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

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TitleAccurate analytical perturbation approach for large amplitude vibration of functionally graded beams
AuthorsLai, SK1
Harrington, J2
Xiang, Y2
Chow, KW1
KeywordsEuler-Bernoulli Beam Theory
Functionally Graded Beam
Geometric Non-Linearity
Non-Linear Differential Equation
Perturbation Approach
Issue Date2012
PublisherPergamon. The Journal's web site is located at http://www.elsevier.com/locate/nlm
CitationInternational Journal Of Non-Linear Mechanics, 2012, v. 47 n. 5, p. 473-480 [How to Cite?]
DOI: http://dx.doi.org/10.1016/j.ijnonlinmec.2011.09.019
AbstractThe 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. © 2011 Elsevier Ltd. All rights reserved.
ISSN0020-7462
2011 Impact Factor: 1.209
2011 SCImago Journal Rankings: 0.064
DOIhttp://dx.doi.org/10.1016/j.ijnonlinmec.2011.09.019
ISI Accession Number IDWOS:000304848200007
Funding AgencyGrant Number
University of Western Sydney20731-80749
Research Grants CouncilHKU7120/08E
University of Hong Kong200911159076
Civionics Research Centre of the University of Western Sydney
Funding Information:

The work described in this paper was supported by the University of Western Sydney through a Research Grant Scheme (Project no. 20731-80749). Partial financial support has been provided by the Research Grants Council contract HKU7120/08E and the University of Hong Kong Seed Funding Program for Basic Research 200911159076. A fellowship offered to the first author for working at the Civionics Research Centre of the University of Western Sydney is also gratefully acknowledged.

ReferencesReferences in Scopus
GrantsCompeting nonlinearities in systems of hydrodynamic waves
DC Field
Value
dc.contributor.authorLai, SK
dc.contributor.authorHarrington, J
dc.contributor.authorXiang, Y
dc.contributor.authorChow, KW
dc.date.accessioned2012-08-08T08:45:43Z
dc.date.available2012-08-08T08:45:43Z
dc.date.issued2012
dc.description.abstractThe 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. © 2011 Elsevier Ltd. All rights reserved.
dc.description.grantCompeting nonlinearities in systems of hydrodynamic waves
dc.description.grantcode101854
dc.description.natureLink_to_subscribed_fulltext
dc.identifier.citationInternational Journal Of Non-Linear Mechanics, 2012, v. 47 n. 5, p. 473-480 [How to Cite?]
DOI: http://dx.doi.org/10.1016/j.ijnonlinmec.2011.09.019
dc.identifier.citeulike9833459
dc.identifier.doihttp://dx.doi.org/10.1016/j.ijnonlinmec.2011.09.019
dc.identifier.epage480
dc.identifier.hkuros204446
dc.identifier.isiWOS:000304848200007
Funding AgencyGrant Number
University of Western Sydney20731-80749
Research Grants CouncilHKU7120/08E
University of Hong Kong200911159076
Civionics Research Centre of the University of Western Sydney
Funding Information:

The work described in this paper was supported by the University of Western Sydney through a Research Grant Scheme (Project no. 20731-80749). Partial financial support has been provided by the Research Grants Council contract HKU7120/08E and the University of Hong Kong Seed Funding Program for Basic Research 200911159076. A fellowship offered to the first author for working at the Civionics Research Centre of the University of Western Sydney is also gratefully acknowledged.

dc.identifier.issn0020-7462
2011 Impact Factor: 1.209
2011 SCImago Journal Rankings: 0.064
dc.identifier.issue5
dc.identifier.scopuseid_2-s2.0-84859743908
dc.identifier.spage473
dc.identifier.urihttp://hdl.handle.net/10722/157186
dc.identifier.volume47
dc.languageeng
dc.publisherPergamon. The Journal's web site is located at http://www.elsevier.com/locate/nlm
dc.publisher.placeUnited Kingdom
dc.relation.ispartofInternational Journal of Non-Linear Mechanics
dc.relation.referencesReferences in Scopus
dc.subjectEuler-Bernoulli Beam Theory
dc.subjectFunctionally Graded Beam
dc.subjectGeometric Non-Linearity
dc.subjectNon-Linear Differential Equation
dc.subjectPerturbation Approach
dc.titleAccurate analytical perturbation approach for large amplitude vibration of functionally graded beams
dc.typeArticle
Author Affiliations
  1. The University of Hong Kong
  2. University of Western Sydney