International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 12, Pages 739-761
doi:10.1155/S0161171202007019
Parametrically excited nonlinear systems: a comparison of two methods
1Mathematics Department, Faculty of Science, Benha University, Benha 13518, Egypt
2Department of Mathematics, Jubail College of Education for Girls, P.O. Box 12020, Jubail Industrial city 31961, Saudi Arabia
Received 9 March 2001
Copyright © 2002 A. F. El-Bassiouny. 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.
Abstract
Subharmonic resonance of two-degree-of-freedom systems with cubic nonlinearities to multifrequency parametric excitations in the
presence of three-to-one internal resonance is investigated. Two
approximate methods (the multiple scales and the generalized
synchronization) are used to construct a first-order nonlinear
ordinary differential equations governing the modulation of the
amplitudes and phases. Steady state solutions and their stability
are computed for selected values of the system parameters. The
results obtained by the two methods are in excellent agreement.
Numerical solutions are carried out and graphical representations
of the results are presented and discussed.