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

Journal of the Egyptian Mathematical Society

I. H. Mustafa¹  M. Sayed²  S. I. El-Bendary³  D. Y. Alzaharani⁴  A. A. Mousa⁵ 

[1] Biomedical Engineering Department, Helwan University, Cairo, Egypt;Chemical Engineering Department, University of Waterloo, Waterloo, Canada;Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, 32952, Menouf, Egypt;Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt;Mathematics Department, Faculty of Arts and Science in Baljurashi, Al-Baha University, Al Baha, Kingdom of Saudi Arabia;Mathematics and Statistics Department, Faculty of Science, Taif University, Taif, Kingdom of Saudi Arabia;Basic Engineering Sciences Department, Faculty of Engineering, Menoufia University, Shibin El-Kom, Egypt;

关键词: Active control;    Stability;    Internal resonance;    Bifurcation analysis;    Point and periodic attractor;    34D10;    34D20;    34H20;   

DOI  :  10.1186/s42787-020-00102-7

来源: Springer

【 摘 要 】

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.

【 授权许可】

CC BY