2D and 3D Visualization for the Static Bifurcations and Nonlinear Oscillations of a Self-Excited System with Time-Delayed Controller

Symmetry

Jan Awrejcewicz¹  Mohamed S. Mohamed²  Nasser A. Saeed³  Mohamed A. Alkashif⁴ 

[1] Department of Automation, Biomechanics, and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90924 Lodz, Poland;Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia;Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt;Department of Science and Technology, University College—Ranyah, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia;

关键词: time delay;    self-excited system;    stability chart;    static bifurcation;    periodic and quasi-periodic oscillations;    bifurcation diagrams;   

DOI  :  10.3390/sym14030621

来源: DOAJ

【 摘 要 】

This research focuses on the nonlinear vibration control of a self-excited single-degree-of-freedom system. The integral resonant controller (IRC) is introduced to stabilize the unstable motion and suppress nonlinear oscillations of the considered system. The nonlinear dynamical equations that govern the vibratory behaviors of the proposed closed-loop control system are investigated using perturbation analysis, where loop delays have been included in the studied model. The system bifurcation behaviors have been visualized in both the two and three-dimensional spaces, and corresponding dynamical behaviors have been explored numerically using the bifurcation diagrams, Poincaré map, time-response, zero-one chaotic test algorithm, and frequency spectrum. The obtained analytical investigations revealed that the uncontrolled system can oscillate with one of four vibration modes depending on the excitation frequency, which are mono-stable periodic motion, bi-stable periodic motion, periodic and quasi-period motion, and quasi-periodic motion only. In addition, it is found that the existence of time delays in the control loop can either improve or degrade the control performance. Therefore, an objective function has been introduced to design the optimum control parameters. Based on the derived objective function, it is found that the performance of the proposed control strategy is proportional to the product of the control and feedback gains and inversely proportional to the internal loop feedback gain when the loop delays are neglected. Moreover, it is reported that the controller performance is a periodic function of the total sum of the loop delays. Accordingly, the optimal operating conditions of the time-delayed integral resonant controller have been explained. Finally, numerical validations for all obtained analytical results have been performed, where an excellent correspondence between the analytical and numerical investigations has been demonstrated.

【 授权许可】

Unknown