Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt.
Department of Mathematics, Faculty of Science, Tanta University, Tanta, 31527, Egypt.
Sci Rep. 2023 Apr 5;13(1):5570. doi: 10.1038/s41598-023-32743-w.
The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge-Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.
本文采用时滞的平方位置和速度研究了摇摆刚杆的稳定性。时滞是考虑到非线性振动系统的额外安全措施。由于时滞技术最近已成为多项研究的核心，因此本研究的主题非常相关。本文通过改进同伦摄动法（HPM）来获得更精确的近似结果。因此，本文的新颖之处在于时滞的耦合及其与改进 HPM 的相关性。通过与四阶龙格-库塔（RK4）技术进行比较，评估了分析和数值解之间的精度。该研究允许对现实近似分析方法的结果进行全面检查。对于不同的物理频率和时滞因素数量，在各种图中描绘了找到的解的时间历程。根据相关参数值，根据显示的曲线对这些图形进行了讨论。通过多时间尺度方法对组织的非线性原型方法进行了一阶近似检验。所得到的结果具有周期性行为和稳定的方式。本研究使得可以仔细检查通过采用实用估计的分析技术获得的发现。此外，与系统的非线性振荡潜力相比，时滞表现为额外的保护。
