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This paper introduces a novel methodology to determine the frequency-amplitude relationship of fractal-order viscoelastic polymer materials using the two-scale fractal dimension transform, the equivalent power-form representation of the conservative restoring forces, and a simple coordinate transformation to eliminate viscoelastic effects. Then, the ancient Chinese algorithm Ying Bu Zu Shu and He's formulation are used for obtaining the frequency-amplitude relationship. Simulation results obtained from the derived expressions exhibit good agreement when compared to numerical integration solutions. This article elucidates how the molecular structure of polymer chains influences the relaxation oscillations as a function of the fractal parameter values. [ABSTRACT FROM AUTHOR]

This paper uses the two-scale fractal dimension transform and He's formula derived from the ancient Chinese algorithm Ying Bu Zu Shu to find the approximate frequency–amplitude expression of the fractal and forced anharmonic oscillator that can be used to study the nonlinear oscillations produced by the plasma physics fractal structures. The results show how the electron frequency and wavelength change as a function of the plasma physics fractal structure. In fact, if the value of the fractal parameter is decreased, the wavelength increases, and consequently, the system frequency decreases. The introduced solution procedure sheds a bright light on the easy-to-follow steps to obtain an accurate steady-state analytical solution of fractal anharmonic nonlinear oscillators. [ABSTRACT FROM AUTHOR]