Multi- stage variational iteration analysis of a spring pendulum system

The current research evaluates the nonlinear motion of a spring pendulum system with a constant mass based on a variational approach. Using first principles, the Lagrangian of the model was created in polar coordinates, which produced the Euler-Lagrange equations that describe the motion of the radial and angular components as functions of each other. Under the premise of small oscillation, the equations were simplified into a more easily solved nonlinear equation. To provide accurate analytical solutions for large timescales, the Ms-VIM) solution technique was used to overcome the limitations associated with the convergence rate of the classical VIM solution technique. In addition to the analytical solutions obtained, the solutions were also evaluated using the fourth order Runge-Kutta (RK4) method. An assessment of the analytical and numerical solutions indicated very good agreement among them, indicating that the proposed solution technique was reliable and efficient. Phase plots and error analyses were also conducted to illustrate the nature of the motion and the stability characteristics of the system. This research provided a viable analytical-numeric solution strategy for evaluating the motion of a constant mass oscillating system and can be applied to more complicated nonlinear mechanical models.