Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt.
Mathematics Department, Faculty of Science, Tanta University, Tanta, 527, Egypt.
Sci Rep. 2023 Apr 20;13(1):6507. doi: 10.1038/s41598-023-33461-z.
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.
这篇论文关注的是带时滞的阻尼非线性激励 Duffing 振子（DO）的控制器。由于时滞技术最近一直是众多研究的焦点，因此本研究的主题非常具有现代性。因此，利用位置和速度的时滞来减少所考虑模型的非线性振荡。利用先进的同伦摄动法（HPM）可以获得更精确的近似解。对于所采用的因素的不同数量，绘制了该解的时间变化。通过将估计解的图与利用四阶龙格-库塔技术（RK4）获得的数值解的图进行比较，验证了模型的组织。结果表明，改进的 HPM 适用于各种阻尼非线性振子，因为它可以在减少解的误差的同时增加验证范围。此外，它提出了一种潜在的模型，可以处理各种非线性问题。利用多重尺度同伦技术来获得所提出的时滞结构的估计公式。获得了在稳态下的振幅振荡的控制非线性代数方程。研究了所提出的控制器的有效性、时滞的影响、控制器增益和反馈增益。所得到的结果表明，控制器的性能受到控制和反馈增益的乘积以及控制回路中的时滞的影响。分析和数值计算表明，对于控制和反馈信号改进的某些数量，所提出的控制器可以完全减少系统的振动。所得到的结果被认为是新颖的，其中所使用的方法应用于带有时滞的 DO。时滞参数的增加导致 DO 处于稳定状态，这与该参数的影响是一致的。这些绘制的曲线表明，系统达到了一个稳定的平衡点，这证明了所提出的讨论。
