NUMERICAL INVESTIGATION OF FRACTAL OSCILLATOR FOR A PENDULUM WITH A ROLLING WHEEL

In this paper, we consider a fractal oscillator for modeling the motion of a pendulum attached to a rolling wheel. It is a fractal oscillation system defined by He’s fractal derivative. The difficulty for solving this fractal differential equation results from its nonlinear parts and fractal operators. By using Taylor series approximation, an approximated fractal oscillation equation is obtained. A combined technique based upon two-scale fractal theory and harmonic method is proposed for solving the corresponding approximated system. By applying the fractal complex transformation, the fractal equation is approximately transformed as an ordinary second-order differential equation. The fractal or conventional approximations are given with the help of the spreading harmonic balance method. Numerical comparisons with Runge–Kutta method and sensitivity analysis of the approximated solutions and frequencies are presented to confirm the stability and efficiency of the proposed approach.