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

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.

Automated summary

This study focuses on the motion of a simple pendulum connected to a wheel and a lightweight spring. A complicated nonlinear ordinary differential equation is obtained from the fundamental equation of motion under restricted conditions. The combination of the Homotopy perturbation method (HPM) and Laplace transforms is used to obtain an approximate regular solution. The validity of the solution is verified using the fourth-order Runge-Kutta method (RK4). The influence of parameters on the motion behavior is displayed through the graphs of the obtained solutions over time and their related phase plane plots.