Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis
Abstract This study analyzes the fourth-order nonlinear free vibration of a Timoshenko beam. We discretize the governing differential equation by Galerkin's procedure and then apply the homotopy analysis method (HAM) to the obtained ordinary differential equation of the generalized coordinate....
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Marcílio Alves
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Series: | Latin American Journal of Solids and Structures |
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doaj-0b6346cecf8442e1a0e8f536b196ac082020-11-25T01:52:55ZengMarcílio AlvesLatin American Journal of Solids and Structures1679-782513101866187710.1590/1679-78252766S1679-78252016001001866Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy AnalysisShahram Shahlaei-FarAirton NabarreteJosé Manoel BalthazarAbstract This study analyzes the fourth-order nonlinear free vibration of a Timoshenko beam. We discretize the governing differential equation by Galerkin's procedure and then apply the homotopy analysis method (HAM) to the obtained ordinary differential equation of the generalized coordinate. We derive novel analytical solutions for the nonlinear natural frequency and displacement to investigate the effects of rotary inertia, shear deformation, pre-tensile loads and slenderness ratios on the beam. In comparison to results achieved by perturbation techniques, this study demonstrates that a first-order approximation of HAM leads to highly accurate solutions, valid for a wide range of amplitude vibrations, of a high-order strongly nonlinear problem.http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1679-78252016001001866&lng=en&tlng=enstrongly nonlinear vibrationhomotopy analysis methodGalerkin methodTimoshenko beamnonlinear natural frequency |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Shahram Shahlaei-Far Airton Nabarrete José Manoel Balthazar |
spellingShingle |
Shahram Shahlaei-Far Airton Nabarrete José Manoel Balthazar Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis Latin American Journal of Solids and Structures strongly nonlinear vibration homotopy analysis method Galerkin method Timoshenko beam nonlinear natural frequency |
author_facet |
Shahram Shahlaei-Far Airton Nabarrete José Manoel Balthazar |
author_sort |
Shahram Shahlaei-Far |
title |
Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis |
title_short |
Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis |
title_full |
Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis |
title_fullStr |
Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis |
title_full_unstemmed |
Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis |
title_sort |
nonlinear vibrations of cantilever timoshenko beams: a homotopy analysis |
publisher |
Marcílio Alves |
series |
Latin American Journal of Solids and Structures |
issn |
1679-7825 |
description |
Abstract This study analyzes the fourth-order nonlinear free vibration of a Timoshenko beam. We discretize the governing differential equation by Galerkin's procedure and then apply the homotopy analysis method (HAM) to the obtained ordinary differential equation of the generalized coordinate. We derive novel analytical solutions for the nonlinear natural frequency and displacement to investigate the effects of rotary inertia, shear deformation, pre-tensile loads and slenderness ratios on the beam. In comparison to results achieved by perturbation techniques, this study demonstrates that a first-order approximation of HAM leads to highly accurate solutions, valid for a wide range of amplitude vibrations, of a high-order strongly nonlinear problem. |
topic |
strongly nonlinear vibration homotopy analysis method Galerkin method Timoshenko beam nonlinear natural frequency |
url |
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1679-78252016001001866&lng=en&tlng=en |
work_keys_str_mv |
AT shahramshahlaeifar nonlinearvibrationsofcantilevertimoshenkobeamsahomotopyanalysis AT airtonnabarrete nonlinearvibrationsofcantilevertimoshenkobeamsahomotopyanalysis AT josemanoelbalthazar nonlinearvibrationsofcantilevertimoshenkobeamsahomotopyanalysis |
_version_ |
1724992114775818240 |