Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System

Usama H. Hegazy^1, and Mousa A. ALshawish¹

¹Department of Mathematics, Faculty of Science, Al-Azhar University, Gaza, Palestine

Cite this paper:
Usama H. Hegazy and Mousa A. ALshawish. Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System. International Journal of Partial Differential Equations and Applications. 2016; 4(1):7-15. doi: 10.12691/ijpdea-4-1-2

Abstract

This paper is concerned with the three dimensional motion of a nonlinear dynamical system. The motion is described by nonlinear partial differential equation, which is converted by Galerkin method to three dimensional ordinary differential equations. The three dimensional differential equations, under the influence of external forces, are solved analytically and numerically by the multiple time scales perturbation technique and the Runge-Kutta fourth order method. Phase plane technique and frequency response equations are used to investigate the stability of the system and the effects of the parameters of the system, respectively.

Keywords:
Galerkin method resonances nonlinearities

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Yaman, M, “Analysis of subcombination internal resonances in a non- linear cantilever beam of varying orientation with tip mass,” Int J Non-Linear Mech, 58. 22-29. 2014.
[2]	Huang, J., Fung, R. and Lin, C, “Dynamic stability of a moving string undergoing three-dimensional vibration,” Int J Mech Sci, 37 (2). 145-160. 1995.
[3]	Pai, P.F. and Nayfeh, A.H, “Three-Dimensional nonlinear vibrations of composite beams-III. Chordwise excitations,” Nonlinear Dyn, 2. 137-156. 1991.
[4]	Hegazy, U.H., 3:1 Internal resonance of a string-beam coupled system with cubic nonlinearities, Commun Nonlinear Sci Numer Simul, 15, 4219-4229, 2010.
[5]	Hegazy, U.H, “Nonlinear vibrations of a thin plate under simultaneous internal and external resonances,” J Vib Acoust, 132. 051004. 9 pages. 2010.
[6]	Srinil, N., Rega, R. and Chucheepsaku, S, “Large amplitude three-dimensional free vibrations of inclined sagged elastic cables,” Nonlinear Dyn, 33. 129-154. 2003.
[7]	Sadri, M. and Younesian, D, “Nonlinear harmonic vibration analysis of a plate-cavity system,” Nonlinear Dyn, 74. 1267-1279. 2013.
[8]	Zhang, G., Ding, H., Chen, L. and Yang, S, “Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds,” J Sound Vibr, 331 (7). 1612-1623. 2012.
[9]	Cveticanin, L, “Analytical methods for solving strongly non-linear differential equations,” J Sound Vibr, 214 (2). 325-338. 1998.
[10]	Hegazy, U.H, “Single-Mode response and control of a hinged-hinged flexible beam,” Arch Appl Mech, 79. 335-345. 2009.