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Advances in Mathematical Physics
Volume 2016, Article ID 8734360, 15 pages
Research Article

On the Motion of Harmonically Excited Spring Pendulum in Elliptic Path Near Resonances

1Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
2Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta University, Tanta 31734, Egypt

Received 30 June 2016; Revised 26 September 2016; Accepted 4 October 2016

Academic Editor: Ciprian G. Gal

Copyright © 2016 T. S. Amer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The response of a nonlinear multidegrees of freedom (M-DOF) for a nature dynamical system represented by a spring pendulum which moves in an elliptic path is investigated. Lagrange’s equations are used in order to derive the governing equations of motion. One of the important perturbation techniques MS (multiple scales) is utilized to achieve the approximate analytical solutions of these equations and to identify the resonances of the system. Besides, the amplitude and the phase variables are renowned to study the steady-state solutions and to recognize their stability conditions. The time history for the attained solutions and the projections of the phase plane are presented to interpret the behavior of the dynamical system. The mentioned model is considered one of the important scientific applications like in instrumentation, addressing the oscillations occurring in sawing buildings and the most of various applications of pendulum dampers.