Analytical solution for the motion of a pendulum with rolling wheel: stability analysis

• Published in: Scientific reports Vol. 12; no. 1; p. 12628
• Main Authors: Moatimid, Galal M., Amer, T. S.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group 24.07.2022; Nature Publishing Group UK; Nature Portfolio
• Subjects: Approximation; Engineering; Mathematics; Mechanics; Partial differential equations; Physics; Stability analysis
• ISSN: 2045-2322; 2045-2322
• DOI: 10.1038/s41598-022-15121-w

Abstract 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.