Moatimid Galal M, Amer T S
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt.
Department of Mathematics, Faculty of Science, Tanta University, Tanta, 31527, Egypt.
Sci Rep. 2022 Jul 24;12(1):12628. doi: 10.1038/s41598-022-15121-w.
The current work focuses on the motion of a simple pendulum connected to a wheel and a lightweight spring. The fundamental equation of motion is transformed into a complicated nonlinear ordinary differential equation under restricted surroundings. To achieve the approximate regular solution, the combination of the Homotopy perturbation method (HPM) and Laplace transforms is adopted in combination with the nonlinear expanded frequency. In order to verify the achievable solution, the technique of Runge-Kutta of fourth-order (RK4) is employed. The existence of the obtained solutions over the time, as well as their related phase plane plots, are graphed to display the influence of the parameters on the motion behavior. Additionally, the linearized stability analysis is validated to understand the stability in the neighborhood of the fixed points. The phase portraits near the equilibrium points are sketched.
当前的工作聚焦于一个与轮子和轻质弹簧相连的单摆的运动。在受限环境下，基本运动方程被转化为一个复杂的非线性常微分方程。为了得到近似正则解，将同伦摄动法（HPM）与拉普拉斯变换相结合，并结合非线性扩展频率。为了验证可得到的解，采用了四阶龙格 - 库塔（RK4）技术。绘制所得到的解随时间的存在情况及其相关的相平面图，以展示参数对运动行为的影响。此外，通过线性化稳定性分析来验证，以了解在固定点邻域内的稳定性。绘制了平衡点附近的相图。
