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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Nonlinear Dynamic Response of a Stiffened Imperfect Beam under Primary Resonance Excitation

Journal of Applied Nonlinear Dynamics 11(3) (2022) 667--702 | DOI:10.5890/JAND.2022.09.010

Osama F. Abdel Aal, Mohammad Abdel Aal, Ahmad Al Qaisia

$^1$ Department of Mechatronics Engineering, University of Jordan, Amman, Jordan

$^2$ Department of Basic Sciences, Middle East University, Amman, Jordan

$^3$ Department of Mechanical Engineering, University of Jordan, Amman, Jordan

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Abstract

This work presents an investigation on the effect of an initial geometric imperfection; wavelength, amplitude, and degree of localization on the in-plane nonlinear resonance responses of an imperfect beam. The beam was modeled as an Euler Bernoulli beam resting on an elastic foundation, hinged at one end and supported by a torsional spring at the other end. The governing model accounts for the effect of the axial force induced by mid-plane stretching. The imperfection was introduced as a rise with different shapes, different amplitudes, and wavelengths. A quantitative and qualitative analysis of the steady-state responses and their stability is obtained by computing force-frequency response curves using the Harmonic Balance method $HB$ and the method of Multiple Scales $MMS$. Results have shown that the behavior of the steady-state responses may change from hardening to softening types depending on the geometrical and physical parameters of the beam under consideration. Results are presented in a dimensionless form for the steady-state forced vibration in the aid of time histories, phase planes, frequency spectrums, and Poincare maps for selected values of physical parameters. It is shown that the beam response may exhibit complicated dynamic behaviors including period multiplying and period de multiplying bifurcations, period three and period six motions, jump phenomenon, and chaos.

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