Stochastically analysis by using fixed point approach of pendulum with rolling wheel via translational and rotational motion

This paper presents an analytical solution for a complex mechanical system consisting of a pendulum with a rolling wheel, which combines translational and rotational motion. The system's dynamics are described by a set of coupled differential equations that are challenging to solve analytically using traditional methods. To overcome this challenge, the variation iteration method (VIM) is employed to derive an analytical solution. VIM is a powerful technique that allows for the approximate solution of differential equations by constructing a series solution iteratively. The Lagrange multiplier, a crucial component in the solution process, is determined for the first time using the Elzaki transformation. Interestingly, the Lagrange multiplier obtained through the Elzaki transformation matches the result obtained from Laplace transformation, which is a fundamental finding of this paper. Also we compared its results with the by employing the VIM and the derived Lagrange multiplier, a comprehensive analytical solution for the complex pendulum with a rolling wheel system is obtained. The solution provides insights into the system's behavior, such as the oscillation amplitudes, angular velocities and other relevant dynamic parameters. The proposed approach demonstrates the efficacy of the VIM in tackling complex mechanical systems and showcases the equivalence between the Lagrange multiplier derived through Elzaki transformation and the well-established Laplace transformation. The results obtained from this study contribute to the understanding and analysis of coupled translational and rotational systems, providing a valuable tool for researchers and engineers working in the field of mechanical dynamics. Sketches are made of the phase portraits close to the equilibrium points.