Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance
Sayed, M., Mousa, A., Alzaharani, D., Mustafa, I., El-Bendary, S. (2020). Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance. EKB Journal Management System, 28(1), 1-21. doi: 10.1186/s42787-020-00102-7
M. Sayed; A. A. Mousa; D. Y. Alzaharani; I. H. Mustafa; S. I. El-Bendary. "Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance". EKB Journal Management System, 28, 1, 2020, 1-21. doi: 10.1186/s42787-020-00102-7
Sayed, M., Mousa, A., Alzaharani, D., Mustafa, I., El-Bendary, S. (2020). 'Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance', EKB Journal Management System, 28(1), pp. 1-21. doi: 10.1186/s42787-020-00102-7
Sayed, M., Mousa, A., Alzaharani, D., Mustafa, I., El-Bendary, S. Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance. EKB Journal Management System, 2020; 28(1): 1-21. doi: 10.1186/s42787-020-00102-7

Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance

Journal of the Egyptian Mathematical Society
Volume 28, Issue 1, June 2020, Page 1-21  XML PDF (1.83 MB)
DOI: 10.1186/s42787-020-00102-7
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Authors
M. Sayed; A. A. Mousa; D. Y. Alzaharani; I. H. Mustafa; S. I. El-Bendary
Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt
Abstract
In this paper, we study a multiple scales perturbation and numerical solution for
vibrations analysis and control of a system which simulates the vibrations of a
nonlinear composite beam model. System of second order differential equations
with nonlinearity due to quadratic and cubic terms, excited by parametric and
external excitations, are presented. The controller is implemented to control one
frequency at primary and parametric resonance where damage in the mechanical
system is probable. Active control is applied to the system. The multiple scales
perturbation (MSP) method is implemented to obtain an approximate analytical
solution. The stability analysis of the system is obtained by frequency response (FR).
Bifurcation analysis is conducted using various control parameters such as natural
frequency (ω1), detuning parameter (σ1), feedback signal gain (β), control signal
gain (γ), and other parameters. The dynamic behavior of the system is predicted
within various ranges of bifurcation parameters. All of the stable steady state (point
attractor), stable periodic attractors, unstable steady state, and unstable periodic
attractors are determined efficiently using bifurcation analysis. The controller’s
influence on system behavior is examined numerically. To validate our results, the
approximate analytical solution using the MSP method is compared with the
numerical solution using the Runge-Kutta (RK) method of order four.

Keywords
Active control; Stability; Internal resonance; Bifurcation analysis; Point and periodic attractor
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