Dynamical analysis of a damped harmonic forced duffing oscillator with time delay

Abstract

This paper is concerned with a time-delayed controller of a damped nonlinear excited Duffing oscillator (DO). Since time-delayed techniques have recently been the focus of numerous studies, the topic of this investigation is quite contemporary. Therefore, time delays of position and velocity are utilized to reduce the nonlinear oscillation of the model under consideration. A much supplementary precise approximate solution is achieved using an advanced Homotopy perturbation method (HPM). The temporal variation of this solution is graphed for different amounts of the employed factors. The organization of the model is verified through a comparison between the plots of the estimated solution and the numerical one which is obtained utilizing the fourth order Runge–Kutta technique (RK4). The outcomes show that the improved HPM is appropriate for a variety of damped nonlinear oscillators since it minimizes the error of the solution while increasing the validation variety. Furthermore, it presents a potential model that deals with a diversity of nonlinear problems. The multiple scales homotopy technique is used to achieve an estimated formula for the suggested time-delayed structure. The controlling nonlinear algebraic equation for the amplitude oscillation at the steady state is gained. The effectiveness of the proposed controller, the time delays impact, controller gains, and feedback gains have been investigated. The realized outcomes show that the controller performance is influenced by the total of the product of the control and feedback gains, in addition to the time delays in the control loop. The analytical and numerical calculations reveal that for certain amounts of the control and feedback signal improvement, the suggested controller could completely reduce the system vibrations. The obtained outcomes are considered novel, in which the used methods are applied on the DO with time-delay. The increase of the time delay parameter leads to a stable case for the DO, which is in harmony with the influence of this parameter. This drawn curves show that the system reaches a stable fixed point which assert the presented discussion.

Mathieu-Hill Equation Stability Analysis for Trapped Ions. Anharmonic Corrections for Nonlinear Electrodynamic Traps

“…Ref. [115] also establishes that for some specific combination of the system parameters, vibration amplitudes and harvested power can be amplified up to three or five times in comparison to the classical broadband nonlinear energy harvester based on the forced (kicked) Duffing oscillator (DO) [15,116,117]. A method to generate stability plots for the Mathieu equation in case of a toroidal ion trap mass analyser is presented in [118].…”

Studying highly nonlinear oscillators using the non-perturbative methodology

“…Analytical approaches for nonlinear issues have garnered escalating interest among scientists and professionals in engineering due to the nonlinear sciences' rapid development during the past two decades. It was developed to study the behavior of these NLDEs using numerical and other approximation methods [6][7][8][9][10][11][12][13][14] . There have been a number of new techniques for finding analytical solutions to the NLDE recently.…”

Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps

“…Amongst other issues, the response of a Duffing oscillator (DO) [15,116] to a harmonic excitation, in presence of (viscous) damping, is known to exhibit hysteretic and chaotic behaviors [1,13,100,117,118]. It is demonstrated that in particular cases, the damping force may induce instability in the ion dynamics [119,120].…”