Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams

Siu Kai Lai, J. Harrington, Y. Xiang, K. W. Chow

Research output: Journal article publicationJournal articleAcademic researchpeer-review

40 Citations (Scopus)


The present work derives the accurate analytical solutions for large amplitude vibration of thin functionally graded beams. In accordance with the Euler-Bernoulli beam theory and the von Kármán type geometric non-linearity, the second-order ordinary differential equation having odd and even non-linearities can be formulated through Hamilton's principle and Galerkin's procedure. This ordinary differential equation governs the non-linear vibration of functionally graded beams with different boundary constraints. Building on the original non-linear equation, two new non-linear equations with odd non-linearity are to be constructed. Employing a generalised Senator-Bapat perturbation technique as an ingenious tool, two newly formulated non-linear equations can be solved analytically. By selecting the appropriate piecewise approximate solutions from such two new non-linear equations, the analytical approximate solutions of the original non-linear problem are established. The present solutions are directly compared to the exact solutions and the available results in the open literature. Besides, some examples are selected to confirm the accuracy and correctness of the current approach. The effects of boundary conditions and vibration amplitudes on the non-linear frequencies are also discussed.
Original languageEnglish
Pages (from-to)473-480
Number of pages8
JournalInternational Journal of Non-Linear Mechanics
Issue number5
Publication statusPublished - 1 Jun 2012
Externally publishedYes


  • Euler-Bernoulli beam theory
  • Functionally graded beam
  • Geometric non-linearity
  • Non-linear differential equation
  • Perturbation approach

ASJC Scopus subject areas

  • Mechanical Engineering
  • Mechanics of Materials
  • Applied Mathematics


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