loaded large deflection beams in compliant mechanisms. Journal of Mechanical Design, Trans. ASME 117(1), 156–165 (1995)

    Article  Google Scholar 

  17. Saxena, A., Kramer, S.N.: A simple and accurate method for determining large deflections in compliant mechanisms subjected to end forces and moments. ASME J. Mech. Des. 120(4), 392–400 (1998)

    Article  Google Scholar 

  18. Kimball, C., Tsai, L.W.: Modeling of flexural beams subjected to arbitrary end loads. ASME J. Mech. Des. 124(2), 223–234 (2002)

    Article  Google Scholar 

  19. Shvartsman, B.S.: Direct method for analysis of flexible cantilever beam subjected to two follower forces. International Journal of Non-Linear Mechanics 44(2), 249–252 (2009)

    Article  MathSciNet  Google Scholar 

  20. Banerjee, A., Bhattacharya, B., Mallik, A.K.: Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches. International Journal of Non-Linear Mechanics 43(5), 366–376 (2008)

    Article  MATH  Google Scholar 

  21. Rahimi, G.H., Davoodinik, A.R.: Large deflection of functionally graded cantilever flexible beam with geometric nonlinearity: analytical and numerical approaches. Scientia Iranica, Transaction B: Mechanical Engineering 17(1), 25–40 (2010)

    MATH  Google Scholar 

  22. Pai, P.F., Anderson, T.J., Wheater, E.A.: Large-deformation tests and total-Lagrangian finite-element analyses of flexible beams. International Journal of Solids and Structures 37(21), 2951–2980 (2000)

    Article  MATH  Google Scholar 

  23. Suresh, S., Mortensen, A.: Fundamentals of Functionally Graded Materials. London, UK: IOM Communications Limited (1998)

    Google Scholar 

  24. Filipich, C.P., Rosales, M.B.: A further study on the postbuckling of extensible elastic rids. International Journal of Non-Linear Mechanics 35(6), 997–1022 (2000)

    Article  MATH  Google Scholar 

  25. Coffin, D.W., Bloom, F.: Elastic solution for the hygrothermal buckling of a beam. International Journal of Non-Linear Mechanics 34(5), 935–947 (1999)

    Article  MathSciNet  Google Scholar 

  26. Li, S.R., Zhou, Y.H., Zheng, X.J.: Thermal post-buckling of a heated elastic rod with pined-fixed ends. Journal of Thermal Stresses 25(1), 45–56 (2002)

    Article  Google Scholar 

  27. Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. John Wiley & Sons, Inc. (2004)

  28. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994)

    MATH  Google Scholar 

  29. El-Wakil, S.A., Abdou, M.A.: New applications of variational iteration method using Adomian polynomials. Nonlinear Dyn. 52(1–2), 41–49 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wazwaz, A.: A reliable algorithm for solving boundary value problems for higher-order integro-differential equations. Appl. Math. Comput. 118(2–3), 327–342 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hashim, I.: Adomian decomposition method for solving boundary value problems for fourth-order integro-differential equations. J. Comput. Appl. Math. 193(2), 658–664 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sadighi, A., Ganji, D.D., Ganjavi, B.: Traveling wave solutions of the Sine-Gordon and the coupled Sine-Gordon equations using the homotopy-perturbationmethod. Scientia Iranica. 16(2), 189–195 (2009)

    MATH  Google Scholar 

  33. Davoodinik, A.R.: Structural behavior analysis of nonhomogenous and non-uniform flexible beam. [Ph.D. Thesis], Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran, In Persian, Ch.3, Sec.3–9 (July 2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran

    A. R. Davoodinik & G. H. Rahimi

Authors
  1. A. R. Davoodinik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. H. Rahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. H. Rahimi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Davoodinik, A.R., Rahimi, G.H. Large deflection of flexible tapered functionally graded beam. Acta Mech Sin 27, 767–777 (2011). https://doi.org/10.1007/s10409-011-0476-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-011-0476-2

Keywords

Search

Navigation