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

Area of Science:

• * Mechanical Engineering
• * Applied Mathematics
• * Nonlinear Dynamics

Background:

• * The motion of a simple pendulum connected to a wheel and a lightweight spring presents complex nonlinear dynamics.
• * Analyzing such systems requires advanced mathematical techniques to derive accurate solutions.

Purpose of the Study:

• * To investigate the motion of a nonlinear pendulum system.
• * To develop and apply a hybrid analytical-numerical method for solving the governing differential equation.
• * To analyze the stability and parameter influence on the system's behavior.

Main Methods:

• * The Homotopy perturbation method (HPM) combined with Laplace transforms was used to find an approximate analytical solution.
• * Nonlinear expanded frequency was incorporated to refine the solution.
• * The Runge-Kutta of fourth-order (RK4) method was employed for numerical verification.

Main Results:

• * The study successfully derived an approximate regular solution for the nonlinear pendulum's motion.
• * Graphical representations (time plots, phase plane plots) illustrated the impact of various parameters on the system's dynamics.
• * Linearized stability analysis confirmed the system's stability near fixed points, supported by phase portraits.

Conclusions:

• * The combined HPM and Laplace transform method provides an effective approach for solving complex nonlinear ordinary differential equations.
• * Parameter variations significantly influence the pendulum's motion and stability characteristics.
• * The findings contribute to a deeper understanding of nonlinear oscillatory systems.