Nonlinear longitudinal vibration of an elastic rod is studied. The motion of a uniform elastic rod is described by a nonlinear partial differential equation, which has a cubic nonlinear term and a Winkler elastic force that acts along the longitudinal axis of the rod. Galerkin method is used to develop the nonlinear differential equation of elastic rod, which resembles similarity with the Duffing equation. Three different types of robust analytical methods are chosen to solve the nonlinear differential equation and obtain the natural frequency of the system. These are the Homotopy analysis method (HAM), Energy balance method (EBM) and Hamiltonian approach (HA). Subsequently, the analytical results are compared with the numerical solution of the exact equation in order to evaluate the correctness of the applied approaches. Moreover, the effects of the constant coefficients of the elastic force on the ratio of the nonlinear to the linear frequencies are studied. The singular points of the nonlinear differential equation of the elastic rod are extracted and the Jacobian matrix is constructed to recognize their types. Finally, phase-plane trajectories of the system are constructed in order to verify the results obtained from the Jacobian matrix.
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ASME 2013 International Mechanical Engineering Congress and Exposition
November 15–21, 2013
San Diego, California, USA
Conference Sponsors:
- ASME
ISBN:
978-0-7918-5625-3
PROCEEDINGS PAPER
Nonlinear Longitudinal Vibration Solutions of an Elastic Rod Available to Purchase
Hassan Askari,
Hassan Askari
University of Ontario Institute of Technology, Oshawa, ON, Canada
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Ebrahim Esmailzadeh,
Ebrahim Esmailzadeh
University of Ontario Institute of Technology, Oshawa, ON, Canada
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Davood Younesian
Davood Younesian
Iran University of Science and Technology, Tehran, Iran
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Hassan Askari
University of Ontario Institute of Technology, Oshawa, ON, Canada
Ebrahim Esmailzadeh
University of Ontario Institute of Technology, Oshawa, ON, Canada
Davood Younesian
Iran University of Science and Technology, Tehran, Iran
Paper No:
IMECE2013-65990, V04BT04A027; 7 pages
Published Online:
April 2, 2014
Citation
Askari, H, Esmailzadeh, E, & Younesian, D. "Nonlinear Longitudinal Vibration Solutions of an Elastic Rod." Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition. Volume 4B: Dynamics, Vibration and Control. San Diego, California, USA. November 15–21, 2013. V04BT04A027. ASME. https://doi.org/10.1115/IMECE2013-65990
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