Analytical approaches to fractal oscillator for a pendulum with a rolling wheel

This study considers two analytical approaches for solving a fractal oscillator modelling a pendulum incorporated with a rolling wheel. By applying fractal variational principle, the approximated frequency and fractal periodic solution are given. Approximations are also derived by using He’s frequency formulation. The accuracy of both methods is validated through comparisons with numerical solutions obtained via Runge-Kutta method and spreading residue harmonic balance method. Numerical results demonstrate that the fractal variational method outperforms He’s frequency formulation and spreading residue harmonic balance method for the conventional oscillators with large parameters. Sensitivity analysis of the approximated frequencies provides insights into the oscillator’s stability characteristics. Furthermore, numerical behaviour of the fractal periodic solutions with various amplitudes and fractional orders is also investigated.