A novel frequency formula and its application for a bead sliding on a wire in fractal space

The present study investigates the frequency-amplitude relationship of a nonlinear oscillator in fractal space, focusing on the dynamics of a bead sliding along a rotating wire with inhomogeneous angular velocity. Utilizing the two-scale fractal theory, the original fractal differential equation is transformed into an equivalent linear damped system in continuous space, thereby enabling the derivation of an exact analytical solution that does not rely on perturbation methods. A novel frequency formula is proposed that integrates fractal parameters and system constants. The establishment of these expressions is achieved through the application of energy conservation principles and Taylor series approximations, thereby providing explicit expressions for the fractal parameters. Numerical simulations were conducted to verify the analytical results and to demonstrate the influence of the parameters on damping behavior and oscillation profiles. The proposed framework is a versatile analytical tool for the study of fractal-mediated dynamics in mechanical systems, with potential applications in resonant engineering and multiscale materials design.